(30 June 2020)
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* Section 4 - Further Information *
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This section of the manual contains literature references
and hints on how to make skillful use of GAMESS.
The following topics are covered:
Computational References
Basis Set References
Spherical Harmonics
How to do RHF, ROHF, UHF, and GVB calculations
general considerations
direct SCF
convergence accelerators
high spin open shell SCF (ROHF)
other open shell SCF cases (GVB)
true GVB perfect pairing runs
the special case of TCSCF
a caution about symmetry
How to do MCSCF (and CI) calculations
MCSCF implementation
orbital updates
CI coefficient optimization
determinant CI
CSF CI
starting orbitals
miscellaneous hints
MCSCF references
Second Order Perturbation Theory
RHF and UHF reference MP2
high spin ROHF reference MP2
GVB based MP2
MCSCF reference perturbation theory
Coupled-Cluster Theory
available computations (ground states)
available computations (excited states)
density matrices and properties
excited state example
resource requirements
restarts in ground-state calculations
initial guesses in excited-state calculations
eigensolvers for excited-state calculations
references and citations required in publications
Density Functional Theory
DFTTYP keywords
grid-free DFT
DFT with grids
Time Dependent Density Functional Theory (TD-DFT)
references for DFT
Multiconfiguration Pair-Density Functional Theory
Summary of excited state methods
Geometry Searches and Internal Coordinates
quasi-Newton Searches
the nuclear Hessian
coordinate choices
the role of symmetry
practical matters
saddle points
mode following
Intrinsic Reaction Coordinate Methods
Gradient Extremals
Continuum Solvation Methods
Self Consistent Reaction Field (SCRF)
Polarizable Continuum Model (PCM)
SVPE and SS(V)PE.
Conductor-like screening model (COSMO)
The Effective Fragment Potential Method
terms in an EFP
constructing an EFP1
constructing an EFP2
current limitations
practical hints for using EFPs
global optimization
QM/MM across covalent bonds
Simpler potentials
references
The Fragment Molecular Orbital method
Surfaces and solids
FMO variants
Effective fragment molecular orbital method (EFMO)
Guidelines for approximations with FMO3
How to perform FMO-MCSCF calculations
How to perform multilayer runs
How to mix basis sets in FMO
How to perform FMO/PCM calculations
How to perform FMO/EFP calculations
Geometry optimization or saddle point search for FMO
FMO hessian calculations
Molecular dynamics with FMO
Pair interaction energy decomposition analysis (PIEDA)
Excited states
Selective and sussystem FMO
Frozen domain
IMOMM with FMO
Analyzing and visualizing the results
Parallelization of FMO runs with GDDI
Limitations of the FMO method in GAMESS
Restarts with the FMO method
Note on accuracy
FMO References
The Cluster-in-Molecules method
sequential and parallel execution
restarts
the cimshell script
CIM references
MOPAC Calculations within GAMESS
Molecular Properties and Conversion Factors
Polarizabilities
Localized Molecular Orbitals
Transition Moments and Spin-Orbit Coupling
states
orbitals
symmetry
spin orbit coupling
input nitty-gritty
references
examples
------------------------------------------------------------
For people who are newcomers to computational chemistry, it
may be helpful to study an introductory book.
First, some texts about quantum chemistry:
"Ab Initio Molecular Orbital Theory"
W.J.Hehre, L.Radom, J.A.Pople, P.v.R.Schleyer
Wiley and Sons, New York, 1986
"Modern Quantum Chemistry" (now a Dover paperback)
A.Szabo, N.S.Ostlund McGraw-Hill, 1989
"Quantum Chemistry, 6th Edition"
I.N.Levine Prentice Hall, 2008
Then, a few books more focused on computation:
"Introduction to Quantum Mechanics in Chemistry"
M.A.Ratner, G.C.Schatz Prentice Hall, 2000
"Introduction to Computational Chemistry, 2nd Edition"
Frank Jensen Wiley and Sons, Chichester, 2006
"Molecular Modeling Basics"
Jan H. Jensen CRC Press, Boca Raton, 2010
Frank's book is an outstanding survey of methods, basis
sets, properties, and other topics.
Jan's book is a good complement to Frank's, staying at a
simpler level, using GAMESS input examples. It has an
accompanying online blog,
http://molecularmodelingbasics.blogspot.com
------------------------------------------------------------
Computational References
GAMESS -
M.W.Schmidt, K.K.Baldridge, J.A.Boatz, S.T.Elbert,
M.S.Gordon, J.H.Jensen, S.Koseki, N.Matsunaga,
K.A.Nguyen, S.Su, T.L.Windus, M.Dupuis, J.A.Montgomery
J.Comput.Chem. 14, 1347-1363 (1993)
M.S.Gordon, M.W.Schmidt pp 1167-1189
in "Theory and Applications of Computational Chemistry,
the first forty years" C.E.Dykstra, G.Frenking, K.S.Kim,
G.E.Scuseria (editors), Elsevier, Amsterdam, 2005.
HONDO -
These papers describes many of the algorithms in detail,
and much of these applies also to GAMESS:
"The General Atomic and Molecular Electronic Structure
System: HONDO 7.0" M.Dupuis, J.D.Watts, H.O.Villar,
G.J.B.Hurst Comput.Phys.Comm. 52, 415-425(1989)
"HONDO: A General Atomic and Molecular Electronic
Structure System" M.Dupuis, P.Mougenot, J.D.Watts,
G.J.B.Hurst, H.O.Villar in "MOTECC: Modern Techniques
in Computational Chemistry" E.Clementi, Ed.
ESCOM, Leiden, the Netherlands, 1989, pp 307-361.
"HONDO: A General Atomic and Molecular Electronic
Structure System" M.Dupuis, A.Farazdel, S.P.Karna,
S.A.Maluendes in "MOTECC: Modern Techniques in
Computational Chemistry" E.Clementi, Ed.
ESCOM, Leiden, the Netherlands, 1990, pp 277-342.
M.Dupuis, S.Chin, A.Marquez in "Relativistic and Electron
Correlation Effects in Molecules", G.Malli, Ed. Plenum
Press, NY 1994, pp 315-338.
sp integrals and gradient integrals -
inner axis sp integration is done by McMurchie/Davidson
J.A.Pople, W.J.Hehre J.Comput.Phys. 27, 161-168(1978)
H.B.Schlegel, J.Chem.Phys. 77, 3676-3681(1982)
spd integrals by rotated axis/McMurchie-Davidson
K.Ishimura, S.Nagase Theoret.Chem.Acc. 120, 185-189(2008)
McMurchie/Davidson integrals -
L.E.McMurchie, E.R.Davidson
J.Comput.Phys. 26, 218-231(1978)
spdfghi integrals -
"Numerical Integration Using Rys Polynomials"
H.F.King and M.Dupuis J.Comput.Phys. 21,144(1976)
"Evaluation of Molecular Integrals over Gaussian
Basis Functions"
M.Dupuis,J.Rys,H.F.King J.Chem.Phys. 65,111-116(1976)
"Molecular Symmetry and Closed Shell HF Calculations"
M.Dupuis and H.F.King Int.J.Quantum Chem. 11,613(1977)
"Computation of Electron Repulsion Integrals using
the Rys Quadrature Method"
J.Rys,M.Dupuis,H.F.King J.Comput.Chem. 4,154-157(1983)
ERIC spdfg integrals -
"Recursion Formula for Electron Repulsion Integrals Over
Hermite Polynomials"
G.D.Fletcher Int.J.Quantum Chem. 106, 355-360(2006)
spdfg gradient integrals -
"Molecular Symmetry. II. Gradient of Electronic Energy
with respect to Nuclear Coordinates"
M.Dupuis and H.F.King J.Chem.Phys. 68,3998(1978)
although the implementation is much newer than this paper.
spd hessian integrals -
"Molecular Symmetry. III. Second derivatives of Electronic
Energy with respect to Nuclear Coordinates"
T.Takada, M.Dupuis, H.F.King
J.Chem.Phys. 75, 332-336 (1981)
the Q matrix, and integral transformation symmetry -
E.Hollauer, M.Dupuis J.Chem.Phys. 96, 5220 (1992)
spdfg effective core potential (ECP) integral/derivatives -
C.F.Melius, W.A.Goddard Phys.Rev.A 10,1528-1540(1974)
L.R.Kahn, P.Baybutt, D.G.Truhlar
J.Chem.Phys. 65, 3826-3853 (1976)
M.Krauss, W.J.Stevens Ann.Rev.Phys.Chem. 35, 357-385(1985)
J.Breidung, W.Thiel, A.Komornicki
Chem.Phys.Lett. 153, 76-81(1988)
B.M.Bode, M.S.Gordon J.Chem.Phys. 111, 8778-8784(1999)
See also the papers listed for SBKJC and HW basis sets.
model core potential (MCP) reviews -
S.Huzinaga Can.J.Chem. 73, 619-628(1995)
M.Klobukowski, S.Huzinaga, Y.Sakai, in Computational
Chemistry: Reviews of current trends, volume 3, pp 49-74,
edited by J.Leszczynski, World Scientific, Singapore, 1999.
Quantum fast multipole method (QFMM) -
E.O.Steinborn, K.Ruedenberg
Adv.Quantum Chem. 7, 1-81(1973)
L.Greengard "The Rapid Evaluation of Potential Fields in
Particle Systems" (MIT, Cambridge, 1987)
C.H.Choi, J.Ivanic, M.S.Gordon, K.Ruedenberg
J.Chem.Phys. 111, 8825-8831(1999)
C.H.Choi, K.Ruedenberg, M.S.Gordon
J.Comput.Chem. 22, 1484-1501(2001)
C.H.Choi J.Chem.Phys. 120, 3535-3543(2004)
RHF -
C.C.J.Roothaan Rev.Mod.Phys. 23, 69-89(1951)
UHF -
J.A.Pople, R.K.Nesbet J.Chem.Phys 22, 571-572(1954)
high-spin coupled ROHF -
C.C.J.Roothaan Rev.Mod.Phys. 32, 179-185(1960)
R.McWeeny, G.Diercksen J.Chem.Phys. 49,4852-4856(1968)
M.F.Guest, V.R.Saunders Mol.Phys. 28, 819-828(1974)
J.S.Binkley, J.A.Pople, P.A.Dobosh
Mol.Phys. 28, 1423-1429(1974)
E.R.Davidson Chem.Phys.Lett. 21,565-567(1973)
K.Faegri, R.Manne Mol.Phys. 31,1037-1049(1976)
H.Hsu, E.R.Davidson, and R.M.Pitzer
J.Chem.Phys. 65,609-613(1976)
B.N.Plakhutin, E.V.Gorelik, N.N.Breslavskaya
J.Chem.Phys. 125, 204110/1-10(2006)
B.N.Plakhutin, E.R.Davidson
J.Phys.Chem.A 113, 12386-12395(2009)
E.R.Davidson, B.N.Plakhutin
J.Chem.Phys. 132, 184110/1-14(2010)
K.R.Glaesemann, M.W.Schmidt
J.Phys.Chem.A 114, 8772-8777(2010)
Constrained UHF (CUHF is equivalent to high spin ROHF) -
G.E.Scuseria, T.Tsuchimochi
J.Chem.Phys. 134, 064101/1-14(2011)
GVB and low-spin coupled ROHF -
F.W.Bobrowicz and W.A.Goddard, in Modern Theoretical
Chemistry, Vol 3, H.F.Schaefer III, Ed., Chapter 4.
DFT and TD-DFT -
All appropriate references are included in the section on
density functional theory included below.
Multiconfiguration Pair-Density Functional Theory (MCPDFT)
G.LiManni, R K.Carlson, S.Luo, D.Ma, J.Olsen,
D.G.Truhlar, L.Gagliardi
J.Chem.TheoryComput. 10, 3669-3680(2014)
R.K.Carlson, D.G.Truhlar, L.Gagliardi
J.Chem.TheoryComput. 11, 4077-4085(2015)
L.Gagliardi, D.G.Truhlar, G.LiManni, R.K.Carlson,
C.E.Hoyer, J.L.Bao
Acc.Chem.Res. 50, 66-73(2016)
MCSCF - see reference list in its own subsection below
determinant CI -
full CI (ALDET) and general CI (GENCI),
J.Ivanic, K.Ruedenberg
Theoret.Chem.Acc. 106, 339-351(2001)
occupation restricted multiple active space (ORMAS),
J.Ivanic J.Chem.Phys. 119, 9364-9376, 9377-9385(2003)
configuration state function CI (GUGA) -
B.Brooks and H.F.Schaefer J.Chem. Phys. 70,5092(1979)
B.Brooks, W.Laidig, P.Saxe, N.Handy, and H.F.Schaefer,
Physica Scripta 21, 312(1980).
CIS energy and gradient -
J.B.Foresman, M.Head-Gordon, J.A.Pople, M.J.Frisch
J.Phys.Chem. 96, 135-149(1992)
R.M.Shroll, W.D.Edwards
Int.J.Quantum Chem. 63, 1037-1049(1997)
the parallel CIS implementation in GAMESS is described in
S.P.Webb Theoret.Chem.Acc. 116, 355-372(2006)
which has a nice review of other excited state methods.
spin-flip CIS:
A.I.Krylov Chem.Phys.Lett. 338, 375(2001)
Closed shell and unrestricted open shell 2nd order
Moller-Plesset perturbation theory (MP2), also known as
2nd order many body perturbation theory (MBPT(2)) -
R.J.Bartlett & D.M.Silver
Phys. Rev. A, 10, 1927 (1974)
J.A.Pople, J.S.Binkley, R.Seeger
Int. J. Quantum Chem. S10, 1-19(1976)
M.J.Frisch, M.Head-Gordon, J.A.Pople,
Chem.Phys.Lett. 166, 275-280(1990)
C.M.Aikens, S.P.Webb, R.L.Bell, G.D.Fletcher, M.W.Schmidt,
M.S.Gordon Theoret.Chem.Acc., 110, 233-253(2003)
with the TCA "overview article" being a thorough review of
the single determinant MP2 gradient equations.
CODE=SERIAL is generally based on the CPL paper above, as
described in the HONDO references given above.
The next two document CODE=DDI for RHF and UHF,
G.D.Fletcher, M.W.Schmidt, M.S.Gordon
Adv.Chem.Phys. 110, 267-294(1999)
C.M.Aikens, M.S.Gordon
J.Phys.Chem.A, 108, 3103-3110(2004)
The next two document CODE=IMS for RHF,
K.Ishimura, P.Pulay, S.Nagase
J.Comput.Chem. 27, 407-413(2006)
K.Ishimura, P.Pulay, S.Nagase
J.Comput.Chem. 28, 2034-2042(2007)
The next documents code=RIMP2 for RHF and UHF,
M.Katouda, S.Nagase
Int.J.Quantum Chem. 109, 2121-2130(2009)
Spin Component Scaled MP2 (SCS-MP2)
S.Grimme
J.Chem.Phys. 118, 9095-9102(2003)
spin restricted open shell MP2, ZAPT energy -
T.J.Lee, D.Jayatilaka
Chem.Phys.Lett. 201, 1-10(1993)
T.J.Lee, A.P.Rendell, K.G.Dyall, D.Jayatilaka
J.Chem.Phys. 100, 7400-7409(1994)
nuclear gradients for ZAPT -
The next two document the CODE=DDI program,
G.D.Fletcher, M.S.Gordon, R.L.Bell
Theoret.Chem.Acc. 107, 57-70(2002)
C.M.Aikens, G.D.Fletcher, M.W.Schmidt, M.S.Gordon
J.Chem.Phys. 124, 014107/1-14(2006)
spin restricted open shell MP2, RMP method -
P.J.Knowles, J.S.Andrews, R.D.Amos, N.C.Handy, J.A.Pople
Chem.Phys.Lett. 186, 130-136 (1991)
W.J.Lauderdale,J.F.Stanton,J.Gauss,J.D.Watts,R.J.Bartlett
Chem.Phys.Lett. 187, 21-28(1991)
CUMP2 is equivalent to RMP2 (See CUHF reference above).
multiconfigurational quasidegenerate perturbation theory -
H.Nakano J.Chem.Phys. 99, 7983-7992(1993)
ORMAS-based multireference perturbation theory -
L.Roskop, M.S.Gordon J.Chem.Phys. 135, 044101/1-11(2012)
Coupled-Cluster -
Equation of Motion Coupled-Cluster (EOMCC) -
this is a subset of the relevant papers:
P.Piecuch, S.A.Kucharski, K.Kowalski, M.Musial,
Comput.Phys.Commun. 149, 71-96(2002)
K.Kowalski, P.Piecuch,
J.Chem.Phys. 120, 1715-1738 (2004)
P.Piecuch, S.A.Kucharski, K.Kowalski, M.Musial
Comput.Phys.Commun. 149, 71-96(2002).
parallel CCSD(T) program -
J.L.Bentz, R.M.Olson, M.S.Gordon, M.W.Schmidt, R.A.Kendall
Comput.Phys.Commun. 176, 589-600(2007)
R.M.Olson, J.L.Bentz, R.A.Kendall, M.W.Schmidt, M.S.Gordon
J.Comput.Theoret.Chem. 3, 1312-1328(2007)
Any publication describing the results of ground-state
and/or excited-state calculations using the equation of
motion coupled-cluster and/or completely renormalized
EOMCCSD(T) options (CCTYP=EOM-CCSD or CR-EOM) obtained with
GAMESS should reference the specific papers appearing in
the printout. For more references to the primary
literature for both types of coupled-cluster methods, see
the section "Coupled-Cluster theory" below.
RHF/ROHF/TCSCF coupled perturbed Hartree Fock -
"Single Configuration SCF Second Derivatives on a Cray"
H.F.King, A.Komornicki in "Geometrical Derivatives of
Energy Surfaces and Molecular Properties" P.Jorgensen
J.Simons, Ed. D.Reidel, Dordrecht, 1986, pp 207-214.
"A parallel Distributed data CPHF algorithm for analytic
Hessians" Y.Alexeev, M.W.Schmidt, T.L.Windus, M.S.Gordon
J.Comput.Chem. 28, 1685-1694(2007).
Y.Osamura, Y.Yamaguchi, D.J.Fox, M.A.Vincent, H.F.Schaefer
J.Mol.Struct. 103, 183-186(1983)
M.Duran, Y.Yamaguchi, H.F.Schaefer
J.Phys.Chem. 92, 3070-3075(1988)
"A New Dimension to Quantum Chemistry" Y.Yamaguchi,
Y.Osamura, J.D.Goddard, H.F.Schaefer Oxford Press, NY 1994
MCSCF coupled perturbed Hartree-Fock -
M.R.Hoffman, D.J.Fox, J.F.Gaw, Y.Osamura, Y.Yamauchi,
R.S.Grev, G.Fitzgerald, H.F.Schaefer, P.J.Knowles,
N.C.Handy J.Chem.Phys. 80, 2660-2668(1984)
the book by Osamura, Goddard, and Schaefer just mentioned.
T.J.Dudley, R.M.Olson, M.W.Schmidt, M.S.Gordon
J.Comput.Chem. 27, 353-362(2006)
non-adiabatic coupling matrix element (NACME) -
J.C.Tully, chapter 5 (pp 217-267) in "Dynamics of Molecular
Collisions - Part B", edited by W.H.Miller, Plenum Press,
NY, 1976.
B.H.Lengsfield, D.R.Yarkony, chapter 1 (pp. 1-71) in
"State-selected and state-to-state in-molecule reaction
dynamics- Part 2, theory", edited by M.Baer and C.-Y.Ng,
John Wiley, NY, 1992.
harmonic vibrational analysis in Cartesian coordinates -
W.D.Gwinn J.Chem.Phys. 55,477-481(1971)
Normal coordinate decomposition analysis -
J.A.Boatz and M.S.Gordon,
J.Phys.Chem. 93, 1819-1826(1989).
Partial Hessian vibrational analysis -
H.Li, J.H.Jensen, Theoret.Chem.Acc. 107, 211-219(2002)
anharmonic vibrational spectra (VSCF) -
a review of VSCF:
R.B.Gerber, J.O.Jung
in "Computational Molecular Spectroscopy"
P.Jensen, P.R.Bunker, eds.
Wiley and Sons, Chichester, 2000, pp 365-390.
the basic method for VSCF and cc-VSCF:
G.M.Chaban, J.O.Jung, R.B.Gerber
J.Chem.Phys. 111, 1823-1829(1999)
the QFF approximation:
K.Yagi, K.Hirao, T.Taketsugu, M.W.Schmidt, M.S.Gordon
J.Chem.Phys. 121, 1383-1389(2004)
the VDPT solver:
N.Matsunaga, G.M.Chaban, R.B.Gerber
J.Chem.Phys. 117, 3541-3547(2002)
solver for larger systems:
L.Pele, B.Brauer, R.B.Gerber
Theoret.Chem.Acc. 117, 69-72(2007)
use of internal coordinates, and thermochemistry
B.Njegic, M.S.Gordon
J.Chem.Phys. 125, 224102/1-12(2006)
applications of RUNTYP=VSCF:
G.M.Chaban, J.O.Jung, R.B.Gerber
J.Phys.Chem.A 104, 2772-2779(2000)
J.Lundell, G.M.Chaban, R.B.Gerber
Chem.Phys.Lett. 331, 308-316(2000)
K.Yagi, T.Taketsugu, K.Hirao, M.S.Gordon
J.Chem.Phys. 113, 1005-1017(2000)
G.M.Chaban, R.B.Gerber, K.C.Janda
J.Phys.Chem.A 105, 8323-8332(2001)
A.T.Kowal Spectrochimica Acta A 58, 1055-1067(2002)
G.M.Chaban, S.S.Xantheas, R.B.Gerber
J.Phys.Chem.A 107, 4952-4956(2003)
G.M.Chaban J.Phys.Chem.A 108, 4551-4556(2004)
Y.Miller, G.M.Chaban, R.B.Gerber
J.Phys.Chem.A 109, 6565-6574(2005)
Y.Miller, G.M.Chaban, R.B.Gerber
Chem.Phys. 313, 213-224(2005)
C.A.Brindle, G.M.Chaban, R.B.Gerber, K.C.Janda
Phys.Chem.Chem.Phys. 7, 945-954(2005)
G.M.Chaban, R.M.Gerber
Theoret.Chem.Acc. 120, 273-279(2008)
Raman spectrum -
A.Komornicki, J.W.McIver J.Chem.Phys. 70, 2014-2016(1979)
G.B.Bacskay, S.Saebo, P.R.Taylor
Chem.Phys. 90, 215-224(1984)
static polarizabilities:
H.A.Kurtz, J.J.P.Stewart, K.M.Dieter
J.Comput.Chem. 11, 82-87 (1990)
dynamic polarizabilities:
P.Korambath, H.A.Kurtz, in "Nonlinear Optical Materials",
ACS Symposium Series 628, S.P.Karna and A.T.Yeates, Eds.
pp 133-144, Washington DC, 1996.
nuclear derivatives of dynamic polarizabilities,
and dynamic Raman and hyper-Raman:
O.Quinet, B.Champagne J.Chem.Phys. 115, 6293-6299(2001)
O.Quinet, B.Champagne B.Kirtman
J.Comput.Chem. 22, 1920-1932(2001)
O.Quinet, B.Champagne J.Chem.Phys. 117, 2481-2488(2002)
O.Quinet, B.Kirtman, B.Champagne
J.Chem.Phys. 118, 505-513(2003)
Geometry optimization and saddle point location -
J.Baker J.Comput.Chem. 7, 385-395(1986).
T.Helgaker Chem.Phys.Lett. 182, 503-510(1991).
P.Culot, G.Dive, V.H.Nguyen, J.M.Ghuysen
Theoret.Chim.Acta 82, 189-205(1992).
Dynamic Reaction Coordinate (DRC) -
J.J.P.Stewart, L.P.Davis, L.W.Burggraf,
J.Comput.Chem. 8, 1117-1123 (1987)
S.A.Maluendes, M.Dupuis, J.Chem.Phys. 93, 5902-5911(1990)
T.Taketsugu, M.S.Gordon, J.Phys.Chem. 99, 8462-8471(1995)
T.Taketsugu, M.S.Gordon, J.Phys.Chem. 99, 14597-604(1995)
T.Taketsugu, M.S.Gordon, J.Chem.Phys. 103, 10042-9(1995)
M.S.Gordon, G.Chaban, T.Taketsugu
J.Phys.Chem. 100, 11512-11525(1996)
T.Takata, T.Taketsugu, K.Hirao, M.S.Gordon
J.Chem.Phys. 109, 4281-4289(1998)
T.Taketsugu, T.Yanai, K.Hirao, M.S.Gordon
THEOCHEM 451, 163-177(1998)
Energy orbital localization -
C.Edmiston, K.Ruedenberg Rev.Mod.Phys. 35, 457-465(1963).
R.C.Raffenetti, K.Ruedenberg, C.L.Janssen, H.F.Schaefer,
Theoret.Chim.Acta 86, 149-165(1993)
Boys orbital localization -
S.F.Boys, "Quantum Science of Atoms, Molecules, and Solids"
P.O.Lowdin, Ed, Academic Press, NY, 1966, pp 253-262.
See the first paper on oriented localized orbitals if you
wish to know the true origin of "Boys localization"
Population orbital localization -
J.Pipek, P.Z.Mezey J.Chem.Phys. 90, 4916(1989).
Oriented localized orbitals -
J.Ivanic, G.M.Atchity, K.Ruedenberg
Theoret.Chem.Acc. 120, 281-294(2008)
J.Ivanic, K.Ruedenberg
Theoret.Chem.Acc. 120, 295-305(2008)
Valence Virtual Orbitals (VVOS) -
W.C.Lu, C.Z.Wang, M.W.Schmidt, L.Bytautas, K.M.Ho,
K.Ruedenberg
J.Chem.Phys. 120, 2629-2637 and 2638-2651(2004)
W.C.Lu, C.Z.Wang, T.L.Chan, K.Ruedenberg, K.M.Ho
Phys.Rev.B 70, 041101-1/4(2004)
Mulliken Population Analysis -
R.S.Mulliken J.Chem.Phys. 23, 1833-1840, 1841-1846,
2338-2342, 2343-2346(1955)
so called "Lowdin Population Analysis" -
This should be described as "a Mulliken population analysis
(ref M1-M4 above) based on symmetrically orthogonalized
orbitals (ref L)", where reference L is
P.-O.Lowdin Adv.Chem.Phys. 5, 185-199(1970)
Lowdin populations are not invariant to rotation if the
basis set used is Cartesian d,f,...:
I.Mayer, Chem.Phys.Lett. 393, 209-212(2004).
Bond orders and valences -
M.Giambiagi, M.Giambiagi, D.R.Grempel, C.D.Heymann
J.Chim.Phys. 72, 15-22(1975)
I.Mayer, Chem.Phys.Lett. 97,270-274(1983), 117,396(1985).
M.S.Giambiagi, M.Giambiagi, F.E.Jorge
Z.Naturforsch. 39a, 1259-73(1984)
I.Mayer, Theoret.Chim.Acta 67, 315-322(1985).
I.Mayer, Int.J.Quantum Chem. 29, 73-84(1986).
I.Mayer, Int.J.Quantum Chem. 29, 477-483(1986).
The same formula (apart from a factor of two) may also be
seen in equation 31 of the second of these papers (the bond
order formula in the 1st of these is not the same formula):
T.Okada, T.Fueno Bull.Chem.Soc.Japan 48, 2025-2032(1975)
T.Okada, T.Fueno Bull.Chem.Soc.Japan 49, 1524-1530(1976)
a review about bond orders:
I. Mayer, J.Comput.Chem. 28, 204-221(2007).
Direct SCF -
J.Almlof, K.Faegri, K.Korsell
J.Comput.Chem. 3, 385-399 (1982)
M.Haser, R.Ahlrichs
J.Comput.Chem. 10, 104-111 (1989)
DIIS (Direct Inversion in the Iterative Subspace) -
P.Pulay J.Comput.Chem. 3, 556-560(1982)
SOSCF -
G.Chaban, M.W.Schmidt, M.S.Gordon
Theor.Chem.Acc. 97, 88-95(1997)
T.H.Fischer, J.Almlof J.Phys.Chem. 96,9768-74(1992)
Modified Virtual Orbitals (MVOs) -
C.W.Bauschlicher, Jr. J.Chem.Phys. 72,880-885(1980)
Thermochemistry (RUNTYP=G3MP2) -
G3(MP2,CCSD(T)) is defined in
L.A.Curtiss, K.Ragavachari, P.C.Redfern, A.G.Baboul,
J.A.Pople Chem.Phys.Lett. 314, 101-107(1999)
based on various other G3 basis set/method papers:
L.A.Curtiss, P.C.Redfern, K.Raghavachari, V.Rassolov,
J.A.Pople J.Chem.Phys. 110, 4703-4709(1999)
L.A.Curtiss, P.C.Redfern, K.Raghavachari, V.Rassolov,
J.A.Pople J.Chem.Phys. 114, 9287-9295(2001)
L.A.Curtiss, P.C.Redfern, K.Raghavachari, V.Rassolov,
J.A.Pople J.Chem.Phys. 109,7764-7776(1998)
L.A.Curtiss, K.Ragavachari
Theoret.Chem.Acc. 108, 61-70(2002)
EVVRSP, in memory diagonalization -
S.T.Elbert Theoret.Chim.Acta 71,169-186(1987)
Davidson eigenvector method -
E.R.Davidson J.Comput.Phys. 17,87(1975)
"Matrix Eigenvector Methods" p. 95-113 in "Methods in
Computational Molecular Physics", edited by G.H.F.Diercksen
and S.Wilson, D.Reidel Publishing, Dordrecht, 1983.
M.L.Leininger, C.D.Sherrill, W.D.Allen, H.F.Schaefer,
J.Comput.Chem. 22, 1574-1589(2001)
RESC (Relativistic Elimination of Small Components) -
T.Nakajima, K.Hirao Chem.Phys.Lett. 302, 383-391(1999)
T.Nakajima, T.Suzumura, K.Hirao
Chem.Phys.Lett. 304, 271(1999)
D.G.Fedorov, T.Nakajima, K.Hirao
Chem.Phys.Lett. 335, 183-187(2001)
DK (Douglas-Kroll relativistic transformation) -
M.Douglas, N.M.Kroll Ann.Phys. 82, 89-155(1974)
B.A.Hess Phys.Rev. A33, 3742-3748(1986)
G.Jansen, B.A.Hess Phys.Rev. A39, 6016-6017(1989)
T.Nakajima, K.Hirao J.Chem.Phys. 113, 7786-7789(2000)
T.Nakajima, K.Hirao Chem.Phys.Lett. 329, 511-516(2000)
W.A.DeJong, R.J.Harrison, D.A.Dixon
J.Chem.Phys. 114, 48-53(2001)
A.Wolf, M.Reiher, B.A.Hess J.Chem.Phys. 117, 9215-26(2002)
T.Nakajima, K.Hirao J.Chem.Phys. 119, 4105-4111(2003)
(and see just below for DK1 during SOC)
IOTC (Infinite-Order Two-Component) relativy correction -
M.Barysz, A.J.Sadlej J.Chem.Phys. 116, 2696-2704(2002)
M.Barysz, Progress in Theoretical Chemistry and Physics,
Kluwer Academic Publishers, 349-397(2002)
D.Kedziera, M.Barysz, A.J.Sadlej
Struct.Chem. 15, 369-377(2004)
D.Kedziera, M.Barysz, J.Chem.Phys. 121, 6719-6727(2004)
M.Barysz, L.Mentel, J.Leszczynski
J.Chem.Phys. 130, 164114/1-7(2009)
LUT-IOTC (local unitary transformation IOTC relativity -
J.Seino, H.Nakai J.Chem.Phys. 136, 244101/1-13(2012)
J.Seino, H.Nakai J.Chem.Phys. 137, 144101/1-15(2012)
Y.Nakajima, J.Seino, H.Nakai
J.Chem.Phys. 139, 244107/1-13(2013)
J.Seino, H.Nakai Int.J.Quantum Chem. 115, 253-257(2014)
NESC (Normalized Elimination of Small Components) -
K.G.Dyall J.Comput.Chem. 23, 786-793(2002)
Spin-orbit coupling and transition moments ?
Many references can be found in the section on this topic
below.
GIAO NMR -
R.Ditchfield Mol.Phys. 27, 789-807(1974)
M.A.Freitag, B.Hillman, A.Agrawal, M.S.Gordon
J.Chem.Phys. 120, 1197-1202(2004)
Solvation models: EFP, SCRF, PCM, or COSMO.
All appropriate references are included in the sections on
these topics included below.
MOPAC 6 -
J.J.P.Stewart
J.Computer-Aided Molecular Design 4, 1-105 (1990)
References for parameters for individual atoms may be found
on the printout from your runs.
MacMolPlt -
B.M.Bode, M.S.Gordon J.Mol.Graphics Mod. 16, 133-138(1998)
quantum chemistry parallelization in GAMESS -
for SCF, see the main GAMESS paper quoted above.
T.L.Windus, M.W.Schmidt, M.S.Gordon,
Chem.Phys.Lett. 216, 375-379(1993)
T.L.Windus, M.W.Schmidt, M.S.Gordon,
Theoret.Chim.Acta 89, 77-88 (1994)
T.L.Windus, M.W.Schmidt, M.S.Gordon, in "Parallel Computing
in Computational Chemistry", ACS Symposium Series 592,
Ed. by T.G.Mattson, ACS Washington, 1995, pp 16-28.
K.K.Baldridge, M.S.Gordon, J.H.Jensen, N.Matsunaga,
M.W.Schmidt, T.L.Windus, J.A.Boatz, T.R.Cundari
ibid, pp 29-46.
G.D.Fletcher, M.W.Schmidt, M.S.Gordon
Adv.Chem.Phys. 110, 267-294 (1999)
H.Umeda, S.Koseki, U.Nagashima, M.W.Schmidt
J.Comput.Chem. 22, 1243-1251 (2001)
C.H.Choi, K.Ruedenberg J.Comput.Chem. 22, 1484-1501(2001)
D.G.Fedorov, M.S.Gordon ACS Symp.Series 828, 1-22(2002)
H.Li, C.S.Pomelli, J.H.Jensen
Theoret.Chem.Acc. 109, 71-84(2003)
C.M.Aikens, M.S.Gordon J.Phys.Chem.A 108, 3103-3110(2004)
H.M.Netzloff, M.S.Gordon J.Comput.Chem. 25, 1926-1936(2004)
T.J.Dudley, R.M.Olson, M.W.Schmidt, M.S.Gordon
J.Comput.Chem. 27, 353-362(2006)
C.M.Aikens, G.D.Fletcher, M.W.Schmidt, M.S.Gordon
J.Chem.Phys. 124, 014107/1-14(2006)
Y.Alexeev, M.W.Schmidt, T.L.Windus, M.S.Gordon
J.Comput.Chem. 28, 1685-1694(2007).
R.M.Olson, J.L.Bentz, R.A.Kendall, M.W.Schmidt, M.S.Gordon
J.Comput.Theoret.Chem. 3, 1312-1328(2007)
J.L.Bentz, R.M.Olson, M.S.Gordon, M.W.Schmidt, R.A.Kendell
Comput.Phys.Commun., 176, 589-600 (2007).
G.D.Fletcher Mol.Phys. 105, 2971-2976(2007)
The Distributed Data Interface (DDI), which is the computer
science layer underneath the parallel quantum chemistry -
G.D.Fletcher, M.W.Schmidt, B.M.Bode, M.S.Gordon
Comput.Phys.Commun. 128, 190-200 (2000)
R.M.Olson, M.W.Schmidt, M.S.Gordon, A.P.Rendell
Proc. of Supercomputing 2003, IEEE Computer Society.
This does not exist on paper, but can be downloaded at
http://www.sc-conference.org/sc2003/tech_papers.php
D.G.Fedorov, R.M.Olson, K.Kitaura, M.S.Gordon, S.Koseki
J.Comput.Chem. 25, 872-880(2004).
Basis Set References
An excellent review of the relationship between the
atomic basis used, and the accuracy with which various
molecular properties will be computed is:
E.R.Davidson, D.Feller Chem.Rev. 86, 681-696(1986).
STO-NG H-Ne Ref. 1 and 2
Na-Ar, Ref. 2 and 3 **
K,Ca,Ga-Kr Ref. 4
Rb,Sr,In-Xe Ref. 5
Sc-Zn,Y-Cd Ref. 6
1) W.J.Hehre, R.F.Stewart, J.A.Pople
J.Chem.Phys. 51, 2657-2664(1969).
2) W.J.Hehre, R.Ditchfield, R.F.Stewart, J.A.Pople
J.Chem.Phys. 52, 2769-2773(1970).
3) M.S.Gordon, M.D.Bjorke, F.J.Marsh, M.S.Korth
J.Am.Chem.Soc. 100, 2670-2678(1978).
** the valence scale factors for Na-Cl are taken
from this paper, rather than the "official"
Pople values in Ref. 2.
4) W.J.Pietro, B.A.Levi, W.J.Hehre, R.F.Stewart,
Inorg.Chem. 19, 2225-2229(1980).
5) W.J.Pietro, E.S.Blurock, R.F.Hout,Jr., W.J.Hehre, D.J.
DeFrees, R.F.Stewart Inorg.Chem. 20, 3650-3654(1980).
6) W.J.Pietro, W.J.Hehre J.Comput.Chem. 4, 241-251(1983).
MINI/MIDI H-Xe Ref. 9
9) "Gaussian Basis Sets for Molecular Calculations"
S.Huzinaga, J.Andzelm, M.Klobukowski, E.Radzio-Andzelm,
Y.Sakai, H.Tatewaki Elsevier, Amsterdam, 1984.
This book is referred to in certain circles as "the green
book" based on the color of its cover.
The MINI bases are three Gaussian expansions of each
atomic orbital. The exponents and contraction coefficients
are optimized for each element, and s and p exponents are
not constrained to be equal. As a result these bases give
much lower energies than does STO-3G. The valence MINI
orbitals of main group elements are scaled by factors
optimized by John Deisz at North Dakota State University.
Transition metal MINI bases are not scaled. The MIDI bases
are derived from the MINI sets by floating the outermost
primitive in each valence orbitals, and renormalizing the
remaining 2 gaussians. MIDI bases are not scaled by
GAMESS. The transition metal bases are taken from the
lowest SCF terms in the s**1,d**n
configurations.
3-21G H-Ne Ref. 10 (also 6-21G)
Na-Ar Ref. 11 (also 6-21G)
K,Ca,Ga-Kr,Rb,Sr,In-Xe Ref. 12
Sc-Zn Ref. 13
Y-Cd Ref. 14
10) J.S.Binkley, J.A.Pople, W.J.Hehre
J.Am.Chem.Soc. 102, 939-947(1980).
11) M.S.Gordon, J.S.Binkley, J.A.Pople, W.J.Pietro,
W.J.Hehre J.Am.Chem.Soc. 104, 2797-2803(1982).
12) K.D.Dobbs, W.J.Hehre J.Comput.Chem. 7,359-378(1986)
13) K.D.Dobbs, W.J.Hehre J.Comput.Chem. 8,861-879(1987)
14) K.D.Dobbs, W.J.Hehre J.Comput.Chem. 8,880-893(1987)
N-31G references for 4-31G 5-31G 6-31G
H 15 15 15
He 23 23 23
Li 19,24 19
Be 20,24 20
B 17 19
C-F 15 16 16
Ne 23 23
Na-Al 22
Si 21 **
P-Cl 18 22
Ar 22
K-Kr 26
15) R.Ditchfield, W.J.Hehre, J.A.Pople
J.Chem.Phys. 54, 724-728(1971).
16) W.J.Hehre, R.Ditchfield, J.A.Pople
J.Chem.Phys. 56, 2257-2261(1972).
17) W.J.Hehre, J.A.Pople J.Chem.Phys. 56, 4233-4234(1972).
18) W.J.Hehre, W.A.Lathan J.Chem.Phys. 56,5255-5257(1972).
19) J.D.Dill, J.A.Pople J.Chem.Phys. 62, 2921-2923(1975).
20) J.S.Binkley, J.A.Pople J.Chem.Phys. 66, 879-880(1977).
21) M.S.Gordon Chem.Phys.Lett. 76, 163-168(1980)
** - Note that the built in 6-31G basis for Si is
not that given by Pople in reference 22.
The Gordon basis gives a better wavefunction,
for a ROHF calculation in full atomic (Kh)
symmetry,
6-31G Energy virial
Gordon -288.828573 1.999978
Pople -288.828405 2.000280
See the input examples for how to run in Kh.
22) M.M.Francl, W.J.Pietro, W.J.Hehre, J.S.Binkley,
M.S.Gordon, D.J.DeFrees, J.A.Pople
J.Chem.Phys. 77, 3654-3665(1982).
23) Unpublished, copied out of GAUSSIAN82.
24) For Li and Be, 4-31G is actually a 5-21G expansion.
25) V.A.Rassolov, J.A.Pople, M.A.Ratner, T.L.Windus
J.Chem.Phys. 109, 1223-1229(1998)
26) A.V.Mitin, J.Baker, P.Pulay
J.Chem.Phys. 118, 7775-7782(2003) - not in GAMESS.
27) V.A.Rassolov, M.A.Ratner, J.A.Pople, P.C.Redfern,
L.A.Curtiss J.Comput.Chem. 22, 976-984(2001).
Note that reference 27 renames basis sets published earlier
as "6-31G*" in references 25 and 32. GAMESS was changed to
use the 6-31G* basis sets from reference 27 for K, Ca, and
Ga-Kr in September 2006. Sc-Zn remain those of ref. 25.
Extended basis sets
--> 6-311G
28) R.Krishnan, J.S.Binkley, R.Seeger, J.A.Pople
J.Chem.Phys. 72, 650-654(1980).
--> valence double zeta "DZV" sets:
"DH" basis - DZV for H, Li-Ne, Al-Ar
30) T.H.Dunning, Jr., P.J.Hay Chapter 1 in "Methods of
Electronic Structure Theory", H.F.Schaefer III, Ed.
Plenum Press, N.Y. 1977, pp 1-27.
Note that GAMESS uses inner/outer scale factors of
1.2 and 1.15 for DH's hydrogen (since at least 1983).
To get Thom's usual basis, scaled 1.2 throughout:
HYDROGEN 1.0 x, y, z
DH 0 1.2 1.2
DZV for K,Ca
31) J.-P.Blaudeau, M.P.McGrath, L.A.Curtiss, L.Radom
J.Chem.Phys. 107, 5016-5021(1997)
"BC" basis - DZV for Ga-Kr
32) R.C.Binning, Jr., L.A.Curtiss
J.Comput.Chem. 11, 1206-1216(1990)
Note, this basis set is available only by GBASIS=DZV, since
it is no longer considered to be the 6-31G substitute.
--> valence triple zeta "TZV" sets:
TZV for H,Li-Ne
40) T.H. Dunning, J.Chem.Phys. 55 (1971) 716-723.
TZV for Na-Ar - also known as the "MC" basis
41) A.D.McLean, G.S.Chandler
J.Chem.Phys. 72,5639-5648(1980).
TZV for K,Ca
42) A.J.H. Wachters, J.Chem.Phys. 52 (1970) 1033-1036.
(see Table VI, Contraction 3).
TZV for Sc-Zn (taken from HONDO 7)
This is Wachters' (14s9p5d) basis (ref 42) contracted
to (10s8p3d) with the following modifications
1. the most diffuse s removed;
2. additional s spanning 3s-4s region;
3. two additional p functions to describe the 4p;
4. (6d) contracted to (411) from ref 43,
except for Zn where Wachter's (5d)/[41]
and Hay's diffuse d are used.
43) A.K. Rappe, T.A. Smedley, and W.A. Goddard III,
J.Phys.Chem. 85 (1981) 2607-2611
Valence only basis sets (ECPs and MCPs)
SBKJC ECP, these are -31G splits for main group, bigger for
transition metals (available Li-Rn):
50) W.J.Stevens, H.Basch, M.Krauss
J.Chem.Phys. 81, 6026-6033 (1984)
51) W.J.Stevens, M.Krauss, H.Basch, P.G.Jasien
Can.J.Chem. 70, 612-630 (1992)
52) T.R.Cundari, W.J.Stevens
J.Chem.Phys. 98, 5555-5565(1993)
HW ECP, these are -21 splits (sp exponents not shared)
transition metals (not built in at present, although
they will work if you type them in):
53) P.J.Hay, W.R.Wadt J.Chem.Phys. 82, 270-283 (1985)
main group (available Na-Xe)
54) W.R.Wadt, P.J.Hay J.Chem.Phys. 82, 284-298 (1985)
see also
55) P.J.Hay, W.R.Wadt J.Chem.Phys. 82, 299-310 (1985)
Model core potentials (MCP):
To understand the model core potential formalism itself,
see the review articles
S.Huzinaga Can.J.Chem. 73, 619-628(1995)
M.Klobukowski, S.Huzinaga, Y.Sakai, in Computational
Chemistry: Reviews of current trends, volume 3, pp 49-74,
edited by J.Leszczynski, World Scientific, Singapore, 1999.
The MCP-xZP,MCP-AxZP,MCP-CxZP, MCP-ACxZP families:
60) Y.Sakai, E.Miyoshi, M.Klobukowski, S.Huzinaga,
"Model potentials for main group elements",
J. Chem. Phys. 106, 8084-8092 (1997).
61) E. Miyoshi, Y. Sakai, K. Tanaka, M. Masamura,
"Relativistic dsp-model core potentials for main group
elements in the fourth, fifth and sixth row
and their applications",
J. Mol. Struct. (THEOCHEM) 451, 73-79 (1998)
62) Y. Sakai, E. Miyoshi, H. Tatewaki,
"Model core potentials for the lanthanides",
J. Mol. Struct. (THEOCHEM) 451, 143-150 (1998)
63) E.Miyoshi, H.Mori, R.Hirayama, Y.Osanai, T.Noro,
H.Honda, M.Klobukowski
"Compact and efficient basis sets of s- and p-block
elements for model core potential method"
J.Chem.Phys. 122, 074104/1-8(2005)
64) M. Sekiya, T. Noro, Y. Osanai, E. Miyoshi, T. Koga,
"Relativistic Correlating Basis Sets for Lanthanide
Atoms from Ce to Lu",
J. Comput. Chem. 27, 463 (2006)
65) H. Anjima, S. Tsukamoto, H. Mori, H. Mine,
M. Klobukowski, E. Miyoshi,
"Revised Model Core Potentials of s-Block Elements",
J. Comput. Chem. 28, 2424-2430 (2007)
66) Y. Osanai, M. S. Mon, T. Noro, H. Mori,
H. Nakashima, M. Klobukowski, E. Miyoshi,
"Revised model core potentials for first-row
transition-metal atoms from Sc to Zn",
Chem. Phys. Lett. 452, 210-214 (2008)
67) Y. Osanai, E. Soejima, T. Noro, H. Mori, M. Ma San,
M. Klobukowski, E. Miyoshi,
"Revised model core potentials for second-row
transition metal atoms from Y to Cd",
Chem. Phys. Lett. 463, 230-234 (2008)
68) H. Mori, K. Ueno-Noto, Y. Osanai, T. Noro, T. Fujiwara,
M. Klobukowski, E. Miyoshi,
"Revised model core potentials for third-row
transition-metal atoms from Lu to Hg",
Chem. Phys. Lett. 476, 317-322 (2009)
the iMCP (improved model core families) are:
71) C.C.Lovallo, M.Klobukowski
J.Comput.Chem. 24, 1009-10015(2003)
72) C.C.Lovallo, M.Klobukowski
J.Comput.Chem. 25, 1206-1213(2004)
the ZFK (Zeng, Fedorov, Klobukowski) family for sp block:
72) T.Zeng, D.G.Fedorov, M. Klobukowski
J.Chem.Phys. 133, 114107/1-11 (2010)
For additional information, see also
T.Zeng, D.G.Fedorov, M. Klobukowski
J.Chem.Phys. 131, 124109/1-17 (2009)
T.Zeng, D.G.Fedorov, M.Klobukowski
J.Chem.Phys. 132, 074102/1-15 (2010)
The MCP family, built into the $DATA group only:
75) Y.Sakai, E.Miyoshi, M.Klobukowski, S.Huzinaga,
"Model potentials for molecular calculations. I.
The sd-MP set for transition metal atoms Sc-Hg",
J. Comput. Chem. 8 (1987) 226-255.
76) Y.Sakai, E.Miyoshi, M.Klobukowski, S.Huzinaga,
"Model potentials for molecular calculations. II.
The spd-MP set for transition metal atoms Sc-Hg",
J. Comput. Chem. 8 (1987) 256-264.
77) Y.Sakai, E.Miyoshi, M.Klobukowski, S.Huzinaga,
"Model potentials for main group elements",
J. Chem. Phys. 106 (1997) 8084-8092.
78) E.Miyoshi, Y.Sakai, K.Tanaka, M.Masamura
"Relativistic dsp-Model Core Potentials for Main Group
Elements in the 4th, 5th, and 6th-Row and Applications"
J. Mol. Struct. (Theochem), 451 (1998) 73-79.
79) Y.Sakai, E.Miyoshi, H.Tatewaki
"Model Core Potentials for the Lanthanides"
J. Mol. Struct. (Theochem), 451 (1998) 143-150.
Systematic basis set families:
Polarization Consistent basis sets (PCseg-n):
The segmented contractions which are internally stored in
GAMESS are described in this paper:
81) F.Jensen, J.Chem.Theory Comp. 10, 1074-1085(2014)
Papers describing the older general contractions are:
F.Jensen J.Chem.Phys. 115, 9113-9125(2001).
erratum J.Chem.Phys. 116, 3502(2002).
F.Jensen J.Chem.Phys. 116, 7372-7379(2002).
F.Jensen J.Chem.Phys. 117, 9234-9240(2002).
F.Jensen J.Chem.Phys. 118, 2459-2463(2003).
F.Jensen, T.Helgaker J.Chem.Phys. 121, 3463-3470(2004).
F.Jensen, J. Phys. Chem. A 111, 11198-11204(2007)
F.Jensen, J. Chem. Phys. 136, 114107(2012)
F.Jensen, J. Chem. Phys. 138, 014107(2013)
Correlation Consistent bases (CCn, ACCn, etc.):
The GAMESS keyword and official names for these "Dunning-
style" basis sets are,
CCn=cc-pVnZ, ACCn=aug-cc-pVnZ, n=D,T,Q,5,...
CCnC=cc-pCVnZ, ACCnC=aug-cc-pCVnZ,
CCnWC=cc-pwCVnZ, ACCnWC=aug-cc-pwCVnZ (w=?omega?).
See $BASIS for important information about Al-Ar?s bases,
where the GAMESS keyword invokes ?tight d? (n+d) sets.
Please see the Pacific Northwest National Laboratory web
page http://www.emsl.pnl.gov/forms/basisform.html for
references to these basis sets. Kirk Peterson's very
thorough bibliography can be found at
http://tyr0.chem.wsu.edu/~kipeters/basis-bib.html
Sapporo (SPK) basis set family
first, the non-relativistic valence sets,
S1. H.Tatewaki, T.Koga J.Chem.Phys. 104, 8493(1996)
S2. H.Tatewaki, T.Koga, H.Takashima
Theoret.Chem.Acc. 96, 243(1997)
S3. T.Koga, H.Tatewaki, Y.Satoh
Theoret.Chem.Acc. 102, 105(1999)
S4. T.Koga, S.Yamamoto, T.Shimazaki, H.Tatewaki,
Theoret.Chem.Acc. 108, 41(2002)
then, the relativistic valence sets,
S6. T.Noro, M.Sekiya, T.Koga, S.L.Saito
Chem.Phys.Lett. 481, 229-233(2009)
core/valence relativistic and non-relativistic:
S7. T.Noro, M.Sekiya, T.Koga (main group)
Theoret.Chem.Acc. 131, 1124(2012)
S8. M.Sekiya, T.Noro, T.Koga, T.Shimuzaki (lanthanides)
Theoret.Chem.Acc. 131, 1247(2012)
Karlsruhe basis sets (group of Reinhart Ahlrichs)
91) A.Schaefer, H.Horn, R.Ahlrichs
J.Chem. Phys. 97,2571 (1992).
92) A.Schaefer, C.Huber, R.Ahlrichs
J.Chem. Phys. 100, 5829 (1994).
Polarization exponents:
STO-NG*
100) J.B.Collins, P. von R. Schleyer, J.S.Binkley,
J.A.Pople J.Chem.Phys. 64, 5142-5151(1976).
3-21G*. See also reference 12.
101) W.J.Pietro, M.M.Francl, W.J.Hehre, D.J.DeFrees, J.A.
Pople, J.S.Binkley J.Am.Chem.Soc. 104,5039-5048(1982)
6-31G* and 6-31G**. See also reference 22 above.
102) P.C.Hariharan, J.A.Pople
Theoret.Chim.Acta 28, 213-222(1973)
multiple polarization, and f functions
103) M.J.Frisch, J.A.Pople, J.S.Binkley J.Chem.Phys.
80, 3265-3269 (1984).
Anion diffuse functions:
3-21+G, 3-21++G, etc.
105) T.Clark, J.Chandrasekhar, G.W.Spitznagel, P. von R.
Schleyer J.Comput.Chem. 4, 294-301(1983)
106) G.W.Spitznagel, Diplomarbeit, Erlangen, 1982.
------------
STO-NG* means d orbitals are used on third row atoms only.
The original paper (ref 100) suggested z=0.09 for
Na and Mg, and z=0.39 for Al-Cl.
We prefer to use the same exponents as are used
in 3-21G* and 6-31G*, so we know we're looking
at changes in the sp basis, not the d exponent.
3-21G* means d orbitals on main group elements in the
third and higher periods. Not defined for the
transition metals, where there are p's already
in the basis. Except for alkalis and alkali
earths, the 4th and 5th row zetas are from
Huzinaga, et al. (ref 9). The exponents are
normally the same as for 6-31G*.
6-31G* means d orbitals on second and third row atoms.
We use Mark Gordon's z=0.395 for Silicon, as well
as his fully optimized sp basis (ref 21).
This is often written 6-31G(d) today.
For the first row transition metals, the *
means an f function is added. The transition
metal 3d 6-31G orbital is NOT of triple zeta
quality, and thus is probably not very accurate.
6-31G** means the same as 6-31G*, except that p functions
are added on hydrogens.
This is often written 6-31G(d,p) today.
6-311G** means p orbitals on H, and d orbitals elsewhere.
The exponents were derived from correlated atomic
states, and so are considerably tighter than the
polarizing functions used in 6-31G**, etc.
This is often written 6-311G(d,p) today.
The exponents for 6-31G* for C-F are disturbing, in
that each atom has exactly the same value. Dunning and Hay
(ref 30) have recommended a better set of exponents for
second row atoms and a slightly different value for H.
2p, 3p, 2d, 3p polarization sets are usually thought of
as arising from applying splitting factors to the 1p and 1d
values. For example, SPLIT2=2.0, 0.5 means to double and
halve the single value. The default values for SPLIT2 and
SPLIT3 are taken from reference 103, and were derived with
correlation in mind. The SPLIT2 values often produce a
higher (!) HF energy than the singly polarized run, because
the exponents are split too widely. SPLIT2=0.4,1.4 will
always lower the SCF energy (the values are the unpublished
personal preference of MWS), and for SPLIT3 we might
suggest 3.0,1.0,1/3.
With all this as background, we are ready to present
the tables of polarization exponents that are built into
GAMESS. Please note that the names associated with each
column are only generally descriptive. The column marked
"COMMON" is obtained from both Pople (mostly his 6-31G, but
using Gordon's value for Silicon) and Huzinaga (from the
"green book"). The exponents for K-Kr under "Dunning" are
from Curtiss, et al., not Thom Dunning, and so on. The
exponents are for d functions unless otherwise indicated.
Polarization exponents, chosen by POLAR= in $BASIS:
COMMON POPN31 POPN311 DUNNING HUZINAGA HONDO7
------ ------ ------- ------- -------- ------
H 1.1(p) 0.75(p) 1.0(p) 1.0(p) 1.0(p)
He 1.1(p) 0.75(p) 1.0(p) 1.0(p) 1.0(p)
Li 0.2 0.200 0.076(p)
Be 0.4 0.255 0.164(p) 0.32
B 0.6 0.401 0.70 0.388 0.50
C 0.8 0.626 0.75 0.600 0.72
N 0.8 0.913 0.80 0.864 0.98
O 0.8 1.292 0.85 1.154 1.28
F 0.8 1.750 0.90 1.496 1.62
Ne 0.8 2.304 1.00 1.888 2.00
Na 0.175 0.061(p) 0.157
Mg 0.175 0.101(p) 0.234
Al 0.325 0.198 0.311
Si 0.395 0.262 0.388
P 0.55 0.340 0.465
S 0.65 0.421 0.542
Cl 0.75 0.514 0.619
Ar 0.85 0.617 0.696
K 0.2 0.04485 0.260 0.039(p)
Ca 0.2 0.0502 0.229 0.059(p)
Sc-Zn N/A 0.8(f) N/A N/A N/A N/A
Ga 0.207 0.2289 0.141
Ge 0.246 0.2772 0.202
As 0.293 0.3277 0.273
Se 0.338 0.3810 0.315
Br 0.389 0.4366 0.338
Kr 0.443 0.4948 0.318
Rb 0.11 0.034(p)
Sr 0.11 0.048(p)
A blank means the value equals the "COMMON" column.
Common d polarization for all sets ("green book"):
In Sn Sb Te I Xe
0.160 0.183 0.211 0.237 0.266 0.297
Tl Pb Bi Po At Rn
0.146 0.164 0.185 0.204 0.225 0.247
see f exponents on next page...
f polarization functions, from reference 103:
Li Be B C N O F Ne
0.15 0.26 0.50 0.80 1.00 1.40 1.85 2.50
Na Mg Al Si P S Cl Ar
0.15 0.20 0.25 0.32 0.45 0.55 0.70 --
Anions usually require diffuse basis functions to
properly represent their spatial diffuseness. The use of
diffuse sp shells on atoms in the second and third rows is
denoted by a + sign, also adding diffuse s functions on
hydrogen is symbolized by ++. These designations can be
applied to any of the Pople bases, e.g. 3-21+G, 3-21+G*,
6-31++G**. The following exponents are for L shells,
except for H. For H-F, they are taken from ref 105. For
Na-Cl, they are taken directly from reference 106. These
values may be found in footnote 13 of reference 103. For
Ga-Br, In-I, and Tl-At these were optimized for the atomic
ground state anion, using ROHF with a flexible ECP basis
set, by Ted Packwood at NDSU.
H
0.0360
Li Be B C N O F
0.0074 0.0207 0.0315 0.0438 0.0639 0.0845 0.1076
Na Mg Al Si P S Cl
0.0076 0.0146 0.0318 0.0331 0.0348 0.0405 0.0483
Ga Ge As Se Br
0.0205 0.0222 0.0287 0.0318 0.0376
In Sn Sb Te I
0.0223 0.0231 0.0259 0.0306 0.0368
Tl Pb Bi Po At
0.0170 0.0171 0.0215 0.0230 0.0294
Additional information about diffuse functions and also
Rydberg type exponents can be found in reference 30.
The following atomic energies are UHF (RHF on 1-S
states), p orbitals are not symmetry equivalent, using the
default scale factors. They may be useful in picking a
basis of the desired accuracy.
Atom state STO-2G STO-3G 3-21G 6-31G
H 2-S -.454397 -.466582 -.496199 -.498233
He 1-S -2.702157 -2.807784 -2.835680 -2.855160
Li 2-S -7.070809 -7.315526 -7.381513 -7.431236
Be 1-S -13.890237 -14.351880 -14.486820 -14.566764
B 2-P -23.395284 -24.148989 -24.389762 -24.519492
C 3-P -36.060274 -37.198393 -37.481070 -37.677837
N 4-S -53.093007 -53.719010 -54.105390 -54.385008
O 3-P -71.572305 -73.804150 -74.393657 -74.780310
F 2-P -95.015084 -97.986505 -98.845009 -99.360860
Ne 1-S -122.360485 -126.132546 -127.803825 -128.473877
Na 2-S -155.170019 -159.797148 -160.854065 -161.841425
Mg 1-S -191.507082 -197.185978 -198.468103 -199.595219
Al 2-P -233.199965 -239.026471 -240.551046 -241.854186
Si 3-P -277.506857 -285.563052 -287.344431 -288.828598
P 4-S -327.564244 -336.944863 -339.000079 -340.689008
S 3-P -382.375012 -393.178951 -395.551336 -397.471414
Cl 2-P -442.206260 -454.546015 -457.276552 -459.442939
Ar 1-S -507.249273 -521.222881 -524.342962 -526.772151
Atom state DH 6-311G MC SCF limit*
H 2-S -.498189 -.499810 -- -0.5
He 1-S -- -2.859895 -- -2.861680
Li 2-S -7.431736 -7.432026 -- -7.432727
Be 1-S -14.570907 -14.571874 -- -14.573023
B 2-P -24.526601 -24.527020 -- -24.529061
C 3-P -37.685571 -37.686024 -- -37.688619
N 4-S -54.397260 -54.397980 -- -54.400935
O 3-P -74.802707 -74.802496 -- -74.809400
F 2-P -99.395013 -99.394158 -- -99.409353
Ne 1-S -128.522354 -128.522553 -- -128.547104
Na 2-S -- -- -161.845587 -161.858917
Mg 1-S -- -- -199.606558 -199.614636
Al 2-P -241.855079 -- -241.870014 -241.876699
Si 3-P -288.829617 -- -288.847782 -288.854380
P 4-S -340.689043 -- -340.711346 -340.718798
S 3-P -397.468667 -- -397.498023 -397.504910
Cl 2-P -459.435938 -- -459.473412 -459.482088
Ar 1-S -- -- -526.806626 -526.817528
* M.W.Schmidt and K.Ruedenberg, J.Chem.Phys. 71,
3951-3962(1979). These are ROHF energies in Kh symmetry.
H-Xe can be found in Phys.Rev.A 46, 3691-3696(1992).
Spherical Harmonics
The implementation of ISPHER in $CONTRL does not rely
on using a spherical harmonic basis set, in fact the atomic
basis remains the Cartesian Gaussians. Instead, certain
MOs formed from particular combinations of the Cartesian
Gaussians (for example, xx+yy+zz) are deleted from the MO
space. Thus a run with ISPHER=1 will have fewer MOs than
AOs. Since neither the occupied nor virtual MOs contain
any admixture of xx+yy+zz, the resulting energy and wave-
function is exactly equivalent to the use of a spherical
harmonic basis.
The log file output will contain expansions of each MO
in terms of 6 d's, 10 f's, and 15 g's, and the $VEC also
contains the same expansion over Cartesian Gaussians. Both
the matrix in your log file and in $VEC will contain fewer
MOs than AOs, the exact number of MOs used is printed in
the initial guess section of the log file. It should be
possible to read such $VEC groups into runs with different
settings of ISPHER, should you choose to do so.
The advantage of this approach is that intelligence in
the generation of symmetry orbitals combined with the
capability to drop linearly dependent MO combinations means
that the details of ISPHER are located only in the orbital
optimization code, where the variational spaces are simply
reduced in size to eliminate the undesired contaminant
functions. This means that none of the integral routines
need be modified, as the atomic basis remains the Cartesian
Gaussians. The disadvantage is that AO integral files run
over the Cartesian Gaussians, and thus are not reduced in
size. Of course transformed MO integrals and various
computations in correlated calculations are reduced in
size, since the number of MOs may be greatly reduced.
Computationally, the advantages of ISPHER=1 are not
limited to the reduced CPU time associated with fewer total
MOs. Questions about d orbital participation as measured
by Mulliken populations are cleanly addressed when the d's
usage in the MOs does not contain any contamination from
the s shape xx+yy+zz. Less obviously, the use of spherical
harmonics frequently greatly reduces problems with linear
dependency, that exhibit as poor SCF convergence.
How to do RHF, ROHF, UHF, and GVB calculations
general considerations
These four SCF wavefunctions are all based on Fock
operator techniques, even though some GVB runs use more
than one determinant. Thus all of these have an intrinsic
N**4 time dependence, because they are all driven by
integrals in the AO basis. This similarity makes it
convenient to discuss them all together. In this section
we will use the term HF to refer generically to any of
these four wavefunctions, including the multi-determinate
GVB-PP functions. $SCF is the main input group for all
these HF wavefunctions.
As will be discussed below, in GAMESS the term ROHF
refers to high spin open shell SCF only, but other open
shell coupling cases are possible using the GVB code.
Analytic gradients are implemented for every possible
HF type calculation possible in GAMESS, and therefore
numerical hessians are available for each.
Analytic hessian calculation is implemented for RHF,
ROHF, and any GVB case with NPAIR=0 or NPAIR=1. Analytic
hessians are more accurate, and much more quickly computed
than numerical hessians, but require additional disk
storage to perform an integral transformation, and also
more physical memory.
The second order Moller-Plesset energy correction (MP2)
is implemented for RHF, UHF, ROHF, and MCSCF wavefunctions.
Analytic gradients may be obtained for MP2 with RHF, UHF,
or ROHF reference wavefunctions, and MP2 level properties
are therefore available for these, see MP2PRP in $MP2. All
other cases give properties for the SCF function.
Direct SCF is implemented for every possible HF type
calculation. The direct SCF method may not be used with
DEM convergence. Direct SCF may be used during energy,
gradient, numerical or analytic hessian, CI or MP2 energy
correction, or localized orbital computations.
direct SCF
Normally, HF calculations proceed by evaluating a large
number of two electron repulsion integrals, and storing
these on a disk. This integral file is read in once during
each HF iteration to form the appropriate Fock operators.
In a direct HF, the integrals are not stored on disk, but
are instead reevaluated during each HF iteration. Since
the direct approach *always* requires more CPU time, the
default for DIRSCF in $SCF is .FALSE.
Even though direct SCF is slower, there are at least
two reasons why you may want to consider using it. The
first is that it may not be possible to store all of the
integrals on the disk drives attached to your computer.
Second, what you are really interested in is reducing the
wall clock time to obtain your answer, not the CPU time.
Workstations, particularly nodes with multiple CPUs and
only one disk subsystem, may have modest hardware I/O
capabilities. Other environments such as a mainframe
shared by many users may also have very poor CPU/wall clock
performance for I/O bound jobs such as conventional HF.
You can estimate the disk storage requirements for
conventional HF using a P or PK file by the following
formulae:
nint = 1/sigma * 1/8 * N**4
Mbytes = nint * x / 1024**2
Here N is the total number of basis functions in your run,
which you can learn from an EXETYP=CHECK run. The 1/8
accounts for permutational symmetry within the integrals.
Sigma accounts for the point group symmetry, and is
difficult to estimate accurately. Sigma cannot be smaller
than 1, in no symmetry (C1) calculations. For benzene,
sigma would be almost six, since you generate 6 C's and 6
H's by entering only 1 of each in $DATA. For water sigma
is not much larger than one, since most of the basis set is
on the unique oxygen, and the C2v symmetry applies only to
the H atoms. The factor x is 12 bytes per integral for
basis sets smaller than 255, and 16 otherwise. Finally,
since integrals that are very close to zero need not be
stored on disk, the actual power dependence is not as bad
as N**4, and in fact in the limit of very large molecules
can be as low as N**2. Thus plugging in sigma=1 should
give you an upper bound to the actual disk space needed.
If the estimate exceeds your available disk storage, your
only recourse is direct HF.
What are the economics of direct HF? Naively, if we
assume the run takes 10 iterations to converge, we must
spend 10 times more CPU time computing the integrals on
each iteration. However, we do not have to waste any CPU
time reading blocks of integrals from disk, or in unpacking
their indices. We also do not have to waste any wall clock
time waiting for a relatively slow mechanical device such
as a disk to give us our data.
There are some less obvious savings too, as first noted
by Almlof. First, since the density matrix is known while
we are computing integrals, we can use the Schwarz
inequality to avoid doing some of the integrals. In a
conventional SCF this inequality is used to avoid doing
small integrals. In a direct SCF it can be used to avoid
doing integrals whose contribution to the Fock matrix is
small (density times integral=small). Secondly, we can
form the Fock matrix by calculating only its change since
the previous iteration. The contributions to the change in
the Fock matrix are equal to the change in the density
times the integrals. Since the change in the density goes
to zero as the run converges, we can use the Schwarz
screening to avoid more and more integrals as the
calculation progresses. The input option FDIFF in $SCF
selects formation of the Fock operator by computing only
its change from iteration to iteration. The FDIFF option
is not implemented for GVB since there are too many density
matrices from the previous iteration to store, but is the
default for direct RHF, ROHF, and UHF.
So, in our hypothetical 10 iteration case, we do not
spend as much as 10 times more time in integral evaluation.
Additionally, the run as a whole will not slow down by
whatever factor the integral time is increased. A direct
run spends no additional time summing integrals into the
Fock operators, and no additional time in the Fock
diagonalizations. So, generally speaking, a RHF run with
10-15 iterations will slow down by a factor of 2-4 times
when run in direct mode. The energy gradient time is
unchanged by direct HF, and this is a large time compared
to HF energy, so geometry optimizations will be slowed down
even less. This is really the converse of Amdahl's law:
if you slow down only one portion of a program by a large
amount, the entire program slows down by a much smaller
factor.
To make this concrete, here are some times for GAMESS
for a job which is a RHF energy for a SbC4O2NH4. These
timings were obtained an extremely long time ago, on a
DECstation 3100 under Ultrix 3.1, which was running only
these tests, so that the wall clock times are meaningful.
This system is typical of Unix workstations in that it uses
SCSI disks, and the operating system is not terribly good
at disk I/O. By default GAMESS stores the integrals on
disk in the form of a P supermatrix, because this will save
time later in the SCF cycles. By choosing NOPK=1 in
$INTGRL, an ordinary integral file can be used, which
typically contains many fewer integrals, but takes more CPU
time in the SCF. Because the DECstation is not terribly
good at I/O, the wall clock time for the ordinary integral
file is actually less than when the supermatrix is used,
even though the CPU time is longer. The run takes 13
iterations to converge, the times are in seconds.
P supermatrix ordinary file
# nonzero integrals 8,244,129 6,125,653
# blocks skipped 55,841 55,841
CPU time (ints) 709 636
CPU time (SCF) 1289 1472
CPU time (job total) 2123 2233
wall time (job total) 3468 3200
When the same calculation is run in direct mode (integrals
are processed like an ordinary integral disk file when
running direct),
iteration 1: FDIFF=.TRUE. FDIFF=.FALSE.
# nonzero integrals 6,117,416 6,117,416
# blocks skipped 60,208 60,208
iteration 13:
# nonzero integrals 3,709,733 6,122,912
# blocks skipped 105,278 59,415
CPU time (job total) 6719 7851
wall time (job total) 6764 7886
Note that elimination of the disk I/O dramatically
increases the CPU/wall efficiency. Here's the bottom line
on direct HF:
best direct CPU / best disk CPU = 6719/2123 = 3.2
best direct wall/ best disk wall= 6764/3200 = 2.1
Direct SCF is slower than conventional disk SCF, but not
outrageously so! From the data in the tables, we can see
that the best direct method spends about 6719-1472 = 5247
seconds doing integrals. This is an increase of about
5247/636 = 8.2 in the time spent doing integrals, in a run
that does 13 iterations (13 times evaluating integrals).
8.2 is less than 13 because the run avoids all CPU charges
related to I/O, and makes efficient use of the Schwarz
inequality to avoid doing many of the integrals in its
final iterations.
convergence accelerators
Generally speaking, the simpler the HF function, the
better its convergence. In our experience, the majority of
RHF and ROHF runs converge readily from GUESS=HUCKEL. UHF
often takes considerably more iterations than either of
these, due to the extremely common occurrence of heavy spin
contamination. GVB runs typically require GUESS=MOREAD,
although the Huckel guess usually works for NPAIR=0. GVB
cases with NPAIR greater than one are particularly
difficult.
Unfortunately, not all HF runs converge readily. The
best way to improve your convergence is to provide better
starting orbitals! In many cases, this means to MOREAD
orbitals from some simpler HF case. For example, if you
want to do a doublet ROHF, and the HUCKEL guess does not
seem to converge, do this: Do an RHF on the +1 cation. RHF
is typically more stable than ROHF, UHF, or GVB, and
cations are usually readily convergent. Then MOREAD the
cation's orbitals into the neutral calculation which you
wanted to do at first.
GUESS=HUCKEL does not always start with the correct
electronic configuration. It may be useful to use PRTMO in
$GUESS during a CHECK run to examine the starting orbitals,
and then reorder them with NORDER if that seems
appropriate.
Of course, by default GAMESS uses the convergence
procedures which are usually most effective. Still, there
are cases which are difficult, so the $SCF group permits
you to select several alternative methods for improving
convergence. Briefly, these are
EXTRAP. This extrapolates the three previous Fock
matrices, in an attempt to jump ahead a bit faster. This
is the most powerful of the old-fashioned accelerators, and
normally should be used at the beginning of any SCF run.
When an extrapolation occurs, the counter at the left of
the SCF printout is set to zero.
DAMP. This damps the oscillations between several
successive Fock matrices. It may help when the energy is
seen to oscillate wildly. Thinking about which orbitals
should be occupied initially may be an even better way to
avoid oscillatory behaviour.
SHIFT. Level shifting moves the diagonal elements of
the virtual part of the Fock matrix up, in an attempt to
uncouple the unoccupied orbitals from the occupied ones.
At convergence, this has no effect on the orbitals, just
their orbital energies, but will produce different (and
hopefully better) orbitals during the iterations.
RSTRCT. This limits mixing of the occupied orbitals
with the empty ones, especially the flipping of the HOMO
and LUMO to produce undesired electronic configurations or
states. This should be used with caution, as it makes it
very easy to converge on incorrectly occupied electronic
configurations, especially if DIIS is also used. If you
use this, be sure to check your final orbital energies to
see if they are sensible. A lower energy for an unoccupied
orbital than for one of the occupied ones is a sure sign of
problems.
DIIS. Direct Inversion in the Iterative Subspace is a
modern method, due to Pulay, using stored error and Fock
matrices from a large number of previous iterations to
interpolate an improved Fock matrix. This method was
developed to improve the convergence at the final stages of
the SCF process, but turns out to be quite powerful at
forcing convergence in the initial stages of SCF as well.
By giving ETHRSH as 10.0 in $SCF, you can practically
guarantee that DIIS will be in effect from the first
iteration. The default is set up to do a few iterations
with conventional methods (extrapolation) before engaging
DIIS. This is because DIIS can sometimes converge to
solutions of the SCF equations that do not have the lowest
possible energy. For example, the 3-A-2 small angle state
of SiLi2 (see M.S.Gordon and M.W.Schmidt, Chem.Phys.Lett.,
132, 294-8(1986)) will readily converge with DIIS to a
solution with a reasonable S**2, and an energy about 25
milliHartree above the correct answer. A SURE SIGN OF
TROUBLE WITH DIIS IS WHEN THE ENERGY RISES TO ITS FINAL
VALUE. However, if you obtain orbitals at one point on a
PES without DIIS, the subsequent use of DIIS with MOREAD
will probably not introduce any problems. Because DIIS is
quite powerful, EXTRAP, DAMP, and SHIFT are all turned off
once DIIS begins to work. DEM and RSTRCT will still be in
use, however.
SOSCF. Approximate second-order (quasi-Newton) SCF
orbital optimization. SOSCF will converge about as well as
DIIS at the initial geometry, and slightly better at
subsequent geometries. There's a bit less work solving the
SCF equations, too. The method kicks in after the orbital
gradient falls below SOGTOL. Some systems, particularly
transition metals with ECP basis sets, may have Huckel
orbitals for which the gradient is much larger than SOGTOL.
In this case it is probably better to use DIIS instead,
with a large ETHRSH, rather than increasing SOGTOL, since
you may well be outside the quadratic convergence region.
SOSCF does not exhibit true second order convergence since
it uses an approximation to the inverse hessian. SOSCF
will work for MOPAC runs, but is slower in this case. SOSCF
will work for UHF, but its convergence may be better than
DIIS. SOSCF will work for non-Abelian cases, but may
encounter problems if the open shell is degenerate.
It should be clear that SOSCF and DIIS are the two
work-horse convergers, with DAMP (and possibly SHIFT)
useful in cases where the initial guess is such that these
two are not engaged immediately. If you compute many
different types of molecules, you will find cases where
SOSCF works but DIIS does not, but also cases where DIIS
works but SOSCF does not (although often both will work).
If you do not obtain convergence with one of these, try the
other one! If you still have problems, attempt to get
better starting orbitals.
DEM. Direct energy minimization should be your last
recourse. It explores the "line" between the current
orbitals and those generated by a conventional change in
the orbitals, looking for the minimum energy on that line.
DEM should always lower the energy on every iteration, but
is very time consuming, since each of the points considered
on the line search requires evaluation of a Fock operator.
DEM will be skipped once the density change falls below
DEMCUT, as the other methods should then be able to affect
final convergence. While DEM is working, RSTRCT is held
to be true, regardless of the input choice for RSTRCT.
Because of this, it behooves you to be sure that the
initial guess is occupying the desired orbitals. DEM is
available only for RHF. The implementation in GAMESS
resembles that of R.Seeger and J.A.Pople, J.Chem.Phys. 65,
265-271(1976). Simultaneous use of DEM and DIIS resembles
the ADEM-DIOS method of H.Sellers, Chem.Phys.Lett. 180,
461-465(1991). DEM does not work with direct SCF.
high spin open shell SCF (ROHF)
Open shell SCF calculations are performed in GAMESS by
both the ROHF code and the GVB code. Note that when the
GVB code is executed with no pairs, the run is NOT a true
GVB run, and should be referred to in publications and
discussion as a ROHF calculation. Low spin couplings are
possible with the GVB program.
The ROHF module in GAMESS can handle any number of open
shell electrons, provided these have a high spin coupling.
For example: one open shell, doublet:
$CONTRL SCFTYP=ROHF MULT=2 $END
two open shells, triplet:
$CONTRL SCFTYP=ROHF MULT=3 $END
m open shells, high spin:
$CONTRL SCFTYP=ROHF MULT=m+1 $END
John Montgomery (who was then at United Technologies)
is responsible for the ROHF implementation in GAMESS. The
following discussion is due to him, dating from 1988 when
his method of forming a combined Fock operator was included
in GAMESS. Other choices (Euler and two "canonical" sets)
were added to the table in 2009/2010.
The Fock matrix in the MO basis has the form
closed open virtual
closed F2 | Fb | (Fa+Fb)/2
-----------------------------------
open Fb | F1 | Fa
-----------------------------------
virtual (Fa+Fb)/2 | Fa | F0
where Fa and Fb are the usual alpha and beta Fock matrices
any UHF program produces. All ROHF methods agree on these,
as they are the variational conditions that separate the
doubly occupied, alpha occupied, and empty orbital spaces.
The diagonal blocks can be written
F2 = Acc*Fa + Bcc*Fb
F1 = Aoo*Fa + Boo*Fb
F0 = Avv*Fa + Bvv*Fb
Some choices for the canonicalization coefficients to
define the diagonal blocks are
Acc Bcc Aoo Boo Avv Bvv
Guest and Saunders 1/2 1/2 1/2 1/2 1/2 1/2
Roothaan single matrix -1/2 3/2 1/2 1/2 3/2 -1/2
Davidson/1988 1/2 1/2 1 0 1 0
Binkley, Pople, Dobosh 1/2 1/2 1 0 0 1
McWeeny and Diercksen 1/3 2/3 1/3 1/3 2/3 1/3
Faegri and Manne 1/2 1/2 1 0 1/2 1/2
GVB program/Euler 1/2 1/2 1/2 0 1/2 1/2
"canonical 1" 0 1 1 0 1 0
"canonical 2" (2S+1)/2S -1/2S 0 1 -1/2S (2S+1)/2S
See below for how these last two rows connect to ionization
events.
The 1988 GAMESS ROHF program using a now deleted
Davidson-type ROHF produced final orbitals matching the
line "1988" above. This differs from the choices made in
Davidson's own MELD program. The MELD program itself has
always done a "cleanup" of the virtual space, after
convergence, using Avv=Bvv=1/2, producing orbitals which
are the same as Faegri/Manne. If MELD's MP2 option is
chosen, the occupied space is also altered after
convergence, using Aoo=Boo=1/2, which is the Guest/Saunders
line above. Thus the term "Davidson orbitals" is used here
to refer to the behavior of the now-deleted 1988 ROHF code
in GAMESS, which didn't have either type of final orbital
cleanup.
The choice of the diagonal blocks is arbitrary, as ROHF
is converged when the off diagonal blocks go to zero. The
exact choice for these blocks can however have an effect on
the convergence rate. This choice also affects the MO
coefficients, and orbital energies, as the different
choices produce different canonical orbitals within the
three subspaces. All methods, however, will give identical
total wavefunctions, and hence identical properties such as
nuclear gradients and hessians. Some of the perturbation
theories for open shell cases are defined in terms of a
particular canonicalization, if so, GAMESS automatically
canonicalizes after convergence so the desired orbitals and
energies are given to the perturbation theory codes.
The default coupling case in GAMESS is the Roothaan
single matrix set. Note that pre-1988 versions of GAMESS
produced "Davidson/1988" orbitals. If you would like to
fool around with any of these other canonicalizations, the
Acc, Aoo, Avv and Bcc, Boo, Bvv parameters can be input as
the first three elements of ALPHA and BETA in $SCF. For
example, the McWeeny/Diercksen canonicalization, in double
precision, is obtained by
$scf couple=.true. alpha(1)=0.333333333333333333,
0.333333333333333333,
0.666666666666666667
beta(1)=0.666666666666666667,
0.333333333333333333,
0.333333333333333333 $end
Here is some idea of the range of eigenvalues that
result from using the various canonicalization schemes in
the table above. The system is 6-31G nitrogen atom, all
runs give E= -54.3820511123 (matching all 10 decimals):
1s 2s 2p "3p" "3s"
Roothaan -15.5514 -0.5306 -0.1774 +0.7666 +0.8704
McWeeny,Diercksen -15.6214 -0.8745 -0.1183 +0.8984 +0.9684
Davidson -15.6355 -0.9432 -0.5657 +0.8457 +0.9292
Guest,Saunders -15.6355 -0.9432 -0.1774 +0.9248 +0.9880
Binkley,Pople,Dob. -15.6355 -0.9432 -0.5657 +1.0039 +1.0469
Faegri,Manne -15.6355 -0.9432 -0.5657 +0.9248 +0.9880
GVB (aka Euler) -15.6355 -0.9432 -0.2828 +0.9248 +0.9880
"canonical 1" -15.5933 -0.7370 -0.5657 +0.8457 +0.9292
"canonical 2" -15.7065 -1.2863 +0.2109 +1.0567 +1.0861
Since the ionization potential (IP) for a 2p electron
in Nitrogen is 0.53 Hartree, it is clear that most of the
orbital energies above do not approximately predict this
IP. Recent work by Boris Plakhutin and co-workers (see the
3 papers in the ROHF references above) leads to two sets of
orbitals and eigenvalues, for the prediction of both IP and
electron affinities (EA), for various ionization events,
starting from a high spin ROHF state:
canonical 1 (to produce high spin final states)
A1 = remove beta e- from filled space,
B1 = remove alpha e- from open shell,
C1 = attach alpha e- to virtual space.
canonical 2 (to produce low spin final states)
A2 = remove alpha e- from filled space,
B2 = attach beta e- to filled space,
C2 = attach beta e- to virtual space.
Correct handling of spin requires the value of the spin S
of the initial state in the 2nd canonicalization set. Two
different ROHF runs are necessary to get all six EA and IP
processes.
Additional discussion about ROHF orbital energies may
be found in
K.R.Glaesemann, M.W.Schmidt
J.Phys.Chem.A 114, 8772-8777(2010) [available on-line]
along with a demonstration of the non-uniqueness of ROHF-
based perturbation theories.
High-spin ROHF results are automatically obtained if a
UHF calculation is constrained so that the occupied beta
orbitals lie entirely within the occupied alpha orbital
space. Such a constraint is called CUHF, and while it will
appear to have different alpha/beta orbitals, and
alpha/beta eigenvalues, its spin expectation value will be
=2S+1, and its energy will be exactly that of ROHF.
The CUHF method is given by
G.E.Scuseria, T.Tsuchimochi
J.Chem.Phys. 134, 064101/1-14(2011)
CUHF runs as a UHF calculation, solving two distinct SCF
equations in the alpha and beta space, so its convergence
behavior and its final eigenvalues will differ from that
for any of the ROHF canonicalizations. See the section on
perturbation theory in this chapter regarding CUHF's
perturbation theory, CUMP2.
other open shell SCF cases (GVB)
Genuine GVB-PP "perfect pairing" runs will be discussed
later in this section. First, we will consider how to do
open shell SCF with the GVB part of the program, with
NPAIR=0. This is an extremely powerful open shell SCF
program, capable of handling low spin couplings and/or
degenerate open shells, with the energy formula
E = 2 sum F-i*h-ii + sum sum ALPHA-ij*J-ij + BETA-ij*Kij
i i j
Here F-i (meaning subscript i) is a fractional occupancy,
and is 1.0 for the filled shell that is usually present
below the open shell(s). A d**2 shell has F-i=0.2. The
h,J,K are the usual one electron, coulomb, and exchange
operators. For a few common cases, the F, ALPHA, BETA are
internally stored in the program,
one open shell, doublet:
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO(1)=1 $END
two open shells, triplet:
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=2 NO(1)=1,1 $END
two open shells, singlet:
$CONTRL SCFTYP=GVB MULT=1 $END
$SCF NCO=xx NSETO=2 NO(1)=1,1 $END
Note that the first two cases duplicate runs which the
high spin ROHF module can readily do. The last case is
also an ROHF calculation, albeit for a low-spin coupling.
One should note that the open shell singlet case is a
variational calculation IF AND ONLY IF the two singly
degenerate orbitals have different space symmetry.
A publication should refer to any run that has NPAIR=0
as being ROHF type, rather than GVB, since there are no
perfect pairs in use. Note, however, that the GVB program
can, if you wish, have both open shells and genuine valence
bond pairs at the same time! See the following section
about the use of GVB's perfect pairs. The use of open
shells and pairs is illustrated in one of the GAMESS
standard example inputs.
If you would like to do any cases other than those
shown above, including many cases with orbital degeneracy,
you must derive the coupling coefficients ALPHA and BETA,
and input them with the occupancies F in the $SCF group.
Fortunately, many cases can be looked up!
Mariusz Klobukowski of the University of Alberta has
shown how to obtain coupling coefficients for the GVB open
shell program for many such open shell states. These can
be obtained from Appendix A of the book "A General SCF
Theory" by Ramon Carbo and Josep M. Riera, Springer-Verlag
(1978). The rule is
(1) F(i) = 1/2 * omega(i)
(2) ALPHA(i) = alpha(i)
(3) BETA(i) = - beta(i),
where omega, alpha, beta are symbols used in these Tables.
The keyword NSETO gives the number of open shells, and
keyword NO gives the degeneracy of each open shell. Thus
the 5-S state of carbon (from configuration s1,p3) would
enter NSETO=2 and NO(1)=1,3. NCO is an easy keyword: that
is the number of filled orbitals. The total number of
electrons in an open shell GVB run is
NE = 2*NCO + sum 2*F(i)*NO(i)
i
and this value will be checked against your ICHARG input.
- - - -
Specific input for all terms found in the atomic p**N
configurations follow. Be sure to enter 14 digits, as
these values are part of a double precision energy formula!
Values for the excited terms in the p**N configurations
were extracted from C.F.Jackels and E.R.Davidson Int. J.
Quantum Chem. 8, 707-714(1974), which explains the concept
of averaging over equivalent determinants to enforce a
symmetric density matrix, which preserves radial symmetry
in the atomic orbitals. The ALPHA and BETA values can also
be extracted from Roothaan's 1960 A and B coefficients by
the prescription detailed below.
! p**1 2-P state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.16666666666667
ALPHA(1)= 2.0 0.33333333333333 0.00000000000000
BETA(1)= -1.0 -0.16666666666667 -0.00000000000000 $END
! p**2 3-P state
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.33333333333333
ALPHA(1)= 2.0 0.66666666666667 0.16666666666667
BETA(1)= -1.0 -0.33333333333333 -0.16666666666667 $END
For the 1-D excited state, change the open-shell parameters
to ALPHA(3)=0.1 and BETA(3)= +0.03333333333333
For the 1-S excited state, change the open-shell parameters
to ALPHA(3)=0.0 and BETA(3)= +0.33333333333333
! p**3 4-S state
$CONTRL SCFTYP=ROHF MULT=4 $END
which is equivalent to
$CONTRL SCFTYP=GVB MULT=4 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.50000000000000
ALPHA(1)= 2.0 1.00000000000000 0.50000000000000
BETA(1)= -1.0 -0.50000000000000 -0.50000000000000 $END
For 2-D, use ALPHA(3)= 0.4 and BETA(3)= -0.2
for 2-P, use ALPHA(3)= 0.33333333333333 and BETA(3)= 0.0
! p**4 3-P state
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.66666666666667
ALPHA(1)= 2.0 1.33333333333333 0.83333333333333
BETA(1)= -1.0 -0.66666666666667 -0.50000000000000 $END
For 1-D, use ALPHA(3)= 0.76666666666667 and BETA(3)= -0.3
for 1-S, use ALPHA(3)= 0.66666666666667 and BETA(3)= 0.0
! p**5 2-P state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.83333333333333
ALPHA(1)= 2.0 1.66666666666667 1.33333333333333
BETA(1)= -1.0 -0.83333333333333 -0.66666666666667 $END
- - - -
Coupling constants for the highest spin state(s) in d**N
configurations are taken from "Handbook of Gaussian Basis
Sets", R.Poirier, R.Kari, I.G.Csizmadia, Elsevier,
Amsterdam, 1985.
! d**1 2-D state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.1
ALPHA(1)= 2.0, 0.20, 0.00
BETA(1)=-1.0,-0.10, 0.00 $END
! d**2 average of 3-F and 3-P states
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.2
ALPHA(1)= 2.0, 0.40, 0.05
BETA(1)=-1.0,-0.20,-0.05 $END
Note: "average" means a degeneracy weighted combination of
determinants with Sz=S: here, 2 electrons in 5 orbitals
coupled as a triplet is ten determinants with Sz=1. The
energy from these parameters is E=[7xE(3-F)+3xE(3-P)]/10.
! d**3 average of 4-F and 4-P states
$CONTRL SCFTYP=GVB MULT=4 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.3
ALPHA(1)= 2.0, 0.60, 0.15
BETA(1)=-1.0,-0.30,-0.15 $END
! d**4 5-D state
$CONTRL SCFTYP=GVB MULT=5 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.4
ALPHA(1)= 2.0, 0.80, 0.30
BETA(1)=-1.0,-0.40,-0.30 $END
! d**5 6-S state
$CONTRL SCFTYP=ROHF MULT=6 $END
! d**6 5-D state
$CONTRL SCFTYP=GVB MULT=5 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.6
ALPHA(1)= 2.0, 1.20, 0.70
BETA(1)=-1.0,-0.60,-0.50 $END
! d**7 average of 4-F and 4-P states
$CONTRL SCFTYP=GVB MULT=4 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.7
ALPHA(1)= 2.0, 1.40, 0.95
BETA(1)=-1.0,-0.70,-0.55 $END
! d**8 average of 3-F and 3-P states
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.8
ALPHA(1)= 2.0, 1.60, 1.25
beta(1)=-1.0,-0.80,-0.65 $end
! d**9 2-D state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.9
ALPHA(1)= 2.0, 1.80, 1.60
BETA(1)=-1.0,-0.90,-0.80 $END
Note that GAMESS can do a proper calculation on the
ground term for the d**2, d**3, d**7, d**8 configurations
only by means of state averaged MCSCF. For d**8, use
$contrl scftyp=mcscf mult=3 $end
$det group=c1 ncore=xx nact=5 nels=8
nstate=10 wstate(1)=1,1,1,1,1,1,1,0,0,0 $end
to correctly average the 7 lowest roots (3-F) with no
weight given to the highest three roots (3-P). Although
this is done with the SCFTYP=MCSCF program, this is still a
SCF calculation since only the orbitals, but not any CI
coefficients, are optimized.
Dirk Andrae in Berlin has provided a great many
examples of using the MCSCF program to do terms found in
the p**n, d**n, and even f**n configurations:
http://userpage.fu-berlin.de/~dandrae/
openshell/openls/openls.html
Type that web reference on just one line, of course.
Open shell cases such as s**1,d**n are probably most
easily tackled with the state-averaged MCSCF program. The
ORMAS CI code may be convenient in fixing the number of
electrons found in each open shell.
If you are a true afficionado of atomic calculations,
including open shell f configurations, look for Russ
Pitzer's version of the famous Roothaan/Bagus ATOM-SCF
program
R.M.Pitzer, Comput.Phys.Commun. 170, 239-264(2005)
- - - -
The literature contains an alternate energy formula,
involving the so-called Roothaan A and B parameters:
E = 2 sum f-i*h-ii + sum sum f-i*f-j[2Aij*Jij - Bij*Kij]
i i j
Comparing this to the GVB program's energy formula above,
we immediately see that its fractional occupancy f is the
same as GVB's F, and that
ALPHA-ij = +2 * f-i * f-j * A-ij
BETA-ij = -1 * f-i * f-j * B-ij
Let c=the closed shell orbitals, and o=a single open shell.
Tables such as Roothaan's 1960 paper give only the A-oo and
B-oo values, since A-cc = B-cc = A-co = B-co = 1.0. It is
easy to convert such A,B values to all ALPHA-cc,-co,-oo and
BETA-cc,-co,-oo values. Degenerate open shells in linear
molecules follow immediately from Roothaan's 1960 table:
! 2-Pi from configuration pi**1:
$contrl scftyp=gvb mult=2 $end
$scf nco=xx nseto=1 no=2 couple=.true. f(1)=1.0,0.25
alpha(1)= 2.0, 0.50, 0.0
beta(1)=-1.0,-0.25, 0.0 $end
! 3-Sigma-minus from configuration pi**2:
$contrl scftyp=gvb mult=3 $end
$scf nco=xx nseto=1 no=2 couple=.true. f(1)=1.0,0.50
alpha(1)= 2.00, 1.00, 0.50
beta(1)=-1.00,-0.50,-0.50 $end
! 1-Delta from configuration pi**2:
$contrl scftyp=gvb mult=1 $end
$scf nco=xx nseto=1 no=2 couple=.true. f(1)=1.0,0.50
alpha(1)= 2.00, 1.00, 0.00
beta(1)=-1.00,-0.50,+0.50 $end
! 1-Sigma-plus from configuration pi**2:
$contrl scftyp=gvb mult=1 $end
$scf nco=xx nseto=1 no=2 couple=.true. f(1)=1.0,0.50
alpha(1)= 2.00, 1.00, 0.25
beta(1)=-1.00,-0.50, 0.00 $end
! 2-Pi from configuration pi**3:
$contrl scftyp=gvb mult=2 $end
$scf nco=xx nseto=1 no=2 couple=.true. f(1)=1.0,0.75
alpha(1)= 2.00, 1.50, 1.00
beta(1)=-1.00,-0.75,-0.50 $end
The same inputs apply to delta**N configurations: only the
names of the energy terms (states) change!
It is possible to do every term arising from the
s**1,p**N configurations. Roothaan-style A,B parameters
are as follows:
Acc = Bcc = 1.0 (as always)
Aco = Bco = 1.0, for o=s and for o=p (as always)
Ass = Bss = 0.0, since there is no other s e- to repel
Asp = 1.0
Bsp = -3K101 + 1
App and Bpp: take from the parent term in config p**N
and then apply the rule turning A,B into ALPHA,BETA, for
Fc=1, Fs=0.5, Fp=N/6. K101 should be taken from Roothaan
and Bagus in Methods Comput. Phys. 2, 49(1963).
- - - -
It may be useful to obtain a "configuration average",
especially whenever a specific term energy cannot be
computed. Boris Plakhutin in Novosibirsk has kindly
provided a recipe for the Roothaan A,B values that give the
average energy of configuration O**N, meaning N electrons
in some open shell O. Let D be the degeneracy of the open
shell O:
atoms: O=p, d, f... has D= 3, 5, 7...
non-linear molecules: O=e, t, g, h has D= 2, 3, 4, 5.
The fractional occupation number F-o = N/(2*D). The open
shell Roothaan couplings are A-oo = B-oo = (N-1)/(N-F).
For example, the case d**2 has F-o=1/5 so A-oo=B-oo=5/9.
Using the formulae to generate all ALPHA and BETA causes
the GVB program to optimize the following spin-and-space
weighted average of all terms appearing in d**2:
E-avg = [21xE(3-F) + 9xE(3-P) +
9xE(1-G) + 5xE(1-D) + E(1-S)]/45
by this input
$contrl scftyp=gvb mult=3 $end
$scf nco=xx nseto=1 no=5 couple=.true.
f(1)= 1.0, 0.2
alpha(1)= 2.00, 0.40, 0.044444444444444444
beta(1)=-1.00,-0.20,-0.022222222222222222 $end
Note that here, as in all other runs with the GVB module,
the spin in $CONTRL is irrelevant: the energy that is
optimized is determined by F, ALPHA, and BETA. In this
last case, these data implicitly include both triplet and
singlet determinants in the energy averaging.
Boris Plakhutin also provided a formula in case you
want to average over a particular spin S within an open
shell O**N with some degeneracy D implying a fractional
occupancy of F. The result is
N**2(N-1) + FxN(N-4) + 4FxS(S+1)
A-oo = --------------------------------
N(N**2-4F**2)
and
4F(N-1) + N(N-4) + 4S(S+1)
B-oo = --------------------------
(N**2-4F**2)
For the d**2 configuration, the space-and-spin degeneracy
weighted average of the 3-F and 3-P terms has Aoo=5/8 and
Boo=10/8, while the average of the 1-G, 1-D, and 1-S terms
has Aoo=5/12 and Boo=-10/12. These optimize orbitals for
the average
E-triplet = [21xE(3-F) + 9xE(3-P)]/30
which is not quite the same as the space degeneracy average
shown above, and for the average
E-singlet = [9xE(1-G) + 5xE(1-D) + E(1-S)]/15
The literature reference for the overall and the spin-
specific configurational averages is
B.N.Plakhutin, J. Mathematical Chem. 22, 203-233(1997)
true GVB perfect pairing runs
True GVB runs are obtained by choosing NPAIR nonzero.
If you wish to have some open shell electrons in addition
to the geminal pairs, you may add the pairs to the end of
any of the GVB coupling cases shown above. The GVB module
assumes that you have reordered your MOs into the order:
NCO double occupied orbitals, NSETO sets of open shell
orbitals, and NPAIR sets of geminals (with NORDER=1 in the
$GUESS group).
Each geminal consists of two orbitals and contains two
singlet coupled electrons (perfect pairing). The first MO
of a geminal is probably heavily occupied (such as a
bonding MO u), and the second is probably weakly occupied
(such as an antibonding, correlating orbital v). If you
have more than one pair, you must be careful that the
initial MOs are ordered u1, v1, u2, v2..., which is -NOT-
the same order that RHF starting orbitals will be found in.
Use NORDER=1 to get the correct order.
These pair wavefunctions are actually a limited form of
MCSCF. GVB runs are much faster than MCSCF runs, because
the natural orbital u,v form of the wavefunction permits a
Fock operator based optimization. However, convergence of
the GVB run is by no means assured. The same care in
selecting the correlating orbitals that you would apply to
an MCSCF run must also be used for GVB runs. In
particular, look at the orbital expansions when choosing
the starting orbitals, and check them again after the run
converges.
GVB runs will be carried out entirely in orthonormal
natural u,v form, with strong orthogonality enforced on the
geminals. Orthogonal orbitals will pervade your thinking
in both initial orbital selection, and the entire orbital
optimization phase (the CICOEF values give the weights of
the u,v orbitals in each geminal). However, once the
calculation is converged, the program will generate and
print the nonorthogonal, generalized valence bond orbitals.
These GVB orbitals are an entirely equivalent way of
presenting the wavefunction, but are generated only after
the fact.
Convergence of true GVB runs is by no means as certain
as convergence of RHF, UHF, ROHF, or GVB with NPAIR=0. You
can assist convergence by doing a preliminary RHF or ROHF
calculation, and use these orbitals for GUESS=MOREAD. Few,
if any, GVB runs with NPAIR non-zero will converge without
using GUESS=MOREAD. Generation of MVOs during the
preliminary SCF can also be advantageous. In fact, all the
advice outlined for MCSCF computations below is germane,
for GVB-PP is a type of MCSCF computation.
The total number of electrons in the GVB wavefunction
is given by the following formula:
NE = 2*NCO + sum 2*F(i)*NO(i) + 2*NPAIR
i
The charge is obtained by subtracting the total number of
protons given in $DATA. The multiplicity is implicit in
the choice of alpha and beta constants. Note that ICHARG
and MULT must be given correctly in $CONTRL anyway, as the
number of electrons from this formula is double checked
against the ICHARG value.
the special case of TCSCF
The wavefunction with NSETO=0 and NPAIR=1 is called
GVB-PP(1) by Goddard, two configuration SCF (TCSCF) by
Schaefer or Davidson, and CAS-SCF with two electrons in two
orbitals by others. Note that this is just semantics, as
these are identical. This is a very important type of
wavefunction, as TCSCF is the minimum acceptable treatment
for singlet biradicals. The TCSCF wavefunction can be
obtained with SCFTYP=MCSCF, but it is usually much faster
to use the Fock based SCFTYP=GVB. Because of its
importance, the TCSCF function (together with open shells,
if desired) permits analytic hessian computation.
a caution about symmetry
Caution! Some exotic calculations with the GVB program
do not permit the use of symmetry. The symmetry algorithm
in GAMESS was "derived assuming that the electronic charge
density transforms according to the completely symmetric
representation of the point group", Dupuis/King, JCP, 68,
3998(1978). This may not be true for certain open shell
cases, and in fact during GVB runs, it may not be true for
closed shell singlet cases!
First, consider the following correct input for the
singlet-delta state of NH:
$CONTRL SCFTYP=GVB NOSYM=1 $END
$SCF NCO=3 NSETO=2 NO(1)=1,1 $END
for the x**1y**1 state, or for the x**2-y**2 state,
$CONTRL SCFTYP=GVB NOSYM=1 $END
$SCF NCO=3 NPAIR=1 CICOEF(1)=0.707,-0.707 $END
Neither gives correct results, unless you enter NOSYM=1.
The electronic term symbol is degenerate, a good tip off
that symmetry cannot be used. However, some degenerate
states can still use symmetry, because they use coupling
constants averaged over all degenerate states within a
single term, as is done in EXAM15 and EXAM16. Here the
"state averaged SCF" leads to a charge density which is
symmetric, and these runs can exploit symmetry.
Secondly, since GVB runs exploit symmetry for each of
the "shells", or type of orbitals, some calculations on
totally symmetric states may not be able to use symmetry.
An example is CO or N2, using a three pair GVB to treat the
sigma and pi bonds. Individual configurations such as
(sigma)**2,(pi-x)**2,(pi-y*)**2 do not have symmetric
charge densities since neither the pi nor pi* level is
completely filled. Correct answers for the sigma-plus
ground states result only if you input NOSYM=1.
Problems of the type mentioned should not arise if the
point group is Abelian, but will be fairly common in linear
molecules. Since GAMESS cannot detect that the GVB
electronic state is not totally symmetric (or averaged to
at least have a totally symmetric density), it is left up
to you to decide when to input NOSYM=1. If you have any
question about the use of symmetry, try it both ways. If
you get the same energy, both ways, it remains valid to use
symmetry to speed up your run.
And beware! Brain dead computations, such as RHF on
singlet O2, which actually is a half filled degenerate
shell, violate the symmetry assumptions, and also violate
nature. Use of partially filled degenerate shells always
leads to very wild oscillations in the RHF energy, which is
how the program tries to tell you to think first, and
compute second. Configurations such as pi**2, e**1, or
f2u**4 can be treated, but require GVB wavefunctions and F,
ALPHA, BETA values from the sources mentioned.
How to do MCSCF (and CI) calculations
Multi-configuration self consistent field (MCSCF)
wavefunctions are the most general SCF possible. MCSCF
allows for a natural description of chemical processes
involving the separation of electrons (bond breaking,
electronic excitation, etc), which are often not well
represented using the single configuration SCF methods.
MCSCF wavefunctions, as the name implies, contain more
than one configuration, each of which is multiplied by a
configuration interaction (CI) coefficient, optimized to
determine its weight. In addition, the shapes of the
orbitals used to form each of the configurations are
optimized, just as in a simpler SCF, to self consistency.
Typically every different chemical problem requires
that an MCSCF wavefunction be designed to treat it, on a
case by case basis, by choosing an "active space". For
example, one may be interested in describing the reactivity
of a particular functional group, instead of elsewhere in
the molecule. So, the active electrons and active orbitals
will be those that are "active" on that functional group.
Orbitals elsewhere in the molecule just remain doubly
occupied, as for RHF. This means some attention must be
paid to orbitals in order to obtain the desired results.
Procedures for the selection of configurations (which
amounts to choosing the number of active electrons and
active orbitals), for the two mathematical optimizations
just mentioned, ways to interpret the resulting MCSCF
wavefunction, and treatments for dynamical electron
correlation of MCSCF wavefunctions are the focus of a
review article:
"The Construction and Interpretation of MCSCF
wavefunctions"
M.W.Schmidt and M.S.Gordon,
Annu.Rev.Phys.Chem. 49,233-266(1998)
One section of this article is devoted to the problem of
designing the correct active space to treat your problem.
Additional reading is listed at the end of this section. A
much newer and very valuable review article is
"Multiconfiguration self-consistent field and
multireference configuration interaction
methods and applications"
P.G.Szalay, T.Muller, G.Gidofalvi, H.Lischka, R.Shepard
Chem.Rev. 112, 108-181(2012)
These pages describe a powerful and mature MCSCF
program, allowing computation of the MCSCF energy, nuclear
gradient, and nuclear hessian for pure states. Practical
procedures for generating starting orbitals are available.
Localized orbital analysis of the final active orbitals is
provided. State-averaged energies and their analytic
gradients can be obtained. Minimum energy crossing points
and conical intersections between surfaces may be found,
see RUNTYP=MEX and CONINT. Non-adiabatic coupling matrix
elements (NACME) can be computed, see RUNTYP=NACME.
Diabatic potential energy surfaces can be obtained, see
DIABAT in $MCSCF. If desired, spin-orbit couplings or
transition dipole moments can be found, see elsewhere in
this chapter. Efficient perturbative treatments of the
dynamical correlation energy for all electrons, whether in
active and filled orbitals, are provided. Of course,
parallel computation has been enabled.
The most efficient technique implemented in GAMESS for
finding the dynamic correlation energy of MCSCF is second
order perturbation theory, in the variant known as MCQDPT
(known as MRMP for one state). MCQDPT is discussed in a
different section of this chapter.
The use of CI, probably in the form of second order CI,
will be described below, en passant, during discussion of
the input defining the configurations for MCSCF. Selection
of a CI following any type of SCF (except UHF) is made with
CITYP in the $CONTRL group, and masterminded by $CIINP.
MCSCF implementation
With the exception of the QUAD converger, the MCSCF
program is of the type termed "unfolded two-step" by Roos.
This means the orbital and CI coefficient optimizations are
separated. The latter are obtained in a conventional CI
diagonalization, while the former are optimized by a
separate orbital improvement step.
Each MCSCF iteration (except for the JACOBI and QUAD
convergers) consists of the following steps:
1) transformation of AO integrals to the current MO basis,
2) generation of the Hamiltonian matrix and optimization
of the CI coefficients by a Davidson diagonalization,
3) generation of the first and second order density matrix,
4) improvement of the molecular orbitals.
During the first iteration at the first geometry, you will
receive verbose output from each of these steps, but each
subsequent iteration produce only a single summary line.
The CI problem in steps two and three has four options
for the many electron basis, namely ALDET, ORMAS, or GENCI
using determinants, or GUGA using CSFs. This choice is
made with the keyword CISTEP in $MCSCF. Much more will be
said below about the differences between determinants and
CSFs. The word "configuration" will be used throughout
this section to refer to either determinants or CSFs, when
a generic term is needed for the many-electron functions.
Most people use CSF and configuration interchangeably, so
please note the distinction made here.
The orbital update in step four has five options,
namely FOCAS, SOSCF, FULLNR, JACOBI, and QUAD, listed here
in roughly the order of their increasing mathematical
sophistication, convergence characteristics, and of course,
their computer resource requirements. Again, these are
chosen by keywords in the $MCSCF group. More will be said
just below about the relative merits of these.
Depending on the converger chosen, the program will
select the appropriate kind of integral transformation at
step one. There's seldom need to try to fine tune this, but
note that the $TRANS group allows you to choose an AO
integral direct transformation, with the DIRTRF flag.
The type of CI and the type of orbital converger are to
some extent "mix and match". This is particularly true for
the two full CI programs, GUGA or ALDET, where either
produces exactly the same CI density matrices. Here is a
chart of the ways to combine CI and orbital optimizers:
parallel run's
orbital transformation CI computation via CISTEP=
converger memory GUGA ALDET GENCI ORMAS
--------- -------------- ---- ----- ----- -----
FOCAS replicated ok ok silly silly
SOSCF replicated ok ok ok ok
FULLNR distributed ok ok ok ok
QUAD serial ok xx xx xx
JACOBI serial ok ok ok ok
"xx" means QUAD converger is coded only for CISTEP=GUGA.
"silly" means that this converger ignores active-active
rotations, so these runs are likely to be divergent,
or perhaps converge to a false solution.
"serial" means this can only run sequentially at present.
The next two sections provide more information on the
two mathematical optimizations, first how the orbital shape
is refined, and then the determinantion of CI coefficients.
orbital updates
There are presently five orbital improvement options,
namely FOCAS, SOSCF, FULLNR, JACOBI, and QUAD. All but the
JACOBI update run in parallel. Each converger is discussed
briefly below, in order of increasing robustness. The most
commonly used convergers are SOSCF and FULLNR.
The input to control the orbital update step is the
$MCSCF group, where you can pick the convergence procedure.
Most of the input in this group is rather specialized, but
note in particular MAXIT and ACURCY, which control the
convergence behavior.
FOCAS is a first order, complete active space MCSCF
optimization procedure. It is based on a novel approach
due to Meier and Staemmler, using very fast but numerous
microiterations to improve the convergence of what is
intrinsically a first order method. Since FOCAS requires
only one virtual orbital in the integral transformation to
compute the Lagrangian (whose asymmetry is the orbital
gradient, and must fall below ACURCY at convergence), the
total MCSCF job may take less time than a second order
method, even though it may require many more iterations to
converge. The use of microiterations is crucial to FOCAS'
ability to converge. It is important to take a great deal
of care choosing the starting orbitals.
SOSCF is a method built upon the FOCAS code, which
seeks to combine the speed of FOCAS with approximate second
order convergence properties. Thus SOSCF is an approximate
Newton-Raphson, based on a diagonal guess at the orbital
hessian, and in fact has much in common with the SOSCF
option in $SCF. Its time requirements per iteration are
like FOCAS, with a convergence rate better than FOCAS but
not as good as true second order. Storage of only the
diagonal of the orbital hessian allows the SOSCF method to
be used with much larger basis sets than exact second order
methods. Because SOSCF usually requires the least CPU
time, disk space, and memory needs, it is the default.
Good convergence by the SOSCF method requires that you
prepare starting orbitals carefully, and read in all MOs in
$VEC, as providing canonicalized virtual orbitals increases
the diagonal dominance of the orbital hessian. Parallel
computations are possible with SOSCF, but only to a modest
number of nodes.
FULLNR means a full Newton-Raphson orbital improvement
step is taken, using the exact orbital hessian. FULLNR is
a robust convergence method, and normally takes the fewest
iterations to converge. Computing the exact orbital
hessian requires two virtual orbital indices be included in
the integral transformation, making this step quite time
consuming, and of course memory for storage of the orbital
hessian must be available. Because both the transformation
and orbital improvement steps of FULLNR are time consuming,
FULLNR is not the default. You may want to try FULLNR when
convergence is difficult, assuming you have already tried
preparing good starting orbitals by the hints below.
The serial FULLNR code uses the augmented hessian
matrix approach to solve the Newton-Raphson equations.
There are two suboptions for computation of the orbital
hessian: DM2 is faster, but takes more memory than TEI.
The parallel implementation of FULLNR avoids explicit
storage of the orbital hessian, by recomputing the product
of the hessian times orbital rotation vector during the
subiterations solving the Newton-Raphson problem. The
partial integral transformation used to set up the FULLNR
converger has been changed to use distributed memory, and
will scale like the MP2 energy/gradient programs, to many
nodes. Parallel FULLNR requires large MEMORY only for the
CI step (if the active space is big), but always requires a
large MEMDDI. The parallel FULLNR program is essentially
diskless, apart from storage of converged CI vectors.
The JACOBI method uses a series of 2 by 2 orbital
rotations by an angle predicted to lower the energy. This
should essentially ensure convergence after sweeping
through all possible orbital pairs enough times. The
procedure was created to converge selected (general)
determinant MCSCF functions, but of course it can be used
will full lists as well in difficult cases. The JACOBI
calculation will consist of a full four index
transformation over all MOs before the iterations begin.
Each iteration consists of
1. a small 4 index transformation over active orbitals
2. optimization of the CI vector
3. generation of the 1e- and 2e- density matrices
4. sweeps over Jacobi rotations, using MO integrals in
memory to generate each rotation, with a subsequent
update after each pair is rotated.
5. when sufficient energy lowering has been achieved,
begin a new iteration.
This procedure never generates the orbital Lagrangian!
Unfortunately this means that at present it is not possible
to compute nuclear gradients. Due to lack of a Lagrangian,
ACURCY is of course irrelevant, so the convergence test is
on ENGTOL.
QUAD uses a fully quadratic, or second order approach
and is thus the most powerful MCSCF converger. The QUAD
code is programmed only for CISTEP=GUGA. QUAD runs begin
with normal unfolded FULLNR iterations, until the orbitals
approach convergence sufficiently. QUAD then begins the
simultaneous optimization of CI coefficients and orbitals,
and convergence should be obtained in 3-4 additional MCSCF
iterations. The QUAD method requires building the full
electronic hessian, including orbital/orbital, orbital/CI,
and CI/CI blocks, which is a rather big matrix. In
principle, this is the most robust method available, but it
is limited to perhaps 50-100 CSFs only, because it is a
memory hog. QUAD may be helpful in converging excited
electronic states, but note that you may not use state
averaging with QUAD. In practice, QUAD has not received
very much use compared to the unfolded convergers.
CI coefficient optimization
Determinants or configuration state functions (CSFs)
may be used to form the many electron basis set. It is
necessary to explain these in a bit of detail so that you
can understand the advantages of each.
A determinant is a simple object: a product of spin
orbitals with a given Sz quantum number, that is, the
number of alpha spins and number of beta spins are a
constant. Matrix elements involving determinants are
correspondingly simple, but unfortunately determinants are
not necessarily eigenfunctions of the S**2 operator.
To expand on this point, consider the four familiar 2e-
functions which satisfy the Pauli principle. Here u, v are
space orbitals, and a, b are the alpha and beta spin
functions. As you know, the singlet and triplets are:
S1 = (uv + vu)/sqrt(2) * (ab - ba)/sqrt(2)
T1 = (uv - vu)/sqrt(2) * aa
T2 = (uv - vu)/sqrt(2) * (ab + ba)/sqrt(2)
T3 = (uv - vu)/sqrt(2) * bb
It is a simple matter to multiply out S1 and T2, and to
expand the two determinants which have Sz=0,
D1 = |ua vb| D2 = |va ub|
This reveals that
S1 = (D1+D2)/sqrt(2) or D1 = (S1 + T2)/sqrt(2)
T2 = (D1-D2)/sqrt(2) D2 = (S1 - T2)/sqrt(2)
Thus, one must take a linear combination of determinants in
order to have a wavefunction with the desired total spin.
There are two important points to note:
a) A two by two Hamiltonian matrix over D1 and D2 has
eigenfunctions with -different- spins, S=0 and S=1.
b) use of all determinants with Sz=0 does allow for the
construction of spin adapted states. D1+D2, or D1-D2,
are -not- spin contaminated.
By itself, a determinant such as D1 is said to be "spin
contaminated", being a fifty-fifty admixture of singlet and
triplet. (It is curious that calculations with just one
such determinant are often called "singlet UHF", when this
is half triplet!). Of course, some determinants are spin
adapted all by themselves, for example the spin adapted
functions T1 and T3 above are single determinants, as are
the closed shells
S2 = (uu) * (ab - ba)/sqrt(2).
S3 = (vv) * (ab - ba)/sqrt(2).
It is possible to perform a triplet calculation, with no
singlet states present, by choosing determinants with Sz=1
such as T1, since then no state with Sz=0 exists in the
determinant basis set (as is required when S=0). To
summarize, the eigenfunctions of a Hamiltonian formed by
determinants with any particular Sz will be spin states
with S=Sz, S=Sz+1, S=Sz+2, ... but will not contain any S
values smaller than Sz.
CSFs are an antisymmetrized combination of a space
orbital product, and a spin adapted linear combination of
simple alpha-beta products. Namely, the following CSF
C1 = A (uv) * (ab-ba)/sqrt(2)
which has a singlet spin function is identical to S1 above
if you write out what the antisymmetrizer A does, and the
CSFs
C2 = A (uv) * aa
C3 = A (uv - vu)/sqrt(2) * (ab + ba)/sqrt(2)
C4 = A (uv) * bb
equal T1-T3. Since the three triplet CSFs have the same
energy, GAMESS works with the simpler form C2. Singlet and
triplet computations using CSFs are done in separate runs,
because when spin-orbit coupling is not considered, the
Hamiltonian is block diagonal in a CSF basis. Technical
information about the CSFs is that they use Yamanouchi-
Kotani spin couplings, and matrix elements are obtained
using a GUGA, or graphical unitary group approach.
Determinant and CSF are both primarily used for MCSCF
wavefunctions, but can be used in CI (see CITYP in
$CONTRL). Other comparisons between the determinant and
CSF implementations, as they exist in GAMESS today, are
determinants CSFs
parallel execution yes yes
direct CI yes no
use Abelian group symmetry yes yes
state average mixed spins yes no
first order density yes yes
state averaged densities yes yes
analytic nuclear hessian yes no
can form CI Lagrangian no yes
In nearly every circumstance the determinant CI will run
faster than GUGA, so it is the default. Here are timings
for N electrons in N orbitals, no symmetry used:
N in N ALDET GENCI --- GUGA ---
8 1 1 1 0
10 8 38 19 33
12 228 3122 534 2209
14 7985 -- 15377 130855
The reason there are two numbers under GUGA is that the
first is for writing the loops (Hamiltonian data) to disk,
and the second is for the actual diagonalization.
Two of the determinant CI programs, namely ALDET or
ORMAS (but not GENCI) have been changed to use replicated
memory parallelism, with modest scaling. The GENCI program
will run as a serial bottleneck (no speedup) in parallel
runs.
The next two sections describe in detail the input for
specification of the configurations, either determinants or
CSFs.
determinant CI
Three determinant CI codes are provided for MCSCF, one
for full CI spaces (ALDET), another named the Occupation
Restricted Multiple Active Spaces (ORMAS), and finally
there is a program for arbitrary (selected) determinant
lists (GENCI). For straight CI, but not MCSCF, there is a
fourth program, the full second order CI (CITYP=FSOCI),
whose purpose is MR-CISD. Lastly, there's also the
spin flip version of ORMAS-CI, called by the SFORM keyword.
ALDET is a full CI within the chosen active space. It
is possible to go up to 16 electrons in 16 orbitals, if
your computer has a lot of memory. ALDET is the only
CISTEP for which analytic nuclear hessian is possible, and
it is also the most scalable CI code (using replicated
memory to store CI vectors). A sample input for ALDET is
$DET STSYM=B1 NSTATE=3 NCORE=xx NELS=8 NACT=6 $END
Keywords in this group actually relate to all determinant
programs, and are described below.
The $DET input group is basic to all determinant CI
codes. Keywords GROUP and STSYM specify the desired
spatial symmetry of the determinants. Most runs need give
only the orbital and electron counts: NCORE, NACT, and
NELS. The number of electrons is 2*NCORE+NELS, and will be
checked against the charge implied by ICHARG. The MULT
given in $CONTRL is used to determine the desired Sz value,
by extracting S from MULT=2S+1, then by default Sz=S. If
you wish to include lower spin multiplicities, which will
increase the CPU time of the run, but will let you know
what the energies of such states are, just input a smaller
value for SZ. The states whose orbitals will be MCSCF
optimized will be those having the requested MULT value,
unless you choose otherwise with the PURES flag.
The remaining parameters in the $DET group give extra
control over the diagonalization process. Most are not
given in normal circumstances, except NSTATE, which you may
need to adjust to produce enough roots of the desired MULT
value. The only important keyword which has not been
discussed is the WSTATE array, giving the weights for each
state in forming the first and second order density matrix
elements, which drive the orbital update methods. Note
that analytic gradients are available only when the WSTATE
array is a unit vector, corresponding to a pure state, such
as WSTATE(1)=0,1,0 which permits gradients of the first
excited state to be computed. When used for state averaged
MCSCF, WSTATE is normalized to a unit sum, thus input of
WSTATE(1)=1,1,1 really means a weight of 0.33333... for
the each of the states being averaged.
ORMAS (Occupation Restricted Multiple Active Space) is
a program designed to limit the size of the full CI
problem, and may be useful when the number of active
orbitals is 10 or higher. By dividing your total active
space into multiple subspaces, and specifying a range of
electrons to occupy each subspace, most of the full CI's
effect can be included. ORMAS generates a full CI within
each orbital subspace, taking the product of each small
full CI to generate the determinant list.
Here are some ideas on how to use ORMAS, which is a
very flexible CI program:
a) single reference, arbitrary excition level CI-X, from
a closed shell reference:
$det ncore=y nact=z nels=10 (y+z=entire basis)
$ormas nspace=2 mstart(1)=y+1,y+6 mine(1)=10-x,0
maxe(1)=10,x
This excites the 5 doubly occupied orbitals, to the
desired excitation level of X.
An open shell example of CI-SD from ...22111 might be
$contrl mult=4
$det ncore=y nact=z nels=7 (y+z=entire basis)
$ormas nspace=3 mstart(1)=y+1,y+3,y+6
mine(1)=2,1,0
maxe(1)=4,5,2
No more than 2e- are allowed to be promoted from the
doubly occupied or singly occupied spaces, and no
more than 2 are allowed to enter the singly occupied
or empty spaces.
b) simple product of active spaces
For example, consider furan, with two active
subspaces. Keeping the 5 true core and the 4 CH
bonds in the core space, the sigma subspace might
contains 5 ring sigma, one oxygen lone pair, and 5
ring sigma antibonds, with a total of 12 e-. The pi
active space contains 5 pi orbitals and 6 e-:
$det ncore=9 nact=16 nels=18
$ormas nspace=2 mstart(1)=10,21 mine(1)=12,6
maxe(1)=12,6
Having the minimum and maximum electron counts the
same is what makes this the simple product of two
separate active spaces. In other words, this is
similar to the QCAS procedure of Nakano and Hirao,
but ORMAS limits only the total electron counts,
not separately the numbers of alpha and beta e-,
in other words all spin couplings are used.
c) flexible occupancy between active subspaces
Imagine that you are interested in excited states of
formaldehyde, some of which will have Rydberg
character, dominated by single excitations into
diffuse orbitals. H2CO's valence states arise from 3
orbitals, the CO pi and pi* and one oxygen lone pair.
Placing the O sp lone pair and three sigma bonds into
the filled space, and centering diffuse s,p,d shells
on the carbon:
$det ncore=6 nact=12 nels=4
$ormas nspace=2 mstart(1)=7,10 mine(1)=3,0
maxe(1)=4,1
This is a 4e-, 3 orbital n,pi,pi* space to describe
valence states, and excites one electron into the 9
diffuse orbitals to describe Rydberg states. It is
many fewer determinants than a 4e- in 12 orbital FCI.
d) RAS-like CI
The previous example is reminiscent of Roos' RAS-SCF.
In fact ORMAS can do RAS-SCF, which is three spaces:
the lowest space is allowed to excite only a few
electrons, a middle space that is the rest, and a top
space into which only a few electrons can be excited.
Suppose there are 10 e-, 10 orbitals, that the bottom
and top spaces involve 3 orbitals, and that a "few"
means specifically 2 e-:
$det ncore=20 nact=10 nels=10 $end
$ormas nspace=3 mstart(1)=21,24,28 mine(1)=4,2,0
maxe(1)=6,6,2
However, ORMAS can use more than 3 orbital subspaces.
e) first or second order CI.
Consider C2H4, with a 4 orbital active space of CC
sigma, pi, pi*, and sigma*. In order to correlate
the four valence CH orbitals by double excitations,
an MCSCF based on $DET, followed by SOCI based on
$CIDET and $ORMAS, is:
$contrl scftyp=mcscf cityp=ormas
$mcscf cistep=aldet
$det ncore=6 nact=4 nels=4
$cidet ncore=2 nact=y nels=12 (y=rest of basis)
$ormas nspace=3 mstart(1)=3,7,11 mine(1)=6,2,0
maxe(1)=8,6,2
which permits singles and doubles out of the CH and
CC spaces, into the CC and external spaces.
f) Spin-flip CI (aka SF-ORMAS).
This may consists of any regular ORMAS-CI calculation.
Unlike normal ORMAS-CI, this is activated by
CITYP=SFORM in $CONTRL. MULSF must be
provided in $CIDET, and it must be less than MUL in
$CONTRL by multiples of two(2). An example SF-ORMAS
calculation on ethylene is:
$CONTRL SCFTYP=rohf mult=3 cityp=sform $end
$BASIS GBASIS=n311 ngauss=6 ndfunc=1 $END
$cidet nstate=6 ncore=0 nact=50 nels=16
mulsf=1 !<-- This is required for SF
$end
$ormas nspace=3 mstart(1)=1,8,10 mine(1)=13,1,0
maxe(1)=14,3,1
$end
ORMAS is a full CI (or several full CI's) within each
orbital subspace. However, ORMAS does not generate all
excitation levels between spaces (just those implied by the
minimum and maximum electron counts you give). This means
ORMAS MCSCF runs must optimize active-active rotations
between the subspaces, and therefore you should expect
better convergence from FULLNR than SOSCF.
ORMAS is sure to require orbital reordering. For the
furan example just mentioned, there is no reason to expect
that the RHF occupied orbitals will not have the filled
sigma and pi orbitals intermingled. You must use the
NORDER and IORDER keywords in $GUESS to carefully partition
starting orbitals into sigma and pi subspaces.
The selected (general) determinant list is used if
CISTEP=GENCI, and the list is controlled by two input
groups. The first is $GEN, which is identical to $DET
except for inclusion of an additional keyword GLIST=INPUT.
This reads the determinants (as space products) from an
additional input group $GCILST. Completely arbitrary
choices for the space products may be made, but peculiar
lists may lead to poor MCSCF convergence. The FOCAS
converger should not be used, as it assumes full CI spaces.
If you are doing straight CI calculations, the required
input for each determinant CITYP is:
ALDET needs $CIDET
ORMAS needs $CIDET and $ORMAS
GENCI needs $CIDET and $CIGEN and probably $GCILST
FSOCI needs $CIDET and $SODET
In other words, $CIDET replaces $DET, and $CIGEN replaces
$GEN, but the keywords in the groups mean the same thing.
The reason for different names is to allow CI calculations
to follow MCSCF in the same run, without clashing input
group names.
CSF CI
The GUGA-based CSF package was originally a set of
different programs, so its input is spread over several
input groups. The CSFs are specified by a $CIDRT group in
the case of CITYP=GUGA, and by a $DRT group for MCSCF
wavefunctions. Thus it is possible to perform an MCSCF
defined by a $DRT input (or perhaps using $DET during the
MCSCF), and follow this with a second order CI defined by a
$CIDRT group, in the same run.
The remaining input groups used by the GUGA CSFs are
$CISORT, $GUGEM, $GUGDIA, and $GUGDM2 for MCSCF runs, with
the latter two being the most important, and in the case of
CI computations, $GUGDM and possibly $LAGRAN groups are
relevant. Perhaps the most interesting variables outside
the $DRT/$CIDRT group are NSTATE in $GUGDIA to include
excited states in the CI computation, IROOT in $GUGDM to
select the CI state for properties, and WSTATE in $GUGDM2
to control which state's orbitals are optimized, and
possible state-averaging.
The $DRT and $CIDRT groups are almost the same, with
the only difference being orbitals restricted to double
occupancy are called MCC in $DRT, and FZC in $CIDET.
Therefore the rest of this section refers only to "$DRT".
The CSFs are specified by giving a reference CSF,
together with a maximum degree of electron excitation from
that single CSF. The MOs in the reference CSF are filled
in the order MCC or FZC first, followed by DOC, AOS, BOS,
ALP, VAL, and EXT (the Aufbau principle). AOS, BOS, and
ALP are singly occupied MOs. ALP means a high spin alpha
coupling, while AOS/BOS are an alpha/beta coupling to an
open shell singlet. This requires the value NAOS=NBOS, and
their MOs alternate. An example is
NFZC=1 NDOC=2 NAOS=2 NBOS=2 NALP=1 NVAL=3
which gives the reference CSF
FZC,DOC,DOC,AOS,BOS,AOS,BOS,ALP,VAL,VAL,VAL
This is a doublet state with five unpaired electrons. VAL
orbitals are unoccupied only in the reference CSF, they
will become occupied as the other CSFs are generated. This
is done by giving an excitation level, either explicitly by
the IEXCIT variable, or implicitly by the FORS, FOCI, or
SOCI flags. One of these four keywords must be chosen, and
during MCSCF runs, this is usually FORS.
Consider another simpler example, for an MCSCF run,
NMCC=3 NDOC=3 NVAL=2
which gives the reference CSF
MCC,MCC,MCC,DOC,DOC,DOC,VAL,VAL
having six electrons in five active orbitals. MCSCF
calculations are usually of the Full Optimized Reaction
Space (FORS) type. Some workers refer to FORS as CASSCF,
complete active space SCF. These are the same, but the
keyword is spelled FORS in GAMESS. In the present
instance, choosing FORS=.TRUE. gives an excitation level of
4, as the 6 valence electrons have only 4 holes available
for excitation. MCSCF runs typically have only a small
number of VAL orbitals. It is common to summarize this
example as "six electrons in five orbitals".
The next example is a first or second order multi-
reference CI wavefunction, where
NFZC=3 NDOC=3 NVAL=2 NEXT=-1
leads to the reference CSF
FZC,FZC,FZC,DOC,DOC,DOC,VAL,VAL,EXT,EXT,...
FOCI or SOCI is chosen by selecting the appropriate flag,
the correct excitation level is automatically generated.
Note that the -1 for NEXT causes all remaining MOs to be
included in the external orbital space. One way of viewing
FOCI and SOCI wavefunctions is as all singles, or all
singles and doubles, from the entire MCSCF wavefunction as
a reference. An equivalent way of saying this is that all
CSFs with N electrons (in this case N=6) distributed in the
valence orbitals in all ways (that is the FORS MCSCF
wavefunction) make up the reference wavefunction. To this,
FOCI adds all CSFs with N-1 electrons in active and 1
electron in external orbitals. SOCI adds all CSFs with N-2
electrons in active orbitals and 2 in external orbitals.
SOCI is often prohibitively large, but is also a very
accurate wavefunction. SOCI can also be performed with
determinants, as CITYP=FSOCI, or CITYP=ORMAS. The latter
may be the most efficient way to generate SOCI energies.
For larger molecules, where SOCI is impractical, the most
effective way to recover dynamic correlation energy is the
multireference perturbation method.
Sometimes people use the CI package for ordinary single
reference CI calculations, such as
NFZC=3 NDOC=5 NVAL=34
which means the reference RHF wavefunction is
FZC FZC FZC DOC DOC DOC VAL VAL ... VAL
and in this case NVAL is a large number conveying the total
number of -virtual- orbitals into which electrons are
excited. The excitation level would be given as IEXCIT=2,
perhaps, to perform a SD-CI. All excitations smaller than
the value of IEXCIT are automatically included in the CI.
Note that NVAL's spelling was chosen to make the most sense
for MCSCF calculations, and so it is a bit of a misnomer
here.
Before going on, there is a quirk related to single
reference CI that should be mentioned. Whenever the single
reference contains unpaired electrons, such as
NFZC=3 NDOC=4 NALP=2 NVAL=33
some "extra" CSFs will be generated. The reference here
can be abbreviated
2222 11 000 000 000 000 000 000 000 000 000 000 000
Supposing IEXCIT=2, the following CSF
2200 22 000 011 000 000 000 000 000 000 000 000 000
will be generated and used in the CI. Most people would
prefer to think of this as a quadruple excitation from the
reference, but acting solely on the reasoning that no more
than two electrons went into previously vacant NVAL
orbitals, the GUGA CSF package decides it is a double. So,
an open shell SD-CI calculation with GAMESS will not give
the same result as other programs, although the result for
any such calculation with these "extras" is correctly
computed. Note that if you also select the INTACT option,
the extra space products are eliminated, but that some of
the spin couplings for the truly IEXCIT'd space products
are also eliminated. Note that this kind of problem does
not arise if you use ORMAS!
As was discussed above, the CSFs are automatically
spin-symmetry adapted, with S implicit in the reference
CSF. The spin quantum number you appear to be requesting
in $DRT (basically, S = NALP/2) will be checked against the
value of MULT in $CONTRL. The total number of electrons,
2*NMCC(or NFZC) + 2*NDOC + NAOS + NBOS + NALP will be
checked against the input given for ICHARG.
The CSF package is also able to exploit spatial
symmetry, which like the spin and charge, is implicitly
determined by the choice of the reference CSF. The keyword
GROUP in $DRT governs the use of spatial symmetry.
The CSF program works with Abelian point groups, which
are D2h and any of its subgroups. However, $DRT allows the
input of some (but not all) higher point groups. For non-
Abelian groups, the program automatically assigns the
orbitals to an irrep in the highest possible Abelian
subgroup. For the other non-Abelian groups, you must at
present select GROUP=C1. Note that when you are computing
a Hessian matrix, many of the displaced geometries are
asymmetric, hence you must choose C1 in $DRT (however, be
sure to use the highest symmetry possible in $DATA!).
The symmetry of the reference CSF given in your $DRT is
one way to determine the symmetry of the CSFs which are
generated. As an example, consider a molecule with Cs
symmetry, and these two reference CSFs
...MCC...DOC DOC VAL VAL
...MCC...DOC AOS BOS VAL
Suppose that the 2nd and 3rd active MOs have symmetries a'
and a". Both of these generate singlet wavefunctions, with
4 electrons in 4 active orbitals, but the former constructs
1-A' CSFs, while the latter generates 1-A" CSFs. However,
if the 2nd and 3rd orbitals have the same symmetry type, an
identical list of CSFs is generated. The alternative is to
enter the spatial symmetry with the STSYM keyword.
In cases with high point group symmetry, it may be
possible to generate correct state degeneracies only by
using no symmetry (GROUP=C1) when generating CSFs. As an
example, consider the 2-pi ground state of NO. If you use
GROUP=C4V, which will be mapped into its highest Abelian
subgroup C2v, the two components of the pi state will be
seen as belonging to different irreps, B1 and B2. The only
way to ensure that both sets of CSFs are generated is to
enforce no symmetry at all, so that CSFs for both
components of the pi level are generated. This permits
state averaging (WSTATE(1)=0.5,0.5) to preserve cylindrical
symmetry. It is however perfectly feasible to use C4v or
D4h symmetry in $DRT when treating sigma states.
The use of spatial symmetry decreases the number of
CSFs, and thus the size of the Hamiltonian that must be
computed. In molecules with high symmetry, this may lead
to faster run times with the GUGA CSF code, compared to the
determinant code.
starting orbitals
The first step is to partition the orbital space into
core, active, and external sets, in a manner which is
sensible for your chemical problem. This is a bit of an
art, and the user is referred to the references quoted at
the end of this section. Having decided what MCSCF to
perform, you now must consider the more pedantic problem of
what orbitals to begin the MCSCF calculation with.
You should always start an MCSCF run with orbitals from
some other run, by means of GUESS=MOREAD. Do not expect to
be able to use HUCKEL! At the start of a MCSCF problem,
use orbitals from some appropriate converged SCF run. A
realistic example of an MCSCF calculation is GAMESS
examples 8 and 9. Once you get an MCSCF to converge, you
can and should use these MCSCF MOs at other nearby
geometries (MOREAD will apply an appropriate Schmidt
orthogonalization).
Starting from SCF orbitals can take a little bit of
care. Most of the time (but not always) the orbitals you
want to correlate will be the highest occupied orbitals in
the SCF. Fairly often, however, the correlating orbitals
you wish to use will not be the lowest unoccupied virtuals
of the SCF. You will soon become familiar with NORDER=1 in
$GUESS, as reordering is needed in 50% or more cases.
The occupied and especially the virtual canonical SCF
MOs are often spread out over regions of the molecule other
than "where the action is". Orbitals which remedy this can
generated by two additional options at almost no CPU cost.
The best way to improve upon the SCF canonical virtual
orbitals as starting MOs is to generate valence virtual
orbitals (VVOs), after any RHF, ROHF, or GVB calculation.
These are constructed by projection of internally stored
atomic core and valence orbitals onto the SCF external
orbitals, so by construction, the resulting VVOs are
valence in character. See VVOS in $SCF.
An alternative choice, usually not as good as VVOs, are
the modified virtual orbitals. MVOs are obtained by
diagonalizing the Fock operator of a very positive ion,
within the virtual orbital space only. As implemented in
GAMESS, MVOs can be obtained at the end of any RHF, ROHF,
or GVB run by setting MVOQ in $SCF nonzero, at the cost of
a single SCF cycle. Typically, we use MVOQ=+6. Generating
MVOs does not change any of the occupied SCF orbitals of
the original neutral, but gives more valence-like LUMOs.
Another way to improve SCF starting orbitals is by a
partial localization of the occupied orbitals. Typically
MCSCF active orbitals are concentrated in the part of the
molecule where bonds are breaking, etc. Canonical SCF MOs
are normally more spread out. By choosing LOCAL=BOYS along
with SYMLOC=.TRUE. in $LOCAL, you can get orbitals which
are localized, but still retain orbital symmetry to help
speed the MCSCF along. In groups with an inversion center,
a SYMLOC Boys localization does not change the orbitals,
but you can instead use LOCAL=POP. LOCAL=RUEDNBRG may also
be used, but requires more machine resources, and the other
two localizations are normally good enough for starting
orbital purposes. Localization tends to order the orbitals
fairly randomly, so be prepared to inspect them, and then
reorder them appropriately.
In case VVOS are generated in the same run as the
localization, the localization is also applied within the
valence virtual space. The effect is to localize occupied
orbitals into lone pairs and bond pairs, and in the VVOs
space, to find localized antibond pairs. When you choose
the bonds and antibonds of chemical interest as the
starting orbitals for the active space, convergence to the
desired MCSCF wavefunction should follow.
If you take the time to design your active space
sensibly, select appropriate starting orbitals from the
occupied and VVO unoccupied spaces, possibly by localizing
these two subspaces, and carefully inspect your converged
results, you will be able to carry out MCSCF computations
correctly.
Convergence of MCSCF is by no means guaranteed. Poor
convergence can invariably be traced back to either a poor
initial selection of orbitals, or poor design of the active
space. The best advice is, before you even start:
"Look at the orbitals."
"Then, look at the orbitals again".
Later, if you have any trouble:
"Look at the orbitals some more".
Few people are able to see the orbital shapes in the LCAO
matrix in a log file, and so need a visualization program.
In particular, you should download a copy of MacMolPlt from
http://code.google.com/p/wxmacmolplt
This runs on all popular desktop operating systems (Apple,
Linux, and Windows), making it easy to see your final MCSCF
orbital shapes.
Even if you don't have any trouble, look at the
orbitals to see if they converged to what you expected, and
have reasonable occupation numbers. It is particularly
useful to check the oriented localized MCSCF orbitals (see
the discussion of this in the section on localized orbitals
in this section for more information). MCSCF is by no
means the sort of "black box" that RHF is these days, so
please look very carefully at your final results.
miscellaneous hints
It is very helpful to execute a EXETYP=CHECK run before
doing any MCSCF or CI run. The CHECK run will tell you the
total number of configurations and check the charge and
multiplicity and electronic state symmetry, based on your
input. The CHECK run also lets the program feel out the
memory that will be required to actually do the run. Thus
the CHECK run can potentially prevent costly mistakes, or
tell you when a calculation is prohibitively large.
A very common MCSCF wavefunction has 2 electrons in 2
active MOs. This is the simplest possible wavefunction
describing a singlet diradical. While this function can be
obtained in an MCSCF run (using NACT=2 NELS=2 or NDOC=1
NVAL=1), it can be obtained much faster by use of the GVB
code, with one GVB pair. This GVB-PP(1) wavefunction is
also known in the literature as two configuration SCF, or
TCSCF. The two configurations of this GVB are equivalent
to the three configurations used in this MCSCF, as orbital
optimization in natural form (configurations 20 and 02)
causes the coefficient of the 11 configuration to vanish.
If you are using a large active space (say, 12 or more
orbitals), the main bottleneck in the MCSCF calculation is
the formation and diagonalization of the Hamiltonian, not
the integral transformation and orbital updates. Of
course, since determinants are much faster than CSFs, and
do not use large disk files, you should use determinants
for large active spaces. In this case, you would be wise
to switch to FULLNR, which will minimize the total number
of iterations, and thus the number of CI calculations.
Note that by selecting ITERMX=5 in $DET or $GEN, you can
avoid fully converging the CI during each MCSCF iteration,
saving a bit of time. Since each iteration's CI
calculation starts with the previous iteration's result,
the CI vectors will become fully converged during the MCSCF
cycles. The total run time may decrease, although a few
additional MCSCF iterations may be required. For small
active spaces, where the CI step takes trivial time, you
should use a bigger ITERMX to ensure fully converged CI
states are generated every iteration.
If you choose to use ORMAS, a general determinant CI,
or if you select an CSF excitation level IEXCIT smaller
than that needed to generate the FORS space, you must use
the SOSCF, JACOBI, or FULLNR method as these can optimize
active-active rotations. Be sure to set FORS=.FALSE. in
$MCSCF when for non-full CI cases, or else very poor
convergence will result. Actually, the convergence for
incomplete active spaces is likely to be poorer than for
full active spaces, anyway.
A good way to check the active space is to localize the
orbitals, to see if they resemble the atomic orbitals which
you imagined formed the bonds, antibonds, and lone pairs in
the active space. The ORIENT keyword in $LOCAL will print
a density matrix analysis, showing active electron bonding
and antibonding patterns (see reference 18/19 below).
- - - - -
The MCSCF technology in GAMESS is the result of some
considerable programming effort: The FOCAS, serial FULLNR,
and QUAD convergers were adapted from Michel Dupuis' HONDO
program. The SOSCF converger was written by Galina Chaban,
the parallel FULLNR converger is due to Graham Fletcher,
and the JACOBI converger is due to Joe Ivanic. The GUGA CI
programs were written by Bernie Brooks and others, while
all determinant CI codes (ALDET, GENCI, ORMAS, and FSOCI)
stem from Joe Ivanic. Analytic nuclear Hessians were
programmed by Tim Dudley. The CSF-based multireference
pertubation program was written by Haruyuki Nakano, with a
determinant implementation provided by Joe Ivanic. Shiro
Koseki and Dmitri Fedorov are responsible for the spin-
orbit coupling and transition moment codes. The expertise
of Klaus Ruedenberg in MCSCF wavefunctions has been the
inspiration for many of these developments!
MCSCF references
There are several review articles about MCSCF listed
below. Of these, the first two are a nice overview of the
subject, the final 3 are more technical.
1. "The Construction and Interpretation of MCSCF
wavefunctions"
M.W.Schmidt and M.S.Gordon,
Ann.Rev.Phys.Chem. 49,233-266(1998)
2a. "The Multiconfiguration SCF Method"
B.O.Roos, in "Methods in Computational Molecular
Physics", edited by G.H.F.Diercksen and S.Wilson
D.Reidel Publishing, Dordrecht, Netherlands,
1983, pp 161-187.
2b. "The Multiconfiguration SCF Method"
B.O.Roos, in "Lecture Notes in Quantum Chemistry",
edited by B.O.Roos, Lecture Notes in Chemistry v58,
Springer-Verlag, Berlin, 1994, pp 177-254.
3. "Optimization and Characterization of a MCSCF State"
J.Olsen, D.L.Yeager, P.Jorgensen
Adv.Chem.Phys. 54, 1-176(1983).
4. "Matrix Formulated Direct MCSCF and Multiconfiguration
Reference CI Methods"
H.-J.Werner, Adv.Chem.Phys. 69, 1-62(1987).
5. "The MCSCF Method"
R.Shepard, Adv.Chem.Phys. 69, 63-200(1987).
There is an entire section on the choice of active
spaces in Reference 1. As this is a matter of great
importance, here are two alternate presentations of the
design of active spaces:
6. "The CASSCF Method and its Application in Electronic
Structure Calculations"
B.O.Roos, in "Advances in Chemical Physics", vol.69,
edited by K.P.Lawley, Wiley Interscience, New York,
1987, pp 339-445.
7. "Are Atoms Intrinsic to Molecular Electronic
Wavefunctions?"
K.Ruedenberg, M.W.Schmidt, M.M.Gilbert, S.T.Elbert
Chem.Phys. 71, 41-49, 51-64, 65-78 (1982).
Two papers germane to the FOCAS implementation are
8. "An Efficient first-order CASSCF method based on
the renormalized Fock-operator technique."
U.Meier, V.Staemmler Theor.Chim.Acta 76, 95-111(1989)
9. "Modern tools for including electron correlation in
electronic structure studies"
M.Dupuis, S.Chen, A.Marquez, in "Relativistic and
Electron Correlation Effects in Molecules and
Solids", edited by G.L.Malli, Plenum, NY 1994
The paper germane to the the SOSCF converger is
10. "Approximate second order method for orbital
optimization of SCF and MCSCF wavefunctions"
G.Chaban, M.W.Schmidt, M.S.Gordon
Theor.Chem.Acc. 97: 88-95(1997)
Two papers germane to the FULLNR converger, and two
discussing implementation details are
11. "General second order MCSCF theory: A Density Matrix
Directed Algorithm"
B.H.Lengsfield, III, J.Chem.Phys. 73,382-390(1980).
12. "The use of the Augmented Matrix in MCSCF Theory"
D.R.Yarkony, Chem.Phys.Lett. 77,634-635(1981).
13. M.Dupuis, P.Mougenot, J.D.Watts, in "Modern Techniques
in Theoretical Chemistry", E.Clementi, editor, ESCOM,
Leiden, 1989, chapter 7.
14. "A parallel multi-configuration self-consistent field
algorithm"
G.D.Fletcher, Mol.Phys. 105, 2971-2976(2007)
The paper describing the JACOBI converger is
15. "A MCSCF method for ground and excited states based on
full optimizatons of successive Jacobi rotations"
J.Ivanic, K.Ruedenberg
J.Comput.Chem. 24, 1250-1262(2003)
For determinant CI codes, see
16. "Identification of deadwood in configuration spaces
through general direct configuration interaction"
J.Ivanic, K.Ruedenberg
Theoret.Chem.Acc. 106, 339-351(2001)
17. "Direct configuration interaction and multi-
configurational self-consistent-field method for
multiple active spaces with variable occupancies.
Part I.Method Part II.Applications"
J.Ivanic
J.Chem.Phys. 119, 9364-9376 and 9377-9385(2003)
For CSFs, see
18. "GUGA approach to the electron correlation problem"
B.R.Brooks, H.F.Schaefer
J.Chem.Phys. 70, 5092-5106(1979)
Orientation of localized MCSCF active orbitals for
bonding analysis:
19. J.Ivanic, G.M.Atchity, K.Ruedenberg
Theoret.Chem.Acc. 120, 281-294(2008)
20. J.Ivanic, K.Ruedenberg
Theoret.Chem.Acc. 120, 295-305(2008)
For SF-ORMAS, see:
21. J. Mato and M. S. Gordon, PCCP, 20, 2615-2626 (2018)
22. J. Mato and M. S. Gordon, JPCA, 123, 1260-1272 (2019)
Second Order Perturbation Theory
The perturbation theory techniques available in GAMESS
expand to the second order energy correction only, but
permit use of nearly any zeroth order SCF wavefunction.
Since MP2 theory for systems well described by the chosen
zeroth order reference recovers about 80-85% of the
dynamical correlation energy (assuming the use of large
basis sets), MP2 is often a computationally effective
theory. For higher accuracy, you can instead choose the
more time consuming coupled cluster theory. When using
MPLEVL=2, it is important to ensure that your system is
well described at zeroth order by your choice of SCFTYP.
The input for second order pertubation calculations
based on SCFTYP=RHF, UHF, or ROHF is found in $MP2, while
for SCFTYP=MCSCF, see $MRMP.
By default, frozen core MP2 calculations are performed.
RHF and UHF reference MP2
These methods are well defined, due to the uniqueness of
the Fock matrix definitions. These methods are also well
understood, so there is little need to say more, except to
point out an overview article on RHF or UHF MP2 gradients:
C.M.Aikens, S.P.Webb, R.L.Bell, G.D.Fletcher,
M.W.Schmidt, M.S.Gordon
Theoret.Chem.Acc. 110, 233-253(2003)
The distributed memory parallel MP2 gradient program is
described in
G.D.Fletcher, M.W.Schmidt, M.S.Gordon
Adv.Chem.Phys. 110, 267-294(1999)
and that for UMP2 in
C.M.Aikens, M.S.Gordon
J.Phys.Chem.A 108, 3103-3110(2004)
One point which may not be commonly appreciated is that
the density matrix for the first order wavefunction for the
RHF and UHF case, which is generated during gradient runs
or if properties are requested in the $MP2 group, is of the
type known as "response density", which differs from the
more usual "expectation value density". The eigenvalues of
the response density matrix (which are the occupation
numbers of the MP2 natural orbitals) can therefore be
greater than 2 for frozen core orbitals, or even negative
values for the highest 'virtual' orbitals. The sum is of
course exactly the total number of electrons. We have seen
values outside the range 0-2 in several cases when the
single configuration HF wavefunction is not an appropriate
description of the system, and thus these occupancies may
serve as a guide to the wisdom of using a HF reference:
M.S.Gordon, M.W.Schmidt, G.M.Chaban, K.R.Glaesemann,
W.J.Stevens, C.Gonzalez J.Chem.Phys. 110,4199-4207(1999)
high spin ROHF reference MP2
There are a number of open shell perturbation theories
described in the literature. It is important to note that
these methods give different results for the second order
energy correction, reflecting ambiguities in the selection
of the zeroth order Hamiltonian and in defining the ROHF
Fock matrices. See
K.R.Glaesemann, M.W.Schmidt
J.Phys.Chem.A 114, 8772-8777(2010)
for a figure showing 4 different ROHF-based perturbation
theory potentials, which are highly parallel, but have
different total energy values.
Two of the perturbation theories mentioned below, RMP
and ZAPT, are available in GAMESS using SCFTYP=ROHF (see
OSPT in $MP2). Nuclear gradients can be obtained for ZAPT.
The OPT1 results can be generated using MPLEVL=2 with
SCFTYP=MCSCF, using an active space where every orbital is
singly occupied with the highest MULT possible (which is
single-determinant).
One theory is known as RMP, which it should be pointed
out, is entirely equivalent to the ROHF-MBPT2 method and to
the CUMP2 method. The perturbation theory is as UHF-like
as possible, and can be chosen in GAMESS by selection of
OSPT=RMP. The second order energy is defined by
1. P.J.Knowles, J.S.Andrews, R.D.Amos, N.C.Handy,
J.A.Pople Chem.Phys.Lett. 186, 130-136(1991)
2. W.J.Lauderdale, J.F.Stanton, J.Gauss, J.D.Watts,
R.J.Bartlett Chem.Phys.Lett. 187, 21-28(1991).
The submission dates are in inverse order of publication
dates, and -both- papers should be cited when using this
method. Here we will refer to the method as RMP in keeping
with much of the literature. The RMP method diagonalizes
the alpha and beta Fock matrices separately, so their
occupied-occupied and virtual-virtual blocks are
canonicalized. This generates two distinct orbital sets,
whose double excitation contributions are processed by the
usual UHF MP2 program, but an additional energy term from
single excitations is required. The recent CUHF method
converges a UHF calculation directly to these semi-
canonical orbitals, rather than generating them after ROHF
converges. The CUMP2 perturbation theory is thus identical
to methods described above:
3. G.E.Scuseria, T.Tsuchimochi
J.Chem.Phys. 134, 064101/1-14(2011)
CUMP2's input is SCFTYP=UHF, MPLEVL=2, CUHF=.TRUE. rather
than SCFTYP=ROHF, MPLEVL=2, OSPT=RMP.
RMP's use of different orbitals for different spins adds
to the CPU time required for integral transformations, of
course. just like UMP2. RMP is invariant under all of the
orbital transformations for which the ROHF itself is
invariant. Unlike UMP2, the second order RMP energy does
not suffer from spin contamination, since the reference
ROHF wave-function has no spin contamination. The RMP
wavefunction, however, is spin contaminated at 1st and
higher order, and therefore the 3rd and higher order RMP
energies are spin contaminated. Other workers have
extended the RMP theory to gradients and hessians at second
order, and to fourth order in the energy,
3. W.J.Lauderdale, J.F.Stanton, J.Gauss, J.D.Watts,
R.J.Bartlett J.Chem.Phys. 97, 6606-6620(1992)
4. J.Gauss, J.F.Stanton, R.J.Bartlett
J.Chem.Phys. 97, 7825-7828(1992)
5. D.J.Tozer, J.S.Andrews, R.D.Amos, N.C.Handy
Chem.Phys.Lett. 199, 229-236(1992)
6. D.J.Tozer, N.C.Handy, R.D.Amos, J.A.Pople, R.H.Nobes,
Y.Xie, H.F.Schaefer Mol.Phys. 79, 777-793(1993)
We deliberately omit references to the ROMP precursor of
the RMP formalism. RMP gradients are not available.
The Z-averaged perturbation theory (ZAPT) formalism for
ROHF perturbation theory is the preferred implementation of
open shell spin-restricted perturbation theory (OSPT=ZAPT
in $MP2). The ZAPT theory has only a single set of
orbitals in the MO transformation, and therefore runs in a
time similar to the RHF perturbation code. The second
order energy is free of spin-contamination, but some spin-
contamination enters into the first order wavefunction (and
hence properties). This should be much less contamination
than for OSPT=RMP. For these reasons, OSPT=ZAPT is the
default open shell method.
References for ZAPT are
7. T.J.Lee, D.Jayatilaka Chem.Phys.Lett. 201, 1-10(1993)
8. T.J.Lee, A.P.Rendell, K.G.Dyall, D.Jayatilaka
J.Chem.Phys. 100, 7400-7409(1994)
The formulae for the seven terms in the ZAPT energy are
clearly summarized in the paper
9. I.M.B.Nielsen, E.T.Seidl
J.Comput.Chem. 16, 1301-1313(1995)
The ZAPT gradient equations are found in
10. G.D.Fletcher, M.S.Gordon, R.L.Bell
Theoret.Chem.Acc. 107, 57-70(2002)
11. C.M.Aikens, G.D.Fletcher, M.W.Schmidt, M.S.Gordon
J.Chem.Phys. 124, 014107/1-14(2006)
We would like to thank Tim Lee for his gracious assistance
in the implementation of the ZAPT energy.
There are a number of other open shell theories, with
names such as HC, OPT1, OPT2, and IOPT. The literature for
these is
12. I.Hubac, P.Carsky Phys.Rev.A 22, 2392-2399(1980)
13. C.Murray, E.R.Davidson
Chem.Phys.Lett. 187,451-454(1991)
14. C.Murray, E.R.Davidson
Int.J.Quantum Chem. 43, 755-768(1992)
15. P.M.Kozlowski, E.R.Davidson
Chem.Phys.Lett. 226, 440-446(1994)
16. C.W.Murray, N.C.Handy
J.Chem.Phys. 97, 6509-6516(1992)
17. T.D.Crawford, H.F.Schaefer, T.J.Lee
J.Chem.Phys. 105, 1060-1069(1996)
The latter two of these give comparisons of the various
high spin methods, and the numerical results in ref. 17 are
the basis for the conventional wisdom that restricted open
shell theory is better convergent with order of the
perturbation level than unrestricted theory. Paper 8 has
some numerical comparisons of spin-restricted theories as
well. We are aware of one paper on low-spin coupled open
shell SCF perturbation theory
18. J.S.Andrews, C.W.Murray, N.C.Handy
Chem.Phys.Lett. 201, 458-464(1993)
but this is not implemented in GAMESS. See the MCSCF
reference perturbation code for this case.
GVB based MP2
This is not implemented in GAMESS. Note that the MCSCF
perturbation program discussed below should be able to
develop the perturbation corrections to open shell
singlets, by using a $DRT input such as
NMCC=N/2-1 NDOC=0 NAOS=1 NBOS=1 NVAL=0
which generates a single CSF if the two open shells have
different symmetry, or for a one pair GVB function
NMCC=N/2-1 NDOC=1 NVAL=1
which generates a 3 CSF function entirely equivalent to
the two configuration TCSCF, a.k.a GVB-PP(1). For the
record, note that if we attempt a triplet state with the
MCSCF program,
NMCC=N/2-1 NDOC=0 NALP=2 NVAL=0
we get a result equivalent to the OPT1 open shell method
described above, not the RMP or ZAPT result. It is
possible to generate the orbitals with a simpler SCF
computation than the MCSCF $DRT examples just given, and
read them into the MCSCF based MP2 program described below,
by RDVECS=.TRUE..
MCSCF reference perturbation theory
Just as for the open shell case, there are several ways
to define a multireference perturbation theory. The most
noteworthy are the CASPT2 method of Roos' group, the MRMP2
method of Hirao, the closely related MCQDPT2 method of
Nakano, and the MROPTn methods of Davidson. Although the
total energies of each method are different, energy
differences should be rather similar. In particular, the
MRMP/MCQDPT method implemented in GAMESS gives results for
the singlet-triplet splitting of methylene in close
agreement to CASPT2, MRMP2(Fav), and MROPT1, and differs by
2 Kcal/mole from MRMP2(Fhs), and the MROPT2 to MROPT4
methods.
The MCQDPT method implemented in GAMESS is a multistate
perturbation theory due to Nakano. If applied to 1 state,
it is the same as the MRMP model of Hirao. When applied to
more than one state, it is of the philosophy "perturb
first, diagonalize second". This means that perturbations
are made to both the diagonal and off-diagonal elements to
give an effective Hamiltonian, whose dimension equals the
number of states being treated. The effective Hamiltonian
is diagonalized to give the second order state energies.
Diagonalization after inclusion of the off-diagonal
perturbation ensures that avoided crossings of states of
the same symmetry are treated correctly. Such an avoided
crossing is found in the LiF molecule, as shown in the
first of the two papers on the MCQDPT method:
H.Nakano, J.Chem.Phys. 99, 7983-7992(1993)
H.Nakano, Chem.Phys.Lett. 207, 372-378(1993)
The closely related single state "diagonalize, then
perturb" MRMP model is discussed by
K.Hirao, Chem.Phys.Lett. 190, 374-380(1992)
K.Hirao, Chem.Phys.Lett. 196, 397-403(1992)
K.Hirao, Int.J.Quant.Chem. S26, 517-526(1992)
K.Hirao, Chem.Phys.Lett. 201, 59-66(1993)
Computation of reference weights and energy contributions
is illustrated by
H.Nakano, K.Nakayama, K.Hirao, M.Dupuis
J.Chem.Phys. 106, 4912-4917(1997)
T.Hashimoto, H.Nakano, K.Hirao
J.Mol.Struct.(THEOCHEM) 451, 25-33(1998)
Single state MCQDPT computations are very similar to MRMP
computations. A beginning set of references to the other
multireference methods used includes:
P.M.Kozlowski, E.R.Davidson
J.Chem.Phys. 100, 3672-3682(1994)
K.G.Dyall J.Chem.Phys. 102, 4909-4918(1995)
B.O.Roos, K.Andersson, M.K.Fulscher, P.-A.Malmqvist,
L.Serrano-Andres, K.Pierloot, M.Merchan
Adv.Chem.Phys. 93, 219-331(1996).
and a review article is available comparing these methods,
E.R.Davidson, A.A.Jarzecki in "Recent Advances in Multi-
reference Methods" K.Hirao, Ed. World Scientific, 1999,
pp 31-63.
The CSF (GUGA-based) MRMP/MCQDPT code was written by
Haruyuki Nakano, and was interfaced to GAMESS by him in the
summer of 1996. This program makes extensive use of disk
files during its specialized transformations and the
perturbation steps. Its efficiency is improved if you can
add extra physical memory to reduce the number of file
reads. In practice we have used this program up to about
12 active orbitals, and with very large disks, to about 500
AOs. In 2005, Joe Ivanic programmed a determinant based
MRMP/MCQDPT program. This uses the normal integral
transformation routines already present in GAMESS, and
direct CI technology to avoid disk I/O. The determinant
program is able to handle larger active spaces than the CSF
program, and has already been used for cases with 16
electrons in 16 orbitals, and basis sets up to 500 AOs.
When proper care is taken with numerical cutoffs, such
as CI vector convergence and the generator cutoff in the
CSF code, both programs produce identical results. Both
are enabled for parallel execution. The more mature CSF
program has several interesting options not found in the
determinant program: perturbative treatment of spin-orbit
coupling, energy denominators which are a band-aid for the
horribly named "intruder states", and the ability to find
the weight of the MCSCF reference in the 1st order
wavefunction. Neither program produces a density matrix
for property evaluation, nor are analytic gradients
programmed.
Second order perturbation corrections are also available
for ORMAS-type references, due to a program by Luke Roskop.
This is accessed by MRPT=DETMRPT, defining the reference by
CISTEP=ORMAS in $MCSCF, with a $ORMAS group. In its
present form, this program is limited to cases with equal
numbers of alpha and beta spins (i.e. singlets, triplets,
... with SZ=0, but not doublets, quartets...). See
L.Roskop, M.S.Gordon J.Chem.Phys. 135, 044101/1-11(2012)
Finally, note the existence of the MRMP=GMCPT program by
Haruyuki Nakano, for other non-CAS references.
- - - - - - - - - - - -
We end with an input example to illustrate open shell
reference and multi-reference pertubation computations on
the ground state of NH2 radical:
! 2nd order perturbation test on NH2, following
! T.J.Lee, A.P.Rendell, K.G.Dyall, D.Jayatilaka
! J.Chem.Phys. 100, 7400-7409(1994), Table III.
! State is 2-B-1, 69 AOs, either 1 or 49 CSFs.
!
! For 1 CSF reference,
! E(ROHF) = -55.5836109825
! E(ZAPT) = -55.7763947115
! [E(ZAPT) = -55.7763947289 at lit's ZAPT geom]
! E(RMP) = -55.7772299958
! E(OPT1) = -55.7830422945
! [E(OPT1) = -55.7830437413 at lit's OPT1 geom]
!
! For 49 CSF full valence MCSCF reference,
! CSFs: E(MRMP2) = -55.7857440268
! dets: E(MRMP2) = -55.7857440267
!
$contrl scftyp=mcscf mplevl=2 runtyp=energy mult=2 $end
$system mwords=1 memddi=1 $end
$guess guess=moread norb=69 $end
$mcscf fullnr=.true. $end
!
! Next set of lines carry out a MRMP computation,
! after a preliminary MCSCF orbital optimization.
!
! using determinants
$det stsym=B1 ncore=1 nact=6 nels=7 $end
! using CSFs, for the very same calculation.
--- $mcscf cistep=guga $end
--- $drt group=c2v stsym=B1 fors=.t.
--- nmcc=1 ndoc=3 nalp=1 nval=2 $end
--- $mrmp mrpt=mcqdpt $end
--- $mcqdpt stsym=B1 nmofzc=1 nmodoc=0 nmoact=6 $end
! Next lines carry out a single reference OPT1.
--- $det stsym=B1 ncore=4 nact=1 nels=1 $end
--- $mrmp mrpt=mcqdpt rdvecs=.true. $end
--- $mcqdpt nmofzc=1 nmodoc=3 nmoact=1 stsym=B1 $end
! Next lines are single reference RMP and/or ZAPT
--- $contrl scftyp=rohf $end
--- $mp2 ospt=rmp $end
$data
2-B-1 state...TZ2Pf basis, RMP geom. of Lee, et al.
Cnv 2
Nitrogen 7.0
S 6
1 13520.0 0.000760
2 1999.0 0.006076
3 440.0 0.032847
4 120.9 0.132396
5 38.47 0.393261
6 13.46 0.546339
S 2
1 13.46 0.252036
2 4.993 0.779385
S 1 ; 1 1.569 1.0
S 1 ; 1 0.5800 1.0
S 1 ; 1 0.1923 1.0
P 3
1 35.91 0.040319
2 8.480 0.243602
3 2.706 0.805968
P 1 ; 1 0.9921 1.0
P 1 ; 1 0.3727 1.0
P 1 ; 1 0.1346 1.0
D 1 ; 1 1.654 1.0
D 1 ; 1 0.469 1.0
F 1 ; 1 1.093 1.0
Hydrogen 1.0 0.0 0.7993787 0.6359684
S 3 ! note that this is unscaled
1 33.64 0.025374
2 5.058 0.189684
3 1.147 0.852933
S 1 ; 1 0.3211 1.0
S 1 ; 1 0.1013 1.0
P 1 ; 1 1.407 1.0
P 1 ; 1 0.388 1.0
D 1 ; 1 1.057 1.0
$end
OPT1 geom: H 1.0 0.0 0.7998834 0.6369401
RMP geom: H 1.0 0.0 0.7993787 0.6359684
ZAPT geom: H 1.0 0.0 0.7994114 0.6357666
E(ROHF)= -55.5836109825, E(NUC)= 7.5835449477, 9 ITERS
$VEC
...omitted...
$END
Coupled-Cluster Theory
The single-reference coupled-cluster (CC) theory, employing
the exponential wave function ansatz
|Psi0> = exp(T) |Phi> = exp(T1+T2+...) |Phi>,
where T1, T2, etc. are the singly excited (1-particle-1-
hole), doubly excited (2-particle-2-hole), etc. components
of the cluster operator T and |Phi> is the single-
determinantal reference state (e.g., the Hartree-Fock
determinant), is widely recognized as one of the most
accurate methods for describing ground electronic states of
atoms and molecules. CC approaches provide the best
compromise between relatively low computer costs and high
accuracy. They are particularly effective in accounting for
the dynamical correlation effects. For example, the
CCSD(T) approach, which is a No**2 * Nu**4 (or N**6)
procedure in the iterative CCSD steps and a No**3 * Nu**4
(or N**7) procedure in the non-iterative steps related to
the calculation of triples (T3) energy corrections, is
capable of providing results of the CISDTQ or better
quality (CISDTQ is an iterative No**4 * Nu**6 or N**10
procedure) when closed-shell molecules are examined. Here
and elsewhere in this section, No and Nu are the numbers of
correlated occupied and unoccupied orbitals. Symbol N
designates a measure of the system size in the following
sense: N=2 means a simultaneous increase of the number of
correlated electrons and basis functions by a factor of
two. Unlike single- and multi-reference CI methods and some
variants of multi-reference perturbation theory, all
standard CC methods, such as CCSD or CCSD(T), provide a
size extensive description of molecular systems, i.e. no
loss of accuracy occurs due to the mere increase of the
system size when CC calculations are performed.
Thanks to numerous advances in both the formal aspects
of CC theory and the development of efficient computer
codes, the single-reference CC approaches, such as CCSD and
CCSD(T), are nowadays routinely used in calculations for
non-degenerate closed- and open-shell electronic ground
states of atomic and molecular systems with up to 50 or so
correlated electrons and up to 200-300 or so basis
functions. The application of the local correlation
formalism within the context of CC theory enables one to
extend the applicability of the CCSD(T) and similar CC
approaches to systems with approximately 100 light atoms
(hundreds of correlated electrons and > 1000 basis
functions). Generalizations of CC theory to open-shell,
quasi-degenerate, and excited states are possible, via the
multi-reference, renormalized, extended, equation-of-
motion, and response CC formalisms, and some of these
extensions (for example, the equation-of-motion CC methods
for excited states) have become as popular as the multi-
reference CI, multi-reference perturbation theory, or
CASSCF methods. We should also add that CC theory is a
fundamental many-body formalism, whose applicability ranges
from electronic structure of atoms and molecules and
nuclear physics to extended systems, phase transitions,
condensed matter theory, theories of homogeneous electron
gas, and relativistic quantum field theory, to mention a
few examples. Examples of applications of quantum chemical
CC methods in ab initio calculations for atomic nuclei
using modern nucleon-nucleon interactions by Piecuch and
co-workers are listed in the reference section below.
A number of review articles have been written over the
years and it is difficult to cite all of them here. We
recommend that users of GAMESS planning to use CC/EOMCC
methods read one or more reviews listed below:
"Coupled-cluster theory"
J. Paldus, in S. Wilson and G.H.F. Diercksen (Eds.),
Methods in Computational Molecular Physics, NATO Advanced
Study Institute, Series B: Physics, Vol. 293, Plenum, New
York, 1992, pp. 99-194.
"Applications of post-Hartree-Fock methods: a tutorial."
R.J. Bartlett and J.F. Stanton, in K.B. Lipkowitz and
D.B.Boyd (Eds.), Reviews in Computational Chemistry,
Vol. 5, VCH Publishers, New York, 1994, pp. 65-169.
"Coupled-Cluster Theory: Overview of Recent Developments"
R.J. Bartlett, in D.R. Yarkony (Ed.), Modern Electronic
Structure Theory, Part I, World Scientific, Singapore,
1995, pp. 1047-1131.
"Achieving chemical accuracy with coupled-cluster theory"
T.J. Lee and G.E. Scuseria, in S.R. Langhoff (Ed.),
Quantum Mechanical Electronic Structure Calculations with
Chemical Accuracy, Kluwer, Dordrecht, The Netherlands,
1995, pp. 47-108.
"Coupled-cluster Theory"
J. Gauss, in Encyclopedia of Computational Chemistry,
P.v.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger,
P.A. Kollman, H.F. Schaefer III, P.R. Schreiner (Eds.)
Wiley, Chichester, U.K., 1998, Vol. 1, pp. 615-636.
"A Critical Assessment of Coupled Cluster Method in Quantum
Chemistry"
J. Paldus and X. Li, Adv. Chem. Phys. 110, 1-175 (1999),
"EOMXCC: A New Coupled-Cluster Method for Electronically
Excited States"
P. Piecuch and R.J. Bartlett, Adv. Quantum Chem. 34,
295-380 (1999).
"An Introduction to Coupled Cluster Theory for
Computational Chemists"
T.D.Crawford, H.F.Schaefer in K.B. Lipkowitz and D.B.Boyd
(Eds.), Reviews in Computational Chemistry, Vol. 14, VCH
Publishers, New York, 2000, pp. 33-136.
"In Search of the Relationship between Multiple Solutions
Characterizing Coupled-Cluster Theories"
P. Piecuch and K. Kowalski, in J. Leszczynski (Ed.),
Computational Chemistry: Reviews of Current Trends,
Vol. 5, World Scientific, Singapore, 2000), pp. 1-104.
"Recent Advances in Electronic Structure Theory: Method of
Moments of Coupled-Cluster Equations and Renormalized
Coupled-Cluster Approaches"
P. Piecuch, K. Kowalski, I.S.O. Pimienta, M.J. McGuire,
Int. Rev. Phys. Chem. 21, 527-655 (2002).
"New Alternatives for Electronic Structure Calculations:
Renormalized, Extended, and Generalized Coupled-Cluster
Theories"
P. Piecuch, I.S.O. Pimienta, P.-F. Fan, and K. Kowalski,
in J. Maruani, R. Lefebvre, and E. Brandas (Eds.),
Progress in Theoretical Chemistry and Physics, Vol. 12,
Advanced Topics in Theoretical Chemical Physics,
Kluwer, Dordrecht, 2003, pp. 119-206.
"Coupled Cluster Methods"
J. Paldus, in Handbook of Molecular Physics and Quantum
Chemistry, edited by S. Wilson (Wiley, Chichester, 2003),
Vol. 2, pp. 272-313.
"Method of Moments of Coupled-Cluster Equations: A New
Formalism for Designing Accurate Electronic Structure
Methods for Ground and Excited States"
P. Piecuch, K. Kowalski, I.S.O. Pimienta, P.-D. Fan, M.
Lodriguito, M.J. McGuire, S.A. Kucharski, T. Kus, and M.
Musial,
Theor. Chem. Acc. 112, 349-393 (2004).
"Noniterative Coupled-Cluster Methods for Excited
Electronic States"
P. Piecuch, M. Wloch, M. Lodriguito, and J.R. Gour,
in Progress in Theoretical Chemistry and Physics, Vol.
15, Recent Advances in the Theory of Chemical and
Physical Systems," edited by S. Wilson, J.-P. Julien, J.
Maruani, E. Brandas, and G. Delgado-Barrio (Springer,
Berlin, 2006), pp. XXX-XXXX, in press.
"Bridging Quantum Chemistry and Nuclear Structure Theory:
Coupled-Cluster Calculations for Closed- and Open-Shell
Nuclei"
P. Piecuch, M. Wloch, J.R. Gour, D.J. Dean, M. Hjorth-
Jensen, and T. Papenbrock, in V. Zelevinsky (Ed.), Nuclei
and Mesoscopic Physics: Workshop on Nuclei and Mesoscopic
Physics WNMP 2004, AIP Conference Proceedings, Vol. 777,
AIP Press, 2005, pp. 28-45.
These reviews point to the other review articles and many
original papers. The list of original papers relevant to
CC/EOMCC methods implemented in GAMESS is provided below.
available computations (ground states)
The CC programs incorporated in GAMESS enable user to
perform conventional LCCD, CCD, CCSD, CCSD[T] (also known
as CCSD+T(CCSD)), CCSD(T), and CCSD(TQ) calculations,
renormalized (R) and completely renormalized (CR) CCSD[T],
CCSD(T), and CCSD(TQ) calculations, and calculations using
the rigorously size extensive completely renormalized CR-
CC(2,3) (or CR-CCSD(T)L) approach for closed-shell RHF
references. Performance of the ground-state CC methods has
been discussed in a number of places (cf. the review
articles mentioned above and references listed at the end
of the "Coupled-Cluster Theory" section). Methods such as,
for example, CCSD(T), CR-CC(2,3), and CCSD(TQ) provide
excellent results for molecules in or near the equilibrium
geometries. Almost all CC methods are excellent in
describing dynamical correlation, while being relatively
inexpensive and easy to use. One must remember, however,
that the conventional single-reference CC methods, such as
CCSD(T), should not be applied to bond breaking,
diradicals, and other quasi-degenerate states, particularly
(but not only) when the RHF determinant is used as a
reference. In some of the most frequent cases of electronic
quasi-degeneracies, including single-bond breaking and
diradicals, the CR-CCSD(T), CR-CCSD(TQ), and CR-CC(2,3)=
CR-CCSD(T)L methods can be used instead. The recently
proposed CR-CC(2,3) approach seems particularly promising
in this regard, although the CR-CCSD(T) and CR-CCSD(TQ)
approaches are very useful as well. The CR-CC(2,3) method
has costs similar to those characterizing the CCSD(T)
approach, while providing the results of the very high,
full CCSDT, quality for diradicals and single-bond breaking
where CCSD(T) fails. At the same time, the accuracy of CR-
CC(2,3) calculations is comparable to or, sometimes, even
better than that obtained with the conventional CCSD(T)
approach for closed-shell molecules near the equilibrium
geometries. Just like CCSD(T), the CR-CC(2,3) approximation
is rigorously size extensive, while working much better
than CCSD(T) when non-dynamical correlation effects become
large. CR-CC(2,3) (CCTYP=CR-CCL) is among the most
attractive ground-state CC options in GAMESS, providing
GAMESS users with the highly accurate energies in the
closed-shell, single-bond breaking, and diradical regions
of molecular potential energy surfaces, and a number of
one-electron properties calculated at the CCSD level at a
price of single, relatively inexpensive calculation of the
CCSD(T) type.
One of the interesting features of GAMESS that can be
particularly useful in high accuracy calculations for
closed-shell systems is the presence of the (TQ)
corrections to CCSD energies among various ground-state CC
options. This includes the factorized CCSD(TQ),b method
suggested by Kowalski and Piecuch, which describes triples
effects at the CCSD(T) level, using noniterative steps that
scale as N**7 with the system size, while providing
information about the dominant effects due to quadruply
excited clusters. The CCSD(TQ),b method is closely related
to its CCSD(TQf) predecessor proposed by Kucharski and
Bartlett. In fact, if desired, one can extract the
CCSD(TQf) energy from the information printed in the GAMESS
output when CCTYP=CCSD(TQ) or CR-CC(Q) as follows:
CCSD + [R1-CCSD(TQ),A ? CCSD] * [CCSD(TQ),A DENOMINATOR]
(the R1-CCSD(TQ),A method in the GAMESS output represents
one of the renormalized CCSD(TQ) approaches, termed R-
CCSD(TQ)-1,a, which are discussed below). The differences
between the CCSD(TQ),b and CCSD(TQf) methods are minimal
and the accuracies and costs of both approaches are
virtually identical. In particular, both methods use
relatively inexpensive noniterative steps that scale as
N**6 or N**7 with the system size to determine the
quadruples corrections.
The unique features of the ground-state CC code in
GAMESS are the renormalized (R) and completely renormalized
(CR) CCSD[T], CCSD(T), and CCSD(TQ) methods [see K.
Kowalski and P. Piecuch, J. Chem. Phys. 113, 18-35 (2000),
idem., ibid. 113, 5644-5652 (2000), and P. Piecuch and K.
Kowalski, in J. Leszczynski (Ed.), Computational Chemistry:
Reviews of Current Trends, Vol. 5, World Scientific,
Singapore, 2000, pp. 1-104], and the most recent (Fall
2005), rigorously size extensive formulation of CR-CCSD(T),
termed CR-CC(2,3) or CR-CCSD(T)L [see P. Piecuch and M.
Wloch, J. Chem. Phys. 123, 224105-1 - 224105-10 (2005) and
P. Piecuch, M. Wloch, J.R. Gour, and A. Kinal, Chem. Phys.
Lett. 418, 467-474 (2006)]. All of these approaches are
based on the more general formalism of the method of
moments of coupled-cluster equations (MMCC; biorthogonal
MMCC in the case of CR-CC(2,3)), developed by the Piecuch
group at Michigan State University. They remove or
considerably reduce the pervasive failing of the
conventional CCSD[T], CCSD(T), and CCSD(TQ) approximations
at larger internuclear separations and for diradical
systems, while preserving the ease of use and the
relatively low cost of the single-reference methods of the
CCSD(T) or CCSD(TQ) type. In analogy to the CCSD[T],
CCSD(T), and CCSD(TQ) methods, the R-CCSD[T], R-CCSD(T), R-
CCSD(TQ)-n,x (n=1,2;x=a,b), CR-CCSD[T], CR-CCSD(T), CR-
CC(2,3), and CR-CCSD(TQ),x (x=a,b) approaches are based on
an idea of improving the CCSD results by adding a
posteriori noniterative corrections to CCSD energies. These
corrections employ the generalized moments of CCSD
equations (projections of the Schroedinger equation for the
CCSD wave function on the triply (T) or triply and
quadruply (TQ) excited determinants) and are designed by
extracting the leading terms that define the theoretical
difference between the CCSD and full CI energies. The CR-
CCSD[T], CR-CCSD(T), and CR-CC(2,3) approaches are capable
of eliminating the unphysical humps on the potential energy
surfaces involving single bond breaking produced by the
conventional CCSD[T] and CCSD(T) methods. They also
significantly improve the poor description of diradical
species (for example, diradical transition states and
intermediates) by the CCSD[T] and CCSD(T) methods. What is
important in practical applications, the CR-CCSD(T) and CR-
CC(2,3) approaches are capable of providing a good balance
between the dynamical and nondynamical correlation effects
when the diradical and closed-shell structures have to be
examined together. The rigorously size extensive CR-CC(2,3)
method is particularly effective in this regard, although
the older and somewhat less expensive CR-CCSD(T) approach
is very useful as well. The R-CCSD[T] and R-CCSD(T)
approaches may improve the CCSD[T] and CCSD(T) results at
intermediate internuclear separations, but they usually
fail at larger distances. The CR-CCSD[T], CR-CCSD(T), and
CR-CC(2,3) methods are better in this regard, since they
often provide a very good description of single bond
breaking at all internuclear separations. This includes
various cases of unimolecular dissociations and exchange
and bond insertion chemical reactions, in which single
bonds break and form. We DO NOT recommend applying the CR-
CCSD[T], CR-CCSD(T), and CR-CC(2,3) approaches to multiple
bond breaking, although some types of multiple bond
stretching can be described by these methods very well if
the relevant stretches of chemical bonds are not too large.
In general, however, multiple bond dissociations require
using the higher-order methods, such as the completely
renormalized CCSD(TQ) and CCSDT(Q) approaches (the CR-
CCSD(TQ) methods are available in GAMESS), the so-called
MMCC(2,6) method, and the more recent generalized and
quadratic MMCC methods, if the single-reference approach is
preferred, or the multi-reference CC methods of the state-
universal and state-specific type (some of the most
promising approaches in these categories, including active-
space and state-universal CC methods, will be included in
GAMESS in the future). In particular, the CR-CCSD(TQ)
approaches available in GAMESS are reasonably accurate in
situations involving double bond dissociations and a
simultaneous stretching or breaking of two single bonds.
They may work reasonably well even when the triple bond
stretching or breaking is examined, but the results for
more complicated cases of bond breaking are not as good as
those that one can obtain with the best multi-reference
approaches. A detailed description of the R-CCSD[T], R-
CCSD(T), CR-CCSD[T], CR-CCSD(T), CR-CC(2,3), R-CCSD(TQ),
and CR-CCSD(TQ) approaches and other MMCC methods can be
found in several papers by Piecuch and coworkers listed at
the very end of the "Coupled-Cluster Theory" section.
Unlike the newest CR-CC(2,3) approximation, the somewhat
older R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), R-
CCSD(TQ), and CR-CCSD(TQ) methods are not strictly size
extensive, i.e. there are unlinked terms in the MBPT (many-
body perturbation theory) expansions of the renormalized
and completely renormalized [T], (T), and (TQ) corrections
to CCSD energies. This has little or no effect on bond
breaking (on the contrary, the CR-CCSD[T], CR-CCSD(T), and
CR-CCSD(TQ) potential surfaces are MUCH better than
potential energy surfaces obtained in the standard and size
extensive CCSD[T], CCSD(T), and CCSD(TQ) calculations), but
lack of strict size extensivity may have an effect on the
results of calculations for larger and extended systems. A
lot depends on the values of T2 amplitudes and the chemical
problem of interest. If the T2 amplitudes are small, then
the overlap denominator expressions which define the
renormalized [T], (T), and (TQ) corrections of the R-
CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), R-CCSD(TQ), and
CR-CCSD(TQ) methods are close to 1, in which case there is
no major problem. If the T2 amplitudes are large, then
these denominators may become significantly greater than 1.
This behavior of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR-
CCSD(T), R-CCSD(TQ), and CR-CCSD(TQ) denominator
expressions is extremely useful for improving the results
for bond breaking, since the denominators defining the
renormalized [T], (T), and (TQ) corrections damp the
unphysical values of the standard [T], (T), and (TQ)
corrections at larger internuclear separations or when the
wave function gains a significant multi-reference
character. The same applies to diradical species, where the
standard [T], (T), and (TQ) corrections produce unphysical
results and need damping that the renormalized methods
provide. However, for larger many-electron systems (with
50 correlated electrons or more), the denominators defining
the renormalized [T], (T), and (TQ) corrections may
"overdamp" the [T], (T), and (TQ) energy corrections. On
the other hand, the renormalized [T], (T), and (TQ) energy
corrections are constructed using the cluster amplitudes
resulting from the size extensive CCSD calculations.
Moreover, it is often the case that the number of
correlated electrons used in CC calculations for larger
molecules (and only these electrons are used in
constructing the renormalized [T], (T), and (TQ)
corrections to CCSD energies) is much smaller than the
total number of electrons. Thus, the consequences of the
lack of strict size extensivity of the R-CCSD[T], R-
CCSD(T), CR-CCSD[T], CR-CCSD(T), R-CCSD(TQ), and CR-
CCSD(TQ) methods do not have to be serious for larger
systems, particularly when one examines, for example, the
relative energies of stationary points along the reaction
pathways relative to the relevant reactants (see comments
below). A number of interesting chemical problems
involving smaller and medium size polyatomic diradical
systems, including, for example, the Cope rearrangement of
1,5-hexadiene, the cycloaddition of cyclopentyne to
ethylene, the isomerizations of bicyclopentene and
tricyclopentane into cyclopentadiene, the thermal
stereomutations of cyclopropane, and the relative
energetics of dicopper systems relevant to molecular oxygen
activation by copper metalloenzymes, where the standard
CCSD(T) approach and, in some cases, the low-order multi-
reference perturbation theory methods encounter serious
difficulties, have been successfully examined with the CR-
CCSD(T) approach, demonstrating that problems of size
extensivity in CR-CCSD(T) calculations are of no major
significance in molecules of these sizes. But one may have
to be more careful when chemical systems have more than 50
correlated electrons. Extensive numerical tests indicate
that lack of strict size extensivity has little (fraction
of a millihartree or so) effect on the results of the CR-
CCSD[T], CR-CCSD(T), and CR-CCSD(TQ) calculations for
smaller systems. For larger systems, such as the glycine
dimer described by the 6-31G basis set, the departure from
rigorous size extensivity, as measured by forming the
difference of the sum of the energies of isolated glycine
molecules from the energy of the dimer consisting of
glycine molecules at very large (200 bohr) distance, is ca.
3 millihartree (2 kcal/mol). The violation of strict size
extensivity by the CR-CCSD(T) methods has been estimated at
approximately 0.5 % of the total correlation energy
(changes in the correlation energy if the relative energies
along reaction pathways are examined), which is often a
small price to pay considering the significant improvements
that the renormalized CC methods offer for potential energy
surfaces and diradicals and the ease with which the CR-CC
calculations can be performed. IMPORTANT PRACTICAL ADVICE:
In studies of reaction pathways with the CR-CCSD(T)
approach, where reactants and products are connected by one
or more transition states and intermediates and where there
are two or more reactants, we STRONGLY RECOMMEND that the
user of CR-CCSD(T) proceeds in a manner similar to multi-
reference CI calculations. Thus, we advise to calculate the
energies of transition states, intermediates, and products
relative to reactants, using the total CR-CCSD(T) energy of
a noninteracting complex formed by reactants (reactants
separated by a large distance, say, 200 Angs.) as the
reference energy of reactants rather than the sum of the
CR-CCSD(T) energies of isolated reactants. This reduces the
possible size extensivity errors in the CR-CCSD(T)
calculations for larger systems to a minimum, since all
species along a reaction pathway (including reactants,
transition states, intermediates, and products) are treated
then in the same, well balanced, manner. Similar remarks
apply to the CR-CCSD(TQ) (and all R-CC) calculations. None
of the above has to be done when the CR-CC(2,3) approach is
employed, since CR-CC(2,3) is size extensive and the CR-
CC(2,3) energy of A+B equals the sum of CR-CC(2,3) energies
of A and B.
The rigorously size extensive modifications of the CR-
CC methods have recently (2005) been developed, using the
idea of locally renormalized methods, such as LR-CCSD(T),
which lead to size extensive results when localized
orbitals are employed, and, in an alternative formulation,
the idea of exploiting the left CC states combined with the
so-called biorthogonal MMCC theory. The latter development
seems particularly attractive. The resulting CR-CC(2,3)
method, also called CR-CCSD(T)L, which combines the best
features of CCSD(T) and CR-CCSD(T) and which we already
mentioned above, satisfies the following criteria: (i) is
at least as accurate as (sometimes more accurate than)
CCSD(T) for nondegenerate ground states, (ii) provides
highly accurate results for single-bond breaking and
diradicals with the noniterative No**3 * Nu**4 steps
similar to those of CCSD(T) and CR-CCSD(T),(iii) is more
accurate than the CR-CCSD(T), LR-CCSD(T), and other non-
iterative triples CC approaches, such as CCSD(2)T, which
all aim at eliminating the failures of CCSD(T) in the
diradical/bond breaking regions, and (iv) is rigorously
size extensive without localizing orbitals. The criterion
(ii) of a highly accurate description at the triples level
of CC theory is defined here by the accuracy provided by
the full CCSDT approach, which is almost exact in studies
of diradicals and single-bond breaking, but also limited to
very small systems with up to 2-3 light atoms due to very
expensive iterative No**3 * Nu**5 steps that it uses. As
demonstrated, for example, in recent studies of the
relative energetics of the Cu2O2 systems with up to six
ammonia ligands and thermal stereomutations of cyclopropane
involving the trimethylene diradical as a transition state,
CR-CC(2,3) has a wide range of applicability that includes
larger polyatomic systems with up to 10-20 light and a few
transition metal atoms. At the same time, CR-CC(2,3)
provides a size extensive, highly accurate, and well
balanced description of dynamical and nondynamical
correlation effects in studies of single bond breaking and
diradicals, particularly when the molecular systems
involving a varying degree of diradical character along the
relevant reaction pathways are examined.
For all these reasons, the CR-CC(2,3) approach has been
recently included in GAMESS. The CR-CC(2,3) method (invoked
by typing CCTYP=CR-CCL in the input) seems to represent the
most accurate non-iterative triples CC approximation
formulated to date. Since the construction of the triples
corrections to CCSD energies in CR-CC(2,3) calculations
requires the determination of the left CCSD eigenstates,
the CCPRP variable from $CCINP is automatically set at
.TRUE. when variable CCTYP in $CONTRL is set at CR-CCL. As
a result, by running the CR-CC(2,3) calculations, the user
of GAMESS obtains a great deal of useful information in
addition to excellent energetics (excellent as long as
multiple bonds are not broken). This information includes
the first-order reduced density matrices (printed in the
PUNCH file), natural occupation numbers, and a variety of
one-electron properties (e.g., electrostatic multipole
moments) calculated at the CCSD level of theory. The
ground-state CR-EOMCCSD(T) energies (cf. the next
subsection), corresponding to CCTYP=CR-EOM calculations
with NSTATE(1)=0,0,0,0,0,0,0,0, are printed as well.
The CR-CC(2,3) approach has several variants, labeled
with an additional letter, A-D (D means a full treatment of
the perturbative denominators that are used to define
triple excitation components, based on the diagonal matrix
elements of the triples-triples block of the CCSD
similarity transformed Hamiltonian; A means the crudest
treatment of these denominators through bare orbital
energies). Of all printed CR-CC(2,3) energies, the CR-
CC(2,3),D value, which corresponds to the most complete
variant of CR-CC(2,3), is the most accurate one and we
STRONGLY RECOMMEND to use it in high accuracy calculations
of molecular energetics. Because of the way the CR-
CC(2,3),D approach is presently implemented in GAMESS, it
is safer, for now, to use the simplified CR-CC(2,3),A or
CR-CC(2,3),B models in numerical derivative calculations if
there are orbital degeneracies (the aforementioned CCSD(2)T
approach is equivalent to the CR-CC(2,3),A approximation).
Because of some small simplifications in the present
computer implementation of the CR-CC(2,3),D method, the CR-
CC(2,3),D energies may slightly depend on the choice of
molecular coordinate system if there are orbital
degeneracies. Although changes in the most accurate CR-
CC(2,3),D energies for systems with orbital degeneracies
due to changes of the coordinate system are minimal (0.1
millihartree or less), it is safer to calculate numerical
CR-CC(2,3) derivatives for systems with orbital
degeneracies using the CR-CC(2,3),A or CR-CC(2,3),B
approximations. For this reason, the CR-CC(2,3),A energy is
automatically passed to the numerical derivative
calculations with GAMESS if they are requested by the user,
with the most complete CR-CC(2,3),D approach providing the
most accurate energetics. We should emphasize, however,
that the above technical issues are only limited to systems
with orbital degeneracies. When there are no orbital
degeneracies (which is the case when the highest molecular
symmetry group is an Abelian group), the present
implementation of the CR-CC(2,3),D approach in GAMESS leads
to perfectly invariant energies. The issue of a slight (0.1
millihartree or less) dependence of the CR-CC(2,3),D (also
CR-CC(2,3),C) energies on the choice of molecular
coordinate system when orbital degeneracies are present is
only temporary and will be eliminated in the future
releases of GAMESS via a suitable modification of the CR-
CC(2,3) code.
Since CR-CC methods can find use in applications
involving bond breaking and reaction pathways, one has to
make sure that the underlying solution of the CCSD
equations, on which the completely renormalized [T], (T),
(2,3), and (TQ) corrections are based, represents the same
physical solution as those defining other regions of a
given molecular potential energy surface. This remark is
quite important, since, for example, diradical regions of
potential energy surface are characterized by larger
cluster amplitudes and one has to make sure that the
properly converged values of these amplitudes are obtained.
GAMESS is equipped with a good algorithm for converging
CCSD equations and a restart option discussed in a later
part of this document that facilitate converging larger
cluster amplitudes in difficult cases.
The user is encouraged to examine various interesting
elements of the CC input and output. In addition to CC
energies, GAMESS prints the largest T1 and T2 cluster
amplitudes obtained in the CCSD calculations, the T1
diagnostic, norms of T1 and T2 vectors, and the R-CCSD[T],
R-CCSD(T), and R-CCSD(TQ) denominators that define the
renormalized and completely renormalized triples and
quadruples corrections. For example, bond breaking and
diradical cases are characterized by larger cluster
amplitudes (particularly, T2) and a significant increase in
the values of the R-CCSD[T], R-CCSD(T), and CR-CCSD(TQ)
denominators, which damp unphysical triples and quadruples
corrections of the standard CCSD[T], CCSD(T), and CCSD(TQ)
approximations, compared to closed-shell regions of
potential energy surface. As already mentioned, the CR-
CC(2,3) calculations provide user with one-particle reduced
density matrices, natural occupation numbers, and a number
of one-electron properties, calculated at the CCSD level,
in addition to the highly accurate CR-CC(2,3) and some
other CR-CC energies.
available computations (excited states)
The equation of motion coupled cluster (EOMCC) method
and the closely related response CC and symmetry-adapted
cluster configuration interaction (SAC-CI) approaches
provide very useful extensions of the ground-state CC
theory to excited states. In the EOMCC theory, the excited
states |PsiK> are obtained by applying the excitation
operator
R = R0 + R1 + R2 + ...,
where R0, R1, R2, etc. are the reference, singly excited
(1-particle-1-hole), doubly excited (2-particle-2-hole),
etc. components of R, to the CC ground state |Psi0>. Thus,
the EOMCC expression for the excited state |PsiK> is
|PsiK> = R |Psi0> = R exp(T) |Phi>
= (R0+R1+R2+...) exp(T1+T2+...) |Phi> .
In practice, the standard EOMCC calculations are performed
by diagonalizing the CC similarity transformed Hamiltonian
H-bar = exp(-T) H exp(T) in the space of excited
determinants included in the cluster operator T and the
excitation operator R. For example, the basic EOMCCSD
calculations defined by the truncation schemes T=T1+T2 and
R=R0+R1+R2 are performed by diagonalizing exp(-T1-T2) H
exp(T1+T2) in the space of singly and doubly excited
determinants defining the CCSD (T=T1+T2) approximation.
The direct result of such diagonalization are the vertical
excitation energies omegaK = EK - E0 (EK and E0 and the
excited- and ground- state energies, respectively).
The EOMCC methods have several advantages. The most
expensive steps of the basic EOMCCSD calculations scale
only as No**2 * Nu**4 and yet the accuracy of the EOMCCSD
results for excited states dominated by one-electron
transitions (single excitations or singles or 1-particle-1-
hole excitations) is very good. The errors in the EOMCCSD
calculations for such states are often on the order of 0.1-
0.3 eV, which is acceptable in many applications. The
EOMCCSD approximation and other standard EOMCC methods have
an ease of application that is not matched by the multi-
reference techniques, since formally the EOMCC theory is a
single-reference formalism. Thus, the EOMCC methods are
particularly well suited for calculations where active
orbital spaces required in CASSCF-related calculations
become very large or difficult to identify. Given
sufficient computational resources, the EOMCCSD
calculations for systems involving up to 10-20 light or a
few heavy atoms are nowadays (meaning year 2004 and on)
routine. The EOMCCSD method works reasonably well for
excited states dominated by singles, but it fails to
describe states dominated by two-electron transitions
(doubles) and potential energy surfaces along bond breaking
coordinates. These failures can be remedied by the CR-
EOMCCSD(T) approximations described below.
The EOMCC programs incorporated in GAMESS enable user to
perform standard EOMCCSD calculations employing the RHF
reference determinant. They also enable to improve the
EOMCCSD results by adding the state-selective noniterative
corrections due to triples to the ground and excited-state
CCSD/EOMCCSD energies via the completely renormalized
EOMCCSD(T) (CR-EOMCCSD(T)) approaches developed by the
Piecuch group. The CR-EOMCCSD(T) approaches represent
extensions of the ground-state CR-CCSD(T) method to excited
states. In particular, in analogy to the CR-CCSD(T)
approximation, the excited-state CR-EOMCCSD(T) approaches
are based on the formalism of the method of moments of
coupled-cluster equations (MMCC). Moreover, the CR-
EOMCCSD(T) methods preserve the relatively low computer
costs and ease of use of the ground-state CCSD(T)
calculations. The most expensive noniterative steps of the
CR-EOMCCSD(T) approach scale as No**3 * Nu**4. The CR-
EOMCCSD(T) option (CCTYP=CR-EOM) is a unique feature of
GAMESS. At this time, the applicability of the EOMCCSD and
CR-EOMCCSD(T) codes in GAMESS is limited to singlet states.
The main advantage of the MMCC-based CR-EOMCCSD(T)
approximations, in addition to their "black-box" character
and relatively low computer costs, is their high (0.1 eV or
so) accuracy in the calculations of excited states
dominated by double excitations and excited-state potential
energy surfaces along bond breaking coordinates, for which
the standard EOMCCSD method fails (producing errors on the
order of 1 eV or even bigger). In this regard, the CR-
EOMCCSD(T) methods are quite similar to the CR-CCSD(T)
approach, which is capable of describing ground-state
potential energy surfaces involving single bond breaking.
As a matter of fact, when limited to the ground-state
problem, the CR-EOMCCSD(T) approximations become
essentially identical to the CR-CCSD(T) method. There are,
however, small differences and the CR-EOMCCSD(T) energies
of the ground state are slightly different than the CR-
CCSD(T) energies discussed in the earlier section. This is
due to the fact that the original CR-CCSD(T) approximation
has been designed for the ground states only, whereas the
CR-EOMCCSD(T) approaches apply to ground and excited states
and this required small modifications in the ground-state
energy equations.
A few different variants of the CR-EOMCCSD(T) method,
termed the CR-EOMCCSD(T),IX, CR-EOMCCSD(T),IIX, and CR-
EOMCCSD(T),III approaches (X=A,B,C,D) have been proposed
and included in GAMESS. Types I, II, and III refer to
three different ways of defining the approximate wave
functions |PsiK> that are used to construct the CR-
EOMCCSD(T) triples corrections to EOMCCSD energies in the
underlying MMCC formalism. Types I and II use perturbative
expressions for |PsiK> in terms of cluster components T1
and T2 and excitation components R0, R1, and R2. Type III
uses additional CISD (CI singles and doubles) calculations
in designing the wave functions |PsiK> that enter the CR-
EOMCCSD(T) triples corrections. Thus, user should be aware
of the fact that CR-EOMCCSD(T),III calculations involve the
single-reference CISD calculations, in addition to the
CCSD, EOMCCSD, and (T) steps common to all CR-EOMCCSD(T)
methods. This increases the CPU timings of the CR-
EOMCCSD(T),III calculations, when compared to CR-
EOMCCSD(T),IX and CR-EOMCCSD(T),IIX (X=A-D) approaches.
Additional letters A-D that label the CR-EOMCCSD(T),I and
CR-EOMCCSD(T),II approximations refer to different ways of
treating perturbative denominators in evaluating the (T)
triples corrections (D means full treatment of these
denominators, based on the diagonal matrix elements of the
triples-triples block of the CCSD similarity transformed
Hamiltonian, A means the crudest treatment through bare
orbital energies). The user interested in further details
is referred to a 2004 paper by Kowalski and Piecuch (J.
Chem. Phys. 120, 1715-1738 (2004)).
Our experience to date indicates that the CR-
EOMCCSD(T),ID and CR-EOMCCSD(T),III methods are the most
accurate ones when it comes to the calculations of excited
states dominated by double excitations and excited-state
potential energy surfaces along bond breaking coordinates,
at least for moderate bond stretches. The CR-EOMCCSD(T),ID
and CR-EOMCCSD(T),III methods are particularly good when
examining the total energies of excited states (for
example, as functions of nuclear geometries). If the user
is only interested in vertical excitation energies rather
than total energies, the good balance between ground and
excited states, particularly when excited states are
dominated by doubles, can be achieved by considering mixed
approximations, such as CR-EOMCCSD(T),ID/IB. The ID/IB
acronym means that the excitation energy is obtained by
subtracting the CR-EOMCCSD(T),IB ground-state energy from
the CR-EOMCCSD(T),ID energy of excited state. Other mixed
approaches (IID/IB, etc.) are obtained in a similar way.
The ID/IB results are particularly good when the excited
states have significant doubly excited character. The fact
that the CR-EOMCCSD(T),ID results for excited states are
usually better than the CR-EOMCCSD(T),IA,IB,IC results is
related to a better treatment of perturbative denominators
in evaluating the (T) triples corrections in the CR-
EOMCCSD(T),ID approximation.
In addition to the total CR-EOMCCSD(T),IX, CR-
EOMCCSD(T),IIX (X=A-D), and CR-EOMCCSD(T),III energies and
vertical excitation energies based on the idea of mixing
different approximations for excited and ground states (the
ID/IA, IID/IA, ID/IB, and IID/IB excitation energies),
GAMESS prints the so-called DELTA-CR-EOMCCSD(T) values (the
del(IA), del(IB), del(IC), del(ID), del(IIA), del(IIB),
del(IIC), del(IID), and del(III) energies). These are the
vertical excitation energies obtained by directly
correcting the EOMCCSD excitation energies rather than the
total CCSD/EOMCCSD energies by triples corrections. For
example, del(ID) refers to the vertical excitation energy
obtained by subtracting the CCSD ground-state energy from
the excited-state CR-EOMCCSD(T),ID energy. The DELTA-CR-
EOMCCSD(T) values may be somewhat worse than the pure CR-
EOMCCSD(T) (e.g., CR-EOMCCSD(T),ID) or CR-EOMCCSD(T),III)
or mixed CR-EOMCCSD(T) (e.g., CR-EOMCCSD(T),ID/IB)) values
of vertical excitation energies for states dominated by
doubles, but they may provide a reasonable balance between
ground and excited states and somewhat bigger improvements
for vertical excitation energies corresponding to states
dominated by singles. The DELTA-CR-EOMCCSD(T) methods
provide a reasonably good balance between improvements in
the results for excited states dominated by singles and
improvements in the results for excited states dominated by
doubles, but one should treat this remark with caution.
In addition to the above CR-EOMCCSD(T) results, GAMESS
also prints the so-called (T)/R excitation energies. These
are the analogs of the EOMCCSD(T~) excitation energies
proposed by Watts and Bartlett, obtained by using the right
eigenvectors of the CCSD similarity transformed and right-
hand moments of EOMCCSD equations rather than the left
eigenstates of EOMCCSD and left-hand analogs of the EOMCCSD
moments (see K. Kowalski and P. Piecuch, J. Chem. Phys.
120, 1715-1738 (2004) for details). Just like the
EOMCCSD(T~) method of Watts and Bartlett, the (T)/R
approach is based on the idea of directly correcting the
EOMCCSD vertical excitation energies by triples. In
analogy to the EOMCCSD(T~) method, the (T)/R corrections
improve the EOMCCSD results for states dominated by
singles, but they may fail to produce reasonable results
for states dominated by doubles and for excited-state
potential energy surfaces along bond breaking coordinates.
The CR-EOMCCSD(T) methods are considerably more robust in
this regard.
In performing the CR-EOMCCSD(T) calculations, user
should realize that the EOMCCSD method can provide a wrong
state ordering if low-lying doubly excited states are mixed
up with singly excited states in the electronic spectrum.
This may require calculating a larger number of EOMCCSD
states before correcting them for triples. An example of
this situation has been described in K. Kowalski and P.
Piecuch, J. Chem. Phys. 120, 1715-1738 (2004). The
EOMCCSD method provides an incorrect ordering of the
singlet A1 states of ozone, so that one must use the third
excited EOMCCSD state of the singlet A1 (1A1) symmetry (the
fourth 1A1 state total, using the CCSD/EOMCCSD energy
ordering of ground and excited states) to calculate the
noniterative CR-EOMCCSD(T) triples correction that
describes the first excited singlet A1 (the second 1A1)
state. Without calculating several states of each symmetry
at the EOMCCSD level prior to CR-EOMCCSD(T) calculations,
one would risk losing information about some important low-
lying doubly excited states. Because of the inherent
limitations of the EOMCCSD approximation, complicated
doubly excited states resulting from the EOMCCSD
calculations may be shifted to high energies, mixing with
the singly excited states that are accurately described by
the EOMCCSD method. After correcting the EOMCCSD energies
for the effect of triples, these doubly excited states may
become low-lying states. This is exactly what we observe in
the case of ozone and other cases of severe quasi-
degeneracies.
The issues of size extensivity in the EOMCCSD and CR-
EOMCCSD(T) calculations are highly complex and much beyond
the scope of this writing. Briefly, none of the EOMCC
methods are rigorously size extensive and yet all EOMCC
methods are very useful in great many applications. The
EOMCCSD approach is size intensive for excited states
dominated by singles and the EOMCCSD energies correctly
separate when the one-electron charge-transfer excitations
are considered. Thus, the EOMCCSD approach correctly
describes the dissociation of a singly excited system (AB)*
into the A* + B, A + B*, (A+) + (B-), and (A-) + (B+)
fragments (* designates a one-electron excitation). We must
remember, however, that the above separability properties
of the EOMCCSD energies are no longer true if the reference
determinant |Phi> does not separate correctly (for example,
the RHF determinant does not correctly separate if the AB
-> A+B fragmentation involves the dissociation of the
closed-shell system AB into open-shell fragments A and B).
As in the case of the ground-state CR-CCSD(T) approach, the
CR-EOMCCSD(T) methods slightly violate the rigorous size
extensivity/intensivity (at the level of 1-2 millihartree
for systems with up to 30-50 correlated electrons), but at
the same time the CR-EOMCCSD(T) approaches significantly
improve a poor description of excited states with
significant double excitation components by the EOMCCSD
method. As a result, lack of strict size extensivity of the
CR-EOMCCSD(T) theories is of relatively minor significance
in applications for systems with up to at least 50
correlated electrons [see M. Wloch, J.R. Gour, K. Kowalski,
and P. Piecuch, J. Chem. Phys. 122, 214107-1 - 214107-15
(2005) for a thorough discussion of the complicated
extensivity issues in EOMCCSD and CR-EOMCCSD(T)
calculations].
The user is encouraged to examine various interesting
elements of the EOMCC input and output. In addition to
EOMCC energies, GAMESS prints the largest R1 and R2
excitation amplitudes and the so-called reduced excitation
level (REL) diagnostic, which provides information about
the character of a given excited state (REL close to 1
means singly excited, REL close to 2 means doubly excited).
GAMESS also prints the R0 value (the coefficient at the
reference in the EOMCCSD wave function). If a molecule has
symmetry and R0 equals 0, user immediately learns the
excited state has a different symmetry than the ground
state. GAMESS provides full information about irreps of
the calculated excited states.
density matrices and properties
One of the major advantages of EOMCC methods, including
EOMCCSD, is a relatively straightforward access to reduced
density matrices and molecular properties that these
methods offer. This is done by considering the left
eigenstates of the similarity transformed Hamiltonian H-bar
= exp(-T) H exp(T) mentioned in the earlier sections. The
similarity transformed Hamiltonian H-bar is not hermitian,
so that, in addition to the right eigenstates R|Phi>, which
define the "ket" CC or EOMCC wave functions discussed in
the previous section, we can also define the left
eigenstates of H-bar, form a
biorthonormal set. We can use these eigenstates to
calculate expectation values and transition matrix elements
of quantum-mechanical operators (observables), involving
the CC and EOMCC ground and excited states, as follows:
= ,
where W-bar = exp(-T) W exp(T) is a similarity transformed
form of the observable W we are interested in and where we
added labels K and M to operators L and R to indicate the
CC/EOMCC electronic states they are associated with. The
operator W could be, for example, a dipole or quadrupole
moment. It could also be a product of creation and
annihilation operators, which we could use to calculate the
reduced density matrices. For example, if the operator
W = (ap-dagger) aq, where ap-dagger and aq are the creation
and annihilation operators associated with the spin-
orbitals p and q, respectively, we can calculate the CC or
EOMCC one-body reduced density matrix in the electronic
state K, Gamma(qp,K), as
Gamma(qp,K)
= .
For the corresponding transition density matrix involving
two different states K and M, say ground and excited states
or some other combination, we can write
Gamma(qp,KM)
= .
By having access to reduced density matrices, we can
calculate various properties analytically. For example, by
calculating the one-body reduced density matrices of ground
and excited states and the corresponding transition density
matrices, we can determine all one-electron properties and
the corresponding transition matrix elements involving one-
electron properties using a single mathematical expression:
= Sum_pq Gamma(qp,KM),
where are matrix elements of the one-body property
operator W in a basis set of molecular spin-orbitals used
in the calculations. The calculation of reduced density
matrices provides the most convenient way of calculating CC
and EOMCC properties of ground and excited states. In
addition, by having reduced density matrices, one can
calculate CC and EOMCC electron densities,
rhoK(x) = Sum_pq Gamma(qp,K) (phi_q(x))* phi_p(x),
where phi_p(x) and phi_q(x) are molecular spin-orbitals and
x represents the electronic (spatial and spin) coordinates.
By diagonalizing Gamma(qp,K), one can determine the natural
occupation numbers and natural orbitals for the CC or EOMCC
state |PsiK>.
The above strategy of handling molecular properties
analytically by determining one-body reduced density
matrices was implemented in the CC/EOMCC programs
incorporated in GAMESS. At this time, the calculations of
reduced density matrices and selected properties are
possible at the CCSD (ground states) and EOMCCSD (ground
and excited states) levels of theory (T=T1+T2, R=R1+R2,
L=L0+L1+L2). Currently, in the main output the program
prints the CCSD and EOMCCSD electric multipole (dipole,
quadrupole, etc.) moments and several other one-electron
properties that one can extract from the CCSD/EOMCCSD
density matrices, the EOMCCSD transition dipole moments and
the corresponding dipole and oscillator strengths, and the
natural occupation numbers characterizing the CCSD/EOMCCSD
wave functions. In addition, the complete CCSD/EOMCCSD one-
body reduced density matrices and transition density
matrices in the RHF molecular orbital basis and the CCSD
and EOMCCSD natural orbital occupation numbers are printed
in the PUNCH output file. The eigenvalues of the density
matrix (natural occupation numbers) are ordered such that
the corresponding eigenvectors (CCSD or EOMCCSD natural
orbitals) have the largest overlaps with the consecutive
ground-state RHF MOs. Thus, the first eigenvalue of the
density matrix corresponds to the CCSD or EOMCCSD natural
orbital that has the largest overlap with the RHF MO 1, the
second with RHF MO 2, etc. This ordering is particularly
useful for analyzing excited states, since in this way one
can easily recognize orbital excitations that define a
given excited state.
One has to keep in mind that the reduced density
matrices originating from CC and EOMCC calculations are not
symmetric. Thus, if we, for example, want to calculate the
dipole strength between states K and M for the x component
of the dipole mu_x, ||**2, we must
write
||**2
= ,
where each matrix element in the above expression is
evaluated using the expression for shown
above. A similar remark applies to the corresponding
component of the oscillator strength,
(2/3)*|EK-EM|*||**2,
which we have to write as
(2/3)*|EK-EM|*.
In other words, both matrix elements
and have to be evaluated, since they
are not identical. This is reflected in the GAMESS output,
where the user can see quantities such as the left and
right transition dipole moments.
From the above description, it follows that in order to
calculate reduced density matrices and properties using CC
and EOMCC methods, one has to determine the left as well as
the right eigenstates of the similarity transformed
Hamiltonian H-bar. For the ground state, this is done by
solving the linear system of equations for the deexcitation
operator Lambda (in the CCSD case, the one- and two-body
components Lambda1 and Lambda2). For excited states, we can
proceed in several different ways. We can solve the linear
system of equations for the amplitudes defining the EOMCC
deexcitation operator L, after determining the
corresponding EOMCC excitation operator R and excitation
energy omega (recommended option, default in GAMESS), or we
can solve for the L and R amplitudes simultaneously in the
process of diagonalizing the similarity transformed
Hamiltonian. These different ways of solving the EOMCC
problem are discussed in section "Eigensolvers for excited-
state calculations."
As already mentioned, the left eigenstates of the
similarity transformed Hamiltonian of the CCSD approach are
also used to construct the triples corrections to CCSD
energies defining the rigorously size extensive completely
renormalized CR-CC(2,3) approximation. This is why the user
gets an immediate access to electrostatic multipole moments
and other one-electron properties calculated at the CCSD
level, when running the CR-CC(2,3) calculations.
excited state example
! excited states of methylidyne cation...CH+
! Basis set and geometry come from a FCI study by
! J.Olsen, A.M.Sanchez de Meras, H.J.Aa.Jensen,
! P.Jorgensen Chem. Phys. Lett. 154, 380-386(1989).
!
! EOMCC methods give:
! STATE EOMCCSD ID/IA IID/IA ID/IB IID/IB FCI
! B1 (1Pi) 3.261 3.226 3.226 3.225 3.224 3.230
! A1 (1Delta) 7.888 6.988 6.963 6.987 6.962 6.964
! A1 (1Sigma+) 9.109 8.656 8.638 8.654 8.637 8.549
! A1 (1Sigma+) 13.580 13.525 13.526 13.524 13.525 13.525
! B1 (1Pi) 14.454 14.229 14.221 14.228 14.219 14.127
! A1 (1Sigma+) 17.316 17.232 17.220 17.231 17.219 17.217
! A2 (1Delta) 17.689 16.820 16.790 16.819 16.789 16.833
! Note the improvements in the EOMCCSD results by the
! CR-EOMCCSD(T) appproaches (e.g., ID/IB) for the Sigma+
! state at 8.549 eV and both Delta states.
!
! The ground state CCSD dipole is z=-0.645, and the
! right/left transition moment to the first pi state
! is x=0.297 and 0.320, with oscillator strength 0.0076
!
$contrl scftyp=rhf cctyp=cr-eom runtyp=energy
icharg=1 units=bohr $end
$system mwords=5 $end
$ccinp ncore=0 $end
$eominp nstate(1)=4,2,2,0 minit=1 noact=3 nuact=7
ccprpe=.true. $end
$data
CH+ at R=2.13713...basis set from CPL 154, 380 (1989)
Cnv 2
Carbon 6.0 0.0 0.0 0.16558134
S 6
1 4231.610 0.002029
2 634.882 0.015535
3 146.097 0.075411
4 42.4974 0.257121
5 14.1892 0.596555
6 1.9666 0.242517
S 1 ; 1 5.1477 1.0
S 1 ; 1 0.4962 1.0
S 1 ; 1 0.1533 1.0
S 1 ; 1 0.0150 1.0
P 4
1 18.1557 0.018534
2 3.9864 0.115442
3 1.1429 0.386206
4 0.3594 0.640089
P 1 ; 1 0.1146 1.0
P 1 ; 1 0.011 1.0
D 1 ; 1 0.75 1.0
Hydrogen 1.0 0.0 0.0 -1.97154866
S 3
1 1.924060D+01 3.282800D-02
2 2.899200D+00 2.312080D-01
3 6.534000D-01 8.172380D-01
S 1 ; 1 1.776D-01 1.0
S 1 ; 1 2.5D-02 1.0
P 1 ; 1 1.0 1.0
$end
resource requirements
User can perform LCCD, CCD, and CCSD calculations, that
is without calculating the [T], (T), (2,3), and (TQ)
corrections, or calculate the entire set of the standard
and renormalized [T], (T), (2,3), and (TQ) ground-state
corrections, in addition to the CCSD energies. User can
also perform the EOMCCSD calculations of excited states and
stop at EOMCCSD or continue to obtain some or all CR-
EOMCCSD(T) triples corrections (cf. the values of input
variable CCTYP in $CONTRL and $EOMINP group). Finally, user
can perform the calculations of ground-state properties at
the CCSD level or calculate ground- and excited-state
properties. It is also possible to combine some of the
above calculations. For example, one can calculate the CCSD
and EOMCCSD properties and obtain triples corrections to
the calculated CCSD and EOMCCSD energies from a single
input (see the example above). The CR-CC(2,3) calculation
produces the MBPT(2) and CCSD energies, and CCSD one-
electron properties and density matrices, in addition to
the CR-CC(2,3) and some other CR-CC triples corrections to
the CCSD energies, again all from a single input (CCTYP=CR-
CCL). The most expensive steps in CC/EOMCC calculations
scale as follows:
LCCD, CCD, CCSD, EOMCCSD
No**2 times Nu**4 (iterative)
CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T],
CR-CCSD(T), CR-CC(2,3) (#1), CR-EOMCCSD(T) (#2)
No**3 times Nu**4 (non-iterative)
plus No**2 times Nu**4 (iterative)
CCSD(TQ), R-CCSD(TQ), CR-CCSD(TQ)
No**2 times Nu**5 or
Nu**6 (#3) (non-iterative) plus
No**3 times Nu**4 (non-iterative)
plus No**2 times Nu**4 (iterative)
----
(#1) In addition to the usual No**2 times Nu**4 iterative
CCSD steps and No**3 times Nu**4 non-iterative steps needed
to determine the (2,3) triples correction, the CR-CC(2,3)
calculations require extra No**2 times Nu**4 iterative
steps needed to obtain the left CCSD state, which enters
the CR-CC(2,3) triples correction formula.
(#2) In addition to the No**2 times Nu**4 iterative
CCSD and EOMCCSD steps and No**3 times Nu**4 non-iterative
(T) steps that are common to all CR-EOMCCSD(T) models,
the CR-EOMCCSD(T),III method requires the iterative
No**2 times Nu**4 steps of CISD. The CR-EOMCCSD(T),IX
and CR-EOMCCSD(T),IIX (X=A-D) methods do not require
these additional CISD calculations.
(#3) To reduce the cost, the program will automatically
choose between the No**2 times Nu**5 and Nu**6 algorithms
in the (Q) part, depending on the ratio of Nu to No.
----
The cost of calculating the standard CCSD[T] and CCSD(T)
energies and the cost of calculating the R-CCSD[T] and R-
CCSD(T) energies are essentially the same. The cost of
calculating the triples corrections of the CR-CCSD[T] and
CR-CCSD(T) approaches is essentially twice the cost of
calculating the standard CCSD[T] and CCSD(T) corrections.
Similar relationships hold between the costs of the
CCSD(TQ), R-CCSD(TQ), and CR-CCSD(TQ) calculations. The
cost of calculating the triples corrections of the CR-
CC(2,3),X (X=A-D) approaches is also twice the cost of
calculating the CCSD[T] and CCSD(T) triples corrections,
but additional No**2 times Nu**4 iterative steps are
required to generate the left CCSD state after converging
the CCSD equations in order to calculate the final CR-
CC(2,3) energies. Although the noniterative triples
corrections may be seen to grow as the seventh power of the
system size, they often require less time than the sixth
power iterations of the CCSD step, while providing a great
increase in accuracy. Similar remarks apply to the CR-
EOMCCSD(T) calculations: The cost of the CR-EOMCCSD(T)
calculation for a single electronic state, in its
noniterative triples part, is twice the cost of computing
the standard (T) corrections of CCSD(T). The total CPU time
of the CR-EOMCCSD(T) calculations scales linearly with the
number of calculated states. In spite of the formal N**6
scaling, the calculations of the CCSD/EOMCCSD properties
per single electronic state are considerably less expensive
than the CCSD calculations for two reasons. First of all,
the process of obtaining the left eigenstates of the
similarity transformed Hamiltonian H-bar can reuse the
intermediates (matrix elements of H-bar) which are obtained
in the prior CCSD calculations. Second, converging left
eigenstates of H-bar is usually much quicker than
converging the CCSD equations when one obtains the left
eigenstates of H-bar by solving the linear system of
equations for the L deexcitation amplitudes after
determining the R excitation amplitudes and excitation
energies. This means that computing properties at the
CCSD/EOMCCSD level is not very expensive once the CCSD and
EOMCCSD right eigenvectors are obtained. Similar remarks
apply to the CR-CC(2,3) calculations, which require the
left CCSD eigenstates in addition to the CCSD T1 and T2
amplitudes: The determination of the left CCSD states that
are needed to determine the non-iterative triples
corrections of the CR-CC(2,3) approach makes the entire
CCSD part of the CR-CC(2,3) calculation only somewhat more
expensive than the regular CCSD iterations needed to obtain
T1 and T2 clusters. The CCSD(TQ), R-CCSD(TQ), and CR-
CCSD(TQ) calculations are more expensive than the CCSD(T)
calculations, in spite of the fact that all of these
methods use non-iterative N**7 steps. This is related to
the fact that the No**2 times Nu**5 steps of the (TQ)
methods are more expensive than the No**3 times Nu**4 steps
of the (T) approaches. On the other hand, the CCSD(TQ), R-
CCSD(TQ), and CR-CCSD(TQ) methods are much less expensive
than the iterative ways of obtaining the information about
quadruply excited clusters. This is a result of an
efficient use of diagram factorization in coding the
CCSD(TQ), R-CCSD(TQ), and CR-CCSD(TQ) methods, which leads
to a reduction of the N**9-type steps in the original (Q)
expressions to N**7 steps.
Rough estimates of the memory required are:
CCSD 4 No**2 times Nu**2 + No times Nu**3
CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T)
4 No**2 times Nu**2 + No times Nu**3
CR-CCSD[T], CR-CCSD(T)
No**2 times Nu**2 + 2 * No times Nu**3 (faster algorithm)
4 No**2 times Nu**2 + No times Nu**3 (slower, less memory)
CR-CC(2,3)
The most expensive routine requires 3 * No * Nu**3 + 3 *
Nu**3 + 5 * No**2 *Nu**2 words
CCSD(TQ),b, R-CCSD(TQ)-n,x (n=1,2;x=a,b), CR-CCSD(TQ),x
(x=a,b)
2 * No times Nu**3 + No**2 times Nu**2 + Nu**3, preceded
and followed by steps that require memories, such as, for
example, 3 * Nu**3 + 5 * No**2 * Nu**2
EOMCCSD No times Nu**3 + 4 No**2 times Nu**2 (MEOM=0,1)
if MEOM=2, add to this
(4 times number of roots + 2) times No**2 times Nu**2
CR-EOMCCSD(T),IX, 2 * No times Nu**3 + 3 No**2 times Nu**2
CR-EOMCCSD(T),IIX(X=A-D) [MTRIP=1 in $EOMINP]
CR-EOMCCSD(T) 3 * No times Nu**3 + 5 No**2 times Nu**2
all variants (faster algorithm) [MTRIP=2 in $EOMINP]
CR-EOMCCSD(T),III 2 * No times Nu**3 + 5 No**2 times Nu**2
[MTRIP=3 in $EOMINP]
CR-EOMCCSD(T) 2 * No times Nu**3 + 5 No**2 times Nu**2
all variants (slower algorithm) [MTRIP=4 in $EOMINP]
The program automatically selects the algorithm for the CR-
CCSD[T] and CR-CCSD(T) calculations, depending on the
amount of available memory. A similar remark applies to the
EOMCCSD calculations, where some additional reductions of
memory requirements are possible if memory is low. The
above estimates are rough.
The time required for calculating the CR-CCSD[T] and CR-
CCSD(T) triples corrections is only twice the time used to
calculate the standard CCSD[T] and CCSD(T) corrections.
Thus, by just doubling the CPU time for the noniterative
triples corrections and by selecting CCTYP=CR-CC, we gain
access to all six noniterative triples corrections (the
CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-
CCSD(T) energies) plus, of course, to the MBPT(2) and CCSD
energies. At the same time, the CR-CCSD[T] and CR-CCSD(T)
results for stretched nuclear geometries and diradicals are
better than the results of the conventional CCSD[T] and
CCSD(T) calculations. In some cases, choosing CCTYP=R-CC
might be reasonable, too. The choice CCTYP=R-CC gives five
different energies (CCSD, CCSD[T], CCSD(T), R-CCSD[T], and
R-CCSD(T)) for the price of three (CCSD, CCSD[T], and
CCSD(T)) as the there is no extra time needed for the R-
theories compared to the standard ones. If we ignore the
iterative CCSD steps and additional iterative steps needed
to determine the left CCSD state, the time required for
calculating the size extensive CR-CC(2,3) triples
corrections is also only twice the time of calculating the
CCSD[T] and CCSD(T) corrections. There is an additional
bonus though: The CR-CC(2,3) calculations automatically
produce a variety of CCSD one-electron properties at no
extra cost. Similar remarks apply to quadruples and excited
state calculations, although in the latter case a lot
depends on user's expectations. If user is only interested
in excited states dominated by singles and if accuracies on
the order of 0.1-0.3 eV (sometimes better, sometimes worse)
are acceptable, EOMCCSD is a good choice. However, it may
be worth improving the EOMCCSD results by performing the
CR-EOMCCSD(T) calculations, which often lower the errors in
calculated excited states to 0.1 eV or less without making
the calculations a lot more expensive (the CR-EOMCCSD(T)
corrections are noniterative, so that the CPU time needed
to calculate them may be comparable to the time spent in
all EOMCCSD iterations). If there is a risk of encountering
low-lying states having significant doubly excited
contributions or multi-reference character, choosing CR-
EOMCCSD(T) is a necessity, since errors obtained in EOMCCSD
calculations for states dominated by doubles can easily be
on the order of 1 eV. The CCSD(T) approach is often fine
for closed-shell molecules, but there are cases, such as
the vibrational frequencies of ozone and properties of
other multiply bonded systems, where inclusion of
quadruples is necessary. The CR-CCSD(T) approach is very
useful in cases involving single bond breaking and
diradicals, but CR-CC(2,3) and CR-CCSD(TQ) should be
better. In addition, the CR-CC(2,3) method provides
rigorously size extensive results. In cases of multiple
bond dissociations, CR-CCSD(TQ) is a better alternative.
The program is organized such that choosing a CR-CCSD(TQ)
option (CCTYP=CR-CC(Q)) produces all energies obtained with
CCTYP=CR-CCSD(T) and all CCSD(TQ), R-CCSD(TQ), and CR-
CCSD(TQ) energies. By selecting CCTYP=CCSD(TQ), the user
can obtain the CCSD(TQ) and R-CCSD(TQ) energies, in
addition to the CCSD, CCSD[T], CCSD(T), R-CCSD[T], and R-
CCSD(T) energies.
We encourage the user to read papers, such as
P.Piecuch, S.A.Kucharski, K.Kowalski, M.Musial
Comput. Phys. Comm., 149, 71-96(2002);
K. Kowalski and P. Piecuch,
J. Chem. Phys., 120, 1715-1738 (2004);
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J.
Chem. Phys. 122, 214107 (2005);
K. Kowalski, P. Piecuch, M. Wloch, S.A. Kucharski, M.
Musial, and M.W. Schmidt, in preparation,
where time and memory requirements for various types of CC
and EOMCC calculations are described in considerable
detail.
restarts in ground-state calculations
The CC code incorporated in GAMESS is quite good in
converging the CCSD equations with the default guess for
cluster amplitudes. The code is designed to converge in
relatively few iterations for significantly stretched
nuclear geometries, where it is not unusual to obtain large
cluster amplitudes whose absolute values are close to 1.
This is accomplished by combining the standard Jacobi
algorithm with the DIIS extrapolation method of Pulay. The
maximum number of amplitude vectors used in the DIIS
extrapolation procedure is defined by the input variable
MXDIIS. The default for MXDIIS is as follows:
MXDIIS = 5, for 5 < No*Nu,
MXDIIS = 3, for 2 < No*Nu < 6,
MXDIIS = 0, for No*Nu < 3.
Thus, in the vast majority of cases, the default value of
MXDIIS is 5. However, for very small problems, when the
DIIS expansion subspace leads to singular systems of linear
equations, it is necessary to reduce the value of MXDIIS to
2-4 (we chose 3) or switch off DIIS altogether (which is
the case when MXDIIS = 0).
It may, of course, happen that the solver for the CCSD
equations does not converge, in spite of increasing the
maximum number of iterations (input variable MAXCC; the
default value is 30) and in spite of changing the default
value of MXDIIS. In order to facilitate the calculations
in all such cases, we included the restart option in the CC
codes incorporated in GAMESS. Thus, user can restart a
CCSD (or (L)CCD) calculation from the restart file created
by an earlier CC calculation. In order to use the restart
option, user must save the disk file CCREST (unit 70) from
the previous CC run (cf. the GAMESS script rungms) and make
sure that this file is copied to scratch directory where
the restarted calculation is carried out. A restart is
invoked by entering a nonzero value for IREST, which should
be the number of the last iteration completed, and must be
some value greater than or equal 3. Examples of using the
restart option include the following situations:
o The CCSD program did not converge in MAXCC iterations,
but there is a chance to converge it if the value of
MAXCC is increased. User restarts the calculation with
the increased value of MAXCC.
o User ran a CCSD calculation, obtaining the converged CCSD
energy, but later decided to run CR-CCSD(T) or CR-CC(2,3)
calculation. Instead of running the entire CCSD --> CR-
CCSD(T) or CCSD --> CR-CC(2,3) task again, user restarts
the calculation after changing the value of input
variable CCTYP to CR-CC (the CR-CCSD(T) case) or CR-CCL
(the CR-CC(2,3) case) and entering IREST to reuse the
previous CCSD amplitudes, proceeding at once to the non-
iterative triples corrections (left CCSD calculations and
triples corrections in the CR-CC(2,3) case).
o The CCSD program diverged for some geometry with a
significantly stretched bond. User performs an extra
calculation for a different nuclear geometry, for which
it is easier to converge the CCSD equations, and restarts
the calculation from the restart file generated by an
extra calculation. This technique of restarting the CC
calculations from the cluster amplitudes obtained for a
neighboring nuclear geometry is particularly useful for
scanning PESs and for calculating energy derivatives by
numerical differentiation.
There also are situations where restart of the ground-
state CCSD calculations is useful for excited-state and
property calculations:
o User ran a CCSD, CCSD(T), or CR-CCSD(T) calculation,
obtaining the converged CC energies for the ground state,
but later decided to run an excited-state EOMCCSD or
CR-EOMCCSD(T) calculations. Instead of running the entire
CCSD --> EOMCCSD or CCSD --> CR-EOMCCSD(T) task,
user restarts the calculation after changing the
value of input variable CCTYP to EOM-CCSD or CR-EOM,
selecting excited-state options in $EOMINP, and entering
IREST greater or equal to 3 to reuse the previously
converged CCSD amplitudes, proceeding at once to the
excited-state (EOMCCSD or CR-EOMCCSD(T)) calculations.
o User ran an EOMCCSD excited-state calculation, obtaining
the converged CCSD amplitudes, but later discovered
(by analyzing R1 and R2 amplitudes and REL values)
that some states are dominated by doubles, so that
the EOMCCSD results need to be improved by the
CR-EOMCCSD(T) triples corrections. Instead of running
the entire CCSD --> CR-EOMCCSD(T) task, user restarts the
calculation after changing the value of input variable
CCTYP from EOM-CCSD to CR-EOM, and entering IREST
greater or equal to 3 to reuse the previously converged
CCSD amplitudes, proceeding at once to the EOMCCSD and
CR-EOMCCSD(T) calculations.
o User ran a CR-CCSD(T) calculation, obtaining the
converged ground-state energies, but later decided to run
CCSD and EOMCCSD properties. Instead of running the CCSD
--> EOMCCSD task again, user restarts the calculation after
changing the value of input variable CCTYP to EOM-CCSD,
adding CCPRPE=.TRUE. and the desired values of NSTATE in
$EOMINP, and entering IREST to reuse the previously
converged CCSD amplitudes, proceeding at once to CCSD and
EOMCCSD properties.
initial guesses in excited-state calculations
The EOMCCSD calculation is an iterative procedure which
needs initial guesses for the excited states of interest.
The popular initial guess for the EOMCCSD calculations is
obtained by performing the CIS calculations (diagonalizing
the Hamiltonian in a space of singles only). This is
acceptable for states dominated by singles, but user may
encounter severe convergence difficulties or even miss some
states entirely if the calculated states have significant
doubly excited character. One possible philosophy is not
to worry about it and use the CIS initial guess only, since
EOMCCSD fails to describe states with large doubly excited
components. This is not the philosophy of the EOMCC
programs in GAMESS. GAMESS is equipped with the CR-
EOMCCSD(T) triples corrections to EOMCCSD energies, which
are capable of reducing the large errors in the EOMCCSD
results for states dominated by two-electron transitions,
on the order of 1 eV, to 0.1 eV or even less. Thus, the
ability to capture states with significant doubly excited
contributions is an important element of the EOMCC GAMESS
codes.
Excited states with significant contributions from
double excitations can easily be found by using the EOMCCSd
(little d) initial guesses provided by GAMESS. In the
EOMCCSd calculations (and analogous CISd calculations used
to initiate the CISD calculations for the CR-EOMCCSD(T),III
method), the initial guesses for the calculated excited
states are defined using all single excitations (letter S
in EOMCCSd and CISd) and a small subset of double
excitations (the little d in EOMCCSd and CISd) defined by
active orbitals or orbital range specified by the user.
The inclusion of a small set of active double excitations
in addition to all singles in the initial guess greatly
facilitates finding excited states characterized by
relatively large doubly excited amplitudes. GAMESS input
offers a choice between the CIS and EOMCCSd/CISd initial
guesses. The use of EOMCCSd/CISd initial guesses is highly
recommended. This is accomplished by setting the input
variable MINIT at 1 and by selecting the orbital range
(active orbitals to define "little doubles" d) through the
numbers of active occupied and active unoccupied orbitals
(variables NOACT and NUACT, respectively) or an array of
active orbitals called MOACT.
eigensolvers for excited-state calculations
The basic eigensolver for the EOMCCSD calculations is
the Hirao and Nakatsuji's generalization of the Davidson
diagonalization algorithm to non-Hermitian problems (the
similarity transformed Hamiltonian H-bar is non-Hermitian).
GAMESS offers the following three choices of EOMCCSD
eigensolvers for the right eigenvalue problem (R amplitudes
and energies only):
o the true multi-root eigensolver based on the
Hirao and Nakatsuji's algorithm, in which all
states are calculated at once using a united
iterative space (variable MEOM=2).
o the single-root eigensolver, in which one calculates
one state at a time, but the iterative subspace
corresponding to all calculated roots remains united
(variable MEOM=0).
o the single-root eigensolver, in which one calculates
one state at a time and each calculated root has a
separate iterative subspace (variable MEOM=1).
The latter option (MEOM=1) leads to the fastest
algorithm, but there is a risk (often worth taking) that
some states will be converged more than once. The true
multi-root eigensolver (MEOM=2) is probably the safest, but
it is also the most expensive solver and there are some
risks associated with using it too. When MEOM=2, there is
a risk that one root, which is difficult to converge, may
cause the entire multi-root procedure fail in spite of the
fact that all other roots participating in the calculation
converged. The EOMCCSD program in GAMESS is prepared to
handle this problem by saving individual roots that
converged during multi-root iterations in case the entire
procedure fails because of one or more roots which are
difficult to converge. In this way, at least some roots
are saved for the subsequent CR-EOMCCSD(T) calculations.
The middle option (MEOM=0) seems to offer the best
compromise. MEOM=0 is a single-root eigensolver, so there
are no risks associated with loosing some states during
multi-root calculations. At the same time, the use of the
united iterative subspace for all calculated roots helps to
eliminate the problem of MEOM=1 of obtaining the same root
more than once. The single-root eigensolver with a united
iterative subspace (MEOM=0) is recommended (and used as a
default), although other ways of converging the right
EOMCCSD equations (MEOM=1,2) are very useful too.
As pointed out earlier, in order to calculate reduced
density matrices and properties using CCSD and EOMCCSD
methods, one has to determine the left as well as the right
eigenstates of the non-Hermitian similarity transformed
Hamiltonian H-bar. For the ground state, this is done by
solving the linear system of equations for the deexcitation
operator Lambda (in the CCSD case, the one- and two-body
components Lambda1 and Lambda2). For the amplitudes
defining the L1 and L2 components of the excited-state
operator L, one can proceed in several different ways and
these different ways are reflected in the EOMCCSD algorithm
incorporated in GAMESS. One can, for example, solve the
linear system of equations for the amplitudes defining the
EOMCCSD deexcitation operator L=L1+L2, after determining
the corresponding excitation operator R=R1+R2 and
excitation energy omega. This is a highly recommended
option, which is also a default in GAMESS. This option is
executed with any choice of MEOM=0,1,2 and when the user
selects CPRPE=.TRUE. In case of unlikely difficulties with
obtaining the L1 and L2 components, one can solve for the
EOMCCSD values of the L1,L2 and R1,R2 amplitudes and
excitation energies simultaneously in the process of
diagonalizing the similarity transformed Hamiltonian H-bar
completely in a single sequence of iterations. This
approach is reflected by the following two additional
choices of the input variable MEOM:
o MEOM=3, one root at a time, separate iterative space for
each computed root, left and right eigenvectors
of the similarity transformed Hamiltonian and
energies (like MEOM=1, but both left and right
eigenvectors are iterated).
o MEOM=4, one root at a time, united iterative spaces
for all calculated roots, left and right
eigenvectors of the similarity transformed
Hamiltonian and energies (like MEOM=0, but both
left and right eigenvectors are iterated).
In both cases, the user has to select CCPRPE=.TRUE. in
order for these two choices of MEOM to work.
references and citations required in publications
Any publication describing the results of CC calculations
obtained using GAMESS should give reference to the relevant
papers. Depending on the specific CCTYP value, these are:
CCTYP = LCCD, CCD, CCSD, CCSD(T)
P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial
Comput. Phys. Commun. 149, 71-96 (2002).
CCTYP = R-CC, CR-CC, CCSD(TQ), CR-CC(Q)
P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial
Comput. Phys. Commun. 149, 71-96 (2002);
K. Kowalski and P. Piecuch
J. Chem. Phys. 113, 18-35 (2000);
K. Kowalski and P. Piecuch
J. Chem. Phys. 113, 5644-5652 (2000).
CCTYP = CR-CCL
P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial
Comput. Phys. Commun. 149, 71-96 (2002);
P. Piecuch and M. Wloch
J. Chem. Phys. 123, 224105/1-10 (2005).
CCTYP = EOM-CCSD, CR-EOM
P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial
Comput. Phys. Commun. 149, 71-96 (2002);
K. Kowalski and P. Piecuch,
J. Chem. Phys. 120, 1715-1738 (2004);
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch,
J. Chem. Phys. 122, 214107-1 - 214107-15 (2005).
CCTYP = CR-EOML
P. Piecuch, J. R. Gour, and M. Wloch
Int. J. Quantum Chem. 109, 3268-3304(2009)
and the first two papers cited for CR-EOM just above
CCTYP = IP-EOM2, EA-EOM2
J. R. Gour, P. Piecuch, M. Wloch
J. Chem. Phys. 123, 134113/1-14(2005)
J. R. Gour, P. Piecuch
J. Chem. Phys. 125, 234107/1-17(2006)
In addition, the explicit use of CCPRP=.TRUE. in $CCINP
and/or the use of CCPRPE=.TRUE. in $EOMINP should reference
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch,
J. Chem. Phys. 122, 214107/1-15 (2005).
---
The rest of this section is a list of references to the
original formulation of various areas in Coupled-Cluster
Theory relevant to methods available in GAMESS:
Electronic structure:
J. Cizek, J. Chem. Phys. 45, 4256 (1966).
J. Cizek, Adv. Chem. Phys. 14, 35 (1969).
J. Cizek, J. Paldus, Int.J.Quantum Chem. 5, 359 (1971).
Nuclear theory (examples):
F. Coester, Nucl. Phys. 7, 421 (1958).
F. Coester, H. Kuemmel, Nucl. Phys. 17, 477 (1960).
K. Kowalski, D.J. Dean, M. Hjorth-Jensen, T. Papenbrock,
P. Piecuch, Phys. Rev. Lett. 92, 132501 (2004).
D.J. Dean, J.R. Gour, G. Hagen, M. Hjorth-Jensen, K.
Kowalski, T. Papenbrock, P. Piecuch, M. Wloch,
Nucl. Phys. A. 752, 299 (2005).
M. Wloch, D.J. Dean, J.R. Gour, P. Piecuch, M. Hjorth-
Jensen, T. Papenbrock, K. Kowalski, Eur. Phys. J. A 25
(Suppl. 1), 485 (2005).
M. Wloch, J.R. Gour, P. Piecuch, D.J. Dean, M. Hjorth-
Jensen, T. Papenbrock, J. Phys. G: Nucl. Phys. 31,
S1291 (2005).
M. Wloch, D.J. Dean, J.R. Gour, M. Hjorth-Jensen, K.
Kowalski, T. Papenbrock, P. Piecuch, Phys. Rev.
Lett. 94, 212501 (2005).
P. Piecuch, M. Wloch, J.R. Gour, D.J. Dean, M. Hjorth-
Jensen, T. Papenbrock, in V. Zelevinsky (Ed.),
Nuclei and Mesoscopic Physics, AIP Conference
Proceedings, Vol. 777 (AIP Press, 2005), p. 28.
D.J. Dean, M. Hjorth-Jensen, K. Kowalski, T. Papenbrock, M.
Wloch, and P. Piecuch, in Key Topics in Nuclear
Structure, Proceedings of the 8th International Spring
Seminar on Nuclear Physics, edited by A. Covello
(World Scientific, Singapore, 2005), p. 147.
Coupled-Cluster Method with Doubles (CCD) -
J. Cizek, J. Chem. Phys. 45, 4256 (1966).
J. Cizek, Adv. Chem. Phys. 14, 35 (1969).
J. Cizek, J. Paldus, Int.J.Quantum Chem. 5, 359 (1971).
J.A. Pople, R. Krishnan, H.B. Schlegel, J.S. Binkley,
Int. J. Quantum Chem. Symp. 14, 545 (1978).
R.J. Bartlett and G.D. Purvis,
Int. J. Quantum Chem. Symp. 14, 561 (1978).
J. Paldus, J. Chem. Phys. 67, 303 (1977)
[orthogonally spin-adapted formulation].
Linearized Coupled-Cluster Method with Doubles (LCCD)
J. Cizek, J. Chem. Phys. 45, 4256 (1966).
J. Cizek, Adv. Chem. Phys. 14, 35 (1969).
R.J. Bartlett, I. Shavitt, Chem.Phys.Lett.50, 190 (1977)
57, 157 (1978) [Erratum].
R. Ahlrichs, Comp. Phys. Commun. 17, 31 (1979).
Coupled-Cluster Method with Singles and Doubles (CCSD) -
G.D.Purvis III, R.J.Bartlett, J.Chem.Phys. 76, 1910 (1982)
[spin-orbital formulation].
P. Piecuch, J. Paldus, Int.J.Quantum Chem. 36, 429 (1989).
[orthogonally spin-adapted formulation].
G.E.Scuseria, A.C.Scheiner, T.J.Lee, J.E.Rice,
H.F.Schaefer III, J. Chem. Phys. 86, 2881 (1987)
[non-orthogonally spin-adapted formulation].
G.E. Scuseria, C.L. Janssen, H.F.Schaefer III
J. Chem. Phys. 89, 7382 (1988)
[non-orthogonally spin-adapted formulation].
T.J. Lee and J.E. Rice, Chem. Phys. Lett. 150, 406 (1988)
[non-orthogonally spin-adapted formulation].
Coupled-Cluster Method with Singles and Doubles and
Noniterative Triples, CCSD[T] = CCSD+T(CCSD) -
M. Urban, J. Noga, S. J. Cole, and R. J. Bartlett,
J. Chem. Phys. 83, 4041 (1985).
P. Piecuch and J. Paldus, Theor. Chim. Acta 78, 65 (1990)
[orthogonally spin-adapted formulation].
P. Piecuch, S. Zarrabian, J. Paldus, and J. Cizek,
Phys. Rev. B 42, 3351-3379 (1990)
[orthogonally spin-adapted formulation].
P. Piecuch, R. Tobola, and J. Paldus,
Int. J. Quantum Chem. 55, 133-146 (1995)
[orthogonally spin-adapted formulation].
Coupled-Cluster Method with Singles and Doubles and
Noniterative Triples, CCSD(T) -
K. Raghavachari, G. W. Trucks, J. A. Pople, M. Head-Gordon,
Chem. Phys. Lett. 157, 479 (1989).
Equation of Motion Coupled-Cluster Method, Response
CC/Time Dependent CC Approaches, SAC-CI (Original Ideas), -
H. Monkhorst, Int. J. Quantum Chem. Symp. 11, 421 (1977).
K. Emrich, Nucl. Phys. A 351, 379 (1981).
H. Sekino and R.J. Bartlett,
Int. J. Quantum Chem. Symp. 18, 255 (1984).
E. Daalgard and H. Monkhorst, Phys. Rev. A 28, 1217 (1983).
M. Takahashi and J. Paldus, J. Chem. Phys. 85, 1486 (1986).
H. Koch and P. Jorgensen, J. Chem. Phys. 93, 3333 (1990).
H. Nakatsuji, K. Hirao, Chem. Phys. Lett. 47, 569 (1977).
H. Nakatsuji, K. Hirao, J.Chem.Phys. 68, 2053, 4279 (1978).
Equation of Motion Coupled-Cluster Method with Singles and
Doubles, EOMCCSD -
J. Geertsen, M. Rittby, and R.J. Bartlett,
Chem. Phys. Lett. 164, 57 (1989).
J.F. Stanton and R.J. Bartlett,
J. Chem. Phys. 98, 7029 (1993).
Method of Moments of Coupled-Cluster Equations and
Renormalized and Completely Renormalized Coupled-Cluster
Methods (Overviews) -
P. Piecuch, K. Kowalski, I.S.O. Pimienta, S.A. Kucharski,
in M.R. Hoffmann, K.G. Dyall (Eds.), Low-Lying
Potential Energy Surfaces, ACS Symposium Series, Vol.
828, Am. Chem. Society, Washington, D.C., 2002, p. 31
[ground and excited states].
P. Piecuch, K. Kowalski, I.S.O. Pimienta, M.J. McGuire,
Int. Rev. Phys. Chem. 21, 527 (2002)
[ground and excited states].
P. Piecuch, I.S.O. Pimienta, P.-F. Fan, K. Kowalski,
in J. Maruani, R. Lefebvre, E. Brandas (Eds.),
Progress in Theoretical Chemistry and Physics,
Vol. 12, Advanced Topics in Theoretical Chemical
Physics, Kluwer, Dordrecht, 2003, p. 119
[ground states].
P. Piecuch, K. Kowalski, I.S.O. Pimienta, P.-D. Fan,
M. Lodriguito, M.J. McGuire, S.A. Kucharski, T. Kus,
M. Musial, Theor. Chem. Acc. 112, 349 (2004)
[ground and excited states].
P. Piecuch, M. Wloch, M. Lodriguito, and J.R. Gour, in S.
Wilson, J.-P. Julien, J. Maruani, E. Brandas, and
G. Delgado-Barrio (Eds.), Progress in Theoretical
Chemistry and Physics, Vol. 15, Recent Advances in the
Theory of Chemical and Physical Systems, Springer,
Berlin, 2006, p. XX, in press
[excited states].
P. Piecuch, I.S.O. Pimienta, P.-D. Fan, and K. Kowalski,
in A.K. Wilson (Ed.), Recent Progress in Electron
Correlation Methodology, ACS Symposium Series, Vol.
XXX, Am. Chem. Society, Washington, D.C., 2006, p. XX
[in press; ground states].
P.-D. Fan and P. Piecuch, Adv. Quantum Chem., in press
(2006).
Renormalized and Completely Renormalized Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
(Initial Original Papers, Ground States) -
P. Piecuch, K. Kowalski,
in J. Leszczynski (Ed.), Computational Chemistry:
Reviews of Current Trends, Vol. 5, World Scientific,
Singapore, 2000, p. 1.
K. Kowalski, P. Piecuch, J. Chem. Phys. 113, 18 (2000).
K. Kowalski, P. Piecuch, J. Chem. Phys. 113, 5644 (2000).
Biorthogonal Method of Moments of Coupled-Cluster Equations
and Size Extensive Completely Renormalized Coupled-Cluster
Singles, Doubles, and Non-iterative Triples Approach (CR-
CC(2,3)=CR-CCSD(T)L; Initial Original Papers) ?
P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105(2005).
P. Piecuch, M. Wloch, J.R. Gour, and A. Kinal, Chem. Phys.
Lett. 418, 467-474 (2006).
Renormalized and Completely Renormalized Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
(Other Original Papers, Higher-Order Methods, Ground-State
Benchmarks) -
K. Kowalski, P. Piecuch, Chem. Phys. Lett. 344, 165 (2001).
P. Piecuch, S.A. Kucharski, K. Kowalski,
Chem. Phys. Lett. 344, 176 (2001).
P. Piecuch, S.A. Kucharski, V. Spirko, K. Kowalski,
J.Chem.Phys. 115, 5796 (2001).
P. Piecuch, K. Kowalski, and I.S.O. Pimienta,
Int. J. Mol. Sci. 3, 475 (2002).
M.J. McGuire, K. Kowalski, P. Piecuch,
J. Chem. Phys. 117, 3617 (2002).
P. Piecuch, S.A. Kucharski, K. Kowalski, M. Musial,
Comput. Phys. Comm., 149, 71 (2002).
I.S.O. Pimienta, K. Kowalski, and P. Piecuch,
J. Chem. Phys. 119, 2951 (2003).
S. Hirata, P.-D. Fan, A.A. Auer, M. Nooijen, P. Piecuch,
J. Chem. Phys. 121, 12197 (2004).
K. Kowalski and P. Piecuch, J. Chem. Phys. 122, 074107
(2005).
P.-D. Fan, K. Kowalski, and P. Piecuch, Mol. Phys. 103,
2191 (2005).
Completely Renormalized Coupled-Cluster Methods, Examples
of Large-Scale Applications to Ground-State Properties -
I. Ozkan, A. Kinal, M. Balci,
J.Phys.Chem. A 108, 507 (2004).
R.L. DeKock, M.J. McGuire, P. Piecuch, W.D. Allen,
H.F. Schaefer III, K. Kowalski, S.A. Kucharski, M. Musial,
A.R. Bonner, S.A. Spronk, D.B Lawson, S.L. Laursen,
J. Phys. Chem. A 108, 2893 (2004).
M.J. McGuire, P. Piecuch, K. Kowalski, S.A. Kucharski,
M. Musial, J. Phys. Chem. A 108, 8878 (2004).
M.J. McGuire, P. Piecuch
J. Am. Chem. Soc. 127, 2608 (2005).
A. Kinal, P. Piecuch, J. Phys. Chem. A 110, 367 (2006).
C.J. Cramer, M. Wloch, P. Piecuch, C. Puzzarini, and L.
Gagliardi, J. Phys. Chem. A 110, 1991 (2006).
Completely Renormalized Equation of Motion Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
for Ground and Excited States (Original Papers) -
K. Kowalski P. Piecuch, J. Chem. Phys. 115, 2966 (2001).
K. Kowalski P. Piecuch, J. Chem. Phys. 116, 7411 (2002).
K. Kowalski P. Piecuch, J. Chem. Phys. 120, 1715 (2004).
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J. Chem.
Phys. 122, 214107 (2005).
Also, multi-reference and other externally corrected MMCC
methods including ground and excited states,
K. Kowalski and P. Piecuch,
J. Molec. Struct.: THEOCHEM 547, 191 (2001).
K. Kowalski and P. Piecuch, Mol. Phys. 102}, 2425 (2004).
M.D. Lodriguito, K. Kowalski, M. Wloch, and P. Piecuch
J. Mol. Struct: THEOCHEM, in press (2006).
Completely Renormalized Equation of Motion Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
for Ground and Excited States (Selected Benchmarks and
Applications)
C.D. Sherrill, P. Piecuch, J.Chem.Phys. 122, 124104 (2005)
R.K. Chaudhuri, K.F. Freed, G. Hose, P. Piecuch,
K. Kowalski, M. Wloch, S. Chattopadhyay, D. Mukherjee,
Z. Rolik, A. Szabados, G. Toth, and P.R. Surjan,
J. Chem. Phys. 122, 134105-1 (2005).
K. Kowalski, S. Hirata, M. Wloch, P. Piecuch, and T.L.
Windus, J. Chem. Phys. 123, 074319 (2005).
S. Nangia, D.G. Truhlar, M.J. McGuire, and P. Piecuch
J. Phys. Chem. A 109, 11643 (2005).
P. Piecuch, S. Hirata, K. Kowalski, P.-D. Fan, and T.L.
Windus, Int. J. Quantum Chem. 106, 79 (2006).
M. Wloch, M.D. Lodriguito, P. Piecuch, and J.R. Gour
Mol. Phys., in press (2006).
S. Coussan, Y. Ferro, A. Trivella, P. Roubin, R. Wieczorek,
C. Manca, P. Piecuch, K. Kowalski, M. Wloch, S.A.
Kucharski, and M. Musial, J. Phys. Chem. A, in press
(2006).
Completely Renormalized Coupled-Cluster and Equation of
Motion Coupled-Cluster Methods, GAMESS Implementations -
P. Piecuch, S.A. Kucharski, K. Kowalski, M. Musial,
Comput. Phys. Comm., 149, 71 (2002).
K. Kowalski and P. Piecuch
J. Chem. Phys. 120, 1715 (2004).
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch,
J. Chem. Phys. 122, 214107 (2005).
P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (2005).
K. Kowalski, P. Piecuch, M. Wloch,S.A. Kucharski,
M. Musial, and M.W. Schmidt, in preparation.
T1 diagnostic:
T.J.Lee, P.R.Taylor Int.J.Quantum Chem., S23, 199-
207(1989).
It is often assumed that T1>0.02 indicates that CCSD may
not be correct for a system which is not very single
reference in nature. (T) corrections tolerate greater
singles amplitudes. However, T1 diagnostic is in many cases
misleading, since one can easily have small (or even
vanishing) T1 cluster amplitudes due to symmetry and a
significant configurational quasi-degeneracy and multi-
reference character. In general, in typical multi-reference
situations, such as bond stretching and diradicals, one
observes a significant increase of T2 cluster amplitudes.
The larger values of T2 amplitudes are a clear signature of
a multi-reference character of the wave function. The CR-
CCSD(T), CR-CCSD(TQ), and CR-CC(2,3) methods tolerate
significant increases of T2 amplitudes in cases of single-
bond breaking and diradicals. CCSD(T) and CCSD(TQ)
approaches cannot do this, when the spin-adapted RHF
references are employed.
Written by Piotr Piecuch, Michigan State University
(updated March 18, 2006)
Density Functional Theory
There are actually two DFT programs in GAMESS, one using
the typical grid quadrature for integration of functionals,
and one using resolution of the identity to avoid the need
or grids. The default METHOD=GRID program is discussed
below, following a short description of METHOD=GRIDFREE.
The final section is references to various functionals, and
other topics of interest.
DFTTYP keywords
Let's begin with a translation table to NWchem's input:
GAMESS NWchem's XC keyword
Slater slater
Gill gill96
SVWN slater vwn_5 (SVWN1RPA=slater vwn_1_rpa, etc)
Becke becke88
BVWN becke88 vwn_5
BLYP becke88 lyp
B97 becke97
B97-1,B97-2,B97-3
becke97-1, becke-2, becke-3
HCTH93,HCTH120,HCTH147,HCTH407
hcth,hcth120,hcth147,hcth407
B98 becke98
B3LYP HFexch 0.20 slater 0.80 \
becke88 nonlocal 0.72 \
lyp 0.81 vwn_5 0.19
B3LYPV1R b3lyp
or, if you like to type:
HFexch 0.20 slater 0.80 \
becke88 nonlocal 0.72 \
lyp 0.81 vwn_1_rpa 0.19
B3P86 HFexch 0.20 slater 0.80 \
becke88 nonlocal 0.72 \
vwn_5 0.19 perdew81 0.81 perdew86 0.81
X3LYP HFexch 0.218 slater 0.782 \
becke88 nonlocal 0.542 \
xperdew91 nonlocal 0.167 \
lyp 0.871 vwn_1_rpa 0.129
PW91 xperdew91 perdew91
B3PW91 HFexch 0.20 slater 0.80 \
becke88 nonlocal 0.72 \
perdew91 0.81 pw91lda 1.00
PBE xpbe96 cpbe96
PBE0 pbe0
revPBE revpbe cpbe96
VS98 vs98
M06 m06 (and similarly for M05-2X, etc.)
PKZB xpkzb99 cpkzb99
TPSS xtpss03 cptss03
TPSSh xctpssh
Note that B3LYP in GAMESS is based in part on the VWN5
electron gas correlation functional. Since there are five
formulae with two possible parameterizations mentioned in
the VWN paper about local correlation, other programs may
use other choices, and therefore generate different B3LYP
energies. For example, NWChem's manual says it uses the
"VWN 1 functional with RPA parameters as opposed to the
prescribed Monte Carlo parameters" as its default. Should
you wish to use this parameterization of the VWN1 formula
in a B3LYP hybrid, simply choose "DFTTYP=B3LYPV1R".
grid-free DFT
The grid-free code is a research tool into the use of
the resolution of the identity to simplify evaluation of
integrals over functionals, rather than quadrature grids.
This trades the use of finite grids and their associated
errors for the use of a finite basis set used to expand the
identity, with an associated truncation error. The present
choice of auxiliary basis sets was obtained by tests on
small 2nd row molecules like NH3 and N2, and hence the
built in bases for the 3rd row are not as well developed.
Auxiliary bases for the remaining elements do not exist at
the present time.
The grid-free Becke/6-31G(d) energy at a C1 AM1 geometry
for ethanol is -154.084592, while the result from a run
using the "army grade grid" is -154.105052. So, the error
using the AUX3 RI basis is about 5 milliHartree per 2nd row
atom (the H's must account for some of the error too). The
energy values are probably OK, the differences noted should
by and large cancel when comparing different geometries.
The grid-free gradient code contains some numerical
inaccuracies, possibly due to the manner in which the RI is
implemented for the gradient. Computed gradients
consequently may not be very reliable. For example, a
Becke/6-31G(d) geometry optimization of water started from
the EXAM08 geometry behaves as:
FINAL E= -76.0439853638, RMS GRADIENT = .0200293
FINAL E= -76.0413274662, RMS GRADIENT = .0231574
FINAL E= -76.0455283912, RMS GRADIENT = .0045887
FINAL E= -76.0457360477, RMS GRADIENT = .0009356
FINAL E= -76.0457239113, RMS GRADIENT = .0001222
FINAL E= -76.0457216186, RMS GRADIENT = .0000577
FINAL E= -76.0457202264, RMS GRADIENT = .0000018
FINAL E= -76.0457202253, RMS GRADIENT = .0000001
Examination shows that the point on the PES where the
gradient is zero is not where the energy is lowest, in fact
the 4th geometry is the lowest encountered.
The behavior for Becke/6-31G(d) ethanol is as follows:
FINAL E= -154.0845920132, RMS GRADIENT = .0135540
FINAL E= -154.0933138447, RMS GRADIENT = .0052778
FINAL E= -154.0885472996, RMS GRADIENT = .0009306
FINAL E= -154.0886268185, RMS GRADIENT = .0002043
FINAL E= -154.0886352947, RMS GRADIENT = .0000795
FINAL E= -154.0885599794, RMS GRADIENT = .0000342
FINAL E= -154.0885514829, RMS GRADIENT = .0000679
FINAL E= -154.0884955093, RMS GRADIENT = .0000205
FINAL E= -154.0886438244, RMS GRADIENT = .0000330
FINAL E= -154.0886596883, RMS GRADIENT = .0000325
FINAL E= -154.0886094081, RMS GRADIENT = .0000120
FINAL E= -154.0886054003, RMS GRADIENT = .0000109
FINAL E= -154.0885939751, RMS GRADIENT = .0000152
FINAL E= -154.0886711482, RMS GRADIENT = .0000439
FINAL E= -154.0886972557, RMS GRADIENT = .0000230
with similar fluctuations through a total of 50 steps
without locating a zero gradient. Note that the second
energy above is substantially below all later points, so
geometry optimizations with the grid-free DFT gradient code
are at this time unsatisfactory.
DFT with grids
METHOD=GRID (the default for DFT) produces good energy
and gradient quantities. Its energy errors should usually
be less than 10 microHartree/atom, using the default grid.
The default grid was changed on April 11, 2008 to use
Lebedev angular grids. This changes all results obtained
prior to that date using the original polar coordinate
angular grid. The old grids can still be used,
$dft nrad=96 nthe=12 nphi=24 $end
$tddft nrad=24 nthe=8 nphi=16 $end
in case you need to reproduce numbers from older versions.
Since April 2008, the default is
$dft nrad=96 nleb=302 $end
$tddft nrad=48 nleb=110 $end
The default for the more accurate meta-GGA functionals was
changed in February 2012 to a more accurate grid
$dft nrad=99 nleb=590 $end
$tddft nrad=48 nleb=110 $end
but all GGA and LDA functionals remain 96/302 and 48/110.
The 96/302 grid settings produce root mean square gradient
vectors accurate to about 0.00010, which matches the
default value for OPTTOL in $STATPT. The "standard grid-
one" contains many fewer points,
$dft sg1=.true. $end
$tddft sg1=.true. $end
so SG1 will produce nuclear gradients accurate only to
about 5 times OPTTOL, namely 0.00050 or so. SG1 is a very
fast grid, and will provide substantial speedups if SG1 is
used for the early steps of geometry optimizations. Rather
high quality results, meaning an OPTTOL near 0.00001 can be
used, may be obtained by
$dft nrad=96 nleb=590 $end
Very accurate (converged) results come from using the "army
grade" grid,
$dft nrad=96 nleb=1202 $end
Turn to the next page to see numerical results.
A numerical demonstration of grid accuracies can be
obtained from ethanol, DFTTYP=BECKE:
energy RMS grad. CPU
sg1=.true. -154.105070 0.010837 11
nrad=96 nthe=12 nphi=24 -154.104863 0.010724 56
nrad=96 nleb=302 -154.105042 0.010704 58
nrad=96 nleb=590 -154.105051 0.0107349 108
nrad=96 nleb=1202 -154.105052 0.0107353 214
Note that the energies are a function of the grid size,
just as they are a function of the basis used, so you must
only compare runs that use the same grid size (and of
course the same basis set). The default grid (and the 590
point grid) will give nuclear gradients which are accurate
enough to lead to satisfactory geometry optimizations.
This means that DFT frequencies obtained by numerical
differentiation should also be OK. RUNTYP=ENERGY,
GRADIENT, HESSIAN, and their chemical combinations for
OPTIMIZE, SADPOINT, IRC, DRC, VSCF, RAMAN, and FFIELD
should all work.
The grid DFT program uses symmetry during the numerical
quadrature in two ways. First, the integration runs only
over grid points placed around the symmetry unique atoms.
Your run should be done in the full non-Abelian group, so
that grid points as well as the usual integrals and the SCF
steps can exploit full molecular symmetry. Symmetry is
turned off during any TD-DFT stages, since excited states
often have different symmetry than the ground state, but
will be used in the ground state DFT.
Secondly, for polar coordinate angular grids only,
"octant symmetry" is implemented using an appropriate
Abelian subgroup of the full group. The grid evaluation
automatically uses an appropriate subgroup to reduce the
number of grid points for atoms that lie on symmetry axes
or planes. For example, in Cs, atoms lying in the xy plane
will be integrated only over the upper hemisphere of their
grid points. Octant symmetry is not used for any of these:
a) if a non-standard axis orientation is input in $DATA
b) if the angular grid size (NTHE,NTHE0,NPHI,NPI0) is not
a multiple of the octant symmetry factors, such as
NTHE=15 in C2v. The permissible values depend on the
group, but NTHE a multiple of 2 and NPHI a multiple of
4 is generally safe.
Time Dependent Density Functional Theory (TD-DFT)
Two review articles are available,
"Single-Reference ab Initio Methods for the Calculation of
Excited States of Large Molecules"
A.Dreuw, M.Head-Gordon
Chem.Rev. 105, 4009-4037(2005)
"Excited states from time-dependent density functional
theory"
P.Elliott, F.Furche, K.Burke
Rev.Comp.Chem. 26, 91-166(2009)
The following article is very informative:
S.Hirata, M.Head-Gordon
Chem.Phys.Lett. 314, 291-299(1999)
It also explains the Tamm/Dancoff approximation which
connects TD-DFT to CIS.
TD-DFT requires higher functional derivatives of the
exchange correlation energy with respect to the density:
2nd derivatives to do TD-DFT excitation energies, and 3rd
derivatives to do TD-DFT nuclear gradients. Consequently,
some of the functionals permit only excitation energies.
To use metaGGAs in TD-DFT, the above functional derivatives
involve a non-trivial differentiation of the kinetic energy
tau's density dependence. The latter is the subject of
F.Zahariev, S.S.Leang, M.S.Gordon
J.Chem.Phys. 138, 244108/1-11(2013)
The TD-DFT nuclear gradient implementation in GAMESS is
M.Chiba, T.Tsuneda, K.Hirao
J.Chem.Phys. 124, 144106/1-11(2006)
and the long-range correction (useful in Rydberg and/or
charge transfer states is
Y.Tawada, T.Tsuneda, S.Yanagisawa, Y.Yanai, K.Hirao
J.Chem.Phys. 120, 8425-8433(2004)
See also
K.A.Nguyen, P.N.Day, R.Pachter
Int.J.Quantum Chem. 110, 2247-2255(2010)
The "lambda diagnostic" is described by
M.J.G.Peach, P.Benfield, T.Helgaker, D.J.Tozer
J.Chem.Phys. 128, 044118/1-8(2008)
This is a criterion for separating valence states from
charge transfer and Rydberg states.
Note that it is possible to do TD-HF excitation energies,
by requesting TDDFT=EXCITE, but leaving DFTTYP=NONE.
* * * * *
Two-photon absorption (TPA) cross-sections are computed by
first evaluating the excitation energy to some desired
number of excited states. The oscillator strengths for
each state give some idea of the intensity of one-photon
absorption (OPA cross-section). Then, the TPA cross-
sections to these same excited states are evaluated. Note
that Franck-Condon factors are not included in either OPA
or TPA oscillator strengths.
Beta hyperpolarizabilities (see BETA in $TDDFT) are
computed by the TDDFT code, but the computation of excited
states is skipped.
TPA and BETA calculations
a) should not use the Tamm/Dancoff approximation,
b) may include solvent effects by EFP,
c) make sense only at a single geometry: use the
ordinary TD-DFT program to optimize excited state
geometries, if needed.
TPA and BETA calculations are described in
F.Zahariev, M.S.Gordon
J.Chem.Phys. 140, 18A523/1-10(2014)
* * * * *
Solvation effects on the excited state energies can be
modeled by PCM or EFP or both, with nuclear gradients.
PCM + TD-DFT gradient:
Y.Wang, H.Li J.Chem.Phys. 133, 034108/1-11(2010)
EFP1 + TD-DFT energy:
S.Yoo, F.Zahariev, S.Sok, M.S.Gordon
J.Chem.Phys. 129, 144112/1-8(2008)
EFP1 + TD-DFT gradient:
N.Minezawa, N.De Silva, F.Zahariev, M.S.Gordon
J.Chem.Phys. 134, 05411(2011)
POL5P + TD-DFT gradient (similar polarizable solvent):
D.Si, H.Li J.Chem.Phys. 133, 144112(2010)
* * * * *
In some cases, a more balanced description of the states
might be obtained if the orbitals are optimized for a
reference with unpaired electrons. This is possible with
spin-flip methods, see TDDFT=SPNFLP. For example, in C2H4,
one might optimize the orbitals for the triplet state
(pi)1(pi*)1, but be interested in the energies of the three
singlets and one triplet states N=(pi)2, T=(pi)1(pi*)1,
V=(pi)1(pi*)1, and Z=(pi*)2. Using the T state as the
reference optimizes the shape of both pi and pi*, since
both are occupied. Flipping one of the two unpaired alpha
spins in the T reference will access all four valence
states (recall that a triplet state with Ms=0 is perfectly
OK, namely ab+ba). For more information, see
Y.Shao, M.Head-Gordon, A.I.Krylov
J.Chem.Phys. 118,4807(2003)
F.Wang, T.Ziegler
J.Chem.Phys. 121, 12191(2004)
J.Chem.Phys. 122, 074109(2005)
O.Vahtras, Z.Rinkevicius
J.Chem.Phys. 126, 114101(2007)
Z.Rinkevicius, H.Agren
Chem.Phys.Lett. 491, 132(2010)
Z.Rinkevicius, O.Vahtras, H.Agren
J.Chem.Phys. 113, 114101(2010
M.Huix-Rotllant, B.Natarajan, A.Ipatov, C.M.Wawire,
T.Deutsch, M.E.Casida
Phys.Chem.Chem.Phys. 12, 12811(2010)
The penalty constrained optimization procedure was used to
find ethylene's conical intersections by
N.Minezawa, M.S.Gordon
J.Phys.Chem.A 113, 12749(2009)
references for DFT
An excellent overview of DFT can be found in Chapter 6 of
Frank Jensen's book. Two other monographs are
"Density Functional Theory of Atoms and Molecules"
R.G.Parr, W.Yang Oxford Scientific, 1989
"A Chemist's Guide to Density Functional Theory"
W.Koch, M.C.Holthausen Wiley-VCH, 2001
If you would like to understand the "theory" of Density
Functional Theory, see Kieron Burke's online book "The ABC
of DFT", at http://dft.uci.edu/dftbook.html; PDF version of
it can be found here: https://dft.uci.edu/doc/g1.pdf.
A good list of literature can be also found at Kieron
Burke group's site: https://dft.uci.edu/learnDFT.php
A delightful and thought provoking paper on the
relationship of DFT to conventional wavefunction theory:
"Obituary: Density Functional Theory (1927-1993)"
P.M.W.Gill Aust.J.Chem. 54, 661-662(2001)
You may also enjoy
"Fourteen easy lessons in Density Functional Theory",
John Perdew and Adrienn Ruzsinszky
Int. J. Quantum Chem. 110, 2801-2807(2010)
"Perspective on density functional theory"
Kieron Burke J.Chem.Phys. 136, 150901/1-9(2012)
"Perspective: fifty years of density-functional theory
in chemical physics"
A.D.Becke J.Chem.Phys. 140, 18A301/1-18(2014)
A paper comparing DFT's approach to correlation to
traditional quantum chemistry methods:
E.J.Baerends, O.V.Gritsenko
J.Phys.Chem.A 101, 5383-5403(1997)
Some philosophy about designing functionals at each rung of
DFT's "Jacob's ladder":
J.P.Perdew, A.Ruzsinszky, J.Tao, V.N.Staroverov,
G.E.Scuseria, G.I.Csonka
J.Chem.Phys. 123, 062201/1-9(2005)
On hybridization:
J.P.Perdew, M.Ernzerhof, K.Burke
J.Chem.Phys. 105, 9982-9985(1996)
G.I.Csonka, J.P.Perdew, A.Ruzsinszky
J.Chem.Theory Comput. 6, 3688-3703(2010)
Some reading on the grid-free approach to density
functional theory is:
Y.C.Zheng, J.Almlof
Chem.Phys.Lett. 214, 397-401(1996)
Y.C.Zheng, J.Almlof
J.Mol.Struct.(Theochem) 288, 277(1996)
K.Glaesemann, M.S.Gordon
J.Chem.Phys. 108, 9959-9969(1998)
K.Glaesemann, M.S.Gordon
J.Chem.Phys. 110, 6580-6582(1999)
K.Glaesemann, M.S.Gordon
J.Chem.Phys. 112, 10738-10745(2000)
References about gridding:
A.D.Becke
J.Chem.Phys. 88, 2547-2553(1988)
C.W.Murray, N.C.Handy, G.L.Laming
Mol.Phys. 78, 997-1014(1993)
P.M.W.Gill, B.G.Johnson, J.A.Pople
Chem.Phys.Lett. 209, 506-512(1993)
A.A.Jarecki, E.R.Davidson
Chem.Phys.Lett. 300, 44-52(1999)
R.Lindh, P.-A.Malmqvist, L.Gagliardi
Theoret.Chem.Acc. 106, 178-187(2001)
S.-H.Chien, P.M.W.Gill
J.Comput.Chem. 27, 730-739(2006)
J.Grafenstein, D.Izotov, D.Cremer
J.Chem.Phys. 127, 164113/1-7(2007)
Gill's 1993 paper is the reference for SG1=.TRUE.
Handy's 1993 paper is a reference for polar coordinates.
Lebedev grids may be referenced as
V.I.Lebedev, D.N.Laikov Doklady Math. 59, 477-481(1999)
GAMESS uses Christoph van Wuellen's FORTRAN translation of
these grids, originally coded in C by Laikov (www.ccl.net).
--- exchange functionals
Slater exchange:
J.C.Slater Phys.Rev. 81, 385-390(1951)
XALPHA is Slater with alpha=0.70
BECKE (often called B88) exchange:
A.D.Becke Phys.Rev. A38, 3098-3100(1988)
GILL (often called G96) exchange:
P.M.W.Gill Mol.Phys. 89, 433-445(1996)
OPTX exchange:
N.C.Handy, A.J.Cohen Mol.Phys. 99, 403-412(2001)
Depristo/Kress exchange:
A.E.DePristo, J.E.Kress J.Chem.Phys. 86, 1425-1428(1987)
--- correlation functionals
VWN local correlation:
S.H.Vosko, L.Wilk, M.Nusair
Can.J.Phys. 58, 1200-1211(1980)
This paper has five formulae in it, and since the 5th is
a good quality fit, it states "since formula 5 is easiest
to implement in LSDA calculations, we recommend its use".
PZ81 correlation:
J.P.Perdew, A.Zunger Phys.Rev.B 23, 5048-5079(1981)
P86 GGA correlation:
J.P.Perdew Phys.Rev.B 33, 8822(1986)
PW local correlation (used in PW91):
J.P.Perdew, Y.Wang Phys.Rev.B 45, 13244-13249(1992)
LYP correlation:
C.Lee, W.Yang, R.G.Parr Phys.Rev. B37, 785-789(1988)
For practical purposes this is always used in a transformed
way, involving the square of the density gradient:
B.Miehlich, A.Savin, H.Stoll, H.Preuss
Chem.Phys.Lett. 157, 200-206(1989)
OP (One-parameter Progressive) correlation:
T.Tsuneda, K.Hirao Chem.Phys.Lett. 268, 510-520(1997)
T.Tsuneda, T.Suzumura, K.Hirao
J.Chem.Phys. 110, 10664-10678(1999)
--- exchange/correlation functionals
PW91 exchange/correlation:
J.P.Perdew, J.A.Chevray, S.H.Vosko, K.A.Jackson,
M.R.Pederson, D.J.Singh, C.Fiolhais
Phys.Rev. B46, 6671-6687(1992)
EDF1 - empirical density functional #1, a tweaked BLYP
developed for use with 6-31+G(d) basis sets,
R.D.Adamson, P.M.W.Gill, J.A.Pople
Chem.Phys.Lett. 284, 6-11(1998)
MOHLYP - metal optimized OPTX exchange,
half LYP correlation
N.E.Schultz, Y.Zhao, D.G.Truhlar
J.Phys.Chem.A 109, 11127-11143(2005)
See also comp.chem.umn.edu/info/MOHLYP_reference.pdf for
information about the related functional MOHLYP2.
PBE exchange/correlation functional:
J.P.Perdew, K.Burke, M.Ernzerhof
Phys.Rev.Lett. 77, 3865-8(1996); Err. 78,1396(1997)
revPBE (revised PBE exchange, but see RPBE below):
Y.Zhang, W.Yang Phys.Rev.Lett. 80, 890(1998)
RPBE (a different revision of PBE exchange):
B.Hammer, L.B.Hansen, J.K.Norskov
Phys.Rev.B 59, 7413-7421(1999)
This revision retains the same increase in accuracy for
atomization energies that revPBE affords, while rigorously
preserving the correct Lieb-Oxford limit, unlike revPBE.
PBEsol (modified PBE parameters, for solid properties):
J.P.Perdew, A.Ruzsinszky, G.I.Csonka, O.A.Vydrov,
G.E.Scuseria, L.A.Constantin, Z.Zhou, K.Burke
Phys.Rev.Lett. 100, 136406/1-7(2008)
APF - A Density Functional with
Spherical Atom Dispersion Terms
Amy Austin, George A. Petersson, Michael J. Frisch†,
Frank J. Dobek, Giovanni Scalmani, Kyle Throssell
J. Chem. Theory Comput. 2012, 8, 12, 4989-5007;
doi: 10.1021/ct300778e
HCTH & HCTH-A - Development and assessment of
new exchange-correlation functionals
Fred A. Hamprecht, Aron J. Cohen,
David J. Tozer, Nicholas C. Handy
J. Chem. Phys. 109, 6264 (1998); doi: 10.1063/1.477267
HCTH p=1/4 & HCTH p=7/6 - Emphasizing the
exchange-correlation potential in functional development
Giuseppina Menconi, Philip J. Wilson, David J. Tozer
J. Chem. Phys. 114, 3958 (2001); doi: 10.1063/1.1342776
CAM-QTP00 - Increasing the applicability of
density functional theory. IV. Consequences of
ionization-potential improved
exchange-correlation potentials
Prakash Verma, Rodney J. Bartlett
J. Chem. Phys. 140, 18A534 (2014);
doi: 10.1063/1.4871409
CAM-QTP01 - The QTP family of consistent functionals and
potentials in Kohn-Sham density functional theory
Yifan Jin, Rodney J. Bartlett
J. Chem. Phys. 145, 034107 (2016);
doi: 10.1063/1.4955497
revM06 - Revised M06 density functional for main-group
and transition-metal chemistry
Ying Wang, Pragya Verma, Xinsheng Jin,
Donald G. Truhlar, Xiao He
PNAS October 9, 2018 115 (41) 10257-10262;
doi: 10.1073/pnas.1810421115
revM06-L - Revised M06-L functional for improved accuracy
on chemical reaction barrier heights,
noncovalent interactions, and solid-state physics
Ying Wang, Xinsheng Jin, Haoyu S. Yu,
Donald G. Truhlar, Xiao He
PNAS August 8, 2017 114 (32) 8487-8492;
doi: 10.1073/pnas.1705670114
The next two occur in the grid-free program only,
various WIGNER exchange/correlation functionals:
Q.Zhao, R.G.Parr Phys.Rev. A46, 5320-5323(1992)
CAMA/CAMB exchange/correlation functionals:
G.J.Laming, V.Termath, N.C.Handy
J.Chem.Phys. 99. 8765-8773(1993)
--- dispersion corrections:
dispersionless Density Functional Theory (dlDF)
K.Pernal, R.Podeszwa, K.Patkowski, K.Szalewicz
Phys.Rev.Lett. 103, 263201/1-4(2009)
This approach recognizes that density functionals may be
optimized to reproduce interaction energies from which the
dispersion energy has been subtracted. dlDF adjusts the
M05-2X parameterization to accomplish this for a training
set. A -D correction specific to dlDF was developed (see
the paper's supplementary material) to address the now
cleanly separated dispersion energy. The dldf-D correction
term is available in the form of a Python script at the
Szalewicz web site. Usage of dlDF by itself is not
sensible.
Local Response Dispersion (LRD)
T.Sato, H.Nakai J.Chem.Phys. 131, 224104/1-12(2009)
T.Sato, H.Nakai J.Chem.Phys. 132, 194101/1-9(2010)
This computes dispersion energies using C6/C8 parameters
evaluated from the final electron density of the molecule's
DFT calculation.
empirical dispersion correction (DC):
This is developed in three successive versions by Grimme
1: S.Grimme J.Comput.Chem. 25, 1463-1473(2004)
2: S.Grimme J.Comput.Chem. 27, 1787-1799(2006)
3: S.Grimme, J.Antony, S.Ehrlich, H.Krieg
J.Chem.Phys. 132, 154104/1-19(2010)
which are applied to different functionals with different
parameterizations of the correction. Setting DC=.TRUE.
thus converts functionals such as BLYP/B3LYP/PBE/BP86/TPSS
to BLYP-D, B3LYP-D, and so forth. See the papers for more
details. The analytic calculation of hessians has been
implemented into GAMESS for these corrections:
H.Nakata, D.G.Fedorov, S.Yokojima, K.Kitaura, S.Nakamura
J.Chem.Theory Comput. 10, 3689-3698(2014)
A functional where the input keyword contains already
the -D, namely B97-D, consists of a revamping of the B97
functional to remove its hybridization with HF exchange and
reparameterization, as well as adding the dispersion
correction:
S.Grimme J.Comput.Chem. 27, 1787-1799(2006)
A somewhat different form for the dispersion correction is
used in the wB97-D functional. Selection of DFTTYP=B97-D
or wB97-D does not require setting DC, as the D is already
present in the name entered for DFTTYP.
--- hybrids with HF exchange
B3PW91 hybrid:
A.D.Becke J.Chem.Phys. 98, 5648-5642(1993)
B3LYP hybrid:
A.D.Becke J.Chem.Phys. 98, 5648-5642(1993)
P.J.Stephens, F.J.Devlin, C.F.Chablowski, M.J.Frisch
J.Phys.Chem. 98, 11623-11627(1994)
R.H.Hertwig, W.Koch Chem.Phys.Lett. 268, 345-351(1997)
The first paper is actually on B3PW91 hybridization, and
optimizes the mixing of five functionals with PW91 as the
correlation GGA. The second paper then proposed use of LYP
in place of PW91, without reoptimizing the mixing ratios of
the hybrid. The final paper discusses the controversy
surrounding which VWN functional is used in the hybrid.
GAMESS uses VWN5 in its B3LYP hybrid, but see also B3LYPV1R
to use the RPA parameterized VWN1 formula.
B97 hybrid:
A.D.Becke J.Chem.Phys. 107, 8554-8560(1997)
B97-1 hybrid, a reparameterization of B97:
F.A.Hamprecht, A.J.Cohen, D.J.Tozer, N.C.Handy
J.Chem.Phys. 109, 6264-6271(1998)
B97-2 hybrid, a reparameterization of B97:
P.J.Wilson, T.J.Bradley, D.J.Tozer
J.Chem.Phys. 115, 9233-9242(2001)
B97-3 hybrid, a reparameterization of B97:
T.W.Keal, D.J.Tozer
J.Chem.Phys. 123, 121103-1/4(2005)
B97-K and BMK hybrids, K=kinetics:
A.D.Boese, J.M.L.Martin
J.Chem.Phys. 121, 3405-3416(2004)
HCTH93, HCTH120, HCTH147, and HCTH407 use training sets
with the indicated number of atoms and molecules used to
adjust the B97 functional:
HCTH93 is defined in the B97-1 paper.
HCTH120 and HCTH147:
A.D.Boese, N.L.Doltsinis, N.C.Handy, M.Sprik
J.Chem.Phys. 112, 1670-1678(2000)
HCTH407:
A.D.Boese, N.C.Handy
J.Chem.Phys. 114, 5497-5503(2001)
B98, Becke's reparameterization of B97:
A.D. Becke J.Chem.Phys. 108, 9624-9631(1998)
...bringing to an end "the B97 family".
X3LYP hybrid:
X.Xu, Q.Zhang, R.P.Muller, W.A.Goddard
J.Chem.Phys. 122, 014105/1-14(2005)
PBE0 hybrid:
C.Adamo, V.Barone J.Chem.Phys. 110, 6158-6170(1999)
in the grid free program only,
HALF exchange:
This is programmed as 50% HF plus 50% B88 exchange.
BHHLYP exchange/correlation:
This is 50% HF plus 50% B88, with LYP correlation.
Note: neither is the HALF-AND-HALF exchange/correlation:
A.D.Becke J.Chem.Phys. 98, 1372-1377(1993)
which he defined as 50% HF + 50% SVWN.
--- meta-GGA functionals
These are pure DFT meta-GGAs, unless the description
explicitly says it is a hybrid!
PKZB (a prototype of the TPSS family):
J.P.Perdew, S.Kurth, A.Zupan, P.Blaha
Phys.Rev.Lett. 82, 2544-2547(1999)
tHCTH and tHCTHhyb=15% HF exchange:
A.D.Boese, N.C.Handy
J.Chem.Phys. 116, 9559-9569(2002)
TPSS:
J.P.Perdew, J.Tao, V.N.Staroverov, G.E.Scuseria
Phys.Rev.Lett. 91, 146401/1-4(2003)
J.P.Perdew, J.Tao, V.N.Staroverov, G.E.Scuseria
J.Chem.Phys. 120, 6898-6911(2004)
TPSSm, a modified TPSS improving atomization energies:
J.P.Perdew, A.Ruzsinszky, J.Tao, G.I.Csonka, G.E.Scuseria
Phys.Rev.A 76, 042506/1-6(2007)
TPSSh, a 10% hybrid using TPSS:
V.N.Staroverov, G.E.Scuseria, J.Tao, J.P.Perdew
J.Chem.Phys. 119, 12129-12137(2003),
erratum is J.Chem.Phys. 121, 11507(2004)
revTPSS, "workhorse functional for CMP and QC"
J.P.Perdew, A.Ruzsinsky, G.I.Csonka, L.A.Constantin, J.Sun
Phys.Rev.Lett. 103, 026403/1-4(2009)
VS98 (whose form is the prototype of the M06 family):
T.V.Voorhis, G.E.Scuseria J.Chem.Phys. 109, 400-410(1998)
U.Minnesota xc family:
M05: Y.Zhao, N.E.Schultz, D.G.Truhlar
J.Chem.Phys. 123, 161103/1-4(2005)
M05-2X: Y.Zhao, D.G.Truhlar
J.Comput.Chem.Theory Comput. 2, 1009-1018(2006)
M06: Y.Zhao, D.G.Truhlar
Theoret.Chem.Acc. 120,215-241(2008)
M06-2X: ibid
M06-HF: Y.Zhao, D.G.Truhlar
J.Phys.Chem.A 110, 13126-13130(2006)
M06-L: Y.Zhao, D.G.Truhlar
J.Chem.Phys. 125, 194101/1-18(2006)
SOGGA: Y.Zhao, D.G.Truhlar
J.Chem.Phys. 128, 184109/1-8(2008)
M08-HX and M08-SO: Y.Zhao, D.G.Truhlar
J.Chem.Theory Comput. 4, 1849-1868(2008)
SOGGA11: R.Peverati, Y.Zhao, D.G.Truhlar
J.Phys.Chem.Lett. 2, 1991-1997(2011)
SOGGA11-X: R.Peverati, D.G.Truhlar
J.Chem.Phys. 135, 191102(2011)
M11: R.Peverati, D.G.Truhlar
J.Phys.Chem.Lett. 2, 2810-2817(2011)
M11-L: R.Peverati, D.G.Truhlar
J.Phys.Chem.Lett. 3, 117-124(2012)
For reviews, please see the paper for M06, and also
Y.Zhao, D.G.Truhlar Acc.Chem.Res. 41, 157-167(2008)
These contain recommendations for choosing the one most
appropriate to your problem.
---- long-range corrected functionals:
LC-BLYP, LC-BOP, LC-BVWN:
Y.Tawada, T.Tsuneda, S.Yanagisawa, Y.Yanai, K.Hirao
J.Chem.Phys. 120, 8425-8433(2004)
CAM-B3LYP:
T.Yanai, D.P.Tew, N.C.Handy
Chem.Phys.Lett. 393, 51-57(2004)
wB97, wB97X, wB97X-D:
J.-D. Chai, M.Head-Gordon
J.Chem.Phys. 128, 084106/1-15(2004)
J.-D. Chai, M.Head-Gordon
Phys.Chem.Chem.Phys. 10, 6615-6620(2008)
A review on the topic of long range corrections, which are
also called 'range separated hybrids', is
D.Jacquemin, E.A.Perpete, G.E.Scuseria, I.Ciofini,
C.Adamo J.Chem.Theory Comput. 4, 123-135(2008)
---- "double-hybrid" ----
The B2PLYP family is a mixture of B88 and HF exchange, and
a mixture of LYP and MP2 correlation:
B2-PLYP: S.Grimme J.Chem.Phys. 124, 034108/1-15(2006)
B2G-PLYP: A.Karton, A.Tarnopolsky, J.F.Lamere,
G.C.Schatz, J.M.L.Martin
J.Phys.Chem. A 112, 12868(2008)
B2K-PLYP, B2T-PLYP: A.Tarnopolsky, A.Karton,
R.Sertchook, D.Vuzman, J.M.L.Martin
J.Phys.Chem. A 112, 3(2008)
Double hybrids which are also "long range corrected" (and
whose parameters depend on the basis set):
wB97X-2, wB97X-2L: J.-D. Chai, M.Head-Gordon
J.Chem.Phys. 131, 174105/1-13(2009)
* * * * *
Some of the functionals now present in GAMESS were made
using code from the "density functional repository",
http://www.cse.clrc.ac.uk/qcg/dft
We thank Huub van Dam for his assistance with this, and
particularly for providing the VWN1RPA functional. The
Minnesota functionals are based on subroutines provided by
the Truhlar group at the University of Minnesota. Some
functionals, and particularly their high derivatives needed
by TDDFT, were created by MAXIMA's algebraic manipulation,
along the lines described by
P.Salek, A.Hesselmann
J.Comput.Chem. 28, 2569-2575(2007)
* * * * *
The paper of Johnson, Gill, and Pople listed below has a
useful summary of formulae, and details about a gradient
implementation. A paper on 1st and 2nd derivatives of DFT
with respect to nuclear coordinates and applied fields is
A.Komornicki, G.Fitzgerald
J.Chem.Phys. 98, 1398-1421(1993)
and see also
P.Deglmann, F.Furche, R.Ahlrichs
Chem.Phys.Lett. 362, 511-518(2002).
A few of the many papers assessing the accuracy of DFT:
B.Miehlich, A.Savin, H.Stoll, H.Preuss
Chem.Phys.Lett. 157, 200-206(1989)
B.G.Johnson, P.M.W.Gill, J.A.Pople
J.Chem.Phys. 98, 5612-5626(1993)
N.Oliphant, R.J.Bartlett
J.Chem.Phys. 100, 6550-6561(1994)
L.A.Curtiss, K.Raghavachari, P.C.Redfern, J.A.Pople
J.Chem.Phys. 106, 1063-1079(1997)
E.R.Davidson Int.J.Quantum Chem. 69, 241-245(1998)
B.J.Lynch, D.G.Truhlar
J.Phys.Chem.A 105, 2936-2941(2001)
R.A.Pascal J.Phys.Chem.A 105, 9040-9048(2001)
A.D.Boese, J.M.L.Martin, N.C.Handy
J.Chem.Phys. 119, 3005-3014(2003)
Y.Zhao, D.G.Truhlar,
J.Phys.Chem.A 109, 5656-5667(2005)
K.E.Riley, B.T.Op't Holt, K.M.Merz
J.Chem.Theory Comput. 3, 407-433(2007)
S.F.Sousa, P.A.Fernandes, M.J.Ramos
J.Phys.Chem.A 111, 10439-10452(2007)
Boese et al. include basis set comparisons, as well as
functional comparisons. The final paper is a review of
reviews, and encourages you to think past B3LYP, which
after all dates from 1993! Of course there are assessments
in many of the functional papers as well.
On the accuracy of DFT for large molecule thermochemistry:
L.A.Curtiss, K.Ragavachari, P.C.Redfern, J.A.Pople
J.Chem.Phys. 112, 7374-7383(2000)
P.C.Redfern, P.Zapol, L.A.Curtiss, K.Ragavachari
J.Phys.Chem.A 104, 5850-5854(2000)
On the accuracy of TD-DFT excitation energies:
S.S.Leang, F.Zahariev, M.S.Gordon
J.Chem.Phys. 136, 104101/1-12(2012)
Spin contamination in DFT:
1. It is empirically observed that the values for
unrestricted DFT are smaller than for unrestricted HF.
2. GAMESS computes the quantity in an approximate
way, namely it pretend that the Kohn-Shan orbitals can be
used to form a determinant (WRONG, WRONG, WRONG, there is
no wavefunction in DFT!!!) and then uses the same formula
that UHF uses to evaluate that determinant's spin
expectation value. See
G.J.Laming, N.C.Handy, R.D.Amos
Mol.Phys. 80, 1121-1134(1993)
J.Baker, A.Scheiner, J.Andzelm
Chem.Phys.Lett. 216, 380-388(1993)
C.Adamo, V.Barone, A.Fortunelli
J.Chem.Phys. 98, 8648-8652(1994)
J.A.Pople, P.M.W.Gill, N.C.Handy
Int.J.Quantum Chem. 56, 303-305(1995)
J.Wang, A.D.Becke, V.H.Smith
J.Chem.Phys. 102, 3477-3480(1995)
J.M.Wittbrodt, H.B.Schlegel
J.Chem.Phys. 105, 6574-6577(1996)
J.Grafenstein, D.Cremer
Mol.Phys. 99, 981-989(2001)
and commentary in Koch & Holthausen, pp 52-54.
Orbital energies:
The discussion on page 49-50 of Koch and Holthausen shows
that although the highest occupied orbital's eigenvalue
should be the ionization potential for exact Kohn-Sham
calculations, the functionals we actually have greatly
underestimate IP values. The 5th reference below shows how
inclusion of HF exchange helps this, and provides a linear
correction formula for IPs. The first two papers below
connect the HOMO eigenvalue to the IP, and the third shows
that while the band gap is underestimated by existing
functionals, the gap's center is correctly predicted.
However, the 5th paper shows that DFT is actually pretty
hopeless at predicting these gaps. The 4th paper uses SCF
densities to generate exchange-correlation potentials that
actually give fairly good IP values:
J.F.Janak Phys.Rev.B 18, 7165-7168(1978)
M.Levy, J.P.Perdew, V.Sahni
Phys.Rev.A 30, 2745-2748(1984)
J.P.Perdew, M.Levy Phys.Rev.Lett. 51, 1884-1887(1983)
A.Nagy, M.Levy Chem.Phys.Lett. 296, 313-315(1998)
G.Zhang, C.B.Musgrave J.Phys.Chem.A 111, 1554-1561(2007)
Summary of excited state methods
This is not a "how to" section, as the actual calculations
will be carried out by means described elsewhere in this
chapter. Instead, a summary of methods that can treat
excited states is given.
The simplest possibility is SCFTYP. For example, a closed
shell molecule's first triplet state can always be treated
by SCFTYP=ROHF MULT=3. Assuming there is some symmetry
present, the GVB program may be able to do excited singlets
variationally, provided they are of a different space
symmetry than the ground state. The MCSCF program gives a
general entree into excited states, since upper roots of a
Hamiltonian are always variational: see for example NSTATE
and WSTATE and IROOT in $DET. Of course, 2nd order
perturbation theory can include correlation energy into
these SCF level calculations. Note in particular the
usefulness of quasi-degenerate multireference perturbation
theory when electronic states have similar energies.
CI calculations also give a simple entree into excitated
states. There are a variety of programs, selected by CITYP
in $CONTRL. Note in particular CITYP=CIS, programmed for
closed shell ground states, with gradient capability for
singlet excited states, and for the calculation of triplet
state energies. The other CI programs can generate very
flexible wavefunctions for the evaluation of the excitation
energy, and property values. Note that the GUGA program
will do nuclear gradients provided the reference is RHF.
The TD-DFT method treats singly excited states, including
correlation effects, and is a popular alternative to CIS.
The program allows for excitation energies from a UHF
reference, but is much more powerful for RHF references:
nuclear gradients and/or properties may be computed. Use
of a "long range corrected" or "range separated" functional
(the two terms are synonymous) is often thought to be
important when treating charge transfer or Rydberg states:
see the LC=.TRUE. flag or CAMB3LYP. Spin-flip TDDFT allows
the users to select as the reference state something more
appropriate to the orbital optimization stage. See $TDDFT
for details.
Equation of Motion (EOM) coupled cluster can give accurate
estimates of excitation energies. There are no gradients,
and properties exist only for the EOM-CCSD level, but
triples corrections to the energy are available. See
$EOMINP for more details.
Most of the runs will predict oscillator strengths, or
Einstein coefficients, or similar data regarding the
electronic transition moments. Full prediction of UV-vis
spectra is not possible without Franck-Condon information.
Excited states frequently come close together, and
crossings between them are of great interest.
See RUNTYP=TRANSITION for spin-orbit coupling, which is
responsible for InterSystem Crossing (ISC) between states
of different spin multiplicity. See RUNTYP=NACME for the
computation of the non-adiabatic coupling matrix elements
that cause Internal Conversion (IC) between states of the
same spin multiplicity. Alternatively, diabatic potential
surfaces may be generated at the MCSCF or MCQDPT levels:
see DIABAT in the $MCSCF group.
It is possible to search for the lowest energy on the
crossing seam between two surfaces. In case those surfaces
have different spins, or different space symmetries (or
both), see RUNTYP=MEX. When the surfaces have the same
symmetry, see RUNTYP=CONINT for location of conical
intersections.
Solvent effects (EFP and/or PCM) can easily be incorporated
when using SCFTYP to generate the states, and nuclear
gradients are available. It is now possible to assess
solvent effects on TD-DFT excitation energies from closed
shell references, using either EFP or PCM.
Excited states often possess Rydberg character, so diffuse
functions in the basis set are likely to be important.
Geometry Searches and Internal Coordinates
Stationary points are places on the potential energy
surface with a zero gradient vector (first derivative of
the energy with respect to nuclear coordinates). These
include minima (whether relative or global), better known
to chemists as reactants, products, and intermediates; as
well as transition states (also known as saddle points).
The two types of stationary points have a precise
mathematical definition, depending on the curvature of the
potential energy surface at these points. If all of the
eigenvalues of the hessian matrix (second derivative
of the energy with respect to nuclear coordinates) are
positive, the stationary point is a minimum. If there is
one, and only one, negative curvature, the stationary
point is a transition state. Points with more than one
negative curvature do exist, but are not important in
chemistry. Because vibrational frequencies are basically
the square roots of the curvatures, a minimum has all
real frequencies, and a saddle point has one imaginary
vibrational "frequency".
GAMESS locates minima by geometry optimization, as
RUNTYP=OPTIMIZE, and transition states by saddle point
searches, as RUNTYP=SADPOINT. In many ways these are
similar, and in fact nearly identical FORTRAN code is used
for both. The term "geometry search" is used here to
describe features which are common to both procedures.
The input to control both RUNTYPs is found in the $STATPT
group.
As will be noted in the symmetry section below, an
OPTIMIZE run does not always find a minimum, and a
SADPOINT run may not find a transtion state, even though
the gradient is brought to zero. You can prove you have
located a minimum or saddle point only by examining the
local curvatures of the potential energy surface. This
can be done by following the geometry search with a
RUNTYP=HESSIAN job, which should be a matter of routine.
quasi-Newton Searches
Geometry searches are most effectively done by what is
called a quasi-Newton-Raphson procedure. These methods
assume a quadratic potential surface, and require the
exact gradient vector and an approximation to the hessian.
It is the approximate nature of the hessian that makes the
method "quasi". The rate of convergence of the geometry
search depends on how quadratic the real surface is, and
the quality of the hessian. The latter is something you
have control over, and is discussed in the next section.
GAMESS contains different implementations of quasi-
Newton procedures for finding stationary points, namely
METHOD=NR, RFO, QA, and the seldom used SCHLEGEL. They
differ primarily in how the step size and direction are
controlled, and how the Hessian is updated. The CONOPT
method is a way of forcing a geometry away from a minimum
towards a TS. It is not a quasi-Newton method, and is
described at the very end of this section.
The NR method employs a straight Newton-Raphson step.
There is no step size control, the algorithm will simply
try to locate the nearest stationary point, which may be a
minimum, a TS, or any higher order saddle point. NR is
not intended for general use, but is used by GAMESS in
connection with some of the other methods after they have
homed in on a stationary point, and by Gradient Extremal
runs where location of higher order saddle points is
common. NR requires a very good estimate of the geometry
in order to converge on the desired stationary point.
The RFO and QA methods are two different versions of
the so-called augmented Hessian techniques. They both
employ Hessian shift parameter(s) in order to control the
step length and direction.
In the RFO method, the shift parameter is determined by
approximating the PES with a Rational Function, instead of
a second order Taylor expansion. For a RUNTYP=SADPOINT,
the TS direction is treated separately, giving two shift
parameters. This is known as a Partitioned Rational
Function Optimization (P-RFO). The shift parameter(s)
ensure that the augmented Hessian has the correct eigen-
value structure, all positive for a minimum search, and
one negative eigenvalue for a TS search. The (P)-RFO step
can have any length, but if it exceeds DXMAX, the step is
simply scaled down.
In the QA (Quadratic Approximation) method, the shift
parameter is determined by the requirement that the step
size should equal DXMAX. There is only one shift
parameter for both minima and TS searches. Again the
augmented Hessian will have the correct structure. There
is another way of describing the same algorithm, namely as
a minimization on the "image" potential. The latter is
known as TRIM (Trust Radius Image Minimization). The
working equation is identical in these two methods.
When the RFO steplength is close to DXMAX, there is
little difference between the RFO and QA methods. However,
the RFO step may in some cases exceed DXMAX significantly,
and a simple scaling of the step will usually not produce
the best direction. The QA step is the best step on the
hypersphere with radius DXMAX. For this reason QA is the
default algorithm.
Near a stationary point the straight NR algorithm is
the most efficient. The RFO and QA may be viewed as
methods for guiding the search in the "correct" direction
when starting far from the stationary point. Once the
stationary point is approached, the RFO and QA methods
switch to NR, automatically, when the NR steplength drops
below 0.10 or DXMAX, whichever is the smallest.
The QA method works so well that we use it exclusively,
and so the SCHLEGEL method will probably be omitted from
some future version of GAMESS.
You should read the papers mentioned below in order to
understand how these methods are designed to work. The
first 3 papers describe the RFO and TRIM/QA algorithms. A
good but slightly dated summary of search procedures is
given by Bell and Crighton, and see also the review by
Schlegel. Most of the FORTRAN code for geometry searches,
and some of the discussion in this section was written by
Frank Jensen of the University of Aarhus, whose paper
compares many of the algorithms implemented in GAMESS:
1. J.Baker J.Comput.Chem. 7, 385-395(1986)
2. T.Helgaker Chem.Phys.Lett. 182, 503-510(1991)
3. P.Culot, G.Dive, V.H.Nguyen, J.M.Ghuysen
Theoret.Chim.Acta 82, 189-205(1992)
4. H.B.Schlegel J.Comput.Chem. 3, 214-218(1982)
5. S.Bell, J.S.Crighton
J.Chem.Phys. 80, 2464-2475(1984).
6. H.B.Schlegel Advances in Chemical Physics (Ab Initio
Methods in Quantum Chemistry, Part I), volume 67,
K.P.Lawley, Ed. Wiley, New York, 1987, pp 249-286.
7. F.Jensen J.Chem.Phys. 102, 6706-6718(1995).
the nuclear Hessian
Although quasi-Newton methods require only an
approximation to the true hessian, the quality of this
matrix has a great affect on convergence of the geometry
search.
There is a procedure contained within GAMESS for
guessing a positive definite hessian matrix, HESS=GUESS.
If you are using Cartesian coordinates, the guess hessian
is based on pairwise atom stretches. The guess is more
sophisticated when internal coordinates are defined, as
empirical rules will be used to estimate stretching and
bending force constants. Other angular force constants are
set to 1/4. The guess often works well for minima, but
cannot possibly find transition states (because it is
positive definite). Therefore, GUESS may not be selected
for SADPOINT runs.
Two options for providing a more accurate hessian are
HESS=READ and CALC. For the latter, the true hessian is
obtained by direct calculation at the initial geometry, and
then the geometry search begins, all in one run. The READ
option allows you to feed in the hessian in a $HESS group,
as obtained by a RUNTYP=HESSIAN job. The second procedure
is actually preferable, as you get a chance to see the
frequencies. Then, if the local curvatures look good, you
can commit to the geometry search. Be sure to include a
$GRAD group (if the exact gradient is available) in the
HESS=READ job so that GAMESS can take its first step
immediately.
Note also that you can compute the hessian at a lower
basis set and/or wavefunction level, and read it into a
higher level geometry search. In fact, the $HESS group
could be obtained at the semiempirical level. This trick
works because the hessian is 3Nx3N for N atoms, no matter
what atomic basis is used. The gradient from the lower
level is of course worthless, as the geometry search must
work with the exact gradient of the wavefunction and basis
set in current use. Discard the $GRAD group from the lower
level calculation!
You often get what you pay for. HESS=GUESS is free, but
may lead to significantly more steps in the geometry
search. The other two options are more expensive at the
beginning, but may pay back by rapid convergence to the
stationary point.
The hessian update frequently improves the hessian for a
few steps (especially for HESS=GUESS), but then breaks
down. The symptoms are a nice lowering of the energy or
the RMS gradient for maybe 10 steps, followed by crazy
steps. You can help by putting the best coordinates into
$DATA, and resubmitting, to make a fresh determination of
the hessian.
The default hessian update for OPTIMIZE runs is BFGS,
which is likely to remain positive definite. The POWELL
update is the default for SADPOINT runs, since the hessian
can develop a negative curvature as the search progresses.
The POWELL update is also used by the METHOD=NR and CONOPT
since the Hessian may have any number of negative
eigenvalues in these cases. The MSP update is a mixture of
Murtagh-Sargent and Powell, suggested by Josep Bofill,
(J.Comput.Chem., 15, 1-11, 1994). It sometimes works
slightly better than Powell, so you may want to try it.
coordinate choices
Optimization in cartesian coordinates has a reputation
of converging slowly. This is largely due to the fact
that translations and rotations are usually left in the
problem. Numerical problems caused by the small eigen-
values associated with these degrees of freedom are the
source of this poor convergence. The methods in GAMESS
project the hessian matrix to eliminate these degrees of
freedom, which should not cause a problem. Nonetheless,
Cartesian coordinates are in general the most slowly
convergent coordinate system.
The use of internal coordinates (see NZVAR in $CONTRL
as well as $ZMAT) also eliminates the six rotational and
translational degrees of freedom. Also, when internal
coordinates are used, the GUESS hessian is able to use
empirical information about bond stretches and bends.
On the other hand, there are many possible choices for the
internal coordinates, and some of these may lead to much
poorer convergence of the geometry search than others.
Particularly poorly chosen coordinates may not even
correspond to a quadratic surface, thereby ending all hope
that a quasi-Newton method will converge.
Internal coordinates are frequently strongly coupled.
Because of this, Jerry Boatz has called them "infernal
coordinates"! A very common example to illustrate this
might be a bond length in a ring, and the angle on the
opposite side of the ring. Clearly, changing one changes
the other simultaneously. A more mathematical definition
of "coupled" is to say that there is a large off-diagonal
element in the hessian. In this case convergence may be
unsatisfactory, especially with a diagonal GUESS hessian,
where a "good" set of internals is one with a diagonally
dominant hessian. Of course, if you provide an accurately
computed hessian, it will have large off-diagonal values
where those are truly present. Even so, convergence may
be poor if the coordinates are coupled through large 3rd
or higher derivatives. The best coordinates are therefore
those which are the most "quadratic".
One very popular set of internal coordinates is the
usual "model builder" Z-matrix input, where for N atoms,
one uses N-1 bond lengths, N-2 bond angles, and N-3 bond
torsions. The popularity of this choice is based on its
ease of use in specifying the initial molecular geometry.
Typically, however, it is the worst possible choice of
internal coordinates, and in the case of rings, is not
even as good as Cartesian coordinates.
However, GAMESS does not require this particular mix
of the common types. GAMESS' only requirement is that you
use a total of 3N-6 coordinates, chosen from these 3 basic
types, or several more exotic possibilities. (Of course,
we mean 3N-5 throughout for linear molecules). These
additional types of internal coordinates include linear
bends for 3 collinear atoms, out of plane bends, and so on.
There is no reason at all why you should place yourself in
a straightjacket of N-1 bonds, N-2 angles, and N-3
torsions.
If the molecule has symmetry, be sure to use internals
which are symmetrically related.
For example, the most effective choice of coordinates
for the atoms in a four membered ring is to define all
four sides, any one of the internal angles, and a dihedral
defining the ring pucker. For a six membered ring, the
best coordinates seem to be 6 sides, 3 angles, and 3
torsions. The angles should be every other internal
angle, so that the molecule can "breathe" freely. The
torsions should be arranged so that the central bond of
each is placed on alternating bonds of the ring, as if
they were pi bonds in Kekule benzene. For a five membered
ring, we suggest all 5 sides, 2 internal angles, again
alternating every other one, and 2 dihedrals to fill in.
The internal angles of necessity skip two atoms where the
ring closes. Larger rings should generalize on the idea
of using all sides but only alternating angles. If there
are fused rings, start with angles on the fused bond, and
alternate angles as you go around from this position.
Rings and more especially fused rings can be tricky.
For these systems, especially, we suggest the Cadillac of
internal coordinates, the "natural internal coordinates"
of Peter Pulay. For a description of these, see
P.Pulay, G.Fogarosi, F.Pang, J.E.Boggs,
J.Am.Chem.Soc. 101, 2550-2560 (1979).
G.Fogarasi, X.Zhou, P.W.Taylor, P.Pulay
J.Am.Chem.Soc. 114, 8191-8201 (1992).
These are linear combinations of local coordinates, except
in the case of rings. The examples given in these two
papers are very thorough.
An illustration of these types of coordinates is given
in the example job EXAM25.INP, distributed with GAMESS.
This is a nonsense molecule, designed to show many kinds
of functional groups. It is defined using standard bond
distances with a classical Z-matrix input, and the angles
in the ring are adjusted so that the starting value of
the unclosed OO bond is also a standard value.
Using Cartesian coordinates is easiest, but takes a very
large number of steps to converge. This however, is better
than using the classical Z-matrix internals given in $DATA,
which is accomplished by setting NZVAR to the correct 3N-6
value. The geometry search changes the OO bond length to
a very short value on the 1st step, and the SCF fails to
converge. (Note that if you have used dummy atoms in the
$DATA input, you cannot simply enter NZVAR to optimize in
internal coordinates, instead you must give a $ZMAT which
involves only real atoms).
The third choice of internal coordinates is the best set
which can be made from the simple coordinates. It follows
the advice given above for five membered rings, and because
it includes the OO bond, has no trouble with crashing this
bond. It takes 20 steps to converge, so the trouble of
generating this $ZMAT is certainly worth it compared to the
use of Cartesians.
Natural internal coordinates are defined in the final
group of input. The coordinates are set up first for the
ring, including two linear combinations of all angles and
all torsions withing the ring. After this the methyl is
hooked to the ring as if it were a NH group, using the
usual terminal methyl hydrogen definitions. The H is
hooked to this same ring carbon as if it were a methine.
The NH and the CH2 within the ring follow Pulay's rules
exactly. The amount of input is much greater than a normal
Z-matrix. For example, 46 internal coordinates are given,
which are then placed in 3N-6=33 linear combinations. Note
that natural internals tend to be rich in bends, and short
on torsions.
The energy results for the three coordinate systems
which converge are as follows:
NSERCH Cart good Z-mat nat. int.
0 -48.6594935049 -48.6594935049 -48.6594935049
1 -48.6800538676 -48.6806631261 -48.6838361406
2 -48.6822702585 -48.6510215698 -48.6874045449
3 -48.6858299354 -48.6882945647 -48.6932811528
4 -48.6881499412 -48.6849667085 -48.6946836332
5 -48.6890226067 -48.6911899936 -48.6959800274
6 -48.6898261650 -48.6878047907 -48.6973821465
7 -48.6901936624 -48.6930608185 -48.6987652146
8 -48.6905304889 -48.6940607117 -48.6996366016
9 -48.6908626791 -48.6949137185 -48.7006656309
10 -48.6914279465 -48.6963767038 -48.7017273728
11 -48.6921521142 -48.6986608776 -48.7021504975
12 -48.6931136707 -48.7007305310 -48.7022405019
13 -48.6940437619 -48.7016095285 -48.7022548935
14 -48.6949546487 -48.7021531692 -48.7022569328
15 -48.6961698826 -48.7022080183 -48.7022570260
16 -48.6973813002 -48.7022454522 -48.7022570662
17 -48.6984850655 -48.7022492840
18 -48.6991553826 -48.7022503853
19 -48.6996239136 -48.7022507037
20 -48.7002269303 -48.7022508393
21 -48.7005379631
22 -48.7008387759
...
50 -48.7022519950
from which you can see that the natural internals are
actually the best set. The $ZMAT exhibits upward burps
in the energy at step 2, 4, and 6, so that for the
same number of steps, these coordinates are always at a
higher energy than the natural internals.
The initial hessian generated for these three columns
contains 0, 33, and 46 force constants. This assists
the natural internals, but is not the major reason for
its superior performance. The computed hessian at the
final geometry of this molecule, when transformed into the
natural internal coordinates is almost diagonal. This
almost complete uncoupling of coordinates is what makes
the natural internals perform so well. The conclusion
is of course that not all coordinate systems are equal,
and natural internals are the best. As another example,
we have run the ATCHCP molecule, which is a popular
geometry optimization test, due to its two fused rings:
H.B.Schlegel, Int.J.Quantum Chem., Symp. 26, 253-264(1992)
T.H.Fischer and J.Almlof, J.Phys.Chem. 96, 9768-9774(1992)
J.Baker, J.Comput.Chem. 14, 1085-1100(1993)
Here we have compared the same coordinate types, using a
guess hessian, or a computed hessian. The latter set of
runs is a test of the coordinates only, as the initial
hessian information is identical. The results show clearly
the superiority of the natural internals, which like the
previous example, give an energy decrease on every step:
HESS=GUESS HESS=READ
Cartesians 65 41 steps
good Z-matrix 32 23
natural internals 24 13
A final example is phosphinoazasilatrane, with three rings
fused on a common SiN bond, in which 112 steps in Cartesian
space became 32 steps in natural internals. The moral is:
"A little brain time can save a lot of CPU time."
In late 1998, a new kind of internal coordinate method
was included into GAMESS. This is the delocalized
internal
coordinate (DLC) of
J.Baker, A. Kessi, B.Delley
J.Chem.Phys. 105, 192-212(1996)
although as is the usual case, the implementation is not
exactly the same. Bonds are kept as independent
coordinates,
while angles are placed in linear combination by the DLC
process. There are some interesting options for applying
constraints, and other options to assist the automatic DLC
generation code by either adding or deleting coordinates.
It is simple to use DLCs in their most basic form:
$contrl nzvar=xx $end
$zmat dlc=.true. auto=.true. $end
Our initial experience is that the quality of DLCs is
not as good as explicitly constructed natural internals,
which benefit from human chemical knowledge, but are almost
always better than carefully crafted $ZMATs using only the
primitive internal coordinates (although we have seen a few
exceptions). Once we have more numerical experience with
the use of DLC's, we will come back and revise the above
discussion of coordinate choices. In the meantime, they
are quite simple to choose, so give them a go.
the role of symmetry
At the end of a succesful geometry search, you will
have a set of coordinates where the gradient of the energy
is zero. However your newly discovered stationary point
is not necessarily a minimum or saddle point!
This apparent mystery is due to the fact that the
gradient vector transforms under the totally symmetric
representation of the molecular point group. As a direct
consequence, a geometry search is point group conserving.
(For a proof of these statements, see J.W.McIver and
A.Komornicki, Chem.Phys.Lett., 10,303-306(1971)). In
simpler terms, the molecule will remain in whatever point
group you select in $DATA, even if the true minimum is in
some lower point group. Since a geometry search only
explores totally symmetric degrees of freedom, the only
way to learn about the curvatures for all degrees of
freedom is RUNTYP=HESSIAN.
As an example, consider disilene, the silicon analog
of ethene. It is natural to assume that this molecule is
planar like ethene, and an OPTIMIZE run in D2h symmetry
will readily locate a stationary point. However, as a
calculation of the hessian will readily show, this
structure is a transition state (one imaginary frequency),
and the molecule is really trans-bent (C2h). A careful
worker will always characterize a stationary point as
either a minimum, a transition state, or some higher order
stationary point (which is not of great interest!) by
performing a RUNTYP=HESSIAN.
The point group conserving properties of a geometry
search can be annoying, as in the preceeding example, or
advantageous. For example, assume you wish to locate the
transition state for rotation about the double bond in
ethene. A little thought will soon reveal that ethene is
D2h, the 90 degrees twisted structure is D2d, and
structures in between are D2. Since the saddle point is
actually higher symmetry than the rest of the rotational
surface, you can locate it by RUNTYP=OPTIMIZE within D2d
symmetry. You can readily find this stationary point with
the diagonal guess hessian! In fact, if you attempt to do
a RUNTYP=SADPOINT within D2d symmetry, there will be no
totally symmetric modes with negative curvatures, and it
is unlikely that the geometry search will be very well
behaved.
Although a geometry search cannot lower the symmetry,
the gain of symmetry is quite possible. For example, if
you initiate a water molecule optimization with a trial
structure which has unequal bond lengths, the geometry
search will come to a structure that is indeed C2v (to
within OPTTOL, anyway). However, GAMESS leaves it up to
you to realize that a gain of symmetry has occurred.
In general, Mother Nature usually chooses more
symmetrical structures over less symmetrical structures.
Therefore you are probably better served to assume the
higher symmetry, perform the geometry search, and then
check the stationary point's curvatures. The alternative
is to start with artificially lower symmetry and see if
your system regains higher symmetry. The problem with
this approach is that you don't necessarily know which
subgroup is appropriate, and you lose the great speedups
GAMESS can obtain from proper use of symmetry. It is good
to note here that "lower symmetry" does not mean simply
changing the name of the point group and entering more
atoms in $DATA, instead the nuclear coordinates themselves
must actually be of lower symmetry.
practical matters
Geometry searches do not bring the gradient exactly to
zero. Instead they stop when the largest component of the
gradient is below the value of OPTTOL, which defaults to
a reasonable 0.0001. Analytic hessians usually have
residual frequencies below 10 cm**-1 with this degree of
optimization. The sloppiest value you probably ever want
to try is 0.0005.
If a geometry search runs out of time, or exceeds
NSTEP, it can be restarted. For RUNTYP=OPTIMIZE, restart
with the coordinates having the lowest total energy
(do a string search on "FINAL"). For RUNTYP=SADPOINT,
restart with the coordinates having the smallest gradient
(do a string search on "RMS", which means root mean
square).
These are not necessarily at the last geometry!
The "restart" should actually be a normal run, that is
you should not try to use the restart options in $CONTRL
(which may not work anyway). A geometry search can be
restarted by extracting the desired coordinates for $DATA
from the printout, and by extracting the corresponding
$GRAD group from the PUNCH file. If the $GRAD group is
supplied, the program is able to save the time it would
ordinarily take to compute the wavefunction and gradient
at the initial point, which can be a substantial savings.
There is no input to trigger reading of a $GRAD group: if
found, it is read and used. Be careful that your $GRAD
group actually corresponds to the coordinates in $DATA, as
GAMESS has no check for this.
Sometimes when you are fairly close to the minimum, an
OPTIMIZE run will take a first step which raises the
energy, with subsequent steps improving the energy and
perhaps finding the minimum. The erratic first step is
caused by the GUESS hessian. It may help to limit the size
of this wrong first step, by reducing its radius, DXMAX.
Conversely, if you are far from the minimum, sometimes you
can decrease the number of steps by increasing DXMAX.
When using internals, the program uses an iterative
process to convert the internal coordinate change into
Cartesian space. In some cases, a small change in the
internals will produce a large change in Cartesians, and
thus produce a warning message on the output. If these
warnings appear only in the beginning, there is probably
no problem, but if they persist you can probably devise
a better set of coordinates. You may in fact have one of
the two problems described in the next paragraph. In
some cases (hopefully very few) the iterations to find
the Cartesian displacement may not converge, producing a
second kind of warning message. The fix for this may
very well be a new set of internal coordinates as well,
or adjustment of ITBMAT in $STATPT.
There are two examples of poorly behaved internal
coordinates which can give serious problems. The first
of these is three angles around a central atom, when
this atom becomes planar (sum of the angles nears 360).
The other is a dihedral where three of the atoms are
nearly linear, causing the dihedral to flip between 0 and
180. Avoid these two situations if you want your geometry
search to be convergent.
Sometimes it is handy to constrain the geometry search
by freezing one or more coordinates, via the IFREEZ array.
For example, constrained optimizations may be useful while
trying to determine what area of a potential energy surface
contains a saddle point. If you try to freeze coordinates
with an automatically generated $ZMAT, you need to know
that the order of the coordinates defined in $DATA is
y
y x r1
y x r2 x a3
y x r4 x a5 x w6
y x r7 x a8 x w9
and so on, where y and x are whatever atoms and molecular
connectivity you happen to be using.
saddle points
Finding minima is relatively easy. There are large
tables of bond lengths and angles, so guessing starting
geometries is pretty straightforward. Very nasty cases
may require computation of an exact hessian, but the
location of most minima is straightforward.
In contrast, finding saddle points is a black art.
The diagonal guess hessian will never work, so you must
provide a computed one. The hessian should be computed at
your best guess as to what the transition state (T.S.)
should be. It is safer to do this in two steps as outlined
above, rather than HESS=CALC. This lets you verify you
have guessed a structure with one and only one negative
curvature. Guessing a good trial structure is the hardest
part of a RUNTYP=SADPOINT!
This point is worth iterating. Even with sophisticated
step size control such as is offered by the QA/TRIM or RFO
methods, it is in general very difficult to move correctly
from a region with incorrect curvatures towards a saddle
point. Even procedures such as CONOPT or RUNTYP=GRADEXTR
will not replace your own chemical intuition about where
saddle points may be located.
The RUNTYP=HESSIAN's normal coordinate analysis is
rigorously valid only at stationary points on the surface.
This means the frequencies from the hessian at your trial
geometry are untrustworthy, in particular the six "zero"
frequencies corresponding to translational and rotational
(T&R) degrees of freedom will usually be 300-500 cm**-1,
and possibly imaginary. The Sayvetz conditions on the
printout will help you distinguish the T&R "contaminants"
from the real vibrational modes. If you have defined a
$ZMAT, the PURIFY option within $STATPT will help zap out
these T&R contaminants).
If the hessian at your assumed geometry does not have
one and only one imaginary frequency (taking into account
that the "zero" frequencies can sometimes be 300i!), then
it will probably be difficult to find the saddle point.
Instead you need to compute a hessian at a better guess
for the initial geometry, or read about mode following
below.
If you need to restart your run, do so with the
coordinates which have the smallest RMS gradient. Note
that the energy does not necessarily have to decrease in a
SADPOINT run, in contrast to an OPTIMIZE run. It is often
necessary to do several restarts, involving recomputation
of the hessian, before actually locating the saddle point.
Assuming you do find the T.S., it is always a good
idea to recompute the hessian at this structure. As
described in the discussion of symmetry, only totally
symmetric vibrational modes are probed in a geometry
search. Thus it is fairly common to find that at your
"T.S." there is a second imaginary frequency, which
corresponds to a non-totally symmetric vibration. This
means you haven't found the correct T.S., and are back to
the drawing board. The proper procedure is to lower the
point group symmetry by distorting along the symmetry
breaking "extra" imaginary mode, by a reasonable amount.
Don't be overly timid in the amount of distortion, or the
next run will come back to the invalid structure.
The real trick here is to find a good guess for the
transition structure. The closer you are, the better. It
is often difficult to guess these structures. One way
around this is to compute a linear least motion (LLM)
path. This connects the reactant structure to the product
structure by linearly varying each coordinate. If you
generate about ten structures intermediate to reactants
and products, and compute the energy at each point, you
will in general find that the energy first goes up, and
then down. The maximum energy structure is a "good" guess
for the true T.S. structure. Actually, the success of
this method depends on how curved the reaction path is.
A particularly good paper on the symmetry which a
saddle point (and reaction path) can possess is by
P.Pechukas, J.Chem.Phys. 64, 1516-1521(1976)
mode following
In certain circumstances, METHOD=RFO and QA can walk
from a region of all positive curvatures (i.e. near a
minimum) to a transition state. The criteria for whether
this will work is that the mode being followed should be
only weakly coupled to other close-lying Hessian modes.
Especially, the coupling to lower modes should be almost
zero. In practise this means that the mode being followed
should be the lowest of a given symmetry, or spatially far
away from lower modes (for example, rotation of methyl
groups at different ends of the molecule). It is certainly
possible to follow also modes which do not obey these
criteria, but the resulting walk (and possibly TS location)
will be extremely sensitive to small details such as the
stepsize.
This sensitivity also explain why TS searches often
fail, even when starting in a region where the Hessian has
the required one negative eigenvalue. If the TS mode is
strongly coupled to other modes, the direction of the mode
is incorrect, and the maximization of the energy along
that direction is not really what you want (but what you
get).
Mode following is really not a substitute for the
ability to intuit regions of the PES with a single local
negative curvature. When you start near a minimum, it
matters a great deal which side of the minima you start
from, as the direction of the search depends on the sign
of the gradient. We strongly urge that you read before
trying to use IFOLOW, namely the papers by Frank Jensen
and Jon Baker mentioned above, and see also Figure 3 of
C.J.Tsai, K.D.Jordan, J.Phys.Chem. 97, 11227-11237 (1993)
which is quite illuminating on the sensitivity of mode
following to the initial geometry point.
Note that GAMESS retains all degrees of freedom in its
hessian, and thus there is no reason to suppose the lowest
mode is totally symmetric. Remember to lower the symmetry
in the input deck if you want to follow non-symmetric
modes. You can get a printout of the modes in internal
coordinate space by a EXETYP=CHECK run, which will help
you decide on the value of IFOLOW.
* * *
CONOPT is a different sort of saddle point search
procedure. Here a certain "CONstrained OPTimization" may
be considered as another mode following method. The idea
is to start from a minimum, and then perform a series of
optimizations on hyperspheres of increasingly larger
radii. The initial step is taken along one of the Hessian
modes, chosen by IFOLOW, and the geometry is optimized
subject to the constraint that the distance to the minimum
is constant. The convergence criteria for the gradient
norm perpendicular to the constraint is taken as 10*OPTTOL,
and the corresponding steplength as 100*OPTTOL.
After such a hypersphere optimization has converged, a
step is taken along the line connecting the two previous
optimized points to get an estimate of the next hyper-
sphere geometry. The stepsize is DXMAX, and the radius of
hyperspheres is thus increased by an amount close (but not
equal) to DXMAX. Once the pure NR step size falls below
DXMAX/2 or 0.10 (whichever is the largest) the algorithm
switches to a straight NR iterate to (hopefully) converge
on the stationary point.
The current implementation always conducts the search
in cartesian coordinates, but internal coordinates may be
printed by the usual specification of NZVAR and ZMAT. At
present there is no restart option programmed.
CONOPT is based on the following papers, but the actual
implementation is the modified equations presented in
Frank Jensen's paper mentioned above.
Y. Abashkin, N. Russo,
J.Chem.Phys. 100, 4477-4483(1994).
Y. Abashkin, N. Russo, M. Toscano,
Int.J.Quant.Chem. 52, 695-704(1994).
There is little experience on how this method works in
practice, experiment with it at your own risk!
Intrinsic Reaction Coordinate Methods
The Intrinsic Reaction Coordinate (IRC) is defined as
the minimum energy path connecting the reactants to
products via the transition state. In practice, the IRC is
found by first locating the transition state for the
reaction. The IRC is then found in halves, going forward
and backwards from the saddle point, down the steepest
descent path in mass weighted Cartesian coordinates. This
is accomplished by numerical integration of the IRC
equations, by a variety of methods to be described below.
The IRC is becoming an important part of polyatomic
dynamics research, as it is hoped that only knowledge of
the PES in the vicinity of the IRC is needed for prediction
of reaction rates, at least at threshhold energies. The
IRC has a number of uses for electronic structure purposes
as well. These include the proof that a certain transition
structure does indeed connect a particular set of reactants
and products, as the structure and imaginary frequency
normal mode at the saddle point do not always unambiguously
identify the reactants and products. The study of the
electronic and geometric structure along the IRC is also of
interest. For example, one can obtain localized orbitals
along the path to determine when bonds break or form.
The accuracy to which the IRC is determined is dictated
by the use one intends for it. Dynamical calculations
require a very accurate determination of the path, as
derivative information (second derivatives of the PES at
various IRC points, and path curvature) is required later.
Thus, a sophisticated integration method (such as GS2), and
small step sizes (STRIDE=0.05, 0.01, or even smaller) may
be needed. In addition to this, care should be taken to
locate the transition state carefully (perhaps decreasing
OPTTOL by a factor of 10, to OPTTOL=1D-5), and in the
initiation of the IRC run. The latter might require a
hessian matrix obtained by double differencing, certainly
the hessian should be PROJCT'd or PURIFY'd. Note also that
EVIB must be chosen carefully, as decribed below.
On the other hand, identification of reactants and
products allows for much larger step sizes, and cruder
integration methods. In this type of IRC one might want to
be careful in leaving the saddle point (perhaps STRIDE
should be reduced to 0.10 or 0.05 for the first few steps
away from the transition state), but once a few points have
been taken, larger step sizes can be employed. In general,
the defaults in the $IRC group are set up for this latter,
cruder quality IRC. The STRIDE value for the GS2 method
can usually be safely larger than for other methods, no
matter what your interest in accuracy is.
The next few paragraphs describe the various
integrators, but note that GS2 is superior to the others.
The simplest method of determining an IRC is linear
gradient following, PACE=LINEAR. This method is also known
as Euler's method. If you are employing PACE=LINEAR, you
can select "stabilization" of the reaction path by the
Ishida, Morokuma, Komornicki method. This type of
corrector has no apparent mathematical basis, but works
rather well since the bisector usually intersects the
reaction path at right angles (for small step sizes). The
ELBOW variable allows for a method intermediate to LINEAR
and stabilized LINEAR, in that the stabilization will be
skipped if the gradients at the original IRC point, and at
the result of a linear prediction step form an angle
greater than ELBOW. Set ELBOW=180 to always perform the
stabilization.
A closely related method is PACE=QUAD, which fits a
quadratic polynomial to the gradient at the current and
immediately previous IRC point to predict the next point.
This pace has the same computational requirement as LINEAR,
and is slightly more accurate due to the reuse of the old
gradient. However, stabilization is not possible for this
pace, thus a stabilized LINEAR path is usually more
accurate than QUAD.
Two rather more sophisticated methods for integrating
the IRC equations are the fourth order Adams-Moulton
predictor-corrector (PACE=AMPC4) and fourth order Runge-
Kutta (PACE=RK4). AMPC4 takes a step towards the next IRC
point (prediction), and based on the gradient found at this
point (in the near vincinity of the next IRC point) obtains
a modified step to the desired IRC point (correction).
AMPC4 uses variable step sizes, based on the input STRIDE.
RK4 takes several steps part way toward the next IRC point,
and uses the gradient at these points to predict the next
IRC point. RK4 is one of the most accurate integration
method implemented in GAMESS, and is also the most time
consuming.
The Gonzalez-Schlegel 2nd order method (PACE=GS2) finds
the next IRC point by a constrained optimization on the
surface of a hypersphere, centered at a point 1/2 STRIDE
along the gradient vector leading from the previous IRC
point. By construction, the reaction path between two
successive IRC points is a circle tangent to the two
gradient vectors. The algorithm is much more robust for
large steps than the other methods, so it has been chosen
as the default method. Thus, the default for STRIDE is too
large for the other methods. The number of energy and
gradients need to find the next point varies with the
difficulty of the constrained optimization, but is
normally not very many points. Taking more than 2-3 steps
in this constrained optimization is indicative of reaction
path curvature, and thus it may help to reduce the step
size. Use a small GCUT (same value as OPTTOL) when trying
to integrate an IRC very accurately, to be sure the
hypersphere optimizations are well converged. Be sure to
provide the updated hessian from the previous run when
restarting PACE=GS2.
The number of wavefunction evaluations, and energy
gradients needed to jump from one point on the IRC to the
next point are summarized in the following table:
PACE # energies # gradients
---- ---------- -----------
LINEAR 1 1
stabilized
LINEAR 3 2
QUAD 1 1 (+ reuse of historical
gradient)
AMPC4 2 2 (see note)
RK4 4 4
GS2 2-4 2-4 (equal numbers)
Note that the AMPC4 method sometimes does more than one
correction step, with each such correction adding one more
energy and gradient to the calculation. You get what you
pay for in IRC calculations: the more energies and
gradients which are used, the more accurate the path found.
A description of these methods, as well as some others
that were found to be not as good is geven by Kim Baldridge
and Lisa Pederson, Pi Mu Epsilon J., 9, 513-521 (1993).
* * *
All methods are initiated by jumping from the saddle
point, parallel to the normal mode (CMODE) which has an
imaginary frequency. The jump taken is designed to lower
the energy by an amount EVIB. The actual distance taken is
thus a function of the imaginary frequency, as a smaller
FREQ will produce a larger initial jump. You can simply
provide a $HESS group instead of CMODE and FREQ, which
involves less typing. To find out the actual step taken
for a given EVIB, use EXETYP=CHECK. The direction of the
jump (towards reactants or products) is governed by FORWRD.
Note that if you have decided to use small step sizes, you
must employ a smaller EVIB to ensure a small first step.
The GS2 method begins by following the normal mode by one
half of STRIDE, and then performing a hypersphere
minimization about that point, so EVIB is irrelevant to
this PACE.
The only method which proves that a properly converged
IRC has been obtained is to regenerate the IRC with a
smaller step size, and check that the IRC is unchanged.
Again, note that the care with which an IRC must be
obtained is highly dependent on what use it is intended
for.
Some key IRC references are:
K.Ishida, K.Morokuma, A.Komornicki
J.Chem.Phys. 66, 2153-2156 (1977)
K.Muller
Angew.Chem., Int.Ed.Engl. 19, 1-13 (1980)
M.W.Schmidt, M.S.Gordon, M.Dupuis
J.Am.Chem.Soc. 107, 2585-2589 (1985)
B.C.Garrett, M.J.Redmon, R.Steckler, D.G.Truhlar,
K.K.Baldridge, D.Bartol, M.W.Schmidt, M.S.Gordon
J.Phys.Chem. 92, 1476-1488(1988)
K.K.Baldridge, M.S.Gordon, R.Steckler, D.G.Truhlar
J.Phys.Chem. 93, 5107-5119(1989)
C.Gonzalez, H.B.Schlegel
J.Chem.Phys. 90, 2154-2161(1989)
The IRC discussion closes with some practical tips:
The $IRC group has a confusing array of variables, but
fortunately very little thought need be given to most of
them. An IRC run is restarted by moving the coordinates of
the next predicted IRC point into $DATA, and inserting the
new $IRC group into your input file. You must select the
desired value for NPOINT. Thus, only the first job which
initiates the IRC requires much thought about $IRC.
The symmetry specified in the $DATA deck should be the
symmetry of the reaction path. If a saddle point happens
to have higher symmetry, use only the lower symmetry in the
$DATA deck when initiating the IRC. The reaction path will
have a lower symmetry than the saddle point whenever the
normal mode with imaginary frequency is not totally
symmetric. Be careful that the order and orientation of
the atoms corresponds to that used in the run which
generated the hessian matrix.
If you wish to follow an IRC for a different isotope,
use the $MASS group. If you wish to follow the IRC in
regular Cartesian coordinates, just enter unit masses for
each atom. Note that CMODE and FREQ are a function of the
atomic masses, so either regenerate FREQ and CMODE, or more
simply, provide the correct $HESS group.
Gradient Extremals
This section of the manual, as well as the source code
to trace gradient extremals was written by Frank Jensen of
the University of Aarhus.
A Gradient Extremal (GE) curve consists of points where
the gradient norm on a constant energy surface is
stationary. This is equivalent to the condition that
the gradient is an eigenvector of the Hessian. Such GE
curves radiate along all normal modes from a stationary
point, and the GE leaving along the lowest normal mode
from a minimum is the gentlest ascent curve. This is not
the same as the IRC curve connecting a minimum and a TS,
but may in some cases be close.
GEs may be divided into three groups: those leading
to dissociation, those leading to atoms colliding, and
those which connect stationary points. The latter class
allows a determination of many (all?) stationary points on
a PES by tracing out all the GEs. Following GEs is thus a
semi-systematic way of mapping out stationary points. The
disadvantages are:
i) There are many (but finitely many!) GEs for a
large molecule.
ii) Following GEs is computationally expensive.
iii) There is no control over what type of
stationary point (if any) a GE will lead to.
Normally one is only interested in minima and TSs, but
many higher order saddle points will also be found.
Furthermore, it appears that it is necessary to follow GEs
radiating also from TSs and second (and possibly also
higher) order saddle point to find all the TSs.
A rather complete map of the extremals for the H2CO
potential surface is available in a paper which explains
the points just raised in greater detail:
K.Bondensgaard, F.Jensen,
J.Chem.Phys. 104, 8025-8031(1996).
An earlier paper gives some of the properties of GEs:
D.K.Hoffman, R.S.Nord, K.Ruedenberg,
Theor. Chim. Acta 69, 265-279(1986).
There are two GE algorithms in GAMESS, one due to Sun
and Ruedenberg (METHOD=SR), which has been extended to
include the capability of locating bifurcation points and
turning points, and another due to Jorgensen, Jensen, and
Helgaker (METHOD=JJH):
J. Sun, K. Ruedenberg, J.Chem.Phys. 98, 9707-9714(1993)
P. Jorgensen, H. J. Aa. Jensen, T. Helgaker
Theor. Chim. Acta 73, 55 (1988).
The Sun and Ruedenberg method consist of a predictor
step taken along the tangent to the GE curve, followed by
one or more corrector steps to bring the geometry back to
the GE. Construction of the GE tangent and the corrector
step requires elements of the third derivative of the
energy, which is obtained by a numerical differentiation
of two Hessians. This puts some limitations on which
systems the GE algorithm can be used for. First, the
numerical differentiation of the Hessian to produce third
derivatives means that the Hessian should be calculated by
analytical methods, thus only those types of wavefunctions
where this is possible can be used. Second, each
predictor/corrector step requires at least two Hessians,
but often more. Maybe 20-50 such steps are necessary for
tracing a GE from one stationary point to the next. A
systematic study of all the GE radiating from a stationary
point increases the work by a factor of ~2*(3N-6). One
should thus be prepared to invest at least hundreds, and
more likely thousands, of Hessian calculations. In other
words, small systems, small basis sets, and simple wave-
functions.
The Jorgensen, Jensen, and Helgaker method consists of
taking a step in the direction of the chosen Hessian
eigenvector, and then a pure NR step in the perpendicular
modes. This requires (only) one Hessian calculation for
each step. It is not suitable for following GEs where the
GE tangent forms a large angle with the gradient, and it
is incapable of locating GE bifurcations.
Although experience is limited at present, the JJH
method does not appear to be suitable for following GEs in
general (at least not in the current implementation).
Experiment with it at your own risk!
The flow of the SR algorithm is as follows: A
predictor geometry is produced, either by jumping away
from a stationary point, or from a step in the tangent
direction from the previous point on the GE. At the
predictor geometry, we need the gradient, the Hessian, and
the third derivative in the gradient direction. Depending
on HSDFDB, this can be done in two ways. If .TRUE. the
gradient is calculated, and two Hessians are calculated at
SNUMH distance to each side in the gradient direction.
The Hessian at the geometry is formed as the average of
the two displaced Hessians. This corresponds to a double-
sided differentiation, and is the numerical most stable
method for getting the partial third derivative matrix.
If HSDFDB = .FALSE., the gradient and Hessian are
calculated at the current geometry, and one additional
Hessian is calculated at SNUMH distance in the gradient
direction. This corresponds to a single-sided differen-
tiation. In both cases, two full Hessian calculations are
necessary, but HSDFDB = .TRUE. require one additional
wavefunction and gradient calculation. This is usually
a fairly small price compared to two Hessians, and the
numerically better double-sided differentiation has
therefore been made the default.
Once the gradient, Hessian, and third derivative is
available, the corrector step and the new GE tangent are
constructed. If the corrector step is below a threshold,
a new predictor step is taken along the tangent vector.
If the corrector step is larger than the threshold, the
correction step is taken, and a new micro iteration is
performed. DELCOR thus determines how closely the GE will
be followed, and DPRED determine how closely the GE path
will be sampled.
The construction of the GE tangent and corrector step
involve solution of a set of linear equations, which in
matrix notation can be written as Ax=B. The A-matrix is
also the second derivative of the gradient norm on the
constant energy surface.
After each corrector step, various things are printed
to monitor the behavior: The projection of the gradient
along the Hessian eigenvalues (the gradient is parallel
to an eigenvector on the GE), the projection of the GE
tangent along the Hessian eigenvectors, and the overlap
of the Hessian eigenvectors with the mode being followed
from the previous (optimzed) geometry. The sign of these
overlaps are not significant, they just refer to an
arbitrary phase of the Hessian eigenvectors.
After the micro iterations has converged, the Hessian
eigenvector curvatures are also displayed, this is an
indication of the coupling between the normal modes. The
number of negative eigenvalues in the A-matrix is denoted
the GE index. If it changes, one of the eigenvalues must
have passed through zero. Such points may either be GE
bifurcations (where two GEs cross) or may just be "turning
points", normally when the GE switches from going uphill
in energy to downhill, or vice versa. The distinction is
made based on the B-element corresponding to the A-matrix
eigenvalue = 0. If the B-element = 0, it is a bifurcation,
otherwise it is a turning point.
If the GE index changes, a linear interpolation is
performed between the last two points to locate the point
where the A-matrix is singular, and the corresponding
B-element is determined. The linear interpolation points
will in general be off the GE, and thus the evaluation of
whether the B-element is 0 is not always easy. The
program additionally evaluates the two limiting vectors
which are solutions to the linear sets of equations, these
are also used for testing whether the singular point is a
bifurcation point or turning point.
Very close to a GE bifurcation, the corrector step
become numerically unstable, but this is rarely a problem
in practice. It is a priori expected that GE bifurcation
will occur only in symmetric systems, and the crossing GE
will break the symmetry. Equivalently, a crossing GE may
be encountered when a symmetry element is formed, however
such crossings are much harder to detect since the GE
index does not change, as one of the A-matrix eigenvalues
merely touches zero. The program prints an message if
the absolute value of an A-matrix eigenvalue reaches a
minimum near zero, as such points may indicate the
passage of a bifurcation where a higher symmetry GE
crosses. Run a movie of the geometries to see if a more
symmetric structure is passed during the run.
An estimate of the possible crossing GE direction is
made at all points where the A-matrix is singular, and two
perturbed geometries in the + and - direction are written
out. These may be used as predictor geometries for
following a crossing GE. If the singular geometry is a
turning point, the + and - geometries are just predictor
geometries on the GE being followed.
In any case, a new predictor step can be taken to trace
a different GE from the newly discovered singular point,
using the direction determined by interpolation from the
two end point tangents (the GE tangent cannot be uniquely
determined at a bifurcation point). It is not possible to
determine what the sign of IFOLOW should be when starting
off along a crossing GE at a bifurcation, one will have to
try a step to see if it returns to the bifurcation point
or not.
In order to determine whether the GE index change it
is necessary to keep track of the order of the A-matrix
eigenvalues. The overlap between successive eigenvectors
are shown as "Alpha mode overlaps".
Things to watch out for:
1) The numerical differentiation to get third derivatives
requires more accuracy than usual. The SCF convergence
should be at least 100 times smaller than SNUMH, and
preferably better. With the default SNUMH of 10**(-4)
the SCF convergence should be at least 10**(-6). Since
the last few SCF cycles are inexpensive, it is a good idea
to tighten the SCF convergence as much as possible, to
maybe 10**(-8) or better. You may also want to increase
the integral accuracy by reducing the cutoffs (ITOL and
ICUT) and possibly also try more accurate integrals
(INTTYP=HONDO). The CUTOFF in $TRNSFM may also be reduced
to produce more accurate Hessians. Don't attempt to use a
value for SNUMH below 10**(-6), as you simply can't get
enough accuracy. Since experience is limited at present,
it is recommended that some tests runs are made to learn
the sensitivity of these factors for your system.
2) GEs can be followed in both directions, uphill or
downhill. When stating from a stationary point, the
direction is implicitly given as away from the stationary
point. When starting from a non-stationary point, the "+"
and "-" directions (as chosen by the sign of IFOLOW)
refers to the gradient direction. The "+" direction is
along the gradient (energy increases) and "-" is opposite
to the gradient (energy decreases).
3) A switch from one GE to another may be seen when two
GE come close together. This is especially troublesome
near bifurcation points where two GEs actually cross. In
such cases a switch to a GE with -higher- symmetry may
occur without any indication that this has happened,
except possibly that a very large GE curvature suddenly
shows up. Avoid running the calculation with less
symmetry than the system actually has, as this increases
the likelihood that such switches occuring. Fix: alter
DPRED to avoid having the predictor step close to the
crossing GE.
4) "Off track" error message: The Hessian eigenvector
which is parallel to the gradient is not the same as
the one with the largest overlap to the previous
Hessian mode. This usually indicate that a GE switch
has occured (note that a switch may occur without this
error message), or a wrong value for IFOLOW when starting
from a non-stationary point. Fix: check IFOLOW, if it is
correct then reduce DPRED, and possibly also DELCOR.
5) Low overlaps of A-matrix eigenvectors. Small overlaps
may give wrong assignment, and wrong conclusions about GE
index change. Fix: reduce DPRED.
6) The interpolation for locating a point where one of the
A-matrix eigenvalues is zero fail to converge. Fix:
reduce DPRED (and possibly also DELCOR) to get a shorther
(and better) interpolation line.
7) The GE index changes by more than 1. A GE switch may
have occured, or more than one GE index change is located
between the last and current point. Fix: reduce DPRED to
sample the GE path more closely.
8) If SNRMAX is too large the algorithm may try to locate
stationary points which are not actually on the GE being
followed. Since GEs often pass quite near a stationary
point, SNRMAX should only be increased above the default
0.10 after some consideration.
Continuum Solvation Methods
In a very thorough 1994 review of continuum solvation
models, Tomasi and Persico divide the possible approaches
to the treatment of solvent effects into four categories:
a) virial equations of state, correlation functions
b) Monte Carlo or molecular dynamics simulations
c) continuum treatments
d) molecular treatments
The Effective Fragment Potential method, documented in the
following section of this chapter, falls into the latter
category, as each EFP solvent molecule is modeled as a
distinct object (discrete solvation). This section
describes the four continuum models which are implemented
in the standard version of GAMESS, and a fifth model which
can be interfaced.
Continuum models typically form a cavity of some sort
containing the solute molecule, while the solvent outside
the cavity is thought of as a continuous medium and is
categorized by a limited amount of physical data, such as
the dielectric constant. The electric field of the charged
particles comprising the solute interact with this
background medium, producing a polarization in it, which in
turn feeds back upon the solute's wavefunction.
Self Consistent Reaction Field (SCRF)
A simple continuum model is the Onsager cavity model,
often called the Self-Consistent Reaction Field, or SCRF
model. This represents the charge distribution of the
solute in terms of a multipole expansion. SCRF usually
uses an idealized cavity (spherical or ellipsoidal) to
allow an analytic solution to the interaction energy
between the solute multipole and the multipole which this
induces in the continuum. This method is implemented in
GAMESS in the simplest possible fashion:
i) a spherical cavity is used
ii) the molecular electrostatic potential of the
solute is represented as a dipole only, except
a monopole is also included for an ionic solute.
The input for this implementation of the Kirkwood-Onsager
model is provided in $SCRF.
Some references on the SCRF method are
1. J.G.Kirkwood J.Chem.Phys. 2, 351 (1934)
2. L.Onsager J.Am.Chem.Soc. 58, 1486 (1936)
3. O.Tapia, O.Goscinski Mol.Phys. 29, 1653 (1975)
4. M.M.Karelson, A.R.Katritzky, M.C.Zerner
Int.J.Quantum Chem., Symp. 20, 521-527 (1986)
5. K.V.Mikkelsen, H.Agren, H.J.Aa.Jensen, T.Helgaker
J.Chem.Phys. 89, 3086-3095 (1988)
6. M.W.Wong, M.J.Frisch, K.B.Wiberg
J.Am.Chem.Soc. 113, 4776-4782 (1991)
7. M.Szafran, M.M.Karelson, A.R.Katritzky, J.Koput,
M.C.Zerner J.Comput.Chem. 14, 371-377 (1993)
8. M.Karelson, T.Tamm, M.C.Zerner
J.Phys.Chem. 97, 11901-11907 (1993)
The method is very sensitive to the choice of the solute
RADIUS, but not very sensitive to the particular DIELEC of
polar solvents. The plots in reference 7 illustrate these
points very nicely. The SCRF implementation in GAMESS is
Zerner's Method A, described in the same reference. The
total solute energy includes the Born term, if the solute
is an ion. Another limitation is that a solute's electro-
static potential is not likely to be fit well as a dipole
moment only, for example see Table VI of reference 5 which
illustrates the importance of higher multipoles. Finally,
the restriction to a spherical cavity may not be very
representative of the solute's true shape. However, in the
special case of a roundish molecule, and a large dipole
which is geometry sensitive, the SCRF model may include
sufficient physics to be meaningful:
M.W.Schmidt, T.L.Windus, M.S.Gordon
J.Am.Chem.Soc. 117, 7480-7486(1995).
Most cases should choose PCM (next section) over SCRF!!!
Polarizable Continuum Model (PCM)
A much more sophisticated continuum method, named the
Polarizable Continuum Model, is also available. The PCM
method places a solute in a cavity formed by a union of
spheres centered on each atom. PCM includes a more exact
treatment of the electrostatic interaction of the solute
with the surrounding medium, on the cavity's surface. The
computational procedure divides this surface into many
small tesserae, each having a different "apparent surface
charge", reflecting the solute's and other tesserae's
electric field at each. These surface charges are the PCM
model's "solvation effect" and make contributions to the
energy and to the gradient of the solute.
Typically the cavity is defined as a union of atomic
spheres, which should be roughly 1.2 times the atomic van
der Waals radii. A technical difficulty caused by the
penetration of the solute's charge density outside this
cavity is dealt with by a renormalization. The solvent is
characterized by its dielectric constant, surface tension,
size, density, and so on. Procedures are provided not only
for the computation of the electrostatic interaction of the
solute with the apparent surface charges, but also for the
cavitation energy, and for the dispersion and repulsion
contributions to the solvation free energy.
Methodology for solving the Poisson equation to obtain
the "apparent surface charges" has progressed from D-PCM to
IEF-PCM to C-PCM over time, with the latter preferred.
Iterative solvers require far less computer resources than
direct solvers. Advancements have also been made in
schemes to divide the surface cavity into tiny tesserae.
As of fall 2008, the FIXPVA tessellation, which has smooth
switching functions for tesserae near sphere boundaries,
together with iterative C-PCM, gives very satisfactory
geometry optimizations for molecules of 100 atoms. The
FIXPVA tessellation was extended to work for cavitation
(ICAV), dispersion (IDP), and repulsion (IREP) options in
fall 2009, and dispersion/repulsion (IDISP) in spring 2010.
Other procedures remain, and make the input seem complex,
but their use is discouraged. Thus
$PCM SOLVNT=WATER $END
chooses iterative C-PCM (IEF=-10) and FIXPVA tessellation
(METHOD=4 in $TESCAV) to do basic electrostatics in an
accurate fashion.
The main input group is $PCM, with $PCMCAV providing
auxiliary cavity information. If any of the optional
energy computations are requested in $PCM, the additional
input groups $IEFPCM, $NEWCAV, $DISBS, or $DISREP may be
required.
It is useful to summarize the various cavities used by
PCM, since as many as three cavities may be used:
the basic cavity for electrostatics,
cavity for cavitation energy, if ICAV=1,
cavity for dispersion/repulsion, if IDISP=1.
The first and second share the same radii (see RADII in
$PCMCAV), which are scaled by ALPHA=1.2 for electrostatics,
but are used unscaled for cavitation. The dispersion
cavity is defined in $DISREP with a separate set of atomic
radii, and even solvent molecule radii! Only the
electrostatics cavity can use any of the GEPOL-GB, GEPOL-AS
or recommended FIXPVA tessellation, while the other two use
only the original GEPOL-GB.
Radii are an important part of the PCM parameterization.
Their values can have a significant impact on the quality
of the results. This is particularly true if the solute is
charged, and thus has a large electrostatic interaction
with the continuum. John Emsley's book "The Elements" is a
useful source of van der Waals and other radii.
PCM is at heart a means of treating the electrostatic
interactions between the solute's wavefunction and a
dielectric model for the bulk solvent. The former is
represented as an electron density from whatever quantum
mechanical treatment is used for the solute, and the latter
is a set of surface charges on the finite elements of the
cavity (tessellation). This leaves out other important
contributions to the solvation energy! These include the
energy needed to make a hole in the solvent (cavitation
energy), dispersion or repulsive interactions between the
solute and solvent, and in the SMD model (see below)
solvent structure changes such as would occur in the first
solvation shell. Some empirical formulae for such "CDS"
corrections are provided as keywords ICAV, IDISP, IREP/IDP,
which may not work with all wavefunctions, and may not be
compatible with gradients.
The SMD model gives an alternative set of such "CDS"
corrections, which are compatible with nuclear gradients:
see SMD=.TRUE. in $PCM. A more detailed description of SMD
is given in the paper cited below. The SMD solvent
parameters is described (as of 2010) in
http://comp.chem.umn.edu/solvation/mnsddb.pdf
This gives numerical parameters, all built into GAMESS, as
the various SOLX values, for SOLVNT=
ACETACID CLPROPAN PHOPH EGME E2PENTEN
ACETONE OCLTOLUE DPROAMIN MEACETAT PENTACET
ACETNTRL M-CRESOL DODECAN MEBNZATE PENTAMIN
ACETPHEN O-CRESOL MEG MEBUTATE PFB
ANILINE CYCHEXAN ETSH MEFORMAT BENZALCL
ANISOLE CYCHEXON ETHANOL MIBK PROPANAL
BENZALDH CYCPENTN ETOAC MEPROPYL PROPACID
BENZENE CYCPNTOL ETOME ISOBUTOL PROPANOL
BENZNTRL CYCPNTON EB TERBUTOL PROPNOL2
BENZYLCL DECLNCIS PHENETOL NMEANILN PROPNTRL
BRISOBUT DECLNTRA C6H5F MECYCHEX PROPENOL
BRBENZEN DECLNMIX FOCTANE NMFMIXTR PROPACET
BRETHANE DECANE FORMAMID ISOHEXAN PROPAMIN
BROMFORM DECANOL FORMACID MEPYRID2 PYRIDINE
BROCTANE EDB12 HEPTANE MEPYRID3 C2CL4
BRPENTAN DIBRMETN HEPTANOL MEPYRID4 THF
BRPROPA2 BUTYLETH HEPTNON2 C6H5NO2 SULFOLAN
BRPROPAN ODICLBNZ HEPTNON4 C2H5NO2 TETRALIN
BUTANAL EDC12 HEXADECN CH3NO2 THIOPHEN
BUTACID C12DCE HEXANE NTRPROP1 PHSH
BUTANOL T12DCE HEXNACID NTRPROP2 TOLUENE
BUTANOL2 DCM HEXANOL ONTRTOLU TBP
BUTANONE ETHER HEXANON2 NONANE TCA111
BUTANTRL ET2S HEXENE NONANOL TCA112
BUTILE DIETAMIN HEXYNE NONANONE TCE
NBA MI C6H5I OCTANE ET3N
NBUTBENZ DIPE IOBUTANE OCTANOL TFE222
SBUTBENZ DMDS C2H5I OCTANON2 TMBEN124
TBUTBENZ DMSO IOHEXDEC PENTDECN ISOCTANE
CS2 DMA CH3I PENTANAL UNDECANE
CARBNTET CISDMCHX IOPENTAN NPENTANE M-XYLENE
CLBENZEN DMF IOPROPAN PENTACID O-XYLENE
SECBUTCL DMEPEN24 CUMENE PENTANOL P-XYLENE
CHCL3 DMEPYR24 P-CYMENE PENTNON2 XYLENEMX
CLHEXANE DMEPYR26 MESITYLN PENTNON3
CLPENTAN DIOXANE METHANOL PENTENE
and provides a translation table to full chemical names, if
you can't guess from the input choices given above. The
translations can also be found in the source code. Two
important things to note about SMD are:
a) the atomic radii are changed, so although the
algorithms for electrostatics are those of standard PCM,
the numerical results for the electrostatics do change.
b) SMD's parameterization was developed for IEF-PCM
using GEPOL tessellation with a fine grid: IEF=-3 and
MTHALL=2, NTSALL=240. However it is considered acceptable
to use SMD's parameters, unchanged, with C-PCM and with the
FIXPVA tessellation, at default coarseness. Hence, input
such as
$pcm solvnt=dmso smd=.true. $end
is enough to carry out a SMD-style C-PCM treatment in DMSO.
c) The CDS correction involves cavitation, dispersion,
and as a collective "solvent structure contribution"
estimates for partial hydrogen bonding, repulsion, and
deviation of the dielectric constant from its bulk value.
d) See also SMVLE in the more sophisticated SS(V)PE
continuum model's description.
Solvation of course affects the non-linear optical
properties of molecules. The PCM implementation extends
RUNTYP=TDHF to include solvent effects. Both static and
frequency dependent hyperpolarizabilities can be found.
Besides the standard PCM electrostatic contribution, the
IREP and IDP keywords can be used to determine the effects
of repulsion and dispersion on the polarizabilities.
The implementation of the PCM model in GAMESS has
received considerable attention from Hui Li and Jan Jensen
at the University of Iowa, Iowa State University, and
University of Nebraska. This includes new iterative
techniques to solving the surface charge problem, new
tessellations that provide for numerically stable nuclear
gradients, the implementation of C-PCM equations, the
extension of PCM to all SCFTYPs and TDDFT, development of
an interface with the EFP model (quo vadis), and
heterogenous dielectric. Dmitri Fedorov at AIST has
interfaced PCM to the FMO method (quo vadis), and reduced
storage requirements.
Due to its sophistication, users of the PCM model are
strongly encouraged to read the primary literature:
Of particular relevance to PCM in GAMESS:
1) "Continuum solvation of large molecules described by
QM/MM: a semi-iterative implementation of the PCM/EFP
interface"
H.Li, C.S.Pomelli, J.H.Jensen
Theoret.Chim.Acta 109, 71-84(2003)
2) "Improving the efficiency and convergence of geometry
optimization with the polarizable continuum model: new
energy gradients and molecular surface tessellation"
H.Li, J.H.Jensen J.Comput.Chem. 25, 1449-1462(2004)
3) "The polarizable continuum model interfaced with the
Fragment Molecular Orbital method"
D.G.Fedorov, K.Kitaura, H.Li, J.H.Jensen, M.S.Gordon
J.Comput.Chem. 27, 976-985(2006)
4) "Energy gradients in combined Fragment Molecular Orbital
and Polarizable Continuum Model (FMO/PCM)"
H.Li, D.G.Fedorov, T.Nagata, K.Kitaura, J.H.Jensen,
M.S.Gordon J.Comput.Chem. 31, 778-790(2010)
5) "Continuous and smooth potential energy surface for
conductor-like screening solvation model using fixed points
with variable area"
P.Su, H.Li J.Chem.Phys. 130, 074109/1-13(2009)
6) "Heterogenous conductorlike solvation model"
D.Si, H.Li J.Chem.Phys. 131, 044123/1-8(2009)
7) "Quantum mechanical/molecular mechanical/continuum style
solvation model: linear response theory, variational
treatment, and nuclear gradients"
H.Li J.Chem.Phys. 131, 184103/1-8(2009)
8) "Smooth potential energy surface for cavitation,
dispersion, and repulsion free energies in polarizable
continuum model"
Y.Wang, H.Li J.Chem.Phys. 131, 206101/1-2(2009)
9) "Excited state geometry of photoactive yellow protein
chromophore: a combined conductorlike polarizable continuum
model and time-dependent density functional study"
Y.Wang, H.Li J.Chem.Phys. 133, 034108/1-11(2010)
Paper number 7 is about the treatment of QM systems with
the solvation models EFP and/or C-PCM.
SMD and its CDS cavitation/dispersion/solvent structure
corrections are described in
"Universal solution model based on solute electron density
and on a continuum model of the solvent defined by the bulk
dielectric constant and atomic surface tensions"
A.V.Marenich, C.J.Cramer, D.G.Truhlar
J.Phys.Chem.B 113, 6378-6396(2009)
General papers on PCM:
10) S.Miertus, E.Scrocco, J.Tomasi
Chem.Phys. 55, 117-129(1981)
11) J.Tomasi, M.Persico Chem.Rev. 94, 2027-2094(1994)
12) R.Cammi, J.Tomasi J.Comput.Chem. 16, 1449-1458(1995)
13) J.Tomasi, B.Mennucci, R.Cammi
Chem.Rev. 105, 2999-3093(2005)
The GEPOL-GB method for cavity construction:
14) J.L.Pascual-Ahuir, E.Silla, J.Tomasi, R.Bonaccorsi
J.Comput.Chem. 8, 778-787(1987)
Charge renormalization (see also ref. 12):
15) B.Mennucci, J.Tomasi J.Chem.Phys. 106, 5151-5158(1997)
Derivatives with respect to nuclear coordinates:
(energy gradient and hessian) See also paper 2 and 3.
16) R.Cammi, J.Tomasi J.Chem.Phys. 100, 7495-7502(1994)
17) R.Cammi, J.Tomasi J.Chem.Phys. 101, 3888-3897(1995)
18) M.Cossi, B.Mennucci, R.Cammi
J.Comput.Chem. 17, 57-73(1996)
Derivatives with respect to applied electric fields:
(polarizabilities and hyperpolarizabilities)
19) R.Cammi, J.Tomasi
Int.J.Quantum Chem. Symp. 29, 465-474(1995)
20) R.Cammi, M.Cossi, J.Tomasi
J.Chem.Phys. 104, 4611-4620(1996)
21) R.Cammi, M.Cossi, B.Mennucci, J.Tomasi
J.Chem.Phys. 105, 10556-10564(1996)
22) B. Mennucci, C. Amovilli, J. Tomasi
Chem.Phys.Lett. 286, 221-225(1998)
Cavitation energy:
23) R.A.Pierotti Chem.Rev. 76, 717-726(1976)
24) J.Langlet, P.Claverie, J.Caillet, A.Pullman
J.Phys.Chem. 92, 1617-1631(1988)
Dispersion and repulsion energies:
25) F.Floris, J.Tomasi J.Comput.Chem. 10, 616-627(1989)
26) C.Amovilli, B.Mennucci
J.Phys.Chem.B 101, 1051-1057(1997)
Integral Equation Formalism PCM. The first of these
deals with anisotropies, the last 2 with nuclear gradients.
27) E.Cances, B.Mennucci, J.Tomasi
J.Chem.Phys. 107, 3032-3041(1997)
28) B.Mennucci, E.Cances, J.Tomasi
J.Phys.Chem.B 101, 10506-17(1997)
29) B.Mennucci, R.Cammi, J.Tomasi
J.Chem.Phys. 109, 2798-2807(1998)
30) J.Tomasi, B.Mennucci, E.Cances
J.Mol.Struct.(THEOCHEM) 464, 211-226(1999)
31) E.Cances, B.Mennucci J.Chem.Phys. 109, 249-259(1998)
32) E.Cances, B.Mennucci, J.Tomasi
J.Chem.Phys. 109, 260-266(1998)
Conductor PCM (C-PCM):
33) V.Barone, M.Cossi J.Phys.Chem.A 102, 1995-2001(1998)
34) M.Cossi, N.Rega, G.Scalmani, V.Barone
J.Comput.Chem. 24, 669-681(2003)
C-PCM with TD-DFT:
35) M.Cossi, V.Barone J.Chem.Phys. 115, 4708-4717(2001)
See also paper #8 above for the coding in GAMESS.
At the present time, the PCM model in GAMESS has the
following limitations:
a) Any SCFTYP may be used (RHF to MCSCF). MP2 or DFT
may be used with any of the RHF, UHF, and ROHF
gradient programs. Closed shell TD-DFT excited
state gradients may also be used.
CI and Coupled Cluster programs are not available.
b) semi-empirical methods may not be used.
c) the only other solvent method that may be used at
used with PCM is the EFP model.
d) point group symmetry is switched off internally
during PCM.
e) The PCM model runs in parallel for IEF=3, -3, 10,
or -10 and for all 5 wavefunctions (energy or
gradient), but not for RUNTYP=TDHF jobs.
f) D-PCM stores electric field integrals at normals to
the surface elements on disk.
IEF-PCM and C-PCM using the explicit solver (+3 and
+10) store electric potential integrals at normals
to the surface on disk.
This is true even for direct AO integral runs, and
the file sizes may be considerable (basis set size
squared times the number of tesserae).
IEF-PCM and C-PCM with the iterative solvers do not
store the potential integrals, when IDIRCT=1 in the
$PCMITR group (this is the default)
g) nuclear derivatives are limited to gradients,
although theory for hessians is given in paper 17.
* * *
The only PCM method prior to Oct. 2000 was D-PCM, which
can be recovered by selecting IEF=0 and ICOMP=2 in $PCM.
The default PCM method between Oct. 2000 and May 2004 was
IEF-PCM, recoverable by IEF=-3 (but 3 for non-gradient
runs) and ICOMP=0. As of May 2004, the default PCM method
was changed to C-PCM (IEF=-10, ICOMP=0). The extension of
PCM to all SCFTYPs as of May 2004 involved a correction to
the MCSCF PCM operator, so that it would reproduce RHF
results when run on one determinant, meaning that it is
impossible to reproduce prior MCSCF PCM calculations.
The cavity definition was GEPOL-GB (MTHALL=1 in $TESCAV)
prior to May 2004, GEPOL-AS (MTHALL=2) from then until
September 2008, and FIXPVA (MTHALL=4) to the present time.
The option for generation of 'extra spheres' (RET in $PCM)
was changed from 0.2 to 100.0, to suppress these, in June
2003.
* * *
In general, use of PCM electrostatics is very simple, as
may be seen from exam31.inp supplied with the program.
The calculation shown next illustrates the use of some
of the older PCM options. Since methane is non-polar, its
internal energy change and the direct PCM electrostatic
interaction is smaller than the cavitation, repulsion, and
dispersion corrections. Note that the use of ICAV, IREP,
and IDP are currently incompatible with gradients, so a
reasonable calculation sequence might be to perform the
geometry optimization with PCM electrostatics turned on,
then perform an additional calculation to include the other
solvent effects, adding extra functions to improve the
dispersion correction.
! calculation of CH4 (metano), in PCM water.
!
! This input reproduces the data in Table 2, line 6, of
! C.Amovilli, B.Mennucci J.Phys.Chem.B 101, 1051-7(1997)
! To do this, we must use many original PCM options.
!
! The gas phase FINAL energy is -40.2075980292
! The FINAL energy in PCM water is -40.2048210283
! (lit.)
! FREE ENERGY IN SOLVENT = -25234.89 KCAL/MOL
! INTERNAL ENERGY IN SOLVENT = -25230.64 KCAL/MOL
! DELTA INTERNAL ENERGY = .01 KCAL/MOL ( 0.0)
! ELECTROSTATIC INTERACTION = -.22 KCAL/MOL (-0.2)
! PIEROTTI CAVITATION ENERGY = 5.98 KCAL/MOL ( 6.0)
! DISPERSION FREE ENERGY = -6.00 KCAL/MOL (-6.0)
! REPULSION FREE ENERGY = 1.98 KCAL/MOL ( 2.0)
! TOTAL INTERACTION = 1.73 KCAL/MOL ( 1.8)
! TOTAL FREE ENERGY IN SOLVENT= -25228.91 KCAL/MOL
!
$contrl scftyp=rhf runtyp=energy $end
$guess guess=huckel $end
$system mwords=2 $end
! the "W1 basis" input here exactly matches HONDO's DZP
$DATA
CH4...gas phase geometry...in PCM water
Td
Carbon 6.
DZV
D 1 ; 1 0.75 1.0
Hydrogen 1. 0.6258579976 0.6258579976 0.6258579976
DZV 0 1.20 1.15 ! inner and outer scale factors
P 1 ; 1 1.00 1.0
$END
! The reference cited used a value for H2O's solvent
! radius that differs from the built in value (RSOLV).
! The IEF, ICOMP, MTHALL, and RET keywords are set to
! duplicate the original code's published results,
! namely D-PCM and GEPOL-GB. This run doesn't put in
! any "extra spheres" but we try that option (RET)
! like it originally would have.
$PCM SOLVNT=WATER RSOLV=1.35 RET=0.2
IEF=0 ICOMP=2 IDISP=0 IREP=1 IDP=1 ICAV=1 $end
$TESCAV MTHALL=1 $END
$NEWCAV IPTYPE=2 ITSNUM=540 $END
! dispersion "W2 basis" uses exponents which are
! 1/3 of smallest exponent in "W1 basis" of $DATA.
$DISBS NADD=11 NKTYP(1)=0,1,2, 0,1, 0,1, 0,1, 0,1
XYZE(1)=0.0,0.0,0.0, 0.0511
0.0,0.0,0.0, 0.0382
0.0,0.0,0.0, 0.25
1.1817023, 1.1817023, 1.1817023, 0.05435467
1.1817023, 1.1817023, 1.1817023, 0.33333333
-1.1817023, 1.1817023,-1.1817023, 0.05435467
-1.1817023, 1.1817023,-1.1817023, 0.33333333
1.1817023,-1.1817023,-1.1817023, 0.05435467
1.1817023,-1.1817023,-1.1817023, 0.33333333
-1.1817023,-1.1817023, 1.1817023, 0.05435467
-1.1817023,-1.1817023, 1.1817023, 0.33333333 $end
SVPE and SS(V)PE.
The Surface Volume Polarization for Electrostatics
(SVPE), and an approximation to SVPE called the Surface and
Simulation of Volume Polarization for Electrostatics
(SS(V)PE) are continuum solvation models. Compared to
other continuum models, SVPE and SS(V)PE pay careful
attention to the problems of escaped charge, the shape of
the surface cavity, and to integration of the Poisson
equation for surface charges.
The original references for what is now called the
SVPE (surface and volume polarization for electrostatics)
method are the theory paper:
"Charge penetration in Dielectric Models of Solvation"
D.M.Chipman, J.Chem.Phys. 106, 10194-10206 (1997)
and two implementation papers:
"Volume Polarization in Reaction Field Theory"
C.-G.Zhan, J.Bentley, D.M.Chipman
J.Chem.Phys. 108, 177-192 (1998)
"New Formulation and Implementation for Volume
Polarization in Dielectric Continuum Theory"
D.M.Chipman, J.Chem.Phys. 124, 224111-1/10 (2006)
which should be cited in any publications that utilize the
SVPE code.
There are two options to include with SS(V)PE or SVPE
additional models that describe short-range solute-solvent
interactions to achieve a more complete description of
solvation energies. Both have their keywords merged into
the $SVP input group for convenience. One option to
include short-range interactions is CMIRS (Composite
Method for Implicit Representation of Solvent)
that combines the SS(V)PE dielectric continuum model with
the DEFESR (Dispersion, Exchange, and Field-Extremum Short-
Range) model.
A complete account of the original version
labeled CMIRS1.0 with application to hydration energies is
given in:
Hydration Energy from a Composite Method for Implicit
Representation of Solvent
A.Pomogaeva, D.M.Chipman
J.Chem.Theory Comput. 10, 211-219(2014)
which should be referenced in any publication that uses the
model. Applications to DMSO and acetonitrile solvents are
reported in
Composite Method for Implicit Representation of
Solvent in Diemthyl Sulfoxide and Acetonitrile
A.Pomogaeva, D.M. Chipman
J.Phys.Chem.A 119, 5173-5180(2015)
References to earlier works that develop the individual
components of the DEFESR parts of the full CMIRS recipe are:
Modeling short-range contributions to hydration
energies with minimal parameterization
A.Pomogaeva, D.W.Thompson, and D.M.Chipman
Chem.Phys.Lett. 511, 161-165(2011).
Field-Extremum Model for Short-Range Contributions
to Hydration Free Energy
A.Pomogaeva and D.M.Chipman
J.Chem.Theory Comput. 7, 3952-3960(2011).
New Implicit Solvation Models for Dispersion and
Exchange Energies
A.Pomogaeva and D.M.Chipman
J.Phys.Chem.A, 117, 5812-5820 (2013).
An error in the original CMIRS1.0 code for dispersion
energy has been reported in
Reparameterization of an Accurate, Few-Parameter
Implicit Solvation Model for Quantum Chemistry:
Composite Method for Implicit Representation of
Solvent, CMIRS v. 1.1
Zhi-Qiang You, John M. Herbert,
J.Chem.Theor.Comp. 12, 4338-4346 (2016)
This paper defines the CMIRS1.1 method giving revised
parameters for use with water, cyclohexane, benzene, DMSO,
and acetonitrile solvents obtained with the B3LYP/6-31+G*
method in conjunction with isodensity contours of 0.005
and 0.001 au. This error has been corrected in the current
Gamess code.
The other option to include short-range interactions is
the SMVLE (solvation model with volume and local
electrostatics) as described in
"Free energies of solvation with surface, volume, and
local electrostatic effects and atomic surface
tensions to represent the first solvation shell"
J.Liu, C.P.Kelly, A.C.Goren, A.V.Marenich,
C.J.Cramer, D.G.Truhlar, C.-G. Zhan
J.Chem.Theory Comput. 6, 1109-1117(2010).
Further information on the performance of SVPE and of
SS(V)PE can be found in:
"Comparison of Solvent Reaction Field Representations"
D.M.Chipman, Theor.Chem.Acc. 107, 80-89 (2002).
Details of the SS(V)PE convergence behavior and programming
strategy are in:
"Implementation of Solvent Reaction Fields for
Electronic Structure" D.M.Chipman, M.Dupuis,
Theor.Chem.Acc. 107, 90-102 (2002).
The SMVLE option (solvation model with volume and
local electrostatics) is described in
"Free energies of solvation with surface, volume, and
local electrostatic effects and atomic surface tensions to
represent the first solvation shell" J.Liu, C.P.Kelly,
A.C.Goren, A.V.Marenich, C.J.Cramer, D.G.Truhlar, C.-G.
Zhan J.Chem.Theory Comput. 6, 1109-1117(2010).
The SVPE and SS(V)PE models are like PCM and COSMO in
that they treat solvent as a continuum dielectric residing
outside a molecular-shaped cavity, determining the apparent
charges that represent the polarized dielectric by solving
Poisson's equation. The main difference between SVPE and
SS(V)PE is in treatment of volume polarization effects that
arise because of the tail of the electronic wave function
that penetrates outside the cavity, sometimes referred to
as the "escaped charge." SVPE treats volume polarization
effects explicitly by including apparent charges in the
volume outside the cavity as well as on the cavity surface.
With a sufficient number of grid points, SVPE can then
provide an exact treatment of charge penetration effects.
SS(V)PE, like PCM and COSMO, is an approximate treatment
that only uses apparent charges located on the cavity
surface. The SS(V)PE equation is particularly designed to
simulate as well as possible the influence of the missing
volume charges. For more information on the similarities
and differences of the SVPE and SS(V)PE models with other
continuum methods, see the paper "Comparison of Solvent
Reaction Field Representations" cited just above.
In addition, the cavity construction and Poisson
solver used in this implementation of SVPE and SS(V)PE also
receive careful numerical treatment. For example, the
cavity may be chosen to be an isodensity contour surface,
and the Lebedev grids for the Poisson solver can be chosen
very densely. The Lebedev grids used for surface
integration are taken from the Fortran translation by C.
van Wuellen of the original C programs developed by D.
Laikov. They were obtained from the CCL web site
www.ccl.net/cca/software/SOURCES/FORTRAN/Lebedev-Laikov-
Grids. A recent leading reference is V. I. Lebedev and D.
N. Laikov, Dokl. Math. 59, 477-481 (1999). All these grids
have octahedral symmetry and so are naturally adapted for
any solute having an Abelian point group. The larger and/or
the less spherical the solute may be, the more surface
points are needed to get satisfactory precision in the
results. Further experience will be required to develop
detailed recommendations for this parameter. Values as
small as 110 are usually sufficient for simple diatomics
and triatomics. The default value of 1202 has been found
adequate to obtain the energy to within 0.1 kcal/mol for
solutes the size of monosubstituted benzenes. The SVPE
method uses additional layers of points outside the cavity.
Typically just two layers are sufficient to converge the
direct volume polarization contribution to better than 0.1
kcal/mol.
The SVPE and SS(V)PE codes both report the amount of
solute charge penetrating outside the cavity as calculated
by Gauss' Law. The SVPE code additionally reports the same
quantity as alternatively calculated from the explicit
volume charges, and any substantial discrepancy between
these two determinations indicates that more volume
polarization layers should have been included for better
precision. The energy contribution from the outermost
volume polarization layer is also reported. If it is
significant then again more layers should have been
included. However, these tests are only diagnostic.
Passing them does not guarantee that enough layers are
included.
Analytic nuclear gradients are not yet available for
the SVPE or SS(V)PE energy, but numerical differentiation
will permit optimization of small solute molecules.
Wavefunctions may be any of the SCF type: RHF, UHF, ROHF,
GVB, and MCSCF, or the DFT analogs of some of these. In the
MCSCF implementation, no initial wavefunction is available
so the solvation code does not kick in until the second
iteration.
We close with a SVPE example. The gas phase energy,
obtained with no $SVP group, is -207.988975, and the run
just below gives the SVPE energy -208.006282. The free
energy of solvation, -10.860 kcal/mole, is the difference
of these, and is quoted at the right side of the 3rd line
from the bottom of Table 2 in the paper cited. The
"REACTION FIELD FREE ENERGY" for SVPE is -12.905 kcal/mole,
which is only part of the solvation free energy. There is
also a contribution due to the SCRF procedure polarizing
the wave function from its gas phase value, causing the
solute internal energy in dielectric to differ from that in
gas. Evaluating this latter contribution is what requires
the separate gas phase calculation. Changing the number of
layers (NVLPL) to zero produces the SS(V)PE approximation
to SVPE, E= -208.006208.
! SVPE solvation test...acetamide
! reproduce data in Table 2 of the paper on SVPE,
! D.M.Chipman J.Chem.Phys. 124, 224111/1-10(2006)
!
$contrl scftyp=rhf runtyp=energy $end
$system mwords=4 $end
$basis gbasis=n31 ngauss=6 ndfunc=1 npfunc=1 $end
$guess guess=moread norb=16 $end
$scf nconv=8 $end
$svp nvlpl=3 rhoiso=0.001 dielst=78.304 nptleb=1202 $end
$data
CH3CONH2 cgz geometry RHF/6-31G(d,p)
C1
C 6.0 1.361261 -0.309588 -0.000262
C 6.0 -0.079357 0.152773 -0.005665
H 1.0 1.602076 -0.751515 0.962042
H 1.0 1.537200 -1.056768 -0.767127
H 1.0 2.002415 0.542830 -0.168045
O 8.0 -0.387955 1.310027 0.002284
N 7.0 -1.002151 -0.840834 -0.011928
H 1.0 -1.961646 -0.589397 0.038911
H 1.0 -0.752774 -1.798630 0.035006
$end
gas phase vectors, E(RHF)= -207.9889751769
$VEC
1 1 1.18951670E-06 1.74015997E-05
...snipped...
$END
The example just above can be changed to a CMIRS run, by
changing NVLPL to zero and also requesting calculation of
the DEFESR contributions. The result should be a CMIRS1.1
total free energy of -208.013009. The input should
(1) in $CONTRL, specify DFTTYP=HFX, which requests a
Hartree-Fock calculation, but sets up DFT grids,
(2) in $DFT, request a very accurate grid
$dft method=grid nrad=96 nleb=1202 nrad0=96 nleb0=1202 $end
(3) in $SVP, invoke the keyword IDEF=1.
Adding the keyword EGAS=-207.9889751769 to any of
these examples allows reporting of the free energy of
solvation (-10.860 kcal/mol for SVPE, -10.814 for SS(V)PE,
-15.081 for CMIRS1.1). Note that this example invokes the
HF/6-31G** method for the wavefunction while inconsistently
utilizing CMIRS1.1 solvation parameters that were instead
optimized for use with the B3LYP/6-31+G* method. The latter
method gives much better agreement with experiment.
Conductor-like screening model (COSMO)
The COSMO (conductor-like screening model) represents a
different approach for carrying out polarized continuum
calculations. COSMO was originally developed by Andreas
Klamt, with extensions to ab initio computation in GAMESS
by Kim Baldridge.
In the COSMO method, the surrounding medium is modeled
as a conductor rather than as a dielectric in order to
establish the initial boundary conditions. The assumption
that the surrounding medium is well modeled as a conductor
simplifies the electrostatic computations and corrections
may be made a posteriori for dielectric behavior.
The original model of Klamt was introduced using a
molecular shaped cavity, which had open parts along the
crevices of intersecting atomic spheres. While having
considerable technical advantages, this approximation
causes artifacts in the context of the more generalized
theory, so the current method for cavity construction
includes a closure of the cavity to eliminate crevices or
pockets.
Two methodologies are implemented for treatment of the
outlying charge errors (OCE). The default is the well-
established double cavity procedure using a second, larger
cavity around the first one, and calculates OCE through the
difference between the potential on the inner and the outer
cavity. The second involves the calculation of distributed
multipoles up to hexadecapoles to represent the entire
charge distribution of the molecule within the cavity.
The COSMO model accounts only for the electrostatic
interactions between solvent and solute. Klamt has
proposed a novel statistical scheme to compute the full
solvation free energy for neutral solutes, COSMO-RS, which
is formulated for GAMESS by Peverati, Potier and Baldridge,
and is available as external plugin to the COSMOtherm
program by COSMOlogic GmbH&Co.
The iterative inclusion of non-electrostatic effects is
also possible right after a COSMO-RS calculation. The
DCOSMO-RS approach was implemented in GAMESS by Peverati,
Potier, and Baldridge, and more information is available on
Baldridge website at:
http://ocikbws.uzh.ch/gamess/
The simplicity of the COSMO model allows computation of
gradients, allowing optimization within the context of the
solvent. The method is programmed for RHF and UHF, all
corresponding kinds of DFT (including DFT-D), and the
corresponding MP2, energy and gradient.
Some references on the COSMO model are:
A.Klamt, G.Schuurman
J.Chem.Soc.Perkin Trans 2, 799-805(1993)
A.Klamt J.Phys.Chem. 99, 2224-2235(1995)
K.Baldridge, A.Klamt
J.Chem.Phys. 106, 6622-6633 (1997)
V.Jonas, K.Baldridge
J.Chem.Phys. 113, 7511-7518 (2000)
L.Gregerson, K.Baldridge
Helv.Chim.Acta 86, 4112-4132 (2003)
R.Peverati, Y.Potier, K.Baldridge
TO BE PUBLISHED SOON
Additional references on the COSMO-RS model, with
explanation of the methodology and program can be found:
A.Klamt, F.Eckert, W.Arlt
Annu.Rev.Chem.Biomol.Eng. 1, (2010)
The Effective Fragment Potential Method
The basic idea behind the effective fragment potential
(EFP) method is to replace the chemically inert part of a
system by EFPs, while performing a regular ab initio
calculation on the chemically active part. Here "inert"
means that no covalent bond breaking process occurs. This
"spectator region" consists of one or more "fragments",
which interact with the ab initio "active region" through
non-bonded interactions, and so of course these EFP
interactions affect the ab initio wavefunction. The EFP
particles can be closed shell or open shell (high spin
ROHF) based potentials. The "active region" can use nearly
every kind of wavefunction available in GAMESS.
A simple example of an active region might be a solute
molecule, with a surrounding spectator region of solvent
molecules represented by fragments. Each discrete solvent
molecule is represented by a single fragment potential, in
marked contrast to continuum models for solvation.
The quantum mechanical part of the system is entered in
the $DATA group, along with an appropriate basis. The EFPs
defining the fragments are input by means of a $EFRAG
group, and one or more $FRAGNAME groups describing each
fragment's EFP. These groups define non-bonded
interactions between the ab initio system and the
fragments, and also between the fragments. The former
interactions enter via one-electron operators in the ab
initio Hamiltonian, while the latter interactions are
treated by analytic functions. The only electrons
explicitly treated (with basis functions used to expand
occupied orbitals) are those in the active region, so there
are no new two electron terms. Thus the use of EFPs leads
to significant time savings, compared to full ab initio
calculations on the same system.
There are two types of EFP available in GAMESS, EFP1 and
EFP2. EFP1, the original method, employs a fitted
repulsive potential. EFP1 is primarily used to model water
molecules to study aqueous solvation effects, at the
RHF/DZP or DFT/DZP (specifically, B3LYP) levels, see
references 1-3 and 26, respectively. EFP2 is a more
general method that is applicable to any species, including
water, and its repulsive potential is obtained from first
principles. EFP2 has been extended to include other
effects as well, such as charge transfer and dispersion.
EFP2 forms the basis of the covalent EFP method described
below for modeling enzymes, see reference 14.
Parallelization of the EFP1 and EFP2 models is described
in reference 32.
MD simulations with EFP are described in reference 31.
The ab initio/EFP1, or pure EFP system can be wrapped in
a Polarizable Continuum Model, see references 23, 43, and
50.
terms in an EFP
The non-bonded interactions currently implemented are:
1) Coulomb interaction. The charge distribution of the
fragments is represented by an arbitrary number of charges,
dipoles, quadrupoles, and octopoles, which interact with
the ab initio hamiltonian as well as with multipoles on
other fragments (see reference 2 and 18). It is possible
to use a screening term that accounts for the charge
penetration (reference 17 and 42). This screening term is
automatically included for EFP1. Typically the multipole
expansion points are located on atomic nuclei and at bond
midpoints.
2) Dipole polarizability. An arbitrary number of dipole
polarizability tensors can be used to calculate the induced
dipole on a fragment due to the electric field of the ab
initio system as well as all the other fragments. These
induced dipoles interact with the ab initio system as well
as the other EFPs, in turn changing their electric fields.
All induced dipoles are therefore iterated to self-
consistency. Typically the polarizability tensors are
located at the centroid of charge of each localized orbital
of a fragment. See reference 41.
3) Repulsive potential. Two different forms are used in
EFP1: one for ab initio-EFP repulsion and one for EFP-EFP
repulsion. The form of the potentials is empirical, and
consists of distributed Gaussian or exponential functions,
respectively. The primary contribution to the repulsion is
the quantum mechanical exchange repulsion, but the fitting
technique used to develop this term also includes the
effects of charge transfer. Typically these fitted
potentials are located on each atomic nucleus within the
fragment (see reference 3). In EFP2, polarization energies
can also be augmented by screening terms, analogous to the
electrostatic screening, to prevent "polarization collapse"
(MS in preparation)
For EFP2, the third term is divided into separate analytic
formulae for different physical interactions:
a) exchange repulsion
b) dispersion
c) charge transfer
A summary of EFP2, and its contrast to EFP1 can be found in
reference 18 and 44. The repulsive potential for EFP2 is
based on an overlap expansion using localized molecular
orbitals, as described in references 5, 6, and 9.
Dispersion energy is described in reference 34, and charge
transfer in reference 39 (which supercedes reference 22's
formulae).
EFP2 potentials have no fitted parameters, and can be
automatically generated during a RUNTYP=MAKEFP job, as
described below.
constructing an EFP1
RUNTYP=MOROKUMA assists in the decomposition of inter-
molecular interaction energies into electrostatic,
polarization, charge transfer, and exchange repulsion
contributions. This is very useful in developing EFPs
since potential problems can be attributed to a particular
term by comparison to these energy components for a
particular system.
A molecular multipole expansion can be obtained using
$ELMOM. A distributed multipole expansion can be obtained
by either a Mulliken-like partitioning of the density
(using $STONE) or by using localized molecular orbitals
($LOCAL: DIPDCM and QADDCM). The dipole polarizability
tensor can be obtained during a Hessian run ($CPHF), and a
distributed LMO polarizability expression is also available
($LOCAL: POLDCM).
In EFP1, the repulsive potential is derived by fitting
the difference between ab initio computed intermolecular
interaction energies, and the form used for Coulomb and
polarizability interactions. This difference is obtained
at a large number of different interaction geometries, and
is then fitted. Thus, the repulsive term is implicitly a
function of the choices made in representing the Coulomb
and polarizability terms. Note that GAMESS currently does
not provide a way to obtain these EFP1 repulsive potential.
Since a user cannot generate all of the EFP1 terms
necessary to define a new $FRAGNAME group using GAMESS, in
practice the usage of EFP1 is limited to the internally
stored H2ORHF or H2ODFT potentials mentioned below.
constructing an EFP2
As noted above, the repulsive potential for EFP2 is
derived from a localized orbital overlap expansion. It is
generally recommended that one use at least a double zeta
plus diffuse plus polarization basis set, e.g. 6-31++G(d,p)
to generate the EFP2 repulsive potential. However, it has
been observed that 6-31G(d) works reasonably well due to a
fortuitous cancellation of errors. The EFP2 potential for
any molecule can be generated as follows:
(a) Choose a basis set and geometry for the molecule of
interest. The geometry is ordinarily optimized at your
choice of Hartree-Fock/MP2/CCSD(T), with your chosen basis
set, but this is not a requirement. It is good to recall,
however, that EFP internal geometries are fixed, so it is
important to give some thought to the chosen geometry.
(b) Perform a RUNTYP=MAKEFP run for the chosen molecule
using the chosen geometry in $DATA and the chosen basis set
in $BASIS. This will generate the entire EFP2 potential in
the run's .efp file. The only user-defined variable that
must be filled in is changing the FRAGNAME's group name, to
$C2H5OH or $DMSO, etc. This step can use RHF or ROHF to
describe the electronic structure of the system.
(c) Transfer the entire fragment potential for the molecule
to any input file in which this fragment is to be used.
Since the internal geometry of an EFP is fixed, one need
only specify the first three atoms of any fragment in order
to position them in $EFRAG. Coordinates of any other atoms
in the rigid fragment will be automatically determined by
the program.
If the EFP contains less than three atoms, you can still
generate a fragment potential. After a normal MAKEFP run,
add dummy atoms (e.g. in the X and/or Y directions) with
zero nuclear charges, and add corresponding dummy bond
midpoints too. Carefully insert zero entries in the
multipole sections, and in the electrostatic screening
sections, for each such dummy point, but don't add data to
any other kind of EFP term such as polarizability. This
trick gives the necessary 3 points for use in $EFRAG groups
to specify "rotational" positions of fragments.
current limitations
1. For EFP1, the energy and energy gradient are programmed,
which permits RUNTYP=ENERGY, GRADIENT, and numerical
HESSIAN. The necessary programing to use the EFP gradients
to move on the potential surface are programmed for
RUNTYP=OPTIMIZE, SADPOINT, IRC, and VSCF, but the other
gradient based potential surface explorations such as DRC
are not yet available. Finally, RUNTYP=PROP is also
permissible.
For EFP2, the gradient terms for ab initio-EFP interactions
have not yet been coded, so geometry optimizations are only
sensible for a COORD=FRAGONLY run; that is, a run in which
only EFP2 fragments are present.
2. The ab initio part of the system must be treated with
RHF, ROHF, UHF, the open shell SCF wavefunctions permitted
by the GVB code, or MCSCF. DFT analogs of RHF, ROHF, and
UHF may also be used. Correlated methods such as MP2 and
CI should not be used.
3. EFPs can move relative to the ab initio system and
relative to each other, but the internal structure of an
EFP is frozen.
4. The boundary between the ab initio system and EFP1's
must not be placed across a chemical bond. However, see
the discussion below regarding covalent bonds.
5. Calculations must be done in C1 symmetry at present.
6. Reorientation of the fragments and ab initio system is
not well coordinated. If you are giving Cartesian
coordinates for the fragments (COORD=CART in $EFRAG), be
sure to use $CONTRL's COORD=UNIQUE option so that the ab
initio molecule is not reoriented.
7. If you need IR intensities, you have to use NVIB=2. The
potential surface is usually very soft for EFP motions, and
double differenced Hessians should usually be obtained.
practical hints for using EFPs
At the present time, we have only two internally stored
EFP potentials suitable for general use. These model
water, using the fragment name H2ORHF or H2ODFT. The
H2ORHF numerical parameters are improved values over the
values which were presented and used in reference 2, and
they also include the improved EFP-EFP repulsive term
defined in reference 3. The H2ORHF water EFP was derived
from RHF/DH(d,p) computations on the water dimer system.
When you use it, therefore, the ab initio part of your
system should be treated at the SCF level, using a basis
set of the same quality (ideally DH(d,p), but probably
other DZP sets such as 6-31G(d,p) will give good results as
well). Use of better basis sets than DZP with this water
EFP has not been tested. Similarly, H2ODFT was developed
using B3LYP/DZP water wavefunctions, so this should be used
(rather than H2ORHF) if you are using DFT to treat the
solute. Since H2ODFT water parameters are obtained from a
correlated calculation, they can also be used when the
solute is treated by MP2.
As noted, effective fragments have frozen internal
geometries, and therefore only translate and rotate with
respect to the ab initio region. An EFP's frozen
coordinates are positioned to the desired location(s) in
$EFRAG as follows:
a) the corresponding points are found in $FRAGNAME.
b) Point -1- in $EFRAG and its FRAGNAME equivalent are
made to coincide.
c) The vector connecting -1- and -2- is aligned with
the corresponding vector connecting FRAGNAME points.
d) The plane defined by -1-, -2-, and -3- is made to
coincide with the corresponding FRAGNAME plane.
Therefore the 3 points in $EFRAG define only the relative
position of the EFP, and not its internal structure. So, if
the "internal structure" given by points in $EFRAG differs
from the true values in $FRAGNAME, then the order in which
the points are given in $EFRAG can affect the positioning
of the fragment. It may be easier to input water EFPs if
you use the Z-matrix style to define them, because then you
can ensure you use the actual frozen geometry in your
$EFRAG. Note that the H2ORHF EFP uses the frozen geometry
r(OH)=0.9438636, a(HOH)=106.70327, and the names of its 3
fragment points are ZO1, ZH2, ZH3.
* * *
Building a large cluster of EFP particles by hand can be
tedious. The RUNTYP=GLOBOP program described below has an
option for constructing dense clusters. The method tries
to place particles near the origin, but not colliding with
other EFP particles already placed there, so that the
clusters grow outwards from the center. Here are some
ideas:
a) place 100 water molecules, all with the same coords
in $EFRAG. This will build up a droplet of water
with particles close together, but not on top of
each other, with various orientations.
b) place 16 waters (same coords, all first) followed by
16 methanols (also sharing their same coords, after
all waters). A 50-50 mixture of 32 molecules will
be created, if you choose the default of picking the
particles randomly from the initial list of 32.
c) to solvate a solute, add the solute in the $DATA
group at or near the origin. Add the solvent
molecules near by (same coords is ok), and run
the globop run with RNDINI as demonstrated below.
(optional, add MCTYP=3 to $GLOBOP input)
Example, allowing the random cluster to have 20 geometry
optimization steps:
$contrl runtyp=globop coord=fragonly $end
$globop rndini=.true. riord=rand mcmin=.true.
mctyp=4 nblock=0 $end
$statpt nstep=20 $end
$efrag
coord=cart
FRAGNAME=WATER
O1 -2.8091763203009 -2.1942725073400 -0.2722207394107
H2 -2.3676165499399 -1.6856118830379 -0.9334073942601
H3 -2.1441965467625 -2.5006167998896 0.3234583094693
...repeat this 15 more times...
FRAGNAME=MeOH
O1 4.9515153249 .4286994611 1.3368662306
H2 5.3392575544 .1717424606 3.0555957053
C3 6.2191743799 2.5592349960 .4064662379
H4 5.7024200977 2.7548960076 -1.5604873643
H5 5.6658856694 4.2696553371 1.4008542042
H6 8.2588049857 2.3458272252 .5282762681
...repeat 15 more times...
$end
$water
...give a full EFP2 potential for water...
$end
$meoh
...give a full EFP2 potential for methanol...
$end
Note that the random cluster generation now proceeds into a
full Monte Carlo simulation.
* * *
The translations and rotations of EFPs with respect to
the ab initio system and one another are automatically
quite soft degrees of freedom. After all, the EFP model is
meant to handle weak interactions! Therefore the
satisfactory location of structures on these flat surfaces
will require use of a tight convergence on the gradient:
OPTTOL=0.00001 in the $STATPT group.
The effect of a bulk continuum surrounding the solute
plus EFP waters can be obtained by using the PCM model, see
reference 23 and 43. To do this, simply add a $PCM group
to your input, in addition to the $EFRAG. The simultaneous
use of EFP and PCM allows for gradients, so geometry
optimization can be performed.
global optimization
If there are a large number of particles to move (EFP
and/or FMO and/or atom groups), it is difficult to locate
the lowest energy structures by hand. Typically these are
numerous, and one would like to have a number of them, not
just the very lowest energy. The RUNTYP of GLOBOP contains
a Monte Carlo procedure to generate a random set of
starting structures to look for those with the lowest
energy at a single temperature. If desired, a simulated
annealing protocol to cool the temperature may be used.
These two procedures may be combined with a local minimum
search, at some or all of the randomly generated
structures. The local minimum search is controlled by the
usual geometry optimizer, namely $STATPT input, and thus
permits the optimization of any ab initio atoms.
The Monte Carlo procedure by default uses a Metropolis
algorithm to move just one of the fragments. The method of
Parks to move all fragments simultaneously is also allowed.
The present program was used to optimize the structure
of water clusters. Let us consider the case of the twelve
water cluster, for which the following ten structures were
published by Day, Pachter, Gordon, and Merrill:
1. (D2d)2 -0.170209 6. (D2d)(C2) -0.167796
2. (D2d)(S4) -0.169933 7. S6 -0.167761
3. (S4)2 -0.169724 8. cage b -0.167307
4. D3 -0.168289 9. cage a -0.167284
5. (C1c)(Cs) -0.167930 10. (C1c)(C1c) -0.167261
A test input using Metropolis style Monte Carlo to examine
300 geometries at each temperature value, using simulated
annealing cooling from 200 to 50 degrees, and with local
minimization every 10 structures was run ten times. Each
run sampled about 7000 geometries. One simulation found
structure 2, while two of the runs found structure 3. The
other seven runs located structures with energy values in
the range -0.163 to -0.164. In all cases the runs began
with the same initial geometry, but produced different
results due to the random number generation used in the
Monte Carlo. Clearly one must try a lot of simulations to
be confident about having found most of the low energy
structures. In particular, it is good to try more than one
initial structure, unlike what was done in this test.
Ab initio atoms can be addressed using FMO, either in
multiple fragments, or perhaps a single large fragment.
Alternatively, ab initio atoms can be put into groups and
used directly in globop, which for small systems has a
lower overhead than FMO. In the case of large molecules
separated into multiple fragments, the keywords NPRBND,
PRSEP, IBNDS, and INDEP are applicable. These specify the
atoms in each set of fragments or groups whose bond is cut
in the fragmentation process. The paired atoms are
constrained during the Monte Carlo procedure to ensure that
the bond is not spacially broken. In the case where a
fragment that is being translated or rotated is paired with
two or more fragments, the movement is repeated on all
attached fragments, after randomly choosing which pair is
the starting point. For example, given a molecule split
into five fragments such that:
A-B-C-D-E
where A,B,C,D,E are the fragments. If C is chosen for a
translation, either B or D will be randomly chosen to be
the starting pair. When B is chosen as the starting pair,
C, D, and E will all be translated by the same amount:
A-B--C-D-E
which maintains the relative position of C, D, and E.
Setting INDEP=1 will not propagate the translation:
A-B--CD-E
So that only C is moved.
The same approach is used for rotations. Since a small
translation or rotation can result in a significant change
in the total system, it is advised that case be taken when
using solvent molecules and to the size of boundary
conditions. If a propagated movement moves a fragment
outside the boundary, a warning will be printed and the
step will be discarded as a proximity alert. Also, the
pair binding is not implemented for RNDINI=.TRUE. To
initialize a set of solvent molecules around pair bonded
fragments, include the pair bonded fragments in IFXFMO.
The Metrpolis Monte Carlo procedure involves the movement
of groups that are internally rigid. To introduce some
internal flexibility for FMO and ab initio groups, a
secondary Monte Carlo search where the entire system is
held rigid while the atoms in one group are moved is
implemented. The secondary Monte Carlo occurs when a FMO
or ab initio group is translated and occurs for that group.
The lowest energy internal configuration for the secondary
Monte Carlo is used when evaluating the step of the primary
Monte Carlo search. The temperature at which the secondary
Monte Carlo is used in the case of simulated annealing is
set by SMTEMP and the number of steps in each secondary
search is given by NSMTP. To turn on this feature, set the
values of SMTEMP and NSMTP to non-zero values.
Monte Carlo references:
N.Metropolis, A.Rosenbluth, A.Teller
J.Chem.Phys. 21, 1087(1953).
G.T.Parks Nucl.Technol. 89, 233(1990).
Monte Carlo with local minimization:
Z.Li, H.A.Scheraga
Proc.Nat.Acad.Sci. USA 84, 6611(1987).
Simulated annealing reference:
S.Kirkpatrick, C.D.Gelatt, M.P.Vecci
Science 220, 671(1983).
The present program is described in reference 15. It is
patterned on the work of
D.J.Wales, M.P.Hodges Chem.Phys.Lett. 286, 65-72 (1998).
QM/MM across covalent bonds
Recent work by Visvaldas Kairys and Jan Jensen has made
it possible to extend the EFP methodology beyond the simple
solute/solvent case described above. When there is a
covalent bond between the portion of the system to be
modeled by quantum mechanics, and the portion which is to
be treated by EFP multipole and polarizability terms, an
additional layer is needed in the model. The covalent
linkage is not so simple as the interactions between closed
shell solute and solvent molecules. The "buffer zone"
between the quantum mechanics and the EFP consists of
frozen nuclei, and frozen localized orbitals, so that the
quantum mechanical region sees a orbital representation of
the closest particles, and multipoles etc. beyond that.
Since the orbitals in the buffer zone are frozen, it need
extend only over a few atoms in order to keep the orbitals
in the fully optimized quantum region within that region.
The general outline of this kind of computation is as
follows:
a) a full quantum mechanics computation on a system
containing the quantum region, the buffer region,
and a few atoms into the EFP region, to obtain the
frozen localized orbitals in the buffer zone.
This is called the "truncation run".
b) a full quantum mechanics computation on a system
with all quantum region atoms removed, and with
the frozen localized orbitals in the buffer zone.
The necessary multipole and polarizability data
to construct the EFP that will describes the EFP
region will be extracted from the wavefunction.
This is called the "MAKEFP run". It is possible
to use several such runs if the total EFP region
is quite large.
c) The intended QM/MM run(s), after combining the
information from these first two types of runs.
As an example, consider a protonated lysine residue
which one might want to consider quantum mechanically in a
protein whose larger parts are to be treated with an EFP.
The protonated lysine is
NH2
+ /
H3N(CH2)(CH2)(CH2)--(CH2)(CH)
\
COOH
The bonds which you see drawn show how the molecule is
partitioned between the quantum mechanical side chain, a
CH2CH group in the buffer zone, and eventually two
different EFPs may be substituted in the area of the NH2
and COOH groups to form the protein backbone.
The "truncation run" will be on the entire system as you
see it, with the 13 atoms in the side chain first in $DATA,
the 5 atoms in the buffer zone next in $DATA, and the
simplified EFP region at the end. This run will compute
the full quantum wavefunction by RUNTYP=ENERGY, followed by
the calculation of localized orbitals, and then truncation
of the localized orbitals that are found in the buffer zone
so that they contain no contribution from AOs outside the
buffer zone. The key input groups for this run are
$contrl
$truncn doproj=.true. plain=.true. natab=13 natbf=5 $end
This will generate a total of 6 localized molecular
orbitals in the buffer zone (one CC, three CH, two 1s inner
shells), expanded in terms of atomic orbitals located only
on those atoms.
The truncation run prepares template input files for
the next run, including adjustments of nuclear charges at
boundaries, etc.
The "MAKEFP" run drops all 13 atoms in the quantum
region, and uses the frozen orbitals just prepared to
obtain a wavefunction for the EFP region. The carbon atom
in the buffer zone that is connected to the now absent QM
region will have its nuclear charge changed from 6 to 5 to
account for a missing electron. The key input for this
RUNTYP=MAKEFP job is the six orbitals in $VEC, plus the
groups
$guess guess=huckel insorb=6 $end
$mofrz frz=.true. ifrz(1)=1,2,3,4,5,6 $end
$stone
QMMMbuf
$end
which will cause the wavefunction optimization for the
remaining atoms to optimize orbitals only in the NH2 and
COOH pieces. After this wavefunction is found, the run
extracts the EFP information needed for the QM/MM third
run(s). This means running the Stone analysis for
distributed multipoles, and obtaining a polarizability
tensor for each localized orbital in the EFP region.
The QM/MM run might be RUNTYP=OPTIMIZE, etc. depending
on what you want to do with the quantum atoms, and its
$DATA group will contain both the 13 fully optimized atoms,
and the 5 buffer atoms, and a basis set will exist on both
sets of atoms. The carbon atom in the buffer zone that
borders the EFP region will have its nuclear charge set to
4 since now two bonding electrons to the EFP region are
lost. $VEC input will provide the six frozen orbitals in
the buffer zone. The EFP atoms are defined in a fragment
potential group.
The QM/MM run could use RHF or ROHF wavefunctions, to
geometry optimize the locations of the quantum atoms (but
not of course the frozen buffer zone or the EFP piece). It
could remove the proton to compute the proton affinity at
that terminal nitrogen, hunt for transition states, and so
on. Presently the gradient for GVB and MCSCF is not quite
right, so their use is discouraged.
Input to control the QM/MM preparation is $TRUNCN and
$MOFRZ groups. There are a number of other parameters in
various groups, namely QMMMBUF in $STONE, MOIDON and POLNUM
in $LOCAL, NBUFFMO in $EFRAG, and INSORB in $GUESS that are
relevant to this kind of computation. For RUNTYP=MAKEFP,
the biggest choices are LOCAL=RUEDENBRG vs. BOYS, and
POLNUM in $LOCAL, otherwise this is pretty much a standard
RUNTYP=ENERGY input file.
Source code distributions of GAMESS contain a directory
named ~/gamess/tools/efp, which has various tools for EFP
manipulation in it, described in file readme.1st. A full
input file for the protonated lysine molecule is included,
with instructions about how to proceed to the next steps.
Tips on more specialized input possibilities are appended
to the file readme.1st.
Simpler potentials
Since the EFP model's electrostatics is a set of
distributed multipoles (monopole to octopole) and
distributed polarizabilities (dipole), it is possible to
generate some water potentials found in the literature by
setting many EFP terms to zero. It is also necessary to
provide a Lennard-Jones 6-12 repulsive potential, and then
make a choice to follow the EFP1 type formula for QM/EFP
repulsion. Accordingly, EFP1 type calculations can be made
with the following water potentials,
FRAGNAME=SPC, SPCE, TIP5P, TIP5PE, or POL5P
The Wikipedia page
http://en.wikipedia.org/wiki/Water_model
defines the first four of these, which are not polarizable
potentials. The same web site references the primary
literature, so that is not repeated here. POL5P is a
polarizable potential, with parameters given by
D.Si and H.Li J.Chem.Phys. 133, 144112/1-8(2010)
references
The first paper is more descriptive, while the second
presents a very detailed derivation of the EFP1 method.
Reference 18 is an overview article on EFP2. Reference 44
is the most recent review.
The model development papers are: 1, 2, 3, 5, 6, 9, 14,
17, 18, 22, 23, 26, 31, 32, 34, 39, 41, 42, 43, 44, 46, 50,
51, 55, 57, 58.
1. "Effective fragment method for modeling intermolecular
hydrogen bonding effects on quantum mechanical
calculations"
J.H.Jensen, P.N.Day, M.S.Gordon, H.Basch, D.Cohen,
D.R.Garmer, M.Krauss, W.J.Stevens in "Modeling the
Hydrogen Bond" (D.A. Smith, ed.) ACS Symposium Series
569, 1994, pp 139-151.
2. "An effective fragment method for modeling solvent
effects in quantum mechanical calculations".
P.N.Day, J.H.Jensen, M.S.Gordon, S.P.Webb,
W.J.Stevens, M.Krauss, D.Garmer, H.Basch, D.Cohen
J.Chem.Phys. 105, 1968-1986(1996).
3. "The effective fragment model for solvation: internal
rotation in formamide"
W.Chen, M.S.Gordon, J.Chem.Phys., 105, 11081-90(1996)
4. "Transphosphorylation catalyzed by ribonuclease A:
Computational study using ab initio EFPs"
B.D.Wladkowski, M. Krauss, W.J.Stevens
J.Am.Chem.Soc. 117, 10537-10545(1995)
5. "Modeling intermolecular exchange integrals between
nonorthogonal orbitals"
J.H.Jensen J.Chem.Phys. 104, 7795-7796(1996)
6. "An approximate formula for the intermolecular Pauli
repulsion between closed shell molecules"
J.H.Jensen, M.S.Gordon Mol.Phys. 89, 1313-1325(1996)
7. "A study of aqueous glutamic acid using the effective
fragment potential model"
P.N.Day, R.Pachter J.Chem.Phys. 107, 2990-9(1997)
8. "Solvation and the excited states of formamide"
M.Krauss, S.P.Webb J.Chem.Phys. 107, 5771-5(1997)
9. "An approximate formula for the intermolecular Pauli
repulsion between closed shell molecules. Application
to the effective fragment potential method"
J.H.Jensen, M.S.Gordon
J.Chem.Phys. 108, 4772-4782(1998)
10. "Study of small water clusters using the effective
fragment potential method"
G.N.Merrill, M.S.Gordon J.Phys.Chem.A 102, 2650-7(1998)
11. "Solvation of the Menshutkin Reaction: A Rigourous
test of the Effective Fragement Model"
S.P.Webb, M.S.Gordon J.Phys.Chem.A 103, 1265-73(1999)
12. "Evaluation of the charge penetration energy between
nonorthogonal molecular orbitals using the Spherical
Gaussian Overlap approximation"
V.Kairys, J.H.Jensen
Chem.Phys.Lett. 315, 140-144(1999)
13. "Solvation of Sodium Chloride: EFP study of NaCl(H2O)n"
C.P.Petersen, M.S.Gordon
J.Phys.Chem.A 103, 4162-6(1999)
14. "QM/MM boundaries across covalent bonds: frozen LMO
based approach for the Effective Fragment Potential
method"
V.Kairys, J.H.Jensen J.Phys.Chem.A 104, 6656-65(2000)
15. "A study of water clusters using the effective fragment
potential and Monte Carlo simulated annealing"
P.N.Day, R.Pachter, M.S.Gordon, G.N.Merrill
J.Chem.Phys. 112, 2063-73(2000)
16. "A combined discrete/continuum solvation model:
Application to glycine" P.Bandyopadhyay, M.S.Gordon
J.Chem.Phys. 113, 1104-9(2000)
17. "Evaluation of charge penetration between distributed
multipolar expansions"
M.A.Freitag, M.S.Gordon, J.H.Jensen, W.J.Stevens
J.Chem.Phys. 112, 7300-7306(2000)
18. "The Effective Fragment Potential Method: a QM-based MM
approach to modeling environmental effects in
chemistry"
M.S.Gordon, M.A.Freitag, P.Bandyopadhyay, J.H.Jensen,
V.Kairys, W.J.Stevens J.Phys.Chem.A 105, 293-307(2001)
19. "Accurate Intraprotein Electrostatics derived from
first principles: EFP study of proton affinities of
lysine 55 and tyrosine 20 in Turkey Ovomucoid"
R.M.Minikis, V.Kairys, J.H.Jensen
J.Phys.Chem.A 105, 3829-3837(2001)
20. "Active site structure & mechanism of Human Glyoxalase"
U.Richter, M.Krauss J.Am.Chem.Soc. 123, 6973-6982(2001)
21. "Solvent effect on the global and atomic DFT-based
reactivity descriptors using the EFP model. Solvation
of ammonia." R.Balawender, B.Safi, P.Geerlings
J.Phys.Chem.A 105, 6703-6710(2001)
22. "Intermolecular exchange-induction and charge transfer:
Derivation of approximate formulas using nonorthogonal
localized molecular orbitals."
J.H.Jensen J.Chem.Phys. 114, 8775-8783(2001)
23. "An integrated effective fragment-polarizable continuum
approach to solvation: Theory & application to glycine"
P.Bandyopadhyay, M.S.Gordon, B.Mennucci, J.Tomasi
J.Chem.Phys. 116, 5023-5032(2002)
24. "The prediction of protein pKa's using QM/MM: the pKa
of Lysine 55 in turkey ovomucoid third domain"
H.Li, A.W.Hains, J.E.Everts, A.D.Robertson, J.H.Jensen
J.Phys.Chem.B 106, 3486-3494(2002)
25. "Computational studies of aliphatic amine basicity"
D.C.Caskey, R.Damrauer, D.McGoff
J.Org.Chem. 67, 5098-5105(2002)
26. "Density Functional Theory based Effective Fragment
Potential" I.Adamovic, M.A.Freitag, M.S.Gordon
J.Chem.Phys. 118, 6725-6732(2003)
27. "Intraprotein electrostatics derived from first
principles: Divid-and-conquer approaches for QM/MM
calculations" P.A.Molina, H.Li, J.H.Jensen
J.Comput.Chem. 24, 1971-1979(2003)
28. "Formation of alkali metal/alkaline earth cation water
clusters, M(H2O)1-6, M=Li+, K+, Mg+2, Ca+2: an
effective fragment potential caase study"
G.N.Merrill, S.P.Webb, D.B.Bivin
J.Phys.Chem.A 107, 386-396(2003)
29. "Anion-water clusters A-(H2O)1-6, A=OH, F, SH, Cl, and
Br. An effective fragment potential test case"
G.N.Merrill, S.P.Webb
J.Phys.Chem.A 107,7852-7860(2003)
30. "The application of the Effective Fragment Potential to
molecular anion solvation: a study of ten oxyanion-
water clusters, A-(H2O)1-4"
G.N.Merrill, S.P.Webb J.Phys.Chem.A 108, 833-839(2004)
31. "The effective fragment potential: small clusters and
radial distribution functions"
H.M.Netzloff, M.S.Gordon J.Chem.Phys. 121, 2711-4(2004)
32. "Fast fragments: the development of a parallel
effective fragment potential method"
H.M.Netzloff, M.S.Gordon
J.Comput.Chem. 25, 1926-36(2004)
33. "Theoretical investigations of acetylcholine (Ach) and
acetylthiocholine (ATCh) using ab initio and effective
fragment potential methods"
J.Song, M.S.Gordon, C.A.Deakyne, W.Zheng
J.Phys.Chem.A 108, 11419-11432(2004)
34. "Dynamic polarizability, dispersion coefficient C6, and
dispersion energy in the effective fragment potential
method"
I.Adamovic, M.S.Gordon Mol.Phys. 103, 379-387(2005)
35. "Solvent effects on the SN2 reaction: Application of
the density functional theory-based effective fragment
potential method"
I.Adamovic, M.S.Gordon J.Phys.Chem.A 109, 1629-36(2005)
36. "Theoretical study of the solvation of fluorine and
chlorine anions by water"
D.D.Kemp, M.S.Gordon J.Phys.Chem.A 109, 7688-99(2005)
37. "Modeling styrene-styrene interactions"
I.Adamovic, H.Li, M.H.Lamm, M.S.Gordon
J.Phys.Chem.A 110, 519-525(2006)
38. "Methanol-water mixtures: a microsolvation study using
the Effective Fragment Potential method"
I.Adamovic, M.S.Gordon
J.Phys.Chem.A 110, 10267-10273(2006)
39. "Charge transfer interaction in the effective fragment
potential method" H.Li, M.S.Gordon, J.H.Jensen
J.Chem.Phys. 124, 214108/1-16(2006)
40. "Incremental solvation of nonionized and zwitterionic
glycine"
C.M.Aikens, M.S.Gordon
J.Am.Chem.Soc. 128, 12835-12850(2006)
41. "Gradients of the polarization energy in the Effective
Fragment Potential method"
H.Li, H.M.Netzloff, M.S.Gordon
J.Chem.Phys. 125, 194103/1-9(2006)
42. "Electrostatic energy in the Effective Fragment
Potential method: Theory and application to benzene
dimer"
L.V.Slipchenko, M.S.Gordon
J.Comput.Chem. 28, 276-291(2007)
43. "Polarization energy gradients in combined Quantum
Mechanics, Effective Fragment Potential, and
Polarizable Continuum Model Calculations"
H.Li, M.S.Gordon J.Chem.Phys. 126, 124112/1-10(2007)
44. "The Effective Fragment Potential: a general method
for predicting intermolecular interactions"
M.S.Gordon, L.V.Slipchenko, H.Li, J.H.Jensen
Annual Reports in Computational Chemistry, Volume 3,
pp 177-193 (2007).
45. "An Interpretation of the Enhancement of the Water
Dipole Moment Due to the Presence of Other Water
Molecules"
D.D.Kemp, M.S.Gordon
J.Phys.Chem.A 112, 4885-4894(2008)
46. "Solvent effects on optical properties of molecules: a
combined time-dependent density functional/effective
fragment potential approach"
S.Yoo, F.Zahariev, S.Sok, M.S.Gordon
J.Chem.Phys. 129, 144112/1-8(2008)
47. "Modeling pi-pi interactions with the effective
fragment potential method: The benzene dimer and
substituents"
T.Smith, L.V.Slipchenko, M.S.Gordon
J.Phys.Chem.A 112, 5286-5294(2008)
48. "Water-benzene interactions: An effective fragment
potential and correlated quantum chemistry study"
L.V.Slipchenko, M.S.Gordon
J.Phys.Chem.A 113, 2092-2102(2009)
49. "Ab initio QM/MM excited-state molecular dynamics study
of Coumarin 151 in water solution"
D.Kina, P.Arora, A.Nakayama, T.Noro, M.S.Gordon,
T.Taketsugu Int.J.Quantum Chem. 109, 2308-2318(2009)
50. "Damping functions in the effective fragment potential
method
L.V.Slipchenko, M.S.Gordon
Mol.Phys. 197, 999-1016 (2009)
51. "A combined effective fragment potential-fragment
molecular orbital method. 1. the energy expression"
T.Nagata, D.G.Fedorov, K.Kitaura, M.S.Gordon
J.Chem.Phys. 131, 024101/1-12(2009)
52. "Alanine: then there was water"
J.M.Mullin, M.S.Gordon
J.Phys.Chem.B 113, 8657-8669(2009)
53. "Water and Alanine: from puddles(32) to ponds(49)"
J.M.Mullin, M.S.Gordon
J.Phys.Chem.B 113, 14413-14420(2009)
54. "Structure of large nitrate-water clusters at ambient
temperatures: simulations with effective fragment
potentials and force fields with implications for
atmospheric chemistry"
Y.Miller, J.L.Thoman, D.D.Kemp, B.J.Finlayson-Pitts,
M.S.Gordon, D.J.Tobias, R.B.Gerber
J.Phys.Chem.A 113, 12805-12814(2009)
55. "Quantum mechanical/molecular mechanical/continuum
style solvation model: linear response theory,
variational treatment, and nuclear gradients"
H.Li J.Chem.Phys. 131, 184103/1-8(2009)
56. "Aqueous solvation of bihalide anions"
D.D.Kemp, M.S.Gordon
J.Phys.Chem.A 114, 1298-1303(2010)
57. "Exchange repulsion between effective fragment
potentials and ab initio molecules"
D.D.Kemp, J.M.Rintelman, M.S.Gordon, J.H.Jensen
Theoret.Chem.Acc. 125, 481-491(2010).
58. Modeling Solvent Effects on Electronic Excited States
A.DeFusco, N.Minezawa, L.V.Slipchenko, F.Zahariev,
M.S.Gordon J.Phys.Chem.Lett. 2, 2184-2192(2011)
The Fragment Molecular Orbital method
coded by D.G. Fedorov, M.Chiba, T. Nagata and K. Kitaura at
Research Institute for Computational Sciences (RICS)
National Institute of Advanced Industrial Science
and Technology (AIST)
AIST Tsukuba Central 2, Umezono 1-1-1,
Tsukuba, 305-8568, Japan.
with code contributions by:
N. Asada (Kyoto U.), C. H. Choi (Kyungpook U.),
C. Steinmann (U. Copenhagen).
The method was proposed by Professor Kitaura and coworkers
in 1999, based on the Energy Decomposition Analysis (EDA,
sometimes called the Morokuma-Kitaura energy
decomposition). The FMO method is completely independent of
and bears no relation to:
1. Frontier molecular orbitals (FMO),
2. Fragment molecular orbitals (FMO).
The latter name is often used for the process of
construction of full molecular orbitals by combining MO
diagrams for parts of a molecule, ala Roald Hoffmann.
The effective fragment molecular orbital method (EFMO) is
closely related to but also bears significant difference to
FMO, and discussed below.
The FMO program was interfaced with GAMESS and follows
general GAMESS guidelines for code distribution and usage.
The users of the FMO program are requested to cite the
FMO3-RHF paper as the basic FMO reference,
D.G. Fedorov, K. Kitaura,
J. Chem. Phys. 120, 6832-6840(2004)
and other papers as appropriate (see below).
The basic idea of the method is to acknowledge the fact the
exchange and self-consistency are local in most molecules
(and clusters and molecular crystals), which permits
treating remote parts with Coulomb operators only, ignoring
the exchange. This idea further evolves into doing
molecular calculations, piecewise, with Coulomb fields due
to the remaining parts. In practice one divides the
molecule into fragments and performs n-mer calculations of
these in the Coulomb field of other fragments (n=1,2,3).
There are no empirical parameters, and the only departure
from ab initio rigor is the subjective fragmentation. It
has been observed that if performed physically reasonably,
the fragmentation scheme alters the results very little.
What changes the accuracy the most is the fragment size,
which also determines the computational efficiency of the
method.
The first question is how to get started. The easiest way
to prepare an FMO input file for GAMESS is to use the free
GUI software Facio, developed by M. Suenaga at Kyushu
University. It can do molecular modeling, automatic
fragmentation of peptides, nucleotides and saccharides and
create GAMESS/FMO input files:
http://www1.bbiq.jp/zzzfelis/Facio.html
A web bsed interface to FMO is maintained by Y. Alexeev
(Argonne National Lab):
http://www.fmo-portal.info (shut down at present)
Alternatively, if you prefer a command line interface, and
your molecule is a protein found in the PDB
http://www.rcsb.org/pdb
you can simply use the fragmentation program "fmoutil" that
is provided with GAMESS in tools/fmo, or the FMO home page
http://staff.aist.go.jp/d.g.fedorov/fmo/main.html
If you have a cluster of identical molecules, you can
perform fragmentation with just one keyword ($FMO NACUT=).
Computationally, it is always better to partition in a
geometrical way (close parts together), so that the
distance-based approximations are more efficient. The
accuracy depends mainly upon the locality of the density
distribution, and the appropriateness of partitioning it
into fragments. There is no simple connexion between the
geometrical proximity of fragmentation and accuracy.
Supposing you know how to fragment, you should choose a
basis set and fragment size. We recommend 2 amino acid
residues or 2-4 water molecules per fragment for final
energetics (or, even better, three-body with 1 molecule or
residue per fragment). For geometry optimizations one may
be able to use 1 res/mol per fragment, especially if
gradient convergence to about 0.001 is desired. Note that
although it was claimed that FMO gradient is analytic
(Chem. Phys. Lett., 336 (2001), 163.) it is not so. Neither
theory nor program for fully analytic gradient has been
developed, to the best of our knowledge up to this day
(December 21, 2006). The gradient implementation is nearly
analytic, meaning three small terms are missing, one which
can now be included using MODGRD=8+2. The magnitude of
these small terms depends upon the fragment size (larger
fragments have smaller errors). It has been our experience
that in proteins with 1 residue per fragment one gets 1e-
3...1e-4 error in the gradient, and with 2 residues per
fragment it is about 1e-4...1e-5. If you experience energy
rising during geometry optimizations, you can consider two
countermeasures:
1. increase approximation thresholds, e.g. RESPPC from
2.0->2.5, RESDIM from 2.0 -> 2.5.
2. increase fragment size (e.g. by merging very small
fragments with their neighbors).
Finally a word of caution: optimizing systems with charged
fragments in the absence of solvent is frequently not a
good idea: oppositely charged fragments will most likely
seek each other, unless there is some conformational
barrier.
For basis sets you should use general guidelines and your
experience developed for ab initio methods. There is a file
provided (HMOs.txt) that contains hybrid molecular orbitals
(HMO) used to divide the MO space along fragmentation
points at covalent bonds. If your basis set is not there
you need to construct your own set of HMOs. See the example
file makeLMO.inp for this purpose.
Next you choose a wave function type. At present one can
use RHF, DFT, MP2, CC, and MCSCF (all except MCSCF support
the 3-body expansion). Geometry optimization can be
performed with all of these methods, except CC.
Note that presence of $FMO turns FMO on.
Surfaces and solids
Until 2008, for treating covalently connected fragments,
FMO had fully relaxed electron density of the detached
bonds. This method is now known as FMO/HOP (HOP=hybrid
orbital projection operator). It allows for a full
polarization of the system and is thus well suited to very
polar systems, such as proteins with charged residues. In
2008, an alternative fragmentation was suggested, based on
adaptive frozen orbitals (AFO), FMO/AFO. In it, the
electron density for each detached bond is first computed
in the automatically generated small model system (with the
bond intact), and in the FMO fragment calculations this
electron density is frozen. It was found that FMO/AFO works
quite well for surfaces and solids, where there is a dense
network of bonds to be detached in order to define
fragments (and the detached bonds interact quite strongly).
In addition, by restricting the polarization, FMO/AFO was
found to give a more balanced properties for large basis
sets (triple-zeta with polarization or larger), or in
comparing different isomers. However, for proteins with
charged residues the original FMO/HOP scheme has a better
accuracy (except large basis sets). At this point, FMO/AFO
was applied to zeolites only, and some more experience is
needed to give more practical advice to applications.
FMO/AFO is turned on by a nonzero rafo(1) parameter (rafo
array provides the thresholds to build model systems).
FMO variants
In 2007, Dahlke et al. introduced the Electrostatically
Embedded Many-Body Expansion method (see E. E. Dahlke and
D. G. Truhlar, J. Chem. Theory Comput. 4, 1-6 (2008) for
more recent work). This method is essentially FMO with the
RESPPC approximation (point charges for the electrostatic
field) applied to all fragments, with the further provision
that these charges may be defined at will (whereas RESPPC
uses Mulliken charges), and they are kept frozen (not
optimized, as in FMO). Next, Kamiya et al. suggested a fast
electron correlation method (M. Kamiya, S. Hirata, M.
Valiev, J. Chem. Phys. 128, 074103 (2008)), where again FMO
with the RESPPC approximation to all fragments is applied
with the further provision that the charges are derived
from the electrostatic potential (so called ESP charges),
and BSSE correction is added. The Dahlke's method was
generalized in GAMESS with the introduction of an arbitrary
hybrid approach, in which some fragments may have fixed and
some variationally optimized charges. This implementation
was employed in FMO-TDDFT calculations of solid state
quinacridone (see Ref. 16 below) by using DFT/PBC frozen
charges. The present energy only implementation is mostly
intended for such cases as that (i.e., TDDFT), and some
more work is needed to finish it for general calculations.
To turn this on, set RESPPC=-1 and define NOPFRG for frozen
charge fragments to 64, set frozen charges in ATCHRG.
Another FMO-like method is EFMO, see its own subsection
below. EFMO itself is related to several methods (PMISP: P.
Soederhjelm, U. Ryde, J. Phys. Chem. A 2009, 113, 617?627;
another is G. J. O. Beran, J. Chem. Phys. 2009, 130,
164115).
Effective fragment molecular orbital method (EFMO)
EFMO has been formulated by combining the physical models
in EFP and FMO, namely, in EFMO, fragments are computed
without the ESP (of FMO), and the polarization is estimated
using EFP models of fragment polarizabilities, which are
computed on the fly, so this can be thought of as
automatically generated potentials in EFP. Consequently,
close dimers are computed quantum-mechanically (without
ESP) and far dimers are computed using the electrostatic
multipole models of EFP. At present, only vacuum closed-
shell RHF and DFT are supported, for energy and gradient;
and only molecular clusters can be computed (no systems
with detached bonds). From the user point of view, EFMO
functionality is very intensively borrowed from FMO, and
the calculation setup is almost identical. Most additional
physical models such as PCM are not supported in EFMO. EFMO
should not be confused with FMO/EFP. The latter uses FMO
for some fragments and EFP for others. EFMO uses the same
model (EFMO), which is neither FMO nor EFP. For
approximations, EFMO at present has only RESDIM.
EFMO references are:
1. Effective Fragment Molecular Orbital Method: A Merger of
the Effective Fragment Potential and Fragment Molecular
Orbital Methods.
C. Steinmann, D. G. Fedorov, J. H. Jensen
J. Phys. Chem. A 114, 8705-8712 (2010).
2. The Effective Fragment Molecular Orbital Method for
Fragments Connected by Covalent Bonds.
C. Steinmann, D. G. Fedorov, J. H. Jensen
PLoS One, 7, e41117(2012).
3. Mapping enzymatic catalysis using the effective fragment
molecular orbital method: towards all ab initio
biochemistry.
C. Steinmann, D. G. Fedorov, J. H. Jensen
PLoS One 8, e60602 (2013).
Guidelines for approximations with FMO3
Three sets are suggested, for various accuracies:
low: resppc=2.5 resdim=2.5 ritrim(1)=0.001,-1,1.25
medium: resppc=2.5 resdim=3.25 ritrim(1)=1.25,-1,2.0
high: resppc=2.5 resdim=4.0 ritrim(1)=2,2,2
For correlated runs, add one more value to ritrim, equal to
the third element (i.e., 1.25 or 2.0). Note that gradient
runs do not support nonzero RESDIM and thus use RESDIM=0 if
gradient is to be computed. The "low" level of accuracy
for FMO3 has an error versus full ab initio similar to
FMO2, except for extended basis sets (6-311G** etc) where
it is substantially better than FMO2. Thus the low level is
only recommended for those large basis sets, and if a
better level cannot be afforded. The medium level is
recommended for production FMO3 runs; the high level is
mostly for accuracy evaluation in FMO development. The
cost is roughly: 3(low), 6(medium), 12(high). This means
that FMO3 with the medium level takes roughly six times
longer than FMO2.
Some of the default tolerances were changed as of January
2009, when FMO 3.2 was included in GAMESS. In general,
stricter parameters are now enforced when using FMO3, which
of course is intended to produce more accurate results. If
you wish to reproduce earlier results with the new code,
use the input to revert to the earlier values:
former -> FMO2 or FMO3 (as of 1/2009)
RESPPC: 2.0 2.0 2.50
RESDIM: 2.0 2.0 3.25
RCORSD: 2.0 2.0 3.25
RITRIM: 2.0,2.0,2.0,2.0 -> 1.25,-1.0,2.0,2.0 (FMO3 only)
MODESP: 1 0 1
MODGRD: 0 10 0
and two other settings which are not strictly speaking FMO
keywords may change FMO results:
MTHALL: 2 -> 4 (FMO/PCM only, see $TESCAV)
DFT grid: spherical -> Lebedev (FMO-DFT only, see $DFT)
Note that FMO2 energies printed during a FMO3 run will
differ from those in a FMO2 run, due to the different
tolerances used.
How to perform FMO-MCSCF calculations
Assuming that you are reasonably acquainted with ab initio
MCSCF, only FMO-specific points are highlighted. The active
space (the number of orbitals/electrons) is specified for
the MCSCF fragment. The number of core/virtual orbitals for
MCSCF dimers will be automatically computed. The most
important issue is the initial orbitals for the MCSCF
monomer. Just as for ab initio MCSCF, you should exercise
chemical knowledge and provide appropriate orbitals. There
are two basic ways to input MCSCF initial orbitals:
A) through the FMO monomer density binary file
B) by providing a text $VEC group.
The former way is briefly described in INPUT.DOC (see
orbital conversion). The latter way is really identical to
ab initio MCSCF, except the orbitals should be prepared for
the fragment (so in many cases you would have to get them
from an FMO calculation). Once you have the orbitals, put
them into $VEC1, and use the IJVEC option in $FMOPRP (e.g.,
if your MCSCF fragment is number 5, you would use $VEC1 and
ijvec(1)=5,0). For two-layer MCSCF the following
conditions apply. Usually one cannot simply use F40
restart, because its contents will be overwritten with RHF
orbitals and this will mess up your carefully chosen MCSCF
orbitals. Therefore, two ways exist. One is to modify A)
above by reordering the orbitals with something like
$guess guess=skip norder=1 iorder(28)=29,30,31,32,28 $end
Then the lower RHF layer will converge RHF orbitals that
you reorder with iorder in the same run (add 512 to nguess
in $FMO). This requires you know how to reorder before
running the job so it is not always convenient. Probably
the best way to run two-layer MCSCF is verbatim B) above,
so just provide MCSCF monomer orbitals in $VEC1. Finally,
it may happen that some MCSCF dimer will not converge.
Beside the usual MCSCF tricks to gain convergence as the
last resort you may be able to prepare good initial dimer
orbitals, put them into $VEC2 ($VEC3 etc) and read them
with ijvec. SOSCF is the preferred converger in FMO, and
the other one (FULLNR) has not been modified to eradicate
the artefacts of convergence (due to detached bonds). In
the bad cases you can try running one or two monomer SCF
iterations with FULLNR, stop the job and use its orbitals
in F40 to do a restart with SOSCF. We also found useful to
set CASDII=0.005 and nofo=10 in some cases running FOCAS
longer to get better orbitals for SOSCF.
How to perform multilayer runs
For some fragments you may like to specify a different
level of electron correlation and/or basis set. In a
typical case, you would use high level for the reaction
center and a lower level for the remaining part of the
system. The set up for multilayer runs is very similar to
the unilayer case. You only have to specify to what layer
each fragment belongs and for each layer define DFTTYP,
MPLEVL, SCFTYP as well as a basis set. If detached bonds
are present, appropriate HMOs should be defined. See the
paragraph above for multilayer MCSCF. Currently geometry
optimizations of multilayer runs require adding 128 to
NGUESS, if basis sets in layers differ from each other.
How to mix basis sets in FMO
You can mix basis sets in three ways.
a) single layer with ESP
You can specify different basis sets using the basis
set library in $DATA (such as C.1 and C.2 for CH3
and COO- carbons.
b) multilayer with ESP
Each layer can have its own basis set.
The difference between a 2-layer run with one basis set per
layer and a 1-layer run with 2-basis sets is significant:
in the former case the lower level densities are converged
with all fragments computed at the lower level. In the
latter case, the fragments are converged simultaneously,
each with its own basis set. In addition, dimer corrections
between layers will be computed differently: with the lower
basis set in the former case and with mixed basis set in
the latter. The latter approach may result in unphysical
polarization, so mixing basis sets is mainly intended to
add diffuse functions to anionic (e.g., carboxyl) groups,
not as a substitute for two-layer runs.
c) dual basis in an additive way
In this case, the whole system is calculated without
ESP and the smaller basis set, then with ESP and the smaller basis set,
and finally without ESP and the larger basis set.
This (c) way is often used with diffuse functions for
fragments with covalent boundaries.
How to perform FMO/PCM calculations
Solvent effects can be taken into account with PCM. PCM in
FMO is very similar to regular PCM. There is one basic
difference: in FMO/PCM the total electron density that
determines the electrostatic interaction is computed using
the FMO density expansion up to n-body terms. The cavity
is constructed surrounding the whole molecule, and the
whole cavity is used in each individual m-mer calculation.
There are several levels of accuracy (determined by the "n"
above), and the recommended level is FMO/PCM<1>,
specified by:
$pcm ief=-10 icomp=2 icav=1 idisp=1 ifmo=-1 $end
Many PCM options can be used as in the regular PCM. The
following restrictions apply:
IEF may be only -3 or -10, IDP must be 0.
Multilayer FMO runs are supported. Restarts are limited to
IREST=2, and in this case PCM charges (the ASCs) are not
recycled. However, the initial guess for the charges is
fairly reasonable, so IREST=2 may be useful although
reading the ASCs may be implemented in future.
Note for advanced users. IFMO < NBODY runs are permitted.
They are denoted by FMOm/PCM[n], where m=NBODY and n=IFMO.
In FMOm/PCM[n], the ASCs are computed with n-body level.
The difference between FMO2/PCM[1] and FMO2/PCM[1(2)] is
that in the former the ASCs are computed at the 1-body
level, whereas for the former at the 2-body level, but
without self-consistency (which would be FMO2/PCM[2]).
Probably, FMO3/PCM[2] should be regarded as the most
accurate and still affordable (with a few thousand nodes)
method. However, FMO3/PCM[1(2)] (specified with NBODY=3,
IFMO=2 and NPCMIT=2) is much cheaper and slightly less
accurate than FMO3/PCM[2]. FMO3/PCM[3] is the most
accurate and expensive level of all.
PCM<1> is very similar to PCM[1] except that the dimer
ES solvent contributions are halved.
How to perform FMO/EFP calculations
Solvent effects can also be taken into account with the
Effective Fragment Potential model. The presence of both
$FMO and $EFRAG groups selects FMO/EFP calculations. See
the $EFRAG group and the $FMO group for details.
In the FMO/EFP method, the Quantum Mechanical part of the
calculation in the usual EFP method is replaced by the FMO
method, which may save time for large molecules such as
proteins.
In the present version, only FMOn/EFP1 (water solvent only)
is available for RHF, DFT and MP2. One can use the MC
global optimization technique for FMO/EFP by RUNTYP=GLOBOP.
Of course, the group DDI (GDDI) parallelization technique
for the FMO method can be used.
Geometry optimization or saddle point search for FMO
The standard optimizers in GAMESS are now well
parallelized, and thus recommended to be used with FMO up
to the limit hardwired in GAMESS (2000 atoms). In practice,
if more than about 1000 atoms are present, numeric Hessian
updates often result in the improper curvature and
optimization stops. One can either do a restart, or use
RUNTYP=OPTFMO (which does not diagonalize the Hessian).
RUNTYP=OPTIMIZE applies to Cartesian coordinates or DLC.
RUNTYP=OPTFMO works only with Cartesian coordinates. If
your system has more than 2000 atoms you can consider
RUNTYP=OPTFMO, which can now use Hessian updates and
provides reasonable way to optimize although it is not as
good as the standard means in RUNTYP=OPTIMIZE.
A transition state search for FMO can be performed with
RUNTYP=SADPOINT using either Cartesian coordinates or DLC.
IRC calculations can be performed.
FMO hessian calculations
Analytic FMO Hessian with RUNTYP=HESSIAN may be computed
for RHF, ROHF, UHF and RDFT, in the gas phase (no
PCM, EFP etc).
Molecular dynamics with FMO
MD can be run for any FMO method, which has the gradient
implemented. However, in many cases the approximations in
the gradient for a particular method may lead to large
discrepancies in MD. The following methods have a fully
analytic gradient (which has to be turned on with $FMO
keyword MODGRD=42): FMO-RHF, FMO-DFT, FMO-MP2, FMO-RHF/EFP;
The RESPPC approximation adds a small error to the gradient,
so in MD larger values such as 3.0 should be used,
or 0.0 to turn RESPPC off.
Pair interaction energy decomposition analysis (PIEDA)
PIEDA can be performed for the PL0 and PL states. The PL0
state is the electronic state in which fragments are
polarised by the environment in its free (noninteracting)
state. The simplest example is that in a water cluster,
each molecule is computed in the electrostatic field
exerted by the electron densities of free water molecules.
The PL state is the FMO converged monomer state, that is,
the state in which fragments are polarised by the self-
consistently converged environment. Namely, following the
FMO prescription, fragments are recomputed in the external
field, until the latter converges. Using the PL0 state
requires a series of separate runs; and it also relies on a
"free state" which can be defined in many ways for
molecules with detached covalent bonds.
What should be done to do the PL0 state analysis?
1. run FMO0.
This computes the free state for each fragment, and those
electron densities are stored on file 30 (to be renamed
file 40 and reused in step 3).
2. compute BDA energies (if detached bonds are present),
using sample files in tools/fmo/pieda. This corrects for
artifacts of bond detaching, and involves running a model
system like H3C-CH3, to amend for C-C bond detaching.
3. Using results of (1) and (2), one can do the PL0
analysis. In addition to pasting the data from the two
punch files in steps 1,2 and the density file in step 1
should be provided.
What should be done to do the PL state analysis? The PL
state itself does not need either the free state or PL0
results. However, if the PL0 results are available,
coupling terms can be computed, and in this case IPIEDA is
set to 2; otherwise to 1.
So the easiest and frequently sufficient way to run PIEDA
is to set IPIEDA=1 and do not provide any data from
preceding calculations. The result of a PIEDA calculation
is a set of pair interaction energies (interfragment
interaction energies), decomposed into electrostatic,
exchange-repulsion, charge transfer and dispersion
contributions.
Finally, PIEDA (especially for the PL state) can be thought
of as FMO-EDA, EDA being the Kitaura-Morokuma decomposition
(RUNTYP=MOROKUMA). In fact, PIEDA (for the PL state) in
the case of just two fragments of standalone molecules is
entirely equivalent to EDA, which can be easily verified,
by running the full PIEDA analysis (ipieda=2). Note that
PIEDA can be run as direct SCF, whereas EDA cannot be, and
for large fragments PIEDA code can be used to perform EDA.
Also, EDA in GAMESS has no explicit dispersion.
In 2012, PIEDA/PCM was developed describing the solvent
screening. RO-PIEDA based on RO-(HF, MP2 or CC) may be
used for radicals. Grimme's dispersion models may be used
in PIEDA.
Excited states
At present, one can use CI, MCSCF, or TDDFT to compute
excited states in FMO. MCSCF is discussed separately
above, so here only TDDFT and CI are explained. They are
enabled by setting the IEXCIT option (EXCIT(1) defines the
excited state's fragment ID).
Two levels are implemented for TDDFT (FMO1-TDDFT and FMO2-
TDDFT). In the former, only monomer TDDFT calculations are
performed, whereas the latter adds pair corrections from
TDDFT dimers. PCM may be used for solvent effects with
TDDFT (PCM[1] is usually sufficient).
CI can only be done for CIS at the monomer level (nbody=1),
FMO1-CIS. The set-up for CI is similar to that for TDDFT.
Selective and sussystem FMO
Sometimes, one is interested only in some pair
interactions, for example, between ligand and protein, or
the opposite, only pair interactions within ligand. This
saves a lot of CPU time by omitting all other pair
calculations, but does not give the total properties. To
use this feature, define MOLFRG and MODMOL. RUNTYP=ENERGY
only is implemented.
In the subsystem analysis, one can divide fragments into
subsystems and obtain various properties of subsystems.
Frozen domain
To accelerate geometry optimisations, one can specify that
the electronic state of the first layer in a 2-layer FMO
can be computed at the initial geometry and consequently be
frozen. One can define the polarizable buffer (equal to
layer 2) and frozen domain (layer 1). Fragments in the
polarizable buffer which contain the atoms active in
geometry optimisation form the active domain. The
fragments in the active domain should have a nonzero
separation from the frozen domain. In FMO/FD all dimers in
the polarizable buffer are computed; in FMO/FDD only those
dimers which have at least one monomer in the active domain
are computed. FMO/FD and FNI/FDD are only implemented for
RUNTYP=OPTIMIZE. MODFD and IACTAT in $FMO specify
FMO/FD(D) atop of the usual multilayer FMO setup with some
atoms frozen in geometry optimization by the standard means
(i.e., IACTAT in $STATPT). Note that in FMO/FD(D) the
Hessian as used in RUNTYP=OPTIMIZE is formed only for the
atoms in the second layer, so this upper layer should not
have more than the GAMESS limit (currently, 2000 atoms).
IMOMM with FMO
IMOMM (namely, SIMOMM) calculations can be performed with
the "MO" in IMOMM treated using FMO, i.e., this is like
QM/MM but without electronic embedding of QM by MM.
This calculation uses Tinker, a plug-in source code,
available from the GAMESS web site. You should compile and
link in the Tinker plug-in by changing a single line in
comp/compall/lked,
set TINKER=false
into
set TINKER=true
In addition, you should change MAXATM=10 (the maximum
number of atoms in the whole system, as used by Tinker) in
several GAMESS source files into MAXATM=12000 (this number
is used inside Tinker). If you need a larger number,
change it within Tinker as well. After changing this,
recompile and link GAMESS.
The input file style is in general like that of SIMOMM
(q.v.). Different from regular FMO, the atomic coordinates
are given in $TINXYZ, not in $FMOXYZ. The fragmentation in
FMO applies to QM atoms only, selected by IQMATM, and
numbered consequently in FMO, so that INDAT in $FMO applies
to the atoms renumbered from 1 (defined in IQMATM). Other
than $FMOXYZ being superceded by $TINXYZ, the rest of FMO
options is like in normal FMO. IMOMM based on FMO is
usually referred to as FMO/MM for short, although "FMO-
based SIMOMM" is probably easier to understand. The
somewhat tautological FMO-IMOMM has also been used by some.
Covalent boundaries between FMO and MM are supported (via
link atoms). FMO/MM can be used to run geometry
optimizations, whichis really what it is designed for.
Analyzing and visualizing the results
Annotated outputs provided in tools/fmo have matching
mathematical formulae added onto the outputs, for easier
reading.
Facio (http://www1.bbiq.jp/zzzfelis/Facio.html) can plot
various FMO properties such as interaction energies, using
interactive GUI viewers.
To plot orbitals for an n-mer, set NPUNCH=2 in $SCF and
PLTORB=.T. There are several ways to produce cube files
with electron densities. They are described in detail in
tools/fmo/fmocube/README. To plot pair interaction maps,
use tools/fmo/fmograbres to generate CSV files from GAMESS
output, which can be easily read into Gnuplot or Excel.
FMO portal offers tools for visialising FMO results:
http://www.fmo-portal.info/ (shut down at present).
Parallelization of FMO runs with GDDI
The FMO method has been developed within a 2-level
hierarchical parallelization scheme, group DDI (GDDI),
allowing massively parallel calculations. Different groups
of processors handle the various monomer, dimer, and maybe
trimer computations. The processor groups should be sized
so that GAMESS' innate scaling is fairly good, and the
fragments should be mapped onto the processor groups in an
intelligent fashion.
This is a very important and seemingly difficult issue. It
is very common to be able to speed up parallel runs at
least several times just by using GDDI better. First of
all, do not use plain DDI and always employ GDDI when
running FMO calculations. Next, learn that you can and
should divide nodes into groups to achieve better
performance. The very basic rule of thumb is to try to have
several times fewer groups than jobs. Since the number of
monomers and dimers is different, group division should
reflect that fact. Ergo, find a small parallel computer
with 8-32 nodes and experiment changing just two numbers:
ngrfmo(1)=N1,N2 and see how performance changes for your
particular system.
Limitations of the FMO method in GAMESS
1. Dimensions: in general none, except that the standard
GAMESS engines RUNTYP=OPTIMIZE and IRC are limited to 2000
atoms (for FD(D), domain B may not exceed this limit). The
limit can be increased by changing the source and
recompiling GAMESS (see elsewhere).
2. CHARMM may not be combined with FMO, and some other
extensions may not work. Not every illegal combination is
trapped, caveat emptor!
3. RUNTYP is limited to ENERGY, GRADIENT, OPTIMIZE, OPTFMO,
IRC, FMO0, MD, GLOBOP, SADPOINT, FMOHESS and RAMAN only:
Do not even try other ones!
4. Three-body FMO-MCSCF and FMO-TDDFT are not implemented.
5. No MOPAC semiempirical methods may be used, but DFTB was
interfaced with FMO..
What will work the same way as ab initio:
The various SCF convergers, all DFT functionals, in-core
integrals, direct SCF.
Restarts with the FMO method
RUNTYP=ENERGY can be restarted from anywhere before
trimers. To restart monomer SCF, copy file F40 with monomer
densities to the grandmaster node. To restart dimers,
provide file F40 and monomer energies ($FMOENM).
Optionally, some dimer energies can be supplied ($FMOEND)
to skip computation of corresponding dimers.
RUNTYP=GRADIENT can be easily restarted from monomer SCF
(which really means it is a restart of RUNTYP=ENERGY, since
gradient is computed at the end of this step). Provide
file F40. There is another restart option (1024 in $FMOPRP
irest=), supporting full gradient restart, requiring,
however, that you set this option in the original run
(whose results you use to restart). To use this option, you
would also need to keep (or save and restore) F38 files on
each node (they are different).
RUNTYP=OPTIMIZE can be restarted from anywhere within the
first RUNTYP=GRADIENT run (q.v.). In addition, by
replacing FMOXYZ group, one can restart at a different
geometry.
RUNTYP=OPTFMO can be restarted by providing a new set of
coordinates in $FMOXYZ and, optionally, by transferring
$OPTRST from the punch into the input file.
Note on accuracy
The FMO method is aimed at computation of large molecules.
This means that the total energy is large, for example, a
6646 atom molecule has a total energy of -165,676 Hartrees.
If one uses the standard accuracy of roughly 1e-9 (that
should be taken relatively), one gets an error as much as
0.001 hartree, in a single calculation. FMO involves many
ab initio single point calculations of fragments and their
n-mers, thus it can be expected that numeric accuracy is 1-
2 orders lower than that given by 1e-9. Therefore, it is
compulsory that accuracy should be raised, which is done by
default.
The following default parameters are reset by FMO:
ICUT/$CONTRL (9->12), ITOL/$CONTRL(20->24),
CONV/$SCF(1e-5 -> 1e-7),
CUTOFF/$MP2 (1e-9->1e-12), CUTTRF/$TRANS(1e-9->1e-10).
CVGTOL/$DET,$GUGDIA (1e-5 -> 1e-6)
This to some extent slows down the calculation (perhaps on
the order of 10-15%). It is suggested that you maintain
this accuracy for all final energetics. However, you may
be able to drop the accuracy a bit for the initial part of
geometry optimization if you are willing to do manual work
of adjusting accuracy in the input. It is recommended to
keep high accuracy at the flat surfaces (the final part of
optimizations) though. For DFT the numeric grid's accuracy
may be increased in accordance with the molecule size, e.g.
extending the default grid of 96*12*24 to 96*20*40.
However, some tests indicate that energy differences are
quite insensitive to this increase.
FMO References
I. Basic FMO papers
A book chapter contains an introduction to FMO basics:
Theoretical development of the fragment molecular
orbital (FMO) method, D. G. Fedorov, K. Kitaura,
in "Modern methods for theoretical physical chemistry of
biopolymers", E. B. Starikov, J. P. Lewis, S. Tanaka,
Eds., pp 3-38, Elsevier, Amsterdam, 2006.
There is now a full FMO book (11 chapters), which contains
an introduction to FMO aimed at general application
chemists, and a wealth of practical advice on doing FMO
calculations:
The Fragment Molecular Orbital Method: Practical
Applications to Large Molecular System,
D. G. Fedorov, K. Kitaura, Eds.,
CRC Press, Boca Raton, FL, 2009.
FMO reviews:
D. G. Fedorov, K. Kitaura (Feature Article)
J. Phys. Chem. A 111, 6904-6914 (2007).
D. G. Fedorov, T. Nagata, K. Kitaura (Perspective)
Phys. Chem. Chem. Phys., 14, 7562-7577 (2012)
A review of FMO in the context of other fragment-based
methods is
M. S. Gordon, D. G. Fedorov, S. R. Pruitt,
L. V. Slipchenko Chem. Rev. 112, 632-672 (2012).
A very concise and detailed mathematical formulation of FMO
including various extensions and property calculations is
published as
Mathematical formulation of the fragment molecular
orbital method.
T. Nagata, D. G. Fedorov, K. Kitaura.
In "Linear-Scaling Techniques in Computational Chemistry
and Physics". R. Zalesny, M. G. Papadopoulos,
P. G. Mezey, J. Leszczynski, Eds., pp. 17-64,
Springer, New York, 2011.
1. Fragment molecular orbital method: an approximate
computational method for large molecules"
K. Kitaura, E. Ikeo, T. Asada, T. Nakano, M. Uebayasi
Chem. Phys. Lett., 313, 701(1999).
2. Fragment molecular orbital method: application to
polypeptides
T. Nakano, T. Kaminuma, T. Sato, Y. Akiyama,
M. Uebayasi, K. Kitaura Chem.Phys.Lett. 318, 614(2000).
3. Fragment molecular orbital method: analytical energy
gradients
K. Kitaura, S.-I. Sugiki, T. Nakano, Y. Komeiji,
M. Uebayasi, Chem. Phys. Lett., 336, 163(2001).
4. Fragment molecular orbital method: use of approximate
electrostatic potential
T. Nakano, T. Kaminuma, T. Sato, K. Fukuzawa,
Y. Akiyama, M. Uebayasi, K. Kitaura
Chem. Phys. Lett., 351, 475(2002).
5. The extension of the fragment molecular orbital method
with the many-particle Green's function,
K. Yasuda, D. Yamaki, J. Chem. Phys. 125, 154101(2006).
6. The role of the exchange in the embedding electrostatic
potential for the fragment molecular orbital method.
D. G. Fedorov, K. Kitaura
J. Chem. Phys. 131, 171106(2009).
7. Analytic second derivatives of the energy in the
fragment molecular orbital method.
H. Nakata, T. Nagata, D. G. Fedorov, S. Yokojima, K.
Kitaura, S. Nakamura, J. Chem. Phys. 138 (2013) 164103.
II. FMO in GAMESS
1. A new hierarchical parallelization scheme: generalized
distributed data interface (GDDI), and an application to
the fragment molecular orbital method (FMO).
D. G. Fedorov, R. M. Olson, K. Kitaura, M. S. Gordon,
S. Koseki J. Comput. Chem. 25, 872-880(2004).
2. The importance of three-body terms in the fragment
molecular orbital method.
D. G. Fedorov and K. Kitaura
J. Chem. Phys. 120, 6832-6840(2004).
3. On the accuracy of the 3-body fragment molecular orbital
method (FMO) applied to density functional theory
D. G. Fedorov and K. Kitaura
Chem. Phys. Lett. 389, 129-134(2004).
4. Second order Moeller-Plesset perturbation theory based
upon the fragment molecular orbital method.
D. G. Fedorov and K. Kitaura
J. Chem. Phys. 121, 2483-2490(2004).
5. Multiconfiguration self-consistent-field theory based
upon the fragment molecular orbital method.
D. G. Fedorov and K. Kitaura
J. Chem. Phys. 122, 054108/1-10(2005).
6. Multilayer Formulation of the Fragment Molecular Orbital
Method (FMO).
D. G. Fedorov, T. Ishida, K. Kitaura
J. Phys. Chem. A. 109, 2638-2646(2005).
7. Coupled-cluster theory based upon the Fragment Molecular
Orbital method.
D. G. Fedorov, K. Kitaura
J. Chem. Phys. 123, 134103/1-11 (2005)
8. The polarizable continuum model (PCM) interfaced with
the fragment molecular orbital method (FMO).
D. G. Fedorov, K. Kitaura, H. Li, J. H. Jensen,
M. S. Gordon, J. Comput. Chem., 27, 976-985(2006)
9. The three-body fragment molecular orbital method for
accurate calculations of large systems,
D. G. Fedorov, K. Kitaura
Chem. Phys. Lett. 433, 182-187(2006).
10. Pair interaction energy decomposition analysis,
D. G. Fedorov, K. Kitaura
J. Comp. Chem. 28, 222-237(2007).
11. On the accuracy of the three-body fragment molecular
orbital method (FMO) applied to Moeller-Plesset
perturbation theory,
D. G. Fedorov, K. Ishimura, T. Ishida, K. Kitaura,
P. Pulay, S. Nagase
J. Comput. Chem., 28, 1476-1484 (2007).
12. The Fragment Molecular Orbital method for geometry
optimizations of polypeptides and proteins,
D.G.Fedorov, T. Ishida, M. Uebayasi, K. Kitaura
J.Phys.Chem.A, 111, 2722-2732(2007).
13. Time-dependent density functional theory with the
multilayer fragment molecular orbital method
M. Chiba, D. G. Fedorov, K. Kitaura
Chem. Phys. Lett. 444, 346-350 (2007).
14. Time-dependent density functional theory based upon the
fragment molecular orbital method
M. Chiba, D. G. Fedorov, K. Kitaura
J. Chem. Phys. 127, 104108(2007).
15. Polarizable continuum model with the fragment molecular
orbital-based time-dependent density functional theory.
M. Chiba, D. G. Fedorov, K. Kitaura
J. Comput. Chem. 29, 2667-2676 (2008).
16. Theoretical Analysis of the Intermolecular Interaction
Effects on the Excitation Energy of Organic Pigments: Solid
State Quinacridone.
H. Fukunaga, D.G.Fedorov, M. Chiba, K. Nii, K. Kitaura
J. Phys. Chem. A 112, 10887-10894 (2008).
17. Covalent Bond Fragmentation Suitable To Describe Solids
in the Fragment Molecular Orbital Method.
D. G. Fedorov, J. H. Jensen, R. C. Deka, K. Kitaura
J. Phys. Chem. A 112, 11808-11816 (2008).
18. Excited state geometry optimizations by time-dependent
density functional theory based on the fragment molecular
orbital method.
M. Chiba, D. G. Fedorov, T. Nagata, K. Kitaura
Chem. Phys. Lett. 474, 227-232 (2009).
19. Derivatives of the approximated electrostatic
potentials in the fragment molecular orbital method.
T. Nagata, D. G. Fedorov, K. Kitaura,
Chem. Phys. Lett. 475, 124-131 (2009).
20. A combined effective fragment potential - fragment
molecular orbital method. I. The energy expression and
initial applications.
T. Nagata, D. G. Fedorov, K. Kitaura, M. S. Gordon,
J. Chem. Phys. 131, 024101 (2009).
21. Analytic gradient for the adaptive frozen orbital bond
detachment in the fragment molecular orbital method.
D. G. Fedorov, P. V. Avramov, J.H. Jensen, K. Kitaura,
Chem. Phys. Lett. 477, 169-175 (2009).
22. Fragment molecular orbital study of the electronic
excitations in the photosynthetic reaction center of
Blastochloris viridis.
T. Ikegami, T. Ishida, D. G. Fedorov, K. Kitaura,
Y. Inadomi, H. Umeda, M. Yokokawa, S. Sekiguchi,
J. Comp. Chem. 31, 447-454 (2010).
23. Open-Shell Formulation of the Fragment Molecular
Orbital Method.
S. R. Pruitt, D. G. Fedorov, K. Kitaura, M. S. Gordon
J. Chem. Theor. Comp. 6, 1-5 (2010)
24. Energy gradients in combined fragment molecular orbital
and polarizable continuum model (FMO/PCM) calculation.
H. Li, D. G. Fedorov, T. Nagata, K. Kitaura,
J. H. Jensen, M. S. Gordon
J. Comput. Chem. 31, 778-790 (2010).
25. Nuclear-Electronic Orbital Method within the Fragment
Molecular Orbital Approach.
B. Auer, M. V. Pak, S. Hammes-Schiffer,
J. Phys. Chem. C 114, 5582-5588 (2010).
26. Importance of the hybrid orbital operator derivative
term for the energy gradient in the fragment molecular
orbital method.
T. Nagata, D. G. Fedorov, K. Kitaura,
Chem. Phys. Lett. 492, 302-308 (2010).
27. Systematic Study of the Embedding Potential Description
in the Fragment Molecular Orbital Method.
D. G. Fedorov, L. V. Slipchenko, K. Kitaura,
J. Phys. Chem. A 114, 8742-8753 (2010).
28. A combined effective fragment potential - fragment
molecular orbital method. II. Analytic gradient and
application to the geometry optimization of solvated
tetraglycine and chignolin.
T. Nagata, D. G. Fedorov, T. Sawada, K. Kitaura,
M. S. Gordon, J. Chem. Phys. 134, 034110 (2011).
29. Geometry optimization of the active site of a large
system with the fragment molecular orbital method.
D. G. Fedorov, Y. Alexeev, K. Kitaura,
J. Phys. Chem. Lett. 2, 282-288 (2011).
30. Fully analytic energy gradient in the fragment
molecular orbital method.
T. Nagata, K. Brorsen, D. G. Fedorov, K. Kitaura,
M. S. Gordon, J. Chem. Phys. 134, 124115(2011).
31. Analytic energy gradient for second-order Moeller-
Plesset perturbation theory based on the fragment molecular
orbital method.
T. Nagata, D. G. Fedorov, K. Ishimura, K. Kitaura,
J. Chem. Phys. 135, 044110 (2011).
32. Large-Scale MP2 Calculations on the Blue Gene
Architecture Using the Fragment Molecular Orbital Method.
G. D. Fletcher, D. G. Fedorov, S.R.Pruitt, T.L.Windus,
M. S. Gordon, J. Chem. Theory Comput. 8, 75-79(2012).
33. Energy decomposition analysis in solution based on the
fragment molecular orbital method.
D.G.Fedorov, K.Kitaura
J.Phys.Chem. A 116, 704-719(2012).
34. Analytic gradient and molecular dynamics simulations
using the fragment molecular orbital method combined with
effective potentials.
T. Nagata, D. G. Fedorov, K. Kitaura
Theor. Chem. Acc. 131, 1136 (2012).
35. Geometry Optimizations of Open-Shell Systems with the
Fragment Molecular Orbital Method.
S. R. Pruitt, D. G. Fedorov, M. S. Gordon,
J. Phys. Chem. A, 116, 4965-4974 (2012).
36. Analytic gradient for second order Moeller-Plesset
perturbation theory with the polarizable continuum model
based on the fragment molecular orbital method.
T. Nagata, D. G. Fedorov, H. Li, K. Kitaura,
J. Chem. Phys., 136, 204112 (2012).
37. Reducing scaling of the fragment molecular orbital
method using the multipole method.
C. H. Choi, D. G. Fedorov
Chem. Phys. Lett. 543, 159-165(2012).
38. Unrestricted Hartree-Fock based on the fragment
molecular orbital method: energy and its analytic gradient.
H. Nakata, D. G. Fedorov, T. Nagata, S. Yokojima, K.
Ogata, K. Kitaura, S. Nakamura
J. Chem. Phys. 137, 044110 (2012).
39. Analytic gradient for the embedding potential with
approximations in the fragment molecular orbital method.
T. Nagata, D. G. Fedorov, K. Kitaura
Chem. Phys. Lett. 544, 87-93 (2012).
40. Analysis of solute-solvent interactions in the fragment
molecular orbital method interfaced with the effective
fragment potentials: theory and application to solvated
griffithsin-carbohydrate complex.
T. Nagata, D. G. Fedorov, T. Sowada, K. Kitaura
J. Phys. Chem. A, 116, 9088-9099 (2012).
41. Open-shell pair interaction energy decomposition
analysis (PIEDA): Formulation and application to the
hydrogen abstraction in tripeptides.
M.C.Green, D.G.Fedorov, K.Kitaura, J.S.Francisco,
L.V.Slipchenko, J.Chem.Phys. 138 (2013) 074111.
Other FMO references including applications can be found
at:
http://staff.aist.go.jp/d.g.fedorov/fmo/main.html
EFMO references are given in its own subsection.
The Cluster-in-Molecules method
sequential and parallel execution
If the user is not interested in parallel CIM calculations,
MTDCIM must be set at 0, which is a default value, and no
additional steps have to be taken. If the user is
interested in a parallel execution, MTDCIM must initially
be set at 1 to prepare for individual subsystem GAMESS
runs, and then, after the individual subsystem runs are
completed, reset to 2 to complete the CIM calculation. When
MTDCIM is initially set at 1, multiple input files
$JOB.Sys-N.inp for individual subsystem calculations with
GAMESS, which can be run independent of one another, and
the $JOB.cim file, which contains the information about all
subsystems needed to complete the CIM calculation, are
automatically generated, and the program stops awaiting
further execution. Each subsystem N has then to be run as
an independent GAMESS calculation using the $JOB.Sys-N.inp
input file. This produces the $JOB.Sys-N.cim files which
contain the information about the correlation energy
contributions due to the occupied LMOs central in
subsystems. All $JOB.Sys-N.cim files resulting from the
individual subsystem calculations with GAMESS and the
$JOB.cim file are used to assemble the final results of a
CIM calculation for the entire system. In order to
accomplish this and complete the CIM run, one has to reset
MTDCIM in the main $JOB.inp input file to 2 and run GAMESS
again.
restarts
If any of the subsystem GAMESS calculations using the
$JOB.Sys-N.inp input files fails, the user can always rerun
it (editing the corresponding $JOB.Sys-N.inp file(s), if
need be), and then use MTDCIM=2 to complete the desired CIM
calculation for the entire system. This applies to
sequential and parallel CIM calculations. In the latter
case, this is a natural consequence of the way the parallel
execution is structured (see note 1). In the former case
(MTDCIM=0), if the entire calculation is completed, the
$JOB.Sys-N.* subsystem files are deleted, but if one of the
subsystem calculations fails, the program aborts, leaving
all $JOB.Sys-N.inp input files, the $JOB.Sys-N.cim output
files from the completed subsystem calculations, and the
$JOB.cim file on the disk. One can rerun the subsystem
GAMESS calculation that failed, editing the corresponding
$JOB.Sys-N.inp file if need be, and the remaining subsystem
calculations, and then, once all subsystem GAMESS runs are
completed, finish the calculation by using MTDCIM=2 in the
main $JOB.inp input. This has an advantage over the more
automated method of restarting the sequential CIM
calculations described below in that the Hartree-Fock and
orbital localization calculations for the entire system do
not have to be repeated.
If the sequential (MTDCIM=0) run does not complete due to
the failure of one of the subsystem GAMESS calculations,
one can also follow a simpler, more automated restart
strategy. At the time of failure, all $JOB.Sys-N.inp input
files, the $JOB.Sys-N.cim output files from the completed
subsystem calculations, and the $JOB.cim file are saved on
the disk. After inspecting the main output file, the user
can simply delete the $JOB.dat file and the $JOB.Sys-N.cim
file resulting from the failed subsystem calculation, edit
the corresponding $JOB.Sys-N.inp file, if necessary, and
rerun the GAMESS calculation with MTDCIM=0. The Hartree-
Fock and orbital localization calculations for the entire
system will be performed again, but the user will avoid the
need for running individual subsystem calculations one-by-
one, as described above.
the cimshell script
The Python script "cimshell" that automatically produces
typical $JOB.sh files for parallel OpenMP and MPI subsystem
calculations using the $JOB.Sys-N.inp files can be found in
$GMS_PATH/tools/cim/, where $GMS_PATH is the GAMESS main
directory. In order to run the subsystem calculations with
OpenMP or MPI, and with the help of the "cimshell" script,
the following steps should be performed:
1. Run GAMESS CIM calculation using MTDCIM=1 to produce
the $JOB.Sys-N.inp and $JOB.cim files.
2. After all subsystem $JOB.Sys-N.inp input files are
generated, use "cimshell" to automatically generate
the OpenMP or MPI script $JOB.sh for parallel
execution. By default, the "cimshell" program must
be run in the directory where the $JOB.inp and all
$JOB.Sys-N.inp files reside. For example, "cimshell
--np 4 $JOB" generates the $JOB.sh script for an
OpenMP parallel calculation on 4 cores, whereas
"cimshell --np 8 --para mpi --submit pbs --MPI_EXEC
mpiexec $JOB" generates the $JOB.sh script for an
MPI parallel calculation on 8 processors using the
PBS queue system for submitting the job. In these
two examples, we are assuming that the "cimshell"
has been copied to the directory where all
$JOB.Sys-N.inp files reside; this can be altered by
redefining the $PATH variable (adding
$GMS_PATH/tools/cim/ to it). Use "cimshell -h" for
more information about the "cimshell" options.
3. Run or submit the $JOB.sh script to have subsystem
calculations performed in parallel. In order to do
this, the user must compile the ompjob.for (the
OpenMP case) or mpijob.for (the MPI case) programs
that reside in $GMS_PATH/tools/cim/. The
corresponding executables used by $JOB.sh are called
ompjob and mpijob, respectively. The example of the
Makefile that can be used to install ompjob and
mpijob can be found in $GMS_PATH/tools/cim/.
4. After all subsystem calculations are completed, run
the final GAMESS CIM calculation using the original
input file $JOB.inp in which with MTDCIM=2. GAMESS
will automatically find the relevant $JOB.Sys-N.cim
and $JOB.cim files to complete the CIM calculation
for the entire system and print the final CIM
energies in the main output.
CIM references
THE FOLLOWING PAPERS SHOULD BE CITED WHEN USING
CLUSTER-IN-MOLECULE OPTIONS:
DUAL-ENVIRONMENT CIM (CIMTYP=DECIM)
W. LI, P. PIECUCH, J.R. GOUR, AND S. LI,
J. CHEM. PHYS. 131, 114109-1 - 114109-30 (2009).
SEE, ALSO, S. LI, J. SHEN, W. LI, AND Y. JIANG,
J. CHEM. PHYS. 125, 074109-1 - 074109-10 (2006)
SINGLE-ENVIRONMENT CIM (CIMTYP=SECIM,GSECIM)
W. LI, P. PIECUCH, J.R. GOUR, AND S. LI,
J. CHEM. PHYS. 131, 114109-1 - 114109-30 (2009);
W. LI AND P. PIECUCH, J. PHYS. CHEM. A 114, 8644-8657
(2010).
IN ADDITION, THE USE OF MULTI-LEVEL CIM SHOULD REFERENCE
W. LI AND P. PIECUCH, J. PHYS. CHEM. A 114, 6721-6727
(2010).
MOPAC Calculations within GAMESS
Parts of MOPAC 6.0 have been included in GAMESS giving
access to four semiempirical wavefunctions: MNDO, AM1,
PM3, and RM1. RM1 is the most recent parameterization,
replacing AM1 data for H, C-F, P-Cl, Br, and I. See G.
Bruno Rocha, R. Oliveira Freire, A. Mayall Simas, and
J.J.P.Stewart, J.Comput.Chem. 27, 1101-1111(2006).
These wavefunctions are quantum mechanical in nature
but neglect most two electron integrals, a deficiency that
is (hopefully) compensated for by introduction of empirical
parameters. The quantum mechanical nature of semiempirical
theory makes it quite compatible with the ab initio
methodology in GAMESS. As a result, very little of MOPAC
6.0 actually is incorporated into GAMESS. The part that
did survive is the code that evaluates
1) the one- and two-electron integrals,
2) the two-electron part of the Fock matrix,
3) the cartesian energy derivatives, and
4) the ZDO atomic charges and molecular dipole.
Everything else is actually GAMESS: coordinate input
(including point group symmetry), the SCF convergence
procedures, the matrix diagonalizer, the geometry searcher,
the numerical hessian driver, and so on. Most of the
output will look like an ab initio output.
It is extremely simple to perform these calculations.
All you need to do is specify GBASIS=MNDO, AM1, PM3, or RM1
in the $BASIS group. Note that this not only selects a
particular "Hamiltonian" (parameter set), it also picks a
Slater Type Orbital (STO) basis.
MOPAC parameters exist for the following elements. The
printout when you run will give you specific references for
each kind of atom. The quote on alkali's below means that
these elements are treated as "sparkles", rather than as
atoms with genuine basis functions.
For MNDO:
H
Li * B C N O F
Na' * Al Si P S Cl
K' * ... Zn * Ge * * Br
Rb' * ... * * Sn * * I
* * ... Hg * Pb *
For AM1: For PM3:
H H
* * B C N O F Li Be * C N O F
Na Mg Al Si P S Cl Na Mg Al Si P S Cl
K Ca ... Zn * Ge * * Br K Ca ... Zn Ga Ge As Se Br
Rb' * ... * * Sn * * I Rb' * ... Cd In Sn Sb Te I
* * ... Hg * * * * * ... Hg Tl Pb Bi
For RM1:
H
C N O F
P S Cl
Br
I
RM1 uses AM1 parameters for any element not listed above.
* * * * *
MOPAC will not work with every option in GAMESS: the
semiempirical wavefunctions must be RHF, UHF, and ROHF in
any combination with run types ENERGY, GRADIENT, OPTIMIZE,
SADPOINT, HESSIAN, and IRC. Note that nuclear hessian runs
use numerical finite differencing of analytic gradients.
MOPAC's CI and half electron methods are not supported.
Because the majority of the implementation is GAMESS
rather than MOPAC6, you will notice a few improvements.
Dynamic memory allocation is used, so GAMESS uses far less
memory for a given size of molecule. The starting orbitals
for SCF calculations are generated by a Huckel initial
guess routine. Spin restricted (high spin) ROHF can be
performed. Converged SCF orbitals will be labeled by their
symmetry type. Numerical hessians will make use of point
group symmetry, so that only the symmetry unique atoms need
to be displaced. Infrared intensities will be calculated
at the end of hessian runs. We have not at present used
the block diagonalizer during intermediate SCF iterations,
so that the run time for a single geometry point in GAMESS
is usually longer than in MOPAC. However, the geometry
optimizer in GAMESS can frequently optimize the structure
in fewer steps than the procedure in MOPAC. Orbitals and
hessians are punched out for convenient reuse in subsequent
calculations. Your molecular orbitals can be drawn with
the PLTORB graphics program, which has been taught about s
and p STO basis sets.
However, because of the STO basis set used in semi-
empirical runs, the various property calculations coded for
Gaussian (GTO) basis sets are unavailable. This means
$ELMOM, $ELPOT, etc. properties are unavailable. Note that
MOPAC6 did not include d STO integrals, so it is quite
impossible to run transition metals.
The PCM solvation model implemented in GAMESS can be
used with MOPAC runs by GAMESS.
To reduce CPU time, by default only the EXTRAP
convergence accelerator is used by the SCF procedures. For
difficult cases, the DIIS, RSTRCT, and/or SHIFT options
will work, but may add significantly to the run time. With
the Huckel guess procedure from GAMESS, most calculations
will converge acceptably without these special options.
The MOPAC implementation is able to run in parallel.
Semiempirical calculations are very fast. One of the
motives for the MOPAC implementation within GAMESS is to
take advantage of this speed. Semiempirical models can
rapidly provide reasonable starting geometries for ab
initio optimizations. Semiempirical hessian matrices are
obtained at virtually no computational cost, and may help
dramatically with an ab initio geometry optimization.
Simply use HESS=READ in $STATPT to use a MOPAC $HESS group
in an ab initio run.
It is important to exercise caution as semiempirical
methods can be dead wrong! The reasons for this are bad
parameters (in certain chemical situations), and the
underlying minimal basis set. A good question to ask
before using MOPAC is "how well is my system modeled by an
ab initio minimal basis set, such as STO-3G"? If the
answer is "not very well", there is a good chance that a
semiempirical description is equally poor.
Molecular Properties and Conversion Factors
These two papers are of general interest:
A.D.Buckingham, J.Chem.Phys. 30, 1580-1585(1959).
D.Neumann, J.W.Moskowitz J.Chem.Phys. 49, 2056-2070(1968).
The first deals with multipoles, and the second with other
properties such as electrostatic potentials.
All units are derived from the atomic units for distance
and the monopole electric charge, as given below.
distance 1 au = 5.291771E-09 cm
monopole 1 au = 4.803242E-10 esu
1 esu = sqrt(g-cm**3)/sec
dipole 1 au = 2.541766E-18 esu-cm
1 Debye = 1.0E-18 esu-cm
quadrupole 1 au = 1.345044E-26 esu-cm**2
1 Buckingham = 1.0E-26 esu-cm**2
octopole 1 au = 7.117668E-35 esu-cm**3
electric potential 1 au = 9.076814E-02 esu/cm
electric field 1 au = 1.715270E+07 esu/cm**2
1 esu/cm**2 = 1 dyne/esu
electric field gradient 1 au = 3.241390E+15 esu/cm**3
The atomic unit for the total electron density is
electron/bohr**3, but 1/bohr**3 for an orbital density.
The atomic unit for spin density is excess alpha spins per
unit volume, h/4*pi*bohr**3. Only the expectation value is
computed, with no constants premultiplying it.
IR intensities are printed in Debye**2/amu-Angstrom**2.
These can be converted into intensities as defined by
Wilson, Decius, and Cross's equation 7.9.25, in km/mole, by
multiplying by 42.255. If you prefer 1/atm-cm**2, use a
conversion factor of 171.65 instead. A good reference for
deciphering these units is A.Komornicki, R.L.Jaffe
J.Chem.Phys. 1979, 71, 2150-2155. A reference showing how
IR intensities change with basis improvements at the HF
level is Y.Yamaguchi, M.Frisch, J.Gaw, H.F.Schaefer,
J.S.Binkley, J.Chem.Phys. 1986, 84, 2262-2278.
Raman activities in A**4/amu multiply by 6.0220E-09 for
units of cm**4/g. One of the many sources explaining how
activity relates to intensity is D.Michalska, R.Wysokinski
Chem.Phys.Lett. 403, 211-217(2005)
Polarizabilities
Static polarizabilities are named alpha, beta, and gamma;
these are called the polarizability, hyperpolarizability,
and second hyperpolarizability. They are the 2nd, 3rd, and
4th derivatives of the energy with respect to uniform
applied electric fields, with the 1st derivative being the
dipole moment.
It is worth mentioning that a uniform (static) electric
field can be applied using $EFIELD, if you wish to develop
custom usages, but $EFIELD input must not be given for any
kind of run discussed below.
A general approach to computing static polarizabilities is
numerical differentiation, namely RUNTYP=FFIELD, which
should work for any energy method provided by GAMESS. A
sequence of computations with fields applied in the x, y,
and/or z directions will generate the three alpha, beta,
and gamma tensors. See $FFCALC for details. Analytic
computation of all three tensors is available for closed
shells only, see RUNTYP=TDHF and $TDHF input, or TDDFT=HPOL
and $TDDFT input. If you need to know just the static
alpha polarizability, see POLAR in $CPHF during any
analytic hessian job.
A break down of the static alpha polarizability in terms of
contributions from individual localized orbitals can be
obtained by setting POLDCM=.TRUE. in $LOCAL. Calculation
will be by analytic means, unless POLNUM in that group is
selected. This option is available only for SCFTYP=RHF.
The keyword LOCHYP in $FFCALC gives a similar analysis for
all three static polarizabilities, determined by numerical
differentiation.
Polarizabilities in a static electric field differ from
those in an oscillating field, such as a laser produces.
These are called frequency dependent alpha, beta, or gamma,
and in the limit of entering a zero frequency, become the
static quantities discussed just above.
For RHF cases, various frequency dependent alpha, beta, and
gamma polarizabilities can be generated, depending on the
experiment. A particularly easy one to understand is
'second harmonic generation', governed by a beta tensor
describing the absorption of two photons with the emission
of one photon at doubled frequency. See RUNTYP=TDHF, and
papers listed under $TDHF, for many other non-linear
optical experiments. A program for the computation of the
frequency dependent beta hyperpolarizability at the DFT
level is also available, for closed shell molecules: see
TDDFT=HPOL and keywords in $TDDFT input.
Nuclear derivatives of the dipole moment and the various
polarizabilities are also of interest. For example,
knowledge of the derivative of the dipole with respect to
nuclear coordinates yields the IR intensity. Similarly,
the nuclear derivative of the static alpha polarizability
gives Raman activities: see RUNTYP=RAMAN. Analytically
computed 1st or 2nd nuclear derivatives of static or
frequency dependent polarizabilities are available for
SCFTYP=RHF, see RUNTYP=TDHFX and $TDHFX, giving rise to
experimental observations such as resonance Raman and
hyper-Raman.
Finally, instead of considering polarizabilities to be a
function of real frequencies, they can be considered to be
dependent on the imaginary frequency. The imaginary
frequency dependent alpha polarizability can be computed
analytically for SCFTYP=RHF only, using POLDYN=.TRUE. in
$LOCAL. Integration of this quantity over the imaginary
frequency domain can be used to extract C6 dispersion
constants.
Polarizabilities are tensor quantities. There are a number
of different ways to define them, and various formulae to
extract "scalar" and "vector" quantites from the tensors.
A good reference for learning how to compare the output of
a theoretical program to experiment is
A.Willetts, J.E.Rice, D.M.Burland, D.P.Shelton
J.Chem.Phys. 97, 7590-7599(1992)
Localized Molecular Orbitals
Three different orbital localization methods are
implemented in GAMESS. The energy and dipole based
methods normally produce similar results, but see
M.W.Schmidt, S.Yabushita, M.S.Gordon in J.Chem.Phys.,
1984, 88, 382-389 for an interesting exception. You can
find references to the three methods at the beginning of
this chapter.
The method due to Edmiston and Ruedenberg works by
maximizing the sum of the orbitals' two electron self
repulsion integrals. Most people who think about the
different localization criteria end up concluding that
this one seems superior. The method requires the two
electron integrals, transformed into the molecular orbital
basis. Because only the integrals involving the orbitals
to be localized are needed, the integral transformation is
actually not very time consuming.
The Boys method maximizes the sum of the distances
between the orbital centroids, that is the difference in
the orbital dipole moments.
The population method due to Pipek and Mezey maximizes
a certain sum of gross atomic Mulliken populations. This
procedure will not mix sigma and pi bonds, so you will not
get localized banana bonds. Hence it is rather easy to
find cases where this method give different results than
the Ruedenberg or Boys approach.
GAMESS will localize orbitals for any kind of RHF, UHF,
ROHF, or MCSCF wavefunctions. The localizations will
automatically restrict any rotation that would cause the
energy of the wavefunction to be changed (the total
wavefunction is left invariant). As discussed below,
localizations for GVB or CI functions are not permitted.
The default is to freeze core orbitals. The localized
valence orbitals are scarcely changed if the core orbitals
are included, and it is usually convenient to leave them
out. Therefore, the default localizations are: RHF
functions localize all doubly occupied valence orbitals.
UHF functions localize all valence alpha, and then all
valence beta orbitals. ROHF functions localize all valence
doubly occupied orbitals, and all singly occupied orbitals,
but do not mix these two orbital spaces. MCSCF functions
localize all valence MCC type orbitals, and localize all
active orbitals, but do not mix these two orbital spaces.
To recover the invariant MCSCF function, you must be using
a FORS=.TRUE. wavefunction, and you must set GROUP=C1 in
$DRT, since the localized orbitals possess no symmetry.
In general, GVB functions are invariant only to
localizations of the NCO doubly occupied orbitals. Any
pairs must be written in natural form, so pair orbitals
cannot be localized. The open shells may be degenerate, so
in general these should not be mixed. If for some reason
you feel you must localize the doubly occupied space, do a
RUNTYP=PROP job. Feed in the GVB orbitals, but tell the
program it is SCFTYP=RHF, and enter a negative ICHARG so
that GAMESS thinks all orbitals occupied in the GVB are
occupied in this fictitous RHF. Use NINA or NOUTA to
localize the desired doubly occupied orbitals. Orbital
localization is not permitted for CI, because we cannot
imagine why you would want to do that anyway.
Boys localization of the core orbitals in molecules
having elements from the third or higher row almost never
succeeds. Boys localization including the core for second
row atoms will often work, since there is only one inner
shell on these. The Ruedenberg method should work for any
element, although including core orbitals in the integral
transformation is more expensive.
The easiest way to do localization is in the run which
generates the wavefunction, by selecting LOCAL=xxx in the
$CONTRL group. However, localization may be conveniently
done at any time after determination of the wavefunction,
by executing a RUNTYP=PROP job. This will require only
$CONTRL, $BASIS/$DATA, $GUESS (pick MOREAD), the converged
$VEC, possibly $SCF or $DRT to define your wavefunction,
and optionally some $LOCAL input.
There is an option to restrict all rotations that would
mix orbitals of different symmetries. SYMLOC=.TRUE. yields
only partially localized orbitals, but these still possess
symmetry. They are therefore very useful as starting
orbitals for MCSCF or GVB-PP calculations. Because they
still have symmetry, these partially localized orbitals run
as efficiently as the canonical orbitals. Because it is
much easier for a user to pick out the bonds which are to
be correlated, a significant number of iterations can be
saved, and convergence to false solutions is less likely.
* * *
The most important reason for localizing orbitals is to
analyze the wavefunction. A simple example is to look at
shapes of the orbitals with the MacMolPlt program. Or, you
might read the localized orbitals in during a RUNTYP=PROP
job to examine their Mulliken populations.
Localized orbitals are a particularly interesting way
to analyze MCSCF computations. The localized orbitals may
be oriented on each atom (see option ORIENT in $LOCAL) to
direct the orbitals on each atom towards their neighbors
for maximal bonding, and then print a bond order analysis.
The orientation procedure is newly programmed by J.Ivanic
and K.Ruedenberg, to deal with the situation of more than
one localized orbital occuring on any given atom. Some
examples of this type of analysis are
D.F.Feller, M.W.Schmidt, K.Ruedenberg
J.Am.Chem.Soc. 104, 960-967 (1982)
T.R.Cundari, M.S.Gordon
J.Am.Chem.Soc. 113, 5231-5243 (1991)
N.Matsunaga, T.R.Cundari, M.W.Schmidt, M.S.Gordon
Theoret.Chim.Acta 83, 57-68 (1992).
In addition, the energy of your molecule can be
partitioned over the localized orbitals so that you may
be able to understand the origin of barriers, etc. This
analysis can be made for the SCF energy, and also the MP2
correction to the SCF energy, which requires two separate
runs. An explanation of the method, and application to
hydrogen bonding may be found in J.H.Jensen, M.S.Gordon,
J.Phys.Chem. 99, 8091-8107(1995).
Analysis of the SCF energy is based on the localized
charge distribution (LCD) model: W.England and M.S.Gordon,
J.Am.Chem.Soc. 93, 4649-4657 (1971). This is implemented
for RHF and ROHF wavefunctions, and it requires use of
the Ruedenberg localization method, since it needs the
two electron integrals to correctly compute energy sums.
All orbitals must be included in the localization, even
the cores, so that the total energy is reproduced.
The LCD requires both electronic and nuclear charges
to be partitioned. The orbital localization automatically
accomplishes the former, but division of the nuclear
charge may require some assistance from you. The program
attempts to correctly partition the nuclear charge, if you
select the MOIDON option, according to the following: a
Mulliken type analysis of the localized orbitals is made.
This determines if an orbital is a core, lone pair, or
bonding MO. Two protons are assigned to the nucleus to
which any core or lone pair belongs. One proton is
assigned to each of the two nuclei in a bond. When all
localized orbitals have been assigned in this manner, the
total number of protons which have been assigned to each
nucleus should equal the true nuclear charge.
Many interesting systems (three center bonds, back-
bonding, aromatic delocalization, and all charged species)
may require you to assist the automatic assignment of
nuclear charge. First, note that MOIDON reorders the
localized orbitals into a consistent order: first comes
any core and lone pair orbitals on the 1st atom, then
any bonds from atom 1 to atoms 2, 3, ..., then any core
and lone pairs on atom 2, then any bonds from atom 2 to
3, 4, ..., and so on. Let us consider a simple case
where MOIDON fails, the ion NH4+. Assuming the nitrogen
is the 1st atom, MOIDON generates
NNUCMO=1,2,2,2,2
MOIJ=1,1,1,1,1
2,3,4,5
ZIJ=2.0,1.0,1.0,1.0,1.0,
1.0,1.0,1.0,1.0
The columns (which are LMOs) are allowed to span up to 5
rows (the nuclei), in situations with multicenter bonds.
MOIJ shows the Mulliken analysis thinks there are four
NH bonds following the nitrogen core. ZIJ shows that
since each such bond assigns one proton to nitrogen, the
total charge of N is +6. This is incorrect of course,
as indeed will always happen to some nucleus in a charged
molecule. In order for the energy analysis to correctly
reproduce the total energy, we must ensure that the
charge of nitrogen is +7. The least arbitrary way to
do this is to increase the nitrogen charge assigned to
each NH bond by 1/4. Since in our case NNUCMO and MOIJ
and much of ZIJ are correct, we need only override a
small part of them with $LOCAL input:
IJMO(1)=1,2, 1,3, 1,4, 1,5
ZIJ(1)=1.25, 1.25, 1.25, 1.25
which changes the charge of the first atom of orbitals
2 through 5 to 5/4, changing ZIJ to
ZIJ=2.0,1.25,1.25,1.25,1.25,
1.0, 1.0, 1.0, 1.0
The purpose of the IJMO sparse matrix pointer is to let
you give only the changed parts of ZIJ and/or MOIJ.
Another way to resolve the problem with NH4+ is to
change one of the 4 equivalent bond pairs into a "proton".
A "proton" orbital AH treats the LMO as if it were a
lone pair on A, and so assigns +2 to nucleus A. Use of
a "proton" also generates an imaginary orbital, with
zero electron occupancy. For example, if we make atom
2 in NH4+ a "proton", by
IPROT(1)=2
NNUCMO(2)=1
IJMO(1)=1,2,2,2 MOIJ(1)=1,0 ZIJ(1)=2.0,0.0
the automatic decomposition of the nuclear charges will be
NNUCMO=1,1,2,2,2,1
MOIJ=1,1,1,1,1,2
3,4,5
ZIJ=2.0,2.0,1.0,1.0,1.0,1.0
1.0,1.0,1.0
The 6th column is just a proton, and the decomposition
will not give any electronic energy associated with
this "orbital", since it is vacant. Note that the two ways
we have disected the nuclear charges for NH4+ will both
yield the correct total energy, but will give very
different individual orbital components. Most people
will feel that the first assignment is the least arbitrary,
since it treats all four NH bonds equivalently.
However you assign the nuclear charges, you must
ensure that the sum of all nuclear charges is correct.
This is most easily verified by checking that the energy
sum equals the total SCF energy of your system.
As another example, H3PO is studied in EXAM26.INP.
Here the MOIDON analysis decides the three equivalent
orbitals on oxygen are O lone pairs, assigning +2 to
the oxygen nucleus for each orbital. This gives Z(O)=9,
and Z(P)=14. The least arbitrary way to reduce Z(O)
and increase Z(P) is to recognize that there is some
backbonding in these "lone pairs" to P, and instead
assign the nuclear charge of these three orbitals by
1/3 to P, 5/3 to O.
Because you may have to make several runs, looking
carefully at the localized orbital output before the
correct nuclear assignments are made, there is an
option to skip directly to the decomposition when the
orbital localization has already been done. Use
$CONTRL RUNTYP=PROP
$GUESS GUESS=MOREAD NORB=
$VEC containing the localized orbitals!
$TWOEI
The latter group contains the necessary Coulomb and
exchange integrals, which are punched by the first
localization, and permits the decomposition to begin
immediately.
SCF level dipoles can also be analyzed using the
DIPDCM flag in $LOCAL. The theory of the dipole
analysis is given in the third paper of the LCD
sequence. The following list includes application of
the LCD analysis to many problems of chemical interest:
W.England, M.S.Gordon J.Am.Chem.Soc. 93, 4649-4657 (1971)
W.England, M.S.Gordon J.Am.Chem.Soc. 94, 4818-4823 (1972)
M.S.Gordon, W.England J.Am.Chem.Soc. 94, 5168-5178 (1972)
M.S.Gordon, W.England Chem.Phys.Lett. 15, 59-64 (1972)
M.S.Gordon, W.England J.Am.Chem.Soc. 95, 1753-1760 (1973)
M.S.Gordon J.Mol.Struct. 23, 399 (1974)
W.England, M.S.Gordon, K.Ruedenberg,
Theoret.Chim.Acta 37, 177-216 (1975)
J.H.Jensen, M.S.Gordon, J.Phys.Chem. 99, 8091-8107(1995)
J.H.Jensen, M.S.Gordon, J.Am.Chem.Soc. 117, 8159-8170(1995)
M.S.Gordon, J.H.Jensen, Acc.Chem.Res. 29, 536-543(1996)
* * *
It is also possible to analyze the MP2 correlation
correction in terms of localized orbitals, for the RHF
case. The method is that of G.Peterssen and M.L.Al-Laham,
J.Chem.Phys., 94, 6081-6090 (1991). Any type of localized
orbital may be used, and because the MP2 calculation
typically omits cores, the $LOCAL group will normally
include only valence orbitals in the localization. As
mentioned already, the analysis of the MP2 correction must
be done in a separate run from the SCF analysis, which must
include cores in order to sum up to the total SCF energy.
* * *
Typically, the results are most easily interpreted
by looking at "the bigger picture" than at "the details".
Plots of kinetic and potential energy, normally as a
function of some coordinate such as distance along an
IRC, are the most revealing. Once you determine, for
example, that the most significant contribution to the
total energy is the kinetic energy, you may wish to look
further into the minutia, such as the kinetic energies
of individual localized orbitals, or groups of LMOs
corresponding to an entire functional group.
Transition Moments and Spin-Orbit Coupling
A review of various ways of computing spin-orbit coupling:
D.G.Fedorov, S.Koseki, M.W.Schmidt, M.S.Gordon,
Int.Rev.Phys.Chem. 22, 551-592(2003)
GAMESS can compute transition moments and oscillator
strengths for the radiative transitions between states
written in terms of CI wavefunctions (GUGA only). The
transition moments are computed using both the "length
form" and the "velocity form". In a.u., where h-bar=m=1, we
start from
[A,q] = -i dA/dp
For A=H, dH/dp=p, and p= -i d/dq,
[H,q] = -i p = -d/dq.
For non-degenerate states,
=
(Ea-Eb) = -
This relates the dipole to the velocity form,
= -1/(Ea-Eb)
but the CI states will give different numbers for each
side, since the states aren't exact eigenfunctions.
Transition moment computation is OPERAT=DM in $TRANST. For
transition moments, the CI is necessarily performed on
states of the same multiplicity.
All other operators are various spin-orbit coupling
options. There are two kinds of calculations possible,
which we will call SO-CI and SO-MCQDPT. Note that there
is a hyphen in "spin-orbit CI" to avoid confusion with
"second order CI" in the sense of the SOCI keyword in $DRT
input. For SO-CI, the initial states may be any CI wave-
function that the GUGA package can generate. For SO-MCQDPT
the initial states for spin-orbit coupling are of CAS type,
and the operator mixing them corresponds to MCQDPT
generalised for spin-dependent operators (with certain
approximations).
GAMESS can compute the "microscopic Breit-Pauli
spin-orbit operator", which includes both a one and two
electron operator. Additional information is given in a
subsection below
states
For transition moments, the states are generated by
CI calculations using the GUGA package. These states are
the final states, and the results are just the transition
moments between these states. The states are defined by
$DRTx input groups.
For SO-CI, the energy of the CI states forms the
diagonal of a spin-orbit Hamiltonian, as in the state basis
the spin-free Hamiltonian is of course diagonal. Addition
of the Pauli-Breit operator does not change the diagonal,
but does add off-diagonal H-so elements. For SO-MCQDPT,
the spin-free MCQDPT matrix elements are expanded into
matrices corresponding to all Ms values for a pair of
multiplicities. These matrices are block-diagonal before
the addition of spin-orbit coupling terms, coupling Ms
values. The diagonalization of this spin-orbit Hamiltonian
gives new energy levels, and spin-mixed final states.
Optionally, the transition dipoles between the final states
can be computed. The input requirements are $DRTx or
$MCQDx groups which define the original pure spin states.
We will call the initial states CAS-CI, since most of
the time they will be MCSCF states. There may be cases
such as the Na example below where SCF orbitals are used,
or other cases where a FOCI or SOCI wavefunction might be
used for the initial states. Please keep in mind that the
term does not imply the states must be MCSCF states, just
that they commonly are.
In the above, x may vary from 1 to 64. The reason for
allowing such a large range is to permit the use of Abelian
point group symmetry during the generation of the initial
states. The best explanation will be an example, but the
number of these input groups depends on both the number of
orbital sets input, and how much symmetry is present. The
next two subsections discuss these points.
orbitals
The orbitals for transition moments or for SO-CI can be
one common set of orbitals used by all CI states. If one
set of orbitals is used, the transition moment or spin-
orbit coupling can be found for any type of GUGA CI wave-
function. Alternatively, two sets of orbitals (obtained by
separate MCSCF orbital optimizations) can be used. Two or
more separate CIs will be carried out. The two MO sets
must share a common set of frozen core orbitals, and the
CI must be of the complete active space type. These
restrictions are needed to leave the CI wavefunctions
invariant under the necessary rotation to corresponding
orbitals. The non-orthogonal procedure implemented is a
GUGA driven equivalent to the method of Lengsfield, et al.
Note that the FOCI and SOCI methods described by these
workers are not available in GAMESS.
If you would like to use separate orbitals during the
CI, they may be generated with the FCORE option in $MCSCF.
Typically you would optimize the ground state completely,
then use these MCSCF orbitals in an optimization of the
excited state, under the constraint of FCORE=.TRUE.
For SO-MCQDPT calculations, only one set of orbitals
may be input to describe all CAS-CI states. Typically that
orbital set will be obtained by state-averaged MCSCF, see
WSTATE in $DET/$DRT, and also in the $MCQDx input. Note
that although the RUNTYP=TRANSITN driver is tied to the
GUGA CI package, there is no reason the orbitals cannot be
obtained using the determinant CI package. In fact, for
the case of spin-orbit coupling, you might want to utilize
the ability to state average over several spins, see PURES
in $DET.
If there is no molecular symmetry present, transition
moment calculations will provide $DRT1 if there is one set
of orbitals, otherwise $DRT1 defines the CI based on $VEC1
and $DRT2 the CI based on $VEC2. Also for the case of no
symmetry, a spin-orbit job should enter one $DRTx or $MCQDx
for every spin multiplicity, and all states of the same
multiplicity have to be generated from $VEC1 or $VEC2,
according to IVEX input.
symmetry
The CAS-CI states are most efficiently generated using
symmetry, since states of different symmetry have zero
Hamiltonian matrix elements. It is probably more efficient
to do four CI calculations in the group C2v on A1, A2, B1,
and B2 symmetry, than one CI with a combined Hamiltonian
in C1 symmetry (unless the active space is very small), and
similar remarks apply to the SO-MCQDPT case. In order to
avoid repeatedly saying $DRTx or $MCQDx, the following few
paragraphs say $DRTx only.
Again supposing the group is C2v, and you are
interested in singlet-triplet coupling. After some
preliminary CI calculations, you might know that the lowest
8 states are two 1-a1, 1-b1, two 1-b2, one 3-a1, and two 3-
b2 states. In this case your input would consist of five
$DRTx, of which you can give the three singlets in any
order but these must preceed the two triplet input groups
to follow the rule of increasing multiplicity. Clearly it
is not possible to write a formula for how many $DRTx there
will be, this depends not only on the point group, but also
the chemistry of the situation.
If you are using two sets of orbitals, the generation
of the corresponding orbitals for the two sets will permute
the active orbitals in an unpredictable way. Use STSYM to
define the desired state symmetry, rather than relying on
the orbital order. It is easy and safer to be explicit
about the spatial orbital symmetry.
The users are encouraged to specify full symmetry in
their $DATA input even though they may choose to set the
symmetry in $DRTx to C1. The CI states will be labelled in
the group given in $DATA. The use of non-Abelian symmetry
is limited by the absence of non-Abelian CI or MCQDPT. In
this case the users can choose between setting full non-
Abelian symmetry in $DATA and C1 in $DRT or else an Abelian
subgroup in both $DATA and $DRT. The latter choice appears
to be most efficient at present.
An example of SO-MCQDPT illustrating how the carbon
atom of Kh symmetry (full rotation-reflection group) can be
entered in D2h, Kh's highest Abelian group. The run time is
considerably longer in C1 symmetry.
As another example, consider an organic molecule with a
singly excited state, where that state might be coupled to
low or high spin, and where these two states might be close
enough to have a strong spin-orbit coupling. If it happens
that the S1 and S0 states possess different symmetry, a
very reaasonable calculation would be to treat the S1 and
T1 state with the same $VEC2 orbitals, leaving the ground
state described by $VEC1. After doing an MCSCF on the S0
ground state for $VEC1, you could do a state-averaged MCSCF
for $VEC2 optimized for T1 and S1 simultaneously, using
PURES. The spin orbit job would obtain its initial states
from three GUGA CI computations, S0 from $VEC1 and $DRT1,
S1 from $VEC2 and $DRT2, and T1 from $VEC2 and $DRT3. Your
$TRANST would be NUMCI=3, IROOTS(1)=1,1,1, IVEX(1)=1,2,2.
Note that the second IROOTS value is 1 because S1 was
presumed to have a different symmetry than S0, so STSYM in
$DRT1 and $DRT2 will differ. The calculation just outlined
cannot be done if S0 and S1 have the same spatial symmetry,
as IROOTS(1)=1,2,1 to obtain S1 during the second CI will
bring in an additional S0 state (one expressed in terms of
the $VEC2, at slightly higher energy). This problem is the
origin of the statement several paragraphs above that a
system with no symmetry will have one $DRTx for every spin
multiplicity included.
For transition moments, which do not diagonalize a
matrix containing these duplicated states, it is OK to
proceed, provided you ignore all transition moments between
the same states obtained in the two different CIs.
spin orbit coupling
Spin-orbit coupling calculations are always performed
in a quasi-degenerate perturbative manner. Typically the
states close in energy are included into the spin-orbit
coupling matrix. "Close" has a easily understandable
meaning, since in the limit of small coupling the quasi-
degenerate treatment is reduced to a second order
perturbative treatment, that is, the affect of a state upon
the state of primary interest is given by the square of the
spin-orbit coupling matrix element divided by the
difference of the adiabatic energies. This is useful to
keep in mind when deciding how many CI states to include in
the matrix. The states that are included are treated in a
fashion that is equivalent to infinite order perturbation
theory (exact) whereas the states that are not included
make no contribution.
Spin-orbit runs can be done for even or odd numbers of
electrons (any spin), for more than two different spin
multiplicities at once, for general active spaces. At
times, when the spatial wavefunction is degenerate, a spin-
orbit run might involve only one spin multiplicity, e.g. a
triplet-pi state in a linear molecule. The most common
case is two different spins, with non-zero spin orbit
coupling possible only for delta-S=1: singlets spin-orbit
couple with triplets, but not with quintets. Use of three
spins, such as S=0,1,2, will generate couplings between
singlets and triplets, and between triplets and quintets,
which together engender an indirect singlet/quintet mixing.
As noted above, the treatment of spin-orbit involves
first obtaining a handful of spin-pure states, whose
energies form the diagonal of a model Hamiltonian. The
spin-orbit operator introduces off-diagonal couplings, and
the resulting small Hamiltonian is diagonalized. This
generates spin-mixed states in the model space. Since the
model states are not fully relaxed (internally contracted),
this is essentially a perturbative treatment: certainly
the spin-orbit effects have no influence on orbital
optimizations or potential energy surfaces when treated in
this manner, at the very last stage.
The Breit-Pauli spin-orbit operator contains a one
electron term arising from Pauli's reduction of the Dirac
hydrogenic equation to a single-component form, and a two
electron term due to Breit. Computation of the full Breit-
Pauli operator is OPERAT=HSO2 (or HSOFF). A close
approximation to the latter is HSO2P (P=partial), which
neglects all active-active two electron terms, which
usually do not contribute very much to the total coupling,
while saving substantial computer time.
HSO1 completely omits the two electron terms, so is
much faster than any of the two electron operators, but
represents a potentially much greater loss of accuracy.
HSO1's error can be remedied to some extent by regarding
the nuclear charge in the one electron term as an
adjustable parameter. In addition, these effective charges
are often used to compensate for missing nodes in valence
orbitals of ECP runs, in which case the ZEFF are typically
very far from the two nuclear charges. ZEFTYP selects some
built in values obtained by S.Koseki et al, but if you have
some favorite parameters, they can be read in as the ZEFF
input array. Effective charges may be used for any OPERAT,
but are most often used with HSO1.
Theoretical considerations indicate that the Breit-
Pauli operator is not variationally stable, if it were to
be used during the SCF iterations determining orbitals.
However, a first order Douglas-Kroll type correction to the
one electron part of the operator reduces its size by means
of certain kinematic factors, removing this problem. This
can produce substantially better results, even if the
operator is being treated pertubatively. This correction
(at first order) is automatically applied to the one
electron part of the Breit-Pauli operator by any run that
selects RELWFN=DK (at any order) in the scalar relativity
treatment during the variational steps prior to the spin-
orbit perturbation. See paper 32 below.
Because the diagonalization of the model spin-orbit
Hamiltonian leads to spin-mixed eigenvectors, approximate
wavefunctions including spin-orbit coupling are generated.
It is now possible to generate the density matrix for the
spin-mixed states, so that property values for spin-mixed
states can be found: see keyword ISTNO. The natural
orbitals of these spin-orbit density matrices turn out to
be good approximations to the two spinors of the large
components of full Dirac four component runs. The total
density of these spinors can be obtained for interpretation
purposes. Recognition that the spin-orbit coupling is a
rotational operator (L dot S, where L = R cross P) when it
acts on an orbital can lead to chemical interpretability of
the spin orbit results. See papers 40 and 41 below.
It is also possible to obtain the dipole transition
moments between the final spin-mixed wavefunctions, which
of course do not any longer have a rigourous S quantum no.
When the run is SO-MCQDPT, the transition moment are first
computed only between CAS states, and then combined with
the spin-mixed SO-MCQDPT coefficients.
technical matters:
The only practical limitation on the computation of
the Breit term is that HSO2FF is limited to 10 active
orbitals on 32 bit machines, and to about 26 active
orbitals on 64 bit machines. The spin-orbit matrix
elements vanish for |delta-S| > 1, but it is possible to
include three or more spins in the computation. Since
singlets interact with triplets, and triplets interact with
pentuplets, inclusion of S=0,1,2 simultaneously lets you
pick up the indirect interaction between singlets and
pentuplets that the intermediate triplets afford.
The choice between HSO2 and HSO2FF is very often in
favor of the former. HSO2 computes the matrix elements in
CSF basis and then contracts them with CI coefficients,
whereas HSO2FF uses a generalized density in AO basis
computed for each pair of states, thus HSO2 is much more
efficient in case of multiple states given in IROOTS.
HSO2FF takes less memory for integral storage, thus it can
be superior in case of small active spaces and large basis
sets, in part because it does not store 2e SOC integrals on
disk and secondly, it does not redundantly treat the same
pair of determinants if they appear in different CSFs. The
numerical results with HSO2 and HSO2FF should be identical
within machine and algorithmic accuracy.
Various symmetries are used to avoid computing zero
spin-orbit matrix elements. NOSYM in $TRANST allows some
control over this: NOSYM=1 gives up point group symmetry
completely, while 2 turns off additional symmetries such
as spin selection rules. HSO1,2,2P compute all matrix
elements in a group (i.e. between two $DRTx groups with
fixed Ms(ket)-Ms(bra)) if at least one of them does not
vanish by symmetry, and HSO2P actually avoids computation
for each pair of states if forbidden by symmetry. Setting
NOSYM=2 will cause HSO2FF to consider the elements between
two singlets, which are always calculated for HSO1,2,2P
when transition dipoles are requested as well.
SYMTOL has a dramatic effect on the run speed. This
cutoff is applied to CSF coefficcients, their products,
and these products times CSF orbital overlaps. The value
permits a tradeoff of accuracy for run time, and since the
error in the spin-orbit properties approaches SYMTOL mainly
for SOCI functions, it may be useful to increase SYMTOL to
save time for CAS or FOCI functions. Some experimenting
will tell you what you can get away with. SYMTOL is also
used during CI state symmetry assignment, for NOIRR=-1
in $DRT.
In case if you do not provide enough storage for the
form factors sorting then some extra disk space will be
used; the extra disk space can be eliminated if you set
SAVDSK=.TRUE. (the amount of savings depends on the active
space and memory provided, it some cases it can decrease
the disk space up to one order of magnitude). The form
factors are in binary format, and so can be transfered
between computers only if they have compatible binary
files. There is a built-in check for consistency of a
restart file DAFL30 with the current run parameters.
input nitty-gritty
The transition moment and spin-orbit coupling driver
is a rather restricted path through the GUGA CI part of
GAMESS. Note that $GUESS is not read, instead the MOs will
be MOREAD in a $VEC1 and perhaps a $VEC2 group. It is not
possible to reorder MOs. For SO-CI,
1) Give SCFTYP=NONE CITYP=GUGA MPLEVL=0.
2) $CIINP is not read. The CI is hardwired to consist
of CI DRT generation, integral transformation/sorting,
Hamiltonian generation, and diagonalization. This
means $DRT1 (and maybe $DRT2,...), $TRANS, $CISORT,
$GUGEM, and $GUGDIA input is read, and acted upon.
3) The density matrices are not generated, and so no
properties (other than the transition moment or the
spin-orbit coupling) are computed.
4) There is no restart capability provided, except for
saving some form-factor information.
5) $DRT1, $DRT2, $DRT3, ... must go from lowest to highest
multiplicity.
6) IROOTS will determine the number of CI states in each
CI for which the properties are calculated. Use
NSTATE to specify the number of CI states for the
CI Hamiltonian diagonalization. Sometimes the CI
convergence is assisted by requesting more roots
to be found in the diagonalization than you want to
include in the property calculation.
For SO-MCQDPT, the steps are
1) Give SCFTYP=NONE CITYP=NONE MPLEVL=2.
2) the number of roots in each MCQDPT is controlled by
$TRANST's IROOTS, and each such calculation is
defined by $MCQD1, $MCQD2, ... input. These must go
from lowest multiplicity to highest.
references
The review already mentioned:
"Spin-orbit coupling in molecules: chemistry beyond the
adiabatic approximation".
D.G.Fedorov, S.Koseki, M.W.Schmidt, M.S.Gordon,
Int.Rev.Phys.Chem. 22, 551-592(2003)
Reference for separate active orbital optimization:
1. B.H.Lengsfield, III, J.A.Jafri, D.H.Phillips,
C.W.Bauschlicher, Jr. J.Chem.Phys. 74,6849-6856(1981)
References for transition moments:
2a. H.C.Longuet-Higgins
Proc.Roy.Soc.(London) A235, 537-543(1956)
2b. F.Weinhold, J.Chem.Phys. 54,1874-1881(1970)
3. C.W.Bauschlicher, S.R.Langhoff
Theoret.Chim.Acta 79:93-103(1991)
4. "Intermediate Quantum Mechanics, 3rd Ed." Hans A.
Bethe, Roman Jackiw Benjamin/Cummings Publishing,
Menlo Park, CA (1986), chapters 10 and 11.
5. S.Koseki, M.S.Gordon
J.Mol.Spectrosc. 123, 392-404(1987)
References for HSO1 spin-orbit coupling, and Zeff values:
6. S.Koseki, M.W.Schmidt, M.S.Gordon
J.Phys.Chem. 96, 10768-10772 (1992)
7. S.Koseki, M.S.Gordon, M.W.Schmidt, N.Matsunaga
J.Phys.Chem. 99, 12764-12772 (1995)
8. N.Matsunaga, S.Koseki, M.S.Gordon
J.Chem.Phys. 104, 7988-7996 (1996)
9. S.Koseki, M.W.Schmidt, M.S.Gordon
J.Phys.Chem.A 102, 10430-10435 (1998)
10. S.Koseki, D.G.Fedorov, M.W.Schmidt, M.S.Gordon
J.Phys.Chem.A 105, 8262-8268 (2001)
11. S.Koseki, T.Matsushita, M.S.Gordon
J.Phys.Chem.A 110, 2560-2570(2006)
References for full Breit-Pauli spin-orbit coupling:
20. T.R.Furlani, H.F.King
J.Chem.Phys. 82, 5577-5583 (1985)
21. H.F.King, T.R.Furlani
J.Comput.Chem. 9, 771-778 (1988)
22. D.G.Fedorov, M.S.Gordon
J.Chem.Phys. 112, 5611-5623 (2000)
Paper 22 contains information on the HSO2P partial two
electron operator method.
Symmetry in spin-orbit coupling:
23. D.G.Fedorov, M.S.Gordon
ACS Symposium Series 828, pp 1-22(2002)
Reference for SO-MCQDPT:
25. D.G.Fedorov, J.P.Finley Phys.Rev.A 64, 042502 (2001)
Reference for Spin-Orbit with Model Core Potentials:
30. D.G.Fedorov, M.Klobukowski
Chem.Phys.Lett. 360, 223-228(2002)
31. T.Zeng, D.G.Fedorov, M.Klobukowski
J.Chem.Phys. 131, 124109/1-17(2009)
32. T.Zeng, D.G.Fedorov, M.Klobukowski
J.Chem.Phys. 132, 074102/1-15(2010)
The last two of these also discuss the 1st order Douglas-
Kroll transformation of the 1e- part of the spin orbit
operator.
Reference for properties, interpretations, and Spin-Orbit
Natural Spinors:
40. T. Zeng, D. G. Fedorov, M. W. Schmidt, M. Klobukowski
J. Chem. Phys. 134, 214107/1-9(2011)
41. T. Zeng, D. G. Fedorov, M. W. Schmidt, M. Klobukowski
J. Chem. Theor. Comput. 7, 2864-2875(2011)
with an application being
42. T.Zeng, D.G.Federov, M.W.Schmidt, M.Klobukowski
J.Chem.Theo.Comput. 8, 3061-3071(2012).
Recent applications (see also 32,40,41):
50. S.P.Webb, M.S.Gordon
J.Chem.Phys. 109, 919-927(1998)
51. D.G.Fedorov, M.Evans, Y.Song, M.S.Gordon, C.Y.Ng
J.Chem.Phys. 111, 6413-6421 (1999)
52. D.G.Fedorov, M.S.Gordon, Y.Song, C.Y.Ng
J.Chem.Phys. 115, 7393-7400 (2001)
53. B.J.Duke J.Comput.Chem. 22, 1552-1556 (2001)
54. C.M.Aikens, M.S.Gordon
J.Phys.Chem.A 107, 104-114(2003)
55. D.G.Fedorov, S.Koseki, M.W.Schmidt, M.S.Gordon
K.Hirao and Y.Ishikawa (eds.) Recent Advances in
Relativistic Molecular Theory, Vol. 5,
(World Scientific, Singapore), 2004, pp 107-136.
* * *
Special thanks to Bob Cave and Dave Feller for their
assistance in performing check spin-orbit coupling runs
with the MELDF programs. Special thanks to Tom Furlani for
contributing his 2e- spin-orbit code and answering many
questions about its interface. Special thanks to Haruyuki
Nakano for explaining the spin functions used in the MCQDPT
package.
examples
We end with 2 examples. Note that you must know what
you are doing with term symbols, J quantum numbers, point
group symmetry, and so on in order to make skillful use of
this part of the program. Seeing your final degeneracies
turn out like a text book says it should is beautiful!
! Compute the splitting of the famous sodium D line.
! Joseph von Fraunhofer (Denkschriften der Koeniglichen
! Akademie der Wissenschf. zu Muenchen, 5, 193(1814-1815))
! observed the sun through good prisms, finding 700 lines,
! and named the brightest ones A, B, C... just in order.
! He was able to resolve the D line into two lines, which
! occur at 5895.940 and 5889.973 Angstroms. It would take
! a century to understand the D line is Na's 3s <-> 3p
! transition, and that spin-orbit coupling is what splits
! the D line into two. Charlotte Moore's Atomic Energy
! Levels, volume 1, gives the experimental 2-P interval
! as 17.1963, since the three relevent levels are at
! 2-S-1/2= 0.0, 2-P-1/2= 16,956.183, 2-P-3/2= 16,973.379.
1. generate ground state 2-S orbitals by conventional ROHF.
the energy of the ground state is -161.8413919816
--- $contrl scftyp=rohf mult=2 $end
--- $system kdiag=3 memory=300000 $end
--- $guess guess=huckel $end
2. generate excited state 2-P orbitals, using a state-
averaged SCF wavefunction to ensure radial degeneracy of
the 3p shell is preserved. The open shell SCF energy is
-161.7682895801. The computation is both spin and space
restricted open shell SCF on the 2-P Russell-Saunders term.
Starting orbitals are reordered orbitals from step 1.
--- $contrl scftyp=gvb mult=2 $end
--- $system kdiag=3 memory=300000 $end
--- $guess guess=moread norb=13
--- norder=1 iorder(6)=7,8,9,6 $end
--- $scf nco=5 nseto=1 no(1)=3 rstrct=.t. couple=.true.
--- f(1)= 1.0 0.16666666666667
--- alpha(1)= 2.0 0.33333333333333 0.0
--- beta(1)= -1.0 -0.16666666666667 0.0 $end
3. compute spin-orbit coupling in the 2-P term. The use of
C1 symmetry in $DRT1 ensures that all three spatial CSFs
are kept in the CI function. In the preliminary CI, the
spin function is just the alpha spin doublet, and all three
roots should be degenerate, and furthermore equal to the
GVB energy at step 2. The spin-orbit coupling code uses
both doublet spin functions with each of the three spatial
wavefunctions, so the spin-orbit Hamiltonian is a 6x6
matrix. The two lowest roots of the full 6x6 spin-orbit
Hamiltonian are the doubly degenerate 2-P-1/2 level, while
the other four roots are the degenerate 2-P-3/2 level.
$contrl scftyp=none cityp=guga runtyp=transitn mult=2 $end
$system memory=2000000 $end
$basis gbasis=n31 ngauss=6 $end
$gugdia nstate=3 $end
$transt operat=hso1 numvec=1 numci=1 nfzc=5 nocc=8
iroots=3 zeff=10.04 $end
$drt1 group=c1 fors=.true. nfzc=5 nalp=1 nval=2 $end
$data
Na atom...2-P excited state...6-31G basis
Dnh 2
Na 11.0
$end
--- GVB ORBITALS --- GENERATED AT 7:46:08 CST 30-MAY-1996
Na atom...2-P excited state
E(GVB)= -161.7682895801, E(NUC)= .0000000000, 5
ITERS
$VEC1
1 1 9.97912679E-01 8.83038094E-03 0.00000000E+00...
... orbitals from step 2 go here ...
13 3-1.10674398E+00 0.00000000E+00 0.00000000E+00
$END
As an example of both SO-MCQDPT, and the use of as much
symmetry as possible, consider carbon. The CAS-CI uses
an active space of 2s,2p,3s,3p orbitals, and the spin-orbit
job includes all terms from the lowest configuration,
2s2,2p2. These terms are 3-P, 1-D, and 1-S. If you look
at table 58 in Herzberg's book on electronic spectra, you
will be able to see how the Kh spatial irreps P, D, S are
partitioned into the D2h irreps input below.
! C SO-MRMP on all levels in the s**2,p**2 configuration.
!
! levels CAS and MCQDPT
! 1 .0000 .0000 cm-1 3-P-0
! 2-4 12.6879-12.8469 13.2721-13.2722 3-P-1
! 5-9 37.8469-37.8470 39.5638-39.5639 3-P-2
! 10-14 12169.1275 10251.7910 1-D-2
! 15 19264.4221 21111.5130 1-S-0
!
! The active space consists of (2s,2p,3s,3p) with 4 e-.
! D2h symmetry speeds up the calculation considerably,
! on the same computer D2h = 78 and C1 = 424 seconds.
$contrl scftyp=none cityp=none mplevl=2
runtyp=transitn $end
$system memory=5000000 $end
!
! below is input to run in C1 subgroup
!
--- $transt operat=hso2 numvec=-2 numci=2 nfzc=1 nocc=9
--- iroots(1)=6,3 parmp=3
--- ivex(1)=1,1 $end
--- $mrmp mrpt=mcqdpt rdvecs=.t. $end
--- $MCQD1 nosym=1 nstate=6 mult=1 iforb=3
--- nmofzc=1 nmodoc=0 nmoact=8
--- wstate(1)=1,1,1,1,1,1 thrcon=1e-8 thrgen=1e-10 $END
--- $MCQD2 nosym=1 nstate=3 mult=3 iforb=3
--- nmofzc=1 nmodoc=0 nmoact=8
--- wstate(1)=1,1,1 thrcon=1e-8 thrgen=1e-10 $END
!
! below is input to run in D2h subgroup
!
$transt operat=hso2 numvec=-7 numci=7 nfzc=1 nocc=9
iroots(1)=3,1,1,1, 1,1,1 parmp=3
ivex(1)=1,1,1,1,1,1,1 $end
$mrmp mrpt=mcqdpt rdvecs=.t. $end
$MCQD1 nosym=-1 mult=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=Ag wstate(1)=1,1,1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD2 nosym=-1 mult=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=B1g wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD3 nosym=-1 mult=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=B2g wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD4 nosym=-1 mult=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=B3g wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD5 nosym=-1 mult=3 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=B1g wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD6 nosym=-1 mult=3 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=B2g wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD7 nosym=-1 mult=3 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
stsym=B3g wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
!
! input to prepare the 3-P ground state orbitals
! great care is taken to create symmetry equivalent p's
!
--- $contrl scftyp=mcscf cityp=none mplevl=0
--- runtyp=energy mult=3 $end
--- $guess guess=moread norb=55 purify=.t. $end
--- $mcscf cistep=guga fullnr=.t. $end
--- $drt group=c1 fors=.true.
--- nmcc=1 ndoc=1 nalp=2 nval=5 $end
--- $gugdia nstate=9 maxdia=1000 $end
--- $gugdm2 wstate(1)=1,1,1 $end
!
$data
C...aug-cc-pvtz (10s,5p,2d,1f) -> [4s,3p,2d,1f]
(1s,1p,1d,1f)
Dnh 2
C 6.0
S 8
1 8236.000000 0.5310000000E-03
2 1235.000000 0.4108000000E-02
3 280.8000000 0.2108700000E-01
4 79.27000000 0.8185300000E-01
5 25.59000000 0.2348170000
6 8.997000000 0.4344010000
7 3.319000000 0.3461290000
8 0.3643000000 -0.8983000000E-02
S 8
1 8236.000000 -0.1130000000E-03
2 1235.000000 -0.8780000000E-03
3 280.8000000 -0.4540000000E-02
4 79.27000000 -0.1813300000E-01
5 25.59000000 -0.5576000000E-01
6 8.997000000 -0.1268950000
7 3.319000000 -0.1703520000
8 0.3643000000 0.5986840000
S 1
1 0.9059000000 1.000000000
S 1
1 0.1285000000 1.000000000
P 3
1 18.71000000 0.1403100000E-01
2 4.133000000 0.8686600000E-01
3 1.200000000 0.2902160000
P 1
1 0.3827000000 1.000000000
P 1
1 0.1209000000 1.000000000
D 1
1 1.097000000 1.000000000
D 1
1 0.3180000000 1.000000000
F 1
1 0.7610000000 1.000000000
S 1
1 0.440200000E-01 1.00000000
P 1
1 0.356900000E-01 1.00000000
D 1
1 0.100000000 1.00000000
F 1
1 0.268000000 1.00000000
$end
--- OPTIMIZED MCSCF MO-S --- GENERATED 22-AUG-2000
E(MCSCF)= -37.7282408589, 11 ITERS
$VEC1
1 1 9.75511467E-01 ...snipped...
$END
Edited by Shiro KOSEKI.