$RELWFN group                              (optional)
 
    This group is relevant if RELWFN in $CONTRL choses any
of the relativistic transformations for elimination of the
small components of relativistic wavefunctions, to produce
corrected single component wavefunctions.  These scalar
relativistic corrections may be included during any self-
consistent method, and any correlation treatment may be
used.  Wavefunctions incorporating scalar relativity may
also be used by the spin-orbit coupling perturbation
program (see RUNTYP=TRANSITN, and NESOC just below).
 
    The RELWFN keywords are intended for use in all-
electron calculations only.  Scalar relativistic effects
may also be treated by the use of ECP-type or MCP-type core
potentials, in which case see the PP keyword in $CONTRL.
 
    One family of ESC methods began with the relativistic
elimination of small components (RESC), continued through
second and third order Douglas/Kroll (DK), reaching an
infinite order two component scheme (IOTC) equivalent to
converging the DK series.  The pinnacle of this line is the
local unitary transformation approximation to full IOTC
(LUT-IOTC).  RELWFN=LUT-IOTC is the most numerically
accurate and fastest running method available, so the use
of LUT-IOTC is recommended.
 
    Within this ESC progression, only one electron kinetic
energy, nuclear attraction, and overlap integrals (and
associated one electron gradient terms) are modified.  Note
that scalar 2e- relativistic corrections exist in nature,
as well as the Dirac-Coulomb equation, but are not treated
by RESC, DK, IOTC, or LUT-IOTC.  One electron effects are
larger by far, being about 1,147 Hartree for a gold atom,
compared to 27 Hartrees for Au's two electron correction.
 
    The Normalized Elimination of Small Components (NESC)
treats corrections to two electron integrals by means of a
relativistically averaged basis set.  This is in addition
to the one electron modifications mentioned above.  All of
the relativistic methods in GAMESS neglect two-electron
corrections coming from pVp integrals.
 
    Analytic gradients are available for any RELWFN choice,
provided the basic quantum chemistry method itself has
gradient programming.  NESC, RESC, and LUT-IOTC have fully
analytical gradients.  For DK and IOTC, the relativistic
gradient contributions are evaluated numerically by a
double difference formula, so that one might think of their
gradients as "semi-analytic".  Relativistic force constant
matrices are evaluated by semi-numerical differencing of
relativistic gradients.  The accuracy of the LUT-IOTC
gradients is similar to non-relativistic runs, and should
be suitable for frequency evaluation.
 
    For NESC, RESC, any order DK, or IOTC (but not LUT-
IOTC), the 1e- part of the Breit-Pauli operator's integrals
are corrected only to first order (DK1): this is keyword
NESOC=1 below.  It has been observed by many people that
even the first order correction is small, and thus should
be sufficient.
 
    Scalar relativity produces great changes in radial
sizes of atomic orbitals, so care must be paid to the basis
set.  Certainly at the bottom of the periodic table, one
must use basis sets which have been contracted using some
kind of relativistic treatment (literature basis sets often
use 2nd order DK when contracting, and these are fine to
use with RELWFN=LUT-IOTC.  The best choices available, at
present, are the Sapporo core/valence type relativistic
bases (see SPKrnDZ, n=D,T,Q in $BASIS), available H-Rn.
Alternatives include the University of Tokyo's DK3 basis
sets for H-Lr obtained at U. of Tokyo which exist in the
form of general contractions.  The web site
            http://www.riken.jp/qcl/
      publications/dk3bs/periodic_table.html
gives the supplemental data from
     T.Tsuchiya, M.Abe, T.Nakajima, K.Hirao
       J.Chem.Phys. 115,4463-4472(2001)
which may be processed into $DATA input with the helper
program dk3.f found in source code distributions of GAMESS.
Using uncontracted WTBS basis sets may be reasonable for
very small molecules.  Finally, one might check the PNNL
web site looking for other relativistic basis sets.
 
    For NESC, you must provide three basis sets, for the
large and small components and an averaged one, which are
given in $DATAL, $DATAS, $DATA, respectively.  The only
possible choice for these basis sets is due to Dyall, and
these are available from
   http://www.emsl.pnl.gov:2080/forms/basisform.html Their
names are similar to cc-pVnZ(pt/sf/lc), pt=point or
fi=finite nucleus, sf for spin-free and the final field is
lc=large component ($DATAL), sc=small component ($DATAS),
and wf is a typo for Foldy-Wouthuysen 2e- basis ($DATA).
In GAMESS you can only use point nucleus approximation, so
do not select any of the 'finite nucleus size' type.  The
need to input three basis sets means that you cannot use
$BASIS input, and you must use COORD=UNIQUE style input in
the various $DATA's.  The three $DATA input groups must
contain identical information except for the primitive
expansion coefficients, as the three basis sets must have
the same exponents.  In case the options below to treat
only some atoms relativistically is chosen, all non-
relativistic atoms must have identical basis input in all
three groups.
 
    During geometry optimizations, in rare cases, the
number of nearly linearly independent functions in the
Resolution of the Identity (RI) used to evaluate the most
difficult integrals may change at some new geometry.  If
so, the job will quit with an error message, and the user
must restart it again manually.
 
 
* * * the next parameter applies only to RELWFN=DK:
 
NORDER gives the order of the DK transformation to be
       applied to the one-electron potential:
     = 1 corresponds to the free particle
     = 2 is the most commonly implemented DK method.  It
         has all relativistic corrections to second order.
         (default)
     = 3 represents 3rd order DK transformation.  It does
         not include all 3rd order relativity corrections,
         in the sense of collecting all terms in the same
         order of c (speed of light), due to using only a
         2nd order form of the Coulomb potential (1/rij).
         However, DK3 gives the closest approximation to
         the Dirac-Coulomb equation of all methods here.
 
* * * the next parameter applies to spin-orbit coupling:
 
NESOC  requests the Douglas-Kroll 1st order relativistic
       corrections for the 1e- SOC integrals.  It has been
       observed that the 1st order correction is often
       sufficient.
       NESOC is relevant only if OPERAT=HSO1, HSO2P, or
       HSO2, for RUNTYP=TRANSITN.
     = 0 no corrections (default for no relativity)
         This is the only choice possible for LUT-IOTC.
     = 1 apply DK1 correction to one-electron spin-orbit
         integrals.  This is the default if any of RESC,
         NESC, DK, or IOTC scalar relativity was chosen).
 
       * * * the next few parameters are used by * * *
                LUT-IOTC, IOTC, DK, and RESC:
 
MODEQR are options for quasi-relativistic calculations.
       The default is 1.  Most runs will select 1, or else
       9 if additional accuracy is needed in generating the
       RI basis due to a large span in Gaussian exponents.
       These are additive (bitwise) options, meaning you
       would enter 11 to request options 1+2+8:
 
       = 0 use the input contracted atomic basis set for
           the Resolution of the Identity (RI) used to
           simplify the pVp relativistic integrals, in
           order to evaluate them in closed form.
           The accuracy of the RI will be severely
           compromised, so this option is not recommended.
       = 1 use the Gaussian primitives constituting the
           input contracted atomic basis set to define the
           RI.  This produces a considerable increase in
           accuracy of the integrals compared to "0".
       = 2 The uncontracted GTO basis set will be used in
           spherical harmonic form, which helps eliminate
           linear dependence cleanly from the RI steps.
           However, this option is not available for
           nuclear gradients, so it is not used by default.
           You might choose to this for extra accuracy,
           when doing final single point energy runs.
           ISPHER=1 to choose spherical harmonics for the
           contracted basis used elsewhere in the run may
           always be used, and should be selected if "2"
           is chosen.
       = 4 avoid redundant exponents when splitting L
           shells into s and p, when generating the
           internally uncontracted basis set.  This is
           necessary if you are using s or p primitives
           with the same exponents as in some L shell.
           This is unlikely to occur, but if so, the L
           shell must be entered before the s or p.
           Option 4 requires option 1.
       = 8 use 128 bit precision in the RIs.  Select this
           option if your exponent range is larger than 64
           bits can handle - it is a little difficult to
           relate Gaussian exponents to overlap matrix
           precision, but if the range of exponents reaches
           ten, one should think about using 128 bit math.
           This is a concern mainly for 6th row elements,
           where it may easily be probed by comparing the
           the energy and gradient for MODEQR=1 to 9.
                  Notes:
           1. 128 bit math can be very slow, depending
           on your CPU and/or compiler's support for it.
           Only relativistic 1e- integrals use 128 bits.
           2. LUT-IOTC's local nature makes "8" much more
           economical than for the other ESC schemes.
           3. If your FORTRAN library does not support the
           REAL*16 data type (128 bits), the code compiles
           itself in 64 bit mode, and will halt if you ask
           for 128 bits.
 
QMTTOL same as in $CONTRL, but used for the preparation of
       the RI space (see MODEQR suboption "1").  LUT-IOTC's
       RI applies to atomic domains, separately, whereas
       RESC, DK, and IOTC use this parameter for the entire
       molecule's uncontracted basis set, where linear
       dependence is an even greater concern.
       Usually values considerably smaller than the QMTTOL
       of $CONTRL, which applies to the contracted working
       basis may be used, improving accuracy.
       The default is 1d-10.
 
QRTOL  accuracy parameter for relativistic gradients.
 
       RESC or LUT-IOTC: tolerance for equating nearly
       degenerate eigenvalues of the kinetic energy and
       overlaps, when evaluating the gradient.
       Values that are too large (>1e-6) cause numerical
       errors in the gradient, approximately on the same
       order as QRTOL.
       Values that are too small can cause large gradient
       errors due to divsion by small numbers not screened
       away by QRTOL.
       (LUT-IOTC default = smaller of 1d-10 or QMTTOL)
           (RESC default = smaller of 1d-08 or QMTTOL)
 
       DK or IOTC:  Coordinate offset in bohr used for the
       numerical differentiation of the relativistic
       contributions to the gradient (analogous to VIBSIZ
       in $HESS).  Only totally symmetric coordinate
       directions are explored (analogous to NUMGRD in
       $CONTRL).  All other gradient terms are still
       computed analytically, but the effect of this single
       numerical step is to make DK or IOTC gradients be
       somewhat less accurate than most analytic gradients.
       See also NVIB.
       Default for DK or IOTC:  0.01 Bohr
 
NVIB   The number of offsets per coordinate (similar to
       NVIB in $FORCE).  NVIB can be 1 or 2 (or -1 or -2).
       This parameter applies only to DK or IOTC gradients,
       as RESC and LUT-IOTC are fully analytic.
       Positive values correspond to the projected mode,
       in which translations, rotations, and any modes
       which are not totally symmetric are projected out.
       Negative values correspond to using Cartesian
       coordinates.
       In most cases projected modes are superior; however
       they can cause slight distortions away from the
       true symmetry -IF- you specify lower symmetry than
       the molecule actually possesses. (default=2)
 
* * * the next parameter applies only to LUT-IOTC:
 
TAU    The distance cutoff to consider "local" for the
       local unitary transformation approximation.  The
       value should include any bonded atom pairs, but is
       chosen to eliminate most next nearest neighbor atom
       pairs.  Increasing TAU causes LUT-IOTC to converge
       to the full IOTC result (apart from some technical
       differences in the RI treatment of integrals).
       The default is 3.5 Angstroms.
 
* * * the next few parameters apply mainly to NESC:
 
NRATOM the number of different elements to be treated
       nonrelativistically.  For example, in Pb(CH3)2, to
       treat only lead relativistically, enter NRATOM=2.
       The elements to be treated nonrelativistically are
       defined by CHARGE.  (default=0)
       For NESC, this parameter affects the choice of the
       basis sets, you should use identical large, small,
       and averaged basis set for such atoms.
       For DK or RESC, MODEQR=1 won't uncontract to the
       primitives of such atoms.
 
CHARGE is an array containing nuclear charges of the atoms
       to be treated nonrelativistically.
       For example, CHARGE(1)=6.0,1.0, to drop all C/H
       atoms in Pb(CH3)2.
 
 
   *** for those who wish to live in other universes ***
 
CLIGHT gives the speed of light (atomic units), introduced
       as a parameter in order to reproduce exactly results
       published with a slightly different choice.
       Default: 137.0359895