# Students' guide for GMCPT/GMCQDPT calculations on GAMESS

This document is a guide for students who perform GMCPT or GMCQDPT (MCSCF+PT) calculations on GAMESS. GMCPT is a single-state theory corresponding to MRMP PT in the CAS reference case and GMCQDPT is a multi-state theory. Use the names properly according to the definition.

## Examples

Set the following namelists first:
```  \$CONTRL SCFTYP=MCSCF RUNTYP=ENERGY MPLEVL=2 ... \$END
\$MRMP   MRPT=GMCPT RDVECS=.T. \$END  *T->F for also MCSCF calculation
\$MCSCF  CISTEP=GMCCI \$END```
• ### GMCPT for RASSCF reference function (RASSCF+GMCPT)

`\$GMCPT  REFTYP=RAS MSTART(1)=30,34,36 NEXCIT=2 NMOCOR=29 NMOACT=10 KSTATE(1)=1 ISTSYM=1 \$END`
Typical input [Input example] [Input file] [Output file]
• ### GMCPT for ORMASSCF reference functions (ORMASSCF+GMCPT)

`\$GMCPT  REFTYP=ORMAS NSPACE=3 MSTART(1)=3,8,10 MINE(1)=4,0,0 MAXE(1)=8,4,4 NMOFZC=0 NMODOC=2 NMOACT=9 \$END`
Typical input [Input example] [Input file] [Output file]
• ### GMCPT for CISD reference functions (CISD+GMCPT)

```\$GMCPT  REFTYP=MRX NPDET=1 NEXCIT=2 NMOCOR=8 NMOACT=16 KSTATE(1)=1 ISTSYM=1 \$END
\$PDET
2222222200000000
\$END```
Typical input [Input example] [Input file] [Output file]
• ### GMCQDPT for MRSD reference function (MRSD+GMCQDPT)

```\$GMCPT  REFTYP=MRX NPDET=13 NEXCIT=2 NMOCOR=3 NMOACT=25 KSTATE(1)=1,1,1,1,1,1,1 KNOSYM=.T. \$END
\$PDET
2222200000000000000000000
222-2+0000000000000000000
222+2-0000000000000000000
2222-00+00000000000000000
2222+00-00000000000000000
2222-+0000000000000000000
2222+-0000000000000000000
2-222+0000000000000000000
2+222-0000000000000000000
22-22+0000000000000000000
22+22-0000000000000000000
2222-0+000000000000000000
2222+0-000000000000000000
\$END```
Typical input [Input example] [Input file] [Output file]
A tip for parallel runs: MWORDS=xxx MEMDDI=0 DDTFPT=.F.

## Manual

Taken from GAMESS document:
```\$GMCPT group        (relevant if CISTEP=GMCCI in \$MCSCF)
(relevant if MRPT=GMCPT in \$MRMP)

This group specifies the determinants to be used in a
general MCSCF wavefunction.  It also gives the necessary
information to compute a 2nd order perturbation energy
correction to the MCSCF energy of such a MCSCF reference,
by choosing MPLEVL=2 in \$CONTRL and MRPT=GMCPT.

The PT is of quasidegenerate type, in which several
MCSCF states can be perturbed simultaneously.  If more than
one state is considered, the unperturbed model Hamiltonian
is diagonal in the MCSCF state basis.  After 2nd order
correction to both its diagonal and off-diagonal matrix
elements, this model Hamiltonian is diagonalized to give
the GMCQDPT energies.  The diagonalization also yields some
information about the remixing of the reference states at
2nd order.  GMCQDPT is therefore analogous to the two
equivalent MCQDPT programs (MRPT=MCQDPT or DETMRPT) for
CAS-references, but allows more general types of MCSCF
reference.  The letters GMCPT should be understood as
standing for GMCQDPT, and have been shortened only because
of the constraints on input group names to 6 or fewer
letters.  Of course, the program can also be used to obtain
the 2nd order correction to the energy of just one state.

At the present time, this program does not support
EXETYP=CHECK.  It is enabled for parallel execution.

1. data to specify active space and electronic state:

NMOFZC: number of frozen core orbitals, during the PT
the shape of these orbitals will be optimized in
the MCSCF stage, so they are "frozen" in the sense
of not being correlated in the PT.  The default
is the number of chemical core orbitals.

NMODOC: number of orbitals restricted to double occupancy
during MCSCF, but which are correlated in the PT
calculation.  In other words, the filled valence
orbitals. (no default).  (It is possible to enter a
different keyword NMOCOR which is the total number
of doubly occupied orbitals, and NMOFZC.  In this
case the program will obtain NMODOC by subtraction,
namely NMODOC = NMOCOR - NMOFZC).

NMOACT: number of active orbitals in the MCSCF (no default)

NMOFZV: number of virtual orbitals to be omitted from the
PT step.  The default is 0, retaining all virtuals.

NELACT: number of active electrons.  Since the default is
computed from the total number of electrons given
in \$DATA and \$CONTRL's ICHARG, minus 2*NMOFZC minus
2*NMODOC, there is little reason to input this.

MULT:   multiplicity of the state, with the default being
taken from MULT in \$CONTRL.

ISTSYM: a set of integers specifying the spatial symmetry
of the electronic state.  Choose from the table
ISTSYM=  1   2   3   4   5   6   7   8
C1   A
Ci   Ag  Au
Cs   A'  A''
C2   A   B
C2v  A1  A2  B1  B2
C2h  Ag  Au  Bu  Bg
D2   A   B1  B3  B2
D2h  Ag  Au  B3u B3g B1g B1u B2u B2g
Caution, this table differs from ISTSYM tables in
other input groups.  Default is ISTSYM(1)=1,0,0

If you are treating a system with degenerate states in an
appropriate Abelian subgroup of the true group, up to three
ISTSYM values can be given, to specify all components of
that originally degenerate state.

2. data to specify the MCSCF CI (and PT's reference CI):

The type of general MCSCF reference is specified by REFTYP,
which can be MRX, ORMAS, or RAS:

REFTYP= MRX means multi-reference determinant list, plus
excitations (default).  The determinants will be
given in a \$PDET group, and the keywords NPDET and
NEXCIT defined below are required.

REFTYP= RAS means the active space is divided into three
subspaces, known as RAS1, RAS2, and RAS3.  Keywords
MSTART and NEXCIT defined below are required.  For
example, MSTART(1)=4,6,9 defines a RAS with three
orbitals in the NMOFZC/NMODOC spaces, while the
RAS1, RAS2, and RAS3 subspaces contain 2, 3, and
NMOACT-5 orbitals.  It remains only to specify the
excitation level NEXCIT between these spaces.

REFTYP= ORMAS defines even more general subspaces than RAS,
and requires inputs NSPACE, MSTART, MINE, and MAXE.
These have the same meaning as the \$ORMAS keywords.

NPDET   is the number of parent determinants, to be given
as NPDET lines in the \$PDET group.  A value is
required for REFTYP=MRX.

NEXCIT  is an excitation level.  A value is required for
REFTYP=MRX or REFTYP=RAS.

NSPACE  is the number of subspaces into which the active
space is divided.  Required for REFTYP=ORMAS.

MSTART  is an array telling the starting MO of each orbital
space.  It is required for REFTYP=RAS and ORMAS.

MINE    is an array giving the minimum number of electrons
occupying each subspace. Required for REFTYP=ORMAS.

MAXE    is an array giving the maximum number of electrons
occupying each subspace. Required for REFTYP=ORMAS.

NSPACE, MSTART, MINE, and MAXE have the same meaning as in
the \$ORMAS group.  See there, and also in the MCSCF/CI
section of REFS.DOC, for help in understanding the power of
the ORMAS type of reference determinant list.

3. data to define the reference CI states:

KSTATE  is an array of states to be used.  As an example,
KSTATE(1)=0,1,0,1 means use states 2 and 4.  The
default is the ground state, KSTATE(1)=1,0,0...

WSTATE  is a set of weights for each state.  The default
is equal weight assigned to every state selected
by KSTATE (WSTATE(1)=1.0, 1.0, 1.0, ...)

0 means off, 1 means on (strictly), -1 means on
(loosely).  Proper spin states are picked up
automatically so this input is usually skipped.
See NSOLUT in this context.

KNOSYM  a flag to turn off space symmetry use, i.e. ISTSYM.
.FALSE. will ignore symmetry (default=.TRUE.)

KNOSPN  a flag to ignore spin symmetry, i.e. MULT.  Give
as .FALSE. to ignore the spin (default=.TRUE.)

The next few influence the Davidson CI diagonalization, and
are quite similar to \$MCQDPT keywords, so the description
here is terse.

NSOLUT  is the number of roots to be obtained.  If there
are not enough states of the correct spin found in
the first NSOLUT states to satisfy KSTATE/WSTATE,
increase this parameter to find enough.

MXITER  is the maximum number of Davidson iterations to
find the states (default=200)

THRCON  is the convergence criterion on the CI coefficient
convergence (default= 1.0d-6)

THRENE  is a convergence criterion on the total energy of
the states.  This is ignored if given as a negative
number.  (default = -1.0d-12 Hartree)

MAXBAS  maximum expansion space size in the Davidson
diagonalization subspace (default=100)

MDI     dimension of the initial guess subspace used to
initiate the Davidson iterative CI solver.  See

4. data to define the perturbation computation:

IWGT    selects wavefunction analysis (default=1)
0 means off, 1 means on (external), -1 means on
(internal orbitals).  This will compute the
approximate weight of the MCSCF reference CI in
the first order wavefunction.  It is therefore
a very useful diagnostic for the quality of the
calculation, as the MCSCF state should be a high
percentage.  The formula for the decomposition is
changed from the original CAS-type MCQDPT (REFWGT
in \$MCQDPT).  The reference for this option is
given below.  Select IWGT=0 if the fastest speed
is desired.

KFORB   flag to request canonicalization (default=.TRUE.)
Canonicalization within the core, virtual, and any
rotationally invariant active subspaces yields a
well defined theoretical model.  You would not
normally turn this option off.

KSZDOE  flag to use spin (Sz) dependent orbital energies.
If .TRUE., alpha and beta orbital canonicalization
will involve unequal energies.  Default=.TRUE.

THRWGT  threshold weight on the square of CI coefficients,
for determinant selection.  The default is 1.0d-8.
Give as a negative number to retain all of the
determinants, even those of very little importance,
in the reference of the perturbation treatment.

THRGEN  threshold on generator constants.  Default=1.0d-9
Raising lowers accuracy but produces speedups.
Lowering to 1.0d-12 should give full accuracy for
benchmarking purposes.

The next two deal with the so-called "intruder states".
There are theoretical difficulties with either one.  THRDE
just drops terms, so the potential surface may have small
discontinuities.  EDSHFT always shifts results a little
bit, even if no small denominators (aka intruder states)
are actually present.  Clearly both are "band-aids"!

THRDE   is a threshold to simply drop out any term whose
energy denominator is too small.  The default for
this is 0.005 Hartree.  Change to zero to turn this
option off.

EDSHFT  is similar to the same keyword in \$MCQDPT.  The
denominators D are changed to D + EDSHFT/D.  Turn
off THRDE if you select this option.  A reasonable
value to try is 0.02, the default is 0.0.

5. miscellaneous data

CEXCEN = string defining the units for the excitation
energy.  Choose from these 4 strings (any case):
eV (default), cm-1, Kcal/mol, KJ/mol

DDTFPT = a flag requesting the distributed data integral
transformation be used, if the run is parallel.
This option requires MEMDDI in \$SYSTEM.  If there
is not enough memory to allow this, turn this
option off to use an alternate parallel
transformation (DEFAULT=.TRUE.).

There are additional technical parameters documented in the
source code file gmcpt.src.
```

## References

Cite the following two papers in your paper (The program is based on the latter, and GMC-QDPT was originally proposed in the former):
1. H. Nakano, R. Uchiyama, and K. Hirao, J. Comput. Chem. 23, 1166-1175 (2002)
2. R. Ebisuzaki, Y. Watanabe, and H. Nakano, Chem. Phys. Lett. 442, 164-169 (2007)
In addition, cite the following paper if you use the results of wavefunction analysis (the approximate weight was defined in this paper):
1. M. Miyajima, Y. Watanabe, and H. Nakano, J. Chem. Phys. 124, 044101/1-9 (2006)

Note:
The present program is based on a scheme that uses matrix elements between the reference and ionized determinants [See Ref.2]. It is different from our original MC-QDPT scheme used in mcqdpt.src, which is based on the Goldstone diagrams, and also different from the original MRMP scheme by Hirao, which uses the matrix elements between CSFs, i.e. the Boys-Reeves formalism with the use of bonded functions. (It has been sometimes confused recent days, but historically the original MRMP and MC-QDPT computational schemes and codes were developed in different groups at Nagoya and Kyoto at that time, and hence quite independent.) The current scheme is also related to the Kozlowski-Davidson (adopted by Dr. Ivanic in demrpt.src) and Celani-Werner schemes.

## Key words

GMCPT, GMCQDPT, RASSCF, RASPT, ORMASSCF, ORMASPT, MCQDPT, MRMP
(GMC-PT, GMC-QDPT, RAS-SCF, RAS-PT, ORMAS-SCF, ORMAS-PT, MC-QDPT, MRMP)

Haruyuki Nakano
Department of Chemistry, Graduate School of Science
Kyushu University
744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan