The perturbation theory techniques available in GAMESS expand to the second order energy correction only, but permit use of a broad range of zeroth order wavefunctions. Since MP2 theory for systems well described as closed shells recovers only about 80% of the correlation energy (assuming the use of large basis sets), it is common to extend the perturbative treatment to higher order, or to use coupled cluster theory. While this is reasonable for systems well described by RHF or UHF with small spin contamination, this is probably a poor approach when the system is not well described at zeroth order by these wave- functions.
The input for second order pertubation calculations based on SCFTYP=RHF, UHF, or ROHF is found in $MP2, while for SCFTYP=MCSCF, see $MCQDPT.
These methods are well defined, due to the uniqueness of the Fock matrix definitions. These methods are also well understood, and there is little need to say more.
One point which may not be commonly appreciated is that the density matrix for the first order wavefunction for the RHF 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 where the single configuration RHF wavefunction was not an appropriate description of the system, and thus these occupancies may serve as a guide to the wisdom of using a RHF reference.
RHF energy corrections are the only method currently programmed for parallel computation. The analytic energy gradient is available only for RHF references, and this does permit frozen cores.
There are a number of open shell perturbation theories
described in the literature, bearing names such as RMP,
ROMP, OPT1, OPT2, IOPT, and ZAPT. These methods give
different results for the 2nd order energy correction,
reflecting ambiguities in the definition of the ROHF Fock
matrices. The ROHF MP2 method which is implemented in
GAMESS is the RMP theory,
P.J.Knowles, J.S.Andrews, R.D.Amos, N.C.Handy,
J.A.Pople Chem.Phys.Lett. 186, 130-136(1991)
which it should be pointed out, is entirely equivalent to
the ROHF-MBPT2 method of
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 that 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.
Besides the obvious fact that RMP's use of different
orbitals for different spins adds to the CPU time required
for integral transformations, the method is otherwise
reasonable. 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 would
be spin contaminated. It is generally thought that all
open shell MPn methods are better convergent than UMPn,
with order of the perturbation level n. These statements
are elaborated on in several papers comparing open shell
MP2 models:
This is not implemented in GAMESS. Note that the MCSCF
MP2 program discussed below should be able to develop the
perturbation correction for open shell singlets, by using
a $DRT input such as
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 MROPTn methods of Davidson, and the MCQDPT2 method of Nakano. Although the results of each method are different, energy differences should be rather similar. In particular, the MCQDPT2 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. If applied to 1 state, it is similar
to 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 offdiagonal elements of an effective
Hamiltonian, whose dimension equals the number of states
being treated. The perturbed Hamiltonian is diagonalized
to give the corrected state energies. Diagonalization
after inclusion of the offdiagonal 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:
The MCQDPT code was written by Haruyuki Nakano, and was interfaced to GAMESS by him in the summer of 1996. At the present time, the Ames group has little experience with the code, and so can offer no advice about its CPU or disk requirements. We therefore close the discussion with an input example illustrating RMP and MCQDPT 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, 49 CSFs.
!
! For 1 CSF reference,
! E(ROHF) = -55.5836109825
! E(RMP) = -55.7772299929 (lit. RMP = -75.777230)
! E(MCQDPT) = -55.7830423024 (lit. OPT1= -75.783044)
! [E(MCQDPT) = -55.7830437413 at the lit's OPT1 geometry]
!
! For 49 CSF reference,
! E(MCSCF) = -55.6323324949
! E(MCQDPT) = -55.7857458575
!
$contrl scftyp=mcscf mplevl=2 runtyp=energy mult=2 $end
$system timlim=60 memory=1000000 $end
$guess guess=moread norb=69 $end
$mcscf fullnr=.true. $end
!
! Next two lines carry out a MCQDPT computation, after
! carrying out a full valence MCSCF orbital optimization.
$drt group=c2v fors=.t. nmcc=1 ndoc=3 nalp=1 nval=2 $end
$mcqdpt inorb=0 mult=2 nmofzc=1 nmodoc=0 nmoact=6
istsym=3 nstate=1 $end
!
! Next two lines carry out a single reference computation,
! using converged ROHF orbitals from the $VEC.
--- $drt group=c2v fors=.t. nmcc=4 ndoc=0 nalp=1 nval=0 $end
--- $mcqdpt inorb=1 nmofzc=1 nmodoc=3 nmoact=1
--- istsym=3 nstate=1 $end
$data
NH2...2-B-1...TZ2Pf basis, RMP geom. used by 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
--- ROHF ORBITALS --- GENERATED AT 17:17:13 CST 28-OCT-1996
E(ROHF)= -55.5836109825, E(NUC)= 7.5835449477, 9 ITERS
$VEC
1 1 5.58322965E-01....
...omitted...
69 14 2.48059888E-02....
$END