6.6 Auxiliary Basis (Resolution of the Identity) MP2 Methods

6.6.5 Orbital-Optimized MP2

Brueckner orbitals (BOs) are highly desirable when one is unsure whether artificial symmetry breaking occurs at the Hartree-Fock (HF) level. It is artificial because this symmetry beraking merely reflects the lack of dynamic correlation at the HF level, not the lack of strong correlation. On the other hand, it is possible for a single-reference approach to attempt to describe strongly correlated systems by essential symmetry breaking. Therefore, it is often crucial to distinguish these two by obtaining orbitals in the presence of electron correlation.543

Since orbital-optimized coupled-cluster doubles (OOCCD) can be computationally demanding (𝒪(o2v4)), Rohini Lochan working with Martin Head-Gordon proposed to obtain orbitals by optimizing MP2 correlation energy. To this end, BOs are introduced into SOSMP2 and MOSMP2 methods to resolve the problems of symmetry breaking and spin contamination that are often associated with Hartree-Fock orbitals. So the molecular orbitals are optimized with the mean-field energy plus a correlation energy taken as the opposite-spin component of the second-order many-body correlation energy, scaled by an empirically chosen parameter. This “optimized second-order opposite-spin” (O2) method586 requires fourth-order computation on each orbital iteration. O2 is shown to yield predictions of structure and frequencies for closed-shell molecules that are very similar to scaled MP2 methods. However, it yields substantial improvements for open-shell molecules, where problems with spin contamination and symmetry breaking are shown to be greatly reduced.

Example 6.11  Example of O2 methodology applied to 𝒪(N4) SOSMP2

$molecule
   1 2
   F
   H 1 1.001
$end

$rem
   UNRESTRICTED      TRUE
   JOBTYPE           FORCE            Options are SP/FORCE/OPT
   EXCHANGE          HF
   DO_O2             1                O2 with O(N^4) SOS-MP2 algorithm
   SOS_FACTOR        1000000          Opposite Spin scaling factor = 1.0
   SCF_ALGORITHM     DIIS_GDM
   SCF_GUESS         GWH
   BASIS             sto-3g
   AUX_BASIS         rimp2-vdz
   SCF_CONVERGENCE   8
   THRESH            14
   SYMMETRY          FALSE
   PURECART          1111
$end

Example 6.12  Example of O2 methodology applied to 𝒪(N4) MOSMP2

$molecule
   1 2
   F
   H 1 1.001
$end

$rem
   UNRESTRICTED      TRUE
   JOBTYPE           FORCE            Options are SP/FORCE/OPT
   EXCHANGE          HF
   DO_O2             2                O2 with O(N^4) MOS-MP2 algorithm
   OMEGA             600              Omega = 600/1000 = 0.6 a.u.
   SCF_ALGORITHM     DIIS_GDM
   SCF_GUESS         GWH
   BASIS             sto-3g
   AUX_BASIS         rimp2-vdz
   SCF_CONVERGENCE   8
   THRESH            14
   SYMMETRY          FALSE
   PURECART          1111
$end

Although O2 (or OOMP2) was successful in numerous applications, there are two limiations of this model. First of all, the energy optimization often runs into a numerical instabillity caused by the singularity of the MP2 energy due to a small energy denominator. Secondly, the disappearance of Coulson-Fischer point hinders the use of essential symmetry breaking. This led David Stück and Martin Head-Gordon to regualrized OOMP2 where they employed a linear level shift parameter, δ, to stabilize small energy denominators.882 The thermochemistry performance of δ-OOMP2 was found to be disappoitning when one wishes to keep δ large enough to recover the Coulson-Fischer point.774

Joonho Lee working with Martin Head-Gordon developed a new regularized OOMP2 suite of methods that utilizes an energy-dependent regularizer (κ-ragularizer) unlike the δ-regularizer.542 The κ-regularizer modifies the MP2 correlation energy as follows:

Eκ-MP2=-14ijab|ij||ab|2Δijab(1-exp(-κΔijab))2 (6.23)

where the energy denominator Δijab=ϵaϵb-ϵi-ϵj and κ controls the regularization strength. Evidently, κ=0 gives zero correlation energy (i.e., HF) and κ recovers the unregularized MP2 energy expression. In κ-OOMP2, orbitals are then determined as a minimizer for EHFEκ-MP2. The κ value of 1.45 Eh-1 is recommended due to its good balance between the Coulson-Fischer point recovery and thermochemistry performance. It should be noted that κ-OOMP2 runs through Q-Chem’s new SCF library, libgscf, and new MP2 library, libgmbpt. The older OOMP2 code (written by Rohini Locan and David Stück) is no longer supported and used with a greater caution. Furthermore, the new OOMP2 code can handle restricted (R), complex, restricted (cR), unrestricted (U), generalized (G), and complex, generalized (cG) orbital types. The complex, unrestricted (cU) orbital type is not yet supported due to its limited applicability.

Summary of rem variables relevant to run κ-OOMP2:

CORRELATION None (default)
JOBTYPE sp (default) single point energy evaluation (force is not yet supported)
BASIS user’s choice (standard or user-defined: GENERAL or MIXED)
GEN_SCFMAN_FINAL TRUE (default if κ-OOMP2 is requested)
FALSE (default for other SCF jobs)
AUX_BASIS corresponding auxiliary basis (standard or user-defined:
AUX_GENERAL or AUX_MIXED)
REGULARIZED_O2 0 (no regularizer; default)
1 (δ-regularizer)
2 (κ-regularizer; recommended)
3 (σ-regularizer)
REG_PARAMETER regularization parameter multipled by 1e3; no default
1450 (Recommended value for κ-OOMP2)
N_FROZEN_CORE 0 (Code supports this functionality but it is not
recommended due to some convergence issues)
N_FROZEN_VIRTUAL 0 (Code supports this functionality but it is not
recommended due to some convergence issues)
SCS 0 (default)
1 Turns on spin-component scaling with SCS-OOMP2,
2 SOS-OOMP2,
3 arbitrary SCS-OOMP2
SSS_FACTOR 1000000 (default) Specify same-spin-component scaling factor (multipled by 1e6)
SOS_FACTOR 1000000 (default) Specify opposite-spin-component scaling factor (multipled by 1e6)
DO_S2 0 (default)
1 (Compute S2 at the MP2 level)

Example 6.13  Example of κ-OOMP2 with the cG orbital type applied to OH

$molecule
0  2
O -2.766559046 0.187082886 0.566917837
H -3.6963043 1.179189102 -0.642506882
$end

$rem
jobtype sp
basis cc-pvdz
aux_basis rimp2-cc-pvdz
exchange hf
thresh 14
input_bohr true
scf_convergence 8
scf_algorithm gdm
maxscf 1000
symmetry false
scf_guess sad
gen_scfman true
gen_scfman_final true
n_frozen_core 0       no frozen core
n_frozen_virtual 0    no frozen virtual
do_o2 3               run OOMP2
REGULARIZED_O2 2      use kappa-regularizer
REG_VARIABLE 1450     set kappa = 1.45
scs 3                 use arbitrary SCS
SOS_FACTOR 883532     use cos = 0.883532
SSS_FACTOR 883532     use css = 0.883532
do_s2 1               compute s^2 at the MP2 level
unrestricted true     use unrestricted
ghf true              use generalized
complex true          use complex
$end

Example 6.14  Example of κ-OOMP2 with the R orbital type applied to a water dimer

$molecule
0  1
O -2.766559046 0.187082886 0.566917837
H -3.6963043 1.179189102 -0.642506882
H -3.395837846 -1.509891173 0.389283582
O 2.587035064 0.275900014 -0.746441819
H 3.57914128 0.918406897 0.633058252
H 0.852266482 0.311804811 -0.156847268
$end

$rem
jobtype sp
basis cc-pvdz
aux_basis_corr rimp2-cc-pvdz
exchange hf
thresh 14
input_bohr true
scf_convergence 8
scf_algorithm gdm
maxscf 1000
scf_guess sad
symmetry false
gen_scfman true
unrestricted false      use restricted
do_o2 3                 run OOMP2
REGULARIZED_O2 2        use kappa-regularizer
REG_VARIABLE 1450       set kappa = 1.45
$end