6.6 Auxiliary Basis (Resolution of the Identity) MP2 Methods 6.6.5 Spin-Biased MP2 Methods (SCS-MP2, SOS-MP2, and MOS-MP2)6.6.7 Brueckner CC2

6.6.6 Orbital-Optimized MP2

(April 13, 2024)

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 breaking 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. ⁷⁰⁸ Lee J., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2019), 21, pp. 47638. Link

Since orbital-optimized coupled-cluster doubles (OOCCD) can be computationally demanding ( $\mathcal{O}(o^{2}v^{4})$ ), Rohini Lochan working with Martin Head-Gordon proposed to obtain orbitals by optimizing the 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. 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) method ⁷⁷¹ Lochan R. C., Head-Gordon M.
J. Chem. Phys.
(2007), 126, pp. 164101. Link 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 ${\cal{O}}({N^{4}})$ SOSMP2

$molecule
   1 2
   F
   H 1 1.001
$end

$rem
   EXCHANGE          HF
   BASIS             sto-3g
   UNRESTRICTED      TRUE
   JOBTYPE           FORCE            Options are SP/FORCE/OPT
   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
   AUX_BASIS         rimp2-vdz
   SCF_CONVERGENCE   8
   THRESH            14
integral_symmetry FALSE
   PURECART          1111
$end

Example 6.12 Example of O2 methodology applied to ${\cal{O}}({N^{4}})$ 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
integral_symmetry FALSE
   PURECART          1111
$end

Although O2 (or OOMP2) was successful in numerous applications, there are two limitations of this model. First of all, the energy optimization often runs into a numerical instability 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 regularized OOMP2 where they employed a linear level shift parameter, $\delta$ , to stabilize small energy denominators. ¹¹⁸⁷ Stück D., Head-Gordon M.
J. Chem. Phys.
(2013), 139, pp. 244109. Link The thermochemistry performance of $\delta$ -OOMP2 was found to be disappointing when one wishes to keep $\delta$ large enough to recover the Coulson-Fischer point. ¹⁰⁴² Razban R. M., Stück D., Head-Gordon Martin
Mol. Phys.
(2017), 115, pp. 21029. Link

Joonho Lee working with Martin Head-Gordon developed a new regularized OOMP2 suite of methods that utilizes an energy-dependent regularizer ( $\kappa$ -regularizer) unlike the $\delta$ -regularizer. ⁷⁰⁷ Lee J., Head-Gordon M.
J. Chem. Theory Comput.
(2018), 14, pp. 5203. Link The $\kappa$ -regularizer modifies the MP2 correlation energy as follows:

E_{\kappa-\text{MP2}}=-\frac{1}{4}\sum_{ijab}\frac{|\langle ij||ab\rangle|^{2}% }{\Delta_{ij}^{ab}}\left(1-\exp(-\kappa\Delta_{ij}^{ab})\right)^{2}

(6.23)

where the energy denominator $\Delta_{ij}^{ab}=\epsilon_{a}\epsilon_{b}-\epsilon_{i}-\epsilon_{j}$ and $\kappa$ controls the regularization strength. Evidently, $\kappa=0$ gives zero correlation energy (i.e., HF) and $\kappa\rightarrow\infty$ recovers the unregularized MP2 energy expression. In $\kappa$ -OOMP2, orbitals are then determined as a minimizer for $E_{\text{HF}}E_{\kappa-\text{MP2}}$ . The $\kappa$ value of 1.45 $E_{h}^{-1}$ is recommended due to its good balance between the Coulson-Fischer point recovery and thermochemistry performance. It should be noted that $\kappa$ -OOMP2 runs through Q-Chem’s new SCF library, libgscf, and new MP2 library, libgmbpt. The older OOMP2 code (written by Rohini Lochan and David Stück) is no longer supported and should be 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 $\kappa$ -OOMP2:

CORRELATION	None (default)
JOBTYPE	sp (default) single point energy evaluation; force (force supported); opt (geometry optimization supported)
BASIS	user’s choice (standard or user-defined: GENERAL or MIXED)
GEN_SCFMAN_FINAL	TRUE (default if $\kappa$ -OOMP2 is requested)
	FALSE (default for other SCF jobs)
SCF_ALGORITHM	GDM (default)
	DIIS
	GDM-LS
AUX_BASIS	corresponding auxiliary basis (standard or user-defined:
	AUX_GENERAL or AUX_MIXED)
REGULARIZED_O2	0 (no regularizer; default)
	1 ( $\delta$ -regularizer)
	2 ( $\kappa$ -regularizer; recommended)
	3 ( $\sigma$ -regularizer)
REG_PARAMETER	regularization parameter multiplied by 1e ${}^{3}$ ; no default
	1450 (Recommended value for $\kappa$ -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 (multiplied by 1e ${}^{6}$ )
SOS_FACTOR	1000000 (default) Specify opposite-spin-component scaling factor (multiplied by 1e ${}^{6}$ )
DO_S2	0 (default)
	1 (Compute $\langle\hat{S}^{2}\rangle$ at the MP2 level)

Example 6.13 Example of $\kappa$ -OOMP2 with the cG orbital type applied to OH

$molecule
   0  2
   O  -2.766559046  0.187082886   0.566917837
   H  -3.696304300  1.179189102  -0.642506882
$end

$rem
   BASIS              cc-pvdz
   AUX_BASIS          rimp2-cc-pvdz
   EXCHANGE           hf
   THRESH             14
   INPUT_BOHR         true
   SCF_CONVERGENCE    8
   SCF_ALGORITHM      gdm
   MAXSCF             1000
integral_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 $\kappa$ -OOMP2 with the R orbital type applied to a water dimer

$molecule
   0  1
   O    -2.766559046    0.187082886    0.566917837
   H    -3.696304300    1.179189102   -0.642506882
   H    -3.395837846   -1.509891173    0.389283582
   O     2.587035064    0.275900014   -0.746441819
   H     3.579141280    0.918406897    0.633058252
   H     0.852266482    0.311804811   -0.156847268
$end

$rem
   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
integral_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

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6.6.6 Orbital-Optimized MP2