6.6.6 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 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.⁶⁵⁰

Since orbital-optimized coupled-cluster doubles (OOCCD) can be computationally demanding ($𝒪(o^2{v}^4)$), 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) method⁷¹¹ 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.

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.¹⁰⁸⁹ 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.⁹⁵⁴

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.⁶⁴⁹ The $\kappa$-regularizer modifies the MP2 correlation energy as follows:

where the energy denominator ${\mathrm{\Delta }}_{ij}^{ab}={ϵ}_a{ϵ}_b-{ϵ}_i-{ϵ}_j$ and $\kappa$ controls the regularization strength. Evidently, $\kappa =0$ gives zero correlation energy (i.e., HF) and $\kappa \to \mathrm{\infty }$ 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: