An Improved Size-Consistent Second-Order Brillouin–Wigner Perturbation Theory: Which Desirable Properties Are Compatible with Unconditional Size-Consistency and Optimized Chemical Accuracy?

Recently, it has been established that a size-consistent second-order Brillouin-Wigner perturbation (BW-s2) theory can be obtained by satisfying a trace condition with a repartitioning of the Hamiltonian between a one-body reference and a correlation-containing perturbation. BW-s2 has the twin advantages of being regular like the conventional second-order BW theory and being conditionally size-consistent like the conventional second-order Mo̷ller-Plesset theory. Conditional size-consistency means that spin-polarized reference orbitals must be used whenever the spin-restricted orbitals are unstable, whereas unconditional size-consistency means that size-consistency follows from the use of orbitals yielding the lowest BW-s2 energy. There is a wide range of possible forms for the partition, and here we present an exploration of the most desirable choices for regularization in the occupied subspace (controlled by a parameter α) and in the virtual subspace (controlled by a second parameter γ). Desirable parameter choices for BW-s2(α, γ) should exhibit (i) numerical stability, (ii) satisfactory accuracy for chemical energy differences, (iii) unconditional size-consistency, (iv) exactness for 2 electrons in 2 orbitals (2-in-2) at dissociation (which requires α + γ = 1), (v) flatness of the 2-in-2 PES approaching dissociation (which requires α + γ = 4), and (vi) particle-hole symmetry (α = γ). Tests on a range of bond-dissociation curves reveal stronger constraints on (α, γ) values that achieve unconditional size-consistency (α + γ > 1) than previously known. We combine this new information with minimizing errors against reference data across 14 data sets of chemical energy differences to recommend the nonempirical choice of BW-s2(α = 2, γ = 2). This choice appears to ensure unconditional size-consistency, as well as satisfactory accuracy on chemical problems, and enforces 2-in-2 flatness toward dissociation and particle-hole symmetry. Our analysis additionally reveals that BW-s2(α = 4, γ = 0), which was previously recommended on the basis of minimizing errors on chemical data, is also an alternative nonempirical choice that sacrifices 2-in-2 exactness and particle-hole symmetry.