6.8.1 Introduction

While second-order Møller-Plesset perturbation theory (MP2) offers the simplest and most cost-effective ab initio correlation energy, it requires a separate formulation for degenerate states because the MP2 energy diverges in the zero-gap limit. Second-order Brillouin-Wigner perturbation theory (BW2) requires no such reformulation, as the second-order energy given by,

is finite when the orbital energy gap ${\mathrm{\Delta }}_{ij}^{ab}={\epsilon }_a+{\epsilon }_b-{\epsilon }_i-{\epsilon }_j$ is zero due to the presence of the correlation energy in the denominator. However, the BW2 correlation energy is not size-consistent in the sense that it does not satisfy $E(A\cup B)=E(A)+E(B)$ for two distant, noninteracting subsystems A and B, severely limiting its applicability to chemistry.

Recently, Carter-Fenk and Head-Gordon introduced a size-consistent-to-second-order Brillouin-Wigner perturbation theory (BW-s2) that retains this essential property while remaining finite for zero-gap systems. ¹⁷⁸ Carter-Fenk K., Head-Gordon M.
J. Chem. Phys.
(2023), 158, pp. 234108. Link This theory, based on a repartitioning of the zeroth-order Hamiltonian, results in a slightly-modified amplitude equation. Whereas the MP2 amplitudes can be found by solving,

the BW-s2 amplitude equation contains a regularizing tensor,

In the above equations ${\mathrm{\Delta }}_{ijkl}^{abcd}$ is composed of Fock matrix elements and reduces to the familiar ${\mathrm{\Delta }}_{ij}^{ab}$ when canonical orbitals are used, and

where,

This form of $𝐖$ was chosen because it is size-consistent, leading to a size-consistent BW-s2 correlation energy. Physically, $𝐖$ represents the correlation energy of a Koopmans’ (static orbital) ionization process. Thus, the occupied orbitals in BW-s2 are imbued with correlation such that the occupied/virtual gap increases. After rotating the occupied orbitals into a basis where $𝚫+𝑹$ is diagonal, the BW-s2 working equation looks like that of MP2,

but the orbitals and corresponding anti-symmetrized two-electron integrals have been rotated into the new basis.

In equation 6.30 there is an implicit parameter $\alpha$ that controls the regularization strength. Initially, this parameter was set to $\alpha =1$ to obtain the exact result for the two-electron two-orbital system of minimal-basis H${}_{\text{2}}$ at the dissociation limit, but $\alpha$ was later tuned to achieve much more accurate results for a wide array of chemical problems ranging from thermochemical properties to noncovalent interaction energies and closed-shell transition-metal reaction energies. ¹⁸⁴ Carter-Fenk K., Shee J., Head-Gordon M.

(2023), 159, pp. 171104. Link The optimal $\alpha$ value for the resultant BW-s2($\alpha$) approach varies somewhat between chemical problems, but was found to be more flexible than gap-dependent regularizer parameters like $\kappa$-, $\sigma$-, or ${\sigma }^2$-MP2. A “universal” parameter of $\alpha =4$ was suggested as a compromise to achieve the best all-around results for a wide array of chemical properties. ¹⁸⁴ Carter-Fenk K., Shee J., Head-Gordon M.

(2023), 159, pp. 171104. Link