6.8 Size-Consistent Brillouin-Wigner Perturbation Theory 6.8 Size-Consistent Brillouin-Wigner Perturbation Theory 6.8.2 BW-s2 Job Control

6.8.1 Introduction

(November 19, 2024)

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,

$E_{\text{c}}^{(2)}=-\frac{1}{4}\sum\limits_{ijab}\frac{|\langle ij||ab\rangle|% ^{2}}{\Delta_{ij}^{ab}+E_{\text{c}}^{(2)}}$

(6.27)

is finite when the orbital energy gap $\Delta_{ij}^{ab}=\varepsilon_{a}+\varepsilon_{b}-\varepsilon_{i}-\varepsilon_{j}$ 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,

$\Delta_{ijkl}^{abcd}\cdot t_{kl}^{cd}=-\langle ij||ab\rangle\;,$

(6.28)

the BW-s2 amplitude equation contains a regularizing tensor,

$\big{(}\Delta_{ijkl}^{abcd}+R_{ijkl}^{abcd}\big{)}\cdot t_{kl}^{cd}=-\langle ij% ||ab\rangle$

(6.29)

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

$R_{ijkl}^{abcd}=\frac{\alpha}{2}\big{(}W_{ik}\delta_{jl}+\delta_{ik}W_{jl}\big% {)}\delta_{ac}\delta_{bd}$

(6.30)

where,

$W_{ij}=\frac{1}{2}\sum\limits_{kab}\big{(}t_{ik}^{ab}\langle jk||ab\rangle+t_{% jk}^{ab}\langle ik||ab\rangle\big{)}$

(6.31)

This form of $\mathbf{W}$ was chosen because it is size-consistent, leading to a size-consistent BW-s2 correlation energy. Physically, $\mathbf{W}$ 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 $\bm{\Delta}+\bm{R}$ is diagonal, the BW-s2 working equation looks like that of MP2,

$E_{\text{c}}^{(2)}=-\frac{1}{4}\sum\limits_{ijab}\frac{|\widetilde{\langle ij|% |ab\rangle}|^{2}}{\varepsilon_{a}+\varepsilon_{b}-\tilde{\varepsilon}_{i}-% \tilde{\varepsilon}_{j}}$

(6.32)

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

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6.8.1 Introduction