6.9.1 Introduction

Maximum Physical Regularisaion can be understood both as an extension to size-consistent Brillouin-Wigner theory (BW-s2) as well as to Møller-Plesset perturbation theory (MP2). In either framework, it introduces additional terms into the unperturbed Hamiltonian which allow the user to manipulate ground state properties and energies with more parametric precision. The second order energy in MPR-BWs(2) is given by an equation superficially reminiscient of both MP2 and BW-s2:

where the first order amplitudes $t_{ij}^{ab}{}^{(1)}$ are determined by their linear amplitude equation. In strong analogy to BW-s2, MPR-BWs2 contains two matrices $W_{ij}$ and $W_{ab}$ which augment and mould the phyiscal description of the unperturbed ground state. The MPR-BWs2 amplitude equation is given by:

The central distinction between MPR-BWs2, BW-s2 and MP2 lies in the definition of $W_{ab}$ and $W_{ij}$. The occupied block in MPR-BWs2 has five distinct contributions. The first term ($A_0$) is identical to BW-s2 and is characterised by the fact that it traces out to the second order energy multiplied by $A_0$:

Analogously, the second term ($A_1$) traces out to the opposite-spin part of the second order correlation energy and can best be written in spatial orbitals as:

The remaining occupied terms contain the pseudo-density matrices ${\rho }_{ij}$ and ${\rho }_{ab}$, which are defined in analogy to MP2, but are notably distinct from the true second order MPR-BWs2 density matrix:

The $A_2$ and $A_3$ terms are constructed as follows:

Finally, the $A_4$ term is given by a symmetric contraction with the fock matrix:

The virtual parameters are labelled according to their occupied counterparts, wherefore the $B_0$ and $B_1$ parameters are:

The $B_2$ parameter, which would contract ${\rho }_{ab}$ into $W_{ab}$ is deliberately not implemented, since its evaluation contains a step of order $O(V^4)$, where $V$ is the number of virtual orbitals. The $B_3$ and $B_4$ terms are given by:

For an interpretation of these parameters and their respective terms, see Ref.  326 Dittmer L. B., Head-Gordon M.
J. Chem. Phys.
(2024), 162, pp. 054109. Link .

Since terms with quadratic dependence on the amplitudes subsequently turn the amplitude equation into a cubic tensor equation, convergence of the self-consistent construction scheme described in section 6.8 is sometimes difficult or impossible to achieve. For this reason, it can be best to approximate the effect of each parameter non-iteratively. As can be seen from the Hylleraas functional, the second order correlation energy can be rewritten as:

where $P_{pq}^{(2)}$ denotes the exact second order density matrix. Replacing $P_{ij}^{(2)}$ with ${\rho }_{ij}$ and $P_{ab}^{(2)}$ with ${\rho }_{ab}$ and subtracting the iterative contribution defines the non-iterative approximation: