Second Quantization-based Symmetry-Adapted Perturbation Theory: Generalizing Exchange Beyond Single Electron Pair Approximation

This paper presents a general second-quantized form of a permutation operator interchanging $n$ pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of single-pair permutation operators. This generalised form of the permutation operator has enabled the derivation of universal formulae for $S^{2n}$ approximations of the exchange energies in the first- and second-order of the interaction operator. We present the expressions for corrections of the $S^4$ approximations, and assess its efficacy on a selection of systems anticipated to exhibit a slowly converging overlap expansion. Additionally, we outline a method to sum the overlap expansion series to infinity in second-quantization, up to the second order in $V$. This new approach offers an alternative to the existing formalism based on the density-matrix formulations. When combined with a symbolic algebra program for automated derivations, it paves the way for advancements in SAPT theory, particularly for intricate wavefunction theories.