First-order symmetry-adapted perturbation theory for multiplet splittings

TL;DR

This paper introduces a novel symmetry-adapted perturbation theory (SAPT) method for calculating spin state splittings in high-spin open-shell molecular complexes, providing a first-order exchange energy approach that simplifies analysis of multiplet interactions.

Contribution

The paper develops and implements a first-order SAPT formalism for multiplet splittings in high-spin open-shell molecules, extending SAPT capabilities beyond high-spin states.

Findings

First-order exchange energies are linear combinations of diagonal and spin-flip matrix elements.

The approach models splittings using a Heisenberg Hamiltonian with a single coupling parameter.

Validated on various open-shell complexes including diatomic and larger systems.

Abstract

We present a symmetry-adapted perturbation theory (SAPT) for the interaction of two high-spin open-shell molecules (described by their restricted open-shell Hartree-Fock determinants) resulting in low-spin states of the complex. The previously available SAPT formalisms, except for some system-specific studies for few-electron complexes, were restricted to the high-spin state of the interacting system. Thus, the new approach provides, for the first time, a SAPT-based estimate of the splittings between different spin states of the complex. We have derived and implemented the lowest-order SAPT term responsible for these splittings, that is, the first-order exchange energy. We show that within the so-called S2 approximation commonly used in SAPT (neglecting effects that vanish as fourth or higher powers of intermolecular overlap integrals), the first-order exchange energies for all multiplets are linear combinations of two matrix elements: a diagonal exchange term that determines the spin-averaged effect and a {\em spin-flip term} responsible for the splittings between the states. The numerical factors in this linear combination are determined solely by the Clebsch-Gordan coefficients: accordingly, the S2 approximation implies a Heisenberg Hamiltonian picture with a single coupling strength parameter determining all the splittings. The new approach is cast into both molecular-orbital and atomic-orbital expressions: the latter enable an efficient density-fitted implementation. We test the newly developed formalism on several open-shell complexes ranging from diatomic systems (Li+H, Mn+Mn,...) to the phenalenyl dimer.