Symmetry in Multiple Self-Consistent-Field Solutions of Transition-Metal Complexes

TL;DR

This paper explores multiple self-consistent-field solutions in transition-metal complexes, classifies their symmetries, and demonstrates how they can be used with NOCI to accurately describe electronic states and vibronic effects.

Contribution

It introduces a group-theoretical classification of symmetry-broken solutions and shows their use as bases for NOCI to obtain physically meaningful multi-determinantal wavefunctions.

Findings

Symmetry-broken solutions can be used to construct correct electronic states.

NOCI restores symmetry and captures vibronic stabilization effects.

Analysis of natural orbitals provides insight into static correlation.

Abstract

We use a method based on metadynamics to locate multiple low-energy Unrestricted Hartree--Fock (UHF) self-consistent-field (SCF) solutions of two model octahedral $d^1$ and $d^2$ transition-metal complexes, $[\mathrm{MF}_6]^{3-} (\mathrm{M} = \mathrm{Ti}, \mathrm{V})$. By giving a group-theoretical definition of symmetry breaking, we classify these solutions in the framework of representation theory and observe that a number of them break spin or spatial symmetry, if not both. These solutions seem unphysical at first, but we show that they can be used as bases for Non-Orthogonal Configuration Interaction (NOCI) to yield multi-determinantal wavefunctions that have the right symmetry to be assigned to electronic terms. Furthermore, by examining the natural orbitals and occupation numbers of these NOCI wavefunctions, we gain insight into the amount of static correlation that they incorporate. We then investigate the behaviors of the most low-lying UHF and NOCI wavefunctions when the octahedral symmetry of the complexes is lowered and deduce that the symmetry-broken UHF solutions must first have their symmetry restored by NOCI before they can describe any vibronic stabilization effects dictated by the Jahn--Teller theorem.