Eigenvalue crossings in equivariant families of matrices

TL;DR

This paper extends the mathematical understanding of eigenvalue crossings in symmetric matrices by incorporating symmetry groups, revealing new types of confluences relevant to molecular chemistry and the structure of energy level crossings.

Contribution

It generalizes Wigner and von Neumann's analysis to include symmetry, classifies confluences in molecules, and predicts new types of eigenvalue crossing phenomena.

Findings

Eigenvalue crossings are affected by symmetry, often not occurring on codimension 2 submanifolds.

Classification of confluences for triatomic and planar molecules is provided.

New types of confluence phenomena are predicted in molecular energy levels.

Abstract

According to a result of Wigner and von Neumann [1], real symmetric matrices with a doubly degenerate lowest eigenvalue form a submanifold of codimension 2 within the space of all real symmetric matrices. This mathematical result has important consequences for chemistry. First, it implies that degeneracies do not occur within generic one-parameter families of real symmetric matrices - this is the famous non-crossing rule, and is responsible for the phenomenon of avoided crossings in the energy levels of diatomic molecules. Second, it implies that energy levels are expected to cross in polyatomic molecules, with crossings taking place on a submanifold of nuclear configuration space which is codimension 2 - this submanifold is the famous conical intersection seam, of central importance in nonadiabatic chemistry. In this paper we extend the analysis of Wigner and von Neumann to include symmetry. We introduce a symmetry group, and consider parametrised families of matrices which respect an action of that symmetry group on both the parameter space and on the space of matrices. A concrete application is to molecules, for which the relevant symmetry group is generated by permutations of atomic nuclei combined with spatial reflections and rotations. In the presence of this extra symmetry, we find that energy level crossings do not typically occur on codimension 2 submanifolds, and connect our findings with the discovery of confluences of conical intersection seams in the chemical literature. We give a classification of confluences for triatomic molecules and planar molecules, unifying the previous literature on this topic, and predict several new types of confluence.