# Continuous vibronic symmetries in Jahn-Teller models

## TL;DR

This paper systematically explores Jahn-Teller models with continuous symmetries, identifying their algebraic structures, orbit spaces, and topological properties, revealing universal spectral features and their implications for molecular conical intersections.

## Contribution

It introduces a comprehensive algebraic and topological framework for Jahn-Teller models with continuous symmetries, linking their geometric properties to electronic spectra and conical intersections.

## Key findings

- Identification of symmetric spaces for JT models with Lie group symmetry
- Proof of universality in electronic spectra of JT minima
- Topological equivalence between JT troughs and projective spaces

## Abstract

We develop a systematic study of Jahn-Teller (JT) models with continuous symmetries by explor- ing their algebraic properties. The compact symmetric spaces corresponding to JT models carrying a Lie group symmetry are identified, and their invariants used to reduce their adiabatic potential energy surfaces into orbit spaces. Each orbit consists of a set of JT distorted molecular structures with equal adiabatic electronic spectrum. Molecular motion may be decomposed into pseudorota- tional and radial. The former preserves the orbit, while the latter maps an orbit into another. The dimensionality and topology of the internal space of each orbit depends on the number of degener- ate states in its adiabatic electronic spectra. Furthermore, qualitatively different pseudorotational modes occur in orbits of different types. We also provide a simple proof that the electronic spectrum for the space of JT minimum-energy structures (trough) displays a universality predicted by the epikernel principle. This result is in turn used to prove the topological equivalence between bosonic (fermionic) JT troughs and real (quaternionic) projective spaces, a conclusion which has outstanding physical consequences, as explained in our work. The relevance of our study for the more common case of JT systems with only discrete point group symmetry, and for generic asymmetric molecular systems with conical intersections involving more than two states is likewise discussed. In particular, we show that JT models with continuous symmetries present the simplest models of conical intersections among an arbitrary number of electronic state crossings.