Exact zero-point energy shift in the $e\otimes (n~E)$, $t\otimes (n~H)$ many modes dynamic Jahn-Teller systems at strong coupling

TL;DR

This paper derives exact semiclassical zero-point energy shifts for complex Jahn-Teller systems with multiple vibrational modes, providing analytical formulas and validating them against numerical methods, with implications for fullerene molecules.

Contribution

It presents the first exact semiclassical zero-point energy shift formulas for multi-mode Jahn-Teller systems at strong coupling, including an analytical normal mode frequency expression.

Findings

Derived exact energy shifts for multi-mode Jahn-Teller systems.

Provided analytical formulas with Slater-Koster form.

Validated results against numerical diagonalizations.

Abstract

We find the exact semiclassical (strong coupling) zero-point energy shifts applicable to the $e\otimes (n E)$ and $t\otimes (n H)$ dynamic Jahn-Teller problems, for an arbitrary number $n$ of discrete vibrational modes simultaneously coupled to one single electronic level. We also obtain an analytical formula for the frequency of the resulting normal modes, which has an attractive and apparently general Slater-Koster form. The limits of validity of this approach are assessed by comparison with O'Brien's previous effective-mode approach, and with accurate numerical diagonalizations. Numerical values obtained for $t\otimes (n H)$ with $n =8$ and coupling constants appropriate to C$_{60}^-$ are used for this purpose, and are discussed in the context of fullerene.