Generalized "Quasi-classical" Ground State for an Interacting Two Level System

TL;DR

This paper develops a simplified, closed-form method to approximate the vibronic ground state of a two-level electronic system coupled to harmonic oscillators, applicable to various symmetries and useful for calculating physical properties.

Contribution

It extends a previous guess-based solution to lower symmetry cases, providing a practical approach for estimating vibronic states and related quantities in complex systems.

Findings

Derived explicit expressions for vibronic ground states.

Calculated q-factors and p-factor for dihedral symmetry.

Estimated hyperfine line narrowing in ESR spectra.

Abstract

We treat a system (a molecule or a solid) in which electrons are coupled linearly to any number and type of harmonic oscillators and which is further subject to external forces of arbitrary symmetry. With the treatment restricted to the lowest pair of electronic states, approximate "vibronic" (vibration-electronic) ground state wave functions are constructed having the form of simple, closed expressions. The basis of the method is to regard electronic density operators as classical variables. It extends an earlier "guessed solution", devised for the dynamical Jahn-Teller effect in cubic symmetry, to situations having lower (e.g., dihedral) symmetry or without any symmetry at all. While the proposed solution is expected to be quite close to the exact one, its formal simplicity allows straightforward calculations of several interesting quantities, like energies and vibronic reduction (or Ham) factors. We calculate for dihedral symmetry two different $q$-factors ("$q_z$" and "$q_x$") and a $p$-factor. In simplified situations we obtain $p=q_z +q_x -1$. The formalism enables quantitative estimates to be made for the dynamical narrowing of hyperfine lines in the observed ESR spectrum of the dihedral cyclobutane radical cation.