An Egorov Theorem for avoided crossings of eigenvalue surfaces

TL;DR

This paper develops a mathematical framework to describe quantum nuclear dynamics near avoided crossings of electron energy levels, using surface hopping and Egorov-type theorems to approximate solutions in the semi-classical limit.

Contribution

It introduces a surface hopping semigroup and proves convergence of the approximation to the true quantum solution near avoided crossings.

Findings

Construction of a surface hopping semigroup for avoided crossings

Proof of convergence to the true solution in the semi-classical limit

Application of microlocal normal forms similar to Landau-Zener theory

Abstract

We study nuclear propagation through avoided crossings of electron energy levels. We construct a surface hopping semigroup, which gives an Egorov-type description of the dynamics. The underlying time-dependent Schroedinger equation has a two-by-two matrix-valued potential, whose eigenvalue surfaces have an avoided crossing. Using microlocal normal forms reminiscent of the Landau-Zener problem, we prove convergence to the true solution in the semi-classical limit.