On the time-dependent Born-Oppenheimer Approximation

TL;DR

This paper develops a quantitative, higher-order approximation of the time-dependent Born-Oppenheimer method for quantum molecules, deriving effective equations for nuclear dynamics that improve upon classical models.

Contribution

It introduces an iterative, higher-order approximation of molecular evolution and derives effective nuclear equations that extend beyond the classical Born-Oppenheimer approximation.

Findings

Derived an arbitrary order approximation of molecular evolution.

Formulated effective equations for nuclei without electron variables.

Provided estimates and tractable approximations for molecular dynamics.

Abstract

In this paper, we consider the time-dependent Born-Oppenheimer approximation (BOA) of a classical quantum molecule involving a possibly large number of nuclei and electrons, described by a Schr\"odinger equation. In the spirit of Born and Oppenheimer's original idea we study quantitatively the approximation of the molecular evolution. We obtain an iterable approximation of the molecular evolution to arbitrary order and we derive an effective equation for the reduced dynamics involving the nuclei equivalent to the original Schr\"odinger equation and containing no electron variables. We estimate the coefficients of the new equation and find tractable approximations for the molecular dynamics going beyond the one corresponding to the original Born and Oppenheimer approximation.