A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates

TL;DR

This paper develops a highly accurate time-dependent Born-Oppenheimer approximation for molecular quantum systems, achieving exponentially small error bounds by optimal asymptotic truncation, with nuclear masses scaled by a small parameter.

Contribution

It introduces a novel construction of an exponentially precise approximation for molecular Schrödinger equations with scaled nuclear masses, improving accuracy over existing methods.

Findings

Approximate solutions match exact solutions with errors bounded by exponential terms.

Optimal truncation of asymptotic expansions yields exponentially small errors.

The method applies to systems with nuclear masses proportional to ^{-4}.

Abstract

We present the construction of an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to $\epsilon^{-4}$, where $\epsilon$ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time-dependent Schr\"odinger equation that agree with exact normalized solutions up to errors whose norms are bounded by $\ds C \exp(-\gamma/\epsilon^2)$, for some C and $\gamma>0$.