On the mathematical treatment of the Born-Oppenheimer approximation

TL;DR

This paper reviews rigorous mathematical results on the Born-Oppenheimer approximation's accuracy in molecular quantum mechanics, highlighting challenges and differences from physical applications.

Contribution

It provides an accessible overview of mathematical approaches and results related to the Born-Oppenheimer approximation for researchers in quantum chemistry and physics.

Findings

Summarizes mathematical results on bound states, scattering, and resonance theory.

Critiques the differences between mathematical and physical uses of the approximation.

Identifies main difficulties and future directions in mathematical analysis.

Abstract

Motivated by a paper by B.T. Sutcliffe and R.G. Woolley, we present the main ideas used by mathematicians to show the accuracy of the Born-Oppenheimer approximation for molecules. Based on mathematical works on this approximation for molecular bound states, in scattering theory, in resonance theory, and for short time evolution, we give an overview of some rigourous results obtained up to now. We also point out the main difficulties mathematicians are trying to overcome and speculate on further developments. The mathematical approach does not fit exactly to the common use of the approximation in Physics and Chemistry. We criticize the latter and comment on the differences, contributing in this way to the discussion on the Born-Oppenheimer approximation initiated by B.T. Sutcliffe and R.G. Woolley. The paper neither contains mathematical statements nor proofs. Instead we try to make accessible mathematically rigourous results on the subject to researchers in Quantum Chemistry or Physics.