On Factorization of Molecular Wavefunctions

TL;DR

This paper analyzes the mathematical foundations of factorizing molecular wavefunctions into electronic and nuclear parts, addressing ambiguities caused by Coulomb potential singularities, and extends the understanding of exact wavefunction representations.

Contribution

It provides a rigorous mathematical analysis of the factorization of molecular wavefunctions, clarifying ambiguities and limitations related to Coulomb singularities.

Findings

Identifies ambiguities in wavefunction factorization due to Coulomb singularities

Clarifies the limitations of exact electronic-nuclear wavefunction separation

Extends the theoretical understanding of molecular wavefunction representations

Abstract

Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions \{$\Phi$\} of the molecular Schr\"odinger equation as originally proposed by Hunter in the 1970s. The idea is to represent $\Phi$ in the form $\varphi\chi$ where $\chi$ is \textit{purely} a function of the nuclear coordinates, while $\varphi$ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an `exact' representation of $\Phi$ can be achieved in this form with $\varphi$ and $\chi$ interpretable as `electronic' and `nuclear' wavefunctions respectively. We offer a mathematical analysis of these proposals that identifies ambiguities stemming mainly from the singularities in the Coulomb potential energy.