# Binding of muonated hydrogen molecules on the occasionof the Born-Oppenheimer approximation 90th anniversary

## TL;DR

This paper examines the stability of muonated hydrogen molecules using variational methods, highlighting the limitations of the Born-Oppenheimer approximation, especially for molecules beyond di-hydrogen.

## Contribution

It provides an exact variational analysis of muonated hydrogen molecules and assesses the validity of the Born-Oppenheimer approximation in these systems.

## Key findings

- Born-Oppenheimer approximation is valid mainly for di-hydrogen.
- Variational solutions confirm stability of muonated hydrogen molecules.
- Limitations of the approximation are demonstrated for more complex molecules.

## Abstract

The stability of four fermionic particles with unit charge, of which, two are positively, and two negatively charged, is discussed. Except for using the simplest approximation of a single Gaussian orbital per particle, the problem is exactly solved variationally and, by varying the masses to simulate molecular di-hydrogen, mono-muonated di-hydrogen and di-muonated di-hydrogen, employed to illustrate the celebrated Born-Oppenheimer approximation on the occasion of its 90th anniversary. It is suggested that it is valid only for di-hydrogen.