On the four-body problem in the Born-Oppenheimer approximation

TL;DR

This paper analyzes the four-body quantum problem with harmonic interactions, demonstrating that the Born-Oppenheimer approximation accurately captures the ground state energy and phase in certain cases, with implications for molecular models like H2 and H2O2.

Contribution

It provides an exact solvability analysis of the four-body problem and validates the Born-Oppenheimer approximation for specific mass configurations, extending to general n-particle systems.

Findings

Exact match of the first two terms of the Puiseux series for phase and energy with Born-Oppenheimer results.

Application to models of H2 molecule and H2O2 (Hydrogen peroxide).

Discussion of generalization to n-particle systems with harmonic interactions.

Abstract

The quantum problem of four particles in $\mathbb{R}^d$ ($d\geq 3$), with arbitrary masses $m_1,m_2,m_3$ and $m_4$, interacting through an harmonic oscillator potential is considered. This model allows exact solvability and a critical analysis of the Born-Oppenheimer approximation. The study is restricted to the ground state level. We pay special attention to the case of two equally heavy masses $m_1=m_2=M$ and two light particles $m_3=m_4=m$. It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter $\sigma=\frac{m}{M}$, of the exact phase $\Phi$ of the wave function $\psi_0=e^{-\Phi}$ and the corresponding ground state energy $E_0$, coincide exactly with the values obtained in the Born-Oppenheimer approximation. A physically relevant rough model of the $H_2$ molecule and of the chemical compound $H_2O_2$ (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles $n$, with $d$ degrees of freedom ($d\geq n-1$), interacting through an harmonic oscillator potential is briefly discussed as well.