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

Abstract

The quantum problem of four particles in ${\mathbb{R}}^d$ ($d\ge 3$), 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 $H_2{O}_2$ (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles $n$, with $d$ degrees of freedom ($d\ge n-1$), interacting through an harmonic oscillator potential is briefly discussed as well.

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