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

Abstract

The quantum problem of four particles in Rd (d≥3), with arbitrary masses m1,m2,m3 and m4, interacting through a 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 m1=m2=M and two light particles m3=m4=m. It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter σ=mM, of the exact phase Φ of the wave function ψ0=e−Φ and the corresponding ground state energy E0, coincide exactly with the values obtained in the Born–Oppenheimer approximation. A physically relevant rough model of the H2 molecule and of the chemical compound H2O2 (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles n, with d degrees of freedom (d≥n−1), interacting through a harmonic oscillator potential is briefly discussed as well.