  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.
