The Born-Oppenheimer approximation makes it possible to separate the Schrodinger equation for a molecule into an electronic and a nuclear equation. The Schrödinger equation for nuclear motion in a given electronic state is given by:

\begin{equation}\label{1} \left(\hat{T}_N + E_e\right)\psi_N=E\psi_N \end{equation}

Where $\hat{T}_N$ represents the kinetic energy of the nuclei. $E_e$ is the nuclear potential (electronic energy and repulsion between nuclei) and depends on the electronic state of the molecule. E is the total energy of the molecule.

Particularizing these results for a diatomic molecule composed of two atoms 1 and 2 of masses $m_1$ and $m_2$

\begin{equation}\label{2} \left[-\frac{\hbar^2}{2m_1}\nabla_{1}^{2}-\frac{\hbar^2}{2m_2}\nabla_{2 }^{2} +E_e(R)\right]\psi_N =E\psi_N \end{equation}

The nuclear potential $E_e(R)$ depends exclusively on the separation between the nuclei, which allows to separate the molecular movement in the translational movements of the center of mass and in the internal movement. Energy can also be separated into translational energy and internal energy $E=E_{tr}+E_{int}$

The Schrödinger equation for the internal movement is obtained using as mass the reduced mass of the system and as coordinates those of the vector that joins both nuclei.

\begin{equation}\label{3} \left[-\frac{\hbar^2}{2\mu}\nabla^{2} +E_e(R)\right]\psi_{int} =E_{int }\psi_{int} \end{equation}

The reduced mass is given by the equation $\mu = \frac{m_1 m_2}{m_1 + m_2}$ and the operator nabla squared by: $\nabla^2=\frac{\partial^2}{\partial x^ 2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$, where x,y,z are the coordinates of the vector that joins both nuclei .