7.1: Introduction

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As the simplest example of covalent bonding, we consider the hydrogen molecular ion.

The hydrogen molecular ion H\(^+_2\) is a system composed of two protons and a single electron. It is useful to use center of mass (CM) coordinates by defining the relative position vector, \({\bf R}\), of proton 2 with respect to proton 1, and the position vector \({\bf r}\) of the electron relative to the center of mass of the two protons.

The Schrödinger equation is

\[\left[-\frac{\hbar^{2}}{2 \mu_{12}} \nabla_{R}^{2}-\frac{\hbar^{2}}{2 \mu_{e}} \nabla_{r}^{2}-\frac{e^{2}}{\left(4 \pi \epsilon_{0}\right) r_{1}}-\frac{e^{2}}{\left(4 \pi \epsilon_{0}\right) r_{2}}+\frac{e^{2}}{\left(4 \pi \epsilon_{0}\right) R}\right] \psi(\mathbf{r}, \mathbf{R})=E \psi(\mathbf{r}, \mathbf{R}) \nonumber\]

where the reduced mass of the two-proton system is \(\mu_{12} = M/2\), with \(M\) the proton mass, and \(\mu_e\) is the reduced mass of the electron/two-proton system:

\[\mu_e = \frac{m(2M)}{m + 2M} \simeq m \nonumber\]

where \(m\) is the electron mass.