7.1: Introduction

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.

Figure $\PageIndex{1}$

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.