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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.

    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.

    This page titled 7.1: Introduction is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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