7.2: Born-Oppenheimer Approximation

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Because nuclei are a great deal more massive than electrons, the motion of the nuclei is much slower than that of the electrons. Thus the nuclear and electronic motions can be treated more or less independently and it is a good approximation to determine the electronic states at any value of \({\bf R}\) by treating the nuclei as fixed. This is the basis of the Born-Oppenheimer approximation.

In this approximation, the electron is described by an eigenfunction \(U_j ({\bf r, R})\) satisfying the Schrödinger equation

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

This is solved keeping \({\bf R}\) constant. For each \({\bf R}\), a set of energy eigenvalues \(E_j ({\bf R})\) and eigenfunctions \(U_j ({\bf r, R})\) is found. The functions \(U_j ({\bf r, R})\) are known as molecular orbitals.

The full wavefunction for the \(j^{th}\) energy level at given \({\bf R}\) is taken to be the simple product

\[\psi({\bf r, R}) = F_j ({\bf R}) U_j ({\bf r, R}) \nonumber\]

where \(F_j ({\bf R})\) is a wavefunction describing the nuclear motion.

Substituting this form into the full Schrödinger equation and using the electronic equation yields

\[ \left[-\frac{\hbar^{2}}{2 \mu_{12}} \nabla_{R}^{2} + E_j ({\bf R}) - E \right] F_j ({\bf R}) U_j ({\bf r, R}) = 0 \nonumber\]

A little vector calculus gives

\[ \nabla_{R}^{2}\left\{F_{j}(\mathbf{R}) U_{j}(\mathbf{r}, \mathbf{R})\right\}= \underline{\nabla}_{R} \cdot\left\{\underline{\nabla}_{R}\left[F_{j}(\mathbf{R}) U_{j}(\mathbf{r}, \mathbf{R})\right]\right\} \\ = \underline{\nabla}_{R} \cdot\left\{U_{j}(\mathbf{r}, \mathbf{R}) \underline{\nabla}_{R} F_{j}(\mathbf{R})+F_{j}(\mathbf{R}) \underline{\nabla}_{R} U_{j}(\mathbf{r}, \mathbf{R})\right\} \\ = U_{j}(\mathbf{r}, \mathbf{R}) \nabla_{R}^{2} F_{j}(\mathbf{R})+F_{j}(\mathbf{R}) \nabla_{R}^{2} U_{j}(\mathbf{r}, \mathbf{R}) \\ + 2\left(\underline{\nabla}_{R} U_{j}(\mathbf{r}, \mathbf{R})\right) \cdot\left(\underline{\nabla}_{R} F_{j}(\mathbf{R})\right)
\nonumber\]

Assuming that the variation of the molecular orbitals with inter-proton separation, \({\bf R}\), is weak, we can neglect the terms involving \(\underline{\nabla}_R U_j ({\bf r, R})\), and \(\nabla^2_R U_j ({\bf r, R})\) leaving a single-particle type Schrödinger equation for the nuclear motion

\[ \left[-\frac{\hbar^{2}}{2 \mu_{12}} \nabla_{R}^{2} + E_j ({\bf R}) - E \right] F_j ({\bf R}) = 0 \nonumber\]

in which \(E_j ({\bf R})\) plays the role of a potential. We will return to this later.