8.4: Appendix 3 - The Born-Oppenheimer Approximation

Consider the hydrogen atom Hamiltonian. Let the electron coordinate be x, and the nuclear coordinate be X. We will assume that the system is one dimensional for the purposes of explaining the approximation.

\[ \hat{H} = -\frac{\hbar^{2}}{2m_{e}} \frac{d^{2}}{dx^{2}} - -\frac{\hbar^{2}}{2m_{N}} \frac{d^{2}}{dX^{2}} - \frac{e^{2}}{4 \pi \epsilon_{0} |x - X |} \label{8.4.1} \]

Now let's separate the solution, \(\psi\), into an electron-only factor \(\varphi\), and the nuclear-dependent factor \(\chi\):

\[ \psi(x,X) = \varphi(x,X) \chi(X) \label{8.4.2} \]

Substituting into Equation \ref{8.4.1} gives:

\[ H \psi = -\frac{\hbar^{2}}{2m_{e}} \frac{d^{2} \varphi}{dx^{2}} \chi - \frac{\hbar^{2}}{2m_{N}} \left( \frac{d^{2} \varphi}{dX^{2}} + 2 \frac{d \varphi}{dX} \frac{d \chi}{dX} + \frac{d^{2} \chi}{dX^{2}} \varphi \right) + V(x,X) \varphi \chi \label{8.4.3} \]

where we have replaced the Coulomb potential by V.

Now using the Born-Oppenheimer approximation, i.e. \(m_{e} \ll m_{N}\), we approximate Equation \ref{8.4.3} by:

\[ H \varphi = -\frac{\hbar^{2}}{2m_{e}} \frac{d^{2} \varphi}{dx^{2}} \chi + V(x,X) \varphi \nonumber \]

This equation is used to solve for the electron coordinates in a given nuclear configuration. The nuclear configuration is then optimized.

\(^{†}\)This Appendix is adapted in part from Molecular Quantum Mechanics by Atkins and Friedman

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