Born–Oppenheimer approximation

Definition

Consider a molecule with nuclei of masses $M_1, \ldots, M_K$ at positions $R = (R_1, \ldots, R_K) \in \R^{3K}$ and $N$ electrons of mass $m$ at positions $r = (r_1, \ldots, r_N) \in \R^{3N}$. Write its Hamiltonian as $$H = -\sum_{k=1}^{K} \frac{\hbar^2}{2M_k} \Delta_{R_k} + H_{\mathrm{el}}(R).$$ The electronic Hamiltonian $H_{\mathrm{el}}(R)$ is the kinetic energy of the electrons plus all Coulomb interactions. It acts on functions of $r$ for fixed $R$. The Born–Oppenheimer approximation consists of two steps:

1. For each $R$, find an eigenvalue $E_j(R)$ and a normalised eigenvector $\phi_j(r; R)$ of $H_{\mathrm{el}}(R)$; the function $R \mapsto E_j(R)$ is the $j$-th potential energy surface.
2. Let $\chi$ be an eigenfunction, with eigenvalue $E$, of the nuclear Hamiltonian $-\sum_k \frac{\hbar^2}{2M_k} \Delta_{R_k} + E_j(R)$. Approximate the eigenfunctions of $H$ by the products $\Psi(r, R) = \chi(R)\, \phi_j(r; R)$, and the eigenvalues of $H$ by these $E$.