= Solution
* The <atomic nuclei> are much heavier than <electrons>, so nuclear motion is slower. In the <Born-Oppenheimer approximation>, first hold the nuclear positions $\mathbf R$ fixed and solve
$$
\hat H_e(\mathbf r;\mathbf R)\psi_n(\mathbf r;\mathbf R)
=U_n(\mathbf R)\psi_n(\mathbf r;\mathbf R).
$$
* Approximate the complete <wavefunction> by $\Psi(\mathbf r,\mathbf R)=\psi_n(\mathbf r;\mathbf R)\chi_n(\mathbf R)$ and neglect the nuclear-derivative couplings involving $\nabla_{\mathbf R}\psi_n$ and $\nabla_{\mathbf R}^2\psi_n$.
* Nuclear motion then occurs on one electronic <potential energy> surface:
$$
\boxed{\left[\hat T_N+U_n(\mathbf R)+V_{NN}(\mathbf R)\right]\chi_n=E\chi_n.}
$$
Here $U_n$ excludes the nuclear repulsion $V_{NN}$, which must be included once.
* The mass separation makes the discarded coupling terms small when electronic <energy> gaps are sufficiently large; near a degeneracy or avoided crossing, a single-surface description can fail.
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