The purpose of this post is to explain the Born-Oppenheimer approximation.
Ignoring relativistic fine/hyperfine structure effects, the gross structure molecular Hamiltonian \(H\) (i.e. the “theory of everything”) is:
\[H=T_{\text n}+H_{\text e}\]
where the nuclear kinetic energy is:
\[T_{\text n}=\sum_{i}\frac{\textbf P_i^2}{2M_i}\]
and, for reasons that will become apparent soon, the electronic Hamiltonian \(\text H_{\text e}:=T_{\text e}+V_{\text{ee}}+V_{\text{ne}}\) is thought of as consisting of the following terms:
\[V_{\text{nn}}=\alpha\hbar c\sum_{i<j}\frac{Z_iZ_j}{|\textbf X_i-\textbf X_j|}\]
\[T_{\text e}=\sum_{i}\frac{\textbf p_i^2}{2m_e}\]
\[V_{\text{ee}}=\alpha\hbar c\sum_{i<j}\frac{1}{|\textbf x_i-\textbf x_j|}\]
\[V_{\text{ne}}=-\alpha\hbar c\sum_{i,j}\frac{Z_i}{|\textbf X_i-\textbf x_j|}\]
For an arbitrary molecule, the corresponding molecular Hamiltonian \(H\) is difficult to diagonalize. Fortunately, the existence of large mass gap \(M_i/m_e\sim 10^3\) between the nuclei and electrons motivates the Born-Oppenheimer approximation. This consists of \(2\) steps:
Step #\(1\): Diagonalize the electronic Hamiltonian \(H_{\text e}\) by itself first, viewing the nuclei positions \(\{\textbf X_i\}\) as fixed parameters:
\[H_{\text e}|E_{\text e}\{\textbf X_i\}\rangle=E_{\text e}\{\textbf X_i\}|E_{\text e}\{\textbf X_i\}\rangle\]
By adiabatically varying \(\{\textbf X_i\}\), one thereby obtains an effective potential energy surface \(E_{\text e}\{\textbf X_i\}\).
Step #\(2\): Solve \((T_{\text n}+E_{\text e}\{\textbf X_i\})|E\rangle=E|E\rangle\) to get the molecular energies \(E\).
in which the nuclei are taken to be roughly clamped \(T_{\text n}\approx 0\) about their equilibrium positions \(\langle{\textbf X}_i\) except that the internuclear potential:
\[V_{\text{nn}}\approx \text{const}+\frac{1}{2}\sum_{i,j}\left(\frac{\partial^2 V_{\text{nn}}}{\partial X_i\partial X_j}\right)_{\text{eq}}\eta_i\eta_j\]
permits small harmonic vibrations of the nuclei about their clamped equilibria, where \(\eta_i=X_i-\bar X_i\). By diagonalizing the Hessian \(\left(\frac{\partial^2 V_{\text{nn}}}{\partial X_i\partial X_j}\right)_{\text{eq}}\) in its eigenbasis of normal modes, one can obtain a set of fictitious nuclei whose vibrations are completely decoupled from each other, i.e. a bunch of independent harmonic oscillators.
Semiclassically, the perturbation due to an external electromagnetic wave \(\gamma\) is \(\Delta H_{\gamma}=-e\sum_{\text{electrons } i}\textbf E_0\cos(\textbf k\cdot\textbf x_i-\omega t)+e\sum_{\text{nuclei }i} Z_i\cos(\textbf k\cdot\textbf X_i-\omega t)\) where \(\omega=c|\textbf k|\).