.

Assume a Hamiltonian

$$\mathcal{H}= \mathcal{H}_{\rm e}(q,p)+ \mathcal{H}_{\rm n}(X,P) + \mathcal{H}_{\rm en}(q,X)$$ (5.1)

for the interaction between electrons $e$ and nuclei $n$ in a molecule, where $X$ stands for the nuclear and $q$ for the electronic coordinates.

a) Write down the Schrödinger equation for i) the electronic wave function $\psi_e(q,X)$, and ii) the nuclear wave function $\phi_n(X)$ in the Born-Oppenheimer approximation.

b) Briefly explain the idea of the Born-Oppenheimer approximation.

c) Assuming a basis of electronic states $\psi_\alpha(q,X)$, write the total wave function of a molecule as $\Psi(q,X)=\sum_\alpha \phi_\alpha (X) \psi_\alpha(q,X)$. Hence derive the Schrödinger equation for the nuclear part,

$$\left[-\frac{\hbar^2}{2M}\nabla_X^2 +E_\alpha(X) -\frac{\hbar^2}{2M}G(X) -\frac{\hbar^2}{M}F(X)\right] \vert\phi_\alpha\rangle_n = \mathcal{E} \vert\phi_\alpha\rangle_n$$

$$G(X)\equiv \langle \psi_\alpha\vert\nabla^2_X\psi_\alpha\rangle,\quad F(X)\equiv \langle \psi_\alpha\vert\nabla_X\psi_\alpha\rangle ,$$ (5.2)