Here's what I've understood so far.
The Hamiltonian for a molecule made of $N$ nuclei, and $n$ electrons is: $${\cal \hat H} = \underbrace{{-\frac {\hbar^2}{2} \sum _{\alpha =1}^N } \frac {\nabla_\alpha ^2}{M_\alpha}}_{\rm {kinetic \ energy \\ of \ all \ nuclei }} \underbrace{{-\frac {\hbar^2}{2m} \sum _{i=1}^n} {\nabla_i ^2}}_{\rm {kinetic \ energy \\ of \ all \ electrons }} \underbrace{+ {\sum _{i<j}} \frac {e^2}{r_{ij}}}_{\rm {\ electric \ repulsion \\ electron-electron }} \underbrace{+ {\sum _{\alpha<\beta}} \frac {Z_{\alpha} Z_{\beta}e^2}{R_{\alpha \beta}}}_{\rm {\ electric \ repulsion \\ nucleus-nucleus }} \underbrace{- {\sum _{\alpha, i}} \frac {Z_\alpha e^2}{r_{\alpha i}}}_{\rm {\ electric \ attraction \\ electron-nucleus }}$$
Or, in short terms: $$ {\cal \hat H}= T_N + T_e + V_{ee} + V_{NN} + V_{Ne}$$
The strategy adopted to find a solution for the time-independent Schrödinger equation ${\cal \hat H} \psi = E \psi $ in the Born-Oppenheimer approximation is the following:
- Assume a factorized eigenstate of the kind $\psi ({\bf r},{\bf R}) = \phi_N ({\bf R}) \phi_e({\bf r}, {\bf R})$. (Here ${\bf r} =\{ {\vec r_i} \}_{i = 1, \dots, n}$, ${\bf R} =\{ {\vec R_\alpha} \}_{\alpha = 1, \dots, N} $)
- Treat ${\bf R} $ as a parameter and optimize $\langle \psi | {\cal \hat H}| \psi \rangle =E $ with respect to $\bf R$. (We are hence using a variational approach)
Let's analyze now the matrix element $\langle \psi | {\cal \hat H}| \psi \rangle =E $ for each term of the Hamiltonian written above.
My troubles start right away with the first term: $T_N$
$$ {T_N}| \psi \rangle = {T_N}| \phi_N \phi_e \rangle = -\frac {\hbar^2}{2} \sum _{\alpha =1}^N \frac {\nabla_\alpha ^2}{M_\alpha} \phi_N \phi_e $$
applying vector identity: $\nabla ^2 (ab) = b \nabla^2 a + 2 {{\vec \nabla} a \cdot {\vec \nabla} b + a \nabla ^2 b}$
$${T_N}| \phi_N \phi_e \rangle = \phi _e T_N \phi_N + \phi_N T_N \phi_e - \hbar^2 \sum_\alpha {1 \over M_\alpha}{\vec \nabla_\alpha} \cdot \{ \phi_e \vec \nabla_\alpha \phi_N + \phi_N \vec \nabla_\alpha \phi_e \} $$
My professor said the second term is negligible, and if you apply the bra to the third term it vanishes. uhm... can you show me?