Born-Oppenheimer approximation

[jaejoon89]

I'm trying to figure out what the Hamiltonian for a simple molecule is using the Born-Oppenheimer approximation.

1) My textbook gives the Hamiltonian for a simple system like H2 when you hold the internuclear distance constant. The only terms that drop out are the ones where you take the Laplacian for the atoms. Since the B-O approximation separates the wavefunction into nuclear and electronic components, I'm guessing this must be the electronic component.

2) But what does the nuclear Hamiltonian look like?

 

[transience]

Are you trying to find the full Hamiltonian? If so you need to add up all contributions to the energy from the different sources and then apply the Born-Oppenheimer approximation.

The Born-Oppenheimer treats the nuclei as fixed, so they have zero kinetic energy and the interaction between them is constant, this will simplify the full Hamiltonian and give you a chance of separating electronic and nuclear potential energy.