Understanding the Born-Oppenheimer Approximation: A Mathematical Proof

[Vicol]

Hello everyone,

In Born-Oppenheimer approximation there is one step, when you divide your wavefunction into two pieces - first dependent on nuclei coordinates only and second dependent on electron coordinates only (the nuclei coordinates are treated as parameter here). The "global" wavefunction is a product of these two. Why "almost" independent movement of nuclei and electrons determine form of global wavefunction as product of electron and nuclei wavefunctions? What is mathematical proof of that?

 

[DrDu]

If you write the Hamiltonian as $H_{tot}=T_N+T_e +V(r,R)$ with $T_{N/e}$ being the kinetic energy operator of the nuclei and electrons, respectively, and V the Coulomb interaction of the electrons with coordinates r and nuclei with coordinates R, then the electronic hamiltonian is
$H_{el}=T_e+ V(r,R)$ with eigenvalues $\psi_n(r;R)$. These eigenvalues form a complete basis in which also the eigenvalues $\Psi_m(r,R)$ of $H_{tot}$ can be developed, namely $\Psi_m=\sum_n \psi_n(r;R) \phi_{nm}(R)$.
Born and Oppenheimer now claim that it is - at least sometimes - a good approximation to keep only a single term of the sum. The condition for this to be approximately true is that the action of $T_N$ on $\psi_n$ can be neglected, which is justified mainly by the dependence of $T_N$ on the small factor $m/M$ where $M$ is the mass of the electron and M a typical mass of the nuclei. A further condition is that the electronic states are energetically well separated - this condition fails for example for Jahn-Teller states, where orbitals become degenerate due to symmetry restrictions.
I invite you to set up an equation for the $\phi_{nm}$ so we can work out the details.