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in the Born Oppenehimer Approximation, one expands the molecular wavefunction $\Psi(x,X)$ in terms of the electronic wavefunctions $\phi(x;X)$: $\Psi(x,X)$ = $ \sum_k(c(X)_k\phi(x;X)_k)$ (x are the electronic coordinates and X are the nucleonic coordinates)

Now, since the electronic wavefunctions are eigenstates of the electronic Hamiltonian, the constitue a complete basis of the electronic space. Thus any electronic wavefunction can be expanded in termes of the eigenfunctions. But, how can we be sure that any molecelar wavefunction can be expanded in terms of the electronic wavefunctions? How can we be sure that the molecular Hilbertspace is not larger than the space which is spanned by the eigenstates of the electronic Hamiltonian?

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I thought the Born Approximation was the latest Matt Damon movie. Thanks for the clarification! –  twistor59 Feb 4 '13 at 7:24

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The key word here is approximation. Its not the that the hilbert space of the full molecule is not larger, its that the velocities of the much more massive nucleons can be neglected and only their positions need to be considered. In fact, the BO approximation breaks down in some cases when non-adiabatic effects are not negligible (e.g. loss or gain of energy due to changes in electronic orbits). An important point that Born and Oppenheimer make in their paper is their approximation is possible because earlier classically based approximations overstated the nuclear contribution because it neglected spin, which is purely a quantum phenomenon with no classical analog. This causes the nuclear contribution to be of the fourth order instead of the second order. This is another excellent case to show where classical mechanics breaks down in the description of the universe.

I guess to answer your question though, the hilbert space of the full system is actually larger then the electronic states, its just that under certain conditions some nuclear degrees of freedom can be ignored.

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