# Born Oppenheimer Approximation: Why can any molecular state be represented as a linearcombination of electronic states?

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