Why we use the multipole expansion

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    By first utilising the multipole expansion on the interaction term, we can seperate the electron nucleus interaction from more complicated interactions due to the shape of the nucleus. Even for a general nucleus there will be some form of deformation thus if we would use perturbation theory without the multipole expansion it would be more difficult to distinguish between different interaction contributions. This becomes problematic as the basic electron nucleus interaction is part of what forms the energy levels thus being unable to distinguish these interactions might lead to energy levels different to what we might expect.

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