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When not using the multi pole expansion first you will have to deal with the difficult nucleus-electron interaction term (Q x V) –> nucleus can have difficult shape. The multipole expansion can manage this complication.
Through the multipole expansion we can also easily recognize the energy scales making perturbation theory very convenient. Every term of the “Q x V = (Q0 x V0) + (Q1 x V1) + …” expansion, starting at term “1” corresponds to at term of the energy shifts of the perturbed Hamiltonian, these are the energy shifts due to the complicated shape of the nucleus.
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