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– Without making a multipole expansion, the potential due to the general shape of the nucleus would be highly complex and difficult to handle mathematically. Perturbation theory relies on treating the perturbing potential as a small deviation from a known, solvable system. If the potential is too complex, it becomes challenging to apply perturbation theory effectively.
– 1. The multipole expansion is valid when the potential in the system varies slowly over the spatial extent of the system itself. So the higher order terms are very small.
2. Perturbation theory is applicable when the perturbation (the additional potential) introduced to the system is small compared to the unperturbed system’s energy.
3. The validity of the multipole expansion typically requires slow variations in the potential, while perturbation theory requires weak interactions. So I think it is difficult but could be the case in some situations.
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