Suppose that you have a nucleus with a complex shape. Using multipole expansion, you can say that its general shape is an overlap of a sphere, an ellipsoid and higher order corrections. Each one of these corrections has a respective hamiltonian. If you truncate the series up to 1st order, you have to think about 2 hamiltonians, an H_0 for the sphere and an H_1 for the ellipsoid. Then, you can for instance use H_1 for your 1st order perturbation theory. If, however, you do not use a multipole expansion, you can’t know what the Hamiltonian responsible for the perturbation is.