Multipole polarizabilities for hydrogenic bound states

The bound-state wave function of a single nonrelativistic particle is written as Fexp(-G), where F contains the nodal information and is restricted to be polynomial and G is the negative of the logarithm of the wave-function envelope which contains the spectral information. As a perturbation is turned on, both F and G respond, but the response in G can be absorbed in F. A perturbative expansion on F and the energy leads to a hierarchy of inhomogeneous differential equations which resemble Gauss's law with a variable dielectric constant. If the perturbation is of polynomial form, one reasonably expects polynomial solutions for the perturbative corrections to F in this hierarchy. This method is used to obtain the first-order wave-function correction for the hydrogenic $2S$ and 2P₀ states in a multipole field and their corresponding multipole polarizabilities. In the dipole case, the method is modified to treat degenerate mixing. Then the first-order correction to the wave function for an arbitrary hydrogenic bound state with azimuthal quantum number $m=0$ in a large-order multipole field where neither degeneracy mixing nor first-order energy shift occurs and its corresponding multipole polarizabilities are calculated in closed forms.