Multipole polarizabilities for hydrogenic bound states

The bound-state wave function of a single nonrelativistic particle is written as $F\exp \mathrm{}(-G)\mathrm{}$, 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 $2P_0$ 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 $m=0\mathrm{}$ 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.