Hydrogen atom under a sequence of static multipole perturbations: Wave-function corrections

A closed-form expression is obtained for 〈r‖$G^j$V‖${\phi }_0$〉, where ${\phi }_0$ is the nonrelativistic hydrogenic ground state, V is a sum of static multipole perturbations of the form $A_{\mathrm{Ll}}$ $r^L$ $P_l$, L≥l, G=Q($E_0$-$H_0$ ${)}^{\mathrm{-}1}$Q is the projected static Green operator, and Q=1-‖${\phi }_0$〉〈${\phi }_0$‖ is the pro- jection operator. This method can be used iteratively to obtain the solution for &V &... &‖${\mathrm{\phi }}_0$〉, where {$V_m$,$V_{m\mathrm{-}1}$,...,$V_1$} is a set of static multipole perturbations. This quantity is of great interest in the computation of sum rules, high-order wave-function corrections due to multipole perturbations, and the adiabatic and nonadiabatic potentials in the scattering of a charged particle by a hydrogenic ion and in the interaction between a Rydberg electron and the core ion in high Rydberg states of heliumlike ions.