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

A closed-form expression is obtained for 〈r{}GʲV{}φ₀〉, where φ₀ is the nonrelativistic hydrogenic ground state, V is a sum of static multipole perturbations of the form ALl{r}LPₗ, L{≥}l, G=Q(E₀-H₀{)}^{{{-}}1}Q is the projected static Green operator, and Q=1-{{φ}}₀$〉〈${{φ}}₀is the pro- jection operator. This method can be used iteratively to obtain the solution for ...{{{φ}}}₀〉, where{V}ₘ$,${V}_{m{{-}}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.