A comparison of the hydrogenlike dipole radial matrix elements with overlap integrals and a step toward explicit expressions of the multipole matrix elements

For nonrelativistic radial hydrogenlike wavefunctions R(nl)(r), the ratio < R(nu lambda)vertical bar r vertical bar R(nl)>/< R(nu lambda)vertical bar R(nl)> of the dipole to the overlap integral is proportional to (l . Delta E)(-1) for l = lambda + 1. Delta E is the energy difference between the shells. and n. This property holds for discrete or continuous states, and can be derived as a particular case of the Feneuille recurrence formula (Feneuille 1970 C. R. Acad. Sci., Paris B 271 992). This relation and Feinberg's comparable one for < R(nu lambda)vertical bar r(-2)vertical bar R(nl)> suggest the existence of explicit expressions for the matrix elements < R(nu lambda)vertical bar r(t)vertical bar R(nl)>. For. nu not equal n we are not yet able to write down a general formula valid for any l and lambda, but we show with examples (lambda = 0, l = 1 and lambda = 1, l = 0) how explicit formulae can be obtained for fixed values of l and lambda.