The exact calculation of long-range forces between atoms by perturbation theory

A general sum rule is described which has many variants; it permits, for example, the exact calculation of the long-range forces between a proton and a hydrogen atom using conventional perturbation theory. The method is exemplified by the calculation of the second-order term and a precise assessment of the significance of the continuum states is made. The basic identity for a given function g is (r$|$g$|$s)/(E$_{r}$ - E$_{s}$) $\equiv $ (r$|$f$|$s), where the function f satisfies a certain differential equation (13). The limitation of the method is the difficulty which may arise in solving the equation. The sum rule technique appears to have applications to many fields.