Range of validity for perturbative treatments of relativistic sum rules

The range of validity of perturbative calculations of relativistic sum rules is investigated by calculating the second-order relativistic corrections to the Bethe sum rule and its small momentum limit, the Thomas-Reiche-Kuhn (TRK) sum rule. For the TRK sum rule and atomic systems, the second-order correction is found to be less than $0.5%$ up to about $Z=70.$ The total relativistic corrections should then be accurate at least through this range of Z, and probably beyond this range if the second-order terms are included. For Rn $(Z=86),$ however, the second-order corrections are nearly 1%. The total corrections to the Bethe sum rule are largest at small momentum, never being significantly larger than the corresponding corrections to the TRK sum rule. The first-order corrections to the Bethe sum rule also give better than $0.5%$ accuracy for $Z<70,$ and inclusion of the second-order corrections should extend this range, as well.