Relativistic extension of the Bethe sum rule

We have derived an expression in terms of eigenvalues and eigenfunctions of the no-pair Hamiltonian for the sum over the generalized oscillator strength distribution for the relativistic, positive-energy solutions for a many-electron system. Formally, it looks very much like the corresponding nonrelativistic sum. However, only in the nonrelativistic limit is the sum equal to the number of electrons (the Bethe sum rule). We have determined the leading (1/c²) relativistic correction to the nonrelativistic Bethe sum rule and we argue that this correction is more in accordance with relativistic quantum mechanics than is a previous suggestion. The same statement holds for relativistic corrections to the Thomas-Reiche-Kuhn sum rule, previously derived only for one-electron systems. We discuss the problem that the explicit summation of the generalized oscillator strengths and calculation of the sum as a ground-state average of a comutator, while formally equivalent, may not give identical results in approximate calculations.