High accuracy for atomic calculations involving logarithmic sums

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Abstract
A method for the calculation of logarithmic sums that yields very high accuracy even for small basis-set dimensions is introduced. The best values achieved are accurate to 23 significant figures without extrapolation. The sums are performed directly on variational intermediate sets. The method automatically rejects any basis functions that could introduce linear dependence, therefore guaranteeing high numerical stability for a wide range of nonlinear parameters. Accurate values for the ordinary and a higher-order version of the Bethe logarithm are presented for a range of energy states and angular momenta. Given that the intermediate basis functions are increasingly confined to extremely small distances from the origin, a discussion of finite nuclear-size effects is given. The contribution to the sums from states with extremely high energies, orders of magnitude larger than the electron rest mass, is discussed.