Calculation of the Bethe logarithm for the Rydberg states of helium

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Abstract
A very efficient technique for the finite-basis-set calculation of logarithmic sums is introduced. The basis sets contain sequences of nonlinear parameters determined by the zeros of Laguerre polynomials. It is shown that one can easily obtain convergence to 12 digits in fast calculations involving small basis sets. The method is then applied to calculations of the asymptotic expansion of the Bethe logarithm for the Rydberg states of helium. As perturbation calculations of the sums involved diverge, the present method is used to obtain accurate results for atoms in dipole and quadrupole fields from which the adiabatic contributions are extracted. A discussion of gauges in the multipole case and the suppression of roundoff errors by working in a mixed gauge is presented. A strategy for handling the nonadiabatic contributions to sums is presented and used to obtain preliminary bounds on these contributions.