Bethe Logarithm and QED Shift for Lithium

Abstract

A novel finite basis set method is used to calculate the Bethe logarithm for the ground $2{}^{2}S_{1/2}$ and excited $3{}^{2}S_{1/2}$ states of lithium. The basis sets are constructed to span a huge range of distance scales within a single calculation, leading to well-converged values for the Bethe logarithm. The results are used to calculate an accurate value for the complete quantum electrodynamic energy shift up to order ${\alpha }^3$ Ry. The calculated $3{}^{2}S_{1/2}–2{}^{2}S_{1/2}$ transition frequency for ${}^7\mathrm{L}\mathrm{i}$ is $27206.0926(9){\mathrm{c}\mathrm{m}}^{-1}$, and the ionization potential for the $2{}^{2}S_{1/2}$ state is $43487.1583(6){\mathrm{c}\mathrm{m}}^{-1}$. The ${}^7\mathrm{L}\mathrm{i}$–${}^6\mathrm{L}\mathrm{i}$ isotope shift is also considered, and all the results compared with experiment.