Calculations of Bethe logarithm for hydrogen and helium using B-splines in different gauges

Abstract

The efficient and simple B-splines variational method is successfully used to calculate Bethe logarithm for the hydrogen atom in the velocity and length gauges. The ground state Bethe logarithm of hydrogen with fourteen accurate figures is obtained in the velocity gauge, and in the length gauge the ground state value has eleven accurate figures. Present velocity- and length-gauge results for the ns, np, nd, and nf states up to n = 200 of hydrogen are above the 10−10 level of relative difference, which represent the successful variational attempt to calculate Bethe logarithm of hydrogen in the velocity and length gauges. In addition, the B-splines variational method is successfully extended to calculate Bethe logarithm for the helium atom combined with configuration interaction. Results of numerical calculations for n1,3S up to n = 8 states are presented in the acceleration gauge, the velocity gauge, and a hybrid of the velocity and acceleration gauges(also called pa-gauge). For the 23S state, the acceleration-gauge value of 4.364 036 7(2) a.u. and the pa-gauge value of 4.364 036 4(2) a.u. both have eight significant digits. For other triplet S states, present results in three different gauges all have five to seven accurate figures. While for the singlet S states, the best convergent values are obtained in the pa-gauge, of which the relative differences with the latest results are at the 10−6 to 10−8 orders of magnitude.