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Calculations of Bethe logarithms for Hydrogen and Helium atoms using B-splines in different gauges

The efficient and simple B-splines variational method of calculating the hydrogen atom Bethe logarithms in the acceleration gauge [Y-B Tang et.al., Phys. Rev. A $\mathbf{87}$, 022510 (2013)] is successfully applied to other gauges. The ground state Bethe logarithm of H 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 H are at the 10$^{-10}$ level of accuracy. This represents the successful variational attempt to calculate Bethe logarithms of the hydrogen atom in the velocity and length gauges. Furthermore, the B-splines variational method is successfully extended to the helium atom by the aid of configuration interaction. Bethe logarithms for $n\,^{1,3}S\,\,(n=1-8)$ states are calculated in three different gauges. The acceleration-gauge value of 4.364 036 7(2) a.u. and the velocity-acceleration gauge value of 4.364 036 4(2) a.u. for the $2\,^3S$ state are achieved, which are both at the 10$^{-7}$ level of accuracy. For other triplet states, present results in three different gauges all have five to seven accurate figures. While for the singlet states, the best convergent values are obtained in the pa-gauge, of which the numerical precision is also at the $10^{-5}$ to $10^{-7}$ level of accuracy.