Selection of a Gaussian basis set for calculating the Bethe logarithm for the ground state of the hydrogen atom

The inclusion of the leading quantum electrodynamic energy correction is unavoidable if one aims for sub 0.01 cm^-1 accuracy in the calculations of atomic and molecular spectra. The important part of the calculation of the correction is the determination of the value of the so-called Bethe logarithm (ln k 0). In this work, we examine how to construct Gaussian basis sets for such calculations. While the testing is performed for the ground state of the hydrogen atom, the results allow us to suggest how the construction of appropriate basis sets can be handled in ln k 0 calculations of atoms and molecules with more than one electron performed with explicitly correlated all-electron Gaussian functions.