An algorithm for calculating the Bethe logarithm for small molecules with all-electron explicitly correlated Gaussian functions

An algorithm for calculating the Bethe logarithm, which is a part of the leading quantum electrodynamics energy correction, for the ground states of light molecules with an arbitrary number of electrons is derived and tested. The tests concern small atoms and one- and two-electron dihydrogen molecular systems. All-electron explicitly correlated Gaussian functions are used in the calculations. Next, the approach is employed to calculate the Bethe logarithm for the LiH molecule. These are the first calculations of the logarithm for a four-electron system. The method developed in this work allows to extend the Bethe-logarithm calculations to a wider range of molecules.