Leading relativistic corrections for atomic $P$ states calculated with a finite-nuclear-mass approach and all-electron explicitly correlated Gaussian functions

In this work we report progress in the development and implementation of quantum-mechanical methods for calculating bound ground and excited states of small atomic systems. The work concerns singlet states with the $L=1$ total orbital angular momentum ($P$ states). The method is based on the finite-nuclear-mass (non-Born-Oppenheimer; non-BO) approach and the use of all-particle explicitly correlated Gaussian functions for expanding the nonrelativistic wave function of the system. The development presented here includes derivation and implementation of algorithms for calculating the leading relativistic corrections for singlet states. The corrections are determined in the framework of the perturbation theory as expectation values of the corresponding effective operators using the non-BO wave functions. The method is tested in the calculations of the ten lowest ${}^{1}P$ states of the helium atom and the four lowest ${}^{1}P$ states of the beryllium atom.