Relativistic two-electron repulsion operator formulas for the Douglas–Kroll method

Abstract Two-electron repulsion operator formulas for the second- and third-order Douglas–Kroll (DK2 and DK3) method were derived and implemented. The formulas were applied to helium-like ions with Z=10, 20, …, 130, and to rare gas atoms. The energies obtained for the DK2 and DK3 two-electron operators showed differences from the four-component method that were one order of magnitude smaller than those of the nonrelativistic and DK1 two-electron operators. Approximation formulas for the small component-type two-electron integrals were also derived; they were applied to the helium-like ions with Z=80 and 130 and were numerically confirmed to reproduce unapproximated values very accurately.