**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.