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.

推导并实现了用于二阶和三阶Douglas-Kroll（DK2和DK3）方法的双电子排斥算子公式。该公式适用于Z  = 10、20，…，130的类氦离子以及稀有气体原子。DK2和DK3双电子算子获得的能量显示出与四分量方法的差异，该差异比非相对论和DK1双电子算子的能量小一个数量级。还推导了小分量型两电子积分的近似公式。将它们应用到Z  = 80和130的氦样离子上，并通过数字证实非常精确地再现未逼近的值。