Expectation values in two-component relativistic theories

We examined numerically the equivalence between the expectation values calculated by the four-component wave function and those calculated by the two-component wave functions generated by the infinite-order Douglas-Kroll (IODK) transformation. We showed the expectation values 〈 r^-1 〉 and 〈 δ (r-R) 〉 in several closed-shell atoms using the so-called picture-changed 2×2 operators at some levels of approximation. The effect of the two-electron Coulomb term was also discussed. The numerical results indicated that the accuracy of 〈 r^-1 〉 mainly depends on the level of the wave functions, while that of 〈 δ (r-R) 〉 is affected by the accuracy of both the wave functions and the picture-changed operators. As expected, the picture-changed operators and the IODK wave functions generate essentially equivalent expectation values in comparison with those calculated by the four-component Dirac-type method.