Calculation of electric-field gradients based on higher-order generalized Douglas–Kroll transformations

In this paper, the calculation of electric-field-like properties based on higher-order Douglas–Kroll–Hess (DKH) transformations is discussed. The electric-field gradient calculated within the Hartree–Fock self-consistent field framework is used as a representative property. The properties are expressed as an analytic first derivative of the four-component Dirac energy and the $nth$-order DKH energy, respectively. The differences between a “forward” transformation of the relativistic energy or the “back transformation” of the wave function is discussed in some detail. Detailed test calculations were carried out on the electric-field gradient at the halogen nucleus in the series $HX$ $(X=F,Cl,Br,I,At)$ for which extensive reference data are available. The DKH method is shown to reproduce (spin-free) four-component Dirac–Fock results to an accuracy of better than 99% which is significantly closer than previous DKH studies. The calculations of both the Hamiltonian and the property operator are shown to be essentially converged after the second-order transformation, even for elements as heavy as At. In addition, we have obtained results within the density-functional framework using the $DKHZ$ and zeroth-order regular approximation (ZORA) methods. The latter results included picture-change effects at the scalar relativistic variant of the ZORA-4 level and were shown to be in quantitative agreement with earlier results obtained by van Lenthe and Baerends. The picture-change effects are somewhat smaller for the ZORA method compared to DKH. For heavier elements significant differences in the field gradients predicted by the two methods were found. Based on comparison with four-component Dirac–Kohn–Sham calculations, the DKH results are more accurate. Compared to the spin-free Dirac–Kohn–Sham reference values, the ZORA-4 formalism did not improve the results of the ZORA calculations.