Analytic evaluation of first-order properties within the mean-field variant of spin-free exact two-component theory

We present a scheme for the calculation of energies and analytic energy gradients within spin-free exact two-component (SFX2C) theory in its mean-field variant, which we refer to as SFX2C-mf. In the presented scheme, the Foldy-Wouthuysen transformation is carried out after the spin-free four-component Hartree-Fock treatment such that in electron-correlated calculations only the non-mean-field part of the two-electron interactions is handled in an untransformed manner. The formulation of analytic gradients requires some adjustments in comparison with the nonrelativistic case, i.e., the additional solution of the spin-free Dirac Coulomb coupled-perturbed Hartee-Fock equations together with a simplified treatment of orbital relaxation at the SFX2C-mf level. The improved accuracy of SFX2C-mf in comparison with SFX2C-1e is demonstrated in the calculation of energies, dipole moments, and electric-field gradients for the hydrogen halides HX, X = F-At. It is shown that the main contribution to the improvement stems from the elimination of the error at the Hartree-Fock (HF) level; however, the corresponding correlation contribution is also improved such that SFX2C-mf can be considered a suitable scheme for the treatment of heavy-element compounds for which the error of SFX2C-1e is rather substantial.