Although the H MF SO operator in eq. ( ) defines an effective oneelectron operator, the number of atomic spin-orbit integrals which must be calculated it is not reduced if the Cholesky representation is not used. A widely used approximation used in the literature to reduce the computational effort is to restrict to calculation of the two-electron integrals on the AO basis to comprise only the one-center terms. This methodology is based on the assumption that the spin-orbit two-electron operator is a short-range operator, which can be justified by the presence of 1/|r i -r j | 3 terms in H BP SO . This is called the atomic approximation. Moreover, the development that has gained larger popularity in the literature is the atomic mean field implementation (AMFI) proposed by Hess and Schimmelpfennig , where the H MF SO operator is used together with two additional approximations: (i) only one-center integrals are retained throughout, and (ii) the molecular orbitals φ a in eq. ( ) are replaced by atomic selfconsistent field (SCF) orbitals obtained from spherically averaged atomic SCF calculations with predetermined valence shell occupations rather than using the complete non-spherical molecular density. While the combination of these approximations reduces the time needed for the SOC integrals evaluation to a negligible amount, it has been shown that the inclusion of all the significant multicenters two-electron SOC integrals is of primary importance to achieve a good accuracy .