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            <title xml:lang="en">On the Accuracy of Mean-Field Spin–Orbit Operators for 3d Transition-Metal Systems</title>
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                <title xml:lang="en">On the Accuracy of Mean-Field Spin–Orbit Operators for 3d Transition-Metal Systems</title>
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              <p>We present an extensive study of the performance of mean-ﬁeld approximations to the spin−orbit operators on realistic molecular systems, as widely used in applications like single-molecule magnets, molecular quantum bits, and molecular spintronic devices. The test systems feature a 3d transition-metal center ion (V, Cr, Mn, Fe, Co, and Ni) in various coordinations and a multitude of energetically close- lying open-shell conﬁgurations that can couple via the spin−orbit operator. We performed complete active space spin−orbit conﬁguration interaction calculations and compared the full two-electron Breit-Pauli spin−orbit operator to diﬀerent approximations: the one-center approximation, the spin−orbit mean-ﬁeld approach with electron densities from diﬀerent state-averaging procedures, and the atomic mean-ﬁeld integral approximation. We show that the mean-ﬁeld approaches can lead to signiﬁcant errors in the spin−orbital coupling matrix elements, which becomes particularly visible for the computed zero-ﬁeld splittings. The one-center approximation, keeping all relevant two-electron terms, seems to be a signiﬁcantly more accurate choice for the examples from our test set.</p>
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