Nonintegral Occupation Numbers in Transition Atoms in Crystals

Abstract

We consider the implications of a nonintegral occupation number of $3d$ and $4s$ electrons in a $3d$ transition element or compound, in a configuration such as $3{d}^{n+x}4{s}^{2\ensuremath{-}x}$, where $x$ is a variable. In the energy-band problem, such fractional occupation numbers are common, but we pay particular attention to the atomic problem. We apply Hartree-Fock procedures to such a problem, using the formula for the average energy of all multiplets associated with the configuration, and vary not only the orbitals but the occupation numbers to minimize the energy. We consider both the non-spin-polarized and spin-polarized cases. This procedure, which is more general than the ordinary Hartree-Fock procedure, we shall call the hyper-Hartree-Fock method (HHF). We have carried through HHF calculations for fractional occupation numbers in the Co and Ni atoms, and have also treated these atoms by several schemes involving approximate statistical exchanges. We compare the results with the atomic spectra of these atoms. We find that the condition for minimum energy, in the HHF scheme, can be put in a form stating that one-electron energies $E_{i}^{}{}_{}{}^{\ensuremath{'}}$ of the $3d$ and $4s$ orbitals must be equal; these quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$, which we call modified one-electron energies, are different from the ordinary one-electron energies ${E}_{i}$ of Hartree-Fock theory, involving only one-half the self-energy correction met with in HF theory. These quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$, rather than the ordinary one-electron energies ${E}_{i}$, are the quantities which have the properties desired for one-electron energies in energy-band theory and Fermi statistics: The change in the total energy of the system, when an infinitesimal fraction of the electrons shifts from one orbital to another, rigorously equals the net change in the quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$ for the electrons which have made the shift. We show that the ordinary one-electron eigenvalues of the Kohn-Sham statistical exchange method form fairly good approximations to these HHF quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$, which explains why energy-band calculations using that exchange have had considerable success in studies of transition-element crystals and compounds. Preliminary mention is made of calculations under way by one of the authors (TMW) on the antiferromagnetic crystals MnO and NiO, in which an exchange potential set up according to the ideas presented here leads to energy bands describing correctly the electrical, magnetic, and optical behavior of these crystals, including the insulating properties and the crystal-field splitting of the $3d$ orbitals into the ${e}_{g}$ and ${t}_{2g}$ components.