Nonintegral Occupation Numbers in Transition Atoms in Crystals

作者：Joseph C. Slater, J. Bryan Mann, Timothy M. Wilson, John H. Wood · 发表于：Physical Review · 年份：1969 · DOI：10.1103/physrev.184.672 · 被引用次数：281 · 研究领域：Advanced Chemical Physics Studies、Thermodynamic and Structural Properties of Metals and Alloys、X-ray Diffraction in Crystallography

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. ...