Exchange in Spin-Polarized Energy Bands

In an atom, or a crystal, with net magnetic moment, the Hartree-Fock equations for electrons with spins up (that is, parallel to the net magnetization) and spins down (opposite) are different. By using different wave functions for the different spins, one attains spin-polarized calculations, which are more accurate than the ordinary restricted Hartree-Fock type. One-electron energies for electrons of spin up and down will differ. The method has been used in recent calculations of energy bands in ferromagnetic and antiferromagnetic crystals. The present paper studies the essential feature of the calculations, the nature of the exchange integrals responsible for the energy difference between spin-up and spin-down energies. This is done both in terms of the rigorous Hartree-Fock method, and in terms of the free-electron exchange method which is used in the energy-band calculations. It is shown that the two methods are consistent with each other, the free-electron exchange method giving a good approximation to the spin-polarization effect. Correlation can be taken into account by decreasing the difference in exchange energy between spin up and spin down, in a way similar to what must be done in the theory of atomic multiplets, in using empirical $F^k$ integrals which are smaller than those found by the Hartree-Fock method. Such a decrease in exchange effect has been found necessary to get agreement with experiment in the spin-polarized energy-band calculations.