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We consider the implications of a nonintegral occupation number of 3d and 4? electrons in a 3d transition element or compound, in a configuration such as 3d n+x 4:S 2~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/' of the 3d and 4s orbitals must be equal; these quantities E/, which we call modified one-electron energies, are different from the ordinary one-electron energies Ei of Hartree-Fock theory, involving only one-half the self-energy correction met with in HF theory. These quantities E/, rather than the ordinary one-electron energies Ei, 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 t f 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/, 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 0 and h a components. r , , rr, . i J ences to all that earlier work here. Their general ","_ .. conclusion is that the occupation 3d n+1 As leads to energy * Work performed under the auspices of the National Science UJ-U^ , .,i . ^ ^ Foundation and the U.S. Atomic Energy Commission. bands ln better agreement with experiment than 1 J. C. Slater, T. M. Wilson, and J. H. Wood, Phys. Rev. 179, 3dHs 2 , but they leave open the question as to whether 28 (1969) The reader's attention is also called to the interesting a nonin...

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Self-consistent calculations of theCu + ion have been carried out using five different methods of approximating the Hartree-Fock exchange. These calculations have been compared with Hartree's Cu + calculation to test the accuracy of the various approximations and to interpret their interrelations. The best results were obtained from two quite different methods. The first, suggested by Uberman with modifications which we have introduced, uses a different local exchange potential for each orbital and gives a very good approximation to the Hartree-Fock method, but with considerable computational difficulty. The second amounts to multiplying the local potential proportional to the J power of the electronic charge density, suggested by the senior author in 1951, by a constant factor a chosen to minimize the total energy. This second method is much simpler to apply than the first and gives very nearly as good orbitals, as well as a very good total energy, but gives poor one-electron energies for the x-ray levels. The reasons for the different results are discussed. The latter method, which has been empirically arrived at by a number of the workers in the energy-band field, is probably the most useful one for practical calculation.

The statistical exchange method is derived directly in terms of a variation of the total energy expressed in terms of the statistical exchange energy. The formulation holds for the so-called Xoc and Xm,8 exchange methods. I t is pointed out that though the orbitals determined by these methods agree well with Hartree-Fock orbitals in cases where the comparison can be made, the eigenvalues are different, because they represent different quantities: in the Hartree-Fock method the energy differences between the energies of the ground state and the ion lacking an electron, in the statistical method the partial derivative of the total energy with respect to the occupation number. The total energy as derived by the statistical method is discussed, as a possible substitute for the exact total energy. I t is shown that the parameters E and ,8 can be chosen so that the statistical total energy agrees exactly with the Hartree-Fock total energy, and this is recommended as the best method for choosing these parameters. I t is shown that the exchange potential encountered in the Xu8 method is much further from the Hartree-Fock exchange potential than that found in the X a method, so that the latter is to be preferred. I n order to test the use of the statistical expression for total energy, we study the statistical total energy as a function of occupation numbers, and discuss a power series expansion of this total energy. I n terms of this expansion, it is simple to discuss the difference between the eigenvalues of the HartreeFock and of the statistical methods. I t is pointed out that when one uses an energy-band approach to a problem, one must shift electrons from one energy band to another in the iteration required to achieve self-consistency, and with the X a method this means filling the lowest energy bands up to the Fermi energy. On the basis of such energy shifts, a qualitative discussion is given of the disappearance of magnetic moments ofsuch atoms as vanadium, when the atoms are combined into a metallic crystal. A discussion is given of the relation between eigenvalues of the statistical or of the Hartree-Fock method in studying the optical absorption by an insulating crystal. From a historical sketch of the development of the theory of the exciton, it is seen that absorption by such an insulating crystal is much more like absorption by an isolated atom than like the transitions between extended or Bloch functions which one meets with a metal.