STATISTICAL EXCHANGE AND THE TOTAL ENERGY OF A CRYSTAL.

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 Xα and Xαβ exchange methods. It 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. It is shown that the parameters α and β 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. It is shown that the exchange potential encountered in the Xαβ method is much further from the Hartree-Fock exchange potential than that found in the Xα method, so that the latter is to be preferred. In 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. In terms of this expansion, it is simple to discuss the difference between the eigenvalues of the Hartree-Fock and of the statistical methods. It 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 Xα 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 of such 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. Consequently, though the eigenvalues of the statistical method are a reliable guide in transitions met in a metallic crystal, the absorption by an insulator should be handled differently, more like an atom, resulting in wider energy gaps than would be found by direct use of the statistical eigenvalues.

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