The self‐consistent field for crystals

Exchange in isolated atoms and ions is investigated, to throw light on the approximate exchange to be used in crystals. An intercomparison, carried out by the author, J. B. Mann, J. H. Wood, and T. M. Wilson, for the case of Cu + is described in detail. Here five different values of exchange are used: V XHF , the Hartree‐Fock value; V XS , a free‐electron exchange suggested by Kohn, Sham, and Gaspar, in which the exchange correct for a free‐electron gas at the Fermi energy is assumed; V XLSW , a modification of an exchange recently suggested by Liberman, proposed by the writer and Wood, in which we use an energy‐dependent exchange suggested by the free‐electron gas theory; and V Xa , in which the exchange V XS is multiplied by such a factor α that the total energy of the atom is minimized. Of these, V XLSW and V Xα both give orbitals which are good approximations to the Hartree‐Fock and V XLSW also gives eigenvalues agreeing well with the Hartree‐Fock. The V Xα method, however, gives eigenvalues numerically smaller than the Hartree‐Fock, by amounts which increase as we go to the inner electrons. In spite of this, the eigenvalues of the V Xα method empirically prove to form very good one‐electron energies to use in the energy‐band problem. The reason for this is then examined. A method for studying atoms with partially filled shells, such as the 3 d transition atoms, is described. We call it the Hyper‐Hartree‐Fock method, since it deals with an average energy over many electronic states, rather than a single state as in the HF method. The criterion for minimum energy is shown to be the equality of one‐electron energies E' i of two shells such as 3 d and 4 s , which can interchange electrons, where E' i differs from the ordinary eigenvalue of the Hartree‐Fock method. The latter represents the energy difference between an atom and an ion lacking an electron; the modified energy E' i is the derivative of the energy with respect to the occupation number. It is shown that these E' i 's are the energies which should be used in an energy‐band calculation, not the ordinary HF energies. The two sets of energies differ by a considerable amount in the transition elements, but the quantities E' i agree well with the eigenvalues of the method using the exchange V Xα . This appears to explain the success of the latter method for energy‐band calculations, and suggests that it is better than a Hartree‐Fock calculation, even if the latter could be carried out. It is pointed out that the quantities E' i are closely related to the electronegativities used by the chemists, and also to one‐particle energies met in the theory of the Fermi liquid.