scholarly journals iOI: An Iterative Orbital Interaction Approach for Solving the Self-Consistent Field Problem

Author(s):  
Zikuan Wang ◽  
Wenjian Liu

Quantitative solutions by Fock's equations for the self-consistent field, including exchange effects, have now been obtained for the normal state of Na + , and for the normal and some excited states of neutral Li and Na, by Fock and Petrashen, and for the normal and the (2 s ) (2 p ) 3 P and 1 P excited states of neutral Be, by the present authors. The work here described was undertaken as the first step in carrying out the solution of Fock's equations for heavier atoms; calculations for Cu + , in progress at the time of writing, will carry this a step further. The effect of the inclusion of exchange terms on the self-consistent field may be expected to be particularly large for a negative ion, on account of the sensiticeness of the wave function of the outer ( nl ) group of such an ion (and, to a less extent, of those of the other groups of the outer shell also). This, of course, is likely to make the process of solution of Fock's equations more than usually difficult and lengthy for a negative ion, and made the project of obtaining such a solution appear somewhat ambitions; but, on the other calculations, are greatest when the difference from the solution of the self-consistent field problem without exchange is greatest, and for this reason it seemed desirable to carry out the solution of Fock's equations for at least one negative ion.


1—The method of the self-consistent field for determining the wave functions and energy levels of an atom with many electrons was developed by Hartree, and later derived from a variation principle and modified to take account of exchange and of Pauli’s exclusion principle by Slater* and Fock. No attempt was made to consider relativity effects, and the use of “ spin ” wave functions was purely formal. Since, in the solution of Dirac’s equation for a hydrogen-like atom of nuclear charge Z, the difference of the radial wave functions from the solutions of Schrodinger’s equation depends on the ratio Z/137, it appears that for heavy atoms the relativity correction will be of importance; in fact, it may in some cases be of more importance as a modification of Hartree’s original self-nsistent field equation than “ exchange ” effects. The relativistic self-consistent field equation neglecting “ exchange ” terms can be formed from Dirac’s equation by a method completely analogous to Hartree’s original derivation of the non-relativistic self-consistent field equation from Schrodinger’s equation. Here we are concerned with including both relativity and “ exchange ” effects and we show how Slater’s varia-tional method may be extended for this purpose. A difficulty arises in considering the relativistic theory of any problem concerning more than one electron since the correct wave equation for such a system is not known. Formulae have been given for the inter-action energy of two electrons, taking account of magnetic interactions and retardation, by Gaunt, Breit, and others. Since, however, none of these is to be regarded as exact, in the present paper the crude electrostatic expression for the potential energy will be used. The neglect of the magnetic interactions is not likely to lead to any great error for an atom consisting mainly of closed groups, since the magnetic field of a closed group vanishes. Also, since the self-consistent field type of approximation is concerned with the interaction of average distributions of electrons in one-electron wave functions, it seems probable that retardation does not play an important part. These effects are in any case likely to be of less importance than the improvement in the grouping of the wave functions which arises from using a wave equation which involves the spins implicitly.


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