Energy partitioning of the self‐consistent field interaction energy of ScCO

1989 ◽  
Vol 90 (10) ◽  
pp. 5555-5562 ◽  
Author(s):  
Regina F. Frey ◽  
Ernest R. Davidson
Keyword(s):  
Interaction Energy ◽  
Energy Partitioning ◽  
The Self ◽  
Field Interaction ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

scholarly journals The relativistic interaction of two electrons in the self-consistent field method

1936 ◽  
Vol 157 (892) ◽  
pp. 680-696 ◽  
Keyword(s):  
Interaction Energy ◽  
Structure Constant ◽  
Field Method ◽  
The Self ◽  
Coulomb Energy ◽  
Relativistic Case ◽  
Consistent Field ◽  
Self Consistent ◽  
The Matrix ◽  
Self Consistent Field

The extension of the self-consistent field method for the relativistic case has been discussed in a previous paper, There no attempt was made to consider the interaction energy of two electrons to a greater degree of accuracy than that given by the Coulomb energy. In the present paper the interaction of the spins and the effect of retardation is introduced. The method is then applied to the evaluation of the separations of the components of the 2 3 P term of helium. The relativistic expression for the interaction of two electrons has been discussed by several authors. The expressions they obtain may be shown to agree as far as terms of the first order in e 2 and the square of the fine structure constant. We shall follow the discussion of Bethe and Fermi since this seems to be most suited for application to the self-consistent field method. We require the matrix elements of the interaction energy I of two electrons 1 and 2, corresponding to given transitions of the two electrons. We denote the states of electron 1 by N 1 , N' 1 , N" 1 , ..., and those of electron 2 by N 2 , N' 2 , N" 2 , ..., where each N stands for the set of four quantum numbers specifying a state of the electron. Then the matrix element (N 1 , N 2 |I| N' 1 , N' 2 ) corresponding to a transition N 1 → N' 1 for electron 1 and N 2 → N' 2 for electron 2 is found as follows. We form the charge and current density corresponding to a transition N 1 → N' 1 of electron 1. In Hartree atomic units these are (N 1 |ρ| N' 1 ) = Ψ* (N' 1 |1) Ψ (N 1 |1) e i (E' 1 -E 1 ) t (1)

scholarly journals The relativistic self-consistent field

1935 ◽  
Vol 152 (877) ◽  
pp. 625-649 ◽  
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Wave Functions ◽  
The Self ◽  
Magnetic Interactions ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Schrödinger's Equation

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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