First-Order geometrical response equations for state-averaged multiconfigurational self-consistent field (SA-MCSCF) wave functions

1991 ◽  
Vol 40 (3) ◽  
pp. 361-378 ◽  
Author(s):  
Keld Lars Bak ◽  
Jerry Boatz ◽  
Jack Simons
Keyword(s):  
Wave Functions ◽  
First Order ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

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.

The Radial Charge Densities for the Ti+2 Argon Core

1960 ◽  
Vol 56 (2) ◽  
pp. 174-175
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Functions ◽  
Charge Densities ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Materials Research ◽  
Research Office

The radial charge densities tabulated opposite were communicated some years ago by the late D. R. Hartree to R. J. Weiss, of the Ordnance Materials Research Office, Watertown Arsenal, Watertown 72, Massachusetts, U.S.A. At his suggestion they are now published, no self-consistent field calculations for Ti being available. It is not known how Hartree obtained the wave functions which he describes as ‘estimated’, but it is presumed that they are interpolated. Radial charge densities for the 3d electrons for Ti+1 have already been published (l).

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