A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level

2015 ◽  
Vol 142 (9) ◽  
pp. 094101 ◽  
Author(s):  
Jörg Kussmann ◽  
Arne Luenser ◽  
Matthias Beer ◽  
Christian Ochsenfeld
Keyword(s):  
Density Matrix ◽  
Hessian Matrix ◽  
The Self ◽  
Consistent Field ◽  
Field Level ◽  
Self Consistent ◽  
Self Consistent Field

scholarly journals A density matrix approach to the convergence of the self-consistent field iteration

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Parikshit Upadhyaya ◽  
◽  
Elias Jarlebring ◽  
Emanuel H. Rubensson ◽  
Keyword(s):  
Density Matrix ◽  
The Self ◽  
Matrix Approach ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

Note on the Interpretation of the Density Matrix in the Many-Electron Problem

1931 ◽  
Vol 27 (2) ◽  
pp. 240-243 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Density Matrix ◽  
The Self ◽  
Transformation Theory ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
The Many

When we treat an atom containing a number n of electrons by the method of the self-consistent field, we assume that each electron has its own particular “orbit,” specified by a wave function (q|r) in four variables q. These four variables are usually taken to be the three coordinates of the electron together with a variable describing the spin, but according to the transformation theory of quantum mechanics, they may be any four independent commuting functions of the coordinates, momenta and spin variables.

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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