Hartree-Fock decoupling

The main problem is to find good approximations to break this hierarchy, in order to find the Wigner distribution function and subsequently the induced electron density. We adopt the Hartree-Fock decoupling, written schematically as ... 

Chemical reaction occurs between reactants in their valence state, which is different from the ground state. It requires excitation by the environment, to the point where a valence electron is decoupled from the atomic or molecular core and set free to establish new liaisons, particularly with other itinerant electrons, likewise decoupled from their cores [114]. The energy required to promote atoms into their valence state has been studied before [24] in terms of the simplest conceivable model of environmental pressure, namely uniform isotropic compression. This was simulated by an atomic Hartree-Fock procedure, subject to the boundary condition that confines all electron density to within an impenetrable sphere of adjustable finite radius. 

The Hartree-Fock (HF) approximation. The HF method is based on the Born-Oppenheimer and orbital approximations. Under the Born-Oppenheimer approximation the nuclear and electronic degrees of freedom of a molecule are decoupled, and the nuclei are held fixed while the electronic contribution to the energy is calculated. In the orbital approximation... 

The expression (11.75) is also valid in the Hartree-Fock approximation, where all the Vkr vanish. In the more general case, this expression may not agree with the direct calculation of the hamiltonian expectation value. This is a difficulty not uncommon in the application of Green s functions. It should have been avoided by some auxiliary constraint on the decoupling procedure . 

As a result of this expansion, followed by a decoupling of four operator products (two magnon terms) in the Hartree-Fock approximation, and subsequent Fourier transformation, Lindgard and Danielsen (1975) deduce the full hamiltonian in the... 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

[Pg.533]

In order to provide an overview on the accuracy of electronic energies obtained within the scalar-relativistic exact-decoupling approaches, we review Hartree-Fock results for the Rn atom in Table 14.4, taken from Ref. [16]. 

Mandd with a quite different method. They applied the decoupling (Eq. 10) in the equation of motion for the double-time-retarded commutator of the charge density fluctuation operators, and imposed conservation of frequency moments to all orders in the Hartree-Fock approximation for the... 

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