Hartree contribution

Efj is the Hartree contribution to the electron-electron interaction energy... 

A proper relativistic description of the Hartree contribution has to be based on the true four-component density of the electrons. The two tasks, construction... 

The correct representation of the electron-electron interaction is important for a quantitative description of the spin-orbit splitting [19,33] and, thus, the anisotropy of g tensors [35]. Therefore, for the calculations of g tensors with the DKH method, it is essential to subject at least the Hartree contribution to the DK transformation [19] (see Sections 2.3 and 3.1). 

Although smaller than the Hartree contribution, the remaining xc part of the electron-electron interaction should also be subjected to the DK transformation to obtain further improved two-component wave functions for calculating g values work in this direction is in progress in our group. We have good reason to assume that the difference of our g values from those calculated with the two-component KS method ZORA [112] can be rationalized by the fact that the xc potential remains untransformed in our present g tensor approach. 

The quantity (f) q) represents a two-body density obtained for a given approximation to calculate < F>. For this two-body density all variables have been integrated except the momentum transfer. Therefore this 4>(q) is a measure to which extent the local interaction at momentum transfer q contributes to the total interaction energy. In the BHF approximation this (q) can be split into a direct part (related to the Hartree contribution) and an exchange contribution (related to the Fock term)... 

The density-functional based calculations described above were done for small apphed potential bias between the electrodes. In contrast, the density functional approach of Hirose and Tsukada [172] calculates the electronic structure of a metal-insulator-metal system under strong apphed bias. The main difference from the density functional approaches described above comes in the way the effective one-electron potential is calculated. The potential used in this work contains the usual contributions from the Coulomb and the exchange[Pg.611]

This is exact for one electron, and for (spin-unpolarized) two-electron exchange. Note that this approximation has a long-ranged contribution, which can cancel exactly the direct hartree contribution to the matrix M in Eq. (2). 

It is tempting to assume that the Kohn-Sham potential depends linearly on the density, so that the unscreening of the pseudo-potential can be performed as in (6.60). Unfortunately, even though the Hartree contribution is indeed linearly dependent on the density, the xc term is not... 

Figure 5.7 Schematic energy level diagram illustrating the various contributions to the barrier for electron removal from the bulk of a metal. The gray curve traces the electrostatic potential step induced by the Friedel oscillations and the ensuing staggered dipole layers at the surface. The dashed curve results from adding the Hartree contribution...

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