Self-Coulomb energy

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

One obvious drawback of the LDA is that, when we replace unknown exchange-correlation energy by the known form of the exchange-correlation for a homogeneous electron gas in Equation (17), we have a problem in that cancelation of self-Coulomb... 

Do so, we use the formalism of the variational density fitting method [55, 56] where the Coulomb self-interaction energy of the error is minimized ... 

Umt[p is the classical Coulomb energy and Exc[p] is the XC energy. It is the functional form of this XC functional, which is usually approximated in absence of an exact expression. The one-electron orbitals ipk(r) are obtained through self-... 

The derivation of these expressions involves lengthy algebra details which can be found in Ghosh and Dhara [14]. Here, the internal energy C/int[pJ] is basically the classical Coulomb energy, while the term xc[p,j] denotes the well-known XC energy density functional. With a suitable chosen form for xc[p,jL Equations 6.19 through 6.21 have to be solved self-consistently for the density and the current density. 

The XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

The third term on the right in Eq. (20) cancels the divergent self energy Zee in (V>. The final term on the right equals (-1/ne) times the internal Coulomb energy of the charge distribution (p (r)) [the second term in Eq. (14)] thus it removes the self-interactions from the static Coulomb energy. 

In addition, the functional must somehow cancel the fictitious repulsion energy between an electron and itself, which arises if the electron density, due to all the electrons, is used to compute the Coulombic energy of a single electron. As discussed in Section 3.2.1, in HF theory cancellation of the self-repulsion energy results from the presence of the exchange operator in T. If this effect of Kj, in the Fock operator is not mirrored exactly by the functional chosen, the cancellation of the self-repulsion energy will not occur. 

In the DFT-LSDA approach, the local approximation to the exchange functional achieves only partial cancellation of the self-Coulomb term. Only for orbitals which are delocalized over the whole system does this self-interaction vanish. In the general case, for finite systems and localized states in extended systems, it leads to systematic errors, which have been summarized in Ref. 25. This work proposes a method for SIC, in which the LSDA exchange-correlation energy functional which de-... 

Edminton-Ruedenberg Maximization of the Coulomb self-repulsion energy [236]... 

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