Fock term

Eq. (5.1) is the same as for G3 theory except for the addition of the Hartree-Fock term. This term extends the HF/G3Large energy, which... 

The energy of the electron gas is composed of two terms, one Hartree-Fock term (T)hp) and one correlation term (Hartree-Fock term comprises the zero-point kinetic energy density and the exchange contribution (first and second terms on the right in equation 1.148, respectively) ... 

The main advantage of such an approach is that it allows for greater flexibility in the choice of appropriate xc functionals. In particular, the OEP method can be used for the treatment of the exact exchange energy functional, defined by inserting KS orbitals in the Fock term, i.e. 

F irst, consider the d-state energy, c,. The only serious discrepancy between Mattheiss s calculation and the experimental optical spectra was that Mattheiss s calculation appeared to overestimate the band gap by about three electron volts. Since this gap is dominated by the energy i — r,p, the discrepancy suggested an overestimate of this difference. In fact, his calculated bands were positioned roughly in accord with the splitting predicted by term values of Herman and Skillman (1963). Finally, the same overestimate applies to the Herman-Skillman term values in comparison to Hartree-Fock term values. This suggested, then, that values for e, should be taken from Hartree-Fock calculations, and those are what appear in the Solid State Table and therefore also in Table 19-3. C. Calandra has suggested independently (unpublished) from consideration of transition metals... 

Consideration of Harlree-Fock term values from Fischer (1972), as discussed in Appendix A, indicates that Hartree-Fock values for valence s- and p-state energies are quite similar to the Herman-Skillman values given in the Solid Slate Table. Thus a more systematic treatment would result from use of the Hartree-Fock values throughout they are more appropriate for the transition metals and it would make little difference which set of values is used for other systems. The reasons for retaining Herman-Skillman values here are largely historical it is also possible that use of Hartree-Fock values would increase dLscrepancics with existing band calculations since, as indicated in Appendix A, the approximations almost universally used in solids are the same as those used in the Herman-Skillman calculation. For. s and p states the differences are small in any case. 

Hartrec-Fock term values after Mann (1967), in eV. In the upper part, , values are given first for each element and Cp values are given next for the transition metals. 

Most recently developed functionals use either the GGA for or a generalization thereof. Of particular importance are hybrid functionals, which express the total exchange-correlation as a sum of an exact exchange (Hartree-Fock) term and a GGA term [29,30]. 

Since hybrid functionals, Meta-GGAs, SIC, the Fock term and all other orbital functionals depend on the density only implicitly, via the orbitals i[n, it is not possible to directly calculate the functional derivative vxc = 5Exc/5n. Instead one must use indirect approaches to minimize E[n and obtain vxc. In the case of the kinetic-energy functional Ts[ 0j[rr] ] this indirect approach is simply the Kohn-Sham scheme, described in Sec. 4. In the case of orbital expressions for Exc the corresponding indirect scheme is known as the optimized effective potential (OEP) [120] or, equivalently, the optimized-potential model (OPM) [121]. The minimization of the orbital functional with respect to the density is achieved by repeated application of the chain rule for functional derivatives,... 

In the case of hybrid functionals, still another mode of implementation has become popular. This alternative, which also avoids solution of Eq. (91), is to calculate the derivative of the hybrid functional with respect to the singleparticle orbitals, and not with respect to the density as in (91). The resulting single-particle equation is of Hartree-Fock form, with a nonlocal potential, and with a weight factor in front of the Fock term. Strictly speaking, the orbital derivative is not what the HK theorem demands, but rather a Hartree-Fock like procedure, but in practice it is a convenient and successful approach. This scheme, in which self-consistency is obtained with respect to the singleparticle orbitals, can be considered an evolution of the Hartree-Fock Kohn-Sham method [6], and is how hybrids are commonly implemented. Recently, it has also been used for Meta-GGAs [2]. For occupied orbitals, results obtained from orbital selfconsistency differ little from those obtained from the OEP. 

Fig. 5. The one-body diagrams through second-order without a spectator valence line. Diagram (a) is the Hartree-Fock term. Diagram (b) is the auxiliary potential U, while (c) and (d) are the 2plh and 3p2h diagrams, respectively.

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)... 

As was shown in Chapter 1, the Madelung potential renormalizes the atomic energies and shifts the anion and cation levels towards lower and higher energies, respectively (Equations (1.4.1) and (1.4.2)). In the surface layer, the effective levels of the cations are thus lower than in the bulk, and the reverse is true for the surface anions. The actual levels also depend upon the intra-atomic Hartree or Hartree-Fock terms, which shift the atomic levels in the opposite direction (Ellialtioglu et al., 1978), but. 

As we have seen in the case of T, it can be much easier to represent a functional in terms of single-particle orbitals than directly in terms of the density. Such functionals are known as orbital functionals, and (45) constitutes a simple example. Another important orbital-dependent functional is the exchange energy (Fock term) of (47). The meta-GGAs and hybrid functionals mentioned above are also orbital functionals, because they depend on the kinetic energy density (79), and on a combination of the orbital functional (47) with ordinary GGAs, respectively. 

The application of the OEP methodology to the Fock term (47), either with or without the JCLl approximation, is also known as the EXX method. The OEP-EXX equations have been solved for atoms " and solids, with very encouraging results. Other orbital-dependent functionals that have been treated within the OEP scheme are the PZ-SIC and the Colle-Salvetti functional. A detailed review of the OEP and its JCLl approximation is given in Ref. [135]. 

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