Exchange-energy functionals

The exchange part is given by the Dirac exchange-energy functional... 

Lembarki, A., F. Regemont, and H. Chermette. 1995. Gradient-corrected exchange potential with the correct asymptotic behavior and the corresponding exchange-energy functional obtained from virial theorem. Phys. Rev. A 52, 3704. 

The universal functional G[p] is written as the sum of kinetic and exchange energy functionals... 

One can improve upon the TF model by incorporating two-electron effects into P nlpl as the approximate, local Dirac exchange energy functional (cx is the Dirac exchange constant)... 

Since the Coulomb, exchange, and correlation energies are all consequences of the interelectronic 1 /r12 operator in the Hamiltonian, one can define the exchange energy functional Ex [p] in the same manner as... 

Note that the energy is minimized with respect to all choices of the orbital basis and subject to the (1, conditions on p, = F, ,- this ensures that there exists an ensemble of Slater determinants with the desired electron density. Because an ensemble average of Slater determinants does not describe electron correlation, these variational energy expressions include a correlation functional, Ec p, which corrects the energy for the effects of electron correlation. Reasonable approximations for Ec[p] exist, though they tend to work only in conjunction with approximate exchange-energy functionals, Ex p. ... 

This proposition has been tested in the exact-exchange limit of the implied linear-response theory [329], The TDFT exchange response kernel disagrees qualitatively with the corresponding expression in Dirac s TDHF theory [79,289]. This can be taken as evidence that an exact local exchange potential does not exist in the form of a Frechet derivative of the exchange energy functional in TDFT theory. 

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. 

In order to prove statement (i) of the lemma we first investigate the asymptotic form of the quantities uxl0.(r). Employing the exact exchange-energy functional (22) we find... 

The kinetic and exchange energy functionals given by Eqs. (8) and (12), respectively, contain universal terms that just depend upon the one-particle density. In the case of the former, such term is p6/3, the Thomas-Fermi term [22,23] and for the latter, the set p(ri)(4+fc 3, where the first term p4 3 (for k = 0) is the Dirac exchange expression [24]. But in addition, in Eq. (8) we observe the presence of a factor, which we call Fis([p]jr) defined as ... 

In the present review of LS-DFT the constructive nature of this approach leading to the actual formulation of density functionals has been emphasized. We have shown explicit expressions for the kinetic and exchange energy functionals and have compared the former to a number of the usual representations advanced in conventional DFT. We have further analyzed the correlation problem from the point of view of local-scaling transformations and have made some... 

B. Physical interpretation of Kohn-Sham theory exchange energy functional and its derivative... 

B. PHYSICAL INTERPRETATION OF KOHN-SHAM THEORY EXCHANGE ENERGY FUNCTIONAL AND ITS DERIVATIVE... 

To determine the correlation-kinetic field and potential we assume the KS exchange-potential vx(r) to bejthat derived6,7 by restricted differentiation of the exchange energy functional Ex [p]. The resulting expression for the potential which is in terms of the density p(r) and Slater potential Vx (r) is... 

V. CONSTRUCTION OF APPROXIMATE KOHN-SHAM EXCHANGE ENERGY FUNCTIONAL AND DERIVATIVE WITH EXACT ASYMPTOTIC STRUCTURE... 

In this section we derive a local density approximation-like expression for the KS exchange energy functional and its derivative such that the latter possesses the correct asymptotic structure both in the classically forbidden and metal-bulk regions. 

The first root of this equation p = -1 gives rise to the local density approximation exchange energy functional and corresponding derivative. The second root is... 

In a manner similar to that described above, it is possible to correct other exchange energy functionals more accurate that the LDA, such as the generalized gradient approximation40. Similarly it is also possible to obtain the correction to... 

---