Exact exchange energy density functional

USING THE EXACT KOHN-SHAM EXCHANGE ENERGY DENSITY FUNCTIONAL AND POTENTIAL TO STUDY ERRORS INTRODUCED BY APPROXIMATE CORRELATION FUNCTIONALS... 

P is an empirical parameter that was determined to 0.0042 by a least-squares fit to the exactly known exchange energies of the rare gas atoms He through Rn. In addition to the sum rules, this functional was designed to recover the exchange energy density asymptotically far from a finite system. 

The fourth rung of the ladder in Fig. 10.2 is important because the most common functionals used in quantum chemistry calculations with localized basis sets lie at this level. The exact exchange energy can be derived from the exchange energy density, which can be written in terms of the Kohn-Sham orbitals as... 

A critical feature of this quantity is that it is nonlocal, that is, a functional based on this quantity cannot be evaluated at one particular spatial location unless the electron density is known for all spatial locations. If you look back at the Kohn-Sham equations in Chapter 1, you can see that introducing this nonlocality into the exchange-correlation functional creates new numerical complications that are not present if a local functional is used. Functionals that include contributions from the exact exchange energy with a GGA functional are classified as hyper-GGAs. 

In practice, only approximate expressions are known for the exchange-correlation density functional xcM and the most important ones stem from two classes the generalized gradient approximation (GGA) functionals also called gradient-corrected functionals and their combination with the exact exchange energy expression 52 =i / known from Hartree-Fock theory denoted hy-... 

The particular form of Eq. (110) was chosen to satisfy a large number of exact constraints. Although PW91 is modeled after the B88 functional, it does not yield the correct asymptotic behavior of the exchange energy density, a property abandoned in favor of more desirable constraints. The function sinh (ci 5) in Eq.(llO) is only a B88 relic. 

Another disadvantage of the LDA is that the Hartree Coulomb potential includes interactions of each electron with itself, and the spurious term is not cancelled exactly by the LDA self-exchange energy, in contrast to the HF method (see A1.3I. where the self-interaction is cancelled exactly. Perdew and Zunger proposed methods to evaluate the self-interaction correction (SIC) for any energy density functional [40]. However, full SIC calculations for solids are extremely complicated (see, for example [41. 42 and 43]). As an alternative to the very expensive GW calculations, Pollmann et al have developed a pseudopotential built with self-interaction and relaxation corrections (SIRC) [44]. 

Table I. Magnitude of the exact exchange energy (Hartrees) and difference between exact exchange energy and various density functional for the exchange energy, for neutral atoms. All calculations employ Hartree-Fock densities for the computed ground-state configuration and term...

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