Exchange-correlation energy quantum chemistry

By defining all these quantities as explicit functions of A, we can relate the density functional quantities to those more familiar from quantum chemistry. The exchange-correlation energy of density functional theory can be shown, via the Hellmann-Feynman theorem [38, 37], to be given by a coupling-constant average, i.e.. 

Quantum chemistry is most simply done with single-particle orbitals o,single-particle equations. The exchange-correlation energy Exc is then constructed from the orbitals, or from the spin densities raj and raj. The Hartree-Fock (HF) approximation neglects correlation but treats exchange exactly ... 

The quest for various approximations for the exchange-correlation energy density/(p) has spanned decades in quantum chemistry and was recently reviewed [92]. Here, we will present the red line of its implementation, as it will be further used for the current applications. The benchmark density functional stands as the Slater exchange approximation, derived within the so-called Xa theory [179] ... 

The local spin density approximation (LSD) for the exchange-correlation energy, (1.11), was proposed in the original work of Kohn and Sham [6], and has proved to be remarkably accurate, useful, and hard to improve upon. The generalized gradient approximation (GGA) of (1.12), a kind of simple extension of LSD, is now more widely used in quantum chemistry, but LSD remains the most popular way to do electronic-structure calculations in solid state physics. Tables 1.1 and 1.2 provide a summary of typical errors for LSD and GGA, while Tables 1.3 and 1.4 make this comparison for a few specific atoms and molecules. The LSD is parametrized as in Sect. 1.5, while the GGA is the non-empirical one of Perdew, Burke, and Ernzerhof [20], to be presented later. 

Kurth, S., Perdew, J. P. (2000). Role of the exchange-correlation energy Nature s glue. International Journal of Quantum Chemistry, 77,... 

The exact energy functional (and the exchange correlation functional) are indeed functionals of the total density, even for open-shell systems [47]. However, for the construction of approximate functionals of closed as well as open-shell systems, it has been advantageous to consider functionals with more flexibility, where the a- and j8-densities can be varied separately, i.e. E[p, p ]. The variational search for a minimum of tire E[p, p ] functional can be carried out by unrestricted and spin-restricted approaches. The two methods differ only by the conditions of constraint imposed in minimization and lead to different sets of Kohn-Sham equations for the spin orbitals. The unrestricted Kohn-Sham approach is the one most commonly used and is implemented in various standard quantum chemistry software packages. However, this method has a major disadvantage, namely a spin contamination problem, and in recent years the alternative spin-restricted Kohn-Sham approach has become a popular contester [48-50]. 

The Hartree-Fock and the Kohn-Sham Slater determinants are not identical, since they are composed of different single-particle orbitals, and thus the definition of exchange and correlation energy in DFT and in conventional quantum chemistry is slightly different [52]. 

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