Correlation energy in DFT

Correlation energies in DFT must be approximated. For this purpose, knowledge of exact properties is necessary. With this in mind, it is now known that the exact correlation energy for use as part of a full DFT calculation, Ec[n] [1-10], satisfies [10-13] the expansion... 

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]. 

Gritsenko, O. V., Schipper, P. R. T., Baerends, E. J., 1997, Exchange and Correlation Energy in Density Functional Theory. Comparison of Accurate DFT Quantities With Traditional Hartree-Fock Based Ones and Generalized Gradient Approximations for the Molecules Li2, N2, F2 , J. Chem. Phys., 107, 5007. 

The two delta terms which have been placed side by side encapsulate the main problem with DFT the sum of the kinetic energy deviation from the reference system and the electron-electron repulsion energy deviation from the classical system, called the exchange-correlation energy. In each term an unknown functional transforms electron density into an energy, kinetic and potential respectively. This exchange-correlation energy is a functional of the electron density function ... 

Mathematical expressions for the functionals which are found in the Kohn-Sham operator are usually derived either from the model of a uniform electron gas or from a fitting procedure to calculated electron densities of noble gas atoms. Two different functionals are then derived. One is the exchange functional Fx and the other the correlation functional Fc, which are related to the exchange and correlation energies in ab initio theory. We point out, however, that the definition of the two terms in DFT is slightly different from ab initio theory, which means that the corresponding energies cannot be directly compared between the two methods. 

Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn-Sham method 1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.7,8 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9... 

Abstract. The 1/Z expansion will first be used to discuss the scaling properties of the ground-state energy of heavy (non-relativistic) neutral atoms with atomic number Z. The question will be addressed as to what order in Z electron correlation first enters the expansion. The density functional theory (DFT) invoked above will be utilized then to treat, but now inevitably more approximately, the correlation energy in a variety of molecules. Finally, recent studies at Hartree-Fock level on almost spherical B and C cages will be reviewed. For buckminsterfullerene, the role of electron correlation will then be assessed using the Hubbard Hamiltonian, as in the study of Flocke et al. 

A pleasant aspect of DFT is that, unlike the case of ab initio methods for including electron correlation, the accuracy with which various properties are predicted does not seem to improve much with increasing basis set size, once one has reached a split-valence basis that includes polarization functions (SVP basis set). To recover a significant amount of electron correlation energy in ab initio calculations, large basis sets are necessary to provide an adequate space of virtual MOs. However, in DFT electron correlation is sensitive to the V,. functional, but not so much to how well the exact electron density is reproduced. Thus, from a B3LYP/SVP calculation, one can generally expect to get better results and usually at smaller computational cost than from an MP2 calculation with the same basis set. 

The construction of the Kohn-Sham and Fock matrices will be reviewed in this section. We will focus on three components of the matrices exchange-correlation energy (for DFT),... 

DFT coding proves more accurate for NMR chemical shifts because it accounts for the majority of the electron correlation energy. In this case, the ETOs are fitted to large Gaussian expansions, following the algorithm in [49] and Gaussian 03 is... 

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