Correlation energy functions

Professor Axel Becke of Queens University, Belfast has been very actively involved in developing and improving exchange-correlation energy functionals. For a good recent overview, see ... 

This result applies when the number of up spins equals the number of down spins and so is not applicable to systems with an odd number of electrons. The correlation energy functional was also considered by Vosko, Wdk and Nusarr [Vosko et al. 1980], whose expression is ... 

Laming, G. J., Termath, V., Handy, N, C., 1993, A General Purpose Exchange-Correlation Energy Functional ,... 

In principle, the KS equations would lead to the exact electron density, provided the exact analytic formula of the exchange-correlation energy functional E was known. However, in practice, approximate expressions of Exc must be used, and the search of adequate functionals for this term is probably the greatest challenge of DFT8. The simplest model has been proposed by Kohn and Sham if the system is such that its electron density varies slowly, the local density approximation (LDA) may be introduced ... 

Laming, G.J., V. Termath, and N.C. Handy. 1993. A general purpose exchange-correlation energy functional. J. Chem. Phys. 99, 8765. 

Weighted-Density Exchange and Local-Density Coulomb Correlation Energy Functionals for Finite Systems—Applications to Atoms. Phys. Rev. A 48, 4197. 

The term Exc[p] is called the exchange-correlation energy functional and represents the main problem in the DFT approach. The exact form of the functional is unknown, and one must resort to approximations. The local density approximation (LDA), the first to be introduced, assumed that the exchange and correlation energy of an electron at a point r depends on the density at that point, instead of the density at all points in space. The LDA was not well accepted by the chemistry community, mainly because of the difficulty in correctly describing the chemical bond. Other approaches to Exc[p] were then proposed and enable satisfactory prediction of a variety of observables [9]. 

The third term of Eq (54) is the electronic Hartree potential, whereas the fourth one represents the exchange-correlation potential. This last term is usually obtained from a model exchange-correlation energy functional xc[pl To a first order approximation, the effective KS potential compatible with the electron density p f) given in Eq (51) may be written as ... 

Equation 4.49 defines the exchange or Fermi hole. It is as if an electron of a given spin digs a hole around itself in space in order to exclude another electron of the same spin from coming near it (Pauli exclusion principle). The integrated hole charge is unity, i.e., there is exactly one electron inside the hole. Likewise, the correlation energy functional can be defined as... 

Uniform Density Limit of Exchange-Correlation Energy Functionals... 

Systematic Approaches to the Exchange-Correlation Energy Functional... 

For the spin-unpolarized uniform electron gas, the LYP correlation energy functional reduces to that of Colie and Salvetti [58], on which LYP is based. McWeeny s work [59] may have contributed to the widespread misimpression that the Colle-Salvetti functional is accurate in this limit. Note however the McWeeny tested Eq. (9) of Ref. [58], and not the further-approximated Eq. (19) of Ref. [58], which is the basis of LYP and other Colle-Salvetti applications, and which is shown in Table 2. 

Functional Taylor series expansion of the functional minimized in Eq. (87), in powers of noK ") = [nGs( ) - gs( )] has been employed first, and Eq. (88) used in the last step. So E " is close to KS correlation energy functional taken for the GS density of HF approximation, corrected by the (much smaller) HF correlation energy, and a small remainder of the second order in the density difference. The last quantity gives an estimate to the large parentheses term of Eq. (28) in [12]. 

Thus Eq. (92) gives an approximation to this functional in terms of KS and HF correlation energy functionals. Rewriting Eq. (27) with the help of (93) and (64) we obtain... 

The form (260) of the correlation force suggests the following approximation for it in terms of the approximation to the correlation energy functional... 

To study the structure of the exchange-correlation energy functional, it is useful to relate this quantity to the pair-correlation function. The pair-correlation function of a system of interacting particles is defined in terms of the diagonal two-particle density matrix (for an extensive discussion of the properties of two-particle density matrices see [30]) as... 

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