Exchange and correlation energy in density functional theory: Comparison of accurate density functional theory quantities with traditional Hartree–Fock based ones and generalized gradient approximations for the molecules Li2, N2, F2

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Abstract
The density functional definition of exchange and correlation differs from the traditional one. In order to calculate the density functional theory (DFT), quantities accurately, molecular Kohn–Sham (KS) solutions have been obtained from ab initio wave functions for the homonuclear diatomic molecules Li2, N2, F2. These afford the construction of the KS determinant Ψs and the calculation of its total electronic energy EKS and the kinetic, nuclear-attraction and Coulomb repulsion components Ts, V, WH as well as the (DFT) exchange energy Ex and correlation energy Ec. Comparison of these DFT quantities has been made on one hand with the corresponding Hartree–Fock (HF) quantities and on the other hand with local density approximation (LDA) and generalized gradient approximation (GGA). Comparison with HF shows that the correlation errors in the components T, V, and WH of the total energy are much larger for HF than KS determinantal wave functions. However, the total energies EKS and EHF appear to be close to each other, as well as the exchange energies Ex and ExHF and correlation energies Ec and EcHF. The KS determinantal wave function and the KS orbitals therefore correspond to much improved kinetic and Coulombic energies, while having only a slightly larger total correlation energy. It is stressed that these properties of the Kohn–Sham orbitals make them very suitable for use in the molecular orbital theories of chemistry. Comparison of the accurate Kohn–Sham exchange and correlation energies with LDA and GGA shows that the GGA exchange energies are consistently too negative, while the GGA correlation energies are not negative enough. It is argued that the GGA exchange functionals represent effectively not only exchange, but also the molecular non-dynamical correlation, while the GGA correlation functionals represent dynamical correlation only.