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

Abstract

The density functional definition of exchange and correlation differs from the traditional one. In order to calculate the density functional theory ͑DFT͒, quantities accurately, molecular Kohn-Sham ͑KS͒ solutions have been obtained from ab initio wave functions for the homonuclear diatomic molecules Li 2 , N 2 , F 2 . These afford the construction of the KS determinant ⌿ s and the calculation of its total electronic energy E KS and the kinetic, nuclear-attraction and Coulomb repulsion components T s , V, W H as well as the ͑DFT͒ exchange energy E x and correlation energy E c . Comparison of these DFT quantities has been made on one hand with the corresponding HartreeFock ͑HF͒ quantities and on the other hand with local density approximation ͑LDA͒ and generalized gradient approximation ͑GGA͒. Comparison with HF shows that the correlation errors in the components T, V, and W H of the total energy are much larger for HF than KS determinantal wave functions. However, the total energies E KS and E HF appear to be close to each other, as well as the exchange energies E x and E x HF and correlation energies E c and E c HF . 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. © 1997 American Institute of Physics. ͓S0021-9606͑97͒00337-1͔

Broken symmetry approach to calculation of exchange coupling constants for homobinuclear and heterobinuclear transition metal complexes

“…Within the density functional framework, Baerends et al indicated that the single-determinant wave function includes all the electron correlation contributions to the energy, even the nondynamic correlation. 35 In DFT calculations the energy of the singlet state can be approximated in principle using that of the single-determinant BS state 1 , BS which was confirmed by the good results obtained by Perdew et al 36 for the atomization energy of H with such approximation to the energy of the 2 singlet. These authors indicated that such a BS state describes the electron density and the on-top electron pair density with remarkable accuracy, even if it gives an unrealistic distribution of the spin density, and propose that the BS single-determinant wave function is the correct solution for the Kohn᎐Sham equations.…”

Empirical analysis of the Lieb–Oxford bound in ions and molecules

“…Approximate correlation energies of H 2 as a function of bond length are available from Ref. [30]. From these data we find that for H 2 at the equilibrium bond length λ(R e = 1.401a.u.)…”

Density functional theory: Its origins, rise to prominence, and future

“…Time-dependent DF theory, based on the theorem of Runge and Gross (1984), has proved to be valuable in predicting the excited states and optical properties in molecules (Burke, Werschnik, and Gross, 2005;Marques et al, 2006;Botti et al, 2007;Isegawa, Peverati, and Truhlar, 2012) and has become an option in most DF program packages. In addition, the exact KS potential can be determined if the exact wave functions are known, and Gritsenko, Schipper, and Baerends (1997) showed that eigenvalue differences provide an excellent approximation to excitation energies in several small molecules when calculated with this potential.…”