Hartree-Fock-Slater exchange

Tschinke, V., andT. Ziegler. 1991. Gradient corrections to the Hartree-Fock-Slater exchange and their influence on bond energy calculations. Theor. Chim. Acta 81, 81. 

Density functional theory, 21, 31, 245-246 B3LYP functional, 246 Hartree-Fock-Slater exchange, 246 Kohn-Sham equations, 245 local density approximation, 246 nonlocal corrections, 246 Density matrix, 232 Determinantal wave function, 23 Dewar benzene, 290 from acetylene + cyclobutadiene, 290 interaction diagram, 297 rearrangement to benzene, 290, 296-297 DFT, see Density functional theory... 

Density functional theory, 21, 31, 245-246 B3LYP functional, 246 Hartree-Fock-Slater exchange, 246 Kohn-Sham equations, 245 local density approximation, 246 nonlocal corrections, 246... 

What does this mean We have replaced the non-local and therefore fairly complicated exchange term of Hartree-Fock theory as given in equation (3-3) by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. As noted above, this formula was originally explicitly derived as an approximation to the HF scheme, without any reference to density functional theory. To improve the quality of this approximation an adjustable, semiempirical parameter a was introduced into the pre-factor Cx which leads to the Xa or Hartree-Fock-Slater (HFS) method which enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry,... 

This simplified model of electronic polarization may be used within a KS like formalism to determine the electron density p(r). For instance, if we place the model within the Hartree-Fock-Slater X — a approximation [33], the exchange-correlation potential reduces to ... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

LCAO expansion of the MOs [15]. In the DV-Xa MO method based on the Hartree-Fock-Slater approach, the exchange-correlation potential is approximated by the simple Slater form [16] Vxc(r) = —3a 3p(r)/47i 1/3, where the coefficient a is a scaling parameter (fixed at 0.7 in the present study) and p(r) is the local electron density at a position r. The basis functions for the MO calculation consisted of atomic orbital wave eigenfunctions obtained in numerical form, which included the ls-6s, ls-5s, ls-6p, ls-4p, and ls-2p orbitals for Ba, Sr, Pb, Ti, and O ions, respectively... 

The wave eigenfunctions and energy eigenvalues were obtained by real space electronic state calculations, which were performed by the use of the program code SCAT of the DV-Xx molecular orbital (MO) cluster method with the Hartree-Fock-Slater approximation [8,9]. In the method, the exchange-correlation term Vxc in the one-electron Hamiltonian was expressed in terms of the statistical local potential (1),... 

The DV-Xa cluster method is a molecular orbital calculating method, assuming a Hartree-Fock-Slater (HFS) approximation. In this calculation, the exchange-correlation between electrons, is given by the following Slater s Xa potential. 

The calculation of x-ray emission spectra of molecules or solids are one of the most successful applications of the discrete variational (DV) Hartree-Fock-Slater (Xa) MO method using cluster approximation [8-10], which was originally coded by Ellis and his CO workers [11-14] based on Slater s Xa exchange potential [15]. The DV-Xa method has several advantages for the calculation of x-ray transition process as follows. 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

Recently, Density Functional investigations of molecular bond energies have gained novel impetus due to the introduction by Becke (7) of a gradient correction to the Hartree-Fock-Slater local exchange expression. 

Instead of relying on experimental data for the ionization potentials, the essential EH energy (H ) and orbital contraction Q parameters can also be deduced from theoretical calculations [115,116]. Recently, a complete set of EH parameters has been derived from atomic Hartree-Fock-Slater calculations (an early form of density-functional theory, see Section 2.12) which were also adjusted to fit some experimental data. The parameter set thus derived [117] includes exchange, some correlation, and also the influences of relativity for convenience, we include these data in Table 2.1. These parameters may be used to study the trends in the periodic table and, also, to perform simple calculations. Other sets of EH parameters, from very different sources, are also available. These then typically include better basis sets (such as double- parameters for d orbitals) although they are less self-consistent for the whole periodic table. 

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