Density functional theory Hartree-Fock-Slater exchange

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... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

Let us introduce another early example by Slater, 1951, where the electron density is exploited as the central quantity. This approach was originally constructed not with density functional theory in mind, but as an approximation to the non-local and complicated exchange contribution of the Hartree-Fock scheme. We have seen in the previous chapter that the exchange contribution stemming from the antisymmetry of the wave function can be expressed as the interaction between the charge density of spin o and the Fermi hole of the same spin... 

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,... 

Koopmans Theorem applies to Hartree-Fock theory by virtue of the particular method for evaluating the quantum mechanical exchange interaction. In Density Functional Theory, a different method is employed. Hence, HF orbitals are not the same as DFT orbitals and Koopmans Theorem does not apply. This can be illustrated with reference to Slater s Xu (i.e. DFT exchange only) model [15]. 

The precursor to Kohn-Sham density-functional theory is Slater theory [12], In the latter theory, the nonlocal exchange operator of Hartree-Fock theory [25] is replaced by the Slater local exchange potential Vf(r) defined in terms of the Fermi hole p,(r, r ) as... 

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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