Xa exchange functional

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

Local exchange and correlation functionals involve only the values of the electron spin densities. Slater and Xa are well-known local exchange functionals, and the local spin density treatment of Vosko, Wilk and Nusair (VWN) is a widely-used local correlation functional. 

Variational fitting enables a completely analytic treatment of the Xa exchange correlation functional. This practical first-principles AIMD method enables studies of detonations, photodissociation, and ultimately the even more complex chemical reactions that can be driven tribologicially. 

The results reported here use the Xa exchange-correlation function, which has historical interest and can be compared to past calculations. Within the local density approximation, parametrizations that include the correlation effects found in a uniform electron gas often give a better account of spin-dependent properties (19). Since correlation effects generally stabilize low spin species more than high-spin states (20), one would expect correlation effects to increase J over the values reported here, and this was indeed found in our earlier studies of oxidized three-iron clusters (9). Calculations on the reduced species using improved exchange-correlation potentials are in progress. 

Figure 1.24 Comparisons of the numerical radial wave function for boron with the Slater and sto-3g) approximations. The extra boron data were obtained by running the Herman-Skillman program using the Schwarz values for the Xa exchange term.

Any exchange functional can be combined with any correlation functional. For example, the notation BLYP/6-31G denotes a DF calculation done with the Becke 1988 exchange functional and the Lee-Yang-Parr correlation functional, with the KS orbitals expanded in a 6-31G basis set. The letter S (which acknowledges Slater s Xa method) denotes the LSDA exchange functional (15.140). VWN denotes the Vosko-Wilk-Nusair expression for the LSDA correlation functional (actually, these workers gave two differ-... 

During the past three deeades, three main versions of the MCP method have been developed [1,53]. Version I is based on the local approximation. The core-valence Coulomb repulsion is a local interaction and can be satisfactorily approximated by a local potential function. For convenience of the integral evaluation, such a local potential function is chosen to be a linear combination of Gaussian type functions. The core-valence exchange operator is not a local operator. However, in Version I, this non-local interaction is also approximated by the local potential function of Gaussian type. This non-local to local approximation for the exchange operator shares the same concept with Slater s Xa density functional model [69]. Under such an approximation, the one-electron hamiltonian for the valence space in an atom (Eq. 8.5) is rewritten as... 

Density functionals can be broken down into several classes. The simplest is called the Xa method. This type of calculation includes electron exchange but not correlation. It was introduced by J. C. Slater, who in attempting to make an approximation to Hartree-Fock unwittingly discovered the simplest form of DFT. The Xa method is similar in accuracy to HF and sometimes better. 

A common problem for both methods lies in the use of potentials that do not possess the correct net attractiveness. This can have the consequence that continuum feamres appear shifted in energy. In particular, there is evidence that the LB94 exchange-correlation potential currently used for the B-spline calculations, although possessing the correct asymptotic behavior for ion plus electron, is too attractive, and near threshold features can then disappear below the ionization threshold. An empirical correction can be made, offsetting the energy scale, but this can mean that dynamics within a few electronvolts of threshold get an inadequate description or are lost. There is limited scope to tune the Xa potential, principally by adjustment of the assumed a parameter, but for the B-spline method a preferable alternative for the future may well be use of the SAOP functional that also has correct asymptotic behavior, but appears to be better calibrated for such problems [79]. 

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,... 

I drew attention in Chapter 12 to the fact that the Xa orbitals did not satisfy the nice properties of standard HF-LCAO ones the Koopmans theorem is not valid, and so on. The same is true of all density functional KS-LCAO calculations. In practice, it usually turns out that the KS-LCAO orbitals are very similar to ordinary HF-LCAO ones, which must mirror the fact that exchange-correlation effects are only a minor part of the total electronic energy. So the orbitals are often analysed as if they were ordinary HF orbitals (Figure 13.4). 

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 Xa 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. 

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