Slater Xa method

I do not recall when I first heard of the Hohenberg-Kohn-Sham papers, but I do know that the quantum chemistry community at first paid little attention to them. In June of 1966 Lu Sham spoke about DFT at a Gordon Conference. But in those days, there was more discussion about another prescription that had been on the scene since 1951, the Slater Xa method. The Xa method was a well-defined, substantial improvement over the Thomas-Fermi method, a sensible approximation to exact Kohn-Sham. Debate over Xa went on for a number of years. Slater may never have recognized DFT as the major contribution to physics that it was. [When I asked John Connolly five or six years ago how he thought Slater had viewed DFT, he replied that he felt that Slater regarded it as obvious. ]... 

These equations are the density functional version of the ab initio Slater Xa method (with a multiplicative parameter a on the exchange term). The exchange potential is similar to that originally given by Caspar [234]. 

Becke A D 1983 Numerical Hartree-Fock-Slater calculations on diatomic molecules J. Chem. Phys. 76 6037 5 Case D A 1982 Electronic structure calculation using the Xa method Ann. 

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. 

In order to give you some background to Slater s Xa method, I would like to describe some very simple models that were used many years ago in order to understand the behaviour of electrons in metallic conductors. 

Slater s Xa method is now regarded as so much history, but it gave an important stepping stone towards modem density functional theory. In Chapter 12, I discussed the free-electron model of the conduction electrons in a solid. The electrons were assumed to occupy a volume of space that we identified with the dimensions of the metal under smdy, and the electrons were taken to be non-interacting. 

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,... 

Methods of density functional theory (DFT) originate from the Xa method originally proposed by Slater [78] on the base of statistical description of atomic electron structure within the Thomas-Fermi theory [79]. From the point of view of Eq. (3), fundamental idea of the DFT based methods consist first of all in approximate treatment of the electron-electron interaction energy which is represented as ... 

The LDA, Slater, and Xa methods can all be extended to the spin-polarized regime using... 

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. 

The Xa (X = exchange, a is a parameter in the Xa equation) method gives much better results [23, 24]. It can be regarded as a more accurate version of the Thomas-Fermi model, and is probably the first chemically useful DFT method. It was introduced in 1951 [25] by Slater, who regarded it [26] as a simplification of the Hartree-Fock (Section 5.2.3) approach. The Xa method, which was developed mainly for atoms and solids, has also been used for molecules, but has been replaced by the more accurate Kohn-Sham type (Section 7.2.3) DFT methods. 

For a personal history of much of the development of quantum mechanics, with significant emphasis on the Xa method, see Slater JC (1975) Solid-state and molecular theory a scientific biography. Wiley, New York... 

The DFT concept of calculating the energy of a system from its electron density seems to have arisen in the 1920s with work by Fermi, Dirac, and Thomas. However, this early work was useless for molecular studies, because it predicted molecules to be unstable toward dissociation. Much better for chemical work, but still used mainly for atoms and in solid-state physics, was the Xa method, introduced by Slater in 1951. Nowadays the standard DFT methodology used by chemists is based on the Hohenberg-Kohn theorems and the Kohn-Sham approach... 

The DV-Xa molecular orbital (MO) calculations were performed to analyze the Li-K XANES spectra of lithium halides powder. The computational details of the DV-Xa method have been previously described [10]. In this method, the exchange-correlation interaction, Vxcr between electrons is given by the Slater s Xa potential,... 

Slater summarized the power of the Xa method at the end of the article from 1972 by stating that The Xa method provides a practical starting point for an a priori theory of such problems, as magnetism, a starting point which has not been practical with earlier techniques. The power of the Xa method was further accentuated in an article by Johnson on Scattering-Wave Theory of the Chemical Bond in 1973 [6], where he gives an overview of the traditional Hartree-Fock, HF, SCF method and also semi-empirical ones. He points out that the new Xa technique based on Scattering Wave Theory has... 

Another reason for the choice of the title is the above-mentioned introduction of the Xa-method and the MS-Xa method by Slater and coworkers. There are, however, in particular two other reasons for choosing the title. The first is the formulation of the Density Functional Theory by Hohenberg and Kohn in 1964 [19], which today is probably one of the most quoted papers in electronic structure calculations. This basic work was followed by another important paper in 1965 by Kohn and Sham [20], where they showed how one could use the method for practical calculations and introduced the Kohn-Sham, KS, exchange potential. Exactly the same expression for the exchange potential had previously been derived by Caspar [21], This exchange potential is therefore often known as the Caspar-Kohn-Sham, GKS, potential. Another very important reason for choice of the title is the introduction of the three dimensional numerical integration method by Ellis and Painter in 1968-1970 [22-24]. This... 

To elucidate the nature of chemical bonding in metal carbides with the NaCl structure, the valence electronic states for TiC and UC have been calculated using the discrete-variational (DV) Xa method. Since relativistic effects on chemical bonding of compounds containing uranium atom become significant, the relativistic Hamiltonian, i.e., the DV-Dirac-Slater method, was used for UC. The results... 

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