Xa approximation

The weaknesses of the Xa approximation itself have been considered by Slater.138 He gives reasons why the present method should be poorer for small atoms than for larger ones and offers suggestions for improvement. 

Other Related Methods.—Baerends and Ros have developed a method suitable for large molecules in which the LCAO form of the wavefunction is combined with the use of the Xa approximation for the exchange potential. The method makes use of the discrete variational method originally proposed by Ellis and Painter.138 The one-electron orbitals are expanded in the usual LCAO form and the mean error function is minimized. 

It has been argued (47) that the Xa approximation for the local exchange potential overestimates the tendency towards spin-polarization. The results for another local density functional will be presented later, elsewhere, but for the moment these Xa calculations (with a = 0.7) seem appropriate because we would like to determine an upper limit on the effect oif isomerization on the magnetic moment of this hopefully representative duster of magnetic atoms. 

We are discussing our manner to calculate the total energy for small molecules within the DV-Xa approximation by using only the monopol part of the potential in the solution of the Poisson equation. A discussion of the relativistic effects, including our results for heavy diatomic molecules, is followed by remarks on the choice of the exchange-correlation potential together with our results of calculations on molecules for the element 106 and their chemical interpretation. We conclude with results on very heavy correlation diagrams for collision systems with a united Z above 110. 

But already at that point we realized that the accuracy of this method was not very good. Many physical and chemical quantities, like binding energy or chemical bond distances, could not be calculated directly with this method. When we finally also started calculations of real molecules at chemical distances this deficiency became even more obvious and we began to improve the method itself in order to calculate the total energy of a system within the Xa-approximation. 

Hohenberg and Kohn and Kohn and Sham in the early 1960s, to a host of applications primarily involving the local (spin) density " (LSD) (or Xa) approximation (for some representative reviews, see Refs 12 and 15-22). 

The first generation is the local density approximation (LDA). This estimation involves the Dirac functional for exchange, which is nothing else than the functional proposed by Dirac [15] in 1927 for the so-called Thomas-Fermi-Dirac model of the atoms. For the correlation energy, some parameterizations have been proposed, and the formula can be considered as the limit of what can be obtained at this level of approximation [16-18], The Xa approximation falls into this category, since a known proportion of the exchange energy approximates the correlation. 

If one adds this term to the Thomas-Fermi expression (35) one obtains the so-called Thomas-Fermi-Dirac approximation to E[n]. It one multiplies it with an adjustable parameter a one obtains the so-called Xa approximation to Exc[n]. These approximations are not much used today in DFT. 

Phase equilibrium concentration at the bottom product concentration Xa (approximate)... 

The exchange potential is further simplified by the Xa approximation of Slater (1951). For the free electron gas of density p one can show that the average exchange energy is... 

We combine extensive spectroscopic data for the series C60 > (n=0,1,2,3) in conjunction with electrochemical data and Molecular Orbital calculations performed within the Multiple Scattering X-Alpha (MS-Xa) approximation, to provide a self consistent map of the valence energy levels of C60 and its derivatives. 

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