Exchange-Correlation Parametrization

A number of different parametrizations of the exchange-correlation term exist. Here we mention a relatively simple scheme by Perdew et al. Thus the GGA method has been parametrized as 

The simple parametrization given by Perdew et al. in ref. [228] for the correlation part in the GGA scheme is expressed as 

The first term is the correlation energy of the uniform electron gas [225] expressed in terms of r) and = (3/(47m)) / the Seitz radius. In ref. [225], a simple parametrization of the correlation energy of the first term for an uniform electron gas has been given. 

The parameter t is expressed in terms of the Thomas-Fermi screening wave number a = /Akp/nao (see also Chapter 9), the spin scaling factor 

The function H should interpolate smoothly between the low gradient or high density limit f - 0 and the opposite limit where the correlation should vanish, this opposite limit being dominated by the exchange term. Both limits could be obtained by the ansatz 

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We have used the basis set of the Linear-Muffin-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

Table 4.1. Acronyms used in text for parametrizations for the exchange-correlation (Exc) functional. The acronyms for separate exchange (Ej and correlation (E ) components of A xc are specified when applicable. Throughout the text, density functional calculations following the Kohn-Sham formalism are referred to as DFT(XXX), where XXX stands either for the acronym of the approximate exchange-correlation functional or for the acronyms of the exchange and correlation functionals, separated by the / symbol.

Analytic or semi-analytic many-body methods provide an independent estimate of ec( .>0- Before the Diffusion Monte Carlo work, the best calculation was probably that of Singwi, Sjblander, Tosi and Land (SSTL) [38] which was parametrized by Hedin and Lundqvist (HL) [39] and chosen as the = 0 limit of Moruzzi, Janak and Williams (MJW) [40]. Table I shows that HL agrees within 4 millihartrees with PW92. A more recent calculation along the same lines, but with a more sophisticated exchange-correlation kernel [42], agrees with PW92 to better than 1 millihartree. 

The answer is yes, in a very general way, as has been discussed before [62,63]. Consider any parameter in the external potential, called y. For definiteness, we choose the internuclear separation in a diatomic molecule. Then the exchange-correlation energy depends parametrically on this quantity. Now imagine making an infinitesimal change in y. The differential change in is... 

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

A system of fundamental theoretical importance in many-body theory is the uniform-density electron gas. After decades of effort, exchange-correlation effects in this special though certainly not trivial system are by now well understood. In particular, sophisticated Monte Carlo simulations have provided very useful information (5) and have been conveniently parametrized by several authors (6). If the exchange-correlation hole function at a given reference point r in an atomic or molecular system is approximated by the hole function of a uniform electron gas with spin-densities given by the local values of p (r) and Pp(C obtain an... 

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. 

We finally mention that an extension of the parametrization (206) to nonvanishing q was given by Dabrowski [102], The spin-dependent case was treated by Liu [103]. A similar interpolation for the exchange-correlation kernel of the 2-dimensional electron gas has been derived by Holas and Singwi [95]. 

There are a number of model exchange-correlation functionals for the ground-state. How do they perform for ensemble states Recently, several local density functional approximations have been tested [24]. The Gunnarsson-Lundqvist-Wilkins (GLW) [26], the von Barth-Hedin (VBH)[25] and Ceperley-Alder [27] local density approximations parametrized by Perdew and Zunger [28] and Vosko, Wilk and Nusair (VWN) [29] are applied to calculate the first excitation energies of atoms. 

Table II presents the first excitation energies obtained from spin- polarized calculations [24]. As ground-state exchange-correlation potentials were used the extra term in Eq.(20) does not appear. This is, certainly, one of the reasons for the difference between the calculated and the experimental excitation energies. There is a definite improvement comparing with the nonspin-polarized results [13]. Still, in most cases the calculated excitation energies are highly overestimated. The results provided by the different local density approximations are quite close to each other. The best one seems to be the Gunnarson-Lundqvist-Wilkins approximation. (In non-spin-polarized case the Perdew-Zunger parametrization gives results closest to the experimental data[30].)...

A. D. Boese and N. C. Handy (2001) A new parametrization of exchange-correlation generalized gradient approximation functionals. J. Chem. Phys. 114, p. 5497... 

A number of exchange-correlation potentials have been proposed over the years including some based on relativistic treatments. Those reported in Refs 31 and 32 are parametrizations of accurate Monte Carlo calculations for the electron gas and are believed to represent closely the limit of the LSD approximation. 

For simplicity, the LDA exchange-correlation functional is used. The correlation is the VWN parametrization of Monte Carlo result of Ceperley and Alder for a free electron gas [48,61]. The calculation is not spin-polarized. The purpose here is to show the mechanism of the divide-and-conquer method. While nonlocal corrections to Exc[p] and spin-polarization are instrumental to get results of chemical accuracy, none of these is expected to affect the basic mechanism of the method. 

Finally, there is an alternative and decidedly different way to incorporate electron correlation in quantum chemical calculations that is growing rapidly in importance DFT [100]. By using the Kohn-Sham formulation, DFT methods have been used extensively in quantum chemistry during the last decade and yield results that are superior to HF-SCF calculations at essentially the same cost. A further advantage seems to be that DFT appears to hold promise in the treatment of transition metal compounds, which is an area where standard methods (except elaborate MCSCF and MR-CI treatments) often fail catastrophically. Concerning the treatment of electron correlation, it should be noted that DFT methods — unlike the more traditional methods discussed so far—are semiempirical in nature and therefore only provide an implicit treatment. Correlation effects are incorporated in DFT (via an adequate parametrization) through the exchange-correlation functional and not explicitly treated in the usual sense. 

For scientific theories, being exact in principle seems to be a nice euphemism for being approximate in practice. Density-functional theory suffers from the same fate, and any DFT calculation can only be as reliable as the incorporated parametrization scheme for exchange and correlation. Indeed, the search for reliable exchange-correlation functionals is the greatest challenge to DFT. 

As has been widely accepted in Car-Parrinello (CP) simulations [38, 39], the exchange-correlation energy and potential are described in the local density approximation (LDA) [49]. A reasonable level of accuracy is achieved with the LDA including the correlation part by Ceperley and Alder [148] as parametrized by Perdew and Zunger [149], applied to... 

Preliminary studies were carried out in order to justify, using the reactivity index machinery, the higher reactivity of Co(II) derivatives with respect to other M(II) transition metal complexes, in particular when M = Mn(II) or Fe(II). Several ab intio smdies of the ground state properties of M-N4 complexes can be found in literature, especially concerning the relative stability of the different spin states (for instance in the case of Fe(II) derivatives). Here we consider only the most stable spin state for each metal complex and analyse the effect of the metal on the reactivity indexes (i.e. hardness, softness and electrophilicity). As already mentioned, and contrary to all other calculations reported in this review, these computations were performed using the parametrized hybrid Becke three-parameter exchange correlation functional (B3LYP " ) and a smaller basis set. The same level of theory was used to compute the donor molecule, i.e. the anionic form of 2-mercaptoethanol. 

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