Vosko, Wilk and Nusair

One approach, using a local density approximation for each part, has E - = Es -1- Evwn, where Eg is a Slater functional and Evwn is a correlation functional from Vosko, Wilk, and Nusair (1980). Both functionals in this treatment assume a homogeneous election density. The result is unsatisfactory, leading to enors of more than 50 kcal mol for simple hydrocarbons. 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

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. 

Which is very closely related to Vosko, Wilk and Nusair s local correlation functional (VWN). 

Accurate values of the correlation functional are available thanks to the quantum Monte Carlo calculations of Ceperley and Alder (1980). These values have been interpolated in order to give an analytic form to the correlation potential (Vosko, Wilk and Nusair, 1980). 

The correlation energy of a uniform electron gas has been determined by Monte Carlo methods for a number of different densities. In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula. This has been constructed by Vosko, Wilk and Nusair (VWN) and is in general considered to be a very accurate fit. It interpolates between die unpolarized ( = 0) and spin polarized (C = 1) limits by the following functional. 

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

Three density functional theories (DFT), namely LDA, BLYP, and B3LYP, are included in this section. The simplest is the local spin density functional LDA (in the SVWN implementation), which uses the Slater exchange functional [59] and the Vosko, Wilk and Nusair [60] correlation functional. The BLYP functional uses the Becke 1988 exchange... 

Local density approximation (LDA) with Slater s Xa functional for exchange (Ref. 57) and the functional of Vosko, Wilk, and Nusair (Ref. 109) for correlation. 

Different fits were obtained by Vosko, Wilk, and Nusair for the spin-compensated and spin-... 

Tables 1-3 show the results of calculations based on Eqs. (362) and (363). The calculation of Table 1 employs the ordinary local density approximation (LDA) for and the adiabatic LDA (188) for /,c (both using the parametriz-ation of Vosko, Wilk and Nusair [90]). In this limit, the kernel G is approximated by [103]...

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. 

The Kohn-Sham-Gaspar potential derived from density-functional theory has a similar expression for V c with Xa = 2/3, and only took into account exchange [20],[40]. To include correlation, several forms were proposed for / , with parameters obtained from fits to RPA calculations or more accurate Monte Carlo simulations [41] and different spin interpolations. The current version of the DVM code contains altogether nine choices of Vje, the preferred form being the Vosko, Wilk and Nusair [42] parametrization of the Ceperley and Alder Monte Carlo simulations [43]. 

The simplest approximation, employed for very many years until the most recent developments, is known as Local Spin Density Approximation (LSDA) and does not depend on the gradients of the electronic density but only on the electronic density itself. One of the variants of LSDA, commonly employed in the applications to molecular systems in the last years, is the one called SVWN. In this exchange-correlation functional, the exchange is provided by Slater s formula (3) for the uniform electron gas, whereas the correlation is evaluated according to the expression derived by Vosko, Wilk and Nusair (4) from an interpolation of previous Monte-Carlo results for the spin-polarized homogeneous electron gas... 

For high density system, the enhancement factor becomes unity, and exchange effects dominate over the correlation effects. When the density becomes lower, the enhancement factor kicks in and includes correlation effects into the exchange energies. The enhancement factor is not unique, but can be derived differently in different approximations. The most reliable ones are parameterizations of molecular Monte-Carlo data. Some well known, and regularly used, parameterizations have been made by Hedin and Lundqvist [29], von Barth and Hedin [22], Gun-narsson and Lundqvist [30], Ceperly and Adler [31], Vosko, Wilk, and Nusair [32], and Perdew and Zunger [27]. 

Later Vosko, Wilk and Nusair (VWN) [32] proposed a correlation functional that was obtained using Pade approximant interpolations of very accurate numerical calculations made by Ceperley and Alder, who used a quantum Monte Carlo method [33], The VWN correlation functional is,... 

All the calculations have been carried out using a modified version of the deMon program [78]. The Vosko, Wilk and Nusair exchange-correlation... 

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