Uniform density limit energy functionals

Uniform Density Limit of Exchange-Correlation Energy Functionals... 

On the practical side, we note that nature provides a number of extended systems like solid metals [29, 30], metal clusters [31], and semiconductors [30, 32]. These systems have much in common with the uniform electron gas, and their ground-state properties (lattice constants [29, 30, 32], bulk moduli [29, 30, 32], cohesive energies [29], surface energies [30, 31], etc.) are typically described much better by functionals (including even LSD) which have the right uniform density limit than by those that do not. There is no sharp boundary between quantum chemistry and condensed matter physics. A good density functional should describe all the continuous gradations between localized and delocalized electron densities, and all the combinations of both (such as a molecule bound to a metal surface a situation important for catalysis). 

How Do Energy Functionals Perform in the Uniform Density Limit ... 

The correlation energy can in principle be resolved as a sum of contributions from tT> ii> ti correlations. Such a resolution even in the uniform density limit, is not really needed for the construction of density functional approximations, and no assumption about the spin resolution has been made in any of the functionals from our research group (which are all correct by construction in the uniform density limit). 

A second observation about the LYP functional is that it predicts no correlation energy for a fully spin-polarized system of electrons. Yet, in the uniform-density limit, the correlation energy at full spin-polarization is about half that of the unpolarized system [3, 4, 57]. Even in the Ne atom, the parallel-spin contribution accounts for about 24% of the total correlation energy (Sect. 3.4). 

Nonempirical GGA functionals satisfy the uniform density limit. In addition, they satisfy several known, exact properties of the exchange-correlation hole. Two widely used nonempirical functionals that satisfy these properties are the Perdew-Wang 91 (PW91) functional and the Perdew-Burke-Ernzerhof (PBE) functional. Because GGA functionals include more physical ingredients than the LDA functional, it is often assumed that nonempirical GGA functionals should be more accurate than the LDA. This is quite often true, but there are exceptions. One example is in the calculation of the surface energy of transition metals and oxides. 

Langreth and Mehl [134] used the sharp cut-olf procedure in momentum space to eliminate spurious contributions to and an empirical exponential function to damp the gradient contribution to the energy. The Langreth-Mehl (LM) functional [134,160,161] has now mostly a historical significance. A few years later, Perdew [162] improved the LM functional by imposing two additional requirements that it recover the correct second-order DGE in the slowly varying density limit and reduce in the uniform density limit to LDA, not to the random-phase approximation (RPA), as the LM functional does. Perdew s 1986 correlation functional is... 

Becke s 1995 correlation functional (B95) [168] was constructed to satisfy the following set of conditions (a) the correct uniform density limit (b) separation of the correlation energy into parallel-spin and opposite-spin components (c) zero correlation energy for one-electron systems (d) good fit to the atomic correlation energies. These requirements are met by the following analytic form... 

In contrast to the gradient expansion of the exchange hole of Perdew [135], the BR functional does not reduce to LSDA in the uniform density limit. To recover this limit approximately, Becke and Roussel multiplied the term t — t ) in Eq. (141) by an adjustment factor of 0.8. At the same time, the BR exchange energy density has the correct — p(r)/2r asymptotic behavior in the r — oo limit. 

The VWN LDA correlation functional (Vosko et al. 1980) was developed to make the RPA expression converge to the high and low density limits. This functional was inductively derived on the basis of the Pad6 interpolation by fitting parameters to the exact correlation energy of a uniform density gas given by the quantum Monte... 

It was obtained by a modified Colle-Salvetti approach. At first, an analytic expression of the kinetic contribution to the correlation energy per electron was determined. Then, the total correlation energy was derived by means of the DFT virial theorem. The value of the only parameter entering in this approach was fixed by applying the resulting expression to the uniform electron gas (UEG) in the low and high density limit cases. Thus, in spite of the constants entering in (1.8), the RC correlation functional is parameter-free. 

In contemporary theories, a is taken to be and correlation energies are explicitly included in the energy functionals [15]. Sophisticated numerical studies have been performed on uniform electron gases resulting in local density expressions of the form F j.[p(r)] = K [p(r)] -l- F. [p(r)] where represents contributions to the total energy beyond the Hartree-Fock limit [22]. It is also possible to describe the role of spin explicitly by considering the charge density for up and down spins p = p -i- p. This approximation is called the local spin density approximation [15]. 

The analytical form for the correlation energy of a uniform electron gas, which is purely dynamical correlation, has been derived in the high and low density limits. For intermediate densities, the correlation energy has been determined to a high precision by quantum Monte Carlo methods (Section 4.16). In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula, and such formulas have been constructed by Vosko, Wilk and Nusair (VWN) and by Perdew and Wang (PW), and are considered to be accurate fits. The VWN parameterization is given in eq. (6.36), where a slightly different spin-polarization function has been used. 

The exact density functionals for arbitrary electron densities are not known. The simplest approximate density functionals are those which are exact for the uniform electron gas [16, 17] (the correlation energy functional is known exactly for the uniform electron gas only in the limits ofhigh and low electron densities the correlation energy for intermediate densities can be obtained by interpolation between these limits [14]). To improve upon the uniform electron gas functionals, Waldman and Gordon introduced scaling coefficients which depend on the number of electrons in the system [18]. Although these scaled functionals increase the accuracy of electron gas model calculations, this increased accuracy is due somewhat to a cancellation of the error due to the approximate electron density [19]. Accurate non-local density functionals have recently been developed [20, 21, 22], which lead to more accurate calculations of interaction energies [19,22]. 

These LDA correlation functionals have been used in various property calculations, especially in solid state calculations. However, we should recall that these LDA correlation functionals are not exact functionals but are inductively derived approximative functionals. Actually, even though the exact correlation energy of a uniform electron gas in a quanmm Monte Carlo calculation has (9(p) and density dependences at the high and low density limits, respectively... 

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