Local density approximation energy models

Here, we pointed to the problem of theoretical representation, in particular, in two aspects of theory (i) the existence of highly mobile atoms at the surface such as hydrogen, which are usually not considered in the atomistic models and (ii) the importance of bandgaps and relative energy levels of electronic states, which is often distorted in local density approximations. In both respects, a quick fix to the problem is not very likely. However, as both theory and experiment continue to be developed and applied in common research projects, it can be expected that the actual understanding of the processes involved in reaction on model catalysts will substantially improve over the next 10 years. After all, the ability to trace reactions and to account for the position and charge state of each reactant is already a realization of what seemed 20 years ago a fiction rather than fact. 

In principle, the KS equations would lead to the exact electron density, provided the exact analytic formula of the exchange-correlation energy functional E was known. However, in practice, approximate expressions of Exc must be used, and the search of adequate functionals for this term is probably the greatest challenge of DFT8. The simplest model has been proposed by Kohn and Sham if the system is such that its electron density varies slowly, the local density approximation (LDA) may be introduced ... 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.

The cornerstone of the field (the "Hartree-Fock" of Density Functional Theory) is the Local Density Approximation (LDA) also called the Local (Spin) Density (LSD) method Here the basic information on electron correlation, how electrons avoid each other, is taken from the uniform density electron gas Essentially exact calculations exist for this system (the Quantum Monte Carlo work of Ceperley and Alder) and this information from the homogeneous model is folded into the inhomogeneous case through the energy integral ... 

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. 

Different DFT models apply different approximations for the T[p] and Vee[p] functionals. Thus, the local density approximation (EDA) implies that for a density p(r) the kinetic energy and exchange-correlation energy functionals in each point of the space r are those of a homogeneous electronic system of the same density. For a discussion of these different DFT models, we refer the interested reader to earlier chapters in this series and to the indispensable book by Parr and Yang. ... 

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