Semilocal functionals

Note added in proof Since this article was written, a method for improving the energy of semilocal functionals, using the ideas of Section 3.1, has appeared [83], and further examples of abnormal systems have been identified [84]. 

Burke K, Perdew JP, Ernzerhof M, Why semilocal functionals work Accuracy of the on-top hole density, in preparation for J Chem Phys... 

To correct for the nonuniformity of the electron density, gradients of the density are introduced into the exchange and correlation functionals, creating in this manner nonlocal or semilocal functionals. The first gradient approximation was not successful since it did not fulfill many of the requirements of the exchange-correlation functional it was even worse than the local approximation. 

This article is organized as follows. Section 2 is a discussion of many conditions which all electronic systems are known to satisfy. Section 3 is a discussion of the LSD approximation and semilocal functionals. Section 4 describes some recent progress made in the study of exact conditions, while section 5 describes results of recent applications of GGA s in real physical and chemical systems. 

In this section we define the local spin density (LSD) approximation, the workhorse of density functional theory. We then examine its extension to semilocal functionals, i.e., those which employ both the local density and its derivatives, also called generalized gradient approximations. We show how the PW91 functional obeys many exact conditions for the inhomogeneous system, as described in section 2, which earlier semilocal functionals do not. 

This constraint cannot be satisfied by any smilocal functional. That is, a semilocal functional is incapable of describing how the kinetic energy of the uniform electron gas changes when it is simultaneously perturbed away from uniformity at two different points. The idea that imposing the correct linear response would improve the description of nonuniform electron densities emerged from the work of Herring " and Chacon et al. ... 

The particular way in which the inhomogeneous many-body problem is decomposed, and the various possible improvements on the LDA, are behind the success of DFT in practical calculations, in particular, materials. The most important improvement of LDA is connected with the attempt to introduce a spatially varying density and include information on the rate of this variation in the functional. The corresponding fimctionals, known as semilocal functionals [330], are considered in the next section. 

Another consequence of the wrong asymptotic behaviour of local and semilocal functionals is that static polarizabilities are frequently overestimated by standard functionals. Much useful information on the performance of approximate density functionals in calculations of electric polarizabilities and hyperpolarizabilities of atoms, molecules and clusters can be found in Schwerdtfeger, " Fuentealba and Pouchen et al. ... 

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