Semilocal Approximations

One of the most intriguing properties of the exact functional, which has resisted all attempts of describing it in local or semilocal approximations, is the derivative discontinuity of the xc functional with respect to the total particle number [50, 58, 59],... 

The dependence of these functionals on n(r), the integral over n(r), instead of on derivatives, as in the GGAs, is the reason why such functionals are called nonlocal. In practice, this integral turns the functionals computationally expensive, and in spite of their great promise they are much less used than GGAs. However, recent comparisons of ADA and WDA with LDA and GGAs for low-dimensional systems [114, 130] and for bulk silicon [131] show that nonlocal integral-dependent density functionals can outperform local and semilocal approximations. 

For finite systems and clusters in particular an important source of problems strictly related to the local and semilocal approximations is the asymptotic behavior of the potential felt by an electron that is well outside the cluster. The exact exchange-correlation potential converges asymptotically (and probably rather quickly [40]) to l)/r,... 

Because of the importance of local and semilocal approximations like (1.11) and (1.12), bounds on the exact functionals are especially useful when the bounds are themselves local functionals. 

Electron affinities are harder to obtain than ionization energies, because within local and semilocal approximations the A-f 1 st electron often is not bound at all, so wrong affinities result both from eigenvalues and from total energies. By exploiting error cancellation... 

The electronic interactions between the MMe3 substituents and the sulphur rm orbital were analysed121 on the basis of the semilocalized orbitals approximation in two series of the structures S(MMe3)2 and MeSMMe3 (M = C, Si, Ge, Sn, Pb). 

In the lineage of the methodology developed above, the ADA and the WDA are nonlocal extensions of the EDA formulation. In this sense, the TF model discussed in Section II. 1 is the EDA counterpart for the OF-KEDF. however, the vW model discussed in Section III. 1 is somewhat different, because the ansatz in Eq. (39) departs from the EDA ansatz in Eq. (23). For later convenience, we name the strategy in Section III. 1 the Semilocal-Density Approximation (SEDA). In the following, through a detailed analysis of the exchange energy density functional (XEDF) and the OF-KEDF, we shall see the classical WDA is actually closely related to the SEDA. 

As all numerical results indicate that the WDA is heading into the right direction, incorporating the correct LR behavior - becomes the next logical step. The ADA immediately comes into mind, but proper modifications have to be made. The Semilocal Average-Density Approximation (SADA) constitutes the first step towards this goal. 

The main issue involved in using DFT and the KS scheme pertains to construction of expressions for the XC functional, Exc[n], containing the many-body aspects of the problems (1.38). The main approaches to this issue are (a) local functionals the Thomas Fermi (TF) and LDA, (b)semilocal or gradient-dependent functionals the gradient-expansion approximation (GEA) and generalized gradient approximation (GGA), and (c) nonlocal functionals hybrids, orbital functionals, and SIC. For detailed discussions the reader is referred to the reviews [257,260-272]. 

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. 

It is apparent that as the momentum p increases, the finite difference spectrum deviates more and more from the correct value. It is usually assumed that acceptable accuracy with the FD method is obtained when at least 10 points are used per wave period. This means also using 10 points per unit volume in phase space. The finite difference algorithms are based on a local polynomial approximation of the wave function and therefore the convergence of the method follows a power law of the form (Aq)n, where n is the order of the finite difference approximation. This semilocal description leads to a poor spectral representation of the kinetic energy operator, which will be true as well, for other banded representations of the kinetic energy operator such as the... 

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