LSD approximation

All calculations presented here are based on density-functional theory [37] (DFT) within the LDA and LSD approximations. The Kohn-Sham orbitals [38] are expanded in a plane wave (PW) basis set, with a kinetic energy cutoff of 70 Ry. The Ceperley-Alder expression for correlation and gradient corrections of the Becke-Perdew type are used [39]. We employ ah initio pseudopotentials, generated by use of the Troullier-Martins scheme [40], The coreradii used, in au, were 1.23 for the s, p atomic orbitals of carbon, 1.12 for s, p of N, 0.5 for the s of H, and 1.9, 2.0, 1.5, 1.97,... 

The first density fiinctional for the exchange-correlation energy was the local spin density (LSD) approximation [1,2]... 

By studying the exchange (i.e., A = 0) hole in Ne, Gunnarson et al. pointed out that the LSD approximation to this quantity was far better than the LSD approximation to the exchange hole prior to spherical-averaging. Then, from Eq. (7), the energy depends only on this spherically-averaged hole, i.e.. 

The LDA or its spin-polarized generalization, the local spin density (LSD) approximation provides a means of folding exchange and correlation effects, calculated on the basis of the local behavior of a uniform electron gas, into a set of self-consistent Hartree-like equations which contain only local operators for the potential. This procedure is represented schematically in Fig. 1. 

Of course, the Kohn-Sham paper was much more than a mere rediscovery of an approximate Hartree-Fock method. It is soundly based on density functional theory and has paved the way for the approximate treatment of correlation effects, for example, through the use of the LSD approximation in conjunction with correlated electron-gas calculations. 

A number of exchange-correlation potentials have been proposed over the years including some based on relativistic treatments. Those reported in Refs 31 and 32 are parametrizations of accurate Monte Carlo calculations for the electron gas and are believed to represent closely the limit of the LSD approximation. 

Perdew and Zunger (PZ) have listed a number of inadequacies of the LSD approximation which they attributed primarily to the spurious selfinteraction terms. They then proposed an orbital self-interaction correction scheme. The list of LSD failures given by PZ is ... 

There is no rigorous separation of exchange and correlation in DFT since one does not work directly with a wavefunction. In the LSD approximation the two can be separated in a natural way by reference to an HF calculation for the homogeneous electron gas. 

Notwithstanding these comments in defence of the LSD approximation, the residual self-interaction certainly represents an error which should be removed as one develops more and more accurate functionals. 

The main source of the error was attributed to a residual self-correlation energy (the electron is treated as a continuous fluid and the correlation energy is calculated between different parts of this density ). Various fixes have been attempted . That of Stoll et al. involves using the LSD approximation ... 

While the SIC, exact exchange plus local correlation, and gradient expansion techniques have offered some improvements and several directions for new research, the LSD approximation remains the workhorse of the field and we turn, in the next section, to a discussion of some of the methods for solving the LSD equations. 

Errors inherent in the LSD approximation Orbital basis-set incompleteness Auxiliary basis-set incompleteness Near-linear dependencies in auxiliary basis sets Orbital basis-set superposition error Auxiliary basis-set superposition error Exchange-correlation sampling grid Model potential for core electrons (if employed)... 

O(1023) electrons. On the other hand, the functional Exc[nhrii] is an extremely sophisticated many-body object, so that robust moderate-accuracy methods (such as the LSD approximation) can be difficult to improve systematically. 

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. 

In terms of the exchange-correlation hole, we may write the LSD approximation as... 

We have made the LSD approximation to the quantities plotted in Figure 2-4, in which we see that LSD works very well. These plots were made using a... 

We note a very important point in density functional theory and the construction of approximate functionals. It is the hole itself which can be well-approximated by, e.g., a local approximation. This is because it is the hole which obeys the exact conditions we have been discussing. To illustrate this point, Figure 6 is a plot of the pair distribution function around the origin in Hooke s atom, both exactly and within the LSD approximation. We see that the two functions are quite different. In particular, the exact pair distribution function has not saturated even far from the center. The corresponding holes of Figure 2, on the other hand, are much more similar. 

To be fair, there are several well-known exact conditions that the LSD approximation does not get right it is not self-interaction free[37], v (r) does not have the correct — 1/r behavior at large r for finite systems[38], it does not contain the integer discontinuity[39—41], etc. These shortcomings may be overcome by other improvements[42], but not by the gradient corrections discussed in this article. 

An obvious way to improve on the LSD approximation is to allow the exchange-correlation energy per particle to depend not only on the (spin) density at the point r, but also on the (spin) density gradients. This generalizes Eq. (57) to the form... 

Figure 7 is a plot of Fxc in the LSD approximation. The curves are horizontal lines in this case, as the LSD approximation is independent of the local gradient. However, only in the high density limit, rs = 0, is Fxc = 1, because this is where exchange dominates. The increase in Fxc beyond 1 for finite values of rB represents the correlation contribution to the exchange-correlation energy. The LSD approximation obeys all the conditions of section 2 except Eqs. (39), (40), (44), (45), and (47), because it approximates the hole by a hole taken from another physical system. 

One way around this problem is to make a true GGA, where the function / is often chosen so as to reproduce the GEA form for slowly-varying densities, but contains all powers of Vna, and the higher powers may be chosen by some criteria to produce (one hopes) an improvement on the LSD approximation. 

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