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148 CHAPTER 18. GENERALIZED GRADIENT APPROXIMATION

Chapter 18

Generalized gradient approximation

1 In this chapter, we first discuss in more detail why the gradient expansion fails for finite
systems, in terms of the exchange-correlation hole. Fixing this leads directly to a generalized
gradient approximation (GGA) that is numerically defined, and corrects many of the limita-
tions of GEA for finite systems. We then discuss how to represent some of the many choices
sph.av. (u) for s = 1.
of forms for GGA, using the enhancement factor. We can understand the enhancement factor Figure 18.1: Spherically-averaged exchange hole density n X
in terms of the holes, and then understand consequences of gradient corrections for chemical
truncate the resulting hole at the first value of u which satisfies the sum-rule. In the figure, for
and physical systems. Finally we mention other popular GGAs not constructed in this fashion.
small z, the GGA hole becomes more negative than GEA, because in some directions, the GEA
hole has become positive, and these regions are simply sliced out of GGA, leading to a more
18.1 Fixing holes negative spherically-averaged hole. Then, at about 2k F u = 6.4, the hole is truncated, because
here the normalization integral equals -1, and its energy density contribution calculated.
We begin with exchange. We have seen that LSD works by producing a good approximation We note the following important points:
to the system- and spherically-averaged hole near u = 0, and then being a controlled extrap-
olation into the large u region. The control comes from the fact that we are modelling the • Simply throwing away positive contributions to the GEA exchange hole looks very ugly
in real space. But remember we are only trying to construct a model for the spherical-
hole by that of another physical system, the uniform gas. Thus the LSD exchange hole is
normalized, and everywhere negative. When we add gradient corrections, the model for the and system-average. Thus although that has happened in some directions for small z,
the spherically-averaged result remains smooth.
hole is no longer that of any physical system. In particular, its behavior at large separations
goes bad. In fact, the gradient correction to the exchange hole can only be normalized with • By throwing out postive GEA contributions, a significantly larger energy density is found
the help of a convergence factor. for moderate values of s. Even for very small s, one finds corrections. Even when GEA
To see this, consider Fig. 18.1, which shows the spherically averaged exchange hole for a corrections to the LSD hole are small, there are always points at which the LSD hole
given reduced gradient of s = 1. Everything is plotted in terms of dimensionless separation, vanishes. Near these points, GEA can make a postive correction, and so need fixing.
z = 2k F u. The region shown is the distance to the first maximum in the LDA hole, where it This means that even as s → 0, the GGA energy differs from GEA.
just touches zero for the first time. The GEA correction correctly deepens the hole at small
• For very large s (e.g. greater than 3), the wild GEA hole produces huge corrections
u, but then contains large oscillations at large u. These oscillations cause the GEA hole to
unphysically become positive after z = 6. These oscillations are so strong that they require to LSD, the normalization cutoff becomes small, and limits the growth of the energy
density. However, this construction clearly cannot be trusted in this limit, since even a
some damping to make them converge. Without a damping factor, the gradient correction
to the exchange energy from this hole is undefined. small distance away from u = 0 may have a completely different density.
The real-space cutoff construction of the generalized gradient approximation (GGA), is The results of the procedure are shown in Fig. 18.2. We see that the LDA underestimate
to include only those contributions from the GEA exchange hole that are negative, and to is largely cured by the procedure, resulting in a system-averaged hole that matches the exact
1 c !2000 by Kieron Burke. All rights reserved. one better almost everywhere.
147