Page 76 - 71 the abc of dft_opt

18.2. VISUALIZING AND UNDERSTANDING GRADIENT CORRECTIONS 151 152 CHAPTER 18. GENERALIZED GRADIENT APPROXIMATION
and all curves initially drop, the GGA correlation hole always produces a negative correlation 1.6
energy density. 1.5

0 F XC (r s , s) 1.4 rs
1.3
2πu%n λ=1 (u)& C -0.04 1.2 0 1/2
2
1.1
4
10
He atom
exact 1 0 0.5 1 1.5 2 2.5 3
-0.08
LDA
GGA s = |∇n|/(2k F n)
0 1 2 3 Figure 18.6: Exchange-correlation enhancement factors for the PBE GGA.
u
These curves contain all the physics (and ultimately) chemistry behind GGA’s. We make
Figure 18.5: System-averaged correlation hole density at full coupling strength (in atomic units) in the He atom. The exact the following observations:
curve (solid) is from a CI calculation, the long dashed is LDA, and the short-dashes are real-space cutoff GGA. The area
under each curve is the potential contribution to correlation energy. • Along the y-axis, we have F unif (r s ), the uniform gas enhancement factor.
XC
In Fig. 18.5, we see the results for the system-averaged hole, which are even more dramatic • The effect of gradients is to enhance exchange. We have already seen this in action in
than the exchange case. At small separations, the GEA correclty makes the LDA hole less the previous sections. Essentially, in the positive direction of the gradient, the density
deep, as expected. But now, instead of producing a wild (i.e., unnormalized) postive peak, is increased, while dropping on the opposite side. This allows the center of the hole to
the postive contribution is much less pronounced, and is controlled by the normalization. move in that direction, becoming deeper due to the higher density, and yielding a higher
The upward bump and rapid turnoff beyond u = 2 are artifacts of the crude real-space cutoff exchange energy density.
procedure, and remind us of how little we know of the details of the corrections to LDA. But
• The effect of gradients is turn off correlation relative to exchange. This is because, in
the overall effect on the energy is very satisfying. The LDA hole spreads out enormously,
regions of high gradient the exchange effect keeps electrons apart, so that their correlation
giving rise to a correlation energy that is at least a factor of 2 too deep, while the real-space energy becomes relatively smaller. In our real-space analysis of the hole, we see that large
cutoff hole has produced something similar to the true hole.
gradients ultimately eliminate correlation.
• A simple picture of the correlation effects is given by the observation that for t up to
18.2 Visualizing and understanding gradient corrections
about 0.5, the GGA make little correction to LDA, but then promptly cuts off correlation
as t grows, as in Fig. ??. But t is the reduced gradient for correlation, defined in terms
To understand gradient corrections for exchange-correlation, we begin with the spin-unpolarized of the screening wavevector, not the Fermi wavevector. For an unpolarized system:
case, where we write
4
5
|∇n| k F 51.5
1 6
3
E GGA [n] = d r e unif (n(r)) F XC (r s (r), s(r)). (18.2) t = = s ≈ s (18.4)
XC X 2k s n k s r s
This enhancement factor contains all our others as special cases: Thus, for r s about 2, t = s, and we see correlation turning off at t about 0.5. For
r s = 10, t ≈ 0.4s, so that correlation turns off about s = 1.2, while for r s = 1/2,
unif
F XC (r s , s = 0) = F XC (r s ), F XC (r s = 0, s) = F X (s). (18.3) t ≈ 1.7s, so that correlation turns off very quickly, at s about 0.3.
We plot it for the case of the PBE functional in Fig 18.6. We will discuss the various kinds of • An interesting consequence of the point above is that pure exchange is the least local
GGA functionals that have been developed and are in use later, but for now we simply note curve, and that as correlation is turned on, the functional becomes more local, i.e., closer
that this GGA was designed to recover the GGA for the real-space correction of the GEA hole to the LDA value, both in the sense that gradient corrections become significant at larger
for moderate s, and includes also several other energetically significant constraints. s values, and also that even when they do, their magnitude is less.