Page 67 - 71 the abc of dft_opt

16.4. OLD FAITHFUL 133 134 CHAPTER 16. EXCHANGE-CORRELATION HOLE
system-averaged density:
1 1 dΩ u 1 3 0
%n(u)& = d r n(r) n(r + u). (16.52)
N 4π
and a system-averaged mean r s value: 2πu%n λ=1 (u)& C -0.04
d r n (r) r s (r)
! 3 2
%r s & = ! . (16.53)
3
2
d r n (r) He atom
-0.08 exact
Clearly, the system-averaged ontop hole is very accurate in LSD. LDA
That LSD is most accurate near u = 0 can be easily understood physically. At u grows r # 0 1 2 3
gets further and further away from r. But in LSD, our only inputs are the spin-densities at r,
u
and so our ability to make an accurate estimate using only this information suffers. Indeed, as
mentioned above, at large separations the pair-correlation function has qualitatively different Figure 16.10: System-averaged correlation hole density at full coupling strength (in atomic units) in the He atom, in LSD,
numerical GGA, and exactly (CI). The area under each curve is the full coupling-strength correlation energy.
behavior in different systems, so that LSD is completely incorrect for this quantity. Thus, we
may expect LSD to be most accurate for small u, as indeed it is. But even at large u, its most systems, this implies that their exact system-averaged hole looks very like its LSD
behavior is constained by the hole normalization sum-rule. approximation, with only small differences in details, but never with very large differences.
0 In Figs. 16.9-16.10, we plot the system-averaged exchange and potential-correlation holes
for the He atom, both exactly and in LSD. Each is multiplied by a factor of 2πu, so that
-0.1 its area is just the exchange or correlation energy contribution (per electron). We see that
2πu%n X (u)& -0.2 LSD is exact for small u exchange, and typically underestimates the hole in the all-important
moderate u region, while finally dying off too slowly at large u. For correlation, we see that
the LSD hole is always negative in the energetically significant regions, whereas the true hole
-0.3
has a significant positive bump near u = 1.5. This is yet another explanation of the large
-0.4 He atom
HF overestimate of LSD correlation. In the next chapter, we will show how to use the failed
LDA
-0.5 gradient expansion to improve the description of the hole, and so construct a generalized
0 1 2 3
gradient approximation.
u
Figure 16.9: System-averaged exchange hole density (in atomic units) in the He atom, in LDA (dashed) and exactly (HF -
solid). The area under each curve is the exchange energy.
Two other points are salient. The exchange-correlation hole in the uniform gas must be
spherical, by symmetry, whereas the true hole is often highly aspherical. But this is irrelevant,
since it is only the spherical-average that occurs in E XC . Furthermore, the accuracy of LSD
can fail in regions of extreme gradient, such as near a nucleus or in the tail of a density. But
in the former, the phase-space in the system-average is small, while in the latter, the density
itself is exponentially small. The same argument applies to large separations: even if g is
badly approximated by LSD, it is −n(r + u)g(r, r + u) that appears in the hole, and this
vanishes rapidly, both exactly and in LSD. Thus limitations of LSD in extreme situations do
not contribute strongly to the exchange-correlation energy.
This then is the explanation of LSD’s reliability. The energy-density of the uniform gas
contains, embedded in it, both the sum-rule and the accurate ontop hole information. For