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16.4. OLD FAITHFUL 131 132 CHAPTER 16. EXCHANGE-CORRELATION HOLE
0 where 1
exp(−2|u|) %n LSD (u)& = N d r n(r) dΩ u n unif (r s (r), ζ(r); u) (16.49)
1
3
XC
4π
XC
-0.25 is the system- and spherically-averaged hole within LSD.
%n X (|u|)& 1-d H-atom
exact
LDA
-0.5
0 0.5 1 1.5
|u|
Figure 16.7: System-averaged symmetrized exhange hole of one-dimensional H-atom, both exactly and in LDA, weighted by
exp(−2|u|).
large u, the oscillating power-law LDA behavior is completely different from the exponentially
decaying exact behavior:
Exercise 80 Show that the exact system-averaged symmetrized hole in the 1-d H atom is:
%n X (u)& = − exp(−2u) (1 + 2u), (= f(πu) LSD), (16.45)
where
2 1 z 2
#
#
f(z) = − dz sin (z )/z . # (16.46)
z 2 0 Figure 16.8: Universal curve for the system-averaged on-top hole density in spin-unpolarized systems. The solid curve is for
But the constraint of the sum-rule and the wonders of system- and spherical- (in the 3d case) the uniform gas. The circles indicate values calculated within LSD, while the crosses indicate essentially exact results, and
the plus signs indicate less accurate CI results.
averaging lead to a very controlled extrapolation at large u. This is the true explanation of
LDA’s success. Notice also that this explanation requires that uniform gas values be used: Why should this hole look similar to the true hole? Firstly, note that, because the uniform
Nothing else implicitly contains the information about the hole. gas is an interacting many-electron system, its hole satisfies the same conditions all holes
satisfy. Thus the uniform gas exchange hole integrates to -1, and its correlation hole integrates
16.4 Old faithful to zero. Furthermore, its exchange hole can never be negative. Also, the ontop hole (u = 0)
is very well approximated within LSD. For example, in exchange,
We are now in a position to understand why LSD is such a reliable approximation. While it ( )
2
2
n X (r, r) = − n (r) + n (r) /n(r) (16.50)
may not be accurate enough for most quantum chemical purposes, the errors it makes are ↑ ↓
very systematic, and rarely very large. i.e., the ontop exchange hole is exact in LSD. Furthermore, for any fully spin-polarized or
We begin from the ansatz that LSD is a model for the exchange-correlation hole, not just highly-correlated system,
the energy density. We denote this hole as n unif (r s , ζ; u), being the hole as a function of
XC
3
separation u of a uniform gas of density (4πr /3) −1 and relative spin-polarization ζ. Then, n XC (r, r) = −n(r) (16.51)
s
by applying the technology of the previous section to the uniform gas, we know the potential i.e., the ontop hole becomes as deep as possible (this makes P(r, r) = 0), again making
exchange-correlation energy density is given in terms of this hole: it exact in LSD. It has been shown to be highly accurate for exchange-correlation for most
1 ∞
u unif (r s , ζ) = 2π duu n unif (r s , ζ; u). (16.47) systems, although not exact in general. Then, with an accurate ontop value, the cusp
XC 0 XC
This then means that, for an inhomogeneous system, condition, which is also satisfied by the uniform gas, implies that the first derivative w.r.t. u
at u = 0 is also highly accurate. In Fig. 16.8, we plot the system-averaged ontop hole for
3
U LSD [n] = 1 d r u unif (r s (r), ζ(r)) = N 1 ∞ duu %n LSD (u)&, (16.48)
XC XC 0 XC several systems where it is accurately known. For these purposes, it is useful to define the