Page 64 - 71 the abc of dft_opt

16.3. TRANSFERABILITY OF HOLES 127 128 CHAPTER 16. EXCHANGE-CORRELATION HOLE
0 −δ(u) as we have used in the past, and since the exact ontop hole is just −n(x), then LDA
would be exact for X in that 1d world. But since the real Coulomb interaction in 3d is not a
-0.2
contact interaction, we will imagine a 1d world in which the repulsion is exp(−2|u|). Then,
n X (0, u) -0.4 because the overall hole shapes are similar, we find &(0) = −0.5 exactly, and = −0.562 in
LDA.
-0.6
0
-0.8 1-d H-atom
exact -0.2
LDA
-1
-3 -2 -1 0 1 2 3 n X (0.5, u) -0.4
u -0.6
Figure 16.2: Exhange hole at center of one-dimensional H-atom, both exactly and in LDA.
-0.8 1-d H-atom
exact
These depend only on the separation between points, as the density is constant in the system. LDA
-1
They also are symmetric in u, as there cannot be any preferred direction. This hole is also -3 -2 -1 0 1 2 3
plotted in Fig. 16.2, for a density of n = 1, the density at the origin in the 1-d H-atom. u
Comparison of the uniform gas hole and the H-atom hole is instructive. They are remarkably
Figure 16.3: Exhange hole at x = 0.5 of one-dimensional H-atom, both exactly and in LDA.
similar, even though their pair densities are utterly different. If the electron-electron repulsion
is taken as δ(u), U = 1/8 for the 1d H-atom, while U diverges for the uniform gas. Why are
We have shown above how to understand qualitatively why LDA would give results in the
they so similar? Because they are both holes of some quantum mechanical system. So both
right ball-park. But can we use this simple picture to understand quantitatively the LDA
holes are normalized, and integrate to -1. They are also equal at the ontop value, u = 0.
exchange? If we calculate the exchange energy per electron everywhere in the system, we
Exercise 79 Ontop exchange hole Show that the ontop exchange hole is a local-spin get Fig. 16.4. This figure resembles that of the real He atom given earlier. Looked at more
density functional, and give an expression for it. 0
Why is this important? We may think of the local approximation as an approximation to
the hole, in the following way: -0.2
n LDA (x, x + u) = n unif (n(x); u), (16.37) & X (x)
X X
i.e., the local approximation to the hole at any point is the hole of a uniform electron gas, -0.4
whose density is the density at the electron point. Fig 16.2 suggests this will be a pretty good 1d H atom
approximation. Then the energy per electron due to the hole is just X
LDA
-0.6
1 ∞
& X (x) = du v ee (u) n X (x, x + u), (16.38) 0 1 2
−∞
x
with
1 ∞
E X = dx n(x) & X (x). (16.39) Figure 16.4: Exhange energy per electron in the one-dimensional H-atom, both exactly and in LDA, for repulsion exp(−2|u|).
−∞
Since, for the uniform gas, closely, our naive hopes are dashed. In fact, our x = 0 case is more an anomaly than typical.
1 ∞
& unif (n) = du v ee (u) n unif (n; u), (16.40) We find, for our exponential repulsion, E X = −0.351 in LDA, and −0.375 exactly, i.e., the
X X
−∞ usual 10% underestimate in magnitude by LDA. But at x = 0, the LDA overestimates the
we may consider E LDA [n] as arising from a local approximation to the hole at x. Thus, the contribution, while for large x, it underestimates it.
X
general similarities tell us that LDA should usually be in the ball park. In fact, if v ee (u) = At this point, we do well to notice the difference in detail between the two holes. The