Page 65 - 71 the abc of dft_opt

16.3. TRANSFERABILITY OF HOLES 129 130 CHAPTER 16. EXCHANGE-CORRELATION HOLE
exact H-atom hole contains a cusp at u = 0, missing from the uniform gas hole. This causes are. To begin with, both positive and negative values contribute equally, so we may write:
it to deviate quickly from the uniform gas hole near u = 0. At large distances, the H-atom 1 ∞ 1 ∞ sym
E X = dx n(x) du n (x, u), (16.41)
hole decays exponentially, because the density does, while the uniform gas hole decays slowly, −∞ 0 X
being a power law times an oscillation. These oscillations are the same Friedel oscillations where
we saw in the surface problem. For an integral weighted exp(−2|u|), the rapid decay of the n sym (x, u) = 1 (n X (x, u) + n X (x, −u)) (16.42)
exact hole leads to a smaller energy density than LDA. X 2
This symmetrizing has no effect on the LDA hole, since it is already symmetric, but Fig.
Now watch what happens when we move the hole point off the origin. In Fig. 16.3, we
?? shows that it improves the agreement with the exact case: both holes are now parabolic
plot the two holes at x = 0.5. The H-atom hole is said to be static, since it does not change around u = 0, and the maximum deviation is much smaller. However, the cusp at the nucleus
position with x. By plotting it as a function of u, there is a simple shift of origin. The hole
is clearly missing from the LDA hole, and still shows up in the exact hole.
remains centered on the nucleus, which is now at u = −0.5. On the other hand, we will
0
see that for most many electron systems, the hole is typically quite dynamic, and follows the
position where the first electron was found. This is entirely true in the uniform gas, whose -0.1
hole is always symetrically placed around the electron point, as seen in this figure. Note that,
although it is still true that both holes are normalized to −1, and that the ontop values still %n X (|u|)& -0.2
agree, the strong difference in shape leads to more different values of &(x) (-0.312 in LDA, -0.3
and -0.368 exactly). In Fig. 16.4, we plot the resulting exchange energy/electron throughout
the atom. The LDA curve only loosely resembles the exact curve. Near the nucleus it has a -0.4 1-d H-atom
exact
cusp, and overestimates & X (x). As x gets even larger, n(x) decays exponentially, so that the LDA
-0.5
LDA hole becomes very diffuse, since its length scale is determined by 1/k F , where k F = πn, 0 0.5 1 1.5 2 2.5 3
while the exact hole never changes, but simply moves further away from the electron position.
|u|
large x. When weighted by the electron density, to deduce the contribution to the exchange
energy, there is a large cancellation of errors between the region near the nucleus and far Figure 16.6: System-averaged symmetrized exhange hole of one-dimensional H-atom, both exactly and in LDA.
away. (To see the similarities with real systems, compare this figure with that of Fig. 17.7). Our last and most important step is to point out that, in fact, it is the system-averaged
symmetrized hole that appears in E X , i.e.,
0 1 ∞
%n X (u)& = dx n(x)n sym (x; u) (16.43)
-0.1 where now u always taken to be positive, since X
−∞
n sym (0.5, u) X -0.2 E X = 1 0 ∞ du v ee (u)%n X (u)& (16.44)
-0.3
-0.4
cusps remain in the exact hole, because of the system-averaging. Note how both rise smoothly
1-d H-atom The system-averaged symmetrized hole is extremely well-approximated by LDA. Note how no
-0.5
exact
LDA from the same ontop value. Note how LDA underestimates the magnitude at moderate u,
-0.6
0 0.5 1 1.5 2 2.5 3 which leads to the characteristic underestimate of LDA exchange energies. Finally, multiplying
by v ee = exp(−2|u|), we find Fig. 16.7. Now the 10% underestimate is clearly seen, with no
u
cancellation of errors throughout the curve.
Figure 16.5: Symmetrized exhange hole at x = 0.5 of one-dimensional H-atom, both exactly and in LDA.
We can understand this as follows. For small u, a local approximation can be very accurate,
as the density cannot be very different at x + u from its value at x. But as u increases, the
But, while all the details of the hole are clearly not well-approximated in LDA, especially as density at x + u could differ greatly from that at x, especially in a highly inhomogeneous
we move around in a finite system, we now show that the important averages over the hole system. So the short-ranged hole is well-approximated, but the long-range is not. In fact, for