Page 36 - 71 the abc of dft_opt

72 CHAPTER 8. THE LOCAL DENSITY APPROXIMATION
Now, in flatland, we wish to approximate the perimeter with a local functional of r, so we
write 1 2π
P loc = dθ f(r) (8.2)
0
To determine the function f, we note that the perimeter of a circle is 2πr. The only place
Chapter 8 our approximation can be exact is for the circle itself, since its r is independent of θ. Thus
we should choose f = r if we want our functional to be correct whenever it can be. Thus
The local density approximation P loc = 1 2π dθ r(θ) (8.3)
0
is the local approximation to the perimeter. Our more advanced mathematicians might be
able to prove simple rules, such as P loc ≤ P, etc. Also, we can test this approximation on
1 √
This chapter deals with the mother of all density functional approximations, namely the squares we really care about, and find, for the unit square, P loc = 4 ln(1 + 2), a 12%
the local approximation, for exchange and correlation.
underestimate, just like the kinetic energy approximation from Chapter 1.

8.1 Local approximations 8.2 Local density approximation
To illustrate why a local approximation is the first thing to try for a functional, we return So now its simple to see how to construct local density approximations.
to the problem in Chapter 2 of the perimeter of a curve r(θ). Suppose we live in a flatland Recall the example of Chapter 1, the non-interacting kinetic energy of same-spin electrons,
in which it is very important to estimate the perimeters of curves, but we haven’t enough as a warm-up exercise. We first write
calculus to figure out the exact formula. loc 1 ∞
T [n] = dx t s (n(x)) (8.4)
To study the problem, we first consider a smoother class of curves, S −∞
3
Next we deduced that t s , the kinetic energy density, must be proportional to n , from dimen-
r(θ) = r 0 (1 + & cos(nθ)). (8.1)
sional analysis, i.e.,
where n = 1, 2, 3... is chosen as an integer to preserve periodicity. Here & determines how t s (n) = a s n 3 (8.5)
large the deviation from a circle is, while n is a measure of how rapidly the radius changes Last, we deduced a s by ensuring T loc is exact for the only system for which it can be exact,
S
with angle. To illustrate, we show in Fig. 8.1 the case where n = 20 and & = 0.1. namely a uniform gas of electrons. We did this by looking at the leading contribution to the
2
energy of the electrons as the number became large. This yielded a s = π /6.
n=20, eps=0.1 At this point, we introduce a useful concept, called the Fermi wavevector. This is the
circle
wavevector of the highest occupied orbital in our system. Since we have N electrons in the
box,
Nπ
k F = = nπ (1D polarized) (8.6)
L
smooth curves Then 1
2
T loc = ∞ dx n(x) τ S (n(x)), τ S (n) = k /6 (8.7)
S F
−∞
Thus, for 3D Coulomb-interacting problems, we write
3
loc
E [n] = 1 d r f(n(r)) (8.8)
XC
Figure 8.1: Smooth parametrized curve, r = 1 + 0.1 cos(20θ). where f(n) is some function of n. To determine this function, we look to the uniform electron
1 c !2000 by Kieron Burke. All rights reserved. gas.
71