Page 61 - 71 the abc of dft_opt

122 CHAPTER 16. EXCHANGE-CORRELATION HOLE
For a single Slater determinant, we have the following simple result:

N σ
-
# ∗ #
iσ
γ S (x, x ) = δ σσ ! φ (r)φ iσ (r ) (16.6)
i=1
Chapter 16 where φ iσ (r) is the i-th Kohn-Sham orbital of spin σ. This is diagonal in spin.
Exercise 70 For same-spin electrons in a large box (from 0 to L, where L → ∞) in one-
Exchange-correlation hole dimension, show that
 
#
#
k F sin(k F (x − x )) sin(k F (x + x ))
γ S (x, x ) =  −  . (16.7)
#
π k F (x − x ) k F (x + x )
#
#
1 In this chapter, we explore in depth the reliability of LSD. Although not accurate enough for
Prove this density matrix has the right density, and extract t S (x) from it.
thermochemistry, LSD has proven remarkably systematic in the errors that it makes. Improved
functionals should incorporate this reliability. We will see that in fact, understanding the We will, however, focus more on the potential energy, since that appears directly in the
inherent reliability of LDA is the first step toward useful generalized gradient approximations. usual adiabatic connection formula. To this end, we define the pair density:
1 1 2
P(x, x ) = N(N − 1) dx 3 . . . dx N |Ψ(x, x , x 3 , . . . , x N )| (16.8)
#
#
16.1 Density matrices and holes
which is also known as the (diagonal) second-order density matrix. This quantity has an
3
We now dig deeper, to understand better why LSD works so reliably, even for highly inho- important physical interpretation: P(rσ, r σ )d rd r is the probability of finding an electron
3 #
# #
3
mogeneous systems. We must first return to the many-body wavefunction. We define the of spin σ in d r around r, and a second electron of spin σ in d r around r . Thus it contains
3 #
#
#
following (first-order) density matrix: information on the correlations among the electrons. To see this, we may write:
1 1 & ' & '
3
3
3 #
3 #
#
#
#
#
∗
γ(x, x ) = N dx 2 . . . dx N Ψ (x, x 2 , . . . , x N )Ψ(x , x 2 , . . . , x N ) (16.1) P(x, x )d rd r = n(x)d r n 2 (x, x )d r (16.9)
#
The diagonal elements of this density matrix are just the spin-densities: where n 2 (x, x )d r is the conditional probability of finding the second electron of spin σ #
3 #
3
in d r around r , given that an electron of spin σ has already been found in d r around r.
3 #
#
γ(x, x) = n(x) (16.2)
Now, if the finding of the second electron were completely independent of the first event,
#
#
and the exact kinetic energy can be extracted from the spin-summed (a.k.a reduced) density then n 2 (x, x ) = n(x ). This can never be true for any wavefunction, since, by definition:
matrix: 1
#
#
dx P(x, x ) = (N − 1) n(x) (16.10)
-
#
γ(r, r ) = γ(rσ, r σ ) (16.3)
# #
σσ ! implying:
1
and dx n 2 (x, x ) = N − 1, (16.11)
#
#
1 1 3 2
#
T = − d r∇ γ(r, r )| r=r !. (16.4)
2 i.e., the fact that we have found one electron already, implies that the remaining conditional
Note that, although γ(x, x ) is a one-body property, only its diagonal elements are deter- probability density integrates up to one less number of electrons.
#
mined by the density. For example, we know that γ S , the density matrix in the Kohn-Sham
system, differs from the true density matrix, since the true kinetic energy differs from the Exercise 71 Show that 1
#
#
non-interacting kinetic energy: dx P(x, x ) = (N − 1) n(x) (16.12)
1 1 and state this result in words. What is the pair density of a one-electron system? Deduce
2
3
#
#
T C = − d r∇ (γ(r, r ) − γ S (r, r )) | r=r ! (16.5)
2 what you can about the parallel and antiparallel pair density of a spin-unpolarized two electron
1 c !2000 by Kieron Burke. All rights reserved. system, like the He atom.
121