Page 62 - 71 the abc of dft_opt

16.1. DENSITY MATRICES AND HOLES 123 124 CHAPTER 16. EXCHANGE-CORRELATION HOLE
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Why is this an important quantity? Just as the kinetic energy can be extracted from
the first-order density matrix, the potential energy of an interacting electronic system is a 4
two-body operator, and is known once the pair density is known: cartoon
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1 1 1
#
V ee = dx dx P(x, x )/|r − r | (16.13) n(x # ) 2
#
#
2
Although it can be very interesting to separate out the parallel and antiparallel spin contribu- 1
tions to the pair density, we won’t do that here, since the Coulomb repulsion does not. The
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reduced pair density is the spin-summed pair density:
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- -6 -4 -2 0 2 4 6
P(r, r ) = P(rσ, r σ ), (16.14)
#
# #
σσ ! x #
and is often called simply the pair density. Then Figure 16.1: Cartoon of hole in one-dimensional exponential density. Plotted are the actual density n(x ) (long dashes), the
!
conditional density n 2 (x = 2, x ) (solid line), and their difference, the hole density, n XC (x = 2, x ) (dashed line).
!
!
1 1 3 1 P(r, r )
#
3 #
V ee = d r d r . (16.15)
2 |r − r | # rapidly with distance, as the electrons avoid the electron at x. The product of densities in
P(r, r ) gives rise to U, the Hartree energy, while
#
As we have seen, a large chunk of V ee is simply the Hartree electrostatic energy U, and
since this is an explicit density functional, does not need to be approximated in a Kohn-Sham U XC = 1 d r n(r) 1 d u n XC (r, u) . (16.21)
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calculation. Thus it is convenient to subtract this off. We define the (potential) exchange- 2u
correlation hole density around an electron at r of spin σ: Thus we may say, in a wavefunction interpretation of density functional theory, the exchange-
correlation energy is simply the Coulomb interaction between the charge density and its
P(x, x ) = n(x) (n(x ) + n XC (x, x )) . (16.16)
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#
#
surrounding exchange-correlation hole.
The hole is usually (but not always) negative, and integrates to exactly −1: The exchange hole is a special case, and arises from the Kohn-Sham wavefunction. For a
single Slater determinant, one can show
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dx n XC (x; x ) = −1 (16.17)
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#
N σ N σ !
- -
∗
∗
#
#
#
#
Since the pair density can never be negative, P X (x, x ) = φ (r)φ jσ !(r ) (φ iσ (r)φ jσ !(r ) − φ iσ (r )φ jσ !(r)) (16.22)
iσ
i=1 j=1
#
n XC (x, x ) ≥ −n(x ). (16.18) #
#
The first (direct) term clearly yields simply the product of the spin-densities, n(x)n(x ), while
This is not a very strong restriction, especially for many electrons. Again, while the spin- the second (exchange) term can be expressed in terms of the density matrix:
# 2
decomposition of the hole is interesting, our energies depend only on the spin-sum, which is P X (x, x ) = n(x)n(x ) − |γ S (x, x )| (16.23)
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#
less trivial for the hole:
  yielding
- # # 2
n XC (r, r ) =  n(x)n XC (x, x ) /n(r). (16.19) n X (x, x ) = −|γ S (x, x )| /n(x). (16.24)
#
# 
σσ !
This shows that the exchange hole is diagonal in spin (i.e., only like-spins exchange) and is
The pair density is a symmetrical function of r and r : everywhere negative. Since exchange arises from a wavefunction, it satisfies the sum-rule, so
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that
P(r , r) = P(r, r ), (16.20) 1 3
#
#
d u n X (r, u) = −1 (16.25)
but the hole is not. It is often useful to define u = r − r as the distance away from the
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The exchange hole gives rise to the exchange energy:
electron, and consider the hole as function of r and u. These ideas are illustrated in Fig.
#
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(16.1). It is best to think of the hole as a function of u = x − x, i.e., distance from the E X = 1 1 d r 1 d r n(r)n X (r, r ) (16.26)
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3 #
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electron point. The hole is often (but not always) deepest at the electron point, and decays 2 |r − r |