Page 55 - 71 the abc of dft_opt

13.4. STATIC CORRELATION 109 110 CHAPTER 13. ADIABATIC CONNECTION
which is just Eq. (12.38). Thus, the adiabatic connection formula may be thought of as an Thus, it has always been found that 0 < b ≤ 1/2, but b is always close to 0.5 for real systems.
integration of this scaling formula between γ = 1 and γ = ∞. An alternative way to write it is
(13.34)
E C = (1 − b)U C
Exercise 61 E C from T C
λ
λ
Derive a relation to extract E C [n] from T C [n γ ] alone. Rewrite this to get E [n] from T [n]. or that if we simplify the adiabatic curve as a step down at some value of λ from E X to
C
C
E X + U C , then the step occurs at λ = b.
Exercise 62 Use Eq. (16.5) and the scaling relations of the previous chapter to write E C in
terms of density matrices of different λ. Considering both the figures and the numbers, we see that LDA clearly significantly un-
derestimates b. This is again because of the logarithmic divergence as r s → 0, and we will
We can easily scale any approximate functional, and so extract the separate contributions see that this does not happen for more sophisticated approximations.
to the correlation energies, and test them against their exact counterparts, check limiting
The reason that most systems have b close to 1/2 is that they are not very far from
values, etc. A simple example is LDA. Then when n → n γ , r S → r S /γ. Thus the high-density limit. As one moves towards lower densities (much lower than in realistic
1 1
3
3
E LDA [n γ ] = d r n γ (r) & unif (r s,γ (r)) = d rn(r)& unif (r s (r)/γ) (13.31) systems), b reduces, even coming close to zero. Thus, even for the uniform gas, T C eventually
C C C
becomes small relative to |U C |, as r s → ∞. Thus we can speak of b as measuring the amount
In the high-density limit, γ → ∞, & unif (r s /γ) → & unif (0). This is singular in true LDA, and is of dynamic correlation, i.e., the fraction of correlation energy that is kinetic, with a value
C C
an error made by LDA. However, in the low density limit, & unif (r s ) = −d 0 /r s + d 1 /r 3/2 + . . ., of 1/2 being the maximum. Typical systems are close to this value, and only for very low
s
C
3
!
so that E LDA [n γ ] = −γd 0 d r n(r)/r s (r), vanishing linearly with γ, correctly. Note that in densities does b become small.
C
this low-density regime, correlation is so strong it scales the same as exchange. -0.4
Exercise 63 Adiabatic connection for He atom:
-0.45
Using the Wigner approximation and the effective exponential density, draw the adiabatic -0.5
connection curve for He. U XC (λ)
-0.55 exact (R=5)
Exercise 64 Adiabatic connection for H atom:
-0.6
Draw the adiabatic connection curve for the H atom.
-0.65
0 0.2 0.4 0.6 0.8 1
Exercise 65 Changing λ:
Derive E LDA,λ [n] and T LDA [n]. λ
XC C
Figure 13.3: Adiabatic decomposition of exchange-correlation energy in stretched H 2 .
13.4 Static correlation
To appreciate why we are introducing this parameter, we next consider what happens to
We have seen how, for the total XC energy of many systems of interest, the adiabatic a H 2 molecule when we stretch it. In Figure 13.3, we plot the adiabatic connection curve
connection curve is close to linear. In fact, it has always been found that the adiabatic at R = 5. It looks very different from that of He, or of H 2 at equilibrium. To understand
connection curve is (slightly) concave upwards: it, we note that at λ = 1, the curve is almost flat, and just about equal to −5/8, i.e., the
exchange(-correlation) energy of two separate H atoms. On the other hand, at λ = 0, we
2
d U C (λ) have the exact exchange value of KS theory. For this problem, the electrons are always in a
dλ 2 ≥ 0 (unproven) (13.32)
singlet and the exchange energy is about -0.42. The value of b is about 0.16, and this system
Given the geometric interpretation of Fig X, this means T C ≤ |E C |. To quantify the fraction has strong static correlation.
of correlation that is kinetic, we define There is no way for LDA to produce a curve that looks anything like this. Because the
average density will be almost the same as a single H atom, the adiabatic curve is very flat,
T C
b = (13.33)
|U C | with essentially no static correlation. Since we are beyond the Coulson-Fischer point, we