Page 48 - 71 the abc of dft_opt

12.2. DENSITY FUNCTIONALS 95 96 CHAPTER 12. SCALING
12.2 Density functionals 12.3 Correlation

Now we turn our attention to density functionals. Functionals defined as explicit operators Levy has shown that for finite systems, E C [n γ ] tends to a negative constant as γ → ∞. From
on the Kohn-Sham wavefunction are far simpler than those including correlation effects. In Fig. 12.2, we see that for the He atom E C varies little with scaling. We may write power
particular, there is both the non-interacting kinetic energy and the exchange energy. We shall 0
see that both their uniform scaling and their spin-scaling are straightforward. He atom
-0.05
Consider uniform scaling of an N-electron wavefunction, as in Eq. (12.9). The density of
the scaled wavefunction is -0.1
1 1 2 E C [n γ ]
3
3
3
n(r) = N d r 2 . . . d r N |Ψ γ (r, r 2 , . . . , r N )| , = γ n(γr) = n γ (r). (12.12) -0.15
Now, a key question is this. If Ψ[n] is the ground-state wavefunction with density n(r), is -0.2
Ψ[n γ ] = Ψ γ [n]? That is, is the scaled wavefunction the same as the wavefunction of the -0.25
exact
scaled density? We show below that the answer is no for the physical wavefunction, but is LDA
-0.3
yes for the Kohn-Sham wavefunction. 1 2 3 4 5 6 7 8 9 10
2
Consider the latter case first. We already know that T S [Φ γ ] = γ T S [Φ]. Thus if Φ minimizes γ
T S and yields density n, then Φ γ also minimizes T S , but yields density n γ . Therefore, Φ γ is
Figure 12.2: Correlation energy of the He atom, both exactly and within LDA, as the density is squeezed.
the Kohn-Sham wavefunction for n γ , or
Φ γ [n] = Φ[n γ ]. (12.13) series for E C [n γ ] around the high-density limit:
(2)
(3)
This result is central to understanding the behaviour of the non-interacting kinetic and ex- E C [n γ ] = E [n] + E [n]/γ + . . . , γ → ∞ (12.17)
C
C
change energies. We can immediately use it to see how they scale, since we can turn density We can easily scale any approximate functional, and so extract the separate contributions
functionals into orbital functionals, and vice versa:
to the correlation energies, and test them against their exact counterparts, check limiting
2
2
T S [n γ ] = T S [Φ[n γ ]] = T S [Φ γ [n]] = γ T S [Φ[n]] = γ T S [n], (12.14) values, etc. A simple example is correlation in LDA. Then when n → n γ , r S → r S /γ. Thus
3
3
using Eq. (12.24), and E LDA [n γ ] = 1 d r n γ (r) & unif (r s,γ (r)) = 1 d rn(r)& unif (r s (r)/γ) (12.18)
C C C
E X [n γ ] = E X [Φ[n γ ]] = E X [Φ γ [n]] = γE X [Φ[n]] = γE X [n], (12.15) unif unif
In the high-density limit, γ → ∞, & C (r s /γ) → & C (0). This is logarithmically singular in
using Eq. (F.15), and the fact that true LDA, as we saw in Chapter 8, and is an error made by LDA.
U[n γ ] = γ U[n] (12.16) From the figure, we see that for the He atom, as is the case with most systems of chemical
interest, E C [n γ ] is almost constant. In fact, it is very close to linear in 1/γ, showing that the
These exact conditions are utterly elementary, saying only that if the length scale of a sys-
tem changes, these functionals should change in a trivial way. Any approximation to these system is close to the high-density limit. The high-density property is violated by LDA (see
the figure), and diverges logarithmically as γ → ∞. This is another way to understand the
functionals must therefore satisfy these relations. However, they are also extremely powerful
LDA overestimate of correlation for finite systems. Since the LDA correlation energy diverges
in limiting the possible forms functional approximations can have.
The next exercise shows us that scaling relations determine the functional forms of the as γ → ∞, it will be overestimated at γ = 1.
On the other hand, the correlation energy is believed to have the following expansion under
local approximation to these simple functionals, and the uniform electron gas is referred to
only for the values of the coefficients. scaling to the low-density limit:
E C [n γ ] = γB[n] + γ 3/2 C[n] + . . . , γ → 0 (12.19)
Exercise 44 Local density approximations for T S and E X :
Show that, in making a local approximation for three-dimensional systems, the power of the Note that in this low-density regime, correlation is so strong it scales the same as exchange.
density in both the non-interacting kinetic and exchange energy approximations are deter- This limit is satisfied within LDA, as seen from Eqs. (??), although it might not be numerically
mined by scaling considerations. terribly accurate here. In the low density limit, & unif (r s ) = −d 0 /r s + d 1 /r 3/2 + . . ., so that
C s