Page 54 - 71 the abc of dft_opt

13.3. RELATION TO SCALING 107 108 CHAPTER 13. ADIABATIC CONNECTION
-0.85 Our argument also yields a simple relation for the energies:
He atom exact
LDA
2
λ
-0.9 F [n] = λ F[n 1/λ ], (13.22)
U XC (λ) -0.95 which shows how the λ-dependence of the universal functional (or, as we shall see, any energy
component) is completely determined by its scaling dependence. Contrast Eq. (13.22) with
-1
Eq. (12.21).
These relations prove to be extremely useful in analyzing functionals and their behavior.
-1.05
First note that we can take any functional and find out its scaling behavior quite easily. Then,
-1.1 through Eq. (13.22), applied to that functional, we can now deduce its coupling-constant
0 0.2 0.4 0.6 0.8 1
dependence. For example, the adiabatic connection integrand of Eq. (13.19) is simply
λ
U XC (λ)[n] = λU XC [n 1/λ ]. (13.23)
Figure 13.2: Adiabatic decomposition of exchange-correlation energy in He atom, both exactly and in LDA.
The simplest example in this regard is the non-interacting kinetic energy, which we know
claim that the LDA error drops with λ, then the usual cancellation of exchange and correlation scales quadraticly with scaling parameter. Then
2
λ
errors follows, since the area under the curve will then be more accurate than the value at T [n] = λ T S [n 1/λ ] = T S [n] (13.24)
S
λ = 0. Why this happens will be discussed later. But for now, we simply note that the
i.e., the kinetic energy is independent of coupling constant, as it should be. Similarly,
statement that LDA improves with λ is equivalent to the cancellation of errors statement,
2
λ
but is much more transparent. E [n] = λ E X [n 1/λ ] = λE X [n] (13.25)
X
We can use the adiabatic decomposition (λ-dependence of U XC (λ)) to analyze any approx- which again makes sense, since E X is constructed from the λ-independent Kohn-Sham orbitals,
imate functional, and find out why it behaves as it does. The next section shows how easy integrated with λ/|r − r |.
#
it is to find U XC (λ) from E XC . In fact, we already know. Much less trivial is the relation between different components of the correlation energy.
Consider Eq. (13.19) in differential form:
λ
13.3 Relation to scaling dE [n] @ @ @ @
λ
λ
XC = %Ψ [n] @ V ee@ Ψ [n]& − U[n] (13.26)
@ˆ @
dλ
A third important concept is the relation between scaling and coupling constant. Consider Then, from Eq. (13.22),
λ ˆ ˆ λ ˆ 2 ˆ
Ψ [n], which minimizes T+λV ee and has density n(r). Then Ψ [n] minimizes T/γ +(λ/γ)V ee @ @
γ λ λ @ @ λ λ
2 ˆ
ˆ
ee
@ˆ @
and has density n γ (r). If we choose γ = 1/λ, we find Ψ λ [n] minimizes λ (T + V ee ) and V [n] = λ%Ψ [n] @ V ee@ Ψ [n]& = λ(U[n] + E X [n]) + U [n], (13.27)
C
1/λ
ˆ
ˆ
ˆ
2 ˆ
has density n 1/λ (r). But if λ (T + V ee ) is minimized, then T + V ee is minimized, so that we where we have used the linear dependence of exchange on λ, and we have written
can identify this wavefunction being simply Ψ[n 1/λ ]. If we scale both wavefunctions by λ, we
E C [n] = (T[n] − T S [n]) + (V ee [n] − U[n]) = T C [n] + U C [n] (13.28)
find the extremely simple but important result
and T C [n] is the kinetic contribution to the correlation energy, while U C [n] is the potential
λ
Ψ [n] = Ψ λ [n 1/λ ], (13.20) contribution. Inserting Eq. (13.27) into Eq. (13.26) and cancelling the trivial exchange
contributions to both sides, we find
which tells us how to construct a wavefunction of coupling constant λ by first scaling the λ
density by 1/λ, finding the ground-state wavefunction for the scaled density, and then scaling dE [n] = U [n]/λ (13.29)
λ
C
that wavefunction back to the original size. dλ C
2
λ
λ
This solves a mystery from the previous chapter: The ground-state wavefunction of a Finally, we rewrite this as a scaling relation. Since E [n] = λ E C [n 1/λ ] and U [n] =
C
C
2
scaled density is the scaled ground-state wavefunction, but only if the coupling constant is λ U C [n 1/λ ] and writing γ = 1/λ, we find, after a little manipulation
changed, i.e., dE C [n γ ]
Ψ[n γ ] = Ψ 1/γ [n]. (13.21) γ dγ = E C [n γ ] + T C [n γ ]. (13.30)
γ