Page 53 - 71 the abc of dft_opt

13.2. ADIABATIC CONNECTION FORMULA 105 106 CHAPTER 13. ADIABATIC CONNECTION
@ @
13.2 Adiabatic connection formula where we have used V ee λ = λ%Ψ [n] @ V ee@ Ψ [n]& and U = λU, and introduced the simple
λ
λ
λ
@
@
@ˆ @
λ
notation U XC (λ) = U /λ for later convenience. This is the celebrated adiabatic connection
Now we apply the same thinking to DFT. When doing so, we always try to keep the density XC
formula. We have written the exchange-correlation energy as a solely potential contribution,
fixed (or altered only in some simple way), and not include the external potential. The
adiabatic connection formula is a method for continuously connecting the Kohn-Sham system but at the price of having to evaluate it at all intermediate coupling constants. We will see
shortly that this price actually provides us with one of our most important tools for analyzing
with the physical system. Very simply, we introduce a coupling constant λ into the universal
functional, to multiply V ee : density functionals.
A cartoon of the integrand in the adiabatic connection formula for a given physical system
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λ @ ˆ ˆ @
@
F [n] = min%Ψ @ T + λV ee @ Ψ& (13.13) is sketched in Fig. 13.1. The curve itself is given by the solid line, and horizontal lines have
@
Ψ→n
been drawn at the value at λ = 0 and at λ = 1. We can understand this figure by making
For λ = 1, we have the physical system. For λ = 0, we have the Kohn-Sham system. We 0
can even consider λ → ∞, which is a highly correlated system, in which the kinetic energy is
negligible. But for all values of λ, the density remains that of the physical system. Note that E
λ
this implies that the external potential is a function of λ: v λ ext (r). We denote Ψ [n] as the U XC (λ) X
minimizing wavefunction for a given λ. We can generalize all our previous definitions, but we
must do so carefully. By a superscript λ, we mean the expecation value of an operator on
the system with coupling-constant λ. Thus E X
E C
λ
ˆ
λ
λ
T [n] = %Ψ [n]|T|Ψ [n]&, (13.14) U XC −T C
but 0 0.2 0.4 0.6 0.8 1
λ
ˆ
λ
λ
V [n] = %Ψ [n]|λV ee |Ψ [n]&. (13.15) λ
ee
The Kohn-Sham quantities are independent of λ, so Figure 13.1: Cartoon of the adiabatic connection integrand.
@ @ @ @
λ λ @ ˆ @ λ @ ˆ @
@
@
E [n] = %Ψ [n] @ T + λV ee@ Ψ [n]& − %Φ[n] @ T + λV ee@ Φ[n]& (13.16) several key observations:
ˆ @
ˆ @
XC
Exercise 60 Coupling constant dependence of exchange • The value at λ = 0 is just E X .
Show that • The value at λ = 1 is just E X + U C .
λ
λ
U [n] = λU[n], E [n] = λE X [n], (13.17)
X
• The area between the curve and the x-axis is just E XC .
i.e., both Hartree and exchange energies have a linear dependence on the coupling-constant.
• The area beneath the curve, between the curve and a horizontal line drawn through the
Our next step is to write the Hellmann-Feynman theorem for this λ-dependence in the
Hamiltonian. We write value at λ = 1, is just T C .
1 λ @ @ λ Thus the adiabatic connection curve gives us a geometrical interpretation of many of the
1 @ @
F[n] = T S [n] + dλ %Ψ [n] @ V ee@ Ψ [n]& (13.18)
@ˆ @
0 energies in density functional theory.
λ
where F[n] = F λ=1 [n], T S [n] = F λ=0 [n]. The derivative w.r.t. λ of F [n] is just the To demonstrate the usefulness of adiabatic decomposition, we show this curve for the He
λ
derivative of the operator w.r.t. λ, because Ψ [n] is a minimizing wavefunction at each λ, atom in Fig. 13.2. Note first the solid, exact line. It is almost straight on this scale. This
just as in the one-electron case. Inserting the definition of the correlation energy, we find is telling us that this system is weakly correlated, as we discuss below. The numbers, from
Table X, are E X = −1.025, E C = −0.042, U C = −0.079, and T C = 0.037. Thus E C is only
1 1 @ @ @ @
λ
λ
E XC [n] = dλ %Ψ [n] @ V ee@ Ψ [n]& − U[n] slightly more negative than −T C .
@ˆ @
0
1 1 dλ λ 1 1 This produces yet another way to understand the cancellation of errors. We see that LDA
= U [n] = dλU XC [n](λ), (13.19)
0 λ XC 0 improves as λ grows from 0 to 1. This is characteristic of a very general trend. But once we