Page 87 - 71 the abc of dft_opt

22.2. RUNGE-GROSS THEOREM 173 174 CHAPTER 22. TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
state contributions in terms of sums over excited states. Thus any approximation for the This is the first part of the Runge-Gross theorem, which establishes a one-to-one correspon-
exchange-correlation kernel yields an approximation to E XC . Such calculations are signifi- dence between current densities and external potentials.
cantly more demanding than regular ground-state DFT calculations. Most importantly, this In the second part, we extend the proof to the densities. From continuity,
produces a natural method for incorporating time-dependent fluctuations in the exchange-
correlation energy. In particular, as a system is pulled apart into fragments, this approach ∆ ˙n(x) = −∇ · ∆j(x) (22.5)
includes correlated fluctuations on the two separated pieces. While in principle all this is so that
(
k
included in the exact ground-state functional, in practice TDDFT provides a natural method- ∂ k+2 ∆n(x) = ∇ · n 0 (r)∇∂ ∆v ext (x) ) . (22.6)
0
0
ology for modeling these fluctuations. Simple ALDA approximations for polarizabilities of
Now, if not for the divergence on the right-hand-side, we would be done, i.e., if f(r) =
atoms and molecules allow C 6 coefficients to be accurately calculated by this method, and ∂ ∆v ext (x) is non-zero for some k, then the density difference must be non-zero, since n 0 (r)
k
attempts exist to generate entire molecular energy curves, from the covalent bond distance 0
is non-zero everywhere.
to dissociation, including van der Waals. Such calculations are much more demanding than Does the divergence allow some escape from this conclusion? The answer is no, for any
simple self-consistent ground-state calculations, but may well be worth the payoff.
physical density. To see this, write
1 3 1 3 C 2 D
d r f(r)∇ · (n 0 (r)∇f(r)) = d r ∇ · (f(r)n 0 (r)∇f(r)) − n 0 (r)|∇f(r)| (22.7)
22.2 Runge-Gross theorem
The first term on the right may be written as a surface integral at r = ∞, and vanishes for
The analog of the Hohenberg-Kohn theorem for time-dependent problems is the Runge-Gross
any realistic potential (which fall off at least as fast as −1/r). If we imagine that ∇f(r)
theorem[?]. We consider N non-relativistic electrons, mutually interacting via the Coulomb
is non-zero somewhere, then the second term on the right is definitely negative, so that the
repulsion, in a time-dependent external potential. The Runge-Gross theorem states that the integral on the left cannot vanish, and its integrand must be non-zero somewhere. Thus
densities n(x) and n (x) evolving from a common initial state Ψ 0 = Ψ(t = 0) under the
#
there is no way for ∇f(r) to be non-zero, and yet have f∇(n 0 ∇f) vanish everywhere.
influence of two potentials v ext (x) and v (x) (both Taylor expandable about the initial time Since the density determines the potential up to a time-dependent constant, the wavefunc-
#
ext
0) are always different provided that the potentials differ by more than a purely time-dependent tion is in turn determined up to a time-dependent phase, which cancels out of the expectation
(r-independent) function:
value of any operator.
#
∆v ext (x) = v(x) − v (x) *= c(t) . (22.1)
22.3 Kohn-Sham equations
If so, there is a one-to-one mapping between densities and potentials, and one can construct
a density functional theory.
We will see below that, under certain quite general conditions, one can establish a one-to-
We prove this theorem by first showing that the corresponding current densities must differ. one correspondence between time-dependent densities n(rt) and time-dependent one-body
Because the Hamiltonians only differ in their external potentials, the equation of motion for potentials v ext (rt), for a given initial state. That is, a given history of the density can be
the difference of the two current densities is: generated by at most one history of the potential. This is the time-dependent analog of the
A B
˙ ˆ ˆ Hohenberg-Kohn theorem. Then one can define a fictious system of non-interacting electrons
∆j(x)| t=0 = −i%Ψ 0 | j(r), ∆H(t 0 ) |Ψ 0 & = −n 0 (r)∇∆v ext (r, 0), (22.2)
that satisfy time-dependent Kohn-Sham equations
where n 0 (r) = n(r, 0) is the initial density. One can go further, by repeatedly using the  2 
˙
equation of motion, and considering t = 0, to find[?] iφ j (rt) = − ∇ + v S [n](x) φ j (x) . (22.8)


2
k
∂ 0 k+1 ∆j(x) = −n 0 (r) ∇∂ ∆v ext (x) , (22.3) whose density is
0
N
k k k -
2
0
where ∂ = (∂ /∂t )| t=0 , i.e., the k-th derivative evaluated at the initial time. If Eq. (22.1) n(x) = |φ j (x)| , (22.9)
holds, and the potentials are Taylor expandable about t o , then there must be some finite k j=1
for which the right hand side of (22.2) does not vanish, so that precisely that of the real system. We define the exchange-correlation potential via:
∆j(x) *= 0. (22.4) v S (x) = v ext (x) + v H (x) + v XC (x) (22.10)