Page 94 - 71 the abc of dft_opt

188 CHAPTER 25. EXOTICA
the ground-state problem, what we really care about is the energy, which will determine the
geometry of our molecules and solids.
Nevertheless, the action is the mathematically analogous quantity, defined as
1 t f ∂
ˆ
Chapter 25 A[Ψ] = 0 dt %Ψ(t)|H − i ∂t |Ψ(t)& (25.1)
The principle of least action says that
Exotica ∂A
= 0 (25.2)
∂Ψ
is satisfied by the wavefunction obeying the time-dependent Schrodinger equation. It would
25.1 Currents be very nice to define an exact A[n] whose variations with respect to the density yielded the
time-dependent Schroedinger equation.
25.2 Initial-state dependence The original Runge-Gross paper defined this action simply as
Not much attention is paid in the literature to the initial-state dependence in the Runge-Gross A[n] = A[Ψ[n]] (25.3)
theorem, and its often not mentioned explicitly. But TDDFT would be useless if we always However, this has since been shown to lead to several inconsistencies. Essentially, variations
had to account for it, since we’d have a different functional for every initial wavefunction. in the density restrict variations in the wavefunction to only those that are v-representable,
In practice, this is not really a problem. Often, one restricts oneself to starting in a i.e., form
non-degenerate ground state. In that case, by virtue of the ground-state Hohenberg-Kohn
theorem, the initial wavefunction is an (implicit) density functional, and so the entire potential
25.4 Solids
is a density functional.
In fact, one can be even more general, and include any initial wavefunction that can be 25.5 Back to the ground state
generated by some time-evolution of the system from a non-degenerate ground-state. Suppose
one can find such a pre-history, using an external potential v(x), with t running from −t P to The last class of application of TDDFT is, perhaps surprisingly, to the ground-state problem.
0. In that case, simply tack on this pre-history, and apply TDDFT with the initial ground- This is because one can extract the ground-state exchange-correlation energy from a response
state functional, begining from −t P . The original initial potential (at t = 0) differs from function, in the same fashion as perturbation theory yields expressions for ground-state con-
what it would be if we began in the ground-state, because of the memory-dependence of the tributions in terms of sums over excited states. To do this, we simply note that the equal-time
functional. susceptibility, χ(rr , t − t = 0), yields the density-density correlation function, i.e., the pair
#
#
Can all initial-states be generated by a pseduo-prehistory begining in a ground-state? This density of chapter X. Thus
seems unlikely, since one only has the freedom to vary v ext (rt), but the wavefunction is a λ λ (3)
#
#
#
#
function of many variables. But presumably most of physical interest are of this kind. n (r, r ) = −χ (rr , t → t )/n(r) − δ (r − r ), (25.4)
XC
We will not discuss further the initial-state dependence, and assume from here on that the where we have now generalized Eq. (23.4) to arbitrary λ by using a kernel λ/|r 1 − r 2 | +
initial state is a non-degenerate ground state. f (r 1 r 2 ω). Inserting this expression for the hole into the adiabatic connection formula for
λ
XC
the energy, we find:
3
1 1 1 ∞ dω
25.3 Lights, camera, and...Action E XC = d r d r .... (25.5)
3 #
0 π
λ
Before discussing the action in time-dependent DFT, I first point out that, for TDDFT, there giving the exchange-correlation energy directly in terms of χ . Any approximation for f XC ,
is no analog of the ground-state energy. That is, there is no one functional in which we have whose λ-dependence can be simply extracted via scaling, yields an approximation to E XC .
an overarching interest. In a time-dependent problem, we wish to know the time-dependent Most importantly, Eq. (25.5) produces a natural method for incorporating time-dependent
expectation value of the density, which is determined by the Kohn-Sham potential. But in fluctuations in the exchange-correlation energy. In particular, as a system is pulled apart
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