Page 88 - 71 the abc of dft_opt

22.4. ADIABATIC APPROXIMATION 175 176 CHAPTER 22. TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
where the Hartree potential has the usual form, but for a time-dependent density. The In practice, the spatial locality of the functional is also approximated. As mentioned above,
exchange-correlation potential is then a functional of the entire history of the density, n(x), ALDA is the mother of all TDDFT approximations, but any ground-state functional, such
the initial interacting wavefunction Ψ(0), and the initial Kohn-Sham wavefunction, Φ(0). as a GGA or hybrid, automatically yields an adiabatic approximation for use in TDDFT
This functional is a very complex one, much more so than the ground-state case. Knowledge calculations. A simple way to go beyond the adiabatic approximation would be to include
of it implies solution of all time-dependent Coulomb interacting problems. some dependence on, say, ˙n(rt).
Bureaucratically, ALDA should only work for systems with very small density gradients in
Exercise 91 Kohn-Sham potential for one electron
space and time. It may seem like a drastic approximation to neglect all non-locality in time.
By inverting the time-dependent Kohn-Sham equation, give an explicit expression for the However, we’ve already seen how well LDA works beyond its obvious range of validity for the
time-dependent Kohn-Sham potential in terms of the density for one electron.
ground-state problem, due to satisfaction of sum rules, etc., and how its success can be built
In ground-state DFT, the exchange-correlation potential is the functional derivative of upon with GGA’s and hybrids. We’ll soon see if some of the same magic applies to TDDFT.
E XC [n]. It would be nice to find a functional of n(rt) for which v XC (rt) was the functional Exercise 92 Continuity in ALDA
derivative, called the exchange-correlation action. However, as shown in Sec. X, this turns
Does a Kohn-Sham ALDA calculation satisfy continuity? Prove your answer.
out not to be possible.
22.5 Questions on general principles of TDDFT
22.4 Adiabatic approximation
1. If a system is in a linear combination of two eigenstates, does its density change?
As we have noted, the exact exchange-correlation potential depends on the entire history of
the density, as well as the initial wavefunctions of both the interacting and the Kohn-Sham 2. In a time-dependent Kohn-Sham calculation starting from the ground state, will electronic
transitions occur at frequencies that are differences between ground-state Kohn-Sham
systems:
orbital energies?
v XC [n; Ψ(0), Φ(0)](rt) = v S [n; Φ(0)](rt) − v ext [n; Ψ(0)](rt) − v H [n](rt). (22.11)
3. Is there anything special about the response of a particle in a harmonic well to an external
However, in the special case of starting from a non-degenerate ground state (both interacting electic field?
and non-), the initial wavefunctions themselves are functionals of the initial density, and so
4. Does the Runge-Gross theorem guarantee that the current density in a TDDFT calcula-
the initial-state dependence disappears. But the exchange-correlation potential at r and t tion is correct?
has a functional dependence not just on n(r t) but on all n(r t ) for 0 ≤ t ≤ t. We call this
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a history-dependence. 5. Can more than one potential produce the same time-dependent density?
The adiabatic approximation is one in which we ignore all dependence on the past, and 6. If a system is subjected to a driving potential that, after a time, returns its density to its
allow only a dependence on the instantaneous density: initial value, will the potential return to its initial value?
v adia [n](rt) = v approx [n(t)](r), (22.12) 7. Repeat question above, but replace density with wavefunction.
XC XC
i.e., it approximates the functional as being local in time. If the time-dependent potential 8. Repeat both questions above, within the adiabatic approximation. What happens if your
changes very slowly (adiabatically), this approximation will be valid. But the electrons will functional has some memory?
remain always in their instantaneous ground state. To make the adiabatic approximation
9. Consider a He atom sitting in its ground-state. Imagine you have a time-dependent
exact for the only systems for which it can be exact, we require
potential that pushes it into its first excited state (1s2s). Now imagine the exact Kohn-
gs
v adia [n](rt) = v [n 0 ](r)| n 0 (r)=n(rt) (22.13) Sham calculation. Describe the final situation.
XC XC
gs
where v [n 0 ](r) is the exact ground-state exchange-correlation potential of the density n 0 (r).
XC
This is the precise analog of the argument made to determine the function used in LDA for
the ground-state energy (see Sec. X).