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172 CHAPTER 22. TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
system at 27 times the frequency of the perturbing electric field, or to cause a specific chemical
reaction to occur (quantum control). Previously, only one and two electron systems could
be handled computationally, as the full time-dependent wavefunction calculations are very
demanding. Crude and unreliable approximations had to be made to tackle larger systems.
Chapter 22 But with the advent of TDDFT, which has become very popular in this community, larger
systems with more electrons can now be tackled.

Time-dependent density functional theory When the perturbing field is weak, as in normal spectroscopic experiments, perturbation
theory applies. Then, instead of needing knowledge of v XC for densities that are chang-
ing significantly with time, we only need know this potential in the vicinity of the initial
state, which we take to be a non-degenerate ground-state. These changes are character-
So far, we have restricted our attention to the ground state of nonrelativistic electrons in time- ized by a new functional, the exchange-correlation kernel. While still a more complex beast
independent potentials. This is where most DFT research has been done, and has achieved its than the ground-state exchange-correlation potential, the exchange-correlation kernel is much
great successes, by providing a general tool for tackling electronic structure problems. In this more manageable than the full time-dependent exchange-correlation potential, because it is a
chapter, we extend our view, to include time-dependent potentials. In subsequent chapters, functional of the ground-state density alone. Analysis of the linear response then shows that
we apply the methods of DFT to such potentials. it is (usually) dominated by the response of the ground-state Kohn-Sham system, but cor-
rected by TDDFT via matrix elements of the exchange-correlation kernel. In the absence of
Hartree-exchange-correlation effects, the allowed transitions are exactly those of the ground-
22.1 Overview
state Kohn-Sham potential. But the presence of the kernel shifts the transition frequencies
away from the Kohn-Sham values to the true values. Moreover, the intensities of the optical
We will see below that, under certain quite general conditions, one can establish a one-to- transitions can be extracted in the same calculations, so these also are affected by the kernel.
one correspondence between time-dependent densities n(rt) and time-dependent one-body
potentials v ext (rt), for a given initial state. That is, a given evolution of the density can be Chemists and physicists have devised separate approaches to extracting excitations from
generated by at most one time-dependent potential. This is the time-dependent analog of TDDFT for atoms, molecules, and clusters. The chemists way is to very efficiently convert
the Hohenberg-Kohn theorem, and is called the Runge-Gross theorem. Then one can define a the search for poles of response functions into a large eigenvalue problem, in a space of
fictitious system of non-interacting electrons moving in a Kohn-Sham potential, whose density the single-excitations of the system. The eigenvalues yield transition frequencies while the
is precisely that of the real system. The exchange-correlation potential (defined in the usual eigenvectors yield oscillator strengths. This allows use of many existing fast algorithms to
way) is then a functional of the entire history of the density, n(x), the initial interacting extract the lowest few excitations. In this way, TDDFT has been programmed into most
wavefunction Ψ(0), and the initial Kohn-Sham wavefunction, Φ(0). This functional is a very standard quantum chemical packages and, after a molecule’s structure has been found, it is
complex one, much more so than the ground-state case. Knowledge of it implies solution of usually not too costly to extract its low-lying spectrum. Physicists, on the other hand, prefer
all time-dependent Coulomb-interacting problems. to simply solve the time-dependent Kohn-Sham equations when a weak field has been turned
An obvious and simple approximation is adiabatic LDA (ALDA), sometimes called time- on. Fourier-transform of the time-dependent dipole matrix element then yields the optical
dependent LDA, in which we use the ground-state potential of the uniform gas with that spectrum. Using either methodology, the number of these TDDFT response calculations
instantaneous and local density, i.e., v XC [n](x) ≈ v unif (n(x)). This gives us a working Kohn- for transition frequencies is growing exponentially at present. Overall, results tend to be
XC
Sham scheme, just as in the ground state. We can then apply this DFT technology to all fairly good (0.1 to 0.2 eV errors, typically), but little is understood about their reliability.
problems of time-dependent electrons. These applications fall into three general categories: Difficulties remain for the application of TDDFT to solids, because the present generation of
non-perturbative regimes, linear (and higher-order) response, and ground-state applications. approximate functionals (local and semi-local) lose important effects in the thermodynamic
limit, but much work is currently in progress.
The first of these involves atoms and molecules in intense laser fields, in which the field
is so intense that perturbation theory does not apply. In these situations, the perturbing The last class of application of TDDFT is, perhaps surprisingly, to the ground-state prob-
electric field is comparable to or much greater than the static electric field due to the nuclei. lem. This is because one can extract the ground-state exchange-correlation energy from a
Experimental aims would be to enhance, e.g., the 27th harmonic, i.e., the response of the response function, in the same fashion as perturbation theory yields expressions for ground-
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