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##### Time-Dependent Density Functional Theory

**Authors**: M.A.L. Marques and E.K.U. Gross

**Ref.**: in *A Primer in Density-Functional Theory*, C. Fiolhais, F. Nogueira, and M.A.L. Marques (ed.), Lecture Notes in Physics, Vol. 620, Springer, Berlin., 144-184 (2003)

**Abstract**: Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body SchrÃ¶dinger equation, its basic variable is the one-body electron density, n(**r**). The advantages are clear: The many-body wave-function, a function in a 3N-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector **r**. The standard way to obtain n(**r**) is with the help of a fictitious system of non-interacting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated.

The scheme is perfectly general, and can be applied to essentially any time-dependent situation. Two regimes can however be observed: If the time-dependent potential is weak, it is sufficient to resort to linear-response theory to study the system. In this way it is possible to calculate e.g. optical absorption spectra. It turns out that, even with the simplest approximation to the Kohn-Sham potential, spectra calculated within this framework are in very good agreement with experimental results. However, if the time-dependent potential is strong, a full solution of the Kohn-Sham equations is required. A canonical example of this regime is the treatment of atoms or molecules in strong laser fields. In this case, TDDFT is able to describe non-linear phenomena like high-harmonic generation, or multi-photon ionization.

Our purpose in this chapter is to provide a pedagogical introduction to TDDFT. With that in mind, we present, in Sect. 4.2, a quite detailed proof of the Runge-Gross theorem [5], i.e. the time-dependent generalization of the Hohenberg-Kohn theorem [6], and the corresponding Kohn-Sham construction [7]. These constitute the mathematical foundations of TDDFT. Several approximate exchange-correlation (xc) functionals are then reviewed. In Sect. 4.3 we are concerned with linear-response theory, and with its main ingredient, the xc kernel. The calculation of excitation energies is treated in the following section. After giving a brief overlook of the competing density-functional methods to calculate excitations, we present some results obtained from the full solution of the Kohn-Sham scheme, and from linear-response theory. Section 4.5 is devoted to the problem of atoms and molecules in strong laser fields. Both high-harmonic generation and ionization are discussed. Finally, the last section is reserved for some concluding remarks.

For simplicity, we will write all formulae for spin-saturated systems. Obviously, spin can be easily included in all expressions when necessary. Hartree atomic units will be used throughout this chapter.

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