Time-dependent density-functional theory (TDDFT) is one of the many methods that can be used to understand the excited electronic structure of atoms, molecules, or materials. It can be viewed as an extension of density functional theory (DFT) to time-dependent problems, and as an alternative formulation of time-dependent quantum mechanics. The basic rationale behind DFT is the reformulation of the many-electron problem (roughly speaking, an equation with 3N variables, where N is the number of electrons) as a problem whose basic variable is the one-particle density, an object depending on the three spatial variables. Alternatively, we can view DFT as a manner to tackle with the interacting many-electron problem by studying a much easier non-interacting one (since in almost all cases it is the Kohn-Sham formulation of DFT that is used). TDDFT is based on the same reduction of complexity. The advantages are clear: a complex function in 3N-dimensional space is replaced by a real function that depends solely on 3-dimensional vector - the density. Usually this is obtained using an auxiliary system of non-interacting electrons that feel an effective time-dependent potential, the time-dependent Kohn-Sham potential. Its exact form is, however, unknown, and has to be approximated.

TDDFT is by no means the only approach to the excitations of many-electron systems. In fact, more accurate (yet more expensive) techniques (based on many-body perturbation theory, for example) exist, and therefore these alternatives will also be covered at the workshop, in particular their relation and comparison to TDDFT. However, TDDFT achieves a good balance between accuracy and computational cost. In consequence, its use is increasing, and it is fast becoming one of the tools of choice to get accurate and reliable predictions for excited-state properties in solid state physics, chemistry and biophysics, both in the linear and non-linear regimes. Around 300 publications every year contain results computed with TDDFT, and this number is increasing. This interest has been motivated by the recent developments of TDDFT (and time-dependent current functional theory) and include the description of photo-absorption cross section of molecules and nanostructures, electron-ion dynamics in the excited state triggered by either a small or high intense laser fields, van der Waals interactions, development of new functionals coping with memory and non-locality effects, applications to biological systems (chromophores), transport phenomena, optical spectra of solids and low- dimensional structures (as nanotubes, polymers, surfaces...). All these possibilities add up to the scientific success of TDDFT.

TDDFT is still a young discipline. For example, pretty much as it happened with ground state DFT, it has taken a few decades since the “official” birth of the theory until we are witnessing fully rigorous mathematical studies of its foundations. Likewise, the youth of the discipline is demonstrated by the fact that it is only recently that somehow “obvious” applications of it, such as real-time simulations of high-intensity laser irradiation of solids, have appeared. Technical and theoretical difficulties slowed the progress of these studies for years.

o Use of TDDFT to study the fast evolving experiments in the area of time-resolved spectroscopies.

o Spin dynamics in molecules and materials: Models based on TDDFT, as compared to other alternatives.

o Non-adiabatic molecular dynamics, coupled to TDDFT: various alternatives.

o The long quest for better approximations to the exchange-and-correlation functionals, both for ground-state and excited-states problems. The problem of non-adiabaticity. Solutions to the charge-transfer excitation problems.

o Modeling materials useful for photo-voltaics applications: what is the current state-of-the-art from an industrial view-point? How can the theory help?

o Coupling of TDDFT to the electromagnetic field: classical and quantum photons, coupled and uncoupled to the electronic currents.

o Relation of TDDFT with more advanced many-body schemes, such as Bethe-Salpeter, etc.