*3.1. Time-Dependent Density Functional Theory*

Density Functional Theory proved an important point of quantum mechanics, i.e., that some exact properties of a many-body interacting system can be extracted from the study of a Doppelgänger of non-interacting particles [40–42,47]. The simplification brought about cannot be underestimated: it is enough to think that an interacting system requires an exponentially large Hilbert's space for its accurate numerical description, while the same system of non-interacting particles requires only a polynomially large (with the number of states) space. This difference separates being able to treat just a few particles from studying complex molecules or crystal unit cells. There are two prices to

pay for this simplification; On the one hand, we are ready to give up the complete information about the many-body system and study only certain quantities. The standard DFT formalism was aimed at the ground-state energy and density. Any other quantity that is then obtained in DFT must then be checked against other models; On the other hand, the particle–particle interaction is *replaced* in the so-called Kohn-Sham (KS) system with a non-linear external potential (known as KS potential) that is assumed to depend only on the single-particle ground-state density [39,47,48]. This potential is clearly unknown and we need to revert to some sort of approximation to make the theory useful. Standard approximations have been developed and admittedly work rather well especially when dealing with atoms and small molecules, but failure is also around the corner. Indeed, some of the standard problems of DFT in dealing with transport calculation derives from a combination of these points: band-gap and transport coefficients are normally not static ground-state properties, and the standard approximations for the KS potential consistently underestimate the electronic band-gap. Finally, there are classes of problems that are outside the realm of standard DFT, such as for example the calculation of spectroscopic quantities. A detailed introduction to DFT would bring us far away from the scope of this review. We can however, recommend excellent introductory material for the interested reader [40–42,47].

There are many ways to go beyond these issues. We can formulate the theory in the time domain, in such a way that we can extract the exact time dynamics of some of the quantities of the many-body interacting system [49–51]. This is the case of Time-Dependent Density Functional Theory. Meanwhile, we can improve the standard approximations of the KS potential, for example introducing corrections that take into account some part of the strong correlation between particles in certain regimes [52].

In the next sections we will introduce the Time-Dependent Density Functional Theory (TDDFT) [40,41] and its further extensions and later the so-called "i-DFT" that can be used to introduce strong correlation effects, such as Coulomb blockade into a TDDFT formulation (see Section 3.2) [53,54].
