3.1.1. Time-Dependent Density Functional Theory—Fundamentals

The standard approach of DFT is based on the existence of a one-to-one mapping between the exact ground-state density and the external potential applied to a quantum system [48]. Furthermore, one can extend this mapping between two systems, the many-body interacting "real" system and a many-body non-interacting "Kohn-Sham" system [39] where we introduce an effective external potential (dubbed KS potential). This second mapping allows for calculating quantities, i.e., the ground-state density and energy, belonging to the real system by using the KS system, bringing about a large numerical simplification. DFT is therefore the actual method of choice for calculating electronic structure of materials as well as the energies of complex atoms and molecules [42]. The KS mapping was later extended to the dynamics of the single-particle density by Runge and Gross [55]. The existence of this second mapping lays the foundations for performing, e.g., numerical spectroscopy with so-called ab-initio methods, i.e., without—in principle—any fitting parameter [41]. TDDFT can theoretically be used for studying electrical transport (see for example Ref. [56–60]), but the information it can provide is only partial [61,62].

It appears natural therefore, from the Runge and Gross' theorem (RG), to establish a mapping between the single-particle current density and the external (vector) potential applied to the real system [63]. In principle, this Time-Dependent Current-Density Functional Theory (TDCDFT) should be the workhorse for charge transport calculations, but the lack of suitable approximations for the KS's potential renders the theory of little use at the moment. One the other hand, addressing thermal transport phenomena within DFT would require instead a non-equilibrium theory based on either a local temperature or a local energy density. Recently, a functional theory based on the excess energy density as the basic variable has been presented [64,65] which is suited to study thermoelectrical phenomena in the static and time-dependent case. In the following, we will look only at standard TDDFT.

The RG theorem proves that for a general time-dependent Hamiltonian *H* that describes the dynamics of a many-body system,

$$H(t) = T + V\_{\text{int}} + V\_{\text{ext}}(t) \tag{35}$$

there exists a one-to-one mapping between the single-particle density *<sup>n</sup>*(**<sup>r</sup>**, *t*) and the external potential *<sup>V</sup>*ext(**<sup>r</sup>**, *t*), given the initial conditions. Here, *T* is the kinetic energy, *V*int the particle-particle interaction, and *V*ext the time-dependent external potential. The proof of the theorem assumes that both density and potential can be expanded in series around the initial time: a detailed proof and the physical assumption on which this is based can be found in [40,41]. It has since been shown that there exists a system of non-interacting particles whose single-particle density is identical at each time to *<sup>n</sup>*(**<sup>r</sup>**, *t*) of the interacting system [66]. The existence of this mapping then allows investigation of the dynamics of a system by looking at its non-interacting Doppelgänger. Clearly, this brought about the same reduction of computational requirements as the standard DFT. However, as was the case with DFT, we are exchanging some of the physical information about the system to reduce the dimensionality of the problem. It should then come as no-surprise that TDDFT cannot for example reproduce the exact single-particle current density **j**(**<sup>r</sup>**, *t*) since some of its contributions are not derivable from the knowledge of the density alone [67]. Indeed, if starting from the continuity equation,

$$
\partial\_l n(\mathbf{r}, t) = -\nabla \cdot \mathbf{j}(\mathbf{r}, t), \tag{36}
$$

we need to conclude that given the density we can determine only the longitudinal part of **j**(**<sup>r</sup>**, *t*) and any term **j***T* such that **j**(**<sup>r</sup>**, *t*) = **j***L*(**<sup>r</sup>**, *t*) + **j***T*(**<sup>r</sup>**, *t*) and ∇ · **j***T* ≡ 0 cannot be obtained from Equation (36). However, some information about the total current can still be obtained from the continuity equation [68].

As with the static DFT, we study the KS Hamiltonian *H*KS,

$$H\_{\rm KS} = T\_{\rm S} + V\_{\rm KS} \tag{37}$$

where *TS* = ∑*i p*2 *i* /2*m* is the kinetic energy of the non-interacting particles of mass *m* and momenta {**<sup>p</sup>***i*}, and *VKS* = *V*ext + *V*Hxc. The potential *V*Hxc is the sum of the Hartree and exchange–correlation (xc) potentials that replace the particle–particle interaction. By the RG theorem *V*Hxc is a functional of the density only, *V*Hxc = *<sup>V</sup>*Hxc[*n*](**<sup>r</sup>**, *t*), and the dynamics of the single-particle density of the real system evolving with Hamiltonian *H* is identical to the dynamics of the single-particle density evolving with *H*KS. Notice that in principle, given the particle–particle interaction of the original many-body problem, *V*Hxc is a universal functional and does not depend on the external potential *V*ext. This implies for example that if we are interested in the dynamics of an electronic system, *V*Hxc is the same either if we are studying an atom, a molecule, a slab, or a bulk, i.e., this potential is transferrable to whatever system we want to study. This high transferability makes finding reliable approximation for *V*Hxc difficult. Indeed, the universal *V*Hxc for the electrons contains the physical information needed to go from the standard weakly interacting Landau's Fermi liquid, to the strongly correlated Wigner's crystal or superconducting phases. Notice that for the electrons, a large part of *V*Hxc is given by the Hartree's (mean-field) interaction, and only a relatively small contribution of the overall energy determines all these interesting phases of matter. Therefore, standard approaches to build some approximation to *V*Hxc are based on interpolating the numerical solution of the many-body problem with some known high- and low-density limits [69,70]. Many of these approximations are static. To apply them to the dynamics, a common method is the so-called adiabatic local density approximation (ALDA). In this approximation, we take a static *<sup>V</sup>*Hxc[*n*] and replace the static density with the instantaneous density *<sup>n</sup>*(**<sup>r</sup>**, *t*). Indeed, this neglects the history of the system—while we generally expect that the xc potential to be history dependent. For example, one can show that in this approximation there is not relaxation induced by particle–particle interaction, in contrast with observation [71,72].
