We introduce a new class of exchange-correlation potentials for a static and time-dependent Density Functional Theory of strongly correlated systems in 3D. The potentials are obtained via Dynamical Mean Field Theory and, for strong enough interactions, exhibit a discontinuity at half filling density, a signature of the Mott transition. For time-dependent perturbations, the dynamics is described in the adiabatic local density approximation. Results from the new scheme compare very favorably to exact ones in clusters. As an application, we study Bloch oscillations in the 3D Hubbard model. 71.27.+a, 31.70.Hq, 71.10. Fd Time-dependent quantum phenomena hold an important place in today's condensed matter research. A major theoretical challenge in this field is to describe strongly correlated systems out of equilibrium.In the last decade, Time-Dependent Density Functional Theory (TDDFT) has gained favor as a computationally viable, in principle exact time-dependent description of materials [1,2]. The basic TDDFT variable is the one-particle density n and a key ingredient is the time-dependent exchange-correlation potential v xc , embodying the complexities of the many-body problem. TDDFT applied to strongly correlated systems is in its beginnings. Describing these systems in equilibrium with static density functional theory (DFT) [3] is already a difficult task [4]. TDDFT retains these difficulties, but also adds another hurdle: Since time enters explicitly the formulation, v xc depends on the history of n (memory effects) [1,2].In equilibrium, an effective ab-initio method to describe strong correlations is the LDA+DMFT [5,6], combining DFT in the local density approximation (LDA) with Dynamical Mean Field Theory (DMFT) [7]. DMFT, which treats correlations nonperturbatively via a local self-energy Σ [8], is also at the core of the DMFT+GW [9, 10], another ab-initio method, which deals with nonlocal correlations within the GW approximation [11]. These DMFT-based methods rely on Green's function formulations, and the practical feasibility (in a foreseeable future) of a nonequilibrium generalization is not easy to assess, since Green's-function propagation scales quadratically [12][13][14] with the simulation time.TDDFT dynamics involves only one time variable. It would thus be useful to have exchange-correlation potentials suitable for strongly correlated systems. In equilibrium, they could offer a better start for Green's function based ab-initio schemes. Out of equilibrium, they could be used for adiabatic LDA [15] dynamics via TDDFT and possibly be improved by including memory effects, absent in the adiabatic LDA.In this Letter we suggest a novel avenue to deal with strongly correlated systems in 3D and out of equilibrium, by combining DMFT with TDDFT. For model strongly correlated systems in 1D, exchange-correlation potentials for DFT were introduced [16,17], and a BetheAnsatz-based LDA (BALDA) for v xc was proposed. Such v BALDA xcwas then used to introduce an adiabatic scheme for the TDDFT of the 1D Hubbard m...

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