Abstract:
In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schr\"odinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.

Abstract:
It is shown that the density-potential mapping and the ${\cal V}$-representability problems in the time-dependent current density functional theory (TDCDFT) are reduced to the solution of a certain many-body nonlinear Schr\"odinger equation (NLSE). The derived NLSE for TDCDFT adds a link which bridges the earlier NLSE-based formulations of the time-dependent deformation functional theory (TDDefFT) and the time-dependent density functional theory (TDDFT). We establish close relations between the nonlinear many-body problems which control the existence of TDCDFT, TDDFT, and TDDefFT, and thus develop a unified point of view on the whole family of the TDDFT-like theories.

Abstract:
We clarify some misunderstandings on the time-dependent current density functional theory for open quantum systems we have recently introduced [M. Di Ventra and R. D'Agosta, Phys. Rev. Lett. {\bf 98}, 226403 (2007)]. We also show that some of the recent formulations attempting to improve on this theory suffer from some inconsistencies, especially in establishing the mapping between the external potential and the quantities of interest. We offer a general argument about this mapping, showing that it must fulfill certain "dimensionality" requirements.

Abstract:
The generalized constrained search formalism is used to address the issues concerning density-to-potential mapping for excited states in time-independent density-functional theory. The multiplicity of potentials for any given density and the uniqueness in density-to-potential mapping are explained within the framework of unified constrained search formalism for excited-states due to G\"orling, Levy-Nagy, Samal-Harbola and Ayers-Levy. The extensions of Samal-Harbola criteria and it's link to the generalized constrained search formalism are revealed in the context of existence and unique construction of the density-to-potential mapping. The close connections between the proposed criteria and the generalized adiabatic connection are further elaborated so as to keep the desired mapping intact at the strictly correlated regime. Exemplification of the unified constrained search formalism is done through model systems in order to demonstrate that the seemingly contradictory results reported so far are neither the true confirmation of lack of Hohenberg-Kohn theorem nor valid representation of violation of Gunnarsson-Lundqvist theorem for excited states. Hence the misleading interpretation of subtle differences between the ground and excited state density functional formalism are exemplified.

Abstract:
The key element in time-dependent density functional theory is the one-to-one correspondence between the one-particle density and the external potential. In most approaches this mapping is transformed into a certain type of Sturm-Liouville problem. Here we give conditions for existence and uniqueness of solutions and construct the weighted Sobolev space they lie in. As a result the class of v-representable densities is considerably widened with respect to previous work.

Abstract:
We propose a practical approximation to the exchange-correlation functional of (time-dependent) density functional theory for many-electron systems coupled to photons. The (time non-local) optimized effective potential (OEP) equation for the electron- photon system is derived. We test the new approximation in the Rabi model from weak to strong coupling regimes. It is shown that the OEP (i) improves the classical description, (ii) reproduces the quantitative behavior of the exact ground-state properties and (iii) accurately captures the dynamics entering the ultra-strong coupling regime. The present formalism opens the path to a first-principles description of correlated electron-photon systems, bridging the gap between electronic structure methods and quantum optics for real material applications.

Abstract:
We show that the time-dependent particle density $n(\vec r,t)$ and the current density ${\vec j}(\vec r,t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\vec r,t)$ and $\vec A(\vec r,t)$ and is initially in the quantum state $|\psi (0)>$, can always be reproduced (under mild assumptions) in another many-particle system, with different two-particle interaction, subjected to external potentials $V'(\vec r,t)$ and $\vec A'(\vec r,t)$, starting from an initial state $|\psi' (0)>$, which yields the same density and current as $|\psi (0)>$. Given the initial state of this other many-particle system, the potentials $V'(\vec r,t)$ and $\vec A'(\vec r,t)$ are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a new and simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.

Abstract:
Time-dependent (TD) density functional theory (TDDFT) promises a numerically tractable account of many-body electron dynamics provided good simple approximations are developed for the exchange-correlation (XC) potential functional (XCPF). The theory is usually applied within the adiabatic XCPF approximation, appropriate for slowly varying TD driving fields. As the frequency and strength of these fields grows, it is widely held that memory effects kick in and the eligibility of the adiabatic XCPF approximation deteriorates irreversibly. We point out however that when a finite system of electrons in its ground-state is gradually exposed to a very a high-frequency and eventually ultra-strong homogeneous electric field, the adiabatic XCPF approximation is in fact rigorously applicable. This result not only helps to explain recent numerical results for a 1D-helium atom subject to a strong linearly-polarized laser pulse (Thiel et al, Phys. Rev. Lett. 100, 153004, (2008)) but also shows that it is applicable to any number of electrons and in full 3D.

Abstract:
In a recent Comment (arXiv:0710.0018), Maitra, Burke, and van Leeuwen (MBL) attempt to refute our criticism of the foundations of TDDFT (see Phys. Rev. A 75, 022513 (2007)). However, their arguments miss the essence of our position. This is mainly due to an ambiguity concerning the meaning of the so-called mapping derivation of time-dependent Kohn-Sham equations. We distinguish two different conceptions, referred to as potential-functional based fixed-point iteration (PF-FPI) and direct Kohn-Sham potential (DKSP) scheme, respectively. We argue that the DKSP scheme, apparently adopted by MBL, is not a density-functional method at all. The PF-FPI concept, on the other hand, while legitimately predicated on the Runge-Gross mapping theorem, is invalid because the convergence of the fixed-point iteration is not assured.

Abstract:
The logical structure and the basic theorems of time-dependent current density functional theory (TDCDFT) are analyzed and reconsidered from the point of view of recently proposed time-dependent deformation functional theory (TDDefFT). It is shown that the formalism of TDDefFT allows to avoid a traditional external potential-to-density/current mapping. Instead the theory is formulated in a form similar to the constrained search procedure in the ground state DFT. Within this formulation of TDCDFT all basic functionals appear from the solution of a constrained universal many-body problem in a comoving reference frame, which is equivalent to finding a conditional extremum of a certain universal action functional. As a result the physical origin of the universal functionals entering the theory, as well as their proper causal structure becomes obvious. In particular, this leaves no room for any doubt concerning predictive power of the theory.