Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory

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 Schrödinger 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.

Exact functionals for correlated electron–photon systems

“…So in practice we have to use approximations. The standard way to devise such approximations is the use of a non-interacting auxiliary system, a so-called Kohn-Sham system [30]. In the Kohn-Sham scheme the difference in forces between the non-interacting and interacting system is subsumed in a mean-field term and the unknown xc potential.…”

“…Especially when going beyond the dipole approximation, the minimal-coupling prescription forces us to use the full current density to describe the coupling to the photon field. In this context a current-density functional (CDFT) scheme becomes unavoidable [14,30]. It seems possible by studying coupled matter-photon systems beyond the dipole approximation that we get novel insight also into CDFT.…”

A theoretical investigation of time-dependent Kohn–Sham equations: new proofs

“…Some maximal regularity results are proved in [7], where the unique solution has the same space regularity of the initial condition. An extreme case is also shown in [22], where the authors prove that, for a simple Schrödinger equation defined on a bounded one-dimensional space domain, an initial condition in L 2 (outside the domain of the Hamiltonian) leads to a solution (so-called 'mild solution') that is continuous but nowhere differentiable in time and continuous but nowhere differentiable in space for almost all times. For these reasons, in this section we prove existence and regularity results in case of a less regular initial condition Ψ 0 ∈ H 1 0 (Ω; C).…”

“…This theory has been extended to time-dependent problems by E. Runge and E. K. U. Gross in 1984 [23]; see also [27,20,22]. These theories allow one to describe the state of a multi-particle physical system, represented by the solution of the multi-particle Schrödinger equation, by a density function corresponding to a system of nonlinear integral one-particle Schrödinger equations.…”

Light-matter interactions within the Ehrenfest–Maxwell–Pauli–Kohn–Sham framework: fundamentals, implementation, and nano-optical applications

“…Here α ∈ N 0 and we assume from now on that all external fields and expectation values are real analytic in time, i.e., that their Taylor series in time has a finite convergence radius. In general one can get away with fewer conditions [79] but for the sake of simplicity we stick with these rather stringent ones. Having assumed that all time-derivatives at zero exist (which implies a rather well-behaved initial state) we can, based on the Eq.…”