Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory
Ruggenthaler, M., Penz, M., & van Leeuwen, R. (2015). Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory. Journal of Physics: Condensed Matter, 27 (20), 203202. doi:10.1088/0953-8984/27/20/203202
Published inJournal of Physics: Condensed Matter
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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 Schrodinger 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. ...