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), Article 203202. https://doi.org/10.1088/0953-8984/27/20/203202
Published inJournal of Physics: Condensed Matter
© 2015 IOP Publishing Ltd. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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. ...
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Except where otherwise noted, this item's license is described as © 2015 IOP Publishing Ltd. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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