Density-potential mappings in quantum dynamics

Abstract

In a recent letter [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed point problem. This idea was used to generalize the existence and uniqueness theorems underlying timedependent density functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case as it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that in the most physically relevant cases the fixed point procedure converges. This is further demonstrated with an example.

Density gradient expansion of correlation functions

“…The situation of time-dependent systems is more delicate. Although the existence of a density-potential mapping has been wellestablished [43][44][45][46] and hence a density functional theory can be formulated, the time variable induces severe complications. As was shown by Vignale and Kohn,47,48 the temporal and spatial nonlocality of time-dependent density functionals are intimately correlated.…”

Nonexistence of a Taylor expansion in time due to cusps

“…The motivation to develop the s-expansion described here was to convincingly show that such effects are generic, rather than unusual, once the ground-state wavefunction contains spatial cusps. In the last few years, much work toward a proof of existence of the KS potential without a Taylor-expansion has been performed [10,11].…”

“…This could only be done convincingly by creating a methodology for explicitly extracting the short-time asymptotic behavior in such cases, and demonstrating the non-expandability of the density. This has reopened the question of the existence of a KS potential in the common case of Coulomb attraction to the nuclei, and recent work has focused on avoiding the Taylor expansion in time [10,11].…”

Time-dependent bond-current functional theory for lattice Hamiltonians: Fundamental theorem and application to electron transport

“…Moreover, for a given initial state the potential v ′ (r, t) is unique up to a purely time-dependent function. Recently, Ruggenthaler and van Leeuwen extended the validity of these results by relaxing the condition of Taylor-expandability in time [5]. The van Leeuwen theorem reduces to the Runge-Gross theorem for w = w ′ and |Ψ(0) = |Ψ ′ (0) , and establishes the existence of a non-interacting KS system for w ′ = 0 and |Ψ ′ (0) = |Φ KS a Slater determinant.…”