Numerical construction of the density-potential mapping

We demonstrate how a recently developed method Nielsen et al. [Nielsen et al., EPL 101, 33001 (2013)] allows for a comprehensive investigation of time-dependent density functionals in general, and of the exact time-dependent exchange-correlation potential in particular, by presenting the first exact results for two- and three-dimensional multi-electron systems. This method is an explicit realization of the Runge-Gross correspondence, which maps time-dependent densities to their respective potentials, and allows for the exact construction of desired density functionals. We present in detail the numerical requirements that makes this method efficient, stable and precise even for large and rapid density changes, irrespective of the initial state and two-body interaction. This includes among others the proper treatment of low density regions, a subtle interplay between numerical time-propagation and zero boundary conditions, the choice of time-stepping strategy, and an error damping mechanism based on both the density and current density residuals. These considerations are also relevant for computing time-independent density-functionals and lead to a more precise implementation of quantum mechanics in general, which is mainly relevant for cases in which there is notable contact with a boundary or when the low density regions matter.