Analytic density functionals with initial-state dependence and memory

Abstract

We analytically construct the wave function that, for a given initial state, produces a prescribed density for a quantum ring with two noninteracting particles in a singlet state. In this case the initial state is completely determined by the initial density, the initial time derivative of the density and a single integer that characterizes the (angular) momentum of the system. We then give an exact analytic expression for the exchange-correlation potential that relates two noninteracting systems with different initial states. This is used to demonstrate how the Kohn-Sham procedure predicts the density of a reference system without the need of solving the reference system's Schr\"odinger equation. We further numerically construct the exchange-correlation potential for an analytically solvable system of two electrons on a quantum ring with a squared cosine two-body interaction. For the same case we derive an explicit analytic expression for the exchange-correlation kernel and analyze its frequency dependence (memory) in detail. We compare the result to simple adiabatic approximations and investigate the single-pole approximation. These approximations fail to describe the doubly excited states, but perform well in describing the singly excited states.