Beyond the Runge–Gross Theorem

Abstract

The Runge–Gross theorem (Runge and Gross, Phys Rev Lett, 52:997–1000, 1984) states that for a given initial state the time-dependent density is a unique functional of the external potential. Let us elaborate a bit further on this point. Suppose we could solve the time-dependent Schrödinger equation for a given many-body system, i.e. we specify an initial state $| \Uppsi_0 \rangle$ at $t=t_0$ and evolve the wavefunction in time using the Hamiltonian ${\hat{H}} (t).$ Then, from the wave function, we can calculate the time-dependent density $n (\user2{r},t).$ We can then ask the question whether exactly the same density $n(\user2{r},t)$ can be reproduced by an external potential $v^{\prime}_{\rm ext} (\user2{r},t)$ in a system with a different given initial state and a different two-particle interaction, and if so, whether this potential is unique (modulo a purely time-dependent function). The answer to this question is obviously of great importance for the construction of the time-dependent Kohn–Sham equations. The Kohn–Sham system has no two-particle interaction and differs in this respect from the fully interacting system.