Beyond the Runge-Gross Theorem

Abstract

The Runge-Gross theorem [Runge 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 timedependent Schrödinger equation (TDSE) for a given many-body system, i.e., we specify an initial state ∣Ψ₀〉 at t = t ₀ and evolve the wave function in time using the Hamiltonian H(t). Then, from the wave function, we can calculate the time-dependent density n(r, t). We can then ask the question whether exactly the same density n(r, t) can be reproduced by an external potential υ^′ _ext(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. It has, in general, also a different initial state. This state is usually a Slater determinant rather than a fully interacting initial state. A time-dependent Kohn-Sham system therefore only exists if the question posed above is answered affirmatively. Note that this is a υ-representability question: Is a density belonging to an interacting system also noninteracting υ-representable? We will show in this chapter that, with some restrictions on the initial states and potentials, this question can indeed be answered affirmatively [van Leeuwen 1999, van Leeuwen 2001, Giuliani 2005]. We stress that we demonstrate here that the interacting-υ-representable densities are also noninteracting-υ-representable rather than aiming at characterizing the set of υ-representable densities. The latter question has inspired much work in ground state density functional theory (for extensive discussion see [van Leeuwen 2003]) and has only been answered satisfactorily for quantum lattice systems [Chayes 1985].