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Beyond the Runge-Gross Theorem

  • R. van Leeuwen
Part of the Lecture Notes in Physics book series (LNP, volume 706)

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 = t0 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 υext(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].

Keywords

External Potential Convergence Radius Linear Response Function Quantum Lattice System Density Response Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 2006

Authors and Affiliations

  • R. van Leeuwen

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