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Beyond the Runge–Gross Theorem

  • Michael Ruggenthaler
  • Robert van Leeuwen
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 837)

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.

Keywords

External Potential Convergence Radius Linear Response Function Allowed Variation 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.

Copyright information

© Springer-Verlag Berlin Heidelberg  2012

Authors and Affiliations

  • Michael Ruggenthaler
    • 1
  • Robert van Leeuwen
    • 1
  1. 1.Department of Physics, Nanoscience CenterUniversity of JyväskyläJyväskylä,Finland

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