Beyond the Runge–Gross Theorem

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


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.


External Potential Convergence Radius Linear Response Function Allowed Variation Density Response Function 
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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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