Beyond the Runge–Gross Theorem

  • Ruggenthaler M
  • van Leeuwen R
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Abstract

The Runge-Gross theorem (Runge and Gross 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 (TDSE) for a given many-body system, i.e. we specify an initial state |Ψ 0 at t = t 0 and evolve the wavefunction in time using the HamiltonianˆH Hamiltonianˆ HamiltonianˆH (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 v 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 v-representability question (see Sect. 4.4.2): is a density belonging to an interacting system also nonin-teracting v-representable? We will show in this chapter that, with some restrictions

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Ruggenthaler, M., & van Leeuwen, R. (2012). Beyond the Runge–Gross Theorem (pp. 187–210). https://doi.org/10.1007/978-3-642-23518-4_9

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