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Corollary to the Hohenberg–Kohn theorem

Authors

  • Xiao-Yin Pan,

    1. Department of Physics, Brooklyn College of the City University of New York, 2900 Bedford Avenue, Brooklyn, New York 11210-2889, USA
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  • Viraht Sahni

    Corresponding author
    1. Department of Physics, Brooklyn College of the City University of New York, 2900 Bedford Avenue, Brooklyn, New York 11210-2889, USA
    • Department of Physics, Brooklyn College of the City University of New York, 2900 Bedford Avenue, Brooklyn, New York 11210-2889, USA
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Abstract

According to the Hohenberg–Kohn theorem, there is an invertible one-to-one relationship between the Hamiltonian equation image of a system and the corresponding ground-state density ρ(r). The extension of the theorem to the time-dependent case by Runge and Gross states that there is an invertible one-to-one relationship between the density ρ(rt) and the Hamiltonian equation image(t). In the proof of the theorem, Hamiltonians equation image/equation image(t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians equation image/equation image(t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian equation image/equation image(t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians {equation image}/{equation image(t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians {equation image}/{equation image(t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians {equation image}/{equation image(t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

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