Corollary to the Hohenberg–Kohn theorem

According to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian 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( r t ) and the Hamiltonian ( t ). In the proof of the theorem, Hamiltonians /( 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 /( t ) is the same . Thus, according to the theorem, the density ρ( r )/ρ( r t ) uniquely determines the physical system as defined by its Hamiltonian /( t ). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians {}/{( 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( r t ). 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( r t ) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians {}/{( 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 {}/{( t )} that represent different physical systems, but differ by a constant C/ function C ( t ), and yet possess the same density ρ( r )/ρ( r t ), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003