Non-Uniqueness of Local Effective Potential Energy in Density Functional Theory

As a consequence of the first Hohenberg-Kohn (HK) theorem, in the mapping from a \emph{ground} state of an interacting system to an S system of noninteracting fermions with equivalent density, the effective potential energy of the latter is \emph{unique}. But it is so \emph{only} if these fermions are in their \emph{ground} state. It can be shown via Quantal Density Functional Theory,\footnote{\emph{Quantal Density Functional Theory}, V. Sahni (Springer-Verlag, 2004)} that the \emph{ground} state density of an interacting system can also be reproduced by S systems that are in an \emph{excited} state. Hence, in principle, there are an infinite number of functions that can reproduce a \emph{ground} state density. Similarly, in the mapping from an \emph{excited} state of the interacting system to an S system with equivalent density, the state of the latter is \emph{arbitrary}. Hence, there are an infinite number of functions that can reproduce the excited state density. The latter proves the lack of a first HK theorem for \emph{excited }states. The difference between the potential energy functions in either case is due solely to Correlation-Kinetic effects.