Hohenberg-Kohn theorem and generalizations Mel Levy Tulane University Chemistry The original Hohenberg-Kohn theorem and its generalizations, to include degeneracies and non-v-representability, shall be proven by emphasizing the transparent constrained-search approach. In this formulation, F[n] is defined simply as the minimum of the sum of the kinetic energy and electron repulsion energy, where the minimizing search is over all wavefunctions that are constrained to integrate to n. With F[n] so defined, the total ground-state energy and corresponding density are then obtained by optimizing n when the one-body external potential is included. Also, it follows that the Euler equation for the ground-state density shows that this density determines the external potential which, in turn, implies that the ground-state density determines all the properties of the system. Next, extensions to ensemble constrained searches and fractional electron number will be given. Then the Legendre transform formulation will be discussed. The lecture will close with a summary of existence theorems for time-independent DFT for excited states.