# On the Full scope of Hohenberg-Kohn Theorem

The Hohenberg-Kohn Theorem (HK Theorem) tells us that, knowing the electronic density in the ground state of a system of electrons, we can reconstruct the external potential up to an additive constant and that, therefore, knowing the ground state multielectronic wavefunction is equivalent to knowing the density, which is quite an impresive statement.

I've read that the theorem cannot be extended easily (although it also holds for the Dirac equation, for instance, which is already interesting), but I have two questions related to precisely this: Under which conditions can the HK theorem be extended:

1) Are there some interesting (not too restrictive) sufficient conditions for the HK Theorem to hold in excited states or in Temporal Dependent DFT?

2) Is there a version of the HK Theorem for Multicomponent DFT, that is, DFT for systems of different species of fermions?

• Are you interested in the HK-theorem as such or rather in whether knowing the density is equivalent to knowing the multi-electron wavefunction? – Norbert Schuch Oct 21 '16 at 16:45