Density functional theory/Hohenberg–Kohn theorems

Hohenberg–Kohn theorems
1. If two systems of electrons, one trapped in a potential $$ v_1(\vec r)$$ and the other in $$ v_2(\vec r)$$, have the same ground-state density $$ n(\vec r)$$ then necessarily $$ v_1(\vec r)-v_2(\vec r) = const$$.

Corollary: the ground state density uniquely determines the potential and thus all properties of the system, including the many-body wave function. In particular, the "HK" functional, defined as $$ F[n]=T[n]+U[n] $$ is a universal functional of the density (not depending explicitly on the external potential).

2. For any positive integer $$ N $$ and potential $$ v(\vec r)$$ it exists a density functional $$F[n]$$ such that $$ E_{(v,N)}[n] = F[n]+\int{v(\vec r)n(\vec r)d^3r} $$ obtains its minimal value at the ground-state density of $$ N $$ electrons in the potential $$ v(\vec r)$$. The minimal value of $$ E_{(v,N)}[n] $$ is then the ground state energy of this system.