Hohenberg–Kohn theorems

1. If two systems of electrons, one trapped in a potential ${\displaystyle v_{1}({\vec {r}})}$ and the other in ${\displaystyle v_{2}({\vec {r}})}$, have the same ground-state density ${\displaystyle n({\vec {r}})}$ then necessarily ${\displaystyle v_{1}({\vec {r}})-v_{2}({\vec {r}})=const}$.

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 ${\displaystyle F[n]=T[n]+U[n]}$ is a universal functional of the density (not depending explicitly on the external potential).

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