# Density functional theory/Hohenberg–Kohn theorems

## 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.