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 $E_{(v,N)}[n]=F[n]+\int {v({\vec {r}})n({\vec {r}})d^{3}r}$ 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.