Density functional theory/Hohenberg–Kohn theorems

Hohenberg–Kohn theorems

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

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