The Hohenberg-Kohn Theorem

Density functional theory (DFT) is founded on the Hohenberg-Kohn
[4]theorem[13]. This comes in two parts, the first of which states that the ground state energy of a system of electrons is a unique functional of the ground state density:

$$E_{GS}=E[\rho_{GS}].$$ (1.26)

In fact all properties of the system, including excited state properties, are, in principle, exact functionals of the ground state density. The reason for this, as was proven by Hohenberg and Kohn, is that there is a one-to-one mapping between the ground state density and the external potential. If we happen to know the ground state density, then, in principle, we know the external potential, and if we know the external potential we can, again in principle, solve the many-electron Schrödinger equation and know everything about the system. Of course, this is not yet of any practical use, because the whole point of using DFT is so that we can avoid having to deal with the many-electron Schrödinger equation. Nevertheless, we are provided, at least in principle, with a means of finding the ground state energy for a given external potential. The internal electronic energy, $F$, of a system in its ground state can be expressed as

$$F=E-V_{ext},$$ (1.27)

where $V_{ext}$ is the external potential energy, given by

$$V_{ext}=\int d\ensuremath{\mathbf{r}}v_{ext}(\ensuremath{\mathbf{r}})\rho(\ensuremath{\mathbf{r}}).$$ (1.28)

Since $E$ and $v_{ext}$ are functionals of the density, it follows that $F$ is also a functional of the density.

Supposing we now have an external potential and a ground state density, which may or may not be the ground state density corresponding to that potential, we can define the variational energy, $E_{var}$, as

$$E_{var}[v_{ext},\rho]=F[\rho]+\int d\ensuremath{\mathbf{r}}v_{ext}(\ensuremath{\mathbf{r}})\rho(\ensuremath{\mathbf{r}}).$$ (1.29)

The true ground state density for $v_{ext}(\ensuremath{\mathbf{r}})$ is the density that minimises this energy - this is the second part of the Hohenberg-Kohn theorem. If we were able to calculate $F[\rho]$ for any given density, then we could perform a search to find the ground state density for any given external potential.

Such a search may be complicated by the fact that we have so far only defined the functional $F[\rho]$ for densities that correspond to the ground state of some external potential; such densities are described as being $V$-representable. It may be the case that, during a search, we would encounter densities that did not correspond to the ground state of any external potential. This problem can be overcome by extending the definition of $F[\rho]$ to include such densities, so long as $E_{var}$ is still minimised by the correct ground state density. We now define $F[\rho]$ as the minimum internal electronic energy of any electronic wavefunction, ground state or otherwise, whose corresponding density is equal to $\rho$, i.e.

$$F[\rho]=\min_{\ensuremath{\mathbf{\Psi}}\rightarrow \rho}\... ...hbf{\Psi}}\vert\hat{F}\vert\ensuremath{\mathbf{\Psi}}\rangle.$$ (1.30)

Essentially all densities, $\rho(\ensuremath{\mathbf{r}})$, that integrate to $N$, correspond to some $N$-electron wavefunction [14]; this property is described as $N$-representability. At this stage we are still no nearer to a practical method because exact evaluation of $F[\rho]$ would require us to solve the many-body Schrödinger equation. But, supposing we have a functional that is a good approximation to $F[\rho]$, but that can be evaluated in a practical manner, then a search should lead us to a good approximation to the ground state energy and density. This is the fundamental principle upon which all practical DFT calculation are founded.