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:

[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:

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, , of a system in its ground state can be expressed as

(1.27) |

where is the external potential energy, given by

(1.28) |

Since and are functionals of the density, it follows that 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*, , as

The true ground state density for
is the
density that minimises this energy - this is the second
part of the Hohenberg-Kohn theorem. If we were able to calculate 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
for densities that correspond to the ground state of
some external potential; such densities are described as being *-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 to include
such densities, so long as is still minimised by the correct
ground state density.
We now define
as the minimum internal electronic energy of
any electronic wavefunction, ground state or
otherwise, whose corresponding density is equal to , i.e.

(1.30) |

Essentially all densities,
, that integrate to , correspond to some -electron
wavefunction [14]; this property is described as *-representability*.
At this stage we are still no nearer to a
practical method because exact evaluation of would require us to solve the many-body Schrödinger
equation. But, supposing we have a functional that is
a good approximation to , 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.