The Hohenberg-Kohn Theorems

If the exact electron density is known, then the cusps in
will provide the positions of the nuclei. The slope of
at the nucleus *A* must obey

(where
denotes the spherical average of the density) giving the charge at the nucleus, *Z*_{A}. Thus the full Schrödinger Hamiltonian is known, as it is completely defined by the nuclear charges and position. Therefore the wavefunction and energy can be found, and hence, the system can be completely described by the electron density.

A Hamiltonian of the form

is completely determined by the external potential,
.
The first Hohenberg and Kohn theorem [70] states that, for non-degenerate ground states, *the external potential
is determined, to within an additive constant, by the electron density,
*. The theorem has since been extended to include degenerate ground states [71].

The proof is based on the minimum energy principle and begins by considering two external potentials,
and
arising from the same density. There will be two Hamiltonians,
and
with the same density, but different wavefunctions,
and .
Now, using the variational principle,

Similarly

which leads to the contradiction

Hence, the external potential is determined by the density and we may thus represent the energy as a functional of the density

where
is the kinetic energy and
is the electron-electron repulsion, including the Coulombic interaction, :

The second Hohenberg-Kohn theorem [70] introduces the variational principle into DFT; *for a trial density
,
such that
and
,
where
is the energy functional from equation* (2.7). The proof is as follows: the first Hohenberg-Kohn theorem allows
to determine its own potential ,
Hamiltonian
and wavefunction
,
which can be used as a trial wavefunction for the problem with external potential

Assuming that
is differentiable, equation (2.9) requires that the ground state density be stationary, subject to the constraint that the integral of the density gives the number of electrons,

which leads to the Euler-Lagrange equation

where a familiar property to chemists, the chemical potential ,
has been introduced.

Equation (2.12) would be an exact equation for if the exact form of and were known. Unfortunately the Hohenberg-Kohn theorems do not provide this, only that they exist. Also and are defined independently of , so once we have a form for these functionals they can be applied to any system.