Editing Density functional theory (section)

==Hohenberg–Kohn theorems==
The Hohenberg–Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential.

'''Theorem 1.''' The external potential (and hence the total energy), is a unique functional of the electron density.

: If two systems of electrons, one trapped in a potential <math>v_1(\mathbf r)</math> and the other in <math>v_2(\mathbf r)</math>, have the same ground-state density <math>n(\mathbf r)</math>, then <math>v_1(\mathbf r) - v_2(\mathbf r)</math> is necessarily a constant.

: '''Corollary 1:''' the ground-state density uniquely determines the potential and thus all properties of the system, including the many-body wavefunction. In particular, the HK functional, defined as <math>F[n] = T[n] + U[n]</math>, is a universal functional of the density (not depending explicitly on the external potential).
:'''Corollary 2:''' In light of the fact that the sum of the occupied energies provides the energy content of the Hamiltonian, a unique functional of the ground state charge density, the spectrum of the Hamiltonian is also a unique functional of the ground state charge density.<ref name="Bagayoko 127104"/>

'''Theorem 2.''' The functional that delivers the ground-state energy of the system gives the lowest energy if and only if the input density is the true ground-state density.

: In other words, the energy content of the Hamiltonian reaches its absolute minimum, i.e., the ground state, when the charge density is that of the ground state.

: For any positive integer <math>N</math> and potential <math>v(\mathbf r)</math>, a density functional <math>F[n]</math> exists such that
:: <math>E_{(v,N)}[n] = F[n] + \int v(\mathbf r) n(\mathbf r) \,\mathrm d^3 \mathbf r</math>
: reaches its minimal value at the ground-state density of <math>N</math> electrons in the potential <math>v(\mathbf r)</math>. The minimal value of <math>E_{(v,N)}[n]</math> is then the ground-state energy of this system.