gives the exact energies and densities of all N eigenstates. The second theorem corresponds to an extension of the Theophilou variational principle for the whole subspace 460 in terms of H[D(r)], ensuring that the energies and vectors of individual states are obtained simultaneously. In these expressions, D(r) is a matrix of electron densities and transition densities of a set of basis states that represent the ensemble density ρ V (r) = tr V [S -1 D(r)] of the subspace V N , where S is the overlap matrix of the basis states; D(r) is not to be confused with the one-electron density matrix.