Spin and orbital degeneracy problems in the DFT method. Relation to the Jahn-Teller effect

The problems in the density functional theory arising when it is applied to the spin- and orbitally-degenerate states are discussed. It is rigorously proved that the electron density of an arbitrary N-electron system cannot, in principle, depend upon the total spin S and for all values of S has the same form as it has for a single-determinantal wave function. It is also proved that the diagonal element of the density matrix is invariant with respect to the symmetry of the state and in the frame of density matrix description there is no difference between degenerate and nondegenerate states. Thus, the problems in DFT connected with the total spin and degenerate states cannot be rigorously solved within the framework of the density matrix formalism.