Energy surface, chemical potentials, Kohn-Sham energies in spin-polarized density functional theory

On the basis of the zero-temperature grand canonical ensemble generalization of the energy E[N,N(s),v,B] for fractional particle N and spin N(s) numbers, the energy surface over the (N,N(s)) plane is displayed and analyzed in the case of homogeneous external magnetic fields B(r). The (negative of the) left-/right-side derivatives of the energy with respect to N, N(↑), and N(↓) give the fixed-N(s), spin-up, and spin-down ionization potentials/electron affinities, respectively, while the derivative of E[N,N(s),v,B] with respect to N(s) gives the (signed) half excitation energy to the lowest-lying state with N(s) increased (or decreased) by 2. The highest occupied and lowest unoccupied Kohn-Sham spin-orbital energies are identified as the corresponding spin-up and spin-down ionization potentials and electron affinities. The excitation energies to the lowest-lying states with N(s)±2 can be obtained as the differences between the lowest unoccupied and the opposite-spin highest occupied spin-orbital energies, if the (N,N(s)) representation of the Kohn-Sham spin-potentials is used. The cases where the convexity condition on the energy does not hold are also discussed. Finally, the discontinuities of the energy derivatives and the Kohn-Sham potential are analyzed and related.