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

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On the basis of the zero-temperature grand canonical ensemble generalization of the energy E[N,Nₛ,v,B] for fractional particle N and spin Nₛ numbers, the energy surface over the (N,Nₛ) 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, Nᵤp, and Ndown give the fixed-Nₛ, spin-up, and spin-down ionization potentials/electron affinities, respectively, while the derivative of E[N,Nₛ,v,B] with respect to Nₛ gives the (signed) half excitation energy to a state with Nₛ 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 states with Nₛ+2, Nₛ-2, can be obtained as the differences between the lowest unoccupied and the opposite-spin highest occupied spin-orbital energies, if the (N,Nₛ) 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.