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On the basis of the zero-temperature grand canonical ensemble generalizationof the energy E[N,N_s,v,B] for fractional particle N and spin N_s numbers, theenergy surface over the (N,N_s) plane is displayed and analyzed in the case ofhomogeneous external magnetic fields B(r). The (negative of the)left/right-side derivatives of the energy with respect to N, N_up, and N_downgive the fixed-N_s, spin-up, and spin-down ionization potentials/electronaffinities, respectively, while the derivative of E[N,N_s,v,B] with respect toN_s gives the (signed) half excitation energy to a state with N_s increased (ordecreased) by 2. The highest occupied and lowest unoccupied Kohn-Shamspin-orbital energies are identified as the corresponding spin-up and spin-downionization potentials and electron affinities. The excitation energies to thestates with N_s+2, N_s-2, can be obtained as the differences between the lowestunoccupied and the opposite-spin highest occupied spin-orbital energies, if the(N,N_s) representation of the Kohn-Sham spin-potentials is used. The caseswhere the convexity condition on the energy does not hold are also discussed.Finally, the discontinuities of the energy derivatives and the Kohn-Shampotential are analyzed and related.

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