Exact exchange-correlation potentials in spin-density functional theory and their discontinuities at unit electron number

The exact exchange-correlation potential of Kohn–Sham density functional theory is known to jump discontinuously by a spatial constant as the average electron number, N, crosses an integer in an open system of fluctuating electron number, with important physical consequences for charge transfers and band gaps. We have recently constructed an essentially exact exchange-correlation potential v xc for N electrons (0 ≤ N ≤ 2) in the presence of a –1/r external potential, i.e., for a ground ensemble of H + ion, H atom, and H – ion densities. That construction illustrates the discontinuity at N = 1, where it equals I H – A H , the positive difference between the ionization energy and the electron affinity of the hydrogen atom. Here we construct the corresponding essentially exact spin-up and spin-down exchange-correlation potentials v xc,↑ and v xc,↓ of the Kohn–Sham spin-density functional theory, more commonly used for electronic structure calculations, for the ground ensemble with most-negative z-component of spin (or equivalently in the presence of a uniform magnetic field of infinitesimal strength). The potentials v xc , v xc,↑ , and v xc,↓ , which vanish as r → ∞ (except when N approaches an integer from above), are identical for 0 ≤ N ≤ 1 and for N = 2 but not for 1 < N < 2. We find that the majority or spin-down potential has a spatially constant discontinuity at N = 1 equal to I H – A H . The minority or spin-up potential has a discontinuity which is this constant in one order of limits, but is a spatially varying function in a different order of limits. This order-of-limits problem is a consequence of a special circumstance: the vanishing of the spin-up density at N = 1.