Many-electron systems with fractional electron number and spin: Exact properties above and below the equilibrium total spin value

The description of many-electron systems with a fractional electron number, Ntot, and fractional z-projection of the spin, Mtot, is of great importance in physical chemistry, solid-state physics, and materials science. In this study, we analyze the fundamental question of what the ensemble ground state of a general, finite, many-electron system at zero temperature is, with a given Ntot and Mtot, distinguishing between low- and high-spin cases (separated by the boundary spin MB). For the low-spin case, the general form of the ensemble ground state has been rigorously derived in Goshen and Kraisler [J. Phys. Chem. Lett. 15, 2337 (2024)], generalizing the piecewise linearity and the flat-plane conditions for many-electron systems. Here, we provide an alternative proof for this case, discuss the ambiguity in the description of the ground state, and show that this ambiguity can be removed via maximization of the system's entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. We prove three general properties that characterize the ground state at high spins and narrow down the list of pure states it may consist of. We illustrate the aforementioned properties of high-spin cases by examining the ensemble ground state when Mtot approaches MB from above during the addition of (a fraction of) an up- and down-electron to a given system. Furthermore, we relate the frontier orbital energies of Kohn-Sham (KS) density functional theory (DFT) to total energy differences at high spin values, particularly the ionization potential (IP), the fundamental gap, and the spin flip energies. Analyzing the frontier energies on both sides of each boundary in the total energy profile, where the energy slope changes abruptly, we derive expressions for new derivative discontinuities, which are predicted to appear as jumps in the corresponding KS potentials. In this way, we generalize the well-known IP theorem of DFT to cases with fractional electron number and to cases with high spin. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this study are instrumental for the development of advanced approximations in DFT and other many-electron methods.