Energy of a many-electron system in an ensemble ground-state, versus electron number and spin: piecewise-linearity and flat plane condition generalized

Description of the total energy of a many-electron system, E, as a function of the total number of electrons N_tot (integer or fractional) is of great importance in atomic and molecular physics, theoretical chemistry and materials science. Equally significant is the correct dependence of the energy on the spin of the system, M_tot. In my talk I present our recent work, where we extend previous studies, allowing both N_tot and M_tot to vary continuously, taking on both integer and fractional values. We describe the ground state of a finite, many-electron system by an ensemble of pure states, and characterize the dependence of the energy and the spin-densities on both N_tot and M_tot. Our findings generalize the piecewise-linearity principle of Phys. Rev. Lett. 49, 1691 (1982) and the flat-plane condition of Phys. Rev. Lett. 102, 066403 (2009). Focusing on the case where the spin is smaller or equal to its equilibrium value, we show which pure states contribute to the ensemble for given N_tot and M_tot, and which states do not. Interestingly, we find a degeneracy of the ground state, where the total energy and density are unique, but the spin-densities are not. Moreover, we find a new type of a derivative discontinuity, which manifests in the case of spin variation at constant N_tot, as a jump in the Kohn-Sham potential at the edges of the variation range. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.