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Density Functional Theory for Fractional Particle Number: Derivative Discontinuity of the Energy at the Maximum Number of Bound Electrons

The derivative discontinuity in the exact exchange-correlation potential of ensemble Density Functional Theory (DFT) is investigated at the specific integer number that corresponds to the maximum number of bound electrons, $J_{max}$. A recently developed complex-scaled analog of DFT is extended to fractional particle numbers and used to study ensembles of both bound and metastable states. It is found that the exact exchange-correlation potential experiences discontinuous jumps at integer particle numbers including $J_{max}$. For integers below $J_{max}$ the jump is purely real because of the real shift in the chemical potential. At $J_{max}$, the jump has a non-zero imaginary component reflecting the finite lifetime of the $(J_{max}+1)$ state.