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Using the discontinuity of the chemical potential as a function of excess charge, the fundamental gaps for finite systemsand the band gaps of extended solids are determined within reduced density matrix functional theory. We also present the necessaryand sufficient conditions for the one-body reduced density matrix of a system with fractional charge to be ensemble  N -representable. The performance of most modern day reduced density matrix functionals is assessed for the gaps and the correlationenergy of finite systems. Our results show that for finite systems the PNOF, BBC3, and power functionals yield very accuratecorrelation energies while for a correct description of the fundamental gap the removal of self-interaction terms is essential.For extended solids we find that the power functional captures the correct band gap behavior for conventional semiconductorsas well as strongly correlated Mott insulators, where a gap is obtained in absence of any magnetic ordering.

Discontinuities of the Chemical Potential in Reduced Density Matrix Functional Theory