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Generalised Kohn-Sham equations with accurate total energy and single-particle eigenvalue spectrum

We propose a new generalised Kohn-Sham or constrained hybrid method, where the exchange potential is the (equally weighted) average of the nonlocal Fock exchange term and the self-interaction-corrected exchange potential, as obtained from our constrained minimisation method of semi-local approximations. The new method gives an accurate single-particle eigenvalue spectrum with an average deviation between (the negative of) the valence orbital eigenvalues and the experimental ionisation potentials of about 0.5eV, while the deviation of core orbitals is within 2eV. The improvement in the eigenvalue spectrum is achieved with a minimal increase in the total energy.