Density-functional theory is an extremely powerful and widely used tool for quantum simulations. It reformulates the electronic-structure problem into a functional minimization with respect to the charge density of interacting electrons in an external potential. While exact in principle, it is approximate in practice, and even in its exact form it is meant to reproduce correctly only the total energy and its derivatives, such as forces, phonons, or dielectric properties. Quasiparticle levels are outside the scope of the theory, with the exception of the highest occupied state, since this is given by the derivative of the energy with respect to the number of electrons. A fundamental property of the exact energy functional is that of piecewise linearity at fractional occupations in between integer fillings, but common approximations do not follow such piecewise behavior, leading to a discrepancy between total and partial electron removal energies. Since the former are typically well described, and the latter provide, via Janak’s theorem, orbital energies, this discrepancy leads to a poor comparison between predicted and measured spectroscopic properties. We illustrate here the powerful consequences that arise from.
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