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The derivative discontinuity is a key concept in electronic structure theory in general and density functional theory in particular. The electronic energy of a quantum system exhibits derivative discontinuities with respect to different degrees of freedom that are a consequence of the integer nature of electrons. The classical understanding refers to the derivative discontinuity of the total energy as a function of the total number of electrons ($N$), but it can also manifest at constant $N$. Examples are shown in models including several Hydrogen systems with varying numbers of electrons or nuclear charge ($Z$), as well as the 1-dimensional Hubbard model (1DHM). Two sides of the problem are investigated: first, the failure of currently used approximate exchange-correlation functionals in DFT and, second, the importance of the derivative discontinuity in the exact electronic structure of molecules, as revealed by full configuration interaction (FCI). Currently, all approximate functionals miss the derivative discontinuity, leading to basic errors that can be seen in many ways: from the complete failure to give the total energy of H$_2$ and H$_2^+$, to the missing gap in Mott insulators such as stretched H$_2$ and the thermodynamic limit of the 1DHM, or a qualitatively incorrect density in the HZ molecule with two electrons and incorrect electron transfer processes. Description of the exact particle behavior of electrons is emphasized, which is key to many important physical processes in real systems, especially those involving electron transfer, and offers a challenge for the development of new exchange-correlation functionals.

Discontinuous changes in the electronic structure upon infinitesimal changes to the Hamiltonian are demonstrated. Remarkably, these are revealed in one and two electron molecular systems if the realm of the nuclear charge is extended to be fractional. Dramatic changes in the electron density from full configuration interaction are observed in real space illustrating key intricacies of electronic structure including the transfer, hopping and removal of electrons. Physically, this is due to the particle nature of electrons and manifests itself theoretically as a diverging linear density response function or an energy derivative discontinuity that occurs at constant number of electrons. This is essential to correctly describe real physical processes, from chemical reactions to electron transport and metal-insulator transitions. The dramatic errors of DFT are seen in real space as this physics is missing from currently used approximations and poses a great challenge for the development of new electronic structure methods., Comment: 5 pages, 4 figures

Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified for states with both fractional charge and fractional spin: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. This sheds light on the nature of the derivative discontinuity and calls for explicitly discontinuous functionals of the density or orbitals that go beyond currently used smooth approximations. This is key for the application of DFT to strongly correlated systems., Comment: 5 pages 2 figures

The band-gap problem and other systematic failures of approximate functionals are explained from an analysis of total energy for fractional charges. The deviation from the correct intrinsic linear behavior in finite systems leads to delocalization and localization errors in large or bulk systems. Functionals whose energy is convex for fractional charges such as LDA display an incorrect apparent linearity in the bulk limit, due to the delocalization error. Concave functionals also have an incorrect apparent linearity in the bulk calculation, due to the localization error and imposed symmetry. This resolves an important paradox and opens the possibility to obtain accurate band-gaps from DFT., Comment: 4 pages 4 figures

The calculation of the band-gap by density-functional theory (DFT) methods is examined by considering the behavior of the energy as a function of number of electrons. It is found that the incorrect band-gap prediction with most approximate functionals originates mainly from errors in describing systems with fractional charges. Formulas for the energy derivatives with respect to number of electrons are derived which clarify the role of optimized effective potentials in prediction of the band-gap. Calculations with a recent functional that has much improved behavior for fractional charges give a good prediction of the energy gap and also $\epsilon_{{\rm homo}}\simeq-I$ for finite systems. Our results indicate it is possible, within DFT, to have a functional whose eigenvalues or derivatives accurately predict the band-gap.