  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.
