  Density functional theory provides the most widespread framework for the
realistic description of the electronic structure of solids, but the
description of strongly-correlated systems has remained so far elusive. Here we
consider a particular limit of electrons in a periodic ionic potential in which
a one-band description becomes exact all the way from the weakly-correlated
metallic regime to the strongly-correlated Mott-Hubbard regime. We provide a
necessary condition a density functional should fulfill to describe
Mott-Hubbard behavior and show that it is not satisfied by standard and widely
used local, semilocal and hybrid functionals. We illustrate the condition in
the case of a few-atom system and provide an analytic approximation to the
exact exchange-correlation potential based on a variational wave function which
shows explicitly the correct behavior providing a robust scheme to combine
lattice and continuum methods.
