  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.
