  Four types of atomic-scale multipoles, electric, magnetic, magnetic toroidal,
and electric toroidal multipoles, give a complete set to describe arbitrary
degrees of freedom for coupled charge, spin, and orbital of electrons. We here
present a systematic classification of these multipole degrees of freedom
towards the application in condensed matter physics. Starting from the
multipole description under the rotation group in real space, we generalize the
concept of multipoles in momentum space with the spin degree of freedom. We
show how multipoles affect the electronic band structures and linear responses,
such as the magneto-electric effect, magneto-current (magneto-gyrotropic)
effect, spin conductivity, Piezo-electric effect, and so on. Moreover, we
exhibit a complete table to represent the active multipoles under 32
crystallographic point groups. Our comprehensive and systematic analyses will
give a foundation to identify enigmatic electronic order parameters and a guide
to evaluate peculiar cross-correlated phenomena in condensed matter physics
from microscopic point of view.
