  The term "long-range interactions" refers to electrostatic and magnetostatic
potential energies between atoms and molecules with mutual distances ranging
from a few tens to a few hundreds Bohr radii. The involved energies are much
smaller than the usual chemical bond energies. However, they are comparable
with the typical kinetic energies of particles in an ultracold gas ($T\ll 1K$),
so that the long-range interactions play a central role in its dynamics. The
progress of research devoted to ultracold gases shed a new light on the
well-established topic of long-range interactions, because: (i) the interacting
atoms and molecules can be prepared in a well-defined quantum (electronic,
vibrational, rotational, fine or hyper-fine), ground or excited level; and (ii)
long-range interactions can be tailored at will using external electromagnetic
fields.
  In this chapter, we present the essential concepts and mathematical relations
to calculate long-range potential energies. We start with deriving the
multipolar expansion of the electrostatic interaction energy between classical
charge distributions, both in Cartesian coordinates for pedagogical purpose,
and in spherical coordinates for practical use. Then we combine multipolar
expansion and quantum perturbation theory, to obtain the general first- and
second-order energy corrections, including the well-known van der Waals energy.
We consider two central examples in the current context of ultracold gases: (i)
a pair of alkali-metal atoms and (ii) a pair of alkali-metal heteronuclear
diatomic molecules submitted to an electric field. We highlight the key role of
the total angular momenta of the interacting particles and of the complex,
irrespective of their electronic or nuclear, orbital or spin nature.
