  Spin-dependent electric dipole operators are investigated group-theoretically
for the emergence of an electric dipole induced by a single spin or by two
spins, where the spin dependences are completely classified up to the quadratic
order. For a single spin, a product of spin operators behaves as an even-parity
electric quadrupole operator, which differs from an odd-parity electric dipole.
The lack of the inversion symmetry allows the even- and odd-parity mixing,
which leads to the electric dipole described by the electric quadruple
operators. Point-group tables are given for classification of the possible
spin-dependent electric dipoles and for the qualitative analysis of
multiferroic properties, such as an emergent electric dipole moment coexisting
with a magnetic moment, electromagnon excitation, and directional dichroism.
The results can be applied to a magnetic ion in crystals or embedded in
molecules at a site without the inversion symmetry. In the presence of an
inversion symmetry, the electric dipole does not appear for a single spin. This
is not the case for the electric dipole induced by two spins with antisymmetric
spin dependence, which is known as vector spin chirality, in the presence of
the inversion center between the two spins. In the absence of the inversion
center, symmetric spin-dependent electric dipoles are also relevant. The
detailed analysis of various symmetries of two-spin states is applied to spin
dimer systems and the related multiferroic properties.
