  We present a general formalism of multipole descriptions under the space-time
inversion group. We elucidate that two types of atomic toroidal multipoles,
i.e., electric and magnetic, are fundamental pieces to express electronic order
parameters in addition to ordinary electric and magnetic multipoles. By
deriving quantum-mechanical operators for both toroidal multipoles, we show
that electric (magnetic) toroidal multipole higher than dipole (monopole) can
become a primary order parameter in a hybridized-orbital system. We also
demonstrate emergent cross-correlated couplings between electric, magnetic, and
elastic degrees of freedom, such as magneto-electric and
magneto(electro)-elastic couplings, under toroidal multipole orders.
