  A whole series of expressions for four species of multipoles (electric,
magnetic, magnetic toroidal, and electric toroidal) is provided as a complete
basis set to describe arbitrary single-centered spinful electron systems. A
compact formula to calculate matrix elements of these multipoles is also
derived. A visualization method of an electronic state characterized in terms
of multipoles is proposed. The complete basis set is useful to narrow down a
candidate order parameter of electron systems in phase transition, to describe
a property of cross-correlated phenomena, to analyze spectra of x-ray
scattering in magnetically ordered state, and so on. We demonstrate a usage of
the complete basis set by taking monopole and toroidal dipole orderings, and
the mutual relationship among three distinct magnetic dipoles (orbital, spin
angular momenta and anisotropic dipole) in a spin-orbit coupled system as prime
examples.
