  We propose a systematic method to generate a complete orthonormal basis set
of multipole expansion for magnetic structures in arbitrary crystal structure.
The key idea is the introduction of a virtual atomic cluster of a target
crystal, on which we can clearly define the magnetic configurations
corresponding to symmetry-adapted multipole moments. The magnetic
configurations are then mapped onto the crystal so as to preserve the magnetic
point group of the multipole moments, leading to the magnetic structures
classified according to the irreducible representations of crystallographic
point group. We apply the present scheme to pyrhochlore and hexagonal ABO3
crystal structures, and demonstrate that the multipole expansion is useful to
investigate the macroscopic responses of antiferromagnets.
