  We have developed a symmetry-adapted modeling procedure for molecules and
crystals. By using the completeness of multipoles to express spatial and
time-reversal parity-specific anisotropic distributions, we can generate
systematically the complete symmetry-adapted multipole basis set to describe
any of electronic degrees of freedom in isolated cluster systems and periodic
crystals. The symmetry-adapted modeling is then achieved by expressing the
Hamiltonian in terms of the linear combination of these bases belonging to the
identity irreducible representation, and the model parameters (linear
coefficients) in the Hamiltonian can be determined so as to reproduce the
electronic structures given by the density-functional computation. We
demonstrate our method for the modeling of graphene, and emphasize usefulness
of the symmetry-adapted basis to analyze and predict physical phenomena and
spontaneous symmetry breaking in a phase transition. The present method is
complementary to de-facto standard Wannier tight-binding modeling, and it
provides us with a fundamental basis to develop a symmetry-based analysis for
materials science.
