Treatment of symmetry in the Ritz method for the Schrödinger equation in crystals with a basis

This paper is devoted to treatment of symmetry in the Schrödinger equation with a periodic potential for crystals with a basis. We present a general group-theoretical approach, which yields the matrix elements of the Hamiltonian in the tight-binding approximation, using the spatial symmetry of the problem, time reversal symmetry, and the Hermitian property of the Hamiltonian. The developed mathematical theory generalizes the well-known result for crystals with two atoms in the unit cell to the case of crystals with several atoms in the unit cell.