Effect of Time-Reversal Symmetry on Energy Bands of Crystals

In the Hartree and Fock approximations the description of the electronic state of a crystal can be made in terms of one-electron wave functions and one-electron energies, which have a band structure. It is known that in addition to the "sticking together" of these energy bands caused by the spatial symmetry of the crystal, additional "sticking" may be necessitated by the fact that the Hamiltonian of the problem is real. In this paper a criterion is developed to facilitate calculation of when and how such additional degeneracy will occur. The consequences of the reality of the Hamiltonian are tabulated for a number of cases. It is pointed out that the same "sticking together" of bands occurs in the theory of the frequency spectrum of the normal modes of vibration of a crystal.