Possible band-structure shapes of quasi-one-dimensional solids

The well-known restrictions on the possible shapes of electron energy bands arising from one-dimensional (1D) periodic potentials do not apply to chainlike physical systems such as polymers or molecular stacks, which are three dimensional objects periodic along a line. For such quasi-one-dimensional (Q1D) crystals, bands can be nondegenerate, but they may also contact one another at the center and/or at the boundaries of the Brillouin zone (BZ), cross at a general $k$ point, or be degenerate throughout the BZ, so that a variety of topologically inequivalent shapes can be formed. These shapes are essentially determined by the spatial symmetry of the Q1D object considered, i.e., by its line group. A complete classification has been worked out, and for every possible Q1D crystal structure the corresponding band shapes are specified. The method focuses on the quantum numbers having clear physical meaning (quasimomentum, quasi angular momentum, etc.); elaborate group-theoretical parlance is avoided. Furthermore, it is studied how this complexity of band shapes which may occur in Q1D crystals affects some predicted physical properties—optical-absorption spectra and vibronic coupling, in particular. First, each singularity of the density-of-states function bears definite symmetry labels, and many direct interband transitions in polymers are forbidden by the line-group selection rules. Next, in the case of electronic degeneracy, many different types of vibronic instabilities are possible. Depending on the line group, the band shapes, and the quantum numbers of the electronic states involved, the active phonons may be longitudinal or transversal, acoustic or optical, single or degenerate, displacive or distortive, etc.; also, various types of symmetry breaking can occur. Finally, applicability to other quasi particles (phonons, etc.) and to solids of higher dimensionality is pointed out.