Self-Consistent-Field Tight-Binding Treatment of Polymers. I. Infinite Three-Dimensional Case

A general self-consistent-field tight-binding linear-combination-of-atomic-orbitals (LCAO) formalism is given for three-dimensional polymers containing many atoms in the elementary cell with all neighbors interacting, taking overlap explicitly into account. This formalism, which corresponds essentially to the formulation given by Roothaan for closed-shell molecules, has been developed with the aid of Hermitian complex matrices. The special cases of nearest-neighbor approximation and of a linear chain are then derived from the general expression obtained. Finally, formulas are given, again in complex-matrix formulation, for the dependence of the energy levels and wave functions of the polymer on the wave number $k$.