Investigation on the eigenvalue‐equation problem of molecular crystals and polymers

One‐dimensional periodic system, such as molecular crystal and polymer, can be expressed as … ABABAB … structure, where A and B stand for a complete molecule or a part of a molecule. When (AB) is taken as a structure unit, one can obtain the complex generalized eigenvalue‐equation H AB ( k ) ‐ C AB ( k ) ‐ S AB ( k ) C AB ( k ) E AB ( k ); and if (BA) is taken as a structure unit, the corresponding eigenvalue‐equation is H BA ( k ) C BA ( k ) ‐ S BA ( k ) C BA ( k ) E BA ( k ). The relationship between the two equations has been investigated. The results of theoretical analysis are \documentclass{article}\pagestyle{empty}\begin{document}$ E_m^{{\rm BA}} (k) = E_m^{{\rm AB}} (k);\quad C_{j_{\rm a} m}^{{\rm BA}} (k) - C_{j_{\rm a} m}^{{\rm AB}} (k) \cdot \exp (i\Delta \phi) $\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ C_{j_{\rm b} m}^{{\rm BA}} (k)\quad C_{j_{\rm b} m}^{{\rm AB}} (k) \cdot \quad \exp [i(k.R_{ - 1} + \Delta \phi)] - C_{j_{\rm b} m}^{{\rm AB}} (k) \times \exp (i\Delta \phi) $\end{document} where j a and j b are the index number of atomic orbitals within (A) and (B) respectively, and m stands for the index number of crystal or polymer orbitals. This result has been verified by the concrete calculation of three periodic systems: (1) hydrogen‐molecular chain, …H (A) H (B) … H (A) H (B) …, (2) polyphenylene, … (A) (B) (A) (B) …, where (A) and (B) stand for =C (CH=) 2 and ( HC) 2 C= respectively, and (3) the TCNQ molecular column, … TCNQ (A). TCNQ (B) … TCNQ (A). TCNQ(B) … . The results can be generalized to two‐ and three‐dimensional systems straightforwardly.