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 $ 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) $ and $ 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) $ 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.