Quantitative Pertubational Crystal Orbital Method and Its Application to Chain Polymer

Quantitative pertubational molecular orbital Method （PMO） has been found to be very useful for interpretation and prediction of the structure-property relations. However, there is yet no such method for polymers and crystal. In this work, we have developed a quantitative pertubational crystal orbital method （PCO） within the framework ofab initio SCF-CO theory. In this procedure, the unit cell is divided into fragments, and the framental crystal orbitals are calculated by ab initio SCF-CO method. Then by means of pertubation theory, the interaction between the fragmental crystal orbitals can be calculated, and the properties of crystal or polymers are analyzed from the point of view of the interactions between the fragmental crystal orbitals.If the composite polymer is （AB）_n and the fragmental polymers are （A）_n and （B）_n, then from pertubation theory, we have the first order and second order correction for energy of thei-th band with wave vector k:\begingatheredE_i^(1) k=\Delta_i i^k \\ E_i^(2) k=-\sum_i(i \neq i) \frac\left|\Delta_i i^k-\tildeS_i i^k E_i^(0) k\right|^2E_i^(0) k-E_i^(0) k\endgatheredwhere △_{i i}^k and \widetilde\boldsymbolS_ii^k are the matrix elements of the followong matrix respectively: and \delta \mathbfF=\left\beginarrayl\mathbfF_\mathrmA^k-\mathbfF_\mathrmA^(0) k \mathbfF_\mathrmAB^k \\ \mathbfF_\mathrmAB^k+\mathbfF_\mathrmB^k-\mathbfF_\mathrmb^(0) k\endarray\right, \quad \mathbfC^(0) k=\left\beginarraylr\mathbfC_\mathrmA^(0) k & \mathbfO \\ \mathbfO & \mathbfC_\mathrmB^(0) k\endarray\right S^k is the overlap matrix when atomic orbitals are used as basis set and the is the overlap matrix when fragmental crystal orbitals are used as basis Set.Analogous to the treatment of PMO, we can define the useful terminology "two-electron stabilization and four-electron destabilization" as follow:If Ψ_i^（0）k is a doubly occupied fragmental crystal orbital, Ψ_i^（0）k is an unccupied fragmental crystal orbital, the the term 2\left|\Delta_i j^k-E_i^(0) ^k \widetildeS_i i^k\right|_2 /\left(E_i^(0) k-E_i^(0) k\right) approximates the two-electon stabilization interaction energy between Ψ_i^（0）k and Ψ_j^（0）k.If both Ψ_i^（0）k and Ψ_j^（0）k are doubly occupied, the term may be rebarded as the four-electron bestabilization interaction energy between Ψ_i^（0）k and Ψ_j^（0）k （where（R）, （I） represent the real part and imaginary part respectively）.For illustration, we have performed ab initio SCF-CO calculation and PCO calculation on a chain polymer, polymethylacetylene （PMA）. The effect of substituent （CH₃） on the π band structure of the polymer from the interaction between the non-bonded π crystal orbital of the substituent With the π crystal orbital of the backbone has been investigated. Besides, the effect of internal rotation upon the stability and band structure of PMA has also been studied and reasonable explanation for some experimental facts obtained.