Abstract: 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 of ab initio SCF-CO theory. In this procedure, the unit cell is divided into fragments, and the fragmental crystal orbitals are calculated by ab initio SCF- CO method. Then by meas of pertubation theory, the interaction between the fragmental crystal orbital 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 the ith band with wave vector k:
E_i~((2)k)=-Σ_(h(≠i) {|Δ_(i j)~k-S_(i j)~kE_i~((0)k)|~2}/{E_j~((0)k)-E_i~((0)k)}
where Δ_(i j)~k and S_(i j)~k are the matrix elements of the followong matrix respectively:
Δ~k=C~((0)k+)δF~kC~((0)k), S~k=C~((0)k+)s~kC~((0)k)
and
δF=[F_A~k-F_A~((0)k) F_(AB)~k C~((0)k)=[C_A~((0)k) O
F_(AB)~(k+) F_B~k-F_B~((0)k)], O C_B~((0)k)]
S~k is the overlap matrix when atomic orbitals are used as basis set and the S~k isthe 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 follows:
If Ψ_i~((0)k) is a doubly occupied fragmental crystal orbital, Ψ_j~((0)k) is an unccupied fragmental crystal orbital, then the term
2|Δ_(i j)~k-E_i~((0)k)S_(i j)~k|~2/(E_i~((0)k)-E_j~((0)k))
approximates the two-electron tabilization interaction energy between Ψ_i~((0)k) and Ψ_j~((0)k).
If both Ψ_i~((0)k) and Ψ~((0)k) are doubly occupied, the term
[2/(1-|S_(i j)~k|~2)[{(E_i~((0)k)+E_j~((0)k))|S_(i j)~k|~2-2[Δ_(i j)~k(R)S_(i j)~k(R)+Δ_(i j)~k(I)S_(i j)~k(I)]}
may be regarded as the four-electron destabilization 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 several one dimensional polymers, they are polymethylacetylene, polymonocyanoacetylenes and polyfluoroacetylene. We have explained quite well the effects of substituents (CH_3, CN F) on the π band structure of the polymers from the interaction between the π crytal orbital of the substituents with the π crystal orbitals of the backbone. The details of this work will be published later.