Abstract:

The eigenvalue problem of Hückel matrix including next nearest interaction has been dealt with using graphic method as well as difference equation method respe- ctively.In difference equation approach, a parametric form of equation (10) is obtained and both the coponents of the molecular orbitals Cn and energy levals x have been written formally out as formulas (11) and (13). With large n and small η(|η|<1) the secular equation can be approximated by eq.(16), from which the Taylor expansion (18) of θ in terms of η can be derived. In graphic approach both the characteristic polynomial P_N (x, η) and the C_n(x) are expanded in powers of η by means of theorems~[1] published previously. When |η|<1 the secular equation is reasonably approximated by eq.(32) where terms higher than η~3 have been neglected. Thus a Taylor expansion of x can be formulated as (34). It is easily seen that both approaches are conformed to each other.
As an application of the present results, the role of the next nearest interaction in linear polyene has been investigated in detail where a simple numerical analysis has been done. We see that the next nearest interaction changes the energe levels somewhat, however, the HOMO-LUMO gap Δθ_F, the total energy of electrons E_H and the bond orders are almost kept unchanged. Only the wavefunctions are infuenced which in turn makes the charge density un-uniform; as a result, terminal wave and a charge density wave (CDW) could be predicted.