Idealized HMO pd Model

Deviations from Optimum ipd) Overlap Caused by Geometry of Polygons 

The replacement of ap orbital of a cyclopolyene by ad orbital, in either radial or tangential orientation, causes a deviation from the optimum overlap of the dv orbital with the neighbouring pn orbital which is proportional to the cosine of the angle of deviation between the most favourable linear arrangement and the actual angle of the regular polygon. 

In an HMO calculation the reductions due to deviation from maximum (pd)v overlap may be introduced by the use of perturbation theory ll9) instead of solving the secular determinant directly. As usual, the Hiickel bond-intergral parameter (3 v is assumed to be proportional to the overlap integral of the corresponding bond 6 7). The reduction is therefore proportional in our case to the cosine of the angle of deviation y (10). 

The resultant change in 7r-bond energy by first-order perturbation theory is given in (11). 

In Fig. 11 the correspondingly obtained corrected specific 77-bond energies are plotted as a function of polygon size N. 

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