An analogue of the Woodward-Hoffmann rule in terms of bond orders

The article contains an application to pericyclic reactions of the non-canonical method of molecular orbitals (MOs) suggested previously [18], [14] and based on obtaining the localized MOs (LMOs) and the respective one-electron density matrix (bond order matrix) directly without any reference to delocalized (canonical) MOs. The thermal electrocyclic closure of polyenes containing N CC bonds (C2NH2N+2) is modelled by emergence of additional resonance parameters between 2pz AOs of the terminal carbon atoms C1 and C2N. At the early stage of the reaction, alterations in the total energies due to the above-specified perturbation are related to bond orders (P1,2N) between the terminal AOs of respective initial (open) polyene chain. As a result, an analogue of the famous Woodward-Hoffmann rule is formulated in terms of alternating signs of bond orders P1,2N for growing N values. Moreover, a relation is established between the signs of P1,2N and those of direct (indirect) interactions of orbitals of terminal CC bonds by means of orbitals of intervening bonds. For later stages of the same process, the relevant alterations in bond orders themselves are studied, and these also are shown to yield an analogous selection rule. An additional insight is given into the mechanism of the closure process, viz. choice between a cyclic- and Möbius-array-like overlap topology of 2pz AOs of carbon atoms is shown to be made at the very early stage of the reaction. Inasmuch as bond order matrices are closely related to respective LMOs and thereby belong to the localized way of representing electronic structures (see the above-cited references), the results obtained may be considered as a localized description of pericyclic reactions.