It has been well accepted that when quantum interference (QI) occurs in a single molecular junction comprised of a π-conjugated molecule, the elastic π-electron transmission is blocked, while the elastic σ-electron transmission remains unchanged. When it comes to inelastic transport, in which passing electrons across the molecule trapped in between two metallic electrodes lose their energy through electron-phonon coupling, it is not necessarily obvious whether vibration affects the QI feature or not. In this paper, on the basis of a Hückel/tight-binding model, we address the inelastic transport through linear and cyclic polyenes which are conditioned to show QI. The zeroth-order Green's function approximated by the negative inverse of the adjacency matrix of a molecular graph is used in conjunction with the lowest order expansion of the self-consistent Born approximation. Owing to the simplification of the model, it just finds the limited applicability for the π-to-π scattering. Only topological aspects of dephasing are included. In such a theoretical construct, the alternant nature of the π-conjugated molecule is found helpful for classifying the dephasing patterns based on the parity of atomic sites. A rule is proposed, and it says that when both starred or both unstarred atoms are connected with the electrodes, QI always occurs, and atoms which belong to a different partite set from that of the atoms connected with the electrodes contribute to the inelastic π-to-π scattering. If QI occurs when a starred atom and an unstarred atom are connected with the electrodes, the contribution of the inelastic π-to-π scattering to the transport is expected to be unimportant.
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