Quantum Interference, Graphs, Walks, and Polynomials
| Autor: | Tsuji, Yuta, Estrada, Ernesto, Movassagh, Ramis, Hoffmann, Roald |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Chemical Reviews Article ASAP, Publication Date (Web): April 9, 2018 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1021/acs.chemrev.7b00733 |
| Popis: | In this paper, we explore quantum interference in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or H{\"u}ckel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For non-alternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for non-alternants, from the cancellation of odd and even-length walk terms. We report some progress, but not a complete resolution of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. And we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function. Comment: 58 pages. 19 Figures. 148 references |
| Databáze: | arXiv |
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