TY - JOUR

T1 - Quantum interference, graphs, walks, and polynomials

AU - Tsuji, Yuta

AU - Estrada, Ernesto

AU - Movassagh, Ramis

AU - Hoffmann, Roald

N1 - Publisher Copyright:
Copyright © 2018, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2018/4/24

Y1 - 2018/4/24

N2 - 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 Hückel 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 cancelled 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 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.

AB - 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 Hückel 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 cancelled 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 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.

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M3 - Article

AN - SCOPUS:85093223139

JO - Quaternary International

JF - Quaternary International

SN - 1040-6182

ER -