TY - JOUR

T1 - Quantum Interference, Graphs, Walks, and Polynomials

AU - Tsuji, Yuta

AU - Estrada, Ernesto

AU - Movassagh, Ramis

AU - Hoffmann, Roald

N1 - Funding Information:
We are grateful to S. Datta and M. H. Garner for their comments on this work, and the reviewers for their suggestions, especially of a summary paragraph for section 5, and the inclusion of a reader’s guide. Y.T. thanks the Research Institute for Information Technology (Kyushu University) for the computer facilities and financial support from JSPS KAKENHI Grant Number JP17K14440 and from Qdai-jump Research Program, Wakaba Challenge of Kyushu University. The work at Cornell was supported by the National Science Foundation through Grant CHE 1305872. E.E. thanks the Royal Society of London for a Wolfson Research Merit Award. R.M. thanks IBM TJ Watson Research for the freedom and support.
Funding Information:
Y.T. thanks the Research Institute for Information Technology (Kyushu University) for the computer facilities and financial support from JSPS KAKENHI Grant Number JP17K14440 and from Qdai-jump Research Program, Wakaba Challenge of Kyushu University. The work at Cornell was supported by the National Science Foundation through Grant CHE 1305872. E.E. thanks the Royal Society of London for a Wolfson Research Merit Award. R.M. thanks IBM TJ Watson Research for the freedom and support.

PY - 2018/5/23

Y1 - 2018/5/23

N2 - In this paper, we explore quantum interference (QI) 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 canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants 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. Furthermore, 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 (QI) 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 canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants 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. Furthermore, 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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U2 - 10.1021/acs.chemrev.7b00733

DO - 10.1021/acs.chemrev.7b00733

M3 - Review article

C2 - 29630345

AN - SCOPUS:85047513333

VL - 118

SP - 4887

EP - 4911

JO - Chemical Reviews

JF - Chemical Reviews

SN - 0009-2665

IS - 10

ER -