### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 4887-4911 |

Number of pages | 25 |

Journal | Chemical Reviews |

Volume | 118 |

Issue number | 10 |

DOIs | |

Publication status | Published - May 23 2018 |

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### All Science Journal Classification (ASJC) codes

- Chemistry(all)

### Cite this

*Chemical Reviews*,

*118*(10), 4887-4911. https://doi.org/10.1021/acs.chemrev.7b00733

**Quantum Interference, Graphs, Walks, and Polynomials.** / Tsuji, Yuta; Estrada, Ernesto; Movassagh, Ramis; Hoffmann, Roald.

Research output: Contribution to journal › Review article

*Chemical Reviews*, vol. 118, no. 10, pp. 4887-4911. https://doi.org/10.1021/acs.chemrev.7b00733

}

TY - JOUR

T1 - Quantum Interference, Graphs, Walks, and Polynomials

AU - Tsuji, Yuta

AU - Estrada, Ernesto

AU - Movassagh, Ramis

AU - Hoffmann, Roald

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.

UR - http://www.scopus.com/inward/record.url?scp=85047513333&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047513333&partnerID=8YFLogxK

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 -