### Abstract

Applications of the Huckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Huckel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)).We then extend the results to d-dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.

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

Number of pages | 1 |

Journal | Journal of Mathematical Physics |

Volume | 58 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*58*(3). https://doi.org/10.1063/1.4977080

**The green's function for the huckel (tight binding) model.** / Movassagh, Ramis; Strang, Gilbert; Tsuji, Yuta; Hoffmann, Roald.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 58, no. 3. https://doi.org/10.1063/1.4977080

}

TY - JOUR

T1 - The green's function for the huckel (tight binding) model

AU - Movassagh, Ramis

AU - Strang, Gilbert

AU - Tsuji, Yuta

AU - Hoffmann, Roald

PY - 2017/3/1

Y1 - 2017/3/1

N2 - Applications of the Huckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Huckel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)).We then extend the results to d-dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.

AB - Applications of the Huckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Huckel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)).We then extend the results to d-dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.

UR - http://www.scopus.com/inward/record.url?scp=85014779087&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014779087&partnerID=8YFLogxK

U2 - 10.1063/1.4977080

DO - 10.1063/1.4977080

M3 - Article

AN - SCOPUS:85014779087

VL - 58

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -