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

Ramis Movassagh, Gilbert Strang, Yuta Tsuji, Roald Hoffmann

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish
Number of pages1
JournalJournal of Mathematical Physics
Volume58
Issue number3
DOIs
Publication statusPublished - Mar 1 2017

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Tight-binding
Green's function
Green's functions
Number theory
number theory
matrices
Small Divisors
Quantum Chemistry
Laplacian Matrix
Matrix Representation
Adjacency Matrix
Bipartite Graph
Conductivity
Corollary
Determinant
Closed-form
Isomorphic
Harmonic
Odd
Physics

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Journal of Mathematical Physics, Vol. 58, No. 3, 01.03.2017.

Research output: Contribution to journalArticle

Movassagh, Ramis ; Strang, Gilbert ; Tsuji, Yuta ; Hoffmann, Roald. / The green's function for the huckel (tight binding) model. In: Journal of Mathematical Physics. 2017 ; Vol. 58, No. 3.
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