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

    All Science Journal Classification (ASJC) codes

    • Statistical and Nonlinear Physics
    • Mathematical Physics

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