The Green’s function for the Hückel (tight binding) model
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Author(s)
Movassagh, Ramis
Tsuji, Yuta
Hoffmann, Roald
Strang, W. Gilbert
Date Issued
March 2017
Journal
Journal of Mathematical Physics
Publisher
AIP Publishing
Citation
Movassagh, Ramis et al. “The Green’s Function for the Hückel (tight Binding) Model.” Journal of Mathematical Physics 58, 3 (March 2017): 033505
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1063/1.4977080
