39
views
0
recommends
+1 Recommend
0 collections
4
shares
• Record: found
• Abstract: found
• Article: found
Is Open Access

# The Green's Function for the H\"uckel (Tight Binding) Model

Preprint

Bookmark
There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

### Abstract

Applications of the H\"uckel (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, $$\mathbf{G}$$, of the $$N\times N$$ H\"uckel 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.

### Author and article information

###### Journal
2014-07-17
2016-02-24
1407.4780