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

TL;DR

This paper analytically derives the Green's function for the Hückel model, extending results to higher dimensions, and establishes a number-theoretic criterion for the inverse's existence, with implications for quantum transport.

Contribution

It provides a closed-form expression for the Green's function of the Hückel model and introduces a new number theory theorem to determine inverse existence in higher dimensions.

Findings

Derived explicit formulas for the Green's function.

Proved inverse existence conditions based on number theory.

Demonstrated applications to transport and conductivity.

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.