The Exact Form of the Green's Function of the Hückel (Tight Binding) Model

The applications of the H\"uckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation 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 for linear chains and cyclic systems. For an open linear chain we prove that $\mathbf{G}$ is a real symmetric matrix whose entries are $G\left(r,s\right)=\left(-1\right)^{\frac{r+s-1}{2}}$ when $ $$r$ is even and $s