[Paper Review] The Exact Form of the Green's Function of the H\"uckel (Tight Binding) Model

Ramis Movassagh, Yuta Tsuji+1 more|arXiv (Cornell University)|Jul 17, 2014

Graph theory and applications3 citations

TL;DR

This paper analytically derives the exact form of the Green's function for H"uckel (tight binding) models in linear chains and cyclic systems, proving that the inverse matrix is real, symmetric, and follows a closed-form expression based on parity and distance. For rings, it shows the inverse is a Toeplitz matrix and establishes number-theoretic conditions for invertibility in d-dimensional lattices, with applications to electronic transport and conductivity.

ABSTRACT

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 $N imes N$ 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 $G\left(r,s ight)=\left(-1 ight)^{\frac{r+s-1}{2}}$ when $ $$r$ is even and $s<r$ is odd; $G\left(r,s ight)=0$ otherwise. A corollary is a closed form expression for a Harmonic sum (Eq. 9). For a ring we calculate the inverse, give formulas for the entries and find that it is always a Toeplitz matrix. 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 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 numerical demonstrations of the entry patterns of the Green's function and discuss applications related to transport and conductivity.

Motivation & Objective

• To derive the exact analytical form of the Green's function for H"uckel matrices in one-dimensional linear chains and cyclic systems.
• To determine the conditions under which the inverse of the H"uckel matrix exists, particularly in higher-dimensional lattices.
• To establish a connection between the invertibility of the H"uckel matrix and number theory, specifically divisibility conditions on system size.
• To provide closed-form expressions for matrix entries that enable direct computation of electronic properties such as conductivity and transport.

Proposed method

• The H"uckel matrix is represented as the adjacency matrix of a bipartite graph, isomorphic to the Laplacian plus twice the identity matrix.
• Analytical computation of the determinant and inverse is performed using combinatorial and algebraic techniques tailored to linear chains and rings.
• For linear chains, the Green's function entries are derived using parity-based sign rules: $ G(r,s) = (-1)^{(r+s-1)/2} $ when $ r $ even and $ s < r $ odd, and zero otherwise.
• For cyclic systems, the inverse is shown to be a Toeplitz matrix, with entries expressible via trigonometric sums and symmetry considerations.
• The analysis is extended to d-dimensional lattices by reducing the invertibility condition to a number-theoretic problem involving $ N+1 $ and the dimension $ d $.
• Numerical verification is used to confirm the entry patterns of the Green's function and validate analytical predictions.

Experimental results

Research questions

• RQ1 What is the exact analytical form of the Green's function for a one-dimensional H"uckel model with open boundary conditions?
• RQ2 How does the structure of the inverse matrix differ between linear chains and cyclic systems in the H"uckel model?
• RQ3 Under what number-theoretic conditions does the H"uckel matrix inverse exist in d-dimensional lattices of size $ N $?
• RQ4 Can a closed-form expression be derived for the entries of the Green's function in cyclic systems, and is it always a Toeplitz matrix?
• RQ5 What are the implications of the derived Green's function for electronic transport and conductivity in such systems?

Key findings

• For an open linear chain, the Green's function is a real symmetric matrix with entries $ G(r,s) = (-1)^{(r+s-1)/2} $ when $ r $ is even and $ s < r $ is odd, and zero otherwise.
• A corollary of this result is a closed-form expression for a specific harmonic sum, as given in Equation (9) of the paper.
• In cyclic systems, the inverse of the H"uckel matrix is always a Toeplitz matrix, with entries that depend on the distance and parity of site indices.
• For d-dimensional lattices, the inverse exists if and only if $ N+1 $ and $ d $ are odd and $ d $ is smaller than the smallest divisor of $ N+1 $.
• Numerical simulations confirm the predicted entry patterns of the Green's function, supporting the analytical derivations.
• The derived Green's function enables direct computation of transport and conductivity properties in H"uckel-model systems.