{
  "id": "1407.4780",
  "version": "v3",
  "published": "2014-07-17T19:12:54.000Z",
  "updated": "2016-02-25T01:07:29.000Z",
  "title": "The Green's Function for the Hückel (Tight Binding) Model",
  "authors": [
    "Ramis Movassagh",
    "Gilbert Strang",
    "Yuta Tsuji",
    "Roald Hoffmann"
  ],
  "comment": "14 + 6 pages, 6 figures. v2: minor typos fixed. The new proof of theorem 1 applies for more general matrices",
  "categories": [
    "math-ph",
    "cond-mat.str-el",
    "math.MP",
    "quant-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-13T21:37:27.000Z",
      "title": "The Exact Form of the Green's Function of the Hückel (Tight Binding) Model",
      "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 $N\\times N$ 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 $s<r$ is odd; $G\\left(r,s\\right)=0$ 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.",
      "comment": "12 + 6 pages, 6 figures. v2: minor typos fixed. The new proof of theorem 1 applies for more general matrices",
      "journal": null,
      "doi": null,
      "authors": [
        "Ramis Movassagh",
        "Yuta Tsuji",
        "Roald Hoffmann"
      ]
    },
    {
      "version": "v3",
      "updated": "2016-02-25T01:07:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "greens function",
      "tight binding",
      "exact form",
      "laplacian matrix plus twice",
      "solid state physics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.4780M"
    }
  }
}