Hückel（紧束缚）模型的格林函数
The Green's Function for the Hückel (Tight Binding) Model
摘要：
Hückel（紧束缚）模型的应用在量子化学和固态物理中无处不在。该模型的矩阵表示同构于二部图的无向顶点邻接矩阵，该矩阵也是拉普拉斯矩阵加两倍单位元。在本文中，我们分析计算的行列式，当它存在时，逆矩阵与格林函数，$\mathbf{G}$，$N\times N$休克尔矩阵。一个推论是一个封闭形式的表达式为调和和（方程。12）。然后我们将结果推广到线性尺寸为$N$的$d-$维格。逆的存在成为数论的一个问题。本文证明了数论中一个新的关于余弦和为零的定理，并用它证明了逆存在的充要条件是$N+1$和$d$是奇数，且$d$小于$N+1$的最小因子.我们证实了我们的结果，通过演示的格林函数的入口模式，并讨论相关的运输和导电性的应用。
英文摘要：
Applications of the Hückel (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 $N\times N$ Hückel 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.
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