MathemaTB: A Mathematica package for tight-binding calculations

MathemaTB is a package developed to enable tight-binding calculations within Mathematica. The package presents 62 functions dedicated to facilitating these quantum mechanical computations. MathemaTB offers functionalities to carry out matrix manipulation, data analysis and visualizations on molecules, wave functions, Hamiltonians, coefficient matrices, and energy spectra, providing a unique insight into the interplay between geometric and electronic structure. Crystal orbitals, projected dispersions and densities of states can be obtained with only a few lines of code. The effect of different structures, heteroatoms and tight-binding parameters can easily be explored. Calculations can be carried out on molecules (Hückel-type calculations) or on systems with periodicities in one, two or three dimensions. Particularly powerful features are the possibility to plot band structures both along paths (one-dimensional) and over planes (two-dimensional) in reciprocal space, where in each case the localization of the wave function onto different sites, symmetries or basis functions can be visualized with color coding. Further features involve crystal orbital plotting with color coding of the complex phase, mean field Hubbard tight-binding and manipulation of the Hamiltonian matrix with numerical and symbolic elements. The conjunction of tight-binding functions, matrix algebra functions for symmetry, overlap and change of basis, wave function- and dispersion functions and a high degree of interactivity and flexibility makes MathemaTB a useful package for electronic structure calculations.MathemaTB是一款专为在Mathematica平台中开展紧束缚（tight-binding）计算而开发的工具包。该工具包内置62个专用函数，用于辅助完成各类量子力学计算。MathemaTB支持对分子、波函数、哈密顿量（Hamiltonian）、系数矩阵及能谱开展矩阵操作、数据分析与可视化工作，可帮助研究者深入剖析几何结构与电子结构间的相互作用关系。仅需数行代码即可获取晶体轨道、投影能带色散与态密度。用户可便捷地探究不同结构、杂原子及紧束缚参数对体系的影响。该工具包可针对分子（休克尔（Hückel）型计算）或具备一维、二维乃至三维周期性的体系开展计算。其尤为突出的功能是可在倒易空间（reciprocal space）中绘制沿路径（一维）及平面（二维）分布的能带结构，且可通过色彩编码直观呈现波函数在不同位点、对称性或基函数上的局域化分布情况。其余功能还包括：对复相位进行色彩编码的晶体轨道绘图、平均场哈伯德（Hubbard）紧束缚计算，以及支持数值与符号元素混合操作的哈密顿量矩阵处理功能。紧束缚相关函数、用于对称性处理、重叠积分与基矢变换的矩阵代数函数、波函数与能带色散函数，结合高度的交互性与灵活性，使MathemaTB成为一款适用于电子结构计算的实用工具包。