Momentum space transformations for incommensurate 2D electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer-MacDonald model. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. Firstly, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straight-forward algorithm for writing the complex momentum space model from the original real space model. Secondly, we implement this for twisted bilayer graphene with mechanical relaxation affects included. We also analyze the convergence rates of the approximations, and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer-MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation.

翻译：非公度二维电子结构计算中的动量空间变换对于降低计算成本以及以更具物理意义的形式表示数据具有重要意义，例如Bistritzer-MacDonald模型所示。然而，在存在机械弛豫图案等复杂系统中，这些变换的实现可能较为困难。本研究旨在实现两个目标：首先，通过严格推导四个相关空间（即实空间、构型空间、动量空间和倒易空间）之间的变换关系，简化对该变换的理解与实现过程。这为从原始实空间模型构建复杂动量空间模型提供了直观算法。其次，我们将该算法应用于包含机械弛豫效应的扭曲双层石墨烯体系。我们分析了近似方法的收敛速率，并表明层间相对扭转角度越小，紧束缚耦合范围越大，从而证明在考虑微小角度下的机械弛豫时，Bistritzer-MacDonald模型的三近邻耦合近似不再充分。我们通过数值模拟对这一结论进行量化验证。