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至关重要，如Bistritzer-MacDonald模型所示。然而，这些变换在更复杂的系统中（例如存在机械弛豫模式时）可能难以实现。本研究旨在达成两个目标。首先，我们通过严谨地描述四个相关空间（即实空间、构型空间、动量空间和倒易空间）之间的变换，简化了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一种直接的算法。其次，我们将该方法应用于包含机械弛豫效应的扭曲双层石墨烯，并分析了近似的收敛速率。结果显示，层间相对扭转角度较小时，紧束缚耦合范围会增大，这表明在极小角度下包含机械弛豫时，Bistritzer-MacDonald模型的三近邻耦合假设并不充分。我们对此进行了量化分析，并通过数值模拟加以验证。