论文标题
对Bloch状态的速度，动量和位置矩阵元素的全面研究，使用局部轨道基础
A comprehensive study of the velocity, momentum and position matrix elements for Bloch states using a local orbital basis
论文作者
论文摘要
我们介绍了速度操作员的全面研究，$ \ hat {\ boldsymbol {v}} = \ frac {i} {\ hbar} [\ hat {h}，\ hat {\ hat {\ boldsymbol {r}}] $速度运算符是评估多种物理特性及其计算的关键，从实际和基本的角度来看，它一直是数十年来的长期辩论。我们的工作总结了文献中发现的不同方法，将它们连接起来并填补了有时不合格的推导中的空白。特别是，我们专注于使用局部轨道基集，其中Bloch Hamiltonian Matrix的$ K $ derivivic无法近似速度运算符。除其他事项外，我们还展示了如何在没有明确的数学步骤的情况下找到正确的表达，浆果连接如何在此表达中脱颖而出，以及如何正确处理文献中共存的两个流行量规选择。最后，我们通过与规范动量操作员的识别与其真实空间评估进行比较，探索了其在密度功能理论计算中的使用。此外，此比较使我们了解了非本地校正的重要性，这可能使天真的动量 - 速度对应关系无效。
We present a comprehensive study of the velocity operator, $\hat{\boldsymbol{v}}=\frac{i}{\hbar} [\hat{H},\hat{\boldsymbol{r}}]$, when used in crystalline solids calculations. The velocity operator is key to the evaluation of a number of physical properties and its computation, both from a practical and fundamental perspective, has been a long-standing debate for decades. Our work summarizes the different approaches found in the literature, connecting them and filling the gaps in the sometimes non-rigorous derivations. In particular we focus on the use of local orbital basis sets where the velocity operator cannot be approximated by the $k$-derivative of the Bloch Hamiltonian matrix. Among other things, we show how the correct expression can be found without unequivocal mathematical steps, how the Berry connection makes its way in this expression, and how to properly deal with the two popular gauge choices that coexist in the literature. Finally, we explore its use in density functional theory calculations by comparing with its real-space evaluation through the identification with the canonical momentum operator. This comparison offers us, in addition, a glimpse of the importance of non-local corrections, which may invalidate the naive momentum-velocity correspondence.
