A comprehensive study of the velocity, momentum and position matrix elements for Bloch states using a local orbital basis

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.