A comprehensive study of the velocity, momentum and position matrix elements for Bloch states: Application to a local orbital basis
A comprehensive study of the velocity, momentum and position matrix elements for Bloch states: Application to a local orbital basis
We present a comprehensive study of the velocity operator, \hat{\boldsymbol{v}}=\frac{i}{\hbar} [\hat{H},\hat{\boldsymbol{r}}]\, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mover><mml:mstyle mathvariant="bold"><mml:mi>𝐯</mml:mi></mml:mstyle><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mfrac><mml:mi>i</mml:mi><mml:mi>ℏ</mml:mi></mml:mfrac><mml:mo stretchy="false" form="prefix">[</mml:mo><mml:mover><mml:mi>H</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mover><mml:mstyle mathvariant="bold"><mml:mi>𝐫</mml:mi></mml:mstyle><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mo stretchy="false" form="postfix">]</mml:mo><mml:mspace width="0.167em" /></mml:mrow></mml:math> , when used in crystalline solids calculations. The velocity operator is key to the evaluation of a number of physical properties and its computation, both …