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

We acknowledge the referee the recommendation for publication. We also include below a small response to the last comment by our referee. We count the equations the referee's last response from 1 to 4, as seen in the webpage for the manuscript submission.

Overall I judge that the authors have done a conscientious job in replying to my comments and making corresponding changes in the manuscript. I still have my reservations about the way my comments 1(b) and 1(d) were answered. (Incidentally, the authors misquoted 1(b) with a cut-and-paste typo; the sentence should have read ¨First, isn't it true that $\langle \boldsymbol{k}|\boldsymbol{v}|n'\boldsymbol{k} \rangle_v$ in Eq. (1) contains a $\delta_{\boldsymbol{k}\boldsymbol{ k}'}$?".) If we accept from other arguments that $\langle n \boldsymbol{k}|\boldsymbol{v}|n'\boldsymbol{k}'\rangle_v$ is diagonal in $\boldsymbol{k}$ (e.g., using that $\boldsymbol{v}$ is a periodic operator), and in view of the $\omega_{n\boldsymbol{k},n'\boldsymbol{k}}$ prefactor, we only have to evaluate $\langle \boldsymbol{k}|\boldsymbol{r}|n'\boldsymbol{k} \rangle_v$ for $\boldsymbol{k}=\boldsymbol{k}'$ and $n\neq n'$. Here I adopt the notation that $\tilde{\psi}$ are the wave functions normaized to volume $L=Na$ while $\psi=\sqrt{N}\tilde{\psi}$ are normalized to a primitive cell. Then in 1D

$$ \begin{split} \langle \boldsymbol{k}|\boldsymbol{r}|n'\boldsymbol{k}\rangle_v&=\int_{0}^{Na}dx \ x \tilde{\psi}_{nk}^{\ast}(x) \tilde{\psi}_{n'k}(x) \\ &=\frac{1}{N}\int_{0}^{Na}dx \ x \psi_{nk}^{\ast}(x) \psi_{n'k}(x) \\ &=\frac{1}{N}\sum_{j=0}^{N-1}\int_{ja}^{(j+1)a}dx \ x \psi_{nk}^{\ast}(x) \psi_{n'k}(x) \\ &=\frac{1}{N}\sum_{j=0}^{N-1}\int_{0}^{a}dx (x+ja) \psi_{nk}^{\ast}(x) \psi_{n'k}(x) \end{split} $$

where I have used that

$$ \psi^{\ast}_{nk}(x)\psi_{n'k}(x) $$

is periodic under $x \rightarrow x+a$. Then

$$ \begin{split} \langle n\boldsymbol{k}|\boldsymbol{r}|n'\boldsymbol{k} \rangle_v=& \Big( \frac{1}{N} \sum_{j=0}^{N-1}\Big) \int_{0}^{a}dx \ x \psi_{nk}^{\ast}(x) \psi_{n'k}(x) \\ & \Big( \frac{a}{N} \sum_{j=0}^{N-1}j\Big) \int_{0}^{a}dx \psi_{nk}^{\ast}(x) \psi_{n'k}(x) \end{split} $$

The second integral vanishes by orthogonality of the wave functions and the first factor in the first term is unity, so

$$ \langle n\boldsymbol{k} |\boldsymbol{r}|n'\boldsymbol{k}\rangle_v=\int_{0}^{a}dx \ x \psi_{nk}^{\ast}(x) \psi_{n' \boldsymbol{k}}(x) $$

I believe another argument along these lines allows to show that $C_{nk,n'k}$ can be similarly writen in terms of the boundaries of the primitive cell. This kind of development is that I had in mind when I wrote ¨I suspect Eqs. (11-12) can be recast in terms of integrals over a the interior and the boundary of a single primitive cell, with wave functions normalized to the unit cell''. I leave it as an option for the authors to discus this somehow in their revised manuscript.

We thank the referee again for providing such detailed feedback. First, let us write the general version of Eq. (3) in this response for the general, nondiagonal case, (we use $\tau$ as the integration variable in the unit cell)

$$ \begin{split} \langle n\boldsymbol{k}|\boldsymbol{r}|n'\boldsymbol{k}' \rangle_v=& \delta_{kk'} \int_{\tau_0}^{\tau_0+a}d\tau \ \tau u_{nk}^{\ast}(\tau) u_{n'k}(\tau) \\ & \Big( \frac{a}{N} \sum_{j=0}^{N-1}j \Big) \int_{\tau_0}^{\tau_0+a}d\tau \psi_{nk}^{\ast}(\tau) \psi_{n'k'}(\tau). \end{split} $$

Note that there is still a dependance with $\tau_0$, that can be related to the original $x_0$ origin for the integration volume $v$. Eq. (3) is a special case of this with $\tau_0=0$. Note also that the second term only includes the orthogonality condition for the $k$-diagonal case, in which we obtain the normalization condition for the periodic part: \begin{equation} \int_{\tau_0}^{\tau_0+a}d\tau \psi_{nk}^{\ast}(\tau) \psi_{n'k}(\tau)= \int_{\tau_0}^{\tau_0+a}d\tau u_{nk}^{\ast}(\tau) u_{n'k}(\tau) =\delta_{nn'}.
\end{equation} With this, we see that the simple form of Eq. (3) can only be obtained for the $k-$diagonal case, in which the position matrix elements still depend on the origin, as in the general case covered with Eqs (11) and (12) in the manuscript. The advantage of this reduction may be in terms of computational terms, where it could be desired to have a smaller integration grid. As nothing fundamental is added in this step, we prefer to avoid including this discussion in the manuscript.