When doing DFT calculations of periodic solids, from either $\mathbf{k \cdot p}$ perturbation theory or simply using the trick $\mathbf{r} \rightarrow i\partial_{\mathbf{k}}$, one can obtain $$ \langle \psi_{c\mathbf{k}}|\partial_\mathbf{k} \psi_{v\mathbf{k}}\rangle = -i \langle \psi_{c\mathbf{k}}| \mathbf{r} | \psi_{v\mathbf{k}}\rangle = -i \langle u_{c\mathbf{k}}|\mathbf{r}| u_{v\mathbf{k}}\rangle \tag{1}\label{1} $$ where $\mathbf{r}$ is the position operator, $\psi_{c\mathbf{k}}=u_{c\mathbf{k}}e^{i\mathbf{k \cdot r}}$ is the Bloch wave function, $c,v$ for conduction and valence bands, respectively. If we insert the Bloch function in the l.h.s of Eq. (1), we alternatively have: $$\tag{2} \begin{align} \langle \psi_{c\mathbf{k}}|\partial_\mathbf{k} \psi_{v\mathbf{k}}\rangle &= \langle u_{c\mathbf{k}}|e^{-i\mathbf{k \cdot r}}\partial_\mathbf{k} (u_{v\mathbf{k}}e^{i\mathbf{k \cdot r}}) \rangle \\ \tag{3}&= \langle u_{c\mathbf{k}}|e^{-i\mathbf{k \cdot r}}(\partial_\mathbf{k} u_{v\mathbf{k}}e^{i\mathbf{k \cdot r}}+i\mathbf{r}e^{i\mathbf{k \cdot r}}u_{v\mathbf{k}}) \rangle \\&= \langle u_{c\mathbf{k}}|\partial_\mathbf{k} u_{v\mathbf{k}}\rangle + i \langle u_{c\mathbf{k}}|\mathbf{r}| u_{v\mathbf{k}}\rangle = 0 \tag{4} \end{align} $$ So we get a different answer for the same initial braket. I believe there is something wrong with the second equation. Any idea?

  • $\begingroup$ could you provide more information? $\endgroup$
    – Cody Aldaz
    Commented Aug 26, 2021 at 17:10
  • 1
    $\begingroup$ I'm sorry but what else information do you need? $\endgroup$ Commented Aug 26, 2021 at 17:54
  • $\begingroup$ Can you give any references or links to the perturbation theory you are referring to? Some background for the setup and what the terms in the above equations specify? $\endgroup$
    – mykd
    Commented Aug 26, 2021 at 21:16
  • $\begingroup$ The $\mathbf{k \cdot p}$ perturbation theory can be found at wiki en.wikipedia.org/wiki/K%C2%B7p_perturbation_theory. Also please refer to Eq. 41 of worldscientific.com/doi/abs/10.1142/S0217979211058912. The matrix elements are used to calculate the optical properties. $\endgroup$ Commented Aug 26, 2021 at 22:25
  • $\begingroup$ The background is doing DFT calculations of periodic solids. $\endgroup$ Commented Aug 26, 2021 at 22:37

1 Answer 1


Expanding my comment to an answer, I believe the "trick" is incorrect.

As far as I'm aware, the relationship between $\boldsymbol{r}$ and $\boldsymbol{\nabla_k}$ is $$\boldsymbol{r}\to-i\exp(i\boldsymbol{k}\cdot \boldsymbol{r})\boldsymbol{\nabla_k}\exp(-i\boldsymbol{k}\cdot \boldsymbol{r})$$

This would mean at least the middle equality of equation [\ref{1}] is incorrect.


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