Transition dipole matrix element

Question

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?

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.