# Transition dipole matrix element

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?

• could you provide more information? Commented Aug 26, 2021 at 17:10
• I'm sorry but what else information do you need? Commented Aug 26, 2021 at 17:54
• 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?
– mykd
Commented Aug 26, 2021 at 21:16
• 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. Commented Aug 26, 2021 at 22:25
• The background is doing DFT calculations of periodic solids. Commented Aug 26, 2021 at 22:37

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