Electric quadrupole operator for periodic systems

Question

Considering one component of the electric quadrupole operator $\hat{r}_x \hat{r}_y$, I'm wondering if the following equalities hold: $$ \langle u_{n\mathbf{k}} | \hat{r}_x \hat{r}_y |u_{m\mathbf{k}} \rangle = -i\langle \partial_{k_x} u_{n\mathbf{k}} | \hat{r}_y |u_{m\mathbf{k}} \rangle = \langle \partial_{k_x} u_{n\mathbf{k}} |\partial_{k_y} u_{m\mathbf{k}} \rangle $$ where $u_{n\mathbf{k}}=e^{-i\mathbf{k \cdot r}}\psi_{n\mathbf{k}}$ is the cell-periodic part of the Bloch function with $n$ and $\mathbf{k}$ the band index and crystal momentum.

Answer 1

To my understanding, the (approximate) substitution to transform the dipole from real to k-space is:^[1] $$\mathbf{r}\to i\exp(i\mathbf{k}\cdot \mathbf{r})\nabla_k\exp(-i\mathbf{k}\cdot \mathbf{r})\tag{1}\label{1}$$ Applying this same identity for the quadrupole operator, you should get something like:

$$\langle\psi_{n\mathbf{k}}|\mathbf{r}\mathbf{r}|\psi_{m\mathbf{k}}\rangle=-\langle u_{n\mathbf{k}}|\nabla_{\mathbf{k}}\nabla_{\mathbf{k}}|u_{m\mathbf{k}}\rangle\tag{2}\label{2}$$

where the main differences between our formulas are the sign of the right hand expression and the use of $\psi_{n\mathbf{k}}$ in the integrals on the lefthand side.

As discussed in [1], this substitution is only valid for insulating systems in weak electric fields, as there should be an additional term in Eq \eqref{1} related to induced current within the bands of the material.

References:

1. Kudin, K. N.; Scuseria, G. E. An efficient finite field approach for calculating static electric polarizabilities of periodic systems. J. Chem. Phys. 2000, 113 (18), 7779–7785. DOI: 10.1063/1.1315999.