# transition dipole moment for periodic systems?

The transition dipole moment $$\mu=\langle\varphi_a|q\hat r|\varphi_b\rangle$$ can be viewed as the off-diagonal matrix element of the position operator $$\hat r$$,multiplied by the charge $$q$$. As we know, the position operator is not well-defined for periodic systems. However, one can use the Berry phase formula as shown in [PRL 80,1800-1803] to define the position expectation value $$\langle X\rangle=\frac{L}{2\pi}\mathrm{Im}\ln\langle\Psi|e^{i\frac{2\pi}{L}\hat{X}}|\Psi\rangle$$where $$\hat{X}$$ is the 1D position operator and $$\Psi$$ is the many-body wave function. To evaluate the above equation the many-body wave function $$\Psi$$ can be replaced by the Slater determinant built from Kohn-Sham one-body occupied orbitals $$\varphi$$, then the above equation can be changed to $$\langle X\rangle=-\frac{L}{2\pi}\mathrm{Im}\ln\det \mathbf{S}$$ where $$S_{ab}=\langle\varphi_a|e^{-i\frac{2\pi}{L}{x}}|\varphi_b\rangle$$

Now I'd like to calculate the transition dipole moment $$\mu$$ between two Kohn-Sham orbitals, one is occupied and the other is empty, say $$\varphi_a,\varphi_b$$. If I understand correctly, $$\mu\not=S_{ab}$$. Is the transition dipole moment $$\mu=\langle\varphi_a|e^{-i\frac{2\pi}{L}\hat{X}}|\varphi_b\rangle$$? I guess no. Since the operator is a many-body operator but the orbital is one-body. Then how to calculate $$\mu$$?