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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 $\mu$?

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