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Below is equation for dipole moment transition: $$\langle \Psi_k| p |\Psi_i \rangle = \langle \Phi \chi_{N,k}|p_e+p_N|\Phi \chi_{N,k} \rangle = \langle \chi_{N,k}|\langle\Phi_k|p_e|\Phi_i\rangle |\chi_{N,i}\rangle + \langle \chi_{N,k}|p_N\langle\Phi_k|\Phi_i\rangle |\chi_{N,i}\rangle$$ Where $\chi$ is a nuclear wave function, $\Phi$ is electronic wave function and $p_e, p_N$ are dipole moment of electron and nucleus, respectively.
My question is why do we put this expression $\langle\Phi_k|p_e|\Phi_i\rangle$ between nuclear wave function $\chi$. As far as I understand $\langle\Phi_k|p_e|\Phi_i\rangle$ doesn't have anything in common. My confusion comes from the fact that we don't put nuclear dipole moment $p_N$ between electronic wave function.

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  • $\begingroup$ Just a hunch, but if you start with something like a Born-Oppenheimer approximation, the nuclear degrees of freedom are much slower than the electronic degrees of freedom. Therefore, as far as the electrons are concerned, the nuclear degrees of freedom are constants, and so the nuclear stuff can be pulled out of any electronic overlap integrals. However, those electronic overlap integrals are functions of nuclear coordinates, and so they have to stay inside the nuclear overlap integrals. $\endgroup$
    – march
    Oct 27, 2021 at 16:50

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