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I have calculated the dipole transition elements of electronic states $\langle a|D_1^m|b \rangle$ following the book of Cohen-Tannoudji (Complement $E_X$) and tried then to calculate from that the Rabi frequency $$\Omega_{a\to b}=\frac{1}{2\hbar}E_m \langle a|D_1^m|b \rangle.$$ But what is the direction of the E-field vector $E_m$ for the $\pi$, $\sigma^+$, $\sigma^-$ transitions?

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Could you please rephrase this question? Direction of electric field acting on a molecule is determined by the electric field vector. This can be related to position and motion of its charged sources, but has nothing to do with various "transitions" in the theory of molecules, because these are merely pairs of Hamiltonian eigenfunctions that have fixed orientation with respect to molecular body (rigid body made of nuclei) but have no fixed orientation with respect to electric field or coordinate system. – Ján Lalinský May 10 at 20:16
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The directions are:

$$\vec{e}_{\sigma^+}=\frac{1}{\sqrt{2}}(\vec{e}_x+i\vec{e}_y)$$ $$\vec{e}_{\pi} =\vec{e}_z$$ $$\vec{e}_{\sigma^-}=-\frac{1}{\sqrt{2}}(\vec{e}_x-i\vec{e}_y)$$

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