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I am trying to compute the oscillator strength in a molecule. Therefore I've got its Energy level values plus both wave functions $\Psi_i$ and $\Psi_j$ for the transition I want to calculate the oscillator strength for.

In general, the oscillator strength of a transition from $E_i \rightarrow E_j$ is defined as

$f_{ij} =\frac{4m_e\pi\nu}{3e^2\hbar}D^2$

where D stands for the transition moment integral which shall be

$D_{ij} = \int\Psi_j\hat{\mu}\Psi_i$.

Now I am wondering how the transition dipole operator $\hat{\mu}$ is defined, since I didn't find helpful.

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  • $\begingroup$ So, the crux of your question is not about calculating $f_{ij}$ per se, but rather about how $\hat{\mu}$ is defined? $\endgroup$ – orthocresol Aug 13 '18 at 6:06
  • $\begingroup$ @orthocresol Well, I'd be extremely glad to get either an answer on how to calculate $f_{ij}$ or how the transition dipole operator is defined. Both would lead to an answer. $\endgroup$ – p_punkt Aug 13 '18 at 6:11
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    $\begingroup$ The $\mu=-e\cdot\bar r$ where the vector $\bar r$ is the sum of all charges on the molecule. (The formula may also assume you average over x, y, z directions as the radiation can be polarised along any axis so divide by 3.) $\endgroup$ – porphyrin Aug 13 '18 at 12:02
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    $\begingroup$ @porphyrin Thanks for your comment! This would mean that the transition moment integral becomes $D_{ij} = -e\int\Psi_j x\Psi_i dx$ if I only respect the x-direction, right? $\endgroup$ – p_punkt Aug 13 '18 at 12:29
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    $\begingroup$ yes where $x$ is the sum of charges in just the $x$ direction, i.e $ex$ is just the dipole component along $x$. $\endgroup$ – porphyrin Aug 13 '18 at 13:59

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