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$$\vec{p}_\text{ind} = \alpha \vec{E}$$

The induced dipole moment is the polarizability times the electric field vector.

$$\vec{P}(\omega) \propto \chi^{(1)}(\omega) \vec{E}(\omega)$$

The polarization is proportional to the susceptibility times the electric field vector.

In spectroscopy we used the dipole approximation. Unfortunately, I often mix up the different formulas. Is there a relationship between both formulas, or do both equations tell us exactly the same thing?

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In fact, the first equation is applicable on a microscopic scale (for an atom or a molecule): $\space$ $\vec{p}_\text{ind}=\alpha \vec{E}_\text{local} $

While the second one is applicable on a macroscopic scale (for the bulk): $\space$ $\vec{P}(\omega) \propto \chi^{(1)}(\omega) \vec{E}_\text{ext}(\omega) $ So, the polarisability is a microscopic quantity, while the susceptibility is a macroscopic quantity.

You have to notice that the electric field in the first equation is the local field, while it's the external field in the second one. For more details, please see this page.

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    $\begingroup$ Indeed, the microscopic / macroscopic versions carry forward also for $\beta$ and $\chi^{(2)}$, $\gamma$, etc. $\endgroup$
    – Geoff Hutchison
    Jun 30, 2015 at 23:33
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    $\begingroup$ I agree with @Geoff Hutchison that the microscopic / macroscopic versions carry forward also for (β and χ(2)), ( γ and χ(3)) etc. $\endgroup$
    – Yomen Atassi
    Jul 1, 2015 at 12:39
  • $\begingroup$ @YomenAtassi I like your answer and as well the additional page. However, I've still confused, because in Spectroscopy both have been used and we only took a look into small molecules, not bulk. I also have no idea to which of the equations (eq. 1 or eq. 2) the formula for the dipole approximation $\hat{V}=-\hat{\mu}\overrightarrow{E}$ is best assigned. (related to my two weak old, but still unsolved question: chemistry.stackexchange.com/questions/33170/…) $\endgroup$
    – laminin
    Jul 1, 2015 at 18:38
  • $\begingroup$ In the scope of our lecture of Spectroscopy (atoms, small molecules, aggregates) we have restricted to linear spectroscopy which means there is no dependence of quadratic terms in $\overrightarrow{E}$ in eq(2). I'm not sure but may be it's possible, you get also quadratic dependency in $\overrightarrow{E}$ for eq(1) if we deal with non-linear spectroscopy? $\endgroup$
    – laminin
    Jul 1, 2015 at 19:49
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    $\begingroup$ Regarding your second comment, you are right! There is also a quadratic dependency in E for eq(1) if we deal with non-linear spectroscopy. As for your first comment, the formula for the dipole approximation is best assigned to the first equation. You said" in Spectroscopy both equations have been used and you only took a look into small molecules, not bulk": I need to look more in depth to the use of these equations in your course in order to reply. But I think they use the second eq when they talk about the emission or absorption of the studied medium (as a bulk). First eq for single molec. $\endgroup$
    – Yomen Atassi
    Jul 2, 2015 at 2:02

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