Relationship between electric dipole moment and polarization? $$\vec{p}_\text{ind} = \alpha \vec{E}$$
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?
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.