# Why do we say “approximation” in the dipole approximation in spectroscopy?

In the dipole approximation, the following relation holds:

$$\hat{V} = -\hat{\mu} \cdot \vec{E}.$$

When we say "approximation", I guess we want to point out that we are dealing with linear spectroscopy, where we have the polarizability of the molecule proportional to the electric susceptibility of a molecule:

$$\vec{p}(\omega) = k\chi \vec{E}.$$

If the wavelength of the light is not larger than the atom size, the dipole approximation is no longer a good approximation. This is probably also true for large molecules. I'm not sure if this has to do with large molecules no longer having symmetry, so transitions might be allowed which otherwise should be forbidden.

Is this correct?

As per your linked reference, it's an approximation because only the first term in the expansion (here, unity) is considered.

The issue with transitions that are allowed or forbidden arises as a result of using such an approximation: "Forbidden" transitions are not strictly forbidden, it's just that they are weak and, in the linear approximation of the transition dipole operator, are usually not observed.

• Well, my question is more, if there is a relation between the "dipole approximation" formula on the top and the "linear spectroscopy" formula? (meaning: Are we dealing with non-linear spectroscopy when we don't make the dipole approximation?) – laminin Jun 24 '15 at 10:54
• The term you have - interaction of a dipole in an electric field - is related to what is called "linear spectroscopy" in that it adequately describes the interaction of the system with a single source of the external electric field: That is, the allowed (and observed) transitions are those that arise from one field and not (as in the non-linear case) from more than one. – Todd Minehardt Jun 24 '15 at 12:51
• So for non-linear spectroscopy the dipole approximation $\hat{V}=-\hat{\mu}\overrightarrow{E}$ is no longer adequate? – laminin Jun 26 '15 at 15:12
• I think it's adequate - but I believe if you want a model that describes the system being studied at a higher degree of accuracy, it would be wise to include higher-order terms when fitting data or modeling a system computationally. – Todd Minehardt Jun 27 '15 at 2:18
• If it's adequate: linear SpY and dipole approximation have not the same meaning/origin. – laminin Jun 27 '15 at 8:48