Why do molecules have to have a change in dipole moment in order for them to be IR active?

My chemistry textbook keeps saying that in order for a molecule to be able to absorb infrared radiation, it has to have a change in dipole moment when the bond vibrates.


I don't understand why that is. Some clarification would be appreciated.




There is an intuitive reason why a vibration is IR-active only if it involves a change in the dipole moment.


Recall that the typical wavelength of IR radiation ($\sim 10\mu\mathrm{m}$) is much larger than the typical size of a molecule ($\sim 1\mathrm{nm}$). Hence, to a very good approximation, the (time-varying) **electric field of the IR radiation is spatially uniform within a molecule.**


Now observe that under a spatially uniform electric field, **all positive charges, regardless of their positions, are pushed to a common direction, and all negative charges are pushed to exactly the opposite direction**. In this case, **can the change in the total dipole moment, defined as $\Delta \vec{\mu} = \sum\_{i} q\_{i} \Delta \vec{r}\_{i}$, be equal to zero?**


Clearly, the answer is **"No."** The displacements $\{\Delta \vec{r}\_{i}\}$ are parallel between like charges and antiparallel between opposite charges. Therefore, $\{q\_{i}\Delta\vec{r}\_{i}\}$ are all parallel to one another, and $\vec{\Delta\mu}$ cannot be equal to zero.


So far, I have argued that a motion induced by a spatially uniform electric field always involves a change in the dipole moment. A corollary is that **any motion that does not involve a change in the dipole moment cannot be induced by a spatially uniform electric field (of IR radiation).** 


As a simple example, imagine applying a spatially uniform electric field to a $\ce{CO\_{2}}$ molecule such that the direction of the field is parallel to the length of the molecule. Can this field ever induce a symmetric stretch, in which the two oxygen atoms move in opposite directions? Certainly no. Both oxygen atoms are negatively charged by exactly the same amount. Under a spatially uniform electric field, they are pushed to the same direction by the same magnitude, and there is no room for a symmetric stretch.


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As a molecule vibrates, if there is a fluctuation in its dipole moment, then this induces an electric field that interacts with the electric field associated with the infra red radiation. If there is a match in frequency of the radiation and the natural vibration of the molecule, absorption occurs. 


For more details, please see [this page](http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Vibrational_Spectroscopy/Infrared_Spectroscopy/Infrared%3A_Theory) on the UC Davis ChemWiki website. 


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