Interaction of the oscillating magnetic field of electromagnetic radiation with a molecule or atom

Whenever we read rotational, vibrational and electronic spectroscopy, we find details on the interaction of the electric dipole moment of the molecule with the oscillating electric field of electromagnetic radiation. The selection rules are also derived on the basis of transition dipole moment operator on wavefunctions. I was talking to spectroscopists today and asked that why do spectroscopy textbooks (and authors) do not mention anything about oscillating magnetic field of light waves. The magnetic field also carries the same energy as the oscillating electric field. Doesn't that interact with electrons as well?* They said that interaction of the magnetic field with a molecule's electric dipole is very weak and hence nobody mentions about it. Does anyone have a good reference or an equation which shows that the oscillating magnetic field interacts very weakly with the molecule's electric dipole moment? They were mentioning Fermi's golden rule and at this stage I lost them.

__ * Electron spin resonance is an example.

• One simple way to intuit the neglect of magnetic field interactions is to note that few molecules have significant magnetic fields to interact with. First order magnetic fields (which would generate a strong interaction with the magnetic field component) are a product are a product of unpaired electrons and these are not that common in stable species. The exceptions are some transition metal compounds and the dioxygen molecule which is a diradical. ESR is an example of a spectroscopic tool that exploits the effect but is often used for exploring short-lived radical intermediates. Apr 11, 2019 at 9:09

The magnetic dipole transition intensity is typically about $$10^5$$ times smaller than the allowed electric dipole but as this can vary by ten orders of magnitude, second order transitions (with terms $$x^2,y^2$$ etc ) as well as magnetic dipole transitions can occasionally be important. The electric dipole matrix element is $$\langle a|\hat r|b\rangle$$, where $$\hat \mu =e\hat r$$, the second order dipole $$\langle a|\hat r \hat r|b\rangle$$ and magnetic dipole $$\langle a|\hat J|b\rangle$$ where $$\hat J$$ is the angular momentum operator $$\displaystyle \sim \langle a|z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z} |b\rangle$$.