Any quadrupole approximation? Any example? In atomic and molecular physics we quite often encounter with electric dipole approximation. The dipole approximation we do when the wave-length of the type of electromagnetic radiation which induces, or is emitted during, transitions between different atomic energy levels is much larger than the typical size of a light atom. This is mostly the case. I have two questions regarding this:
1) Is there any case where we use quadrupole approximation or higher?
2) In the case of transition in molecules (for eg. large organic molecules or polymers) the size of the molecule is larger than the EM radiation. This case how we choose the approximation?
 A: 
The tides are essentially caused by the quadrupole component of the moon's gravitational field, as shown in this picture.
If you think about it, the classic Stern-Gerlach experiment also depends on a quadrupole field. It's a basic fact of magnetostatics that you can't have a field gradient in the x direction without a complimentary gradient in the y direction. This has consequences: most importantly, it means the usual description of the experiment, with "two dots on the screen", is nonsense. You can't split a ray of silver atoms into two paths, up-and-down, without also splitting them  into two paths, left-and-right. Because there's just as much field gradient in the y direction as the x direction. 
I discuss this in my blogpost here:http://marty-green.blogspot.ca/2011/12/quantization-of-spin-revisited.html . The actual pattern you would get for a ray of silver atoms would be a donut shape, not two dots. And here is the pattern you would get on the screen for a polarized ray, if you could create a ray with all spins up:

A: First, take a look at the comment made by @Qmechanic.
I think the relation between the size of the molecule and the wavelength of the EM mode interacting with it is more of a Near-Field issue.
The general, Multipole method, simply addresses the case where (in the Classical picture) the distribution of charges in interaction with the EM field is not only a point charge, a couple of charges, etc. Any complex molecule with non-trivial symmetry will have higher rank EM multipoles. With increasing multipole rank, the interaction of these higher multipoles weakens by orders of magnitude. In other words, the Multipole expanssion is a convergent one.
Notice that I've mentioned the molecular symmetry. Some symmetries will cause lower rank multipoles to be zero, but higher order ones not to. In these cases, in order to characterize the molecular interaction with the EM field, a higher order mutlipole expansion is necessary.
