Raman: Theory

Introduction

Being different from absorption spectrum such as infrared spectrum, we detect the scattering light from molecules after the interaction between the incident light and molecules. 

For absorption spectrum, only when the energy of incident photons matches the energy difference between the ground state and one of the excited states, absorption can happen; on the other hand, this is not the case for Raman scattering. When a molecule absorbs a photon, it can be excited to a virtual state which does not have to be an eigenstate of the molecule, so the energy of the photon does not need to match the energy difference between the ground state and one of the excited states. The life of this virtual state is very short, it will decay very quickly. Most of the time, it will go back to its initial ground state, this is called Rayleigh scattering; very little of them (about one in 10(6))will go to one of the vibrational excited states, then the energy of the scattering photons will be a little smaller than the incident photons, this is called Stokes scattering; sometimes if the molecules are excited from one of the vibrational excited states then go back to the ground state, the energy of the scattering photons will be larger than the incident photons, this is called anti-Stokes scattering. See the following diagram for Raman scattering:

Figure: Diagram of Raman Scattering

Classical Theory

In fact, Raman scattering is due to the oscillation of the induced electronic dipole moment when the molecules are put into an oscillating electric field. The relationship between the oscillating dipole moment and the field is described by the following equation:

Where µ is the induced electronic dipole moment, E is the external field, ? is the polarizability of the molecules. This polarizability which is a kind of tensor is determined by the shape and also size of the electronic cloud of that molecule. So only the vibrational modes which can change the shape and size of the electronic cloud can be possibly Raman active. 

Now, we come to a classical explanation of the Raman scattering. Assume that we have an incident light whose electric field is like:

? is the frequency of the light. Then the induced dipole moment is:  

If the polarizability changes as the following expression due to the oscillation of the molecule:

where ?0  is the equilibrium polarizabilty, ?0 is the frequency of that vibrational mode. So

we can rearrange this equation into

This equation predicts that the induced dipole moment will oscillate with the following three frequencies: ?, ?+?0, ?-?0. So ? will be the Rayleigh scattering, ?+?0 is the anti-Stokes scattering, and ?-?0 is the Stokes scattering.

The upper theory is just the classical wave theory. According to quantum theory, the virtual state is in fact the mixture of all the available vibronic states of the molecules, so Raman scattering is contributed by all the available vibronic states of the molecules. The quantum theory is much more complicated than classical theory, I will not talk about it here.

Selection Rules of Raman Scattering

We all know that whether a kind of transition is allowed or not is determined by the transition moment integral:

The transition is allowed only when this M is non-zero. According to point group theory, this integral is non-zero only when the product of the irreducible representations of ?i, µ and ?f contains the totally symmetric representation.  µ is determined by the polarizability which has the following tensor expression:

This polarizabilty has the same irreducible representation with the following terms:

And at the same time, the polarizability should change with the oscillation of the nuclei.

We can take the molecule of water as a simple example. According to the group theory, water molecule belongs to the C2v group. It has three vibrational modes, their representations are A1, A1, B2, respectively. According to the characteristic table of C2v, xy, yz, xz, x^2, y^2, z^2 have the following representations:

?i and ?f have the following representations:

 so, the representation of M will have the following form:

We can find that all the three modes contain the totally symmetric representation A₁, so all the three modes of water are Raman active.

Outside links

• http://en.wikipedia.org/wiki/Raman_spectrum

References

1. <Modern Raman Spectroscopy-A Practical Approach >   edited by  Ewen Smith, Strathclyde University , Glasgow; Geoffrey Dent, Intertek ASG and UMIST, Manchester
2. <Symmetry and Spectroscopy_ An Introduction to Vibrational and Electronic Spectroscopy>  written by   Daniel C.Harris, Michelson Laboratory China Lake, California  and  Michael D.Bertolucci
3. <Raman Spectroscopy for Catalysis> written by John M.Stencel, University of Kentucky

Problems

Raman spectroscopy now is a very popular method in all kinds of research fields and also industry. For me, I have just mentioned the classical theory about it. You will get a much clearer image of Raman scattering if you go into the quantum theory of it. Besides the normal Raman spectroscopy, people have got a lot kinds of other Raman spectroscopy, like Resonance Raman spectroscopy, surface enhanced Raman and so on. To get a good understanding of those kinds of Raman scattering, you will have to refer to the quantum theory of Raman scattering which is not been talked about here. 

Contributors

• Hong Gao (Chemistry Department of UCDavis)