The Chemical Enhancement in SERS

When electromagnetic radiation interacts with matter, the oscillating electric field causes the matter to oscillate at the same frequency as the radiation. This response is expressed in terms of the induced dipole moment, which may be written 

Here r is the displacement of the electron from the origin, m its mass, and k is the force constant binding the electron. Letting coq = V(/c/m), the steady-state solution to this equation is 

We may use second-order perturbation theory to obtain the expression for the p(T component of the polarizability (p and cr refer to x, y, or z) for transitions between states I and F, in terms of the quantum mechanical wave functions associated with the molecular states as 

In this equation, and iip are the dipole moment operators for the polarization 

For most molecules of interest to us, we may make the zero-order Born-Oppenheimer approximation, which takes advantage of the fact that nuclei 

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