The Kramers-Heisenberg formula clearly allows for the **initial and final state to be different**.
The atom changes state and the scattered photon energy is not equal to the initial photon energy.
This is called the **Raman effect**.
Of course, total energy is still conserved.
A given initial photon frequency will produce scattered photons with definite frequencies, or lines in the spectrum.

Jim Branson
2013-04-22