Suppose we have a photons with energies less that the one to excite the hydrogen atom. If the hydrogen atom is scattered by these photon it gains energy. Can we excite the hydrogen atom by this process?
You already asked this question here and got an answer: Absorption cross section for photon with energy less than the necessary to excite the hydrogen atom
The answer is no. The atom's center-of-mass motion can be affected, but it can't be put into an excited electronic state (with the caveats stated in the answer to your previous question).
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