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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?

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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).

If you want clarification or more details on an answer, please request that in comments, not by asking another almost identical question.

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