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I was studying the coupling of Vibrational and Electronic energies and was left confused at one aspect of the subject.

Consider the image showing a transition from one electronic state to a higher excited state:

enter image description here

With vibrionic coupling, my question was about the transition from $\nu_0$ to $\nu_{1}'$.

Does the transition cause a change in vibronic energy as well as electronic energy of the molecule?

And also, by my interpretation, is it correct to say that the transition from $\nu_0$ to $\nu_{0}'$ signify only the energy needed to change the molecule from the electronic ground state to the electronic excited state i.e. the vibrational energy of the molecule is the same?

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Every transition will result in some change in the vibrational energy. Even a transition like the one shown on the left (with no vibrational coupling) will result in some change, because the effective potential for the vibrational motion depends on the electronic state. (This is not evident in the figure, unfortunately.) So even a transition from the lowest vibrational substate of one electronic state to the lowest vibrational substate of a different electronic state ($\nu_{0}\rightarrow \nu_{0}'$) will change the zero-point energy of the vibrational mode.

However, there are certainly situations where the sharpness of the effective potential does not change all that much between different electronic states. In that case, your description is approximately accurate. The transition $\nu_{0}\rightarrow\nu_{1}'$ represents a change in the electronic quantum state and a change (by 1) in the number of quanta present in the vibrational mode.

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