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Recently I was studying spectroscopy in Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy p.357-358 as shown in the image http://i.stack.imgur.com/LSLyZ.jpg

The author said the internal conversion (laballed IC in that energy level diagram) is isoenergetic, which means there is no energy change in that transition

a. If we use what is discussed earlier in the chat, that a state can be changed by a measurement (i.e. a suitable operator acting on the state which changes it), and energy does not necessary need to change in the process

how does the molecule knows it has to transit to a lower electronic state (but vibrationally excited) as after the absorption of the photon, the molecule is not interacting with anything else

Or in other words, in more physics oriented terms, what is the operator that allows the internal conversion to occur/describes internal conversion?

b. Since the process is isoenergetic, how does the state knows which vibronic state is lower?

While quantum chemists have used operators to describe transitions in terms of the transition moment integral, they don't seemed to give similar mathematical treatment to relaxation pathways (which is why putting this in chem stack exchange might not give relevant answers). I tried to analyse the problem by using bra-kets and anaogous to transition moment integrals

$$\text{Probability} \propto \langle\psi_{\text{excited}}\lvert\hat{O}\rvert\psi_{0}\rangle$$

but I am clueless on what properties the operator $\hat{O}$ must have to describe internal conversion (and other relaxation pathways, thus I don't know how to resolve it

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Isoenergetic transition from (electronic excited state + vibrational ground state) to (electronic ground state + vibrational excited state) occurs because the electronic and ionic degrees of freedom are coupled. The terminology for this phenomenon is vibronic coupling or electron-phonon coupling depending on whether one considers a molecule or a crystal.

In Born-Oppenheimer approximation, electrons move in a static potential of ions. Vibronic coupling is an example of physics beyond this approximation scheme. An intuitive picture is that if ions move (albeit much slower than electrons), the potential they generate correspondingly changes, thereby disturbing the motion of electrons.

To derive a simple vibronic coupling Hamiltonian, notice that the potential of ions is of the form \begin{equation} V(\textbf{r}) = \sum_{i}U(\textbf{r}-\textbf{R}_{i}). \end{equation} Here, $U$ is the potential of a single ion, and $\{\textbf{R}_{i}\}$ are ion positions. If the ions are slightly displaced by $\{\textbf{u}_i\}$, the change in the potential is given by \begin{equation} \Delta V(\textbf{r},\{\textbf{u}_i\}) = \sum_{i}\Big[U(\textbf{r}-\textbf{R}_{i}-\textbf{u}_{i}) - U(\textbf{r}-\textbf{R}_{i})\Big] \approx -\sum_{i}\textbf{u}_{i} \cdot \nabla U(\textbf{r} - \textbf{R}_{i}), \end{equation} where $\textbf{r}$ and $\{\textbf{u}_{i}\}$ respectively act on electronic and ionic degrees of freedom. It is an operator that induces isoenergetic transitions OP described.

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