# 13.6: Electronic Spectra Contain Electronic, Vibrational, and Rotational Information

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Molecules can also undergo changes in electronic transitions during microwave and infrared absorptions. The energy level differences are usually high enough that it falls into the visible to UV range; in fact, most emissions in this range can be attributed to electronic transitions.

## Electron Transitions are not Purely Electronic

We have thus far studied *rovibrational** transitions*--that is, transitions involving both the vibrational and rotational states. Similarly, electronic transitions tend to accompany both rotational and vibrational transitions. These are often portrayed as an electronic potential energy cure with the vibrational level drawn on each curve. Additionally, each vibrational level has a set of rotational levels associated with it.

Recall that in the Born-Oppenheimer approximation, nuclear kinetic energies can be ignored (e.g., fixed) to solve for electronic wavefunctions and energies, which are much faster than rotation or vibration. As such, it is important to note that unlike rovibrational transitions, electronic transitions aren't dependent on rotational or transitional terms and are assumed to be separate. Therefore, when using an anharmonic oscillator-nonrigid rotator approximation (and excluding translation energy), the total energy of a diatomic is:

\[ \tilde{E}_{total} = \tilde{\nu}_{el} + G(v) + F(J) \label{Eqa1}\]

where \( \tilde{\nu}_{el}\) is the electronic transition energy change in wavenumbers, \(G(n)\) is the vibrational energy with energy level \(v\) (assuming anharmonic oscillator), and \(F(J)\) is the rotational energy, assuming a nonrigid rotor. Equation \(\ref{Eqa1}\) can be expanded accordingly:

\[ = \underbrace{\tilde{\nu}_{el}}_{\text{electronic}} + \underbrace{\tilde{\nu}_e \left (v + \dfrac{1}{2} \right) - \tilde{\chi}_e \tilde{\nu}_e \left (v + \dfrac{1}{2} \right)^2}_{\text{vibrational}} + \underbrace{\tilde{B} J(J + 1) - \tilde{D} J^2(J + 1)^2}_{\text{rotational}} \label{Eqa2}\]

Notice that both the vibration constant (\( \tilde{\nu}_e\)) and anharmonic constant ( \(\tilde{\chi}_e\)) are electronic state dependent (and hence the rotational constants would be too, but are ignored here). Since rotational energies tend to be so small compared to electronic, their effects are minimal and are typically ignored when we do calculations and are referred to as *vibronic* transitions.

The eigenstate-to-eigenstate transitions (e.g., \(1 \rightarrow 2\)) possible are numerous and have absorption lines at

\[ \tilde{\nu}_{obs} = \tilde{E}_{2} - \tilde{E}_{1} \label{Eqa21}\]

and for simplification, we refer to constants associated with these states as \(| ' \rangle \) and \(| '' \rangle \), respectively. So Equation \(\ref{Eqa21}\) is

\[ \tilde{\nu}_{obs} = \tilde{E''(v'')} - \tilde{E'(v')} \]

Also important to note that typically vibronic transitions are usually the result of the vibrational \(v'=0\) vibratonal state. Within this assumption and excluding the rotational contributions (due to their low energies), Equation \(\ref{Eqa2}\) can be used with Equation \(\ref{Eqa21}\) to get

\[ \tilde{\nu}_{obs} = \tilde{T}_{el} + \left( \dfrac{1}{2} \tilde{\nu}'_e - \dfrac{1}{4} \tilde{\chi}'_e \tilde{\nu}_e' \right) - \left( \dfrac{1}{2} \tilde{\nu}''_e - \dfrac{1}{4} \tilde{\chi}''_e \tilde{\nu}_e'' \right) + \tilde{\nu}'_e v'' - \tilde{\chi}'_e \tilde{\nu}_e' v''(v''+1) \label{Eqa3}\]

A common transition of importance is the \( \tilde{\nu}_{00}\), which is the \(0 \rightarrow 0\) transition and include no vibrational change. For this case, equation \(\ref{Eqa3}\) is then

\[ \tilde{\nu}_{00} = \tilde{T}_{el} + \left( \dfrac{1}{2} \tilde{\nu}'_e - \dfrac{1}{4} \tilde{\chi}'_e \tilde{\nu}_e' \right) - \left( \dfrac{1}{2} \tilde{\nu}''_e - \dfrac{1}{4} \tilde{\chi}''_e \tilde{\nu}_e'' \right) \]

This is the lowest energy possible to observe in an electronic transition although it may be of low intensity as discussed in the following section.