Generation of vibronic spectra

The FC, HT and FCHT approximations of the electronic transition dipole moment have been implemented in GAUSSIAN package. The vibrationally-resolved electronic spectrum can be generated by specifying a single keyword (FC or HT or FCHT) representing the terms of Taylor series (Eq. 85) which must be used in the calculation. An example of a simulated vibronic spectra (correspond to a one-photon absorption transition) using FC, HT and FCHT approximations with AH model is represented in Figure 31.

Figure 31 Vibrationally-resolved absorption spectra of firefly natural oxyluciferin generated by using FC, HT and FCHT approximations in water (PCM) within TI framework and AH model. The emitted intensity by second is given in µJ/mol, before being normalised to one.

The vibronic spectra can be simulated within TI or TD frameworks depending on whether or not temperature effects are included in the simulation. Furthermore, different models can be applied to describe the electronic transition such as AH, VG models. Each model recovers the frequencies and normal modes from the initial state. Depending on the features of the chosen model, the normal modes and frequencies from the final state can be ignored (e.g., VG model) or considered (e.g., AH model).

In order to generate the vibrationally-resolved spectrum, one needs to prepare correctly the data files of the initial and the final states (equilibrium geometry, frequencies, normal modes…) and to specify keywords of the adequate approach. Whatever the chosen approach, Gaussian package extracts the information from the data files. In addition, it maximises the overall superposition of the initial and final state structures in order to reduce the translation

and rotation effects. The translation can be removed by superposing the center of masses of the two equilibrium geometries. The rotation is then minimised by maximising the superposition.

This procedure avoids numerical problems when the initial and final state geometries are slightly different. In a case where the difference between the two geometries is important (Figure 32), the superposition cannot be done correctly and hence, the vibronic spectrum simulation will not be generated when using some model such as AH model.

Figure 32 Representation of the initial (in blue) and final (in red) state geometries of the natural oxyluciferin (left) and its analogue (right) obtained after the superposition procedure within AH model. The initial and final state geometries of Natural oxyluciferin (left) is superposed correctly. The initial and final state geometries of the oxyluciferin analogue (right) is superimposed, which explains why the spectrum cannot be generated.

It should be noted that the simulation of vibronic spectrum of superimposition geometries using the Adiabatic Hessian (AH) model can be done by forcing the calculations (by using specific commands in Gaussian programme). Therefore, the final AH vibronic spectrum cannot be much reliable as shown in the vibrationally-resolved absorption spectrum of protonated phenol-cycle where the electronic ground and first excited states minimum energy structures are superimposed (Figure 33a). On the other hand, for the deprotonated phenol-cycle, both electronic ground and first excited states minimum energy structures are superposed and thus, AH method generates correctly the vibronic spectrum (Figure 33b).

Figure 33 Vibrationally-resolved absorption spectra computed with the FCHT approximation in gas phase for Phenol-cycle analogue (a) protonated, b) deprotonated) using the Adiabatic Hessian (AH) model. In the case of the protonated phenol-cycle analogue, the superimposition of the electronic ground and first excited states minimum energy structures is provided and explains why the spectrum cannot be correctly simulated (The spectrum calculation has been forced).

If the superposition of the geometries is done successfully (natural oxyluciferin in Figure 32 and deprotonated phenol-cycle in Figure 33b), the relative transition energy with respect to the transition energy between the vibrational ground states of the initial and the final states (〈 0^′ | 0^′′ 〉), the square of the transition dipole moment integral and the intensity of the corresponding spectral line will be obtained. Furthermore, the broadening of the spectral line will be done by Gaussian functions with a certain HWHM value.

In case where the relative position of the ground state geometry of a given compound is not the same in the excited state, it is possible to apply VG model to generate vibronic spectrum. In order to get an idea on the involved effects when applying AH or VG models, we have performed vibronic calculations of two oxyluciferin analogues, phenol-OMe and OMe-enol using FCHT approximation with AH and VG models (Figure 34). The spectra shapes obtained with VG and AH models are very different. The spectra obtained with AH model represent a larger and more intense amount of vibronic transitions in the spectral region around 390–450 nm (Figure 34). In the vibronic spectra obtained with VG model these vibronic transitions (around 410–430 nm) are less intense. Furthermore, in both AH and VG models, the predicted 0-0’

vibronic transition is the most intense one, except for the predicted one of phenol-OMe obtained with AH model. However, it is still close to the wavelength maximum of vibronic spectrum. Moreover, the 0-0’ bands predicted with VG model are blue-shifted around 0.05 eV comparing with those obtained with AH model. This difference in energy is rather small, notwithstanding the VG model weaknesses.

Figure 34 Vibrationally-resolved absorption spectra computed with the FCHT approximation in water (PCM) for a) protonated phenol-OMe (left: with VG model, right: with AH model) and b) protonated OMe-enol (left: with VG model, right: with AH model).