Francisco J. Avila Ferrer

Computing the inhomogeneous broadening of electronic transitions in solution: a first-principle quantum mechanical approach

Starting from Marcuss relationship connecting the inhomogeneous broadening with the solvent reorganization energy and exploiting recent state-specific developments in PCM/TD-DFT calculations, we propose a procedure to estimate the polar broadening of optical transitions. When applied to two representative molecular probes, coumarin C153 and 4-aminophthalimide, in different solvents, our approach provides for the polar broadening values fully consistent with the experimental ones. Thanks to these achievements, for the first time fully ab initio vibrationally resolved absorption spectra in solution are computed, obtaining spectra for coumarin C153 in remarkable agreement with experiments.

Comparison of vertical and adiabatic harmonic approaches for the calculation of the vibrational structure of electronic spectra

The calculation of the vibrational structure associated to electronic spectra in large molecules requires a Taylor expansion of the initial and final state potential energy surface (PES) around some reference nuclear structure. Vertical (V) and adiabatic (A) approaches expand the final state PES around the initial-state (V) or final-state (A) equilibrium structure. Simplest models only take into account displacements of initial- and final-state minima, intermediate ones also allow for difference in frequencies and more accurate models introduce the Dushinsky effect through the computation of the Hessians of both the initial and final state. In this contribution we summarize and compare the mathematical expressions of the complete hierarchy of V and A harmonic models and we implement them in a numerical code, presenting a detailed comparison of their performance on a number of prototypical systems. We also address non-Condon effects through linear expansions of the transition dipole as a function of nuclear coordinates (Herzberg-Teller effect) and compare the results of expansions around initial and final state equilibrium geometries. By a throughout analysis of our results we highlight a number of general trends in the relative performance of the models that can provide hints for their proper choice. Moreover we show that A and V models including final state PES Hessian outperform the simpler ones and that discrepancies in their predictions are diagnostic for failure of harmonic approximation and/or of Born-Oppenheimer approximation (existence of remarkable geometry-dependent mixing of electronic states).

Harmonic Models in Cartesian and Internal Coordinates to Simulate the Absorption Spectra of Carotenoids at Finite Temperatures

When large structural displacements take place between the ground state (GS) and excited state (ES) minima of polyatomic molecules, the choice of a proper set of coordinates can be crucial for a reliable simulation of the vibrationally resolved absorption spectrum. In this work, we study two carotenoids that undergo structural displacements from GS to ES minima of different magnitude, from small displacements for violaxanthin to rather large ones for β-carotene isomers. Their finite-temperature (77 and 300 K) spectra are simulated at the harmonic level, including Duschinsky effect, by time-dependent (TD) and time-independent (TI) approaches, using (TD)DFT computed potential energy surfaces (PES). We adopted two approaches to construct the harmonic PES, the Adiabatic (AH) and Vertical Hessian (VH) models and, for AH, two reference coordinate frames: Cartesian and valence internal coordinates. Our results show that when large displacements take place, Cartesian coordinates dramatically fail to describe curvilinear displacements and to account for the Duschinsky matrix, preventing a realistic simulation of the spectra within the AH model, where the GS and ES PESs are quadratically expanded around their own equilibrium geometry. In contrast, internal coordinates largely amend such deficiencies and deliver reasonable spectral widths. As expected, both coordinate frames give similar results when small displacements occur. The good agreement between VH and experimental line shapes indicates that VH model, in which GS and ES normal modes are both evaluated at the GS equilibrium geometry, is a good alternative to deal with systems exhibiting large displacements. The use of this model can be, however, problematic when imaginary frequencies arise. The extent of the nonorthogonality of the Dushinsky matrix in internal coordinates and its correlation with the magnitude of the displacement of the GS and ES geometries is analyzed in detail.

Duschinsky, Herzberg-Teller, and Multiple Electronic Resonance Interferential Effects in Resonance Raman Spectra and Excitation Profiles. The Case of Pyrene.

We show that a recently developed time-independent approach for the calculation of vibrational resonance Raman (vRR) spectra is able to describe Duschinsky and Herzberg-Teller (HT) effects acting on a single resonant state, together with interferential contributions arising from multiple electronic resonances, allowing us to investigate in detail how their interplay determines both the vRR spectra at selected wavelengths and the Raman excitation profiles. We apply this methodology to the study of the spectra of pyrene in acetonitrile, an ideal system since it exhibits three close-lying electronic transitions that are bright but also subjected to HT effects. To single out the different contributions to vRR line shapes we adopted two different adiabatic models for resonant-state potential energy surfaces, namely, Adiabatic Shift (only accounting from equilibrium geometry displacements) and Adiabatic Hessian (AH, including also the Duschinsky effects), and Franck-Condon (FC) or HT approximations for the transition dipole. We show that, on balance, FC+HT calculations within the AH model provide the best agreement with experiment. Moreover, our methodology permits to individuate bands in the experimental spectra due to the simultaneous contribution of more than one resonant state and to point out and analyze interferential effects between the FC and HT terms in each resonance Raman process, together with FC-HT and HT-HT interferences between different electronic states.

Absorption and emission spectral shapes of a prototype dye in water by combining classical/dynamical and quantum/static approaches.

We study the absorption and emission electronic spectra in an aqueous solution of N-methyl-6-oxyquinolinium betaine (MQ), an interesting dye characterized by a large change of polarity and H-bond ability between the ground (S0) and the excited (S1) states. To that end we compare alternative approaches based either on explicit solvent models and density functional theory (DFT)/molecular-mechanics (MM) calculations or on DFT calculations on clusters models embedded in a polarizable continuum (PCM). In the first approach (ClMD), the spectrum is computed according to the classical Franck-Condon principle, from the dispersion of the time-dependent (TD)-DFT vertical transitions at selected snapshots of molecular dynamics (MD) on the initial state. In the cluster model (Qst) the spectrum is simulated by computing the quantum vibronic structure, estimating the inhomogeneous broadening from state-specific TD-DFT/PCM solvent reorganization energies. While both approaches provide absorption and emission spectral shapes in nice agreement with experiment, the Stokes shift is perfectly reproduced by Qst calculations if S0 and S1 clusters are selected on the grounds of the MD trajectory. Furthermore, Qst spectra better fit the experimental line shape, mostly in absorption. Comparison of the predictions of the two approaches is very instructive: the positions of Qst and ClMD spectra are shifted due to the different solvent models and the ClMD spectra are narrower than the Qst ones, because MD underestimates the width of the vibrational density of states of the high-frequency modes coupled to the electronic transition. On the other hand, both Qst and ClMD approaches highlight that the solvent has multiple and potentially opposite effects on the spectral width, so that the broadening due to solute-solvent vibrations and electrostatic interaction with bulk solvent is (partially) counterbalanced by a narrowing of the contribution due to the solute vibrational modes. Qst analysis evidences a pure quantum broadening effect of the spectra in water due to vibronic progressions along the solute/solvent H-bonds.

Modeling Solvent Broadening on the Vibronic Spectra of a Series of Coumarin Dyes. From Implicit to Explicit Solvent Models.

We present a protocol to estimate the solvent-induced broadening of electronic spectra based on a model that explicitly takes into account the environment embedding the solute. Starting from a classical approximation of the solvent contribution to the spectrum, the broadening arises from the spread of the excitation energies due to the fluctuation of the solvent coordinates, and it is represented as a Gaussian line shape that convolutes the vibronic spectrum of the solute. The latter is computed in harmonic approximation at room temperature with a time-dependent approach. The proposed protocol for the computation of spectral broadening exploits molecular dynamics (MD) simulations performed on the solute-solvent system, keeping the solute degrees of freedom frozen, followed by the computation of the excitation properties with a quantum mechanics/molecular mechanics (QM/MM) approach. The factors that might influence each step of the protocol are analyzed in detail, including the selection of the empirical force field (FF) adopted in the MD simulations and the QM/MM partition of the system to compute the excitation energies. The procedure is applied to a family of coumarin dyes, and the results are compared with experiments and with the predictions of a very recent work (Cerezo et al., Phys. Chem. Chem. Phys. 2015, 17, 11401-11411), where an implicit model was adopted for the solvent. The final spectra of the considered coumarins were obtained without including ad hoc phenomenological parameters and indicate that the broadenings computed with explicit and implicit models both follow the experimental trend, increasing as the polarity change from the initial to the final state increases. More in detail, the implicit model provides larger estimations of the broadening that are closer to the experimental evidence, while explicit models appear to better capture relative differences arising from different solvents or different solutes. Possible inaccuracies of the adopted FF that may lead to the observed underestimation are analyzed in detail.

Quantum-Classical Calculation of the Absorption and Emission Spectral Shapes of Oligothiophenes at Low and Room Temperature by First-Principle Calculations

We report a thorough computational characterization of the low- and room-temperature absorption and emission spectra of a series of oligothiophenes that contain between three and seven thiophene units. Our computational approach is based on time-dependent (TD) density functional calculations with the CAM-B3LYP functional. The effect of vibrations is included without resorting to any empirical parameters either at a fully quantum level or with a hybrid quantum-classical protocol. This latter approach is introduced to describe the relevant broadening effects in absorption at room temperature and is based on the partition of the vibrational modes into two sets: the inter-ring torsions treated at the anharmonic level in a classical way and the remaining modes described at the quantum level. The contribution of the quantum modes to the spectrum is computed by using a harmonic approximation, which accounts for Duschinsky mixing and changes in the vibrational frequencies associated with the electronic transition; a path-integral TD approach is adopted to account for the effect of temperature. The spectra simulated at low temperatures are in very good agreement with their experimental counterparts, which indicates that our calculations can quantitatively reproduce the effect of chain lengthening on the position and the shape of the spectra. Good agreement is also obtained at room temperature, for which we show that the classical description of the broadening, owing to the inter-ring torsions, reproduces the loss of the vibronic structure observed in the experiment and introduces only a slight overestimation of the spectral width.

Quantum-classical effective-modes dynamics of the ππ* → nπ* decay in 9H-adenine. A quadratic vibronic coupling model

We present mixed quantum-classical simulation of the internal conversion between the lowest energy ππ* (SLa) and nπ* (Sn) excited electronic states in adenine in the gas phase, adopting a quadratic vibronic model (QVC), parametrized with the help of PBE0 density functional calculations. Our approach is based on a hierarchical representation of the QVC Hamiltonian and a subsequent treatment of the most relevant coordinates at accurate time-dependent quantum level and of the other ‘bath’ modes at classical level. We predict an ultrafast transfer (∼30 fs) of ≈75% of the initial population excited on SLa to Sn. Within an adiabatic picture, on the same timescale the wave packet concentrates almost completely on the lowest S1 state, where however it shows a very broad distribution with different characteristics (due to the different ‘diabatic’ character). It is shown that the proposed methodology offers a practicable route to describe the quantum dynamics of internal conversion processes in large semi-rigid systems.

The lineshape of the electronic spectrum of the green fluorescent protein chromophore, part I: gas phase.

The vibronic spectra of the green fluorescent protein chromophore analogues p-hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI) and 3,5-tert-butyl-HBDI (35Bu) are similar in the vacuum, but very different in water or ethanol. To understand this difference, we have computed the vibrationally resolved solution spectra of these chromophores, using the polarizable continuum model (PCM) to account for solvent effects on the (harmonic) potential energy surfaces (PES). In agreement with experiment, we found that the vibrational progression increases with the polarity of the solvent, but we could neither reproduce the broadening, nor the large difference between the absorption spectra of HBDI and 35Bu. To account for the inhomogeneous broadening of the solution spectra, we used two approaches. In the first, we estimated the polar broadening from the solvent reorganization energy upon photo-excitation, using the state-specific PCM implementation. In the second, we estimated the broadening from the variance of the vertical excitation energies in molecular dynamics trajectories. Although we found good agreement for the lineshape of 35Bu in ethanol, and to a lesser extent in water, we highly underestimated the broadening for HBDI. To resolve this discrepancy, we explored the PES of HBDI in water and found that in contrast to the PCM result, the ground-state geometry is not planar in explicit solvent. We furthermore found that nonplanar geometries enhance the intramolecular charge transfer upon excitation. Therefore, the solvent reorganization and broadening are much larger and we speculate that the much broader spectrum of HBDI in water is due to the population of nonplanar geometries.

The absorption and emission spectra in solution of oligothiophene-based push–pull biomarkers: a PCM/TD-DFT vibronic study

We here report a thorough quantum mechanical study of the optical properties of several N-succinimidyl-ester and methyl-amide derivatives of bi- and terthiophene, which have shown promising performances as biomarkers. We study in particular the dependence of the absorption and emission spectra on the substituents and on the nature of the embedding medium, from the gas phase to polar solvents. Our approach is based on time-dependent density functional theory calculations, by comparing the performances of different functionals and, finally, using long-range-corrected CAM-B3LYP as reference functional. Solvent effects are included by the Polarizable Continuum Model, exploiting both its linear response and state-specific implementations. In order to simulate the absorption spectra at room temperature, a hybrid quantum/classical approach is adopted where the broadening effects due to the torsional flexibility of the system are taken into account within the classical Franck–Condon principle, starting from a relaxed three-dimensional potential energy surface, while the quantum vibronic contribution of the remaining degrees of freedom is described in harmonic approximation according to a time-dependent approach. The theoretical predictions are in good agreement with experiments, providing interesting indications on the accuracy of different functionals and on the main chemical-physical effects modulating the excited state properties of these compounds.