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).

In this work we carefully investigate the relationship between computed data and experimental electronic spectra. To that end, we compare both vertical transition energies, EV, and characteristic frequencies of the spectrum like the maximum, ν(max), and the center of gravity, M(1), taking advantage of an analytical expression of M(1) in terms of the parameters of the initial- and final-state potential energy surfaces. After pointing out that, for an accurate comparison, experimental spectra should be preliminarily mapped from wavelength to frequency domain and transformed to normalized lineshapes, we simulate the absorption and emission spectra of several prototypical chromophores, obtaining lineshapes in very good agreement with experimental data. Our results indicate that the customary comparison of experimental ν(max) and computational EV, without taking into account vibrational effects, is not an adequate measure of the performance of an electronic method. In fact, it introduces systematic errors that, in the investigated systems, are on the order of 0.1-0.3 eV, i.e., values comparable to the expected accuracy of the most accurate computational methods. On the contrary, a comparison of experimental and computed M(1) and/or 0-0 transition frequencies provides more robust results. Some rules of thumbs are proposed to help rationalize which kind of correction one should expect when comparing EV, M(1), and ν(max).

We present a general mixed quantum classical method that couples classical Molecular Dynamics (MD) and vibronic models to compute the shape of electronic spectra of flexible molecules in condensed phase without, in principle, any phenomenological broadening. It is based on a partition of the nuclear motions of the solute+solvent system in "soft" and "stiff" vibrational modes, and an adiabatic hypothesis that assumes that stiff modes are much faster than soft ones. In this framework the spectrum is rigorously expressed as a conformational integral of quantum vibronic spectra along the stiff coordinates only. Soft modes enter at classical level through the conformational distribution that is sampled with classical MD runs. At each configuration, reduced-dimensionality quadratic Hamiltonians are built in the space of the stiff coordinates only, thanks to a generalization of the Vertical Hessian harmonic model and an iterative application of projectors in internal coordinates to remove soft modes. Quantum vibronic spectra, specific for each sampled configuration of the soft coordinates, are then computed at the desired temperature with efficient time-dependent techniques, and the global spectrum simply arises from their average. For consistency of the whole procedure, classical MD runs are performed with quantum-mechanically derived force fields, parameterized at the same level of theory selected for generating the quadratic Hamiltonians along the stiff coordinates. Application to N-methyl-6-oxyquinolinium betaine in water, dithiophene in ethanol, and a flexible cyanidine in water are presented to show the performance of the method.

Starting from Marcus's 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.

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.