Revisiting Vertical Models To Simulate the Line Shape of Electronic Spectra Adopting Cartesian and Internal Coordinates

Vertical models for the simulation of spectroscopic line shapes expand the potential energy surface (PES) of the final state around the equilibrium geometry of the initial state. These models provide, in principle, a better approximation of the region of the band maximum. At variance, adiabatic models expand each PES around its own minimum. In the harmonic approximation, when the minimum energy structures of the two electronic states are connected by large structural displacements, adiabatic models can breakdown and are outperformed by vertical models. However, the practical application of vertical models faces the issues related to the necessity to perform a frequency analysis at a nonstationary point. In this contribution we revisit vertical models in harmonic approximation adopting both Cartesian (<b>x</b>) and valence internal curvilinear coordinates (<b>s</b>). We show that when <b>x</b> coordinates are used, the vibrational analysis at nonstationary points leads to a deficient description of low-frequency modes, for which spurious imaginary frequencies may even appear. This issue is solved when <b>s</b> coordinates are adopted. It is however necessary to account for the second derivative of <b>s</b> with respect to <b>x</b>, which here we compute analytically. We compare the performance of the vertical model in the <b>s</b>-frame with respect to adiabatic models and previously proposed vertical models in <b>x</b>- or <b>Q</b>₁-frame, where <b>Q</b>₁ are the normal coordinates of the initial state computed as combination of Cartesian coordinates. We show that for rigid molecules the vertical approach in the <b>s</b>-frame provides a description of the final state very close to the adiabatic picture. For sizable displacements it is a solid alternative to adiabatic models, and it is not affected by the issues of vertical models in <b>x</b>- and <b>Q</b>₁-frames, which mainly arise when temperature effects are included. In principle the <b>G</b> matrix depends on <b>s</b>, and this creates nonorthogonality problems of the Duschinsky matrix connecting the normal modes of initial and final states in adiabatic approaches. We highlight that such a dependence of <b>G</b> on <b>s</b> is also an issue in vertical models, due to the necessity to approximate the kinetic term in the Hamiltonian when setting up the so-called GF problem. When large structural differences exist between the initial and the final-state minima, the changes in the <b>G</b> matrix can become too large to be disregarded

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

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., <i>Phys. Chem. Chem. Phys.</i> <b>2015</b>, <i>17</i>, 11401–11411), where an implicit model was adopted for the solvent. The final spectra of the considered coumarins were obtained without including <i>ad hoc</i> 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

Optical Properties of Diarylethenes with TD-DFT: 0–0 Energies, Fluorescence, Stokes Shifts, and Vibronic Shapes

This contribution is an investigation of both the structures and optical properties of a set of 14 diverse, recently synthesized diarylethenes using Time-Dependent Density Functional Theory (TD-DFT) at the ωB97X-D/6-31G­(d) level of theory. The linear response (LR) and state-specific (SS) versions of the Polarizable Continuum Model (PCM) have been adopted to account for the bulk solvation effects and their relative performances were critically accessed. It is shown, for the first time in the case of nontrivial diarylethenes, that TD-DFT provides good agreement between the experimental absorption-fluorescence crossing points (AFCPs) and their theoretical counterparts when a robust model accounting for both geometrical relaxation and vibrational corrections is used instead of the vertical approximation. On the other hand, the theoretical estimates for the Stokes shifts based on the vertical transition energies were found to be in disagreement with respect to experiment, prompting us to simulate the absorption/emission vibronic band shapes. It is proved that difficulties associated with the breakdown of the harmonic approximation in Cartesian coordinates exist for the investigated system, and we show how they can be at least partially overcome by means of a vertical approach including Duschinsky effects. Our results provide a valuable basis to rationalize the experimental vibronic structure of both emission and absorption bands and are expected to be a significant asset to the understanding of the optical properties of diarylethene derivatives

First Principles Studies of the Vibrationally Resolved Magnetic Circular Dichroism Spectra of Biphenylene

We present density-functional response theory calculations of the one-photon absorption and magnetic circular dichroism spectral bandshapes of biphenylene. The effects from the surrounding solvent environment and molecular vibrations have been included. The solvent is described by the Polarizable Continuum Model (PCM), while the vibrational structures of the spectra have been computed including both Franck–Condon and Herzberg–Teller contributions in the vibronic model. This is the first study of vibronic effects on magnetic circular dichroism spectra including non-Franck–Condon contributions. A detailed comparison with experimental data has been performed, revealing that the B3LYP functional in combination with PCM gives the best agreement with experimental data. Our calculations indicate that nonadiabatic vibronic coupling may play a role, and even small computational inaccuracies might cause significant changes in the calculated HT term, which raises concerns about the inclusion of HT contributions in the calculations of vibronic MCD in systems that have close-lying excited states

Electronic Circular Dichroism in Exciton-Coupled Dimers: Vibronic Spectra from a General All-Coordinates Quantum-Dynamical Approach

We present a computational approach of general applicability to simulate the vibronic line shapes of absorption and electronic circular dichroism (ECD) spectra in rigid exciton-coupled dimers based on a time-dependent expression of the spectra and quantum dynamical calculations. We adopt a diabatic model of interacting states localized on the monomers whose electronic potential energy surfaces are described within harmonic approximation, including the effect of displacements, frequency changes, and normal-mode mixings. Spectra that fully account for the effect of all nuclear degrees of freedom of the system are obtained through a hierarchical representation of the Hamiltonian in blocks, defined so that few blocks accurately describe the short-time dynamics of the system. With this approach, on the ground of time-dependent density functional theory calculations, we simulate the absorption and ECD spectra of a covalent compound representing a “dimer” of anthracene, in the spectral region of the ¹L_a monomer transition, obtaining results in good agreement with the experiment

Vibronic Coupling Dominates the Electronic Circular Dichroism of the Benzene Chromophore ¹L_b band

The alkylbenzene derivatives (<i>R</i>)-PhCH­(CH₃)^tBu (<b>1</b>) and (<i>R</i>)-PhCH­(CH₃)ⁱPr (<b>2</b>) were taken as paradigms of chiral benzene compounds and their vibronic electronic circular dichroism (ECD) spectrum in the ¹L_b band region analyzed in detail. The ¹L_b ECD band of chiral benzene compounds is often used to assign absolute configurations on the basis of sector rules. However, ¹L_b ECD bands of several benzene derivatives are associated with a forbidden character and show marked vibrational progressions strongly modulating their shape. This is also true for compounds <b>1</b> and <b>2</b>, the latter also showing a peculiar thermochromism. The low-temperature ECD spectrum of <b>2</b> displays in fact an alternation of positive and negative ECD maxima. Vibronic ECD calculations performed within a TDDFT scheme allowed a full rationalization of the observed ECD spectra of <b>1</b> and <b>2</b>. Especially in the case of <b>2</b>, the ECD spectrum in the ¹L_b band region results from a complex balance of Franck–Condon and Herzberg–Teller effects, as well as of conformational factors. Therefore, straightforward sector rules cannot be safely used to assign the absolute configuration of even these simple derivatives

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

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

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

Insights for an Accurate Comparison of Computational Data to Experimental Absorption and Emission Spectra: Beyond the Vertical Transition Approximation

In this work we carefully investigate the relationship between computed data and experimental electronic spectra. To that end, we compare both vertical transition energies, <i>E</i>_V, and characteristic frequencies of the spectrum like the maximum, ν^max, and the center of gravity, <i>M</i>¹, taking advantage of an analytical expression of <i>M</i>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 <i>E</i>_V, 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 <i>M</i>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 <i>E</i>_V, <i>M</i>¹, and ν^max