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

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

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

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

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

Structure, Spectra, and DFT Simulation of Nickel Benzazolate Complexes with Tris(2-aminoethyl)amine Ligand

Benzazolate complexes of Ni­(II), [Ni­(pbz)­(tren)]­ClO₄ (pbz = 2-(2′-hydroxyphenyl)-benzimidazole (pbm), <b>1</b>, 2-(2′-hydroxyphenyl)-benzoxazole (pbx), <b>2</b>, 2-(2′-hydroxyphenyl)-benzothiazole (pbt), <b>3</b>; tren = tris­(2-aminoethyl)­amine), are prepared by self-assembly reaction and structurally characterized. Theoretical DFT simulations are carried out to reproduce the features of their crystal structures and their spectroscopic and photophysic properties. The three complexes are moderately luminescent at room temperature both in acetonitrile solution and in the solid state. The simulations indicate that the absorption spectrum is dominated by two well-defined transitions, and the electronic density concentrates in three MOs around the benzazole ligands. The Stokes shifts of the emission spectra of complexes <b>1</b>–<b>3</b> are determined by optimizing the electronic excited state

Structure and Spectroscopic Properties of Nickel Benzazolate Complexes with Hydrotris(pyrazolyl)borate Ligand

The reaction of benzazole ligands 2-(2′-hydroxylphenyl)­benzimidazole (Hpbm), 2-(2′-hydroxylphenyl)­benzoxazole (Hpbx), and 2-(2′-hydroxylphenyl)­benzothiazole (Hpbt), with [Ni­(Tp*)­(μ-OH)]₂ (Tp* = hydrotris­(3,5-dimethylpyrazolyl)­borate), leads to pentacoordinate nickel complexes [Ni­(Tp*)­(pbz)] (pbz = pbm (<b>1</b>), pbx (<b>2</b>), pbt (<b>3</b>)). The structures of <b>1</b>, <b>2</b>, and <b>3</b> were determined by X-ray crystallography. The pentacoordinate nickel complexes have distorted trigonal bipyramidal geometries with Addison’s τ parameter values of 0.63, 0.73, and 0.61 for <b>1</b>, <b>2</b> and <b>3</b>, respectively. The benzazolates are bonded in an η²(N,O) fashion to the nickel atoms. DFT calculations are carried out to optimize the structures of the three complexes giving a good agreement with the X-ray structures. The ¹H NMR spectra of complexes <b>1</b>–<b>3</b> exhibit sharp isotropically shifted signals. The complete assignment of these signals required an application of two-dimensional {¹H–¹H}-COSY techniques. The experimental absorption spectra of the three complexes in chloroform solution each show an intense absorption band in the ultraviolet region ca. 240 nm, followed by three less intense bands, the first two at ∼295 and ∼340 nm, and the last more disperse one, at wavelengths between 360 and 410 nm. The absorption spectra are simulated by TD-DFT and reproduce the main features of the experimental spectra well. The analysis of the electronic transitions by inspection of the frontier molecular orbitals and also the natural transition orbitals allowed us to characterize and assign the observed bands properly. The three complexes are moderately blue luminescent at room temperature, both in the solid state and in solution. Emission spectra at room temperature display broad structureless bands in chloroform solution at 460, 482, and 512 nm for complexes <b>1</b>, <b>2</b> and <b>3</b>, respectively, and structured emission in solid state with λ_max values of 473, 486, and 516 nm. Complexes containing different donor atoms in the benzazole ligand are furthermore observed to give different luminescence responses in the presence of Zn­(II), Cd­(II), Hg­(II), and Cu­(II)