Herzberg-Teller Effects Research Articles

Influence of Duschinsky and Herzberg-Teller effects on S₀→ S₁ vibrationally resolved absorption spectra of several porphyrin-like compounds.

The S0 → S1 (Q band) high-resolved absorption spectra of three porphyrin-like compounds, porphycene, magnesium porphyrin, and zinc tetraazaporphyrin, were simulated in the framework of the Franck-Condon approximation including the Duschinsky and Herzberg-Teller (HT) contributions. Substitution of meso-aza on porphyrin macrocycle framework could change severely the absorption energy, vibrational intensity, and spectral profile of Q band. Therefore, we focused attention on the spectral similarities and contrasts among the three compounds based on the density functional theory and its time-dependent extension calculations. The simulated spectra agreed well with the experimental ones and further confirmed that the HT and Duschinsky effects have significant influence on the weakly allowed or forbidden transition of sizable organic molecules. The pure HT and Duschinsky effects were explored separately to clarify their contributions on changing vibrational intensities of different modes. Moreover, we tentatively assigned most of the vibrational modes which appeared in the experimental spectra but corresponding assignments were not given. The present work provided a useful method to simulate and interpret the absorption spectra of porphyrin-like compounds.

A general time-dependent route to resonance-Raman spectroscopy including Franck-Condon, Herzberg-Teller and Duschinsky effects.

We present a new formulation of the time-dependent theory of Resonance-Raman spectroscopy (TD-RR). Particular attention has been devoted to the generality of the framework and to the possibility of including different effects (Duschinsky mixing, Herzberg-Teller contributions). Furthermore, the effects of different harmonic models for the intermediate electronic state are also investigated. Thanks to the implementation of the TD-RR procedure within a general-purpose quantum-chemistry program, both solvation and leading anharmonicity effects have been included in an effective way. The reliability and stability of our TD-RR implementation are validated against our previously proposed and well-tested time-independent procedure. Practical applications are illustrated with some closed- and open-shell medium-size molecules (anthracene, phenoxyl radical, benzyl radical) and the simulated spectra are compared to the experimental results. More complex and larger systems, not limited to organic compounds, can be also studied, as shown for the case of Tris(bipyridine)ruthenium(II) chloride.

Unified theory of electron–phonon renormalization and phonon-assisted optical absorption

We present a theory of electronic excitation energies and optical absorption spectra which incorporates energy-level renormalization and phonon-assisted optical absorption within a unified framework. Using time-independent perturbation theory we show how the standard approaches for studying vibronic effects in molecules and those for addressing electron–phonon interactions in solids correspond to slightly different choices for the non-interacting Hamiltonian. Our present approach naturally leads to the Allen–Heine theory of temperature-dependent energy levels, the Franck–Condon principle, the Herzberg–Teller effect and to phonon-assisted optical absorption in indirect band gap materials. In addition, our theory predicts sub-gap phonon-assisted optical absorption in direct gap materials, as well as an exponential edge which we tentatively assign to the Urbach tail. We also consider a semiclassical approach to the calculation of optical absorption spectra which simultaneously captures energy-level renormalization and phonon-assisted transitions and is especially suited to first-principles electronic structure calculations. We demonstrate this approach by calculating the phonon-assisted optical absorption spectrum of bulk silicon.

New developments of a multifrequency virtual spectrometer: stereo-electronic, dynamical, and environmental effects on chiroptical spectra.

Computational spectroscopy has recently evolved from a field reserved for specialists toward a general tool allowing interpretations and analyses of experimental results. However, the current practice of providing tables of transitions for rigid geometries, possibly tuned by phenomenological broadening, is by far too naive. In order to improve this situation in the last few years we have been developing a general, robust, and user-friendly virtual spectrometer (VS) able to complement experimental studies for complex systems in condensed phases. The VS is based on flexible graphical pre- and postprocessing tools interfaced with general number-crunching software. This last tool is rooted in several electronic structure methodologies (DFT, TD-DFT, post-Hartree-Fock), powerful discrete/continuum models for describing environmental effects, and general vibrational and vibronic models. These last topics are the main focus of this work, which sketches our latest developments related to effective inclusion of anharmonic contributions, together with time-independent and/or time-dependent descriptions of vibronic transitions including Franck-Condon, Herzberg-Teller, and Duschinsky effects. Some test cases are described in some detail with the aim of showing the role of different effects in ruling vibrational (VCD) and electronic (ECD, CPL) chiral spectroscopies.

First-principle computation of absorption and fluorescence spectra in solution accounting for vibronic structure, temperature effects and solvent inhomogenous broadening

We compute the line shape of absorption and emission electronic spectra of two different dyes, Coumarin C153 and N-methyl-6-Quinolinium betaine accounting for the vibronic structure, temperature effects and polar solvent inhomogeneous broadening, without using any phenomenological parameter. We exploit a number of recent developments including a time-dependent (TD) approach to the computation of vibronic spectra that provides fully converged line shapes at finite temperature accounting for both Duschinsky and Herzberg–Teller effects, and the state-specific (SS) implementation of Polarizable Continuum Model (PCM). This latter is adopted to compute the solvent reorganization energy connected to inhomogenoeus broadening. We compute the absorption and fluorescence spectra in the gas-phase, non-polar and polar solvents analyzing the relative importance of different sources of broadening. To this end we investigate the performance of TD Density Functional Theory, Complete Active Space Self Consistent Field (CASSCF) and Complete Active Space second-order Perturbation Theory (CASPT2) methods in the computation of inhomogeneous broadening.

Assessment of mode-mixing and Herzberg-Teller effects on two-photon absorption and resonance hyper-Raman spectra from a time-dependent approach

A time-dependent approach is presented to simulate the two-photon absorption (TPA) and resonance hyper-Raman scattering (RHRS) spectra including Duschinsky rotation (mode-mixing) and Herzberg-Teller (HT) vibronic coupling effects. The computational obstacles for the excited-state geometries, vibrational frequencies, and nuclear derivatives of transition dipole moments, which enter the expressions of TPA and RHRS cross sections, are further overcome by the recently developed analytical excited-state energy derivative approaches in the framework of time-dependent density functional theory. The excited-state potential curvatures are evaluated at different levels of approximation to inspect the effects of frequency differences, mode-mixing and HT on TPA and RHRS spectra. Two types of molecules, one with high symmetry (formaldehyde, p-difluorobenzene, and benzotrifluoride) and the other with non-centrosymmetry (cis-hydroxybenzylidene-2,3-dimethylimidazolinone in the deprotonated anion state (HDBI(-))), are used as test systems. The calculated results reveal that it is crucial to adopt the exact excited-state potential curvatures in the calculations of TPA and RHRS spectra even for the high-symmetric molecules, and that the vertical gradient approximation leads to a large deviation. Furthermore, it is found that the HT contribution is evident in the TPA and RHRS spectra of HDBI(-) although its one- and two-photon transitions are strongly allowed, and its effect results in an obvious blueshift of the TPA maximum with respect to the one-photon absorption maximum. With the HT and solvent effects getting involved, the simulated blueshift of 1291 cm(-1) agrees well with the experimental measurement.

A robust and effective time-independent route to the calculation of Resonance Raman spectra of large molecules in condensed phases with the inclusion of Duschinsky, Herzberg-Teller, anharmonic, and environmental effects.

We present an effective time-independent implementation to model vibrational resonance Raman (RR) spectra of medium-large molecular systems with the inclusion of Franck-Condon (FC) and Herzberg-Teller (HT) effects and a full account of the possible differences between the harmonic potential energy surfaces of the ground and resonant electronic states. Thanks to a number of algorithmic improvements and very effective parallelization, the full computations of fundamentals, overtones, and combination bands can be routinely performed for large systems possibly involving more than two electronic states. In order to improve the accuracy of the results, an effective inclusion of the leading anharmonic effects is also possible, together with environmental contributions under different solvation regimes. Reduced-dimensionality approaches can further enlarge the range of applications of this new tool. Applications to imidazole, pyrene, and chlorophyll a1 in solution are reported, as well as comparisons with available experimental data.

General Time Dependent Approach to Vibronic Spectroscopy Including Franck-Condon, Herzberg-Teller, and Duschinsky Effects.

An effective time-dependent (TD) approach to compute vibrationally resolved optical spectra from first principles is presented for the computation of one-photon electronic spectra induced by either electric or magnetic transition dipoles or by their mutual interaction, namely absorption, emission, and circular dichroism. Particular care has been devoted to generality, modularity, and numerical stability including all the contributions that play a role at the harmonic level of approximation, namely Franck-Condon, Herzberg-Teller, and Dushinsky (i.e., mode mixing) effects. The implementation shares the same general framework of our previous time-independent (TI) model, thus allowing an effective integration between both approaches with the consequent enhancement of their respective strengths (e.g., spectrum completeness and straightforward account of temperature effects for the TD route versus band resolution and assignment for the TI route) using a single set of starting data. Implementation of both models in the same general computer program allows comprehensive studies using several levels of electronic structure description together with effective account of environmental effects by atomistic and/or continuum models of different sophistication. A few medium-size molecules (furan, phenyl radical, anthracene, dimethyloxirane, coumarin 339) have been studied in order to fully validate the approach.

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

The alkylbenzene derivatives (R)-PhCH(CH₃)(t)Bu (1) and (R)-PhCH(CH₃)(i)Pr (2) were taken as paradigms of chiral benzene compounds and their vibronic electronic circular dichroism (ECD) spectrum in the (1)L(b) band region analyzed in detail. The (1)L(b) ECD band of chiral benzene compounds is often used to assign absolute configurations on the basis of sector rules. However, (1)L(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 1 and 2, the latter also showing a peculiar thermochromism. The low-temperature ECD spectrum of 2 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 1 and 2. Especially in the case of 2, the ECD spectrum in the (1)L(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.

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.

Quantum Dynamics Study of Photoexcited Aniline

A model Hamiltonian based on the quadratic vibronic coupling model is developed to describe the photoinduced dynamics of aniline excited to the manifold of states comprising its first six singlet electronic states. The model Hamiltonian is parametrized by fitting to the results of extensive EOM-CCSD calculations and its validity tested through the calculation of the first two bands in the electronic absorption spectrum of aniline. It is found that two previously neglected 3p Rydberg states play an important role in the dynamics of aniline following excitation into the first two (1)ππ* states. Assignments of the vibrational structure seen in the experimental spectrum is made, and the role played by the Herzberg-Teller effect in excitation to the first (1)ππ* state is analyzed.

Photodissociation of carbon dioxide in singlet valence electronic states. II. Five state absorption spectrum and vibronic assignment

The absorption spectrum of CO2 in the wavelength range 120-160 nm is analyzed by means of quantum mechanical calculations performed using vibronically coupled potential energy surfaces of five singlet valence electronic states and the coordinate dependent transition dipole moment vectors. The thermally averaged spectrum, calculated for T = 190 K via Boltzmann averaging of optical transitions from many initial rotational states, accurately reproduces the experimental spectral envelope, consisting of a low and a high energy band, the positions of the absorption maxima, their FWHMs, peak intensities, and frequencies of diffuse structures in each band. Contributions of the vibronic interactions due to Renner-Teller coupling, conical intersections, and the Herzberg-Teller effect are isolated and the calculated bands are assigned in terms of adiabatic electronic states. Finally, diffuse structures in the calculated bands are vibronically assigned using wave functions of the underlying resonance states. It is demonstrated that the main progressions in the high energy band correspond to consecutive excitations of the pseudorotational motion along the closed loop of the CI seam, and progressions differ in the number of nodes along the radial mode perpendicular to the closed seam. Irregularity of the diffuse peaks in the low energy band is interpreted as a manifestation of the carbene-type "cyclic" OCO minimum.

Intermolecular exciton coupling and vibronic effects in solid-state circular dichroism: a case study

The electronic circular dichroism (CD) spectrum of the sesquiterpenoid 1β,10β-epoxydesacetoxymatricarin (1) measured in the microcrystalline solid state differs from the solution one in the appearance of a pronounced vibrational fine structure in the long-wavelength region (n-π* enone transition) and of a new moderately intense band in the π-π* region. TDDFT CD calculations were run on input structures derived from the X-ray geometry of 1 to reproduce the impact of exciton-type couplings between proximate molecules in the crystals. The vibrational structure of the CD spectrum was also reproduced for the isolated molecule by modelling the potential energy surfaces at the harmonic level and taking into account Duschinsky and Herzberg-Teller effects.

Time-Dependent Approach to Resonance Raman Spectra Including Duschinsky Rotation and Herzberg-Teller Effects: Formalism and Its Realistic Applications.

Efficient quantum dynamical and electronic structure approaches are presented to calculate resonance Raman spectroscopy (RRS) with inclusion of Herzberg-Teller (HT) contribution and mode-mixing (Duschinsky) effect. In the dynamical method, an analytical expression for RRS in the time domain is proposed to avoid summation over the large number of intermediate vibrational states. In the electronic structure calculations, the analytic energy-derivative approaches for the excited states within the time-dependent density functional theory (TDDFT), developed by us, are adopted to overcome the computational bottleneck of excited-state gradient and Hessian calculations. In addition, an analytic calculation to the geometrical derivatives of the transition dipole moment, entering the HT term, is also adopted. The proposed approaches are implemented to calculate RR spectra of a few of conjugated systems, phenoxyl radical, 2-thiopyridone in water solution, and free-base porphyrin. The calculated RR spectra show the evident HT effect in those π-conjugated systems, and their relative intensities are consistent with experimental measurements.

Generating Function Approach to the Calculation of Spectral Band Shapes of Free-Base Chlorin Including Duschinsky and Herzberg–Teller Effects

A generating function approach to the calculation of spectral band shapes including Duschinsky and Herzberg-Teller effects is proposed and applied to the computation of the free-base chlorin Q absorption bands, using molecular geometries and normal vibrations obtained by density functional theory computations. The results clearly show that non-Condon effects can significantly affect the relative intensities of the weakest Q(y) and, to a lesser extent, Q(x) bands. The proposed approach is extremely powerful and can be used in the cases where the molecular size makes the direct calculation of Franck-Condon integrals by recurrence formulas prohibitive.

Franck-Condon simulation of vibrationally resolved optical spectra for zinc complexes of phthalocyanine and tetrabenzoporphyrin including the Duschinsky and Herzberg-Teller effects

High resolved absorption and fluorescence spectra of zinc complexes of phthalocyanine (ZnPc) and tetrabenzoporphyrin (ZnTBP) in the region of Q states were reported. Few theoretical investigations were performed to simulate the well-resolved spectra and assigned the vibrational bands of the large molecules, especially for high symmetrical characteristic molecules, on account of the difficulties to optimize the excited states and analyze a large number of final vibrational-normal modes. In the present work, the S(0) ↔ S(1) absorption and fluorescence spectra (that is, the Q band) of ZnPc and ZnTBP were simulated using time-dependent density functional theory with the inclusions of Duschinsky and Herzberg-Teller contributions to the electronic transition dipole moments. The theoretical results provide a good description of the optical spectra and are proved to be in excellent agreement with experimental spectra in inert-gas matrices or in supersonic expansion. This study focused attentions on the optical spectral similarities and contrasts between ZnPc and ZnTBP, in particular the noticeable Duschinsky and Herzberg-Teller effects on the high-resolved absorption and fluorescence spectra were considered. Substitution of meso-tetraaza on the porphyrin macrocycle framework could affect the ground state geometry and alter the electron density distributions, the orbital energies that accessible in the Q band region of the spectrum. The results were used to help interpret both the nature of the electronic transitions in Q band region, and the spectral discrepancies between phthalocyanine and porphyrin systems.

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).

Theoretical investigation on sum-frequency vibrational spectroscopy off electronic resonance for limonene chiral liquids: The second-order Herzberg–Teller effect

We theoretically study the influence of the second-order electric transition dipole moment derivatives on antisymmetric nonresonant vibrational Raman polarizabilities and the sum-frequency vibrational spectroscopy off electronic resonance. Time-dependent density functional theory computation for limonene chiral liquids indicates that the contribution from the second-order derivatives may be detectable in experiments.

Effective Time-Independent Calculations of Vibrational Resonance Raman Spectra of Isolated and Solvated Molecules Including Duschinsky and Herzberg-Teller Effects.

We present a method of modeling vibrational resonance Raman scattering (RRS) spectra of isolated and solvated systems with the inclusion of Franck-Condon (FC) and Herzberg-Teller (HT) effects and a full account for possible differences between the harmonic potential energy surfaces of the initial and resonant electronic states. It describes fundamentals, overtones, and combination bands and computes the RRS spectrum as a two-dimensional function of the incident and scattered frequencies. The theoretical foundations of the method are described and the differences with other currently available methodologies are outlined. Applications to the phenoxyl radical in the gas phase and indolinedimethine-malononitrile (IDMN) in acetonitrile and cyclohexane solution are reported, as well as comparisons with available experimental data.

Vibration correlation function formalism of radiative and non-radiative rates for complex molecules

General radiative and non-radiative rates formalisms are derived using the vibration correlation function method for the transition from the excited singlet to ground singlet states by considering the Duschinsky rotation and Herzberg–Teller effects at finite temperature. For the non-radiative transition process, the conventional assumption of (single) “promoting-mode” is abandoned and a promoting-mode free formula is presented. Using this new rate formalism, we re-examine the well-established photophysical properties of anthracene. Both the calculated radiative and non-radiative rates are in good agreement with the available experimental measurements and previous theoretical values. Furthermore, we rationalize the exotic aggregation induced emission phenomenon in 9-[(o-Aminophenyl)phenylmethylene]-9H-fluorene molecule: the roles of low-frequency phenyl ring twist motions and their Duschinsky mode mixings are found to be crucial, especially for the temperature dependence. The present rate theory can quantitatively describe the excited-states dynamic processes in large molecules and is a powerful tool for the design of new high-efficiency light-emitting materials.