Duschinsky Effect Research Articles

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.

Normal modes and the Duschinsky mixing of the ground- and excited-state vibrations of the green fluorescent protein chromophore

Ground- and excited-state vibrational frequencies were calculated for the chromophore of the green fluorescent protein (GFP) using the complete active space self-consistent field (CASSCF) method and detailed normal-mode analyses were carried out for ground and excited states. The mixing of the vibrational modes between the different states was studied by applying the Duschinsky effect by the expressing excited-state normal modes in terms of the ground-state normal modes. It was found that the low-frequency vibrational modes in the vertical excited state play a significant role in structural adjustment.

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.

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.

Effective time-independent studies on resonance Raman spectroscopy of trans-stilbene including the Duschinsky effect

We simulate the resonance Raman spectra of trans-stilbene using a recently developed time-independent method that allows computations of the full two-dimensional spectrum as a function of the incident and scattered frequencies, including both the Franck–Condon and the Herzberg–Teller contributions. The potential energy surfaces (PESs) of the ground and resonant states are described in the harmonic approximation using density functional theory PBE0/6-31+G(d,p) calculations in gas phase and in cyclohexane. The simulated spectra are in good agreement with the experimental data [J. Chem. Phys. 83, 5000 (1985)] measured at four different excitation wavelengths, and allow us to unambiguously assign the main experimental bands. We perform an extensive comparison of the performance of four different vertical or adiabatic models for the PES of the resonant state, dissecting the effects of nuclear displacements and Duschinsky mixings on the spectra.

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.

An analytical approach for computing Franck‐Condon integrals of harmonic oscillators with arbitrary dimensions

We have developed an analytical approach for computing Franck-Condon integrals (FCIs) of harmonic oscillators (HOs) with arbitrary dimensions in which the mode-mixing Duschinsky effect is taken into account. A general formula of FCIs of HOs was obtained and was applied to study the photoelectron spectroscopy of vinyl alcohol and ovalene (C(32)H(14)). The equilibrium geometries, harmonic vibrational frequencies and normal modes of vinyl alcohol, ovalene, and their cations were computed at the B3LYP/aug-cc-pVTZ or the B3LYP/6-31G(d) level, from which Franck-Condon factors were calculated and photoelectron spectra were simulated. The adiabatic ionization energies of vinyl alcohol were also computed by extrapolating the CCSD(T) energies to the complete basis set limit with aug-cc-pVXZ (X = D, T, Q, 5). The simulated photoelectron spectra of both vinyl alcohol and ovalene are in agreement with the experiments. The computed adiabatic ionization energies of syn- and anti-vinyl alcohol are in consistent with the experiment within 0.008 eV and 0.014 eV, respectively. We show, for the first time, that the analytical approach of computing FCIs is also efficient and promising for the studies of vibronic spectra of macrosystems.

Vibrationally Resolved Absorption and Emission Spectra of Dithiophene in the Gas Phase and in Solution by First-Principle Quantum Mechanical Calculations.

The absorption and emission spectra of dithiophene have been computed in different environments (gas phase, apolar, and polar solvents) and at different temperatures, including Duschinsky, temperature and solvent effects at full ab initio level, and considering the anharmonicity of the double well potential associated with the inter-ring torsional mode. The computed spectra are in very good agreement with the experimental ones, allowing for a complete assignment of the main vibrational features. Five different density functionals (BLYP, B3LYP, CAM-B3LYP, BHLYP, and PBE0) have been tested, and CAM-B3LYP and PBE0 are the most accurate.

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.

Coherent state-based generating function approach for Franck–Condon transitions and beyond

One-photon and multi-photon absorption, spontaneous and stimulated photon emission, resonance Raman scattering and electron transfer are important molecular processes that commonly involve combined vibrational-electronic (vibronic) transitions. The corresponding vibronic transition profiles in the energy domain are usually determined by Franck-Condon factors (FCFs), the squared norm of overlap integrals between vibrational wavefunctions of different electronic states. FC profiles are typically highly congested for large molecular systems and the spectra usually become not well-resolvable at elevated temperatures. The (theoretical) analyses of such spectra are even more difficult when vibrational mode mixing (Duschinsky) effects are significant, because contributions from different modes are in general not separable, even within the harmonic approximation. A few decades ago Doktorov, Malkin and Man'ko [1979 J. Mol. Spectrosc. 77, 178] developed a coherent state-based generating function approach and exploited the dynamical symmetry of vibrational Hamiltonians for the Duschinsky relation to describe FC transitions at zero Kelvin. Recently, the present authors extended the method to incorporate thermal, single vibronic level, non-Condon and multi-photon effects in energy, time and probability density domains for the efficient calculation and interpretation of vibronic spectra. Herein, recent developments and corresponding generating functions are presented for single vibronic levels related to fluorescence, resonance Raman scattering and anharmonic transition.

Unified analytical treatment for calculation of the two-dimensional Franck-Condon factors using the duschinsky transformation

Including binomial expansion theorems, we present an analytical formula for calculating Franck–Condon (FC) factors of two-dimensional (2D) harmonic oscillators including the Duschinsky effect. The FC principle has various practical applications in quantum modeling of electronic spectra of polyatomic molecules. The 2D FC factors are expressed through the binomial coefficients. Use of the memory of the computer for the calculation of binomial coefficients may extend the limits to large arguments for users and result in speeder calculation, should such limits be required in practice. Accurate numerical results are provided to validate the proposed algorithm. © 2012 Wiley Periodicals, Inc.

Franck–Condon analysis of the photoelectron spectra of HCCl−: Considering Duschinsky effects

Franck–Condon analysis and spectral simulations were carried out on the HCCl (X˜1A′)←HCCl-(X˜2A″) photo-detachment process. The theoretical spectra obtained by using CCSD/6-311+G(2p,d) values are in agreement with the observed one. Otherwise, the equilibrium geometry parameters, r(CCl)=1.950±0.002Å of the X˜2A″ state of HCCl− are derived, by using the spectral simulation and employing an iterative Franck–Condon (IF–C) analysis procedure. Furthermore, through the different methods, our best calculated electron affinity is 1.2106eV, which is in fair agreement with the experimental value.

Vibronic fine-structure in the S0 → S1 absorption spectrum of zinc porphyrin: A Franck–Condon simulation incorporating Herzberg–Teller theory and the Duschinsky effect

Although zinc porphyrin is a suitable specimen for studying the electronic structures and fine optical spectra of large molecules, few theoretical investigations have been performed to simulate and assign the vibrational bands in highly resolved spectra. On the basis of density functional theory and its TD extension, the S0→S1 absorption spectrum (that is, the Q band) of zinc porphyrin was simulated using the Franck–Condon (FC) approximation and incorporating the Herzberg–Teller (HT) and Duschinsky contributions to electronic transition dipole moments. Our theoretical results fit well with the optical spectrum obtained by experimental observations. The spectral profile of the Q band of zinc porphyrin is primarily described by the transition of the vibrational normal modes v8, v18, v49, v56, v57, v63, v64 and v87. Comparison of the FC and FCHT spectra with experimental results indicate that the S0→S1 absorption spectrum of zinc porphyrin is primarily due to the HT mechanism. In this regard, for the vibronically allowed or very weak transitions like those of the Q band in zinc porphyrin, the results indicate that the HT effect is more dominant than the FC contribution. In addition, we have tentatively assigned the vibrational lines in the simulated absorption spectrum. When compared with the available experimental spectra, our simulated results accurately reproduce most of the dominant transitions.

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.

A combined ab initio and Franck–Condon simulation study of the photodetachment spectrum of the [formula omitted] anion

A general analytical expression for the calculation of the two-dimensional four-mode Franck–Condon overlap integrals was presented and applied to scrutinize the photoelectron spectroscopy of ClO 2 - . Geometry optimization and harmonic vibrational frequency calculations have been performed on the X ∼ 2B 1 state of ClO 2 and X ∼ 1A 1 state of ClO 2 - by using the density functional theory (B2PLYP functional) and the coupled cluster singles and doubles with perturbative triples [CCSD(T)] methods with various basis sets up to aug-cc-pVQZ. The adiabatic electron affinity of ClO 2 was computed by using the CCSD(T) method extrapolated to the complete basis set limit with aug-cc-pVXZ (X = D, T, Q). Franck–Condon analyses and spectral simulation were carried out on the ClO 2( X ∼ 2B 1)–( ClO 2 - X ∼ 1A 1) photodetachment process. The simulated photoelectron spectrum of ClO 2 - are in accord with the experiment. While the Duschinsky effect plays a minor role and can be neglected. In addition, the equilibrium geometry parameters, R(Cl–O) = 1.572 ± 0.005 Å and ∠ (O–Cl–O) = 112.5 ± 0.5°, of the X ∼ 1A 1 state of ClO 2 - , are obtained by employing an iterative Franck–Condon analysis procedure in the spectral simulations.

Simulated Photoelectron Spectra of the Cyanide-Water Anion via Quasiclassical Molecular Dynamics

We present the simulated photoelectron spectrum (PES) for cyanide-water CN(H(2)O)(-) based on quasiclassical trajectory molecular dynamics (QCT-MD). Using density functional theory to generate trajectories and to calculate vertical detachment energies, we obtain simulated spectra that are in qualitative agreement with experiment. We obtain a theoretical 12 → 300 K temperature red shift of 0.1 eV as compared to an experimental redshift of 0.25 eV. The calculated linewidths of 0.3 eV are in excellent agreement with experiment. Our trajectories show that the temperature red shift as being dominated by dynamics within the basin of the N-bound minimum, however, at 300 K we predict conversion into the basin of the C-bound minimum, equilibrating at a 80:20 ratio of N- vs C-bound mixture. We discuss the potential advantages of QCT-MD over anharmonic Franck-Condon analysis such as natural incorporation of anharmonicity (as necessary for weakly bound systems), and reduced computational scaling, but also drawbacks such as neglect of final-state (e.g., Duschinsky) effects.

Determination of the tautomeric composition of adenine in the gas phase by vibrational spectroscopy methods: II. Analysis of resonance Raman spectra

The resonance Raman spectra of adenine in the gas phase under excitation with laser radiation at wavelengths of 266, 218, and 200 nm have been investigated experimentally. The quantum-mechanical calculations of the intensity distribution in the resonance Raman spectra of three adenine tautomers are performed in the Herzberg-Teller approximation with the inclusion of the Duschinsky and frequency effects. Conclusions regarding the tautomeric composition of adenine in the gas phase are drawn from comparison of the results of quantum-mechanical calculations with experimental data.

Fluorescence spectroscopy of jet-cooled o-fluoroanisole: Mixing through Duschinsky effect and Fermi resonance

Laser-induced fluorescence (LIF) excitation, UV-UV hole burning, and single vibronic level fluorescence (SVLF) spectra of jet-cooled o-fluoroanisole (o-FA) were measured. The most intense lowest-frequency band at 36 612 cm(-1) was assigned to the origin band of the most stable trans conformer. The UV-UV hole-burning spectrum demonstrated that the prominent bands in the LIF excitation spectrum were responsible for the trans conformer. The metastable non-planar conformer was not observed in the spectra. The vibrational band assignments were performed with the aid of quantum chemical calculations at the B3LYP/cc-pVTZ and CIS/6-311G(d,p) levels. The precise analysis of the SVLF spectra indicated that strong vibrational mixing through the Duschinsky effect and the Fermi resonance occurs in the S(1) state.

Calculation of Franck–Condon factors and simulation of photoelectron spectroscopy of D 2CO: Including Duschinsky effects

Calculations of Franck–Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. On the base of the closed form expression of the Franck–Condon integrals between arbitrary multidimensional harmonic oscillators under the Duschinsky mixing effects, a more general algebraic expression for the calculation of the three-dimensional four-mode Franck–Condon factors was derived straightforwardly and applied to study the photoelectron spectrum of D 2 CO + ( B ∼ 2 A 1 ) . Geometry optimization and harmonic vibrational frequency calculations were performed on the X ∼ 1 A 1 state of D 2CO at B3LYP, QCISD, CCSD and CASSCF levels, and the B ∼ 2 A 1 state of D 2CO + at CIS, TD-B3LYP, and CASSCF levels. Franck–Condon analyses and spectral simulations were carried out on the D 2 CO + ( B ∼ 2 A 1 ) - D 2 CO ( X ∼ 1 A 1 ) photoionization process. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectrum. In addition, the equilibrium geometric parameters of the B ∼ 2 A 1 state of D 2CO + were obtained in the spectral simulations.