Duschinsky Rotation Research Articles

Vibrational response functions for multidimensional electronic spectroscopy: From Duschinsky rotations to multimode squeezed coherent states.

Multidimensional spectroscopy unveils the interplay of nuclear and electronic dynamics, which characterizes the ultrafast dynamics of various molecular and solid-state systems. In a class of models widely used for the simulation of such dynamics, field-induced transitions between electronic states result in linear transformations (Duschinsky rotations) between the normal coordinates of the vibrational modes. Here, we present an approach for the calculation of the response functions, based on the explicit derivation of the vibrational state. This can be shown to coincide with a multimode squeezed coherent state, whose expression we derive within a quantum-optical formalism, and specifically by the sequential application to the initial state of rotation, displacement, and squeeze operators. The proposed approach potentially simplifies the numerical derivation of the response functions, avoiding the time integration of the Schrödinger equation, the Hamiltonian diagonalization, and the sum over infinite vibronic pathways. In addition, it quantitatively substantiates in the considered models the intuitive interpretation of the response functions in terms of the vibrational wave packet dynamics.

Topological aspects of system-bath Hamiltonians and a vector model for multisite systems coupled to local, correlated, or common baths.

Some topological features of multisite Hamiltonians consisting of harmonic potential surfaces with constant site-to-site couplings are discussed. Even in the absence of Duschinsky rotation, such a Hamiltonian assumes the system-bath form only if severe constraints exist. The simplest case of a common bath that couples to all sites is realized when the potential minima are collinear. The bath reorganization energy increases quadratically with site distance in this case. Another frequently encountered situation involves exciton-vibration coupling in molecular aggregates, where the intramolecular normal modes of the monomers give rise to local harmonic potentials. In this case, the reorganization energy accompanying excitation transfer is independent of site-to-site separation, thus this situation cannot be described by the usual system-bath Hamiltonian. A vector system-bath representation is introduced, which brings the exciton-vibration Hamiltonian in system-bath form. In this, the system vectors specify the locations of the potential minima, which in the case of identical monomers lie on the vertices of a regular polyhedron. By properly choosing the system vectors, it is possible to couple each bath to one or more sites and to specify the desired initial density. With a collinear choice of system vectors, the coupling reverts to the simple form of a common bath. The compact form of the vector system-bath coupling generalizes the dissipative tight-binding model to account for local, correlated, and common baths. The influence functional for the vector system-bath Hamiltonian is obtained in a compact and simple form.

Vibrationally resolved two-photon electronic spectra including vibrational pre-excitation: Theory and application to VIPER spectroscopy with two-photon excitation.

Following up on our previous work on vibrationally resolved electronic absorption spectra including the effect of vibrational pre-excitation [von Cosel et al., J. Chem. Phys. 147, 164116 (2017)], we present a combined theoretical and experimental study of two-photon-induced vibronic transitions in polyatomic molecules that are probed in the VIbrationally Promoted Electronic Resonance experiment using two-photon excitation (2P-VIPER). In order to compute vibronic spectra, we employ time-independent and time-dependent methods based on the evaluation of Franck-Condon overlap integrals and Fourier transformations of time-domain correlation functions, respectively. The time-independent approach uses a generalized version of the FCclasses method, while the time-dependent approach relies on the analytical evaluation of Gaussian moments within the harmonic approximation, including Duschinsky rotation effects. For the Coumarin 6 dye, two-dimensional 2P-VIPER experiments involving excitation to the lowest-lying singlet excited state (S1) are presented and compared with corresponding one-photon VIPER spectra. In both cases, coumarin ring modes and a CO stretch mode show VIPER activity, albeit with different relative intensities. Selective pre-excitation of these modes leads to a pronounced redshift of the low-frequency edge of the electronic absorption spectrum, which is a prerequisite for the VIPER experiment. Theoretical analysis underscores the role of interference between Franck-Condon and Herzberg-Teller effects in the two-photon experiment, which is at the root of the observed intensity distribution.

From Benzonitrile to Dicyanobenzenes: The Effect of an Additional CN Group on the Thermochemistry and Negative Ion Photoelectron Spectra of Dicyanobenzene Radical Anions.

The negative ion photoelectron spectra of 1,2-dicyanobenzene (o-DCNB), 1,3-dicyanobenzene (m-DCNB), and 1,4-dicyanobenzene (p-DCNB) radical anions (DCNB·-), acquired through the computation of Frack-Condon (FC) factors, are presented. The FC calculations utilize harmonic frequencies and normal mode vectors derived from density functional theory at the B3LYP/aug-cc-pVQZ basis set. All the totally symmetric vibrational modes are treated with Duschinsky rotations to yield neutral DCNBs in their singlet (So) and lowest triplet (T1) states, following an electron removal from the doublet anionic ground state. For the So state, the adiabatic electron affinities (EAs) for o-, m-, and p-DCNB are 1.179, 1.103, and 1.348 eV. The EAs for the lowest T1 state in o-, m-, and p-DCNB are 4.151, 4.185, and 4.208 eV, resulting in an So-T1 energy difference (ΔEST) of 2.973, 3.082, and 2.860 eV. A vibrational analysis reveals evidence of FC activity involving ring distortion, C-N bending, and ring C═C stretching vibrational progressions in both the So and T1 states. With the detection of cyanonaphthalene (C10H7CN) and cyanoindene (C9H7CN) in the interstellar medium (ISM), our results highlight the extent to which replacing a single hydrogen on an aromatic molecule with a cyano group, C≡N, can alter the vibrational structure of the molecule/radical anion. As such, dicyano-polyaromatic hydrocarbons may be reasonably robust in the ISM, making it appealing to search for them in future interstellar detection missions.

Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra

The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is common to apply the quantum harmonic-oscillator model, whose eigenfunctions intimately involve combinatorics. Each electronic state has distinct force constants; therefore, each normal-mode basis is distinct. Duschinsky proposed a linearized approximation to the transformation between the normal-mode bases of two electronic states using a rotation matrix. The rotation angles are typically obtained by using quantum-chemical computations or via gas-phase spectroscopy measurements. Quantifying the rotation angles in the condensed phase remains a challenge. Here, we apply a two-dimensional harmonic model that includes a Duschinsky rotation to condensed-phase femtosecond coherence spectra (FCS), which are created in transient–absorption spectroscopy measurements through impulsive excitation of coherent vibrational wavepackets. Using the 2D model, we simulate spectra to identify the signatures of Duschinsky rotation. The results suggest that peak multiplicities and asymmetries may be used to quantify the rotation angle, which is a key advance in condensed-phase molecular spectroscopy.

Vibronic fine structure in the nitrogen 1s photoelectron spectra of molecules from Franck-Condon simulations: Azines

Vibronic coupling plays a pivotal role in molecular spectroscopy. We present a theoretical study on vibrationally resolved x-ray photoelectron spectra (XPS) of seven azines (${\mathrm{C}}_{x}{\mathrm{H}}_{y}{\mathrm{N}}_{z}$; pyridine, three diazines, two triazines, and one tetrazine) at the nitrogen $1s$ edge, to explore the vibronic coupling effects as influenced by consecutive replacement of the CH group with a N atom. Franck-Condon simulations were performed with the Duschinsky rotation effect included, where the electronic structure was calculated by the density functional theory. Validations on pyrimidine show good agreement with the experiment, weak functional dependence, and weak mode mixing effect. We observed an evident blue shift in binding energies with the increasing number of N atoms in this series, together with molecule-dependent vibronic fine structures. These molecules have either ${\mathrm{C}}_{\text{2v}}$ or ${\mathrm{C}}_{\text{s}}$ molecular symmetry at the optimized core-ionized geometries. Franck-Condon-active vibrational modes were identified to be low frequency (500--1650 ${\mathrm{cm}}^{\ensuremath{-}1}$), totally symmetric (${A}_{1}$ or ${A}^{\ensuremath{'}}$), in-plane ring deformation modes. Core ionization on ${\mathrm{N}}^{*}$ always leads to elongation of the ${\mathrm{N}}^{*}\ensuremath{-}\mathrm{N}$ bond length, accompanied by an increase of the $\ensuremath{\angle}\mathrm{C}\ensuremath{-}{\mathrm{N}}^{*}\ensuremath{-}X$ bond angle ($X=\mathrm{C}$, N). Our study predicts accurate theoretical reference spectra for the azine family and provides useful information on the properties of the core-ionized states as influenced by the structural change of $\mathrm{CH}\ensuremath{\leftrightarrow}\mathrm{N}$ replacement.

Vibrationally-Resolved X-ray Photoelectron Spectra of Six Polycyclic Aromatic Hydrocarbons from First-Principles Simulations.

Vibrationally resolved C 1s X-ray photoelectron spectra (XPS) of a series of six polycyclic aromatic hydrocarbons (PAHs; phenanthrene, coronene, naphthalene, anthracene, tetracene, and pentacene) were computed by combining the full core hole density functional theory and the Franck-Condon simulations with the inclusion of the Duschinsky rotation effect. Simulated spectra of phenanthrene, coronene, and naphthalene agree well with experiments both in core binding energies (BEs) and profiles, which validate the accuracy of our predictions for the rest molecules with no high-resolution experiments. We found that three types of carbons i (inner C), p (peripheral C bonded to three C atoms), and h (peripheral C bonded to an H atom) show decreasing BEs. In linear PAHs (the latter four), h-type carbons further split into h1 or h2 (on inner or edge benzene ring) subtypes with chemical shifts of ca. 0.2-0.4 eV. All major Franck-Condon-active modes are characterized to be in-plane vibrations: low-frequency (<800 cm-1) C-C ring deformation modes play an essential role in determining the peak asymmetries; and for each h-type carbon a high-frequency (ca. 3600 cm-1) C*-H stretching mode is responsible for the high-energy tail. We found that core ionization leads to reduction of all C*-C and C*-H bond lengths and ring deformation with a definite direction. Based on theoretical spectra of four linear PAHs, we found asymptotic relations and anticipated possible spectral features for even larger linear PAHs. Our calculations provide accurate reference spectra for XPS characterizations of PAHs, which are useful in understanding the vibronic coupling effects in this family.

Excited State Properties of Aggregation‐Induced Delayed Fluorescence Molecules: A Microscopic Insight

AbstractPurely organic fluorescence emitters with ultrahigh exciton utilization, such as thermally‐activated delayed fluorescence (TADF) materials, are pursued as the cornerstone of the new‐generation organic light‐emitting diodes (OLEDs). However, most TADF emitters suffer from the dilemma of the aggregation‐caused fluorescence quenching effects, severely limiting their applications. Excitingly, the recently proposed aggregation‐induced delayed fluorescence (AIDF) strategy holds the potential to tackle this problem thoroughly. Here, the recently reported AIDF emitters CP‐BP‐PXZ and its halogenated counterparts, 3‐CCP‐BP‐PXZ and 3‐BCP‐BP‐PXZ, are investigated with a united theoretical and experimental insight. Based on first‐principles calculations combined with the polarizable continuum model (PCM) and the quantum mechanics/molecular mechanics (QM/MM) model, the photophysical mechanisms of these novel materials are proposed. It is found that both reverse intersystem crossing (RISC) and the subsequent radiative transition of CP‐BP‐PXZ are promoted via aggregation, owing to the increased effective RISC channels and boosted oscillator strength. Meanwhile, the unfavorable internal conversion rate is greatly slumped, ascribed to the suppressed Duschinsky rotation effect (DRE). The unique heavy‐atom effect is disclosed in 3‐CCP‐BP‐PXZ and 3‐BCP‐BP‐PXZ that the halogens act as steric blockers in crystal, rather than spin–orbit coupling enhancers. It is hoped that this work can shed new light on the designation of high‐efficient solid‐state fluorescence emitters.

Hybrid Quantum-Classical Boson Sampling Algorithm for Molecular Vibrationally Resolved Electronic Spectroscopy with Duschinsky Rotation and Anharmonicity.

Using a photonic quantum computer for boson sampling has demonstrated a tremendous advantage over classical supercomputers. It is highly desirable to develop boson sampling algorithms for realistic scientific problems. In this work, we propose a hybrid quantum-classical sampling (HQCS) algorithm to calculate the optical spectrum for complex molecules considering Duschinsky rotation effects and anharmonicity. The classical sum-over-states method for this problem has a computational complexity that exponentially increases with system size. The HQCS algorithm creates an intermediate harmonic potential energy surface (PES) to bridge the initial and final PESs. The magnitude and sign of the overlap between the initial and the intermediate state are estimated by boson sampling and classical algorithms, respectively. The overlap between the intermediate and the final state is efficiently evaluated by classical algorithms. The feasibility of HQCS is demonstrated in calculations of the emission spectrum of a Morse model as well as the pyridine molecule.

Origin of Nonmonotonical Variation of Luminescence Efficiency under Pressure in Organic Molecule.

The nonmonotonical variation of luminescence efficiency under extra pressure occurs frequently in organic molecules; however, the mechanism behind this is still elusive. Using a theoretical protocol combining thermal vibration function rate formalism coupled quantum mechanics/molecular mechanics models, we explored the relationship between extra pressure, molecular packing, and fluorescent quantum efficiency (FQE) of the representative 1,2,3,4-tetraphenyl-1,3-cyclopentadiene (TPC). It is found that the first increase and then decrease of FQE in TPC crystalline aggregates upon pressurization is cooperatively attributed to the continuous reduction of the radiative decay rate constant and nonmonotonical change of the nonradiative decay rate constant (kic). The initial decrease of kic originates from the effective suppression of electron-vibration coupling and the Duschinsky rotation effect by extra pressure, whereas the following increase of kic comes from the surge of nonadiabatic electronic coupling and the reduction of adiabatic excitation energy upon further compression. This study can provide a theoretical basis for the rational design and performance control of the piezochromic luminescent materials.

Theoretical Studies on the Photophysical Properties of the Ag(I) Complex for Thermally Activated Delayed Fluorescence Based on TD-DFT and Path Integral Dynamic Approaches.

Theoretical calculation not only is a powerful tool to deeply explore photophysical processes of the emitters but also provides a theoretical basis for material renewal and design strategy in the future. In this work, the interconversion and decay rates of the thermally activated delayed fluorescence (TADF) process of the rigid Ag(dbp)(P2-nCB) complex are quantitatively calculated by employing the optimally tuned range-separated hybrid functional (ω*B97X-D3) method combined with the path integral approach to dynamics considering the Herzberg–Teller and the Duschinsky rotation effects within a multimode harmonic oscillator model. The calculated results show that the small energy splitting ΔE(S1–T1) = 742 cm–1 (experimental value of 650 cm–1) of the lowest singlet S1 and triplet T1 state and proper vibrational spin–orbit coupling interactions facilitate the reverse intersystem crossing (RISC) processes from the T1 to S1 states. The kRISC rate is estimated to be 1.72 × 108 s–1 that is far more than the intersystem crossing rate kISC of 7.28 × 107 s–1, which will greatly accelerate the RISC process. In addition, the multiple coupling routes of zero-field splitting (ZFS) interaction can provide energetically nearby lying states, to speed up the RISC pathway, and restrict the phosphorescence decay rate. A smaller ZFS D-tensor of 0.143 cm–1, E/D ≈ 0.094 ≪ 1/3, and Δg > 0 are obtained, indicating that the excited singlet states are hardly mixed into the T1 state; thus, a lower phosphorescence decay rate (kp = 9.29 × 101 s–1) is expected to occur, and the T1 state has a long lifetime, which is helpful for the occurrence of the RISC process. These works are in excellent agreement with the experimental observation and are useful for improving and designing efficient TADF materials.

Vibronic Coupling Effect on the Vibrationally Resolved Electronic Spectra and Intersystem Crossing Rates of a TADF Emitter: 7-PhQAD.

Assessing and improving the performance of organic light-emitting diode (OLED) materials require quantitative prediction of rate coefficients for the intersystem crossing (ISC) and reverse ISC (RISC) processes, which are determined not only by the energy gap and the direct spin-orbit coupling (SOC) between the first singlet and triplet excited-states at a thermal equilibrium position of the initial electronic state but also by the non-Condon effects such as the Herzberg-Teller-like vibronic coupling (HTVC) and the spin-vibronic coupling (SVC). Here we apply the time-dependent correlation function approaches to quantitatively calculate the vibrationally resolved absorption and fluorescence spectra and ISC/RISC rates of a newly synthesized multiple-resonance-type (MR-type) thermally activated delayed fluorescence (TADF) emitter, 7-phenylquinolino[3,2,1-de]acridine-5,9-dione (7-PhQAD), with the inclusion of the Franck-Condon (FC), HTVC, and Duschinsky rotation (DR) effects. The SVC effect on the rates has also been approximately evaluated. We find that the experimentally measured ISC rates of 7-PhQAD originate predominantly from the vibronic coupling, consistent with the previous reports on other MR-type TADF emitters. The SVC effect on ISC rates is about 10 times larger than the HTVC effect, and the latter increases the ISC rates by more than 1 order of magnitude while it slightly affects the vibrationally resolved absorption and fluorescence spectra. The discrepancy between the theoretical and experimental results is attributed to inaccurately describing excited-states calculated by the time-dependent density functional theory as well as to not fully accounting for the complex experimental conditions. This work provides a demonstration of what proportion of ISC and RISC rate coefficients of a MR-type TADF emitter can be covered by the HTVC effect, and it opens design routes that go beyond the FC approximation for the future development of high-performance OLED devices.

QM/MM Studies on Thermally Activated Delayed Fluorescence of a Dicopper Complex in the Solid State

Dicopper complexes with thermally activated delayed fluorescence (TADF) phenomena are important in enriching the arsenal of organic light-emitting diodes materials. However, the TADF mechanism is still elusive, especially in the solid state. Herein, we chose a TADF dicopper complex and investigated its geometric and electronic structures and absorption and emission spectra using DFT, TD-DFT, and QM/MM methods. On the basis of these results, we further estimate the fluorescence emission rate from the S1 state, phosphorescence emission rate from the T1 state, and forward and reverse intersystem crossing (ISC and rISC) rates between S1 and T1. The present work shows that both the S1 and the T1 states have mixed metal-to-ligand and interligand charge-transfer character. Good spatial separation between the HOMO and the LUMO makes the S1–T1 energy gap small, ca. 2.8 kcal/mol. This small energy gap leads to efficient ISC and rISC processes, whose rates are much larger than the fluorescence and phosphorescence emission rates at 300 K, therefore enabling TADF. In contrast, at 77 K, the rISC process is blocked because its rate is much smaller than the phosphorescence emission. Thus, TADF disappears at 77 K. Further analysis shows that high-frequency deformation and low-frequency torsional vibrational modes make a large contribution to the Huang–Rhys factors, but Duschinsky rotation effects are essentially negligible for both the ISC and the rISC processes.

The spectral-shapes of absorption, emission, ECD and CPL of a fluorene-fused [7]helicene: Vibronic effect and solvent inhomogenous broadening

We compute the line shapes of absorption (ABS), emission (EMI), electronic circular dichroism (ECD), and circularly polarized luminescence (CPL) of a fluorene-fused [7]helicene. To elucidate different physico-chemical factors responsible for the spectral shapes, we account for Franck-Condon (FC) contribution, Herzberg-Teller (HT) effect and Duschinsky rotation, as well as the solvent inhomogenous broadening. The latter is connected to the solvent reorganization energy which is calculated with the state-specific (SS) implementation of Polarizable Continuum Model (PCM). To understand the relative importance of different effects of equilibrium geometry displacements, frequencies changes, and Duschinsky mixing, we carefully compare and discuss the results of Adiabatic Hessian (AH), Adiabatic Shift and Frequencies (ASF), and Adiabatic Shift (AS) models in details. Our results agree with the experimental spectra very well. They show that FC contribution dominates the spectral shapes and HT effect leads to the break of mirror images between absorptive (ABS + ECD) and emissive (EMI + CPL) spectra.

On-the-fly ab initio semiclassical evaluation of third-order response functions for two-dimensional electronic spectroscopy.

Ab initio computation of two-dimensional electronic spectra is an expanding field, whose goal is improving upon simple, few-dimensional models often employed to explain experiments. Here, we propose an accurate and computationally affordable approach, based on the single-trajectory semiclassical thawed Gaussian approximation, to evaluate two-dimensional electronic spectra. Importantly, the method is exact for arbitrary harmonic potentials with mode displacement, changes in the mode frequencies, and inter-mode coupling (Duschinsky effect), but can also account partially for the anharmonicity of the involved potential energy surfaces. We test its accuracy on a set of model Morse potentials and use it to study anharmonicity and Duschinsky effects on the linear and two-dimensional electronic spectra of phenol. We find that in this molecule, the anharmonicity effects are weak, whereas the Duschinsky rotation and the changes in the mode frequencies must be included in accurate simulations. In contrast, the widely used displaced harmonic oscillator model captures only the basic physics of the problem but fails to reproduce the correct vibronic lineshape.

Nonlinear spectroscopy in the condensed phase: The role of Duschinsky rotations and third order cumulant contributions.

First-principles modeling of nonlinear optical spectra in the condensed phase is highly challenging because both environment and vibronic interactions can play a large role in determining spectral shapes and excited state dynamics. Here, we compute two dimensional electronic spectroscopy (2DES) signals based on a cumulant expansion of the energy gap fluctuation operator, with specific focus on analyzing mode mixing effects introduced by the Duschinsky rotation and the role of the third order term in the cumulant expansion for both model and realistic condensed phase systems. We show that for a harmonic model system, the third order cumulant correction captures effects introduced by a mismatch in curvatures of ground and excited state potential energy surfaces, as well as effects of mode mixing. We also demonstrate that 2DES signals can be accurately reconstructed from purely classical correlation functions using quantum correction factors. We then compute nonlinear optical spectra for the Nile red and methylene blue chromophores in solution, assessing the third order cumulant contribution for realistic systems. We show that the third order cumulant correction is strongly dependent on the treatment of the solvent environment, revealing the interplay between environmental polarization and the electronic-vibrational coupling.

Thermal Vibration Correlation Function Formalism for Molecular Excited State Decay Rates

Thermal Vibration Correlation Function Formalism for Molecular Excited State Decay Rates

Probing the Endo/Exo Isomeric Variants of a Binary Complex of Cyclopropylamine with p-Fluorophenol by LIF Spectroscopy: A Comparative Study with Ammonia Complex.

Conformational isomers of an O-H···N hydrogen-bonded binary complex between para-fluorophenol (pFP) and a nonrigid primary amine base, cyclopropylamine (CPA), have been probed by means of laser-induced fluorescence (LIF) spectroscopy in a supersonic jet expansion. Two closely spaced electronic origin bands have been identified in the measured LIF excitation spectrum, and their assignments have been corroborated by making comparisons with the spectra of the parent pFP-NH3 complex recorded under the same expansion condition. The observation is consistent with the presence of endo and exo isomeric variants of the complex predicted by electronic structure theory methods, and the endo isomer is stabilized by ∼2 kcal/mol additionally owing to the formation of a C-H···O and a C-H···π type of weak hydrogen bonds between the two moieties. In the fluorescence excitation (FE) spectrum, the low-frequency bands for different intermolecular modes gain substantial intensity, and this spectral feature is in contrast to that of the pFP-NH3 complex. The Franck-Condon intensity of the bands has been simulated invoking Duschinsky rotation scheme, taking into consideration the ground- and excited-state geometries.

Variations in Duschinsky rotations in m-fluorotoluene and m-chlorotoluene during excitation and ionization.

We investigate Duschinsky rotation/mixing between three vibrations for both m-fluorotoluene (mFT) and m-chlorotoluene (mClT), during electronic excitation and ionization. In the case of mFT, we investigate both the S1 → S0 electronic transition and the D0 + ← S1 ionization, by two-dimensional laser-induced fluorescence (2D-LIF) and zero-electron-kinetic energy (ZEKE) spectroscopy, respectively; for mClT, only the D0 + ← S1 ionization was investigated, by ZEKE spectroscopy. The Duschinsky mixings are different in the two molecules, owing to shifts in vibrational wavenumber and variations in the form of the fundamental vibrations between the different electronic states. There is a very unusual behavior for two of the mFT vibrations, where apparently different conclusions for the identity of two S1 vibrations arise from the 2D-LIF and ZEKE spectra. We compare the experimental observations to the calculated Duschinsky matrices, finding that these successfully pick up the key geometric changes associated with each electronic transition and so are successful in qualitatively explaining the vibrational activity in the spectra. Experimental values for a number of vibrations across the S0, S1, and D0 + states are reported and found to compare well to those calculated. Assignments are made for the observed vibration-torsion ("vibtor") bands, and the effect of vibrational motion on the torsional potential is briefly discussed.

Predicting fluorescence quantum yields for molecules in solution: A critical assessment of the harmonic approximation and the choice of the lineshape function.

For the rational design of new fluorophores, reliable predictions of fluorescence quantum yields from first principles would be of great help. However, efficient computational approaches for predicting transition rates usually assume that the vibrational structure is harmonic. While the harmonic approximation has been used successfully to predict vibrationally resolved spectra and radiative rates, its reliability for non-radiative rates is much more questionable. Since non-adiabatic transitions convert large amounts of electronic energy into vibrational energy, the highly excited final vibrational states deviate greatly from harmonic oscillator eigenfunctions. We employ a time-dependent formalism to compute radiative and non-radiative rates for transitions and study the dependence on model parameters. For several coumarin dyes, we compare different adiabatic and vertical harmonic models (AS, ASF, AH, VG, VGF, and VH), in order to dissect the importance of displacements, frequency changes, and Duschinsky rotations. In addition, we analyze the effect of different broadening functions (Gaussian, Lorentzian, or Voigt). Moreover, to assess the qualitative influence of anharmonicity on the internal conversion rate, we develop a simplified anharmonic model. We address the reliability of these models considering the potential errors introduced by the harmonic approximation and the phenomenological width of the broadening function.