Optical spectra in the condensed phase: Capturing anharmonic and vibronic features using dynamic and static approaches

JOYCE3.0: A General Protocol for the Specific Parametrization of Accurate Intramolecular Quantum Mechanically Derived Force Fields

While the intrinsically multiscale nature of most advanced materials necessitates the use of cost-effective computational models based on classical physics, a reliable description of the structure and dynamics of their components often requires a quantum-mechanical treatment. In this work, we present JOYCE3.0, a software package for the parametrization of accurate, quantum-mechanically derived force fields (QMD-FFs). Since its original release, the code has been extensively automated and expanded, with all novel implementations thoroughly discussed. To illustrate its general applicability, QMD-FFs are parametrized for seven benchmark cases, encompassing molecules with diverse structures and properties. These range from exotic stiff scaffolds, flexible polymeric chains, and polyenes of biological interest to transition-metal complexes. On the one hand, JOYCE3.0 FFs consistently outperform available general-purpose descriptions, achieving excellent agreement with higher-level theoretical methods or available experimental validation data. On the other hand, the remarkable accuracy found in the description of the molecular structures extends to electronic excited states, enabling the integration of the JOYCE3.0 QMD-FFs into multilevel protocols aimed at reliably predicting selected properties and spectral line shapes in advanced optoelectronic materials. The high quality of the results─spanning molecular structures, condensed-phase properties, and spectroscopic features─in combination with the enhanced interface with popular quantum-mechanical codes and molecular dynamics engines, as well as its applicability to chemically diverse species, strongly suggests that JOYCE3.0 could play a pivotal role in the rational design of functionalized materials and heterogeneous systems.

Deciphering the Luminescence Spectral Shape of an Oxyluciferin Analogue through a Mixed Quantum-Classical Approach

In this contribution, we present a computational study on the absorption and emission spectra of the cproxy- anion in water, an analogue of the firefly oxyluciferin phenolate keto form. This compound displays a broad absorption spectrum and a large Stokes shift, two features that remain elusive to computational approaches, preventing a complete understanding of the photophysics behind this molecule. Here we attempt a fully first-principles computation of both absorption and emission spectral shapes and positions, explicitly including the effect of soft molecular flexible modes and of the stiff vibrational motions as well as those of the solvent. Namely, we adopt a recently developed mixed-quantum classical approach, the so-called Adiabatic Molecular Dynamics-generalized vertical Hessian (Ad-MD|gVH) method, which has been revealed to be well suited to reproduce band shapes in condensed phases. We also explore the performance of DFT functionals to build the potential energy surfaces and investigate the possible role of interstate couplings. By this means, we are able to obtain a first-principles simulation of the emission band shape close to the experimental one, and we correctly reproduce the two-peak shape of the absorption spectrum, both in terms of their spacing and relative intensity. However, the low-energy band of the computed absorption spectrum is too narrow, and the Stokes shift is remarkably underestimated. Through a careful analysis of different computational settings, we are able to identify some key aspects that partly explain these discrepancies, including the limitations of TD-DFT to properly describe the electronic energy along the flexible torsional degree of freedom in the lowest-excited state and the key role of mutual polarization of the solvent and the dye.

Initial Conditions for Excited-State Dynamics in Solvated Systems: A Case Study

Simulating excited-state dynamics or computing spectra for molecules in condensed phases requires sampling the ground state to generate initial conditions. Initial conditions (or snapshots for spectra) are typically produced by QM/MM Boltzmann sampling following MM equilibration or optimization. Given the switch from a MM to a QM/MM potential energy surface, one should discard a set period of time (which we call the "healing time") from the beginning of the QM/MM trajectory. Ideally, the healing time is as short as possible (to avoid unnecessary computational effort), but long enough to equilibrate to the QM/MM ground state distribution. Healing times in previous studies range from tens of femtoseconds to tens of picoseconds, suggesting the need for guidelines to choose a healing time. We examine the effect of healing time on the nonadiabatic dynamics and spectrum of a first-generation Donor-Acceptor Stenhouse Adduct in chloroform. Insufficient healing times skew the branching ratio of ground state products and alter the relaxation time for one pathway. The influence of the healing time on the absorption spectrum is less pronounced, warning that the spectrum is not a sensitive indicator for the quality of a set of initial conditions for dynamics. We demonstrate that a reasonable estimate for the healing time can be obtained by monitoring the solute temperature during the healing trajectory. We suggest that this procedure should become standard practice for determining healing times to generate initial conditions for nonadiabatic QM/MM simulations in large molecules and condensed phases.

Mechanism of the Non-Kasha Fluorescence in Pyrene

The high-energy shoulder in the gas-phase fluorescence emission spectrum of pyrene is a well-known example of non-Kasha emission. We comparatively assess two approaches, vibronic perturbation theory and nonadiabatic dynamics, in their ability to predict and explain the gas-phase fluorescence spectrum of pyrene. While both methods qualitatively capture the non-Kasha emission, they differ in their computational requirements, accuracy, and physical interpretation. Vibronic perturbation theory and nonadiabatic dynamics are complementary and can be combined in a two-step approach to non-Kasha fluorescence.

Efficient Polarizable QM/MM Using the Direct Reaction Field Hamiltonian with Electrostatic Potential Fitted Multipole Operators

Electronic polarization and dispersion are decisive actors in determining interaction energies between molecules. These interactions have a particularly profound effect on excitation energies of molecules in complex environments, especially when the excitation involves a significant degree of charge reorganization. The direct reaction field (DRF) approach, which has seen a recent revival of interest, provides a powerful framework for describing these interactions in quantum mechanics/molecular mechanics (QM/MM) models of systems, where a small subsystem of interest is described using quantum chemical methods and the remainder is treated with a simple MM force field. In this paper we show how the DRF approach can be combined with the electrostatic potential fitted (ESPF) multipole operator description of the QM region charge density, which significantly improves the efficiency of the method, particularly for large MM systems, and for typical calculations effectively eliminates the dependence on MM system size. We also show how the DRF approach can be combined with fluctuating charge descriptions of the polarizable environment, as well as previously used atom-centered dipole-polarizability based models. We further show that the ESPF-DRF method provides an accurate description of molecular interactions in both ground and excited electronic states of the QM system and apply it to predict the gas to aqueous solution solvatochromic shifts in the UV/visible absorption spectrum of acrolein.

Simulating temperature and tautomeric effects for vibrationally resolved XPS of biomolecules: Combining time-dependent and time-independent approaches to fingerprint carbonyl groups

Carbonyl groups (C=O) play crucial roles in the photophysics and photochemistry of biological systems. O1s x-ray photoelectron spectroscopy allows for targeted investigation of the C=O group, and the coupling between C=O vibration and O1s ionization is reflected in the fine structures. To elucidate its characteristic vibronic features, systematic Franck-Condon simulations were conducted for six common biomolecules, including three purines (xanthine, caffeine, and hypoxanthine) and three pyrimidines (thymine, 5F-uracil, and uracil). The complexity of simulation for these biomolecules lies in accounting for temperature effects and potential tautomeric variations. We combined the time-dependent and time-independent methods to efficiently account for the temperature effects and to provide explicit assignments, respectively. For hypoxanthine, the tautomeric effect was considered by incorporating the Boltzmann population ratios of two tautomers. The simulations demonstrated good agreement with experimental spectra, enabling differentiation of two types of carbonyl oxygens with subtle local structural differences, positioned between two nitrogens (O1) or between one carbon and one nitrogen (O2). The analysis provided insights into the coupling between C=O vibration and O1s ionization, consistently showing an elongation of the C=O bond length (by 0.08-0.09 Å) upon O1s ionization.

Explicit Modelling of Spectral Bandshapes by a Mixed Quantum-Classical Approach: Solvent Order and Temperature Effects in the Optical Spectra of Distyrylbenzene

The absorption and emission spectral shapes of a flexible organic probe, the distyrylbenzene (DSB) dye, are simulated accounting for the effect of different environments of increasing complexity, ranging from a homogeneous, low-molecular- weight solvent, to a long-chain alkane, and, eventually, a channel-forming organic matrix. Each embedding is treated explicitly, adopting a mixed quantum-classical approach, the Adiabatic Molecular Dynamics - generalized vertical Hessian (Ad-MD|gVH) model, which allows a direct simulation of the environment-induced constraining effects on the vibronic spectral shapes. In such a theoretical framework, the stiff modes of the dye are described at a quantum level within the harmonic approximation, including Duschinsky mixing effects, while flexible degrees of freedom of the solute (e. g. torsions) and those of the solvent are treated classically by means of molecular dynamics sampling. Such a setup is shown to reproduce the distinct effects exerted by the different environments in varied thermodynamic conditions. Besides allowing for a first-principles rationale on the supramolecular mechanism leading to the experimental spectral features, this result represents the first successful application of the Ad-MD|gVH method to complex embeddings and supports its potential application to other heterogeneous environments, such as for instance, pigment-protein complexes or organic dyes adsorbed into metal-organic frameworks.

Absorption Intensities of Organic Molecules from Electronic Structure Calculations versus Experiments: the Effect of Solvation, Method, Basis Set, and Transition Moment Gauge

Recently, we derived experimental oscillator strengths (OSs) from well-defined UV-visible absorption spectral peaks of 100 molecules in solution. Here, we focus on a subset of transitions with the highest reliability to further benchmark the OSs from several wave function methods and density functionals. We consider multiple basis sets, transition moment gauges (length, velocity, and mixed), and solvent corrections. Most transitions in the comparison set come from conjugated molecules and have π → π* character. We use an automated algorithm to assign computed transitions to experimental bands. OSs computed using the Tamm-Dancoff approximation (TDA), CIS, or EOM-CCSD exhibited a strong gauge dependence, which is diminished in linear response theories (TD-DFT, TD-HF, and to a smaller degree LR-CCSD). OSs calculated from TD-DFT with PCM solvent models are systematically larger than apparent OSs derived from experimental spectra. For example, fcomp from hybrid functionals and PCM have mean absolute errors that are ∼10% of n·fexp, where n is a solvent refractive index factor that arises from the energy flux of the radiation field in a dielectric (solvent). Theoretical cavity field corrections considering spherical cavities do not improve the agreement between computed and experimental data. Corrections that account for the molecular shape and the direction of transition dipole moments, or that explicitly account for the effect of solvent molecules on the local field, should be more appropriate.

Calculating absorption and fluorescence spectra for chromophores in solution with ensemble Franck-Condon methods

Accurately modeling absorption and fluorescence spectra for molecules in solution poses a challenge due to the need to incorporate both vibronic and environmental effects, as well as the necessity of accurate excited state electronic structure calculations. Nuclear ensemble approaches capture explicit environmental effects, Franck-Condon methods capture vibronic effects, and recently introduced ensemble-Franck-Condon approaches combine the advantages of both methods. In this study, we present and analyze simulated absorption and fluorescence spectra generated with combined ensemble-Franck-Condon approaches for three chromophore-solvent systems and compare them to standard ensemble and Franck-Condon spectra, as well as to the experiment. Employing configurations obtained from ground and excited state ab initio molecular dynamics, three combined ensemble-Franck-Condon approaches are directly compared to each other to assess the accuracy and relative computational time. We find that the approach employing an average finite-temperature Franck-Condon line shape generates spectra nearly identical to the direct summation of an ensemble of Franck-Condon spectra at one-fourth of the computational cost. We analyze how the spectral simulation method, as well as the level of electronic structure theory, affects spectral line shapes and associated Stokes shifts for 7-nitrobenz-2-oxa-1,3-diazol-4-yl and Nile red in dimethyl sulfoxide and 7-methoxy coumarin-4-acetic acid in methanol. For the first time, our studies show the capability of combined ensemble-Franck-Condon methods for both absorption and fluorescence spectroscopy and provide a powerful tool for simulating linear optical spectra.

First-Principles Modeling of the Absorption Spectrum of Crystal Violet in Solution: The Importance of Environmentally Driven Symmetry Breaking

Theoretical spectroscopy plays a crucial role in understanding the properties of the materials and molecules. One of the most promising methods for computing optical spectra of chromophores embedded in complex environments from the first principles is the cumulant approach, where both (generally anharmonic) vibrational degrees of freedom and environmental interactions are explicitly accounted for. In this work, we verify the capabilities of the cumulant approach in describing the effect of complex environmental interactions on linear absorption spectra by studying Crystal Violet (CV) in different solvents. The experimental absorption spectrum of CV strongly depends on the nature of the solvent, indicating strong coupling to the condensed-phase environment. We demonstrate that these changes in absorption line shape are driven by an increased splitting between absorption bands of two low-lying excited states that is caused by a breaking of the D3 symmetry of the molecule and that in polar solvents, this symmetry breaking is mainly driven by electrostatic interactions with the condensed-phase environment rather than distortion of the structure of the molecule, in contrast with conclusions reached in a number of previous studies. Our results reveal the importance of explicitly including a counterion in the calculations in nonpolar solvents due to electrostatic interactions between CV and the ion. In polar solvents, these interactions are strongly reduced due to solvent screening effects, thus minimizing the symmetry breaking. Computed spectra in methanol are found to be in reasonable agreement with the experiment, demonstrating the strengths of the outlined approach in modeling strong environmental interactions.

Axial H-Bonding Solvent Controls Inhomogeneous Spectral Broadening, While Peripheral H-Bonding Solvent Controls Vibronic Broadening: Cresyl Violet in Methanol

The dynamics of the nuclei of both a chromophore and its condensed-phase environment control many spectral features, including the vibronic and inhomogeneous broadening present in spectral line shapes. For the cresyl violet chromophore in methanol, we here analyze and isolate the effect of specific chromophore-solvent interactions on simulated spectral densities, reorganization energies, and linear absorption spectra. Employing both chromophore and its condensed-phase environment control many spectral features, including the vibronic and inhomogeneous broadening present in spectral line shapes. For the cresyl violet chromophore in methanol, we here analyze and isolate the effect of specific chromophore-solvent interactions on simulated spectral densities, reorganization energies, and linear absorption spectra. Employing both force field and ab initio molecular dynamics trajectories along with the inclusion of only certain solvent molecules in the excited-state calculations, we determine that the methanol molecules axial to the chromophore are responsible for the majority of inhomogeneous broadening, with a single methanol molecule that forms an axial hydrogen bond dominating the response. The strong peripheral hydrogen bonds do not contribute to spectral broadening, as they are very stable throughout the dynamics and do not lead to increased energy-gap fluctuations. We also find that treating the strong peripheral hydrogen bonds as molecular mechanical point charges during the molecular dynamics simulation underestimates the vibronic coupling. Including these peripheral hydrogen bonding methanol molecules in the quantum-mechanical region in a geometry optimization increases the vibronic coupling, suggesting that a more advanced treatment of these strongly interacting solvent molecules during the molecular dynamics trajectory may be necessary to capture the full vibronic spectral broadening.

Protein Effects on the Excitation Energies and Exciton Dynamics of the CP24 Antenna Complex

In this study, the site energy fluctuations, energy transfer dynamics, and some spectroscopic properties of the minor light-harvesting complex CP24 in a membrane environment were determined. For this purpose, a 3 μs-long classical molecular dynamics simulation was performed for the CP24 complex. Furthermore, using the density functional tight binding/molecular mechanics molecular dynamics (DFTB/MM MD) approach, we performed excited state calculations for the chlorophyll a and chlorophyll b molecules in the complex starting from five different positions of the MD trajectory. During the extended simulations, we observed variations in the site energies of the different sets as a result of the fluctuating protein environment. In particular, a water coordination to Chl-b 608 occurred only after about 1 μs in the simulations, demonstrating dynamic changes in the environment of this pigment. From the classical and the DFTB/MM MD simulations, spectral densities and the (time-dependent) Hamiltonian of the complex were determined. Based on these results, three independent strongly coupled chlorophyll clusters were revealed within the complex. In addition, absorption and fluorescence spectra were determined together with the exciton relaxation dynamics, which reasonably well agrees with experimental time scales.

2-in-1 Phase Space Sampling for Calculating the Absorption Spectrum of the Hydrated Electron

The investigation of vibrational effects on absorption spectrum calculations often employs Wigner sampling or thermal sampling. While Wigner sampling incorporates zero-point energy, it may not be suitable for flexible systems. Thermal sampling is applicable to anharmonic systems yet treats nuclei classically. The application of generalized smoothed trajectory analysis (GSTA) as a postprocessing method allows for the incorporation of nuclear quantum effects (NQEs), combining the advantages of both sampling methods. We demonstrate this approach in computing the absorption spectrum of a hydrated electron. Theoretical exploration of the hydrated electron and its embryonic forms, such as water cluster anions, poses a significant challenge due to the diffusivity of the excess electron and the continuous motion of water molecules. In many previous studies, the wave nature of atomic nuclei is often neglected, despite the substantial impact of NQEs on thermodynamic and spectroscopic properties, particularly for hydrogen atoms. In our studies, we examine these NQEs for the excess electrons in various water systems. We obtained structures from mixed classical-quantum simulations for water cluster anions and the hydrated electron by incorporating the quantum effects of atomic nuclei with the filtration of the classical trajectories. Absorption spectra were determined at different theoretical levels. Our results indicate significant NQEs, red shift, and broadening of the spectra for hydrated electron systems. This study demonstrates the applicability of GSTA to complex systems, providing insights into NQEs on energetic and structural properties.

Beyond Explored Functionals: A Computational Journey of Two-Photon Absorption

We present a thorough investigation into the efficacy of 19 density functional theory (DFT) functionals, relative to RI-CC2 results, for computing two-photon absorption (2PA) cross sections (σ2PA) and key dipole moments (|μ00|, |μ11|, |Δμ|, |μ01|) for a series of coumarin dyes in the gas-phase. The functionals include different categories, including local density approximation (LDA), generalized gradient approximation (GGA), hybrid-GGA (H-GGA), range-separated hybrid-GGA (RSH-GGA), meta-GGA (M-GGA), and hybrid M-GGA (HM-GGA), with 14 of them being subjected to analysis for the first time with respect to predicting σ2PA values. Analysis reveals that functionals integrating both short-range (SR) and long-range (LR) corrections, particularly those within the RSH-GGA and HM-GGA classes, outperform the others. Furthermore, the range-separation approach was found more impactful compared to the varying percentages of Hartree-Fock exchange (HF Ex) within different functionals. The functionals traditionally recommended for 2PA do not appear among the top 9 in our study, which is particularly interesting, as these top-performing functionals have not been previously investigated in this context. This list is dominated by M11, QTP variants, ωB97X, ωB97X-V, and M06-2X, surpassing the performance of other functionals, including the commonly used CAM-B3LYP.

Unraveling the mechanisms of triplet state formation in a heavy-atom free photosensitizer

Triplet excited state generation plays a pivotal role in photosensitizers, however the reliance on transition metals and heavy atoms can limit the utility of these systems. In this study, we demonstrate that an interplay of competing quantum effects controls the high triplet quantum yield in a prototypical boron dipyrromethene-anthracene (BD-An) donor-acceptor dyad photosensitizer, which is only captured by an accurate treatment of both inner and outer sphere reorganization energies. Our ab initio-derived model provides excellent agreement with experimentally measured spectra, triplet yields and excited state kinetic data, including the triplet lifetime. We find that rapid triplet state formation occurs primarily via high-energy triplet states through both spin-orbit coupled charge transfer and El-Sayed's rule breaking intersystem crossing, rather than direct spin-orbit coupled charge transfer to the lowest lying triplet state. Our calculations also reveal that competing effects of nuclear tunneling, electronic state recrossing, and electronic polarizability dictate the rate of non-productive ground state recombination. This study sheds light on the quantum effects driving efficient triplet formation in the BD-An system, and offers a promising simulation methodology for diverse photochemical systems.

Environmentally Driven Symmetry Breaking Quenches Dual Fluorescence in Proflavine

Nonadiabatic couplings between several electronic excited states are ubiquitous in many organic chromophores and can significantly influence optical properties. A recent experimental study demonstrated that the proflavine molecule exhibits surprising dual fluorescence in the gas phase, which is suppressed in polar solvent environments. Here, we uncover the origin of this phenomenon by parametrizing a linear-vibronic coupling Hamiltonian from spectral densities of system-bath coupling constructed along molecular dynamics trajectories, fully accounting for interactions with the condensed-phase environment. The finite-temperature absorption, steady-state emission, and time-resolved emission spectra are then computed using powerful, numerically exact tensor network approaches. We find that the dual fluorescence in vacuum is driven by a single well-defined coupling mode but is quenched in solution due to dynamic solvent-driven symmetry breaking that mixes the two low-lying electronic states. We expect the computational framework developed here to be widely applicable to the study of non-Condon effects in complex condensed-phase environments.

Taming the third order cumulant approximation to linear optical spectroscopy

The second order cumulant method offers a promising pathway to predicting optical properties in condensed phase systems. It allows for the computation of linear absorption spectra from excitation energy fluctuations sampled along molecular dynamics (MD) trajectories, fully accounting for vibronic effects, direct solute-solvent interactions, and environmental polarization effects. However, the second order cumulant approximation only guarantees accurate line shapes for energy gap fluctuations obeying Gaussian statistics. A third order correction has recently been derived but often yields unphysical spectra or divergent line shapes for moderately non-Gaussian fluctuations due to the neglect of higher order terms in the cumulant expansion. In this work, we develop a corrected cumulant approach, where the collective effect of neglected higher order contributions is approximately accounted for through a dampening factor applied to the third order cumulant term. We show that this dampening factor can be expressed as a function of the skewness and kurtosis of energy gap fluctuations and can be parameterized from a large set of randomly sampled model Hamiltonians for which exact spectral line shapes are known. This approach is shown to systematically remove unphysical contributions in the form of negative absorbances from cumulant spectra in both model Hamiltonians and condensed phase systems sampled from MD and dramatically improves over the second order cumulant method in describing systems exhibiting Duschinsky mode mixing effects. We successfully apply the approach to the coumarin-153 dye in toluene, obtaining excellent agreement with experiment.

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Beyond the Condon limit: Condensed phase optical spectra from atomistic simulations

While dark transitions made bright by molecular motions determine the optoelectronic properties of many materials, simulating such non-Condon effects in condensed phase spectroscopy remains a fundamental challenge. We derive a Gaussian theory to predict and analyze condensed phase optical spectra beyond the Condon limit. Our theory introduces novel quantities that encode how nuclear motions modulate the energy gap and transition dipole of electronic transitions in the form of spectral densities. By formulating the theory through a statistical framework of thermal averages and fluctuations, we circumvent the limitations of widely used microscopically harmonic theories, allowing us to tackle systems with generally anharmonic atomistic interactions and non-Condon fluctuations of arbitrary strength. We show how to calculate these spectral densities using first-principles simulations, capturing realistic molecular interactions and incorporating finite-temperature, disorder, and dynamical effects. Our theory accurately predicts the spectra of systems known to exhibit strong non-Condon effects (phenolate in various solvents) and reveals distinct mechanisms for electronic peak splitting: timescale separation of modes that tune non-Condon effects and spectral interference from correlated energy gap and transition dipole fluctuations. We further introduce analysis tools to identify how intramolecular vibrations, solute-solvent interactions, and environmental polarization effects impact dark transitions. Moreover, we prove an upper bound on the strength of cross correlated energy gap and transition dipole fluctuations, thereby elucidating a simple condition that a system must follow for our theory to accurately predict its spectrum.

Origin of Vibronic Coherences During Carrier Cooling in Colloidal Quantum Dots

Recent two-dimensional electronic spectroscopy experiments [Tilluck et al. J. Phys. Chem. Lett. 2021, 12 (39), 9677-9683] indicate the creation of coherent vibronic wavepackets in the first femtoseconds of hot carrier cooling in hexadecylamine-passivated CdSe quantum dots. Here we present a quantum chemical study of the origin of these coherences in a CdSe nanocrystal. We find that coherent wavepacket motions along vibrational coordinates with alkylamine character promote nonradiative relaxation through conical intersections between the exciton states of the inorganic core. Electronic excitations in the core are found to pass energy to the vibrations of the ligands via two distinct mechanisms: excitation of core phonon modes that are coupled to the ligand vibrations and direct excitation of ligand vibrations by delocalization of the exciton onto the ligands, both of which naturally arise within a photochemical framework based on many-electron potential energy surfaces. If these findings are demonstrated to be general, vibronic coherences may be leveraged to control photophysical outcomes in colloidal quantum dots.

Unravelling the Origin of the Vibronic Spectral Signatures in an Excitonically Coupled Indocarbocyanine Cy3 Dimer

The indocarbocyanine Cy3 dye is widely used to probe the dynamics of proteins and DNA. Excitonically coupled Cy3 dimers exhibit very unique spectral signatures that depend on the interchromophoric geometrical orientation induced by the environment, making them powerful tools to infer the dynamics of their surroundings. Understanding the origin of the dimeric spectral signatures is a necessity for an accurate interpretation of the experimental results. In this work, we simulate the vibronic spectrum of an experimentally well-studied Cy3 dimer, and we explain the origin of the experimental signatures present in its linear absorption spectrum. The Franck-Condon harmonic approximations, among other tests, are used to probe the factors contributing to the spectrum. It is found that the first peak in the absorption spectrum originates from the lower energy excitonic state, while the next two peaks are vibrational progressions of the higher energy excitonic state. The polar solvent plays a crucial role in the appearance of the spectrum, being responsible for the localized S1 minimum, which results in an increased intensity of the first peak.

Pitfall in simulations of vibronic TD-DFT spectra: diagnosis and assessment

In this Communication, we study the effect of spurious oscillations in the profiles of energy derivatives with respect to nuclear coordinates calculated with density functional approximations (DFAs) for formaldehyde, pyridine, and furan in their ground and electronic excited states. These spurious oscillations, which can only be removed using extensive integration grids that increase enormously the CPU cost of DFA calculations, are significant in the case of third- and fourth-order energy derivatives of the ground and excited states computed by M06-2X and ωB97X functionals. The errors in question propagate to anharmonic vibronic spectra computed under the Franck-Condon approximation, i.e., positions and intensities of vibronic transitions are affected to a large extent (shifts as significant as hundreds of cm-1 were observed). On the other hand, the LC-BLYP and CAM-B3LYP functionals show a much less pronounced effect due to spurious oscillations. Based on the results presented herein, we recommend either LC-BLYP or CAM-B3LYP with integration grids (250, 974) (or larger) for numerically stable simulations of vibronic spectra including anharmonic effects.

Multifidelity Machine Learning for Molecular Excitation Energies

The accurate but fast calculation of molecular excited states is still a very challenging topic. For many applications, detailed knowledge of the energy funnel in larger molecular aggregates is of key importance, requiring highly accurate excitation energies. To this end, machine learning techniques can be a very useful tool, though the cost of generating highly accurate training data sets still remains a severe challenge. To overcome this hurdle, this work proposes the use of multifidelity machine learning where very little training data from high accuracies is combined with cheaper and less accurate data to achieve the accuracy of the costlier level. In the present study, the approach is employed to predict vertical excitation energies to the first excited state for three molecules of increasing size, namely, benzene, naphthalene, and anthracene. The energies are trained and tested for conformations stemming from classical molecular dynamics and density functional based tight-binding simulations. It can be shown that the multifidelity machine learning model can achieve the same accuracy as a machine learning model built only on high-cost training data while expending a much lower computational effort to generate the data. The numerical gain observed in these benchmark test calculations was over a factor of 30 but certainly can be much higher for high-accuracy data.

Elucidating the Role of Hydrogen Bonding in the Optical Spectroscopy of the Solvated Green Fluorescent Protein Chromophore: Using Machine Learning to Establish the Importance of High-Level Electronic Structure

Hydrogen bonding interactions with chromophores in chemical and biological environments play a key role in determining their electronic absorption and relaxation processes, which are manifested in their linear and multidimensional optical spectra. For chromophores in the condensed phase, the large number of atoms needed to simulate the environment has traditionally prohibited the use of high-level excited-state electronic structure methods. By leveraging transfer learning, we show how to construct machine-learned models to accurately predict the high-level excitation energies of a chromophore in solution from only 400 high-level calculations. We show that when the electronic excitations of the green fluorescent protein chromophore in water are treated using EOM-CCSD embedded in a DFT description of the solvent the optical spectrum is correctly captured and that this improvement arises from correctly treating the coupling of the electronic transition to electric fields, which leads to a larger response upon hydrogen bonding between the chromophore and water.

Non-Phenomenological Description of the Time-Resolved Emission in Solution with Quantum-Classical Vibronic Approaches-Application to Coumarin C153 in Methanol

We report a joint experimental and theoretical work on the steady-state spectroscopy and time-resolved emission of the coumarin C153 dye in methanol. The lowest energy excited state of this molecule is characterized by an intramolecular charge transfer thus leading to remarkable shifts of the time-resolved emission spectra, dictated by the methanol reorganization dynamics. We selected this system as a prototypical test case for the first application of a novel computational protocol aimed at the prediction of transient emission spectral shapes, including both vibronic and solvent effects, without applying any phenomenological broadening. It combines a recently developed quantum-classical approach, the adiabatic molecular dynamics generalized vertical Hessian method (Ad-MD|gVH), with nonequilibrium molecular dynamics simulations. For the steady-state spectra we show that the Ad-MD|gVH approach is able to reproduce quite accurately the spectral shapes and the Stokes shift, while a ∼0.15 eV error is found on the prediction of the solvent shift going from gas phase to methanol. The spectral shape of the time-resolved emission signals is, overall, well reproduced, although the simulated spectra are slightly too broad and asymmetric at low energies with respect to experiments. As far as the spectral shift is concerned, the calculated spectra from 4 ps to 100 ps are in excellent agreement with experiments, correctly predicting the end of the solvent reorganization after about 20 ps. On the other hand, before 4 ps solvent dynamics is predicted to be too fast in the simulations and, in the sub-ps timescale, the uncertainty due to the experimental time resolution (300 fs) makes the comparison less straightforward. Finally, analysis of the reorganization of the first solvation shell surrounding the excited solute, based on atomic radial distribution functions and orientational correlations, indicates a fast solvent response (≈100 fs) characterized by the strengthening of the carbonyl-methanol hydrogen bond interactions, followed by the solvent reorientation, occurring on the ps timescale, to maximize local dipolar interactions.

Photogrammetry of Ultrafast Excited-State Intramolecular Proton Transfer Pathways in the Fungal Pigment Draconin Red

Proton transfer processes of organic molecules are key to charge transport and photoprotection in biological systems. Among them, excited-state intramolecular proton transfer (ESIPT) reactions are characterized by quick and efficient charge transfer within a molecule, resulting in ultrafast proton motions. The ESIPT-facilitated interconversion between two tautomers (PS and PA) comprising the tree fungal pigment Draconin Red in solution was investigated using a combination of targeted femtosecond transient absorption (fs-TA) and excited-state femtosecond stimulated Raman spectroscopy (ES-FSRS) measurements. Transient intensity (population and polarizability) and frequency (structural and cooling) dynamics of -COH rocking and -C=C, -C=O stretching modes following directed stimulation of each tautomer elucidate the excitation-dependent relaxation pathways, particularly the bidirectional ESIPT progression out of the Franck-Condon region to the lower-lying excited state, of the intrinsically heterogeneous chromophore in dichloromethane solvent. A characteristic overall excited-state PS-to-PA transition on the picosecond timescale leads to a unique "W"-shaped excited-state Raman intensity pattern due to dynamic resonance enhancement with the Raman pump-probe pulse pair. The ability to utilize quantum mechanics calculations in conjunction with steady-state electronic absorption and emission spectra to induce disparate excited-state populations in an inhomogeneous mixture of similar tautomers has broad implications for the modeling of potential energy surfaces and delineation of reaction mechanisms in naturally occurring chromophores. Such fundamental insights afforded by in-depth analysis of ultrafast spectroscopic datasets are also beneficial for future development of sustainable materials and optoelectronics.

Cost-Effective Simulations of Vibrationally-Resolved Absorption Spectra of Fluorophores with Machine-Learning-Based Inhomogeneous Broadening

The results of electronic and vibrational structure simulations are an invaluable support for interpreting experimental absorption/emission spectra, which stimulates the development of reliable and cost-effective computational protocols. In this work, we contribute to these efforts and propose an efficient first-principle protocol for simulating vibrationally-resolved absorption spectra, including nonempirical estimations of the inhomogeneous broadening. To this end, we analyze three key aspects: (i) a metric-based selection of density functional approximation (DFA) so to benefit from the computational efficiency of time-dependent density function theory (TD-DFT) while safeguarding the accuracy of the vibrationally-resolved spectra, (ii) an assessment of two vibrational structure schemes (vertical gradient and adiabatic Hessian) to compute the Franck-Condon factors, and (iii) the use of machine learning to speed up nonempirical estimations of the inhomogeneous broadening. In more detail, we predict the absorption band shapes for a set of 20 medium-sized fluorescent dyes, focusing on the bright ππ^★ S₀ → S₁ transition and using experimental results as references. We demonstrate that, for the studied 20-dye set which includes structures with large structural variability, the preselection of DFAs based on an easily accessible metric ensures accurate band shapes with respect to the reference approach and that range-separated functionals show the best performance when combined with the vertical gradient model. As far as band widths are concerned, we propose a new machine-learning-based approach for determining the inhomogeneous broadening induced by the solvent microenvironment. This approach is shown to be very robust offering inhomogeneous broadenings with errors as small as 2 cm^-1 with respect to genuine electronic-structure calculations, with a total CPU time reduced by 98%.

Recent progress in atomistic modeling of light-harvesting complexes: a mini review

In this mini review, we focus on recent advances in the atomistic modeling of biological light-harvesting (LH) complexes. Because of their size and sophisticated electronic structures, multiscale methods are required to investigate the dynamical and spectroscopic properties of such complexes. The excitation energies, in this context also known as site energies, excitonic couplings, and spectral densities are key quantities which usually need to be extracted to be able to determine the exciton dynamics and spectroscopic properties. The recently developed multiscale approach based on the numerically efficient density functional tight-binding framework followed by excited state calculations has been shown to be superior to the scheme based on pure classical molecular dynamics simulations. The enhanced approach, which improves the description of the internal vibrational dynamics of the pigment molecules, yields spectral densities in good agreement with the experimental counterparts for various bacterial and plant LH systems. Here, we provide a brief overview of those results and described the theoretical foundation of the multiscale protocol.

Unraveling the contributions to the spectral shape of flexible dyes in solution: insights on the absorption spectrum of an oxyluciferin analogue

We present a computational investigation of the absorption spectrum in water of 5,5-spirocyclopropyl-oxyluciferin (5,5-CprOxyLH), an analogue of the emitter compound responsible for the bioluminescence in fireflies. Several factors participate in determining the 5,5-CprOxyLH's spectral shape: (i) the contribution of the four close-energy excited states, which show significant non-adiabatic couplings, (ii) the flexible molecular structure and (iii) the specific interactions established with the surrounding environment, which strongly couple the protic solvent dynamics with the dye's spectral response. To tackle the challenge to capture and dissect the role of all these effects we preliminarily investigate the role of non-adiabatic couplings with quantum dynamics simulations and a linear vibronic coupling model in the gas phase. Then, we account for both the molecular flexibility and solvent interactions by resorting to a mixed quantum classical protocol, named Adiabatic Molecular Dynamics generalized Vertical Gradient (Ad-MD|gVG), which is built on a method recently proposed by some of us. It is rooted in the partition between stiff degrees of freedom of the dye, accounted for at the vibronic level within the harmonic approximation, and flexible degrees of freedom of the solute (and of the solvent), described classically through a sampling based on Molecular Dynamics (MD). Ad-MD|gVG avoids spurious effects arising in the excited state Hessians due to non-adiabatic couplings, and can therefore be applied to account for the contributions of the first four excited states to the 5,5-CprOxyLH absorption spectrum. The final simulated spectrum is in very good agreement with the experiment, especially when the MD is driven by a refined quantum-mechanically derived force-field. More importantly, the origin of each separate contribution to the spectral shape is appropriately accounted for, paving the way to future applications of the method to more complex systems or alternative spectroscopies, as emission or circular dichroism.

Wave Packet Calculation of Absolute UV Cross Section of Criegee Intermediates

Criegee intermediates, R₁R₂COO, are reactive species formed in the atmosphere through the ozonolysis of alkenes. They have an intense ultraviolet (UV) adsorption between 300 to 400 nm. However, experimentally determining the absolute cross sections is not easy. We used wave packet propagation on an one-dimensional adiabatic potential energy curve (PEC) along the OO bond to simulate the UV spectra for various Criegee intermediates. Our results showed a very fast, ∼20 fs, decay out of the Franck-Condon region. This gives justification for using the semiclassical approach which was utilized in previous studies. From the comparison of various quantum chemistry methods, we found that multireference methods can give spectra with a width and cross section reproducing the experimental results, while single reference methods tend to give narrower skewed peaks with a larger cross section. From the test using wave packet propagation on various approximated PECs and transition moment functions, we show that the Gaussian approximation within the reflection method is valid. In addition, we found that we can obtain peak positions that reproduce the experimental results by shifting those obtained by MRCI+Q, CASSCF, EOMCCSD, and TDCAMB3LYP by -0.2, -1.0, -0.3, and -0.5 eV, respectively. The Gaussian approximation using peak position, oscillator strength, and peak width from MRCI+Q is a cost-effective way to simulate the UV spectra of Crigee intermediates for which experimental determination may be hard.

Comparison among several vibronic coupling methods

A comparison of four approaches to account the vibronic coupling in photoabsorption is performed. The methods considered are nuclear ensemble (NE), direct vibronic coupling (DVC), adiabatic Hessian (AH), and vertical gradient (VG). The case study is the symmetry-forbidden [Formula: see text] [Formula: see text]A[Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text]A[Formula: see text] (n [Formula: see text] [Formula: see text]) transition in formaldehyde. Being forbidden in the equilibrium geometry, this transition is entirely induced by vibronic coupling and constitutes an appropriate case to study the performance of different methods. From DVC, it is found that mode 1 (C=O out-of-plane bending) is the most inducing, followed by mode 6 (in-plane C-H asymmetric stretching) and finally by mode 2 (in-plane C-H asymmetric bending). We were able to correlate 17 out of 20 structures obtained from NE with these modes, showing that these two methods, although different in principle, give comparable results. The simulated spectra were obtained for all methods and compared, and each one has its own advantage. In what concerns the transition studied, NE gives the best description of the spectrum, DVC is the only one that easily gives an absolute value for OOS, and AH and VG are the computationally less expensive methods. From the latter two, VG is the less demanding on computational grounds, since it does not require the excited state Hessian.

UV-Visible Absorption Spectra of Solvated Molecules by Quantum Chemical Machine Learning

Predicting UV-visible absorption spectra is essential to understand photochemical processes and design energy materials. Quantum chemical methods can deliver accurate calculations of UV-visible absorption spectra, but they are computationally expensive, especially for large systems or when one computes line shapes from thermal averages. Here, we present an approach to predict UV-visible absorption spectra of solvated aromatic molecules by quantum chemistry (QC) and machine learning (ML). We show that a ML model, trained on the high-level QC calculation of the excitation energy of a set of aromatic molecules, can accurately predict the line shape of the lowest-energy UV-visible absorption band of several related molecules with less than 0.1 eV deviation with respect to reference experimental spectra. Applying linear decomposition analysis on the excitation energies, we unveil that our ML models probe vertical excitations of these aromatic molecules primarily by learning the atomic environment of their phenyl rings, which align with the physical origin of the π →π* electronic transition. Our study provides an effective workflow that combines ML with quantum chemical methods to accelerate the calculations of UV-visible absorption spectra for various molecular systems.

Modeling the Electronic Absorption Spectra of the Indocarbocyanine Cy3

Accurate modeling of optical spectra requires careful treatment of the molecular structures and vibronic, environmental, and thermal contributions. The accuracy of the computational methods used to simulate absorption spectra is limited by their ability to account for all the factors that affect the spectral shapes and energetics. The ensemble-based approaches are widely used to model the absorption spectra of molecules in the condensed-phase, and their performance is system dependent. The Franck-Condon approach is suitable for simulating high resolution spectra of rigid systems, and its accuracy is limited mainly by the harmonic approximation. In this work, the absorption spectrum of the widely used cyanine Cy3 is simulated using the ensemble approach via classical and quantum sampling, as well as, the Franck-Condon approach. The factors limiting the ensemble approaches, including the sampling and force field effects, are tested, while the vertical and adiabatic harmonic approximations of the Franck-Condon approach are also systematically examined. Our results show that all the vertical methods, including the ensemble approach, are not suitable to model the absorption spectrum of Cy3, and recommend the adiabatic methods as suitable approaches for the modeling of spectra with strong vibronic contributions. We find that the thermal effects, the low frequency modes, and the simultaneous vibrational excitations have prominent contributions to the Cy3 spectrum. The inclusion of the solvent stabilizes the energetics significantly, while its negligible effect on the spectral shapes aligns well with the experimental observations.

Spectral densities and absorption spectra of the core antenna complex CP43 from photosystem II

Besides absorbing light, the core antenna complex CP43 of photosystem II is of great importance in transferring excitation energy from the antenna complexes to the reaction center. Excitation energies, spectral densities, and linear absorption spectra of the complex have been evaluated by a multiscale approach. In this scheme, quantum mechanics/molecular mechanics molecular dynamics simulations are performed employing the parameterized density functional tight binding (DFTB) while the time-dependent long-range-corrected DFTB scheme is applied for the excited state calculations. The obtained average spectral density of the CP43 complex shows a very good agreement with experimental results. Moreover, the excitonic Hamiltonian of the system along with the computed site-dependent spectral densities was used to determine the linear absorption. While a Redfield-like approximation has severe shortcomings in dealing with the CP43 complex due to quasi-degenerate states, the non-Markovian full second-order cumulant expansion formalism is able to overcome the drawbacks. Linear absorption spectra were obtained, which show a good agreement with the experimental counterparts at different temperatures. This study once more emphasizes that by combining diverse techniques from the areas of molecular dynamics simulations, quantum chemistry, and open quantum systems, it is possible to obtain first-principle results for photosynthetic complexes, which are in accord with experimental findings.

Applicability of the Thawed Gaussian Wavepacket Dynamics to the Calculation of Vibronic Spectra of Molecules with Double-Well Potential Energy Surfaces

Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-well potential energy surfaces, often requires expensive quantum dynamics methods. Here, we explore the applicability and limitations of the recently proposed single-Hessian thawed Gaussian approximation for the simulation of spectra of systems with double-well potentials, including 1,2,4,5-tetrafluorobenzene, ammonia, phosphine, and arsine. This semiclassical wavepacket approach is shown to be more robust and to provide more accurate spectra than the conventional harmonic approximation. Specifically, we identify two cases in which the Gaussian wavepacket method is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at the top of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket has enough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requires only a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. We also present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step.

Comparison of Linear Response Theory, Projected Initial Maximum Overlap Method, and Molecular Dynamics-Based Vibronic Spectra: The Case of Methylene Blue

The simulation of optical spectra is essential to molecular characterization and, in many cases, critical for interpreting experimental spectra. The most common method for simulating vibronic absorption spectra relies on the geometry optimization and computation of normal modes for ground and excited electronic states. In this report, we show that the utilization of such a procedure within an adiabatic linear response (LR) theory framework may lead to state mixings and a breakdown of the Born-Oppenheimer approximation, resulting in a poor description of absorption spectra. In contrast, computing excited states via a self-consistent field method in conjunction with a maximum overlap model produces states that are not subject to such mixings. We show that this latter method produces vibronic spectra much more aligned with vertical gradient and molecular dynamics (MD) trajectory-based approaches. For the methylene blue chromophore, we compare vibronic absorption spectra computed with the following: an adiabatic Hessian approach with LR theory-optimized structures and normal modes, a vertical gradient procedure, the Hessian and normal modes of maximum overlap method-optimized structures, and excitation energy time-correlation functions generated from an MD trajectory. Because of mixing between the bright S₁ and dark S₂ surfaces near the S₁ minimum, computing the adiabatic Hessian with LR theory and time-dependent density functional theory with the B3LYP density functional predicts a large vibronic shoulder for the absorption spectrum that is not present for any of the other methods. Spectral densities are analyzed and we compare the behavior of the key normal mode that in LR theory strongly couples to the optical excitation while showing S₁/S₂ state mixings. Overall, our study provides a note of caution in computing vibronic spectra using the excited-state adiabatic Hessian of LR theory-optimized structures and also showcases three alternatives that are less sensitive to adiabatic state mixing effects.

The Influence of Electronic Polarization on Nonlinear Optical Spectroscopy

The environment surrounding a chromophore can dramatically affect the energy absorption and relaxation process, as manifested in optical spectra. Simulations of nonlinear optical spectroscopy, such as two-dimensional electronic spectroscopy (2DES) and transient absorption (TA), will be influenced by the computational model of the environment. We here compare a fixed point charge molecular mechanics model and a quantum mechanical (QM) model of the environment in computed 2DES and TA spectra of Nile red in water and the chromophore of photoactive yellow protein (PYP) in water and protein environments. In addition to simulating these nonlinear optical spectra, we directly juxtapose the computed excitation energy correlation function to the dynamic Stokes shift function often used to analyze environment dynamics. Overall, we find that for the three systems studied here the mutual electronic polarization provided by the QM environment manifests in broader 2DES signals, as well as a larger reorganization energy and a larger static Stokes shift due to stronger coupling between the chromophore and the environment.

Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: Methylene blue

Modeling linear absorption spectra of solvated chromophores is highly challenging as contributions are present both from coupling of the electronic states to nuclear vibrations and from solute-solvent interactions. In systems where excited states intersect in the Condon region, significant non-adiabatic contributions to absorption line shapes can also be observed. Here, we introduce a robust approach to model linear absorption spectra accounting for both environmental and non-adiabatic effects from first principles. This model parameterizes a linear vibronic coupling (LVC) Hamiltonian directly from energy gap fluctuations calculated along molecular dynamics (MD) trajectories of the chromophore in solution, accounting for both anharmonicity in the potential and direct solute-solvent interactions. The resulting system dynamics described by the LVC Hamiltonian are solved exactly using the thermalized time-evolving density operator with orthogonal polynomials algorithm (T-TEDOPA). The approach is applied to the linear absorption spectrum of methylene blue in water. We show that the strong shoulder in the experimental spectrum is caused by vibrationally driven population transfer between the bright S₁ and the dark S₂ states. The treatment of the solvent environment is one of many factors that strongly influence the population transfer and line shape; accurate modeling can only be achieved through the use of explicit quantum mechanical solvation. The efficiency of T-TEDOPA, combined with LVC Hamiltonian parameterizations from MD, leads to an attractive method for describing a large variety of systems in complex environments from first principles.

Multiscale Modeling of Electronic Spectra Including Nuclear Quantum Effects

Theoretical prediction of electronic absorption spectra without input from experiments is no easy feat, as it requires addressing all of the factors that affect line shapes. In practice, however, the methodologies are limited to treat these ingredients only to a certain extent. Here, we present a multiscale protocol that addresses the temperature, solvent, and nuclear quantum effects as well as anharmonicity and the reconstruction of the final spectra from individual transitions. First, quantum mechanics/molecular mechanics (QM/MM) molecular dynamics is conducted to obtain trajectories of solute-solvent configurations, from which the corresponding quantum-corrected ensembles are generated through the generalized smoothed trajectory analysis (GSTA). The optical spectra of the ensembles are then produced by calculating vertical transitions using time-dependent density-functional theory (TDDFT) with implicit solvation. To obtain the final spectral shapes, the stick spectra from TDDFT are convoluted with Gaussian kernels where the half-widths are determined by a statistically motivated strategy. We have tested our method by calculating the UV-vis spectra of a recently discovered acridine photocatalyst in two redox states. Vibronic progressions and broadenings due to the finite lifetime of the excited states are not included in the methodology yet. Nuclear quantization affects the relative peak intensities and widths, which is necessary to reproduce the experimental spectrum. We have also found that using only the optimized geometry of each molecule works surprisingly well if a proper empirical broadening factor is applied. This is explained by the rigidity of the conjugated chromophore moieties of the selected molecules, which are mainly responsible for the excitations in the spectra. In contrast, we have also shown that other parts of the molecules are flexible enough to feature anharmonicities that impair the use of other techniques such as Wigner sampling.

Optimal Representation of the Nuclear Ensemble: Application to Electronic Spectroscopy

Nuclear densities are frequently represented by an ensemble of nuclear configurations or points in the phase space in various contexts of molecular simulations. The size of the ensemble directly affects the accuracy and computational cost of subsequent calculations of observable quantities. In the present work, we address the question of how many configurations do we need and how to select them most efficiently. We focus on the nuclear ensemble method in the context of electronic spectroscopy, where thousands of sampled configurations are usually needed for sufficiently converged spectra. The proposed representative sampling technique allows for a dramatic reduction of the sample size. By using an exploratory method, we model the density from a large sample in the space of transition properties. The representative subset of nuclear configurations is optimized by minimizing its Kullback-Leibler divergence to the full density with simulated annealing. High-level calculations are then performed only for the selected subset of configurations. We tested the algorithm on electronic absorption spectra of three molecules: (E)-azobenzene, the simplest Criegee intermediate, and hydrated nitrate anion. Typically, dozens of nuclear configurations provided sufficiently accurate spectra. A strongly forbidden transition of the nitrate anion presented the most challenging case due to rare geometries with disproportionately high transition intensities. This problematic case was easily diagnosed within the present approach. We also discuss various exploratory methods and a possible extension to dynamical simulations.

Time-dependent atomistic simulations of the CP29 light-harvesting complex

Light harvesting as the first step in photosynthesis is of prime importance for life on earth. For a theoretical description of photochemical processes during light harvesting, spectral densities are key quantities. They serve as input functions for modeling the excitation energy transfer dynamics and spectroscopic properties. Herein, a recently developed procedure is applied to determine the spectral densities of the pigments in the minor antenna complex CP29 of photosystem II, which has recently gained attention because of its active role in non-photochemical quenching processes in higher plants. To this end, the density functional-based tight binding (DFTB) method has been employed to enable simulation of the ground state dynamics in a quantum-mechanics/molecular mechanics (QM/MM) scheme for each chlorophyll pigment. Subsequently, the time-dependent extension of the long-range corrected DFTB approach has been used to obtain the excitation energy fluctuations along the ground-state trajectories also in a QM/MM setting. From these results, the spectral densities have been determined and compared for different force fields and to spectral densities from other light-harvesting complexes. In addition, time-dependent and time-independent excitonic Hamiltonians of the system have been constructed and applied to the determination of absorption spectra as well as exciton dynamics.

Modeling Molecular Emitters in Organic Light-Emitting Diodes with the Quantum Mechanical Bespoke Force Field

Combined molecular dynamics (MD) and quantum mechanics (QM) simulation procedures have gained popularity in modeling the spectral properties of functional organic molecules. However, the potential energy surfaces used to propagate long-time scale dynamics in these simulations are typically described using general, transferable force fields designed for organic molecules in their electronic ground states. These force fields do not typically include spectroscopic data in their training, and importantly, there is no general protocol for including changes in geometry or intermolecular interactions with the environment that may occur upon electronic excitation. In this work, we show that parameters tailored for thermally activated delayed fluorescence (TADF) emitters used in organic light-emitting diodes (OLEDs), in both their ground and electronically excited states, can be readily derived from a small number of QM calculations using the QUBEKit (QUantum mechanical BEspoke toolKit) software and improve the overall accuracy of these simulations.

Multiconfigurational Quantum Chemistry Determinations of Absorption Cross Sections (σ) in the Gas Phase and Molar Extinction Coefficients (ε) in Aqueous Solution and Air-Water Interface

Theoretical determinations of absorption cross sections (σ) in the gas phase and molar extinction coefficients (ε) in condensed phases (water solution, interfaces or surfaces, protein or nucleic acids embeddings, etc.) are of interest when rates of photochemical processes, J = ∫ ϕ(λ) σ(λ) I(λ) dλ, are needed, where ϕ(λ) and I(λ) are the quantum yield of the process and the irradiance of the light source, respectively, as functions of the wavelength λ. Efficient computational strategies based on single-reference quantum-chemistry methods have been developed enabling determinations of line shapes or, in some cases, achieving rovibrational resolution. Developments are however lacking for strongly correlated problems, with many excited states, high-order excitations, and/or near degeneracies between states of the same and different spin multiplicities. In this work, we define and compare the performance of distinct computational strategies using multiconfigurational quantum chemistry, nuclear sampling of the chromophore (by means of molecular dynamics, ab initio molecular dynamics, or Wigner sampling), and conformational and statistical sampling of the environment (by means of molecular dynamics). A new mathematical approach revisiting previous absolute orientation algorithms is also developed to improve alignments of geometries. These approaches are benchmarked through the nπ* band of acrolein not only in the gas phase and water solution but also in a gas-phase/water interface, a common situation for instance in atmospheric chemistry. Subsequently, the best strategy is used to compute the absorption band for the adduct formed upon addition of an OH radical to the C6 position of uracil and compared with the available experimental data. Overall, quantum Wigner sampling of the chromophore with molecular dynamics sampling of the environment with CASPT2 electronic-structure determinations arise as a powerful methodology to predict meaningful σ(λ) and ε(λ) band line shapes with accurate absolute intensities.

Vibronic and Environmental Effects in Simulations of Optical Spectroscopy

Including both environmental and vibronic effects is important for accurate simulation of optical spectra, but combining these effects remains computationally challenging. We outline two approaches that consider both the explicit atomistic environment and the vibronic transitions. Both phenomena are responsible for spectral shapes in linear spectroscopy and the electronic evolution measured in nonlinear spectroscopy. The first approach utilizes snapshots of chromophore-environment configurations for which chromophore normal modes are determined. We outline various approximations for this static approach that assumes harmonic potentials and ignores dynamic system-environment coupling. The second approach obtains excitation energies for a series of time-correlated snapshots. This dynamic approach relies on the accurate truncation of the cumulant expansion but treats the dynamics of the chromophore and the environment on equal footing. Both approaches show significant potential for making strides toward more accurate optical spectroscopy simulations of complex condensed phase systems.

Finite-Temperature, Anharmonicity, and Duschinsky Effects on the Two-Dimensional Electronic Spectra from Ab Initio Thermo-Field Gaussian Wavepacket Dynamics

Accurate description of finite-temperature vibrational dynamics is indispensable in the computation of two-dimensional electronic spectra. Such simulations are often based on the density matrix evolution, statistical averaging of initial vibrational states, or approximate classical or semiclassical limits. While many practical approaches exist, they are often of limited accuracy and difficult to interpret. Here, we use the concept of thermo-field dynamics to derive an exact finite-temperature expression that lends itself to an intuitive wavepacket-based interpretation. Furthermore, an efficient method for computing finite-temperature two-dimensional spectra is obtained by combining the exact thermo-field dynamics approach with the thawed Gaussian approximation for the wavepacket dynamics, which is exact for any displaced, distorted, and Duschinsky-rotated harmonic potential but also accounts partially for anharmonicity effects in general potentials. Using this new method, we directly relate a symmetry breaking of the two-dimensional signal to the deviation from the conventional Brownian oscillator picture.

Explicit environmental and vibronic effects in simulations of linear and nonlinear optical spectroscopy

Accurately simulating the linear and nonlinear electronic spectra of condensed phase systems and accounting for all physical phenomena contributing to spectral line shapes presents a significant challenge. Vibronic transitions can be captured through a harmonic model generated from the normal modes of a chromophore, but it is challenging to also include the effects of specific chromophore-environment interactions within such a model. We work to overcome this limitation by combining approaches to account for both explicit environment interactions and vibronic couplings for simulating both linear and nonlinear optical spectra. We present and show results for three approaches of varying computational cost for combining ensemble sampling of chromophore-environment configurations with Franck-Condon line shapes for simulating linear spectra. We present two analogous approaches for nonlinear spectra. Simulated absorption spectra and two-dimensional electronic spectra (2DES) are presented for the Nile red chromophore in different solvent environments. Employing an average Franck-Condon or 2DES line shape appears to be a promising method for simulating linear and nonlinear spectroscopy for a chromophore in the condensed phase.

Simulating Quantum Vibronic Dynamics at Finite Temperatures With Many Body Wave Functions at 0 K

For complex molecules, nuclear degrees of freedom can act as an environment for the electronic “system” variables, allowing the theory and concepts of open quantum systems to be applied. However, when molecular system-environment interactions are non-perturbative and non-Markovian, numerical simulations of the complete system-environment wave function become necessary. These many body dynamics can be very expensive to simulate, and extracting finite-temperature results—which require running and averaging over many such simulations—becomes especially challenging. Here, we present numerical simulations that exploit a recent theoretical result that allows dissipative environmental effects at finite temperature to be extracted efficiently from a single, zero-temperature wave function simulation. Using numerically exact time-dependent variational matrix product states, we verify that this approach can be applied to vibronic tunneling systems and provide insight into the practical problems lurking behind the elegance of the theory, such as the rapidly growing numerical demands that can appear for high temperatures over the length of computations.