General formulation of vibronic spectroscopy in internal coordinates

Quantum chemical package Jaguar: A survey of recent developments and unique features

This paper is dedicated to the quantum chemical package Jaguar, which is commercial software developed and distributed by Schrödinger, Inc. We discuss Jaguar's scientific features that are relevant to chemical research as well as describe those aspects of the program that are pertinent to the user interface, the organization of the computer code, and its maintenance and testing. Among the scientific topics that feature prominently in this paper are the quantum chemical methods grounded in the pseudospectral approach. A number of multistep workflows dependent on Jaguar are covered: prediction of protonation equilibria in aqueous solutions (particularly calculations of tautomeric stability and pKa), reactivity predictions based on automated transition state search, assembly of Boltzmann-averaged spectra such as vibrational and electronic circular dichroism, as well as nuclear magnetic resonance. Discussed also are quantum chemical calculations that are oriented toward materials science applications, in particular, prediction of properties of optoelectronic materials and organic semiconductors, and molecular catalyst design. The topic of treatment of conformations inevitably comes up in real world research projects and is considered as part of all the workflows mentioned above. In addition, we examine the role of machine learning methods in quantum chemical calculations performed by Jaguar, from auxiliary functions that return the approximate calculation runtime in a user interface, to prediction of actual molecular properties. The current work is second in a series of reviews of Jaguar, the first having been published more than ten years ago. Thus, this paper serves as a rare milestone on the path that is being traversed by Jaguar's development in more than thirty years of its existence.

Harmonic and anharmonic vibrational computations for biomolecular building blocks: Benchmarking DFT and basis sets by theoretical and experimental IR spectrum of glycine conformers

Advanced vibrational spectroscopic experiments have reached a level of sophistication that can only be matched by numerical simulations in order to provide an unequivocal analysis, a crucial step to understand the structure-function relationship of biomolecules. While density functional theory (DFT) has become the standard method when targeting medium-size or larger systems, the problem of its reliability and accuracy are well-known and have been abundantly documented. To establish a reliable computational protocol, especially when accuracy is critical, a tailored benchmark is usually required. This is generally done over a short list of known candidates, with the basis set often fixed a priori. In this work, we present a systematic study of the performance of DFT-based hybrid and double-hybrid functionals in the prediction of vibrational energies and infrared intensities at the harmonic level and beyond, considering anharmonic effects through vibrational perturbation theory at the second order. The study is performed for the six-lowest energy glycine conformers, utilizing available "state-of-the-art" accurate theoretical and experimental data as reference. Focusing on the most intense fundamental vibrations in the mid-infrared range of glycine conformers, the role of the basis sets is also investigated considering the balance between computational cost and accuracy. Targeting larger systems, a broad range of hybrid schemes with different computational costs is also tested.

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.

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.

First-Principles Calculations of Excited-State Decay Rate Constants in Organic Fluorophores

In this Perspective, we discuss recent advances made to evaluate from first-principles the excited-state decay rate constants of organic fluorophores, focusing on the so-called static strategy. In this strategy, one essentially takes advantage of Fermi's golden rule (FGR) to evaluate rate constants at key points of the potential energy surfaces, a procedure that can be refined in a variety of ways. In this way, the radiative rate constant can be straightforwardly obtained by integrating the fluorescence line shape, itself determined from vibronic calculations. Likewise, FGR allows for a consistent calculation of the internal conversion (related to the non-adiabatic couplings) in the weak-coupling regime and intersystem crossing rates, therefore giving access to estimates of the emission yields when no complex photophysical phenomenon is at play. Beyond outlining the underlying theories, we summarize here the results of benchmarks performed for various types of rates, highlighting that both the quality of the vibronic calculations and the accuracy of the relative energies are crucial to reaching semiquantitative estimates. Finally, we illustrate the successes and challenges in determining the fluorescence quantum yields using a series of organic fluorophores.

Theoretical Insight into the Effect of Phosphorus Oxygenation on Nonradiative Decays: Comparative Analysis of P-Bridged Stilbene Analogs

Incorporation of the phosphorus element into a π-conjugated skeleton offers valuable prospects for adjusting the electronic structure of the resulting functional π-electron systems. Trivalent phosphorus has the potential to decrease the LUMO level through σ*-π* interaction, which is further enhanced by its oxygenation to the pentavalent P center. This study shows that utilizing our computational analysis to examine excited-state dynamics based on radiative/nonradiative rate constants and fluorescence quantum yield (ΦF) is effective for analyzing the photophysical properties of P-containing organic dyes. We theoretically investigate how the trivalent phosphanyl group and pentavalent phosphine oxide moieties affect radiative and nonradiative decay processes. We evaluate four variations of P-bridged stilbene analogs. Our analysis reveals that the primary decay pathway for photoexcited bis-phosphanyl-bridged stilbene is the intersystem crossing (ISC) to the triplet state and nonradiative. The oxidation of the phosphine moiety, however, suppresses the ISC due to the relative destabilization of the triplet states. The calculated rate constants match an increase in experimental ΦF from 0.07 to 0.98, as simulated from 0.23 to 0.94. The reduced HOMO-LUMO gap supports a red shift in the fluorescence spectra relative to the phosphine analog. The thiophene-fused variant with the nonoxidized trivalent P center exhibits intense emission with a high ΦF, 0.95. Our prediction indicates that the ISC transfer is obstructed owing to the relatively destabilized triplet state induced by the thiophene substitution. Conversely, the thiophene-fused analog with the phosphine oxide moieties triggers a high-rate internal conversion mediated by conical intersection, leading to a decreased ΦF.

Modified Fermi's golden rule rate expressions

Fermi's golden rule (FGR) serves as the basis for many expressions of spectroscopic observables and quantum transition rates. The utility of FGR has been demonstrated through decades of experimental confirmation. However, there still remain important cases where the evaluation of a FGR rate is ambiguous or ill-defined. Examples are cases where the rate has divergent terms due to the sparsity in the density of final states or time dependent fluctuations of system Hamiltonians. Strictly speaking, assumptions of FGR are no longer valid for such cases. However, it is still possible to define modified FGR rate expressions that are useful as effective rates. The resulting modified FGR rate expressions resolve a long standing ambiguity often encountered in using FGR and offer more reliable ways to model general rate processes. Simple model calculations illustrate the utility and implications of new rate expressions.

FCclasses3: Vibrationally-resolved spectra simulated at the edge of the harmonic approximation

We introduce FCclasses3, a code to carry out vibronic simulations of electronic spectra and nonradiative rates, based on the harmonic approximation. Key new features are: implementation of the full family of vertical and adiabatic harmonic models, vibrational analysis in curvilinear coordinates, extension to several electronic spectroscopies and implementation of time-dependent approaches. The use of curvilinear valence internal coordinates allows the adoption of quadratic model potential energy surfaces (PES) of the initial and final states expanded at arbitrary configurations. Moreover, the implementation of suitable projectors provides a robust framework for defining reduced-dimensionality models by sorting flexible coordinates out of the harmonic subset, so that they can then be treated at anharmonic level, or with mixed quantum classical approaches. A set of tools to facilitate input preparation and output analysis is also provided. We show the program at work in the simulation of different spectra (one and two-photon absorption, emission and resonance Raman) and internal conversion rate of a typical rigid molecule, anthracene. Then, we focus on absorption and emission spectra of a series of flexible polyphenyl molecules, highlighting the relevance of some of the newly implemented features. The code is freely available at http://www.iccom.cnr.it/en/fcclasses/.

Perturb-Then-Diagonalize Vibrational Engine Exploiting Curvilinear Internal Coordinates

The present paper is devoted to the implementation and validation of a second-order perturbative approach to anharmonic vibrations, followed by variational treatment of strong couplings (GVPT2) based on curvilinear internal coordinates. The main difference with respect to the customary Cartesian-based formulation is that the kinetic energy operator is no longer diagonal, and has to be expanded as well, leading to additional terms which have to be taken into proper account. It is, however, possible to recast all the equations as well-defined generalizations of the corresponding Cartesian-based counterparts, thus achieving a remarkable simplification of the new implementation. Particular attention is paid to the treatment of Fermi resonances with significant number of test cases analyzed fully, validating the new implementation. The results obtained in this work confirm that curvilinear coordinates strongly reduce the strength of inter-mode couplings compared to their Cartesian counterparts. This increases the reliability of low-order perturbative treatments for semi-rigid molecules and paves the way toward the reliable representation of more flexible molecules where small- and large-amplitude motions can be safely decoupled and treated at different levels of theory.

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.

On the Importance of Well-Defined Thermal Correlation Functions in Simulating Vibronic Spectra

Two difficulties associated with the computations of thermal vibrational correlation functions are discussed. The first one is the lack of a well-behaved expression that is valid at both high-temperature and T → 0 K limits. Specifically, if the partition function and the propagator are considered separately, then thermal vibrational correlation functions may have an indeterminate form 0/0 in the limit T → 0 K. This difficulty is resolved when the partition function and the propagator are jointly considered in the harmonic approximation, which allows a problematic term that emanates from the zero-point energy to be canceled out, thereby producing a thermal correlation function with a determinate form in the T → 0 K limit. The second difficulty is related to the multivaluedness of the vibrational correlation function. We show numerically that an improper selection of branch leads to discontinuities in the computed correlation function and incorrect vibronic spectra. We propose a phase tracking procedure that ensures continuity of both real and imaginary parts of the correlation function to recover the correct spectra. We support our findings by simulating the UV-vis absorption spectra of pentacene at 4 K and benzene at 298 K. Both are found to be in good agreement with their experimental counterparts.

Accounting for molecular flexibility in photoionization: case of tert-butyl hydroperoxide

tert-Butyl hydroperoxide (tBuOOH) is a common intermediate in the oxidation of organic compounds that needs to be accurately quantified in complex gas mixtures for the development of chemical kinetic models of low temperature combustion. This work presents a combined theoretical and experimental investigation on the synchrotron-based VUV single photon ionization of gas-phase tBuOOH in the 9.0 - 11.0 eV energy range, including dissociative ionization processes. Computations consist of the determination of the structures, vibrational frequencies and the energetics of neutral and ionic tBuOOH. The Franck-Condon spectrum for the tBuOOH⁺ (X⁺) + e^- ← tBuOOH (X) + hν transition is computed, where special treatment is undertaken because of the flexibility of tBuOOH, in particular regarding the OOH group. Through comparison of the experimental mass-selected threshold photoelectron spectra with explicitly correlated coupled cluster calculations and Franck-Condon simulations that account for the flexibility of the molecule, an estimation of the ionization energy is given. The appearance energy of the only fragment observed within the above-mentioned energy range, identified as the tert-butyl C₄H₉⁺, is also reported. Finally, the signal branching ratio between the parent and the fragment ions is provided as a function of photon energy, essential to quantify tBuOOH in gas-phase oxidation/combustion experiments via advanced mass spectrometry techniques.

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.

Integration of theory, simulation, artificial intelligence and virtual reality: a four-pillar approach for reconciling accuracy and interpretability in computational spectroscopy

The established pillars of computational spectroscopy are theory and computer based simulations. Recently, artificial intelligence and virtual reality are becoming the third and fourth pillars of an integrated strategy for the investigation of complex phenomena. The main goal of the present contribution is the description of some new perspectives for computational spectroscopy, in the framework of a strategy in which computational methodologies at the state of the art, high-performance computing, artificial intelligence and virtual reality tools are integrated with the aim of improving research throughput and achieving goals otherwise not possible. Some of the key tools (e.g., continuous molecular perception model and virtual multifrequency spectrometer) and theoretical developments (e.g., non-periodic boundaries, joint variational-perturbative models) are shortly sketched and their application illustrated by means of representative case studies taken from recent work by the authors. Some of the results presented are already well beyond the state of the art in the field of computational spectroscopy, thereby also providing a proof of concept for other research fields.

General Perturb-Then-Diagonalize Model for the Vibrational Frequencies and Intensities of Molecules Belonging to Abelian and Non-Abelian Symmetry Groups

In this paper, we show that the standard second-order vibrational perturbation theory (VPT2) for Abelian groups can be used also for non-Abelian groups without employing specific equations for two- or threefold degenerate vibrations but rather handling in the proper way all the degeneracy issues and deriving the peculiar spectroscopic signatures of non-Abelian groups (e.g., l -doubling) by a posteriori transformations of the eigenfunctions. Comparison with the results of previous conventional implementations shows a perfect agreement for the vibrational energies of linear and symmetric tops, thus paving the route to the transparent extension of the equations already available for asymmetric tops to the energies of spherical tops and the infrared and Raman intensities of molecules belonging to non-Abelian symmetry groups. The whole procedure has been implemented in our general engine for vibro-rotational computations beyond the rigid rotor/harmonic oscillator model and has been validated on a number of test cases.

Interplay of stereo-electronic, vibronic and environmental effects in tuning the chiroptical properties of an Ir(III) cyclometalated N-heterocyclic carbene

Chiroptical spectra are among the most suitable techniques for investigating the ground and excited electronic states of chiral systems, but their interpretation is not straightforward and strongly benefits from quantum chemical simulations, provided that the employed computational model is sufficiently accurate and deals properly with stereo-electronic, vibrational averaging and environmental effects. Since the synergy among all these effects is only rarely accounted for, especially for large and flexible organometallic systems, the main aim of this contribution is to illustrate the latest developments of computational approaches rooted into the density functional theory for describing stereo-electronic effects and complemented by effective techniques to deal with vibrational modulation effects and solvatochromic shifts. In this connection, chiral iridium complexes offer an especially suitable case study in view of their bright phosphorescence, which is particularly significant for building effective light emitting diodes (OLEDs) and biomarkers and can be finely tuned by the nature of the metal ligands. For instance, a recently synthesized family of cycloiridiated complexes, KC and KD, bearing a pentahelicenic N-heterocyclic carbene (KB), has shown an enhanced long-lasting, bright phosphorescence. Deeper insights into the still unclear nature and origin of the enhancement could be gained by the interpretation of the chiroptical spectra, which is quite challenging in view of the presence of two sources of chirality, the chiral center on Ir and the chiral axis related to the helicene ligand, in addition to the relativistic effects related to the presence of the Ir center. At the same time, the large dimensions of KC and KD hamper the use of the most sophisticated (but prohibitively expensive) computational models, so that more approximate approaches must be validated on a suitable model compound. To this end, after optimizing the computational scheme on a model system devoid of the helicene moiety (KA), we have performed a comprehensive investigation of the KC and KD spectra, whose interpretation is further aided by novel graphical tools. The discussion and analysis of the results will not be focused on the theoretical background, but, rather, on practical details (specific functional, basis set, vibronic model, solvent regime) with the aim of providing general guidelines for the use of last-generation computational spectroscopy tools also by non-specialists.

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.

Theoretical Investigation of the Circularly Polarized Luminescence of a Chiral Boron Dipyrromethene (BODIPY) Dye

Over the last decade, molecules capable of emitting circularly polarized light have attracted growing attention for potential technological and biological applications. The efficiency of such devices depend on multiple parameters, in particular the magnitude and wavelength of the peak of emitted light, and also on the dissymmetry factor for chiral applications. In light of these considerations, molecular systems with tunable optical properties, preferably in the visible spectral region, are particularly appealing. This is the case of boron dipyrromethene (BODIPY) dyes, which exhibit large molecular absorption coefficients, have high fluorescence yields, are very stable, both thermally and photochemically, and can be easily functionalized. The latter property has been extensively exploited in the literature to produce chromophores with a wide range of optical properties. Nevertheless, only a few chiral BODIPYs have been synthetized and investigated so far. Using a recently reported axially chiral BODIPY derivative where an axially chiral BINOL unit has been attached to the chromophore unit, we present a comprehensive computational protocol to predict and interpret the one-photon absorption and emission spectra, together with their chiroptical counterparts. From the physico-chemical properties of this molecule, it will be possible to understand the origin of the circularly polarized luminescence better, thus helping to fine-tune the properties of interest. The sensitivity of such processes require accurate results, which can be achieved through a proper account of the vibrational structure in optical spectra. Methodologies to compute vibrationally-resolved electronic spectra can now be applied on relatively large chromophores, such as BODIPYs, but require more extensive computational protocols. For this reason, particular attention is paid in the description of the different steps of the protocol, and the potential pitfalls. Finally, we show how, by means of appropriate tools and approaches, data from intermediate steps of the simulation of the final spectra can be used to obtain further insights into the properties of the molecular system under investigation and the origin of the visible bands.

Interplay of Stereoelectronic and Vibrational Modulation Effects in Tuning the UPS Spectra of Unsaturated Hydrocarbon Cage Compounds

The UPS spectra of six hydrocarbon cage compounds have been investigated by a Green-function approach in conjunction with a full harmonic treatment of vibrational modulation effects. The remarkable agreement with experimental results points out the reliability of the proposed computational approach and the strong interplay of stereoelectronic and vibrational effects in tuning the overall spectra.

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.

CPL Spectra of Camphor Derivatives in Solution by an Integrated QM/MD Approach

We extend a recently proposed computational strategy for the simulation of absorption spectra of semi-rigid molecular systems in condensed phases to the emission spectra of flexible chromophores. As a case study, we have chosen the CPL spectrum of camphor in methanol solution, which shows a well-defined bisignate shape. The first step of our approach is the quantum mechanical computation of reference spectra including vibrational averaging effects and taking bulk solvent effects into account by means of the polarizable continuum model. In the present case, the large amplitude inversion mode is explicitly treated by a numerical approach, whereas the other small-amplitude vibrational modes are taken into account within the harmonic approximation. Next, the snapshots of classical molecular dynamics computations are clusterized and one representative configuration from each cluster is used to compute a reference spectrum. In the present case, different clusters correspond to the two stable conformers of camphor in the S1 excited electronic state and, for each of them, to different numbers of strong solute-solvent hydrogen bonds. Finally, local fluctuation effects within each cluster are taken into account by means of the perturbed matrix model. The overall procedure leads to good agreement with experiment for absorption and emission spectra together with their chiral counterparts, thus allowing to analyze the role of different effects (stereo-electronic, vibrational, environmental) in tuning the overall experimental spectra.

Geometry optimization using Gaussian process regression in internal coordinate systems

Locating the minimum energy structure of molecules, typically referred to as geometry optimization, is one of the first steps of any computational chemistry calculation. Earlier research was mostly dedicated to finding convenient sets of molecule-specific coordinates for a suitable representation of the potential energy surface, where a faster convergence toward the minimum structure can be achieved. More recent approaches, on the other hand, are based on various machine learning techniques and seem to revert to Cartesian coordinates instead for practical reasons. We show that the combination of Gaussian process regression with those coordinate systems employed by state-of-the-art geometry optimizers can significantly improve the performance of this powerful machine learning technique. This is demonstrated on a benchmark set of 30 small covalently bonded molecules.

Adiabatic-Molecular Dynamics Generalized Vertical Hessian Approach: A Mixed Quantum Classical Method To Compute Electronic Spectra of Flexible Molecules in the Condensed Phase

We present a general mixed quantum classical method that couples classical molecular dynamics (MD) and vibronic models to compute the shape of electronic spectra of flexible molecules in the condensed phase without, in principle, any phenomenological broadening. It is based on a partition of the nuclear motions of the solute + solvent system in "soft" and "stiff" vibrational modes and an adiabatic hypothesis that assumes that stiff modes are much faster than soft ones. In this framework, the spectrum is rigorously expressed as a conformational integral of quantum vibronic spectra along the stiff coordinates only. Soft modes enter at the classical level through the conformational distribution that is sampled with classical MD runs. In each configuration, reduced-dimensionality quadratic Hamiltonians are built in the space of the stiff coordinates only, thanks to a generalization of the Vertical Hessian harmonic model and an iterative application of projectors in internal coordinates to remove soft modes. Quantum vibronic spectra, specific for each sampled configuration of the soft coordinates, are then computed at the desired temperature with efficient time-dependent techniques, and the global spectrum simply arises from their average. For consistency of the whole procedure, classical MD runs are performed with quantum-mechanically derived force fields, parameterized at the same level of theory selected for generating the quadratic Hamiltonians along the stiff coordinates. Application to N-methyl-6-oxyquinolinium betaine in water, dithiophene in ethanol, and cyanidine in water is presented to show the performance of the method.

Role of specific solute–solvent interactions on the photophysical properties of distyryl substituted BODIPY derivatives

The role played by specific solute–solvent interactions on the spectroscopic properties of experimentally available BODIPY derivatives has been investigated.

Electronic transitions for a fully polarizable QM/MM approach based on fluctuating charges and fluctuating dipoles: Linear and corrected linear response regimes

The fully polarizable Quantum Mechanics/Molecular Mechanics (QM/MM) approach based on fluctuating charges and fluctuating dipoles, named QM/FQFμ [T. Giovannini et al., J. Chem. Theory Comput. 15, 2233 (2019)], is extended to the calculation of vertical excitation energies of solvated molecular systems. Excitation energies are defined within two different solvation regimes, i.e., linear response (LR), where the response of the MM portion is adjusted to the QM transition density, and corrected-Linear Response (cLR) in which the MM response is adjusted to the relaxed QM density, thus being able to account for charge equilibration in the excited state. The model, which is specified in terms of three physical parameters (electronegativity, chemical hardness, and polarizability) is applied to vacuo-to-water solvatochromic shifts of aqueous solutions of para-nitroaniline, pyridine, and pyrimidine. The results show a good agreement with their experimental counterparts, thus highlighting the potentialities of this approach.

On the simulation of vibrationally resolved electronic spectra of medium-size molecules: the case of styryl substituted BODIPYs

BODIPY dyes are used in a variety of applications because of their peculiar spectroscopic and photo-physical properties that vary depending on the stereochemistry of the functional groups attached to the boron-dipyrromethene core structure. In this work, we have applied several computational methods, adapted for semi-rigid molecules based on the Franck-Condon principle, for the study of the optical properties of BODIPY systems and for the understanding of the influence of functional groups on their spectroscopic features. We have analyzed the electronic spectra of two styryl substituted BODIPY molecules of technological interest, properly taking into account the vibronic contribution. For comparison with recently recorded experimental data in methanol, the vibrationally resolved electronic spectra of these systems were computed using both Time-Independent (TI) and Time-Dependent (TD) formalisms. The first step toward the analysis of optical properties of the styryl modified BODIPYs was a benchmark of several density functionals, to select the most appropriate one. We have found that all benchmarked functionals provide good results in terms of band shape but some of them show strong discrepancies in terms of band position. Beyond the issue of the electronic structure calculation method, different levels of sophistication can be adopted for the calculation of vibronic transitions. In this study, the effect of mode couplings and the influence of the Herzberg-Teller terms on the theoretical spectra has been investigated. It has been found that all levels of theory considered give reproducible results for the investigated systems: band positions and shapes are similar at all levels and little improvements have been found in terms of band shape with the inclusion of Herzberg-Teller effect. Inclusion of temperature effects proved to be challenging due to the important impact of large amplitude motions. Better agreement can be achieved by adopting a suitable set of coordinates coupled with a reduced-dimensionality scheme.

Time-Dependent Formulation of Resonance Raman Optical Activity Spectroscopy

In this work, we extend the theoretical framework recently developed for the simulation of resonance Raman (RR) spectra of medium-to-large sized systems to its chiral counterpart, namely, resonance Raman optical activity (RROA). The theory is based on a time-dependent (TD) formulation, with the transition tensors obtained as half-Fourier transforms of the appropriate cross-correlation functions. The implementation has been kept as general as possible, supporting adiabatic and vertical models for the PES representation, both in Cartesian and internal coordinates, with the possible inclusion of Herzberg-Teller (HT) effects. Thanks to the integration of this TD-RROA procedure within a general-purpose quantum-chemistry program, both solvation and leading anharmonicity effects can be included in an effective way. The implementation is validated on one of the smallest chiral molecule (methyloxirane). Practical applications are illustrated with three medium-size organic molecules (naproxen-OCD₃, quinidine and 2-Br-hexahelicene), whose simulated spectra are compared to the corresponding experimental data.

A 4,4'-bis(2-benzoxazolyl)stilbene luminescent probe: assessment of aggregate formation through photophysics experiments and quantum-chemical calculations

A combination of experimental and quantum mechanical investigations is applied to the study of the optical features of 4,4'-bis(2-benzoxazolyl)stilbene (BBS) dissolved in solution or in a poly(l-lactic acid) (PLA) thermoplastic matrix at different concentrations. The experimental analyses allow the characterization of BBS solutions and dispersions in terms of absorption and emission features, along with the collection of some key parameters such as fluorescence quantum yield, anisotropy and lifetime, while the computational approach gives a detailed description of the photophysical behavior of BBS in the different environments. For the 10^-5 M BBS solution, the fluorescence spectra show the expected peaks at 425 and 455 nm of the non-interacting BBS molecules with a single fluorescence lifetime of 0.85 ns without revealing any aggregation phenomena, prevented by the short lifetime and fast diffusion rate of the monomer. Moreover, the calculated spectra are in excellent agreement with the experiments, thus showing the reliability of the computational approach. In time-resolved emission experiments (TRES) on more concentrated solutions (10^-4 M) and on BBS crystals, the presence of an excimer is revealed by the appearance of a broad peak around 540 nm, followed by the disappearance of the two main peaks at 460 nm on a time scale of about 10 ns. The computational analysis attributes this behavior to the formation of aggregates of different geometries. The BBS dispersions in PLA reveal the presence of different BBS architectures depending on the fluorophore content. Even at low concentrations, BBS is mainly dispersed as a monomer in the matrix, spheroid aggregates of about 800-900 nm in diameter are also present and the relevant fluorescence spectra arise from the combination of monomer and aggregate contributions. At higher concentrations, BBS starts forming crystals of a peculiar helicoidal shape, with a diameter of about 2 μm, variable length up to several hundreds of μm and emission spectra similar to those of isolated BBS crystals.

A computational study of the vibrationally-resolved electronic circular dichroism spectra of single-chain transoid and cisoid oligothiophenes in chiral conformations

We simulate the vibronic profile of the electronic circular dichroism (ECD) spectra of oligothiophenes in cisoid and transoid chiral arrangements. We consider oligomers of different lengths, from two to fifteen units, and investigate extensively how the ECD spectral shapes depend on the inter-ring torsions. In general, the molecular structures we consider are not stationary points of the ground state potential energy surface. Therefore, in order to perform vibronic calculations, we present a new computational protocol able to define reduced-dimensionality models where the effect of the off-equilibrium modes is removed. This is done adopting a description of the vibrational motions in curvilinear internal coordinates, and vertical harmonic models coupled with an iterative application of projectors to define energy Hessians, and therefore effective normal modes, in the space complementary to the one of the off-equilibrium coordinates. Although we consider both Franck-Condon and Herzberg-Teller contributions, the results show that transoid twisted ribbons always give rise to monosignated ECD spectra, while bi-signated and multi-signated spectra are expected for cisoid helices. These findings are explained on the basis of the different transition strengths of the lowest electronic states imparted by the different spatial arrangement, that is almost linear for transoid structures and more globular for cisoid ones. We predicted the chiroptical response of a large number of possible molecular arrangements. These data are employed to critically discuss the experimental ECD of polythiophenes in different experimental conditions, forming either aggregates or host-guest complexes. The method here proposed to perform vibronic calculations in reduced-dimensionality models is of general applicability and its potential interest goes beyond the practical application presented here.

Computational simulation of vibrationally resolved spectra for spin-forbidden transitions

In this computational study, we illustrate a method for computing phosphorescence and circularly polarized phosphorescence spectra of molecular systems, which takes into account vibronic effects including both Franck-Condon and Herzberg-Teller contributions. The singlet and triplet states involved in the phosphorescent emission are described within the harmonic approximation, and the method fully takes mode-mixing effects into account when evaluating Franck-Condon integrals. Spin-orbit couplings, which are responsible for these otherwise forbidden phenomena, are accounted for by means of a relativistic two-component time-dependent density functional theory method. The model is applied to two types of chiral systems: camphorquinone, a rigid organic system that allows for an extensive benchmark, and some members of a class of iridium complexes. The merits and shortcomings of the methods are discussed, and some perspectives for future developments are offered.

High-Resolution Absorption and Electronic Circular Dichroism Spectra of (R)-(+)-1-Phenylethanol. Confident Interpretation Based on the Synergy between Experiments and Computations

Using density functional theory and its time-dependent extension for excited states, the S₀ →S₁ high-resolution vibronic absorption and electronic circular dichroism spectra of (R)-(+)-1-phenylethanol are computed and compared to experimental spectra measured in jet-cooled conditions in the region within 1000 cm^-1 of the 0-0 transition. The agreement between theory and computation is satisfactory and allows a confident assignment of several experimental bands in terms of fundamentals of different modes. Cases are documented for which the analysis of optical anisotropy factors, owing to their signed nature, remarkably enhances the possibility of a robust assignment of the experimental absorption bands. Computational analysis shows that the experimental spectra are dominated by Herzberg-Teller contributions and that the electronic circular dichroism spectrum and the anisotropy factors are also strongly modulated by the effect of Duschinsky mixings.

Towards the SMART workflow system for computational spectroscopy

Is it possible to convert highly specialized research in the field of computational spectroscopy into robust and user-friendly aids to experiments and industrial applications?

Theoretical studies of atmospheric molecular complexes interacting with NIR to UV light

Theoretical structural and spectroscopic data for weakly bonded atmospheric complexes of formaldehyde interacting with Earth’s electromagnetic spectrum.

Assessment of Electron Propagator Methods for the Simulation of Vibrationally Resolved Valence and Core Photoionization Spectra

The analysis of photoelectron spectra is usually facilitated by quantum mechanical simulations. Because of the recent improvement of experimental techniques, the resolution of experimental spectra is rapidly increasing, and the inclusion of vibrational effects is usually mandatory to obtain a reliable reproduction of the spectra. With the aim of defining a robust computational protocol, a general time-independent formulation to compute different kinds of vibrationally resolved electronic spectra has been generalized to also support photoelectron spectroscopy. The electronic structure data underlying the simulation are computed using different electron propagator approaches. In addition to the more standard approaches, a new and robust implementation of the second-order self-energy approximation of the electron propagator based on a transition operator reference (TOEP2) is presented. To validate our implementation, a series of molecules has been used as test cases. The result of the simulations shows that, for ultraviolet photoionization spectra, the more accurate nondiagonal approaches are needed to obtain a reliable reproduction of vertical ionization energies but that diagonal approaches are sufficient for energy gradients and pole strengths. For X-ray photoelectron spectroscopy, the TOEP2 approach, besides being more efficient, is also the most accurate in the reproduction of both vertical ionization energies and vibrationally resolved bandshapes.

Simulation of Vibronic Spectra of Flexible Systems: Hybrid DVR-Harmonic Approaches

Our general framework for the simulation of vibrational signatures in electronic spectra has been extended to treat one large-amplitude motion (LAM) at the anharmonic level, coupled to the other small-amplitude motions (SAM) treated as harmonic. The coupling between LAM and SAM is minimized thanks to the use of delocalized internal coordinates, which are built automatically from the molecular topology. General LAMs can be employed, ranging from intrinsic reaction coordinates to rigid or flexible paths based on the distinguished coordinate approach. The anharmonic model is based on a fully numerical method based on the discrete variable representation (DVR) theory, supporting different types of boundary conditions. The inclusion of this model in a general-purpose electronic structure code makes available to the user a large panel of quantum chemistry models, for both isolated and condensed phases. The flexibility and reliability of the new framework are illustrated by some case studies, covering various types of LAMs, ranging from a small test case, the photoelectron spectrum of ammonia, to larger systems, such as phenylanthracene and cyclobutanone.

Effective Inclusion of Mechanical and Electrical Anharmonicity in Excited Electronic States: VPT2-TDDFT Route

We present a reliable and cost-effective procedure for the inclusion of anharmonic effects in excited-state energies and spectroscopic intensities by means of second-order vibrational perturbation theory. This development is made possible thanks to a recent efficient implementation of excited-state analytic Hessians and properties within the time-dependent density functional theory framework. As illustrated in this work, by taking advantage of such algorithmic developments, it is possible to perform calculations of excited-state infrared spectra of medium-large isolated molecular systems, with anharmonicity effects included in both the energy and property surfaces. We also explore the use of this procedure for the inclusion of anharmonic effects in the simulation of vibronic bandshapes of electronic spectra and compare the results with previous, more approximate models.

Development and Implementation of Advanced Fitting Methods for the Calculation of Accurate Molecular Structures

The determination of accurate equilibrium molecular structures plays a fundamental role for understanding many physical-chemical properties of molecules, ranging from the precise evaluation of the electronic structure to the analysis of dynamical and environmental effects in tuning their overall behavior. For this purpose the so-called semiexperimental approach, based on a nonlinear least-squares fit of the moments of inertia associated with a set of available isotopologues, allows one to obtain very accurate results, without the unfavorable computational cost characterizing high-level quantum chemical methods. In the present work the MSR (Molecular Structure Refinement) software for the determination of equilibrium structures by means of the semiexperimental approach is presented, and its implementation is discussed in some detail. The software, which is interfaced with a powerful graphical user interface, includes different optimization algorithms, an extended error analysis, and a number of advanced features, the most remarkable ones concerning the choice of internal coordinates and the method of predicate observations. In particular, a new black-box scheme for defining automatically a suitable set of nonredundant internal coordinates of A₁ symmetry in place of the customary Z-matrix has been designed and tested. Finally, the implementation of the method of the predicate observations is discussed and validated for a set of test molecules. As an original application, the method is employed for the determination of the semiexperimental structure for the most stable conformer of glycine.

Understanding the interplay between the solvent and nuclear rearrangements in the negative solvatochromism of a push–pull flexible quinolinium cation

The main effects (solvation, vibronic progression) affecting the band position and shape of a push–pull flexible quinolinium cation OPA are highlighted.