Toward Fully Nonempirical Simulations of Optical Band Shapes of Molecules in Solution: A Case Study of Heterocyclic Ketoimine Difluoroborates

The role of microenvironments on computed vibrationally-resolved emission spectra: The case of oxazines

Oxazine dyes act as reporters of their near environment by the response of their fluorescence spectra. At the same time, their fluorescence spectra exhibit a pronounced vibrational progression. In this work, we computationally investigate the impact of near-environment models consisting of aggregated water as well as betaine molecules on the vibrational profile of fluorescence spectra of different oxazine derivatives. For aggregated betaine and a water molecule located above the plane of the dyes, we observe a distinct modification of the vibrational profile, which is more pronounced than the effect of a continuum description of a solvent environment. Our analysis shows that this effect cannot be explained by a pure change in the electronic structure, but that also vibrational degrees of freedom of the environment can be decisive for the vibrational profile and should, hence, not generally be neglected.

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.

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%.

Tailored anharmonic-harmonic vibrational profiles for fluorescent biomarkers

We propose a hybrid anharmonic-harmonic scheme for vibrational broadenings, which embeds a reduced-space vibrational configuration interaction (VCI) anharmonic wave function treatment in the independent-mode displaced harmonic oscillator (IMDHO) model. The resulting systematically-improvable VCI-in-IMDHO model allows including the vibronic effects of all vibrational degrees of freedom, while focusing the effort on the important degrees of freedom with minimal extra computational effort compared to a reduced-space VCI treatment. We show for oligothiophene examples that the VCI-in-IMDHO approach can yield accurate vibrational profiles employing smaller vibrational spaces in the VCI part than the reduced-space VCI approach. By this, the VCI-in-IMDHO model enables accurate calculation of vibrational profiles of common fluorescent dyes with more than 100 vibrational degrees of freedom. We illustrate this for three examples of fluorescent biomarkers of current interest. These are the oligothiophene-based fluorescent dye called HS84, 1,4-diphenylbutadiene, and an anthracene diimide. For all examples, we assess the impact of the anharmonic treatment on the vibrational broadening, which we find to be more pronounced for the intensities than for the peak positions.

Much of a Muchness: On the Origins of Two- and Three-Photon Absorption Activity of Dipolar Y-Shaped Chromophores

The molecular origin of two- (2PA) and three-photon absorption (3PA) activity in three experimentally studied chromophores, prototypical dipolar systems, is investigated. To that end, a generalized few-state model (GFSM) formula is derived for the 3PA transition strength for nonhermitian theories and employed at the coupled-cluster level of theory. Using various computational techniques such as molecular dynamics, linear and quadratic response theories, and GFSM, an in-depth analysis of various optical channels involved in 2PA and 3PA processes is presented. It is found that the four-state model involving the second and third excited singlet states as intermediates is the smallest model among all considered few-state approximations that produces 2PA and 3PA transition strengths (for S₀ → S₁ transition) close to the reference results. By analyzing various optical channels appearing in these models and involved in studied multiphoton processes, we found that the 2PA and 3PA activities in all the three chromophores are dominated and hence controlled by the dipole moment of the final excited state. The similar origins of the 2PA and the 3PA in these prototypical dipolar chromophores suggest transferability of structure–property relations from the 2PA to the 3PA domain.

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.

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.

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.

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.

Stimuli-responsive luminochromic polymers consisting of multi-state emissive fused boron ketoiminate

Both thermochromic luminescence in solution and mechanochromic luminescence were each observed from conjugated polymers containing a fused boron complex.

Photostability of oxazoline RNA-precursors in UV-rich prebiotic environments

Pentose aminooxazolines and oxazolidinone thiones are considered as the key precursors which could have enabled the formation of RNA nucleotides under the conditions of early Earth. UV-irradiation experiments and quantum-chemical calculations demonstrate that these compounds are remarkably photostable and could accumulate over long periods of time in UV-rich prebiotic environments to undergo stereoisomeric purification.

Influence of the Nature of the Amino Group in Highly Fluorescent Difluoroborates Exhibiting Intramolecular Charge Transfer

A series of difluoroborates were synthesized from CH acids. All compounds were substituted with dialkylamino groups (NR₂). The lone electron pair of the nitrogen atom of this donor moiety is variably delocalized toward the difluoroborate core that acts as the electron acceptor. This was rationalized in light of the various geometries of the amino group. The degree of charge transfer was quantified on the basis of the results of time-dependent density functional theory calculations.

The importance of nuclear quantum effects in spectral line broadening of optical spectra and electrostatic properties in aromatic chromophores

In this work, we examine the importance of nuclear quantum effects on capturing the line broadening and vibronic structure of optical spectra. We determine the absorption spectra of three aromatic molecules indole, pyridine, and benzene using time dependent density functional theory with several molecular dynamics sampling protocols: force-field based empirical potentials, ab initio simulations, and finally path-integrals for the inclusion of nuclear quantum effects. We show that the absorption spectrum for all these chromophores are similarly broadened in the presence of nuclear quantum effects regardless of the presence of hydrogen bond donor or acceptor groups. We also show that simulations incorporating nuclear quantum effects are able to reproduce the heterogeneous broadening of the absorption spectra even with empirical force fields. The spectral broadening associated with nuclear quantum effects can be accounted for by the broadened distribution of chromophore size as revealed by a particle in the box model. We also highlight the role that nuclear quantum effects have on the underlying electronic structure of aromatic molecules as probed by various electrostatic properties.

Combining the ensemble and Franck-Condon approaches for calculating spectral shapes of molecules in solution

The correct treatment of vibronic effects is vital for the modeling of absorption spectra of many solvated dyes. Vibronic spectra for small dyes in solution can be easily computed within the Franck-Condon approximation using an implicit solvent model. However, implicit solvent models neglect specific solute-solvent interactions on the electronic excited state. On the other hand, a straightforward way to account for solute-solvent interactions and temperature-dependent broadening is by computing vertical excitation energies obtained from an ensemble of solute-solvent conformations. Ensemble approaches usually do not account for vibronic transitions and thus often produce spectral shapes in poor agreement with experiment. We address these shortcomings by combining zero-temperature vibronic fine structure with vertical excitations computed for a room-temperature ensemble of solute-solvent configurations. In this combined approach, all temperature-dependent broadening is treated classically through the sampling of configurations and quantum mechanical vibronic contributions are included as a zero-temperature correction to each vertical transition. In our calculation of the vertical excitations, significant regions of the solvent environment are treated fully quantum mechanically to account for solute-solvent polarization and charge-transfer. For the Franck-Condon calculations, a small amount of frozen explicit solvent is considered in order to capture solvent effects on the vibronic shape function. We test the proposed method by comparing calculated and experimental absorption spectra of Nile red and the green fluorescent protein chromophore in polar and non-polar solvents. For systems with strong solute-solvent interactions, the combined approach yields significant improvements over the ensemble approach. For systems with weak to moderate solute-solvent interactions, both the high-energy vibronic tail and the width of the spectra are in excellent agreement with experiments.

Mixed Quantum/Classical Method for Nonadiabatic Quantum Dynamics in Explicit Solvent Models: The ππ*/nπ* Decay of Thymine in Water as a Test Case

We present a novel mixed quantum classical dynamical method to include solvent effects on internal conversion (IC) processes. All the solute degrees of freedom are represented by a wavepacket moving according to nonadiabatic quantum dynamics, while the motion of an explicit solvent model is described by an ensemble of classical trajectories. The mutual coupling of the solute and solvent dynamics is included within a mean-field framework and the quantum and classical equations of motions are solved simultaneously. As a test case we apply our method to the ultrafast ππ* → nπ* decay of thymine in water. Solvent dynamical response modifies IC yield already on the 50 fs time scale. This effect is due to water librational motions that stabilize the most populated state. Pure static disorder, that is, the existence of different solvent configurations when photoexcitation takes place, also has a remarkable impact on the dynamics.

Nonempirical Simulations of Inhomogeneous Broadening of Electronic Transitions in Solution: Predicting Band Shapes in One- and Two-Photon Absorption Spectra of Chalcones

We have examined several approaches relying on the Polarizable Embedding (PE) scheme to predict optical band shapes for two chalcone molecules in methanol solution. The PE-TDDFT and PERI-CC2 methods were combined with molecular dynamics simulations, where the solute geometry was kept either as rigid, flexible or partly-flexible (restrained) body. The first approach, termed RBMD-PE-TDDFT, was employed to estimate the inhomogeneous broadening for subsequent convolution with the vibrationally-resolved spectra of the molecule in solution determined quantum-mechanically (QM). As demonstrated, the RBMD-PE-TDDFT/QM-PCM approach delivers accurate band widths, also reproducing their correct asymmetric shapes. Further refinement can be obtained by the estimation of the inhomogeneous broadening using the RBMD-PERI-CC2 method. On the other hand, the remaining two approaches (FBMD-PE-TDDFT and ResBMD-PE-TDDFT), which lack quantum-mechanical treatment of molecular vibrations, lead to underestimated band widths. In this study, we also proposed a simple strategy regarding the rapid selection of the exchange-correlation functional for the simulations of vibrationally-resolved one- and two-photon absorption spectra based on two easy-to-compute metrics.

The shape of the electronic circular dichroism spectrum of (2,6-dimethylphenyl)(phenyl)methanol: interplay between conformational equilibria and vibronic effects

Analysis of the interplay between conformational equilibria, solvent effects and vibronic contributions in the ECD spectra.

Quantum-Classical Calculation of Vibronic Spectra along a Reaction Path: The Case of the ECD of Easily Interconvertible Conformers with Opposite Chiral Responses

We present a novel mixed method suitable for computing the low-to-middle resolution spectra of systems characterized by a large-amplitude motion s (defined by either a reaction or an internal path) coupled to a bath of harmonic oscillators r, which change with s. The method is based on an adiabatic approximation whereby s is considered much slower than r degrees of freedom and is treated classically. We show that the spectrum can be obtained by a suitable average of quantum vibronic spectra of the fast coordinates computed at representative values of the slow coordinate along the path. By our method we calculate the electronic circular dichroism (ECD) of 2,2,2-trifluoro-anthryl-ethanol, which possesses two conformers with opposite chirality separated by a low-energy barrier. Path-averaged spectra are significantly different from those obtained only at the stable minima: Agreement with experiment is improved. The method represents a generalization of those currently based on harmonic approximation.

Revisiting Vertical Models To Simulate the Line Shape of Electronic Spectra Adopting Cartesian and Internal Coordinates

Vertical models for the simulation of spectroscopic line shapes expand the potential energy surface (PES) of the final state around the equilibrium geometry of the initial state. These models provide, in principle, a better approximation of the region of the band maximum. At variance, adiabatic models expand each PES around its own minimum. In the harmonic approximation, when the minimum energy structures of the two electronic states are connected by large structural displacements, adiabatic models can breakdown and are outperformed by vertical models. However, the practical application of vertical models faces the issues related to the necessity to perform a frequency analysis at a nonstationary point. In this contribution we revisit vertical models in harmonic approximation adopting both Cartesian (x) and valence internal curvilinear coordinates (s). We show that when x coordinates are used, the vibrational analysis at nonstationary points leads to a deficient description of low-frequency modes, for which spurious imaginary frequencies may even appear. This issue is solved when s coordinates are adopted. It is however necessary to account for the second derivative of s with respect to x, which here we compute analytically. We compare the performance of the vertical model in the s-frame with respect to adiabatic models and previously proposed vertical models in x- or Q₁-frame, where Q₁ are the normal coordinates of the initial state computed as combination of Cartesian coordinates. We show that for rigid molecules the vertical approach in the s-frame provides a description of the final state very close to the adiabatic picture. For sizable displacements it is a solid alternative to adiabatic models, and it is not affected by the issues of vertical models in x- and Q₁-frames, which mainly arise when temperature effects are included. In principle the G matrix depends on s, and this creates nonorthogonality problems of the Duschinsky matrix connecting the normal modes of initial and final states in adiabatic approaches. We highlight that such a dependence of G on s is also an issue in vertical models, due to the necessity to approximate the kinetic term in the Hamiltonian when setting up the so-called GF problem. When large structural differences exist between the initial and the final-state minima, the changes in the G matrix can become too large to be disregarded.

Elucidating the Mechanism of Zn(2+) Sensing by a Bipyridine Probe Based on Two-Photon Absorption

In this work, we examine, by means of computational methods, the mechanism of Zn(2+) sensing by a bipyridine-centered, D-π-A-π-D-type ratiometric molecular probe. According to recently published experimental data [Divya, K. P.; Sreejith, S.; Ashokkumar, P.; Yuzhan, K.; Peng, Q.; Maji, S. K.; Tong, Y.; Yu, H.; Zhao, Y.; Ramamurthy, P.; Ajayaghosh, A. A ratiometric fluorescent molecular probe with enhanced two-photon response upon Zn(2+) binding for in vitro and in vivo bioimaging. Chem. Sci. 2014, 5, 3469-3474], after coordination to zinc ions the probe exhibits a large enhancement of the two-photon absorption cross section. The goal of our investigation was to elucidate the mechanism behind this phenomenon. For this purpose, linear and nonlinear optical properties of the unbound (cation-free) and bound probe were calculated, including the influence of solute-solvent interactions, implicitly using a polarizable continuum model and explicitely employing the QM/MM approach. Because the results of the calculations indicate that many conformers of the probe are energetically accessible at room temperature in solution and hence contribute to the signal, structure-property relationships were also taken into account. Results of our simulations demonstrate that the one-photon absorption bands for both the unbound and bound forms correspond to the bright π → π* transition to the first excited state, which, on the other hand, exhibits negligible two-photon activity. On the basis of the results of the quadratic response calculations, we put forward a notion that it is the second excited state that gives the strong signal in the experimental nonlinear spectrum. To explain the differences in the two-photon absorption activity for the two lowest-lying excited states and nonlinear response enhancement upon binding, we employed the generalized few-state model including the ground, first, and second excited states. The analysis of the optical channel suggests that the large two-photon response is due to the coordination-induced increase of the transition moment from the first to the second excited state.

Theoretical investigation of the broad one-photon absorption line-shape of a flexible symmetric carbazole derivative

The one-photon absorption spectrum of a carbazole derivative has been studied by employing density functional response theory combined with a mixed quantum/classical approach to simulate the spectral shape.