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

Vibronic Coupling and Multiple Electronic States Effect in ABS and ECD Spectra: Three [7]Helicene Derivatives

Vibronic coupling and multiple electronic states effect play a pivotal role in the molecular spectroscopy of large systems. Herein, we present a detailed theoretical study on the absorption (ABS) and electronic circular dichroism (ECD) spectra of three [7]helicene derivatives in chloroform, with a particular emphasis on the significance of vibronic coupling and the multiple electronic states effect in spectral simulations. The vertical gradient (VG) and vertical Hessian (VH) models, incorporating the Franck-Condon (FC) effect and Herzberg-Teller (HT) contribution, are considered in the vibronic calculations. The results indicate that the simulated vibrationally resolved spectra obtained by the VG model combined with the FC effect are more reliable, showing advantages in the rationality of spectral shapes, the accuracy of the relative heights of absorption peaks, and the correctness of positive and negative signals. Reliable predictions of the three [7]helicene derivatives allowed us to further explore the importance of the multiple electronic states effect in the vibrationally resolved spectra, demonstrating that the high-energy electronic excited states, particularly the fifth (S5) and sixth (S6) excited states, are essential for accurately capturing the fine structures observed in the experimental spectra. Our study predicts reliable theoretical reference spectra for the family of [7]helicene derivatives and provides a fundamental understanding of the vibronic coupling of chiral organic molecules with multiple electronic states.

Anisotropy Factor Spectra for Weakly Allowed Electronic Transitions in Chiral Ketones

Quantum chemical calculations of one-photon absorption, electronic circular dichroism and anisotropy factor spectra for the A-band transition of fenchone, camphor and 3-methylcyclopentanone (3MCP) are reported. While the only weakly allowed nature of the transition leads to comparatively large anisotropies, a proper theoretical description of the absorption for such a transition requires to account for non-Condon effects. We present experimental data for the anisotropy of 3MCP in the liquid phase and show that corresponding Herzberg-Teller corrections are critical to reproduce the main experimental features. The results obtained with our comprehensive theoretical model highlight the importance of the vibrational degree of freedom, paving the way for a deeper understanding of the dynamics in electronic circular dichroism.

Time-Resolved X-ray Absorption Spectroscopy: An MCTDH Quantum Dynamics Protocol

Expressions for linear and nonlinear spectroscopy simulation in the X-ray window in which the time evolution of a photoexcited molecular system is treated via quantum dynamics are derived. By leveraging on the peculiar properties of core-excited/ionized states, first- and third-order response functions are recast in the limit of time-scale separation between the extremely short core-state lifetime and the (comparably longer) electronic-state transfer and nuclear vibrational motion. This work is a natural extension of Segatta et al. (J. Chem. Theory Comput. 2023, 19, 2075-2091), in which some of the present authors coupled MCTDH quantum dynamics to spectroscopy simulation at different levels of sophistication. Full quantum dynamics and approximate expressions are compared by simulating X-ray transient absorption spectroscopy at the carbon K-edge in the pyrene molecule.

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.

The Resonance Raman Spectrum of Cytosine in Water: Analysis of the Effect of Specific Solute-Solvent Interactions and Non-Adiabatic Couplings

In this contribution, we report a computational study of the vibrational Resonance Raman (vRR) spectra of cytosine in water, on the grounds of potential energy surfaces (PES) computed by time-dependent density functional theory (TD-DFT) and CAM-B3LYP and PBE0 functionals. Cytosine is interesting because it is characterized by several close-lying and coupled electronic states, challenging the approach commonly used to compute the vRR for systems where the excitation frequency is in quasi-resonance with a single state. We adopt two recently developed time-dependent approaches, based either on quantum dynamical numerical propagations of vibronic wavepackets on coupled PES or on analytical correlation functions for cases in which inter-state couplings were neglected. In this way, we compute the vRR spectra, considering the quasi-resonance with the eight lowest-energy excited states, disentangling the role of their inter-state couplings from the mere interference of their different contributions to the transition polarizability. We show that these effects are only moderate in the excitation energy range explored by experiments, where the spectral patterns can be rationalized from the simple analysis of displacements of the equilibrium positions along the different states. Conversely, at higher energies, interference and inter-state couplings play a major role, and the adoption of a fully non-adiabatic approach is strongly recommended. We also investigate the effect of specific solute-solvent interactions on the vRR spectra, by considering a cluster of cytosine, hydrogen-bonded by six water molecules, and embedded in a polarizable continuum. We show that their inclusion remarkably improves the agreement with the experiments, mainly altering the composition of the normal modes, in terms of internal valence coordinates. We also document cases, mostly for low-frequency modes, in which a cluster model is not sufficient, and more elaborate mixed quantum classical approaches, in explicit solvent models, need to be applied.

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

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

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/.

Nonadiabatic Absorption Spectra and Ultrafast Dynamics of DNA and RNA Photoexcited Nucleobases

We have recently proposed a protocol for Quantum Dynamics (QD) calculations, which is based on a parameterisation of Linear Vibronic Coupling (LVC) Hamiltonians with Time Dependent (TD) Density Functional Theory (TD-DFT), and exploits the latest developments in multiconfigurational TD-Hartree methods for an effective wave packet propagation. In this contribution we explore the potentialities of this approach to compute nonadiabatic vibronic spectra and ultrafast dynamics, by applying it to the five nucleobases present in DNA and RNA. For all of them we computed the absorption spectra and the dynamics of ultrafast internal conversion (100 fs timescale), fully coupling the first 2-3 bright states and all the close by dark states, for a total of 6-9 states, and including all the normal coordinates. We adopted two different functionals, CAM-B3LYP and PBE0, and tested the effect of the basis set. Computed spectra are in good agreement with the available experimental data, remarkably improving over pure electronic computations, but also with respect to vibronic spectra obtained neglecting inter-state couplings. Our QD simulations indicate an effective population transfer from the lowest energy bright excited states to the close-lying dark excited states for uracil, thymine and adenine. Dynamics from higher-energy states show an ultrafast depopulation toward the more stable ones. The proposed protocol is sufficiently general and automatic to promise to become useful for widespread applications.

Vibronic Spectra of π-Conjugated Systems with a Multitude of Coupled States: A Protocol Based on Linear Vibronic Coupling Models and Quantum Dynamics Tested on Hexahelicene

Hexahelicene is a prototype of an extended π-conjugated system with axial chirality. Its absorption (ABS) and electronic circular dichroism (ECD) spectra show vibronic features and strong nonadiabatic effects, challenging currently available computational methods. Here, we compute the nonadiabatic ABS and ECD vibronic spectra of hexahelicene in the full energy range, covering ∼2 eV and 14-18 coupled electronic states, including all of the relevant nuclear coordinates. To this end, we exploit a recently proposed protocol that uses time-dependent density functional theory to parameterize linear vibronic coupling models comprising several electronic states. Spectra are computed through quantum dynamical propagations with multiconfigurational time-dependent Hartree methods. Our results nicely reproduce the experimental spectra providing an assignment of the main observed bands. On the contrary, we document that the application of the Herzberg-Teller intensity-borrowing theory leads to large artifacts. The proposed approach is of general applicability for rigid systems and represents a viable tool for studying the photophysical properties of π-conjugated systems characterized by a dense manifold of interacting electronic states.

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.

Generating Function Approach to Single Vibronic Level Fluorescence Spectra

An efficient time-dependent generating function method to compute vibronic emission and absorption spectra arising from transitions from a singly excited vibrational initial state is presented. In contrast to existing finite temperature approaches that intrinsically contain these transitions weighted by a Boltzmann factor, the current approach allows one to calculate these transitions individually. Using vibrational frequencies and normal modes computed by the second-order approximate coupled cluster (CC2) method, this formalism is used to compute the single vibronic level (SVL) fluorescence spectra of anthracene. Calculated spectra are in excellent agreement with spectra measured in jet-cooled expansion experiments. Duschinsky mixing is necessary to explain intensities of certain peaks. In a few cases, CC2, however, underestimates Duschinsky mixing, leading to too low peak intensities. An empirical correction of the Duschinsky matrix is presented. The presented method has the potential to facilitate the assignment and interpretation of SVL fluorescence spectra.

Entropic bounds between two thermal equilibrium states

The positivity conditions of the relative entropy between two thermal equilibrium states ρ[over ̂]_{1} and ρ[over ̂]_{2} are used to obtain upper and lower bounds for the subtraction of their entropies, the Helmholtz potential and the Gibbs potential of the two systems. These limits are expressed in terms of the mean values of the Hamiltonians, number operator, and temperature of the different systems. In particular, we discuss these limits for molecules that can be represented in terms of the Franck-Condon coefficients. We emphasize the case where the Hamiltonians belong to the same system at two different times t and t^{'}. Finally, these bounds are obtained for a general qubit system and for the harmonic oscillator with a time-dependent frequency at two different times.

Cumulant expansion for fast estimate of non-Condon effects in vibronic transition profiles

When existing, cumulants can provide valuable information about a given distribution and can in principle be used to either fully reconstruct or approximate the parent distribution function. A previously reported cumulant expansion approach for Franck–Condon profiles [Faraday Discuss., 150, 363 (2011)] is extended to describe also the profiles of vibronic transitions that are weakly allowed or forbidden in the Franck–Condon approximation (non-Condon profiles). In the harmonic approximation the cumulants of the vibronic profile can be evaluated analytically and numerically with a coherent state-based generating function that accounts for the Duschinsky effect. As illustration, the one-photon 1 ¹A_g → 1 ¹B_2u UV absorption profile of benzene in the electric dipole and (linear) Herzberg–Teller approximation is presented herein for zero Kelvin and finite temperatures.

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.

A Gaussian Wave Packet Propagation Approach to Vibrationally Resolved Optical Spectra at Non-Zero Temperatures

An effective time dependent approach based on a method that is similar to the Gaussian wave packet propagation (GWP) technique of Heller is developed for the computation of vibrationally resolved electronic spectra at finite temperatures in the harmonic, Franck-Condon/Hertzberg-Teller approximations. Since the vibrational thermal density matrix of the ground electronic surface and the time evolution operator on that surface commute, it is possible to write the spectrum generating correlation function as a trace of the time evolved doorway state. In the stated approximations, the doorway state is a superposition of the harmonic oscillator zero and one quantum eigenfunctions and thus can be propagated by the GWP. The algorithm has an O(N(3)) dependence on the number of vibrational modes. An application to pyrene absorption spectrum at two temperatures is presented as a proof of the concept.

Polyatomic Candidates for Cooling of Molecules with Lasers from Simple Theoretical Concepts

A rational approach to identify polyatomic molecules that appear to be promising candidates for direct Doppler cooling with lasers is outlined. First-principles calculations for equilibrium structures and Franck-Condon factors of selected representatives with different point-group symmetries (including the chiral nonsymmetric C1) have been performed and a high potential for laser cooling of these molecules is indicated.

General Time Dependent Approach to Vibronic Spectroscopy Including Franck-Condon, Herzberg-Teller, and Duschinsky Effects

An effective time-dependent (TD) approach to compute vibrationally resolved optical spectra from first principles is presented for the computation of one-photon electronic spectra induced by either electric or magnetic transition dipoles or by their mutual interaction, namely absorption, emission, and circular dichroism. Particular care has been devoted to generality, modularity, and numerical stability including all the contributions that play a role at the harmonic level of approximation, namely Franck-Condon, Herzberg-Teller, and Dushinsky (i.e., mode mixing) effects. The implementation shares the same general framework of our previous time-independent (TI) model, thus allowing an effective integration between both approaches with the consequent enhancement of their respective strengths (e.g., spectrum completeness and straightforward account of temperature effects for the TD route versus band resolution and assignment for the TI route) using a single set of starting data. Implementation of both models in the same general computer program allows comprehensive studies using several levels of electronic structure description together with effective account of environmental effects by atomistic and/or continuum models of different sophistication. A few medium-size molecules (furan, phenyl radical, anthracene, dimethyloxirane, coumarin 339) have been studied in order to fully validate the approach.

Franck–Condon factors—Computational approaches and recent developments

Algorithms and methodologies for the calculation of Franck–Condon integrals are reviewed. Starting from the standard approach based on the Cartesian representation of the normal modes and the use of Duschinsky's transformation and recursion formulas, methods for treating large displacements of the equilibrium positions and anharmonic and non-Condon effects are considered. Finally, focusing attention on problems arising from the use of recurrence relations, some of the proposed solutions are critically reviewed along with new alternative approaches.