Ab Initio Computation of the Duschinsky Mixing of Vibrations and Nonlinear Effects

Simulating Vibronic Spectra by Direct Application of Doktorov Formulas on a Superconducting Quantum Simulator

In this work, a direct quantum implementation of the Doktorov formulas for calculating the vibronic spectrum of molecules under the harmonic approximation is presented. It is applied to the three-atom molecules H2O, SO2, ClO2, HS2, and ZnOH. The method solves the classically hard problem of estimating the Franck-Condon (FC) factors by using the Duschinsky matrices as the only input via the Doktorov quantum circuit. This has the advantage of avoiding basis changes, artificial squeezing parameters, and symmetry dependencies. In other words, it is a general method for three-atom molecules that can easily be generalized to bigger molecules. The results are compared with other quantum algorithms and classical anharmonic algorithms. Furthermore, the circuit requirements are studied in order to estimate its applicability on real superconducting quantum hardware.

Bismuth Infrared Star: Being at a Glance

Bismuth polycations have garnered significant attention from researchers due to their extraordinary and counter-intuitive structures and stoichiometries. Despite extensive experimental and theoretical investigations, understanding of the bonding in such clusters remains insufficient. An AdNDP bonding analysis was conducted to elucidate the bonding characteristics using both homoatomic and heteroatomic bismuth clusters with various stoichiometries. Analysis of the calculated nucleus-independent chemical shift data confirmed the aromatic nature of these species. Universal bonding patterns were identified that can be applied to a range of homoatomic and heteroatomic bismuth clusters. Additionally, calculations of absorbance and fluorescence spectra were performed to gain insights into the near-infrared emission and establish a potential correlation between absorbance and the identified bonding patterns.

Estimation of Excited-State Geometries of Benzene and Fluorobenzene through Vibronic Analyses of Absorption Spectra

The parameters used in theoretical modeling of vibrational patterns within Franck-Condon (FC) approximation can be adjusted to match the vibrationally well-resolved experimental absorption spectrum of molecules. These simulation parameters can then be used to reveal the structural changes occurring between the initial and final states assuming the harmonic oscillator approximation holds for both states. Such a theoretical approach has been applied to benzene and fluorobenzene to disclose the first excited-state geometries of both compounds. The carbon-carbon bond length of benzene in the ¹B_2u state has been calculated as 1.430 Å, which is in very good agreement with the experimental bond length of 1.432 Å. The FC spectral fit method has been exploited to reveal the ¹B₂ state of fluorobenzene as well. Commonly employed density functional theory (DFT) and time-dependent DFT methods have been used to calculate the ground- and excited-state geometries of both compounds, respectively. The comparison of geometrical parameters and vibrational frequencies at the relevant states shows that frequently used hybrid functionals perform quite well in the ground state, whereas their performances drop considerably while predicting the excited-state properties. Among the hybrid functionals studied, TD-B3LYP with 6-31+G(d) basis set can be chosen to calculate the excited-state properties of molecules, albeit with much less anticipation of accuracy from the performance that B3LYP usually shows at the ground state.

Gas-Phase Optical Spectra of the Indenyl Radical and Its Quantitative Detection in a Jet-Stirred Reactor

Resonance-stabilized radicals (RSRs), such as the indenyl radical (C₉H₇), are proposed to be initiator radicals in soot inception and growth in hydrocarbon combustion processes, but spectroscopic data for many RSRs are still lacking. In this work, the gas-phase optical absorption spectra of the B̃²A₂-X̃²A₂ electronic transition of indenyl were identified in a supersonic indene/argon plasma jet. Spectroscopic parameters, including the transition energy, rotational constants, and upper-state lifetime broadening, were obtained from analysis of the experimental spectra. The results were readily applied to the quantitative detection of indenyl produced from high-temperature reactions in a jet-stirred reactor. This study now makes indenyl optically accessible in further reaction kinetics studies and in situ spectroscopic diagnostics of hydrocarbon combustion processes.

How to Predict Excited State Geometry by Using Empirical Parameters Obtained from Franck-Condon Analysis of Optical Spectrum

Excited state geometries of molecules can be calculated with highly reliable wavefunction schemes. Most of such schemes, however, are applicable to small molecules and can hardly be viewed as error-free for excited state geometries. In this study, a theoretical approach is presented in which the excited state geometries of molecules can be predicted by using vibrationally resolved experimental absorption spectrum in combination with the theoretical modelling of vibrational pattern based on Franck-Condon approximation. Huang-Rhys factors have been empirically determined and used as input for revealing the structural changes occurring between the ground and the excited state geometries upon photoexcitation. Naphthalene molecule has been chosen as a test case to show the robustness of the proposed theoretical approach. Predicted 1B_2u excited state geometry of the naphthalene has similar but slightly different bond length alternation pattern when compared with the geometries calculated with CIS, B3LYP, and CC2 methods. Excited state geometries of perylene and pyrene molecules are also determined with the presented theoretical approach. This powerful method can be applied to other molecules and specifically to relatively large molecules rather easily as long as vibrationally resolved experimental spectra are available to use.

Variational Energy Decomposition Analysis of Charge-Transfer Interactions between Metals and Ligands in Carbonyl Complexes

Variational energy decomposition analyses have been presented to quantify the σ-dative, ligand-to-metal forward charge transfer (CT) and the π-conjugative, metal-to-ligand backward charge delocalization on a series of isolelectronic transition-metal carbonyl complexes M(CO)₆, including M = Ti^2-, V^-, Cr, Mn⁺, and Fe²⁺. Although the qualitative features of these energy terms are understood, well-defined quantitative studies have been scarce. Consistent with early findings, electrostatic and Pauli exchange effects play a key role in σ-donation, resulting in blue shifts in ligand vibrational frequency in the complex geometries. Excluding chemical bonding interactions between the CO ligand and the metal fragments in the energy decomposition analysis, we found that loosely bound electrostatic complexes can be formed at a longer metal-to-ligand distance due to the exponential decay of Pauli exchange. In all complexes, the overall binding stabilization can be attributed to CT effects, with opposing trends between σ-donation and π-back bonding that follows an order of Ti^2- (4.4) > V^1- (2.6) > Cr (1.5) > Mn¹⁺ (1.1) > Fe²⁺ (0.5) in π-to-σ CT ratio. These electronic and energetic features are mirrored in the vibrational frequency shifts induced by different factors. The present investigation may help stimulate the use of energy decomposition techniques to understand the structure and activity of metallocatalysts using density functional theory.

Polymorphism Dependent 9-Phosphoanthracene Derivative Exhibiting Thermally Activated Delayed Fluorescence: A Computational Investigation

Polymorphs of anthracene derivatives exhibit diverse photophysical properties that can help to develop efficient organic-based photovoltaic devices. 10-Anthryl-9-phosphoanthracene (10-APA) shows different photophysical behaviors for the solid state due to its variety in crystalline arrangement. Herein, we investigate the ground and excited-state properties of the monomer and two different polymorphs of 10-APA from first-principles. Calculations reveal that strong spin-orbit coupling (SOC) between first excited singlet state (S₁) and triplet manifolds at their S₁-optimized geometries enabling the reverse intersystem crossing (RISC). The electron-vibration coupling (Huang-Rhys factor) in the excited state is the most relevant factor here. For both ISC and RISC, a similarity in Huang-Rhys factors for the molecular vibration along the π···π stacking at low-frequency region makes the rates effective. On the other side, the nonvanishing vibronic relaxation modes provide a relatively slower RISC rate in the red crystal. However, for the red crystal, small reorganization energy (λ) and large Huang-Rhys factor toward S₁ → S₀ conversion reduce nonradiative decay, leading to a prompt fluorescence. As the feasibility of S₁ ↔ T₁ conversion increases in the yellow dimer, it allows a delay in fluorescence emission, leading to thermally activated delayed fluorescence (TADF).

The influence of spin-orbit coupling, Duschinsky rotation and displacement vector on the rate of intersystem crossing of benzophenone and its fused analog fluorenone: a time dependent correlation function based approach

To understand the effect of structural rigidity or flexibility on the intersystem crossing rate, herein we have adopted a time dependent correlation function based approach, an appropriate method for a harmonic oscillator under Condon approximation. Following this technique, we have developed generalized codes for calculating the rate of intersystem crossing (ISC) both at 0 K and at finite temperature. Since the rate of ISC is a measurable quantity, we have separated the real and imaginary parts of the complex correlation function carefully and eliminated the imaginary part by exploiting the odd nature of this function. Using this simplified method, we have calculated the ISC rate constant (kISC) of two molecules, namely, benzophenone and its fused analog, fluorenone. The calculations clearly elucidate that kISC of benzophenone is 103 times larger compared to that of fluorenone. Interestingly, our analyses reveal that the combined effect of spin-orbit coupling and the number of normal modes could increase the rate of ISC of benzophenone by three orders in comparison to that of fluorenone. Furthermore, the Duschinsky rotation matrix (J) and displacement vectors (D) could influence the rate of ISC by one order each, indicating that the overall rate of ISC of benzophenone could have been 105 times higher than that of fluorenone if the latter two factors, namely, J and D have practically no impact on the rate of ISC of fluorenone. However, it has been found that albeit J can't alter the rate of ISC of fluorenone, D indeed can change the rate by two orders, thereby keeping the overall ratio of the rate of ISC of benzophenone and fluorenone as 103. The present study elucidates that none of the above mentioned factors alone can explain the relative rate of ISC of the studied systems; rather a complex interplay between all these factors makes the rate of ISC of benzophenone 103 times higher than that of fluorenone.

Exploring the mechanism of olfactory recognition in the initial stage by modeling the emission spectrum of electron transfer

Olfactory sense remains elusive regarding the primary reception mechanism. Some studies suggest that olfaction is a spectral sense, the olfactory event is triggered by electron transfer (ET) across the odorants at the active sites of odorant receptors (ORs). Herein we present a Donor-Bridge-Acceptor model, proposing that the ET process can be viewed as an electron hopping from the donor molecule to the odorant molecule (Bridge), then hopping off to the acceptor molecule, making the electronic state of the odorant molecule change along with vibrations (vibronic transition). The odorant specific parameter, Huang–Rhys factor can be derived from ab initio calculations, which make the simulation of ET spectra achievable. In this study, we revealed that the emission spectra (after Gaussian convolution) can be acted as odor characteristic spectra. Using the emission spectrum of ET, we were able to reasonably interpret the similar bitter-almond odors among hydrogen cyanide, benzaldehyde and nitrobenzene. In terms of isotope effects, we succeeded in explaining why subjects can easily distinguish cyclopentadecanone from its fully deuterated analogue cyclopentadecanone-d28 but not distinguishing acetophenone from acetophenone-d8.

On the theoretical prediction of fluorescence rates from first principles using the path integral approach

In this work, we present and implement the theory for calculating fluorescence rates and absorption and emission spectra from first principles, using the path integral approach. We discuss some approximations and modifications to the full set of equations that improve speed and numerical stability for the case when a large number of modes are considered. New methods to approximate the excited state potential energy surface are also discussed and it is shown that for most purposes, these can be used instead of a full geometry optimization to obtain the rates mentioned above. A few examples are presented and the overall performance of the method is discussed. It is shown that the rates and spectra computed in this way are well within the acceptable range of errors and can be used in future predictions, particularly for screening purposes, with the only limitation on size being that of the electronic structure calculation itself.

Ro-vibronic transition intensities for triatomic molecules from the exact kinetic energy operator; electronic spectrum for the C̃ 1B2 ← X̃ 1A1 transition in SO2

A procedure for calculating ro-vibronic transition intensities for triatomic molecules within the Born-Oppenheimer approximation is reported. Ro-vibrational energy levels and wavefunctions are obtained with the DVR3D suite, which solves the nuclear motion problem with an exact kinetic energy operator. Absolute transition intensities are calculated both with the Franck-Condon approximation and with a full transition dipole moment surface. The theoretical scheme is tested on C̃ ¹B₂ ← X̃ ¹A₁ ro-vibronic transitions of SO₂. Ab initio potential energy and dipole moment surfaces are generated for this purpose. The calculated ro-vibronic transition intensities and cross sections are compared with the available experimental and theoretical data.

Assessment of Franck-Condon Methods for Computing Vibrationally Broadened UV-vis Absorption Spectra of Flavin Derivatives: Riboflavin, Roseoflavin, and 5-Thioflavin

We address the performance of the vertical and adiabatic Franck-Condon (VFC/AFC) approaches combined with time-independent or time-dependent (TI/TD) formalisms in simulating the one-photon absorption spectra of three flavin compounds with distinct structural features. Calculations were done in the gas phase and in two solvents (water, benzene) for which experimental reference measurements are available. We utilized the independent mode displaced harmonic oscillator model without or with frequency alteration (IMDHO/IMDHO-FA) and also accounted for Duschinsky mixing effects. In the initial validation on the first excited singlet state of riboflavin, the range-separated functionals, CAM-B3LYP and ωB97xD, showed the best performance, but B3LYP also gave a good compromise between peak positions and spectral topology. Large basis sets were not mandatory to obtain high-quality spectra for the selected systems. The presence of a symmetry plane facilitated the computation of vibrationally broadened spectra, since different FC variants yield similar results and the harmonic approximation holds rather well. Compared with the AFC approach, the VFC approach performed equally well or even better for all three flavins while offering several advantages, such as avoiding error-prone geometry optimization procedures on excited-state surfaces. We also explored the advantages of curvilinear displacements and of a Duschinsky treatment for the AFC spectra in cases when a rotatable group is present on the chromophore. Taken together, our findings indicate that the combination of the VFC approach with the TD formalism and the IMDHO-FA model offers the best overall performance.

Vibrational control of electron-transfer reactions: a feasibility study for the fast coherent transfer regime

Electron donors are connectedvialeft and right bridges to electron acceptors. Following electron-transfer initiation, the IR excitation of selected bridge vibrational modes can tune the directionality of electron transfer.

Vibrational excitations in weakly coupled single-molecule junctions: a computational analysis

In bulk systems, molecules are routinely identified by their vibrational spectrum using Raman or infrared spectroscopy. In recent years, vibrational excitation lines have been observed in low-temperature conductance measurements on single-molecule junctions, and they can provide a similar means of identification. We present a method to efficiently calculate these excitation lines in weakly coupled, gateable single-molecule junctions, using a combination of ab initio density functional theory and rate equations. Our method takes transitions from excited to excited vibrational state into account by evaluating the Franck-Condon factors for an arbitrary number of vibrational quanta and is therefore able to predict qualitatively different behavior from calculations limited to transitions from ground state to excited vibrational state. We find that the vibrational spectrum is sensitive to the molecular contact geometry and the charge state, and that it is generally necessary to take more than one vibrational quantum into account. Quantitative comparison to previously reported measurements on pi-conjugated molecules reveals that our method is able to characterize the vibrational excitations and can be used to identify single molecules in a junction. The method is computationally feasible on commodity hardware.

Dynamics of Radiationless Transitions: Effects of Displacement−Distortion−Rotation of Potential Energy Surfaces on Internal Conversion Decay Rate Constants,

General expressions for calculating the internal conversion decay rate constants between two adiabatic electronic states and between two diabatic electronic states are derived. The expressions include the displacements, distortions, and rotations of potential energy surfaces as well as the temperature. For illustration, internal conversion rate constants between various singlet electronic states of ethylene and between the first excited S1 and the ground S0 singlet electronic states of azulene are calculated.

The Use of Multidimensional Franck−Condon Simulations to Assess Model Chemistries: A Case Study on Phenol

Multidimensional Franck-Condon simulations of the dispersed fluorescence spectra of phenol generated with geometries obtained from the highly correlated post-Hartree-Fock methods CASSCF, MRCI, and SACCI are presented. While the simulations based on CASSCF and MRCI optimized geometries are very similar to each other and fail to reproduce the experimentally measured intensities faithfully, the simulations obtained from SACCI optimized geometries are very close to the experimental spectra. The code developed for the multidimensional Franck-Condon simulations is described. It is shown that the integral storage problem common to the evaluation of multidimensional Franck-Condon integrals can be overcome by saving all quantities needed to disk. This strategy allows the code to run on computers with limited resources and is very well suited for application to molecules with a very large number of vibrational modes.

First principles simulation of the UV absorption spectrum of ethylene using the vertical Franck-Condon approach

A new method which we refer to as vertical Franck-Condon is proposed to calculate electronic absorption spectra of polyatomic molecules. In accord with the short-time picture of spectroscopy, the excited-state potential energy surface is expanded at the ground-state equilibrium geometry and the focus of the approach is more on the overall shape of the spectrum and the positions of the band maxima, rather than the precise position of the 0-0 lines. The Born-Oppenheimer approximation and the separability of the excited-state potential energy surface along the excited-state normal mode coordinates are assumed. However, the potential surface is not necessarily approximated as harmonic oscillator potentials along the individual normal modes. Instead, depending upon the nature of the potential surface along a particular normal mode, it is treated either in the harmonic approximation or the full one-dimensional potential is considered along this mode. The vertical Franck-Condon approach is applicable therefore even in cases where the excited state potential energy surface is highly anharmonic and the conventional harmonic Franck-Condon approach is inadequate. As an application of the method, the ultraviolet spectrum of ethylene between 6.2 eV (50,000 cm(-1)) and 8.7 eV (70,000 cm(-1)) is simulated, using the Similarity Transformed Equation of Motion Coupled-Cluster method to describe the required features of the potential energy surfaces. The spectrum is shown to be a result of sharp doublet structures stemming from the pi --> 3s (Rydberg) state superimposed on top of a broad band resulting from the pi --> pi* (valence) state. For the Rydberg state, the symmetric C=C stretch and the torsion mode contribute to the spectrum, while the broad valence band results from excitation into the C=C stretch, CH2 scissors, and the torsion mode. For both states, the potential along the torsion mode is highly anharmonic and the full treatment of the potential along this mode in the vertical Franck-Condon method is required.

The Jahn–Teller effect in the lower electronic states of benzene cation. III. The ground-state vibrations of C6H6+ and C6D6+

New mass analyzed threshold ionization (MATI) spectra of the molecules C(6)H(6) (+) and C(6)D(6) (+) have been collected using tunable vacuum ultraviolet (VUV) single photon excitation from the neutral ground state and also using two-photon excitation through the 6(1) vibration of the (1)B(2u) S(1) state. Emphasis was placed on obtaining accurate relative intensities of the vibrational lines in order to use this information in the vibronic analysis. The MATI spectra collected from VUV (S(0) originating state), triplet (T(1)), and resonant two photon (S(1)) excitation schemes were compared with Jahn-Teller calculations employing the classical model of Longuet-Higgins and Moffitt to obtain the Jahn-Teller coupling parameters of 3 of the 4 linearly active modes (e(2g) modes 6-9 in Wilson's notation). Franck-Condon factors, including the effects of geometry changes, were calculated from the vibronic wave functions and used to identify the lines in the various spectra. It is found that most of the lines with substantial intensity can be understood using only the modes 1, 6, 8, and 9. Weaker peaks are due to various non-e(2g) modes, but these do not derive intensity through Jahn-Teller coupling. When the effects of geometry change were included, simulations of the spectra from the calculated vibrational energies and intensities were close to the experimental spectra. This verifies the applicability of the model to the understanding of the vibrational structure of this type of molecule, but some variations indicate directions for further improvement of the model.

A multimode analysis of the gas-phase photoelectron spectra in oligoacenes

We present a multimode vibrational analysis of the gas-phase ultraviolet photoelectron spectra of the first ionization in anthracene, tetracene, and pentacene, using electron-vibration constants computed at the density functional theory level. The first ionization of each molecule exhibits a high-frequency vibronic structure; it is shown that this regularly spaced feature is actually the consequence of the collective action of several vibrational modes rather than the result of the interaction with a single mode. We interpret this feature in terms of the missing mode effect. We also discuss the vibronic coupling constants and relaxation energies obtained from the fit of the photoelectron spectra with the linear vibronic model.

Density functional calculations of the vibronic structure of electronic absorption spectra

Calculations of the vibronic structure in electronic spectra of large organic molecules based on density functional methods are presented. The geometries of the excited states are obtained from time-dependent density functional (TDDFT) calculations employing the B3LYP hybrid functional. The vibrational functions and transition dipole moment derivatives are calculated within the harmonic approximation by finite difference of analytical gradients and the transition dipole moment, respectively. Normal mode mixing is taken into account by the Duschinsky transformation. The vibronic structure of strongly dipole-allowed transitions is calculated within the Franck-Condon approximation. Weakly dipole-allowed and dipole-forbidden transitions are treated within the Franck-Condon-Herzberg-Teller and Herzberg-Teller approximation, respectively. The absorption spectra of several organic pi systems (anthracene, pentacene, pyrene, octatetraene, styrene, azulene, phenoxyl) are calculated and compared with experimental data. For dipole-allowed transitions in general a very good agreement between theory and experiment is obtained. This indicates the good quality of the optimized geometries and harmonic force fields. Larger errors are found for the weakly dipole-allowed S0 --> S1 transition of pyrene which can tentatively be assigned to TDDFT errors for the relative energies of excited states close to the target state. The weak bands of azulene and phenoxyl are very well described within the Franck-Condon approximation which can be explained by the large energy gap (>1.2 eV) to higher-lying excited states leading to small vibronic couplings. Once corrections are made for the errors in the theoretical 0-0 transition energies, the TDDFT approach to calculate vibronic structure seems to outperform both widely used ab initio methods based on configuration interaction singles or complete active space self-consistent field wave functions and semiempirical treatments regarding accuracy, applicability, and computational effort. Together with the parallel computer implementations employed, the present approach appears to be a valuable tool for a quantitative description and detailed understanding of electronic excitation processes in large molecules.

Calculation of triplet-triplet energy transfer rates from emission and absorption spectra. The quenching of hemicarcerated triplet biacetyl by aromatic hydrocarbons

The kinetics of triplet-triplet (T-T) energy transfer have been analysed with a view to linking theories of chemical reactions (involving the rupture and formation of bonds) with theories of processes, such as electron transfer or energy transfer, which preserve chemical bonding. As for the latter, our analysis does not support the claim that, of the two rival expressions for T-T energy transfer, both rooted in the golden rule, only one is applicable to electron transfer or T-T transfer. Though the two expressions do reflect different standpoints, the distinction is eroded by the assumption of a delta-function distribution for the vibrational spectrum. It is shown that theories of chemical reactions also furnish estimates of Franck-Condon factors; rates of chemical reactions and chemical processes are both related to the properties (strengths and lengths) of the reactive bonds, but differ in the mode of energy dissipation. The relationship between the rates of reactions and processes presents new possibilities for a unified view of chemical reactivity.