Study of the franck-condon and herzberg-teller approximations

Significance of Vibronic Coupling that Shapes Circularly Polarized Luminescence of Double Helicenes

The circularly polarized luminescence (CPL) spectra of S- and X-shaped double helicenes exhibit distinct vibrational structures and overall shape variations. In this study, we conducted an in-depth investigation into the vibronic effects influencing the CPL spectra of two double helicenes, namely DPC and DNH. Employing state-of-the-art computations utilizing the FC-HT1|VH model at the CAM-B3LYP/def2-TZVP level, we unveiled the paramount impact of Franck-Condon (FC), Herzberg-Teller (HT), and Duschinsky effects on their chiroptical responses. Our research underscores the pivotal role of structural deformations associated with the S1-to-S0 electronic transition in molding CPL spectra and wavelength-dependent dissymmetry (g) factor values, as well as the significance of HT effects in shaping and enhancing CPL responses. This extensive investigation not only advances our comprehension of the vibronic characteristics in configurationally distinct double helicenes but also offers valuable insights for the design of chiral molecules featuring controllable or finely-tunable CPL responses.

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.

Herzberg-Teller Effect on the Vibrationally Resolved Absorption Spectra of Single-Crystalline Pentacene at Finite Temperatures

The line shape of an electronic spectrum conveys the coupling between electronic and vibrational degrees of freedom. In the present study, the light absorption spectra of single-crystalline pentacene were measured by polarized UV-vis microscopy at 77, 185, and 293 K. The vibronic coupling encoded in each spectrum was resolved by the Herzberg-Teller theory that considers the contributions from the Franck-Condon (FC) factor, Franck-Condo/Herzberg-Teller (FC/HT) interference, and Herzberg-Teller (HT) coupling. Specifically, excitation energies, electronic transition dipole moments, and their nuclear gradients were evaluated by the GW method to ensure numerical accuracy, while the computationally efficient density function theory was employed to determine the optimized structures and vibrational normal modes. For every pair of electronic transition and normal mode that gives rise to a strong vibronic transition intensity, we examined their spatial characteristics by projecting them onto the three crystal axes. It was found that all normal modes strongly coupled to the lowest-lying a-polarized electronic transitions oscillate along axis a, whereas none of their counterparts for the lowest-lying b-polarized electronic transitions is predominantly along axis b. This notable difference on the alignment between the electronic transition and molecular vibration could help the directional control of charge dissociation and/or spin separation. Moreover, a significant variance of the destructive FC/HT interference was discovered with increasing temperatures that can well explain the a-polarized fading tableland near 650 nm. Finally, the importance of HT coupling was corroborated by comparing its intensity with those of FC factor and FC/HT interference. Taken all together, the vibrational dependence of the electronic wave function is critical to resolve the light absorption spectra of single-crystalline pentacene and its temperature effects, facilitating the systematic design of functional optical materials based on pentacene and its derivatives.

Fingerprinting the Excited-State Dynamics in Methyl Ester and Methyl Ether Anions of Deprotonated para-Coumaric Acid

Chromophores based on the para-hydroxycinnamate moiety are widespread in the natural world, including as the photoswitching unit in photoactive yellow protein and as a sunscreen in the leaves of plants. Here, photodetachment action spectroscopy combined with frequency- and angle-resolved photoelectron imaging is used to fingerprint the excited-state dynamics over the first three bright action-absorption bands in the methyl ester anions (pCEs^-) of deprotonated para-coumaric acid at a temperature of ∼300 K. The excited states associated with the action-absorption bands are classified as resonances because they are situated in the detachment continuum and are open to autodetachment. The frequency-resolved photoelectron spectrum for pCEs^- indicates that all photon energies over the S₁(ππ*) band lead to similar vibrational autodetachment dynamics. The S₂(nπ*) band is Herzberg-Teller active and has comparable brightness to the higher lying 2¹(ππ*) band. The frequency-resolved photoelectron spectrum over the S₂(nπ*) band indicates more efficient internal conversion to the S₁(ππ*) state for photon energies resonant with the Franck-Condon modes (∼80%) compared with the Herzberg-Teller modes (∼60%). The third action-absorption band, which corresponds to excitation of the 2¹(ππ*) state, shows complex and photon energy-dependent dynamics, with 20-40% of photoexcited population internally converting to the S₁(ππ*) state. There is also evidence for a mode-specific competition between prompt autodetachment and internal conversion on the red edge of the 2¹(ππ*) band. There is no evidence for recovery of the ground electronic state and statistical electron ejection (thermionic emission) following photoexcitation over any of the three action-absorption bands. The photoelectron spectra for the deprotonated methyl ether derivative (pCEt^-) at photon energies over the S₁(ππ*) and S₂(nπ*) bands indicate diametrically opposed dynamics compared with pCEs^-, namely, intense thermionic emission due to efficient recovery of the ground electronic state.

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.

Ab initio simulation of pyrene spectra in water matrices

The absorption and emission spectra of free pyrene and pyrene in a water ice matrix were simulated ab initio with their vibronic profiles. Water ice was mimicked by a large cluster of explicit water molecules.

Classification of dynamic vibronic couplings in vibrational real-time spectra of a thiophene derivative by few-cycle pulses

Pump-probe spectroscopy was performed with a few cycle pulses of 6.7 fs duration. The electronic transition intensity modulation was induced by molecular vibration in a quinoid thiophene molecule in solution. The real-time vibrational features were analyzed in terms of dependence of vibrational amplitude and phase on probe photon energy. The electronic transition probability is modulated by molecular vibration via vibronic coupling. Changes in the spectral shape and intensity of the time-resolved spectrum were studied by tracking characteristic spectral features including the peak frequency and intensity, spectral bandwidth, and band-integrated intensity. From the analysis the modulation mechanisms were classified into two groups: (1) Condon type and (2) non-Condon type. The features of the wave packet motions were also classified into zeroth-order derivatives due to quasi-pure non-Condon type and first- and second-order derivative types due to the displacement of the potential minimum and the potential curvature change associated with the relevant vibronic transition, respectively.

Laser cooling of vibrational degrees of freedom of a molecular system

We consider the cooling of vibrational degrees of freedom in a photoinduced excited electronic state of a model molecular system. For the various parameters of the potential surfaces of the ground and excited electronic states and depending on the excitation frequency of a single-mode laser light, the average energy or average vibrational temperature of the excited state passes through a minimum. The amount of cooling is quantified in terms of the overlap integral between the ground and excited electronic states of the molecule. We have given an approach to calculate the Franck-Condon factor for a multimode displaced-distorted-rotated oscillator surface of the molecular system. This is subsequently used to study the effect of displacement, distortion, and Duschinsky rotation on the vibrational cooling in the excited state. The absorption spectra and also the average energy or the effective temperature of the excited electronic state are studied for the above model molecular system. Considering the non-Condon effect for the symmetry-forbidden transitions, we have discussed the absorption spectra and average temperature in the excited-state vibrational manifold.

Study of vibronic, spin-orbit and vibronic-spin-orbit couplings of formaldehyde with applications to radiative and non-radiative processes

With the recent availability of large amounts of experimental data, theoretical models of radiative and nonradiative processes can be tested in detail. For this purpose, the absolute calculation of radiative and non-­radiative rate constants is required. The expression for the non-radiative rate constant consists of two parts, one from the electronic motion (the pro­moting part) and the other from the nuclear motion (the statistical part). All recent theoretical work has been focused on the calculation of the non-radiative rate constant of one single vibronic state relative to another so that the promoting part of the rate constant is cancelled and hence has been concerned mainly with the Franck–Condon factor calculation (the statistical part of the rate constant). That the calculation of the statistical part of the rate constant cannot provide a critical test for a theory of radiationless transitions is obvious. In the present investigation, the theory of radiationless transitions to be tested is that originally proposed by Robinson and Frosch and later developed by Lin & Bersohn, Siebrand & Henry, Freed, Nitzan & Jortner and Fischer. According to this theory, one way to calculate the promoting part of the non-radiative rate constant is to invoke the vibronic coupling for the internal conversion and to invoke the vibronic coupling plus the spin-orbit coupling and/or the vibronicspin-orbit coupling for the intersystem crossing. A numerical calculation is carried out for the internal conversion 1 A 2 → 1 A 1 and the intersystem crossing 1 A 2 → 3 A 2 of formaldehyde by using simple m. os. For radiative transitions, we calculate the lifetimes of the states, 3 A 2 ( n ↔ π * ), 1 A 2 ( n ↔ π * ), 1 B 1 ( n ↔σ*), and 1 A 1 ( π ↔ π * ). The corresponding transition moments are given. For symmetry forbidden transitions, we compared the Herzberg–Teller theory with the importance of the correc­tion to the breakdown of the adiabatic approximation. It is shown that for formaldehyde, the B.–O. correction is approximately one order of magnitude smaller than the first order vibronic coupling.