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Vibrational normal modes of terrylene in the first excited electronic singlet state ( 1 B 3u ) with the corresponding harmonic wavenumbers (a) 239, (b) 1263, (c) 1364, (d) 1538 cm À1 . The figures were produced with the program Jmol 34 (H: light grey, C: dark black).
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In this contribution, advantages and disadvantages of the time-independent and time-dependent approaches for Franck-Condon profile calculations are discussed within the displaced-distorted-rotated harmonic oscillator approximation. Particular strengths and prospects of a previously developed time-independent cumulant expansion in the calculation of...
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... al., 26 which can be compared to the computed FC profile in Fig. 3(b). Main features of this spectrum include excitations in totally symmetric in-plane ring bending and stretching modes with calculated harmonic wavenumbers of 239, 1263, 1364 and 1538 cm À1 (experimental fundamental wavenumbers of 241, 1311, 1398 and 1634 cm À1 , respectively; see Fig. 4 for the corresponding normal modes) in the excited state. However, the calculated progressions are shorter than the experimental ones. Returning now to Fig. 1(b), we observe that the TCF-FFT curve (solid line) looks like a combination of a dominant, a moderately strong and a weak Gaussian function, respectively, one at the 0 0 -0 ...
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Citations
... Cumulants of the vibronic spectrum can be obtained from the CSGF directly without computing the total spectrum in frequency domain. This method was exploited already in ref. 29 for FC-allowed transitions, and we report herein an extension of this method to incorporate non-Condon transitions. To illustrate the performance of the approach, we present the profile of the 1 A 1g → 1 B 2u transition of benzene, which frequently served as a prototypical example for multiple authors (see e.g. ...
... Thus, cumulants can be evaluated analytically or numerically by evaluating partial derivatives of χ in Eq. (4) with respect to the time variable at t = 0. Analytic evaluation of the cumulants to arbitrary order within the linear HT approximation can be performed along the lines of the development in refs 16,28,29 for the cumulants of FC profiles to arbitrary order. For numerical evaluation of low-order cumulants one needs to compute χ at the first few time steps. ...
... The vibronic profiles for benzene's 1 1 A g → 1 1 B 2u transition at zero Kelvin and finite temperatures are calculated with the two methods, namely TCF and time-independent cumulant expansion (CE). We use herein the term time-independent CE, which was employed in ref. 29 , to distinguish this CE from the conventional (time-dependent) CE (see e.g. refs [20][21][22][23][24] ) which involves time integration for the cumulant calculation. ...
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 spectral
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 $1 ^{1}\mathrm{A_{g}}\rightarrow1
^{1}\mathrm{B_{2u}}$ UV absorption spectrum of benzene in the electric dipole
and (linear) Herzberg-Teller approximation is presented herein for zero Kelvin
and finite temperatures.
... The computation of FC spectra requires the overlap between the ground-and excited-state vibrational wave functions, which affects the absorption or fluorescence intensity of the vibrational bands. In this work, we will use the time-independent FC method (TI-FC), [16][17][18][19][20][21] but time-dependent approaches (TD-FC) are available as well. 22 For both TI-FC and TD-FC, one can employ Duschinsky rotation, 23,24 which allows for mode mixing by constructing the excited-state modes from a linear combination of the ground-state modes. ...
We address the effects of using Cartesian or internal coordinates in the adiabatic Franck-Condon (AFC) and vertical Franck-Condon (VFC) approaches to electronic spectra. The adopted VFC approach is a simplified variant of the original approach [A. Hazra, H. H. Chang, and M. Nooijen, J. Chem. Phys. 151, 2125 (2004)], as we omit any contribution from normal modes with imaginary frequency. For our test molecules ranging from ethylene to flavin compounds, VFC offers several advantages over AFC, especially by preserving the properties of the FC region and by avoiding complications arising from the crossing of excited-state potential surfaces or from the failure of the harmonic approximation. The spectral quality for our target molecules is insensitive to the chosen approach. We also explore the effects of Duschinsky rotation and relate the need for internal coordinates to the absence of symmetry elements. When using Duschinsky rotation and treating larger systems without planar symmetry, internal coordinates are found to outperform Cartesian coordinates in the AFC spectral calculations.
Ultraviolet resonance Raman (UVRR) scattering is a highly sensitive and selective vibrational spectroscopic technique with a broad range of applications from polyaromatic hydrocarbons (PAHs) to biomolecular systems (peptides/proteins and nucleic acids) and catalysts. The interpretation of experimental UVRR spectra is not as straightforward as in purely vibrational Raman scattering (Placzek approximation) due to the involvement of higher lying electronic states and vibronic coupling. This necessitates the comparison with theoretical UVRR spectra computed by electronic structure calculations. Anthracene is an ideal model system for such a comparison between experiment and theory because it is rigid, symmetric, and of moderate size. By taking into account Herzberg–Teller contributions including Duschinsky effects, bulk solvent effects, and anharmonic contributions, a good qualitative agreement close to the resonance condition is achieved. The present study shows that within the framework of time‐dependent density functional theory (TD‐DFT), a general and robust approach for the analysis and interpretation of resonance Raman spectra of medium‐ to large‐size molecules is available. Anthracene is an ideal model system for a comparison between experiment and theory because it is rigid, symmetric, and of moderate size. A good qualitative agreement between the experimental UVRR spectrum excited at 251 nm and the theoretical UVRR spectrum, computed within the framework of time‐dependent density functional theory (TD‐DFT), was obtained. Herzberg–Teller contributions including Duschinsky effects, bulk solvent effects, and anharmonic contributions were considered.
The probabilities of vibronic transitions in molecules are referred to as Franck-Condon factors (FCFs). Although several approaches for calculating FCFs have been developed, such calculations are still challenging. Recently it was shown that there exists a correspondence between the problem of calculating FCFs and boson sampling. However, if the output photon number distribution of boson sampling is sparse, then it can be classically simulated. Exploiting these results, we develop a method to approximately reconstruct the distribution of FCFs of certain molecules. We demonstrate its proof of concept by applying it to formic acid and thymine at 0 K. In our method, we first obtain the marginal photon number distributions for pairs of modes of a Gaussian state associated with the molecular transition. We then apply a compressive sensing method called polynomial-time matching pursuit to recover FCFs.
Vibrationally resolved electronic absorption spectra including the effect of vibrational pre-excitation are computed in order to interpret and predict vibronic transitions that are probed in the Vibrationally Promoted Electronic Resonance (VIPER) experiment [L. J. G. W. van Wilderen et al., Angew. Chem. Int. Ed. 53, 2667 (2014)]. To this end, we employ time-independent and time-dependent methods based on the evaluation of Franck-Condon overlap integrals and Fourier transformation of time-domain wavepacket autocorrelation functions, respectively. The time-independent approach uses a generalized version of the FCclasses method [F. Santoro et al., J. Chem. Phys. 126, 084509 (2007)]. In the time-dependent approach, autocorrelation functions are obtained by wavepacket propagation and by evaluation of analytic expressions, within the harmonic approximation including Duschinsky rotation effects. For several medium-sized polyatomic systems, it is shown that selective pre-excitation of particular vibrational modes leads to a red-shift of the low-frequency edge of the electronic absorption spectrum, which is a prerequisite for the VIPER experiment. This effect is typically most pronounced upon excitation of ring distortion modes within an aromatic pi-system. Theoretical predictions as to which modes show the strongest VIPER effect are found to be in excellent agreement with experiment.
Kubo’s generating function approach has been employed for simulating from first principles the spectral band shapes of singlet-triplet transitions of some aromatic compounds, representative for their rigid structures and aromatic characters of those used in solid state opto-electronic devices. That approach yields Franck-Condon weighted densities of states in excellent agreement with experimental results, opening the way to reliable calculations of the rates of non-radiative singlet-triplet transitions, an important step for the in silico optimization of the molecular structure of dyes for solar energy conversion cells and for light emitting diodes.
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.
An algorithm to compute vibronic spectra of harmonic surfaces including Dushinsky rotation and Hertzberg-Teller terms is described. The method, inspired by thermo field dynamics, maps the thermal density matrix onto the vacuum state and uses the time-dependent coupled cluster ansatz to propagate it in time. In the Franck-Condon approximation where the dipole matrix elements are taken to be constants, this reduces to the auto correlation function of the new vacuum. In the Hertzberg-Teller approximation, the full time evolution operator is needed. This too is governed by a closed set of equations. The theoretical development is presented along with an application to anthracene.
Vibronic (vibrational-electronic) transition is one of the fundamental processes in molecular physics. Indeed, vibronic transition is essential both in radiative and nonradiative photophysical or photochemical properties of molecules such as absorption, emission, Raman scattering, circular dichroism, electron transfer, internal conversion, etc. A detailed understanding of these transitions in varying systems, especially for (large) biomolecules, is thus of particular interest. Describing vibronic transitions in polyatomic systems with hundreds of atoms is, however, a difficult task due to the large number of coupled degrees of freedom. Even within the relatively crude harmonic approximation, such as for Born-Oppenheimer harmonic potential energy surfaces, the brute-force evaluation of Franck-Condon intensity profiles in a time-independent sum-over-states approach is prohibitive for complex systems owing to the vast number of multi-dimensional Franck-Condon integrals. The main goal of this thesis is to describe a variety of molecular vibronic transitions, with special focus on the development of approaches that are applicable to extended molecular systems. We use various representations of Fermi’s golden rule in frequency, time and phase spaces via coherent states to reduce the computational complexity. Although each representation has benefits and shortcomings in its evaluation, they complement each other. Peak assignment of a spectrum can be made directly after calculation in the frequency domain but this sum-over-states route is usually slow. In contrast, computation is considerably faster in the time domain with Fourier transformation but the peak assignment is not directly available. The representation in phase space does not immediately provide physically-meaningful quantities but it can link frequency and time domains. This has been applied to, herein, for example (non-Condon) absorption spectra of benzene and electron transfer of bacteriochlorophyll in the photosynthetic reaction center at finite temperature. This work is a significant step in the treatment of vibronic structure, allowing for the accurate and efficient treatment of complex systems, and provides a new analysis tool for molecular science.
An analytical expression for the population dynamics of electronic radiationless transitions has been derived from the second order expansion of the quantum evolution operator in the Liouville space and the cumulant theory. The expression includes the effect of both normal mode displacements and Duschinsky rotations and allows to take into account both equilibrium and nonequilibrium initial conditions. The methodology has been applied to model the electron-transfer process between the accessory bacteriochlorophyll and the bacteriopheophytine in bacterial reactions centers, providing a rate in good agreement with experimental findings.
