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Galiana J, Lasorne B. Excitation energy transfer and vibronic relaxation through light-harvesting dendrimer building blocks: A nonadiabatic perspective. J Chem Phys 2024; 160:104104. [PMID: 38465688 DOI: 10.1063/5.0193264] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2023] [Accepted: 02/20/2024] [Indexed: 03/12/2024] Open
Abstract
The light-harvesting excitonic properties of poly(phenylene ethynylene) (PPE) extended dendrimers (tree-like π-conjugated macromolecules) involve a directional cascade of local excitation energy transfer (EET) processes occurring from the "leaves" (shortest branches) to the "trunk" (longest branch), which can be viewed from a vibronic perspective as a sequence of internal conversions occurring among a connected graph of nonadiabatically coupled locally excited electronic states via conical intersections. The smallest PPE building block that is able to exhibit EET, the asymmetrically meta-substituted PPE oligomer with one acetylenic bond on one side and two parallel ones on the other side (hence, 2-ring and 3-ring para-substituted pseudo-fragments), is a prototype and the focus of the present work. From linear-response time-dependent density functional theory electronic-structure calculations of the molecule as regards its first two nonadiabatically coupled, optically active, singlet excited states, we built a (1 + 2)-state-8-dimensional vibronic-coupling Hamiltonian model for running subsequent multiconfiguration time-dependent Hartree wavepacket relaxations and propagations, yielding both steady-state absorption and emission spectra as well as real-time dynamics. The EET process from the shortest branch to the longest one occurs quite efficiently (about 80% quantum yield) within the first 25 fs after light excitation and is mediated vibrationally through acetylenic and quinoidal bond-stretching modes together with a particular role given to the central-ring anti-quinoidal rock-bending mode. Electronic and vibrational energy relaxations, together with redistributions of quantum populations and coherences, are interpreted herein through the lens of a nonadiabatic perspective, showing some interesting segregation among the foremost photoactive degrees of freedom as regards spectroscopy and reactivity.
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2
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Xu C, Lin C, Peng J, Zhang J, Lin S, Gu FL, Gelin MF, Lan Z. On-the-fly simulation of time-resolved fluorescence spectra and anisotropy. J Chem Phys 2024; 160:104109. [PMID: 38477337 DOI: 10.1063/5.0201204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2024] [Accepted: 02/23/2024] [Indexed: 03/14/2024] Open
Abstract
We combine on-the-fly trajectory surface hopping simulations and the doorway-window representation of nonlinear optical response functions to create an efficient protocol for the evaluation of time- and frequency-resolved fluorescence (TFRF) spectra and anisotropies of the realistic polyatomic systems. This approach gives the effective description of the proper (e.g., experimental) pulse envelopes, laser field polarizations, and the proper orientational averaging of TFRF signals directly from the well-established on-the-fly nonadiabatic dynamic simulations without extra computational cost. To discuss the implementation details of the developed protocol, we chose cis-azobenzene as a prototype to simulate the time evolution of the TFRF spectra governed by its nonadiabatic dynamics. The results show that the TFRF is determined by the interplay of several key factors, i.e., decays of excited-state populations, evolution of the transition dipole moments along with the dynamic propagation, and scaling factor of the TFRF signals associated with the cube of emission frequency. This work not only provides an efficient and effective approach to simulate the TFRF and anisotropies of realistic polyatomic systems but also discusses the important relationship between the TFRF signals and the underlining nonadiabatic dynamics.
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Affiliation(s)
- Chao Xu
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education and Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety; School of Environment, South China Normal University, Guangzhou 510006, People's Republic of China
| | - Congru Lin
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education and Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety; School of Environment, South China Normal University, Guangzhou 510006, People's Republic of China
| | - Jiawei Peng
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education and Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety; School of Environment, South China Normal University, Guangzhou 510006, People's Republic of China
| | - Juanjuan Zhang
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education and Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety; School of Environment, South China Normal University, Guangzhou 510006, People's Republic of China
| | - Shichen Lin
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga-Park, Fukuoka 816-8580, Japan
| | - Feng Long Gu
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education and Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety; School of Environment, South China Normal University, Guangzhou 510006, People's Republic of China
| | - Maxim F Gelin
- School of Science, Hangzhou Dianzi University, Hangzhou 310018, People's Republic of China
| | - Zhenggang Lan
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education and Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety; School of Environment, South China Normal University, Guangzhou 510006, People's Republic of China
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Kaczun T, Dempwolff AL, Huang X, Gelin MF, Domcke W, Dreuw A. Tuning UV Pump X-ray Probe Spectroscopy on the Nitrogen K Edge Reveals the Radiationless Relaxation of Pyrazine: Ab Initio Simulations Using the Quasiclassical Doorway-Window Approximation. J Phys Chem Lett 2023:5648-5656. [PMID: 37310800 DOI: 10.1021/acs.jpclett.3c01018] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Transient absorption UV pump X-ray probe spectroscopy has been established as a versatile technique for the exploration of ultrafast photoinduced dynamics in valence-excited states. In this work, an ab initio theoretical framework for the simulation of time-resolved UV pump X-ray probe spectra is presented. The method is based on the description of the radiation-matter interaction in the classical doorway-window approximation and a surface-hopping algorithm for the nonadiabatic nuclear excited-state dynamics. Using the second-order algebraic-diagrammatic construction scheme for excited states, UV pump X-ray probe signals were simulated for the carbon and nitrogen K edges of pyrazine, assuming a duration of 5 fs of the UV pump and X-ray probe pulses. It is predicted that spectra measured at the nitrogen K edge carry much richer information about the ultrafast nonadiabatic dynamics in the valence-excited states of pyrazine than those measured at the carbon K edge.
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Affiliation(s)
- Tobias Kaczun
- Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg D-69120, Germany
| | - Adrian L Dempwolff
- Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg D-69120, Germany
| | - Xiang Huang
- Department of Chemistry, Technical University of Munich, Garching D-85747, Germany
| | - Maxim F Gelin
- School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Wolfgang Domcke
- Department of Chemistry, Technical University of Munich, Garching D-85747, Germany
| | - Andreas Dreuw
- Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg D-69120, Germany
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Zhan S, Gelin MF, Huang X, Sun K. Ab initio simulation of peak evolutions and beating maps for electronic two-dimensional signals of a polyatomic chromophore. J Chem Phys 2023; 158:2890773. [PMID: 37191214 DOI: 10.1063/5.0150387] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2023] [Accepted: 04/28/2023] [Indexed: 05/17/2023] Open
Abstract
By employing the doorway-window (DW) on-the-fly simulation protocol, we performed ab initio simulations of peak evolutions and beating maps of electronic two-dimensional (2D) spectra of a polyatomic molecule in the gas phase. As the system under study, we chose pyrazine, which is a paradigmatic example of photodynamics dominated by conical intersections (CIs). From the technical perspective, we demonstrate that the DW protocol is a numerically efficient methodology suitable for simulations of 2D spectra for a wide range of excitation/detection frequencies and population times. From the information content perspective, we show that peak evolutions and beating maps not only reveal timescales of transitions through CIs but also pinpoint the most relevant coupling and tuning modes active at these CIs.
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Affiliation(s)
- Siying Zhan
- School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Maxim F Gelin
- School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Xiang Huang
- Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany
| | - Kewei Sun
- School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
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Cerezo J, Gao S, Armaroli N, Ingrosso F, Prampolini G, Santoro F, Ventura B, Pastore M. Non-Phenomenological Description of the Time-Resolved Emission in Solution with Quantum-Classical Vibronic Approaches-Application to Coumarin C153 in Methanol. Molecules 2023; 28:molecules28093910. [PMID: 37175320 PMCID: PMC10180259 DOI: 10.3390/molecules28093910] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2023] [Revised: 04/27/2023] [Accepted: 04/27/2023] [Indexed: 05/15/2023] Open
Abstract
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.
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Affiliation(s)
- Javier Cerezo
- Departamento de Química and Institute for Advanced Research in Chemical Sciences (IAdChem), Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Institute of Chemistry of OrganoMetallic Compounds (ICCOM), National Research Council of Italy (CNR), Area di Ricerca di Pisa, Via Moruzzi 1, I-56124 Pisa, Italy
| | - Sheng Gao
- Institute for Organic Synthesis and Photoreactivity (ISOF), National Research Council of Italy (CNR), Via P. Gobetti 101, I-40129 Bologna, Italy
| | - Nicola Armaroli
- Institute for Organic Synthesis and Photoreactivity (ISOF), National Research Council of Italy (CNR), Via P. Gobetti 101, I-40129 Bologna, Italy
| | - Francesca Ingrosso
- Université de Lorraine & CNRS, Laboratoire de Physique et Chimie Théoriques (LPCT), F-54000 Nancy, France
| | - Giacomo Prampolini
- Institute of Chemistry of OrganoMetallic Compounds (ICCOM), National Research Council of Italy (CNR), Area di Ricerca di Pisa, Via Moruzzi 1, I-56124 Pisa, Italy
| | - Fabrizio Santoro
- Institute of Chemistry of OrganoMetallic Compounds (ICCOM), National Research Council of Italy (CNR), Area di Ricerca di Pisa, Via Moruzzi 1, I-56124 Pisa, Italy
| | - Barbara Ventura
- Institute for Organic Synthesis and Photoreactivity (ISOF), National Research Council of Italy (CNR), Via P. Gobetti 101, I-40129 Bologna, Italy
| | - Mariachiara Pastore
- Université de Lorraine & CNRS, Laboratoire de Physique et Chimie Théoriques (LPCT), F-54000 Nancy, France
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Hu D, Huo P. Ab Initio Molecular Cavity Quantum Electrodynamics Simulations Using Machine Learning Models. J Chem Theory Comput 2023; 19:2353-2368. [PMID: 37000936 PMCID: PMC10134431 DOI: 10.1021/acs.jctc.3c00137] [Citation(s) in RCA: 7] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/03/2023]
Abstract
We present a mixed quantum-classical simulation of polariton dynamics for molecule-cavity hybrid systems. In particular, we treat the coupled electronic-photonic degrees of freedom (DOFs) as the quantum subsystem and the nuclear DOFs as the classical subsystem and use the trajectory surface hopping approach to simulate non-adiabatic dynamics among the polariton states due to the coupled motion of nuclei. We use the accurate nuclear gradient expression derived from the Pauli-Fierz quantum electrodynamics Hamiltonian without making further approximations. The energies, gradients, and derivative couplings of the molecular systems are obtained from the on-the-fly simulations at the level of complete active space self-consistent field (CASSCF), which are used to compute the polariton energies and nuclear gradients. The derivatives of dipoles are also necessary ingredients in the polariton nuclear gradient expression but are often not readily available in electronic structure methods. To address this challenge, we use a machine learning model with the Kernel ridge regression method to construct the dipoles and further obtain their derivatives, at the same level as the CASSCF theory. The cavity loss process is modeled with the Lindblad jump superoperator on the reduced density of the electronic-photonic quantum subsystem. We investigate the azomethane molecule and its photoinduced isomerization dynamics inside the cavity. Our results show the accuracy of the machine-learned dipoles and their usage in simulating polariton dynamics. Our polariton dynamics results also demonstrate the isomerization reaction of azomethane can be effectively tuned by coupling to an optical cavity and by changing the light-matter coupling strength and the cavity loss rate.
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Gelin MF, Chen L, Domcke W. Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals. Chem Rev 2022; 122:17339-17396. [PMID: 36278801 DOI: 10.1021/acs.chemrev.2c00329] [Citation(s) in RCA: 16] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.
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Affiliation(s)
- Maxim F Gelin
- School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Lipeng Chen
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany
| | - Wolfgang Domcke
- Department of Chemistry, Technical University of Munich, D-85747 Garching,Germany
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8
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Avagliano D, Bonfanti M, Nenov A, Garavelli M. Automatized protocol and interface to simulate QM/MM time-resolved transient absorption at TD-DFT level with COBRAMM. J Comput Chem 2022; 43:1641-1655. [PMID: 35815854 PMCID: PMC9544370 DOI: 10.1002/jcc.26966] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2022] [Revised: 06/07/2022] [Accepted: 06/10/2022] [Indexed: 12/24/2022]
Abstract
We present a series of new implementations that we recently introduced in COBRAMM, the open-source academic software developed in our group. The goal of these implementations is to offer an automatized workflow and interface to simulate time-resolved transient absorption (TA) spectra of medium-to-big chromophore embedded in a complex environment. Therefore, the excited states absorption and the stimulated emission are simulated along nonadiabatic dynamics performed with trajectory surface hopping. The possibility of treating systems from medium to big size is given by the use of time-dependent density functional theory (TD-DFT) and the presence of the environment is taken into account employing a hybrid quantum mechanics/molecular mechanics (QM/MM) scheme. The full implementation includes a series of auxiliary scripts to properly setup the QM/MM system, the calculation of the wavefunction overlap along the dynamics for the propagation, the evaluation of the transition dipole moment at linear response TD-DFT level, and scripts to setup, run and analyze the TA from an ensemble of trajectories. Altogether, we believe that our implementation will open the door to the easily simulate the time-resolved TA of systems so far computationally inaccessible.
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Affiliation(s)
- Davide Avagliano
- Dipartimento di Chimica Industriale "Toso Montanari", Università degli Studi di Bologna, Bologna, Italy
| | - Matteo Bonfanti
- Dipartimento di Chimica Industriale "Toso Montanari", Università degli Studi di Bologna, Bologna, Italy.,Fondazione Human Technopole - Viale Rita Levi-Montalcini, 1 - Area MIND - Cargo 6 - 20157, Milano, Italy
| | - Artur Nenov
- Dipartimento di Chimica Industriale "Toso Montanari", Università degli Studi di Bologna, Bologna, Italy
| | - Marco Garavelli
- Dipartimento di Chimica Industriale "Toso Montanari", Università degli Studi di Bologna, Bologna, Italy
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Xu C, Lin K, Hu D, Gu FL, Gelin MF, Lan Z. Ultrafast Internal Conversion Dynamics through the on-the-Fly Simulation of Transient Absorption Pump-Probe Spectra with Different Electronic Structure Methods. J Phys Chem Lett 2022; 13:661-668. [PMID: 35023755 DOI: 10.1021/acs.jpclett.1c03373] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
An on-the-fly surface-hopping simulation protocol is developed for the evaluation of transient absorption (TA) pump-probe (PP) signals of molecular systems exhibiting internal conversion to the electronic ground state. We study the nonadiabatic dynamics of azomethane and the associating TA PP spectra at three levels of the electronic-structure theory, OM2/MRCI, SA-CASSCF, and XMS-CASPT2. The impact of these methods on the population dynamics and time-resolved TA PP signals is substantially different. This difference is attributed to the strong non-Condon effects that must be taken into account for the proper understanding and interpretation of time-resolved TA PP signals of nonadiabatic polyatomic systems. This shows that the combination of the dynamical and spectral simulations definitely provides more accurate and detailed information on the microscopic mechanisms of photophysical and photochemical processes. Hence the simulation of time-resolved spectroscopic signals provides another important dimension to examine the accuracy of quantum chemistry methods.
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Affiliation(s)
- Chao Xu
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education; School of Chemistry, South China Normal University, Guangzhou, 510006, P. R. China
| | - Kunni Lin
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education; School of Chemistry, South China Normal University, Guangzhou, 510006, P. R. China
| | - Deping Hu
- Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety and MOE Key Laboratory of Environmental Theoretical Chemistry, SCNU Environmental Research Institute, School of Environment, South China Normal University, Guangzhou, 510006, P. R. China
| | - Feng Long Gu
- Key Laboratory of Theoretical Chemistry of Environment, Ministry of Education; School of Chemistry, South China Normal University, Guangzhou, 510006, P. R. China
| | - Maxim F Gelin
- School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, P. R. China
| | - Zhenggang Lan
- Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety and MOE Key Laboratory of Environmental Theoretical Chemistry, SCNU Environmental Research Institute, School of Environment, South China Normal University, Guangzhou, 510006, P. R. China
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Huang X, Xie W, Došlić N, Gelin MF, Domcke W. Ab Initio Quasiclassical Simulation of Femtosecond Time-Resolved Two-Dimensional Electronic Spectra of Pyrazine. J Phys Chem Lett 2021; 12:11736-11744. [PMID: 34851116 DOI: 10.1021/acs.jpclett.1c03589] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Two-dimensional (2D) electronic spectroscopy is a powerful nonlinear technique which provides spectroscopic information on two frequency axes as well as dynamical information as a function of the so-called waiting time. Herein, an ab initio theoretical framework for the simulation of electronic 2D spectra has been developed. The method is based on the classical approximation to the doorway-window representation of three-pulse photon-echo signals and the description of nuclear motion by classical trajectories. Nonadiabatic effects are taken into account by a trajectory surface-hopping algorithm. 2D electronic spectra were simulated with ab initio on-the-fly trajectory calculations using the ADC(2) electronic-structure method for the pyrazine molecule, which is a benchmark system for ultrafast radiationless decay through conical intersections. It is demonstrated that 2D spectroscopy with subfemtosecond UV pulses can provide unprecedented detailed information on the ultrafast photodynamics of polyatomic molecules.
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Affiliation(s)
- Xiang Huang
- Department of Chemistry, Technical University of Munich, Garching, D-85747, Germany
| | - Weiwei Xie
- Institute of Physical Chemistry, Karlsruhe Institute of Technology, Karlsruhe, 76131, Germany
| | - Nađa Došlić
- Department of Physical Chemistry, Ruder Boscovic Institute, Zagreb, HR-10000, Croatia
| | - Maxim F Gelin
- School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, China
| | - Wolfgang Domcke
- Department of Chemistry, Technical University of Munich, Garching, D-85747, Germany
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