1
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Polonius S, Lehrner D, González L, Mai S. Resolving Photoinduced Femtosecond Three-Dimensional Solute-Solvent Dynamics through Surface Hopping Simulations. J Chem Theory Comput 2024; 20:4738-4750. [PMID: 38768386 PMCID: PMC11171268 DOI: 10.1021/acs.jctc.4c00169] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2024] [Revised: 04/22/2024] [Accepted: 05/07/2024] [Indexed: 05/22/2024]
Abstract
Photoinduced dynamics in solution is governed by mutual solute-solvent interactions, which give rise to phenomena like solvatochromism, the Stokes shift, dual fluorescence, or charge transfer. Understanding these phenomena requires simulating the solute's photoinduced dynamics and simultaneously resolving the three-dimensional solvent distribution dynamics. If using trajectory surface hopping (TSH) to this aim, thousands of trajectories are required to adequately sample the time-dependent three-dimensional solvent distribution functions, and thus resolve the solvent dynamics with sub-Ångstrom and femtosecond accuracy and sufficiently low noise levels. Unfortunately, simulating thousands of trajectories with TSH in the framework of hybrid quantum mechanical/molecular mechanical (QM/MM) can be prohibitively expensive when employing ab initio electronic structure methods. To tackle this challenge, we recently introduced a computationally efficient approach that combines efficient linear vibronic coupling models with molecular mechanics (LVC/MM) via electrostatic embedding [Polonius et al., JCTC 2023, 19, 7171-7186]. This method provides solvent-embedded, nonadiabatically coupled potential energy surfaces while scaling similarly to MM force fields. Here, we employ TSH with LVC/MM to unravel the photoinduced dynamics of two small thiocarbonyl compounds solvated in water. We describe how to estimate the number of trajectories required to produce nearly noise-free three-dimensional solvent distribution functions and present an analysis based on approximately 10,000 trajectories propagated for 3 ps. In the electronic ground state, both molecules exhibit in-plane hydrogen bonds to the sulfur atom. Shortly after excitation, these bonds are broken and reform perpendicular to the molecular plane on timescales that differ by an order of magnitude due to steric effects. We also show that the solvent relaxation dynamics is coupled to the electronic dynamics, including intersystem crossing. These findings are relevant to advance the understanding of the coupled solute-solvent dynamics of solvated photoexcited molecules, e.g., biologically relevant thio-nucleobases.
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Affiliation(s)
- Severin Polonius
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
- Vienna
Doctoral School in Chemistry (DoSChem), University of Vienna, Währinger Str. 42, 1090 Vienna, Austria
| | - David Lehrner
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
| | - Leticia González
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
- Vienna
Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria
| | - Sebastian Mai
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
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2
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Bao S, Raymond N, Nooijen M. Time dependent vibrational electronic coupled cluster (VECC) theory for non-adiabatic nuclear dynamics. J Chem Phys 2024; 160:094105. [PMID: 38426527 DOI: 10.1063/5.0190034] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2023] [Accepted: 01/28/2024] [Indexed: 03/02/2024] Open
Abstract
A time-dependent vibrational electronic coupled-cluster (VECC) approach is proposed to simulate photo-electron/UV-VIS absorption spectra as well as time-dependent properties for non-adiabatic vibronic models, going beyond the Born-Oppenheimer approximation. A detailed derivation of the equations of motion and a motivation for the ansatz are presented. The VECC method employs second-quantized bosonic construction operators and a mixed linear and exponential ansatz to form a compact representation of the time-dependent wave-function. Importantly, the method does not require a basis set, has only a few user-defined inputs, and has a classical (polynomial) scaling with respect to the number of degrees of freedom (of the vibronic model), resulting in a favorable computational cost. In benchmark applications to small models and molecules, the VECC method provides accurate results compared to multi-configurational time-dependent Hartree calculations when predicting short-time dynamical properties (i.e., photo-electron/UV-VIS absorption spectra) for non-adiabatic vibronic models. To illustrate the capabilities, the VECC method is also successfully applied to a large vibronic model for hexahelicene with 14 electronic states and 63 normal modes, developed in the group by Aranda and Santoro [J. Chem. Theory Comput. 17, 1691, (2021)].
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Affiliation(s)
- Songhao Bao
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Neil Raymond
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Marcel Nooijen
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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3
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Polonius S, Zhuravel O, Bachmair B, Mai S. LVC/MM: A Hybrid Linear Vibronic Coupling/Molecular Mechanics Model with Distributed Multipole-Based Electrostatic Embedding for Highly Efficient Surface Hopping Dynamics in Solution. J Chem Theory Comput 2023; 19:7171-7186. [PMID: 37788824 PMCID: PMC10601485 DOI: 10.1021/acs.jctc.3c00805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2023] [Indexed: 10/05/2023]
Abstract
We present a theoretical framework for a hybrid linear vibronic coupling model electrostatically embedded into a molecular mechanics environment, termed the linear vibronic coupling/molecular mechanics (LVC/MM) method, for the surface hopping including arbitrary coupling (SHARC) molecular dynamics package. Electrostatic embedding is realized through the computation of interactions between environment point charges and distributed multipole expansions (DMEs, up to quadrupoles) that represent each electronic state and transition densities in the diabatic basis. The DME parameters are obtained through a restrained electrostatic potential (RESP) fit, which we extended to yield higher-order multipoles. We also implemented in SHARC a scheme for achieving roto-translational invariance of LVC models as well as a general quantum mechanics/molecular mechanics (QM/MM) interface, an OpenMM interface, and restraining potentials for simulating liquid droplets. Using thioformaldehyde in water as a test case, we demonstrate that LVC/MM can accurately reproduce the solvation structure and energetics of rigid solutes, with errors on the order of 1-2 kcal/mol compared to a BP86/MM reference. The implementation in SHARC is shown to be very efficient, enabling the simulation of trajectories on the nanosecond time scale in a matter of days.
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Affiliation(s)
- Severin Polonius
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
- Vienna
Doctoral School in Chemistry (DoSChem), University of Vienna, Währinger Str. 42, 1090 Vienna, Austria
| | - Oleksandra Zhuravel
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
| | - Brigitta Bachmair
- Vienna
Doctoral School in Chemistry (DoSChem), University of Vienna, Währinger Str. 42, 1090 Vienna, Austria
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
| | - Sebastian Mai
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria
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4
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Steenbakkers K, Marimuthu AN, Redlich B, Groenenboom GC, Brünken S. A vibrational action spectroscopic study of the Renner-Teller- and spin-orbit-affected cyanoacetylene radical cation HC 3N . J Chem Phys 2023; 158:084305. [PMID: 36859081 DOI: 10.1063/5.0135000] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
The linear radical cation of cyanoacetylene, HC3N+ (2Π), is not only of astrophysical interest, being the, so far undetected, cationic counterpart of the abundant cyanoaceteylene, but also of fundamental spectroscopic interest due to its strong spin-orbit and Renner-Teller interactions. Here, we present the first broadband vibrational action spectroscopic investigation of this ion through the infrared pre-dissociation (IRPD) method using a Ne tag. Experiments have been performed using the FELion cryogenic ion-trap instrument in combination with the FELIX free-electron lasers and a Laservision optical parametric oscillator/optical parametric amplifier system. The vibronic splitting patterns of the three interacting bending modes (ν5, ν6, ν7), ranging from 180 to 1600 cm-1, could be fully resolved revealing several bands that were previously unobserved. The associated Renner-Teller and intermode coupling constants have been determined by fitting an effective Hamiltonian to the experimental data, and the obtained spectroscopic constants are in reasonable agreement with previous photoelectron spectroscopy (PES) studies and ab initio calculations on the HC3N+ ion. The influence of the attached Ne atom on the infrared spectrum has been investigated by ab initio calculations at the RCCSD(T)-F12a level of theory, which strongly indicates that the discrepancies between the IRPD and PES data are a result of the effects of the Ne attachment.
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Affiliation(s)
- Kim Steenbakkers
- FELIX Laboratory, Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands
| | - Aravindh N Marimuthu
- FELIX Laboratory, Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands
| | - Britta Redlich
- FELIX Laboratory, Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands
| | - Gerrit C Groenenboom
- Radboud University, Institute for Molecules and Materials, Nijmegen, The Netherlands
| | - Sandra Brünken
- FELIX Laboratory, Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands
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5
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Dvořák J, Ranković M, Houfek K, Nag P, Čurík R, Fedor J, Čížek M. Vibronic Coupling through the Continuum in the e+CO_{2} System. PHYSICAL REVIEW LETTERS 2022; 129:013401. [PMID: 35841574 DOI: 10.1103/physrevlett.129.013401] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2022] [Revised: 04/20/2022] [Accepted: 05/20/2022] [Indexed: 06/15/2023]
Abstract
We report two-dimensional electron energy-loss spectra of CO_{2}. The high-resolution experiment reveals a counterintuitive fine structure at energy losses where CO_{2} states form a vibrational pseudocontinuum. Guided by the symmetry of the system, we constructed a four-dimensional nonlocal model for the vibronic dynamics involving two shape resonances (forming a Renner-Teller Π_{u} doublet at the equilibrium geometry) coupled to a virtual Σ_{g}^{+} state. The model elucidates the extremely non-Born-Oppenheimer dynamics of the coupled nuclear motion and explains the origin of the observed structures. It is a prototype of the vibronic coupling of metastable states in continuum.
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Affiliation(s)
- Jan Dvořák
- Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics, V Holešovičkách 2, 180 00 Prague 8, Czech Republic
| | - Miloš Ranković
- J. Heyrovský Institute of Physical Chemistry, Czech Academy of Sciences, Dolejškova 2155/3, 182 23 Prague 8, Czech Republic
| | - Karel Houfek
- Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics, V Holešovičkách 2, 180 00 Prague 8, Czech Republic
| | - Pamir Nag
- J. Heyrovský Institute of Physical Chemistry, Czech Academy of Sciences, Dolejškova 2155/3, 182 23 Prague 8, Czech Republic
| | - Roman Čurík
- J. Heyrovský Institute of Physical Chemistry, Czech Academy of Sciences, Dolejškova 2155/3, 182 23 Prague 8, Czech Republic
| | - Juraj Fedor
- J. Heyrovský Institute of Physical Chemistry, Czech Academy of Sciences, Dolejškova 2155/3, 182 23 Prague 8, Czech Republic
| | - Martin Čížek
- Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics, V Holešovičkách 2, 180 00 Prague 8, Czech Republic
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6
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Hauser AW, Pototschnig JV. Vibronic Coupling in Spherically Encapsulated, Diatomic Molecules: Prediction of a Renner-Teller-like Effect for Endofullerenes. J Phys Chem A 2022; 126:1674-1680. [PMID: 35258966 PMCID: PMC8935370 DOI: 10.1021/acs.jpca.1c10970] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
![]()
In the year 1933,
Herzberg and Teller realized that the potential
energy surface of a triatomic, linear molecule splits into two as
soon as the molecule is bent. The phenomenon, later dubbed the Renner–Teller
effect due to the detailed follow-up work of Renner on the subject,
describes the coupling of a symmetry-reducing molecular vibration
with degenerate electronic states. In this article, we show that a
very similar type of nonadiabatic coupling can occur for certain translational
degrees of freedom of diatomic, electronically degenerate molecules
when trapped in a nearly spherical or cylindrical quantum confinement,
e.g., realized through electromagnetic fields or molecular encapsulation.
We illustrate this on the example of fullerene-encapsulated nitric
oxide, and provide a prediction of its interesting, perturbed vibronic
spectrum.
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Affiliation(s)
- Andreas W Hauser
- Institute of Experimental Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria
| | - Johann V Pototschnig
- Institute of Experimental Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria
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7
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Zhang C, Augenbraun BL, Lasner ZD, Vilas NB, Doyle JM, Cheng L. Accurate prediction and measurement of vibronic branching ratios for laser cooling linear polyatomic molecules. J Chem Phys 2021; 155:091101. [PMID: 34496585 DOI: 10.1063/5.0063611] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We report a generally applicable computational and experimental approach to determine vibronic branching ratios in linear polyatomic molecules to the 10-5 level, including for nominally symmetry-forbidden transitions. These methods are demonstrated in CaOH and YbOH, showing approximately two orders of magnitude improved sensitivity compared with the previous state of the art. Knowledge of branching ratios at this level is needed for the successful deep laser cooling of a broad range of molecular species.
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Affiliation(s)
- Chaoqun Zhang
- Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218, USA
| | | | - Zack D Lasner
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Nathaniel B Vilas
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - John M Doyle
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Lan Cheng
- Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218, USA
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8
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Poluyanov LV, Ushakov VG. Singlet-Triplet Interaction of the 1Π And 3Σ– Terms in Linear Triatomic Molecules. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2021. [DOI: 10.1134/s1990793121030106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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9
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Choi S, Vaníček J. How important are the residual nonadiabatic couplings for an accurate simulation of nonadiabatic quantum dynamics in a quasidiabatic representation? J Chem Phys 2021; 154:124119. [PMID: 33810696 DOI: 10.1063/5.0046067] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Diabatization of the molecular Hamiltonian is a standard approach to remove the singularities of nonadiabatic couplings at conical intersections of adiabatic potential energy surfaces. In general, it is impossible to eliminate the nonadiabatic couplings entirely-the resulting "quasidiabatic" states are still coupled by smaller but nonvanishing residual nonadiabatic couplings, which are typically neglected. Here, we propose a general method for assessing the validity of this potentially drastic approximation by comparing quantum dynamics simulated either with or without the residual couplings. To make the numerical errors negligible to the errors due to neglecting the residual couplings, we use the highly accurate and general eighth-order composition of the implicit midpoint method. The usefulness of the proposed method is demonstrated on nonadiabatic simulations in the cubic Jahn-Teller model of nitrogen trioxide and in the induced Renner-Teller model of hydrogen cyanide. We find that, depending on the system, initial state, and employed quasidiabatization scheme, neglecting the residual couplings can result in wrong dynamics. In contrast, simulations with the exact quasidiabatic Hamiltonian, which contains the residual couplings, always yield accurate results.
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Affiliation(s)
- Seonghoon Choi
- Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Jiří Vaníček
- Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
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10
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Volokhov VM, Poluyanov LV. The Singlet-Triplet Interaction of 3Π and 1Σ+ States in Linear Triatomic Molecules. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2021. [DOI: 10.1134/s1990793121020111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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11
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Nandipati KR, Vendrell O. On the generation of electronic ring currents under vibronic coupling effects. J Chem Phys 2020; 153:224308. [PMID: 33317290 DOI: 10.1063/5.0031389] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study the generation of electronic ring currents in the presence of nonadiabatic coupling using circularly polarized light. For this, we introduce a solvable model consisting of an electron and a nucleus rotating around a common center and subject to their mutual Coulomb interaction. The simplicity of the model brings to the forefront the non-trivial properties of electronic ring currents in the presence of coupling to the nuclear coordinates and enables the characterization of various limiting situations transparently. Employing this model, we show that vibronic coupling effects play a crucial role even when a single E degenerate eigenstate of the system supports the current. The maximum current of a degenerate eigenstate depends on the strength of the nonadiabatic interactions. In the limit of large nuclear to electronic masses, in which the Born-Oppenheimer approximation becomes exact, constant ring currents and time-averaged oscillatory currents necessarily vanish.
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Affiliation(s)
- Krishna Reddy Nandipati
- Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuneheimer Feld 229, 69120 Heidelberg, Germany
| | - Oriol Vendrell
- Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuneheimer Feld 229, 69120 Heidelberg, Germany
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12
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Abstract
This work examines the electronic structure and apparent instability of ethylenedione (OCCO), including an analysis of the singlet and triplet potential energy surfaces along the bending vibrations. While the singlet state is inherently unstable due to the Renner-Teller effect, theory predicts the triplet state to have a stable minimum on the potential energy surface. The stability of the triplet state is examined in detail, taking into account spin-orbit interactions. Using multireference quantum chemical methods, the lifetime of the triplet state is estimated to be in the picosecond range, significantly lower than previously computed. A quasi-atomic molecular orbital (QUAO) analysis is also used to elucidate the nature of bonding along the potential energy surface in both the singlet and triplet states. These calculations confirm the transient nature of the OCCO molecule, although they do not fully explain the lack of experimental detection via spectroscopy, which is known have the capability to probe even shorter lifetimes.
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Affiliation(s)
- Joani Mato
- Department of Chemistry and Ames Laboratory, Iowa State University, Ames, Iowa 50011, United States
| | - David Poole
- Department of Chemistry and Ames Laboratory, Iowa State University, Ames, Iowa 50011, United States
| | - Mark S Gordon
- Department of Chemistry and Ames Laboratory, Iowa State University, Ames, Iowa 50011, United States
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13
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Volokhov V, Poluyanov L. Relativistic pseudo Renner effect of (3Π + 1Σ+) × π type in linear triatomic molecules. Chem Phys Lett 2020. [DOI: 10.1016/j.cplett.2020.137739] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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14
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Volokhov VM, Poluyanov LV. Vibronic and Spin-Orbit Couplings of 3Π and 3Σ+ Electronic States in Linear Triatomic Molecules. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2020. [DOI: 10.1134/s1990793120020141] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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15
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Ghosh A, Reddy SR, Mahapatra S. Electronic spectroscopy of carbon chains (C 2n+1, n = 7–10) of astrophysical importance. II. Quantum dynamics. J Chem Phys 2019. [DOI: 10.1063/1.5108726] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Arpita Ghosh
- School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
| | | | - Susanta Mahapatra
- School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
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16
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Rabidoux SM, Cave RJ, Stanton JF. Nonadiabatic Investigation of the Electronic Spectroscopy of trans-1,3-Butadiene. J Phys Chem A 2019; 123:3255-3271. [DOI: 10.1021/acs.jpca.9b01021] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Scott M. Rabidoux
- Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, United States
| | - Robert J. Cave
- Department of Chemistry, Harvey Mudd College, 241 Platt Boulevard, Claremont, California 91711, United States
| | - John F. Stanton
- Department of Chemistry and Biochemistry, The University of Texas at Austin, 1 University Avenue, Austin, Texas 78712, United States
- The Department of Chemistry, The University of Florida, Gainesville, Florida 32611, United States
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17
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Scheit S, Goswami S, Meyer HD, Köppel H. Fully quantal treatment of nonadiabatic molecular photodynamics: General considerations and application to the benzene cation. COMPUT THEOR CHEM 2019. [DOI: 10.1016/j.comptc.2019.01.011] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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18
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Gaynor JD, Khalil M. Signatures of vibronic coupling in two-dimensional electronic-vibrational and vibrational-electronic spectroscopies. J Chem Phys 2018; 147:094202. [PMID: 28886647 DOI: 10.1063/1.4991745] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
Two-Dimensional Electronic-Vibrational (2D EV) spectroscopy and Two-Dimensional Vibrational-Electronic (2D VE) spectroscopy are new coherent four-wave mixing spectroscopies that utilize both electronically resonant and vibrationally resonant field-matter interactions to elucidate couplings between electronic and vibrational degrees of freedom. A system Hamiltonian is developed here to lay a foundation for interpreting the 2D EV and 2D VE signals that arise from a vibronically coupled molecular system in the condensed phase. A molecular system consisting of one anharmonic vibration and two electronic states is modeled. Equilibrium displacement of the vibrational coordinate and vibrational frequency shifts upon excitation to the first electronic excited state are included in our Hamiltonian through linear and quadratic vibronic coupling terms. We explicitly consider the nuclear dependence of the electronic transition dipole moment and demonstrate that these spectroscopies are sensitive to non-Condon effects. A series of simulations of 2D EV and 2D VE spectra obtained by varying parameters of the system, system-bath, and interaction Hamiltonians demonstrate that one of the following conditions must be met to observe signals: (1) non-zero linear and/or quadratic vibronic coupling in the electronic excited state, (2) vibrational-coordinate dependence of the electronic transition dipole moment, or (3) electronic-state-dependent vibrational dephasing dynamics. We explore how these vibronic interactions are manifested in the positions, amplitudes, and line shapes of the peaks in 2D EV and 2D VE spectroscopies.
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Affiliation(s)
- James D Gaynor
- Department of Chemistry, University of Washington, P.O. Box 351700, Seattle, Washington 98195, USA
| | - Munira Khalil
- Department of Chemistry, University of Washington, P.O. Box 351700, Seattle, Washington 98195, USA
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19
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Baeck KK, An H. Practical approximation of the non-adiabatic coupling terms for same-symmetry interstate crossings by using adiabatic potential energies only. J Chem Phys 2018; 146:064107. [PMID: 28201877 DOI: 10.1063/1.4975323] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Abstract
A very simple equation, FijApp=[(∂2(Via-Vja)/∂Q2)/(Via-Vja)]1/2/2, giving a reliable magnitude of non-adiabatic coupling terms (NACTs, Fij's) based on adiabatic potential energies only (Via and Vja) was discovered, and its reliability was tested for several prototypes of same-symmetry interstate crossings in LiF, C2, NH3Cl, and C6H5SH molecules. Our theoretical derivation starts from the analysis of the relationship between the Lorentzian dependence of NACTs along a diabatization coordinate and the well-established linear vibronic coupling scheme. This analysis results in a very simple equation, α=2κ/Δc, enabling the evaluation of the Lorentz function α parameter in terms of the coupling constant κ and the energy gap Δc (Δc=|Via-Vja|Qc ) between adiabatic states at the crossing point QC. Subsequently, it was shown that QC corresponds to the point where FijApp exhibit maximum values if we set the coupling parameter as κ=[(Via-Vja)⋅(∂2(Via-Vja)/∂Q2)]Qc1/2/2. Finally, we conjectured that this relation could give reasonable values of NACTs not only at the crossing point but also at other geometries near QC. In this final approximation, the pre-defined crossing point QC is not required. The results of our test demonstrate that the approximation works much better than initially expected. The present new method does not depend on the selection of an ab initio method for adiabatic electronic states but is currently limited to local non-adiabatic regions where only two electronic states are dominantly involved within a nuclear degree of freedom.
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Affiliation(s)
- Kyoung Koo Baeck
- Department of Chemistry, Gangneung-Wonju National University, Gangwon-do 25457, South Korea
| | - Heesun An
- Department of Chemistry, Gangneung-Wonju National University, Gangwon-do 25457, South Korea
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20
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Osherov V, Osherov M, Poluyanov L. Singlet-triplet interaction in linear triatomic molecules. Chem Phys Lett 2018. [DOI: 10.1016/j.cplett.2017.12.038] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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21
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Generating symmetry-adapted bases for non-Abelian point groups to be used in vibronic coupling Hamiltonians. Chem Phys 2015. [DOI: 10.1016/j.chemphys.2015.07.034] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Egidi F, Segado M, Koch H, Cappelli C, Barone V. A benchmark study of electronic excitation energies, transition moments, and excited-state energy gradients on the nicotine molecule. J Chem Phys 2015; 141:224114. [PMID: 25494739 DOI: 10.1063/1.4903307] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this work, we report a comparative study of computed excitation energies, oscillator strengths, and excited-state energy gradients of (S)-nicotine, chosen as a test case, using multireference methods, coupled cluster singles and doubles, and methods based on time-dependent density functional theory. This system was chosen because its apparent simplicity hides a complex electronic structure, as several different types of valence excitations are possible, including n-π(*), π-π(*), and charge-transfer states, and in order to simulate its spectrum it is necessary to describe all of them consistently well by the chosen method.
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Affiliation(s)
- Franco Egidi
- Scuola Normale Superiore, Piazza dei Cavalieri, 7 I-56126 Pisa, Italy
| | - Mireia Segado
- Scuola Normale Superiore, Piazza dei Cavalieri, 7 I-56126 Pisa, Italy
| | - Henrik Koch
- Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway
| | - Chiara Cappelli
- Dipartimento di Chimica e Chimica Industriale, Università di Pisa, via G. Moruzzi, 3 I-56124 Pisa, Italy
| | - Vincenzo Barone
- Scuola Normale Superiore, Piazza dei Cavalieri, 7 I-56126 Pisa, Italy
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23
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León I, Yang Z, Wang LS. Probing the electronic structure and Au-C chemical bonding in AuC2(-) and AuC2 using high-resolution photoelectron spectroscopy. J Chem Phys 2014; 140:084303. [PMID: 24588165 DOI: 10.1063/1.4865978] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We report photoelectron spectroscopy (PES) and high-resolution PE imaging of AuC2(-) at a wide range of photon energies. The ground state of AuC2(-) is found to be linear (C∞v, (1)Σ(+)) with a …8π(4)4δ(4)17σ(2)9π(4)18σ(2) valence configuration. Detachments from all the five valence orbitals of the ground state of AuC2(-) are observed at 193 nm. High-resolution PE images are obtained in the energy range from 830 to 330 nm, revealing complicated vibronic structures from electron detachment of the 18σ, 9π, and 17σ orbitals. Detachment from the 18σ orbital results in the (2)Σ(+) ground state of neutral AuC2, which, however, is bent due to strong vibronic coupling with the nearby (2)Π state from detachment of a 9π electron. The (2)Σ(+)-(2)Π vibronic and spin-orbit coupling results in complicated vibronic structures for the (2)Σ(+) and (2)Π3/2 states with extensive bending excitations. The electron affinity of AuC2 is measured accurately to be 3.2192(7) eV with a ground state bending frequency of 195(6) cm(-1). The first excited state ((2)A') of AuC2, corresponding to the (2)Π3/2 state at the linear geometry, is only 0.0021 eV above the ground state ((2)A') and has a bending frequency of 207(6) cm(-1). The (2)Π1/2 state, 0.2291 eV above the ground state, is linear with little geometry change relative to the anion ground state. The detachment of the 17σ orbital also results in complicated vibronic structures, suggesting again a bent state due to possible vibronic coupling with the lower (2)Π state. The spectrum at 193 nm shows the presence of a minor species with less than 2% intensity relative to the ground state of AuC2(-). High-resolution data of the minor species reveal several vibrational progressions in the Au-C stretching mode, which are assigned to be from the metastable (3)Π2,1,0 spin-orbit excited states of AuC2(-) to the (2)Π3/2,1/2 spin-orbit states of neutral AuC2. The spin-orbit splittings of the (3)Π and (2)Π states are accurately measured at the linear geometry. The current study provides a wealth of electronic structure information about AuC2(-) and AuC2, which are ideal systems to investigate the strong Σ-Π and spin-orbit vibronic couplings.
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Affiliation(s)
- Iker León
- Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
| | - Zheng Yang
- Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
| | - Lai-Sheng Wang
- Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
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24
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Douguet N, Orel AE, Greene CH, Kokoouline V. Dissociative recombination of highly symmetric polyatomic ions. PHYSICAL REVIEW LETTERS 2012; 108:023202. [PMID: 22324682 DOI: 10.1103/physrevlett.108.023202] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2011] [Indexed: 05/31/2023]
Abstract
A general first-principles theory of dissociative recombination is developed for highly symmetric molecular ions and applied to H(3)O(+) and CH(3)(+), which play an important role in astrophysical, combustion, and laboratory plasma environments. The theoretical cross sections obtained for the dissociative recombination of the two ions are in good agreement with existing experimental data from storage ring experiments.
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Affiliation(s)
- Nicolas Douguet
- Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
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25
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Kokoouline V, Douguet N, Greene CH. Breaking bonds with electrons: Dissociative recombination of molecular ions. Chem Phys Lett 2011. [DOI: 10.1016/j.cplett.2011.03.062] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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26
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Johnson MA, Rostas J. Vibronic structure of the CO2+ion: reinvestigation of the antisymmetric stretch vibration in the [Xtilde], Ã, and [Btilde] states. Mol Phys 2008. [DOI: 10.1080/00268979500101521] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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27
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Perić M, Peyerimhoff SD, Buenker RJ. Ab initiotreatment of the Renner-Teller effect and application to various AH2and HAB molecules. INT REV PHYS CHEM 2008. [DOI: 10.1080/01442358509353356] [Citation(s) in RCA: 50] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Sioutis I, Mishra S, Poluyanov LV, Domcke W. Renner-Teller and spin-orbit vibronic coupling effects in linear triatomic molecules with a half-filled π shell. J Chem Phys 2008; 128:124318. [DOI: 10.1063/1.2840356] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
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30
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Jankowiak HC, Stuber JL, Berger R. Vibronic transitions in large molecular systems: Rigorous prescreening conditions for Franck-Condon factors. J Chem Phys 2007; 127:234101. [DOI: 10.1063/1.2805398] [Citation(s) in RCA: 142] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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31
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Mishra S, Poluyanov LV, Domcke W. Spin-orbit vibronic coupling in Π3 states of linear triatomic molecules. J Chem Phys 2007; 126:134312. [PMID: 17430037 DOI: 10.1063/1.2713382] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Abstract
The Renner-Teller vibronic-coupling problem of a 3Pi electronic state of a linear molecule is analyzed with the inclusion of the spin-orbit coupling of the 3Pi electronic state, employing the microscopic (Breit-Pauli) spin-orbit coupling operator for the two unpaired electrons. The 6x6 Hamiltonian matrix in a diabatic spin-electronic basis is obtained by an expansion of the molecular Hamiltonian in powers of the bending amplitude. The symmetry properties of the Hamiltonian with respect to the time-reversal operator and the relativistic vibronic angular momentum operator are analyzed. It is shown that there exists a linear vibronic-coupling term of spin-orbit origin, which has not been considered so far in the Renner-Teller theory of 3Pi electronic states. While two of the six adiabatic electronic wave functions do not exhibit a geometric phase, the other four carry nontrivial topological phases which depend on the radius of the integration contour. The spectroscopic effects of the linear spin-orbit vibronic-coupling mechanism have been analyzed by numerical calculations of the vibronic spectrum for selected model examples.
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Affiliation(s)
- Sabyashachi Mishra
- Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany.
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32
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Köuppel H, Domcke W, Cederbaum LS. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. ADVANCES IN CHEMICAL PHYSICS 2007. [DOI: 10.1002/9780470142813.ch2] [Citation(s) in RCA: 887] [Impact Index Per Article: 52.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/03/2022]
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33
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Pacher T, Cederbaum LS, Köppel H. Adiabatic and Quasidiabatic States in a Gauge Theoretical Framework. ADVANCES IN CHEMICAL PHYSICS 2007. [DOI: 10.1002/9780470141427.ch4] [Citation(s) in RCA: 180] [Impact Index Per Article: 10.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
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Perić M, Jerosimić S, Ranković R, Krmar M, Radić-Perić J. An ab initio model for handling the Renner–Teller effect in tetra-atomic molecules. I. Introduction of coordinates and the Hamiltonian. Chem Phys 2006. [DOI: 10.1016/j.chemphys.2006.07.035] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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35
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Mishra S, Vallet V, Poluyanov LV, Domcke W. Calculation of the vibronic structure of the photodetachment spectra of CCCl- and CCBr-. J Chem Phys 2006; 125:164327. [PMID: 17092093 DOI: 10.1063/1.2363193] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
The vibronic structure of the closely spaced and strongly coupled X 2Sigma+ and A 2Pi states in the photodetachment spectra of CCCl- and CCBr- has been calculated by considering Sigma-Pi vibronic coupling together with spin-orbit coupling. The stretching modes are treated within the so-called linear-vibronic-coupling model. The vibronic and spin-orbit parameters have been determined by accurate ab initio electronic-structure calculations. While the nonrelativistic vibronic-coupling parameters are of approximately equal strength in CCCl and CCBr, the vibronic-coupling parameters of spin-orbit origin are found to be larger in the latter. The calculated photodetachment spectra of both systems are shown to exhibit a complicated vibronic structure due to strong Sigma-Pi vibronic coupling. The spectral envelopes of the calculated photodetachment spectra exhibit a double-hump reminiscent of strongly coupled Exe Jahn-Teller systems.
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Affiliation(s)
- Sabyashachi Mishra
- Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany.
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36
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Mishra S, Domcke W, Poluyanov LV. Study of strong Σ–Π and spin–orbit vibronic coupling effects in linear triatomic molecules. Chem Phys 2006. [DOI: 10.1016/j.chemphys.2006.05.023] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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37
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Perić M, Peyerimhoff SD, Buenker RJ. Use of the vibronic CI method in accurate calculations of the Renner-Teller effect. Mol Phys 2006. [DOI: 10.1080/00268978300101241] [Citation(s) in RCA: 49] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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38
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Osherov V, Poluyanov L, Voronin A. Quantum and semiclassical dynamics of quasilinear triatomic system near the adiabatic term intersection point of Σ+ = Π = Σ- type. Mol Phys 2006. [DOI: 10.1080/00268978900100711] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Affiliation(s)
- V.I. Osherov
- a Institute of Chemical Physics, Academy of Sciences , 142 432 Chernogolovka, Moscow , USSR
| | - L.V. Poluyanov
- a Institute of Chemical Physics, Academy of Sciences , 142 432 Chernogolovka, Moscow , USSR
| | - A.I. Voronin
- a Institute of Chemical Physics, Academy of Sciences , 142 432 Chernogolovka, Moscow , USSR
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Mishra S, Vallet V, Poluyanov LV, Domcke W. Spectroscopic effects of first-order relativistic vibronic coupling in linear triatomic molecules. J Chem Phys 2005; 123:124104. [PMID: 16392472 DOI: 10.1063/1.2018702] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023] Open
Abstract
It has recently been shown that there exists, in addition to the well-known nonrelativistic Renner-Teller coupling, a linear (that is, of the first order in the bending distortion) vibronic-coupling mechanism of relativistic (that is, spin-orbit) origin in 2II electronic states of linear molecules [L. V. Poluyanov and W. Domcke, Chem. Phys. 301, 111 (2004)]. The generic aspects of the relativistic linear vibronic-coupling mechanism have been analyzed in the present work by numerical calculations of the vibronic spectrum for appropriate models. The vibronic and spin-orbit parameters have been determined by accurate ab initio electronic-structure calculations for the X 2II states of a series of triatomic radicals and radical cations. It is shown for the example of GeCH that the relativistic linear vibronic-coupling mechanism provides a quantitative explanation of the pronounced perturbations in the vibronic spectrum of the X 2II state of GeCH, which previously have been termed "Sears resonances" [S.-G. He, H. Li, T. C. Smith, D. J. Clouthier, and A. J. Merer, J. Chem. Phys. 119, 10115 (2003)]. The X 2II vibronic spectra of the series BS2, CS2+, OCS+, and OBS illustrate the interplay of nonrelativistic and relativistic vibronic-coupling mechanisms in Renner-Teller systems.
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Affiliation(s)
- Sabyashachi Mishra
- Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany
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41
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Paterson MJ, Bearpark MJ, Robb MA, Blancafort L, Worth GA. Conical intersections: A perspective on the computation of spectroscopic Jahn–Teller parameters and the degenerate ‘intersection space’. Phys Chem Chem Phys 2005; 7:2100-15. [DOI: 10.1039/b416538a] [Citation(s) in RCA: 64] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
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44
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Bevilacqua G, Martinelli L, Pastori Parravicini G. Renner–Teller Interaction Matrices and Green's Function Formalism. ADVANCES IN QUANTUM CHEMISTRY 2003. [DOI: 10.1016/s0065-3276(03)44004-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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45
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Poluyanov L, Domcke W. Four-channel semiclassical S-matrix for a Σ+–Π–Σ−-type conical intersection in triatomic systems. Chem Phys 2002. [DOI: 10.1016/s0301-0104(02)00449-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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46
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Ding H, Pino T, Güthe F, Maier JP. Gas phase electronic spectrum of C3H in the visible. J Chem Phys 2001. [DOI: 10.1063/1.1404391] [Citation(s) in RCA: 39] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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47
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Theoretical investigation of vibronic-coupling and ultrafast internal-conversion dynamics in the acetylene cation. Chem Phys 2001. [DOI: 10.1016/s0301-0104(01)00448-7] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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48
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Bersuker IB. Modern aspects of the Jahn-Teller effect theory and applications to molecular problems. Chem Rev 2001; 101:1067-114. [PMID: 11709858 DOI: 10.1021/cr0004411] [Citation(s) in RCA: 426] [Impact Index Per Article: 18.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Affiliation(s)
- I B Bersuker
- Institute for Theoretical Chemistry, Department of Chemistry & Biochemistry, University of Texas at Austin, Austin, Texas 78712, USA.
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PERIĆ MILJENKO, MARIAN CHRISTELM, ENGELS BERND. Theoretical investigation of the Renner-Teller effect in Δ electronic states of tetra-atomic molecules. 1. Variational calculation of vibronic structure in the 11Δgstate of B2H2. Mol Phys 1999. [DOI: 10.1080/00268979909482873] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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50
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Perić M. Perturbative and variational handling of the Renner–Teller effect in Δ electronic states of triatomic molecules. Chem Phys Lett 1999. [DOI: 10.1016/s0009-2614(99)00008-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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