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Papp D, Tajti V, Avila G, Mátyus E, Czakó G. CH 4·F − revisited: full-dimensional ab initio potential energy surface and variational vibrational states. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2113565] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/15/2022]
Affiliation(s)
- Dóra Papp
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Szeged, Hungary
| | - Viktor Tajti
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Szeged, Hungary
| | - Gustavo Avila
- ELTE, Eötvös Loránd University, Institute of Chemistry, Budapest, Hungary
| | - Edit Mátyus
- ELTE, Eötvös Loránd University, Institute of Chemistry, Budapest, Hungary
| | - Gábor Czakó
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Szeged, Hungary
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2
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Wodraszka R, Carrington T. A rectangular collocation multi-configuration time-dependent Hartree (MCTDH) approach with time-independent points for calculations on general potential energy surfaces. J Chem Phys 2021; 154:114107. [PMID: 33752363 DOI: 10.1063/5.0046425] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We introduce a collocation-based multi-configuration time-dependent Hartree (MCTDH) method that uses more collocation points than basis functions. We call it the rectangular collocation MCTDH (RC-MCTDH) method. It does not require that the potential be a sum of products. RC-MCTDH has the important advantage that it makes it simple to use time-independent collocation points. When using time-independent points, it is necessary to evaluate the potential energy function only once and not repeatedly during an MCTDH calculation. It is inexpensive and straightforward to use RC-MCTDH with combined modes. Using more collocation points than basis functions enables one to reduce errors in energy levels without increasing the size of the single-particle function basis. On the contrary, whenever a discrete variable representation is used, the only way to reduce the quadrature error is to increase the basis size, which then also reduces the basis-set error. We demonstrate that with RC-MCTDH and time-independent points, it is possible to calculate accurate eigenenergies of CH3 and CH4.
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Affiliation(s)
- Robert Wodraszka
- Chemistry Department, Queen's University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Chemistry Department, Queen's University, Kingston, Ontario K7L 3N6, Canada
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Wodraszka R, Carrington T. A collocation-based multi-configuration time-dependent Hartree method using mode combination and improved relaxation. J Chem Phys 2020; 152:164117. [DOI: 10.1063/5.0006081] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Robert Wodraszka
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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Avila G, Papp D, Czakó G, Mátyus E. Exact quantum dynamics background of dispersion interactions: case study for CH 4·Ar in full (12) dimensions. Phys Chem Chem Phys 2020; 22:2792-2802. [PMID: 31957778 DOI: 10.1039/c9cp04426d] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A full-dimensional ab initio potential energy surface of spectroscopic quality is developed for the van-der-Waals complex of a methane molecule and an argon atom. Variational vibrational states are computed on this surface including all twelve (12) vibrational degrees of freedom of the methane-argon complex using the GENIUSH computer program and the Smolyak sparse grid method. The full-dimensional computations make it possible to study the fine details of the interaction and distortion effects and to make a direct assessment of the reduced-dimensionality models often used in the quantum dynamics study of weakly-bound complexes. A 12-dimensional (12D) vibrational computation including only a single harmonic oscillator basis function (9D) to describe the methane fragment (for which we use the ground-state effective structure as the reference structure) has a 0.40 cm-1 root-mean-square error (rms) with respect to the converged 12D bound-state excitation energies, which is less than half of the rms of the 3D model set up with the 〈r〉0 methane structure. Allowing 10 basis functions for the methane fragment in a 12D computation performs much better than the 3D models by reducing the rms of the bound state vibrational energies to 0.07 cm-1. The full-dimensional potential energy surface correctly describes the dissociation of the system, which together with further development of the variational (ro)vibrational methodology opens a route to the study of the role of dispersion forces in the excited methane vibrations and the energy transfer from the intra- to the intermolecular vibrational modes.
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Affiliation(s)
- Gustavo Avila
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary.
| | - Dóra Papp
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Rerrich Béla tér 1, Szeged H-6720, Hungary.
| | - Gábor Czakó
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Rerrich Béla tér 1, Szeged H-6720, Hungary.
| | - Edit Mátyus
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary.
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Avila G, Matyus E. Full-dimensional (12D) variational vibrational states of CH 4·F -: Interplay of anharmonicity and tunneling. J Chem Phys 2019; 151:154301. [PMID: 31640378 DOI: 10.1063/1.5124532] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The complex of a methane molecule and a fluoride anion represents a 12-dimensional (12D), four-well vibrational problem with multiple large-amplitude motions, which has challenged the quantum dynamics community for years. The present work reports vibrational band origins and tunneling splittings obtained in a full-dimensional variational vibrational computation using the GENIUSH program and the Smolyak quadrature scheme. The converged 12D vibrational band origins and tunneling splittings confirm complementary aspects of the earlier full- and reduced-dimensionality studies: (1) the tunneling splittings are smaller than 0.02 cm-1; (2) a single-well treatment is not sufficient (except perhaps the zero-point vibration) due to a significant anharmonicity over the wells; and thus, (3) a full-dimensional treatment appears to be necessary. The present computations extend to a higher energy range than earlier work, show that the tunneling splittings increase upon vibrational excitation of the complex, and indicate non-negligible "heavy-atom" tunneling.
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Affiliation(s)
- Gustavo Avila
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
| | - Edit Matyus
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
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Schäpers D, Manthe U. Vibronic coupling in the F·CH4 prereactive complex. J Chem Phys 2019; 151:104106. [DOI: 10.1063/1.5110246] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Daniela Schäpers
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, D-33615 Bielefeld, Germany
| | - Uwe Manthe
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, D-33615 Bielefeld, Germany
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Wang H, Liu X, Liu J. Accurate calculation of equilibrium reduced density matrix for the system-bath model: A multilayer multiconfiguration time-dependent Hartree approach and its comparison to a multi-electronic-state path integral molecular dynamics approach. CHINESE J CHEM PHYS 2018. [DOI: 10.1063/1674-0068/31/cjcp1805122] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Affiliation(s)
- Haobin Wang
- Department of Chemistry, University of Colorado Denver, Denver CO 80217-3364, USA
| | - Xinzijian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Jian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
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9
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Wodraszka R, Carrington T. A new collocation-based multi-configuration time-dependent Hartree (MCTDH) approach for solving the Schrödinger equation with a general potential energy surface. J Chem Phys 2018; 148:044115. [DOI: 10.1063/1.5018793] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Robert Wodraszka
- Department of Chemistry, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Department of Chemistry, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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Manthe U. Wavepacket dynamics and the multi-configurational time-dependent Hartree approach. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:253001. [PMID: 28430111 DOI: 10.1088/1361-648x/aa6e96] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Multi-configurational time-dependent Hartree (MCTDH) based approaches are efficient, accurate, and versatile methods for high-dimensional quantum dynamics simulations. Applications range from detailed investigations of polyatomic reaction processes in the gas phase to high-dimensional simulations studying the dynamics of condensed phase systems described by typical solid state physics model Hamiltonians. The present article presents an overview of the different areas of application and provides a comprehensive review of the underlying theory. The concepts and guiding ideas underlying the MCTDH approach and its multi-mode and multi-layer extensions are discussed in detail. The general structure of the equations of motion is highlighted. The representation of the Hamiltonian and the correlated discrete variable representation (CDVR), which provides an efficient multi-dimensional quadrature in MCTDH calculations, are discussed. Methods which facilitate the calculation of eigenstates, the evaluation of correlation functions, and the efficient representation of thermal ensembles in MCTDH calculations are described. Different schemes for the treatment of indistinguishable particles in MCTDH calculations and recent developments towards a unified multi-layer MCTDH theory for systems including bosons and fermions are discussed.
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Affiliation(s)
- Uwe Manthe
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, D-33615 Bielefeld, Germany
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Eisfeld W, Viel A. Vibronic eigenstates and the geometric phase effect in the2E″ state of NO3. J Chem Phys 2017; 146:034303. [DOI: 10.1063/1.4973983] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Xie C, Jiang B, Yang M, Guo H. State-to-State Mode Specificity in F + CHD3 → HF/DF + CD3/CHD2 Reaction. J Phys Chem A 2016; 120:6521-8. [DOI: 10.1021/acs.jpca.6b06450] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Changjian Xie
- Department of Chemistry
and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States
| | - Bin Jiang
- Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China
| | - Minghui Yang
- Key Laboratory of Magnetic Resonance in
Biological Systems, Wuhan Center for Magnetic Resonance, State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
| | - Hua Guo
- Department of Chemistry
and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States
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13
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Wodraszka R, Carrington T. Using a pruned, nondirect product basis in conjunction with the multi-configuration time-dependent Hartree (MCTDH) method. J Chem Phys 2016; 145:044110. [DOI: 10.1063/1.4959228] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Robert Wodraszka
- Department of Chemistry, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Department of Chemistry, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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Qi J, Song H, Yang M, Palma J, Manthe U, Guo H. Communication: Mode specific quantum dynamics of the F + CHD3 → HF + CD3 reaction. J Chem Phys 2016; 144:171101. [DOI: 10.1063/1.4948547] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Ji Qi
- Key Laboratory of Magnetic Resonance in Biological Systems, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, National Center for Magnetic Resonance in Wuhan, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
| | - Hongwei Song
- Key Laboratory of Magnetic Resonance in Biological Systems, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, National Center for Magnetic Resonance in Wuhan, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
| | - Minghui Yang
- Key Laboratory of Magnetic Resonance in Biological Systems, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, National Center for Magnetic Resonance in Wuhan, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
| | - Juliana Palma
- Departamento de Ciencia y Tecnología, Universidad Nacional de Quilmes, CONICET, Sáenz Peña 352, Bernal B1876BXD, Argentina
| | - Uwe Manthe
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, D-33615 Bielefeld, Germany
| | - Hua Guo
- Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA
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Wodraszka R, Manthe U. CH+5: Symmetry and the Entangled Rovibrational Quantum States of a Fluxional Molecule. J Phys Chem Lett 2015; 6:4229-4232. [PMID: 26538038 DOI: 10.1021/acs.jpclett.5b01869] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Protonated methane, CH5+, is the prototypical example of a fluxional molecular system. The almost unconstrained angular motion of its five hydrogen atoms results in dynamical phenomena not found in rigid or semirigid molecules. Here it is shown that standard concepts to describe rotational quantum states of molecules can not be applied to CH5+ or any other fluxional system of the type ABn or Bn with n > 4 due to fundamental symmetry reasons. Instead, the ro-vibrational states of CH5+ display a unique level scheme, which results from a complex entanglement of rotational and tunneling motions. A detailed analysis of the ro-vibrational quantum states of CH5+ based on full-dimensional quantum dynamics simulations is presented, and the effects of the Pauli principle are considered. The consequences for the interpretation of recent experimental results are highlighted
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Affiliation(s)
- Robert Wodraszka
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld , Universitätsstraβe 25, D-33615 Bielefeld, Germany
| | - Uwe Manthe
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld , Universitätsstraβe 25, D-33615 Bielefeld, Germany
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Abstract
Multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory is a rigorous and powerful method to simulate quantum dynamics in complex many-body systems. This approach extends the original MCTDH theory of Meyer, Manthe, and Cederbaum to include dynamically contracted layers in a recursive way, within which the equations of motion are determined from the Dirac-Frenkel variational principle. This paper presents the general derivation of the theory and analyzes the important features that make the ML-MCTDH method numerically efficient. Furthermore, we discuss the generalization of the theory to treat many-body identical particles (fermions or bosons) as well as calculating energy eigenstates via the improved relaxation method.
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Affiliation(s)
- Haobin Wang
- Department of Chemistry, University of Colorado Denver, Denver, Colorado 80217-3364, United States
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17
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Eisfeld W, Vieuxmaire O, Viel A. Full-dimensional diabatic potential energy surfaces including dissociation: The 2E″ state of NO3. J Chem Phys 2014; 140:224109. [DOI: 10.1063/1.4879655] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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18
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Otto F. Multi-layer Potfit: An accurate potential representation for efficient high-dimensional quantum dynamics. J Chem Phys 2014; 140:014106. [DOI: 10.1063/1.4856135] [Citation(s) in RCA: 47] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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19
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Westermann T, Kim JB, Weichman ML, Hock C, Yacovitch TI, Palma J, Neumark DM, Manthe U. Resonances in the Entrance Channel of the Elementary Chemical Reaction of Fluorine and Methane. Angew Chem Int Ed Engl 2013. [DOI: 10.1002/ange.201307822] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Westermann T, Kim JB, Weichman ML, Hock C, Yacovitch TI, Palma J, Neumark DM, Manthe U. Resonances in the entrance channel of the elementary chemical reaction of fluorine and methane. Angew Chem Int Ed Engl 2013; 53:1122-6. [PMID: 24307593 DOI: 10.1002/anie.201307822] [Citation(s) in RCA: 63] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2013] [Indexed: 11/08/2022]
Abstract
Extending the fully quantum-state-resolved description of elementary chemical reactions beyond three or four atom systems is a crucial issue in fundamental chemical research. Reactions of methane with F, Cl, H or O are key examples that have been studied prominently. In particular, reactive resonances and nonintuitive mode-selective chemistry have been reported in experimental studies for the F+CH4 →HF+CH3 reaction. By investigating this reaction using transition-state spectroscopy, this joint theoretical and experimental study provides a clear picture of resonances in the F+CH4 system. This picture is deduced from high-resolution slow electron velocity-map imaging (SEVI) spectra and accurate full-dimensional (12D) quantum dynamics simulations in the picosecond regime.
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Affiliation(s)
- Till Westermann
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld (Germany)
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Fábri C, Császár AG, Czakó G. Reduced-dimensional quantum computations for the rotational-vibrational dynamics of F(-)-CH4 and F(-)-CH2D2. J Phys Chem A 2013; 117:6975-83. [PMID: 23402210 DOI: 10.1021/jp312160n] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Variational rotational-vibrational quantum chemical computations are performed for the F(-)-CH4 and F(-)-CH2D2 anion complexes using several reduced-dimensional models in a curvilinear polyspherical coordinate system and utilizing an accurate ab initio potential energy surface (PES). The implementation of the models is made practical by using the general rovibrational code GENIUSH, which constructs the complicated form of the exact rovibrational kinetic energy operator in reduced and full dimensions in any user-specified coordinates and body-fixed frames. A one-dimensional CF stretch, 1D(RCF), a two-dimensional intermolecular bend, 2D(θ,φ), and a three-dimensional intermolecular, 3D(RCF,θ,φ), rigid methane model provide vibrational energies for the low-frequency, large-amplitude modes in good agreement with full-dimensional MCTDH results for F(-)-CH4. The 2D(θ,φ) and 3D(RCF,θ,φ) four-well computations, describing equally the four possible CH-F(-) bonds, show that the ground-state tunneling splitting is less than 0.01 cm(-1). For the hydrogen-bonded CH stretching fundamental a local-mode model is found to have almost spectroscopic accuracy, whereas a harmonic frequency analysis performs poorly. The 2D(θ,φ) and 3D(RCF,θ,φ) rotational-vibrational computations on the Td-symmetric four-well PES reveal that in most cases F(-)-CH4 behaves as a semirigid C3v symmetric top. For the degenerate intermolecular bending vibrational states substantial splittings of the rigid rotor levels are observed. For F(-)-CH2D2 the rotational levels guide the assignment of the vibrational states to either F(-)-H or F(-)-D connectivity.
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Affiliation(s)
- Csaba Fábri
- Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, Eötvös University, H-1518 Budapest 112, P.O. Box 32, Hungary
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Westermann T, Eisfeld W, Manthe U. Coupled potential energy surface for the F(2P) + CH4 → HF + CH3 entrance channel and quantum dynamics of the CH4 · F− photodetachment. J Chem Phys 2013; 139:014309. [DOI: 10.1063/1.4812251] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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23
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Wodraszka R, Manthe U. Iterative Diagonalization in the Multiconfigurational Time-Dependent Hartree Approach: Ro-vibrational Eigenstates. J Phys Chem A 2013; 117:7246-55. [DOI: 10.1021/jp401129t] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Robert Wodraszka
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstraße.
25, D-33615 Bielefeld, Germany
| | - Uwe Manthe
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstraße.
25, D-33615 Bielefeld, Germany
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