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Lepetit B. Computation and analysis of bound vibrational spectra of the neon tetramer using row orthonormal hyperspherical coordinates. J Chem Phys 2020; 153:104302. [PMID: 32933308 DOI: 10.1063/5.0022655] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
This paper presents the first implementation of the row-orthonormal hyperspherical coordinate formalism for the computation of the vibrational spectrum of a tetratomic system. The wavefunction of Ne4 is expanded on a large basis set of hyperspherical harmonics generated numerically. This method not only provides spectra with reasonable accuracy, but also gives physical insight into the vibrational dynamics of the system. The characteristics of the spectra are related to the symmetry and localization of the wavefunction in configuration space.
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Affiliation(s)
- Bruno Lepetit
- Laboratoire Collisions, Agrégats, Réactivité, UMR 5589, CNRS-Université Paul Sabatier Toulouse 3, IRSAMC, 31062 Toulouse Cedex 9, France
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2
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Oka Y, Yanao T, Koon WS. Roles of dynamical symmetry breaking in driving oblate-prolate transitions of atomic clusters. J Chem Phys 2015; 142:134105. [PMID: 25854226 DOI: 10.1063/1.4915928] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper explores the driving mechanisms for structural transitions of atomic clusters between oblate and prolate isomers. We employ the hyperspherical coordinates to investigate structural dynamics of a seven-atom cluster at a coarse-grained level in terms of the dynamics of three gyration radii and three principal axes, which characterize overall mass distributions of the cluster. Dynamics of gyration radii is governed by two kinds of forces. One is the potential force originating from the interactions between atoms. The other is the dynamical forces called the internal centrifugal forces, which originate from twisting and shearing motions of the system. The internal centrifugal force arising from twisting motions has an effect of breaking the symmetry between two gyration radii. As a result, in an oblate isomer, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two largest gyration radii is crucial in triggering structural transitions into prolate isomers. In a prolate isomer, on the other hand, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two smallest gyration radii is crucial in triggering structural transitions into oblate isomers. Activation of a twisting motion that switches the movement patterns of three principal axes is also important for the onset of structural transitions between oblate and prolate isomers. Based on these trigger mechanisms, we finally show that selective activations of specific gyration radii and twisting motions, depending on the isomer of the cluster, can effectively induce structural transitions of the cluster. The results presented here could provide further insights into the control of molecular reactions.
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Affiliation(s)
- Yurie Oka
- Department of Applied Mechanics and Aerospace Engineering, Waseda University, Tokyo 169-8555, Japan
| | - Tomohiro Yanao
- Department of Applied Mechanics and Aerospace Engineering, Waseda University, Tokyo 169-8555, Japan
| | - Wang Sang Koon
- Control and Dynamical Systems, California Institute of Technology, Pasadena, California 91125, USA
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3
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Kalus R, Karlický F, Lepetit B, Paidarová I, Gadea FX. Photoabsorption spectrum of helium trimer cation--theoretical modeling. J Chem Phys 2013; 139:204310. [PMID: 24289357 DOI: 10.1063/1.4832736] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The photoabsorption spectrum of He3(+) is calculated for two semiempirical models of intracluster interactions and compared with available experimental data reported in the middle UV range [H. Haberland and B. von Issendorff, J. Chem. Phys. 102, 8773 (1995)]. Nuclear delocalization effects are investigated via several approaches comprising quantum samplings using either exact or approximate (harmonic) nuclear wavefunctions, as well as classical samplings based on the Monte Carlo methodology. Good agreement with the experiment is achieved for the model by Knowles et al., [Mol. Phys. 85, 243 (1995); Mol. Phys. 87, 827 (1996)] whereas the model by Calvo et al., [J. Chem. Phys. 135, 124308 (2011)] exhibits non-negligible deviations from the experiment. Predictions of far UV absorption spectrum of He3(+), for which no experimental data are presently available, are reported for both models and compared to each other as well as to the photoabsorption spectrum of He2(+). A simple semiempirical point-charge approximation for calculating transition probabilities is shown to perform well for He3(+).
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Affiliation(s)
- René Kalus
- Centre of Excellence IT4Innovations and Department of Applied Mathematics, VSB-Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic
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4
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Marquardt R. Theoretical methods for ultrafast spectroscopy. Chemphyschem 2013; 14:1350-61. [PMID: 23606322 DOI: 10.1002/cphc.201201096] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2012] [Indexed: 11/07/2022]
Abstract
Time-resolved spectroscopy in the femtosecond and attosecond time domain is a tool to unravel the dynamics of nuclear and electronic motion in molecular systems. Theoretical insight into the underlying physical processes is ideally gained by solving the time-dependent Schrödinger equation. In this work, methods currently used to solve this equation are reviewed in a compact presentation. These methods involve numerical representations of wavefunctions and operators, the calculation of time evolution operators, the setting up of the Hamiltonian operators and the types of coordinates to be used hereto. The advantages and disadvantages of some methods are discussed.
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Affiliation(s)
- Roberto Marquardt
- Laboratoire de Chimie Quantique, Institut de Chimie, UMR 7177 CNRS/UdS, Université de Strasbourg, 4, rue Blaise Pascal-CS90032, 67081 Strasbourg-Cedex, France.
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Karlický F, Lepetit B, Kalus R, Gadéa FX. Vibrational spectrum of Ar3(+) and relative importance of linear and perpendicular isomers in its photodissociation. J Chem Phys 2011; 134:084305. [PMID: 21361537 DOI: 10.1063/1.3555275] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The photodissociation dynamics of the argon ionized trimer Ar(3)(+) is revisited in the light of recent experimental results of Lepère et al. [J. Chem. Phys. 134, 194301 (2009)], which show that the fragment with little kinetic energy is always a neutral one, thus the available energy is shared by a neutral and ionic fragments as in Ar(2)(+). We show that these results can be interpreted as the photodissociation of the linear isomer of the system. We perform a 3D quantum computation of the vibrational spectrum of the system and study the relative populations of the linear (trimer-core) and perpendicular (dimer-core) isomers. We then show that the charge initially located on the central atom in the ground electronic state of the linear isomer migrates toward the extreme ones in the photoexcitation process such that photodissociation of the linear isomer produces a neutral central atom at rest in agreement with measured product state distributions.
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Affiliation(s)
- Frantisek Karlický
- Regional Centre of Advanced Technologies and Materials, Department of Physical Chemistry, Faculty of Science, Palacký University, Olomouc, Czech Republic.
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Kuppermann A. Reactive scattering with row-orthonormal hyperspherical coordinates. 4. Four-dimensional-space Wigner rotation function for pentaatomic systems. Phys Chem Chem Phys 2011; 13:8259-68. [DOI: 10.1039/c0cp02907f] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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7
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Wang D, Kuppermann A. Analytical Derivation of Row-Orthonormal Hyperspherical Harmonics for Triatomic Systems. J Phys Chem A 2009; 113:15384-410. [DOI: 10.1021/jp906473n] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Desheng Wang
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
| | - Aron Kuppermann
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
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Kuppermann A. Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 3. Hamiltonian and Transformation Properties for Pentaatomic Systems. J Phys Chem A 2009; 113:4518-33. [DOI: 10.1021/jp811171p] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Aron Kuppermann
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
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9
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Yanao T, Koon WS, Marsden JE. Intramolecular energy transfer and the driving mechanisms for large-amplitude collective motions of clusters. J Chem Phys 2009; 130:144111. [DOI: 10.1063/1.3098141] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Velilla L, Lepetit B, Aguado A, Beswick JA, Paniagua M. The H(3) (+) rovibrational spectrum revisited with a global electronic potential energy surface. J Chem Phys 2009; 129:084307. [PMID: 19044823 DOI: 10.1063/1.2973629] [Citation(s) in RCA: 79] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
In this paper, we have computed the rovibrational spectrum of the H(3) (+) molecule using a new global potential energy surface, invariant under all permutations of the nuclei, that includes the long range electrostatic interactions analytically. The energy levels are obtained by a variational calculation using hyperspherical coordinates. From the comparison with available experimental results for low lying levels, we conclude that our accuracy is of the order of 0.1 cm(-1) for states localized in the vicinity of equilateral triangular configurations of the nuclei, and changes to the order of 1 cm(-1) when the system is distorted away from equilateral configurations. Full rovibrational spectra up to the H(+)+H(2) dissociation energy limit have been computed. The statistical properties of this spectrum (nearest neighbor distribution and spectral rigidity) show the quantum signature of classical chaos and are consistent with random matrix theory. On the other hand, the correlation function, even when convoluted with a smoothing function, exhibits oscillations which are not described by random matrix theory. We discuss a possible similarity between these oscillations and the ones observed experimentally.
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Affiliation(s)
- Luis Velilla
- Departamento de Química Física, Facultad de Ciencias C-XIV, Universidad Autónoma de Madrid, 28049, Madrid, Spain
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11
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Karlický F, Lepetit B, Kalus R, Paidarová I, Gadéa FX. Modeling of HeN+ clusters. II. Calculation of He3+ vibrational spectrum. J Chem Phys 2008; 128:124303. [PMID: 18376915 DOI: 10.1063/1.2841019] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We have computed the vibrational spectrum of the helium ionized trimer He(3)(+) using three different potential energy surfaces [D. T. Chang and G. L. Gellene, J. Chem. Phys. 119, 4694 (2003); E. Scifoni et al., ibid. 125, 164304 (2006); I. Paidarova et al., Chem. Phys. 342, 64 (2007)]. Differences in the details of these potential energy surfaces induce discrepancies between bound state energies of the order of 0.01 eV. The effects of the geometric phase induced by the conical intersection between the ground electronic potential energy surface and the first excited one are studied by computing vibrational spectra with and without this phase. The six lowest vibrational bound states are negligibly affected by the geometric phase. Indeed, they correspond to wavefunctions localized in the vicinity of the linear symmetric configurations and can be assigned well defined vibrational quantum numbers. On the other hand, higher excited states are delocalized, cannot be assigned definite vibrational quantum numbers, and the geometric phase shifts their energies by approximately 0.005 eV.
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Affiliation(s)
- Frantisek Karlický
- Department of Physics, Faculty of Science, University of Ostrava, 30 dubna 22, 701 03 Ostrava, Czech Republic
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12
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Karlický F, Lepetit B, Kalus R, Gadéa FX. Calculation of argon trimer rovibrational spectrum. J Chem Phys 2007; 126:174305. [PMID: 17492861 DOI: 10.1063/1.2721564] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
Rovibrational spectra of Ar3 are computed for total angular momenta up to J=6 using row-orthonormal hyperspherical coordinates and an expansion of the wave function on hyperspherical harmonics. The sensitivity of the spectra to the two-body potential and to the three-body corrections is analyzed. First, the best available semiempirical pair potential (HFDID1) is compared with our recent ab initio two-body potential. The ab initio vibrational energies are typically 1-2 cm-1 higher than the semiempirical ones, which is related to the slightly larger dissociation energy of the semiempirical potential. Then, the Axilrod-Teller asymptotic expansion of the three-body correction is compared with our newly developed ab initio three-body potential. The difference is found smaller than 0.3 cm-1. In addition, we define approximate quantum numbers to describe the vibration and rotation of the system. The vibration is represented by a hyper-radial mode and a two-degree-of-freedom hyperangular mode, including a vibrational angular momentum defined in an Eckart frame. The rotation is described by the total angular momentum quantum number, its projection on the axis perpendicular to the molecular plane, and a hyperangular internal momentum quantum number, related to the vibrational angular momentum by a transformation between Eckart and principal-axes-of-inertia frames. These quantum numbers provide a qualitative understanding of the spectra and, in particular, of the impact of the nuclear permutational symmetry of the system (bosonic with zero nuclear spin). Rotational constants are extracted from the spectra and are shown to be accurate only for the ground hyperangular mode.
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Affiliation(s)
- Frantisek Karlický
- Department of Physics, University of Ostrava, 30 dubna 22, 701 03 Ostrava, Czech Republic
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Yanao T, Koon WS, Marsden JE, Kevrekidis IG. Gyration-radius dynamics in structural transitions of atomic clusters. J Chem Phys 2007; 126:124102. [PMID: 17411103 DOI: 10.1063/1.2710272] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper is concerned with the structural transition dynamics of the six-atom Morse cluster with zero total angular momentum, which serves as an illustrative example of the general reaction dynamics of isolated polyatomic molecules. It develops a methodology that highlights the interplay between the effects of the potential energy topography and those of the intrinsic geometry of the molecular internal space. The method focuses on the dynamics of three coarse variables, the molecular gyration radii. By using the framework of geometric mechanics and hyperspherical coordinates, the internal motions of a molecule are described in terms of these three gyration radii and hyperangular modes. The gyration radii serve as slow collective variables, while the remaining hyperangular modes serve as rapidly oscillating "bath" modes. Internal equations of motion reveal that the gyration radii are subject to two different kinds of forces: One is the ordinary force that originates from the potential energy function of the system, while the other is an internal centrifugal force. The latter originates from the dynamical coupling of the gyration radii with the hyperangular modes. The effects of these two forces often counteract each other: The potential force generally works to keep the internal mass distribution of the system compact and symmetric, while the internal centrifugal force works to inflate and elongate it. Averaged fields of these two forces are calculated numerically along a reaction path for the structural transition of the molecule in the three-dimensional space of gyration radii. By integrating the sum of these two force fields along the reaction path, an effective energy curve is deduced, which quantifies the gross work necessary for the system to change its mass distribution along the reaction path. This effective energy curve elucidates the energy-dependent switching of the structural preference between symmetric and asymmetric conformations. The present methodology should be of wide use for the systematic reduction of dimensionality as well as for the identification of kinematic barriers associated with the rearrangement of mass distribution in a variety of molecular reaction dynamics in vacuum.
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Affiliation(s)
- Tomohiro Yanao
- Control and Dynamical Systems, MC 107-81, California Institute of Technology, Pasadena, California 91125, USA.
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Lombardi A, Palazzetti F, Peroncelli L, Grossi G, Aquilanti V, Sevryuk MB. Few-body quantum and many-body classical hyperspherical approaches to reactions and to cluster dynamics. Theor Chem Acc 2006. [DOI: 10.1007/s00214-006-0195-0] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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15
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Lepetit B, Wang D, Kuppermann A. Numerical generation of hyperspherical harmonics for tetra-atomic systems. J Chem Phys 2006; 125:133505. [PMID: 17029488 DOI: 10.1063/1.2218515] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.
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Affiliation(s)
- Bruno Lepetit
- Laboratoire Collisions, Agrégats, Réactivité, UMR 5589, CNRS, IRSAMC, Université Paul Sabatier Toulouse 3, 31062 Toulouse Cedex 9, France.
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16
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Kuppermann A. Incorporating the Geometric Phase Effect in Triatomic and Tetraatomic Hyperspherical Harmonics. J Phys Chem A 2006; 110:809-16. [PMID: 16405357 DOI: 10.1021/jp054597m] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Hyperspherical harmonics in the democratic row-orthonormal hyperspherical coordinates are very appropriate basis sets for performing reactive scattering calculations for triatomic and tetraatomic systems. The mathematical conditions for incorporating the geometric phase effect in these harmonics are given. These conditions are implemented for triatomic systems, and their explicit analytical expressions in terms of Jacobi polynomials, in both the absence and presence of the geometric phase effect, are given.
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Affiliation(s)
- Aron Kuppermann
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA.
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17
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Teramoto H, Takatsuka K. Dynamical and statistical effects of the intrinsic curvature of internal space of molecules. J Chem Phys 2005; 122:074101. [PMID: 15743215 DOI: 10.1063/1.1850467] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The Hamilton dynamics of a molecule in a translationally and/or rotationally symmetric field is kept rigorously constrained in its phase space. The relevant dynamical laws should therefore be extracted from these constrained motions. An internal space that is induced by a projection of such a limited phase space onto configuration space is an intrinsically curved space even for a system of zero total angular momentum. In this paper we discuss the general effects of this curvedness on dynamics and structures of molecules in such a manner that is invariant with respect to the selection of coordinates. It is shown that the regular coordinate originally defined by Riemann is particularly useful to expose the curvature correction to the dynamics and statistical properties of molecules. These effects are significant both qualitatively and quantitatively and are studied in two aspects. One is the direct effect on dynamics: A trajectory receives a Lorentz-like force from the curved space as though it was placed in a magnetic field. The well-known problem of the trapping phenomenon at the transition state is analyzed from this point of view. By showing that the trapping force is explicitly described in terms of the curvature of the internal space, we clarify that the physical origin of the trapped motion is indeed originated from the curvature of the internal space and hence is not dependent of the selection of coordinate system. The other aspect is the effect of phase space volume arising from the curvedness: We formulate a general expression of the curvature correction of the classical density of states and extract its physical significance in the molecular geometry along with reaction rate in terms of the scalar curvature and volume loss (gain) due to the curvature. The transition state theory is reformulated from this point of view and it is applied to the structural transition of linear chain molecules in the so-called dihedral angle model. It is shown that the curvature effect becomes large roughly linearly with the size of molecule.
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Affiliation(s)
- Hiroshi Teramoto
- Department of Basic Science, Graduate School of Arts and Science, University of Tokyo, Komaba, 153-8902 Tokyo, Japan
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18
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19
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Affiliation(s)
- Desheng Wang
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
| | - Aron Kuppermann
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
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Saito K. Flux Correlation Approach to Thermal Reactions and Recombination Rates. BULLETIN OF THE CHEMICAL SOCIETY OF JAPAN 2003. [DOI: 10.1246/bcsj.76.271] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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21
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AQUILANTI VINCENZO, BEDDONI ANDREA, CAVALLI SIMONETTA, LOMBARDI ANDREA, LITTLEJOHN ROBERT. Collective hyperspherical coordinates for polyatomic molecules and clusters. Mol Phys 2000. [DOI: 10.1080/00268970009483380] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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22
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Blume D, Greene CH, Esry BD. Comparative study of He3, Ne3, and Ar3 using hyperspherical coordinates. J Chem Phys 2000. [DOI: 10.1063/1.482027] [Citation(s) in RCA: 65] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- D. Blume
- Department of Physics and JILA, University of Colorado, Boulder, Colorado 80309-0440
| | - Chris H. Greene
- Department of Physics and JILA, University of Colorado, Boulder, Colorado 80309-0440
| | - B. D. Esry
- Department of Physics, Kansas State University, Manhattan, Kansas 66506
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23
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Parker GA, Keil M, Morrison MA, Crocchianti S. Quantum reactive scattering in three dimensions: Using tangent-sphere coordinates to smoothly transform from hyperspherical to Jacobi regions. J Chem Phys 2000. [DOI: 10.1063/1.481876] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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24
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Miller WH. “Direct” and “Correct” Calculation of Canonical and Microcanonical Rate Constants for Chemical Reactions. J Phys Chem A 1998. [DOI: 10.1021/jp973208o] [Citation(s) in RCA: 167] [Impact Index Per Article: 6.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- William H. Miller
- Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California 94720
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25
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Kuppermann A. Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 2. Transformation Properties and Hamiltonian for Tetraatomic Systems. J Phys Chem A 1997. [DOI: 10.1021/jp9708207] [Citation(s) in RCA: 46] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Aron Kuppermann
- Arthur Amos Noyes Laboratory of Chemical Physics, Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
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