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Mellor TM. Molecular frames for a symmetry-adapted rotational basis set. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2118638] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
Affiliation(s)
- Thomas M. Mellor
- Department of Physics and Astronomy, University College London, London, UK
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Abstract
Umbrella inversion of ammonia is a prototypical example of large-amplitude vibrational motion, described with a symmetric double-well potential. The transition state of the latter corresponds to a planar (D3h) molecular geometry, whereas the two equilibrium configurations are equivalent (C3v) pyramidal structures, with the nitrogen atom being either 'above' or 'below' the plane of the hydrogen atoms. As commonly understood, inversion motion of ammonia corresponds to the coherent, anharmonic, vibrational motion of the molecule, which shuttles back and forth between the two potential wells; that is, oscillation of the nitrogen atom from one side of the H3 plane to the other, via coherent tunneling. However, this intuitively appealing view of umbrella inversion results from a reduced description of the dynamics, which includes only the inversion vibrational coordinate and fully neglects all the other molecular degrees of freedom. As such, this textbook picture of inversion motion ignores the fact that the two equilibrium structures of ammonia are superimposable, and can only be distinguished by labelling the identical hydrogen nuclei. A correct description of umbrella inversion, which incorporates nuclear permutations, requires the inclusion of other molecular modes. Indeed, it is well known that the quantum symmetrization postulate engenders entanglement between ammonia's nuclear-spin, inversion, and rotation. Using the explicit expressions of the corresponding zeroth-order eigenstates, we clearly show that the inversion density of any multilevel wavepacket of ammonia, including the case of perfectly aligned molecules, is symmetrically distributed between the two potential wells, at all times. This follows from a rigorous demonstration based on the evaluation of the expectation values of the inversion coordinate or equivalent projection operators. However, provided that these wavepackets involve inversion-rotation levels with opposite parity, the inversion density may exhibit dynamical spatial localization. In the latter case, the space-fixed inversion density or, equivalently, the expectation values of the projections of the inversion coordinate on the space-fixed axes, may oscillate between opposite directions in the space-fixed frame. Nevertheless, in all cases, localization of ammonia in a single potential well is impossible, even partially or transiently. This is equivalent to saying that the nitrogen atom has the same probability (one-half) to be on either side of the H3 plane, for any wavepacket of the molecule and at all times-a conclusion which is in perfect accord with the principle of the indistinguishability of identical particles (nuclei).
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Affiliation(s)
- Foudhil Bouakline
- Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam-Golm, Germany.
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Sarka J, Poirier B, Szalay V, Császár AG. On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: Considerations of symmetry, accuracy, and simplicity. II. Case studies for H 2O isotopologues, H 3+, O 3, and NH 3. SPECTROCHIMICA ACTA. PART A, MOLECULAR AND BIOMOLECULAR SPECTROSCOPY 2021; 250:119164. [PMID: 33493950 DOI: 10.1016/j.saa.2020.119164] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2020] [Accepted: 10/26/2020] [Indexed: 06/12/2023]
Abstract
For centuries, it has been known that vibrational and rotational degrees of freedom are in general not separable. Nevertheless, surprisingly little is known about the best strategies for approximately separating these degrees of freedom in practice-even in the case of semirigid molecules, where the separation is most meaningful. There is also some confusion in the literature about the proper way to quantify the magnitude of the Coriolis (i.e., rotation-vibration) coupling in rovibrational Hamiltonians or its effect on the rovibrational eigenenergies. In this study, a vibrational-coordinate-independent metric is proposed to quantify the magnitude of the Coriolis contribution to the rovibrational Hamiltonian. The impact of Coriolis coupling on the rovibrational eigenenergies is computed numerically exactly, using both full and various truncated Hamiltonians. The role played by the choice of the vibrational coordinate system-and especially by the choice of "embedding" or body-fixed frame-is examined extensively, both numerically and analytically. This investigation targets several molecular prototypes, all of which serve as important benchmarks for the high-resolution spectroscopic community. Most of these are triatomic molecules, including water (H216O), its deuterated isotopologues (D216O and HD16O), H3+, and ozone (16O3), but the tetratomic ammonia molecule (14NH3) is also investigated. These studies provide important insight into the nature of Coriolis coupling under various circumstances. The findings of this study also have significant practical ramifications, vis-à-vis the use of simplifying numerical approximation techniques in nuclear-motion computations.
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Affiliation(s)
- János Sarka
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX 79409, USA.
| | - Bill Poirier
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX 79409, USA.
| | - Viktor Szalay
- Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Attila G Császár
- MTA-ELTE Complex Chemical Systems Research Group and Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University, H-1117 Budapest, Pázmány Péter sétány 1/A, Hungary.
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Sarka J, Poirier B, Szalay V, Császár AG. On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: considerations of symmetry, accuracy, and simplicity. Sci Rep 2020; 10:4872. [PMID: 32184431 PMCID: PMC7078231 DOI: 10.1038/s41598-020-60971-x] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2019] [Accepted: 02/19/2020] [Indexed: 11/20/2022] Open
Abstract
The rotation-vibration (Coriolis) coupling contribution to variationally computed rovibrational energy levels is investigated, employing triatomic AB[Formula: see text] molecules as models. In particular, calculations are performed for H[Formula: see text][Formula: see text]O, across a range of vibrational and rotational excitations, both with and without the Coriolis contribution. A variety of different embedding choices are considered, together with a hierarchy of increasingly severe approximations culminating in a generalized version of the so-called "centrifugal sudden" method. Several surprising and remarkable conclusions are found, including that the Eckart embedding is not the best embedding choice.
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Affiliation(s)
- János Sarka
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas, 79409, USA
| | - Bill Poirier
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas, 79409, USA.
| | - Viktor Szalay
- Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, H-1525, Budapest, Hungary
| | - Attila G Császár
- MTA-ELTE Complex Chemical Systems Research Group and Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University, H-1117, Budapest, Pázmány Péter sétány 1/A, Hungary.
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Krasnoshchekov SV, Schutski RS, Craig NC, Sibaev M, Crittenden DL. Comparing the accuracy of perturbative and variational calculations for predicting fundamental vibrational frequencies of dihalomethanes. J Chem Phys 2018; 148:084102. [DOI: 10.1063/1.5020295] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sergey V. Krasnoshchekov
- Department of Chemistry, Lomonosov Moscow State University, Leninskiye Gory 1-3, Moscow 119991, Russian Federation
| | | | - Norman C. Craig
- Department of Chemistry and Biochemistry, Oberlin College, Oberlin, Ohio 44074, USA
| | - Marat Sibaev
- Research School of Chemistry, Australian National University, Canberra, ACT 2601, Australia
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Szalay V. Eckart ro-vibrational Hamiltonians via the gateway Hamilton operator: Theory and practice. J Chem Phys 2017; 146:124107. [DOI: 10.1063/1.4978686] [Citation(s) in RCA: 5] [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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Abstract
It is shown that vibrational displacements satisfying the Eckart-Sayvetz conditions can be constructed by projection of unconstrained displacements. This result has a number of interesting direct and indirect ramifications: (i) The normal coordinates corresponding to an electronic state or an isotopologue of a molecule are transformed to those of another state or isotopologue by a linear and, in general, non-orthogonal transformation. (ii) Novel interpretation of axis switching. (iii) One may enhance the separation of rotational-large-amplitude internal motions and the vibrational motions beyond that offered by the standard use of the Eckart-Sayvetz conditions. (iv) The rotational-vibrational Hamiltonian given in terms of curvilinear internal coordinates may be derived with elementary mathematical tools while taking into account the Eckart conditions with or without enhancement.
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Affiliation(s)
- Viktor Szalay
- Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
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Fábri C, Mátyus E, Császár AG. Numerically constructed internal-coordinate Hamiltonian with Eckart embedding and its application for the inversion tunneling of ammonia. SPECTROCHIMICA ACTA. PART A, MOLECULAR AND BIOMOLECULAR SPECTROSCOPY 2014; 119:84-89. [PMID: 23702049 DOI: 10.1016/j.saa.2013.03.090] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2013] [Revised: 03/26/2013] [Accepted: 03/28/2013] [Indexed: 06/02/2023]
Abstract
It is shown that the use of an Eckart-frame embedding with a kinetic energy operator expressed in curvilinear internal coordinates becomes feasible and straightforward to implement for arbitrary molecular compositions and internal coordinates if the operator is defined numerically over a (discrete variable representation) grid. The algorithm proposed utilizes the transformation method of Dymarsky and Kudin to maintain the rotational Eckart condition. In order to demonstrate the applicability and flexibility of our approach the non-rigid ammonia molecule is considered and the corresponding rotational-vibrational energy levels and wave functions are computed using kinetic energy operators with three different embeddings. Two of them fulfill the Eckart conditions corresponding to a trigonal pyramidal (C3v) and a trigonal planar (D3h) reference structure and the third one is a non-Eckart frame. The computed energy levels are, of course, identical, and the structure of the three different wave-function representations are analyzed in terms of the rigid rotor functions for a symmetric top. The possible advantages of one frame representation over another are discussed concerning the interpretation of the rovibrational states in terms of the traditional rigid rotor labels.
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Affiliation(s)
- Csaba Fábri
- Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, Eötvös University, P.O. Box 32, H-1518 Budapest 112, Hungary
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Wang XG, Carrington T. Computing rovibrational levels of methane with curvilinear internal vibrational coordinates and an Eckart frame. J Chem Phys 2013; 138:104106. [DOI: 10.1063/1.4793474] [Citation(s) in RCA: 66] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Ramesh SG, Sibert EL. A study of the vibrations of fluoroform with a sixth order nine-dimensional potential: a combined perturbative-variational approach. Mol Phys 2005. [DOI: 10.1080/00268970512331316247] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Ramesh SG, Sibert EL. Combined perturbative-variational investigation of the vibrations of CHBr3 and CDBr3. J Chem Phys 2004; 120:11011-25. [PMID: 15268131 DOI: 10.1063/1.1738643] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
A full dimensional vibrational treatment of CHBr(3) and CDBr(3) using Van Vleck perturbation theory followed by a variational calculation is presented. The calculation of a force field, and its adjustment for better match with experiment, is discussed. The computed eigenstates and spectral features are compared to experiment. Changes in intensities of the nu(1) and 2nu(4) bands upon simple alterations of the dipole moment expansion are described.
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Affiliation(s)
- Sai G Ramesh
- Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
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Meremianin AV. Body frames in the separation of collective angles in quantum N-body problems. J Chem Phys 2004; 120:7861-76. [PMID: 15267701 DOI: 10.1063/1.1687672] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The application of the concept of body-fixed reference frames, proposed by C. Eckart [Phys. Rev. 47, 552 (1935)], to the problem of the separation of three collective angles in quantum N-body problems is analyzed based on the technique recently developed by Meremianin and Briggs [Phys. Rep. 384, 121 (2003)]. Special attention is paid to the body frame defined by the "second Eckart condition" which minimizes vibro-rotational couplings near the equilibrium position. The important case of the Eckart frame for three-body systems is considered in detail. The connection of the basis vectors of the Eckart frame with Jacobi vectors is derived. All results of this work are valid for an arbitrary choice of internal (body-frame) coordinates.
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Affiliation(s)
- A V Meremianin
- Theoretische Quantendynamik, Fakultat fur Physik, University of Freiburg, Freiburg, Germany.
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Wei H. An Eckart-frame kinetic energy operator for tetra-atomic planar molecules. J Chem Phys 2003. [DOI: 10.1063/1.1562625] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
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Gatti F, Muñoz C, Iung C. A general expression of the exact kinetic energy operator in polyspherical coordinates. J Chem Phys 2001. [DOI: 10.1063/1.1361069] [Citation(s) in RCA: 71] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Wei H, Carrington Jr. T. An exact Eckart-embedded kinetic energy operator in Radau coordinates for triatomic molecules. Chem Phys Lett 1998. [DOI: 10.1016/s0009-2614(98)00186-9] [Citation(s) in RCA: 47] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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Wei H, Carrington T. The triatomic Eckart-frame kinetic energy operator in bond coordinates. J Chem Phys 1997. [DOI: 10.1063/1.475291] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
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Wei H, Carrington T. Explicit expressions for triatomic Eckart frames in Jacobi, Radau, and bond coordinates. J Chem Phys 1997. [DOI: 10.1063/1.474639] [Citation(s) in RCA: 46] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
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