General Internal Motion of Molecules, Classical and Quantum-Mechanical Hamiltonian

Meyer, Rolf; Günthard, Hs. H.

doi:10.1063/1.1670272

Cited by 236 publications

(72 citation statements)

References 29 publications

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“…10,12 In the geometric algebra approach, it is also unnecessary to calculate the inverse of the inertia tensor to obtain the rotational measuring vectors, unlike in the approach where the rotational basis vectors are read off from the components of the rotational velocity. 13 Finally, as the reader has no doubt noticed, the current method applies straightforwardly to any system: there is no restriction in the number of particles involved.…”

Section: Comparison To Other Approachesmentioning

confidence: 99%

Vibration–rotation kinetic energy operators: A geometric algebra approach

Pesonen

2001

The Journal of Chemical Physics

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The elements of the reciprocal metric tensor g (q i q j ) , which appear in the exact internal kinetic energy operators of polyatomic molecules can, in principle, be written as the mass-weighted sum of the inner products of measuring vectors associated to the nuclei of the molecule. In the case of vibrational degrees of freedom, the measuring vectors are simply the gradients of the vibrational coordinates. It is more difficult to find these vectors for the rotational degrees of freedom, because the components of the total angular momentum operator are not conjugated to any rotational coordinates. However, by the methods of geometric algebra, the rotational measuring vectors are easily calculated for any geometrically defined body-frame, without any restrictions to the number of particles in the system. In order to show that the rotational measuring vectors produced by the present method agree with the known results, the general formulas are applied to the triatomic bond-z, and to the triatomic angle bisector frame. All the rotational measuring vectors are also explicitly derived for a new triatomic body frame defined in terms of two Jacobi vectors. As a final application, all the rotational measuring vectors are presented for a new N-atomic frame defined in terms of NϪ1 Jacobi vectors, and for a simple N-atomic frame defined in terms of N nuclear position vectors (Nϭ3,4,5,6, . . . ).

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Section: Comparison To Other Approachesmentioning

confidence: 99%

Vibration–rotation kinetic energy operators: A geometric algebra approach

Pesonen

2001

The Journal of Chemical Physics

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“…To confirm the assignment of the vibrational levels presented in Subsection III B 2 and to be able to calculate the Franck-Condon factors of the X + ←X transitions, model potentials for the X + 2 g, ( = 3/2, 1/2) electronic states were determined in a direct fit based on the solution of the Schrödinger equation using a discrete variable representation (DVR). 30,31 The model potential we used (see Ref. 32) consists of three contributions: a repulsive short-range contribution V short (R) that also contains the effect of chemical bonding, an attractive long-range contribution V attr (R), and a constant E diss that defines the origin of the energy scale:…”

Section: The CL + ( 1 D 2 ) + Cl − ( 1 S 0 ) Ion-pair Dissociation Thmentioning

confidence: 99%

The X+ 2Πg, A+ 2Πu, B+ 2Δu, and $\mathrm{a}^+\ ^4\Sigma _{\mathrm{u}}^-$a+Σu−4 electronic states of ${\rm Cl}_2^+$ Cl 2+ studied by high-resolution photoelectron spectroscopy

Merkt

2013

The Journal of Chemical Physics

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Photoelectron spectroscopic study of the E ⊗ e Jahn-Teller effect in the presence of a tunable spin-orbit Partially rotationally resolved pulsed-field-ionization zero-kinetic-energy photoelectron spectra of the three isotopomers ( 35 Cl 2 , 35 Cl 37 Cl, and 37 Cl 2 ) of Cl 2 have been recorded in the wavenumber ranges 92 500-96 500 cm −1 , corresponding to transitions to the low vibrational levels of the X + 2 g ( = 3/2, 1/2) ground state of Cl + 2 , and 106 750-115 500 cm −1 , where the a + 4 − u ← X 1 + g , A + 2 u ← X 1 + g , and B + 2 u ← X 1 + g band systems overlap with transitions to high vibrational levels (v + > 25) of the X + state. The observation of Franck-Condon-forbidden transitions to vibrational levels of the X + state of the cation with v + ≥ 25 is rationalized by a mechanism involving vertical excitation of predissociative Rydberg states of mixed singlet-triplet character with an A + ion core which are coupled to Rydberg states converging to high-v + levels of the X + state. The same mechanism is proposed to also be responsible for the observation of Cl + − Cl − ion pairs and quartet states in the photoionization of Cl 2 . The potential energy function of the X + state of Cl + 2 was determined in a direct fit to the experimental data. Transitions to vibrational levels of the A + 2 u, 3/2 and B + 2 u, 3/2 states of Cl + 2 could be identified using the results of a recent analysis of the strong perturbation between the A + 2 u, 3/2 and B + 2 u, 3/2 states of Cl + 2 observed in the A + − X + band system [Gharaibeh et al., J. Chem. Phys. 137, 194317 (2012)], and transitions to several vibrational levels of the upper spin-orbit component ( 2 u, 1/2 ) of the A + state were detected in the photoelectron spectrum of Cl + 2 . The a + 4 − u ← X 1 + g photoelectron band system, which is nominally forbidden by single-photon ionization from the ground state was also observed for the first time and its vibrational and spin-orbit structures were analyzed. The 4 − u state is split into two spin-orbit components with = 1/2 and = 3/2, separated by 37.5 cm −1 . The vibrational energy level structure of both components is regular, which indicates that the splitting results from the interaction with one or more distant ungerade = 1/2 or = 3/2 electronic states. © 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.

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“…In terms of internal displacement coordinates, q k , the quantum mechanical Hamiltonian that describes the vibrational excitations of a molecule takes the form [50,51] …”

Section: Hamiltonian Of Triatomic Xy Moleculesmentioning

confidence: 99%

A novel connection between algebraic spectroscopic parameters and force constants in the description of vibrational excitations of linear triatomic molecules

Sánchez-Castellanos

Lemus

Carvajal

et al. 2009

Journal of Molecular Spectroscopy

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A connection between an algebraic approach to the dynamics of triatomic molecules based on the U (2) × U (3) × U (2) Lie algebra and the traditional description in configuration space is presented. The connection is established in four steps. First, the molecular Hamiltonian is expanded in symmetrized local coordinates. Second, the Hamiltonian is transformed into an algebraic representation by introducing the realization of coordinates and momenta in terms of bosonic creation and annihilation operators of normal character. The third step is to perform a canonical transformation applied to the bosons associated with the stretching degrees of freedom in order to obtain a unified representation in a local scheme. Finally, an anharmonization procedure is applied to identify the U (2) × U (3) × U (2) dynamical algebra. The main advantage of the proposed approach is that it provides relations between the spectroscopic parameters and the molecular structure and force constants. As an application, the analysis of the vibrational excitations of CO 2 in its ground electronic state is considered. In this scheme, each stretching degree of freedom is identified as an interacting Morse oscillator, with an associated U (2) dynamical algebra, and the doubly degenerate bending degree of freedom is modelled with a U (3) dynamical algebra, obtaining as a final result a reasonable set of force constants.

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General Internal Motion of Molecules, Classical and Quantum-Mechanical Hamiltonian

Cited by 236 publications

References 29 publications

Vibration–rotation kinetic energy operators: A geometric algebra approach

Vibration–rotation kinetic energy operators: A geometric algebra approach

The X+ 2Πg, A+ 2Πu, B+ 2Δu, and $\mathrm{a}^+\ ^4\Sigma _{\mathrm{u}}^-$a+Σu−4 electronic states of ${\rm Cl}_2^+$ Cl 2+ studied by high-resolution photoelectron spectroscopy

A novel connection between algebraic spectroscopic parameters and force constants in the description of vibrational excitations of linear triatomic molecules

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