Symmetric Top Research Articles

Rotational properties of two interacting cold polar molecules: Linear, symmetric, and asymmetric tops

We examine the potential-energy curves and polarization of the dipole moments of two static polar molecules under the influence of an external dc electric field and their anisotropic dipole-dipole interaction. We model the molecules as quantum rigid rotors to take their rotational degrees of freedom into account and consider a selection of linear, symmetric, and asymmetric top molecules. We provide a comprehensive examination of the energy curves and polarization of the dipoles for varying intermolecular separation and direction of the electric field and find that the properties of the molecules depend strongly on the field's direction at short separations, showing the importance of accounting for molecular rotation. The latter provides insight into the possible effects of accounting for rotational degrees of freedom in molecular dipolar gases. Published by the American Physical Society 2024

Euler–Poisson equations of a dancing spinning top, integrability and examples of analytical solutions

Equations of a rotating body with one point constrained to move freely on a plane (dancing top) are deduced from the Lagrangian variational problem. They formally look like the Euler–Poisson equations of a heavy body with fixed point, immersed in a fictitious gravity field. Using this analogy, we have found examples of analytical solutions for the case of a heavy symmetrical dancing top. They describe the motions with center of mass keeping its height fixed above the supporting plane. General solution to equations of a dancing top in terms of exponential of Hamiltonian field is given. An extra constraint, that take into account the reaction of supporting plane, leads to modification of the canonical Poisson structure and therefore the integrability according to Liouville is under the question.

Effective Dipole Moment Model for Axially Symmetric C3v Molecules: Application to the Precise Study of Absolute Line Strengths of the ν6 Fundamental of CH335Cl.

The effective dipole moment model for molecules of axial C3v symmetry is derived on the basis of the symmetry properties of a molecule which, on the one hand, is of the same order of efficiency (but much simpler and clearer in applications) as the analogous models derived on the basis of the irreducible tensorial sets theory, and, on the other hand, mathematically more correct in comparison with concepts like the Herman-Walles function used in the models. As an application of the general results obtained, we discuss high-resolution infrared spectra of CH335Cl, recorded with the Zürich prototype ZP2001 (Bruker IFS125 HR) Fourier transform infrared spectrometer at a resolution of 0.001 cm-1 and analyzed in the region of 880-1190 cm-1 (ν6 bending fundamental centered at ν0 = 1018.070790 cm-1). Absolute strengths of more than 2800 transitions (2081 lines) were obtained from the fit of their shapes both with Voigt and Hartmann-Tran profiles, and parameters of the effective dipole moment of the ν6 band were determined by the computer code SYMTOMLIST (SYMmetric TOp Molecules: LIne STrengths), created on the basis of a derived theoretical model. As the first step of the analysis of the experimental data, assignments of the recorded lines were made. A total of 5124 transitions with Jmax = 68, Kmax = 21 were assigned to the ν6 band. The weighted fit of 2077 upper energy values obtained from the experimentally recorded transitions was made with a Hamiltonian which takes into account different types of ro-vibrational effects in doubly degenerate vibrational states of the C3v-symmetric molecule. As the result, a set of 25 fitted parameters was obtained which reproduces the initial 2077 upper "experimental" ro-vibrational energy values with a root mean square deviation drms=4.7×10-5 cm-1. At the second step of the analysis, the computer code SYMTOMLIST was used for determination of the parameters of the derived effective dipole moment model. Six effective dipole moment parameters were obtained from the weighted fit procedure which reproduces absolute experimental strengths of the 2804 initial experimental transitions with a relative drms=3.4%.

Heavy symmetric tops and the Hannay angle

The dynamics of a heavy symmetric top are studied in connection with the Hannay angle. When the top undergoes a steady precession due to gravity without nutation, the Hannay angle has a geometric nature such that it is identical to the solid angle subtended by the loop swept out by the symmetry axis of the top. Here, we show that the Hannay angle can also be described by the angle between two radial vectors on the disk of the top corresponding to the pure spinning motion and the coupled motion of spin and precession for one period of the precession. The geometric nature of the angle between the two radial vectors is verified by demonstrating, via parallel transport, that the magnitude of the angle is the same as that of the solid angle. In the presence of nutation, the path constructed by the symmetry axis is not closed, and the steady precession appears in the limit of infinite initial spin angular velocity. As a consequence, in an ideal situation of no friction, the Hannay angle as a pure geometric effect does not exist in the superposed motion of precession and nutation.

Pbqff: Push-Button Quartic Force Fields.

pbqff is an open-source program for fully automating the production of quartic force fields (QFFs) and their corresponding anharmonic spectroscopic data. Rather than being a monolithic piece of code, it consists of several key modules including a generic interface to quantum chemistry codes and, notably, queuing systems; a molecular point group symmetry library; an internal-to-Cartesian coordinate conversion module; a module for the ordinary least-squares fitting of potential energy surfaces; and an improved second-order rotational and vibrational perturbation theory package for asymmetric and symmetric tops that handles type-1 and -2 Fermi resonances, Fermi resonance polyads, and Coriolis resonances. All of these pieces are written in Rust, a modern, safe, and performant programming language that has much to offer for scientific programming. This work introduces pbqff and its surrounding ecosystem, in addition to reporting new anharmonic vibrational data for c-(C)C3H2 and describing how the components of pbqff can be leveraged in other projects.

Geodesics as products of one‐parameter subgroups in compact lie groups and homogeneous spaces

AbstractWe study the geodesic equation for compact Lie groups G and homogeneous spaces , and we prove that the geodesics are orbits of products of one‐parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group , we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.

Revisiting dissipative motion of a spinning heavy symmetric top and the rise of the top by friction

The dissipative motion and the rise of a heavy symmetrical top with a hemispherical peg are studied. A model taking the fixed point of the top as the center of the peg is considered when the top completely slips and the rolling motion is ignored. This is different from existing models like Jellet's one. Jellett's model and pure slipping are compared for different tops for the rise of the top, and an experimental method to determine the better model is proposed.

The Integrable Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint

This paper addresses the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that there is no slipping of the sphere as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. It is also assumed that, in the direction perpendicular to the above-mentioned one, the sphere can slip relative to the plane. Examples of realization of the above-mentioned nonholonomic constraint are given. Equations of motion are obtained and their first integrals are found. It is shown that the system under consideration admits a redundant set of first integrals, which makes it possible to perform reduction to a system with one degree of freedom.

The Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint

This paper investigates the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that the sphere can slip in the direction of the projection of the symmetry axis onto the supporting plane. Equations of motion are obtained and their first integrals are found. It is shown that in the general case the system considered is nonintegrable and does not admit an invariant measure with smooth positive density. Some particular cases of the existence of an additional integral of motion are found and analyzed. In addition, the limiting case in which the system is integrable by the Euler – Jacobi theorem is established.

Stability and bifurcations of symmetric tops

Stability and bifurcations of symmetric tops

Calculations and measurements for the rotational spectrum and structure of the cyclopentadienyl thallium – benzene complex

Calculations on the structure of the cyclopentadienyl thallium – benzene complex were made using Gaussian-G16 and Spartan programs. Calculations converged to symmetric top and asymmetric top structures. Rotational transition frequencies for the cyclopentadienyl thallium – benzene complex were measured in the 4.5–8.3 GHz range using a Flygare-Balle pulsed beam microwave spectrometer. The basic spectrum from the preliminary experiments exhibited splittings of the observed transitions which would not be expected for a rigid symmetric top molecule. 11 of the measured transitions were assigned to an asymmetric top structure and fit, yielding rotational constants A = 1136.8(11) B = 378.44(11) MHz and C = 375.173(79) MHz.

High-order contact transformations of molecular Hamiltonians: general approach, fast computational algorithm and convergence of ro-vibrational polyad models

The paper describes methods and fast computational algorithm for building effective Hamiltonians in molecular physics using perturbative approach. Separations of fast and slow variables are considered in the framework of contact transformations (CT). The particular focus is on a systematic derivation of effective models for rovibrational spectroscopy from ab initio-based potential energy surfaces with an exhaustive review of previous studies in this field. We consider applications to several types of polyads coupled by Fermi, Coriolis, Darling-Dennison and other types of resonance interactions with examples for asymmetric top, symmetric top and spherical top molecules. A flexible choice of the modelling operator accounts for strong couplings of various types of nuclear motion in molecules among closely lying levels including vibrational resonance schemes (2:1:2 . ), (2:1:2:1), (4:2:6:3), (3:2:1:2:1:1), etc. that occur for C2v, C3v and Td molecules and their isotopic species. The method is implemented in the MOL_CT programme suite, which offers a complementary tool to variational methods in terms of convergence and computational time. The range of applications is also different. The goal of the CT method is providing mathematical models for analyses of molecular spectra with the high-resolution accuracy using physically meaningful parameters derived from ab initio functions.

A simple model for high rotational excitations of molecules in a superfluid

Recently it became possible to study highly excited rotational states of molecules in superfluid helium through nonadiabatic alignment experiments (Cherepanov et al 2021 Phys. Rev. A 104 L061303). This calls for theoretical approaches that go beyond explaining renormalized values of molecular spectroscopic constants, which suffices when only the lowest few rotational states are involved. As the first step in this direction, here we present a basic quantum mechanical model describing highly excited rotational states of molecules in superfluid helium nanodroplets. We show that a linear molecule immersed in a superfluid can be seen as an effective symmetric top, similar to the rotational structure of radicals, such as OH or NO, but with the angular momentum of the superfluid playing the role of the electronic angular momentum in free molecules. The simple theory sheds light onto what happens when the rotational angular momentum of the molecule increases beyond the lowest excited states accessible by infrared spectroscopy. In addition, the model allows to estimate the effective rotational and centrifugal distortion constants for a broad range of species and to explain the crossover between light and heavy molecules in superfluid 4He in terms of the many-body wavefunction structure. Some of the above mentioned insights can be acquired by analyzing a simple 2 × 2 matrix.

Rotational Mode-Specificity in the Cl + C2H6 → HCl + C2H5 Reaction.

We perform rotational mode-specific quasi-classical trajectory simulations using a high-quality ab initio analytical potential energy surface for the Cl(2P3/2) + C2H6 → HCl + C2H5 reaction. As ethane, being a prolate-type symmetric top, can be characterized by the J and K rotational quantum numbers, the excitation of two rotational modes, the tumbling (J, K = 0) and spinning (J, K = J) rotations of the reactant is carried out with J = 10, 20, 30, and 40 at a wide range of collision energies. The impacts of rotational excitation on the reactivity, the mechanism, and the post-reaction distribution of energy are investigated: (1) exciting both rotational modes enhances the reactivity with the spinning rotation being more effective due to its coupling to the C–H stretching vibrational normal modes (C–H bond elongating effect) and larger rotational energies, (2) rotational excitation increases the dominance of direct rebound over the stripping mechanism, while collision energy favors the latter, (3) investing energy in tumbling rotation excites the translational motion of the products, while the excess spinning rotational energy readily flows into the internal degrees of freedom of the ethyl radical or, less significantly, into the HCl vibration, probably due to the pronounced rovibrational coupling in this case. We also study the relative efficiency of vibrational and rotational excitation on the reactivity of the barrierless and thus translationally hindered title reaction.

On the free rotation of a polarized spinning-top as a test of the correct radiation reaction torque

The formula for dipole radiation reaction torque acting on a system of charges, and the Larmor-like formula for the angular momentum loss by this system, differ in the time derivative term which is the analogue of the Schott term in the energy loss problem. In the well-known textbooks this discrepancy is commonly avoided via neglect of the Schott term, and the Larmor-like formula is preferred. In the present paper both formulae are used to derive two different equations of motion of a polarized spinning-top. Both equations are integrable for the symmetric top and lead to quite different solutions. That one following from the Larmor-like formula is physically unplausible, in contrast to another one. This result is accorded with the reinterpretation of Larmor’s formula discussed recently in the pedagogical literature. It is appeared, besides, that the Schott term is of not only academic significance, but it may determine the behavior of polarized micro- and nanoparticles in nature or future experiments.

Parent, 34S, and deuterated triflic acid: Microwave spectra and tunneling splittings due to hydroxyl torsion

Spectra for the parent, 34S, and deuterated forms of triflic acid (CF3SO3H) have been observed by chirped-pulse and cavity Fourier transform microwave spectroscopy. The observation of a-, b-, and c-type transitions definitively establishes that the molecule does not have Cs symmetry, and M06-2X/6-311++G(3df,3pd) calculations concur, yielding a structure in which the OH bond is nearly perpendicular to the C-S-O(H) plane. The rotational spectrum for each isotopologue exhibits a pair of tunneling states resulting from large amplitude motion of the hydroxyl hydrogen between two equivalent structures with opposite directions of the OH bond. The experimentally determined tunneling energies, ΔE, for the parent and 34S species are 52.96784(65) MHz and 52.8761(16) MHz, respectively. For the -OD isotopologue, the tunneling energy decreases significantly, with a value of only ΔE = 0.2460(20) MHz. Curiously, we observe that b-type transitions cross between tunneling states in the parent and 34S spectra, while c-type transitions cross in the spectra of the deuterated species. This likely arises because the molecule is close to a symmetric top, with only the location of the hydrogen defining the orientation of the b- and c-inertial axes, enabling slight structural changes to switch the axis orientations at the transition state. Calculations at the M06-2X/6-311++G(3df,3pd) level of theory predict a 2.8 kcal/mol barrier for the large amplitude motion of the hydroxyl hydrogen rotating around the S-O bond through a Cs symmetric transition state in which the O-H is oriented anti with respect to the S-C bond. A complete scan of the hydroxyl proton around the S-O bond shows an additional transition state of Cs symmetry in the syn orientation with a 6.2 kcal/mol barrier relative to the equilibrium configuration.

Teaching the Fixed Spinning Top Using Four Alternative Formulations

This paper discusses four different approaches that can be followed to derive the equations of motion for a fixed and symmetrical spinning top. Starting from the usual Euler equations in the body-fixed system, after manipulation it is shown that identical equations are derived for the space-fixe system as well. All the three Cartesian components of the angular momentum vector are calculated for both the body- and the space-systems and they are formulated so that they can be used for further numerical analysis. In addition to the classical set, the Euler equations are also easily derived using a rotating system originated at the pivot but not spinning. Moreover, Lagrange equations are derived and the latter are proven to be equivalent with the Euler equations. The best way among these four methods for teaching students is probably the instructor’s preference. Moreover, using commercial software, an adequately accurate numerical solution is derived. Not only the position of the spinning top is calculated but also the support forces at the pivot are predicted

Deterministic inertial dynamics of the magnetization of nanoscale ferromagnets

Analytic solutions for the longitudinal and transverse components of the magnetization of a single-domain ferromagnetic nanoparticle with the simplest possible form of uniaxial magnetocrystalline anisotropy and Zeeman energy are obtained in terms of the appropriate Jacobi elliptic functions and elliptic integrals via the undamped limit of the inertial Landau-Lifshitz-Gilbert (ILLG) equation. Moreover, the nutation frequency is also determined in terms of the inverse period of a Jacobi elliptic function. All the results originate by starting from the analogy of the ILLG equation to a symmetric top with an electric dipole lying along its axis of symmetry as used to model inertial, i.e., high-frequency, effects in the Debye theory of dielectric relaxation of assemblies of noninteracting polar molecules, an analogy which allows one to simultaneously obtain closed-form solutions for typical THz or ultrahigh-frequency magnetization observables as well as the GHz ones associated with the usual ferromagnetic resonance due to precessional motion.

General Perturb-Then-Diagonalize Model for the Vibrational Frequencies and Intensities of Molecules Belonging to Abelian and Non-Abelian Symmetry Groups.

In this paper, we show that the standard second-order vibrational perturbation theory (VPT2) for Abelian groups can be used also for non-Abelian groups without employing specific equations for two- or threefold degenerate vibrations but rather handling in the proper way all the degeneracy issues and deriving the peculiar spectroscopic signatures of non-Abelian groups (e.g., -doubling) by a posteriori transformations of the eigenfunctions. Comparison with the results of previous conventional implementations shows a perfect agreement for the vibrational energies of linear and symmetric tops, thus paving the route to the transparent extension of the equations already available for asymmetric tops to the energies of spherical tops and the infrared and Raman intensities of molecules belonging to non-Abelian symmetry groups. The whole procedure has been implemented in our general engine for vibro-rotational computations beyond the rigid rotor/harmonic oscillator model and has been validated on a number of test cases.

Photofragment ion imaging in vibrational predissociation of the H2O+Ar complex ion.

Vibrational predissociation processes of the H2O+Ar complex ion following mid-infrared excitations of the OH stretching modes and bending overtone of the H2O+ unit were studied by photofragment ion imaging. The anisotropy parameters, β, of the angular distributions of the photofragment ions were clearly dependent on the type (branch) of rotational excitation, β > 0 for the P-branch excitations, while β < 0 for the Q-branch excitations, which were consistent with the previous theoretical predictions for the rotationally resolved optical transition of a prolate symmetric top. The translational energy distributions had a similar form, irrespective of the excitation modes. This result suggests that the prepared excited states underwent a common relaxation pathway via the bending or bending overtone state of the H2O+ unit. In addition, the available energy was preferentially distributed into the rotational energy of the H2O+ fragment ions rather than the translational energy. The mechanism of the rotational excitations of the H2O+ fragment ions was discussed based on the steric configuration of the H2O+ and Ar units at the moment of dissociation.