Space-fixed Frame Research Articles

Alignment of ND3 molecules in dc-electric fields.

The control of movement and orientation of gas-phase molecules has become the focus of many research areas in molecular physics. Here, ND3 molecules are polarized in a segmented, curved electrostatic guide and adiabatically aligned inside a rotatable mass spectrometer (MS). Alignment is probed by photoionization using a linearly polarized laser. Rotation of the polarization at fixed MS orientation has the same effect as the rotation of the MS at fixed polarization, proving that the molecular alignment adiabatically follows the MS axis. Polarization-dependent ion signals reveal state-specific populations and allow for a quantification of the aligned sample in the space-fixed reference frame.

Toward Development of a Plate Discrete Element Method: Geometry and Kinematics

Shapes of constituent particles have a prominent effect on the macroscopic responses of granular assemblies. Clayey minerals often possess a plate-shaped geometry with a large surface-to-volume ratio. It is difficult to model such a geometry with spheres or clusters of spheres in a conventional discrete element method (DEM). In this study, we present a new DEM for plate-shaped particles with a focus on particle geometry and kinematics. The moment of inertia for a general convex plate is given and unit quaternions are adopted to represent the angular degrees of freedom. In addition, the equation of motion for rotation is proposed to be solved in a body-fixed rather than in a space-fixed reference frame. We present simulations of the rotation of a system of plate-shaped particles under the conservation of angular momentum without external torque. The results demonstrate the necessity and importance of enforcing the unity constraint on the quaternions numerically solved from the equation of motion for rotation.

Modeling Photoassociative Spectra of Ultracold NaK + K.

A model for photoassociation of ultracold atoms and molecules is presented and applied to the case of 39K and 23Na39K bosonic particles. The model relies on the assumption that photoassociation is dominated by long-range atom-molecule interactions well outside the chemical bond region. The frequency of the photoassociation laser is chosen close to a bound-bound rovibronic transition from the X1Σ+ ground state toward the metastable b3Π lowest excited state of 23Na39K, allowing us to neglect any other excitation, which could hinder the photoassociation detection. The energy level structure of the long-range 39K···23Na39K excited super-dimer is computed in the space-fixed frame by solving coupled-channel equations, involving the coupling between the 23Na39K internal rotation and the mechanical rotation of the super-dimer complex. A quite rich structure is obtained, and the corresponding photoassociation rates are presented. Other possible photoassociation transitions are discussed in the context of the proposed model.

Rotational dynamics and transitions between Λ-type doubling of NO induced by an intense two-color laser field.

We numerically investigate the rotational dynamics of NO in the electronic ground X2Π state induced by an intense two-color laser field (10 TW/cm2) as a function of pulse duration (0.3-25ps). In the short pulse duration of less than 12ps, rotational Raman excitation is effectively induced and results in molecular orientation. On the contrary, when the pulse duration is longer than 15ps, the rotational excitation is suppressed. In addition to the rotational excitation, we find that transitions between Λ-type doubling are induced. Significantly, the maximum coherent wave packet between Λ-type doubling in J = 0.5 is generated using the pulse duration of 19.8ps. The wave packet changes to the eigenstates of Λ = +1 or -1 alternatively, where Λ is the projection of the electronic orbital angular momentum on the N-O axis, which is regarded as the unidirectional rotation of an unpaired 2π electron around the N-O axis in a space-fixed frame as well as in a molecule-fixed frame. The experimental method to observe the alternation of the rotational direction of the electron around the N-O axis is proposed.

Does nuclear permutation symmetry allow dynamical localization in symmetric double-well achiral molecules?

We discuss the effect of molecular symmetry on coherent tunneling in symmetric double-well potentials whose two molecular equilibrium configurations are interconverted by nuclear permutations. This is illustrated with vibrational tunneling in ammonia molecules, electronic tunneling in the dihydrogen cation, and laser-induced rotational tunneling of homonuclear diatomics. In this contribution, we reexamine the textbook picture of coherent tunneling in such potentials, which is depicted with a wavepacket shuttling back and forth between the two potential-wells. We show that the common application of this picture to the aforementioned molecules contravenes the principle of the indistinguishability of identical particles. This conflict originates from the sole consideration of the dynamics of the tunneling-mode, connecting the double-well energy minima, and complete omission of all the remaining molecular degrees of freedom. This gives rise to double-well wavepackets that are nonsymmetric under nuclear permutations. To obey quantum statistics, we show that the double-well eigenstates composing these wavepackets must be entangled with the wavefunctions that describe all the omitted molecular modes. These wavefunctions have compensating and opposite nuclear permutation symmetry. This in turn leads to complete quenching of interference effects behind localization in one potential-well or another. Indeed, we demonstrate that the reduced density of probability of the symmetrized molecular wavefunction, where all the molecular coordinates but the tunneling-mode are integrated out, is symmetrically distributed over the two potential-wells, at all times. This applies to any multilevel wavepacket of isotropic or fully aligned symmetric double-well achiral molecules. However, in the case of coherent electronic or vibrational tunneling, fully aligned molecules may exhibit dynamical localization in the space-fixed frame, where the tunneling-mode density shuttles between the opposite directions of the alignment axis. This dynamical spatial-localization results from linear combinations of molecular states that have opposite parity. In summary, this study shows that dynamical localization of the tunneling-mode density on either of the two indistinguishable molecular equilibrium configurations of symmetric double-well achiral molecules is forbidden by quantum statistics, whereas its dynamical localization in the space-fixed frame is allowed by parity. The subtle distinction between these two types of localization has far-reaching implications in the interpretation of many ultrafast molecular dynamics experiments.

Accurate Calculations of OH-Stretching Intensities with a Reduced-Dimensional Local Mode Model Including Eckart Axis Embedding.

Absolute OH- and OD-stretching transition intensities have been calculated for a series of alcohols (methanol, ethanol, 2-propanol, 1-propanol, and tert-butanol) with one-dimensional (1D) and three-dimensional (3D) local mode models. We compare the calculated intensities for the ΔvOH = 1-5 and ΔvOD = 1-3 transitions with experimental values. Potential energy and dipole moment surfaces are calculated at the CCSD(T)-F12a/VDZ-F12 level of theory. The 1D local mode model includes only the OH(D)-stretching mode, whereas the 3D local mode model also includes the CO-stretching and COH(D)-bending modes. We analyze the effect on vibrational intensities of using either a molecule-fixed Eckart frame or a space-fixed Cartesian frame. We find that both Eckart embedding and inclusion of the CO-stretching and COH(D)-bending modes in the local mode model are important for the OH/OD-stretching fundamental transition intensities, but have a minor effect on overtone intensities. The 3D reduced-dimensional local model, when combined with coupled cluster surfaces, accurately predicts OH/OD-stretching transition intensities and wavenumbers, for all alcohols included in this work.

Moving mode shape function approach for spinning disk and asymmetric disc brake squeal

The solution approach of an asymmetric spinning disk under stationary friction loads requires the mode shape function fixed in the disk in the assumed mode method when the equations of motion is described in the space-fixed frame. This model description will be termed the 'moving mode shape function approach' and it allows us to formulate the stationary contact load problem in both the axisymmetric and asymmetric disk cases.Numerical results show that the eigenvalues of the time-periodic axisymmetric disk system are time-invariant. When the axisymmetry of the disk is broken, the positive real parts of the eigenvalues highly vary with the rotation of the disk in the slow speeds in such application as disc brake squeal. By using the Floquet stability analysis, it is also shown that breaking the axisymmetry of the disc alters the stability boundaries of the system.

Pose Changes From a Different Point of View

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semidirect product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here, we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semidirect product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing pose changes of its hands.

Second rank direction cosine spherical tensor operators and the nuclear electric quadrupole hyperfine structure Hamiltonian of rotating molecules

Transformations of vector or tensor properties from a space-fixed to a molecule-fixed axis system are often required in the study of rotating molecules. Spherical components λμ,ν of a first rank irreducible tensor can be obtained from the direction cosines between the two axis systems, and a second rank tensor with spherical components λμ,ν(2) can be built from the direct product λ × λ.It is shown that the treatment of the interaction between molecular rotation and the electric quadrupole of a nucleus is greatly simplified, if the coefficients in the axis-system transformation of the gradient of the electric field of the outer charges at the coupled nucleus are arranged as spherical components λμ,ν(2). Then the reduced matrix elements of the field gradient operators in a symmetric top eigenfunction basis, including their dependence on the molecule-fixed z-angular momentum component k, can be determined from the knowledge of those of λ(2). The hyperfine structure Hamiltonian Hq is expressed as the sum of terms characterized each by a value of the molecule-fixed index ν, whose matrix elements obey the rule Δk = ν. Some of these terms may vanish because of molecular symmetry, and the specific cases of linear and symmetric top molecules, orthorhombic molecules, and molecules with symmetry lower than orthorhombic are considered.Each ν-term consists of a contraction of the rotational tensor λ(2) and the nuclear quadrupole tensor in the space-fixed frame, and its matrix elements in the rotation-nuclear spin coupled representation can be determined by the standard spherical tensor methods.

Ionospheric corrections for single-frequency tracking of GNSS satellites by VLBI based on co-located GNSS

Tracking L-band signals of GNSS satellites by radio telescopes became a new observation type in recent years and will be used to improve reference system realizations and links between Earth- and space-fixed frames. First successful test observations were done, with the drawback of being single-frequency only. In order to correct the ionospheric delay by using GNSS phase observations from co-located receivers, the L4R approach was developed. Based on residuals derived by a least-squares processing of the GNSS geometry-free linear combination corresponding corrections could be derived. As a first validation step L4R corrections were applied to GNSS $$L_1$$ data analysis. Station coordinate repeatibilities at the 1-cm level were obtained for baselines of a few thousand kilometers. Comparing the derived delay corrections to VLBI ionospheric delays for quasars located in same directions, differences with a standard deviation of 2.2 TECU could be achieved.

Calculation of state-to-state cross sections for triatomic reaction by the multi-configuration time-dependent Hartree method.

A framework for quantum state-to-state integral and differential cross sections of triatomic reactive scattering using the Multi-Configuration Time-Dependent Hartree (MCTDH) method is introduced, where a modified version of the Heidelberg MCTDH package is applied. Parity of the system is adopted using only non-negative helicity quantum numbers, which reduces the basis set size of the single particle functions in angular degree of freedom almost by half. The initial wave packet is constructed in the space-fixed frame, which can accurately account for the centrifugal potential. By using the reactant-coordinate-based method, the product state-resolved information can be accurately extracted. Test calculations are presented for the H + H2 reactive scattering. This work demonstrates the capability of the MCTDH method for extracting accurate state-to-state integral and differential cross sections. As an efficient scheme for high-dimensional problems, the MCTDH method may be promising for the study of product state-resolved cross sections for polyatomic reactive systems.

Rotational quenching of H2O by He: Mixed quantum/classical theory and comparison with quantum results

The mixed quantum/classical theory (MQCT) formulated in the space-fixed reference frame is used to compute quenching cross sections of several rotationally excited states of water molecule by impact of He atom in a broad range of collision energies, and is tested against the full-quantum calculations on the same potential energy surface. In current implementation of MQCT method, there are two major sources of errors: one affects results at energies below 10 cm(-1), while the other shows up at energies above 500 cm(-1). Namely, when the collision energy E is below the state-to-state transition energy ΔE the MQCT method becomes less accurate due to its intrinsic classical approximation, although employment of the average-velocity principle (scaling of collision energy in order to satisfy microscopic reversibility) helps dramatically. At higher energies, MQCT is expected to be accurate but in current implementation, in order to make calculations computationally affordable, we had to cut off the basis set size. This can be avoided by using a more efficient body-fixed formulation of MQCT. Overall, the errors of MQCT method are within 20% of the full-quantum results almost everywhere through four-orders-of-magnitude range of collision energies, except near resonances, where the errors are somewhat larger.

Mixed quantum/classical theory of rotationally and vibrationally inelastic scattering in space-fixed and body-fixed reference frames

We formulated the mixed quantum/classical theory for rotationally and vibrationally inelastic scattering process in the diatomic molecule + atom system. Two versions of theory are presented, first in the space-fixed and second in the body-fixed reference frame. First version is easy to derive and the resultant equations of motion are transparent, but the state-to-state transition matrix is complex-valued and dense. Such calculations may be computationally demanding for heavier molecules and/or higher temperatures, when the number of accessible channels becomes large. In contrast, the second version of theory requires some tedious derivations and the final equations of motion are rather complicated (not particularly intuitive). However, the state-to-state transitions are driven by real-valued sparse matrixes of much smaller size. Thus, this formulation is the method of choice from the computational point of view, while the space-fixed formulation can serve as a test of the body-fixed equations of motion, and the code. Rigorous numerical tests were carried out for a model system to ensure that all equations, matrixes, and computer codes in both formulations are correct.

Semiclassical description of quantum rotator in terms of SU(2) coherent states

We introduce coordinates of the rigid body (rotator) using mutual positions between body-fixed and space-fixed reference frames. Wave functions that depend on such coordinates can be treated as scalar functions of the group SU(2). Irreducible representations of the group SU(2) × SU(2) in the space of such functions describe their possible transformations under independent rotations of the both reference frames. We construct sets of the corresponding group SU(2) × SU(2) Perelomov coherent states (CS) with a fixed angular momentum j of the rotator as special orbits of the latter group. Minimization of different uncertainty relations is discussed. The classical limit corresponds to the limit j → ∞. Considering Hamiltonians of rotators with different characteristics, we study the time evolution of the constructed CS. In some cases, the CS time evolution is completely or partially reduced to their parameter time evolution. If these parameters are chosen as Euler angles, then they obey the Euler equations in the classical limit. Quantum corrections to the motion of the quantum rotator can be found from exact equations on the CS parameters.

Nuclear monopole charge form factor calculation for relativistic models including center-of-mass corrections

Relativistic models for finite nuclei contain spurious center-of-mass motion in most applications for the nuclear many-body problem, where the nuclear wave function is taken as a single Slater determinant within a space-fixed frame description. We use the Peierls-Yoccoz projection method, previously developed for relativistic approaches together with a reparametrization of the coupling constants that fits binding energies and charge radius and apply our results to calculate elastic electron scattering monopole charge form factors for light nuclei.

Simulating Asymmetric Top Impurities in Superfluid Clusters: A para-Water Dopant in para-Hydrogen.

We present the first simulation study of bosonic clusters doped with an asymmetric top molecule. The path-integral Monte Carlo method with the latest methodological advance in treating rigid-body rotation [Noya, E. G.; Vega, C.; McBride, C. J. Chem. Phys.2011, 134, 054117] is employed to study a para-water impurity in para-hydrogen clusters with up to 20 para-hydrogen molecules. The growth pattern of the doped clusters is similar in nature to that of pure clusters. The para-water molecule appears to rotate freely in the cluster. The presence of para-water substantially quenches the superfluid response of para-hydrogen with respect to the space-fixed frame.

Electron scattering cross sections from HCN over a broad energy range (0.1–10 000 eV): Influence of the permanent dipole moment on the scattering process

We report theoretical integral and differential cross sections for electron scattering from hydrogen cyanide derived from two ab initio scattering potential methods. For low energies (0.1-100 eV), we have used the symmetry adapted-single centre expansion method using a multichannel scattering formulation of the problem. For intermediate and high energies (10-10,000 eV), we have applied an optical potential method based on a screening corrected independent atom representation. Since HCN is a strong polar molecule, further dipole-induced excitations have been calculated in the framework of the first Born approximation and employing a transformation to a space-fixed reference frame of the calculated K-matrix elements. Results are compared with experimental data available in the literature and a complete set of recommended integral elastic, inelastic, and total scattering cross sections is provided from 0.1 to 10,000 eV.

Interaction of orientable object fields with gauge fields

We consider a scalar field f(g) on the Poincaré group M(3, 1). This scalar field describes objects that are characterized by a position x and an orientation z, g=(x,z). The field f(x, z) admits two kinds of transformations, corresponding to a change of the space-fixed reference frame, as well as to a change of the body-fixed reference frame, which form the group M(3, 1)ext×M(3, 1)int, and also phase transformations U(1)ch of orientational variables z. Elementary particles considered as elementary orientable objects are described by the scalar functions transforming according to irreps of the group M(3, 1)ext×M(3, 1)int×U(1)ch. Correspondingly, their continuous symmetries can be divided into external, which form the Poincaré group M(3, 1)ext, and internal M(3, 1)int×U(1)ch. The assumption that the internal symmetries in the theory of orientable objects are gauge ones allows one to obtain important features of the known fundamental interactions—the electroweak and the gravitational. Localization of the group of the right translations T(4)int leads to the teleparallel theory of gravity, which is equivalent to general relativity. Localization of the compact subgroup SU(2)int×U(1)ch leads to the theory of electroweak interactions. Thus, the suggested approach can be considered as a possible way to gravitational–electroweak unification.

Vestibulo-Cerebellar Disease Impairs the Central Representation of Self-Orientation

Transformation of head-fixed otolith signals into a space-fixed frame of reference is essential for perception of self-orientation and ocular motor control. In monkeys the nodulus and ventral uvula of the vestibulo-cerebellum facilitate this transformation by computing an internal estimate of direction of gravity. These experimental findings motivated the hypothesis that degeneration of the vestibulo-cerebellum in humans alter perceptual and ocular motor functions that rely on accurate estimates of gravity, such as subjective visual vertical (SVV), static ocular counterroll (OCR), and gravity-dependent modulation of vertical ocular drifts. We assessed the SVV, OCR, and spontaneous vertical ocular drifts in 12 patients with chronic vestibulo-cerebellar disease and in 10 controls. Substantially increased variability in estimated SVV was noted in the patients. Furthermore, gravity-dependent modulation of spontaneous vertical ocular drifts along the pitch plane was significantly (p < 0.05) larger in the patients. However, the gain and variability of static OCR and errors in SVV were not significantly different. In conclusion, in chronic vestibulo-cerebellar disease SVV and OCR remain intact except for an abnormal variability in the perception of verticality and impaired stabilization of gaze mediated by the otoliths. These findings suggest that OCR and perceived vertical are relatively independent from the cerebellum unless there is a cerebellar imbalance like an acute unilateral cerebellar stroke. The increased trial-to-trial SVV variability may be a general feature of cerebellar disease since a function of the cerebellum may be to compensate for such. SVV variability might be useful to monitor disease progression and treatment response in patients.

Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body

A set of four quaternion matrices is used to represent the equations of finite rotation theory and to describe the kinematics and nonlinear dynamics of an asymmetric rigid body in space. The results obtained are tested in setting up direction-cosine matrices, calculating three-index symbols, establishing the relationship between the components of angular velocity in body-fixed and space-fixed frames of reference, and using a set of three independent rotations. Euler–Lagrange equations and a set of four quaternion matrices are used to construct a block-matrix model describing the nonlinear dynamics of a free asymmetric rigid body in three-dimensional space. The model gives the matrix Euler’s equations of motion and other special cases. Algorithms adapted to use in a numerical experiment are developed