Space-fixed Frame Research Articles

Collisions of paramagnetic molecules in magnetic fields: An analytic model based on Fraunhofer diffraction of matter waves

We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions of open-shell molecules ($^{2}\ensuremath{\Sigma}$, $^{3}\ensuremath{\Sigma}$, and $^{2}\ensuremath{\Pi}$) with closed-shell atoms. Our treatment makes use of the Fraunhofer model of matter wave scattering and its recent extension to collisions in electric [M. Lemeshko and B. Friedrich, J. Chem. Phys. 129, 024301 (2008)] and radiative fields [M. Lemeshko and B. Friedrich, Int. J. Mass. Spec. 280, 19 (2009)]. A magnetic field aligns the molecule in the space-fixed frame and thereby alters the effective shape of the diffraction target. This significantly affects the differential and integral scattering cross sections. We exemplify our treatment by evaluating the magnetic-field-dependent scattering characteristics of the He-CaH $(X\phantom{\rule{0.2em}{0ex}}^{2}\ensuremath{\Sigma}^{+})$, He-${\mathrm{O}}_{2}$ $(X\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{--})$, and He-OH $(X\phantom{\rule{0.2em}{0ex}}^{2}\ensuremath{\Pi}_{\ensuremath{\Omega}})$ systems at thermal collision energies. Since the cross sections can be obtained for different orientations of the magnetic field with respect to the relative velocity vector, the model also offers predictions about the frontal-versus-lateral steric asymmetry of the collisions. The steric asymmetry is found to be almost negligible for the He-OH system, weak for the He-CaH collisions, and strong for the He-${\mathrm{O}}_{2}$. While odd $\ensuremath{\Delta}M$ transitions dominate the He-OH $[J=3∕2,f\ensuremath{\rightarrow}{J}^{\ensuremath{'}},e/f]$ integral cross sections in a magnetic field parallel to the relative velocity vector, even $\ensuremath{\Delta}M$ transitions prevail in the case of the He-CaH $({X}^{2}{\ensuremath{\Sigma}}^{+})$ and He-${\mathrm{O}}_{2}$ $(X\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{\ensuremath{-}})$ collision systems. For the latter system, the magnetic field opens inelastic channels that are closed in the absence of the field. These involve the transitions $N=1,J=0\ensuremath{\rightarrow}{N}^{\ensuremath{'}}$, ${J}^{\ensuremath{'}}$ with ${J}^{\ensuremath{'}}={N}^{\ensuremath{'}}$.

Few-cycle laser pulses to obtain spatial separation of OHF− dissociation products

In a two-part theoretical study, field-free orientation of OHF(-) is achieved by means of moderately intense half-cycle, infrared laser pulses. In the first step, a short linearly polarized pulse excites a superposition of rigid rotor rotational eigenstates via interaction with the permanent dipole moment of OHF(-). After the field has been switched off, pronounced molecular orientation is observed for several picoseconds. In the second step, femtosecond few-cycle laser pulses are applied to the oriented system to steer vibrational dynamics, modeled by anharmonic vibrational wave functions calculated on a potential energy surface obtained with unrestricted fourth order Moller-Plesset ab initio calculations. The result is selective bond breaking of OHF, followed by the spatial separation of dissociation products in the space-fixed frame. Due to revivals in the rotational wavepacket, product yields can be enhanced over long times.

Quantum theory of chemical reactions in the presence of electromagnetic fields

We present a theory for rigorous quantum scattering calculations of probabilities for chemical reactions of atoms with diatomic molecules in the presence of an external electric field. The approach is based on the fully uncoupled basis set representation of the total wave function in the space-fixed coordinate frame, the Fock-Delves hyperspherical coordinates, and the adiabatic partitioning of the total Hamiltonian of the reactive system. The adiabatic channel wave functions are expanded in basis sets of hyperangular functions corresponding to different reaction arrangements, and the interactions with external fields are included in each chemical arrangement separately. We apply the theory to examine the effects of electric fields on the chemical reactions of LiF molecules with H atoms and HF molecules with Li atoms at low temperatures and show that electric fields may enhance the probability of chemical reactions and modify reactive scattering resonances by coupling the rotational states of the reactants. Our preliminary results suggest that chemical reactions of polar molecules at temperatures below 1 K can be selectively manipulated with dc electric fields and microwave laser radiation.

An analytic model of rotationally inelastic collisions of polar molecules in electric fields

We present an analytic model of thermal state-to-state rotationally inelastic collisions of polar molecules in electric fields. The model is based on the Fraunhofer scattering of matter waves and requires Legendre moments characterizing the "shape" of the target in the body-fixed frame as its input. The electric field orients the target in the space-fixed frame and thereby effects a striking alteration of the dynamical observables: both the phase and amplitude of the oscillations in the partial differential cross sections undergo characteristic field-dependent changes that transgress into the partial integral cross sections. As the cross sections can be evaluated for a field applied parallel or perpendicular to the relative velocity, the model also offers predictions about steric asymmetry. We exemplify the field-dependent quantum collision dynamics with the behavior of the Ne-OCS((1)Sigma) and Ar-NO((2)Pi) systems. A comparison with the close-coupling calculations available for the latter system [Chem. Phys. Lett. 313, 491 (1999)] demonstrates the model's ability to qualitatively explain the field dependence of all the scattering features observed.

The effect of a nonresonant radiative field on low-energy rotationally inelastic [formula omitted] collisions

We examine the effects of a linearly polarized nonresonant radiative field on the dynamics of rotationally inelastic Na + + N 2 collisions at eV collision energies. Our treatment is based on the Fraunhofer model of matter wave scattering and its recent extension to collisions in electric fields [M. Lemeshko, B. Friedrich, J. Chem. Phys. 129 (2008) 024301]. The nonresonant radiative field changes the effective shape of the target molecule by aligning it in the space-fixed frame. This markedly alters the differential and integral scattering cross-sections. As the cross-sections can be evaluated for a polarization of the radiative field collinear or perpendicular to the relative velocity vector, the model also offers predictions about steric asymmetry of the collisions.

Ab initio rovibrational spectra of , BeHD2+ and

An IC-MRCI ansatz has been used in conjunction with augmented quadruple-ζ basis sets to construct an 89-point potential energy surface of the ground electronic state of . The equilibrium structure was predicted to be of C2v symmetry, possessing Be–H bond lengths of 1.609 Å and a H–Be–H bond angle of 29.4°. A rational Padé analytical function was fitted to the discrete IC-MRCI grid yielding a value of 5.20 cm−1. It was embedded in the t-coordinate Eckart–Watson Hamiltonian to calculate vibrational and rovibrational wave functions. For the vibrational ground state, the vibration-averaged bond lengths and angle were 1.647 Å and 29.2°, respectively. An embedded analytical IC-MRCI dipole moment function fitted to 73 discrete points was transformed to the space-fixed frame. Rovibrational radiative properties were calculated using this surface in conjunction with a novel integration scheme.

Nonequilibrium distribution functions for general rigid bodies in axially symmetric environments

A vector $\mathbf{\ensuremath{\rho}}$ is introduced in such a manner that the equilibrium rotational distribution function for a general rigid body has a simple quadratic form both from a body-fixed and space-fixed frame of reference. It is shown that when considering nonequilibrium distribution functions, representations employing the components of $\mathbf{\ensuremath{\rho}}$ generalize more easily than those employing the components of the angular momentum or angular velocity, and lead to forms with greater accuracy. The behavior of $\mathbf{\ensuremath{\rho}}$ and its relation to the angular momentum of the system is explored in some detail. Comparisons are made with distribution functions generated from molecular dynamics simulations of ${\mathrm{H}}_{2}{\mathrm{O}}^{+}$ drifting in a helium bath gas under the influence of a uniform electric field.

Vibrational inelastic electron-H2 scattering revisited: numerically converged coupled channels space frame calculations with model interactions

The collisional excitation of the lower vibrational levels of H2(1Σg+) molecules by low-energy electron impact is computed using an empirical model potential and by solving the coupled-channels scattering equations within a space-fixed (SF) frame of reference formulation. Numerically converged partial, integral inelastic and elastic cross-sections are obtained from what is an essentially exact treatment of the dynamics and the results are compared with measurements and with earlier calculations on the same system. The usefulness of the SF method for handling excitation processes at near-threshold collision energies is discussed and analyzed through the calculations of collisional superelastic partial cross-sections down to 10-2 meV of collision energy.

Forward Symplectic Integrators for Solving Gravitational Few-Body Problems

We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in additional to the force. We demonstrate the efficiency of these Forward Symplectic Integrators by solving the circularly restricted three-body problem in the space-fixed frame where the force on the third body is explicitly time-dependent. These algorithms can achieve accuracy of Runge-Kutta, backward time step and corrector symplectic algorithms at step sizes five to ten times as large.

Plane wave packet formulation of atom-plus-diatom quantum reactive scattering.

We recently interpreted several reactive scattering experiments using a plane wave packet (PWP) formulation of quantum scattering theory [see, e.g., S. C. Althorpe, F. Fernandez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper presents the first derivation of this formulation for atom-plus-diatom reactive scattering, and explains its relation to conventional time-independent reactive scattering. We generalize recent results for spherical-particle scattering [S. C. Althorpe, Phys. Rev. A 69, 042702 (2004)] to atom-rigid-rotor scattering in the space-fixed frame, atom-rigid-rotor scattering in the body-fixed frame, and finally A+BC rearrangement scattering. The reactive scattering is initiated by a plane wave packet, describing the A+BC reagents in center-of-mass scattering coordinates, and is detected by projecting onto a series of AC+B (or AB+C) plane wave "probe" packets. The plane wave packets are localized at the closest distance from the scattering center at which the interaction potential can be neglected. The time evolution of the initial plane wave packet provides a clear visualization of the scattering into space of the reaction products. The projection onto the probe packets yields the time-independent, state-to-state scattering amplitude, and hence the differential cross section. We explain how best to implement the PWP approach in a numerical computation, and illustrate this with a detailed application to the H+D2 reaction.

Rotationally inelastic collisions of electrons with $\mathsf{H_2}$ and $\mathsf{N_2}$ molecules: converged space-frame calculations at low energies

The quantum treatment of the collisional excitation of rotations in H2 and N2 gases as test molecules is carried out within a space-fixed frame of reference by solving the multichannel close-coupled equations using an approach newly developed in our group, i.e. the modified variable phase approximation (MVPA). The interaction potentials contain the static, exchange and correlation-polarisation contributions and are obtained via a local, multipolar expansion formulation from ab initio data. The results are compared with existing experiments and produce numerically converged rotationally inelastic cross-sections obtained from an exact molecular-frame treatment of the problem.

Study of Translational and Rotational Mobility and Orientational Preference of Ethane in One-Dimensional Channels

Structural and dynamical properties of ethane in one-dimensional channels of AlPO4-5 and carbon nanotube have been investigated at dilute concentration with the help of molecular dynamics simulation. Density distributions and orientational structure of ethane have been analyzed. Repulsive interactions seem to play an important role when ethane is located in the narrow part of the AlPO4-5 channel. In AlPO4-5, parallel orientation is predominant over perpendicular orientation except when ethane is located in the broader part of the channel. Unlike in the case of single-file diffusion, our results in carbon nanotube show that at dilute concentrations the mean squared displacement, mu(2)(t) approximate to t(alpha), alpha = 1.8. The autocorrelation function for the z-component of angular velocity of ethane in space-fixed frame of reference shows a pronounced negative correlation. This is attributed to the restriction in the movement of ethane along the x- and y- directions. It is seen that the ratio of reorientational correlation times does not follow the Debye model for confined ethane but it is closer to the predictions of the Debye model for bulk ethane.

Earth rotation as an interdisciplinary topic shared by astronomers, geodesists and geophysicists

Earth rotation is characterized by its angular speed and by the position of the rotation axis both in space and in a frame tied to the Earth. The changes of the angular velocity induce changes in the length-of-day, and the variation of the position of the rotation axis is associated with polar motion (in the reference frame tied to the Earth) and with precession-nutation (in the space-fixed reference frame). These effects can be decomposed into components varying at different time scales. The study of their causes involves in different fields of physics, from astronomy for the rigid Earth nutation, through internal geophysics for the length-of-day variation at the decadal time scale, to external geophysics for seasonal polar motion. In this paper, we show how each discipline brings its own light into the observation, understanding and modeling of the Earth's rotation.

Invariant spinor representations of finite rotation matrices

Our recent results [N. L. Manakov et al., Phys. Rev. A 57, 3233 (1998); 61, 022103 (2000)] on the invariant representations of finite rotation matrices (FRM's) of integer rank j (in terms of tensor products of vectors connected with a space-fixed reference frame) are generalized here for the general case of arbitrary (i.e., integer or half-integer) rank j. This extension is carried out by using new spinor representations of FRM's in terms of specially introduced spinor-annihilation operators. We demonstrate that all widely used, standard representations of FRM's follow as special cases of our invariant representation for particular parametrizations of the rotation parameters. As the simplest application of invariant spinor representations of FRM's, the factorized form of Wigner ${\mathbf{d}}^{j}(\ensuremath{\beta})$ matrices with an arbitrary rank j is obtained as a product of two triangular matrices composed of various powers of $\mathrm{cos}(\ensuremath{\beta}/2).$

Modeling and Analysis for Active Control of Circular Saw Vibrations

Addressed in this paper is modeling and analysis for active control of circular saw vibrations. A method is put forward based on finite element modeling. It starts with modeling of a circular saw in the body- fixed frame, where vibration analyses including natural frequencies and modes can be readily obtained by a standard finite element analysis package. To consider the rotation of a circular saw, the finite element model is transformed to the space-fixed frame, where vibration mode shapes remain the same while natural frequencies are affected by the product of the rotation speed and the order of the mode shape. Upon normalization, the vibration equations can be expressed in terms of natural frequencies and modes, from which a state-space control model can be obtained. The model developed has shown a good agreement between simulation and experimental results. Furthermore, the model is used to design an online state estimator using the Kalman filter as well as an active vibration controller-based linear-quadratic-Gaussian method.

Rotational dynamics of a discotic liquid crystal studied by deuteron spin relaxation

We report on measurements of deuteron quadrupolar splitting and spin-lattice relaxation times T1Q and T1Z in the columnar phase of a ring-deuteriated hexaoctyloxyrufigallol at 46 MHz as a function of temperature. To describe small-step rotations of these molecules within each column in the columnar phase, a space-fixed frame is used to diagonalize the molecular diffusion tensor. The principal diffusion constants in this so called 'anisotropic viscosity' model Dalpha and Dbeta are for rotations of a molecule around and perpendicular to the columnar axis, respectively. A global target analysis of the spectral densities at seven temperatures in a minimization procedure was carried out. We found that Dalpha > Dbeta, which is consistent with the picture that the motion towards or away from the local director tends to disrupt the packing of molecules within the column.

Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parametrization. I. Orthogonal representations

The interdependence of the description of the internal geometry and the corresponding kinetic energy operator T̂ is investigated in detail for a general n-atomic molecule. For both space-fixed and body-fixed reference frames compact expressions of T̂ are derived which are applicable to any set of n−1 translationally and rotationally invariant internal vectors in a spherical polar parametrization. Simple analytical forms are given for reduced masses and kinetic coupling constants, which are the only vector specific parameters in the final rovibrational kinetic energy expression. The kinetic energy assumes the most separable form for an entirely orthogonal set of internal vectors. A highly efficient computer program for the calculation of rovibrational spectra of tetratomic molecules has been developed on the basis of this formulation. Calculations on the HF dimer and the metastable molecule HOCO illustrate the accuracy and flexibility of this approach.

Deuteron NMR study of dynamics and order in the columnar phase of a fluorinated discotic liquid crystal

Measurements have been made of deuteron quadrupolar splitting and spin—lattice relaxation times T 1Q and T 1z in the columnar phase of a ring deuterated 1-fluoro-2,3,6,7,10,11-hexahex-yloxytriphenylene (F-HAT6) as a function of temperature and at two different Larmor frequencies. The disc-like molecules are stacked, with only short range positional order, into columns which are arranged on a 2-dimensional hexagonal lattice. To describe small step reorientations of these molecules, a rotational diffusion model which uses a space-fixed frame to diagonalize the molecular diffusion tensor is adopted. The principal diffusion constants in this so called ‘anisotropic viscosity model’ Dα and Dβ are for rotations of a molecule around and perpendicular to the columnar axis, respectively. A global target analysis of the spectral densities J 1(ω0) and J 2(2ω0) at seven temperatures and two Larmor frequencies in a single minimization procedure was carried out. It was found that Dα Dβ, which is consistent with the picture that the motion towards or away from the local director (i.e., the columnar axis) tends to disrupt the packing of molecules within the column. Furthermore, the activation energies Eαβ and Eβa for these motions were found to be 67.7 kJmol−1 and 56.3kJmol−1. That these values are higher in comparison with those for HAT6 is interesting, and believed to be a result of a strong lateral dipole in F-HAT6.

The effect of mean flow/source motion on sound-pressure fields with absorption

An analysis of sound-pressure fields relative to both uniformly translating and space-fixed reference frames is presented. By using Fourier transforms and separation of variables, a solution is determined in the source frame. Far-field approximations are transformed to emission coordinates which model the pressure field observed in a space-fixed reference frame. The influence of mean flow/forward flight on source-fixed directivity patterns is shown. A finite cylindrical source was considered and directivity patterns were constructed which show the effect of Mach number, frequency, and displacement pattern on the sound field. The influence of flow on absorption due to viscosity is also presented.

Birefringence induced by a static electric field gradient

A quantum theory of the birefringence induced in a molecular gas by a static electric field gradient is presented; it is an application of the finite temperature perturbation theory to molecular light scattering described in the preceding paper. The kinematics of the experiment are dealt with explicitly through the use of a space-fixed frame based on a set of canonically conjugate variables for the centre-of-mass and internal motions of the molecule. This leads to a definition of the molecular multipoles that is different from the conventional one and there are no ‘origin-dependence problems’ at any stage of the calculation. The susceptibility that governs the birefringence is analysed in the spirit of Van Vleck's quantum mechanical account of polarity in terms of the magnitude, relative to k B T, of the energy level separations associated with the dominant matrix elements of the external field V and the current operator j(k). As in previous treatments, the main contribution is inversely proportional to the temperature, and there is also a temperature independent term. The birefringence depends on the matrix elements of the molecular quadrupole moment operator; in the framework used here, this has a definition independent of the polarity of the molecule.