Space-fixed Axis Research Articles

Seemingly asymmetric atom-localized electronic densities following laser-dissociation of homonuclear diatomics.

Recent experiments on laser-dissociation of aligned homonuclear diatomic molecules show an asymmetric forward-backward (spatial) electron-localization along the laser polarization axis. Most theoretical models attribute this asymmetry to interference effects between gerade and ungerade vibronic states. Presumably due to alignment, these models neglect molecular rotations and hence infer an asymmetric (post-dissociation) charge distribution over the two identical nuclei. In this paper, we question the equivalence that is made between spatial electron-localization, observed in experiments, and atomic electron-localization, alluded by these theoretical models. We show that (seeming) agreement between these models and experiments is due to an unfortunate omission of nuclear permutation symmetry, i.e., quantum statistics. Enforcement of the latter requires mandatory inclusion of the molecular rotational degree of freedom, even for perfectly aligned molecules. Unlike previous interpretations, we ascribe spatial electron-localization to the laser creation of a rovibronic wavepacket that involves field-free molecular eigenstates with opposite space-inversion symmetry i.e., even and odd parity. Space-inversion symmetry breaking would then lead to an asymmetric distribution of the (space-fixed) electronic density over the forward and backward hemisphere. However, owing to the simultaneous coexistence of two indistinguishable molecular orientational isomers, our analytical and computational results show that the post-dissociation electronic density along a specified space-fixed axis is equally shared between the two identical nuclei-a result that is in perfect accordance with the principle of the indistinguishability of identical particles.

Anomalous diffusion of molecules with rotating polar groups: The joint role played by inertia and dipole coupling in microwave and far-infrared absorption.

Budó's generalization [A. Budó, J. Chem. Phys. 17, 686 (1949)10.1063/1.1747370] of the Debye rotational diffusion model of dielectric relaxation of polar molecules to an assembly with internal interacting polar groups is extended to inertial anomalous diffusion. Thus, the theory can be applied both in the GHz and the THz regions, accounting for anomalous behavior as well as the necessary return to optical transparency at very high frequencies. The linking of both dispersion regions in a single model including anomalous effects is accomplished via a fractional Fokker-Planck equation in phase space based on the continuous time random walk ansatz. The latter is written via the Langevin equations for the stochastic dynamics of pairs of interacting heavy polar groups embedded in the frame of reference of a particular molecule or molecular dimer rotating about a space-fixed axis. The fractional Fokker-Planck equation is then converted to a three-term matrix differential recurrence equation for the statistical moments. This is solved in the frequency domain for the linear dielectric response using matrix continued fractions. Thus, one has the complex susceptibility χ(ω) for extensive ranges of damping, group dipole moment ratio, and friction. The susceptibility, as inferred from the small oscillation limit, inherently comprises a low frequency (GHz) band with width depending on the anomalous parameter and a far-infrared (THz) or Poley peak of resonant character with a comblike structure of harmonic peaks. This behavior is due to the double transcendental nature of the after-effect function.

Measurement of the orientation of buffer-gas-cooled, electrostatically-guided ammonia molecules

The extent to which the spatial orientation of internally and translationally cold ammonia molecules can be controlled as molecules pass out of a quadrupole guide and through different electric field regions is examined. Ammonia molecules are collisionally cooled in a buffer gas cell, and are subsequently guided by a three-bend electrostatic quadrupole into a detection chamber. The orientation of ammonia molecules is probed using (2+1) resonance-enhanced multiphoton ionisation (REMPI), with the laser polarisation axis aligned both parallel and perpendicular to the time-of-flight axis. Even with the presence of a near-zero field region, the ammonia REMPI spectra indicate some retention of orientation. Monte Carlo simulations propagating the time-dependent Schrödinger equation in a full basis set including the hyperfine interaction enable the orientation of ammonia molecules to be calculated – with respect to both the local field direction and a space-fixed axis – as the molecules pass through different electric field regions. The simulations indicate that the orientation of ∼95% of ammonia molecules in JK=11 could be achieved with the application of a small bias voltage (17V) to the mesh separating the quadrupole and detection regions. Following the recent combination of the buffer gas cell and quadrupole guide apparatus with a linear Paul ion trap, this result could enable one to examine the influence of molecular orientation on ion-molecule reaction dynamics and kinetics.

Hyperfine and vibrational structure of weakly bound levels of the lowest1gstate of molecular87Rb2

Photoassociation resonances in the ${}^{87}$Rb${}_{2}$ ${1}_{g}$ state dissociating to $5{\phantom{\rule{0.16em}{0ex}}}^{2}S+5{}^{2}{P}_{1/2}$ were produced by the excitation of colliding ${}^{87}$Rb atoms in a far-off resonance trap. Levels down to 31 cm${}^{\ensuremath{-}1}$ below the dissociation limit were measured with resonance linewidths of 15 to 20 MHz, and have been located to a one-sigma combined systematic and statistical uncertainty of 50 MHz relative to the $5{\phantom{\rule{0.16em}{0ex}}}^{2}S+5{}^{2}{P}_{1/2}$ limit. Electron and nuclear spins were fixed to a space-fixed axis by circularly polarized optical pumping, so that only states with total nuclear spin $I=3$ were excited, thereby greatly simplifying the spectrum. The analysis of the data yielded hyperfine coupling parameters $A(v)$, vibrational $G(v)$, and rotational $B(v)$ parameters. The $G(v)$ parameters could be fit to an rms accuracy of about 0.01 cm${}^{\ensuremath{-}1}$ to a potential constructed from current ${C}_{3}$ and ${C}_{6}$ long-range dispersion parameters plus an ab initio potential that was adjusted in depth and with quadratic terms in the internuclear distance added to the inner wall.

Dipolar radicals in crossed electric and magnetic fields

Paramagnetic, dipolar Hund’s case-a radicals are considered in the presence of arbitrary, non-collinear combinations of electric and magnetic fields. The field-dependent part of the Hamiltonian is found to exhibit a good quantum number, consisting of the projection of the molecule’s total angular momentum along a space-fixed axis that is determined by both the fields and the electric and magnetic dipole moments of the molecule. This quantity remains good even when the fields are non-collinear. Exploiting this feature identifies a set of quantum numbers for the molecule in crossed fields. We dub this set a ‘Hund’s case-X’ basis.

Spectroscopic signatures of bond-breaking internal rotation. II. Rotation-vibration level structure and quantum monodromy in HCP

The rotation-vibration level structure of ground electronic state HCP is investigated at vibrational energies approaching and exceeding that of the linear CPH saddle point. With respect to energies above the saddle point, we investigate possible spectroscopic manifestations of strong Coriolis interactions between the hindered, bond-breaking internal rotation of the hydrogen about the CP core and the rotation of the molecule in the space-fixed axis system. With respect to energies below the saddle point, we provide new interpretations, from quantum and semiclassical points of view, of previously observed anomalously large B (rotational) and g22 (energy dependence on the vibrational angular momentum) constants for the large-amplitude pure bending states of HCP (referred to elsewhere as “isomerization” or saddle node states). We also predict similar anomalies in other spectroscopic constants, including the “centrifugal distortion” constant D and the “rotational l-resonance” parameter q2. These changes in the effective spectroscopic rotation-vibration constants are shown to be a direct consequence of the spherical pendulum topology of the HCP bend/internal rotor system, which is associated with a phenomenon called quantum monodromy, defined as the absence of a smoothly valid set of quantum numbers for all states. Our semiempirical model for the HCP bend/internal rotor mode is derived using principles of semiclassical inversion and provides new insights into the breakdown in the ability of rovibrational effective Hamiltonians to model highly vibrationally excited states of HCP.

Comparison of rotation models for describing DNA conformations: application to static and polymorphic forms

A new method, based on a space-fixed rotation axis, or local helix axis, is proposed for the calculation of the relative orientation variables for a sequence of base pairs. With this method, orientation variables are determined through the rotation of a base pair about this axis. These variables uniquely determine a set of helical variables, similar to the roll, tilt, and twist, commonly used for a description of spatial orientations of internally rigid base pairs. The proposed identification of roll and tilt with the direction cosines of the space-fixed rotation axis agrees well with their customary definitions as the openings of the angles between adjoining base pairs toward the minor groove and toward the ascending (5' to 3') backbone strand, respectively. These new variables permit a more direct physical comprehension of DNA conformations and also the behavior of self-complementary sequences. These direction cosines, together with the rotation angle about the space-fixed axis, form a set of three independent orientation variables of the bases that afford some advantages over the variously defined twist, roll, and tilt angles, either for static or average forms. An example for the static form of these variables is shown through their use to interpret crystal coordinates. An example for the average of orientation variables is based on statistical calculations. In this example, the orientation variables, together with the translational variables that describe the relative displacements of a pair of adjacent base pairs, form a canonically distributed ensemble in phase space spanned by these variables. Two sets of conformational variables are generated by using two different methods for performing rotation operations on the sequences of base pairs. The first method is based on the new single rotation about a space-fixed axis of rotation. This space-fixed axis of rotation is, in fact, the local helical axis as constructed previously by others. The second method is based on three consecutive rotations by Euler angles. Because of large flexibilities and anisotropies along various conformational variables of DNA base pairs, the two sets of generated conformational variables, based on these two different methods of performing rotation operations, lead to slightly different sets of structurally different, but energetically equivalent, spatial arrangements of the base pairs.

A Theoretical Study of the Stark Effect in Triatomic Molecules: Application to H2O

We report here the development of a theoretical method to calculate Stark splittings and state-dependent dipole moments for a triatomic molecule based on the MORBID approach [P. Jensen, J. Mol. Spectrosc.128, 478-501 (1988)]. The Stark parameters are obtained directly from the potential energy surface and dipole moment function of the molecule. The transformation of the dipole moment from molecule-fixed axes to space-fixed axes is carried out by means of irreducible tenser formalism. We apply the new method to H2O, since for this molecule we have an accurate potential energy surface [P.Jensen, J. Mol. Spectrosc.133, 438-460 (1989)] obtained by fitting to experimental data, and a high-quality ab initio dipole moment surface [U. G. Jørgensen and P. Jensen, J. Mol. Spectrosc.161, 219-242 (1993)]. The results of calculations on the basis of these input data are in good agreement with experiment. However, we have improved this agreement further by refining the dipole moment surface in least-squares fits of experimental Stark data, in particular, of the very accurate MBER measurements by S. Shostak, W. Ebenstein, and J. S. Muenter [J. Chem. Phys.94, 5875-5882 (1991)]. With the refined dipole moment surface, we have calculated state-dependent dipole moment values. The variation of these values with vibrational excitation is in close agreement with that found in the experimentally derived values of Shostak et al.

Study of Trajectory of Spin-Stabilised Artillery Projectiles

Equations of motion for conventional spin-stabilised artillery projectile have been derived using a pseudo-stability axes system in addition to body-fixed and space-fixed axes systems. The aerodynamic forces and moments have been represented by their respective coefficients and the effects of Mach number and Reynolds number have been suitably accounted. The magnus terms which are significant at high rates of spin are estimated using a simple model. The set of equations have been partly linearised and solved numerically for a typical projectile using NAG system routines. Various trajectory parameters are computed and compared with the range-table data for the projectile. A parametric study had been carried out varying the aerodynamic coefficients to understand the sensitivity of the results obtained.

Figure adjustment and tectonics during polar wander: the flowing-apart of Gondwana

The effects to be expected if the northward motion of Gondwana represented wander of the Earth relative to its space-fixed axis of rotation (as against motion of Gondwana over a fixed Earth) are reviewed. To render the model plausible, this has been based on the Goldreich and Toomre (1969) development of the Liouville equations. Like the spin axis, the orientation of the Equatorial bulge is space-fixed. In passing from polar latitudes northward, a lithospheric region must accommodate itself to the Equatorial bulge and decrease in gravity. The pressure on material initially in P T equilbrium decreases under the influence of the axifugal forces responsible for bulge formation. The figure adjustment is likely to be accomplished through re-patterning of the convection in the mantle. If wander is as fast as suggested in recent paleomagnetic studies, up to 1°/Ma, during fast intervals bulge-motion may have pre-empted a major fraction of the convection. During polar-wander, the Earth's surface is divided into four quadrants. In the two moving towards the Equator, upwelling convection is favored; but in the two moving poleward, figure-adjustment favors subduction. Gondwana “flowed apart” during its residence in Quadrant 1, affected by motion towards the Equatorial bulge. Quadrant II, in which the mantle is experiencing motion towards the North Pole, is the site of the Mesozoic-Tertiary concentration of subduction represented by the Tethys tectonic belt. Owing to rapid longitudinal motion unrecorded in paleomagnetism, events in the Pacific hemisphere cannot be re-constructed.

Hindered rotational energy levels of symmetric tops in trigonal crystalline fields

The energy levels for the hindered rotation of symmetric top molecules or ions in trigonal crystalline fields have been computed. The symmetry group of the Hamiltonian is C̄3v×C3 , where C̄3v is the trigonal group of rotations about the body-fixed axes and C3 is the trigonal group of rotations about the space-fixed axes. The tunneling frequencies of CH3D on a Vulcan III powdered sample have been computed and compared to the observed results.

Viewing the molecular display: Hints from the artificial intelligence of vision

Abstract In this paper I describe a number of “depth cues” which permit the viewer of computer graphics displays to deduce three-dimensional information from the two-dimensional display. These may be divided into “bottom-up” and “top-down” aids to interpretation. These depth cues can be emphasized in the computer graphics displays to assist the novice viewer, with an expert system adapted to the purpose. I provide a brief and general description of expert systems, including a caricature which deals with some of the problems encountered in molecular modeling with computer graphics. The central construct of the expert system DOCENT is its capacity to represent a macromolecule by “generalized cylinders”, and to permit manipulation of the cylinders directly, instead of the usual internal coordinates or space-fixed axes.

A density functional approach to freezing transitions in molecular fluids: Dipolar hard spheres

We extend a density functional theory of atomic systems to simple molecules using first-order perturbation theory. We model the hydrogen halides as dipolar hard spheres and consider freezing into two orientationally different fcc lattices. In one of these, the dipoles may align parallel to a single, space-fixed axis while in the other, the dipoles in adjacent faces of the fcc unit cell are constrained to lie perpendicular to one another. The second structure approximates the experimentally observed molecular crystals of HF and HCl, and the phase diagram calculated from this model agrees qualitatively with experiment. At high temperatures, or for weak enough dipole moments, a plastic (orientationally disordered) solid precedes the molecular crystal. The plastic phase is unaffected by the dipolar perturbation but eventually becomes a molecular crystal upon lowering the temperature or increasing the density. We also discuss second-order perturbation theory, the effects of higher multipoles, and some general consequences of the density functional theory of molecular systems.

Collisional relaxation in resonantly pumped14NH3inversion systems

The recent experiments of Steward and Parsons (1983) on the absorption spectrum of the resonantly pumped (J,K)=(3,3) inversion transition in 14NH3 have been repeated at lower pressures and using a more sensitive detection technique. In both studies, amplification of the probe radiation field under certain pump and probe field conditions predicted by Mollow (1972) has been observed. The theoretical model of Steward and Parsons predicts fine structure on spectra that are otherwise qualitatively the same as those predicted by Mollow, but their experimental results were inconclusive in deciding on the presence, or otherwise, of this structure. Such fine structure is not observed. A theoretical model, incorporating a full master equation approach, is developed to calculate the absorption spectrum of a simplified, resonantly pumped, degenerate system based on the (J,K)=(3,3) inversion system. Several different collisional relaxation models are used. For this system, an isotropic collision process cannot change the component of the total angular momentum of a molecule on a space-fixed axis. In agreement with the experimental results, both isotropic and non-isotropic collisional relaxation models produce spectra that do not show any of the fine structure predicted by Steward and Parsons, Also, all the collision models produce spectra that agree closely with each other showing that, for the energy level scheme considered, the spectra resulting from the theoretical model used are insensitive to collisional selection rules.

Molecular orientation at a gas–liquid interface

In a fluid of molecules with orientation-dependent intermolecular forces there will be preferred molecular orientation near a gas–liquid interface. The present knowledge of such effects is discussed for some simple model fluids, for the case of a plane surface. The property of direct theoretical interest is the density–orientation profile, rho;(z, ω), which gives the density of molecules at height z in the interface having an orientation ω relative to space-fixed axes. It has been studied both theoretically and via computer simulation for several model fluids, including those with site–site Lennard-Jones, site–site Lennard-Jones with distributed point charges, and Lennard-Jones plus multipole intermolecular potentials. Theoretical calculations can be based on perturbation theory or integral equation methods. In the latter approach there are several alternative integro-differential equations for rho;(z1ω–) which provide rigorous starting points for the theory. Results from perturbation theory are compared with those from molecular dynamics simulation for several model potentials. Preliminary results from integral equation theory are given.

Raman intensities and cartesian polarizability derivatives

Cartesian polarizability derivatives are defined as derivatives of the molecular polarizability with respect to the displacement of an atom in a space-fixed axis system. These derivatives are used to describe the intensity of molecular vibrational and rotational Raman scattering. The number of independent derivatives is easily determined and is shown to equal the number of intensity parameters which may be obtained from experiment. The cartesian polarizability derivatives thus form a complete, minimal set of parameters for describing Raman intensities.

Hindered rotational energy levels of a tetrahedron in an orthorhombic crystalline field

The hindered rotational energy levels for a tetrahedron in an orthorhombic crystalline field have been computed. The symmetry group of the Hamiltonian is ?×C2, where ? is the tetrahedral group of rotations about body-fixed axes and C2 represents 180° rotations about the space-fixed axes. The rotational heat capacity of the ammonium ion has been computed using this model. For low barriers to rotation the maximum in the rotational heat capacity is very broad.

Hindered rotational energy levels of a tetrahedron in a trigonal crystalline field

The energy levels for the motion of a rigid tetrahedral molecule in a trigonal crystalline field have been computed. The symmetry group of the Hamiltonian is ?×C3, where ? is the tetrahedral group of rotations about body-fixed axes and C3 is the trigonal group of rotations about space-fixed axes. The effect on the tunneling frequencies, heat capacity, and activation energy of lowering the symmetry of the crystalline field from tetrahedral to trigonal is discussed. The energy levels of a symmetric top molecule in a tetrahedral crystalline field have been computed and used to determine tunneling frequencies.

Ordered spans of unrestricted and self-avoiding random-walk models of polymer chains. I. Space-fixed axes

An N-step random walk on a cubic lattice is adopted as a model of a random polymer chain. The spans, or extents, of each random walk configuration in the principal lattice directions are arranged in order of magnitude, ξ3?ξ2⩾ξ1. In the case of the unrestricted random walk, the average values of the ordered spans 〈ξi,u〉 and 〈ξ2i,u〉, i=1, 2, and 3, are calculated analytically in the limit of large N. The limiting relative values of the first moments, 〈ξi,u〉 are 1.637 : 1.267 : 1; and the limiting values of the second moments 〈ξ2i,u〉 are 2.710 : 1,600 : 1. In the case of the restricted or self-avoiding walk, the corresponding average spans 〈ξi,r〉 and 〈ξ2i,r〉 are estimated for N?150 by using a Monte Carlo procedure. The same Monte Carlo procedure is used to estimate the values of 〈ξi,u〉 and 〈ξ2i,u〉 for N?1000. On the assumption that the rate of approach of the average ordered spans of the self-avoiding walks to their asymptotic forms is similar to the rate of approach of the average ordered spans of the unrestricted walks to their asymptotic forms, the following estimates are obtained for the ordered spans of the self-avoiding walks: 〈ξi,r(N) 〉=?i,rN0.61 for N≫1 and ?3,r : ?2,r : ?1,r =1.75 : 1.31 : 1. The relative values of the estimates of the second moments of these ordered spans for N≫1 are 〈ξ23,r〉 : 〈ξ22,r〉 : 〈ξ21,r〉 =3.08 : 1.71 : 1. A simple ellipsoidal model is analyzed in order to obtain estimates of the intrinsic spans of the two kinds of random walks. It is assumed that all random walks have the same intrinsic dimensions and form (ellipsoidal), but that their principal axes are oriented at random. The average ordered spans with respect to a set of orthogonal space-fixed axes of this randomly oriented ensemble of ellipsoids depend uniquely on the principal diameters of the ellipsoid (intrinsic spans). A procedure is devised to solve the inversion problem of determining the relative lengths of the principal diameters which correspond to a given set of relative ordered moments with respect to space-fixed axes. The relative values obtained for the principal diameters of the ellipsoid are 2.59 : 1.61 : 1 in the case of the unrestricted random walk and 3.05 : 1.77 : 1 in the case of the self-avoiding walk.

General calculation of the collision integral for the linearized Boltzmann transport equation

We develop and describe a general method for evaluating collision integrals in the linearized Boltzmann transport equation which eliminates the necessity to repeat similar integration steps for each force law. Integrations not dependent on scattering cross-section variables have been carried out once and for all. The two mathematical innovations which facilitate these general integrations are (i) the development of an expansion of the Burnett functionXNML(x+y) into products of Burnett functions of argument x with other functions; and (ii) the use of representations of the full rotation group to transform from space-fixed axes to axes aligned with the relative velocity vector of colliding atoms. The relations so derived allow rapid evaluation of the collision integral from a knowledge of the scattering cross section.