Body-fixed Axes Research Articles

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.

Pseudomolecular atoms: geometry of two-electron intrashell excited states

Supermultiplet patterns indicating pronounced collective motion of electrons are prominent among intrashell excited states of helium or alkaline earth atoms. Many aspects of these patterns have previously been discussed in terms of an empirical rovibrator model, e-core-e, analogous to alinear triatomic molecule. However, this has appeared incompatible with a dimensional scaling treatment that predicts pseudomolecular features and relates properties of some excited states to the ground state, but corresponds to abent equilibrium geometry. We examine both the full two-electron Hamiltonian, including angular momentum, and a prototype model for angular correlation which restricts the two particles to the surface of a sphere. By virtue of the boundary conditions imposed by the Jacobian volume element, we find that alinear equilibrium geometry is not allowed. If the Hamiltonian is transformed to reduce the Jacobian to unity, an infinite barrier appears in the effective potential when the electrons are 180° apart and equidistant from the nucleus. We find the supermultiplet patterns, including some “antimolecular” features, are consistent with a floppy butbent geometry, anasymmetric rotor model. The most probable interelectron angle ϑ m varies markedly with the principal quantum number and also with the space-quantization of the angular momentum with respect to body-fixed axes.

The calculation of the vibration-rotation energies of triatomic molecules using scattering coordinates

Department of Physics and Astronomy, University College London, Gower Street, London WCIE 6BT, England Received 25 January 1986 This paper is a review of work I have been involved with concerning the calculation of ro-vibrational energy levels and wavefunctions, mainly of triatomic systems. The method reviewed is that of Tennyson and Sutcliffe for the calculation of spectra of triatomics as atom-diatom collision complexes, which is particularly ap- propriate for molecules with one or more large amplitude vibrational mode. The method uses polynomial basis functions to carry the nuclear motions and is variational for a given potential. Hamiltonians for four different sets of body-fixed axes are derived. The computational implementation of the method is discussed, including improvements to some published algorithms. The use of a two step (secondary) variational procedure for highly-excited rotational states is discussed and sample results given. Extensions to non-Born-Oppenheimer problems, transition intensities and polyatomic molecules are also considered. Comparisons are made with other techniques for calculating ro-vibrational spectra with some specific molecular examples.

Analytical formula for direction cosines of the Eckart frame of a planar molecule

An analytical formula is suggested for calculating direction cosines of the Eckart frame of a planar molecule. As one application of the formula, we give a simple way to find the matrix switching of the geometrical body-fixed axes of the formaldehyde molecule to axes of its Eckart frame.

Rotational diffusion of deformable macromolecules with mean local cylindrical symmetry

A simple diffusion equation is derived for the rotational brownian motions of a single rod element in a deformable macromolecule with mean local cylindrical symmetry. The effective diffusion coefficients for rotations about the local symmetry and transverse axes are expressed in terms of the time-dependent mean-squared angular displacements about those body-fixed axes. The Green's function of the diffusion operator and the correlation functions of the Wigner rotation matrix elements are likewise evaluated in terms of those same quantities. A simple general expression is presented for the fluorescence polarization anisotropy of any macromolecule with parallel absorption and emission dipoles, and its relation to the correlation functions measured in other incoherent, and coherent experiments, including transient photodichroism, is discussed. The fluorescence polarization anisotropy of macromolecules with mean local cylindrical symmetry is expressed in terms of its twisting, tumbling, and internal correlation functions. The anisotropy formula of Barkley and Zimm is shown to be incorrect unless either the helix axis is immobile, or the time is very small and the transition dipole lies along the helix axis. The internal and twisting correlation functions are formulated in a computationally practical way for models of uniform filaments of arbitrary length and uniform rigidity. Both empirical and theoretical approaches to the tumbling correlation function are also described. The twisting correlation function for a filament constrained to girdle the equator of a sphere is presented in a computationally practical form.

Line frequency expressions for triply degenerate fundamentals of spherical top molecules appropriate for large angular momentum

For states of high angular momentum we obtain expressions for the transition frequencies in the triply degenerate fundamental of a “heavy” spherical top, that are accurate to spectroscopic resolution, without tedious calculation of octahedral (or tetrahedral) vector coupling coefficients or diagonalization of matrices. Our calculations are based upon the physical assumption that as the molecule rotates with larger angular momentum it behaves like a symmetric top with the coupled pure rotational angular momentum quantized along either the four- or threefold body-fixed axes. We find that the approximate calculation which involves only 3- J symbols quite accurately reproduces the result of a complete diagonalization of the Hamiltonian, so that for all but the most accurate saturated absorption spectra theoretical analysis may be carried out without reference to the octahedral (or tetrahedral) subgroup of the rotation group.

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.

Hindered rotational energy levels of a tetrahedron in a tetrahedral crystal field

The energy levels for the motion of a rigid tetrahedral molecule in a tetrahedral crystal field have been computed by the method developed by King and Hornig. The symmetry group of the Hamiltonian is T̄× T, where T̄ is the tetrahedral group of rotations about body-fixed axes and T is the tetrahedral group of rotations about space-fixed axes.

Rotational States of a Tetrahedron in a Cubic Crystal Field

The Schrödinger equation is solved for the rotational states of a rigid cube whose center of mass is fixed at a point of symmetry in an external field. This is shown to be equivalent to the equation of motion for a regular tetrahedron in a field with symmetry Oh. The quantum states are classified under the direct product group Ō×O, where Ō is an octahedral group of rotations about body-fixed axes and O is a similar but distinct group of rotations about space-fixed axes. The wavefunctions are obtained by an expansion in a series of symmetry-adapted linear combinations of spherical-top functions. Projection operators are defined which generate the various linear combinations. Closed-form expressions are obtained for the matrix elements of the Hamiltonian and of the projection operators. Upper and lower bounds are computed for representative energy levels. The results demonstrate that using basis sets of practical size, high accuracy can be obtained. This analysis forms the basis of a more general theory of rotation of small molecules and polyatomic ions in crystals.

Celestial Rate Sensing

r T 1 H E ANGULAR velocity of a space vehicle can be imJportant to the operations of attitude control and data compensation. Components of angular velocity relative to inertial space resolved along body-fixed axes can be measured by rate gyroscopes fixed in the vehicle with sensitive axes along the prescribed body axes. However, exactly the same components can be determined from the measured velocity of drift of the star field across the fields of view of two or three body-mounted telescopes. This may be called rate sensing/' and may be an attractive alternative to mechanical rate gyros on mechanization grounds. The purpose of this note is to describe the principle of operation of celestial rate sensors of a particular type, and to derive the relationship between the vehicle's angular velocity in inertial space and the output of such an instrument. Mechanization details are not included, largely because these are proprietary features from the viewpoints of the several companies concerned with such developments. For this analysis the telescope and its field are analogous to a pinhole camera and its field. The camera's photographic plate or photocathode (reference surface) and the image of a star produced on it are shown schematically in Fig. 1. A vector directed perpendicularly from the pinhole to the reference surface is denoted by f0. The position of a star image on the plate at time t is given by the vector h. The position of the image, a small time increment At, later is given by f2. The star field is rotating at an angular velocity -co with respect to the camera pinhole and plate, where co is the vehicle's angular velocity in inertial space. The vector velocity of the star image at h is given by (-co X fi). The projection of this velocity onto the plate produces v, the velocity of the star image in the plane of the plate. Note that the projection to the plate is not a perpendicular one but is directed from the pinhole along fx and r2. A base vector reference system eh e2, e3 is presented in the figure; £3 is along fo while ei and e2 lie in the plane of the reference surface. Polar coordinates p and 6 may be used to represent the position of the star image in the plane. The first step is to determine v in terms of p, 6, rQ and co. From the figure it is clear that (f2 — fo)-r0 = 0 and that [f + (-co X fi)Af] X f2 = 0. Expanding the several vectors in the base system and substituting them into these relations, and using the fact that vAt = r2 — h} it can be shown that in the limit as At -> 0

Operational derivation of wave functions for a symmetrical rigid rotator

An operational procedure is presented for obtaining the normalized eigenfunctions Ψ(JKM) of a rigid symmetrical rotator and the matrix elements of the rectangular components of its angular momentum in both body-fixed and space-fixed axes.