Space-fixed Frame Research Articles

Rotational excitations of HF and HCl molecules by collision with electrons

HCl and HF molecules treated as rigid rotors are examined when colliding with electrons at energies below molecular vibrational thresholds. The overall interaction includes ab initio computed static contributions from ground-state, near Hartree-Fock, wavefunctions and the long-range contributions from the two lowest multipoles and dipole polarisabilities. The close-coupling (CC) equations are solved in the space-fixed (SF) frame of reference and the five lowest-lying rotational states are included in the expansion for each molecular target. The energy dependence of elastic and inelastic cross sections is discussed.

On three-dimensional rotational averages

In theories which describe the response of freely rotating molecules to externally imposed stimuli it is frequently necessary to average rotationally a product of direction cosines relating space-fixed and molecular coordinate frames. In this paper a systematic method for deriving the required tensor averages is presented, and results up to the seventh rank are explicitly shown. Where appropriate both reducible and irreducible expressions are given and their equivalence is demonstrated. Finally, some useful identities relating rotational averages of different ranks are noted.

Nonadiabatic effects in the collision of F(2P) with H2(1Σg+). III. Scattering theory and coupled-channel computations

The theory of nonreactive atom–diatom collisions in the presence of multiple electronic surfaces is developed in both space-fixed and body-fixed coordinate frames. The formalism is applied to the scattering of fluorine atoms by para- and ortho-hydrogen molecules. Coupled-channel computations of integral cross sections for fine structure and rotational transitions are carried out in the rigid rotor approximation using ab initio self-consistent-field potential energy surfaces and are facilitated by the use of a diabatic representation of the molecular channel states. The magnitudes of the cross sections at a specific translational energy are found in general to decrease with increasing energy defect. For F(2P1/2)+p-H2 the cross section for a near resonant electronic-to-rotational energy transfer process dominates other inelastic transitions by at least an order of magnitude.

Fitting an a b i n i t i o HF–HF potential surface

The angular dependence of the ab initio rigid-rotor HF–HF surface of Yarkony et al [J. Chem. Phys. 60, 855 (1974)] is fitted with a form appropriate for future quantum and semiclassical scattering calculations, namely a triple series in Legendre polynomials in the molecule–molecule body–fixed angles. The grid of ab initio points is not dense enough to permit a precise fit to the surface, particularly at small center-of-mass separations. Nevertheless, a reasonably accurate fit, which also gives a good description of the HF dimer, can be attained with ∼30 angular terms. The final body-fixed expansion coefficients were transformed to give 20 angular terms in a space-fixed (SF) frame. For the six largest SF terms, exponentials of various arguments as well as inverse powers were used to fit the dependence on the center-of-mass scattering coordinate. A total of 18 parameters were involved. Asymptotically, the interaction potential goes to the expected dipole–dipole plus dipole–quadrupole form. A simple, empirical modification allows the surface to be used for ro-vibrationally inelastic collisions.

On the decoupling of angular momenta in molecular collisions

It is pointed out that the more exact treatment of the jz-conserving coupled states (R̂-helicity) approximation is quite closely related to the P̂-helicity method of Tamir and Shapiro, and the two methods lead to approximate equations identical in form. The difference in interpretation of the magnetic quantum numbers in the two methods is used to understand the accuracy of the crudest approximations obtained by neglecting the transformation from rotating to space-fixed coordinate frames. Further it is expected that the two approaches lead to similar results in the case where the exact diagonal elements of the l2 operator are retained and the correct transformations to non-helicity frames are used. Finally, the two methods should lead to identical results for integral cross sections and opacities.

Body-fixed equations for atom-molecule scattering: Exact and centrifugal decoupling methods

We demonstrate an exact method for solving the close-coupled equations for rotationally inelastic scattering in the body-fixed (BF) coordinate representation, using a slight modification of the exponential method which allows application of the known boundary conditions for the space-fixed (SF) coordinate frame. The nature of the coupling in the BF frame suggests approximations which reduce the problem to a set of problems each of a smaller dimensionality. Using an atom-rigid rotor model potential, we have compared the decoupling approximation suggested by Pack and by McGuire and Kouri to the fully coupled problem. We find the centrifugal decoupling (CD) approximation to work best at low total angular momenta J, and for low rotor states j. Opacity functions for inelastic transitions are generally better than ones for elastic transitions, due to errors in the phases of elastic S-matrix elements. However, at high J, CD often severely underestimates the inelastic transition probabilities.

The Improvement of a Low-Cost Inertial Navigator Through Conditional Feedback

A concept for providing an accurate low-cost inertial reference was presented in a previous paper. Low-cost platform gyros, poor in terms of their drift rates, furnish the space-fixed reference frame. A single high-accuracy gyro, mounted on the platform, monitors the performance of the platform gyros. Consequently, drift rates are estimated, predicted, and applied as signals to correct the platform for gyro uncertainties. This paper features new estimation and predictive techniques.

Operational procedure for the determination of the matrix elements of the direction cosines for a rigid symmetrical rotator

An operational procedure is presented for obtaining the matrix elements ( J′ K′ M′| λ Aα | JKM), where λ Aα is the direction cosine between the A-axis ( A = X, Y, Z) and the α-axis ( α = x, y, z) of a space-fixed XYZ reference frame and a body-fixed xyz reference frame used to describe the pure rotation of a rigid symmetrical rotator. Ψ( J, K, M) are the normalized symmetrical rotator wave functions.