Michael A. Morrison

Molecular properties of N2 and CO2 as functions of nuclear geometry: Polarizabilities, quadrupole moments, and dipole moments

Ab initio static polarizabilities, quadrupole moments, and electronic energies of ground state N2 and CO2 have been calculated self‐consistently as functions of nuclear geometry in the Hartree–Fock approximation using the finite‐field method. In addition, permanent dipole moments for the infrared‐active vibrational modes of CO2 have been determined. For N2, the effects of electron correlation on these molecular properties have been studied by performing generalized valence bond (perfect pairing) calculations with the same basis sets as were used in the Hartree–Fock calculations. These basis sets are constructed from energy‐optimized contracted sets of nucleus‐centered Gaussian‐orbitals to provide a flexible representation of the entire molecular charge cloud, including the highly polarizable outer regions. Results of these calculations are compared with experimental and other theoretical values where possible.

ABUNDANCES OF GALACTIC ANTICENTER PLANETARY NEBULAE AND THE OXYGEN ABUNDANCE GRADIENT IN THE GALACTIC DISK

We have obtained spectrophotometric observations of 41 anticenter planetary nebulae (PNe) located in the disk of the Milky Way. Electron temperatures and densities, as well as chemical abundances for He, N, O, Ne, S, Cl, and Ar were determined. Incorporating these results into our existing database of PN abundances yielded a sample of 124 well-observed objects with homogeneously determined abundances extending from 0.9 to 21 kpc in galactocentric distance. We performed a detailed regression analysis which accounted for uncertainties in both oxygen abundances and radial distances in order to establish the metallicity gradient across the disk to be 12 + log(O/H) = (9.09 ± 0.05) – (0.058 ± 0.006) × Rg , with Rg in kpc. While we see some evidence that the gradient steepens at large galactocentric distances, more objects toward the anticenter need to be observed in order to confidently establish the true form of the metallicity gradient. We find no compelling evidence that the gradient differs between Peimbert Types I and II, nor is oxygen abundance related to the vertical distance from the galactic plane. Our gradient agrees well with analogous results for H II regions but is steeper than the one recently published by Stanghellini & Haywood over a similar range in galactocentric distance. A second analysis using PN distances from a different source implied a flatter gradient, and we suggest that we have reached a confusion limit which can only be resolved with greatly improved distance measurements and an understanding of the natural scatter in oxygen abundances.

A guide to rotations in quantum mechanics

To lay a foundation for the study and use of rotation operators in graduate quantum mechanics and in research, a thorough discussion is presented of rotations in Euclidean three space (R3 ) and of their effect on kets in the Hilbert space of a single particle. The Wigner D-matrices are obtained and used to rotate spherical harmonics. An extensive ready-reference appendix of the properties of these matrices, expressed in a consistent notation, is provided. Careful attention is paid throughout to various conventions (e.g. active versus passive viewpoints) that are used in the literature.

Near-Threshold Electron-Molecule Scattering

Publisher Summary This chapter discusses the scattering of the near-threshold electron-molecule. Low-energy electron-molecule scattering is rich in fundamental physics and a fascinating phenomenon. It is a field whose applications range from pollution control to astrophysics. At energies near zero for elastic scattering and a few meV above threshold for inelastic processes, the calculation, understanding, and measurement of cross sections encounter unique challenges and bizarre phenomena. Frame transformations are an invaluable conceptual tool. But the few existing applications of this procedure bring into question their practical utility. Use of frame transformations in day-to-day scattering calculations, indeed seem feasible only if there exists a wide, energy-independent band of radii at which the transformation leads to accurate cross sections. In any case, strong sensitivity of the approximate cross sections to the transformation radii poses a serious problem. The body-frame fixed-nuclei (BF-FN) approximation can be considered a composite of the FNO and RR approximations in the BODY representation.

Near-threshold rotational and vibrational excitation of H 2 by electron impact: Theory and experiment

A joint experimental-theoretical attack on low-energy e-H2 scattering is described. The cross sections calculated from a highly converged numerical solution of the nonrelativistic Schrodinger equation, using a parameter-free interaction potential, are first compared with results from swarm experiments, and are later used to improve the accuracy of the swarm analysis at energies above the first vibrational threshold. To provide further perspective, the theoretical results are compared with a variety of other experimental data. The theoretical results for the momentum-transfer and rotational-excitation cross sections are in excellent agreement with the results from swarm experiments, but there is an unresolved and significant difference in the threshold behaviour of the vibrational-excitation cross sections. Both the theoretical and experimental approaches are subjected to close scrutiny in an attempt to uncover possible sources of error that could explain this difference. The failure to locate likely sources points to the need for further independent theoretical and experimental work to resolve a problem that has serious implications.

How to Calculate Rotational and Vibrational Cross Sections for Low-Energy Electron Scattering from Diatomic Molecules using Close-Coupling Techniques

This chapter is not a review; electron-molecule dynamics is already replete with fine reviews, many of which appear in books devoted entirely to this topic.1–6 These reviews discuss the applied importance of this field,7–10 survey the status of electron-molecule collision data,11–40 and address specialized topics such as resonance scattering,15 vibrational excitation,16 near-threshold scattering,17–18 particular theoretical approaches such as the R-matrix method,19 numerical methods for solving the Schrodinger equation,20–22 and scattering from polar23,24 and polyatomic25,26 targets. Neither is this chapter primarily pedagogical; readers can find elsewhere a wealth of pedagogically useful tutorial introductions and reviews that narrate the major developments in the field’s long rich history and survey recent advances that have made it the focus of intense activity during the last 20 years.27–30 Rather, this chapter is a “ready reference” of the key equations for the application of one very widely used theoretical strategy—the eigenfunction-expansion or “close-coupling“ method— to one very important class of problems: quantum scattering (at incident energies less than about 10 eV) from a closed-shell diatomic molecule accompanied, perhaps, by rotational and/or vibrational (but not electronic) excitation of the target. As exemplary of an extremely powerful method for reducing multi-variable integro-differential equations to more tractable sets of fewer-variable equations, this class of problems illustrates strategies used in many other theoretical contexts, both within and outside of electron-molecule collisions.

Rotational and vibrational excitation of nitrogen by electron impact

Rotational excitation of nitrogen by low-energy electron impact has posed an unsolved problem for more than three decades. Early analysis of the results of swarm experiments in nitrogen found that the data could be matched remarkably well by assuming that the energy dependences of the Δj = 2 cross sections from threshold to a few tenths of an eV are given by a simple formula based on the Born approximation. Moreover, the quadrupole moment (the only adjustable parameter in the formula) which gave the best fit to the data was commensurate with existing experimental values. This finding posed an enigma, since the quadrupole Born expression is known to incorrectly represent the interaction potential for scattering except within a few meV of threshold. We have analysed new swarm data, taken in a dilute mixure of nitrogen in neon, using theoretical rotational and momentum transfer cross sections based on a solution of the Schrodinger equation using static, exchange, and polarisation potentials. This work explains the long-standing enigma and provides the basis for a subsequent analysis in which theoretical vibrational excitation cross sections are also investigated using the new swarm data for the mixture.

Non-adiabaticity and other aspects of polarisation in positron-molecule scattering

The authors have investigated various aspects of polarisation/correlation effects at low energies for the prototypical positron-N2 system. First, they have found that, like positron-H2 cross sections, those for this system are highly sensitive to terms in the (adiabatic) polarisation potential that explicitly depend on the sign of the charge of the projectile. Insight into the nature of these sign-dependent distortions is afforded by three-dimensional graphs of polarised N2 density functions. Second, they have explored and characterised nonadiabatic (correlation) effects in this system using a simple two-parameter model, they have found that the onset of these effects in the near-target region in positron scattering is much more gradual than in electron scattering. By including in scattering calculations adiabatic and non-adiabatic effects, they obtain theoretical positron-N2 cross sections in good agreement with recent experimental data.

Polarisation potentials for positron-molecule collisions: positron-H, scattering

Model polarisation potentials for use in calculating positron-molecule cross sections are often based on approximations in which terms that depend on the sign of the charge of the projectile are neglected. Thus, these potentials do not fully distinguish between electron and positron scattering. The validity of using such potentials for positron- H2 collisions is investigated by performing scattering calculations using electron- and positron-H, model polarisation potentials. Results show that, because of differences in the way electrons and positrons distort the target charge cloud, sign-dependent terms in the adiabatic potential can significantly influence the total cross section. Low-energy cross sections are also shown to be rather sensitive to how non-adiabatic polarisation effects are taken into account. Using a cut-off fully adiabatic positron-H, polarisation potential, theoretical total cross sections in reasonably good agreement with recent measurements are obtained.

The separable representation of exchange in electron - molecule scattering: I. Elastic scattering and rotational excitation

We have investigated a systematic procedure for representing, as a separable expansion, the exchange interaction in electron - molecule scattering. Illustrative calculations of scattering quantities (eigenphase sums, total cross sections and rotationally-resolved differential cross sections) have been performed for electron scattering from molecular hydrogen in the fixed-nuclear-orientation approximation. The exchange basis can be constructed from the same Cartesian Gaussian functions used to generate the near-Hartree - Fock static interaction, supplemented by an even-tempered series of Gaussian functions located on the molecular centre of mass. Particular emphasis is placed on examining the convergence properties of this series.