Computational Methods for Electron—Molecule Collisions PDF
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Computational Methods for Electron-Molecule Collisions Computational Methods for Electron-Molecule Collisions Edited by Winifred M. Huo NASA Ames Research Center Mollet Field, California and Franco A. Gianturco UniversityolRome, City of Rome Rome, Italy Springer Science +B usiness Media, LLC Library of Congress Cataloging-in-Publication Data On file Proceedings of the Workshop on Comparative Study of Current Methodologies in Electron—Molecule Scattering, held March 11-13, 1993, in Cambridge, Massachusetts ISBN 978-1-4757-9799-2 ISBN 978-1-4757-9797-8 (eBook) D O I 10.1007/978-1-4757-9797-8 © 1995 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1995 Softcover reprint of the hardcover 1st edition 1995 109 8 7 6 5 4 3 2 1 A l l rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher PREFACE The collision of electrons with molecules and molecular ions is a fundamental pro- cess in atomic and molecular physics and in chemistry. At high incident electron en- ergies, electron-molecule collisions are used to deduce molecular geometries, oscillator strengths for optically allowed transitions, and in the case of electron-impact ionization, to probe the momentum distribution of the molecule itself. When the incident electron energy is comparable to or below those of the molecular valence electrons, the physics involved is particularly rich. Correlation and exchange effects necessary to describe such collision processes bear a close resemblance to similar efft:cts in the theory of electronic structure in molecules. Compound state formations, in the form of resonances and vir- tual states, manifest themselves in experimental observables which provide details of the electron-molecule interactions. Ro-vibrational excitations by low-energy electron collisions exemplify energy transfer between the electronic and nuclear motion. The role of nonadiabatic interaction is raised here. When the final vibrational state is in the continuum, molecular dissociation occurs. Dissociative recombination and dissociative attachment are examples of such fragmentation processes. In addition to its fundamental nature, the study of electron-molecule collisions is also motivated by its relation to other fields of study and by its technological appli- cations. The study of planetary atmospheres and the interstellar medium necessarily involve collision processes of electrons with molecules and molecular ions. For example, the fine-structure changing transitions of the oxygen atom by electron impact and the resonant electron-impact vibrational excitation of N2 are two major cooling mechanisms of electrons in the earth's ionosphere. Electron-atom and electron-molecule collisions in intense laser fields are used to probe the properties of atoms and molecules under high field conditions. Electron scattering is also employed as a research tool in material sci- ence to investigate the properties of condensed matter, from simple crystals to molecules physisorbed or chemisorbed on the surface of a metal or molecular crystal. The differ- ence between the condensed phase versus gas phase spectrum provides information such as the change of bonding in the condensed environment. Recently, this technique was employed in the study of new materials formed by clusters of atoms or molecules. Electron-molecule collisional cross sections are required to input data for the design and modeling of plasma processes, including plasma etching, chemical vapor deposition, and advanced laser developments. The operating condition of these processes can be optimized, and new processes devised, based on the knowledge of electron excitation and fragmentation rates. Such data are also employed to model the weakly ionized flow field experienced by high-speed space vehicles upon re-entry into the planetary atmosphere. Here the actual environment in space is difficult to duplicate in a laboratory and much of the input is from theoretical studies. v The combined effort of theory and experiment contributes to advances in the field of electron-molecule collisions. While the theory of high-energy electron collisions is well understood and cross section calculations using the Born approximation are read- ily available, the situation in the low-energy regime has lagged behind. Much of the earlier development in this area was devoted to qualitative predictions involving sym- metry selections, categorizing resonances, and descriptions of threshold behavior. The development of ab initio methods for low-energy electron-molecule collision came rather late. Stimulated by the availability of modern-day high-speed computers, the advance in molecular electronic structural theory, and access to accurate experimental data, ac- tivities in the development of ab initio computational methods for low-energy electron- molecules collisions have increased significantly in the last ten to fifteen years. At this point the computed cross sections for small polyatomic molecules can be of comparable accuracy as the best experimental data. In some cases, theoretical studies actually were carried out before experiment. While the starting point of all ab initio methods is the Schrodinger equation or Lippmann-Schwinger equation, each method approaches the problem with different em- phasis on the physics. Also, as a method is refined, it is frequently found that certain modificatiOJ.l in its implementation may expedite the convergence of the calculation. In view of the growth in this area, it was deemed timely for a gathering of the practi- tioners of the field to discuss the interplay between fundamental theory and practical implementation, to compare how each method treats various physical aspects of the scattering process, and also the effect of approximate treatments. Under the sponsor- ship of the Institute for Theoretical Atomic and Molecular Physics (ITAMP) at the Harvard-Smithsonian Center for Astrophysics, a workshop on "Comparative Study of Current Methodologies in Electron-Molecule Scattering" was held March 11-13,1993, at ITAMP in Cambridge, Massachusetts. The last day of the workshop, March 13, happened to be the time of the great bliz- zard of '93 in the eastern part of United States. However, the wind and snow outdoors did not diminish the heated discussion inside. During one of the discussion sections, the participants agreed to prepare a book based on the proceedings of the workshop, with the specification that the book should be useful not only to practitioners in the field, but also should serve as an introduction to graduate students and postdoctoral researchers new to the field. Such a book should also offer opportunities for cross fertilization for researchers from different disciplines. The present volume is a result of that effort. It covers the time-independent ab initio methods for low-energy electron-molecule collisions currently in use. Electron impact ionization is not considered here. Care is taken throughout the book to demonstrate how these methods are implemented in actual computations. While this book is concerned with technical aspects of ab initio methods, it is not designed as a review book for cross section data. Other conference proceedings, such as the biennial ICPEAC and its associated satellite meetings, are good sources for such data. The book is organized according to the methods. Some methods are described in a single chapter, and some have multiple contributions. The numbering of equations starts afresh in each chapter except the R-matrix chapters where the authors number their equations sequentially for easy references. The description of each method is self- contained. The reader is assumed to have a general background in scattering theory, but needs not be well versed in electron-molecule collisions. vi We thank Prof. Alex Dalgarno and Dr. Kate Kirby of ITAMP, for their generous support and help in organizing the workshop and for their hospitality to those partici- pants stranded in Cambridge over the snowy weekend. We are grateful to Ms. Valerie Sorenson and Ms. Verity Parris, for arranging housing for the participants from over- seas and for their secretarial help. Special mention should be made of a conscientious caterer, who delivered our coffee and cookies under weather conditions that stopped the Boston subway from running. We also appreciate the help from Professor Jonathan Tennyson, who edited the R-matrix section of the book, and the editorial assistance of Dr. Helmar Thiimmel. Winifred M. Huo Franco A. Gianturco September 1994 vii CONTENTS THE COMPLEX KOHN VARIATIONAL METHOD Chapter 1. THE COMPLEX KOHN VARIATIONAL METHOD T.N. Rescigno, C.W. McCurdy, A.E. Orel, and B.B. Lengsfield, III 1. Introduction ............................................................. 1 2. Theoretical Foundation .................................................. 3 2.1. The Kohn Variational Method for Neutral and Ionic Scattering ...... 3 2.2. Continuum Basis Functions ......................................... 6 2.3. Trial Wave Function for Electron-Molecule Scattering ............... 7 2.4. Feshbach Partitioning .............................................. 9 3. Approximation Schemes ................................................ 13 3.1. Primitive Separable Expansions .................................... 13 3.2. Adaptive Quadrature .............................................. 14 3.3. Pseudoresonances and Intermediate Energy Scattering .............. 19 4. Applications and Extensions ............................................ 22 4.1. Off-Shell Extension to Threshold Vibrational Excitation ............ 22 4.2. Polar Molecules ................................................... 29 4.3. Photoionization ................................................... 33 4.4. Dissociative Recombination ........................................ 37 THE LINEAR ALGEBRAIC METHOD Chapter 2. THE LINEAR ALGEBRAIC METHOD FOR ELECTRON- MOLECULE COLLISIONS Lee A. Collins and Barry 1. Schneider 1. Basic Concepts ......................................................... 45 2. General Formalism ..................................................... 46 3. Numerical Techniques .................................................. 48 3.1. General Remarks .................................................. 48 3.2. Variation-Iteration Method ........................................ 49 3.3. Further Developments ............................................. 51 4. Electron-Atom Scattering in Intense Fields .............................. 51 ix 5. Applications ........................................................... 55 5.1. Scattering in Intense Fields ........................................ 55 5.2. Collisions of Electrons with Ht .................................... 56 THE MULTICHANNEL QUANTUM DEFECT METHOD Chapter 3. ANALYSIS OF DISSOCIATIVE RECOMBINATION OF ELECTRONS WITH ArXe+ USING ArXe* CALCULATIONS A.P. Hickman, D.L. Huestis, and R.P. Saxon 1. Introduction ........................................................... 59 2. Description of DR ...................................................... 60 3. Role of DR in the Atomic Xenon Laser ................................. 61 4. Potential Curves and Matrix Elements .................................. 62 5. Scattering Calculations for Xe* + Ar ................................... 63 6. Potential Curves for DR ................................................ 64 7. Estimate of Rate for DR of ArXe+ ...................................... 70 8. Conclusions ............................................................ 72 METHOD BASED ON SINGLE-CENTER EXPANSION OF THE TARGET Chapter 4. ELECTRON-SCATTERING FROM POLYATOMIC MOLECULES USING A SINGLE-CENTER-EXPANSION FORMULATION F.A. Gianturco, D.G. Thompson, and A. Jain 1. Introduction ........................................................... 75 2. Formulation of The Interaction Forces .................................. 77 2.1. Definition of SE and SEP Approximations ......................... 77 2.2. The SE Equation for F(r) ......................................... 78 2.3. The Iterative Exchange Method ................................... 80 2.4. Local Exchange: The Free Electron Gas ............................ 82 2.5. The Local Exchange Semiclassical Approximation .................. 83 2.6. The Local Exchange Separable Approximation ..................... 85 2.7. The Correlation-Polarisation Potentials ............................ 87 3. The SCE Radial Equations ............................................. 89 3.1. Expansion of the Bound Orbitals .................................. 89 3.2. Continuum Functions and Radial Equations ........................ 92 3.3. Single Centre Expansion of the Static Potential .................... 93 3.4. The Symmetry-Adapted Coefficients ............................... 96 4. Solution ofthe SE and SEP Radial Equations ........................... 97 4.1. An Iterative Method .............................................. 97 4.2. Linear Algebraic Method .......................................... 98 4.3. Sand K Matrices, and Scattering Amplitudes ...................... 99 4.4. The Total Cross Sections ......................................... 101 4.5. The Differential Cross Sections ................................... 102 x 4.6. Transitions Involving Nuclear Motion ............................. 104 5. Examples of Specific Calculations ...................................... 106 6. Conclusions ........................................................... 115 Chapter 5. A STUDY OF THE PORTING ON SIMD AND MIMD MACHINES OF A SINGLE CENTRE EXPANSION CODE TO TREAT ELECTRON SCATTERING FROM POLYATOMIC MOLECULES F.A. Gianturco, N. Sanna, and R. Sarno 1. Introduction .......................................................... 119 2. The Single Centre Expansion (SCE) Method ........................... 119 3. Computational Details ................................................ 121 3.1. Code Description and Parallel Strategies Adopted ................. 121 3.2. SIMD Hardware and Software Description ........................ 121 3.3. MIMD Hardware and Software Description ....................... 122 4. Results and Discussion ................................................ 122 4.1. The Test Case ................................................... 122 4.2. SIMD Version .................................................... 123 4.3. MIMD Version ................................................... 125 5. Future Developments .................................................. 127 ROTATIONAL AND VIBRATIONAL CLOSE COUPLING Chapter 6. HOW TO CALCULATE ROTATIONAL AND VIBRATIONAL CROSS SECTIONS FOR LOW-ENERGY ELECTRON SCATTERING FROM DIATOMIC MOLECULES USING CLOSE- COUPLING TECHNIQUES Michael A. Morrison and Weiguo Sun 1. Introduction .......................................................... 131 2. Theoretical Concerns .................................................. 133 2.1. The Target Molecule ............................................. 134 2.2. The Electron-Molecule Schrodinger Equation ...................... 136 2.3. Boundary Conditions ............................................. 138 2.4. Into the BODY Frame ............................................ 140 2.5. Coupled Equations ............................................... 141 a. Boundary Conditions at Last .................................. 145 b. Nuts and Bolts: Convergence Matters .......................... 146 2.6. S, T, and K Matrices and Relationships between Them ........... 148 a. Rotational and Vibrational Frame Transformations ............. 150 2.7. Integral Scattering Equations and Their Solution .................. 152 a Integral Equations Strategies in Separable Methods ............. 155 2.8. The Interaction Potential and Its Matrix Elements ................ 156 a. Long-Range Behavior .......................................... 156 b. Vibrational Averaging of the Static Potential .................. 157 c. The Long-Range Polarization Potential ........................ 158 d. Coupling Matrix Elements I: Vibrational Coupling ............. 159 xi
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