Lecture Notes in Chemistry
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1 Lecture Notes in Chemistry 59 Edited by: Prof. Dr. Gaston Berthier Universite de Paris Prof. Dr. Michael J. S. Dewar The University of Texas Prof. Dr. Hanns Fischer Universitat Ziirich Prof. Dr. Kenichi Fukui Kyoto University Prof. Dr. George G. Hall University of Nottingham Prof. Dr. Jiirgen Hinze Universitat Bielefeld Prof. Dr. Hans Jaffe University of Cincinnati Prof. Dr. Joshua Jortner Tel-Aviv University Prof. Dr. Werner Kutzelnigg Universitat Bochum Prof. Dr. Klaus Ruedenberg Iowa State University Prof Dr. Jacopo Tomasi Universita di Pisa
2 S. Fraga M. Klobukowski J. Muszynska E. San Fabian K. M. S. Saxena 1. A. Sordo T. L. Sordo Research in Atomic Structure Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
3 Authors S. Fraga M. Klobukowski 1. Muszynska E. San Fabian K. M. S. Saxena 1. A. Sordo T. L. Sordo Department of Chemistry, University of Alberta Edmonton, Alberta, Canada T6G 2G2 ISBN-13 : e - I S B N : ~ DOl: / This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1993 Typesetting: Camera ready by author 52/ Printed on acid-free paper
4 To Harry E. Gunning
5 Preface Impressive advances have been made in the study of atomic structures, at both the experimental and theoretical levels. And yet, the scarcity of information on atomic energy levels is evident At the same time there exists a need for data, because of the developments in such diverse fields as astrophysics and plasma and laser research, all of them of fundamental importance as well as practical impact. This project of research in atomic structure, consisting of three components (formulation, computer program, and numerical results), constitutes a basic and comprehensive work with a variety of uses. In its most practical application, it will yield a rather accurate prediction of the energy levels of any atomic system, of use per se or in the interpretation and confirmation of experimental results. On the other hand, it will also be of use in the comparative study of the appropriateness of the various levels of approximation and as a point of reference. Work on this project has extended over several years and a number of researchers have collaborated, at different stages of its development, during their stays at this laboratory: M. Klobukowski organized the generation of the SL-functions, J. Muszynska developed most of the basic formulation for the matrix elements, K.M.S. Saxena contributed his expertise on numerical integration, and E. San Fabian, J.A. Sordo and T.L. Sordo participated in the study of hyperfinestructure interactions. In addition J.D. Climenhaga and P. Clark adapted the original program, written for IBM-type mainframes, for use on CDC and CRA Y machines, respectively, and J. Jorgensen has been in charge of the word-processing component for the complete project. Serafm Fraga Edmonton, Alberta Canada
6 Table of Contents Introduction... 1 Theoretical Foundation Hamiltonian Operator and Eigenvalue Equations Hamiltonian operator Extended Breit Hamiltonian operator Generalized Hamiltonian operator Eigenvalue equations Basic Theoretical Formulation Angular Functions: Coupling of Angular Momenta One-electron functions SL-functions JM]- and FMF-functions Selection of functions Tensor-Operator Formulation Tensor operators Wigner-Eckart theorem Reduced matrix elements Matrix elements Application of the Basic Formulation... 45
7 x 4 Transformation of Operators to Tensor Form Basic operators Operators s(1),,1(1) and C(k) Other common operators Transformation rules Application Summary Matrix Elements General formulation General expressions SMsLML-coupling M]-coupling..., FMF-coupling Examples for specific interactions..., Summary of Theoretical Results Electronic energy Mass variation Specific mass effect One-electron Darwin correction Two-electron Darwin correction Electron spin-spin contact interaction Orbit-orbit interaction Spin-orbit coupling Spin-spin dipole interaction Magnetic dipole and Fermi contact interactions Electric quadrupole coupling Magnetic octupole coupling Zeeman effect (low field) Zeeman effect (high field) Zeeman effect (very high field) Stark effect Nuclear-mass dependent orbit-orbit interaction Nuclear-mass dependent spin-orbit coupling (electron spin) Nuclear-mass dependent spin-orbit coupling (nuclear spin) Implementation
8 XI 7 Practical Details Selection of configurations Detennination of radial functions Selection rules Masscorrections Numerical Examples Accurate energies SLJ energy levels Hyperfine-structure splittings Nuclear-mass dependent corrections Bibliography References Reference texts Data sources Units and Constants Constants Units Notation and Symbols
9 Acknowledgements The authors acknowledge the kindness of the following organizations for the use of copyright material: American Physical Society; John Wiley & Sons, Inc.; National Research Council (Canada); Springer Verlag; Argonne National Laboratory (managed by The University of Chicago for the U.S. Department of Energy under Contract No. W Eng-38). Specific mention must be made, in particular, of the following works from which copyright material has been taken: S. Fraga and J. Karwowski. Relativistic Treatments for Bound-State Atomic Energies. Theoretica Chimica Acta (Berlin) 35, (1974). S. Fraga, M. Klobukowski, 1. Muszynska, K.M.S. Saxena and J.A. Sordo. Matrix Elements of the Breit Hamiltonian. Physical Review A34, (1986). S. Fraga, E. San Fabian, J.A. Sordo, M. Campillo, J.D. Climenhaga and M. Klobukowski. Atomic Energy Levels from Configuration Interaction Calculations with Relativistic Corrections. International Journal of Quantum Chemistry XXXV, (1989). R.A. Hegstrom. Nuclear-mass and Anomalous-moment Corrections to the Hamiltonianfor an Atom in a Constant External Magnetic Field. Physical Review A7, (1973). M. Klobukowski and S. Fraga. Accurate Theoretical Prediction of the Experimental Ground-state Total Atomic Energies. Physical Review A38, (1988). W.C. Martin. 'Energy Levels of Highly Ionized Atoms'. In 'Proceedings of the Workshop on Foundations of the Relativistic Theory of Atomic Structure held at Argonne National Laboratory, December 4-5, 1980'. Argonne National Laboratory, Report ANL , March E. San Fabian and S. Fraga. Hyperfine-structure Interactions: Preliminary Results. Canadian Journal of Physics 66, (1988). T.L. Sordo, J.A. Sordo and S. Fraga. Nuclear-mass Dependent Spin-orbit Interaction. Canadian Journal of Physics 69, (1991).
Editors. Prof. Dr. Hans H. Jaffe. University of Cincinnati Cincinnati, Ohio 45221/USA
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