Electronic, Atomic and Molecular Calculations: Applying the Generator Coordinate Method

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Electronic Atomic and Molecular Calculations Applying the Generator Coordinate Method


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Cover Illustration The R10(r) and R42(r) radial functions for Xe. The values in R10(r) are 785.7768, 782.8561, and 766.5602 for numerical HF, STOs, and GTOs, respectively.


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Electronic Atomic and Molecular Calculations Applying the Generator Coordinate Method

Milan Trsic and

Albérico B.F. da Silva Universidade de São Paulo São Carlos, SP, Brazil

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo


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Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK

First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-52781-3

For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in The Netherlands 07 08 09 10 11

10 9 8 7 6 5 4 3 2 1

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Dedication Electronic, Atomic and Molecular Calculations – Applying the Generator Coordinate Method

I dedicate this book to my daughter Carmina and my sons Marcos and Manuel. – Milan Trsic I dedicate this book to my wife Sandra, my daughters Priscilla, Helen, and Vanessa, and my son Kevin. – Albérico Borges Ferreira da Silva

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Table of Contents PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 2: THE GENERATOR COORDINATE METHOD . . . . . . . . . . . . . . . . .

3

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Background for the Formulation of the Method . . . . . . . . . . . . . . . . . . . . . . . . . 3. Formulation of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Applications in Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Some Alternative Proposals to the Generator Coordinate Method . . . . . . . . . . .

3 3 4 5 6

CHAPTER 3: ANALYTICAL AND NUMERICAL EXPERIMENTS FOR SIMPLE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Analytical Solutions for the Griffin–Hill–Wheeler Equation . . . . . . . . . . . . . . . 3. Numerical Experiments for the Griffin–Hill–Wheeler Equation . . . . . . . . . . . . .

9 9 14

CHAPTER 4: THE GENERATOR COORDINATE HARTREE–FOCK FORMALISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Background of the Hartree–Fock Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Generator Coordinate Hartree–Fock Method . . . . . . . . . . . . . . . . . . . . . . . . 4. Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. First Applications to the He and Be Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The He Atom with a Slater Orbital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The He Atom with a Gaussian Orbital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The Be Atom with GTOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 22 25 25 26 26

CHAPTER 5: DISCRETIZATION TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2. A Model Problem: The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. Discretization of the Griffin–Hill–Wheeler Equation for the Harmonic Oscillator Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


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4. The Integral Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The Harmonic Oscillator with Translated Gaussians . . . . . . . . . . . . . . . . . . 4.3. The Hydrogen Atom with a Gaussian Generator Function . . . . . . . . . . . . . . 5. A New Proposal for the Discretization of the Griffin–Hill– Wheeler–Hartree–Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The Polynomial Integral Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 38 42 44 46 46 49

CHAPTER 6: ROLE OF THE WEIGHT FUNCTION IN THE DESIGN OF EFFICIENT BASIS SETS FOR ATOMIC AND MOLECULAR NONRELATIVISTIC CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Weight Function and the Generation of Universal Basis Sets . . . . . . . . . . . . . . . 2.1. Slater and Gaussian Universal Basis Sets for the Ground and Certain Low-lying Excited States of the Neutral Atoms from Hydrogen to Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Slater and Gaussian Universal Basis Sets for the Ground and Certain Low-lying Excited States of Positive and Negative Ions of the Atoms from Hydrogen to Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Role of the Weight Functions in the Evaluation of Total Electronic Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Is the Generator Coordinate Weight Function a Distribution? . . . . . . . . . . . . . . . 4. The Future of Generating Basis Sets for Atomic and Molecular Calculations Using the GCHF Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 56

56 61

65 72 75 76

CHAPTER 7: THE GENERATOR COORDINATE DIRAC– FOCK METHOD AND RELATIVISTIC CALCULATIONS FOR ATOMS AND MOLECULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Generator Coordinate Dirac–Fock–Coulomb Formalism . . . . . . . . . . . . . . . 3. The Generator Coordinate Dirac–Fock Method and the Generation of a Universal Gaussian Basis Set for the Relativistic Closed-Shell Atoms from Zinc to Nobelium . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The Generator Coordinate Dirac–Fock Method and the Generation of a Relativistic Universal Gaussian Basis Set for Atoms from Hydrogen to Nobelium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Variational Prolapse Analysis for the Relativistic Universal Gaussian Basis Set Generated with the Generator Coordinate Dirac–Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Generator Coordinate Dirac–Fock–Breit Formalism . . . . . . . . . . . . . . . . . .

79 80

87

93

100 108


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Table of Contents 6. A Polynomial Version of the Generator Coordinate Dirac–Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The Polynomial Version of the Generator Coordinate Dirac–Fock Method and the Generation of Relativistic Adapted Gaussian Basis Sets . . . . . . . . . . . . 7.1. Relativistic Adapted Gaussian Basis Sets for Hydrogen through Xenon . . . 7.2. Relativistic Adapted Gaussian Basis Sets for Cesium through Radon . . . . .

ix

113 119 119 135

CHAPTER 8: THE GENERATOR COORDINATE METHOD AND CONNECTIONS WITH NATURAL ORBITALS AND DENSITY FUNCTIONAL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Natural Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. An Integral Transform View of Natural Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . 4. Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. First Applications of the Generator Coordinate Method to Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 151 156 157

FINAL REMARKS AND PERSPECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 APPENDIX: SELECTED UNIVERSAL AND ATOM-ADAPTED SLATER AND GAUSSIAN BASIS SETS FOR ATOMIC AND MOLECULAR CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Appendix 1 Appendix 2 Appendix 3 Appendix 4 Appendix 5

...................................................... ...................................................... ...................................................... ...................................................... ......................................................

167 169 171 175 179

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299


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Preface In 1975, during my first visit to Brazil (at that time I was a Research Associate at the University of Calgary, Calgary, Canada), I was a Visiting Professor for two months at the Federal University of Pernambuco, Recife, Brazil. I collaborated with Ron Adler, from the United States of America, and we published a paper in The Journal of Chemical Physics on perturbation theory. At that time, Reiner Dreizler, from the University of Frankfurt/M, was also visiting Recife. Reiner being a physicist tried to get me interested in the hot fields in physics. Among other subjects he told me that the generator coordinate (GC) method raised interest in the physics community. We agreed that the application of the GC method should be feasible for electronic structure as well. So, in 1978, Reiner invited me as a Visiting Professor for six months to Frankfurt/M (1978 was also the year I moved to a permanent position in São Carlos, São Paulo, Brazil). At that time, Reiner was the Head of the Institut für Theoretische Physik. As a result, together with Chattopadhyay and Fink we published applications of the GC method to various model problems, with emphasis in discretization techniques for the solution of the Griffin–Hill–Wheeler (GHW) equation, and later on the study of one- and two-electron atoms in an electric field. At that time, the GC method was not a high-priority research subject for either of us and the mutual visits also served the additional purpose of cultivating our new friendship. Our collaboration culminated in our communication (with my student Mohallem) to the 1986 Sanibel Symposia in Florida. That paper is the origin of what we now call the generator coordinate Hartree–Fock (GCHF) method. I must acknowledge that in those days there were other groups applying the GHW equation or other integral transform methods to nonnuclear problems. Hoping not to omit anybody, I remember Thakkar and Smith in Kingston, Canada; Lathouwers and Van Leuven in Antwerp, Belgium; Laskowski and Brändas in Uppsala, Sweden; Galetti and Toledo Pizza in São Paulo, Brazil; and Somorjai and Bishop in Ottawa, Canada. At this point, Reiner decided that he was not interested in getting involved further in electronic structures, so I dare to say that the home of the GCHF method stayed in São Carlos. In the following years, several bright graduate students were captivated by the GC theory and its applications, one of them being my colleague and coauthor of this book, Professor Albérico Borges Ferreira da Silva. The emphasis of our work is on the role of the GC weight function in the design of atomic Slater-type functions (STFs) and Gaussian-type functions (GTFs) basis

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sets, both universal and atom-adapted. In the Appendix, at the end of the book, we present our best STF and GTF basis sets, either for relativistic or nonrelativistic calculations. Along the chapters of the book, the reader will note that there are open questions, as is normal in any field of knowledge. One of them is certainly about a better understanding of the weight function itself. I would like to conclude with some other personal memories. Two scientists who had a profound influence on my career were Professor Raymond Daudel, my doctorate supervisor in Paris in 1966, and Professor Per-Olov Löwdin, my supervisor during my post-doctorate in Uppsala, 1972–1973. Both were very considerate men and inspired scientists and great educators of many generations. Daudel and others started the Centre de Mécanique Ondulatoire Appliqué at Rue du Maroc (Daudel had been a student of de Broglie) under poor circumstances after the Second World War (the three of them used to sit on a board supported by two chairs) but with deep knowledge of physics, chemistry, and mathematics. Löwdin at that time created the Quantum Chemistry Group at the Uppsala University, where many students, post-doctoral fellows and Visiting Professors staged and spent sabbatical leaves. Both of them had a very special consideration for young scientists coming from what at that time we called the Third World. Other than my one-year stay in Uppsala, I had the privilege to participate in many of the marvelous summer or winter institutes organized by Per-Olov (Pelle for many of us) in Sweden, Norway, and Florida. Devoted to the GC method, I used to tire Pelle claiming that we could surpass Rayleigh–Ritz. To this, Pelle always said: “when you discretize, you fall back in variation”; and I always replied: “not if you use integral discretization”. M. Trsic São Carlos, October 2006.

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Acknowledgements We wish to acknowledge the substantial contributions to the present day status of the generator coordinate Hartree–Fock (GCHF) and Dirac–Fock (GCDF) theories and applications from master’s and PhD students, PD fellows, and collaborators. We are grateful for the recognition given to our GCHF work, which may be verified by the citations. We would like to acknowledge Professor José Rachid Mohallem, Professor José Ciríaco Pinheiro, Professor Francisco Elias Jorge, Dr. Hebert Florey Martins da Costa, Dr. Rugles Cesar Barbosa, Dr. Moacyr Comar Jr., Dr. Luiz Guilherme Machado de Macedo, and Dr. Roberto Luiz Andrade Haiduke for their very special collaborations in the development and initial application of the GCHF method (J. R. Mohallem, H. F. M. da Costa, and J. C. Pinheiro), the GCDF method (F. E. Jorge) and the polynomial generator coordinate Hartree–Fock (pGCHF) and Dirac–Fock (pGCDF) methods (R. C. Barbosa, M. Comar, Jr., L. G. M. de Macedo, and R. L. A. Haiduke). We also wish to state the efficient, helpful, and friendly dialog with representatives of Elsevier. We ought to mention Ms. Joan Annuels, who was our contact and adviser for several months and when submitting the final manuscript. Ms. Angela Marcia Deriggi Silva, our secretary of many years, typed most of the chapters efficiently, but, for family reasons, moved to another city before completion of the project. Then, our former master’s student Wagner Fernando Delfino Angelotti, became responsible for finishing the typing and also drawing the figures in TIF format. Wagner merits some further comments. When Wagner entered the graduate programs in Physical Chemistry at the Institute of Chemistry of São Carlos, University of São Paulo, under the supervision of one of us (MT), he had received a bachelor’s degree in Applied Mathematics from the Federal University of São Carlos. My God, if that is what mathematicians call “applied”, we can only hint at what they would consider “pure” mathematics. Anyway, Wagner had a solid background in Mathematics, was creative, and willing to learn. He passed the entrance examination in General Chemistry with such good marks that he received a fellowship during all the time he worked on his master’s degree, and later approved Chemical Thermodynamics and Quantum Chemistry subjects at the graduate level. Our research had much to gain from his insights of Mathematical Physics. We thank Dr. Flávia Pirola Rosselli and the graduate students Francisco das Chagas Alves Lima and Ranylson Marcello Leal Savedra for helping Wagner with the typing in the rush of the last days and for the final preparation of the appendix and figures.

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Acknowledgements

We certainly appreciate that Professor Roy Edward Bruns made the final reading of the English language. The Brazilian agencies CNPq, FAPESP, and CAPES have generously supported our research for long many years. The Instituto de Química de São Carlos and the University of São Paulo have provided excellent conditions for the development of our research work.

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Chapter 1

Introduction

What we now call Quantum Chemistry was initiated in the thirties of the last century by Hartree, Fock, Born, Oppenheimer, Slater, and, perhaps, a few others. This was possible in the background of Quantum Mechanics, created in a spectacular intellectual explosion between 1926 and 1930 by a group of scientists in various countries in Europe. Since then, the evolution has been spectacular in new methodologies, concepts, and calculational capability: configuration interaction, many body perturbation theory (MBPT), density functional theory (DFT), etc. Everyday, more powerful calculational accuracy and prediction ability has closely followed the evolution of the speed and capacity of computers. Mainframes and personal computers are competing for the market (the last gaining territory in some areas, as Quantum Chemistry) and, recently, clusters may be making the bridge between these two types of hardware. Perhaps the main representative of the enormous progress in computational capacity was the late Nobel Prize winner Professor Pople and the Gaussian package created by him and his large group of students and collaborators in Pittsburgh. In the middle of the last century, an innovative concept was introduced by Wheeler and collaborators in the context of Nuclear Physics. They were trying to understand the collective motion of nucleons within the nuclei, so they introduced an parameter controlling the limits of the confinement of the intranuclear particles. This procedure is known as the generator coordinate method (GCM), being the generator coordinate, an integration parameter. Soon it was understood that the GCM was a variational procedure and that the integral transform equation obtained by Wheeler and coworkers led to a more powerful tool than the Rayleigh–Ritz (RR) variational method (well known and used to optimize exponents for Slater-type or Gaussian-type functions, linear combination coefficients for Roothaan expansions for atoms and, mainly, molecules, and configuration interaction coefficients for different states). In the RR variation, the trial function depends on one or various parameters, , and the RR variational procedure would find the best value of , say 0, so as to attain the lower energy value of the system, within this framework.


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The method developed by Wheeler and collaborators is more powerful than the former. One starts again with a trial function, , but the GCM algorithm is an integro-differential equation, often called the Griffin–Hill–Wheeler (GHW) equation, leading to an integral transform of the function, say,

GHW ansatz

(1.1)

thus, a new function is obtained. This function provides an energy that is a lower bound to the RR minimum and the function ⌽ is, in principle, a better approximation than 0 to the exact solution of that particular Hamiltonian. It is apparent that such a powerful tool requires nontrivial mathematics, so it is not surprising that very few analytical solutions for the GCM have been worked out (in fact, there are not many analytical solutions for RR either). In general, there is a need to resort to some kind of approximation scheme. In this book we present the GCM and its applications in Quantum Chemistry. Chapter 2 presents the method as it was introduced in the context of Nuclear Physics. Chapter 3 describes some analytical and numerical experiments for simple systems. Chapter 4 introduces the generator coordinate Hartree–Fock method formulation and some of its applications. Chapter 5 shows discretization techniques and other approximate schemes for the solution of the GHW equation. Chapter 6 discusses technical aspects of the application of the weight function for atomic basis set design and shows the applications for molecular systems. Chapter 7 introduces the generator coordinate Dirac–Fock formalism and its applications in atomic and molecular relativistic calculations. Chapter 8 recognizes connections of the generator coordinate with natural orbitals and the DFT. Finally, in the Appendix we show some selected basis sets, both relativistic and nonrelativistic, for quantum chemical calculations.


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Chapter 2

The Generator Coordinate Method

1.

Introduction

The Hartree–Fock theory [1], at its limit, may provide about 98% of the energy of an atom or molecule. Still, we wish for better, not only in the search for testing Quantum Mechanics, but also because the “small” 2% error may have the magnitude of an ionization potential or an electronic transition. Several alternatives are available if we endeavor to recover the missing portion of energy (correlation energy), such as many body perturbation theory (MBPT), density functional theory (DFT), and the variational configuration interaction (CI) method, often at the cost of nontrivial computational efforts. Wheeler and collaborators [2], in the context of Nuclear Physics, showed in 1953–1957 that the limit in the variational procedure capacity itself was not reached. As we indicated in the Introduction (Chapter 1), the generator coordinate method (GCM) introduces an integral transform capable, in principle, of finding the best functional form for a given trial function through the Griffin–Hill–Wheeler (GHW) integral equation defined below.

2.

Background for the Formulation of the Method

The GCM was introduced [2] in the field of Nuclear Physics. The proposition of Wheeler and collaborators was one of the first attempts to incorporate collective and single-particle nuclear motions into a single coherent quantum-mechanical formulation. The parameter, which plays an important role in the method, is initially introduced as a shape parameter of the nuclear liquid drop model, defining the size and shape of the drop. Physics and chemistry have been interacting and feeding each other with questions and answers for centuries. However, the speed of interpenetration is variable. Thus, until the introduction of what we now call the generator coordinate Hartree–Fock (GCHF) method for atoms and molecules in 1986 [3] (see Chapter 4), the major part of the literature on the GCM dealt with the collective aspects of nuclei. It started with the classical paper of Hill and Wheeler [2a], which aimed at relating collective and single-particle aspects in the fission problem. The direct


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Chapter 2

application to the bound state case was pioneered by Griffin and Wheeler [2b], who clearly recognized the GCM as a variational procedure. In Section 4 we review some of the literature in Nuclear Physics.

3.

Formulation of the Method

Here, we shall follow closely the method presented by Griffin and Wheeler in 1957 [2b]. We search for a solution to the Schrödinger equation: H ( x ) ( x ) ⫽ E ( x ),

(2.1)

where H is the Hamiltonian operator, the eigenfunction, E the energy of the system, and x represents the space and spin coordinates. Then, a trial function ⌽(x ; ) is chosen, where represents one or several generator coordinates. The trial function ⌽ may be an approximate solution of Equation (2.1), or the exact solution of a problem similar to Equation (2.1), or some other function appropriate for the case. Next, the integral transform function is built: ( x ) ⫽ ∫ d f () ( x ; ),

(2.2)

where f () is the weight function that needs to be determined. If the exact f () can be determined, then the integral transform in Equation (2.2) leads to the exact solution . The energy functional E can now be written, E ⫽ ∫ dx ⴱ ( x ) H ( x ) ( x ) 冫 ∫ dx ⴱ ( x ) ( x ).

(2.3)

The use of Equation (2.2) in Equation (2.3) gives E ⫽ ∫ f ⴱ () H (, ) f ( ) d d 冫 f ⴱ () S (, ) f ( ) d d ,

(2.4)

with the energy H(, ) and overlap S (, ) kernels defined as: H (, ) ⫽⬍ ( x, ) 冷 H 冨 ( x, ) ⬎⫽ ∫ dx ⴱ ( x, ) H ( x, )

(2.5)

S (, ) ⫽⬍ ( x, ) ( x, ) ⬎⫽ ∫ dx ⴱ ( x, ) ( x, ) ,

(2.6)

and

respectively.


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5

The generator wave function must now be chosen to make the integral an extreme value [see Equation (2.7)]:

∫ d f 0 ⫽ E ⫽

ⴱ

()∫ d 关 H (, ) ⫺ ES (, )兴 f ( ) ⫹ comp.conj.

∫f

ⴱ

() S (, ) f ( )d d

(2.7)

The coefficients of f ⴱ() and f ⴱ() must vanish separately, because these are two linearly independent variations. Thus, one arrives at the generator wave equation (often called GHW equation):

∫ 关 H (, ) ⫺ ES (, )兴 f ()d ⫽ 0.

(2.8)

The analytical solution of the GHW equation for many electron atoms and molecules (or many particle systems in general) is beyond present mathematical capabilities. Thus most applications have relied on either approximations, which is the case for nuclei, or discretization techniques, as in the case of atoms and molecules (see Chapter 5). There seems to be only a few analytical solutions for the GCM, which we comment upon in the next chapter.

4.

Applications in Nuclear Physics

Certainly the early and numerous applications of the GCM arose in the field of Nuclear Physics. From the very beginning, the Nuclear Physics community gave preference to the Gaussian overlap approximation (GOA) for the solution of the GHW equation. In the GOA [4], the overlap kernel [Equation (2.6)] is replaced by a Gaussian function of the form ⎧⎪ 1 ⎡ ⫺ ⎤ ⎫⎪ S (, ) ⬵ SGaussian (, ) ⫽ exp ⎨⫺ ⎢ ⎥⎬, ⎩⎪ 2 ⎣ a( ) ⎦ ⎭⎪

(2.9)

where the width is often chosen as a function of the average of the GCM generator coordinate 苶 ⫽ ( ⫹ )/2. In the seventies, attempts were made to mobilize the GCM for the scattering problem of complex particles as an alternative to the resonating group method [5]. Considerable literature for the bound state has been reviewed by Klein [6], Villars [7], Brink [8], Mihailovich and Rosina [9], Wong [10], and, with specific emphasis on the scattering aspects, Giraud et al. [11]. We also refer to a conference held in Belgium in 1975 summarizing this topic [12].


6

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Chapter 2

More recent revisions may be found in the book by Ring and Schuck [13] and the review by Bender et al. [4].

5. Some Alternative Proposals to the Generator Coordinate Method In 1968, Somorjai and, later, Somorjai and Bishop [14] introduced an integral transform method, closely related to GCM, for atomic and molecular systems. Somorjai was aware of the work of Wheeler and collaborators, but was critical of the use of Gaussian distributions for solving the integral equation. The procedure to solve the integral transform employed by Somorjai and Bishop was to choose a definite form for the weight function and then to use the upper and lower integration limits as variational parameters. In his first work on the subject [14a], Somorjai employed Hulthen functions to represent 1s orbitals following a proposition by Parr and Wave [15]. The two-parameter Hulthen function, F (r), has the form F ⫽ r⫺1 冤e⫺r ⫺ e⫺r 冥 .

(2.10)

The identity,

F ( r ) ⫽ r⫺1 冤e⫺ r ⫺ e⫺r 冥 ⫽ ∫ e⫺rx dx ,

(2.11)

shows that F is a linear combination of an infinite number of screened 1s functions, with orbital exponents ranging continuously in a variationally optimized interval [, ]. Each 1s orbital has the same weight. Further, Somorjai rewrites the finite integral transform [Equation (2.10)] as a general Laplace transform -

F ( r ) ⫽ ∫ f ( x ) e⫺r x dx ,

(2.12)

0

thus recovering the form of the trial function of the GCM. In work to follow, Somorjai and other authors did not enter the GCM ansatz, but treated the function f (x) in Equation (2.12) as having adjustable parameters. An extensive review of this method was later organized by Bishop and Schneider [16]. Also, in this context, accurate correlated functions for two- and three-electron atomic systems were obtained by Thakkar and Smith [17]. While the former solutions were certainly original and innovative, they masked to some extent the full power of the integral transform. Still, they brought forward the possibility of the weight function being a distribution (the distribution character


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7

of the weight function will be discussed in Chapter 6 of this book). Furthermore, their proposal of a weight function for the transform of a Gaussian orbital to a Slater orbital [14c] provided the path that facilitated one of the very few analytical solutions for the HW equation (see Chapter 3). Some findings are often rediscovered, bringing always some new insight or perspective. Thus, in 1991 Flores put forward a proposition to calculate orbitals as integrals over the exponential parameters [18] with a new set of basis functions, which, as the authors recognize, are too complicated to be competitive with current bases employed in ab initio calculations. Still, the proposal is valid as an experiment, although, as one may expect, numerical problems are not eliminated. In 1999 we proposed [19] that any Roothaan type expansion could be regarded as the discretization of an integral formulation. Let us mention one parallel development of the GCM by Laskowski and Brändas [20] who introduced a self-consistent GCM in quantum lattice dynamics; applications show an interesting parallelism with the self-consistent random phase approximation.

References 1. D. R. Hartree, Proc. Cambridge Phys. Soc., 1928, 24, 89; D. R. Hartree, Proc. Cambridge Phys. Soc., 1928, 24, 111; D. R. Hartree, Proc. Cambridge Phys. Soc., 1928, 24, 426; V. Fock, Z. Phys., 1930, 81, 126. 2. (a) D. L. Hill, and J. A. Wheeler, Phys. Rev., 1953, 89, 1102; (b) J. J. Griffin, and J. A. Wheeler, Phys. Rev., 1957, 108, 311. 3. J. R. Mohallem, R. M. Dreizler, and M. Trsic, Int. J. Quantum Chem. Symp., 1986, 20, 45. 4. M. Bender, P. H. Heenen, and P. G. Reinhard, Rev. Modern Phys., 2003, 75, 121. 5. H. Horiuch, Prog. Theor. Phys. (Kyoto), 1970, 43, 375; D. Zaikin, Nucl. Phys. A, 1971, 170, 584; T. Yukawa, Phys. Lett. B, 1972, 38, 1; T. Yukawa, Nucl. Phys. A, 1972, 186, 127; T. Yukawa, Phys. Rev. C, 1973, 8, 1593; N. de Takacsy, Phys. Rev. C, 1972, 5, 1883; F. Tabakin, Nucl. Phys. A, 1972, 182, 497; T. Fliebach, Nucl. Phys. A, 1972, 194, 625; C. W. Wong, Nucl. Phys. A, 1972, 197, 193; B. Giraud, and J. Letourneux, Nucl. Phys. A, 1972, 197, 410; B. Giraud, and J. Letourneux, Phys. Rev. Lett., 1973, 31, 399; P. Boche, and B. Giraud, Phys. Rev. Lett., 1972, 28, 1720; P. Boche, and B. Giraud, Nucl. Phys. A, 1983, 199, 160; W. Glöckle, Nucl. Phys. A, 1973, 211, 372. 6. A. Klein, In: Lectures in Theoretical Phsycis, Vol. 11 B, p. 1, eds. K. T. Mahanthappa, and W. E. Griffin, New York: Gordon and Breach, 1969; and in: Dynamic Structure of Nuclear States (Proc. Mont Tremblant Int. Summer School, 1971), p. 38, eds. D. J. Rowe et al., Toronto: University of Toronto Press, 1972. 7. F. Villars, School of Physics E. Fermi, Course 36, p. 14, 1966; and in: Dynamic Structure of Nuclear States (Proc. Mont Tremblant Int. Summer School, 1971), p. 3, eds. D. J. Rowe et al., Toronto: University of Toronto Press, 1972. 8. D. M. Brink, Proc. Int. School of Physics E. Fermi, ed. C. Bloch, New York, Academic Press, Course 36, p. 247, 1966.


8 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20.

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Chapter 2 M. V. Mihailovich, and M. Rosina (eds), Fizika, 1973, 5 supplement, p. 1. C. W. Wong, Phys. Rep., 1975, 15, 283. B. Giraud, J. LeTourneux, and E. Osnes, Ann. Phys., 1975, 89, 359. P. Van Leuven, and M. Bouten (eds), Proceedings of the 2nd International Seminar on the Generator Coordinate Method, Mol (Belgium), 1975. P. Ring, and P. Schuk, The Nuclear Many-Body Problem, New York: Springer, Inc., 1980. (a) R. L. Somarjai, Chem. Phys. Lett., 1968, 2, 399; (b) R. L. Somorjai, Phys. Rev. Lett., 1969, 23, 329; (c) D. M. Bishop, and R. L. Somorjai, J. Math. Phys., 1970, 11, 1150. R. G. Parr, and J. H. Wave, Prog. Theor. Phys. (Kyoto), 1966, 36, 854. D. M. Bishop, and B. E. Schneider, Int. J. Quantum Chem., 1975, 9, 67. A. J. Thakkar, and V. H. Smith, Phys. Rev. A, 1977, 15, 1; A. J. Thakkar, and V. H. Smith, Phys. Rev. A, 1977, 15, 16; A. J. Thakkar, and V. H. Smith, Phys. Rev. A, 1977, 15, 2143. J. R. Flores, Chem. Phys. Lett., 1991, 182, 200; J. R. Flores, J. Com. Chem., 1992, 13, 1199. H. F. M. da Costa, M. Trsic, A. B. F. da Silva, and A. M. Simas, Eur. Phys. J. D, 1999, 5, 375. B. Laskowski, and E. Brändas, Phys. Rev., 1976, C13, 1741.


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Chapter 3

Analytical and Numerical Experiments for Simple Systems

1.

Introduction

This chapter belongs to the continuation of our presentation on the generator coordinate method (GCM) before we enter into the applications for atomic or molecular systems; indeed, these last applications rely, in one way or another, on approximate schemes, even if often providing reliable results. The very simple problems solved analytically have the merit to show in this chapter the bare ansatz of the solution of the Griffin–Hill–Wheeler (GHW) equation, from which the energy eigenvalue(s) emerges as well as the weight function to be employed in the integral transform in Equation (2.11), allowing to obtain the exact function of the system. As for the numerical experiments shown in this chapter, the purpose is to raise interest in the implementation of approximations other than the discretization techniques preferred in quantum chemical applications or the Gaussian overlap approximation (GOA), which has dominated applications in nuclear physics.

2.

Analytical Solutions for the Griffin–Hill–Wheeler Equation

Perhaps, the first analytical solution for the GCM was a solution for the harmonic oscillator by Lathouwers and collaborators in 1976–1977 [1]. Soon afterwards, in a study of model problems, Dreizler, Trsic, and collaborators presented a trivial analytical solution for the para-helium independent particle case [2]. The Hamiltonian of the system in atomic units is

1 Z Z 1 H ( x1 , x2 ) ⫽⫺ (12 ⫹ 22 ) ⫺ ⫺ ⫹ , 2 r1 r2 r12

(3.1)

where Z is the nuclear charge, r1 and r2 the distance of the electrons to the nucleus, and r12 the inter-electronic distance.


10

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Chapter 3

The most naïve model problem for the generator coordinate approach would be a screened noninteracting two-electron system ⎛ 1 ⎞ ⎛ 1 ⎞ h( x1 , x2 ) ⫽ ⎜⫺ 12 ⫺ ⎟ ⫹ ⎜⫺ 22 ⫺ ⎟ , r1 ⎠ ⎝ 2 r2 ⎠ ⎝ 2

(3.2)

where is the screening parameter. The space symmetric ground-state function is ( x1 , x2 ; ) ⫽

3 ⫺ ( r1 ⫹r2 ) e

(3.3)

and the corresponding generator coordinate ansatz for the L ⫽ 0, S ⫽ 0 state would be ⬁

( x1 , x2 ) ⫽ ∫ d f () ( x1 , x2 ; ).

(3.4)

0

The calculation of the Hamiltonian and overlap kernels is straightforward and can be found in the basic literature on Quantum Mechanics or in References [3,4]. It is amusing that in this case one can show directly that the ground-state solution of the GHW equation is ⎛ 5 ⎞⎞ ⎛ f () ⫽ ⎜ ⫺ ⎜ Z ⫺ ⎟ ⎟ , ⎝ ⎝ 16 ⎠ ⎠

(3.5)

2

5⎞ ⎛ E ⫽⫺ ⎜ Z ⫺ ⎟ . ⎝ 16 ⎠ The solution of the GHW problem coincides thus with the result of the straightforward variational approach using Equation (3.3) as a trial function. This result can be understood in terms of the following statement. Writing the Hamiltonian in Equation (3.1) as

H (Z ) ⫽ H0 (Z ) ⫹

1 , r12

(3.6)

one directly observes that the Hamiltonian kernel satisfies the relation 5⎞ ⎛ H (Z ) ⫽ H0 ⎜ Z ⫺ ⎟ . ⎝ 16 ⎠

(3.7)


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11

The ground-state solution of the independent particle problem with the 5 Hamiltonian H0 冢Z⫺ ᎏ 16 冣 in terms of the GCM is expressed by Equation (3.5). This observation seems to be initially due to Somorjai [5] in connection with integral transform functions. We now focus on another system: the exact solution of the hydrogen atom ground state with a Gaussian trial function [6]; in spite of its seeming simplicity, this is probably the most elaborate analytical mathematical case solved so far for the GCM. The “wrong” Gaussian trial function is intentionally chosen so as to illustrate the power of the GCM, which will bring the exact ground-state hydrogen atom function and energy. Nonetheless, the mathematical effort for this achievement is expressive indeed, indicating that, with present mathematical tools, hardly more complex systems may have such exact kind of solution. Thus, we start with a (unnormalized) Gaussian as trial function, i.e., 2

( r , ) ⫽ e⫺r .

(3.8)

This case was tested before by a discretization technique [2] leading to the value of ⫺0.4994 a.u. and also with an algorithm for the optimal selection of the discretization points [7], which converges to a quasi-exact numerical solution. The ordinary Rayleigh–Ritz (RR) method gives ⫺0.4243 a.u. It is straightforward to calculate the kernels for this case (a.u.)1 H (, ) ⫽ 3 3 Ⲑ 2 ( ⫹ )⫺5 Ⲑ 2 ⫺ 2( ⫹ )⫺1

(3.9)

S(, ) ⫽ 3 Ⲑ 2 ( ⫹ )⫺3 Ⲑ 2 .

(3.10)

and

Lathouwers and Van Leuven [8] concluded that a direct solution of the GHW equation with the kernels above was not feasible. Through an integral Hellmann–Feynman formulation, Hurley [9] arrived to an integral equation equivalent to the GHW equation, with kernels Equations (3.9) and (3.10), although resorting to a numerical scheme for its solution. Being the Laplace operator Hermitian, we can apply it for the calculation of H(, ) in the following three equivalent manners:

1

∫ ⴱ ( r, )(

2

( r , )) d ⫽ ∫ (( r , ) ) ⴱ ( ( r , )) d ⫽ ∫ (( r , ) 2 ) ⴱ ( r , ) d

Curiously, three different explicit expressions (strictly symmetric) in and will be obtained. It is an interesting exercise in quantum mechanics to verify that in all three cases the very same final solution will be obtained.


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Page 12

Chapter 3

In their integral-transform treatment for Gaussian functions for He-like atoms, Bishop and Somorjai [5c] arrived at the Laplace transform ⬁

g ( r ) ⫽ ∫ e⫺r G ( x ) dx , 2

(3.11)

0

where 2

e⫺q Ⲑ 4 x . x u⫹1

G( x) ⫽

(3.12)

Mohallem and Trsic [6] took advantage of the Laplace transform, Equations (3.11) and (3.12), and demonstrated that with the values q ⫽ 1 and u ⫽ ⫺ ᎏ21ᎏ the exact solution for the weight function and the ground-state wave function were found: ⬁

e⫺r ⫽ ∫ f () e⫺r d , 2

(3.13)

0

where f () ⫽

1 2

⫺3 Ⲑ 2 e⫺(1Ⲑ 4 ) .

(3.14)

The result for the ground-state energy requires some additional effort too. One can now use Equations (3.14), (3.9), and (3.10) to integrate the GHW equation [see Equation (2.11)] explicitly: ⬁

∫ H ( , ) f ( ) d ⫽ 0

⬁

3 ⫺1Ⲑ 2 ( ⫹ )⫺5 Ⲑ 2 e⫺1Ⲑ 4 d 2 ∫0

⫺

⫺1 Ⲑ 2

⬁

∫

⫺3 Ⲑ 2

(3.15)

( ⫹ )⫺1 e⫺1Ⲑ 4 d

0

and ⬁

∫ S ( , ) f ( ) d ⫽ 0

⬁

⫺3 Ⲑ 2 ( ⫹ )⫺3 Ⲑ 2 e⫺1Ⲑ 4 d . ∫ 20

(3.16)

⫽ Ⲑy,

(3.17)

With the change of variables


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13

the GHW equation becomes ⬁

⎛ ⎞ 3 ∫ ( y ⫹1)⫺5 Ⲑ 2 ye⫺ y Ⲑ 4 dy ⫺ 2 ⎜ ⎟ ⎝ ⎠ 0 ⬁

⫽ ∫ ( y ⫹1)

⫺3 Ⲑ 2

⫺ y Ⲑ 4

ye

1Ⲑ 2 ⬁

∫ ( y ⫹1)

⫺1

y1Ⲑ 2 e⫺ y Ⲑ 4 dy

0

(3.18)

dy .

0

The above equation for can now be solved with definite integral formulae [10]: ⬁

∫X

v⫺1

( X ⫹ )⫺v⫹1Ⲑ 2 e⫺ X dX ⫽ 2v⫺1Ⲑ 2 Γ( v ) ⫺1Ⲑ 2 e Ⲑ 2 D1⫺2 v 共 2 兲 (3.19)

0

and ⬁

∫X

v⫺1

( X ⫹ )⫺v⫺1Ⲑ 2 e⫺ X dX ⫽ 2v Γ( v ) ⫺1Ⲑ 2 e Ⲑ 2 D⫺2 v 共 2 兲.

(3.20)

0

The Dp(z) are parabolic-cylinder functions and obey the recursion relation D p⫹1 ( z ) ⫺ zD p ( z ) ⫹ pD p⫺1 ( z ) ⫽ 0.

(3.21)

Application of Equations (3.19) and (3.20) leads to the integration of Equation (3.18), i.e., 3D⫺4 共1冫 2 兲 ⫺ D⫺2 共1冫 2 兲 ⫺ 2 Ⲑ D⫺3 共1冫 2 兲 ⫽ 0.

(3.22)

1 The relation Equation (3.21) for p ⫽⫺ 3 and z ⫽ ᎏ gives 兹苶2苶 3D⫺4 共1冫 2 兲 ⫺ D⫺2 共1冫 2 兲 ⫹1冫 2 D⫺3 共1冫 2 兲 ⫽ 0.

(3.23)

Comparison of Equations (3.22) and (3.23) gives ⫽⫺

1 . 2

On the one hand, the seemingly trivial transform of a Gaussian to an exponential and the correct hydrogen atom ground-state eigenvalue gives beautiful mathematical work with tools from the last century and perhaps from the 1800s.


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Chapter 3

But the task was hard and one can feel that for more complicated systems we will have to recur to other schemes. On the other hand, in the prize of the GCM we comment that the common RR variation of a Gaussian gives a very poor approximation for the ground-state energy and not even a hint for the discovery of a different functional form.

3.

Numerical Experiments for the Griffin–Hill–Wheeler Equation

In the subsequent chapters, we show what has been achieved with our present mathematical capacities. Still, in what follows of this chapter we put forward some nonconclusive ideas for possible alternative paths of numerical or analytical attempts to the solution of the GCM problem. Let us first examine where integration by parts could lead us. We consider that in Equations (3.9) and (3.10) H(, ) and S(, ) are Hermitian and the former as well as f () belong to class C1 of functions (meaning basically continuous and that the first derivative exists). Integration by parts of Equation (2.11) gives

∫ d f () 关 H (, ) ⫺ ES (, )兴 ⫽ f ()关 H (, ) ⫺ ES (, )兴

f ( ) ⫺∫ d 关 H (, ) ⫺ ES (, )兴 ,

(3.24)

which we choose to write as 关 H (, ) ⫺ ES (, )兴 f ( ) ⫹∫ d 关 H (, ) ⫺ ES (, )兴 .

f ( )关 H (, ) ⫺ ES (, )兴 ⫽ ∫ d f ( )

(3.25)

On the other hand, the derivation of Equation (2.11) leads to

∫ d

f ( ) 关 H (, ) ⫺ ES (, )兴 ⫹ ∫ d f ( ) 关 H (, ) ⫺ ES (, )兴 ⫽ 0. (3.26)

Comparison of Equations (3.25) and (3.26) gives f ( )关 H (, ) ⫺ ES (, )兴 ⫽ 0,

(3.27)

H (, ) ⫺ ES (, ) ⫽ 0

(3.28)

thus

otherwise the trivial solution f() ⫽ 0 would prevail.


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15

Equation (3.28) may be regarded as the continuous generalization of the secular equation. In fact, in 1999 da Costa and collaborators [11] put forward the notion that any liner combination of functions, e.g., Roothaan-type expansion, may be regarded as an implicit numerical integration of the GHW equation; this observation will be further considered in Chapter 6, where we discuss the role of the weight function. Thus, me may write E⫽

H ( , ) S ( , )

(3.29)

with the understanding that S(, ) ⫽ 0 and that E ought to be independent from and at the end of any process in the search of the exact solution. In what follows, we perform a numerical experiment with the same model case described in Section 2 for the hydrogen atom. With Equations (3.9) and (3.10) in this chapter for H(, ) and S(, ), we obtain the explicit expression E(, ) ⫽

3 ⫺ 2( ⫹ )1Ⲑ 2 ⫺ 2( ⫹ )1Ⲑ 2 ( ⫹ )

.

(3.30)

As a further experiment, we test the form of E(, ) for arbitrary positive values of and , i.e., (0.0,1.0) and (0.0,1.4). In Fig. 3.1, we present the resulting graph that appears as a saddle-like surface. The saddle critical point

Fig. 3.1 The E(, ) surface for arbitrary positive values for and . Energy is expressed in atomic units (a.u.).


16

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Chapter 3

leads to the values 0 ⫽ 0.2829421209 and 0 ⫽ 0.2829421210 and E( 0, 0) = ⫺0.4244131813 a.u. This value is very close to the RR value of ⫺0.4243 a.u. To this point, we have relied largely on our recent work published in Advances in Quantum Chemistry [12]. Nonetheless, it is legitimate to inquire whether in Fig. 3.1 of Reference [12] the full potential of Equation (3.30) was achieved. This equation is supposed to be exact for the ground-state energy of the hydrogen atom, thus there should be a manner to extract the exact value from it. We may further examine Equation (3.30). The previous paragraphs indicate that values for between 0.0 and 1.0 produce negative values for the energy. We arbitrarily set ⫽ 1.0 and in Fig. 3.2 we plot the exact energy En ⫽ ⫺1/2n2 and E(1.0, ). The similarity between the two curves suggests that not only the ground-state energy may be generated by this experiment, but excited-state energies as well. As a continuation of our experiments, we set the following equation in a.u., i.e., 1 3 ⫺ 2(1⫹ )1Ⲑ 2 ⫺ 2(1⫹ )1Ⲑ 2 ⫽ E (1, ) ⫺ 2⫽ 2n (1⫹ )

(3.31)

so that we force E(1, ) to be equal to the exact expression for the energy. We then scan values of n ⫽ 1, 2, … and find the respective optimal values for . These values are then used in Equation (3.30) to obtain approximate eigenvalues for the excited states. We remark that for n ; ⬁, converges to a value of 1.4. Table 3.1 shows the exact energies for the ground and the first 11 excited levels of the hydrogen atom together with the values calculated through Equations (3.30)

Fig. 3.2

Plot of E(1, ) and En for various values of and n.


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17

Table 3.1 Comparison between exact and calculated hydrogen atom eigenvalues. Atomic units are employed throughout Eexact (a.u.)

Ecalc (a.u.)

⫺0.500000000000 ⫺0.125000000000 ⫺0.055555555556 ⫺0.031250000000 ⫺0.020000000000 ⫺0.013888888889 ⫺0.010204081632 ⫺0.007812500000 ⫺0.006172839506 ⫺0.005000000000 ⫺0.004132231405 ⫺0.003472222222

⫺0.5000000000 ⫺0.1250000000 ⫺0.0555555556 ⫺0.0312500000 ⫺0.0200000000 ⫺0.0138888889 ⫺0.0102040816 ⫺0.0078125000 ⫺0.0061728395 ⫺0.0050000000 ⫺0.0041322314 ⫺0.0034722222

0.3795056641 0.9225340173 1.128946441 1.225896227 1.278060772 1.308955429 1.328611873 1.341829809 1.351116767 1.357877624 1.362945568 1.366838592

The purpose of this chapter is to suggest to the interested reader that there might still be open routes for further mathematical insights into GCM.

and (3.31), i.e., E(1, ), and the corresponding values of for each case. As may be observed, E(1, ) mimics very accurately the exact values.

References 1. (a) L. Lathouwers, Ann. Phys., 1976, 102, 347; (b) L. Lathouwers, P. Van Leuven, and M. Bouten, Chem. Phys. Lett., 1977, 52, 439. 2. P. Chattopadhyay, R. M. Dreizler, M. Trsic, and M. Fink, Z. Phys. A, 1978, 285, 7. 3. E. A. Hylleraas, Z. Phys., 1929, 54, 347. 4. C. Eckart, Phys. Rev., 1930, 36, 878. 5. (a) R. L. Somorjai, Chem. Phys. Lett., 1968, 2, 399; (b) R. L. Somorjai, Phys. Rev. Lett., 1969, 23, 329; (c) D. M. Bishop, and R. L. Somorjai, J. Math. Phys., 1970, 11, 1150. 6. J. R. Mohallem, and M. Trsic, Z. Phys. A, 1985, 322, 535. 7. F. Arickx, J. Broekhove, E. Deumens, and P. Van Leuven, J. Chem. Phys., 1981, 39, 272. 8. L. Lathouwers, and P. Van Leuven, Adv. Chem. Phys., 1982, 49, 115. 9. A. C. Hurley, Int. J. Quant. Chem. Symp., 1967, 1, 677. 10. I. S. Gradshtein, and I. M. Ryzhik, Tables of Integrals, Series and Products, New York: Academic Press, 1965. 11. H. F. M. da Costa, M. Trsic, A. B. F. da Silva, and A. M. Simas, Eur. Phys. J. D, 1999, 5, 375. 12. M. Trsic, W. F. D. Angelotti, and F. A. Molfetta, Adv. Quantum Chem., 2004, 47, 315.



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Chapter 4

The Generator Coordinate Hartree–Fock Formalism

1.

Introduction

In spite of its birth in nuclear physics [1], the generator coordinate method (GCM) found application also in other areas of physics [2–5]. It was probably only a matter of time that somebody would extend the integral transform idea of Griffin, Hill, and Wheeler to the one-electron functions of the Hartree–Fock (HF) scheme. So, at the 1986 Sanibel Symposia in Florida, Mohallem, Dreizler, and Trsic [6] presented the Griffin–Hill–Wheeler (GHW) version of the Hartree–Fock (GHWHF) equations [7]. This proposal opened a new route for very accurate calculations for atomic and molecular systems, including ions. In this work, example applications were for the He and Be atoms; it is interesting that from the beginning, the authors emphasized the role of the weight function, which actually has characterized the applications of the GCM. We also remark that the kernels in Reference [6] were calculated analytically. Below we show, in part, the original proposition of Reference [6]. In Reference [6], the authors advocated for what was called variational discretization (see Chapter 5), which consisted in finding first the variational minimum for the kernel, and then to choose discretization points on the left and right of the optimal value. The strategy in Chapter 5 is to attempt the best numerical integration [6,8] of the GHW equation [see Equation (2.11)] in what since 1986 we call integral discretization.

2.

The Background of the Hartree–Fock Scheme

In 1913, Bohr [9] presented his solution for the hydrogen atom as one of the most important pillars of what we now call Old Quantum Theory. The energies calculated with the Bohr model coincide with the exact values as obtained from the Schrödinger equation presented some 10 years later [10]. Of course, both Bohr and Schrödinger were aware of the ideas of the discrete character of spectra brought out by Planck [11]. Schrödinger, in particular, had


20

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Chapter 4

the inspiration of the attribution of wave character to the electron by Louis de Broglie in his thesis in 1924 [12]. These few years at the beginning of the 20th century were of an intense intellectual activity in physics, mainly in Europe. Thus, since the Old Quantum Theory required several numerical adjustments to meet the criteria for the experiment, in 1925 Heisenberg [13] presented the formalism of his Quantum Mechanics, which allowed the treatment of both intensities and frequencies of spectral lines of atomic systems. In the same year, Pauli [14] formulated his exclusion principle, which required many-electron functions to be antisymmetric. It seems that Slater [15] was the first to write a many-electron wave function as a determinant, thus providing a well-known and easy-to-handle algebraic structure for these functions. In 1928, Hartree [7a] created the self-consistent field method, in which many-electron systems are represented by one-electron functions, each depending on the average field produced by the other electrons. Nevertheless, Hartree employed a simple product of one-electron functions, which lacks antisymmetry. Fock [7b] modified the self-consistent field method so as to include exchange, which originates from antisymmetry of the many-electron function. Thus, this is the origin of the so-called Hartree–Fock scheme, which here we rewrite in the framework of the GCM. The purpose of this section is to provide a background for the theory to be introduced in the following section, and certainly not to discuss in full the history or formulation of Quantum Mechanics; the interested reader is referred to some appropriate literature [16]. Nevertheless, we do not wish to abandon this section without acknowledging the probabilistic interpretation of Quantum Mechanics as developed by Born [17] and the uncertainty principle enounced by Heisenberg [18]. The HF one-electron eigenvalue equation may be written as ⎪⎧ ⎪⎫ ⎨h0 (1) ⫹ ∑ ⎡⎣ J j (1) ⫺ K j (1) ⎤⎦ ⎬ i (1) ⫽ i i (1), j ⎩⎪ ⎭⎪

(4.1)

where h0(1) is the sum of the one-electron kinetic energy operator and attractive interactions of the electron with the atomic nucleus or the molecular nuclei, with the Coulomb Jj (1) and exchange Kj (1) operators defined as ⎤ ⎡ e2 J j (1) i (1) ⫽ ⎢ ∫ ⴱj (2) j (2) d 2 ⎥ i (1) r12 ⎦ ⎣

(4.2)


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21

and ⎤ ⎡ e2 K j (1) i (1) ⫽ ⎢ⴱj (2) i (2) d 2 ⎥ j (1). r12 ⎦ ⎣

(4.3)

The i and j are one-electron spin-functions. The i Lagrange multipliers acquire in the HF theory the significance of one-electron energy eigenvalues, with e being the electron charge. Equation (4.1) results from expressing the expectation value for the energy with a Slater determinant for the ground state and imposing the variational principle. In the following section, we extend the HF one-electron functions to integral transform one-electron functions.

3.

The Generator Coordinate Hartree–Fock Method

We choose the spatial one-electron functions for a 2n-electron system in the form i (1) ⫽ ∫ i (1, ) f i () d , i ⫽ 1,… , n ,

(4.4)

where i (1, ) (STOs or GTOs) depend on the generator coordinate, , and fi () are the unknown weight functions. With the trial function written as a Slater determinant of the above orbitals, the minimization of the total energy with respect to the weight functions fi () leads to the GHWHF equations

∫ 关 F (, ) ⫺ S (, )兴 f ( ) d ⫽ 0, i

i

i ⫽ 1,… , n .

(4.5)

The overlap and Fock kernels are, respectively, S (, ) ⫽ i (1, ) 冏 i (1, )

(4.6)

and n

F (, ) ⫽ h(, ) ⫹ ∑ ⎡⎣ 2 J j (, ) ⫺ K j (, ) ⎤⎦ , j

(4.7)


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Chapter 4

where h (, ) ⫽ i (1, ) h0 (1) i (1, )

(4.8)

and the explicit expressions for the Coulomb, Jj (, ), and exchange, Kj (, ), kernels are J j (, ) ⫽ ∫∫ d ⬘ d ⬘ f j (⬘) f j (⬘) V (, ⬘; ⬘, )

(4.9)

K j (, ) ⫽ ∫∫ d ⬘ d ⬘ f j (⬘) f j (⬘) V (, ⬘; , ⬘)

(4.10)

V (, ⬘; ⬘, ) ⫽ i (1, ) j (2, ⬘) r12⫺1 j (2, ⬘) i (1, )

(4.11a)

V (, ⬘; , ⬘) ⫽ i (1, ) j (2, ⬘) r12⫺1 j (2, ) i (1, ⬘) .

(4.11b)

and

where

and

4.

Numerical Integration

Equation (4.5) is solved by an iterative procedure, starting with an initial fi () [for instance, fi () ⫽ 0 or the solution of Equation (4.5) without the repulsion terms], with an arbitrary numerical criterion for convergence. During each iteration, the integrations are carried out using a discretization technique. In contrast, with usual discretization procedures, which lead formally to the Roothaan equations, we propose to explore the continuous representation of the GCM, i.e., an accurate numerical integration of Equation (4.5) is attempted. This, besides formal refinements, allows us to avoid the optimization of the nonlinear parameters . The straightforward procedure is to take a mesh of N equally spaced points in -space. Actually, we have used this rule for the He atom with STOs. Nevertheless, for the case of GTOs, the weight functions fi () become very broad and we need a more sophisticated rule that permits us to use increasing increments, , for increasing . To attain this we relabel the generator coordinate space [8] ⫽ (log ) ⲐA,

A ⬎1

(4.12a)


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23

and k ⫽ min ⫹ ( k ⫺ 1), k ⫽ 1,… , N ,

(4.12b)

where N is the number of discretization points, an option defining the size of the basis set and is the integration interval, also a matter of numerical choice. The values of min and max ⫽ min ⫹ (N ⫺ 1) are chosen so as to adequately cover the integration range of f (). The new generator coordinate now spans the interval [⫺⬁, ⫹⬁], but the weight function becomes narrow and liable to discretization. An equally spaced mesh in the new interval is now enough for our purposes. This procedure is optimal with the use of GTOs as will be shown in Chapter 6. This is clearly visualized by sketching the weight functions from a preliminary calculation with arbitrary discretization parameters. We illustrate this point by the plots of the 2s weight functions for Li, Be, Ne, and Ar in Fig. 4.1. This matter will be further discussed in detail in Chapter 6. The discretized version of the GHWHF equations becomes N

∑ 关 F (

Fig. 4.1

k

, ) ⫺ i S ( k , )兴 f i ( ) ⫽ 0.

The 2s Gaussian weight functions for Li, Be, Ne, and Ar atoms.

(4.13)


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Chapter 4

The matrix elements h(k , ) and S(k , ) have obvious definitions. The Jj(k , ) and Kj(k , ) are discretized at the same points: N

N

m

n

N

N

m

n

J j ( k , ) ⫽ ∑ ∑ f j (⬘m ) f j (⬘n ) V ( k , ⬘m ; ⬘n , )

(4.14a)

and

K j ( k , ) ⫽ ∑ ∑ f j (⬘m ) f j (⬘n ) V ( k , ⬘m ; , ⬘n ).

(4.14b)

The matrix elements of F are then set up and the diagonalization is made by the method used in Reference [6]. The total energy is calculated from either n

E ⫽ ∑ (i ⫹ hii )

(4.15a)

i

or n

n

n

i

i

j

E ⫽ 2∑ hii ⫹ ∑ ∑ (2 J ii ⫺ K ij ),

(4.15b)

the obtained values being identical in all digits, where N

N

k

N

N

k

N

N

k

hii ⫽ ∑ ∑ f i ( k ) f i ( ) h( k , ),

J ij ⫽ ∑ ∑ f i ( k ) f i ( ) J j ( k , ),

(4.16a)

(4.16b)

and

K ij ⫽ ∑ ∑ f i ( k ) f i ( ) K j ( k , ) .

(4.16c)


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25

The mean value of r for the ith orbital is evaluated from

ri ⫽

N

N

k

∑ ∑ f ( i

k

) f i ( ) ri ( k , ),

(4.17)

where ri (, ) ⫽ i (i, ) ri i (i, ) .

5. 5.1.

(4.18)

First Applications to the He and Be Atoms The He Atom with a Slater Orbital

The generator functions are

( r , ) ⫽ exp(⫺r ),

(4.19)

for which the kernels are easily calculated as S(, ) ⫽ 8Ⲑ ( ⫹ )3

(4.20)

h(, ) ⫽ 关S (, ) Ⲑ 2兴关 ⫺ Z ( ⫹ )兴 ;

(4.21)

and

Z is the nuclear charge and the Coulomb V(, ⬘; ⬘, ) and exchange V(, ⬘; , ⬘) potentials are given by

V ( a, b; c, d ) ⫽

96 2 ⎡ 1 u ⫹ v 1 u ⫹ v ⎤ ⫹ ⫹ ⫹ , (u ⫹ v ) 4 ⎢⎣ u 3u 2 v 3v 2 ⎥⎦

(4.22)

where u ⫽ a ⫹ d and v ⫽ b ⫹ c. We choose the same discretization parameters for , ⬘, and ⬘ (min and are the lowest value and the increment of the generator coordinate, respectively). The HF limit [7,19] is attained with few (nonoptimized) points.


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26

5.2.

Chapter 4

The He Atom with a Gaussian Orbital

The generator functions are

( r , ) ⫽ exp(⫺r 2 ),

(4.23)

S(, ) ⫽ 关Ⲑ ( ⫹ )兴1.5

(4.24)

h(, ) ⫽ ⎡⎣31.5 Ⲑ ( ⫹ ) 2.5 ⎤⎦ ⫺ (2Z ) Ⲑ ( ⫹ ).

(4.25)

with

and

The Coulomb and exchange potentials are given by V ( a, b; c, d ) ⫽ (2 2.5 ) 冫 ⎡⎣( a ⫹ d )(b ⫹ c)( a ⫹ b ⫹ c ⫹ d )0.5 ⎤⎦ .

5.3.

(4.26)

The Be Atom with GTOs

As a test of the continuous representation of the GCM, we employ the same Gaussian form for the 1s and 2s orbitals: ⬁

1s ( r ) ⫽ ∫ f1 ()exp(⫺r 2 ) d 0

(4.27)

and ⬁

2s ( r ) ⫽ ∫ f 2 ()exp(⫺r 2 ) d .

(4.28)

0

Except for the value of the nuclear charge, the kernels are the same as for He. Even in this early work, the authors remarked on the importance of the weight function. However, we shall leave the discussion of its role until after the new concepts of basis set design are presented in Chapter 6. In Table 4.1, we show the main results obtained by Mohallem et al. [6] and compare with the best calculations for He and Be available at the time, i.e.,


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27

Table 4.1 Ground-state energies (atomic units) for He and Be for Slater (STO) and Gaussian (GTO) basis functions

He

STO GTO GTO

Be

GHW Equation

HF

⫺2.861679995612 ⫺2.866167957 ⫺14.572780

⫺2.861679995612a ⫺28616692b ⫺14.572368c

a

Reference [20]. Reference [21]. c Reference [21]. b

Fig. 4.2 The weight functions for the (a) 1s and (b) 2s orbitals of the Be atom for Gaussian generator functions.


28

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Chapter 4

Fig. 4.3 The weight function for the 1s orbital of the He atom for the case of a Gaussian generator function.

by Davies et al. [20] for Slater-type basis functions (STOs) and by Huzinaga [21] for Gaussian-type basis functions (GTOs). The values in Reference [6] were obtained with trivial size matrices of 13 ⫻ 13 for the STOs and a 24 ⫻ 24 for the GTOs. One can see that in this introductory work presented at the 1986 Sanibel Symposia, the results were either competitive with or better than the alternative results available at the time. Figs. 4.2 and 4.3 are likely to be the first GCM weight functions ever plotted.

References 1. D. L. Hill, and J. A. Wheeler, Phys. Rev., 1953, 89, 1102; J. J. Griffin, and J. A. Wheeler, Phys. Rev., 1957, 108, 311. 2. A. J. Thakkar, and V. M. Smith, Phys. Rev. A, 1977, 15, 1; A. J. Thakkar, and V. M. Smith, Phys. Rev., A, 1977, 15, 16. 3. L. Lathouwers, Ann. Phys., 1976, 102, 347; L. Lathouwers, P. Van Leuven, and M. Bouten, Chem. Phys. Lett., 1977, 52, 439. 4. B. Laskowski, and E. Brändas, Phys. Rev. C, 1976, 13, 1741. 5. D. Galleti, and A. F. R. de Toledo Pizza, Phys. Rev. C, 1978, 17, 774; J. G. R. Tostes, and A. F. R. de Toledo Pizza, Phys. Rev. A, 1983, 28, 538. 6. J. R. Mohallem, R. M. Dreizler, and M. Trsic, Int. J. Quantum Chem. Symp., 1986, 20, 45. 7. (a) D. R. Hartree, Proc. Cambridge Phys. Soc., 1928, 24, 89; D. R. Hartree, Proc. Cambridge Phys. Soc., 1928, 24, 111; D. R. Hartree, Proc. Cambridge Phys. Soc., 1928, 24, 426; (b) V. Fock, Z. Phys., 1930, 61, 126.


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29

8. J. R. Mohallem, Z. Phys. D, 1986, 3, 339; H. F. M. DaCosta, M. Trsic, A. B. F. da Silva, and A. M. Simas, Eur. Phys. J. D, 1999, 5, 375. 9. N. Bohr, Phil. Mag., 1913, 26, 1; N. Bohr, Phil. Mag., 1914, 27, 506. 10. E. Schrödinger, Ann. Phys., 1926, 79, 361; E. Schrödinger, Ann. Phys., 1926, 80, 437; E. Schrödinger, Ann. Phys., 1926, 81, 109. 11. M. Planck, Ann. Phys., 1901, 4, 553. 12. L. de Broglie, Thesis, Paris, 1924; L. de Broglie, Ann. Phys., 1925, 3, 22. 13. W. Heisenberg, Z. Phys., 1925, 33, 879. 14. W. Pauli, Z. Phys., 1925, 31, 765. 15. J. C. Slater, Phys. Rev., 1929, 34, 1293. 16. For a discussion of the Hartree–Fock formulation, see, for instance, M. Weissbluth, Atoms and Molecules, New York: Academic Press, 1978; for the fundation of Quantum Mechanics, see, for example, E. Antletta, Fundation and Interpretation of Quantum Mechanics: In the Light of a Critical-Historical Analysis of the Problems and of a Synthesis of the Results, Hackensack: World Scientific Publishing Company, Inc., 2002. 17. M. Born, Z. Phys., 1926, 37, 863; M. Born, Z. Phys., 1926, 38, 803. 18. W. Heisenberg, Z. Phys., 1927, 43, 172. 19. P. Chattopadhyay, R. M. Dreizler, M. Trsic, and M. Fink, Z. Phys., 1978, A285, 7. 20. C. L. Davies, H. J. Jensen, and H. J. Mohkhorst, J. Chem. Phys., 1984, 80, 840. 21. S. Huzinaga, J. Chem. Phys., 1965, 42, 1293.



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Chapter 5

Discretization Techniques

1.

Introduction

Since the generator coordinate method (GCM) was introduced by Griffin, Hill, and Wheeler in the context of nuclear physics [1], it is not surprising that early applications between the fifties and the seventies of the last century were mainly in that field. As we have commented and exemplified in the previous pages, our present mathematical capabilities prevent exact or analytical solutions for any nontrivial physical or chemical system. Thus, from the initial days of the applications of the GCM, resort to approximations was mandatory. In this chapter, we shall first discuss early studies on the discretization techniques for the Griffin–Hill–Wheeler (GHW) equation in physical and chemical problems. These studies were presented in the fifties to seventies of the last century, before the introduction of the generator coordinate Hartree–Fock (GCHF) method (Chapter 4). The interest in the preliminary approaches presented below focuses on model problems and emphasizes mathematical refinements. Also, the concept of integral discretization (ID) and the recognition of the importance of the role of the weight function were recognized later. Thus, in this chapter we shall also discuss the origin and meaning of ID. Finally, at the end of this chapter, we describe a very recent discretization technique, which seems to be the most efficient proposed to date. Perhaps the first attempts to solve the GHW integral equation via discretization techniques were due to Justin et al. [2]. A number of difficulties were reported concerning singularities of the weight function and negative eigenvalues of the overlap matrix. The first problem seems to be linked with the use of the Gaussian overlap approximation (GOA) (see for instance Reference [3]). In 1978, Chattopadhyay et al. [4] applied the discretization technique to simple model systems to obtain a more direct understanding of the intricacies of this methodology. This work stimulated further developments relying on discretization techniques as are described in this book. Here we reproduce the findings of Reference [4] and comment on the context of present understanding of discretization techniques.


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32

Chapter 5

We note that orthogonality of the eigenfunctions of the GCM problem [Equation (2.4)] is expressed through the relations

∫ dx ( x) ⴱ i

j

( x ) ⫽ ∫ f i ⴱ () S (, ) f j ( ) d d ⫽ ij ,

(5.1)

while completeness of the basis requires

∫ d d ( x , ) S ⴱ

⫺1

(, ) ( x⬘, ) ⫽ ( x ⫺ x⬘)

(5.2a)

with

∫ d S

⫺1

(, ) S (, ⬘) ⫽ ( ⫺ ⬘) .

(5.2b)

The problem we choose for the illustration of practical rules in the numerical treatment of the GHW equations is the harmonic oscillator in one dimension. We describe the oscillator problem via a “superposition” of square-well solutions. The kernels of the GHW equation can be obtained analytically. We use this setup as a vehicle to investigate various points of the numerical approach (Section 3).

2.

A Model Problem: The Harmonic Oscillator

The functions to be used for the generator coordinate ansatz of the harmonic oscillator problem are the solutions to the square-well problem ⎧⎪⬁ V ( x) ⫽ ⎨ ⎩⎪0

x ⱖ x ⬍ .

(5.3)

The eigenfunctions of this simple problem can be classified according to parity and energy. We note that the wave functions are

n ( x ) ⫽

1

cos

nx ( 2 ⫺ x 2 ), 2

(5.4)

sin

nx ( 2 ⫺ x 2 ), 2

(5.5)

n ⫽ 1, 3, … for positive parity,

n ( x ) ⫽

1


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33

n ⫽ 2, 4, … for negative parity, and the energy ( ⫽ m ⫽ 1) as

n ⫽

2 n2 8 2

(5.6)

in both cases. For convenience, the step function

(t ) ⫽ 1 for t ⱖ 0 (t ) ⫽ 0 for t ⬍ 0

(5.7)

has been used to represent the appropriate boundary conditions. The width of the square well is our generator coordinate. We thus describe the well-known oscillator wave functions by superposition of the wave functions of square wells with different widths. The Hamiltonian and normalization kernels are easily evaluated. Writing the oscillator Hamiltonian as 1 1 H ⫽⫺ ⵜ2 ⫹ k x 2 2 2

(5.8)

with k being the force constant, we find, e.g., for the negative parity solutions, H ( , ) ⫽

2 n2 2 2 ⫹ k 8 2 n2 2

H (, ) ⫽ (⫺) nⲐ 2⫹1

⎡ n2 2 1 ⎤ ⫺ ⎥ ⎢ 2⎦ ⎣ 12

(5.9)

n n sin 2 2 2 2 ( ⫺ )

(⫺) nⲐ 2 2 2 ⎧ 8 n cos ⎨ 2 2 2 2 2 2 n ⎩ ( ⫺ ) ⎡ 8 2 ( 2 + 3 2 ) n ⎤ n ⎫ ⫹⎢ ⫺ 2 ⎬ ⎥ si n 2 2 3 2 ⎭ ( ⫺ 2 ) ⎦ ⎣ n( ⫺ ) ⫹k

⫽ H ( , )

for ⬍ .

(5.10)


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Page 34

Chapter 5

S(, ) ⫽ 1 S (, ) ⫽ (⫺) nⲐ 2⫹1

4 ⎛ ⎞ n sin ⫽ S ( , ) 2 2⎟ ⎜ n ⎝ ⫺ ⎠ 2

(5.11)

(5.12)

for ⬍ . We consider only the case k ⫽ 1 throughout. One observes that a GOA would not provide a good representation of these kernels.

3. Discretization of the Griffin–Hill–Wheeler Equation for the Harmonic Oscillator Problem Since the GHW equation cannot, in general, be solved exactly, the solution proceeds, in practice, by replacing the integration by summation. This discretization normally relies on application of the simplest form of Simpson’s rule, although more advanced numerical integration techniques are possible. Since the range of the generator coordinate includes, in most cases, infinite intervals, truncation of the sums involved is a second feature of the numerical approach. With the definitions f ( i )⌬ ⫽ Fi H ( j , i ) ⫽ H ji S ( j , i ) ⫽ S ji ,

(5.13)

the GHW equation reduces to a standard (symmetric) eigenvalue problem in a nonorthogonal representation N

∑ [H

ij

⫺ ESij ] Fj ⫽ 0, i ⫽1,..., N

(5.14)

j ⫽1

The solution of this problem involves two steps. One first solves the eigenvalue problem

∑S b ij

j

⫽ bi .

(5.15)

j

This step corresponds to solving the Fredhom equation of the second kind

∫ S ( , ) b ( ) d ⫽ b ( )

(5.16)


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35

by a simple quadrature method, thus determining the geometrical components of the biorthogonal natural state expansion, as pointed out in detail by Lathouwers [5]. The second step involves diagonalization of the Hamiltonian in the basis of Equation (5.15), respectively, Equation (5.16). Starting from the discretized eigenvalue problem of the overlap kernel B S B% ⫽ ,

(5.17)

one arrives at an orthogonal representation ⫺1Ⲑ 2 B S B% ⫺1Ⲑ 2 ⫽ 1,

(5.18) ~

where B is the matrix constructed from the eigenvectors, B its transpose, and the diagonal matrix of eigenvalues. If we multiply the matrix equation corresponding to Equation (5.14) from the left with the matrix D ⫽ ⫺1Ⲑ 2 B

(5.19)

( A ⫺ E1)C ⫽ 0,

(5.20)

A ⫽ D H D% C⫽DF.

(5.21)

we obtain the eigenvalue problem

where

F is the matrix of the discretized weight functions of Equation (5.13). Solution of the eigenvalue problem in Equation (5.15) readily gives the eigenvalues of the Hamiltonian, E, and the eigenvector matrix, C, from which the weight functions of the various eigenstates can be recovered by inversion % . F ⫽ DC

(5.22)

The actual eigenfunctions are then determined by integrating Equation (2.5) with the same numerical method as used in the discretization of the GHW equation. We return to a closer look at the first step as expressed by Equations (5.15) and (5.16). Some general properties of the eigenvalue spectrum of Equation (5.16) can be stated under the assumption that the kernel S(, ) is Hermitian, nonnegative, and square integrable [5].


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36

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Chapter 5

A(1): The spectrum of eigenvalues is positive

1 ⱖ ... ⬎ 0. ⬁

2

A(2): The series n⫽1 n converges and 0 is the only accumulation point of the

n eigenvalues. A(3): The sum rule ⬁

∑

n

n⫽1

⫽ ∫ S ( , ) d

(5.23)

holds. The question arises about how far these properties are maintained in the numerical approximation in Equation (5.15). In this case, one can view the overlap kernel as a Gram-matrix to be constructed from a set of N vectors i in N-space Sij ⫽ ( vi ⭈ v j ).

(5.24)

The following theorems [6] are available for the Gram-matrix: B(1): A necessary and sufficient condition for the linear dependence of the vectors i is the vanishing of the Gram determinant S ⫽ 0. B(2): In case of linear dependence of the vectors, none of the eigenvalues are negative. In the case of linear dependence, say there are k-independent vectors i, one has N–K eigenvalues, which are zero. B(3): We have N

∑

n

⫽ trace( S ).

(5.25)

n ⫽1

Comparison of the statements (A) and (B) invites the following remarks: Negative eigenvalues of S, which are reported in the literature as a result of carrying through the numerical procedure indicated here, are always a consequence of numerical errors. For the harmonic oscillator problem, this could be verified in each case by increasing the precision of the calculation. The inherent structure of the exact spectrum with an accumulation point of eigenvalues at the zero value will, however, eventually lead to numerical difficulties, without necessarily indicating proper linear dependence. This problem is solved when we reject eigensolutions, which fall below a numerically acceptable limit, and not reducing the dimensionality of the vector-space used in the numerical solution


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37

of Equation (5.15) as was repeatedly suggested in the literature [4]. The cost of following the former prescription is a loss of accuracy in all of the final solutions of the problems. The cost of following the latter prescription is mainly the loss of highlying (in energy) solutions, which can be accepted in view of the general aim of reproduction of the low-lying states of a given spectral problem. The truncation of eigenstates of the overlap matrix can be discussed from a more quantitative point of view. Introducing the coordinate overlap [5] X ( x, x⬘) ⫽ ∫ ( x, ) ⴱ ( x⬘, ) d ,

(5.26)

one considers the eigenvalue problem

∫ X ( x, x⬘) y( x⬘) dx⬘⫽ y( x),

(5.27)

which has the same eigenvalues as the problem of Equation (5.16). The truncation prescription amounts to the statement

bn () b () ( n ⫺ c ) n 1 Ⲑ 2 1Ⲑ 2

n

n

(5.28)

yn ( x ) ( n ⫺ c ) yn ( x ).

(5.29)

or equivalently

The final eigenstate (x) can be expanded directly in terms of the solutions of the eigenvalue problem of Equation (5.27) ⬁

( x ) ⫽ ∑ cn yn ( x ) n⫽1

⎡⬁ c ⎤ ⫽ ∫ ⎢ ∑ 1nⲐ 2 bn () ⎥ ( x, ) d ⎣ n⫽1 n ⎦ ⫽ ∫ f () ( x, ) d . From the relations ⬁

cn b () 1Ⲑ 2 n n⫽1 n

f () ⫽ ∑ or f

2

⬁

⫽∑

n⫽1

cn

n

2

,

(5.30)


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Chapter 5

Table 5.1 Results for the positive parity states of the harmonic oscillator problem. The generating function is given by Equation (5.4) with n ⫽ 1 (0 ⫽ 0.2095, ⫽ 0.125, n ⫽ 3.9595)

GCM Exact

E1

E2

E3

E4

E5

E6

0.500 0.5

2.504 2.5

4.525 4.5

6.677 6.5

9.248 8.5

12.47 10.5

2

one concludes, in the case || f || is finite, that 2

cn ⱕ n f

2

.

(5.31)

The relative contribution of the eigensolutions of Equation (5.27) to the representation of the final eigenstates is governed by the factor n, which justifies the introduction of the c value in Equations (5.28) and (5.29). In Table 5.1, we present some results for the positive parity states of the harmonic oscillator. The generating functions in this case are given by the lowest positive parity solutions (n ⫽ 1) of the square-well problem. The results are comparable to those of negative parity states based on the generating function with n ⫽ 2.

4. 4.1.

The Integral Discretization Introduction

From the initial days of the GCHF method [7], we attempted the best numerical integration of the Griffin–Hill–Wheeler–Hartree–Fock (GHWHF) equations, abandoning the variational discretization (VD), advocated earlier [4]. Nevertheless, it was Mohallem [8] who provided a formal background for the use of the ID. It is assumed that the kernel S(, ) is square-integrable. The space spanned by the generator function for all values of is called HGC. Typically, VD or ID implies truncation of HGC , and diagonalizing the H operator in this subspace of finite dimension N leads to the secular equation that provides the optimum coefficients F Ni for the ⏐ N 〉 approximation to ⏐ 〉: N

N ⫽ ∑ Fi N (i ) . i ⫽1

(5.32)


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39

Canonical orthonormalization [9] applied on the N vectors of this discrete basis leads to N

N ⫽ ∑ cn n ,

(5.33)

n⫽1

N

∑c

2

⫽ 1,

n

(5.34)

n⫽1

n ⫽

1

n

N

∑u

ni

(i ) ,

i ⫽1

(5.35)

where n and u ni are respectively eigenvalues and eigenvectors of the matrix S ⫽ S (i, j) (the bar indicates that the quantities are obtained from the discrete basis), i.e., N

∑ S ( , ) u i

j

nj

⫽ n uni .

j ⫽1

(5.36)

It is straightforward to verify that the coefficients Fi N are given by N

Fi N ⫽ ∑

cn uni

n

n⫽1

.

(5.37)

Now, consider the vectors of an orthonormal basis, in the continuous representation, for HGC n ⫽

1

n

∫ d u () () n

(5.38)

with

∫ S (, )u () d ⫽ u (). n

n n

(5.39)

One can verify that Equation (5.39) is a Fredholm equation of the first kind with a square-integrable kernel. It admits a numerical solution that may be written in the form of Equation (5.36). Then vectors ⏐n〉 and ⏐n 〉 may become arbitrarily close as N increases. In view of this, one can assume that a positive number can be found so that the “strong” condition is fulfilled: n ⫺n

⬍ Ⲑ N 2.

(5.40)


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Chapter 5

Then, N

∑c ( n ⫺ n ) n

n⫽1 N

⫽ ∑ cn

n ⫺n

n⫽1

N

ⱕ ∑ cn ( n ⫺ n

)

n⫽1 N

(5.41)

⬍∑ n ⫺ n

⬍ Ⲑ N ,

n⫽1

where Equation (5.34) and the generalized triangular inequality were employed. This implies that the first term of inequality in Equation (5.41) may become as small as desired. Expansion of this term using Equations (5.35) and (5.38) gives 1

n

N

∫∑

cn un ()

n

n⫽1

N ⎡ N cu ⎤

() d ⯝ ∑ ⎢ ∑ n ni ⎥ (i ) i ⫽1 ⎢ n⫽1

n ⎥⎦ ⎣

(5.42)

or, using Equation (5.37)

∫

N

f N () () d ⯝ ∑ Fi N (i ) , i ⫽1

(5.43)

where the functions N

f N () ⫽ ∑

cn un ()

n⫽1

n

(5.44)

are square-integrable (L2). Actually, using the orthonormality of the un() functions, it appears that the norm of the fN () functions is

∫

N

f N () d ⫽ ∑ 2

n⫽1

cn

n

2

⬍ ⫹⬁

(5.45)

since N is always finite. The convergence for ⏐ 〉 is given by

⫽ lim ∫ f N () () d , N ⬁

(5.46)


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41

where whether the limit and the integral commute or not depends on the relative decrease of ⏐cn⏐2 and n for the sum showed in Equation (5.45) [10]. Equation (5.43) is fundamental in drawing the following conclusions. First, the right-hand side does necessarily represent the discretization of the integral on the left, but both sides merely generate approximations to the same vector ⏐ N 〉. There are two cases to consider: (i) In favorable cases, when ⏐cn⏐2 decreases faster than n, f() is (L2) and the GHW equation has Equation (5.14) as its real discretized version with a stable solution for the discrete weight function. But to find the weight function through discretization is often a task requiring intuition and the use of physical arguments. This is clearly only a technical problem that depends on the quality of the integration scheme. For a poor integration, the coefficients should be worse than the VD values, whereas for a good integration it is expected that ID surpasses VD. This point is clarified in the application below. (ii) In the opposite case, if n decreases faster than ⏐cn⏐2 in Equation (5.45), a norm of the weight function increases without bound as N increases. When the density of the points in the mesh is raised the vectors show approximate linear dependence (ALD). This is often irrelevant in the first case, but very important here, since it implies numerical errors. The usual procedure is to establish a c cut-off value for the eigenvalues of S below which the corresponding ⏐n 〉 is set equal to zero. The discrete version of the GHW equation in this case becomes N

∑ [ H ( , i

j ⫽1

j

) ⫺ ES (i , j )] f N ⬘ ( j ) ⫽ 0,

(5.47)

where N⬘ ⫽ N ⫺ Nc, Nc denoting the number of vectors eliminated by the truncation. The solution for the weight functions tends to stabilize in some squareintegrable fN⬘(), in agreement with the results in Reference [10]. As argued in Reference [10], the singular character of f() is concomitant with the importance of the vectors associated with small , which is a consequence of ALD, and truncation should be performed with care. The point here is that in this case, VD seems to be just as incapable as ID in obtaining highlying states. In conclusion, in both the above cases, the only advantage of a full VD procedure seems to be possibly faster convergence to the energy of the state considered, e.g., the ground state. The possible advantages of ID are explored below. The conclusions above are now applied to two problems having exact solutions. Both have appeared in the literature [11,12] as examples of VD and so are suitable for comparison with ID. A simple Simpson rule is first considered for the discretization of Equation (2.8). The disseminated rule of point picking around the variational


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Chapter 5

point 0 may be taken as a starting point, although in some cases it does not include many relevant states of HGC (as in the second application below). Then one should appeal to physical arguments for inferring the approximate form of the weight function [13] as a guide for choosing a mesh adequate for spanning the integration interval. The main advantages expected in using the ID procedure are (1) economy of computational time by avoiding optimizing nonlinear parameters (or generator coordinates); (2) an adequate representation of various excited states with a single diagonalization; and (3) construction of better wave functions by fully spanning the integration interval.

4.2.

The Harmonic Oscillator with Translated Gaussians

This problem has been considered in many papers [1,3,11]. The generator function is (unnormalized) 2

( x, ) ⫽ e[⫺2 s ( x⫺ ) ] ,

(5.48)

where the s parameters control the width of the translated Gaussian. The kernels are given in Reference [3]. To generate the ground state of the harmonic oscilla2 tor, (x) ⫽ e⫺k x , through Equation (2.2), one has to consider the value of the ratio ⫽ k/2s. For ⬍ 1, f() has a (L2 ) Gaussian form; for ⫽ 1 it is a delta distribution and for ⬎ 1 a “worse” distribution [3]. Owing to the symmetrical character of the points ⫾ i in the wave function, one way is to simply take a sufficiently large interval around 0 ⫽ 0, with equally spaced points. The results for the weight function are (a) ⬍ 1, the coefficients converge quickly to f(): with decreasing the weight function becomes narrower. (b) ⫽ 1, the coefficients fall on a delta sequence for increasing N (or decreasing 苶c), Fig. 5.1. (c) ⬎ 1, the coefficients fall on a weight function that shows singular behavior at the origin and oscillates for increasing |i|. For a fixed 苶c the solution is stable, as indicated by Equation (5.47). Table 5.2 shows the results for the harmonic oscillator spectrum for various cases. The quality of the ID results clearly improves with decreasing . Table 5.2 also clearly shows that the diagonalization of larger matrices is compensated for


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43

by recovering the N⬘ (⫽ N ⫺ Nc) states (it was impossible to compare the results with those of Reference [11] because the latter should be non-L2 although the text claims the opposite. Also, the dimension of the matrix diagonalized in the last iteration in Reference [11] seems to be 2n ⫺1 and not n).

Fig. 5.1 Approximations fN⬘() for the delta function with N ⫽ 11, (a) (b)

苶c ⫽ 10⫺8, for the harmonic oscillator problem.

苶c ⫽ 10⫺7 and

Table 5.2 Various ID calculations for the harmonic oscillator. Energies are in h units. The set of points includes 0 ⫽ 0. For all calculations k ⫽ 0.5 and Nc ⫽ 0 except for c (Nc ⫽ 5,

苶c ⫽ 10⫺7) and d (Nc ⫽ 4, 苶c ⫽ 10⫺8) Calc. N

E0

E1

E2

E3

E4

(a) (b) (c) (d) (e) (f) (g)

5/8 5/8 1 1 5/4 5/3 3

0.5 0.5 0.2 0.2 0.2 0.5 0.6

0.5000002 0.5a 0.50000008 0.5a 0.5a 0.500002 0.501 0.5

1.50002 1.5a 1.5000001 1.5000001 1.5a 1.5002 1.53 1.5

2.5002 2.5a 2.5002 2.5000002 2.500002 2.5009 2.6 2.5

3.506 3.5a 3.502 3.5002 3.5003 3.52 4 3.5

4.53 4.500001 4.54 4.5002 4.5004 4.5 5 4.5

a

7 11 11(N⬘ ⫽ 6) 11(N⬘ ⫽ 7) 11 11 11 Exact

Exact within 8 decimals.


44

4.3.

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Chapter 5

The Hydrogen Atom with a Gaussian Generator Function

This problem is the starting point for calculations with Gaussian-type orbitals (GTOs) in quantum chemistry [14]. The calculations in Reference [14] are comparable to the GCM except for the “brute force” optimization of the nonlinear parameters (in what follows, for the sake of comparison, these calculations will be considered as versions of VD). Applied explicitly in the context of the GCM, this problem has been employed as a first test for discretization [4], VD [12], and, more recently, as an example of analytical integration of the GHW equation [15]; for details see Chapter 3. The (unnormalized) generator function for this case is

( r , ) ⫽ e⫺r

2

(5.49)

and the weight functions for all the s states are obtained from a well-known integral transform between exponentials and Gaussians [16]. Since the weight function in this case is L2, comparison with VD is of interest. The kernels are trivially obtained and are shown in any of the GCM calculations for this problem. Before presenting the calculations, it must be noted that the limitation of the results in Reference [5] (E0 ⫽ ⫺0.4994 for N ⫽ 50) should not be connected with the ID approach, as proposed in Reference [12], but rather, owing to the choice of a high value for 苶c (approximately 10⫺8) and mainly to the choice of the mesh points around 0, which restricts the integration interval to [0, 20], thus eliminating many relevant vectors for > 20. By simply lowering 苶c (⬃10⫺15) and enlarging the integration interval with different i (min ⫽ 0.01, i ⫽ 10i⫺3, Ni ⫽ 10, i ⫽ 1,...,5), excellent results were obtained for the ground and first excited state in a 50-points calculations: respectively ⫺0.4999998 and ⫺0.12499996 (a.u.). Nevertheless, this procedure does not allow the weight function to be obtained. The coefficients oscillate between positive and negative values with no definite convergence pattern. As the weight function is L2 this problem is obviously due to the integration technique. Actually, ID must be accomplished by relabeling log , A

A ⱖ 1,

(5.50)

where a good (not optimal) value obtained for A in this case is 8.0. The discretization interval becomes [⫺⬁,⫹⬁] but the weight function becomes narrow and liable to discretization in an equally spaced mesh in the new interval. This is valid also for various excited states (for the ground and first excited states see Fig. 5.2). Also, this procedure eliminates the numerical problems due to ALD and one may use, in general, 苶c⫽ 0.


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45

Fig. 5.2 Weight functions for the (a) ground state and (b) first excited state of a hydrogen atom with Gaussian orbitals.

Table 5.3 shows the results for energies from various VD and ID calculations: these results are unquestionably the strongest argument in support of ID. The description of the H spectrum in the 50-point calculation seems to be unique. Energies are included up to E6 but the diagonalization provides results up to E12, exact to at least five decimal places. Further, the 30-point calculation surpasses the variational result of Arickx et al. [12] for E0. Although this is quite surprising, it is easily explained since the latter are not fully optimized. The computational time used by ID is only a small fraction of that used by VD in this case.


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Table 5.3 Various VD and ID calculations for the H atom. Energies are in a.u. The set of points includes 0 ⫽ ⫺01578, with ⫽ 0.1 for (a) and ⫽ 0.08 for (b). First point (min) is chosen to optimize (a) the spectrum and (b) the ground state

⫺E0 ⫺E1 ⫺E2 ⫺E3 ⫺E4 ⫺E5 ⫺E6

VD Ref. [12] N = 30

ID N = 30

ID N = 50

Exact

0.49999999 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺

0.4999999994 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺

0.49999998 0.12499998 0.05555555 0.031249996 0.019999997 0.01388887 0.010204080

0.5 0.125 0.0555... 0.01325 0.02 0.013888... 0.010204082

The fact that one obtains discretized weight functions has some implications for quantum chemical techniques. It is a common belief in quantum chemistry that Gaussians do not provide a good description of the wave function, in spite of providing accurate values for energy. However, the present approach seems to produce a better wave function. This assumption is supported by results obtained for the cusp (0), mean values 〈r n〉, and lower bounds to E0 in Reference [17].

5. A New Proposal for the Discretization of the Griffin–Hill– Wheeler–Hartree–Fock Equations 5.1.

Introduction

Ab initio electronic structure calculations for atoms and especially for molecules are mostly carried out within the finite basis set expansion method of Roothaan [18]. Since the introduction of the basis set expansion method in Hartree–Fock (HF) theory, made by Roothaan [18], the search for ever more efficient basis sets has been a constant quest. There is a considerable degree of freedom in choosing the basis functions for atomic and molecular calculations, as any complete set of functions can be employed. Although Slater-type functions (STFs) had been widely used as basis sets for calculations with atoms and diatomic molecules, Gaussinan-type functions (GTFs) are invariably the popular choice as basis sets for polyatomic calculations. The prime reason for the preference of GTFs as basis functions (basis sets) in polyatomic calculations lies in the fact that all the multicenter integrals can be evaluated exactly by closed analytical formulas. However, a much larger basis set of GTFs is needed than for the STF basis set because the GTFs behave incorrectly both in the region near the nucleus (if a point nucleus model is used) and at long range.


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47

The deficiencies cited above are relevant in nonrelativistic calculations based on the Hartree–Fock–Roothaan (HFR) method [18], where the point nucleus approximation is usually employed. Otherwise, when one employs the finite nucleus approximation, as commonly used in relativistic calculations, the use of GTFs is the right choice, since relativistic calculations are based on the Dirac equation whose solutions for an electron in a finite nucleus have been shown to be Gaussian [19,20]. Owing to the popularity of the use of GTF basis sets, several groups have attempted to develop techniques that are able to produce efficient GTF basis sets for atomic and molecular calculations. Two of the most popular techniques in the design of GTF basis sets were the even-tempered [21] and the well-tempered [22] techniques. In 1986 another interesting and powerful technique to tailor basis sets was presented [7]. Initially, it was called as the generator coordinate version of the HF equations, and later became widely known as the generator coordinate Hartree–Fock (GCHF) method. Shortly, since the GCHF method was already presented in Chapter 4, the GCHF method is the result of employing the generator coordinate ansatz [1b] in the independent particle model

k (1) ⫽ ∫ k (1, ) f k () d , k ⫽ 1,..., n ,

(5.51)

where k are the generator functions (they can be either STFs or GTFs), f k are the weight functions, is the generator coordinate, and n is the number of particles. The k are then used to build the Slater determinant and calculate the mean value of the total energy (the energy expectation value). The application of the variational principle to the energy expectation value (built with the one-electron functions k) leads to the GHWHF equations

∫ [ F ( , ) ⫺ S ( , ) ] f k

k

( ) d = 0, k ⫽ 1,..., n ,

(5.52)

where the k are the orbital eigenvalues and the Fock, F(, ), and overlap, S(, ), kernels are N

F (, ) ⫽ h(, ) ⫹ ∑ [2 J j (, ) ⫺ K j (, )]

(5.53)

S (, ) ⫽ k (1, ) k (1, ) .

(5.54)

j

and


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Chapter 5

The one-electron kernel, h(, ), and the two-electron kernels, Coulomb Jk(, ) and exchange Kk(, ), are defined as h(, ) ⫽ k (1, ) h(1) k (1, ) ,

(5.55)

J j (, ) ⫽ ∫∫ V (, ; , ) f jⴱ () f j () d d ,

(5.56)

K j (, ) ⫽ ∫∫ V (, ; , ) f jⴱ () f j () d d ,

(5.57)

V (, ; , ) ⫽ k (1, ) j (2, ) r12⫺1 j (2, ) k (1, )

(5.58)

V (, ; , ) ⫽ k (1, ) j (2, ) r12⫺1 k (2, ) j (1, ) .

(5.59)

and

where

and

The GHWHF equations are integrated numerically through discretization with a technique that preserves the continuous representation (the integral character) of the GCHF method [8]. This technique is ID and was originally implemented in the GCHF method through a relabeling of the GC space, namely ⫽

ln , A

A ⬎ 1,

(5.60)

where A is a scaling parameter determined numerically and the new coordinate space is discretized for each w atomic orbital symmetry (w = s, p, d, f, etc.) taking into account an equally spaced numerical mesh, (w) k , so that

(w) k

In (kw ) w) ⫽ ⫽ (min ⫹ ( k ⫺ 1)( w ) , k ⫽ 1,K , N . A

(5.61)

In Equation (5.61), N is the number of discretization points and min and (w) are the initial mesh point and the increment used to obtain the subsequent (w) (w) mesh points, respectively. The lowest ( min ) and highest ( max ) points of the (w) mesh ( k ) are chosen so as to embrace the adequate integration range for the weight function f k.


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49

The choice of the generator coordinate depends on the physical system under study. For atomic systems, can represent the atomic GTF exponents and according to Equation (5.61) the choice of the discretization points (w) k deter(w) mines the number of exponents k of the GTF basis set for each w atomic orbital symmetry. The number of discretization points, N, defines the size of the GTF basis set.

5.2.

The Polynomial Integral Discretization

For over 10 years, the GCHF method was used successfully in the generation of GTF basis sets of good quality for atoms with many electrons. Recently, a modification in Equation (5.61) was proposed [23] to improve the way of obtaining GTF basis sets through the GCHF method. This modification permitted the use of the GCHF method to generate GTF basis sets of smaller size that were as accurate as those obtained with the original method. Now, (w) k in Equation (5.61) is discretized for each w atomic orbital symmetry through the polynomial expansion (kw ) ⫽

ln wk w) ⫽ (min ⫹ ⌬1(w ) ( k ⫺ 1) ⫹ ⌬(2w ) ( k ⫺ 1) 2 ⫹ ⌬(3w ) ( k ⫺ 1)3 A ⫹ ... ⫹⌬(qw ) ( k ⫺ 1) q ,

(5.62)

where each GTF exponent (w) k is now determined by using the expression w) (kw ) ⫽ exp A{(min ⫹ ⌬1(w ) ( k ⫺ 1) ⫹ ⌬(2w ) ( k ⫺ 1) 2 ⫹ ⌬(3w ) ( k ⫺ 1)3 (5.63) ⫹... ⫹⌬(qw ) ( k ⫺ 1) q }.

In Equations (5.62) and (5.63), A is a scaling parameter to be determined numerically and k = 1, ..., N, where N is the number of discretization points, i.e., the size of the GTF basis set, exactly as in the original GCHF method. With the implementation of Equations (5.62) and (5.63) in the original GCHF method, the discretization of the GHWHF equations is now performed through a numerical mesh that is not equally spaced. Then, contrary to the original GCHF method, the GTF exponents are now generated through the numerical mesh, (w) k , with the intervals, (w) , not being equally spaced. This procedure makes the q optimization of the GTF exponents, (w) k , in Equation (5.63) more flexible and more efficient. Although it is practical to perform numerical integration, the condition of an equally spaced numerical mesh in the original GCHF method always imposes some restrictions in the generation of exponents. For heavy atoms (atoms of the third and fourth rows of the periodic table), the GCHF method was always able


50

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Chapter 5

to generate accurate GTF basis sets of large size (very competitive with other accurate GTF basis sets published in the literature). In contrast, for light atoms (atoms of the first and second rows of the periodic table) the GCHF method was never able to generate accurate GTF basis sets of small and medium size [24]. The accuracy of the GTF basis sets, generated with the original GCHF method, was always proportional to their size. This is the reason why other techniques (methods) were always able to present, mainly for light atoms, better GTF basis sets of smaller size (and similar accuracy) than those obtained with the GCHF method [24]. The main goal of any method that is used for the generation of basis sets is the design of accurate basis sets of small size, as the computational cost of molecular calculations depends directly on the size of the basis set employed. With the polynomial expansion presented in Equation (5.62), we will still be able to generate accurate basis sets, but now they could be smaller than those obtained previously with the original GCHF method. Another interesting aspect on the implementation of Equations (5.62) and (5.63) in the original GCHF method is that with an unequally spaced numerical mesh we can also segment Equation (5.62) before optimizing the GTF exponents, k(w). This is a very practical alternative as with the segmentation of Equation (w) (5.62) we can use different sets of min and (w) q for different regions in the (w) atom, i.e., a set with min and (w) to describe the inner shell electrons of the q atom (the largest exponents), another set to describe the intermediate shell electrons, and another to describe the outer shell electrons (the smallest exponents). In practice, we can use as many segmentations as necessary depending only on the total number of electrons in the atom and on the accuracy we are trying to attain for the HF energy. In fact, we use the segmentation of Equation (5.62) as a complement to our polynomial expansion, since the use of segmentation comes from the necessity of truncating Equation (5.62). In practice, if one uses a very large number of terms (complete set) in Equation (5.62) there will be no necessity of employing the segmentation alternative. Table 5.4 presents the ground-state HF energy results obtained for atoms of the first, second, and third rows of the periodic table with GTF basis sets generated with the polynomial expansion implemented in the GCHF method. Here it is interesting to explain how the basis set exponents are generated from Equation (5.63) before discussing the results of Table 5.4. The letter k in Equation (5.63) represents the size of the basis set, i.e., the number of exponents of the basis set, and it varies from 1 to the desired number of points (exponents) of the basis set for each w atomic orbital symmetry (w = s, p, d, f, etc.). The letter q represents the degree of the polynomial and, in practice, q values from 3 to 4 are enough to generate basis sets for atoms of the first, second, and third rows. In Equation (5.63), each w atomic orbital symmetry will have a starting point


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51

Table 5.4 Comparison between the ground-state HF energies (in hartree) obtained with the polynomial expansion and other calculations for atoms of the first, second and third rows of the periodic table Atom (State)

GTF Size (Polynomial)

HF Energy (Polynomial)

GTF Size (GCHF)a [26]

HF Energy (GCHF)a [26]

NHFb [27]

C(3P) O(3P) F(2P) Ne(1S) Mg(1S) Al(2P) Si(3P) S(3P) Cl(2P) Mn(6S) Fe(5D) Zn(1S)

18s13p 18s13p 18s13p 18s13p 20s12p 20s15p 20s15p 20s15p 20s15p 24s16p12d 24s16p12d 24s16p12d

⫺37.68861769 ⫺74.80939608 ⫺99.40934613 ⫺128.5470938 ⫺199.6146249 ⫺241.8766933 ⫺288.8543456 ⫺397.5048770 ⫺459.4820498 ⫺1149.866230 ⫺1262.443642 ⫺1777.848083

23s14p 23s14p 23s15p 23s14p 25s15p 26s18p 26s18p 26s18p 26s18p 29s19p13d 30s19p14d 30s19p14d

⫺37.68861786 ⫺74.80939595 ⫺99.40934657 ⫺128.5470930 ⫺199.6146261 ⫺241.8766995 ⫺288.8543540 ⫺397.5048844 ⫺459.4820589 ⫺1149.866215 ⫺1262.443633 ⫺1777.848065

⫺37.68861896 ⫺74.80939847 ⫺99.40934939 ⫺128.5470981 ⫺199.6146364 ⫺241.8767072 ⫺288.8543625 ⫺397.5048959 ⫺459.4820724 ⫺1149.866252 ⫺1262.443665 ⫺1777.848116

a

Best HF energy reported in the literature obtained with the original GCHF method. Best numerical Hartree–Fock (NHF) energy reported in the literature.

b

(w) ( min ) and a set of increments ( (w) q ) that are determined by optimization from (w) (w) (w) a trial set of min and (w) q , i.e., the initial values for min and q . In our first applications with the polynomial expansion, we always used as trial functions the universal GTF basis set generated previously with the original GCHF method [25]. The number of increments ( (w) q ) to be determined depends on the choice of the degree of the polynomial, i.e., depends on where the polynomial is truncated for each w atomic orbital symmetry. With the truncation of the polynomial, we realized that better GTF basis sets could be obtained when each w atomic orbital symmetry was also segmented individually. In all calculations performed, we noticed that three to four segmentations in each w atomic orbital symmetry are enough to attain very accurate HF energies. From Table 5.4, we can see that the GCHF method with polynomial expansion considerably reduced the GTF basis set size (and maintained the high accuracy) when compared to the best GTF basis sets generated with the original GCHF method. For all atoms studied, the reduction in the total number of GTF exponents, taking into account the s, p, and d atomic orbital symmetries, varies from 6 to 11. Even with this substantial reduction in the number of GTF exponents, the HF energies obtained with the polynomial expansion are slightly larger, and sometimes better (see, for instance, the results for O, Ne, Mn, Fe, and Zn), than the HF energies obtained with the original GCHF method [26], and very close to the best NHF energies reported in the literature [27].


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Chapter 5

In Table 5.4, we can also see that the reduction in the total number of GTF exponents increases with the size of the atom, i.e., for the first-row atoms (C, O, F, and Ne), the reduction in the total number of GTF exponents, when compared to the GTF size of the best GTF basis sets, obtained with the original GCHF method [17], varies between 6 and 7; for the second-row atoms (Mg, Al, Si, S, and Cl), this reduction varies between 8 and 9; and for the third-row atoms (Mn, Fe, and Zn), it varies from 9 to 11. This shows that the GCHF method with the polynomial expansion works very well for light atoms, but its performance gets better when we are working with heavier atoms. This is the reason why the GCHF method with polynomial expansion has become extremely useful in the design of relativistic GTF basis sets, since in these cases we very often work with heavy atoms. The applications of the polynomial expansion in relativistic calculations will be presented in Chapter 7. Except for the cases where the polynomial energies are better (O, Ne, Mn, Fe, and Zn), we can see in Table 5.4 that the energy differences between our results and those obtained with the best original GCHF method [26] vary within an accuracy of 10⫺6 and 10⫺7 hartree. When now we compare our results with the numerical values [27] (last column in Table 5.4), we can see that the energy differences vary within the accuracy of 1015 to 10⫺7 hartree. At this point it is interesting to mention that other authors have also employed the idea of segmenting the space of the basis set exponents [28,29], but in a different way compared to the methodology described above. One proposed the use of basis set segmentation in the even-tempered methodology with STFs [28] and the other one was the first to propose the segmentation of basis set in the original GCHF method by using GTFs [29]. In both cases, the segmentation improved the energy results; and particularly for the work with GTFs [29], the HF energies obtained for the atoms studied were not as accurate as the HF energy results obtained with our polynomial expansion. In fact, the difference between our methodology and the others is that we also use the idea of basis set segmentation in the polynomial expansion proposed in Equation (5.62). The results displayed in Table 5.4 showed that the GCHF method with the polynomial expansion is a very powerful methodology to be employed in the design of compact and highly accurate GTF basis sets for atomic and molecular calculations. For sure, one of the great advantages of this methodology was the generation of GTF basis sets to be used in relativistic atomic and molecular calculations, as we will see in Chapter 7. The GTF basis set size is more critical in relativistic calculations than in the nonrelativistic case, since it is not easy to have a good balance between size and accuracy when we are trying to generate relativistic GTF basis sets by using the GC method [30–33], i.e., when the accuracy is quite satisfactory, the size is too large. Thus, the polynomial expansion became the basis of a methodology capable of generating compact and highly accurate relativistic GTF basis sets by using the GC method, as we will show in Chapter 7.


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53

Concluding, we can say that the modification in the way of discretizing the GHWHF equations by the polynomial expansion presented in Equation (5.62) substantially improved the generation of Gaussian basis sets to be used in ab initio atomic and molecular calculations when we are employing the GCHF method. This new way of discretizing the GHWHF equations made the optimization of Gaussian exponents in the environment of the GCHF method more flexible and more efficient, since it was able to considerably reduce the size of the Gaussian basis set to be generated with the GCHF method.

References 1. (a) D. L. Hill, and J. A. Wheeler, Phys. Rev., 1953, 89, 1102; (b) J. J. Griffin, and J. A. Wheeler, Phys. Rev., 1957, 108, 311. 2. D. Justin, M. V. Mihailovic, and M. Rosina, Nucl. Phys. A, 1971, 182, 54. 3. M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Modern Phys., 2003, 75, 121. 4. P. Chattopadhyay, R. M. Dreizler, M. Trsic, and M. Fink, Z. Phys. A, 1978, 285, 7. 5. L. Lathouwers, Ann. Phys., 1976, 102, 347. 6. R. Courant, and D. Hilbert, Methods of Mathematical Physics, New York: Interscience, 1962. 7. J. R. Mohallem, R. M. Dreizler, and M. Trsic, Int. J. Quantum. Chem. Symp., 1986, 20, 45. 8. J. R. Mohallem, Z. Phys. D, 1986, 3, 339. 9. P. O. Löwdin, Adv. Phys., 1956, 5, 1; L. Lathouwers, Int. J. Quantum Chem. Symp., 1976, 10, 413. 10. D. Galetti, and A. F. R. de Toledo Piza, Phys. Rev. C, 1978, 17, 774. 11. J. Broeckhove, and E. Deumens, Z. Phys. A, 1979, 292, 243. 12. F. Arickx, J. Broeckhove, E. Deumens, and P. Van Leuven, J. Comp. Phys., 1981, 39, 272. 13. D. M. Bishop, and R. L. Somorjai, J. Math. Phys., 1970, 11, 1150. 14. (a) S. Huzinaga, J. Chem. Phys., 1965, 42, 1293; (b) H. Sambe, J. Chem. Phys., 1965, 42, 1732. 15. J. R. Mohallem, and M. Trsic, Z. Phys. A, 1985, 322, 535. 16. L. Lathouwers, and P. Van Leuven, Adv. Chem. Phys., 1982, 49, 115. 17. J. R. Mohallem, and M. Trsic, Int. J. Quantum. Chem., 1988, 33, 555. 18. C. C. J. Roothaan, Rev. Mod. Phys., 1951, 23, 69. 19. Y. Ishikawa, R. Baretty, and R. C. Binning, Jr., Chem. Phys. Lett., 1985, 121, 130. 20. Y. Ishikawa, and H. M. Quincy, Int. J. Quantum Chem. Symp., 1987, 21, 523. 21. K. Ruedenberg, R. C. Raffanetti, and R. D. Bardo, Energy, Structure and Reactivity – Proceedings of the 1972 Boulder Seminar Research Conference on Theoretical Chemistry, New York: Wiley, 1973. 22. S. Huzinaga, M. Klobukowski, and H. Tatewaki, Can. J. Chem., 1985, 63, 1812. 23. R. C. Barbosa, and A. B. F. da Silva, Mol. Phys., 2003, 101, 1073. 24. A. B. F. da Silva, and M. Trsic, Can. J. Chem., 1996, 74, 1526. 25. A. B. F. da Silva, H. F. M. da Costa, and M. Trsic, Mol. Phys., 1989, 68, 433.


54

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26. F. E. Jorge, and E. P. Muniz, Int. J. Quantum Chem., 1999, 71, 307. 27. T. Koga, S. Watanabe, K. Kanayama, R. Yasuda, and A. J. Thakkar, J. Chem. Phys., 1995, 103, 3000. 28. T. Koga, H. Tatewaki, and A. J. Thakkar, Theor. Chim. Acta, 1994, 88, 273. 29. F. E. Jorge, and E. V. R. de Castro, Chem. Phys. Lett., 1999, 302, 454. 30. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, Phys. Rev. A, 1993, 47, 143. 31. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, J. Chem. Phys., 1994, 101, 6829. 32. F. E. Jorge, and A. B. F. da Silva, J. Chem. Phys., 1996, 104, 6278. 33. F. E. Jorge, and A. B. F. da Silva, J. Chem. Phys., 1996, 105, 5503.


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Chapter 6

Role of the Weight Function in the Design of Efficient Basis Sets for Atomic and Molecular Nonrelativistic Calculations

1.

Introduction

In this chapter, we will discuss the importance of the weight function of the generator coordinate Hartree–Fock (GCHF) method in the generation of basis sets for atomic and molecular nonrelativistic calculations. The weight function has a central role in finding basis set functions of good quality. In the GCHF method, the minimization of the total electronic energy is performed through the description of the weight function, i.e., better described the weight function, the better is the total electronic energy of the atomic or molecular system being studied. The weight function of the GCHF method also had a central role in the generation of universal basis sets. The idea of universal basis sets was first introduced in the literature in 1978 when a Slater universal basis set was presented by using the even-tempered formula [1,2]. Afterwards, other works (the first ones) were published applying the concept of universal basis sets [3–7]. The main interest in developing a universal basis set arose with the idea of transferring integrals from one calculation to another [1]. This idea of transferring integrals from one system to another is interesting but not practical. In fact, with the improvement of computers, it is easier to perform another calculation than to store integrals and try to transfer them to other calculations. Moreover, transferring integrals from one system to another is only possible due to certain similarities between the atomic and molecular systems under study. As one will see along this chapter, the GCHF is extremely useful in the generation of large universal basis sets. The first works with universal basis sets were able to generate a unique set of Slater-type functions (STFs) that was able to describe the atoms from Helium (Z ⫽ 2) to Argon (Z ⫽ 18) [1,2]. Later, with the GCHF method we were able to generate a unique set of Slater- and Gaussian-type functions (GTFs) to describe the neutral atoms and respective positive and negative ions from Hydrogen (Z ⫽ 1) to Xenon (Z ⫽ 54) [8,9]. Actually, the works of References [8,9] present the largest universal Slater and Gaussian basis sets published so


56

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Chapter 6

far in the literature that were only generated from a unique set of basis function exponents. After this achievement, we changed the idea of universal basis sets a bit; i.e., since it is not easy to generate a unique set of basis functions exponents that can describe the total electronic energy for a large number of atoms, we realized that this concept of universal basis set can be made flexible by initially generating a unique set of basis set exponents and from it choose only the exponents that are really important to describe the total electronic energy of a particular atom [10–12]. Here, we would like to remark that the most important work published so far with the GCHF method and that uses this idea of making universal basis sets flexible is the work of de Castro and Jorge [12]. They generated a universal Gaussian basis set for the atoms from H (Z ⫽ 1) to No (Z ⫽ 102). The universal Gaussian basis set exponents of Reference [12] are presented in Appendix 3. Since the idea of this chapter is to show the role of the weight function of the GCHF method in the generation of basis sets, we will present this subject using two classical examples: generation of the STF and GTF universal basis sets for the neutral and ionic atomic species for the atoms from H to Xe [8,9]. In Appendixes 1 and 2, we present the GTF and STF universal basis set exponents of Reference [8].

2.

Weight Function and the Generation of Universal Basis Sets

2.1. Slater and Gaussian Universal Basis Sets for the Ground and Certain Low-lying Excited States of the Neutral Atoms from Hydrogen to Xenon As already commented in the previous section, the GCHF method is tailored to generate universal basis sets. This is true as with the integral discretization (ID) technique of the GCHF method [see Equations (4.12) in Chapter 4] we have a lot of flexibility to choose a universal mesh of the GCHF method that can describe the total electronic energy for a large number of atoms. Since with the ID technique we do not have the cumbersome procedure of optimizing all the basis function exponents, the search for a set of exponents that can describe the energy of a large number of atoms becomes easier. The first universal basis set generated with the GCHF method was presented by Mohallem and Trsic in 1987 [13]. In that work they generated a universal Gaussian basis set for the neutral atoms from Li (Z ⫽ 3) to Ne (Z ⫽ 10) [13]. Afterwards, universal Slater and Gaussian basis sets generated with the GCHF method were presented for the neutral atoms from He (Z ⫽ 2) to Ag (Z ⫽ 18) [14] and for the neutral atoms and respective positive and negative ions from H (Z ⫽ 1) to Xe (Z ⫽ 5 4) [8,9]. These last ones [8,9] are the largest universal


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Role of the Weight Function in the Design of Efficient Basis Sets

57

STFs and GTFs presented in the literature so far, and the only case where, in fact, a unique set of STFs and GTFs exponents is able to describe a very large number of atoms. The (12s, 10p, 10d) universal Slater basis set of Reference [8] is generated with the following discretization parameters (i.e., orbital exponents): Symmetry s p d

min ⫺0.11 0.00 ⫺0.11

0.07 0.07 0.07

N 12 10 10

The (18s, 12p, 11d) universal Gaussian basis set of Reference [8] is generated with the following discretization parameters: Symmetry s p d

min ⫺0.55 ⫺0.40 ⫺0.55

0.16 0.16 0.16

N 18 12 11

min, , and N are the discretization parameters of Equations (4.12). In Table 6.1 we show the total ground state Hartree–Fock (HF) energies for both bases (Slater and Gaussian) and when possible we compare them with the Table 6.1 Total HF ground state energies (a.u.) obtained with the STF and GTF universal bases for the neutral atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) Atom

State

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar

2

S S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 1

STF (12s, 10p, 10d) ⫺0.4999990 ⫺2.8616794 ⫺7.4327054 ⫺14.573019 ⫺24.528977 ⫺37.688610 ⫺54.400921 ⫺74.809384 ⫺99.409341 ⫺128.54710 ⫺161.85889 ⫺199.61457 ⫺241.87368 ⫺288.85383 ⫺340.71873 ⫺397.50482 ⫺459.48187 ⫺526.81717

GTF (18s, 12p, 11d) ⫺0.4999972 ⫺2.8616747 ⫺7.4325533 ⫺14.572976 ⫺24.528672 ⫺37.688474 ⫺54.400709 ⫺74.809108 ⫺99.408974 ⫺128.54656 ⫺161.85776 ⫺199.61356 ⫺241.87044 ⫺288.85101 ⫺340.71600 ⫺397.50163 ⫺459.47810 ⫺526.81241

Numerical [15] ⫺0.5 ⫺2.8616800 ⫺7.4327269 ⫺14.573023 ⫺24.529061 ⫺37.688619 ⫺54.400934 ⫺74.809398 ⫺99.409349 ⫺128.54710 ⫺161.85891 ⫺199.61463 ⫺241.87671 ⫺288.85436 ⫺340.71878 ⫺397.50490 ⫺459.48207 ⫺526.81751 (continued )


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Page 58

Chapter 6

Table 6.1

Continued

Atom

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)

Numerical [15]

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

2

⫺599.16429 ⫺676.75775 ⫺759.73544 ⫺848.40567 ⫺942.88403 ⫺1043.3558 ⫺1149.8653 ⫺1262.4424 ⫺1381.4131 ⫺1506.8693 ⫺1638.9620 ⫺1777.8463 ⫺1923.2563 ⫺2075.3572 ⫺2234.2367 ⫺2399.8655 ⫺2572.4388 ⫺2752.0517 ⫺2938.3531 ⫺3131.5401 ⫺3331.6766 ⫺3538.9995 ⫺3753.5846 ⫺3975.5329 ⫺4204.7400 ⫺4441.5154 ⫺4685.8544 ⫺4937.8915 ⫺5197.6682 ⫺5465.1036 ⫺5740.1368 ⫺6022.9043 ⫺6313.4600 ⫺6611.7553 ⫺6917.9409 ⫺7232.0742

⫺599.15586 ⫺676.74917 ⫺759.72531 ⫺848.39417 ⫺942.87129 ⫺1043.3418 ⫺1149.8491 ⫺1262.4234 ⫺1381.3905 ⫺1506.8427 ⫺1638.9315 ⫺1777.8115 ⫺1923.2153 ⫺2075.3125 ⫺2234.1882 ⫺2399.8123 ⫺2572.3800 ⫺2751.9861 ⫺2938.2756 ⫺3131.4544 ⫺3331.5793 ⫺3538.8884 ⫺3753.4577 ⫺3975.3890 ⫺4204.6061 ⫺4441.3345 ⫺4685.6532 ⫺4937.6684 ⫺5197.4204 ⫺5464.8281 ⫺5739.8269 ⫺6022.5586 ⫺6313.0749 ⫺6611.3266 ⫺6917.4681 ⫺7231.5613

⫺599.16479 ⫺676.75818 ⫺759.73572 ⫺848.40600 ⫺942.88433 ⫺1043.3552 [16] ⫺1149.8662 ⫺1262.4437 ⫺1381.4146 ⫺1506.8709 ⫺1638.9628 [16] ⫺1777.8481 ⫺1923.2610 ⫺2075.3597 ⫺2234.2386 ⫺2399.8676 ⫺2572.4413 ⫺2752.0550 ⫺2938.3574 ⫺3131.5457 ⫺3331.6842 ⫺3539.0096 [17] ⫺3753.5977 [17] ⫺3975.5495 [17] ⫺4204.7887 ⫺4441.5395 [17] ⫺4685.8817 [17] ⫺4937.9210 [17] ⫺5197.6985 [17] ⫺5465.1331 ⫺5740.1691 ⫺6022.9317 ⫺6313.4853 ⫺6611.7840 ⫺6917.9809 ⫺7232.1384

S S 2 D 3 F 4 F 7 S 6 S 5 D 4 F 3 F 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 D 5 F 6 D 7 S 6 S 5 F 4 F 1 S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 1

numerical HF values of Froese–Fisher [15], otherwise we show the best STF atom-optimized values of Clementi and Roetti [16] or the numerical results of Huzinaga and Klobukowski [17]. In Table 6.2 we show the results for various low-lying excited states for the atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) and compare them with the values of Clementi and Roetti [16].


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59

Table 6.2 Some low-lying HF-excited state energies (a.u.) obtained with the STF and GTF universal bases Atom

Configuration

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)

Optimized STF [16]

C C N N O O Si Si P P S S Sc Ti Ti Ti Ti Ti V V V V V V V Cr Cr Cr Cr Cr Cr Cr Cr Cr Cr Cr Mn Mn Mn Mn Mn Mn Mn Mn

|He| 2s22p2 |He| 2s22p2 |He| 2s22p3 |He| 2s22p3 |He| 2s22p4 |He| 2s22p4 |Ne| 3s23p2 |Ne| 3s23p2 |Ne| 3s23p3 |Ne| 3s23p3 |Ne| 3s23p4 |Ne| 3s23p4 |Ar| 4s13d2 |Ar| 4s23d2 |Ar| 4s23d2 |Ar| 4s23d2 |Ar| 4s23d2 |Ar| 4s13d3 |Ar| 4s23d3 |Ar| 4s23d3 |Ar| 4s23d3 |Ar| 4s23d3 |Ar| 4s23d3 |Ar| 4s23d3 |Ar| 4s13d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d4 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5

1

⫺37.631327 ⫺37.549530 ⫺54.296155 ⫺54.228090 ⫺74.729248 ⫺74.611002 ⫺288.81397 ⫺288.75584 ⫺340.64877 ⫺340.60310 ⫺397.45229 ⫺397.37463 ⫺759.68851 ⫺848.33198 ⫺848.35829 ⫺848.34823 ⫺848.23057 ⫺848.38595 ⫺942.79865 ⫺942.81973 ⫺942.73615 ⫺942.72243 ⫺942.82008 ⫺942.76040 ⫺942.87949 ⫺1043.1790 ⫺1043.2221 ⫺1043.1990 ⫺1043.1084 ⫺1043.1418 ⫺1043.1108 ⫺1043.3092 ⫺1043.1669 ⫺1043.0455 ⫺1043.1418 ⫺1043.0003 ⫺1149.6626 ⫺1149.6165 ⫺1149.7225 ⫺1149.5512 ⫺1149.6265 ⫺1149.6100 ⫺1149.6877 ⫺1149.5132

⫺37.631141 ⫺37.549172 ⫺54.295947 ⫺54.227878 ⫺74.728976 ⫺74.610735 ⫺288.81068 ⫺288.75168 ⫺340.64581 ⫺340.59993 ⫺397.44905 ⫺397.37127 ⫺759.68849 ⫺848.32046 ⫺848.34677 ⫺848.33672 ⫺848.21902 ⫺848.37459 ⫺942.78589 ⫺942.80698 ⫺942.72338 ⫺942.70966 ⫺942.80733 ⫺942.74764 ⫺942.86689 ⫺1043.1648 ⫺1043.2079 ⫺1043.1848 ⫺1043.0942 ⫺1043.1276 ⫺1043.0965 ⫺1043.2951 ⫺1043.1527 ⫺1043.0313 ⫺1043.1276 ⫺1042.9860 ⫺1149.6464 ⫺1149.6002 ⫺1149.7062 ⫺1149.5349 ⫺1149.6103 ⫺1149.5937 ⫺1149.6714 ⫺1149.4969

⫺37.631325 ⫺37.549582 ⫺54.296158 ⫺54.228089 ⫺74.729223 ⫺74.610978 ⫺288.81506 ⫺288.75853 ⫺340.64875 ⫺340.60320 ⫺397.45221 ⫺397.37450 ⫺759.69860 ⫺848.33198 ⫺848.35823 ⫺848.34818 ⫺848.23057 ⫺848.38600 ⫺942.79882 ⫺942.81991 ⫺942.73633 ⫺942.72261 ⫺942.82008 ⫺942.79882 ⫺942.87951 ⫺1043.1793 ⫺1043.2223 ⫺1043.1993 ⫺1043.1087 ⫺1043.1421 ⫺1043.1110 ⫺1043.3095 ⫺1043.1672 ⫺1043.0458 ⫺1043.1421 ⫺1043.0005 ⫺1149.6629 ⫺1149.6167 ⫺1149.7228 ⫺1149.5515 ⫺1149.6268 ⫺1149.6103 ⫺1149.6879 ⫺1149.5134 (continued )

D S 2 D 2 P 1 D 1 S 1 D 1 S 2 D 2 P 1 D 1 S 4 F 1 G 1 D 3 P 1 S 5 F 2 H 2 G 2 F 2 D 4 P 2 P 6 D 1 I 3 H 3 G 1 G 3 F 1 F 5 D 3 D 1 D 3 P 1 S 2 I 2 H 4 G 2 D 4 F 2 F 4 D 2 D 1


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60

Page 60

Chapter 6

Table 6.2

Continued

Atom

Configuration

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)

Optimized STF [16]

Mn Mn Mn Mn Fe Fe Fe Fe Fe Fe Fe Fe Fe Fe Fe Co Ni Ni Ni Ni Ni Cu Ge As As Se Se Y Zr Zr Nb Nb Mo Mo Tc Tc Ru Ru Rh Rh Pd Pd Ag

|Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s23d5 |Ar| 4s13d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s23d6 |Ar| 4s13d7 |Ar| 4s13d8 |Ar| 4s23d8 |Ar| 4s23d8 |Ar| 4s23d8 |Ar| 4s23d8 |Ar| 4s13d9 |Ar| 4s23d9 |Ar| 4s23d104p2 |Ar| 4s23d104p3 |Ar| 4s23d104p3 |Ar| 4s23d104p4 |Ar| 4s23d104p3 |Kr| 5s14d2 |Kr| 5s04d4 |Kr| 5s24d2 |Kr| 5s04d5 |Kr| 5s24d3 |Kr| 5s04d6 |Kr| 5s24d4 |Kr| 5s04d7 |Kr| 5s14d6 |Kr| 5s04d8 |Kr| 5s24d6 |Kr| 5s04d9 |Kr| 5s24d7 |Kr| 5s14d9 |Kr| 5s24d8 |Kr| 5s24d9

4

⫺1149.7007 ⫺1149.4123 ⫺1149.5594 ⫺1149.7431 ⫺1262.2936 ⫺1262.3430 ⫺1262.3166 ⫺1262.2122 ⫺1262.2507 ⫺1262.2149 ⫺1262.2797 ⫺1262.1392 ⫺1262.2507 ⫺1262.0864 ⫺1262.3764 ⫺1381.3569 ⫺1506.7548 ⫺1506.7961 ⫺1506.7805 ⫺1506.5882 ⫺1506.8224 ⫺1638.9483 ⫺2075.3179 ⫺2234.1700 ⫺2234.1265 ⫺2399.8168 ⫺2399.7448 ⫺3331.6612 ⫺3538.9221 ⫺3538.9850 ⫺3753.5415 ⫺3753.5389 ⫺3975.3890 ⫺3975.4267 ⫺4204.6717 ⫺4204.7609 ⫺4441.4531 ⫺4441.4633 ⫺4685.8197 ⫺4685.7740 ⫺4937.8640 ⫺4937.7536 ⫺5197.4877

⫺1149.6845 ⫺1149.3960 ⫺1149.5431 ⫺1149.7270 ⫺1262.2746 ⫺1262.3239 ⫺1262.2975 ⫺1262.1932 ⫺1262.2317 ⫺1262.1958 ⫺1262.2607 ⫺1262.1202 ⫺1262.2317 ⫺1262.0673 ⫺1262.3575 ⫺1381.3345 ⫺1506.7282 ⫺1506.7695 ⫺1506.7539 ⫺1506.5616 ⫺1506.7960 ⫺1638.9176 ⫺2075.2726 ⫺2234.1213 ⫺2234.0774 ⫺2399.7635 ⫺2399.6913 ⫺3331.5642 ⫺3538.8112 ⫺3538.8735 ⫺3753.4149 ⫺3753.4116 ⫺3975.2455 ⫺3975.2824 ⫺4204.5102 ⫺4204.5991 ⫺4441.2726 ⫺4441.2821 ⫺4685.6187 ⫺4685.5725 ⫺4937.6406 ⫺4937.5300 ⫺5197.2396

⫺1149.7010 ⫺1149.4123 ⫺1149.5597 ⫺1149.7432 ⫺1262.2944 ⫺1262.3437 ⫺1262.3173 ⫺1262.2129 ⫺1262.2514 ⫺1262.2156 ⫺1262.2805 ⫺1262.1399 ⫺1262.2514 ⫺1262.0871 ⫺1262.3763 ⫺1381.3750 ⫺1506.7560 ⫺1506.7973 ⫺1506.7817 ⫺1506.5894 ⫺1506.8224 ⫺1638.9496 ⫺2075.3208 ⫺2234.1718 ⫺2234.1283 ⫺2399.8185 ⫺2399.7466 ⫺3331.6550 ⫺3538.9172 ⫺3538.9821 ⫺3753.5403 ⫺3753.5394 ⫺3975.3889 ⫺3975.4280 ⫺4204.6768 ⫺4204.7669 ⫺4441.4632 ⫺4441.4746 ⫺4685.8308 ⫺4685.7892 ⫺4937.8815 ⫺4937.7709 ⫺5197.5029

P P 2 S 6 D 1 I 3 H 3 G 1 G 3 F 1 F 3 D 1 D 3 P 1 S 5 F 4 F 1 G 1 D 3 P 1 S 3 D 2 D 1 D 2 D 2 P 1 D 1 S 4 F 5 D 3 F 6 S 4 F 5 D 5 D 4 F 6 D 3 F 5 D 2 D 4 F 3 D 3 F 2 D 2


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61

From Tables 6.1 and 6.2, we can see that the universal STF total energy values are very close to the numerical HF results and competitive with the excited state values of Clementi and Roetti. The performance of the universal GTF bases is almost as good as in the STF case, at the expense of a somewhat larger basis set, as expected. It is also interesting to notice from Tables 6.1 and 6.2 that both STF and GTF universal bases perform somewhat better for excited states than for ground states when compared to the Clementi and Roetti results. The explanation is simple: the exponents of Clementi and Roetti are optimized for the ground state energy and are not optimal for excited states. However, the GCHF exponents generated through the ID technique [see Equations (4.12)] are not biased and are equally adequate for any property. In particular, the capacity of attaining excited states with the GC method was noticed in previous tests on model problems [18,19].

2.2.

The Weight Functions

Here, it is relevant to discuss the behavior of the weight functions f() in the generation of the universal Slater and Gaussian basis sets of Reference [8]. In Figs. 6.1–6.6 we have plotted the GTF weight functions for the noble gases. The corresponding weight functions of the other atoms are comprised between the two

Fig. 6.1 The GTF 1s weight functions for the noble gases from He to Xe, and 2s from Ne to Xe.


62

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Chapter 6

Fig. 6.2

The GTF 3s weight functions for the noble gases from Ar to Xe.

noble gases. The GTF weight functions exhibit a smooth and regular character. The number of nodes increases with the principal quantum number, as expected. As the weight functions approach zero, at the left and right sides, they provide a guidance for the adequate integration range. While the behavior of the weight functions seems an adequate criterion in the present case, in general one should also take into account the character of the kernels [20]. The behavior of the STF weight functions is different. This is illustrated in Fig. 6.7 for the 1s and 4d weight functions of Xe. The 4d weight function is regular and similar to that for the GTF case (see Fig. 6.6) although clearly narrower. In general, the smaller the principal quantum number, the less regular is the STF weight function. This behavior for STF weight functions is understood to be a consequence of a high number of exponents than really needed for the inner orbital wave functions, i.e., the 1s generator functions are so close to the HF functions that very few exponents are needed. This can be visualized for the extended STF bases of Clementi and Roetti [16] (indeed, from Reference [16],


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Fig. 6.3

The GTF 4s weight functions for Kr and Xe, and 5s for Xe.

Fig. 6.4

The GTF 2p weight functions for the noble gases from Ne to Xe.

63

one will see for the 1s case that at most two or three exponents have nonnegligible weights for the atoms from H to Xe). In our case, the universal character of the basis forces into the 1s wave functions more exponents than are needed. That is the reason why we abandoned this restricted way of generating a universal


64

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Chapter 6

Fig. 6.5 The GTF 3p weight functions for the noble gases from Ar to Xe, 4p for Kr and Xe, and 5p for Xe.

Fig. 6.6

The GTF 3d weight functions for Kr and Xe, and 4d for Xe.

basis set, since the light atoms do not need so many exponents to describe the inner orbitals as the heavy ones. Then we decided to flexibilize the way of getting a universal basis set by generating a large set of universal exponents and then choosing only those (from the original universal set of exponents) that are really important to describe the occupied orbitals of each specific atom [10–12].


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Fig. 6.7

65

The STF 1s and 4d weight functions for Xe.

Certainly, the oscillating character of the STF 1s weight functions does not hamper the regular character of the wave functions. In Fig. 6.8 we plot the radial R10(r) and R42(r) functions for Xe with STFs, GTFs, and numerical HF functions [15]. At the scales employed, the three R42(r) functions are not distinguishable while for R10(r) they can be recognized at the origin. It is interesting to remark that the Gaussian R42(r) wave function, generated with the ID technique of the GCHF method [Equations (4.12)], has an adequate behavior at the origin.

2.3. Slater and Gaussian Universal Basis Sets for the Ground and Certain Low-lying Excited States of Positive and Negative Ions of the Atoms from Hydrogen to Xenon After generating the STF and GTF universal basis sets for the neutral atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) [8], we generated STF and GTF universal basis sets for the positive and negative ions for the atoms from H to Xe as well [9]. The (12s, 10p, 10d) STF universal basis set for the positive ions from H to Xe is generated with the following discretization parameters (i.e., orbital exponents): Symmetry s p d

min ⫺0.11 0.00 ⫺0.11

0.07 0.07 0.07

N 12 10 10


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Chapter 6

Fig. 6.8 The R10(r) and R42(r) radial functions for Xe. The values in R10(r) are 785.7768, 782.8561, and 766.5602 for numerical HF, STOs (Slater-type orbitals), and GTOs (Gaussiantype orbitals), respectively.

The (18s, 12p, 11d) GTF universal basis set for the positive ions from H to Xe is generated with the following discretization parameters: Symmetry s p d

min ⫺0.55 ⫺0.40 ⫺0.55

0.16 0.16 0.16

N 18 12 11

The (13s, 11p, 10d) STF universal basis set for the negative ions from H to Xe is generated with the following discretization parameters: Symmetry s p d

min ⫺0.21 ⫺0.10 ⫺0.05

0.07 0.07 0.07

N 13 11 10


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67

The (18s, 12p, 11d) GTF universal basis set for the negative ions from H to Xe is generated with the following discretization parameters: Symmetry s p d

min ⫺0.75 ⫺0.55 ⫺0.75

0.16 0.16 0.16

N 18 12 11

The basis (discretization parameters) for the STF and GTF positive ions from H (Z ⫽ 1) to Xe (Z ⫽ 54) are the same as those we previously employed for the neutral atoms from H to Xe [8]. This coincidence was to be expected, as ionization produces orbital contraction and thus the bases for the neutral atoms were adequate for describing the total energies for ground and low-lying excited states of the positive ions. For the negative ions, therefore, it was not possible to describe the total energies by employing the same basis set as for the neutral atoms. This is a consequence of the entry of one electron in the frontier atomic orbital, making it more diffusive; thus, lower values for min are required for the STF and GTF universal bases. In Tables 6.3–6.6 we present the total STF and GTF HF energies for the ground and low-lying excited states of the positive and negative ions from H to Xe. Our results are compared with the numerical results of Koga et al. [21] and only when numerical results are not available they are compared with the atom-optimized values of Clementi and Roetti [16]. One can see from Tables 6.3–6.6 that our STF energy values (ground and low-lying excited states) compare very favorably with the numerical results and the atom-optimized results of Clementi and Roetti, often being lower than those obtained with the latter. Our GTF energies are only slightly above the STF values in spite of a rather modest increase in the number of basis functions. In the case of positive ions (Tables 6.3 and 6.4), we notice a better performance for both STF and GTF universal bases for the excited states than for the ground states when compared with the Clementi and Roetti results [16]. Also, for the positive and negative ions, the STF energy values are close to the numerical results and competitive with the excited state values of Clementi and Roetti. This capacity to describe excited states with the GC method has already been commented in other works [18,19]. We would also like to make a comparison between the STF energy results obtained for the ground state of cations and anions, and the STF energy results obtained by Koga et al. [22] using doubly even-tempered basis set for the same ionic species (the work of Koga et al. was published basically at the same time we published our work [9] and Koga’s results are not included in Tables 6.3–6.6). Here it is important to pay attention to the fact that we are working with universal STF basis set and thus the sizes of our basis sets are always the same, namely (12, 10p, 10d) for the cations and (13s, 11p, 10d) for the anions. In their work, Koga et al. worked with atom-adapted basis sets and thus their STF basis set size increases with the increasing Z (atomic nuclear charge).


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Chapter 6

Table 6.3 Certain total HF ground state energies (a.u.) obtained with the STF and GTF universal bases for the positive ions of the atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) Ion

Configuration

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)

Numerical [21]

He⫹ Li⫹ Be⫹ B⫹ C⫹ N⫹ O⫹ F⫹ Ne⫹ Na⫹ Mg⫹ Al⫹ Si⫹ P⫹ S⫹ Cl⫹ Ar⫹ K⫹ Ca⫹ Sc⫹ Ti⫹ V⫹ Cr⫹ Mn⫹ Fe⫹ Co⫹ Ni⫹ Cu⫹ Zn⫹ Ga⫹ Ge⫹ As⫹ Se⫹ Br⫹ Kr⫹ Rb⫹ Sr⫹ Y⫹ Zr⫹ Nb⫹ Mo⫹ Tc⫹ Ru⫹ Rh⫹ Pd⫹

1s1 1s2 |He| 2s1 |He| 2s2 |He| 2s22p1 |He| 2s22p2 |He| 2s22p3 |He| 2s22p4 |He| 2s22p5 |He| 2s22p6 |Ne| 3s1 |Ne| 3s2 |Ne| 3s23p1 |Ne| 3s23p2 |Ne| 3s23p3 |Ne| 3s23p4 |Ne| 3s23p5 |Ne| 3s23p6 |Ar| 4s1 |Ar| 4s13d1 |Ar| 4s13d2 |Ar| 4s03d4 |Ar| 4s03d5 |Ar| 4s13d5 |Ar| 4s13d6 |Ar| 4s03d8 |Ar| 4s03d9 |Ar| 4s03d10 |Ar| 4s13d10 |Ar| 4s23d10 |Ar| 4s23d104p1 |Ar| 4s23d104p2 |Ar| 4s23d104p3 |Ar| 4s23d104p4 |Ar| 4s23d104p5 |Ar| 4s23d104p6 |Kr| 5s1 |Kr| 5s2 |Kr| 5s14d2 |Kr| 5s04d4 |Kr| 5s04d5 |Kr| 5s14d5 |Kr| 5s04d7 |Kr| 5s04d8 |Kr| 5s04d9

2

⫺1.99999961 ⫺7.2364143 ⫺14.277376 ⫺24.237574 ⫺37.292210 ⫺53.887994 ⫺74.372585 ⫺98.831700 ⫺127.81780 ⫺161.676950 ⫺199.371747 ⫺241.674587 ⫺288.573013 ⫺340.349622 ⫺397.173104 ⫺459.048438 ⫺526.274229 ⫺599.01715 ⫺676.56959 ⫺759.53890 ⫺848.21678 ⫺942.67613 ⫺1043.1389 ⫺1149.6485 ⫺1262.1502 ⫺1381.1273 ⫺1506.5895 ⫺1638.7265 ⫺1777.5658 ⫺1923.0579 ⫺2075.0846 ⫺2233.8863 ⫺2399.5563 ⫺2572.0426 ⫺2751.5642 ⫺2938.2157 ⫺3131.3680 ⫺3331.4653 ⫺3538.7993 ⫺3753.3764 ⫺3975.3171 ⫺4204.5740 ⫺4441.2978 ⫺4685.6369 ⫺4937.6465

⫺1.99999304 ⫺7.2363955 ⫺14.277325 ⫺24.237467 ⫺37.292075 ⫺53.887733 ⫺74.372265 ⫺98.831309 ⫺127.81723 ⫺161.676170 ⫺199.370717 ⫺241.673312 ⫺288.571210 ⫺340.347128 ⫺397.169882 ⫺459.044563 ⫺526.269378 ⫺599.01087 ⫺676.56129 ⫺759.52883 ⫺848.20529 ⫺942.66338 ⫺1043.1249 ⫺1149.6322 ⫺1262.1314 ⫺1381.1049 ⫺1506.5631 ⫺1638.6960 ⫺1777.5309 ⫺1923.0188 ⫺2075.0411 ⫺2233.8383 ⫺2399.5033 ⫺2571.9838 ⫺2751.4985 ⫺2938.1416 ⫺3131.2834 ⫺3331.3679 ⫺3538.6879 ⫺3753.2498 ⫺3975.1735 ⫺4204.4118 ⫺4441.1171 ⫺4685.4357 ⫺4937.4232

⫺2.0 ⫺7.2364152 ⫺14.277395 ⫺24.237575 ⫺37.292224 ⫺53.888005 ⫺74.372606 ⫺98.831720 ⫺127.81781 ⫺161.676963 ⫺199.371810 ⫺241.674670 ⫺288.573131 ⫺340.349776 ⫺397.173183 ⫺459.048591 ⫺526.27534 ⫺599.01758 ⫺676.57001 ⫺759.53914 ⫺848.20340 ⫺942.67078 ⫺1043.1394 ⫺1149.6494 ⫺1262.2130 ⫺1381.1288 ⫺1506.5911 ⫺1638.7282 ⫺1777.5675 ⫺1923.0597 ⫺2075.0865 ⫺2233.8883 ⫺2399.5586 ⫺2572.0452 ⫺2751.5674 ⫺2938.2199 ⫺3131.3738 ⫺3331.4729 ⫺3538.8093 ⫺3753.3895 ⫺3975.3337 ⫺4204.5944 ⫺4441.3220 ⫺4685.6642 ⫺4937.6759

S S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 3 D 4 F 5 D 6 S 7 S 6 D 3 F 2 D 1 S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 4 F 5 D 6 S 7 S 4 F 3 F 2 D 1

(continued )


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Role of the Weight Function in the Design of Efficient Basis Sets Table 6.3

69

Continued

Ion

Configuration

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)

Numerical [21]

Ag⫹ Cd⫹ In⫹ Sn⫹ Sb⫹ Te⫹ I⫹ Xe⫹

|Kr| 5s04d10 |Kr| 5s14d10 |Kr| 5s24d10 |Kr| 5s24d105p1 |Kr| 5s24d105p2 |Kr| 5s24d105p3 |Kr| 5s24d105p4 |Kr| 5s24d105p5

1

⫺5197.4511 ⫺5464.8491 ⫺5739.9507 ⫺6022.6520 ⫺6313.1407 ⫺6611.4745 ⫺6917.5871 ⫺7231.6447

⫺5197.2034 ⫺5464.5736 ⫺5739.6435 ⫺6022.3091 ⫺6312.7568 ⫺6611.0467 ⫺6917.1148 ⫺7231.1320

⫺5197.4813 ⫺5464.8786 ⫺5739.9784 ⫺6022.6783 ⫺6313.1659 ⫺6611.5034 ⫺6917.6273 ⫺7231.7090

S S 1 S 2 P 3 P 4 S 3 P 2 P 2

Table 6.4 Certain HF low-lying excited state energies (a.u.) obtained with the STF and GTF universal bases for the positive ions of atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) Ion

Configuration

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)

Clementi–Roetti (CR) [16]

Sc⫹ Sc⫹ Ti⫹ Ti⫹ V⫹ Cr⫹ Cr⫹ Mn⫹ Mn⫹ Fe⫹ Co⫹ Ni⫹ Cu⫹ As⫹ As⫹ Se⫹ Se⫹ Br⫹ Br⫹ Zr⫹ Zr⫹ Nb⫹ Nb⫹ Mo⫹ Mo⫹ Tc⫹ Tc⫹ Ru⫹ Ru⫹ Rh⫹

|Ar| 4s03d2 |Ar| 4s2 |Ar| 4s03d2 |Ar| 4s23d1 |Ar| 4s23d2 |Ar| 4s13d4 |Ar| 4s23d3 |Ar| 4s03d6 |Ar| 4s23d4 |Ar| 4s23d5 |Ar| 4s23d6 |Ar| 4s23d7 |Ar| 4s23d8 |Ar| 4s23d104p2 |Ar| 4s23d104p2 |Ar| 4s23d104p3 |Ar| 4s23d104p3 |Ar| 4s23d104p4 |Ar| 4s23d104p4 |Kr| 5s04d3 |Kr| 5s24d1 |Kr| 5s14d3 |Kr| 5s24d2 |Kr| 5s14d4 |Kr| 5s24d3 |Kr| 5s04d6 |Kr| 5s24d4 |Kr| 5s14d6 |Kr| 5s24d5 |Kr| 5s14d7

3

⫺759.50957 ⫺759.46186 ⫺848.18675a ⫺848.05617 ⫺942.49804 ⫺1043.0965a ⫺1042.8897 ⫺1149.5206a ⫺1149.3684 ⫺1262.1248 ⫺1381.0080 ⫺1506.4310 ⫺1638.4780 ⫺2233.8375 ⫺2233.7662 ⫺2399.4756 ⫺2399.4226 ⫺2571.9849 ⫺2571.8993 ⫺3538.8015a ⫺3538.6994a ⫺3753.3496 ⫺3753.2047a ⫺3975.2345 ⫺3975.0440 ⫺4204.5435a ⫺4204.3038 ⫺4441.2545 ⫺4441.1075 ⫺4685.5523

⫺759.49979 ⫺759.45138 ⫺848.17553 ⫺848.04427 ⫺942.48494 ⫺1043.0823 ⫺1042.8752 ⫺1149.5046 ⫺1149.3519 ⫺1262.1054 ⫺1380.9850 ⫺1506.4040 ⫺1638.4468 ⫺2233.7895 ⫺2233.7182 ⫺2399.4227 ⫺2399.3697 ⫺2571.9262 ⫺2571.8406 ⫺3538.6906 ⫺3538.5875 ⫺3753.2223 ⫺3753.0767 ⫺3975.0902 ⫺3974.8990 ⫺4204.3818 ⫺4204.1409 ⫺4441.0734 ⫺4440.9258 ⫺4685.3508

⫺759.50974 ⫺759.46197 ⫺848.18639 ⫺848.05622 ⫺942.49822 ⫺1043.0963 ⫺1042.8897 ⫺1149.5205 ⫺1149.3687 ⫺1262.1252 ⫺1381.0089 ⫺1506.4317 ⫺1638.4791 ⫺2233.8390 ⫺2233.7677 ⫺2399.4774 ⫺2399.4245 ⫺2571.9871 ⫺2571.9014 ⫺3538.7663 ⫺3538.6948 ⫺3753.3505 ⫺3753.2028 ⫺3975.2362 ⫺3975.0459 ⫺4204.5228 ⫺4204.3129 ⫺4441.2678 ⫺4441.1216 ⫺4685.5672

F S 4 F 2 D 3 F 6 D 4 F 5 D 5 D 6 S 5 D 4 F 3 F 1 D 1 S 2 D 2 P 1 D 1 S 4 F 2 D 5 F 3 F 6 D 4 F 5 D 5 D 6 D 6 S 5 F 1

(continued )


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70

Page 70

Chapter 6

Table 6.4

Continued

Ion

Configuration

State

STF (12s, 10p, 10d)

GTF (18s, 12p, 11d)


Rh⫹ Pd⫹ Pd⫹ Ag⫹ Ag⫹ Cd⫹

|Kr| 5s24d6 |Kr| 5s14d8 |Kr| 5s24d7 |Kr| 5s14d9 |Kr| 5s24d8 |Kr| 5s24d9

5

⫺4685.3299 ⫺4937.5200 ⫺4937.2579 ⫺5197.2430 ⫺5196.9397 ⫺5464.4605

⫺4685.1280 ⫺4937.2964 ⫺4937.0339 ⫺5196.9950 ⫺5196.6912 ⫺5464.1844

⫺4685.3450 ⫺4937.5510 ⫺4937.2737 ⫺5197.2612 ⫺5196.9579 ⫺5464.4767

a

D F 4 F 3 D 3 F 2 D 4

Calculated energy is lower than CR (last column) energy.

Table 6.5 Certain total HF ground state energies (a.u.) obtained with the STF and GTF universal bases for the negative ions of atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) Ion

Configuration

State

STF (13s, 11p, 10d)

GTF (18s, 12p, 11d)

Numerical [21]

H⫺ Li⫺ B⫺ C⫺ N⫺ O⫺ F⫺ Na⫺ Al⫺ Si⫺ P⫺ S⫺ Cl⫺ K⫺ Sc⫺ Ti⫺ V⫺ Cr⫺ Mn⫺ Fe⫺ Co⫺ Ni⫺ Cu⫺ Ga⫺ Ge⫺ As⫺ Se⫺ Br⫺ Rb⫺ Y⫺

1s2 |He| 2s2 |He| 2s22p2 |He| 2s22p3 |He| 2s22p4 |He| 2s22p5 |He| 2s22p6 |Ne| 3s2 |Ne| 3s23p2 |Ne| 3s23p3 |Ne| 3s23p4 |Ne| 3s23p5 |Ne| 3s23p6 |Ar| 4s2 |Ar| 4s23d2 |Ar| 4s23d3 |Ar| 4s23d4 |Ar| 4s23d5 |Ar| 4s23d6 |Ar| 4s23d7 |Ar| 4s23d8 |Ar| 4s23d9 |Ar| 4s23d10 |Ar| 4s23d104p2 |Ar| 4s23d104p3 |Ar| 4s23d104p4 |Ar| 4s23d104p5 |Ar| 4s23d104p6 |Kr| 5s2 |Kr| 5s24d2

1

⫺0.48792967 ⫺7.4281908 ⫺24.519013 ⫺37.708843 ⫺54.321952 ⫺74.808340 ⫺99.459452 ⫺161.855037 ⫺241.87740 ⫺288.88960 ⫺340.698757 ⫺397.538229 ⫺459.576707 ⫺599.16161 ⫺759.68823 ⫺848.37177 ⫺942.86207 ⫺1042.6537 ⫺1149.7279 ⫺1262.3660 ⫺1381.3509 ⫺1506.8203 ⫺1638.9631 ⫺1923.2340 ⫺2075.3910 ⫺2234.2175 ⫺2399.9090 ⫺2572.5262 ⫺2938.3404 ⫺3331.6436

⫺0.48792905 ⫺7.4278782 ⫺24.518089 ⫺37.708650 ⫺54.321738 ⫺74.808065 ⫺99.459020 ⫺161.853801 ⫺241.87313 ⫺288.88694 ⫺340.695544 ⫺397.534358 ⫺459.571765 ⫺599.15179 ⫺759.67445 ⫺848.35574 ⫺942.84330 ⫺1043.3133 ⫺1149.7003 ⫺1262.3317 ⫺1381.3089 ⫺1506.7697 ⫺1638.9034 ⫺1923.1484 ⫺2075.2975 ⫺2234.1096 ⫺2399.7836 ⫺2572.3792 ⫺2938.1316 ⫺3331.2480

⫺0.48792973 ⫺7.4282321 ⫺24.519221 ⫺37.708844 ⫺54.321959 ⫺74.789746 ⫺99.459454 ⫺161.855126 ⫺241.87827 ⫺288.88966 ⫺340.698874 ⫺397.538430 ⫺459.576925 ⫺599.16192 ⫺759.68877 ⫺848.37255 ⫺942.86313 ⫺1043.3371 ⫺1149.7291 ⫺1262.3671 ⫺1381.3518 ⫺1506.8211 ⫺1638.9641 ⫺1923.2604 ⫺2075.3947 ⫺2234.2229 ⫺2399.9047 ⫺2572.5363 ⫺2938.3549 ⫺3331.6592 (continued )

S S 3 P 4 S 3 P 2 P 1 S 1 S 3 P 4 S 3 P 2 P 1 S 1 S 3 F 4 F 5 D 6 S 5 D 4 F 3 F 2 D 1 S 3 P 4 S 3 P 2 P 1 S 1 S 3 F 1


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Role of the Weight Function in the Design of Efficient Basis Sets Table 6.5

71

Continued

Ion

Configuration

State

STF (13s, 11p, 10d)

GTF (18s, 12p, 11d)

Numerical [21]

Zr⫺ Nb⫺ Mo⫺ Tc⫺ Ru⫺ Rh⫺ Pd⫺ Ag⫺ In⫺ Sn⫺ Sb⫺ Te⫺ I⫺

|Kr| 5s24d3 |Kr| 5s24d4 |Kr| 5s24d5 |Kr| 5s24d6 |Kr| 5s24d7 |Kr| 5s24d8 |Kr| 5s24d9 |Kr| 5s24d10 |Kr| 5s24d105p2 |Kr| 5s24d105p3 |Kr| 5s24d105p4 |Kr| 5s24d105p5 |Kr| 5s24d105p6

4

⫺3538.9803 ⫺3753.5658 ⫺3975.5151 ⫺4204.6782 ⫺4441.5087 ⫺4685.8375 ⫺4937.8172 ⫺5197.5622 ⫺5739.7796 ⫺6022.3540 ⫺6312.5504 ⫺6610.4740 ⫺6916.1643

⫺3538.6225 ⫺3753.1405 ⫺3975.0143 ⫺4204.1687 ⫺4440.8376 ⫺4685.0774 ⫺4936.9716 ⫺5196.6415 ⫺5738.7722 ⫺6021.3631 ⫺6311.6303 ⫺6609.6989 ⫺6915.6287

⫺3538.9945 ⫺3753.5782 ⫺3975.5263 ⫺4204.7646 ⫺4441.5285 ⫺4685.8756 ⫺4937.8915 ⫺5197.7001 ⫺5740.1751 ⫺6022.9727 ⫺6313.4815 ⫺6611.8280 ⫺6918.0759

F D 6 S 5 D 4 F 3 F 2 D 1 S 3 P 4 S 3 P 2 P 1 P 5

Table 6.6 Certain HF low-lying excited state energies (a.u.) obtained with the STF and GTF universal bases for the negative ions of atoms from H (Z ⫽ 1) to Xe (Z ⫽ 54) Ion

Configuration

State

STF (13s, 11p, 10d)

GTF (18s, 12p, 11d)


B⫺ B⫺ C⫺ C⫺ N⫺ N⫺ Al⫺ Al⫺ Si⫺ Si⫺ P⫺ P⫺ Sc⫺ Ti⫺ V⫺ Cr⫺ Ni⫺ Ge⫺ As⫺ As⫺ Y⫺ Nb⫺ Mo⫺ Tc⫺

|He| 2s22p2 |He| 2s22p2 |He| 2s22p3 |He| 2s22p3 |He| 2s22p4 |He| 2s22p4 |Ne| 3s23p2 |Ne| 3s23p2 |Ne| 3s23p3 |Ne| 3s23p3 |Ne| 3s23p4 |Ne| 3s23p4 |Ar| 4s13d3 |Ar| 4s13d4 |Ar| 4s13d5 |Ar| 4s13d6 |Ar| 4s13d10 |Ar| 4s23d104p3 |Ar| 4s23d104p4 |Ar| 4s23d104p4 |Kr| 5s14d3 |Kr| 5s14d5 |Kr| 5s14d6 |Kr| 5s14d7

1

⫺24.489544 ⫺24.450587a ⫺37.642570a ⫺37.600823 ⫺54.266933a ⫺54.186896a ⫺241.85429 ⫺241.82218 ⫺288.84149a ⫺288.81094 ⫺340.65994a ⫺340.60333a ⫺759.58831 ⫺848.25723a ⫺942.76553 ⫺1043.0966a ⫺1506.6623a ⫺2075.3438 ⫺2234.1804 ⫺2234.1262 ⫺3331.5878a ⫺3753.5505a ⫺3975.1645 ⫺4204.7521a

⫺24.487437 ⫺24.445582 ⫺37.642208 ⫺37.600207 ⫺54.266698 ⫺54.186587 ⫺241.84809 ⫺241.81222 ⫺288.83827 ⫺288.80710 ⫺340.65664 ⫺340.59977 ⫺759.58211 ⫺848.24311 ⫺942.73458 ⫺1043.0736 ⫺1506.2314 ⫺2075.2497 ⫺2234.0724 ⫺2234.0179 ⫺3331.2945 ⫺3753.1256 ⫺3974.8965 ⫺4204.0940

⫺24.490501 ⫺24.444257 ⫺37.642523 ⫺37.600849 ⫺54.266877 ⫺54.186826 ⫺241.85645 ⫺241.83001 ⫺288.84143 ⫺288.81109 ⫺340.65980 ⫺340.60316 ⫺759.59416 ⫺848.25608 ⫺942.76611 ⫺1043.0956 ⫺1506.6533 ⫺2075.3470 ⫺2234.1854 ⫺2234.1313 ⫺3331.5782 ⫺3753.5385 ⫺3975.3877 ⫺4204.6719

a

D S 2 D 2 P 1 D 1 S 1 D 1 S 2 D 2 P 1 D 1 S 5 F 6 D 7 S 6 D 2 S 2 D 1 D 1 S 5 F 7 S 6 D 5 F 1

Calculated energy is lower than CR (last column) energy.


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For the cations with s and p orbitals, Koga’s basis set size varies from (7s, 5p) to (11s, 7p) for C⫹ through Ca⫹. For the cations with s, p, and d orbitals, Koga’s basis size varies from (11s, 7p, 5d) to (13s, 12p, 8d) for Sc⫹ through Xe⫹. In general, for the cations, our STF energy results are on average one decimal figure more accurate than Koga’s results but, although our results are better for the cations lighter than Zr⫹, one has to recognize that Koga et al. used smaller basis sets. From Zr⫹ onwards, Koga’s basis set size becomes more similar to our universal basis size (12s, 10p, 10d). The average errors in our STF energies with respect to the numerical HF energies for the cations are, respectively, 0.000012, 0.000121, 0.00136, and 0.02387 millihartrees for the first, second, third, and fourth rows. The average errors in the energies found by Koga et al. are 0.0004, 0.0063, 0.061, and 0.113 millihartrees for the respective first-, second-, third-, and fourth-row cations. For the anions, Koga’s basis set size varies from (8s, 6p) to (11s, 9p) for B⫺ through Cl⫺, and for K⫺ it is (12s, 8p). For Sc⫺ through I⫺, it varies from (12s, 8p, 6d) to (14s, 13p, 9d). When we compare our STF energy results with Koga’s results, we noticed that our results are on average one decimal figure more accurate than Koga’s results from H⫺ to Se⫺. From Br⫺ to Ag⫺, both STF basis set sizes becomes more similar and, in general, our results have the same accuracy as Koga’s results. From In⫺ onwards, Koga’s results become more accurate than our results, but we have to bear in mind that he worked with a slightly larger basis set (14s, 13p, 9d) than our universal basis set (13s, 11p, 10d). The average errors in our STF energies with respect to the numerical HF limits for the anions are, respectively, 0.000044, 0.001757, 0.0026, and 0.3784 millihartrees for the first, second, third, and fourth rows. Koga et al. found, for the respective first-, second-, third-, and fourth-row anions, the following average errors in the energies: 0.00068, 0.0063, 0.087, and 0.112 millihartrees. In conclusion, we would like to say that a comparison of our results with Koga’s brings to attention the fact that when we are developing a universal basis set, instead of a fully optimized basis set, we face the penalty of using, mainly for lighter atomic systems (from H through Ca), a larger number of basis functions than a fully optimized basis set needs to obtain the same degree of accuracy. Indeed, our experience in developing universal basis sets always showed that this penalty is reduced when we work with atomic systems from the third row on [8, 23–26].

2.4. Role of the Weight Functions in the Evaluation of Total Electronic Energies Since the Griffin–Hill–Wheeler–Hartree–Fock (GHWHF) equations (see Chapter 4) are obtained from the minimization of the functional E with respect to the weight functions, fi, the description of fi governs the quest for the total


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73

energies for any atomic system. Achieving the best HF energy for an atomic system means obtaining the best description of fi through numerical integration, i.e., the ID technique outlined in Chapter 4 (Section 4). The quest for the best weight function associated with any atomic orbital is implemented by the integral discretization described by Equations (4.12), and the discretization parameters min, , and N are responsible for attaining the best weight function. When the GCHF method is employed to generate basis sets, the number of points N determines the size of the basis and, certainly, the larger the value of N, the lower is the ground state energy obtained. But the chosen value for N is a compromise between accuracy and size of the basis set. In the section on “Weight Functions,” we presented a series of weight functions generated by the integration mesh of the neutral atoms from H through Xe and discussed some practical aspects and properties of the weight functions. All the observations we brought about in the aforesaid section are valid with respect to the weight functions of the positive and negative ions presented here but, at this time, we add a few relevant remarks especially related to the ionic species and focus on the Gaussians weight functions. In Fig. 6.9 we have plotted the Gaussian 2s weight functions for the isoelectronic species Be⫹ and Li, and Be and Li⫺. It appears that the integration interval with min ⫽ ⫺0.55 is satisfactory for the Be⫹ and Be species, but it is not adequate for the alkaline atom, and it is particularly insufficient for the negative ion of Li. This is a feature that also appears for the highest occupied orbital of the other alkaline atoms considered, i.e., Na through Cs. For this reason, it was necessary to shift

Fig. 6.9 The 2s Gaussian weight functions for Be⫹, Be, Li, and Li⫺. The point min ⫽ ⫺0.55 for s functions is indicated.


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Chapter 6

min for the GTF universal basis set of the negative ions to min ⫽ ⫺0.75 for the s symmetry (min for s symmetry STF weight functions was shifted to ⫺0.21). The p and d symmetry weight functions also required a revision of the integration limits for the negative ions. In Fig. 6.10, we show how the Gaussian 2s weight functions obtained for the species Li and Li⫺ compare with the corresponding fully integrated weight functions. For both Li and Li⫺, the need for higher values of the weight functions to compensate the truncation of the integration range surfaces. In view of our goal of generating “universal” (a unique set of exponents to be used for all atoms under consideration) bases of tractable size, we opted to retain min values that, in a few cases, did not accomplish a complete numerical integration. It is relevant to point out that this limitation of the min values has little effect on the total HF energy, even for the alkaline atoms and negative ions. Of course, the present selection of lower limits for the numerical integration range should be reconsidered with caution if properties demanding very diffuse orbitals were of interest [20]. Otherwise, for most of the atoms and ions considered, the numerical integration ranges are adequate. This is illustrated in Fig. 6.11 with the example of the Gaussian 2p weight functions for neutral and charged fluorine atoms. Owing to the novelty of our procedure for the selection of basis set exponents, we provide some additional details on how a basis set is initially tailored with the GCHF method. Let us take as an example the tailoring of a universal basis set, nonetheless this recipe can be applied to any kind of basis set (universal or atomadapted STF or GTF basis sets).

Fig. 6.10 Comparison of the Gaussian 2s weight function for Li and Li⫺ and the complete integration range.


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Fig. 6.11

75

Plot of the 2p Gaussian weight functions for F⫹, F, and F⫺.

Since a universal basis set for H to Xe is desired, GCHF calculations for these two and a few more sample atoms are performed with some small and arbitrary values for N. The plots of the weight functions indicate adequate ranges for the numerical integration, i.e., the lowest (min) and the highest (max) values outside which the weight functions for most atoms and occupied orbitals are negligible (a few very diffuse orbitals may demand even lower values for min and max but the inclusion of these would significantly enlarge the basis with little or no influence for the total energy). Within this range, one chooses a constant increment and the number of points N, which are mutually dependent. It is the value of N that determines the size of the basis set, and certainly the larger the value of N, the lower is the ground state energy. The chosen value of N is a compromise between accuracy and the basis set size to be handle in molecular calculations. Also, we would like to remark that since Equation (4.4) represents a continuous and infinite superposition, in the GCHF method we do not need to specify the principal quantum numbers, n, for the various symmetries of the generator functions, i, as the weight functions, fi, distinguish the n states. Thus, all our universal bases presented here consist of the simplest 1s, 2p, and 3d STFs or GTFs.

3.

Is the Generator Coordinate Weight Function a Distribution?

Here, we would like to bring to the attention of the reader to the fact that the weight function of the GCHF method can be considered as a distribution


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function [27,28]. In Chapter 3 we showed that the trivial independent particle model for para-helium [see Equation (3.2)] has a Dirac distribution [Equation (3.5)] [19] as a solution for the weight function. One may wonder whether all weight functions are distributions. Nonetheless, there is no unique notion of the distribution character of a function among mathematicians (see for instance References [27,28]). Following Reference [28], we may define a distribution f (x) symbolically by

∫ f ( x)( x) dx ,

(6.1)

provided that the integral exists. In this definition, we call (x) a test function if (x) is infinitely derivable belonging to C ⬁ (basically meaning that the derivatives of all orders exist and are continuous), and if (x) should tend to zero outside a finite interval (a, b). Before we proceed further with our analogies, we stumble on the requirement (x) ⱖ 0 everywhere. While our 1s, 2p, … GTF weight functions do obey this requirement, 2s, 3p, … have nodes being nonpositive in some segments. It is not difficult to understand this behavior since in our algorithm, we enter only with 1s, 2p, … functions and the integration range given by the discretization of the exponents (i.e., the discretization parameters min, , and N). Thus, the burden of guarantying orthogonality between 1s and 2s, 2p and 3p, and so on is left for the weight functions. In an experiment in progress, entering initially in our algorithm with orthogonal bases we may expect all weight functions to be positive everywhere. We leave this question open for further considerations, either by us or other interested workers.

4. The Future of Generating Basis Sets for Atomic and Molecular Calculations Using the GCHF Method Along the years, the GCHF method became very popular and was the object of hundreds of papers. Other researches have also contributed to the improvement of the method as well as in generating a large number of basis sets for atomic and molecular nonrelativistic calculations. We would like to cite some of these works (the main ones) and say that before the polynomial generator coordinate Hartree–Fock (pGCHF) method presented in Chapter 5 (Section 5), other colleagues tried to improve the way of generating basis sets with the GCHF method by segmenting the integration space , i.e., for the occupied s, p, d, etc. atomic orbitals they used different sets of min and for a given Ns, Np, Nd, etc. (i.e., the basis set size N for each atomic symmetry s, p, d, etc.). Then, for instance,


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we could use for a certain atom five different sets of min and to describe the s orbital (i.e., the s exponents), four to the p orbital (the p exponents), three to the d orbital (the d exponents), etc. Among the works published in the literature with the aim to improve the way of obtaining basis sets with the GCHF method, by segmenting the space, we highlight those in References [29–32]. Other works can also be interesting for the reader since they apply the GCHF method in different ways and cases [33–44]. In conclusion, we would like to say that we believe that the future of the GCHF method in the generation of basis sets hinges on the pGCHF method. So far, it has only been applied in the generation of relativistic Gaussian basis sets with an excellent success (see Chapter 7) rate. We hope that it would also be successful in the generation of basis sets for nonrelativistic atomic and molecular calculations.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

D. M. Silver, S. Wilson, and W. C. Nieuwpoort, Int. J. Quantum Chem., 1978, 14, 635. D. M. Silver, and W. C. Nieuwpoort, Chem. Phys. Lett., 1978, 57, 421. S. Wilson, and D. M. Silver, Chem. Phys. Lett., 1978, 63, 367. D. M. Silver, and S. Wilson, J. Chem. Phys., 1978, 69, 3787. D. L. Cooper, and S. Wilson, J. Chem. Phys., 1982, 76, 6088. D. L. Cooper, and S. Wilson, J. Phys. B, 1982, 15, 493. D. L. Cooper, and J. Gerrat, J. Phys. B, 1983, 16, 3703. A. B. F. da Silva, H. F. M. da Costa, and M. Trsic, Mol. Phys., 1989, 68, 433. A. B. F. da Silva, and M. Trsic, Can. J. Chem., 1996, 74, 1526. F. E. Jorge, E. V. R. de Castro, and A. B. F. da Silva, Chem. Phys., 1997, 216, 317. F. E. Jorge, E. V. R. de Castro, and A. B. F. da Silva, J. Comput. Chem., 1997, 18, 1565. E. V. R. de Castro, and F. E. Jorge, J. Chem. Phys., 1998, 108, 5225. J. R. Mohallem, and M. Trsic, J. Chem. Phys., 1987, 86, 5043. H. F. M. da Costa, M. Trsic, and J. R. Mohallem, Mol. Phys., 1987, 62, 91. C. F. Fisher, The Hartree-Fock Method for Atoms, New York, Wiley, 1977. E. Clementi, and C. Roetti, At. Data Nucl. Data Tables, 1974, 14, 177. S. Huzinaga, and M. Klobukowski, J. Mol. Struct. (Theochem), 1986, 135, 403. P. Chattopadhyay, R. M. Dreizler, M. Trsic, and M. Fink, Z. Phys. A, 1978, 285, 7. J. R. Mohallem, Z. Phys. D, 1986, 3, 339. J. R. Mohallem, and M. Trsic, Int. J. Quantum Chem., 1988, 33, 555. T. Koga, H. Tatewaki, and A. J. Thakkar, J. Chem. Phys., 1994, 100, 8140. T. Koga, E. Shibata, and A. J. Thakkar, Theor. Chim. Acta, 1995, 91, 47. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, Chem. Phys. Lett., 1993, 201, 37. A. B. F. da Silva, G. L. Malli, and Y. Ishikawa, Chem. Phys. Lett., 1993, 203, 201. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, Phys. Rev. A, 1993, 47, 143. A. B. F. da Silva, G. L. Malli, and Y. Ishikawa, Can. J. Chem., 1993, 71, 1713. R. S. Strichartz, A Guide to Distribution Theory and Fourier Transform, Boca Raton, FL: CRC Press, 1994.


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28. E. Butkov, Mathematical Physics, Reading, MA: Addison–Wesley Publishing Co., 1968. 29. F. E. Jorge, and E. V. R. de Castro, Chem. Phys. Lett., 1999, 302, 454. 30. E. V. R. de Castro, F. E. Jorge, and J. C. Pinheiro, Chem. Phys., 1999, 243, 1. 31. J. C. Pinheiro, F. E. Jorge, and E. V. R. de Castro, Int. J. Quantum Chem., 2000, 78, 15. 32. A. C. Neto, F. E. Jorge, and M. de Castro, Int. J. Quantum Chem., 2002, 88, 252. 33. R. Custodio, M. Giordan, N. H. Morgon, and J. D. Goddard, Int. J. Quantum Chem., 1992, 42, 411. 34. R. Custodio, J. D. Goddard, M. Giordan, and N. H. Morgon, Can. J. Chem., 1992, 70, 580. 35. N. H. Morgon, J. Phys. Chem. A, 1998, 102, 2050. 36. I. N. Jardim, O. Treu, M. A. U. Martines, M. R. Davolos, M. Jafelicci, and J. C. Pinheiro, J. Mol. Struct. (Theochem), 1999, 464, 15. 37. J. C. Pinheiro, A. B. F. da Silva, and M. Trsic, J. Mol. Struct. (Theochem), 1997, 394, 107. 38. J. C. Pinheiro, A. B. F. da Silva, and M. Trsic, Int. J. Quantum Chem., 1997, 63, 927. 39. J. C. Pinheiro, F. E. Jorge, and E. V. R. de Castro, J. Mol. Struct. (Theochem), 1999, 491, 81. 40. J. C. Pinheiro, M. Trsic, and A. B. F. da Silva, J. Mol. Struct. (Theochem), 2001, 539, 29. 41. F. R. Sensato, R. Custodio, Q. B. Cass, E. Longo, M. Z. Hernandes, R. L. Longo, and J. Andres, J. Mol. Struct. (Theochem), 2002, 589, 251. 42. J. M. D. Trevas, and R. Custodio, J. Mol. Struct. (Theochem), 2001, 539, 17. 43. H. F. M. da Costa, A. B. F. da Silva, J. R. Mohallem, A. M. Simas, and M. Trsic, Chem. Phys., 1991, 154, 379. 44. A. B. F. da Silva, and M. Trsic, Mol. Phys., 1993, 78, 1301.


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Chapter 7

The Generator Coordinate Dirac–Fock Method and Relativistic Calculations for Atoms and Molecules

1.

Introduction

In the recent years there has been a great interest in using Gaussian-type functions (GTFs) as basis set for relativistic calculations [1–5]. This is mainly due to the work of Ishikawa et al. [1,4–7] who have emphasized that the imposition of finite nuclear boundary conditions for solutions of the Dirac–Fock (DF) equations results in a solution that is Gaussian at the origin, and therefore the GTFs of integer power of r are appropriate basis functions for the finite nuclear model. The GTFs that satisfy the boundary conditions for the finite nucleus automatically satisfy the condition of the so-called kinetic balance for a finite speed of light [7]. In the beginning of the 1990s, da Silva et al. [8–12] presented universal Gaussian basis sets for relativistic calculations. In that papers, the generator coordinate Hartree–Fock (GCHF) method (developed initially for a nonrelativistic environment, see Chapter 6) was employed to generate universal Gaussian function exponents and afterward use them in a relativistic code to assess their efficiency in providing Dirac–Fock–Coulomb (DFC) and Dirac–Fock–Breit (DFB) atomic energies. This procedure actually was first employed by Matsuoka and Huzinaga [13] who showed that orbital exponents optimized through the well-tempered scheme for nonrelativistic atoms (i.e., orbital exponents obtained from a nonrelativistic environment) can be carried over to relativistic calculations to produce wave functions close to the relativistic Hartree–Fock (HF) limit (or DF limit). Thus, instead of optimizing the four well-tempered parameters , , , and anew in a DF code they simply carried over the same optimized well-tempered exponents for a nonrelativistic atom to the relativistic calculation since this procedure turned out to be good enough to produce DFC energy results close to those obtained by numerical solution of the DF equations. After this pioneer work [13], instead of going through the cumbersome procedure of optimizing orbital exponents variationally inside of a relativistic environment (code), that is far more expensive than their nonrelativistic counterpart, one could find several relativistic atomic calculations in the literature following the Matsuoka–Huzinaga recipe [1–5,8–12].


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The idea of Matsuoka and Huzinaga of employing GTFs exponents (originally optimized in a nonrelativistic environment and afterward use them to perform relativistic atomic and molecular calculations just adding to the original set of nonrelativistic GTFs exponents a couple of larger exponents with the aim to better describe the behavior of the inner electrons that are closer to the finite nucleus) was extremely practical, but the idea of generating GTFs exponents in the DF environment could not be forsaken. In fact, through the generation of GTFs exponents directly from the DF environment we could obtain a more adequate set of GTFs exponents to be used in relativistic atomic and molecular calculations since they would be the GTFs exponents directly extracted (generated) from the DF formalism. The idea of generating GTFs exponents directly from the DF environment was achieved in 1996 when Jorge and da Silva published two papers introducing the generator coordinate version of the DF equations that came to be named as “the generator coordinate Dirac–Fock (GCDF) method” [14,15]. What we call the GCDF method comprises actually two formalisms, namely: “the generator coordinate Dirac–Fock–Coulomb (GCDFC) formalism” [14] and “the generator coordinate Dirac–Fock–Breit (GCDFB) formalism” [15].

2.

The Generator Coordinate Dirac–Fock–Coulomb Formalism

For a close-shell (2M-electron) atom of nuclear charge Z, the unperturbed Dirac–Coulomb (DC) Hamiltonian, HDC , in atomic units is

H DC ∑ hD (i ) ∑Vij , i j , i

(7.1)

i, j

where hD(i) is the Dirac Hamiltonian of the ith electron and Vij is the instantaneous Coulomb interaction between electrons i and j, namely Vij 1 / rij .

(7.2)

In Equation (7.2), rij is the distance between the ith and jth electrons and the Dirac Hamiltonian, hD, in Equation (7.1) has the following form hD pc + c 2 Vnuc ,

(7.3)

where and are the Dirac matrices in conventional representation, namely ⎛ 0 ⎞ ⎜ ⎝ 0⎟⎠

⎛0 0 ⎞ and ⎜ . ⎝ 0 2I⎟⎠

(7.4)


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81

In the Dirac matrices and of Equation (7.4), the are the 2 2 Pauli matrices and I is a 2 2 unit matrix. In Equation (7.3), p is the momentum operator and Vnuc is the potential energy due to the interaction between the nucleus and the ith electron. Thus, the total energy is given by E H DC ,

(7.5)

where is the normalized total wave function. The total wave function is an antisymmetrized combination of one-electron orbitals defined as ⎛ r1 P ( r ) m (, ) ⎞ nm ( r , , ) ⎜ 1 n , ⎝ ir Qn ( r )m (, )⎟⎠

(7.6)

where Pn(r) and Qn(r) are the large and small radial wave functions, respectively, and satisfy the orthonormality condition

∫ 关P

n

( r ) Pn ( r ) Qn ( r )Qn ( r )兴dr nn ,

(7.7)

0

where nn is the Kronecker delta. The orbitals nm(r, , ) in Equation (7.6) form an orthonormal set and the choice of the phase in Equation (7.6) enables us to use real radial functions for both large and small components [16]. The angular functions m(, ) in Equation (7.6) are expressed in terms of the Clebsch–Gordon coefficients, the normalized spherical harmonics and the two-component Pauli spinors [16]. The quantum number classifies the orbitals according to their symmetry species analogous to the orbital angular momentum in the nonrelativistic case. In the DFC self-consistent-field (SCF) scheme, the behavior of an electron in a central field potential, V, is described by a radial equation of the form F n n n ,

(7.8)

⎛ V c ⎞ F ⎜ ⎝ c V 2c 2 ⎟⎠

(7.9)

where

with

d dr r

(7.10)


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and d . dr r

(7.11)

⎛ Pn ( r ) ⎞

n ⎜ , ⎝ Qn ( r )⎟⎠

(7.12)

The n in Equation (7.8) is

where Pn(r) and Qn(r) are the radial large and small components, respectively [see Equation (7.6)]. The GCDF method is a result of employing the generator coordinate ansatz [17] in the independent particle model. In this approach, the radial functions are written as continuous superpositions, namely Pn ( r ) ∫ X L ( r , ) f nL () d

(7.13)

Qn ( r ) ∫ X S ( r , ) f nS () d ,

(7.14)

and

where X L and X S are the large and small generator functions, respectively (they may be Slater-type functions (STFs), GTFs, or other type of functions), f nL and f nS are the large and small weight functions, respectively, and is the generator coordinate. The variation of the energy expectation value of Equation (7.5) with respect to the weight functions f nT , where T is either L (large) or S (small), produces the integral DF equations

∫ 关F

(, ) f nL ( ) FLS (, ) f nS ( )兴 d ∫ n SLL (, ) f nL ( ) d

(7.15a)

∫ 关F

(, ) f nL ( ) FSS (, ) f nS ( )兴 d ∫ n SSS (, ) f nS ( ) d .

(7.15b)

LL

and SL

In the nonrelativistic limit, these equations reduce to the Griffin–Hill– Wheeler–Hartree–Fock (GHWHF) equations of Reference [17]. The integration of Equations (7.15) is performed by the same technique we have used to solve the GHWHF equations of the nonrelativistic case [17], i.e., the integral discretization (ID) technique [18] (see Chapter 6). Although we resorted


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to a discretization technique for the solution of Equations (7.15), the continuous character of the original generator coordinate method (GCM) was retained. This was achieved by simply choosing equally spaced discretization points in an attempt to find an adequate numerical integration of Equations (7.15). This procedure characterizes the ID technique and it is implemented through a relabeling of the generator coordinate space, namely i

ln i min (i 1), A

A 1.

(7.16)

To solve Equations (7.15) by discretization, and Pn(r) and Qn(r) become Pn ( r ) ⬵ ∑ X L ( r , w ) f nL ( w ) w

(7.17)

Qn ( r ) ⬵ ∑ X S ( r , w ) f nS ( w ) w .

(7.18)

w

and

w

The term f Tn(w)w, where T stands for L or S in Equations (7.17) and (7.18), may be regarded as expansion coefficients while {w} becomes the set of basis exponents. Thus, basis set exponents are interpreted as a mesh of ID points. The lowest (min) and the highest (max) values for the generator coordinate are chosen so as to embrace the adequate integration range for the f Tn . The former values are related through max min ( N 1) ,

(7.19)

where N is the number of discretization points. In fact, the choice of the discretization points determines the exponents of the basis functions and with the ID technique the cumbersome procedure of optimizing orbital exponents variationally is avoided since the GCDF method needs only a good numerical integration to generate a set of exponents to adequately simulate Equations (7.15). It is import to mention that at each iteration of the SCF procedure the integrations are implemented numerically through discretization. This procedure leads formally to the Dirac–Fock–Roothaan (DFR) equations and therefore can be easily implemented in any DFR code. After discretization, Equations (7.15) can be written in the matrix form as F f S f E ,

(7.20)


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where the overlap matrix S is given in a block-diagonal form ⎛ S LL 0⎞ S ⎜ SS ⎟ ⎝ 0 S ⎠

(7.21)

and the superscripts LL and SS are related to the large and small components, respectively. The DF matrix can be written as F o t ,

(7.22)

where the one-electron part o is ⎛ V LL o ⎜ SL ⎝ c

c

LS

V 2c S SS

2

SS

⎞ ⎟⎠ .

(7.23)

The two-electron part t , which consists of the matrices of two-electron Coulomb and exchange interactions, is given by ⎛ J LL K LL t ⎜ ⎝ K SL

K LS ⎞ . J SS K SS ⎟⎠

(7.24)

The one-electron matrix elements are given in terms of the generator coordinate by

T T VTT ij ∫ X ( r , i )Vnuc ( r ) X ( r , j ) dr ,

(7.25)

0

STTij ∫ X T ( r , i ) X T ( r , j ) dr ,

(7.26)

0

where TT is either LL or SS,

⎛ d ⎞ SLij ∫ X S ( r , i ) ⎜ ⎟ X L ( r , j ) dr , ⎝ dr r ⎠ 0

(7.27)

and

⎛ d ⎞ LSij ∫ X L ( r , i ) ⎜ ⎟ X S ( r , j ) dr . ⎝ dr r ⎠ 0

(7.28)


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The two-electron matrices, J TT and KTT苶, where the superscripts T and T 苶 are either L or S, have matrix elements of the form J TTij

∑ (2 j1)共 D

TT k l

, k ,l

J 0,ijTT,,kTTl DTTk l J 0,ijT,T,kTTl 兲

(7.29)

and KTTij ∑ ∑ (2 j1)b ( jj) DTTk l Kij,TT,,kTTl . k l

(7.30)

苶 represent a pair LS or SL. The Coulomb and exchange inteThe superscripts TT grals in Equations (7.29) and (7.30) are given in terms of the generator coordinate and in terms of the GTFs {X L} and {X S} as

J ij,TT,,kTTl ∫ ∫ X T ( r , i ) X T ( r , j )U ( r , s) X T ( s, k ) X T ( s, l ) dr ds

(7.31)

0 0

and

Kij,TT,,kTTl ∫ ∫ X T ( r , i ) X T ( r , k )U ( r , s) X T ( s, j ) X T ( s, l ) dr ds , (7.32) 0 0

where

⎧ r 冒s1 , r s U ( r , s) ⎨ 1 ⎩ s 冒r , s r .

(7.33)

The density matrices are defined in terms of the generator coordinate as DTTij fT (i ) fT ( j ),

(7.34)

where f T are the weight functions. The GTFs generated with the GCDFC formalism satisfy the relativistic boundary conditions associated with the finite nuclear model for a finite speed of light and conform to the so-called kinetic balance at the nonrelativistic limit. In the uniform charge approximation of the finite nucleus, the potential inside the nucleus is represented by (see Reference [7] and references therein) r2 ⎞ ⎛ Z ⎞⎛ in ⎜ ⎟ ⎜ 3 2 ⎟ . Vnuc ⎝ 2R ⎠ ⎝ R ⎠

(7.35)


86

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Chapter 7

In Equation (7.35), R is the nuclear radius given by [19] R 2.2677 105 M 1 3 ,

(7.36)

where M is the atomic mass number. The potential outside the nucleus is given by the Coulomb potential, V out nuc Z/r. With this representation of the potential, the s1/2 solutions near the origin may be given as [5] P(r) 1 g2 r 2 g4 r 4 + K r

(7.37a)

Q( r ) b1r b3 r 3 K , r

(7.37b)

and

so that for w arbitrary parameters we have P ( r ) r g2 r 3 L⬵ r exp( w r 2 )

(7.38)

Q ( r ) b1r 2 b3 r 4 L⬵ r 2 exp( w r 2 ).

(7.39)

and

In Equations (7.37), P(r)/r has a finite value and zero slope at the origin whereas Q(r)/r vanishes there. Thus, in the finite nuclear model the GTF of integer power of r are appropriate basis functions since the imposition of the finite nuclear boundary results in a solution that is Gaussian at the origin. If, for instance, we choose for the s1/2 states the radial large-component basis set {X L} as a GTF of the form X L ( r , w ) N L r exp( w r 2 ),

(7.40)

then the condition of kinetic balance imposes the radial small-component basis set {X L} to be [20–22]

⎡ d ⎤ X S ( r , w ) ⎢ ⎥ X L ( r , w ) N S r 2 exp( w r 2 ). ⎣ dr r ⎦

(7.41)

In Equations (7.40) and (7.41), NL and NS are normalization constants and w is the generator coordinate.


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87

These kinetically balanced Gaussian basis functions are precisely of the form given in Equations (7.38) and (7.39). This is a consequence of the fact that the exponent of r in the Gaussian basis functions does not depend on the speed of light. The kinetic balance simply guarantees that the solution of matrix DF equations approaches the correct nonrelativistic limit when c is taken to infinity [7], and by itself does not guarantee an upper bound on the atomic relativistic energy. But the combination of the kinetic balance condition with the choice of the power of r in the Gaussian basis functions as an integer, as required by the boundary conditions when one uses the finite nuclear model, is capable of reproducing the correct relativistic kinematics of an electron near the nucleus [23].

3. The Generator Coordinate Dirac–Fock Method and the Generation of a Universal Gaussian Basis Set for the Relativistic Closed-Shell Atoms from Zinc to Nobelium The first application of the GCDF method in DFC calculations was in the generation of a relativistic universal Gaussian basis set (RUGBS) for the closed-shell atoms from Zinc (Zn) through Nobelium (No) [14]. Here it is interesting to remember, as we have already mentioned in Chapter 6, that a universal basis set represents an unique set of basis function exponents that is able to describe a large number of atoms. On the contrary, an adapted basis set means that for each atom we have a different set of basis function exponents. In that work [14], were performed DF-SCF calculations on all relativistic closed-shell atoms with occupied d and f orbitals, i.e., all relativistic closed-shell atoms from Zn (Z 30) up to No (Z 102), and the finite nucleus model of uniform proton-charge distribution and the restricted kinetic balance condition (oneto-one correspondence between the basis functions of large and small components) were employed in all of the calculations. The nuclear radius used was R 2.2677 105 M1/3, where M is the atomic mass number, and the speed of light, c, was assumed as 137.0370 a.u. The 32 relativistic universal Gaussian exponents generated in this pioneer work with the GCDFC formalism can be found in Reference [14]. Just for comparison, we listed in Table 7.1 the DFC energies attained with the RUGBS, geometrical Gaussian basis sets (GGBS) [3], and numerical-finite-difference programs [3,10,11,19]. The third column in Table 7.1 (RUGBS size) indicates the number of exponents in each relativistic symmetry (s1/2, p1/2, p3/2, d3/2, d5/2, f5/2, and f7/2) taken from the 32 relativistic universal Gaussian exponents. For each atomic system studied the number of exponents for each s, p, d, and f orbital symmetry was increased until the total DFC energy value was attained within an accuracy comparable (or better) to that obtained with numerical-finite-difference programs.


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Chapter 7

Table 7.1 Total Dirac–Fock–Coulomb (EDFC) energies (in hartree) for Zn (Z 30) through No (Z 102) Atom

Atomic Mass (M)

RUGBS Size

EDFC (RUGBS)a

Zn Ge Kr Sr Pd Cd Sn Xe Ba Sm Yb W Hg Pb Rn Ra Pu No

65.37 72.59 83.80 87.62 106.40 112.40 118.69 131.30 137.34 150.35 173.04 183.85 200.59 207.19 222.00 226.00 242.00 259.00

32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f

1794.61338 2097.46672 2788.86168 3178.08131 5044.40343 5593.32279 6176.14096 7446.90026 8135.65000 10429.3726 14067.6811 16156.2494 19648.8712 20913.7211 23601.9780 25028.0218 29656.5171 36740.2908

EDFC (GGBS)b 1794.61331 2097.46666 2788.86149 3178.08115 5593.32242 6176.14055 7446.89956 8135.64919 16156.2463 19648.8665 20913.7151 23601.9706

Numerical-FiniteDifference (EDFC) 1794.61340c 2097.46675c 2788.86168c 3178.08133c 5044.40d 5593.32286c 6176.14105c 7446.90018c 8135.65006c 10429.2d 14067.7d 16156.2484c 19648.8692c 20913.7184c 23601.9742c 25028.0165e 29656.5122e 36740.2857f

a

Total DFC energies obtained by the relativistic universal Gaussian basis set (RUGBS). EDFC obtained by using geometric Gaussian basis sets (Reference [3]). c EDFC obtained from Reference [3]. d EDFC obtained from Reference [19]. e EDFC obtained from Reference [10]. f EDFC obtained from Reference [11]. b

The 32 relativistic universal Gaussian exponents of Reference [14] were generated by employing the ID technique of the GCDF method, i.e., in the new generator coordinate space, , an equally spaced N-point mesh, i, was chosen so as to obtain an adequate numerical integration range for the relativistic s1/2, p1/2, p3/2, d3/2, d5/2, f5/2, and f7/2 symmetries for all atoms studied. The integration range is characterized by the starting point min, the increment , and the maximum number of points N of the mesh i. The optimum scaling parameter A found for all calculations was 6.0. From the results displayed in Table 7.1 we can see how powerful is the GCDF method in the generation of Gaussian basis functions exponents for relativistic calculations. For the atoms Xe, Sm, W, Hg, Pb, Rn, Ra, Pu, and No, the total DFC energies computed with the RUGBS are lower than those obtained by using numerical-finite-difference DF calculations (last column in Table 7.1). For the atoms Kr, Pd, and Yb, the total DFC energies were the same as those attained by using numerical-finite-difference DF methods. For the atoms Zn, Ge, Sr, Cd, Sn,


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89

and Ba, the total DFC energies were slightly above those attained with the numerical-finite-difference DF results. The total DFC energies computed with the RUGBS were consistently more accurate than those obtained by using the GGBS developed by Mohanti and Clementi [3] although our RUGBS be smaller in size than the GGBS [3]. Also, it is interesting to mention that in contrast to the RUGBS, the GGBS are not universal since for each atomic system displayed in Table 7.1 there is a different set of geometrical Gaussian exponents. The surprising DFC energy results obtained for Xe, Sm, W, Hg, Pb, Rn, Ra, Pu, and No, when compared to numerical-finite-difference calculations, were attributed to the proper representation of the wave function inside the nucleus due to the higher numerical accuracy of the relativistic Gaussian basis functions generated with the GCDF method. This evidence was not completely new since in previous works Malli et al. [12] and Parpia and Mohanty [24] had already verified the possibility of basis set functions be able to better represent the wave function inside the nucleus than the numerical-finite-difference methods available by that time. Parpia and Mohanty [24] have also commented that the lower accuracy of the finite-difference procedure is likely due to the far smaller number of tabulation points in the region of the nuclear “skin,” and the problem gets worse when the atomic number increase since the orbital overlap with the atomic nucleus is known to increase rapidly with the increase of the atomic number. Although the atomic systems studied [14] were different from those studied by Parpia and Mohanty [24], the rapid contraction of the orbital wave function for very large atomic number was qualitatively similar. Parpia and Mohanty [24] also verified that their basis set calculations for Z 70 provided lower DFC energy results than their corresponding numerical-finite-difference calculations. From Table 7.1 we can see that a similar behavior was also found in our DFC calculations since for Z 74 we always were able to obtain DFC energy results lower than the corresponding ones attained with numerical-finite-difference calculations. It is also interesting to mention that Malli et al. [12] and Parpia and Mohanty [24] came up with the same evidence despite each one had used different charge distribution. Malli et al. [12] represented the nucleus as a finite body of uniform proton charge distribution, and Parpia and Mohanty [24] assumed that the nuclear charge density was distributed in the form of a Fermi function. In our work[14] we have also used the same charge distribution as Malli et al. [12]. That pioneer work [14] was really important since it reinforced the fact that some improvements were needed to be made in relativistic numerical codes to correct their bad description for the relativistic kinematics of an electron inside the nucleus. In 1997, Visscher and Dyall [25] presented in the literature the new relativistic numerical results for finite-size nucleus models that seem to have corrected this problem.


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Chapter 7

That time we attributed the high numerical accuracy of our RUGBS to the ID technique of the GCDF method, as with the ID technique the relativistic Gaussian function exponents were generated by the discretization of the integral DF equations [Equations (7.15)] with the best numerical integration as a goal. In fact, a careful numerical integration of Equations (7.15) is able to provide very accurate Gaussian basis sets for relativistic calculations. Also, it is important to bring attention to the fact that the GCDF method is an algorithm whereby Gaussian basis sets can be generated directly from a relativistic environment (code) without variational exponent optimization. Table 7.2 shows the convergence pattern for the Xe atom and from there we can see that the use of the kinetic balance condition, along with the proper boundary conditions imposed on the GTFs, accelerates the convergence and also contributes to the DFC energy stability. Also, we can see that the increasing number of points in each s1/2, p1/2, p3/2, d3/2, and d5/2 symmetry rapidly favor the convergence of the total DFC energy to a value of 7446.90026 a.u., and by the time we have reached Ns 32 (number of basis functions for s1/2 symmetry), Np 30 (number of basis functions for p1/2 and p3/2 symmetries), and Nd 20 (number of basis functions for d3/2 and d5/2 symmetries) the total DFC energy has already converged to the energy limit of 7446.90026 a.u. Even if we go beyond Ns 32, Np 30, and Nd 20 there is no substantial improvement in the total DFC energy. Tables 7.3–7.5 display the DFC orbital energies for Xe (Z 54), Rn (Z 86), and No (Z 102), respectively, obtained with the RUGBS. The third column in Tables 7.3–7.5 shows the DFC orbital energies for Xe, Rn, and No obtained by numerical-finite-difference DF calculations. From Tables 7.3–7.5 we can see that the DFC orbital energies obtained with the RUGBS for Xe, Rn, and No are in good agreement when compared to the corresponding numerical-finite-difference DF orbital energies. Table 7.2 Convergence pattern (in hartree) for the ground state of Xe (Z 54) RUGBS Size

(EDFC)a

30s27p18d 30s27p20d 30s28p20d 31s28p20d 31s29p20d 31s30p20d 32s30p20d 32s31p20d 32s32p20d

7446.892020 7446.892026 7446.892201 7446.899062 7446.899132 7446.899153 7446.900256 7446.900259 7446.900259

a

EDFC represents the Dirac–Fock–Coulomb (DFC) energies.


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Dirac–Fock Method and Relativistic Calculations Table 7.3 Dirac–Fock–Coulomb ( DFC) orbital energies (in hartree) for Xe (Z 54) Orbital

DFC (RUGBS)a

DFCb

1s1/2 2s1/2 3s1/2 4s1/2 5s1/2 2p1/2 3p1/2 4p1/2 5p1/2 2p3/2 3p3/2 4p3/2 5p3/2 3d3/2 4d3/2 3d5/2 4d5/2 Total DFC energy

1277.259 202.4652 43.01052 8.429914 1.010136 189.6796 37.65997 6.452486 0.4925703 177.7046 35.32531 5.982789 0.4398034 26.02343 2.711334 25.53717 2.633763 7446.90026

1277.259 202.4651 43.01043 8.429881 1.010123 189.6795 37.65989 6.452445 0.4925637 177.7047 35.32536 5.982828 0.4398111 26.02344 2.711356 25.53723 2.633820 7446.90018

a

Using the relativistic universal Gaussian basis set (RUGBS). Numerical DFC orbital energies obtained from Reference [3].

b

Table 7.4 Dirac–Fock–Coulomb ( DFC) orbital energies (in hartree) for Rn (Z86) Orbital 1s1/2 2s1/2 3s1/2 4s1/2 5s1/2 6s1/2 2p1/2 3p1/2 4p1/2 5p1/2 6p1/2 2p3/2 3p3/2 4p3/2 5p3/2 6p3/2 3d3/2 4d3/2

DFC (RUGBS)a 3641.154 668.8040 166.8316 41.31328 8.409008 1.071465 642.3282 154.8946 36.01954 6.408996 0.540383 541.1029 131.7312 30.12079 5.175916 0.384030 112.5673 21.54834

DFCb 3641.152 668.8032 166.8311 41.31299 8.408858 1.071426 642.3275 154.8941 36.01923 6.408880 0.540362 541.1030 131.7312 30.12084 5.175965 0.384038 112.5673 21.54831

(continued )

91


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Chapter 7 Table 7.4

Continued

Orbital 5d3/2 3d5/2 4d5/2 5d5/2 4f5/2 4f7/2 Total DFC energy

DFC (RUGBS)a 2.189721 107.7594 20.43896 2.016612 9.194217 8.928509 23601.9780

DFCb 2.189770 107.7595 20.43913 2.016730 9.194232 8.928566 23601.9742

a


b

Table 7.5 Dirac–Fock–Coulomb ( DFC) orbital energies (in hartree) for No (Z 102) Orbital

DFC (RUGBS)a

1s1/2 2s1/2 3s1/2 4s1/2 5s1/2 6s1/2 7s1/2 2p1/2 3p1/2 4p1/2 5p1/2 6p1/2 2p3/2 3p3/2 4p3/2 5p3/2 6p3/2 3d3/2 4d3/2 5d3/2 3d5/2 4d5/2 5d5/2 4f5/2 5f5/2 4f7/2 5f7/2 Total DFC energy

5526.516 1082.793 285.3977 78.61882 18.80527 2.795032 0.2092464 1047.357 269.1022 70.97112 15.54233 1.727430 808.7878 212.1871 55.21117 11.43694 1.049245 187.1341 43.23725 6.610122 176.4444 40.45725 5.985614 24.68849 0.5665813 23.91509 0.4690872 36740.2908

a


b

DFCb 5526.510 1082.790 285.3967 78.61809 18.80506 2.795121 0.2093179 1047.355 269.1013 70.97035 15.54224 1.727562 808.7879 212.1871 55.21111 11.43731 1.049447 187.1341 43.23699 6.610473 176.4445 40.45733 5.986144 24.68830 0.5669459 23.91498 0.4694412 36740.2857


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The GCDF method was the first algorithm capable of generating Gaussian basis sets directly from the relativistic environment (code). In the GCDF method, the Gaussian function exponents are not parameters to be variationally optimized as usual but they are generated by some criterion (the ID technique) for the integration of the integral DF equations. The implementation of the ID technique (through the relabeling of the generator coordinate space [Equation (7.16)] for solving the integral DF equations [Equations (7.15)]) other than allowing to span the relevant integration range (so that we can obtain universal basis sets) has the advantage of avoiding variational exponent optimization. This pioneer work with the GCDF method [14] had the merit to show that a careful numerical integration of the integral DF equations is able to provide highly accurate GTFs to be used in atomic and molecular relativistic calculations.

4. The Generator Coordinate Dirac–Fock Method and the Generation of a Relativistic Universal Gaussian Basis Set for Atoms from Hydrogen to Nobelium Another important application of the GCDF method was in the generation of a RUGBS for the atoms from Hydrogen (Z 1) through Nobelium (Z 102) [26]. In this application, the DFC calculations were performed by using the DFRATOM atomic program [27] and two nuclear models, namely: (a) the finite nucleus of uniform proton-charge distribution and (b) the finite nucleus of Gaussian protoncharge distribution. In the finite nucleus model of uniform proton-charge distribution, the nuclear radius was given by [28]

R

5 共0.836 M 1 3 0.570兲 (fm), 3

(7.42)

where M is the atomic mass number. The speed of light, c, was assumed as 137.0359895 a.u. [26]. The restricted kinetic balance condition [20] was assumed in all calculations, and the SCF equations for the average energy of configurations were solved using the Matsuoka’s method [29]. Also, the same Gaussian functions were employed to represent the following pairs of relativistic atomic orbital symmetries: p1/2 and p3/2; d3/2 and d5/2; f5/2 and f7/2. The RUGBS was obtained by searching the desired accuracy for the heaviest atom (Nobelium) with a basis set error lesser than 10 millihartree for this atom. The optimum discretization parameters found for Nobelium and applied to describe all of the 102 atoms under study, i.e., from Hydrogen (H) through Nobelium (No), were min and equal to 0.710 and 0.115, respectively. The exponents of the RUGBS, obtained from these discretization parameters [see Equation (7.16)], are shown in Appendix 4. Only the exponents of the basis set


94

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Chapter 7

presented in Appendix 4 are used to describe all of the atomic orbital symmetries (s, p, d, and f). The RUGBS functions for each atom were used to calculate the DFC total energies from H through No with the two most popular nuclear models, i.e., the finite nucleus of uniform proton-charge distribution and the finite nucleus of Gaussian proton-charge distribution. The electronic configuration of each atom studied, the basis set size, the DFC energies attained, and their errors with respect to numerical DFC calculations [25] (i.e., the basis set error) are presented in Table 7.6. The choice of the basis set functions for each atom studied (i.e., the basis set truncation process from H to No) was carried out trying to avoid the variational prolapse and attain an accurate basis set (definition and more details on the variational prolapse will be presented in Section 4.1). To avoid variational prolapse, the Gaussian functions with the largest exponents were determined by means of an indirect analysis of the weight functions [see Equations (7.13) and (7.14)]. In the case of the GCM, the weight function presents a simple relationship with the atomic orbital coefficients [30]. The atomic orbital coefficient analysis permitted a good description of the weight function behavior of the innermost atomic orbitals in regions close to the nucleus. Once the function with the largest exponent has been defined, the size of the basis set is finally determined by the diffuseness of the electronic cloud for each atom in its electronic configuration. Hence, different electronic configurations of an atom may be represented by different sets of Gaussian function exponents based on its valence region characteristics. Fig. 7.1 presents the basis set error of the RUGBS as a function of the atomic number for the electronic configuration related to the lowest DFC total energy of each atom from H to No. From Fig. 7.1 we can notice that the basis set error obtained with both nuclear models is almost the same for all atoms studied, which is a clear indication that the RUGBS describes very well the region near the nucleus. The small deviations from a smooth increase of the basis set error with the atomic number are only due to the basis set truncation. Actually, we believe that the basis set error behavior showed in Fig. 7.1 appears to be a characteristic of universal Gaussian basis sets obtained by the GCDF method due to the use of a common ID interval, , for all atoms. The largest basis set error observed in Table 7.6 is 8.8 millihartree for No (the atom with the largest atomic number). Such an error is very small considering that this atom presents a DFC total energy of approximately 36740 millihartree. However, the RUGBS error for lighter atoms is much smaller (for instance, it is 3.6, 0.5, 0.1, 0.08, 0.008, and 0.0004 millihartree for Rn, Xe, Kr, Ar, Ne, and He, respectively). Therefore, these RUGBS errors assure that the RUGBS generated with the GCDF method is very accurate. The percentual errors observed for these illustrative atoms are all around 0.00001–0.00002%, a fact that clearly reinforces the high accuracy of the RUGBS obtained with the GCDF method.


Table 7.6 Total DFC energies (in hartree) and basis set errors (in millihartree) for H (Z 1) through No (Z 102) Atom

Z

M

48

V

23

51

Cr

24

52

Errora

EDFC

Errora

0.50000660 2.86181294 7.43353311 14.57589163 24.53655361 37.67603986 54.32772018 74.82498431 99.50160985 128.6919224 162.0780736 199.9350572 242.3307185 289.4613330 341.4946524 398.5979070 460.9383709 528.6836783 601.5259384 679.7101100 763.3787110 763.3016904 852.8197644 852.7557010 948.1886159 948.1359293 1049.638423 1049.595992

0.00006 0.00038 0.00003 0.00007 0.00063 0.00087 0.00172 0.00179 0.00543 0.008 0.014 0.010 0.031 0.005 0.016 0.023 0.013 0.084 0.015 0.051 0.027 0.043 0.059 0.055 0.040 0.059 0.054 0.042

0.50000660 2.86181294 7.43353311 14.57589163 24.53655361 37.67603986 54.32772018 74.82498431 99.50160985 128.6919224 162.0780736 199.9350572 242.3307185 289.4613331 341.4946525 398.5979071 460.9383709 528.6836784 601.5259388 679.7101106 763.3787119 763.3016913 852.8197657 852.7557023 948.1886177 948.1359311 1049.638426 1049.595994

0.00006 0.00038 0.00003 0.00007 0.00063 0.00088 0.00172 0.00179 0.00544 0.008 0.014 0.010 0.031 0.005 0.016 0.023 0.013 0.084 0.015 0.051 0.027 0.044 0.060 0.055 0.041 0.059 0.054 0.043

(continued )

Page 95

22

20s 19s 28s 27s 26s18p 26s17p 25s17p 25s17p 25s16p 24s16p 31s15p 30s15p 29s25p 29s25p 29s24p 28s24p 28s24p 28s23p 33s23p 32s23p 32s23p16d 32s23p17d 32s22p16d 32s23p16d 32s22p16d 32s22p16d 32s22p15d 32s22p16d

EDFC

07:52

Ti

1s1 1s2 [He]2s1 [He]2s2 [He]2s22p1 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 [He]2s22p6 [Ne]3s1 [Ne]3s2 [Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6 [Ar]4s1 [Ar]4s2 [Ar]4s23d1 [Ar]4s13d2 [Ar]4s23d2 [Ar]4s13d3 [Ar]4s23d3 [Ar]4s13d4 [Ar]4s23d4 [Ar]4s13d5

Gaussian Model

5/4/2007

1 4 7 9 11 12 14 16 19 20 23 24 27 28 31 32 35 40 39 40 45

Uniform Sphere Model

95

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Basis Set Size


H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc

Configuration

Z

M

26

56

Co

27

59

Ni

28

58

Cu

29

63

Zn Ga Ge As Se Br Kr Rb Sr Y

30 31 32 33 34 35 36 37 38 39

64 69 74 75 80 79 84 85 88 89

Zr

40

90

Nb

41

93

Mo

42

98

Tc

43

98

32s22p15d 32s22p16d 32s22p15d 32s22p16d 32s22p15d 32s22p16d 32s22p15d 32s22p15d 32s21p15d 32s22p15d 32s21p15d 31s29p14d 31s29p14d 31s28p14d 31s28p13d 30s28p13d 30s28p13d 34s27p13d 34s27p12d 33s27p20d 33s27p21d 33s27p20d 33s27p21d 33s27p20d 33s27p20d 33s26p20d 33s26p20d 33s26p20d 33s26p20d

EDFC

Errora

EDFC

Errora

1157.321872 1157.288739 1271.391943 1271.367364 1392.001787 1391.985140 1519.305579 1519.296293 1653.457268 1653.454996 1794.612879 1942.563666 2097.470280 2259.441797 2428.588115 2605.023340 2788.860474 2979.804821 3178.079680 3383.761550 3383.699737 3597.083062 3597.041249 3818.167951 3818.148273 4047.136951 4047.141104 4284.112918 4284.142306

0.041 0.037 0.037 0.035 0.036 0.035 0.039 0.078 0.101 0.067 0.089 0.077 0.055 0.083 0.119 0.097 0.089 0.118 0.199 0.165 0.146 0.145 0.147 0.154 0.185 0.241 0.262 0.225 0.234

1157.321875 1157.288743 1271.391947 1271.367368 1392.001792 1391.985145 1519.305586 1519.296300 1653.457277 1653.455005 1794.612890 1942.563682 2097.470300 2259.441822 2428.588147 2605.023379 2788.860523 2979.804894 3178.079770 3383.761658 3383.699845 3597.083191 3597.041379 3818.168107 3818.148430 4047.137142 4047.141295 4284.113144 4284.142532

0.043 0.037 0.038 0.036 0.038 0.037 0.041 0.081 0.104 0.070 0.093 0.082 0.061 0.090 0.127 0.106 0.101 0.119 0.199 0.166 0.146 0.146 0.147 0.155 0.185 0.242 0.262 0.224 0.233

Page 96

Fe

[Ar]4s23d5 [Ar]4s13d6 [Ar]4s23d6 [Ar]4s13d7 [Ar]4s23d7 [Ar]4s13d8 [Ar]4s23d8 [Ar]4s13d9 [Ar]4s23d9 [Ar]4s13d10 [Ar]4s23d10 [Zn]4p1 [Zn]4p2 [Zn]4p3 [Zn]4p4 [Zn]4p5 [Zn]4p6 [Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s14d2 [Kr]5s24d2 [Kr]5s14d3 [Kr]5s24d3 [Kr]5s14d4 [Kr]5s24d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d6

Gaussian Model

07:52

55


5/4/2007

25

Basis Set Size

Chapter 7

Mn

Configuration


Atom

96

Continued

Table 7.6

Pd

46

106

Ag

47

107

Cd In Sn Sb Te I Xe Cs Ba La

48 49 50 51 52 53 54 55 56 57

114 115 120 121 130 127 132 133 138 139

Ce

58

140

Pr Nd Pm Sm Eu Gd

59 60 61 62 63 64

141 144 145 152 153 158

Tb Dy Ho Er Tm

65 66 67 68 69

159 162 162 168 169

0.219 0.224 0.245 0.227 0.251 0.317 0.304 0.281 0.314 0.329 0.564 0.449 0.555 0.511 0.550 0.511 0.553 0.611 0.691 0.683 0.751 0.808 0.835 0.932 1.16 1.24 1.29 1.66 1.31 1.60 1.54 1.49 1.68 1.64

4529.213625 4529.269487 4782.562573 4782.646060 5044.280326 5044.392386 5044.400790 5314.492238 5314.633975 5593.318510 5880.431022 6176.127644 6480.518078 6793.698462 7115.793633 7446.894938 7786.771114 8135.644398 8493.542884 8493.645031 8860.996943 8861.070677 9238.147690 9625.130931 10022.09419 10429.16187 10846.50378 11274.24120 11274.22889 11712.54371 12161.54423 12621.41183 13092.26885 13574.31541

0.219 0.223 0.244 0.227 0.251 0.317 0.303 0.280 0.314 0.327 0.560 0.445 0.549 0.505 0.542 0.502 0.554 0.613 0.693 0.685 0.753 0.811 0.837 0.936 1.17 1.24 1.30 1.66 1.31 1.60 1.55 1.50 1.69 1.66

(continued )

Page 97

103

4529.213354 4529.269215 4782.562252 4782.645740 5044.279945 5044.392005 5044.400409 5314.491791 5314.633528 5593.317969 5880.430389 6176.126892 6480.517203 6793.697409 7115.792434 7446.893528 7786.769498 8135.642505 8493.540705 8493.642851 8860.994438 8861.068172 9238.144815 9625.127610 10022.09039 10429.15743 10846.49871 11274.23533 11274.22302 11712.53701 12161.53656 12621.40314 13092.25880 13574.30401

07:52

45

33s26p20d 33s26p20d 33s26p19d 33s26p20d 33s26p19d 33s26p19d 29s26p20d 33s26p19d 33s26p19d 33s26p19d 32s32p18d 32s32p18d 32s31p18d 32s31p18d 31s31p18d 31s31p18d 35s31p17d 35s30p17d 35s30p17d15f 35s30p20d 35s30p17d15f 35s30p20d14f 35s30p17d15f 35s30p17d15f 34s30p17d15f 34s30p17d15f 34s30p17d15f 34s30p16d15f 34s30p20d15f 34s30p16d15f 34s30p16d15f 34s30p16d15f 34s30p16d14f 34s30p16d14f

5/4/2007

Rh

[Kr]5s24d6 [Kr]5s14d7 [Kr]5s24d7 [Kr]5s14d8 [Kr]5s24d8 [Kr]5s14d9 [Kr]5s04d10 [Kr]5s24d9 [Kr]5s14d10 [Kr]5s24d10 [Cd]5p1 [Cd]5p2 [Cd]5p3 [Cd]5p4 [Cd]5p5 [Cd]5p6 [Xe]6s1 [Xe]6s2 [Xe]6s24f1 [Xe]6s25d1 [Xe]6s24f2 [Xe]6s24f15d1 [Xe]6s24f3 [Xe]6s24f4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f8 [Xe]6s24f75d1 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13


102

97

44


Ru

Z

M

Ta

73

181

W

74

184

Re

75

187

Os

76

192

Ir

77

193

Pt

78

195

Au

79

197

Hg Tl Pb Bi Po

80 81 82 83 84

202 205 208 209 209

[Xe]6s24f14 [Xe]6s24f145d1 [Xe]6s24f145d2 [Xe]6s14f145d3 [Xe]6s24f145d3 [Xe]6s14f145d4 [Xe]6s24f145d4 [Xe]6s14f145d5 [Xe]6s24f145d5 [Xe]6s14f145d6 [Xe]6s24f145d6 [Xe]6s14f145d7 [Xe]6s24f145d7 [Xe]6s14f145d8 [Xe]6s24f145d8 [Xe]6s14f145d9 [Xe]6s24f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10 [Hg]6p1 [Hg]6p2 [Hg]6p3 [Hg]6p4

34s30p16d14f 34s29p22d14f 34s29p22d14f 34s29p22d14f 34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p21d13f 34s29p22d13f 34s29p21d13f 34s29p21d13f 34s29p21d13f 34s29p21d13f 33s29p21d13f 33s29p21d13f 33s29p21d12f 33s33p21d12f 33s33p20d12f 33s33p20d12f 32s32p20d12f

Gaussian Model

EDFC

Errora

EDFC

Errora

14067.66251 14572.51658 15088.76788 15088.68427 15616.60963 15616.54304 16156.16160 16156.11404 16707.59320 16707.56626 17271.05161 17271.04689 17846.75413 17846.77222 18434.83544 18434.87813 19035.48265 19035.55096 19648.84622 20274.79451 20913.65088 21565.63513 22230.93365

1.64 1.81 1.69 1.93 1.87 2.01 1.87 1.92 1.98 2.01 2.35 2.17 2.44 2.76 2.64 2.77 2.85 2.96 3.03 3.01 3.31 3.21 3.44

14067.67561 14572.53143 15088.78491 15088.70130 15616.62891 15616.56232 16156.18354 16156.13598 16707.61814 16707.59120 17271.08010 17271.07538 17846.78631 17846.80439 18434.87186 18434.91454 19035.52384 19035.59214 19648.89313 20274.84766 20913.71105 21565.70290 22231.00977

1.65 1.82 1.70 1.93 1.88 2.02 1.87 1.93 1.99 2.02 2.35 2.17 2.45 2.76 2.58 2.77 2.84 2.96 3.03 2.98 3.28 3.18 3.41

Page 98

174 175 180


07:52

70 71 72

Basis Set Size

Chapter 7

Yb Lu Hf

Configuration

5/4/2007

Atom


98

Continued

Table 7.6

231

U

92

238

Np

93

237

Pu Am Cm

94 95 96

244 243 247

Bk Cf Es Fm Md No

97 98 99 100 101 102

247 251 252 257 258 259

a

RUGBS error with respect to the numerical DFC results [25].

22909.71859 23602.00190 24308.07824 25028.05840 25762.22312 26510.54805 26510.74363 27274.09094 27274.19563 28052.55688 28052.63210 28846.72775 28846.77372 29656.35375 30482.32932 31324.46068 31324.42329 32183.46709 33059.30244 33952.68429 34863.56918 35792.84713 36740.68303

3.38 3.62 4.07 4.02 3.94 4.40 4.06 4.50 4.39 5.12 4.65 5.24 4.95 5.49 5.80 6.16 6.22 6.52 6.92 7.33 7.77 8.25 8.78

22909.80428 23602.10068 24308.18927 25028.18378 25762.36408 26510.70776 26510.90335 27274.26971 27274.37440 28052.76008 28052.83531 28846.95517 28847.00115 29656.61203 30482.61836 31324.78703 31324.74964 32183.83294 33059.71538 33953.14801 34864.09331 35793.43570 36741.34402

3.34 3.57 4.08 4.03 3.95 4.40 4.06 4.51 4.41 5.12 4.65 5.24 4.94 5.48 5.80 6.14 6.21 6.51 6.90 7.31 7.73 8.20 8.71

Page 99

91

32s32p20d12f 32s32p20d11f 36s32p19d11f 36s31p19d11f 35s31p23d12f 35s31p19d17f 35s31p23d12f 35s31p19d17f 35s31p23d16f 35s31p19d16f 35s31p23d16f 35s31p19d16f 35s31p23d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p22d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f

07:52

Pa

[Hg]6p5 [Hg]6p6 [Rn]7s1 [Rn]7s2 [Rn]7s26d1 [Rn]7s25f 2 [Rn]7s26d2 [Rn]7s25f 3 [Rn]7s25f 26d1 [Rn]7s25f 4 [Rn]7s25f 36d1 [Rn]7s25f 5 [Rn]7s25f 46d1 [Rn]7s25f 6 [Rn]7s25f 7 [Rn]7s25f 8 [Rn]7s25f 76d1 [Rn]7s25f 9 [Rn]7s25f 10 [Rn]7s25f 11 [Rn]7s25f 12 [Rn]7s25f 13 [Rn]7s25f 14

5/4/2007

210 222 223 226 227 232


85 86 87 88 89 90


At Rn Fr Ra Ac Th

99


100

5/4/2007

07:52

Page 100

Chapter 7

Fig. 7.1 results.

Total DFC energy error between the calculations with the RUGBS and numerical

4.1. Variational Prolapse Analysis for the Relativistic Universal Gaussian Basis Set Generated with the Generator Coordinate Dirac–Fock Method Variational prolapse is understood as the result of a poor representation of the atomic inner orbitals (those close to the atomic nucleus) and it becomes more acute as the atomic number, Z, increases. It has been more often detected in the s1/2 and p1/2 relativistic atomic orbital symmetries [31,32] and one way of testing the presence of variational prolapse in basis sets is adding tight functions to the original basis set and observing the total energy behavior, i.e., by adding tight functions we must have a correct convergence behavior for the total atomic energy since basis sets containing prolapse present a total energy increase when we add tight functions [31,32]. As the variational prolapse is mainly detected in the s1/2 and p1/2 relativistic orbital symmetries [31,32], the prolapse problem was tested for the RUGBS generated with the GCDF method for all atoms from H to No by adding one tight s or one tight p Gaussian function. These tight functions are also subsequent members of the RUGBS (the next s or p higher Gaussian function exponent than those already included in the basis set of each atom) obtained with the same GCDF discretization parameters. The results for the prolapse analysis are summarized in Fig. 7.2, which presents the difference of the DFC total energies obtained between calculations with the original RUGBS and the RUGBS augmented with


5/4/2007

07:52

Page 101


101

Fig. 7.2 Total DFC energy differences between the calculations with the original RUGBS tight augmented with one tight s or one tight p function (EDFC –E 1 . DFC )

tight a tight s or a tight p Gaussian function (EDFC –E 1 DFC ). From Fig. 7.2 we can see that the addition of a tight function almost always results in a correct convergence of the DFC total energy for both nuclear models studied. However, a very small increase in the DFC total energy of 2 microhartree (at most) was observed with the finite nucleus of uniform proton-charge distribution when we added tight s functions [from Cs (Z 55) to Dy (Z 66)] and tight p functions [from Yb (Z 70) to Hg (Z 80)]. The DFC total energy results with the Gaussian nucleus model did not present any evidence of variational prolapse (see Fig. 7.2). To investigate the possibility of variational prolapse in more details, convergence tests were carried out for some representative atoms: No (Z 102), Rn (Z 86), Hg (Z 80), Yb (Z 70), and Ba (Z 56). These results are presented in Tables 7.7–7.11, which show the DFC total energies and the 1s1/2 and 2p1/2 orbital energies ( 1s1/2 and 2p1/2) obtained with the addition of up to three tight functions to the s, p, d, or f symmetries in the basis set of these atoms. As can be seen from Tables 7.7–7.11, the DFC total energies always reduce or eventually converge with the addition of tight d or f functions showing no sign of variational prolapse. These results were already expected since d and f functions represent electrons that are mainly in distant regions from the atomic nucleus [31]. Also, these results are in agreement with the idea that the variational prolapse only possibly occurs in s1/2 and p1/2 relativistic orbital symmetries,

Gaussian Model 2p1/2

EDFC

1s1/2

2p1/2

36740.683034 36740.683119 36740.683132 36740.683134 36740.683038 36740.683048 36740.683052 36740.683131 36740.683158 36740.683164 36740.683070 36740.683072 36740.683073

5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497 5526.6497

1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650 1047.3650

36741.344020 36741.344020 36741.344020 36741.344020 36741.344020 36741.344020 36741.344020 36741.344117 36741.344143 36741.344150 36741.344055 36741.344058 36741.344058

5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983 5526.8983

1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698 1047.3698

Page 102

1s1/2

07:52

EDFC

Chapter 7

35s31p19d16f 1 tight s 2 tight s 3 tight s 1 tight p 2 tight p 3 tight p 1 tight d 2 tight d 3 tight d 1 tight f 2 tight f 3 tight f


5/4/2007

Calculation


102

Table 7.7 Convergence pattern (in hartree) for No (Z 102)


Gaussian Model 1s1/2

2p1/2

EDFC

1s1/2

2p1/2

23602.001898 23602.001968 23602.001970 23602.001967 23602.001900 23602.001901 23602.001902 23602.001909 23602.001912 23602.001913 23602.001927 23602.001928 23602.001929

3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577 3641.1577

642.32986 642.32986 642.32986 642.32986 642.32986 642.32986 642.32986 642.32986 642.32986 642.32986 642.32985 642.32985 642.32985

23602.100685 23602.100684 23602.100684 23602.100684 23602.100685 23602.100685 23602.100685 23602.100696 23602.100699 23602.100699 23602.100714 23602.100715 23602.100715

3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969 3641.1969

642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011 642.33011

Page 103

EDFC

07:52



5/4/2007

Calculation


Table 7.8 Convergence pattern (in hartree) for Rn (Z 86)

103


EDFC

1s1/2

2p1/2

19648.846222 19648.846248 19648.846245 19648.846242 19648.846220 19648.846221 19648.846222 19648.846227 19648.846228 19648.846229 19648.846230 19648.846231 19648.846231

3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217 3074.2217

526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407 526.85407

19648.893132 19648.893132 19648.893132 19648.893132 19648.893143 19648.893143 19648.893143 19648.893138 19648.893139 19648.893139 19648.893141 19648.893141 19648.893141

3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406 3074.2406

526.85413 526.85413 526.85413 526.85413 526.85414 526.85414 526.85414 526.85413 526.85413 526.85413 526.85413 526.85413 526.85413

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104

Table 7.9 Convergence pattern (in hartree) for Hg (Z 80)


Gaussian Model 1s1/2

2p1/2

EDFC

1s1/2

2p1/2

14067.662509 14067.662510 14067.662508 14067.662507 14067.662508 14067.662508 14067.662509 14067.662528 14067.662532 14067.662533 14067.662509 14067.662509 14067.662509

2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459 2267.6459

370.05413 370.05413 370.05413 370.05413 370.05413 370.05413 370.05413 370.05413 370.05414 370.05413 370.05413 370.05413 370.05414

14067.675613 14067.675612 14067.675612 14067.675612 14067.675616 14067.675616 14067.675616 14067.675632 14067.675636 14067.675637 14067.675613 14067.675613 14067.675613

2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513 2267.6513

370.05414 370.05413 370.05413 370.05413 370.05414 370.05414 370.05414 370.05414 370.05414 370.05414 370.05414 370.05414 370.05414

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Table 7.10 Convergence pattern (in hartree) for Yb (Z 70)

105

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EDFC

1s1/2

2p1/2

8135.642505 8135.642503 8135.642503 8135.642503 8135.642504 8135.642504 8135.642504 8135.642506 8135.642506 8135.642506

1383.8336 1383.8336 1383.8336 1383.8336 1383.8336 1383.8336 1383.8336 1383.8336 1383.8336 1383.8336

209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784

8135.644398 8135.644398 8135.644398 8135.644398 8135.644399 8135.644399 8135.644399 8135.644399 8135.644400 8135.644400

1383.8344 1383.8344 1383.8344 1383.8344 1383.8344 1383.8344 1383.8344 1383.8344 1383.8344 1383.8344

209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784 209.08784

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Chapter 7

35s30p17d 1 tight s 2 tight s 3 tight s 1 tight p 2 tight p 3 tight p 1 tight d 2 tight d 3 tight d



106

Table 7.11 Convergence pattern (in hartree) for Ba (Z 56)


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107

which can present an appreciable penetration into the nuclear region for atomic systems with high Z [33]. The addition of tight d or f functions appears to only slightly disturb the energy of the 2p1/2 orbital just when the uniform sphere nucleus model is employed, as noticed in Tables 7.8 and 7.10 for Rn (addition of tight f functions) and Yb (addition of tight d functions). The convergence patterns for the addition of tight s or p functions show only a small DFC total energy increase of 3 microhartree (at most) between two subsequent calculations when the uniform sphere nucleus model is adopted. The DFC energy results obtained with the Gaussian nucleus model show an even lower DFC energy increase with the addition of tight s functions to Rn and Yb (see Tables 7.8 and 7.10). For the orbital energies, only the 2p1/2 values are slightly disturbed by the addition of tight s functions to the Yb atom when the Gaussian nucleus model is used. However, the increase in the DFC total energy with the addition of tight functions for all atoms studied is much smaller than the corresponding RUGBS error. The smallest basis set error for the atoms in Tables 7.7–7.11 is 0.6 millihartree (for Ba atom). Thus, a DFC total energy increase of 3 microhartree is 2000 times smaller than the basis set error, which is a convincing argument in the direction to consider the RUGBS as free of variational prolapse and apt to be employed in relativistic quantum chemical calculations. Moreover, the increase observed in the DFC total energies in all cases when the Gaussian nucleus model was used is insignificant being equal to 0.1 microhartree at most. Besides, convergence tests for the DFC total energies with the addition of tight s and p Gaussian functions for both nuclear models were also carried out for some lighter atomic systems (Xe, Cd, Kr, Zn, Ar, and Ne) with similar results. In fact, no variational prolapse was found indeed with the RUGBS generated with the GCDF method. The good results obtained with the RUGBS are actually directly linked to the GCDF method. The great advantage of the GCDF method is that the ID technique applied to Equations (7.15) works very well after a satisfactory choice of the GCDF parameters, its interval and limits. Hence, once these discretization parameters are defined, the integral characteristics of Equations (7.15) are kept and the variational prolapse can be easily avoided by ensuring that the entire discretization range for a certain atom is completely covered by the basis set functions we are generating. Concluding, although the basis set size of the RUGBS can be considered large for light atomic systems, it becomes much more competitive for heavier atoms. Only for comparison, the RUGBS size for No is 35s1/2, 31p1/2, 31p3/2, 19d3/2, 19d5/2, 16f5/2, and 16f7/2 whereas the basis set developed by Tatewaki and Watanabe for H through Bi [34], and also without variational prolapse, has a size of 44s1/2, 44p1/2, 44p3/2, 36d3/2, 36d5/2, 32f5/2, and 32f7/2 functions and the Gaussian exponents are different for each pair of relativistic atomic orbital symmetries: p1/2 and p3/2; d3/2 and d5/2; f5/2 and f7/2.


108

5.

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Chapter 7

The Generator Coordinate Dirac–Fock–Breit Formalism

Atomic and molecular DF calculations are usually based on an approximate relativistic many-electron Hamiltonian, the DC Hamiltonian [16,35]. In the DC Hamiltonian, the one-electron interactions are treated relativistically as a sum of Dirac’s one-electron Hamiltonians whereas the two-electron interaction is the usual nonrelativistic instantaneous electrostatic (Coulomb) repulsion. The instantaneous Coulomb interaction between two electrons is not covariant and relativistic corrections to this interaction are provided by quantum electrodynamics (QED) [36,37]. Addition of the frequency-independent Breit interaction B12 (1 2r12 ) 兵1 2 关(1 r12 )( 2 r12 ) 冒r122 兴其

(7.43)

to the instantaneous Coulomb interaction “partially remedies” the lack of covariance of the DC Hamiltonian. Some studies [5,38,39] demonstrated that the Breit interaction term can be included in the SCF procedure of the DF matrix calculations without any of the possible difficulties that had been suggested [26]. Inclusion of the Breit term in the SCF process has the advantage that both Coulomb and Breit interactions are included to the same order in the SCF potentials within the algebraic approximation [40]. Besides, adding the Breit operator to the instantaneous Coulomb operator has the advantage that all effects through order 2 (where is the fine-structure constant) are included in the zero-order Hamiltonian [41]. The use of such a zero-order Hamiltonian in variational calculations naturally leads to the DFB equations that include the Breit interaction as part of the twobody interactions. In Section 2, we presented the integral version of the closed-shell DF equations, known as the generator coordinate version of the closed-shell DF equations or the GCDF method [14]. In the GCDF method, the integral DF equations are solved through numerical integration and this is accomplished by discretization preserving the integral character of the GCDF method. This approach leads to a new view of basis set exponents to be used in relativistic atomic and molecular calculations since in the GCDF method the basis set exponents are generated by discretization in a numerical integration procedure. This procedure is referred to as ID technique [18] in contrast to the procedure of optimizing orbital exponents. Since in Section 2 we have presented the GCDFC formalism, it would be natural to extend it to include the Breit interaction term [showed in Equation (7.43)] and consequently come up with a GCDFB formalism as well. This was done by Jorge and da Silva in 1996 [15] and the GCDFB formalism made possible the GCDF method to perform DFB calculations. To obtain the GCDFB equations, the frequency-independent Breit interaction (the Breit interaction term or simply the Breit term) is added to the unperturbed


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109

DC Hamiltonian, HDC [Equation (7.1)]. As already commented, this “partially remedy” the lack of covariance of HDC leading to the Dirac–Coulomb–Breit Hamiltonian (HDCB)

H DCB ∑ hD (i ) ∑Vij ∑ Bij , i j . i, j

i

i, j

(7.44)

This approach inside the SCF treatment leads to the DFB-SCF equations [5] exactly as we did in Section 2 with GCDFC formalism [see Equations (7.8) to (7.21)]. Now with the inclusion of the Breit term [Equation (7.43)] in the GCDFC-SCF procedure [14], the DFC matrix becomes the DFB matrix that can be written as F o t b ,

(7.45)

where the one-electron part, o, and the two-electron part, t, are the same matrices showed in Section 2 [see Equations (7.23)–(7.34)] and b is ⎡ B LL b ⎢ SL ⎣ B

BLS ⎤ ⎥. BSS ⎦

(7.46)

The Breit interaction matrices are given by , SS BLLij ∑ ∑ (2 j1) eLL (, ) DSSk l Kij, LL , k l , k l

, LS BSLij ∑ ∑ (2 j1)关 f (, ) DLSk l Kij, SL,,kLSl g (, ) DLSk lWij, SL , k l 兴 ,

(7.47) (7.48)

k l

and BSSij ∑ ∑ (2 j1) eSS (, ) DLLk l Kij, SS,,kLLl ,

(7.49)

k l

v,TT苶,T苶T where W ij,l in Equation (7.48) is given in terms of the generator coordinate and in terms of the GTFs {X L} and {X S} as r

,TT T T T T Wij,TT ,k l ∫ ∫ X ( r , i ) X ( r , k )U ( r , s) X ( s, j ) X ( s, l ) ds dr 0 0

∫ ∫ X ( r , i ) X ( r , k )U ( r , s) X ( s, j ) X ( s, l ) ds dr . T

0 r

T

T

T

(7.50)


110

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Chapter 7

The relativistic angular coefficients e vLL(, ), e vSS(, ), f v (, ) and gv (, ) are evaluated by using the technique described by Grant and Pyper [42]. The inclusion of the Breit term in the GCDF method enables us to apply it in DFB calculations. The Breit term leads to changes in the orbitals and their energies, which in turn modify the Coulomb interaction among the electrons in the SCF process. Now the “interference” between the Coulomb and Breit interactions and the resulting orbital reorganization is taken into account in the GCDF-SCF procedure. The first application of the GCDF method in DFB calculations was with a RUGBS obtained for the closed-shell atoms from Zinc (Zn) through Nobelium (No) [15]. That work reported DFB-SCF calculations on all relativistic closedshell atoms with occupied d and f orbitals, i.e., all relativistic closed-shell atoms from Zn (Z 30) to No (Z 102). The finite nucleus model of uniform proton-charge distribution and the restricted kinetic balance condition were also employed in all DFB calculations. The nuclear radius employed was R 2.2677 105 M1/3, where M is the atomic mass number, and the speed of light, c, used was 137.0370 a.u. The 32 relativistic universal Gaussian exponents generated to perform this pioneer work with the GCDFB formalism can be found in Reference [15]. They were generated by employing the ID technique [Equation (7.16)] with the scaling parameter A 6.0 [15]. With the aim of comparison, Table 7.12 shows the DFC and DFB energies and also the variational Breit interaction energy, EB, attained by using the RUGBS and another universal GTF basis set [11]. The EB is the level shift in the total SCF energy due to the inclusion of the Breit term in the GCDF-SCF process and it is computed as the difference EDFB EDFC. The third column in Table 7.12 (RUGBS size) indicates the number of exponents in each s1/2, p1/2, p3/2, d3/2, d5/2, f5/2, and f7/2 symmetry taken from the 32 relativistic universal Gaussian exponents generated with the GCDF method. From Table 7.12 we can see the good performance of the GCDF method with the RUGBS in providing DFB atomic energies. The EDFB obtained with the RUGBS were better than those (last column) obtained in a previous work by using a universal GTF exponents generated from a nonrelativistic environment (code) [11]. Also in Table 7.12 (sixth column) we present the variational Breit interaction (EB) energy for the atomic systems studied. The EDFC displayed in Table 7.12 (fifth column), used to assess EB, correspond to EDFC results obtained previously with the GCDF method by using the GCDFC formalism [14]. The good numerical accuracy of the RUGBS in providing DFB energies was attributed to the implementation of the ID technique inside the relativistic environment (the GCDF method). With the ID technique, the Gaussian function exponents are generated by discretization of Equations (7.15) (the integral DF equations) through a numerical integration procedure. In fact, a careful numerical integration of Equations (7.15) will provide accurate Gaussian basis sets for relativistic atomic


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Table 7.12 Total Dirac–Fock–Breit (EDFB), Dirac–Fock–Coulomb (EDFC) and variational Breit interaction (EB) energies (in hartree) for Zn (Z 30) through No (Z 102) Atom Atomic Mass

RUGBS Size

EDFB (RUGBS)a

EDFC (RUGBS)b

EBa

Zn Ge Kr Sr Pd Cd Sn Xe Ba Sm Yb W Hg Pb Rn Ra Pu No

32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f 32s30p20d15f

1793.85233 2096.51666 2787.43578 3176.36275 5041.08851 5589.48390 6171.72062 7441.13047 8129.10458 10420.0788 14053.5505 16139.0774 19626.2386 20888.9979 23572.6264 24996.1148 29615.8972 36685.1457

1794.61338 2097.46672 2788.86168 3178.08131 5044.40343 5593.32279 6176.14096 7446.90026 8135.65000 10429.3726 14067.6811 16156.2494 19648.8712 20913.7211 23601.9780 25028.0218 29656.5171 36740.2908

0.76105 0.95006 1.42590 2787.43571 1.71856 3176.36267 3.31492 3.83889 4.42034 5.76979 7441.12988 6.54542 8129.10407 9.2938 14.1306 17.1720 22.6326 19626.2346 24.7232 29.3516 23572.6206 31.9070 40.6199 55.1451 36685.1365

65.37 72.59 83.80 87.62 106.40 112.40 118.69 131.30 137.34 150.35 173.04 183.85 200.59 207.19 222.00 226.00 242.00 259.00

EDFB (UGBS)c

a

Using the relativistic universal Gaussian basis set (RUGBS). Total DFC energies computed by using the RUGBS. c Total DFB energies obtained from Reference [11]. b

and molecular calculations. This is accomplished since the discretization of Equations (7.15) is done in such a way that the integral character of the GCDF method is preserved. Table 7.13 shows the convergence pattern for the DFC, DFB, and EB energies for the Xe atom. From Table 7.13 we can see that the use of the kinetic balance condition along with the proper boundary conditions imposed on the GTFs contributes to the DFC and DFB energy stability. We can also see that the increasing number of points in each s1/2, p1/2, p3/2, d3/2, and d5/2 symmetries rapidly favors the convergence of the total DFC and DFB energies to a value of 7446.90026 and 7441.13047 a.u., respectively. By the time we have reached Ns 32 (number of basis functions for s1/2 symmetry), Np 30 (number of basis functions for p1/2 and p3/2 symmetries), and Nd 20 (number of basis functions for d3/2 and d5/2 symmetries), the total DFC and DFB energies have already converged to the energy limit of 7446.90026 and 7441.13047 a.u., respectively. Even if we go beyond Ns 32, Np 30, and Nd 20, there is no substantial improvement in the DFC and DFB energies. It is interesting to see that for Ns 31, Np 28, and Nd 20, the EB has already converged to the value limit of 5.76979 a.u. whereas the DFC and DFB total energies have not converged yet to their values limit of 7446.90026 and 7441.13047 a.u., respectively.


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Chapter 7

Table 7.13 Convergence pattern (in hartree) for the DFC, DFB, and EB energies for Xe (Z 54) RUGBS Size

(EDFC)a

(EDFB)b

(EB)c

30s27p18d 30s27p20d 30s28p20d 31s28p20d 31s29p20d 31s30p20d 32s30p20d 32s31p20d 32s32p20d

7446.892020 7446.892026 7446.892201 7446.899062 7446.899132 7446.899153 7446.900256 7446.900259 7446.900259

7441.122254 7441.122261 7441.122435 7441.129277 7441.129346 7441.129367 7441.130467 7441.130470 7441.130470

5.769766 5.769766 5.769766 5.769785 5.769786 5.769786 5.769789 5.769789 5.769789

a

EDFC represents the Dirac–Fock–Coulomb energy. EDFB represents Dirac–Fock–Breit energy. c EB represents the variational Breit interaction energy. b

Table 7.14 Dirac–Fock–Breit ( DFB) orbital energies (in hartree) for No (Z 102) Orbital

DFB (RUGBS)a

DFB (UGBS)b

1s1/2 2s1/2 2p1/2 2p3/2 3s1/2 3p1/2 3p3/2 3d3/2 3d5/2 4s1/2 4p1/2 4p3/2 4d3/2 4d5/2 4f5/2 4f7/2 5s1/2 5p1/2 5p3/2 5d3/2 5d5/2 5f5/2 5f7/2 6s1/2 6p1/2 6p3/2 7s1/2 Total DFB energy

5500.677 1079.534 1041.879 805.7813 284.7202 267.9741 211.5868 186.6466 176.1145 78.45119 70.68985 55.07783 43.14690 40.40652 24.67776 23.92310 18.76340 15.47428 11.40887 6.596247 5.980731 0.5698279 0.4748396 2.787234 1.716591 1.046202 0.2090072 36685.1457

5500.676 1079.535 1041.880 805.7818 284.7204 267.9745 211.5872 186.6471 176.1149 78.45109 70.68978 55.07775 43.14694 40.40653 24.67797 23.92330 18.76318 15.47406 11.40868 6.596152 5.980649 0.5697671 0.4747903 2.787064 1.716418 1.046053 0.2089819 36685.1365

a

Using the universal Gaussian basis set (RUGBS). Total DFB orbital energies obtained from Reference [11].

b


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113

In Table 7.14 we display the DFB orbital energies, DFB, obtained for No by using the RUGBS generated by employing the GCDF method. There we compare the DFB, as an illustration, with the DFB obtained in a previous work [11] in which the universal GTF exponents were generated inside a nonrelativistic environment and afterwards transferred to perform DFB calculations, following therefore the Matsuoka–Huzinaga recipe [13]. From Table 7.14 we can see that the DFB obtained with the RUGBS are in general slightly lower than those obtained in Reference [11]. The inclusion of the Breit interaction term in the GCDF method allows us to use also this method in the generation of GTF exponents capable of performing accurate DFB atomic and molecular calculations.

6. A Polynomial Version of the Generator Coordinate Dirac–Fock Method In 2004, a polynomial version of the generator coordinate Dirac–Fock (pGCDF) method was presented in the literature that has the great advantage of performing the ID of Equations (7.15) using no longer a set of equally spaced discretization points [43]. The merit of this resides in the fact that doing this way we are able to generate Gaussian basis sets of much smaller size than those generated with the original GCDF method [14,15]. This is really interesting since relativistic calculations are very expensive computationally and the use of small basis sets is of great interest to those that perform relativistic calculations. In fact, the condition of using an equally spaced numerical mesh in the original GCDF method, despite being practical to perform the numerical integration of Equations (7.15), always imposed some restrictions in the generation of Gaussian exponents since accurate relativistic Gaussian basis sets were obtained only when a large Gaussian basis set size was used. With the pGCDF method this problem was overcome. In the pGCDF method, Equation (7.16) is now defined as a polynomial expansion for each atomic orbital symmetry ( s, p, d, f, etc.), namely ) (min

ln (i ) ) (min 1( ) (i 1) (2 ) (i 1) 2 A L (q ) (i 1) q ,

(7.51)

where i 1, 2, …, N and N is the number of discretization points. The ( ) min and ( ) parameters are the initial point of the mesh and the increment of order q q used to obtain the discretization points, respectively. Each GTF exponent ( ) is i now determined by using the expression ) (i ) exp A 兵(min 1( ) (i 1) (2 ) (i 1) 2 L (q ) (i 1) q 其 ,

(7.52)


114

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Chapter 7

where A is still a scaling parameter assumed here as being equal to 6.0 (exactly as in the original GCDF method [14,15]). Actually, the pGCDF method makes the generation of GTF exponents from the integral DF equations [Equations (7.15)] more flexible and more efficient allowing us to obtain smaller relativistic Gaussian basis sets without losing quality. Before presenting and discussing the first applications with the pGCDF method, it is interesting to explain how the Gaussian basis set exponents are generated from this methodology. The letter i in Equation (7.52) represents the index of the basis set function, and it can vary from 1 to the number of points (size) of the basis set for each atomic orbital symmetry. The letter q in Equation (7.52) represents the degree of the polynomial and, in practice, accurate relativistic energies with small basis set sizes are obtained when q varies from 3 onward. Each atomic orbital symmetry in Equation (7.52) has a starting point (( ) min) ( ) and a set of increments (q ), which are determined variationally. The number of increments (( ) q ) to be determined variationally depends on the choice of the polynomial degree, i.e., depends on where the polynomial is truncated for each atomic orbital symmetry. In fact, the truncation of the polynomial always implies in a less computational demanding process since we will have less discretization parameters to optimize. The first application of the pGCDF method was in the generation of Gaussian basis sets for Helium- and Beryllium-like atomic species and for Kr and Xe atoms [43]. The basis sets developed to study these atomic systems were generated employing the finite nucleus model of uniform proton-charge distribution and the restricted kinetic balance condition. The nuclear radius used was R 2.2677 105 M1/3, where M is the atomic mass number. The speed of light, c, was assumed as 137.0370 a.u. The SIMPLEX algorithm [44] was employed for ( ) the optimization of the discretization parameters ( ) min and q and the polynomial used in this first application of the pGCDF method had only the first three ( ) i parameters. The energy results obtained with the Gaussian basis sets developed by using the pGCDF method for He- and Be-like species are presented in Tables 7.15 and 7.16 along with previous results obtained with the original GCDF method [45], the even-tempered scheme, and numerical finite-difference calculations [37,39]. From Tables 7.15 and 7.16 we can see that the relativistic energies obtained with the pGCDF basis sets present an excellent agreement with the corresponding ones obtained with other basis sets and numerical finite-difference calculations. The DFC energy values (EDFC) obtained with the pGCDF basis sets are almost always equal to or smaller than the numerical finite-difference results, except for Ar14 and Sn46 where the numerical finite-difference DFC energies are smaller by only 1 microhartree. The DFC results obtained for He, Sn48 and Be systems are not surprising because energies smaller than finite-difference results have also been found in previous calculations [12,25]. The 1s and 2s orbital energies obtained in


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Table 7.15 Calculated DF energies for He and He-like ions (in hartree) Specie M He

Ne8

Ar16

Sn48

4.000

20.179

C 39.948

118.690

pGCDF

GCDF [45]

Even-tempered [37]

Numerical [37]

Size EDFB 1s(DFB) EDFC 1s(DFC) EB

(N 16)a 2.8617497 0.9179447 2.8618135 0.9179907 0.0000638

(N 22) 2.8617499 0.9179449 2.8618137 0.9179909 0.0000638

(N 16) 2.8617491 0.9179447 2.8618129 0.9179907 0.0000638

2.8618134 0.9179907 0.0000638


(N 18)b 93.970660 43.96039 93.982768 43.97190 0.012108

(N 26) 93.970660 43.96039 93.982768 43.97190 0.012108

(N 24) 93.970659 43.96039 93.982767 43.97190 0.012108

93.982768 43.97190 0.012110


(N 19)c 314.125616 151.4794 314.199594 151.5514 0.073978

(N 31) 314.125617 151.4794 314.199594 151.5514 0.073977

(N 25) 314.125615 151.4794 314.199592 151.5514 0.073977

314.199594 151.5514 0.073998


(N 19)d 2554.613472 1260.137 2556.314089 1261.821 1.700617

(N 31) 2554.613472 1260.137 2556.314089 1261.821 1.700617

(N 26) 2554.613450 1260.137 2556.314067 2556.314071 1261.821 1261.821 1.70062 1.701930

(s) (s) (s) 3 4 (s) min 0.396081326, 1 0.140248553, 2 5.27053896 10 , and 3 4.53948113 10 . 4 (s) 1(s) 0.116728474, 2(s) 4.68915507 103, and (s) min 0.220040763, 3 4.06654503 10 . c (s) 4 min 0.429481230, 1(s) 0.112178238, 2(s) 3.32833538 103, and (s) 3 2.92784498 10 . d (s) 4 min 0.768410105, 1(s) 0.108423878, 2(s) 2.01693410 103, and (s) 2.04724523 10 . 3 a

b

the DFC calculations, 1s(DFC) and 2s(DFC), are also presented in Tables 7.15 and 7.16. The 1s(DFC) and 2s(DFC) obtained with the pGDCF basis sets are practically equal to the numerical finite-difference results. In Tables 7.15 and 7.16 the Breit energies (EB), which are additional corrections to the DC energies [15], are also computed (EB EDFB EDFC). However, the direct comparison of EB is not so simple because numerical results are referring to a first-order perturbative approach while the basis sets results are calculated by means of the SCF procedure. Hence, the numerical values are almost always slightly larger than those calculated with the pGCDF basis sets for at most 1.6 millihartree (Sn46 specie) except for the Be atom. Moreover, the difference between the numerical perturbative and the SCF Breit energies increases with the atomic number, which was also noticed in


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Chapter 7

Table 7.16 Calculated DF energies for Be and Be-like ions (in hartree) Specie M Be

9.000

Ne6

20.000

Ar14

39.948

Sn46 118.690

Size EDFB 1s(DFB) 2s(DFB) EDFC 1s(DFC) 2s(DFC) EB Size EDFB 1s(DFB) 2s(DFB) EDFC 1s(DFC) 2s(DFC) EB Size EDFB 1s(DFB) 2s(DFB) EDFC 1s(DFC) 2s(DFC) EB Size EDFB 1s(DFB) 2s(DFB) EDFC 1s(DFC) 2s(DFC) EB

pGCDF

GCDF [45]

Even-tempered Numerical [37,39] [37,39]

(N 19)a 14.5751900 4.732941 0.3093103 14.5758932 4.733500 0.3093233 0.0007032 (N 21)b 110.242107 40.57650 7.500461 110.255974 40.58839 7.501339 0.013867 (N 23)c 379.111136 144.8599 31.24757 379.198185 144.9363 31.25440 0.087049 (N 22)d 3157.780379 1239.498 297.7360 3159.833054 1241.327 297.9260 2.052675

(N 28) 14.5751899 4.732941 0.3093100 14.5758931 4.733500 0.3093230 0.0007032 (N 31) 110.242108 40.57650 7.500461 110.255974 40.58839 7.501339 0.013866 (N 32) 379.111136 144.8599 31.24757 379.198185 144.9363 31.25440 0.087048 (N 31) 3157.780379 1239.498 297.7360 3159.833054 1241.327 297.9260 2.052675

(N 20) 14.5751891 4.732935 0.3093084 14.5758916 14.5758920 4.733493 4.733498 0.3093211 0.3093221 0.0007024 0.0007025 (N 26) 110.242107 40.57650 7.500461 110.255973 110.255974 40.58839 40.58839 7.501339 7.501339 0.013866 0.013868 (N 28) 379.111135 144.8599 31.24757 379.198184 379.198186 144.9363 144.9363 31.25440 31.25440 0.087049 0.087073 (N 30) 3157.780375 1239.498 297.7360 3159.833050 3159.833055 1241.327 1241.327 297.9260 297.9260 2.052675 2.054246

(s) (s) (s) 3 4 (s) min 0.624833376, 1 0.137382190, 2 4.99453152 10 , and 3 3.33086799 10 . (s) (s) (s) 3 4 (s) min 0.116383875, 1 0.114958902, 2 3.66998448 10 , and 3 2.69545268 10 . c (s) 4 min 0.0811216009, 1(s) 0.106378239, 2(s) 3.19868012 103, and (s) 3 2.08709916 10 . d (s) 4 min 0.503904776, 1(s) 0.102278649, 2(s) 1.92093673 103, and (s) 3 1.57358600 10 . a

b

previous calculations [37–39]. Quiney et al. has pointed out that this difference may be due to higher order covariant contributions that are included in the SCF treatment and absent in the first-order perturbative approach [38]. Total and orbital energies obtained in the DFB calculations performed with the pGCDF basis sets (EDFB, 1s(DFB), and 2s(DFB)) always show an increase when compared with the respective DFC values (EDFC, 1s(DFC), and 2s(DFC)).


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117

For the case of Be-like species, the effect of the Breit term over the 1s orbitals is larger than for the 2s orbitals (see Table 7.16) as the major effect of the Breit interaction is a large shift in the innershell orbital energies [37]. From Tables 7.15 and 7.16 we can see that the sizes of the pGDCF Gaussian basis sets developed in this first application of the pGCDF method were always smaller than those obtained with the original GCDF method (the basis set size reduction varies from 6 to 12). The same occurs when the comparison is made with the even-tempered basis sets, except for the He atom where the basis set sizes are equal. For all atomic systems, except the He atom, the pGCDF basis sets are smaller than the even-tempered ones by 1 to 8 functions. Also, from Tables 7.15 and 7.16 we can notice that the pGCDF method becomes more efficient as the atomic number increases (as can be seen for the Sn ions), which is a great advantage because even larger basis set size reductions can be expected for the heavier atoms in which the relativistic effects are far more important. In fact, the great advantage of using the pGCDF method is its efficacy in generating compact and accurate Gaussian basis sets for relativistic calculations. The basis set size reductions obtained in this first application of the pGDCF method may not appear so significant in absolute terms at a first glance, but considering that relativistic calculations employ a set of radial large and small components [see Equation (7.6)] the effect of any reduction in the basis set size is much more important than it should be in the nonrelativistic case. This can be easily understood because the small component set is generated from the large component set [as required by the kinetic balance condition implicit in Equations (7.8)–(7.12) and exemplified in Equations (7.40) and (7.41)] since a linear combination of two small functions is originated from each single large function (one of larger and another of smaller angular momentum than the original function). The optimized parameters of the discretization grid appear to be similar for alike species (see footnotes of Tables 7.15 and 7.16) as could be expected for such simple systems. However, it must be noticed that the optimized discretization parameters depend on the number of basis set functions and these function numbers are different from one system to another. The ( ) min parameter represents the most diffuse basis set function while the first three ( ) q parameters are responsible for a more effective discretization process. Hence, the general behavior of positive 1( ) and 3( ) and negative 2( ) parameters (see footnotes of Tables 7.15 and 7.16) ensures that the intermediate region is treated with a greater density of discretization points (basis functions) than the extreme regions (the closer and the more distant regions from the atomic nucleus, i.e., the regions of large and small exponents, respectively). To test the ability of the pGCDF method in obtaining basis sets for neutral atoms with high atomic number, we also generated relativistic Gaussian basis


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Chapter 7

sets for Krypton (Kr) and Xenon (Xe) atoms. The results presented in Table 7.17 show that a basis set of similar quality to that obtained with the original GCDF method [46] to Kr and Xe can be reached with a basis set size reduction of 6 or 7 functions in the s symmetry, 9 or 10 functions in the p symmetry, and 4 or 5 functions in the d symmetry. Table 7.17 Calculated DF energies for Kr and Xe (in hartree) Atom

M

Kr

83.80

Xe

131.30

Size EDFB EDFC EB Size EDFB EDFC EB

pGCDF

GCDF [46]

25s21p12da 2787.43580 2788.86169 1.42589 26s23p14db 7441.13067 7446.90046 5.76979

32s30p16d 2787.43583 2788.86173 1.42590 32s33p19d –7441.13089 7446.90068 5.76979

Numerical [47]

2788.86168

7446.90018

(s) (s) (s) 3 4 (s) min 0.33092470, 1 0.153672842, 2 6.57488711 10 , and 3 2.47981003 10 ; (p) (p) (p) 3 4 (p) 0.41419745, 0.154598157, 8.10877365 10 , and 3.74648596 10 ; min 1 2 3 (d) (d) (d) 2 4 (d) min 0.16325250, 1 0.151242915, 2 1.05801395 10 , and 3 8.01145811 10 . b (s) 4 min 0.34224311, 1(s) 0.163510823, 2(s) 6.97515448 103, and (s) 3 2.46105938 10 ; (p) (p) (p) 3 4 (p) min 0.43610911, 1 0.162083531, 2 8.15678377 10 , and 3 3.37646697 10 ; (d) (d) (d) 3 4 (d) 0.18026969, 0.145420061, 8.81986701 10 , and 5.61867429 10 . min 1 2 3 a

Table 7.18 DFC orbital energies for Xe (in hartree) Orbital

pGCDF

Numerical [47]

1s 2s 3s 4s 5s 2p 3p 4p 5p 2p 3p 4p 5p 3d 4d 3d 4d

1277.259 202.4652 43.01050 8.42995 1.01016 189.6796 37.65996 6.45253 0.49260 177.7046 35.32530 5.98284 0.43983 26.02340 2.71138 25.53714 2.63381

1277.259 202.4651 43.01043 8.42988 1.01012 189.6795 37.65989 6.45244 0.49256 177.7047 35.32536 5.98283 0.43981 26.02345 2.71136 25.53723 2.63382


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119

Table 7.18 presents the DFC orbital energies for Xe. The agreement between the numerical values and the results for the orbital energies of Xe is quite satisfactory (with deviations of less than 1 millihartree). These results reinforce the fact that the pGCDF method is able to generate compact relativistic basis sets that represent not only the core region but also the valence region with similar quality in terms of energy, which is extremely important in relativistic molecular calculations. In fact, with the pGCDF method we are able to design accurate adapted Gaussian basis sets (AGBSs) to be used in DFC and DFB calculations with smaller basis set sizes than those obtained with the original GCDF method [45] and the even-tempered formula [37,39]. The pGCDF method makes the generation of Gaussian basis sets in the environment of the DF theory more flexible and more efficient by allowing the use of a not equally spaced grid of points in the ID technique [Equation (7.16)] of the GCDF method. The results obtained for Helium- and Beryllium-like atomic species and for Kr and Xe atoms showed that the pGCDF method is a very powerful methodology to be used in the generation of compact and highly accurate relativistic Gaussian basis sets. The pGCDF method is certainly computationally cheaper than the procedure of optimizing all of the exponents since with the pGCDF method we need only to optimize four or maybe six discretization parameters for each atomic orbital symmetry to reach an accurate relativistic energy.

7. The Polynomial Version of the Generator Coordinate Fock Method and the Generation of Relativistic Adapted Dirac Gaussian Basis Sets 7.1. Relativistic Adapted Gaussian Basis Sets for Hydrogen through Xenon In this section, we present accurate relativistic adapted Gaussian basis sets (RAGBSs) for H (Z 1) through Xe (Z 54), generated with the pGCDF method, that can be used with two finite nuclear models: the uniform sphere and Gaussian nucleus models [48]. The RAGBSs were also analyzed for the variational prolapse by means of the addition of tight functions and no sign of variational prolapse was found after a slight adjustment of the preoptimized pGCDF discretization parameters. The DFC calculations were performed by using the DFRATOM atomic program [27] and two nuclear models: the finite nucleus of uniform proton-charge distribution and the finite nucleus of Gaussian proton-charge distribution. In the finite nucleus model of uniform proton-charge distribution, the nuclear radius is given by Equation (7.42) where M is the atomic mass number. The speed of light, c, used was 137.0359895 a.u. The restricted kinetic balance condition [20] was


120

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Chapter 7

employed in all calculations and the SCF equations for the average energy of configurations were solved using the Matsuoka’s method [29]. Also, the same Gaussian functions were employed to represent the following pairs of relativistic atomic orbital symmetries: p1/2 and p3/2; d3/2 and d5/2; f5/2 and f7/2. The RAGBSs for atoms from Hydrogen (H) to Xenon (Xe) were generated by means of the pGCDF method including the first four parameters of the polyno( ) mial expansion in Equation (7.51), ( ) min and q , where q 1, 2, and 3. The use of only these four parameters has revealed to be a satisfactory alternative in terms of computational effort and variational flexibility [43,49]. The ( ) min parameter is related to the most diffuse basis set function while each subsequent discretization point i (that corresponds to a GTF exponent) is generated through ( ) a combination of the increments 1( ), ( ) 2 , and 3 . The RAGBS parameter optimization process was carried out with the SIMPLEX algorithm [44] and the RAGBSs were generated for every electronic configuration available for comparison with the numerical DFC energy results of Visscher and Dyall [25]. The RAGBSs were also tested for variational prolapse problems with the two finite nucleus models (the uniform sphere and Gaussian models) by the addition of one tight s or one tight p Gaussian function for every atom from H to Xe. The RAGBSs did not show any variational prolapse problem for atoms from H to Kr. However, a small prolapse of 0.1 millihartree, at most, was found initially in the s1/2 orbital symmetry of the remaining atoms (from Rb to Xe) with both nuclear models. Thus, to solve this problem the preoptimized (S) 1 parameter was increased slightly so that the RAGBSs were developed to reach a satisfactory accuracy and at the same time ensure that variational prolapse problems were overcome. The optimal pGCDF parameters from H to Xe are presented in Table 7.19 and in Appendix 5 all of the Gaussian function exponents obtained with the pGCDF method for H through Xe are presented. Table 7.20 presents the RAGBS sizes, the DFC total energies, and the basis set errors with respect to the numerical DFC results [25], i.e., the RAGBS errors. The RAGBS errors are also illustrated in Fig. 7.3 as a function of the atomic number. These errors show, in general, a periodic behavior that increase with the atomic number and the largest RAGBS errors were found for Ag (Z = 47) and Cd (Z = 48) atoms (1.5 millihartree). Moreover, the error behavior observed with both nucleus models was similar although a slight difference is noticed for atoms from Ga (Z =31) onward. In addiction, the RAGBS errors for the He, Ne, Ar, Kr, and Xe noble gas atoms are 0.06, 0.22, 0.32, 0.9, and 1.2 millihartree, respectively. The largest error increase along the noble gas sequence is that between Ar and Kr and this happens probably due to the filling of the first d subshell. A size comparison between the RAGBSs and a RUGBS developed previously [26], which is also free of variational prolapse, shows that the basis set sizes for H through Xe are much smaller for the case of the RAGBSs. For example, the


Table 7.19 The pGCDF parameters for H (Z 1) through Xe (Z 54)a

1( )

( ) 2

( ) 3

H He Li Be B

1 2 3 4 5

1s1 1s2 [He]2s1 [He]2s2 [He]2s22p1

C

6

[He]2s22p2

N

7

[He]2s22p3

O

8

[He]2s22p4

F

9

[He]2s22p5

Ne

10

[He]2s22p6

Na

11

[Ne]3s1

Mg

12

[Ne]3s2

Al

13

[Ne]3s23p1

Si

14

[Ne]3s23p2

P

15

[Ne]3s23p3

S

16

[Ne]3s23p4

s s s s s p s p s p s p s p s p s p s p s p s p s p s p

4.30047807(–01) 2.99298897(–01) 7.23041451(–01) 6.35904109(–01) 4.39039747(–01) 5.66616276(–01) 3.72472592(–01) 5.23640104(–01) 3.19192728(–01) 4.80283925(–01) 2.73411046(–01) 3.56895137(–01) 2.33117368(–01) 3.24151914(–01) 1.96977916(–01) 2.90116228(–01) 5.99576487(–01) 2.46892482(–01) 5.42395766(–01) 1.68241163(–01) 4.75376487(–01) 5.46222921(–01) 4.24683919(–01) 4.99520927(–01) 3.81379202(–01) 4.62009824(–01) 3.45367712(–01) 4.32850598(–01)

1.58732967(–01) 1.70182197(–01) 1.79283569(–01) 1.90764722(–01) 1.57686364(–01) 1.62721207(–01) 1.60793104(–01) 1.80846959(–01) 1.63066776(–01) 1.85563204(–01) 1.63174236(–01) 1.66594506(–01) 1.66570120(–01) 1.64649546(–01) 1.65512282(–01) 1.68122052(–01) 1.83399268(–01) 1.75092001(–01) 1.76526053(–01) 1.59271627(–01) 1.67778526(–01) 1.77264296(–01) 1.67775010(–01) 1.76416646(–01) 1.69512837(–01) 1.65768573(–01) 1.65899855(–01) 1.68963813(–01)

6.46752224(–03) 4.90207613(–03) 9.66393960(–03) 1.21337206(–02) 5.93691241(–03) 1.16744268(–02) 6.78833274(–03) 1.49904932(–02) 7.12497870(–03) 1.59779723(–02) 7.08758509(–03) 1.21120339(–02) 7.80829833(–03) 9.59494109(–03) 7.39599943(–03) 1.03548020(–02) 9.28504958(–03) 1.44876394(–02) 8.46490055(–03) 9.62837111(–03) 7.14546669(–03) 1.14775447(–02) 7.26549368(–03) 1.23456269(–02) 8.32596160(–03) 9.65060844(–03) 7.73642086(–03) 1.03982936(–02)

1.89660077(–03) 1.48926909(–03) 7.35623572(–04) 8.64987373(–04) 5.59773729(–04) 1.44305258(–03) 6.04142160(–04) 1.66039065(–03) 6.27449454(–04) 1.73692053(–03) 6.25123928(–04) 1.49161699(–03) 6.65674292(–04) 1.22896228(–03) 6.43274099(–04) 1.26914562(–03) 4.98880067(–04) 1.68194488(–03) 4.70427984(–04) 1.29088640(–03) 4.19150184(–04) 7.91942357(–04) 4.28675923(–04) 8.63253368(–04) 4.86915608(–04) 7.01968717(–04) 4.66189764(–04) 7.54953208(–04)

(continued )

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( ) min

07:52

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Configuration

121

Z


Atom

Continued

( ) min

1( )

( ) 2

( ) 3

Cl

17

[Ne]3s23p5

Ar

18

[Ne]3s23p6

K

19

[Ar]4s1

Ca

20

[Ar]4s2

Sc

21

[Ar]4s23d1

s p s p s p s p s p d s p d s p d s p d s p d s p d s p d

3.23059659(–01) 3.88090196(–01) 2.84492100(–01) 3.69108236(–01) 6.40383127(–01) 3.38672371(–01) 5.89982118(–01) 2.79319818(–01) 5.72086781(–01) 2.50197788(–01) 4.96718683(–01) 5.90430487(–01) 2.64421741(–01) 5.47571330(–01) 5.59456240(–01) 2.29925854(–01) 4.69663151(–01) 5.81607722(–01) 2.37026408(–01) 4.92868997(–01) 5.52230184(–01) 2.07908399(–01) 3.82512106(–01) 5.67614644(–01) 2.23053888(–01) 4.58856970(–01) 5.37501656(–01) 1.88733186(–01) 4.17328440(–01)

1.59594229(–01) 1.66343507(–01) 1.66764082(–01) 1.66177234(–01) 1.78771507(–01) 1.65524805(–01) 1.76163430(–01) 1.59310661(–01) 1.76694745(–01) 1.58912820(–01) 1.76083038(–01) 1.81195122(–01) 1.63454186(–01) 1.77032835(–01) 1.78789629(–01) 1.62571949(–01) 1.75382507(–01) 1.81809644(–01) 1.61904212(–01) 1.76971381(–01) 1.76296964(–01) 1.60443088(–01) 1.70679514(–01) 1.85652857(–01) 1.64210639(–01) 1.77235725(–01) 1.79073686(–01) 1.60276147(–01) 1.80954013(–01)

6.47143213(–03) 1.06172117(–02) 8.30178967(–03) 1.04557188(–02) 7.28531617(–03) 1.02504929(–02) 7.74966557(–03) 9.91846593(–03) 7.42702927(–03) 9.59532576(–03) 1.44839820(–02) 7.51118959(–03) 1.07238850(–02) 1.22894383(–02) 7.50002077(–03) 1.08898166(–02) 1.25234694(–02) 7.74400573(–03) 1.02998566(–02) 1.34363987(–02) 7.09759141(–03) 1.05256926(–02) 1.41271881(–02) 8.33647255(–03) 1.07441986(–02) 1.36316375(–02) 7.50271976(–03) 9.99353776(–03) 1.52250530(–02)

4.07924773(–04) 7.83767270(–04) 4.91742232(–04) 7.69098226(–04) 3.27562992(–04) 7.60603098(–04) 3.57974237(–04) 7.68964061(–04) 3.43352457(–04) 7.49931723(–04) 1.33618353(–03) 3.37537150(–04) 8.16194157(–04) 1.02252946(–03) 3.39258998(–04) 8.41296361(–04) 1.10819753(–03) 3.48485403(–04) 7.94564208(–04) 1.13717297(–03) 3.26013885(–04) 8.25924227(–04) 1.24084144(–03) 3.67764404(–04) 8.20903568(–04) 1.15539223(–03) 3.41163033(–04) 7.94253065(–04) 1.29156413(–03)

[Ar]4s13d2

Ti

22

[Ar]4s23d2

[Ar]4s13d3

V

23

[Ar]4s23d3

[Ar]4s13d4

Cr

24

[Ar]4s23d4

Page 122

07:52

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5/4/2007

Z

Chapter 7

Atom


122

Table 7.19

Mn

25

[Ar]4s23d5

[Ar]4s13d6

[Ar]4s23d6

27

[Ar]4s23d7

[Ar]4s13d8

Ni

28

[Ar]4s23d8

[Ar]4s13d9

Cu

29

[Ar]4s23d9

[Ar]4s13d10

(continued )

123

3.48038351(–04) 7.84365279(–04) 1.23090252(–03) 3.55642888(–04) 7.86003584(–04) 1.26340154(–03) 3.73259775(–04) 7.77934497(–04) 1.14868549(–03) 3.51607280(–04) 8.68382388(–04) 1.18043769(–03) 3.64589318(–04) 8.37572740(–04) 1.09993168(–03) 3.42333085(–04) 9.12604243(–04) 1.27519947(–03) 3.44095930(–04) 8.46087590(–04) 1.18212093(–03) 3.32555854(–04) 8.67213003(–04) 1.18600016(–03) 3.67353378(–04) 8.31932543(–04) 1.01133120(–03) 3.42759221(–04) 8.44269316(–04) 1.16581739(–03) 3.45111261(–04) 8.10488344(–04) 1.11386082(–03)

Page 123

Co

7.73048820(–03) 1.00488533(–02) 1.45509577(–02) 7.83569594(–03) 9.71222870(–03) 1.54414654(–02) 8.57188379(–03) 9.87002971(–03) 1.37464629(–02) 7.71638202(–03) 1.11449176(–02) 1.35630919(–02) 8.32017692(–03) 1.07776934(–02) 1.33473995(–02) 7.56210765(–03) 1.17885915(–02) 1.46415213(–02) 7.68834079(–03) 1.08104241(–02) 1.42836881(–02) 7.19286565(–03) 1.09515438(–02) 1.33691652(–02) 8.32496819(–03) 1.05419388(–02) 1.20229629(–02) 7.56819564(–03) 1.05385039(–02) 1.33045747(–02) 7.71801410(–03) 1.01283588(–02) 1.34885703(–02)

07:52

[Ar]4s13d7

1.82633967(–01) 1.61430059(–01) 1.78261839(–01) 1.80529993(–01) 1.58857197(–01) 1.83385691(–01) 1.88917185(–01) 1.60605149(–01) 1.78815719(–01) 1.80012923(–01) 1.64668877(–01) 1.71742687(–01) 1.86941238(–01) 1.63852940(–01) 1.78533675(–01) 1.80790272(–01) 1.66057532(–01) 1.73578716(–01) 1.82293628(–01) 1.64314160(–01) 1.80563389(–01) 1.76730483(–01) 1.62533463(–01) 1.70850997(–01) 1.86930020(–01) 1.63622746(–01) 1.74378063(–01) 1.79869516(–01) 1.62603446(–01) 1.70653351(–01) 1.83397050(–01) 1.61416434(–01) 1.79838871(–01)

5/4/2007

26

5.60233802(–01) 1.98231318(–01) 4.33861747(–01) 5.28947477(–01) 1.69063586(–01) 4.04209498(–01) 5.52106423(–01) 1.80543821(–01) 4.17537491(–01) 5.22725816(–01) 1.63374158(–01) 3.46498319(–01) 5.42185812(–01) 1.64660597(–01) 3.97273410(–01) 5.16514187(–01) 1.43001921(–01) 3.23518507(–01) 5.39701524(–01) 1.48672287(–01) 3.74286963(–01) 5.11168226(–01) 1.24713700(–01) 3.07728692(–01) 5.29083449(–01) 1.34129744(–01) 3.60268208(–01) 5.00031119(–01) 1.10436150(–01) 2.92687788(–01) 5.25261580(–01) 1.16983310(–01) 3.39631141(–01)


Fe

s p d s p d s p d s p d s p d s p d s p d s p d s p d s p d s p d


[Ar]4s13d5

Continued

( ) min

1( )

( ) 2

( ) 3

Zn

30

[Ar]4s23d10

Ga

31

[Zn]4p1

Ge

32

[Zn]4p2

As

33

[Zn]4p3

Se

34

[Zn]4p4

Br

35

[Zn]4p5

Kr

36

[Zn]4p6

Rb

37

[Kr]5s1

Sr

38

[Kr]5s2

Y

39

[Kr]5s24d1

s p d s p d s p d s p d s p d s p d s p d s p d s p d s p d

4.92899897(–01) 9.42575773(–02) 2.80639563(–01) 4.38113822(–01) 5.25040264(–01) 2.31682708(–01) 4.01533948(–01) 4.87065931(–01) 1.95942198(–01) 3.71939006(–01) 4.54278283(–01) 1.57909942(–01) 3.44580809(–01) 4.23100297(–01) 1.34900991(–01) 3.20227475(–01) 4.01012717(–01) 1.08986297(–01) 3.02549095(–01) 3.78349192(–01) 9.94031089(–02) 6.55054412(–01) 3.36722384(–01) 6.14847465(–02) 6.07465366(–01) 3.08632340(–01) 6.67111402(–02) 5.97323644(–01) 2.79229724(–01) 4.61990520(–01)

1.81970290(–01) 1.63557999(–01) 1.74377816(–01) 1.79760353(–01) 1.82778828(–01) 1.57647291(–01) 1.77998150(–01) 1.77873222(–01) 1.55397414(–01) 1.70630869(–01) 1.73187451(–01) 1.55206801(–01) 1.67829557(–01) 1.70558730(–01) 1.56597938(–01) 1.67033525(–01) 1.71061593(–01) 1.58173004(–01) 1.64823519(–01) 1.69109433(–01) 1.61401958(–01) 1.89000000(–01) 1.63028719(–01) 1.56391854(–01) 1.83000000(–01) 1.62435084(–01) 1.58882700(–01) 1.68000000(–01) 1.53978071(–01) 1.72380791(–01)

7.65424987(–03) 1.10763080(–02) 1.38388004(–02) 7.64809307(–03) 9.74902493(–03) 9.78419272(–03) 7.76095836(–03) 9.34068865(–03) 9.79471652(–03) 6.80685473(–03) 8.96261822(–03) 1.08237652(–02) 6.61582373(–03) 8.92301000(–03) 1.18485387(–02) 6.67829527(–03) 9.14136299(–03) 1.33006093(–02) 6.53012477(–03) 9.41905626(–03) 1.40845385(–02) 7.39945603(–03) 8.64755426(–03) 1.37884760(–02) 7.16820023(–03) 9.02269594(–03) 1.39339392(–02) 4.97875512(–03) 7.17005460(–03) 8.99953359(–03)

3.43945738(–04) 8.79715889(–04) 1.18580238(–03) 3.46348428(–04) 4.87086184(–04) 9.00050604(–04) 3.53203556(–04) 4.80490682(–04) 9.08916758(–04) 3.23139213(–04) 4.74657104(–04) 1.01233602(–03) 3.16215484(–04) 4.84411436(–04) 1.11339990(–03) 3.21074322(–04) 4.99473257(–04) 1.24195838(–03) 3.17960504(–04) 5.23282131(–04) 1.29037323(–03) 2.57248753(–04) 4.96189040(–04) 1.29918254(–03) 2.54055499(–04) 5.27017766(–04) 1.28814808(–03) 1.87946436(–04) 4.35793619(–04) 5.06293274(–04)

Page 124

07:52

Configuration

5/4/2007

Z

Chapter 7

Atom


124

Table 7.19

Zr

40

[Kr]5s24d2

[Kr]5s14d3

[Kr]5s24d3

42

[Kr]5s24d4

[Kr]5s14d5

Tc

43

[Kr]5s24d5

[Kr]5s14d6

Ru

44

[Kr]5s24d6

[Kr]5s14d7

(continued )

125

1.76745758(–04) 4.34241969(–04) 4.91590361(–04) 1.96539557(–04) 4.51344099(–04) 5.30420482(–04) 1.91157680(–04) 4.50241751(–04) 4.89818487(–04) 2.03314374(–04) 4.55822482(–04) 5.71487586(–04) 2.04157368(–04) 4.62558600(–04) 5.12726705(–04) 2.05863252(–04) 4.52433991(–04) 5.79186724(–04) 1.97366481(–04) 4.73195104(–04) 5.45519683(–04) 2.09304402(–04) 4.76793522(–04) 5.59823282(–04) 2.00163693(–04) 4.55920038(–04) 5.65836767(–04) 2.08967502(–04) 4.78333397(–04) 5.97690340(–04) 2.06956098(–04) 4.70269724(–04) 6.09788838(–04)

Page 125

Mo

4.55386619(–03) 7.13821188(–03) 9.11865311(–03) 5.23854457(–03) 7.31227719(–03) 8.88998316(–03) 5.09756006(–03) 7.35026734(–03) 8.71540378(–03) 5.47762126(–03) 7.32374528(–03) 9.51321722(–03) 5.49225223(–03) 7.49215909(–03) 9.02974769(–03) 5.54384659(–03) 7.15288361(–03) 9.55767941(–03) 5.37776167(–03) 7.63197613(–03) 9.24936020(–03) 5.75640547(–03) 7.60282596(–03) 9.24081975(–03) 5.48765796(–03) 7.21532151(–03) 9.38068637(–03) 5.75985046(–03) 7.56639114(–03) 9.67476355(–03) 5.72974944(–03) 7.47140326(–03) 1.01044015(–02)

07:52

[Kr]5s14d4

1.64000000(–01) 1.53810966(–01) 1.79344985(–01) 1.69000000(–01) 1.52760327(–01) 1.68464955(–01) 1.69000000(–01) 1.54267024(–01) 1.73716534(–01) 1.70000000(–01) 1.52904468(–01) 1.68988202(–01) 1.71000000(–01) 1.53756392(–01) 1.73155526(–01) 1.70000000(–01) 1.50430215(–01) 1.67662100(–01) 1.71000000(–01) 1.53581250(–01) 1.69834171(–01) 1.72000000(–01) 1.52756968(–01) 1.65920173(–01) 1.71000000(–01) 1.51580119(–01) 1.68611286(–01) 1.72000000(–01) 1.51819331(–01) 1.66176918(–01) 1.73000000(–01) 1.52574579(–01) 1.70995238(–01)

5/4/2007

41

6.02762905(–01) 2.82719439(–01) 5.23723761(–01) 5.84492154(–01) 2.57395075(–01) 4.36161217(–01) 5.87186631(–01) 2.64301951(–01) 4.80902177(–01) 5.68260973(–01) 2.42971327(–01) 4.08271343(–01) 5.80325818(–01) 2.46271146(–01) 4.53193759(–01) 5.56975262(–01) 2.23298306(–01) 3.91575130(–01) 5.76034413(–01) 2.31394318(–01) 4.31222298(–01) 5.50864794(–01) 2.08126906(–01) 3.75778218(–01) 5.69469947(–01) 2.17189261(–01) 4.07174194(–01) 5.48196791(–01) 1.98950281(–01) 3.59802225(–01) 5.62670378(–01) 2.02989625(–01) 3.91126334(–01)


Nb

s p d s p d s p d s p d s p d s p d s p d s p d s p d s p d s p d


[Kr]5s14d2

5/4/2007

Continued

( ) min

1( )

( ) 2

( ) 3

Rh

45

[Kr]5s24d7

s p d s p d s p d s p d s p d s p d s p d

5.40901100(–01) 1.78867904(–01) 3.46136513(–01) 5.54909755(–01) 1.88669479(–01) 3.70862069(–01) 5.30989658(–01) 1.69528927(–01) 3.26200172(–01) 5.50623231(–01) 1.77803707(–01) 3.57332522(–01) 2.71010269(–01) 1.84903122(–01) 4.04521346(–01) 5.22777712(–01) 1.57178549(–01) 3.13852760(–01) 5.45161950(–01) 1.63102535(–01) 3.41346577(–01)

1.73000000(–01) 1.52324793(–01) 1.65883860(–01) 1.74000000(–01) 1.50437089(–01) 1.69374807(–01) 1.74000000(–01) 1.51565848(–01) 1.65039344(–01) 1.74000000(–01) 1.51034189(–01) 1.69932556(–01) 1.54000000(–01) 1.55791503(–01) 1.76359834(–01) 1.74000000(–01) 1.51467996(–01) 1.64880256(–01) 1.76000000(–01) 1.53070111(–01) 1.67031925(–01)

5.87771778(–03) 7.73814458(–03) 9.78535992(–03) 5.86268246(–03) 7.16816770(–03) 1.01292049(–02) 5.94979663(–03) 7.77080004(–03) 9.92229627(–03) 5.95148492(–03) 7.31407125(–03) 9.98471793(–03) 5.72517580(–03) 8.15539887(–03) 1.06267567(–02) 6.06468515(–03) 7.76354471(–03) 1.00274394(–02) 6.04374831(–03) 7.86677735(–03) 9.86168273(–03)

2.11498425(–04) 4.90619852(–04) 6.03897294(–04) 2.09553996(–04) 4.62328029(–04) 6.14375470(–04) 2.11943394(–04) 4.95594430(–04) 6.21247300(–04) 2.13234919(–04) 4.70532808(–04) 6.01313141(–04) 2.19406375(–04) 5.08284232(–04) 6.21035842(–04) 2.15487397(–04) 4.96719260(–04) 6.23439992(–04) 2.13146069(–04) 4.97630937(–04) 6.07943576(–04)

Pd

46

[Kr]5s24d8

[Kr]5s14d9

[Kr]5s04d10

Ag

47

[Kr]5s24d9

[Kr]5s14d10

Page 126

Configuration

07:52

Z

Chapter 7

Atom

[Kr]5s14d8


126

Table 7.19

[Cd]5p1

Sn

50

[Cd]5p2

Sb

51

[Cd]5p3

Te

52

[Cd]5p4

I

53

[Cd]5p5

Xe

54

[Cd]5p6

a

The numbers in parentheses are the powers of 10.

1.71000000(–01) 1.50664713(–01) 1.65345827(–01) 1.70000000(–01) 1.65615838(–01) 1.46766410(–01) 1.68000000(–01) 1.64553742(–01) 1.46733369(–01) 1.70000000(–01) 1.64289048(–01) 1.47156529(–01) 1.70000000(–01) 1.65615838(–01) 1.46766410(–01) 1.68000000(–01) 1.64553742(–01) 1.46733369(–01) 1.70000000(–01) 1.64289048(–01) 1.47156529(–01)

5.75314141(–03) 7.66176988(–03) 1.04070719(–02) 6.80312206(–03) 7.91063745(–03) 8.06082946(–03) 6.78318972(–03) 8.19232737(–03) 8.34136078(–03) 7.15039603(–03) 8.25319019(–03) 8.50816670(–03) 6.80312206(–03) 7.91063745(–03) 8.06082946(–03) 6.78318972(–03) 8.19232737(–03) 8.34136078(–03) 7.15039603(–03) 8.25319019(–03) 8.50816670(–03)

2.06685840(–04) 4.96472542(–04) 6.58620823(–04) 2.40352261(–04) 3.57062246(–04) 5.66795512(–04) 2.41640849(–04) 3.74055661(–04) 5.85554765(–04) 2.51759121(–04) 3.79921194(–04) 5.93980660(–04) 2.40352261(–04) 3.57062246(–04) 5.66795512(–04) 2.41640849(–04) 3.74055661(–04) 5.85554765(–04) 2.51759121(–04) 3.79921194(–04) 5.93980660(–04)

Page 127

49

5.16558138(–01) 1.43675376(–01) 2.94167020(–01) 3.85181208(–01) 4.59403524(–01) 1.89530135(–01) 3.59999716(–01) 4.33812203(–01) 1.79155627(–01) 3.42746530(–01) 4.13211855(–01) 1.70621528(–01) 3.85181208(–01) 4.59403524(–01) 1.89530135(–01) 3.59999716(–01) 4.33812203(–01) 1.79155627(–01) 3.42746530(–01) 4.13211855(–01) 1.70621528(–01)

07:52

In

s p d s p d s p d s p d s p d s p d s p d

5/4/2007

[Kr]5s24d10


48


Cd

127

Atom

Z

M

Configuration

Basis Sets Size

Uniform Sphere Model EDFC

22

48

V

23

51

8s 8s 14s 14s 14s9p 14s9p 14s9p 14s9p 14s9p 14s9p 18s9p 18s9p 18s13p 18s13p 18s13p 18s13p 18s13p 18s13p 21s13p 21s13p 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d

–0.500000 2.861753 7.433518 14.575859 24.536522 37.675986 54.327633 74.824873 99.501458 128.691711 162.077859 199.934840 242.330593 289.461147 341.494459 398.597689 460.938114 528.683440 601.525662 679.709852 763.378392 763.301374 852.819418 852.755350 948.188201 948.135523

0.006 0.060 0.015 0.033 0.033 0.055 0.089 0.113 0.157 0.220 0.229 0.226 0.156 0.191 0.210 0.241 0.269 0.322 0.291 0.309 0.346 0.359 0.405 0.406 0.456 0.465

0.500000 2.861753 7.433518 14.575859 24.536522 37.675986 54.327633 74.824873 99.501458 128.691711 162.077859 199.934840 242.330593 289.461147 341.494459 398.597689 460.938114 528.683440 601.525662 679.709852 763.378392 763.301375 852.819418 852.755350 948.188201 948.135524

0.006 0.060 0.015 0.033 0.033 0.055 0.089 0.113 0.157 0.220 0.229 0.227 0.156 0.191 0.210 0.241 0.270 0.323 0.292 0.310 0.347 0.360 0.407 0.407 0.458 0.467

Page 128

Ti

1s1 1s2 [He]2s1 [He]2s2 [He]2s22p1 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 [He]2s22p6 [Ne]3s1 [Ne]3s2 [Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6 [Ar]4s1 [Ar]4s2 [Ar]4s23d1 [Ar]4s13d2 [Ar]4s23d2 [Ar]4s13d3 [Ar]4s23d3 [Ar]4s13d4

Errora

07:52

1 4 7 9 11 12 14 16 19 20 23 24 27 28 31 32 35 40 39 40 45

EDFC

5/4/2007

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Errora

Chapter 7


Gaussian Model


128

Table 7.20 Total DFC energies (in hartree) and the basis set errors (in millihartree) for H (Z 1) through Xe (Z 54)a

55

Fe

26

56

Co

27

59

Ni

28

58

Cu

29

63


30 31 32 33 34 35 36 37 38 39

64 69 74 75 80 79 84 85 88 89

Zr

40

90

Nb

41

93

Mo

42

98

Tc

43

98

Ru

44

102

0.517 0.536 0.585 0.597 0.665 0.687 0.770 0.798 0.848 0.873 0.928 1.004 1.046 0.737 0.736 0.756 0.780 0.823 0.881 0.797 0.819 0.922 1.036 0.944 1.066 0.972 1.064 1.033 1.129 1.095 1.165 1.154 1.227

1049.637961 1049.595498 1157.321329 1157.288180 1271.391317 1271.366714 1392.001056 1391.984379 1519.304773 1519.295501 1653.456445 1653.454063 1794.611928 1942.563015 2097.469611 2259.441142 2428.587476 2605.022644 2788.859722 2979.804220 3178.079156 3383.760907 3383.698961 3597.082399 3597.040465 3818.167298 3818.147557 4047.136358 4047.140437 4284.112285 4284.141613 4529.212704 4529.268498

0.519 0.540 0.589 0.600 0.668 0.690 0.774 0.803 0.854 0.880 0.936 1.012 1.055 0.749 0.750 0.770 0.798 0.841 0.902 0.793 0.813 0.917 1.030 0.938 1.061 0.964 1.058 1.026 1.120 1.083 1.152 1.140 1.212

(continued )

129

1049.637960 1049.595498 1157.321328 1157.288179 1271.391315 1271.366712 1392.001053 1391.984377 1519.304770 1519.295498 1653.456441 1653.454059 1794.611922 1942.563006 2097.469599 2259.441124 2428.587454 2605.022614 2788.859682 2979.804142 3178.079060 3383.760793 3383.698847 3597.082263 3597.040330 3818.167133 3818.147394 4047.136159 4047.140237 4284.112048 4284.141375 4529.212419 4529.268212

Page 129

25

21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s13p10d 21s17p10d 21s17p10d 21s17p10d 21s17p10d 21s17p10d 21s17p10d 25s17p10d 25s17p10d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d

07:52

Mn

[Ar]4s23d4 [Ar]4s13d5 [Ar]4s23d5 [Ar]4s13d6 [Ar]4s23d6 [Ar]4s13d7 [Ar]4s23d7 [Ar]4s13d8 [Ar]4s23d8 [Ar]4s13d9 [Ar]4s23d9 [Ar]4s13d10 [Ar]4s23d10 [Zn]4p1 [Zn]4p2 [Zn]4p3 [Zn]4p4 [Zn]4p5 [Zn]4p6 [Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s14d2 [Kr]5s24d2 [Kr]5s14d3 [Kr]5s24d3 [Kr]5s14d4 [Kr]5s24d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d6 [Kr]5s24d6 [Kr]5s14d7

5/4/2007

52


24


Cr

Z

45

Pd

46

103

106 107

Cd In Sn Sb Te I Xe

48 49 50 51 52 53 54

114 115 120 121 130 127 132

a

The RAGBS error with respect to the numerical DFC results [25].

25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s17p13d 25s21p13d 25s21p13d 25s21p13d 25s21p13d 25s21p13d 25s21p13d


Gaussian Model

EDFC

Errora

EDFC

Errora

4782.561261 4782.644650 5044.278850 5044.390952 5044.399478 5314.490674 5314.632339 5593.316844 5880.429953 6176.126359 6480.516731 6793.696839 7115.791875 7446.892822

1.236 1.317 1.346 1.370 1.235 1.398 1.503 1.454 1.000 0.982 1.027 1.081 1.109 1.217

4782.561597 4782.644988 5044.279249 5044.391352 5044.399875 5314.491144 5314.632808 5593.317410 5880.430616 6176.127146 6480.517653 6793.697948 7115.793138 7446.894310

1.220 1.299 1.328 1.351 1.218 1.374 1.481 1.427 0.966 0.943 0.974 1.019 1.037 1.130

Page 130

47

[Kr]5s24d7 [Kr]5s14d8 [Kr]5s24d8 [Kr]5s14d9 [Kr]5s04d10 [Kr]5s24d9 [Kr]5s14d10 [Kr]5s24d10 [Cd]5p1 [Cd]5p2 [Cd]5p3 [Cd]5p4 [Cd]5p5 [Cd]5p6

Basis Sets Size

Chapter 7

Ag

Configuration

07:52

Rh

M

5/4/2007

Atom


130

Continued

Table 7.20


5/4/2007

07:52

Page 131


131

Fig. 7.3 Total DFC energy error between the calculations with the RAGBSs and numerical results.

RAGBS sizes for He, Be, Ne, Ar, Kr, and Xe are 8s, 14s, 14s9p, 18s13p, 21s17p10d, and 25s21p13d, respectively, while the corresponding RUGBS sizes are 19s, 27s, 24s16p, 28s23p, 30s28p13d, and 31s31p18d. The largest RUGBS error from H to Xe was only 0.5 millihartree, which is around three times smaller than the largest RAGBS error. The slight lower accuracy of the RAGBSs with respect to the RUGBS is, in fact, compensated by the small RAGBS sizes. However, a minor modification of the relativistic molecular calculation codes can certainly turn the RUGBS more competitive since there is no need of reevaluating some (same) integrals involving primitive functions of a certain pair of atoms (such as overlap and two-electron integrals) that will frequently appear with universal exponents. The RAGBSs can also be compared with the well-known relativistic triple-zeta basis sets of Dyall [50] that present basis set sizes for the 4p and 5p elements of 23s16p9d and 28s21p14d, respectively, while the corresponding RAGBS sizes are 21s17p10d and 25s21p13d. Since some of the Dyall’s basis sets are slightly larger than the RAGBSs, it is not surprising that they also show lower total energy errors than the RAGBSs for these 4p and 5p elements. However, a straightforward comparison between Dyall’s basis sets and the RAGBSs cannot be done because, as commented by Dyall [50], his basis sets show clear signs of variational prolapse.


132

5/4/2007

07:52

Page 132

Chapter 7

The energy difference results between the calculations with the original RAGBS and the RAGBS augmented with one tight s or one tight p Gaussian tight function (EDFC –E 1 DFC ) are presented in Fig. 7.4. The tight s or p function added was always the next function right after those already chosen for the considered atom, i.e., the next one from the set given by the pGCDF parameters in Table 7.19. From Fig. 7.4 we can see that the addition of one tight function with both nuclear models always results in the lowering or convergence of the DFC total energy, including the atoms that had first presented signs of variational prolapse in its s1/2 orbital symmetry (from Rb to Xe) and for which the 1(s) parameter was slightly increased. These results show that it is possible to take advantage of the characteristics of the pGCDF method so as to eliminate the variational prolapse problem from a basis set. This happens because 1(s) is a positive parameter that has the aim of ensuring a primary separation between the ID points (Gaussian function exponents) that is adjusted by the combina(s) tion of the (s) 2 and 3 parameters depending on the atomic region (closer or more distant regions from the nucleus). Thus, any increase of the 1(s) parameter will result in a corresponding separation increase of all discretization points into the atomic nucleus that will eventually overcome any previous bad description of the innermost atomic region responsible for the variational prolapse.

Fig. 7.4 Total DFC energy differences between the calculations with the RAGBS and the tight RAGBS augmented with one tight s or one tight p function (EDFC –E 1 DFC ).


5/4/2007

07:52

Page 133


133

The DFC total energy reduction for H through Xe, upon the addition of one tight p function, is larger than the one for the addition of one tight s function, as shown in Fig. 7.4. Moreover, the energy difference behavior observed in Fig. 7.4, between the two nuclear models for the addition of one tight s function, becomes visually distinguished for the atoms from Ca (Z = 20) onward. When one tight p function is added, the distinct behavior between the two nuclear models is only seen for the atoms from Sn (Z = 50) onward. Hence, we suspect that Ca and Sn represent the limits where the s1/2 and p1/2 electronic cloud, respectively, begins to present a significant penetration into the nuclear region resulting in the different behavior observed in Fig. 7.4 between the two finite nucleus models studied (i.e., depending on the proton-charge distribution characteristics of the nucleus model). More detailed variational prolapse tests are shown in Tables 7.21–7.24 for some selected atomic systems: Kr (Z = 36), Sr (Z = 38), Cd (Z = 48), and Xe (Z = 54). The DFC total energy behavior for these convergence patterns was analyzed by the addition of up to three tight Gaussian functions for s, p, or d symmetries. As expected, the addition of tight d functions showed no sign of variational prolapse due to the negligible penetration of electrons in such orbitals into the atomic nucleus [26,31,51]. Moreover, the addition of up to three tight s or p functions, with both nuclear models, also did not present any sign of prolapse. The addition of up to three tight s functions was particularly important in the case of Sr, Cd, and Xe since these atoms had previously presented prolapse in its s1/2 orbital symmetry. All these results allow us to assure that our RAGBSs are totally free of variational prolapse and apt to be used in relativistic atomic and molecular calculations with both uniform sphere and Gaussian nucleus models. Table 7.21 Convergence pattern (in hartree) for Kr (Z 36) Calculation



Gaussian Model

EDFC

EDFC

2788.859682 2788.859690 2788.859690 2788.859690 2788.859798 2788.859814 2788.859815 2788.859712 2788.859713 2788.859713

2788.859722 2788.859753 2788.859753 2788.859753 2788.859837 2788.859854 2788.859855 2788.859752 2788.859752 2788.859752


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Page 134

Chapter 7 Table 7.22 Convergence pattern (in hartree) for Sr (Z 38) Calculation



Gaussian Model

EDFC

EDFC

3178.079060 3178.079062 3178.079062 3178.079062 3178.079169 3178.079183 3178.079183 3178.079100 3178.079100 3178.079100

3178.079156 3178.079156 3178.079156 3178.079156 3178.079265 3178.079279 3178.079279 3178.079196 3178.079196 3178.079196

Table 7.23 Convergence pattern (in hartree) for Cd (Z 48) Calculation



Gaussian Model

EDFC

EDFC

5593.316844 5593.316859 5593.316861 5593.316861 5593.317104 5593.317121 5593.317121 5593.316899 5593.316901 5593.316901

5593.317410 5593.317412 5593.317412 5593.317412 5593.317671 5593.317690 5593.317690 5593.317465 5593.317467 5593.317467

Table 7.24 Convergence pattern (in hartree) for Xe (Z 54) Calculation



Gaussian Model

EDFC

EDFC

7446.892822 7446.892872 7446.892876 7446.892876 7446.892915 7446.892915 7446.892915 7446.892898 7446.892901 7446.892902

7446.894310 7446.894311 7446.894311 7446.894311 7446.894407 7446.894407 7446.894407 7446.894386 7446.894390 7446.894390


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135

7.2. Relativistic Adapted Gaussian Basis Sets for Cesium through Radon RAGBSs for Cs (Z 55) through Rn (Z 86) were obtained by employing the pGCDF method with four parameters for each atomic orbital symmetry (( ) min and q( ) with q = 1, 2, and 3, where s, p, d, and f) [52] and the SIMPLEX algorithm [44] was used to optimize the pGCDF discretization parameters. The basis set sizes for Cs through Rn were chosen by means of a preliminary analysis on the noble gas atom Rn and also taking into account the RAGBS size of Xe (25s21p13d) [48]. As expected, the initial optimized pGCDF parameters for Cs through Rn furnished basis sets with prolapse in the s1/2 relativistic atomic orbital symmetry once this problem was detected in the atomic range from Rb (Z 37) to Xe (Z 54) with a prolapse of at most 0.1 millihartree [48]. Thus, since the atomic numbers for Cs through Rn are larger than the ones studied before [48], the variational prolapses found from Cs to Rn were also more pronounced and reached 5.5 millihartree for Ra with the uniform sphere model. As before [48], the prolapse problem was solved by a slight increase of the (s) 1 parameter that improves the description of the electrons closer to the nucleus (the innermost region of the atom). Furthermore, a very small prolapse (0.01 millihartree for Ra with the uniform sphere model) was also detected in the p1/2 symmetry from Tl (Z 81) to Rn (Z 86) and a subtle increase of the 1(p) parameter was also able to surpass the problem. The optimal pGCDF discretization parameters, obtained after the adjustments of 1(s) and 1(p) with the aim to overcome the variational prolapse, are given in Table 7.25. All of the RAGBS exponents from Cs to Rn are presented in Appendix 5. Table 7.26 presents the DFC total energies, the RAGBS sizes, and its errors with respect to numerical results [25]. The RAGBS errors are illustrated in Fig. 7.5 as a function of the atomic number. From Fig. 7.5 we can see that the RAGBS errors behave in a periodic way and also increase with the atomic number as the subshells are being filled, likewise to the behavior for the atoms from H to Xe [48]. We can also notice from Fig. 7.5 that the RAGBS for Cs through Rn, as for H through Xe [48], showed lower errors with the Gaussian nucleus model than with the uniform sphere model as the atomic numbers are approximately greater than 40, reaching a relative error of 1.3 millihartree for the heaviest atoms. This can be rationalized since adapted basis sets are developed for each atom with its own valence characteristics to attain the lowest basis set error with the possible smallest size. A small basis set requires that the most diffuse and the innermost regions in the atom be treated by a reduced number of Gaussian functions when compared to the intermediate region (this will be more or less successful depending on the nuclear model adopted). The RAGBS sizes from Cs to Rn are smaller than the ones of our prolapse-free RUGBS that was obtained with the original GCDF method [26]. The RUGBS


136

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07:52

Page 136

Chapter 7

Fig. 7.5 results.

Total DFC energy error between the calculations with the RAGBSs and numerical

sizes for Ba, Yb, Hg, and Rn are 35s30p17d, 34s30p16d14f, 33s29p21d12f, and 32s32p20d11f, respectively, while the corresponding RAGBS sizes are 30s21p13d, 30s21p13d11f, 30s22p17d11f, and 30s27p17d11f. The size reduction of the basis sets achieved by the pGCDF method in relation to the original GCDF method for these illustrative atoms is in the range of 25 s functions, 59 p functions, 34 d functions, and 0–3 f functions. In fact, the size difference between the RUGBS and RAGBSs tends to reduce as the atomic number increases. Furthermore, the RUGBS errors for Ba, Yb, Hg, and Rn are 0.6, 1.6, 3.0, and 3.6 millihartree, respectively, for the uniform sphere model, and can be compared with the RAGBS errors that are 0.9, 2.5, 2.9, and 4.5 millihartree, respectively, for the same nucleus model. Thus, such errors between both approaches are similar or, in other words, the RUGBS and the RAGBSs can be considered as having the same accuracy. Another remark on the RAGBS sizes is that one more p function (22 instead of 21) was added for the atoms from Lu (Z 71) to Hg (Z 80) to keep the basis set error below 5 millihartree. The electronic configurations with one electron in the 5d subshell for La (Z 57), Ce (Z 58), and Gd (Z 64) required 15d instead of 13d functions once the d electronic cloud is more diffuse in such configurations. The largest RAGBS errors are those for Rn (Z 86), i.e., 4.5 and 3.2 millihartree for the uniform sphere and the Gaussian models, respectively.


Table 7.25 The pGCDF parameters for Cs (Z 55) through Rn (Z 86)a (1 )

(2 )

(3 )

Cs

55

[Xe] 6s1

Ba

56

[Xe] 6s2

La

57

[Xe] 6s24f 1

s p d s p d s p d f s p d s p d f s p d f s p d f s p d f

6.89309456(01) 3.73954597(01) 1.33627171(01) 6.43427931(01) 3.59625691(01) 1.22352674(01) 6.31540168(01) 3.30933707(01) 9.32342456(02) 4.49989016(01) 6.23817882(01) 3.27841746(01) 4.55912743(01) 6.25910415(01) 3.42177883(01) 6.70688664(02) 4.40600400(01) 6.10850205(01) 3.32111124(01) 4.46240513(01) 3.50968316(01) 6.20008675(01) 3.25339574(01) 7.23137980(02) 4.21335688(01) 6.25819228(01) 3.22727881(01) 7.35878161(02) 4.07525146(01)

1.84500000(01) 1.54182625(01) 1.40010433(01) 1.76500000(01) 1.58413113(01) 1.41353310(01) 1.77300000(01) 1.57782436(01) 1.38683782(01) 1.46815349(01) 1.75900000(01) 1.56745218(01) 1.68587229(01) 1.76300000(01) 1.59174006(01) 1.30990723(01) 1.45481734(01) 1.76500000(01) 1.58451270(01) 1.69350312(01) 1.34271521(01) 1.78100000(01) 1.58518979(01) 1.31745194(01) 1.47026246(01) 1.78700000(01) 1.57981471(01) 1.31115562(01) 1.47278947(01)

7.27838702(03) 7.10237528(03) 7.74851547(03) 7.02749031(03) 7.83219906(03) 8.23617429(03) 7.22990967(03) 7.71570635(03) 7.62257186(03) 4.38026134(03) 7.22526889(03) 7.67231040(03) 8.47568117(03) 7.20484855(03) 7.82366859(03) 6.23869509(03) 4.23799360(03) 7.17401355(03) 7.83409698(03) 8.89767181(03) 3.79771186(03) 7.29246162(03) 7.70667605(03) 6.06630476(03) 4.45187556(03) 7.26198080(03) 7.65907905(03) 6.00293700(03) 4.61416931(03)

1.88656757(04) 3.43631331(04) 5.76592752(04) 1.87869591(04) 3.71877331(04) 6.14459218(04) 1.93851253(04) 3.67462664(04) 5.59506740(04) 3.10417177(04) 1.95049252(04) 3.68657674(04) 4.25949785(04) 1.93959558(04) 3.65879826(04) 4.88970430(04) 3.23719350(04) 1.92132944(04) 3.69753823(04) 4.53370582(04) 3.06995978(04) 1.94619959(04) 3.67393670(04) 4.90444807(04) 3.21449354(04) 1.93059453(04) 3.63804882(04) 4.87121220(04) 3.20791228(04)

[Xe] 6s25d1

Ce

58

[Xe] 6s24f 2

[Xe] 6s24f15d1

Pr

59

[Xe] 6s24f 3

Nd

60

[Xe] 6s24f 4

(continued )

Page 137

) (min

07:52

5/4/2007

Configuration

137

Z


Atom

Continued

Pm

61

[Xe] 6s24f 5

Sm

62

[Xe] 6s24f 6

Eu

63

[Xe] 6s24f 7

Gd

64

[Xe] 6s24f 8

s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p

[Xe] 6s24f 75d1

Tb

65

[Xe] 6s24f 9

Dy

66

[Xe] 6s24f 10

Ho

67

[Xe] 6s24f 11

) (min

6.24292825(01) 3.17981400(01) 6.34675207(02) 3.94105026(01) 6.24665154(01) 3.10611106(01) 5.65387358(02) 3.75031681(01) 6.19958906(01) 3.09193799(01) 5.13423063(02) 3.46318132(01) 6.03213313(01) 3.04243681(01) 2.28591779(02) 2.85459648(01) 5.93031341(01) 2.90022468(01) 4.37798776(01) 2.41595047(01) 6.00636455(01) 2.84282739(01) 2.32279878(02) 2.66901117(01) 6.00436041(01) 2.94004205(01) 2.21146294(02) 2.86089999(01) 5.95940748(01) 2.89428253(01)

(1 )

(2 )

(3 )

1.74300000(01) 1.57194411(01) 1.30031680(01) 1.47958763(01) 1.73800000(01) 1.60435984(01) 1.30142741(01) 1.51064017(01) 1.76900000(01) 1.58949390(01) 1.32520135(01) 1.48917212(01) 1.77500000(01) 1.56783061(01) 1.33774694(01) 1.50937479(01) 1.78700000(01) 1.57129731(01) 1.90831332(01) 1.33850772(01) 1.77600000(01) 1.56374730(01) 1.35509067(01) 1.51545336(01) 1.77600000(01) 1.56309714(01) 1.35143879(01) 1.51360957(01) 1.76900000(01) 1.56607553(01)

6.59756092(03) 7.59651469(03) 6.04293610(03) 5.20613342(03) 6.44921441(03) 7.66762180(03) 6.02615538(03) 6.12961434(03) 6.52966007(03) 7.61749980(03) 5.84039491(03) 6.06505697(03) 6.75107568(03) 7.08189014(03) 6.86008754(03) 7.38125161(03) 6.74089038(03) 7.08802955(03) 1.23005179(02) 3.87083591(03) 6.87613940(03) 7.06413913(03) 7.04765091(03) 7.23364106(03) 6.88913657(03) 7.05405761(03) 7.02102552(03) 7.24815041(03) 6.93934557(03) 7.06649068(03)

1.75407365(04) 3.63591770(04) 4.91178568(04) 3.69779272(04) 1.70898671(04) 3.61896639(04) 4.94350579(04) 4.34841334(04) 1.69790644(04) 3.63995874(04) 4.60620984(04) 4.21986624(04) 1.75941762(04) 3.41377839(04) 5.33697952(04) 5.07305790(04) 1.73836839(04) 3.40421690(04) 6.11623021(04) 3.01316523(04) 1.80027566(04) 3.41946454(04) 5.45865122(04) 5.13619927(04) 1.80513074(04) 3.42021565(04) 5.53864950(04) 5.04203875(04) 1.82863328(04) 3.40302523(04)

Page 138

07:52

Configuration

5/4/2007

Z

Chapter 7

Atom


138

Table 7.25

[Xe] 6s24f13

Yb

70

[Xe] 6s24f14

Lu

71

[Xe] 6s24f145d1

Hf

72

[Xe] 6s24f145d2

[Xe] 6s14f145d3

Ta

73

[Xe] 6s24f145d3

[Xe] 6s14f145d4

139

(continued )

Page 139

69

5.39112469(04) 4.96550879(04) 1.83234525(04) 3.41170302(04) 5.54335932(04) 5.05928860(04) 1.82227177(04) 3.40994132(04) 5.62463163(04) 5.15341382(04) 1.93933315(04) 3.29283189(04) 5.72736797(04) 4.76510982(04) 1.89644729(04) 3.13508924(04) 3.44770646(04) 5.06336533(04) 1.87850347(04) 3.09753650(04) 3.18551448(04) 6.80048021(04) 1.79673807(04) 3.11899328(04) 3.56226701(04) 6.39265251(04) 1.87278576(04) 3.04998747(04) 3.59599333(04) 6.05125752(04) 1.87328960(04) 3.07237653(04) 3.50489830(04)

07:52

Tm

6.91538754(03) 7.24726108(03) 6.90243734(03) 7.07336758(03) 7.20919689(03) 7.41688605(03) 6.85438448(03) 7.10811764(03) 7.10417079(03) 7.94004377(03) 7.13629598(03) 6.81925073(03) 7.32663964(03) 7.21869079(03) 7.19504924(03) 6.84593447(03) 8.08701979(03) 6.58016448(03) 7.17449235(03) 6.85873109(03) 7.28731870(03) 9.31285305(03) 6.97397997(03) 6.88337425(03) 8.06810313(03) 8.79450113(03) 7.06465401(03) 6.87048030(03) 7.94327230(03) 8.28727662(03) 7.11918834(03) 6.92853055(03) 8.07946158(03)

5/4/2007

[Xe] 6s24f12

1.36221543(01) 1.51761849(01) 1.75200000(01) 1.57176352(01) 1.36766975(01) 1.53359991(01) 1.74800000(01) 1.56516855(01) 1.35541332(01) 1.58659080(01) 1.73000000(01) 1.57203350(01) 1.37776507(01) 1.56793397(01) 1.82600000(01) 1.50978899(01) 1.70193513(01) 1.42012116(01) 1.82800000(01) 1.53543320(01) 1.63170149(01) 1.52137449(01) 1.83600000(01) 1.53152048(01) 1.68737020(01) 1.48309179(01) 1.79600000(01) 1.55842630(01) 1.64962678(01) 1.49936407(01) 1.81400000(01) 1.55393654(01) 1.69260535(01)


68

2.31567398(02) 2.80357653(01) 5.87524295(01) 2.87510824(01) 2.21570953(02) 2.68523479(01) 5.91380595(01) 2.78213410(01) 2.12816248(02) 2.74386254(01) 5.88186810(01) 2.81562563(01) 2.39761361(02) 2.96757230(01) 5.64926686(01) 2.50600848(01) 4.66318374(01) 1.95473248(01) 5.46315490(01) 2.36815975(01) 4.42499286(01) 1.54788545(01) 5.37530956(01) 2.49821181(01) 5.11978984(01) 1.69114075(01) 5.32169635(01) 2.25487824(01) 4.30272436(01) 1.26677376(01) 5.38656171(01) 2.31030902(01) 4.63392071(01)


Er

d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d

Z

Configuration

W

74

[Xe] 6s24f145d4

[Xe] 6s24f145d5

76

[Xe] 6s24f145d6

[Xe] 6s14f145d7

Ir

77

[Xe] 6s24f145d7

[Xe] 6s14f145d8

(3 )

f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s

1.34369914(01) 5.26373310(01) 2.13021676(01) 4.17825325(01) 9.13016434(02) 5.27645066(01) 2.29520324(01) 4.63555144(01) 1.33424042(01) 5.21572883(01) 2.06073075(01) 3.99540165(01) 7.19160739(02) 5.09772397(01) 2.10625658(01) 4.35469256(01) 7.82232257(02) 5.02894669(01) 1.82452891(01) 3.77734184(01) 6.96513836(02) 4.97929096(01) 1.90763741(01) 4.09963048(01) 6.96347760(02) 5.06128998(01) 1.94024323(01) 3.57001660(01) 6.88565771(02) 5.09637806(01)

1.39061265(01) 1.83100000(01) 1.51599407(01) 1.70687761(01) 1.43596835(01) 1.84300000(01) 1.53956859(01) 1.69561457(01) 1.40174047(01) 1.85000000(01) 1.51912738(01) 1.69533950(01) 1.42293401(01) 1.80900000(01) 1.53700260(01) 1.72982063(01) 1.40596036(01) 1.80700000(01) 1.51589494(01) 1.68733971(01) 1.39616585(01) 1.80400000(01) 1.51841007(01) 1.71305282(01) 1.40512958(01) 1.80300000(01) 1.50300332(01) 1.65921105(01) 1.40978489(01) 1.80800000(01)

7.52960011(03) 7.23842661(03) 6.90857249(03) 8.71140035(03) 7.93358341(03) 7.17973059(03) 6.82521549(03) 8.08780486(03) 7.54791762(03) 7.23905153(03) 6.84705926(03) 9.46000618(03) 8.01388343(03) 7.15701246(03) 6.89987393(03) 9.45747392(03) 7.45287329(03) 7.21629482(03) 6.91405064(03) 9.58427247(03) 8.43237097(03) 7.18263452(03) 6.93277577(03) 9.53563698(03) 8.28041747(03) 7.22799350(03) 6.87409349(03) 9.71126311(03) 8.85412587(03) 7.19935170(03)

5.88738068(04) 1.88872912(04) 3.13778804(04) 3.84054028(04) 5.81795040(04) 1.85502971(04) 3.08752549(04) 3.51201651(04) 5.96438558(04) 1.86503966(04) 3.10483132(04) 4.27609929(04) 6.07199793(04) 1.88100290(04) 3.10023808(04) 4.25023030(04) 5.82420231(04) 1.91084121(04) 3.16685768(04) 1.90094680(04) 3.13669040(04) 4.50902790(04) 6.80843975(04) 1.88994629(04) 3.12068862(04) 4.35274176(04) 6.62274502(04) 4.59985286(04) 6.88731831(04) 1.89604896(04)

Page 140

Os

(2 )

Chapter 7

[Xe] 6s14f145d6

(1 )

07:52

75

) (min

5/4/2007

[Xe] 6s14f145d5


Continued

Atom

Re

140

Table 7.25

78

[Xe] 6s24f145d8

[Xe] 6s24f145d9

Hg

80

[Xe] 6s24f145d10

Tl

81

[Hg] 6p1

Pb

82

[Hg] 6p2

Bi

83

[Hg] 6p3

141

(continued )

Page 141

[Xe] 6s14f145d10

3.09134981(04) 4.40854149(04) 6.58860288(04) 1.89804311(04) 3.11968000(04) 4.66656919(04) 7.32810627(04) 1.88809583(04) 3.07308029(04) 4.75003925(04) 6.91278328(04) 1.88936628(04) 3.13603202(04) 4.83295214(04) 6.98335743(04) 1.89824828(04) 3.07277630(04) 4.76392610(04) 6.95626485(04) 1.89363281(04) 3.14204061(04) 4.82645741(04) 6.32138122(04) 2.09008309(04) 2.14353804(04) 4.62094365(04) 6.47775140(04) 2.09648775(04) 2.11741788(04) 4.55875951(04) 6.60475488(04) 2.09226021(04) 2.13949775(04)

07:52

79

6.83396876(03) 9.72486919(03) 7.98275953(03) 7.22951206(03) 6.86400909(03) 1.00470449(02) 8.91019882(03) 7.18400144(03) 6.75922139(03) 9.99374703(03) 8.99490059(03) 7.20534446(03) 6.79558423(03) 1.00976982(02) 9.09683872(03) 7.19965286(03) 6.75113422(03) 1.02593641(02) 9.18053274(03) 7.19752491(03) 6.93060787(03) 1.02397693(02) 8.34068952(03) 7.71529866(03) 6.88160798(03) 9.43338242(03) 8.35880991(03) 7.71482568(03) 6.79694338(03) 9.40617385(03) 8.61367150(03) 7.71715770(03) 6.89030506(03)

5/4/2007

Au

1.51484342(01) 1.71521193(01) 1.37411518(01) 1.80600000(01) 1.50683221(01) 1.72268691(01) 1.41250411(01) 1.80800000(01) 1.50919706(01) 1.68439871(01) 1.44174983(01) 1.80500000(01) 1.48085005(01) 1.68796391(01) 1.42982319(01) 1.80300000(01) 1.50587479(01) 1.71944916(01) 1.43027363(01) 1.79900000(01) 1.50429755(01) 1.68619962(01) 1.46835859(01) 1.76900000(01) 1.74400000(01) 1.60119984(01) 1.41045304(01) 1.75700000(01) 1.72900000(01) 1.62006084(01) 1.41169458(01) 1.75500000(01) 1.72800000(01)


[Xe] 6s14f145d9

1.96608142(01) 3.82785717(01) 6.63860490(02) 4.83846242(01) 1.67050078(01) 3.57568745(01) 6.37198257(02) 5.04126276(01) 1.93971400(01) 3.64995839(01) 8.10937994(02) 4.78856882(01) 1.60139319(01) 3.31355363(01) 7.20832800(02) 4.99055379(01) 1.81636017(01) 3.60937937(01) 8.22280622(02) 4.77793648(01) 1.72108243(01) 3.30024559(01) 8.38778018(02) 4.29587199(01) 5.77606365(01) 2.91642516(01) 2.51189732(02) 4.03315235(01) 5.48935567(01) 2.89092235(01) 2.01671420(02) 3.79820685(01) 5.22991552(01)


Pt

p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p

Z

Configuration

At

85

[Hg] 6p5

Rn

86

[Hg] 6p6

a

The numbers in parentheses are the powers of 10.

(2 )

(3 )

d f s p d f s p d f s p d f

2.91527320(01) 1.00985935(02) 3.64017592(01) 4.93510013(01) 2.26298477(01) 5.00568905(03) 3.52727536(01) 4.72411339(01) 2.18559791(01) 1.62596305(03) 3.34453526(01) 4.56637626(01) 2.19218026(01) 1.31091802(03)

1.62141461(01) 1.41944170(01) 1.79400000(01) 1.74900000(01) 1.55315397(01) 1.42011883(01) 1.80200000(01) 1.73900000(01) 1.54287616(01) 1.43425593(01) 1.80100000(01) 1.72000000(01) 1.56024563(01) 1.44417385(01)

9.48940053(03) 8.71357002(03) 7.73032648(03) 6.80541893(03) 9.59666420(03) 8.38077916(03) 7.72522903(03) 6.88179644(03) 9.51650406(03) 8.43292357(03) 7.73485697(03) 6.87350932(03) 9.90932638(03) 8.47418585(03)

4.61517961(04) 6.61010739(04) 2.04762261(04) 2.05926336(04) 4.83788384(04) 6.26379954(04) 2.03454543(04) 2.09220729(04) 4.83509387(04) 6.10117280(04) 2.03326769(04) 2.10783041(04) 4.97949780(04) 6.03346573(04)

Page 142

[Hg] 6p4

(1 )

07:52

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) (min

Chapter 7

Po

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Atom

Continued


142

Table 7.25


Table 7.26 Total DFC energies (in hartree) and the basis set errors (in millihartree) for Cs (Z 55) through Rn (Z 86) Atom

Z

M

58

140

Pr Nd Pm Sm Eu Gd

59 60 61 62 63 64

141 144 145 152 153 158

Tb Dy Ho Er Tm Yb Lu Hf

65 66 67 68 69 70 71 72

159 162 162 168 169 174 175 180

Ta

73

181

W

74

184

Errora

EDFC

Errora

7786.769132 8135.642200 8493.540441 8493.642397 8860.994060 8861.067756 9238.144498 9625.127308 10022.09029 10429.15735 10846.49858 11274.23544 11274.22261 11712.53697 12161.53641 12621.40280 13092.25853 13574.30353 14067.66165 14572.51662 15088.76778 15088.68430 15616.60969 15616.54312 16156.16146 16156.11398

0.919 0.916 0.955 1.137 1.129 1.224 1.152 1.234 1.261 1.318 1.416 1.551 1.723 1.643 1.694 1.828 1.951 2.122 2.501 1.773 1.793 1.896 1.811 1.934 2.006 1.981

7786.770800 8135.644148 8493.542669 8493.644626 8860.996619 8861.070309 9238.147431 9625.130691 10022.09414 10429.16184 10846.50371 11274.24136 11274.22852 11712.54372 12161.54414 12621.41157 13092.26864 13574.31500 14067.67484 14572.53160 15088.78496 15088.70147 15616.62915 15616.56257 16156.18360 16156.13612

0.868 0.863 0.908 1.090 1.077 1.179 1.096 1.176 1.216 1.267 1.373 1.495 1.678 1.587 1.644 1.758 1.896 2.069 2.419 1.650 1.649 1.757 1.640 1.771 1.809 1.786

(continued )

143

30s21p13d 30s21p13d 30s21p13d11f 30s21p15d 30s21p13d11f 30s21p15d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p15d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s21p13d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f

EDFC

Page 143

Ce

[Xe] 6s1 [Xe] 6s2 [Xe] 6s24f1 [Xe] 6s25d1 [Xe] 6s24f 2 [Xe] 6s24f15d1 [Xe] 6s24f 3 [Xe] 6s24f 4 [Xe] 6s24f 5 [Xe] 6s24f 6 [Xe] 6s24f 7 [Xe] 6s24f 8 [Xe] 6s24f 75d1 [Xe] 6s24f 9 [Xe] 6s24f10 [Xe] 6s24f11 [Xe] 6s24f12 [Xe] 6s24f13 [Xe] 6s24f14 [Xe] 6s24f145d1 [Xe] 6s24f145d2 [Xe] 6s14f145d3 [Xe] 6s24f145d3 [Xe] 6s14f145d4 [Xe] 6s24f145d4 [Xe] 6s14f145d5

Gaussian Model

07:52

133 138 139


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55 56 57

Basis Sets Size


Cs Ba La

Configuration

Atom

Continued Z

M

76

192

Ir

77

193

Pt

78

195

Au

79

197

Hg Tl Pb Bi Po At Rn

80 81 82 83 84 85 86

202 205 208 209 209 210 222

a

The RAGBS error with respect to the numerical DFC results [25].

30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s22p17d11f 30s27p17d11f 30s27p17d11f 30s27p17d11f 30s27p17d11f 30s27p17d11f 30s27p17d11f

EDFC

Errora

EDFC

Errora

16707.59299 16707.56614 17271.05175 17271.04682 17846.75432 17846.77274 18434.83555 18434.87853 19035.48280 19035.55137 19648.84638 20274.79483 20913.65117 21565.63502 22230.93324 22909.71775 23602.00097

2.192 2.127 2.211 2.245 2.251 2.242 2.478 2.367 2.699 2.555 2.865 2.687 3.020 3.319 3.853 4.220 4.548

16707.61817 16707.59134 17271.08056 17271.07558 17846.78687 17846.80524 18434.87236 18434.91531 19035.52444 19035.59300 19648.89380 20274.84893 20913.71252 21565.70414 22231.01065 22909.80475 23602.10105

1.965 1.878 1.890 1.968 1.882 1.914 2.083 2.000 2.243 2.096 2.356 1.707 1.809 1.943 2.525 2.865 3.205

Page 144

Os

[Xe] 6s24f145d5 [Xe] 6s14f145d6 [Xe] 6s24f145d6 [Xe] 6s14f145d7 [Xe] 6s24f145d7 [Xe] 6s14f145d8 [Xe] 6s24f145d8 [Xe] 6s14f145d9 [Xe] 6s24f145d9 [Xe] 6s14f145d10 [Xe] 6s24f145d10 [Hg] 6p1 [Hg] 6p2 [Hg] 6p3 [Hg] 6p4 [Hg] 6p5 [Hg] 6p6

Gaussian Model

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75

Basis Sets Size

Chapter 7

Re

Configuration


144

Table 7.26


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Page 145


145

The variational prolapse analysis for the RAGBS obtained from Cs to Rn was also performed by the addition of up to three tight s or tight p functions for every atom in each electronic configuration. These tight functions are obtained by the same pGCDF discretization parameters presented in Table 7.25 and used to generate the RAGBSs from Cs to Rn. The energy differences between the calculations with the nonaugmented and augmented (by one tight s or tight p function) tight RAGBSs (EDFC –E 1 DFC ) are shown in Fig. 7.6. It should be reinforced that positive values for these energy differences indicate that the basis set does not exhibit prolapse. As illustrated in Fig. 7.6, the RAGBSs from Cs to Rn did not show variational prolapse after the pGCDF parameter adjustments since the addition of tight functions always causes a DFC total energy decrease or convergence. The same pattern continued to be noticed with the augmentation of up to three tight s or tight p functions (with further DFC total energy decrease until the eventual convergence). Moreover, we can also see from Fig. 7.6 that the energy behavior with the tight s addition is a characteristic of each nuclear model studied although this is not surprising since it was first detected for the lighter atoms [48]. In fact, it happens because the tight functions have larger exponents and are more compact for high Z systems in a way that the chosen nuclear proton-charge distribution has a major effect on the prolapse test energy differences (given the considerable inner-core

Fig. 7.6 Total DFC energy differences between the calculations with the RAGBS and tight the RAGBS augmented with one tight s or one tight p function (EDFC –E 1 ). DFC


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Chapter 7

electronic penetration into the finite nucleus for heavy atoms). Alike differences are also observed in the augmentation of tight p functions and become more and more prominent as the atomic number increases. Our prolapse-free RUGBS [26] is another example of such energy behavior distinctions (that depend on the atomic number for each nuclear model with the addition of tight functions). Similarly, this fact was first noticed for the RUGBS with the augmentation of tight s functions from Ca (Z = 20) onward and with tight p functions from In (Z = 49) onward [26]. Table 7.27 Convergence pattern (in hartree) for Ba (Z 56) Calculation



Gaussian Model

EDFC

EDFC

8135.642200 8135.642226 8135.642228 8135.642229 8135.642284 8135.642285 8135.642285 8135.642250 8135.642252 8135.642252

8135.644148 8135.644153 8135.644153 8135.644153 8135.644239 8135.644239 8135.644239 8135.644198 8135.644200 8135.644200

Table 7.28 Convergence pattern (in hartree) for Yb (Z 70) Calculation



Gaussian Model

EDFC

EDFC

14067.661649 14067.661686 14067.661690 14067.661690 14067.661779 14067.661781 14067.661781 14067.661773 14067.661779 14067.661779 14067.661755 14067.661758 14067.661758

14067.674841 14067.674864 14067.674866 14067.674867 14067.675030 14067.675031 14067.675031 14067.674966 14067.674971 14067.674972 14067.674948 14067.674950 14067.674950


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147

The magnitude of the energy decrease observed in a tight function augmentation for each nucleus model (Fig. 7.6) depends, among other factors, on the degree of penetration of such tight function into the nuclear region in comparison with the functions already presented in the basis set, i.e., the functions of the basis set before the addition of a tight function. As the uniform sphere model is characterized by a proton-charge distribution with uniform density strictly Table 7.29 Convergence pattern (in hartree) for Hg (Z 80) Calculation



Gaussian Model

EDFC

EDFC

19648.846385 19648.846497 19648.846509 19648.846510 19648.846433 19648.846435 19648.846435 19648.846458 19648.846462 19648.846462 19648.846408 19648.846409 19648.846409

19648.893804 19648.893809 19648.893810 19648.893810 19648.893986 19648.893987 19648.893987 19648.893878 19648.893881 19648.893882 19648.893827 19648.893828 19648.893828

Table 7.30 Convergence pattern (in hartree) for Rn (Z 86) Calculation



Gaussian Model

EDFC

EDFC

23602.000972 23602.001089 23602.001102 23602.001103 23602.000991 23602.000993 23602.000993 23602.001055 23602.001059 23602.001059 23602.001031 23602.001033 23602.001033

23602.101045 23602.101064 23602.101066 23602.101066 23602.101050 23602.101050 23602.101050 23602.101128 23602.101132 23602.101132 23602.101104 23602.101106 23602.101106


148

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Chapter 7

restricted inside the nuclear radius (after which this charge distribution goes suddenly to zero) while the Gaussian nucleus shows a smooth reduction of the proton-charge density with the radius, larger energy decrease values in the tight s addition are more likely to arise with the uniform sphere nucleus model (where the the proton-charge distribution is noncontinuous) than with the Gaussian nucleus model. However, for the tight p case (in which the exponents involved are not so large as the s ones for any given atom and the energy differences are more comparable) it is difficult to draw an explanation although from Hg (Z 80) onward the same behavior seen for the tight s appears to be present. Furthermore, some atomic systems (Ba, Yb, Hg, and Rn) were selected to illustrate the prolapse analysis in more details by the addition of up to three tight s, p, d, and f functions (see Tables 7.27–7.30). These tables show once more that the augmentation of tight d or f functions does not exhibit any sign of variational prolapse since these kind of functions represent electrons that are too far from the atomic nucleus [26,31,48]. The already mentioned tight s or p addition pattern is also shown in Tables 7.27–7.30 and from there we can see that the correct DFC total energy convergence shows that our RAGBSs generated from Cs to Rn do not present any sign of variational prolapse and are also apt to be used in relativistic atomic and molecular calculations with both uniform sphere and Gaussian nucleus models. Other than the applications we have presented in this chapter with the GCDF and pGCDF methods for DFC and DFB calculations, the reader will find in the literature other applications with the relativistic generator coordinate formalism in atomic and molecular DFC and DFB calculations [5361]. Since the GCDF and pGCDF methods have been widely employed in the generation of Gaussian basis sets to be used in relativistic atomic and molecular calculations [62–65], in Appendixes 4 and 5 one will find the ultimate universal and AGBSs generated with the GCDF and pGCDF methods for accurate relativistic atomic and molecular calculations. Also, in the website www.iqsc.usp.br/basis-sets, one will find a whole description of the relativistic Gaussian basis sets presented in this chapter.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Y. Ishikawa, H. Sekino, and R. C. Binning, Jr., Chem. Phys. Lett., 1989, 160, 206. S. Okada, and O. Matsuoka, J. Chem. Phys., 1989, 91, 4193. A. K. Mohanty, and E. Clementi, J. Chem. Phys., 1990, 93, 1829. Y. Ishikawa, Phys. Rev. A, 1990, 42, 1142. Y. Ishikawa, H. M. Quiney, and G. L. Malli, Phys. Rev. A, 1991, 43, 3270. Y. Ishikawa, R. Baretty, and R. C. Binning, Jr., Chem. Phys. Lett., 1985, 121, 130. Y. Ishikawa, and H. M. Quiney, Int. J. Quant. Chem. Symp., 1987, 21, 523. A. B. F. da Silva, G. L. Malli, and Y. Ishikawa, Chem. Phys. Lett., 1993, 203, 201. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, Chem. Phys. Lett., 1993, 201, 37.


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Dirac–Fock Method and Relativistic Calculations 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

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A. B. F. da Silva, G. L. Malli, and Y. Ishikawa, Can. J. Chem., 1993, 71, 1713. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, Phys. Rev. A, 1993, 47, 143. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, J. Chem. Phys., 1994, 101, 6829. O. Matsuoka, and S. Huzinaga, Chem. Phys. Lett., 1987, 140, 567. F. E. Jorge, and A. B. F. da Silva, J. Chem. Phys., 1996, 104, 6278. F. E. Jorge, and A. B. F. da Silva, J. Chem. Phys., 1996, 105, 5503. Y.-K. Kim, Phys. Rev., 1967, 154, 17. A. B. F. da Silva, H. F. M. da Costa, and M. Trsic, Mol. Phys., 1989, 68, 433. J. R. Mohallem, Z. Phys. D, 1986, 3, 339. J. P. Desclaux, At. Data Nucl. Data Tables, 1973, 12, 311. Y.-S. Lee, and A. D. McLean, J. Chem. Phys., 1982, 76, 735; R. E. Stanton, and S. Havriliak, J. Chem. Phys., 1984, 81, 1910. K. G. Dyall, I. P. Grant, and S. Wilson, J. Phys. B, 1984, 17, 1201. P. J. C. Aerts, and W. C. Nieuwpoort, Chem. Phys. Lett., 1986, 125, 83. A. Farazdel, W. M. Westgate, A. M. Simas, R. P. Sagar, and V. H. Smith, Jr., Int. J. Quant. Chem. Symp., 1985, 19, 61. F. A. Parpia, and A. K. Mohanty, Phys. Rev. A, 1992, 46, 3735. L. Visscher, and K. G. Dyall, At. Data Nuc. Data Tables, 1997, 67, 207. R. L. A. Haiduke, L. G. M. de Macedo, and A. B. F. da Silva, J. Comput. Chem., 2005, 26, 932. O. Matsuoka, and Y. Watanabe, Comput. Phys. Commun., 2001, 139, 218. W. R. Johnson, and G. Soff, At. Data Nuc. Data Tables, 1985, 33, 405. O. Matsuoka, J. Phys. Soc. Jpn., 1982, 51, 2263. H. F. M. da Costa, M. Trsic, A. B. F. da Silva, and A. M. Simas, Eur. Phys. J. D, 1999, 5, 375. H. Tatewaki, T. Koga, and Y. Mochizuki, Chem. Phys. Lett., 2003, 375, 399. H. Tatewaki, and Y. Mochizuki, Theor. Chem. Acc., 2003, 109, 40. H. Tatewaki, and Y. Watanabe, J. Comput. Chem., 2003, 24, 1823. H. Tatewaki, and Y. Watanabe, J. Chem. Phys., 2004, 121, 4528. G. W. F. Drake, and S. P. Goldman, Adv. At. Mol. Phys., 1988, 25, 393. H. Bethe, and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Systems, Berlim: Springer-Verlag, 1957. Y. Ishikawa, Int. J. Quant. Chem. Symp., 1990, 24, 383. H. M. Quiney, I. P. Grant, and S. Wilson, J. Phys. B, 1987, 20, 1413. Y. Ishikawa, Chem. Phys. Lett., 1990, 166, 321. H. M. Quiney, In: Methods in Computational Chemistry, Vol. 2, ed. S. Wilson, New York: Plenum, 1988.s J. Sucher, Phys. Scr., 1987, 36, 271; J. Sucher, Phys. Rev. A, 1980, 22, 348; M. Mittleman, Phys. Rev. A, 1971, 4, 893. I. P. Grant, and N. C. Pyper, J. Phys. B, 1976, 9, 761. R. L. A. Haiduke, L. G. M. de Macedo, R. C. Barbosa, and A. B. F. da Silva, J. Comput. Chem., 2004, 25, 1904. J. A. Nelder, and R. Mead, Comp. J., 1965, 7, 308. F. E. Jorge, M. T. Barreto, and A. B. F. da Silva, J. Mol. Struct. (Theochem), 1999, 464, 1. F. E. Jorge, T. B. Bobbio, and A. B. F. da Silva, Chem. Phys. Lett., 1996, 263, 775.


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47. A. K. Mohanty, F. A. Parpia, and E. Clementi, In: Modern Techniques in Computational Chemistry: MOTECC-91, Chap. 4, Netherlands: Ed. ESCOM, 1991. 48. R. L. A. Haiduke, and A. B. F. da Silva, J. Comput. Chem., 2006, 27, 61. 49. R. L. A. Haiduke, L. G. M. de Macedo, R. C. Barbosa, N. H. Morgon, and A. B. F. da Silva, Int. J. Quantum Chem., 2005, 103, 529. 50. K. G. Dyall, Theor. Chem. Acc., 2002, 108, 335. 51. P. Pyykkö, Chem. Rev., 1988, 88, 563. 52. R. L. A. Haiduke, and A. B. F. da Silva, J. Comput. Chem., 2006, 27, 1970. 53. G. L. Malli, A. B. F. da Silva, and Y. Ishikawa, Can. J. Chem., 1992, 70, 1822. 54. G. L. Malli, J. Styszynski, and A. B. F. da Silva, Int. J. Quantum Chem., 1995, 55, 213. 55. F. E. Jorge, and A. B. F. da Silva, Can. J. Chem., 1996, 74, 1748. 56. F. E. Jorge, and A. B. F. da Silva, J. Mol. Struct. (Theochem), 1997, 394, 101. 57. F. E. Jorge, and A. B. F. da Silva, Z. Phys. D, 1997, 41, 235. 58. F. E. Jorge, M. T. Barreto, and A. B. F. da Silva, Chem. Phys., 1997, 221, 45. 59. L. G. M. de Macedo, R. C. Barbosa, and A. B. F. da Silva, Int. J. Quantum Chem., 2005, 102, 1. 60. L. G. M. de Macedo, N. H. Morgon, R. L. A. Haiduke, R. C. Barbosa, and A. B. F. da Silva, Int. J. Quantum Chem., 2005, 103, 523. 61. L. G. M. de Macedo, R. L. A. Haiduke, M. Comar, Jr., and A. B. F. da Silva, Int. J. Quantum Chem., 2006, 106, 2790. 62. G. L. Malli, and J. Styszynski, J. Chem. Phys., 1994, 101, 10736. 63. J. Styszynski, and G. L. Malli, Int. J. Quantum Chem., 1995, 55, 227. 64. G. L. Malli, and J. Styszynski, J. Chem. Phys., 1998, 109, 4448. 65. R. L. A. Haiduke, M. Comer, Jr., and A. B. F. da Silva, Chem. Phys., 2006, 331, 173.


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Chapter 8

The Generator Coordinate Method and Connections with Natural Orbitals and Density Functional Theory

1.

Introduction

As was shown in Chapter 2, Griffin, Hill, and Wheeler [1] introduced the notion of continuity for the mathematical description of the structure of nuclei. As we have been emphasizing in the chapters of this book, it appears that the mathematical description of continuity can be also applied to electronic systems as well. These characteristics are slowly getting their space in the description of the electronic structure of atoms and molecules, although they are not yet fully understood. Also, there are areas in modern Quantum Chemistry in which this notion of continuity has not been tested or has had merely exploratory incursions. This is the case for natural orbitals and density functional theory (DFT), which will have our attention in this chapter.

2.

Natural Orbitals

The idea of natural orbitals was introduced by the late Professor Per-Olov Löwdin in 1955 [2], as one of the many refinements brought by him in Quantum Chemistry. Basically, the notion is to perform a configuration interaction (CI) calculation and, through a linear transform, cumulate the obtained correlation information into a single determinant. The early work in natural orbitals can be attributed to Löwdin and Shull [3]. In 1972, Davidson [4] discussed and updated the literature on natural orbitals. For the reader interested in natural orbitals, we include a list of the latest applications in both electronic and nuclear structures [5].

3.

An Integral Transform View of Natural Orbitals

In 1990, da Costa et al. [6] published the first application of the generator coordinate method (GCM) to natural orbitals with the title “Generator Coordinate Gaussian expanded Natural Orbitals”. The atomic species which illustrated this application were: Li (1S), Li0 (2S), Li (1S), Be (2S), Be0 (1S), and B (1S).


152

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Chapter 8

The first applications of the generator coordinate Hartree–Fock (GCHF) method were in the generation of universal atomic basis sets [7,8] culminating in the very accurate Slater type (STO) and Gaussian (GTO) universal bases for the atoms H to Xe [9]. In these applications, we employed a numerical integration scheme and have retained the same in this chapter. The integration of the Griffin–Wheeler–Hartree–Fock (GWHF) equations is performed by discretization [10]. In later studies, the discretization technique was implemented using a well-defined technique, including relabeling of the generator coordinate space [11], i.e.,

ln , A 1 A

(8.1)

where A is the scaling factor. Relabeling narrows the weight function in the new space and makes simple integration rules efficient. Thus, a given integration range is characterized by min (lowest value), (constant increment), and M (the number of discretization points). Although the complexity of the GC equations forces the use of numerical integration, the continuous nature of the GCM was preserved in both the previous and present calculations. Thus the values of the integration parameters ( min, , and M) are chosen with the criterion of the best possible numerical integration. Note that discretization leads to a set of values of the generator coordinate {i}, which corresponds to a set of generator coordinates (exponents) in the original space {i}. The standard definition [2] of the first-order reduced density matrix is ( X 1 , X 1 ) N ∫ dX 2 K dX N ⴱ (1,K , N ) (1,K , N ),

(8.2)

where is a normalized wave function. The first-order reduced density matrix can be expanded in terms of continuous HF orbitals, i.e., ( X 1 , X 1 ) ∑ ⴱi ( X 1 )ij j ( X 1 ) ij

(8.3)

with ij ∫ dX 1 dX 1 1ⴱ ( X 1 ) ( X 1 , X 1 ) j ( X 1 ) .

(8.4)


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Connections with Natural Orbitals and Density Functional Theory

153

With a unitary transformation [2] it is possible to obtain a diagonal form of , i.e., ( X 1 , X 1 ) ∑ k ⴱk ( X 1 ) k ( X 1 ),

(8.5)

k

where k is a natural orbital and k its occupation number. The natural orbital may be written as: i (1) ∫ d i (1; ) f i ( N ) ().

(8.6)

In the present application, the generator function i is an atomic (Gaussian) orbital and fi (N ) is a continuous natural weight function. The discretized form of the natural orbital i (1) is: M

i (1) ∑ i (1; l ) f i ( N ) ( l ) .

(8.7)

l

Configuration interaction calculations were performed for various 2–4-electron atomic species with charges q 0, 1 and then the natural orbitals i, the occupation numbers i, and the natural weight functions fi (N ) were obtained. The basis sets were selected as described in Chapter 6, which ensures the integral character of the i terms and the continuous behavior of the fi (N ) terms. The CI calculations and further generation of the natural orbitals were performed with the MELD codes [12]. In Table 8.1 we show the trend in the occupation numbers, i, for the lower natural orbitals of the atoms selected as examples. All values shown correspond to the basis defined by min 1.46, 0.16, M 18, and A 6.0. Note that the occupation of Li is more widespread than Be (compare, for instance, 2 in

Table 8.1 Occupation numbers i of the first five natural orbitalsa Atom (state) 1

Li ( S) Li0 ( 2S) Li (1S) Be (2S) Be0 (1S)

Symmetry

i

S S S S S

1.996997 1.997135 1.997307 1.998613 1.998733

0.002936 1.000032 1.906331 1.000026 1.992011

0.000062 0.002762 0.093184 0.001323 0.007984

0.000004 0.000063 0.002615 0.000033 0.001138

0.000001 0.000006 0.000471 0.000004 0.000106

a The values provided correspond to the basis with min 1.46. When all the natural orbitals are considered, one verifies numerically i N for all the cases. Since the calculation is spin-restricted, one has i 2. i


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both cases). This is consistent with the demand for lower values of min (more diffuse orbitals) for Li, as discussed below. Figs. 8.1–8.3 show the lower natural weight functions fi (N ) as a function of the generator coordinate in the space for Li, Li, and Be. The regular and continuous

Fig. 8.1

The natural weight functions fn(N ), n 1,..., 4 in space for the Li atom.

Fig. 8.2

The natural weight functions fn(N ), n 1,..., 4 in space for the Li ion.


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Fig. 8.3

155

The natural weight functions fn(N ), n 1,..., 4 in space for the Be atom.

behavior of fi (N ) is well characterized. We have chosen to plot only the lower fi (N ) values for better viewing, although higher weight functions have the same continuous character. Since the generator function in Equation (8.6) is a 1s GTO, the orthogonality of the various natural orbitals i is ensured through the weight functions fi (N ). Thus, one observes zero nodes for f1(N ), one node for f2(N ), and so on. There is some sort of a “structure” in the natural weight functions, which is not encountered in the corresponding HF weight functions (for plots of HF weight functions see Chapter 6). For instance, f1(N ) for Be (Fig. 8.3) basically shows the behavior of a 1s weight function in the region between 0.1 and 1.0; but it has a second maximum at ca. 0.35. Thus, the n 1 natural orbital requires more diffuse (lower values of ) basis components than an HF 1s orbital. The demand for contracted basis components can be seen from the right-hand side of Figs. 8.1–8.3, which our basis sets with max min (M–1) min 17 0.16 reasonably satisfy. The left-hand side of Figs. 8.1 and 8.3 show that for diffuse orbitals all fi(N ) seem to terminate together at ca. min 0.80. As for Li (Fig. 8.2), there is a clear demand for very diffuse orbitals, i.e. ca. min 1.5. [Recall that from Equation (8.1) the generator coordinate (exponent) in space is exp(A) exp(6.0 )].


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Curiously, for the Li atom (Fig. 8.1), the n 2 natural weight function on the left-hand side is more contracted than the 2s HF weight function. For comparison with the HF weight function, see Fig. 4.1 in Chapter 4. Thus, atomic natural orbitals present a continuous character when expanded in a set of Gaussian functions selected through a discretized integration grid. This opens a wide gamut of applications of the GCM in post-HF calculations, which should bring formal refinements and practical improvements. At this stage, a clear technique for basis set selection for natural orbitals is already available. Indeed, a test run allows one to plot the natural weight function and select min, , and M and, thereby, perform the best numerical integration.

4.

Density Functional Theory

In 1964, Hohenberg and Kohn [13] demonstrated that the exact ground state energy of a many-electron system could be expressed as a function of the oneelectron density (it is to be noted that solely the existence of the function was demonstrated, not its explicit form). Soon afterwards, Kohn and Sham [14] put this theory in terms of orbitals, which is a more familiar formulation nowadays. At this point, we wish to remark that even the HF orbitals have a very ephemeral physical significance. This could merit a long discussion in other opportunities, but let us only say that any linear transformation that preserves energy is still a valid solution of the Schrödinger equation, although the form and orientation of the HF orbitals might differ. This is more serious in the case of the Kohn–Sham orbitals, an auxiliary for the calculation of the electron density, which was never meant to have a physical meaning. This point was sharply stressed recently by Schirmer and Dreuw [15]. Nevertheless, we need to admit that this is a controversial matter since other authors, pioneered by Wolfe et al. [16], claim exactly the contrary, i.e., that the Kohn–Sham orbitals do have a physical meaning. In this field, as in several other fields, the late Professor Slater was a pioneer. Indeed, his X method [17], better known later as the Hartree–Fock–Slater (HFS) method, contained the local one-electron Hamiltonian with the term 1/3(1), which happens to be now known as the first term of the exchange-correlation term in the Kohn and Sham scheme. The extension of the Hohenberg and Kohn expansion to excited states is not a trivial task and is the object of many theoretical efforts (see, for instance, Reference [18]). Also, in this context, the early proposal of Slater’s transition state method [19] is illuminating. DFT is in the way of becoming, if it has not already become, the preferred method to obtain correlated atomic and molecular energies and other properties. We also refer the interested reader to two relatively recent books on the subject [20] and the Gaussian 03 Manual (www.gaussian.com), which describes one of the most popular computer codes for quantum chemical calculations.


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157

5. First Applications of the Generator Coordinate Method to Density Functional Theory As we noted above, in 1951 Slater set the first step of the DFT by replacing the exchange term of the Hartree–Fock formulation with the exchange-correlation term VX 3关(3 8) (1)兴1 3 ,

(8.8)

where (1) is the one-electron density, in many applications represented by the occupied atomic or molecular orbitals, i.e., for an even number of electrons ( n) 2∑

OCC

iⴱ (1) i (1),

(8.9)

where i are the one-electron functions (the expansion could alternatively be in terms of spin-orbitals). The value of the parameter in the applications assumed two possible values depending on the origin of the deduction, i.e., 1.0 or 2/3. In 1972 [17], Slater raised the point that neither the value 2/3 nor the value 1.0 were necessarily optimal. One method to determine the value of was to force the HFS total energy for the ground state of an atomic system to equal the HF total energy [17]. Actually, in a popular version of the HFS equations implemented in the seventies, the value employed was 0.7 [see, for instance, Reference 21]. This issue will be addressed in the context of the GCM below. In 2004, Trsic et al. [22] applied the GCM in an innovative strategy, choosing the parameter in VX in Equation (8.8) as the generator coordinate (in the standard method the generator coordinate is part of the trial function, in general as an exponent of Gaussian- or Slater-type orbitals), thus, the generator coordinate is part of the Hamiltonian. The idea of the application of the GCM in the DFT was proposed initially by Klaus Capelle in 2003 [23]. Trsic et al. [22] initiated a preliminary application of the GCM in DFT with a self-consistent field HF calculation for the He atom. We first obtain the best HF calculation and then replace the exchange term by Slater’s X potential [17] VX, the simpler expression for DFT, but sufficient for this experiment. We interpret as the generator coordinate and weight the exchange-correlation term for different values of the parameter , i.e., i 0 ( N i ) , with N 4, 0 0.25, and 0.25 in our first experiment.

(8.10)


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With the one-electron functions i so generated, we construct the determinants

i i i .

(8.11)

We then perform a CI calculation leading to the wave function 0 ∑ Ci i , i .

(8.12)

Table 8.2 shows the results for the total energies (E0 and E1) for the 11S0 and 2 S0 states for N 4. It is at this point that we bring in the GCM interpretation: thus, we interpret the Ci coefficients (squared) as sample points of a continuous weight function. The behavior of this function leads us to locate the optimal interval for the i values. Fig. 8.4a shows a plot of the weight function for the various values of the parameter for the 11S0 ground state. One can see that f () increases sharply for close to 1.0. To ensure an understanding of the behavior of this function beyond 1.0 we repeated the calculation for N5, which shows clearly that the values of 1.0 do not improve the energy values (Fig. 8.4b). Within the limits of the particular strategy here presented, values in the region of 1.0 seem optimal for the ground state energy. It may be observed that there is a secondary peak ca. 0.4 [Figs. 8.4(a) and 8.4(b)] so that the interval chosen in Equation (8.10) is shown to be efficient. However, Fig. 8.4c shows the shape of the weight function for the excited state, pointing at optimal values close to 0.7, although in this case also there is a secondary peak for low values of . To return to the discussion of Slater in 1972 [17], it seems that, as our results suggest, ⬵ 1.0 would be best for the ground state, while ⬵ 0.7 would be indicated for the first excited state with the same symmetry. One may also speculate along the following lines: the explicit expansion for the DFT energy is not known and several formulae are used, in most cases with great numerical success (see, for instance, www.gaussian 03 manual), but leaving a flavor of hidden parametrization. Could the generator coordinate integral view of the electronic structure allow some further insights into the density 1

Table 8.2 Ground and first excited (11S0 and 21S0) state energies for the He atom (hartrees)

N4 Exact [24] Capelle [23]

E0

E1

2.903 2.904 2.871

2.099 2.146 1.788


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159

Fig. 8.4 The weight functions for the ground and the first excited states for the CI calculation for N 4, (a) ground state for N 4 [Equation (8.10)]; (b) ground state for N 5; (c) excited state for N 4.

functional properties? Also, since in DFT the excited states are a source of additional difficulties while GCM seems to generate excited states naturally, as shown in the very trivial example above, some understanding may be gained in this case as well.


160

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Chapter 8

References 1. (a) D. L. Hill, and J. A. Wheeler, Phys. Rev., 1953, 89, 1102; (b) J. J. Griffin and J. A. Wheeler, Phys. Rev., 1957, 108, 311. 2. P. O. Löwdin, Phys. Rev., 1955, 97, 1474. 3. P. O. Löwdin, and H. Shull, Phys. Rev., 1956, 101, 1730; H. Shull, and P. O. Löwdin, J. Chem. Phys., 1955, 23, 1565; H. Shull, and P. O. Löwdin, J. Chem. Phys., 1959, 30, 617; H. Shull, J. Chem. Phys., 1959, 30, 1405. 4. E. R. Davidson, Adv. Quantum Chem., 1972, 6, 235. 5. A. I. Ermakov, A. E. Merkulov, A. A. Svechnikova, and V. V. Belouskov, J. Struct. Chem., 2004, 45, 923; N. N. Lathiotakis, N. Helbig, and E. K. U. Gross, Phys. Rev. A, 2005, 72, 030501/1; B. Blaive, A. Julg, and A. Pellegatti, European Phys. J. B, 2005, 47, 177; X. Z. Li, and J. Paldus, Int. J. Quantum Chem., 2005, 105, 672; S. Yamanaka, R. Takeda, M. Shoji, Y. Kitqgawa, H. Honda, and K. Yamaguchi, Int. J. Quantum Chem., 2005, 105, 605; D. Bressanini, G. Morosi, and S. Tarasco, J. Chem. Phys., 2005, 123, 204109/1; D. Van Neck, S. Rombouts, and S. Verdonck, Phys. Rev. C., 2005, 72, 054318/1; J. Ma, S. H. Li, and W. Li, J. Comp. Chem., 2006, 27, 39; J. Cioslowski, and M. Buchowiecki, J. Chem. Phys., 2005, 123, 234102/1; L. Abrams, and C. D. Sherrill, Mol. Phys., 2005, 103, 3315; D. C. Thompson, and P. W. Ayers, Int. J. Quantum Chem., 2006, 106, 787; O. Gritsenko, and E. J. Baerends, J. Chem. Phys., 2006, 124, 054115/1; M. Sekiya, T. Noro, E. Miyoshi, Y. Osanai, and T. Koga, J. Comp. Chem., 2006, 27, 463. 6. H. F. M. da Costa, A. B. F. da Silva, M. Trsic, A. M. Simas, and A. J. A. Aquino, J. Mol. Struct. (Theochem)., 1990, 210, 63. 7. J. R. Mohallem, and M. Trsic, J. Chem. Phys., 1987, 86, 5043. 8. H. F. M. da Costa, M. Trsic, and J. R. Mohallem, Mol. Phys., 1987, 62, 91. 9. A. B. F. da Silva, H. F. M. da Costa, and M. Trsic, Mol. Phys., 1989, 68, 433. 10. P. Chattopadyay, R. M. Dreizler, M. Trsic, and M. Fink, Z. Phys. A, 1978, 285, 7. 11. J. R. Mohallem, Z. Phys. D, 1986, 3, 339. 12. The MELD codes were developed by E. R. Davidson, S. R. Langhoff, S. T. Elbert, and L. E. Murchie, University of Indiana; the version employed in Reference [5] was modified by D. Feller, and D. C. Rawlings. 13. P. Hohenberg, and W. Kohn, Phys. Rev. B, 1964, 136, 864. 14. W. Kohn, and L. J. Sham, Phys. Rev. A, 1965, 140, 1133. 15. J. Schirmer, and A. Dreuw, Phys. Rev., Phys. Rev., 2007, 75, 022513. 16. S. Wolfe, Z. Shi, C. E. Brion, J. Rolke, Y. Zeng, G. Cooper, D. P. Chong, and C. Hu, Can. J. Chem., 2002, 80, 222. 17. (a) J. C. Slater, Phys. Rev., 1951, 81, 385; (b) ibid., Phys. Rev., 1951, 82, 538; (c) ibid., Adv. Quantum Chem., 1972, 6, 1. 18. E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev. A, 1988, 37, 2805; ibid., 37, 2809; ibid., 37, 2821; M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett., 1996, 76, 1212; A. Nagy, Chem. Phys. Lett., 1998, 296, 489; A. Görling, Phys. Rev. A, 1999, 59, 3359. 19. J. C. Slater, and J. H. Wood, Int. J. Quantum Chem. Symp., 1971, 4, 3.


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161

20. R. M. Dreizler, and E. K. U. Gross, Density Functional Theory: An Approach to the Quantum Many-Body Problem, Berlin: Springer, 1990; E. S. Kryachko, and E. V. Ludeña, Energy Density Functional Theory of Many-Electron Systems, Dordrecht: Kluwer Academic, 1990. 21. E. J. Baerendes, D. E. Ellis, and P. Ros, Chem. Phys., 1973, 2, 41; E. J. Baerends, and P. Ros, Chem. Phys., 1973, 2, 52; ibid, Chem. Phys., 1975, 8, 412; M. Trsic, T. Ziegler, and W. G. Laidlaw, Chem. Phys., 1976, 15, 383; T. Ziegler, A. Rauk, and E. J. Baerends, 1976, 16, 209; T. Ziegler, A. Rauk, and E. J. Baerends, Theoret. Chim. Acta (Berlin), 1977, 43, 261. 22. M. Trsic, W. F. D. Angelotti, and F. A. Molfetta, Adv. Quantum Chem., 2004, 47, 315. 23. K. Capelle, J. Chem. Phys., 2003, 119, 1285. 24. A. J. Takkar, and V. H. Smith, Phys. Rev. A, 1977, 15, 1; ibid., Phys. Rev. A, 1977, 15, 16.


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Final Remarks and Perspectives

Along the pages of this book, we can witness an intimate connection between chemistry and physics through the generator coordinate (GC) ansatz. Of course, this interaction is present since the birth of both sciences and occurs in several areas, although physics and chemistry are certainly distinct disciplines. One area that is borderline indeed is Quantum Chemistry. Actually, quantum chemists do not feel at home (neither are they particularly welcome) in either community. When one of us (MT) was a post-doctoral fellow in the Quantum Chemistry group in Uppsala in the early seventies, the late Professor Per-Olov Löwdin used to tell us how Professor John C. Slater died without ever being honored with the Nobel Prize. Per-Olov (Pelle) had the clear notion that Slater was one (I mention briefly: Slater determinants, Slater orbitals, Slater’s rules, Xα method of the builders of Quantum Chemistry in the thirties, as soon as Quantum Mechanics was well established). Pelle was a member of the Nobel Committee in Stockholm and would present Slater’s achievements to the Chemistry Committee. They would say: “well, this is really interesting, but this is physics”. So, the following year, Pelle would present his candidate to the Physics Committee and they said: “we can see the relevance of his work, but this is chemistry”. After 10 years, Pelle gave up. However, since mathematical intricacies of the GC algorithm require inevitable approximate schemes for nontrivial systems, both communities are clearly differentiated by their preferences. The physicists preferred the Gaussian overlap approximation, thus simplifying the model and then applying powerful mathematical tools. The chemists opted to preserve the model and introduce simplifications in the algorithm (discretization techniques). Somebody once said: “the table of the chemist has four legs, and the table of the physicist has one leg or an infinite number of legs”. What is really interesting is that the GC method has become one important tool in the generation of accurate basis sets that have been currently used in relativistic and nonrelativistic calculations of atoms and molecules. The secret of the GC method in the generation of accurate basis sets lies on how one discretizes the integral equation of the method, i.e., the Griffin-Hill-Wheeler (GHW) equation. The innovation in discretizing the GHW equation through an integral discretization, instead of the usual variational discretization, allowed us to vary the space of the Slater- and Gaussian-function exponents in such a way that it was possible to obtain universal as well as atom-adapted basis sets of high quality.

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Final Remarks and Perspectives

It is for sure that the way of generating Slater- and Gaussian-function exponents employing the GC method was an innovation when compared to the traditional way of optimizing all of the function exponents. The manner in which the Slater- and Gaussian-function exponents are generated by using the GC method is also a step forward when we compare our methodology with the two other relevant methodologies developed to reach such a goal: the even-tempered and welltempered formulas. Certainly, the applications of the GC method in relativistic and nonrelativistic calculations for atomic and, mainly, molecular systems may gain new perspectives with further improvements or, perhaps, with new methodological insights. However, new analytical and numerical approaches, as exemplified in Chapter 3, may bring unexpected new possibilities. It is good to remember that new findings in Mathematics and numerical techniques require a time gap before being applied in chemistry and physics. At the end of Chapter 6, we also raise the need for a better understanding of the weight function itself. Also, adopting an optimistic perspective, the GC method may open a new vision of the elusive, and multiple, higher correction terms in DFT.

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APPENDIX SELECTED UNIVERSAL AND ATOM-ADAPTED SLATER AND GAUSSIAN BASIS SETS FOR ATOMIC AND MOLECULAR CALCULATIONS



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Appendix 1 Universal Gaussian basis set for the ground [1] and certain low-lying excited [2] states of the neutral atoms and positive ions from Hydrogen (Z ⫽ 1) through Xenon (Z ⫽ 54). Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S

0.036883 0.096328 0.251579 0.657047 1.716007 4.481689 11.704812 30.569415 79.838033 208.512710 544.571910 1422.256537 3714.502383 9701.152773 25336.466485 66171.160168 172818.985654 451350.735373

P

0.090718 0.236928 0.618783 1.616074 4.220696 11.023176 28.789191 75.188628 196.369875 512.858511 1339.430764 3498.186604

D

0.036883 0.096328 0.251579 0.657047 1.716007 4.481689 11.704812 30.569415 79.838033 208.512710 544.571910

References 1. A. B. F. da Silva, H. F. M. da Costa, and M. Trsic, Mol. Phys., 1989, 68, 433. 2. A. B. F. da Silva, and M. Trsic, Can. J. Chem., 1996, 74, 1526.



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Appendix 2 Universal Slater basis set for the ground [1] and certain low-lying excited [2] states of the neutral atoms and positive ions from Hydrogen (Z ⫽ 1) through Xenon (Z ⫽ 54). Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

1 2 3 4 5 6 7 8 9 10 11 12

S

0.516851 0.786628 1.197217 1.822119 2.773195 4.220696 6.423737 9.776680 14.879732 22.646380 34.466919 52.457326

P

1.000000 1.521962 2.316367 3.525421 5.365556 8.166170 12.428597 18.915846 28.789191 43.816042

D

0.516851 0.786628 1.197217 1.822119 2.773195 4.220696 6.423737 9.776680 14.879732 22.646380

References 1. A. B. F. da Silva, H. F. M. DaCosta, and M. Trsic, Mol. Phys., 1989, 68, 433. 2. A. B. F. da Silva, and M. Trsic, Can. J. Chem., 1996, 74, 1526.



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Appendix 3 Universal Gaussian basis set for the ground states [1] of the neutral atoms from Hydrogen (Z ⫽ 1) through Lawrence (Z ⫽ 103). Exp. #

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

0.02000046 0.03916390 0.07668877 0.15016809 0.29405160 0.57579706 1.12749685 2.20780763 4.32321786 8.46550778 16.57673158 32.45972208 63.56099525 124.46194419 243.71511946 477.23068961 934.48913473 1829.86962477 3583.15866841 7016.36109438 13739.08541667 26903.18607430 52680.46591150 103156.23885545 201995.35882388 395537.15256683 774520.95915273 1516627.98873367 2969784.65079438 5815282.94190191 11387194.58507857 22297831.73302219 43662492.66045550 85497697.18196450


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172 Z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Page 172

Appendix 3 Atom

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn

Configuration

1

1s 1s2 [[He]]2s1 [He]2s2 [He]2s22p1 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 [He]2s22p6 [Ne]3s1 [Ne]3s2 [Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6 [Ar]4s1 [Ar]4s2 [Ar]4s23d1 [Ar]4s23d2 [Ar]4s23d3 [Ar]4s13d5 [Ar]4s23d5 [Ar]4s23d6 [Ar]4s23d7 [Ar]4s23d8 [Ar]4s13d10 [Ar]4s23d10 [Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p3 [Ar]4s23d104p4 [Ar]4s23d104p5 [Ar]3s23d104p6 [Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s24d2 [Kr]5s14d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d7 [Kr]5s14d8 [Kr]4d10 [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 [Kr]5s24d105p2

State

2

S S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 D 3 F 4 F 7 S 6 S 5 D 4 F 3 F 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 D 3 F 6 D 7 S 6 S 5 F 4 F 1 S 2 S 1 S 2 P 3 P

1

UGBS size

20s 21s 25s 25s 25s15p 23s15p 23s16p 23s16p 24s16p 23s16p 27s16p 27s16p 27s19p 27s19p 27s19p 27s19p 27s19p 26s19p 29s19p 29s19p 30s20p14d 30s20p14d 30s20p14d 30s20p14d 30s20p15d 31s20p15d 31s20p15d 30s20p15d 30s20p15d 30s21p15d 30s22p14d 30s22p14d 30s23p15d 30s21p14d 30s21p14d 30s20p14d 30s20p14d 30s21p14d 30s22p18d 30s22p18d 31s23p18d 30s23p18d 32s22p17d 31s22p17d 32s21p17d 29s22p18d 31s21p17d 31s20p16d 30s22p16d 30s22p16d

Gaussian exponentsa s

p

d

2-21 3-23 1-25 1-25 2-26 3-25 3-25 4-26 4-27 4-26 1-27 1-27 2-28 2-28 3-29 3-29 3-29 4-29 1-29 1-29 1-30 1-30 1-30 1-30 1-30 1-31 1-31 2-31 2-31 2-31 2-31 2-31 3-32 3-32 3-32 3-32 1-30 1-30 1-30 1-30 1-31 2-31 1-32 1-31 1-32 4-32 1-31 2-32 2-31 2-31

2-16 2-16 3-18 3-18 3-18 3-18 4-19 4-19 2-20 2-20 2-20 2-20 3-21 3-21 3-21 3-21 3-22 3-22 3-22 4-23 4-23 4-23 4-23 4-23 4-23 4-24 2-23 2-23 2-23 2-22 3-23 3-22 3-22 3-23 3-24 3-24 3-25 3-25 3-24 4-25 4-24 4-25 4-24 5-24 2-23 2-23

3-16 3-16 3-16 3-16 3-17 3-17 3-17 3-17 3-17 4-18 4-17 4-17 5-18 5-18 5-18 5-18 5-18 5-18 2-19 2-19 2-19 2-19 3-19 3-19 3-19 3-20 3-19 4-19 4-19 4-19

f

(continued )


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Appendix 3 Z

51 52 53 54 55 56 57 59 60 61 62 63 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 94 95 98 99 100 101 102 103

Atom

Sb Te I Xe Cs Ba La Pr Nd Pm Sm Eu Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pu Am Cf Es Fm Md No Lr

Configuration

[Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6 [Xe]6s1 [Xe]6s2 [Xe]6s25d1 [Xe]6s24f3 [Xe]6s24f 4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13 [Xe]6s24f14 [Xe]6s24f145d1 [Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10 [Xe]6s24f145d106p1 [Xe]6s24f145d106p2 [Xe]6s24f145d106p3 [Xe]6s24f145d106p4 [Xe]6s24f145d106p5 [Xe]6s24f145d106p6 [Rn]7s1 [Rn]7s2 [Rn]7s26d1 [Rn]7s26d2 [Rn]7s25f6 [Rn]7s25f7 [Rn]7s25f10 [Rn]7s25f11 [Rn]7s25f12 [Rn]7s25f13 [Rn]7s25f14 [Rn]7s25f146d1

State

4

S P 2 P 1 S 2 S 1 S 2 D 4 I 5 I 6 H 7 F 8 S 6 H 5 I 4 I 3 H 2 F 1 S 2 D 3 F 4 F 5 D 6 S 5 D 4 F 3 D 2 S 1 S 2 P 3 P 4 S 3 P 2 P 1 S 2 S 1 S 2 D 3 F 7 F 8 S 5 I 4 I 3 H 2 F 1 S 2 D 3

UGBS size

30s22p16d 30s23p16d 30s22p16d 31s22p16d 32s22p15d 32s22p16d 32s24p19d 32s22p16d12f 32s22p16d12f 31s23p16d13f 33s22p16d12f 32s23p16d13f 33s23p16d13f 32s22p16d14f 32s23p16d14f 33s22p16d13f 32s22p17d14f 32s22p17d13f 32s23p19d12f 32s22p18d12f 31s23p19d13f 31s23p19d12f 31s23p19d12f 32s23p18d12f 32s22p18d13f 32s23p18d13f 32s22p18d12f 32s23p17d13f 32s23p18d13f 32s23p17d13f 32s25p17d12f 32s23p18d13f 32s24p18d13f 32s25p17d13f 33s24p18d14f 33s23p18d14f 33s24p20d13f 33s25p20d14f 33s25p19d15f 33s25p19d14f 33s25p19d15f 33s25p19d15f 33s25p19d15f 33s25p19d15f 32s25p19d15f 32s25p20d15f

173 Gaussian exponentsa s

p

d

f

3-32 3-32 3-32 3-33 1-32 1-32 1-32 1-32 1-32 1-31 1-33 1-32 1-33 1-32 1-32 1-33 1-32 1-32 1-32 2-33 2-32 2-32 2-32 1-32 2-33 2-33 2-33 2-33 2-33 2-33 2-33 2-33 3-34 2-33 1-33 1-33 1-33 1-33 1-33 1-33 1-33 1-33 1-33 1-33 2-33 2-33

2-23 2-24 3-24 3-24 3-24 3-24 3-26 3-24 3-24 3-25 3-24 3-25 3-25 3-24 3-25 4-25 4-25 4-25 4-26 4-25 4-26 4-26 4-26 4-26 3-24 4-26 4-25 4-26 2-24 2-24 2-26 3-25 3-26 2-26 3-26 3-25 3-26 3-27 3-27 3-27 3-27 3-27 3-27 3-27 3-27 3-27

4-19 4-19 4-19 4-19 5-19 5-20 2-20 5-20 5-20 5-20 5-20 5-20 5-20 5-20 5-20 5-20 5-21 5-21 2-20 3-20 3-21 3-21 3-21 3-20 4-21 3-20 4-21 4-20 4-21 4-20 4-20 4-21 4-21 5-21 4-21 4-21 3-22 3-22 5-23 5-23 5-23 5-23 5-23 5-23 5-23 3-22

4-15 4-15 4-16 4-15 4-16 4-16 4-17 4-17 4-16 4-17 4-16 5-16 5-16 5-17 6-17 6-17 6-17 6-18 6-18 6-17 6-18 6-18 6-18 7-18 6-18 6-18 7-19 7-20 5-18 7-19 7-20 4-18 4-17 4-18 4-18 4-18 4-18 4-18 5-19

a

These numbers refer to the universal Gaussian exponents labels (Exp. #) shown in the previous table of this Appendix.

References 1. E. V. R. de Castro, and F. E. Jorge, J. Chem. Phys., 1998, 108, 5225.



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Appendix 4 Relativistic Universal Gaussian basis set (RUGBS) for Hydrogen (Z ⫽ 1) through Nobelium (Z ⫽ 102) without variational prolapse and to be used with both uniform sphere and Gaussian nucleus models [1]. Exp. #

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

434627543.306 217998774.679 109342968.465 54843816.3026 27508345.7936 13797527.9497 6920509.83183 3471162.11738 1741051.84993 873269.942927 438011.305207 219695.988672 110194.250388 55270.7989432 27722.5100681 13904.9476246 6974.38897011 3498.18660376 1754.60668558 880.068724108 441.421411146 221.406416204 111.052159906 55.7011058268 27.9383417032 14.0132036077 7.02868758059 3.52542148737 1.76826705143 0.886920436717 0.444858066223 0.223130160148 0.111916748617 0.0561347628341 0.0281558536803 0.0141223024102


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Appendix 4

Atom Z

M


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

1 4 7 9 11 12 14 16 19 20 23 24 27 28 31 32 35 40 39 40 45

Ti

22

48

V

23

51

Cr

24

52

Mn

25

55

Fe

26

56

Co

27

59

Ni

28

58

Cu

29

63


30 31 32 33 34 35 36 37 38 39

64 69 74 75 80 79 84 85 88 89

Configuration

1s1 1s2 [He]2s1 [He]2s2 [He]2s22p1 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 [He]2s22p6 [Ne]3s1 [Ne]3s2 [Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6 [Ar]4s1 [Ar]4s2 [Ar]4s23d1 [Ar]4s13d2 [Ar]4s23d2 [Ar]4s13d3 [Ar]4s23d3 [Ar]4s13d4 [Ar]4s23d4 [Ar]4s13d5 [Ar]4s23d5 [Ar]4s13d6 [Ar]4s23d6 [Ar]4s13d7 [Ar]4s23d7 [Ar]4s13d8 [Ar]4s23d8 [Ar]4s13d9 [Ar]4s23d9 [Ar]4s13d10 [Ar]4s23d10 [Zn]4p1 [Zn]4p2 [Zn]4p3 [Zn]4p4 [Zn]4p5 [Zn]4p6 [Kr]5s1 [Kr]5s2 [Kr]5s24d1

RUGBS size

20s 19s 28s 27s 26s18p 26s17p 25s17p 25s17p 25s16p 24s16p 31s15p 30s15p 29s25p 29s25p 29s24p 28s24p 28s24p 28s23p 33s23p 32s23p 32s23p16d 32s23p17d 32s22p16d 32s23p16d 32s22p16d 32s22p16d 32s22p15d 32s22p16d 32s22p15d 32s22p16d 32s22p15d 32s22p16d 32s22p15d 32s22p16d 32s22p15d 32s22p15d 32s21p15d 32s22p15d 32s21p15d 31s29p14d 31s29p14d 31s28p14d 31s28p13d 30s28p13d 30s28p13d 34s27p13d 34s27p12d 33s27p20d


p

d

15-34 15-33 9-36 9-35 9-34 9-34 9-33 9-33 9-33 9-32 6-36 6-35 6-34 6-34 6-34 6-33 6-33 6-33 4-36 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-35 4-34 4-34 4-34 4-34 4-33 4-33 3-36 3-36 3-35

18-35 18-34 18-34 18-34 18-33 18-33 18-32 18-32 11-35 11-35 11-34 11-34 11-34 11-33 11-33 11-33 11-33 11-33 11-32 11-33 11-32 11-32 11-32 11-32 11-32 11-32 11-32 11-32 11-32 11-32 11-32 11-32 11-31 11-32 11-31 7-35 7-35 7-34 7-34 7-34 7-34 7-33 7-33 7-33

19-34 19-35 19-34 19-34 19-34 19-34 19-33 19-34 19-33 19-34 19-33 19-34 19-33 19-34 19-33 19-33 19-33 19-33 19-33 19-32 19-32 19-32 19-31 19-31 19-31 19-31 19-30 15-34

f

(continued )


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Appendix 4 Atom Z

M

Zr

40

90

Nb

41

93

Mo

42

98

Tc

43

98

Ru

44

102

Rh

45

103

Pd

46

106

Ag

47

107

Cd In Sn Sb Te I Xe Cs Ba La

48 49 50 51 52 53 54 55 56 57

114 115 120 121 130 127 132 133 138 139

Ce

58

140

Pr Nd Pm Sm Eu Gd

59 60 61 62 63 64

141 144 145 152 153 158

Tb 65 Dy 66 Ho 67 Er 68 69 Tm 70 Yb Lu 71 Hf 72

159 162 162 168 169 174 175 180

Ta

181

73

Configuration

[Kr]5s14d2 [Kr]5s24d2 [Kr]5s14d3 [Kr]5s24d3 [Kr]5s14d4 [Kr]5s24d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d6 [Kr]5s24d6 [Kr]5s14d7 [Kr]5s24d7 [Kr]5s14d8 [Kr]5s24d8 [Kr]5s14d9 [Kr]5s04d10 [Kr]5s24d9 [Kr]5s14d10 [Kr]5s24d10 [Cd]5p1 [Cd]5p2 [Cd]5p3 [Cd]5p4 [Cd]5p5 [Cd]5p6 [Xe]6s1 [Xe]6s2 [Xe]6s24f 1 [Xe]6s25d1 [Xe]6s24f 2 [Xe]6s24f 15d1 [Xe]6s24f 3 [Xe]6s24f 4 [Xe]6s24f 5 [Xe]6s24f 6 [Xe]6s24f 7 [Xe]6s24f 8 [Xe]6s24f 75d1 [Xe]6s24f 9 [Xe]6s24f 10 [Xe]6s24f 11 [Xe]6s24f 12 [Xe]6s24f 13 [Xe]6s24f 14 [Xe]6s24f 145d1 [Xe]6s24f 145d2 [Xe]6s14f 145d3 [Xe]6s24f 145d3

RUGBS size

33s27p21d 33s27p20d 33s27p21d 33s27p20d 33s27p20d 33s26p20d 33s26p20d 33s26p20d 33s26p20d 33s26p20d 33s26p20d 33s26p19d 33s26p20d 33s26p19d 33s26p19d 29s26p20d 33s26p19d 33s26p19d 33s26p19d 32s32p18d 32s32p18d 32s31p18d 32s31p18d 31s31p18d 31s31p18d 35s31p17d 35s30p17d 35s30p17d15f 35s30p20d 35s30p17d15f 35s30p20d14f 35s30p17d15f 35s30p17d15f 34s30p17d15f 34s30p17d15f 34s30p17d15f 34s30p16d15f 34s30p20d15f 34s30p16d15f 34s30p16d15f 34s30p16d15f 34s30p16d14f 34s30p16d14f 34s30p16d14f 34s29p22d14f 34s29p22d14f 34s29p22d14f 34s29p22d13f

177 Gaussian exponentsa s

p

d

3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-35 3-31 3-35 3-35 3-35 3-34 3-34 3-34 3-34 3-33 3-33 2-36 2-36 2-36 2-36 2-36 2-36 2-36 2-36 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35

7-33 7-33 7-33 7-33 7-33 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 7-32 4-35 4-35 4-34 4-34 4-34 4-34 4-34 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-32 4-32 4-32 4-32

15-35 15-34 15-35 15-34 15-34 15-34 15-34 15-34 15-34 15-34 15-34 15-33 15-34 15-33 15-33 15-34 15-33 15-33 15-33 15-32 15-32 15-32 15-32 15-32 15-32 15-31 15-31 15-31 15-34 15-31 15-34 15-31 15-31 15-31 15-31 15-31 15-30 15-34 15-30 15-30 15-30 15-30 15-30 15-30 13-34 13-34 13-34 13-34

f

19-33 19-33 19-32 19-33 19-33 19-33 19-33 19-33 19-33 19-33 19-33 19-33 19-33 19-32 19-32 19-32 19-32 19-32 19-32 19-31

(continued )


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Appendix 4

(Continued) Atom Z

M

W

74

184

Re

75

187

Os

76

192

Ir

77

193

Pt

78

195

Au

79

197

Hg Tl Pb Bi Po At Rn Fr Ra Ac Th

80 81 82 83 84 85 86 87 88 89 90

202 205 208 209 209 210 222 223 226 227 232

Pa

91

231

U

92

238

Np

93

237

Pu Am Cm

94 95 96

244 243 247

Bk Cf Es Fm Md No

97 98 99 100 101 102

247 251 252 257 258 259

Configuration

[Xe]6s14f 145d4 [Xe]6s24f 145d4 [Xe]6s14f 145d5 [Xe]6s24f 145d5 [Xe]6s14f 145d6 [Xe]6s24f 145d6 [Xe]6s14f 145d7 [Xe]6s24f 145d7 [Xe]6s14f 145d8 [Xe]6s24f 145d8 [Xe]6s14f 145d9 [Xe]6s24f 145d9 [Xe]6s14f 145d10 [Xe]6s24f 145d10 [Hg]6p1 [Hg]6p2 [Hg]6p3 [Hg]6p4 [Hg]6p5 [Hg]6p6 [Rn]7s1 [Rn]7s2 [Rn]7s26d1 [Rn]7s25f 2 [Rn]7s26d2 [Rn]7s25f 3 [Rn]7s25f 26d1 [Rn]7s25f 4 [Rn]7s25f 36d1 [Rn]7s25f 5 [Rn]7s25f 46d1 [Rn]7s25f 6 [Rn]7s25f 7 [Rn]7s25f 8 [Rn]7s25f 76d1 [Rn]7s25f 9 [Rn]7s25f 10 [Rn]7s25f 11 [Rn]7s25f 12 [Rn]7s25f 13 [Rn]7s25f 14

RUGBS size

34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p22d13f 34s29p21d13f 34s29p22d13f 34s29p21d13f 34s29p21d13f 34s29p21d13f 34s29p21d13f 33s29p21d13f 33s29p21d13f 33s29p21d12f 33s33p21d12f 33s33p20d12f 33s33p20d12f 32s32p20d12f 32s32p20d12f 32s32p20d11f 36s32p19d11f 36s31p19d11f 35s31p23d12f 35s31p19d17f 35s31p23d12f 35s31p19d17f 35s31p23d16f 35s31p19d16f 35s31p23d16f 35s31p19d16f 35s31p23d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p22d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f 35s31p19d16f


p

d

f

2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-35 2-34 2-34 2-34 2-34 2-34 2-34 2-33 2-33 2-33 1-36 1-36 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35 1-35

4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 4-32 3-35 3-35 3-35 3-34 3-34 3-34 3-34 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33 3-33

13-34 13-34 13-34 13-34 13-34 13-33 13-34 13-33 13-33 13-33 13-33 13-33 13-33 13-33 13-33 13-32 13-32 13-32 13-32 13-32 13-31 13-31 13-35 13-31 13-35 13-31 13-35 13-31 13-35 13-31 13-35 13-31 13-31 13-31 13-34 13-31 13-31 13-31 13-31 13-31 13-31

19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-31 19-30 19-30 19-30 19-30 19-30 19-30 19-29 19-29 19-29 18-29 18-34 18-29 18-34 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33 18-33

a

These numbers refer to the universal Gaussian exponents labels (Exp. #) shown in the previous table of this Appendix.

Reference 1. R. L. A. Haiduke, L. G. M. de Macedo, and A. B. F. da Silva, J. Comput. Chem., 2005, 26, 932.


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Appendix 5 Relativistic adapted Gaussian basis sets for Hydrogen (Z ⫽1) through Radon (Z⫽86) without variational prolapse and to be used with both uniform sphere and Gaussian nucleus models [1,2]. Atom

Exp. #

Symm.

Exponents

H

1 2 3 4 5 6 7 8

S⫹

0.4406498716E⫹03 0.6636101523E⫹02 0.1392971613E⫹02 0.3806525865E⫹01 0.1264794537E⫹01 0.4772690339E⫹00 0.1910319536E⫹00 0.7575227197E⫺01

Atom

Exp. #

Symm.

Exponents

He

1 2 3 4 5 6 7 8

S⫹

0.1070160908E⫹04 0.1816369667E⫹03 0.4009745741E⫹02 0.1091195927E⫹02 0.3469587274E⫹01 0.1221681357E⫹01 0.4515012930E⫹00 0.1659957021E⫹00

Atom

Exp. #

Symm.

Exponents

Li

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.1395544240E⫹05 0.2559250298E⫹04 0.5742911768E⫹03 0.1535679448E⫹03 0.4765592080E⫹02 0.1671397863E⫹02 0.6451910321E⫹01 0.2669568833E⫹01 0.1153020595E⫹01 0.5062615733E⫹00 0.2200663508E⫹00 0.9222993773E⫺01 0.3629347927E⫺01 0.1305937900E⫺01


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180

Appendix 5 Atom

Exp. #

Symm.

Exponents

Be

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.2592322398E⫹05 0.4466184818E⫹04 0.9665712381E⫹03 0.2547163426E⫹03 0.7922856412E⫹02 0.2819584147E⫹02 0.1112866072E⫹02 0.4722060844E⫹01 0.2087984916E⫹01 0.9326238210E⫹00 0.4078920936E⫹00 0.1693245671E⫹00 0.6467059123E⫺01 0.2202835804E⫺01

Atom Exp. # Symm. Exponents B

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

Symm. Exponents

0.6134113997E⫹05 P⫺ 0.1200911743E⫹05 0.2788380198E⫹04 0.7525280608E⫹03 0.2313509990E⫹03 0.7940469679E⫹02 0.2981915260E⫹02 0.1200789433E⫹02 0.5081700112E⫹01 0.2214980887E⫹01 0.9745359636E⫹00 0.4241708856E⫹00 0.1789975554E⫹00 0.7177360550E⫺01

Atom Exp. # Symm. Exponents C

Page 180

0.7833526734E⫹02 P⫹ 0.1953316458E⫹02 0.6090838463E⫹01 0.2254806069E⫹01 0.9408218567E⫹00 0.4200588497E⫹00 0.1905267086E⫹00 0.8334566678E⫺01 0.3338336169E⫺01

Symm. Exponents

0.8820854459E⫹05 P⫺ 0.1699873764E⫹05 0.3920047431E⫹04 0.1058495674E⫹04 0.3274646663E⫹03 0.1135721574E⫹03 0.4320816818E⫹02 0.1764415487E⫹02 0.7567133099E⫹01 0.3335123101E⫹01 0.1478076207E⫹01 0.6445264189E⫹00 0.2705814623E⫹00 0.1070097259E⫹00

Symm. Exponents 0.7833526734E⫹02 0.1953316458E⫹02 0.6090838463E⫹01 0.2254806069E⫹01 0.9408218567E⫹00 0.4200588497E⫹00 0.1905267086E⫹00 0.8334566678E⫺01 0.3338336169E⫺01

Symm. Exponents

0.1320504565E⫹03 P⫹ 0.3193206288E⫹02 0.9801835374E⫹01 0.3597672662E⫹01 0.1487339650E⫹01 0.6523991511E⫹00 0.2860037943E⫹00 0.1180390844E⫹00 0.4320320382E⫺01

0.1320504565E⫹03 0.3193206288E⫹02 0.9801835374E⫹01 0.3597672662E⫹01 0.1487339650E⫹01 0.6523991511E⫹00 0.2860037943E⫹00 0.1180390844E⫹00 0.4320320382E⫺01


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Appendix 5 Atom Exp. # Symm. Exponents N

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹


Atom Exp. # Symm. Exponents O

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.1859743474E⫹03 0.4421259099E⫹02 0.1344192809E⫹02 0.4909573736E⫹01 0.2023655418E⫹01 0.8842678572E⫹00 0.3847951163E⫹00 0.1566453985E⫹00 0.5603921587E⫺01

Symm. Exponents

0.3258352357E⫹03 P⫹ 0.7860287126E⫹02 0.2387824255E⫹02 0.8657032665E⫹01 0.3549912629E⫹01 0.1560369119E⫹01 0.6967504630E⫹00 0.2995343169E⫹00 0.1174936697E⫹00

Symm. Exponents


Symm. Exponents

0.1859743474E⫹03 P⫹ 0.4421259099E⫹02 0.1344192809E⫹02 0.4909573736E⫹01 0.2023655418E⫹01 0.8842678572E⫹00 0.3847951163E⫹00 0.1566453985E⫹00 0.5603921587E⫺01

Symm. Exponents


Atom Exp. # Symm. Exponents F

Symm. Exponents

181

0.3258352357E⫹03 0.7860287126E⫹02 0.2387824255E⫹02 0.8657032665E⫹01 0.3549912629E⫹01 0.1560369119E⫹01 0.6967504630E⫹00 0.2995343169E⫹00 0.1174936697E⫹00

Symm. Exponents

0.4237660227E⫹03 P⫹ 0.1076258801E⫹03 0.3320498487E⫹02 0.1190615440E⫹02 0.4746866855E⫹01 0.2013240100E⫹01 0.8690059309E⫹00 0.3652365399E⫹00 0.1429998806E⫹00

0.4237660227E⫹03 0.1076258801E⫹03 0.3320498487E⫹02 0.1190615440E⫹02 0.4746866855E⫹01 0.2013240100E⫹01 0.8690059309E⫹00 0.3652365399E⫹00 0.1429998806E⫹00


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Atom Exp. # Symm. Exponents

Symm. Exponents

Ne

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹


Atom Exp. # Symm. Exponents Na

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S⫹

Atom Exp. # Symm. Exponents Mg

1 2 3 4 5 6 7

S⫹

0.5189110622E⫹03 P⫹ 0.1326904158E⫹03 0.4125911242E⫹02 0.1490360466E⫹02 0.5974633793E⫹01 0.2539435597E⫹01 0.1093267587E⫹01 0.4554456020E⫹00 0.1753980410E⫹00

Symm. Exponents

0.9043548154E⫹06 P⫺ 0.1639818742E⫹06 0.3545358381E⫹05 0.8977015057E⫹04 0.2614637520E⫹04 0.8603971384E⫹03 0.3141915985E⫹03 0.1250542711E⫹03 0.5328574608E⫹02 0.2387439317E⫹02 0.1104746440E⫹02 0.5185636456E⫹01 0.2425218762E⫹01 0.1109965236E⫹01 0.4882897963E⫹00 0.2027943936E⫹00 0.7809882000E⫺01 0.2739324245E⫺01

0.5189110622E⫹03 0.1326904158E⫹03 0.4125911242E⫹02 0.1490360466E⫹02 0.5974633793E⫹01 0.2539435597E⫹01 0.1093267587E⫹01 0.4554456020E⫹00 0.1753980410E⫹00

Symm. Exponents

0.6831690212E⫹03 P⫹ 0.1599059339E⫹03 0.4805891231E⫹02 0.1745660153E⫹02 0.7213156019E⫹01 0.3191348865E⫹01 0.1423016805E⫹01 0.6019161957E⫹00 0.2273294727E⫹00

Symm. Exponents

0.1133047393E⫹07 P⫺ 0.2092484084E⫹06 0.4577626972E⫹05 0.1166347144E⫹05 0.3403059182E⫹04 0.1117919316E⫹04 0.4065329498E⫹03

Symm. Exponents

0.6831690212E⫹03 0.1599059339E⫹03 0.4805891231E⫹02 0.1745660153E⫹02 0.7213156019E⫹01 0.3191348865E⫹01 0.1423016805E⫹01 0.6019161957E⫹00 0.2273294727E⫹00

Symm. Exponents

0.9961517770E⫹03 P⫹ 0.2461311345E⫹03 0.7500780754E⫹02 0.2691304102E⫹02 0.1085307903E⫹02 0.4695637400E⫹01 0.2080678800E⫹01

0.9961517770E⫹03 0.2461311345E⫹03 0.7500780754E⫹02 0.2691304102E⫹02 0.1085307903E⫹02 0.4695637400E⫹01 0.2080678800E⫹01 (continued )


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Appendix 5 Atom Exp. # Symm. Exponents 8 9 10 11 12 13 14 15 16 17 18

0.1609053265E⫹03 0.6815214285E⫹02 0.3037162862E⫹02 0.1400171272E⫹02 0.6565439559E⫹01 0.3078655265E⫹01 0.1419443040E⫹01 0.6326734047E⫹00 0.2680346462E⫹00 0.1061199614E⫹00 0.3860495681E⫺01

Atom Exp. # Symm. Exponents Al

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S⫹

1 2 3 4 5 6 7 8 9 10

S⫹

Symm. Exponents

0.9013675903E⫹00 0.3644204564E⫹00

Symm. Exponents

0.1508491365E⫹07 P⫺ 0.2907088879E⫹06 0.6546164074E⫹05 0.1696583787E⫹05 0.4985059449E⫹04 0.1635756778E⫹04 0.5904283831E⫹03 0.2309205537E⫹03 0.9639434861E⫹02 0.4230395085E⫹02 0.1922637514E⫹02 0.8913489892E⫹01 0.4152204892E⫹01 0.1914419743E⫹01 0.8605372931E⫹00 0.3714694688E⫹00 0.1516855479E⫹00 0.5771380254E⫺01

Atom Exp. # Symm. Exponents Si

Symm. Exponents

183

Symm. Exponents

0.2392158852E⫹04 P⫹ 0.6102448199E⫹03 0.1856099736E⫹03 0.6541840254E⫹02 0.2596679062E⫹02 0.1128170215E⫹02 0.5214195289E⫹01 0.2491578429E⫹01 0.1196340306E⫹01 0.5609783590E⫹00 0.2496703544E⫹00 0.1025027795E⫹00 0.3772857455E⫺01

Symm. Exponents

0.2198070137E⫹07 P⫺ 0.4140071972E⫹06 0.9148390725E⫹05 0.2335338346E⫹05 0.6781426055E⫹04 0.2205749493E⫹04 0.7913217553E⫹03 0.3083256150E⫹03 0.1284764456E⫹03 0.5637579098E⫹02

0.9013675903E⫹00 0.3644204564E⫹00

0.2392158852E⫹04 0.6102448199E⫹03 0.1856099736E⫹03 0.6541840254E⫹02 0.2596679062E⫹02 0.1128170215E⫹02 0.5214195289E⫹01 0.2491578429E⫹01 0.1196340306E⫹01 0.5609783590E⫹00 0.2496703544E⫹00 0.1025027795E⫹00 0.3772857455E⫺01

Symm. Exponents

0.2946762025E⫹04 P⫹ 0.7186607004E⫹03 0.2127286194E⫹03 0.7408913388E⫹02 0.2943154513E⫹02 0.1292721388E⫹02 0.6086021892E⫹01 0.2977159949E⫹01 0.1466946534E⫹01 0.7057863265E⫹00

2.95E⫹03 0.7186607004E⫹03 0.2127286194E⫹03 0.7408913388E⫹02 0.2943154513E⫹02 0.1292721388E⫹02 0.6086021892E⫹01 0.2977159949E⫹01 0.1466946534E⫹01 0.7057863265E⫹00 (continued )


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Appendix 5


Symm. Exponents

11 12 13 14 15 16 17 18

0.2565162221E⫹02 0.1191757507E⫹02 0.5566851736E⫹01 0.2574411301E⫹01 0.1160621348E⫹01 0.5022793711E⫹00 0.2054660716E⫹00 0.7822988729E⫺01

Atom Exp. # Symm. Exponents P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S⫹

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.3214270411E⫹00 0.1343213887E⫹00 0.4993038409E⫺01

Symm. Exponents

0.3012000519E⫹07 P⫺ 0.5205900218E⫹06 0.1077827623E⫹06 0.2626653106E⫹05 0.7403606729E⫹04 0.2371692830E⫹04 0.8484676466E⫹03 0.3330896577E⫹03 0.1410013575E⫹03 0.6324249455E⫹02 0.2953286033E⫹02 0.1410912629E⫹02 0.6776104396E⫹01 0.3214642506E⫹01 0.1480284436E⫹01 0.6501369167E⫹00 0.2676078902E⫹00 0.1014412757E⫹00


Symm. Exponents

Symm. Exponents

0.3304909198E⫹04 P⫹ 0.8697861010E⫹03 0.2692129424E⫹03 0.9555093579E⫹02 0.3791879190E⫹02 0.1640512622E⫹02 0.7544582215E⫹01 0.3596220805E⫹01 0.1732359557E⫹01 0.8223094017E⫹00 0.3750274063E⫹00 0.1602309056E⫹00 0.6253311991E⫺01

Symm. Exponents


0.3214270411E⫹00 0.1343213887E⫹00 0.4993038409E⫺01

0.3304909198E⫹04 0.8697861010E⫹03 0.2692129424E⫹03 0.9555093579E⫹02 0.3791879190E⫹02 0.1640512622E⫹02 0.7544582215E⫹01 0.3596220805E⫹01 0.1732359557E⫹01 0.8223094017E⫹00 0.3750274063E⫹00 0.1602309056E⫹00 0.6253311991E⫺01

Symm. Exponents

0.4498593800E⫹04 P⫹ 0.1135017372E⫹04 0.3408587880E⫹03 0.1185739666E⫹03 0.4649900658E⫹02 0.2000479394E⫹02 0.9188765715E⫹01 0.4385403624E⫹01 0.2116351828E⫹01 0.1005052331E⫹01 0.4570972433E⫹00 0.1937511351E⫹00 0.7448901695E⫺01

0.4498593800E⫹04 0.1135017372E⫹04 0.3408587880E⫹03 0.1185739666E⫹03 0.4649900658E⫹02 0.2000479394E⫹02 0.9188765715E⫹01 0.4385403624E⫹01 0.2116351828E⫹01 0.1005052331E⫹01 0.4570972433E⫹00 0.1937511351E⫹00 0.7448901695E⫺01 (continued )


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Appendix 5 Atom Exp. # Symm. Exponents 15 16 17 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S⫹

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Symm. Exponents

Symm. Exponents

Symm. Exponents


Atom Exp. # Symm. Exponents Ar

Symm. Exponents

0.1771359150E⫹01 0.7829354670E⫹00 0.3261400044E⫹00 0.1259076884E⫹00

Atom Exp. # Symm. Exponents Cl

185

S⫹

0.5437199854E⫹04 P⫹ 0.1341039060E⫹04 0.3971626299E⫹03 0.1373100290E⫹03 0.5387523555E⫹02 0.2332250840E⫹02 0.1082945494E⫹02 0.5243597096E⫹01 0.2573887239E⫹01 0.1245183385E⫹01 0.5771740891E⫹00 0.2492046580E⫹00 0.9743778804E⫺01

Symm. Exponents


0.5437199854E⫹04 0.1341039060E⫹04 0.3971626299E⫹03 0.1373100290E⫹03 0.5387523555E⫹02 0.2332250840E⫹02 0.1082945494E⫹02 0.5243597096E⫹01 0.2573887239E⫹01 0.1245183385E⫹01 0.5771740891E⫹00 0.2492046580E⫹00 0.9743778804E⫺01

Symm. Exponents

0.5945445814E⫹04 P⫹ 0.1486551983E⫹04 0.4445856390E⫹03 0.1546984168E⫹03 0.6091817359E⫹02 0.2640668595E⫹02 0.1225638527E⫹02 0.5924720148E⫹01 0.2901387045E⫹01 0.1400075122E⫹01 0.6475604412E⫹00 0.2792338335E⫹00 0.1091917884E⫹00

0.5945445814E⫹04 0.1486551983E⫹04 0.4445856390E⫹03 0.1546984168E⫹03 0.6091817359E⫹02 0.2640668595E⫹02 0.1225638527E⫹02 0.5924720148E⫹01 0.2901387045E⫹01 0.1400075122E⫹01 0.6475604412E⫹00 0.2792338335E⫹00 0.1091917884E⫹00


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Appendix 5


Symm. Exponents

K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

0.7637240356E⫹07 P⫺ 0.1526038293E⫹07 0.3495680665E⫹06 0.9072231089E⫹05 0.2636277785E⫹05 0.8476989267E⫹04 0.2980877831E⫹04 0.1132863241E⫹04 0.4598542871E⫹03 0.1970385145E⫹03 0.8807422351E⫹02 0.4058747349E⫹02 0.1905721321E⫹02 0.9010100407E⫹01 0.4239175679E⫹01 0.1961523106E⫹01 0.8821538921E⫹00 0.3810771220E⫹00 0.1562707432E⫹00 0.6011978808E⫺01 0.2144424942E⫺01

Atom Exp. # Symm. Exponents Ca

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

0.7444708484E⫹04 P⫹ 0.1853667521E⫹04 0.5515745035E⫹03 0.1908415772E⫹03 0.7470446460E⫹02 0.3219090139E⫹02 0.1485733827E⫹02 0.7146254066E⫹01 0.3485408888E⫹01 0.1677162453E⫹01 0.7747309688E⫹00 0.3342638142E⫹00 0.1310686266E⫹00

Symm. Exponents


Symm. Exponents 0.7444708484E⫹04 0.1853667521E⫹04 0.5515745035E⫹03 0.1908415772E⫹03 0.7470446460E⫹02 0.3219090139E⫹02 0.1485733827E⫹02 0.7146254066E⫹01 0.3485408888E⫹01 0.1677162453E⫹01 0.7747309688E⫹00 0.3342638142E⫹00 0.1310686266E⫹00

Symm. Exponents

0.9872550466E⫹04 P⫹ 0.2389227675E⫹04 0.6960528282E⫹03 0.2374438434E⫹03 0.9225527827E⫹02 0.3971098732E⫹02 0.1842030650E⫹02 0.8956272083E⫹01 0.4439964701E⫹01 0.2182886334E⫹01 0.1035284343E⫹01 0.4607259169E⫹00 0.1871361396E⫹00

0.9872550466E⫹04 0.2389227675E⫹04 0.6960528282E⫹03 0.2374438434E⫹03 0.9225527827E⫹02 0.3971098732E⫹02 0.1842030650E⫹02 0.8956272083E⫹01 0.4439964701E⫹01 0.2182886334E⫹01 0.1035284343E⫹01 0.4607259169E⫹00 0.1871361396E⫹00


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Appendix 5 Atom Exp. # Symm. Exponents Sc

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.1240018858E⫹05 P⫹ 0.3010337959E⫹04 0.8765356301E⫹03 0.2979657625E⫹03 0.1151014362E⫹03 0.4917992071E⫹02 0.2262365275E⫹02 0.1090637849E⫹02 0.5363106943E⫹01 0.2618457438E⫹01 0.1235500334E⫹01 0.5483825387E⫹00 0.2228655224E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.2071459179E⫹03 D⫹ 0.5540004700E⫹02 0.1829724487E⫹02 0.7112337932E⫹01 0.3100983718E⫹01 0.1445294371E⫹01 0.6862654930E⫹00 0.3163853328E⫹00 0.1349702816E⫹00 0.5077698400E⫺01

0.2071459179E⫹03 0.5540004700E⫹02 0.1829724487E⫹02 0.7112337932E⫹01 0.3100983718E⫹01 0.1445294371E⫹01 0.6862654930E⫹00 0.3163853328E⫹00 0.1349702816E⫹00 0.5077698400E⫺01

Atom Exp. # Symm. Exponents Sc

Symm. Exponents

187

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.1293904319E⫹08 P⫺ 0.2509083751E⫹07 0.5600815314E⫹06 0.1421782570E⫹06 0.4054933169E⫹05 0.1283587673E⫹05 0.4455346015E⫹04 0.1675228849E⫹04 0.6741032084E⫹03 0.2867879715E⫹03 0.1274384552E⫹03

0.1240018858E⫹05 0.3010337959E⫹04 0.8765356301E⫹03 0.2979657625E⫹03 0.1151014362E⫹03 0.4917992071E⫹02 0.2262365275E⫹02 0.1090637849E⫹02 0.5363106943E⫹01 0.2618457438E⫹01 0.1235500334E⫹01 0.5483825387E⫹00 0.2228655224E⫹00

Symm. Exponents

0.1183758983E⫹05 P⫹ 0.2790655680E⫹04 0.7991547895E⫹03 0.2699456738E⫹03 0.1044435874E⫹03 0.4494550478E⫹02 0.2088954339E⫹02 0.1018237212E⫹02 0.5054584303E⫹01 0.2481289948E⫹01 0.1169672582E⫹01

0.1183758983E⫹05 0.2790655680E⫹04 0.7991547895E⫹03 0.2699456738E⫹03 0.1044435874E⫹03 0.4494550478E⫹02 0.2088954339E⫹02 0.1018237212E⫹02 0.5054584303E⫹01 0.2481289948E⫹01 0.1169672582E⫹01 (continued )


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Appendix 5


Symm. Exponents

12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10

D⫺

0.5843429843E⫹02 0.2731402363E⫹02 0.1285809941E⫹02 0.6022320589E⫹01 0.2772488139E⫹01 0.1239415715E⫹01 0.5315309000E⫹00 0.2160364307E⫹00 0.8221199158E⫺01 0.2893848462E⫺01

0.5141442170E⫹00 0.2046343715E⫹00

0.1184181440E⫹03 D⫹ 0.3787208488E⫹02 0.1403033622E⫹02 0.5803339846E⫹01 0.2583224488E⫹01 0.1192706740E⫹01 0.5505613716E⫹00 0.2449015306E⫹00 0.1011822004E⫹00 0.3742456470E⫺01

0.1184181440E⫹03 0.3787208488E⫹02 0.1403033622E⫹02 0.5803339846E⫹01 0.2583224488E⫹01 0.1192706740E⫹01 0.5505613716E⫹00 0.2449015306E⫹00 0.1011822004E⫹00 0.3742456470E⫺01

Atom Exp. # Symm. Exponents Ti

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

Symm. Exponents


Symm. Exponents 0.5141442170E⫹00 0.2046343715E⫹00

Symm. Exponents

0.1535788547E⫹05 P⫹ 0.3507922979E⫹04 0.9810760917E⫹03 0.3259387686E⫹03 0.1247948607E⫹03 0.5342338463E⫹02 0.2480771611E⫹02 0.1212298366E⫹02 0.6048480364E⫹01 0.2989122518E⫹01 0.1419542861E⫹01 0.6285029537E⫹00 0.2516904992E⫹00

0.1535788547E⫹05 0.3507922979E⫹04 0.9810760917E⫹03 0.3259387686E⫹03 0.1247948607E⫹03 0.5342338463E⫹02 0.2480771611E⫹02 0.1212298366E⫹02 0.6048480364E⫹01 0.2989122518E⫹01 0.1419542861E⫹01 0.6285029537E⫹00 0.2516904992E⫹00

(continued )


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Appendix 5 Atom Exp. # Symm. Exponents 1 2 3 4 5 6 7 8 9 10

D⫺

0.2244215807E⫹03 D⫹ 0.6640364786E⫹02 0.2326283217E⫹02 0.9271507192E⫹01 0.4039512601E⫹01 0.1848724702E⫹01 0.8539922777E⫹00 0.3826025312E⫹00 0.1597451581E⫹00 0.5972653374E⫺01

Atom Exp. # Symm. Exponents Ti

Symm. Exponents

189 Symm. Exponents

0.2244215807E⫹03 0.6640364786E⫹02 0.2326283217E⫹02 0.9271507192E⫹01 0.4039512601E⫹01 0.1848724702E⫹01 0.8539922777E⫹00 0.3826025312E⫹00 0.1597451581E⫹00 0.5972653374E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.1438455617E⫹05 P⫹ 0.3398921375E⫹04 0.9722039038E⫹03 0.3271316006E⫹03 0.1258385883E⫹03 0.5377849982E⫹02 0.2481328939E⫹02 0.1201207721E⫹02 0.5929080027E⫹01 0.2899807958E⫹01 0.1365655618E⫹01 0.6018392293E⫹00 0.2411928524E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.1549566491E⫹03 D⫹ 0.4800141881E⫹02 0.1755931844E⫹02 0.7281011420E⫹01 0.3284936661E⫹01 0.1547869249E⫹01 0.7311968019E⫹00 0.3323898752E⫹00 0.1395711111E⫹00 0.5196348690E⫺01

0.1549566491E⫹03 0.4800141881E⫹02 0.1755931844E⫹02 0.7281011420E⫹01 0.3284936661E⫹01 0.1547869249E⫹01 0.7311968019E⫹00 0.3323898752E⫹00 0.1395711111E⫹00 0.5196348690E⫺01

0.1438455617E⫹05 0.3398921375E⫹04 0.9722039038E⫹03 0.3271316006E⫹03 0.1258385883E⫹03 0.5377849982E⫹02 0.2481328939E⫹02 0.1201207721E⫹02 0.5929080027E⫹01 0.2899807958E⫹01 0.1365655618E⫹01 0.6018392293E⫹00 0.2411928524E⫹00


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Appendix 5


Symm. Exponents

V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

0.1402990488E⫹08 P⫺ 0.2752073598E⫹07 0.6196134971E⫹06 0.1582486835E⫹06 0.4531277461E⫹05 0.1437690565E⫹05 0.4995480969E⫹04 0.1878707708E⫹04 0.7558124667E⫹03 0.3214731553E⫹03 0.1428742968E⫹03 0.6557603176E⫹02 0.3071996354E⫹02 0.1451722804E⫹02 0.6839699859E⫹01 0.3175291751E⫹01 0.1435576612E⫹01 0.6246951934E⫹00 0.2585901086E⫹00 0.1006376552E⫹00 0.3639291726E⫺01

0.1756100190E⫹05 P⫹ 0.4007694684E⫹04 0.1117964263E⫹04 0.3700293487E⫹03 0.1410608132E⫹03 0.6012091214E⫹02 0.2780870137E⫹02 0.1355060948E⫹02 0.6752225347E⫹01 0.3339888805E⫹01 0.1591846085E⫹01 0.7096449700E⫹00 0.2872362032E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.2406270350E⫹03 D⫹ 0.7257124067E⫹02 0.2640954068E⫹02 0.1109004373E⫹02 0.5139036698E⫹01 0.2513079153E⫹01 0.1240242310E⫹01 0.5907215086E⫹00 0.2596778040E⫹00 0.1007540746E⫹00

0.2406270350E⫹03 0.7257124067E⫹02 0.2640954068E⫹02 0.1109004373E⫹02 0.5139036698E⫹01 0.2513079153E⫹01 0.1240242310E⫹01 0.5907215086E⫹00 0.2596778040E⫹00 0.1007540746E⫹00

Atom Exp. # Symm. Exponents V

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.1491401970E⫹08 P⫺ 0.2777091106E⫹07 0.6017095180E⫹06 0.1497045927E⫹06 0.4220705068E⫹05 0.1330719978E⫹05 0.4630099109E⫹04 0.1754470165E⫹04 0.7145027866E⫹03 0.3086130285E⫹03 0.1395169989E⫹03

0.1756100190E⫹05 0.4007694684E⫹04 0.1117964263E⫹04 0.3700293487E⫹03 0.1410608132E⫹03 0.6012091214E⫹02 0.2780870137E⫹02 0.1355060948E⫹02 0.6752225347E⫹01 0.3339888805E⫹01 0.1591846085E⫹01 0.7096449700E⫹00 0.2872362032E⫹00

Symm. Exponents

0.1653084457E⫹05 P⫹ 0.3846914035E⫹04 0.1089220150E⫹04 0.3643091662E⫹03 0.1397466084E⫹03 0.5968905466E⫹02 0.2756105712E⫹02 0.1335704189E⫹02 0.6596346351E⫹01 0.3222855693E⫹01 0.1512474339E⫹01



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Appendix 5 Atom Exp. # Symm. Exponents 12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10

D⫺

0.6514673960E⫹02 0.3100709539E⫹02 0.1484507772E⫹02 0.7055162263E⫹01 0.3284617696E⫹01 0.1478311962E⫹01 0.6347474205E⫹00 0.2565898894E⫹00 0.9636781804E⫺01 0.3318398616E⫺01

0.6619274877E⫹00 0.2622850661E⫹00

0.1898619291E⫹03 D⫹ 0.5849779789E⫹02 0.2134578752E⫹02 0.8848950904E⫹01 0.3997737106E⫹01 0.1888058394E⫹01 0.8941929908E⫹00 0.4073780752E⫹00 0.1712578537E⫹00 0.6372732656E⫺01

0.1898619291E⫹03 0.5849779789E⫹02 0.2134578752E⫹02 0.8848950904E⫹01 0.3997737106E⫹01 0.1888058394E⫹01 0.8941929908E⫹00 0.4073780752E⫹00 0.1712578537E⫹00 0.6372732656E⫺01

Atom Exp. # Symm. Exponents Cr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

Symm. Exponents

Symm. Exponents


191 Symm. Exponents 0.6619274877E⫹00 0.2622850661E⫹00

Symm. Exponents

0.2220077820E⫹05 P⫹ 0.5081811896E⫹04 0.1413134320E⫹04 0.4639219732E⫹03 0.1747371519E⫹03 0.7338149296E⫹02 0.3339111986E⫹02 0.1599929132E⫹02 0.7844729963E⫹01 0.3825118654E⫹01 0.1802534786E⫹01 0.7977676079E⫹00 0.3222591911E⫹00

0.2220077820E⫹05 0.5081811896E⫹04 0.1413134320E⫹04 0.4639219732E⫹03 0.1747371519E⫹03 0.7338149296E⫹02 0.3339111986E⫹02 0.1599929132E⫹02 0.7844729963E⫹01 0.3825118654E⫹01 0.1802534786E⫹01 0.7977676079E⫹00 0.3222591911E⫹00

(continued )


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192

Appendix 5


Symm. Exponents

1 2 3 4 5 6 7 8 9 10

D⫺

0.2490047092E⫹03 D⫹ 0.7392762983E⫹02 0.2652191317E⫹02 0.1097510463E⫹02 0.5000634966E⫹01 0.2394755672E⫹01 0.1150598361E⫹01 0.5294435775E⫹00 0.2227183088E⫹00 0.8175971486E⫺01

Atom Exp. # Symm. Exponents Cr

Symm. Exponents

0.2490047092E⫹03 0.7392762983E⫹02 0.2652191317E⫹02 0.1097510463E⫹02 0.5000634966E⫹01 0.2394755672E⫹01 0.1150598361E⫹01 0.5294435775E⫹00 0.2227183088E⫹00 0.8175971486E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.1960747797E⫹05 P⫹ 0.4598427188E⫹04 0.1304135080E⫹04 0.4348087610E⫹03 0.1656811891E⫹03 0.7014295992E⫹02 0.3207507680E⫹02 0.1540142174E⫹02 0.7549186872E⫹01 0.3672168102E⫹01 0.1723315160E⫹01 0.7585136517E⫹00 0.3044075325E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.2075082306E⫹03 D⫹ 0.6325426617E⫹02 0.2308175730E⫹02 0.9645567899E⫹01 0.4415927640E⫹01 0.2118881346E⫹01 0.1019382562E⫹01 0.4704017897E⫹00 0.1991852906E⫹00 0.7403846807E⫺01

0.2075082306E⫹03 0.6325426617E⫹02 0.2308175730E⫹02 0.9645567899E⫹01 0.4415927640E⫹01 0.2118881346E⫹01 0.1019382562E⫹01 0.4704017897E⫹00 0.1991852906E⫹00 0.7403846807E⫺01

0.1960747797E⫹05 0.4598427188E⫹04 0.1304135080E⫹04 0.4348087610E⫹03 0.1656811891E⫹03 0.7014295992E⫹02 0.3207507680E⫹02 0.1540142174E⫹02 0.7549186872E⫹01 0.3672168102E⫹01 0.1723315160E⫹01 0.7585136517E⫹00 0.3044075325E⫹00


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Appendix 5 Atom Exp. # Symm. Exponents Mn

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.2640361549E⫹05 P⫹ 0.5979786067E⫹04 0.1645393078E⫹04 0.5347202165E⫹03 0.1995113505E⫹03 0.8308140382E⫹02 0.3753586296E⫹02 0.1788572265E⫹02 0.8737661868E⫹01 0.4254257562E⫹01 0.2006796222E⫹01 0.8915482823E⫹00 0.3626266393E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.2444807265E⫹03 D⫹ 0.7586102116E⫹02 0.2814111223E⫹02 0.1192503244E⫹02 0.5515956573E⫹01 0.2661167960E⫹01 0.1279561406E⫹01 0.5859149762E⫹00 0.2441405426E⫹00 0.8845538415E⫺01

0.2444807265E⫹03 0.7586102116E⫹02 0.2814111223E⫹02 0.1192503244E⫹02 0.5515956573E⫹01 0.2661167960E⫹01 0.1279561406E⫹01 0.5859149762E⫹00 0.2441405426E⫹00 0.8845538415E⫺01

Atom Exp. # Symm. Exponents Mn

Symm. Exponents

193

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.1791899293E⫹08 P⫺ 0.3329567732E⫹07 0.7205581515E⫹06 0.1791931382E⫹06 0.5052532380E⫹05 0.1593662884E⫹05 0.5548143246E⫹04 0.2103424654E⫹04 0.8568377133E⫹03 0.3700213119E⫹03 0.1671381871E⫹03

0.2640361549E⫹05 0.5979786067E⫹04 0.1645393078E⫹04 0.5347202165E⫹03 0.1995113505E⫹03 0.8308140382E⫹02 0.3753586296E⫹02 0.1788572265E⫹02 0.8737661868E⫹01 0.4254257562E⫹01 0.2006796222E⫹01 0.8915482823E⫹00 0.3626266393E⫹00

Symm. Exponents

0.2243162332E⫹05 P⫹ 0.5237608246E⫹04 0.1478280948E⫹04 0.4904219636E⫹03 0.1859555361E⫹03 0.7836315006E⫹02 0.3568745151E⫹02 0.1707880055E⫹02 0.8351696106E⫹01 0.4057932227E⫹01 0.1904956178E⫹01



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Appendix 5


Symm. Exponents

12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10

D⫺

0.7791295063E⫹02 0.3698221034E⫹02 0.1763557228E⫹02 0.8336120096E⫹01 0.3853721384E⫹01 0.1719106728E⫹01 0.7301225742E⫹00 0.2912885100E⫹00 0.1077084572E⫹00 0.3641995144E⫺01

0.8401417904E⫹00 0.3384892564E⫹00

0.2433238431E⫹03 D⫹ 0.7579631204E⫹02 0.2787053110E⫹02 0.1160691855E⫹02 0.5252957790E⫹01 0.2478831054E⫹01 0.1170271823E⫹01 0.5303519034E⫹00 0.2213705474E⫹00 0.8165722745E⫺01

0.2433238431E⫹03 0.7579631204E⫹02 0.2787053110E⫹02 0.1160691855E⫹02 0.5252957790E⫹01 0.2478831054E⫹01 0.1170271823E⫹01 0.5303519034E⫹00 0.2213705474E⫹00 0.8165722745E⫺01

Atom Exp. # Symm. Exponents Fe

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

Symm. Exponents



Symm. Exponents

0.2828520028E⫹05 P⫹ 0.6195586774E⫹04 0.1674435613E⫹04 0.5411784202E⫹03 0.2027317113E⫹03 0.8531694325E⫹02 0.3909349157E⫹02 0.1890394278E⫹02 0.9349799015E⫹01 0.4584344982E⫹01 0.2159735819E⫹01 0.9475354796E⫹00 0.3752191784E⫹00

0.2828520028E⫹05 0.6195586774E⫹04 0.1674435613E⫹04 0.5411784202E⫹03 0.2027317113E⫹03 0.8531694325E⫹02 0.3909349157E⫹02 0.1890394278E⫹02 0.9349799015E⫹01 0.4584344982E⫹01 0.2159735819E⫹01 0.9475354796E⫹00 0.3752191784E⫹00

(continued )


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D⫺

0.3194775408E⫹03 D⫹ 0.9778159372E⫹02 0.3573014347E⫹02 0.1493886511E⫹02 0.6849359735E⫹01 0.3300473109E⫹01 0.1601919176E⫹01 0.7505636161E⫹00 0.3253578201E⫹00 0.1250564662E⫹00

Atom Exp. # Symm. Exponents Fe

Symm. Exponents

195 Symm. Exponents

0.3194775408E⫹03 0.9778159372E⫹02 0.3573014347E⫹02 0.1493886511E⫹02 0.6849359735E⫹01 0.3300473109E⫹01 0.1601919176E⫹01 0.7505636161E⫹00 0.3253578201E⫹00 0.1250564662E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.2640939689E⫹05 P⫹ 0.5946635945E⫹04 0.1639314127E⫹04 0.5368288617E⫹03 0.2026269755E⫹03 0.8553646167E⫹02 0.3918346140E⫹02 0.1889977775E⫹02 0.9313606019E⫹01 0.4549788181E⫹01 0.2137871158E⫹01 0.9375486155E⫹00 0.3723341475E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.2654543806E⫹03 D⫹ 0.8473804319E⫹02 0.3163606712E⫹02 0.1327715654E⫹02 0.6020730666E⫹01 0.2835423059E⫹01 0.1332950691E⫹01 0.6012302975E⫹00 0.2500928033E⫹00 0.9221426338E⫺01

0.2654543806E⫹03 0.8473804319E⫹02 0.3163606712E⫹02 0.1327715654E⫹02 0.6020730666E⫹01 0.2835423059E⫹01 0.1332950691E⫹01 0.6012302975E⫹00 0.2500928033E⫹00 0.9221426338E⫺01

0.2640939689E⫹05 0.5946635945E⫹04 0.1639314127E⫹04 0.5368288617E⫹03 0.2026269755E⫹03 0.8553646167E⫹02 0.3918346140E⫹02 0.1889977775E⫹02 0.9313606019E⫹01 0.4549788181E⫹01 0.2137871158E⫹01 0.9375486155E⫹00 0.3723341475E⫹00


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196

Appendix 5


Symm. Exponents

Co

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.3203769680E⫹05 P⫹ 0.6845355291E⫹04 0.1822402477E⫹04 0.5849759246E⫹03 0.2190830997E⫹03 0.9263790963E⫹02 0.4279661287E⫹02 0.2090271158E⫹02 0.1044480251E⫹02 0.5166947023E⫹01 0.2448700965E⫹01 0.1075814170E⫹01 0.4240043663E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.3628838287E⫹03 D⫹ 0.1083900092E⫹03 0.3921064543E⫹02 0.1640871582E⫹02 0.7586890666E⫹01 0.3701983265E⫹01 0.1820745459E⫹01 0.8621263972E⫹00 0.3753724008E⫹00 0.1435443773E⫹00

0.3628838287E⫹03 0.1083900092E⫹03 0.3921064543E⫹02 0.1640871582E⫹02 0.7586890666E⫹01 0.3701983265E⫹01 0.1820745459E⫹01 0.8621263972E⫹00 0.3753724008E⫹00 0.1435443773E⫹00

Atom Exp. # Symm. Exponents Co

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.1792331541E⫹08 P⫺ 0.3440895104E⫹07 0.7622066307E⫹06 0.1924165451E⫹06 0.5467649925E⫹05 0.1727300042E⫹05 0.5991890379E⫹04 0.2254286577E⫹04 0.9084975869E⫹03 0.3873718124E⫹03 0.1726004942E⫹03

0.3203769680E⫹05 0.6845355291E⫹04 0.1822402477E⫹04 0.5849759246E⫹03 0.2190830997E⫹03 0.9263790963E⫹02 0.4279661287E⫹02 0.2090271158E⫹02 0.1044480251E⫹02 0.5166947023E⫹01 0.2448700965E⫹01 0.1075814170E⫹01 0.4240043663E⫹00

Symm. Exponents

0.3190811065E⫹05 P⫹ 0.7052859632E⫹04 0.1914261061E⫹04 0.6188431971E⫹03 0.2311401370E⫹03 0.9675137782E⫹02 0.4402483687E⫹02 0.2112364767E⫹02 0.1036674450E⫹02 0.5047658368E⫹01 0.2365284295E⫹01



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D⫺

0.7937541947E⫹02 0.3721175975E⫹02 0.1756486261E⫹02 0.8245174982E⫹01 0.3801594276E⫹01 0.1700445115E⫹01 0.7288038124E⫹00 0.2956174136E⫹00 0.1120833100E⫹00 0.3923409484E⫺01

0.1034652083E⫹01 0.4098214418E⫹00

0.3090261061E⫹03 D⫹ 0.9633774175E⫹02 0.3556425757E⫹02 0.1489927154E⫹02 0.6788414265E⫹01 0.3223603836E⫹01 0.1528984298E⫹01 0.6941773388E⫹00 0.2891091886E⫹00 0.1058511118E⫹00

0.3090261061E⫹03 0.9633774175E⫹02 0.3556425757E⫹02 0.1489927154E⫹02 0.6788414265E⫹01 0.3223603836E⫹01 0.1528984298E⫹01 0.6941773388E⫹00 0.2891091886E⫹00 0.1058511118E⫹00

Atom Exp. # Symm. Exponents Ni

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

Symm. Exponents

Symm. Exponents


197 Symm. Exponents 0.1034652083E⫹01 0.4098214418E⫹00

Symm. Exponents

0.3571255933E⫹05 P⫹ 0.7736220974E⫹04 0.2071601623E⫹04 0.6646523954E⫹03 0.2476482890E⫹03 0.1038651818E⫹03 0.4752695668E⫹02 0.2299783020E⫹02 0.1140649010E⫹02 0.5620526121E⫹01 0.2666878078E⫹01 0.1181060299E⫹01 0.4731786813E⫹00

0.3571255933E⫹05 0.7736220974E⫹04 0.2071601623E⫹04 0.6646523954E⫹03 0.2476482890E⫹03 0.1038651818E⫹03 0.4752695668E⫹02 0.2299783020E⫹02 0.1140649010E⫹02 0.5620526121E⫹01 0.2666878078E⫹01 0.1181060299E⫹01 0.4731786813E⫹00

(continued )


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198

Appendix 5


Symm. Exponents

1 2 3 4 5 6 7 8 9 10

D⫺

0.4325647814E⫹03 D⫹ 0.1295554220E⫹03 0.4650796008E⫹02 0.1917448069E⫹02 0.8699652636E⫹01 0.4162167525E⫹01 0.2012027458E⫹01 0.9416777586E⫹00 0.4088670813E⫹00 0.1578086348E⫹00

Atom Exp. # Symm. Exponents Ni

Symm. Exponents

0.4325647814E⫹03 0.1295554220E⫹03 0.4650796008E⫹02 0.1917448069E⫹02 0.8699652636E⫹01 0.4162167525E⫹01 0.2012027458E⫹01 0.9416777586E⫹00 0.4088670813E⫹00 0.1578086348E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.3607302845E⫹05 P⫹ 0.7979958513E⫹04 0.2162495751E⫹04 0.6966899705E⫹03 0.2589682887E⫹03 0.1077877520E⫹03 0.4875297333E⫹02 0.2325598465E⫹02 0.1135437817E⫹02 0.5506555771E⫹01 0.2574410868E⫹01 0.1126029777E⫹01 0.4471869836E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.3421361829E⫹03 D⫹ 0.1097856105E⫹03 0.4080635851E⫹02 0.1694082296E⫹02 0.7574494428E⫹01 0.3517009330E⫹01 0.1635243207E⫹01 0.7341222552E⫹00 0.3068455783E⫹00 0.1151396836E⫹00

0.3421361829E⫹03 0.1097856105E⫹03 0.4080635851E⫹02 0.1694082296E⫹02 0.7574494428E⫹01 0.3517009330E⫹01 0.1635243207E⫹01 0.7341222552E⫹00 0.3068455783E⫹00 0.1151396836E⫹00

0.3607302845E⫹05 0.7979958513E⫹04 0.2162495751E⫹04 0.6966899705E⫹03 0.2589682887E⫹03 0.1077877520E⫹03 0.4875297333E⫹02 0.2325598465E⫹02 0.1135437817E⫹02 0.5506555771E⫹01 0.2574410868E⫹01 0.1126029777E⫹01 0.4471869836E⫹00


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Appendix 5 Atom Exp. # Symm. Exponents Cu

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.4404416285E⫹05 P⫹ 0.9514682989E⫹04 0.2530325972E⫹04 0.8035908742E⫹03 0.2956446863E⫹03 0.1222311516E⫹03 0.5508973202E⫹02 0.2625648174E⫹02 0.1283747781E⫹02 0.6245976813E⫹01 0.2933585179E⫹01 0.1290253058E⫹01 0.5155005544E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.4424892568E⫹03 D⫹ 0.1353235337E⫹03 0.4935411966E⫹02 0.2058381547E⫹02 0.9413554696E⫹01 0.4526678296E⫹01 0.2194707924E⫹01 0.1028768984E⫹01 0.4470704163E⫹00 0.1727125326E⫹00

0.4424892568E⫹03 0.1353235337E⫹03 0.4935411966E⫹02 0.2058381547E⫹02 0.9413554696E⫹01 0.4526678296E⫹01 0.2194707924E⫹01 0.1028768984E⫹01 0.4470704163E⫹00 0.1727125326E⫹00

Atom Exp. # Symm. Exponents Cu

Symm. Exponents

199

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.2181663277E⫹08 P⫺ 0.4160661003E⫹07 0.9158663753E⫹06 0.2298271819E⫹06 0.6493434833E⫹05 0.2040120038E⫹05 0.7039623302E⫹04 0.2634874324E⫹04 0.1056553518E⫹04 0.4482791378E⫹03 0.1987628938E⫹03

0.4404416285E⫹05 0.9514682989E⫹04 0.2530325972E⫹04 0.8035908742E⫹03 0.2956446863E⫹03 0.1222311516E⫹03 0.5508973202E⫹02 0.2625648174E⫹02 0.1283747781E⫹02 0.6245976813E⫹01 0.2933585179E⫹01 0.1290253058E⫹01 0.5155005544E⫹00

Symm. Exponents

0.3903931661E⫹05 P⫹ 0.8698937845E⫹04 0.2366106282E⫹04 0.7630176900E⫹03 0.2833320087E⫹03 0.1176645633E⫹03 0.5307782098E⫹02 0.2525958274E⫹02 0.1231723665E⫹02 0.5977268485E⫹01 0.2803644498E⫹01



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200

Appendix 5


Symm. Exponents

12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10

D⫺

0.9096119828E⫹02 0.4243414308E⫹02 0.1993046468E⫹02 0.9308216164E⫹01 0.4269400208E⫹01 0.1899428574E⫹01 0.8095414384E⫹00 0.3264521557E⫹00 0.1230179995E⫹00 0.4278492406E⫺01

0.1234531470E⫹01 0.4956427569E⫹00

0.3994522561E⫹03 D⫹ 0.1260461873E⫹03 0.4662475556E⫹02 0.1942274783E⫹02 0.8753812800E⫹01 0.4100733516E⫹01 0.1918177447E⫹01 0.8607245350E⫹00 0.3559368766E⫹00 0.1303168031E⫹00

0.3994522561E⫹03 0.1260461873E⫹03 0.4662475556E⫹02 0.1942274783E⫹02 0.8753812800E⫹01 0.4100733516E⫹01 0.1918177447E⫹01 0.8607245350E⫹00 0.3559368766E⫹00 0.1303168031E⫹00

Atom Exp. # Symm. Exponents Zn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

Symm. Exponents



Symm. Exponents

0.4717383569E⫹05 P⫹ 0.1002957946E⫹05 0.2645049372E⫹04 0.8383017278E⫹03 0.3093345654E⫹03 0.1287549830E⫹03 0.5856702145E⫹02 0.2820601415E⫹02 0.1393399929E⫹02 0.6840712277E⫹01 0.3233436472E⫹01 0.1425643053E⫹01 0.5680502114E⫹00

0.4717383569E⫹05 0.1002957946E⫹05 0.2645049372E⫹04 0.8383017278E⫹03 0.3093345654E⫹03 0.1287549830E⫹03 0.5856702145E⫹02 0.2820601415E⫹02 0.1393399929E⫹02 0.6840712277E⫹01 0.3233436472E⫹01 0.1425643053E⫹01 0.5680502114E⫹00

(continued )


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D⫺

0.4896090922E⫹03 D⫹ 0.1506536280E⫹03 0.5524649362E⫹02 0.2313584152E⫹02 0.1060185332E⫹02 0.5093949595E⫹01 0.2459032731E⫹01 0.1142802660E⫹01 0.4899320994E⫹00 0.1856601591E⫹00

Atom Exp. # Symm. Exponents Ga

Symm. Exponents

201 Symm. Exponents

0.4896090922E⫹03 0.1506536280E⫹03 0.5524649362E⫹02 0.2313584152E⫹02 0.1060185332E⫹02 0.5093949595E⫹01 0.2459032731E⫹01 0.1142802660E⫹01 0.4899320994E⫹00 0.1856601591E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹


0.8866794085E⫹05 P⫹ 0.2207419386E⫹05 0.6359554354E⫹04 0.2083419469E⫹04 0.7626399619E⫹03 0.3065065821E⫹03 0.1328991064E⫹03 0.6108729953E⫹02 0.2924899880E⫹02 0.1433464349E⫹02 0.7065830310E⫹01 0.3442108276E⫹01 0.1628378915E⫹01 0.7350902013E⫹00 0.3111462844E⫹00 0.1213422458E⫹00 0.4284177573E⫺01

1 2 3 4 5 6 7 8 9 10

D⫺

0.5470614554E⫹03 D⫹ 0.1785375907E⫹03 0.6714412694E⫹02 0.2817080571E⫹02 0.1276531244E⫹02 0.6048290480E⫹01 0.2900887663E⫹01 0.1363497443E⫹01 0.6080383309E⫹00 0.2490513328E⫹00

0.5470614554E⫹03 0.1785375907E⫹03 0.6714412694E⫹02 0.2817080571E⫹02 0.1276531244E⫹02 0.6048290480E⫹01 0.2900887663E⫹01 0.1363497443E⫹01 0.6080383309E⫹00 0.2490513328E⫹00

0.8866794085E⫹05 0.2207419386E⫹05 0.6359554354E⫹04 0.2083419469E⫹04 0.7626399619E⫹03 0.3065065821E⫹03 0.1328991064E⫹03 0.6108729953E⫹02 0.2924899880E⫹02 0.1433464349E⫹02 0.7065830310E⫹01 0.3442108276E⫹01 0.1628378915E⫹01 0.7350902013E⫹00 0.3111462844E⫹00 0.1213422458E⫹00 0.4284177573E⫺01


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202

Appendix 5


Symm. Exponents

Ge

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹



1 2 3 4 5 6 7 8 9 10

D⫺

0.6209042224E⫹03 D⫹ 0.2032514375E⫹03 0.7685666788E⫹02 0.3249060810E⫹02 0.1486113361E⫹02 0.7117923198E⫹01 0.3455026360E⫹01 0.1644885189E⫹01 0.7433535527E⫹00 0.3086173287E⫹00

0.6209042224E⫹03 0.2032514375E⫹03 0.7685666788E⫹02 0.3249060810E⫹02 0.1486113361E⫹02 0.7117923198E⫹01 0.3455026360E⫹01 0.1644885189E⫹01 0.7433535527E⫹00 0.3086173287E⫹00

Atom Exp. # Symm. Exponents As

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.3670050657E⫹08 P⫺ 0.7096475829E⫹07 0.1577357540E⫹07 0.3983659049E⫹06 0.1129917201E⫹06 0.3557716761E⫹05 0.1229148224E⫹05 0.4605673332E⫹04 0.1850060103E⫹04 0.7874631209E⫹03 0.3510541707E⫹03


Symm. Exponents

0.1331081188E⫹06 P⫹ 0.3199940702E⫹05 0.8926607652E⫹04 0.2840645374E⫹04 0.1013707303E⫹04 0.3987971425E⫹03 0.1700256068E⫹03 0.7722864010E⫹02 0.3673864422E⫹02 0.1799398989E⫹02 0.8920112952E⫹01



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D⫺

0.1620188050E⫹03 0.7651594282E⫹02 0.3654950605E⫹02 0.1745428272E⫹02 0.8236852018E⫹01 0.3796707424E⫹01 0.1689613835E⫹01 0.7175465616E⫹00 0.2874371317E⫹00 0.1073528683E⫹00

0.4399777304E⫹01 0.2122694145E⫹01 0.9847351935E⫹00 0.4318229631E⫹00 0.1759645440E⫹00 0.6550232125E⫺01

0.7360628185E⫹03 D⫹ 0.2341676268E⫹03 0.8756895589E⫹02 0.3711567484E⫹02 0.1719178430E⫹02 0.8390995314E⫹01 0.4161093417E⫹01 0.2021507912E⫹01 0.9276623855E⫹00 0.3877247292E⫹00

0.7360628185E⫹03 0.2341676268E⫹03 0.8756895589E⫹02 0.3711567484E⫹02 0.1719178430E⫹02 0.8390995314E⫹01 0.4161093417E⫹01 0.2021507912E⫹01 0.9276623855E⫹00 0.3877247292E⫹00

Atom Exp. # Symm. Exponents Se

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

Symm. Exponents

Symm. Exponents

0.3505498077E⫹08 P⫺ 0.6911812576E⫹07 0.1562750843E⫹07 0.4005891628E⫹06 0.1151002888E⫹06 0.3665029629E⫹05 0.1278670139E⫹05 0.4832541699E⫹04 0.1956072050E⫹04 0.8383827473E⫹03 0.3761873280E⫹03 0.1747134876E⫹03 0.8303577479E⫹02 0.3992792004E⫹02 0.1920511288E⫹02 0.9135700861E⫹01 0.4249206169E⫹01 0.1910604337E⫹01 0.8210825748E⫹00 0.3334357248E⫹00 0.1265035568E⫹00

203 Symm. Exponents 0.4399777304E⫹01 0.2122694145E⫹01 0.9847351935E⫹00 0.4318229631E⫹00 0.1759645440E⫹00 0.6550232125E⫺01

Symm. Exponents



(continued )


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Appendix 5


Symm. Exponents

1 2 3 4 5 6 7 8 9 10

D⫺

0.8613526999E⫹03 0.2644945669E⫹03 0.9708763909E⫹02 0.4092748054E⫹02 0.1903541733E⫹02 0.9384228044E⫹01 0.4711030303E⫹01 0.2313699078E⫹01 0.1067981960E⫹01 0.4451224144E⫹00

Exponents

D⫹

Symm.

Symm. Exponents

0.8613526999E⫹03 0.2644945669E⫹03 0.9708763909E⫹02 0.4092748054E⫹02 0.1903541733E⫹02 0.9384228044E⫹01 0.4711030303E⫹01 0.2313699078E⫹01 0.1067981960E⫹01 0.4451224144E⫹00

Atom

Symm.

Exponents

Symm.

Br

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

0.4007715215E⫹08 0.7793199815E⫹07 0.1742240559E⫹07 0.4426433146E⫹06 0.1263378002E⫹06 0.4004298600E⫹05 0.1393200026E⫹05 0.5259863784E⫹04 0.2130049307E⫹04 0.9146166077E⫹03 0.4116261029E⫹03 0.1919381305E⫹03 0.9166304357E⫹02 0.4431819634E⫹02 0.2144395083E⫹02 0.1026461302E⫹02 0.4804802525E⫹01 0.2174121545E⫹01 0.9400426156E⫹00 0.3839253963E⫹00 0.1464070021E⫹00

P⫺


1 2 3 4 5 6 7 8 9 10

D⫺

0.9492329793E⫹03 0.2832269677E⫹03 0.1030196347E⫹03 0.4368290404E⫹02 0.2064864558E⫹02 0.1040501435E⫹02 0.5344995194E⫹01 0.2676627603E⫹01 0.1249531069E⫹01 0.5200045164E⫹00

D⫹

0.9492329793E⫹03 0.2832269677E⫹03 0.1030196347E⫹03 0.4368290404E⫹02 0.2064864558E⫹02 0.1040501435E⫹02 0.5344995194E⫹01 0.2676627603E⫹01 0.1249531069E⫹01 0.5200045164E⫹00

Exponents 0.2089306875E⫹06 0.4723830957E⫹05 0.1253377572E⫹05 0.3833150596E⫹04 0.1327110763E⫹04 0.5108891993E⫹03 0.2147857514E⫹03 0.9685774280E⫹02 0.4601551487E⫹02 0.2262070976E⫹02 0.1130137027E⫹02 0.5635985280E⫹01 0.2755582389E⫹01 0.1297336658E⫹01 0.5776684554E⫹00 0.2389363495E⫹00 0.9016839494E⫺01


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Appendix 5 Atom Exp. # Symm. Exponents Kr

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

S⫹

0.4200639615E⫹08 P⫺ 0.8167577619E⫹07 0.1825121075E⫹07 0.4633823275E⫹06 0.1321494758E⫹06 0.4185028586E⫹05 0.1455014203E⫹05 0.5490358094E⫹04 0.2222939064E⫹04 0.9547215105E⫹03 0.4300083951E⫹03 0.2007969227E⫹03 0.9610497644E⫹02 0.4660926135E⫹02 0.2264461277E⫹02 0.1089563949E⫹02 0.5132920224E⫹01 0.2340610133E⫹01 0.1021351518E⫹01 0.4216302889E⫹00 0.1627899483E⫹00

0.2325642133E⫹06 P⫹ 0.5053851611E⫹05 0.1301169781E⫹05 0.3894897434E⫹04 0.1330231877E⫹04 0.5086823116E⫹03 0.2137334885E⫹03 0.9683310490E⫹02 0.4642150096E⫹02 0.2310879126E⫹02 0.1172240450E⫹02 0.5946429749E⫹01 0.2960156615E⫹01 0.1419092112E⫹01 0.6429284965E⫹00 0.2701404693E⫹00 0.1033023502E⫹00

1 2 3 4 5 6 7 8 9 10

D⫺

0.1010594323E⫹04 D⫹ 0.3007980499E⫹03 0.1096394627E⫹03 0.4671733172E⫹02 0.2221436683E⫹02 0.1125278848E⫹02 0.5796718461E⫹01 0.2898849209E⫹01 0.1343432765E⫹01 0.5507806445E⫹00

0.1010594323E⫹04 0.3007980499E⫹03 0.1096394627E⫹03 0.4671733172E⫹02 0.2221436683E⫹02 0.1125278848E⫹02 0.5796718461E⫹01 0.2898849209E⫹01 0.1343432765E⫹01 0.5507806445E⫹00

Atom Exp. # Symm. Exponents Rb

Symm. Exponents

205

1 2 3 4 5 6 7 8 9 10 11

S⫹

Symm. Exponents

0.1877072370E⫹09 P⫺ 0.3771142117E⫹08 0.8578430665E⫹07 0.2189091295E⫹07 0.6208962017E⫹06 0.1939327926E⫹06 0.6609043898E⫹05 0.2434776926E⫹05 0.9607068606E⫹04 0.4022645846E⫹04 0.1770923970E⫹04

0.2325642133E⫹06 0.5053851611E⫹05 0.1301169781E⫹05 0.3894897434E⫹04 0.1330231877E⫹04 0.5086823116E⫹03 0.2137334885E⫹03 0.9683310490E⫹02 0.4642150096E⫹02 0.2310879126E⫹02 0.1172240450E⫹02 0.5946429749E⫹01 0.2960156615E⫹01 0.1419092112E⫹01 0.6429284965E⫹00 0.2701404693E⫹00 0.1033023502E⫹00

Symm. Exponents

0.2798814008E⫹06 P⫹ 0.6144392290E⫹05 0.1589580691E⫹05 0.4760222471E⫹04 0.1620896305E⫹04 0.6164655436E⫹03 0.2572351173E⫹03 0.1156809548E⫹03 0.5507391010E⫹02 0.2726620398E⫹02 0.1378924383E⫹02



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Appendix 5


Symm. Exponents

12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10

D⫺

0.8121444152E⫹03 0.3844057322E⫹03 0.1860577281E⫹03 0.9123991155E⫹02 0.4491374816E⫹02 0.2198918148E⫹02 0.1060845568E⫹02 0.4996738549E⫹01 0.2276619548E⫹01 0.9941265866E⫹00 0.4122098150E⫹00 0.1608045741E⫹00 0.5847361128E⫺01 0.1963726049E⫺01

0.6997380796E⫹01 0.3499871783E⫹01 0.1694855239E⫹01 0.7805832328E⫹00 0.3358568736E⫹00 0.1326111252E⫹00

0.1161742987E⫹04 D⫹ 0.3417961449E⫹03 0.1238975621E⫹03 0.5280616899E⫹02 0.2525344661E⫹02 0.1293178735E⫹02 0.6766843706E⫹01 0.3452957489E⫹01 0.1639689483E⫹01 0.6914886933E⫹00

0.1161742987E⫹04 0.3417961449E⫹03 0.1238975621E⫹03 0.5280616899E⫹02 0.2525344661E⫹02 0.1293178735E⫹02 0.6766843706E⫹01 0.3452957489E⫹01 0.1639689483E⫹01 0.6914886933E⫹00

Atom Exp. # Symm. Exponents Sr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S⫹

Symm. Exponents


Symm. Exponents 0.6997380796E⫹01 0.3499871783E⫹01 0.1694855239E⫹01 0.7805832328E⫹00 0.3358568736E⫹00 0.1326111252E⫹00

Symm. Exponents



(continued )


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Appendix 5 Atom Exp. # Symm. Exponents 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Symm. Exponents

0.5607654173E⫹01 0.2613053325E⫹01 0.1169548636E⫹01 0.4982177654E⫹00 0.2001607796E⫹00 0.7514940620E⫺01 0.2612683855E⫺01 D⫺

0.1143517377E⫹04 D⫹ 0.3412655823E⫹03 0.1248655031E⫹03 0.5347527531E⫹02 0.2559083652E⫹02 0.1306463104E⫹02 0.6792839420E⫹01 0.3434056192E⫹01 0.1611482579E⫹01 0.6701412009E⫹00

Atom Exp. # Symm. Exponents Y

Symm. Exponents

207

S⫹

0.1143517377E⫹04 0.3412655823E⫹03 0.1248655031E⫹03 0.5347527531E⫹02 0.2559083652E⫹02 0.1306463104E⫹02 0.6792839420E⫹01 0.3434056192E⫹01 0.1611482579E⫹01 0.6701412009E⫹00

Symm. Exponents

0.1767964500E⫹09 P⫺ 0.4054630069E⫹08 0.1023450253E⫹08 0.2824109781E⫹07 0.8461693332E⫹06 0.2734358095E⫹06 0.9465366441E⫹05 0.3486301480E⫹05 0.1357061243E⫹05 0.5545011940E⫹04 0.2362302923E⫹04 0.1042222117E⫹04 0.4729749186E⫹03 0.2192959433E⫹03 0.1031808919E⫹03 0.4893343273E⫹02 0.2323332386E⫹02 0.1096927185E⫹02 0.5115246609E⫹01 0.2340125815E⫹01 0.1043175134E⫹01 0.4500721761E⫹00 0.1866704422E⫹00 0.7392629349E⫺01 0.2776603233E⫺01

Symm. Exponents



(continued )


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208

Appendix 5


Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.1227979574E⫹04 D⫹ 0.4524991053E⫹03 0.1829007818E⫹03 0.7962857810E⫹02 0.3666588971E⫹02 0.1753394619E⫹02 0.8550777236E⫹01 0.4175652055E⫹01 0.2005026458E⫹01 0.9295597561E⫹00 0.4085829371E⫹00 0.1671910989E⫹00 0.6254036317E⫺01

Atom Exp. # Symm. Exponents Y

Symm. Exponents

0.1227979574E⫹04 0.4524991053E⫹03 0.1829007818E⫹03 0.7962857810E⫹02 0.3666588971E⫹02 0.1753394619E⫹02 0.8550777236E⫹01 0.4175652055E⫹01 0.2005026458E⫹01 0.9295597561E⫹00 0.4085829371E⫹00 0.1671910989E⫹00 0.6254036317E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4

D⫺

0.1084407387E⫹04 D⫹ 0.4034791886E⫹03 0.1634826444E⫹03 0.7086930822E⫹02

0.1084407387E⫹04 0.4034791886E⫹03 0.1634826444E⫹03 0.7086930822E⫹02


(continued )


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Appendix 5 Atom Exp. # Symm. Exponents 5 6 7 8 9 10 11 12 13

0.3229199959E⫹02 0.1519483781E⫹02 0.7253970181E⫹01 0.3451820553E⫹01 0.1608524138E⫹01 0.7211540770E⫹00 0.3056073768E⫹00 0.1202677186E⫹00 0.4318152376E⫺01

Atom Exp. # Symm. Exponents Zr

Symm. Exponents

209 Symm. Exponents

0.3229199959E⫹02 0.1519483781E⫹02 0.7253970181E⫹01 0.3451820553E⫹01 0.1608524138E⫹01 0.7211540770E⫹00 0.3056073768E⫹00 0.1202677186E⫹00 0.4318152376E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8

D⫺

0.1526852285E⫹04 D⫹ 0.5356762263E⫹03 0.2084007678E⫹03 0.8820535192E⫹02 0.3984706396E⫹02 0.1884996601E⫹02 0.9161039549E⫹01 0.4487515624E⫹01

0.1526852285E⫹04 0.5356762263E⫹03 0.2084007678E⫹03 0.8820535192E⫹02 0.3984706396E⫹02 0.1884996601E⫹02 0.9161039549E⫹01 0.4487515624E⫹01


(continued )


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210

Appendix 5


Symm. Exponents

9 10 11 12 13

0.2173708510E⫹01 0.1021498754E⫹01 0.4569015881E⫹00 0.1908371279E⫹00 0.7302398706E⫺01

Atom Exp. # Symm. Exponents Zr

Symm. Exponents

0.2173708510E⫹01 0.1021498754E⫹01 0.4569015881E⫹00 0.1908371279E⫹00 0.7302398706E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.1300503262E⫹04 D⫹ 0.4754137488E⫹03 0.1900424811E⫹03 0.8161879767E⫹02 0.3700259001E⫹02 0.1739876281E⫹02 0.8336635391E⫹01 0.3999367631E⫹01 0.1887390694E⫹01 0.8608830112E⫹00 0.3728895345E⫹00 0.1506995183E⫹00 0.5583172281E⫺01

0.1300503262E⫹04 0.4754137488E⫹03 0.1900424811E⫹03 0.8161879767E⫹02 0.3700259001E⫹02 0.1739876281E⫹02 0.8336635391E⫹01 0.3999367631E⫹01 0.1887390694E⫹01 0.8608830112E⫹00 0.3728895345E⫹00 0.1506995183E⫹00 0.5583172281E⫺01



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Appendix 5 Atom Exp. # Symm. Exponents Nb

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.1674527139E⫹04 D⫹ 0.5787655273E⫹03 0.2237790645E⫹03 0.9482167120E⫹02 0.4313523108E⫹02 0.2063754553E⫹02 0.1017303107E⫹02 0.5061440408E⫹01 0.2489971264E⫹01 0.1186522591E⫹01 0.5365178517E⫹00 0.2255193704E⫹00 0.8632568868E⫺01

0.1674527139E⫹04 0.5787655273E⫹03 0.2237790645E⫹03 0.9482167120E⫹02 0.4313523108E⫹02 0.2063754553E⫹02 0.1017303107E⫹02 0.5061440408E⫹01 0.2489971264E⫹01 0.1186522591E⫹01 0.5365178517E⫹00 0.2255193704E⫹00 0.8632568868E⫺01

Atom Exp. # Symm. Exponents Nb

Symm. Exponents

211

1 2 3 4

S⫹

Symm. Exponents

0.1961617470E⫹09 P⫺ 0.4346787392E⫹08 0.1067858127E⫹08 0.2887073744E⫹07


Symm. Exponents

0.5103594434E⫹06 P⫹ 0.1105082306E⫹06 0.2807671187E⫹05 0.8231881217E⫹04

0.5103594434E⫹06 0.1105082306E⫹06 0.2807671187E⫹05 0.8231881217E⫹04 (continued )


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212

Appendix 5


Symm. Exponents

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.8527258707E⫹06 0.2731343519E⫹06 0.9418171919E⫹05 0.3470470748E⫹05 0.1356593490E⫹05 0.5584171091E⫹04 0.2402832134E⫹04 0.1072881991E⫹04 0.4934604712E⫹03 0.2320773483E⫹03 0.1107901600E⫹03 0.5329245883E⫹02 0.2564097532E⫹02 0.1224942258E⫹02 0.5767891328E⫹01 0.2657339099E⫹01 0.1189086431E⫹01 0.5130084128E⫹00 0.2118303823E⫹00 0.8310225204E⫺01 0.3074724421E⫺01

0.2739184313E⫹04 0.1017374406E⫹04 0.4148067670E⫹03 0.1825931202E⫹03 0.8534235464E⫹02 0.4165379223E⫹02 0.2087960444E⫹02 0.1057148171E⫹02 0.5316960267E⫹01 0.2612600492E⫹01 0.1233481702E⫹01 0.5503140702E⫹00 0.2281785423E⫹00

0.1425508386E⫹04 D⫹ 0.5170506326E⫹03 0.2061657840E⫹03 0.8871642796E⫹02 0.4044633440E⫹02 0.1917898745E⫹02 0.9285970295E⫹01 0.4506794294E⫹01 0.2152438416E⫹01 0.9931160685E⫹00 0.4345695384E⫹00 0.1770485086E⫹00 0.6592994408E⫺01

0.1425508386E⫹04 0.5170506326E⫹03 0.2061657840E⫹03 0.8871642796E⫹02 0.4044633440E⫹02 0.1917898745E⫹02 0.9285970295E⫹01 0.4506794294E⫹01 0.2152438416E⫹01 0.9931160685E⫹00 0.4345695384E⫹00 0.1770485086E⫹00 0.6592994408E⫺01

Atom Exp. # Symm. Exponents Mo

1 2 3 4 5 6 7 8 9

S⫹

Symm. Exponents

0.1883356285E⫹09 P⫺ 0.4188374968E⫹08 0.1033461372E⫹08 0.2808406536E⫹07 0.8343036535E⫹06 0.2689479074E⫹06 0.9338421177E⫹05 0.3466741180E⫹05 0.1365819992E⫹05


Symm. Exponents

0.5588851874E⫹06 P⫹ 0.1210943119E⫹06 0.3074339110E⫹05 0.8997731984E⫹04 0.2986716265E⫹04 0.1106269369E⫹04 0.4498421952E⫹03 0.1975687482E⫹03 0.9220647890E⫹02

0.5588851874E⫹06 0.1210943119E⫹06 0.3074339110E⫹05 0.8997731984E⫹04 0.2986716265E⫹04 0.1106269369E⫹04 0.4498421952E⫹03 0.1975687482E⫹03 0.9220647890E⫹02 (continued )


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Appendix 5 Atom Exp. # Symm. Exponents 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.5668537259E⫹04 0.2460001944E⫹04 0.1108072583E⫹04 0.5142220474E⫹03 0.2440418517E⫹03 0.1175683640E⫹03 0.5707035754E⫹02 0.2770805728E⫹02 0.1335544415E⫹02 0.6343789730E⫹01 0.2947533970E⫹01 0.1329749587E⫹01 0.5781803464E⫹00 0.2405030629E⫹00 0.9499989696E⫺01 0.3537140435E⫺01

0.4499001454E⫹02 0.2257920831E⫹02 0.1146741535E⫹02 0.5798464551E⫹01 0.2871953975E⫹01 0.1370835446E⫹01 0.6203873464E⫹00 0.2619007049E⫹00

0.1753542601E⫹04 D⫹ 0.6035070644E⫹03 0.2329416325E⫹03 0.9875412327E⫹02 0.4503512508E⫹02 0.2163615919E⫹02 0.1072472756E⫹02 0.5371731798E⫹01 0.2662618664E⫹01 0.1279130540E⫹01 0.5832794228E⫹00 0.2472515261E⫹00 0.9542155525E⫺01

0.1753542601E⫹04 0.6035070644E⫹03 0.2329416325E⫹03 0.9875412327E⫹02 0.4503512508E⫹02 0.2163615919E⫹02 0.1072472756E⫹02 0.5371731798E⫹01 0.2662618664E⫹01 0.1279130540E⫹01 0.5832794228E⫹00 0.2472515261E⫹00 0.9542155525E⫺01

Atom Exp. # Symm. Exponents Mo

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

Symm. Exponents

Symm. Exponents


213 Symm. Exponents 0.4499001454E⫹02 0.2257920831E⫹02 0.1146741535E⫹02 0.5798464551E⫹01 0.2871953975E⫹01 0.1370835446E⫹01 0.6203873464E⫹00 0.2619007049E⫹00

Symm. Exponents

0.5749058874E⫹06 P⫹ 0.1221468969E⫹06 0.3057504265E⫹05 0.8864463139E⫹04 0.2926444705E⫹04 0.1081514900E⫹04 0.4398765142E⫹03 0.1935694916E⫹03 0.9060511039E⫹02 0.4434859967E⫹02 0.2231620515E⫹02 0.1134947515E⫹02 0.5735192429E⫹01 0.2830992871E⫹01

0.5749058874E⫹06 0.1221468969E⫹06 0.3057504265E⫹05 0.8864463139E⫹04 0.2926444705E⫹04 0.1081514900E⫹04 0.4398765142E⫹03 0.1935694916E⫹03 0.9060511039E⫹02 0.4434859967E⫹02 0.2231620515E⫹02 0.1134947515E⫹02 0.5735192429E⫹01 0.2830992871E⫹01 (continued )


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214

Appendix 5


Symm. Exponents

15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.1169019333E⫹03 0.5611944346E⫹02 0.2692466741E⫹02 0.1281878674E⫹02 0.6013361805E⫹01 0.2759788737E⫹01 0.1230372698E⫹01 0.5290712129E⫹00 0.2178825404E⫹00 0.8532490913E⫺01 0.3154921795E⫺01

0.1341991936E⫹01 0.6005970146E⫹00 0.2494826493E⫹00

0.1488058285E⫹04 D⫹ 0.5248991670E⫹03 0.2056587928E⫹03 0.8776172593E⫹02 0.3999636478E⫹02 0.1908817442E⫹02 0.9354224636E⫹01 0.4615531708E⫹01 0.2248418983E⫹01 0.1060340442E⫹01 0.4746757042E⫹00 0.1977895277E⫹00 0.7522032630E⫺01

0.1488058285E⫹04 0.5248991670E⫹03 0.2056587928E⫹03 0.8776172593E⫹02 0.3999636478E⫹02 0.1908817442E⫹02 0.9354224636E⫹01 0.4615531708E⫹01 0.2248418983E⫹01 0.1060340442E⫹01 0.4746757042E⫹00 0.1977895277E⫹00 0.7522032630E⫺01

Atom Exp. # Symm. Exponents Tc

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S⫹

Symm. Exponents


Symm. Exponents 0.1341991936E⫹01 0.6005970146E⫹00 0.2494826493E⫹00

Symm. Exponents



(continued )


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Symm. Exponents

0.6784809115E⫹01 0.3152919084E⫹01 0.1419875514E⫹01 0.6150034839E⫹00 0.2542859433E⫹00 0.9961212644E⫺01 0.3669228500E⫺01 D⫺

0.1829435397E⫹04 D⫹ 0.6377687147E⫹03 0.2483872236E⫹03 0.1059162483E⫹03 0.4846303461E⫹02 0.2331957801E⫹02 0.1156485581E⫹02 0.5793174308E⫹01 0.2872745775E⫹01 0.1382067976E⫹01 0.6322089082E⫹00 0.2694872367E⫹00 0.1049082303E⫹00

Atom Exp. # Symm. Exponents Tc

Symm. Exponents

215

S⫹

0.1829435397E⫹04 0.6377687147E⫹03 0.2483872236E⫹03 0.1059162483E⫹03 0.4846303461E⫹02 0.2331957801E⫹02 0.1156485581E⫹02 0.5793174308E⫹01 0.2872745775E⫹01 0.1382067976E⫹01 0.6322089082E⫹00 0.2694872367E⫹00 0.1049082303E⫹00

Symm. Exponents

0.1527345126E⫹09 P⫺ 0.3516999525E⫹08 0.8949244028E⫹07 0.2498330136E⫹07 0.7596844875E⫹06 0.2498084738E⫹06 0.8819461531E⫹05 0.3319009469E⫹05 0.1321834096E⫹05 0.5531182223E⫹04 0.2414360188E⫹04 0.1091439720E⫹04 0.5073195137E⫹03 0.2407232399E⫹03 0.1157656281E⫹03 0.5601920300E⫹02 0.2708073329E⫹02 0.1298436262E⫹02 0.6130389849E⫹01 0.2829653862E⫹01 0.1267730142E⫹01 0.5473172332E⫹00 0.2260688645E⫹00

Symm. Exponents



(continued )


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216

Appendix 5


Symm. Exponents

24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

0.8869563093E⫺01 0.3281663649E⫺01 D⫺

0.1734804498E⫹04 D⫹ 0.5980616418E⫹03 0.2304976751E⫹03 0.9731184542E⫹02 0.4409575618E⫹02 0.2101421517E⫹02 0.1031972721E⫹02 0.5116998495E⫹01 0.2510196726E⫹01 0.1193710444E⫹01 0.5391906608E⫹00 0.2266685308E⫹00 0.8689583609E⫺01

Atom Exp. # Symm. Exponents Ru

Symm. Exponents

0.1734804498E⫹04 0.5980616418E⫹03 0.2304976751E⫹03 0.9731184542E⫹02 0.4409575618E⫹02 0.2101421517E⫹02 0.1031972721E⫹02 0.5116998495E⫹01 0.2510196726E⫹01 0.1193710444E⫹01 0.5391906608E⫹00 0.2266685308E⫹00 0.8689583609E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹

0.1623681020E⫹09 P⫺ 0.3676656664E⫹08 0.9237014505E⫹07 0.2555460890E⫹07 0.7726784359E⫹06 0.2534271629E⫹06 0.8948808704E⫹05 0.3376508571E⫹05 0.1351120036E⫹05 0.5690839459E⫹04 0.2504078461E⫹04 0.1142461585E⫹04 0.5364018055E⫹03 0.2572330005E⫹03 0.1250500416E⫹03 0.6116381669E⫹02 0.2987388966E⫹02 0.1446133213E⫹02 0.6886158878E⫹01 0.3201333083E⫹01 0.1442124150E⫹01 0.6247773482E⫹00 0.2583637723E⫹00 0.1012172539E⫹00 0.3728438227E⫺01


1 2

D⫺

0.2087625311E⫹04 D⫹ 0.7049592191E⫹03

0.2087625311E⫹04 0.7049592191E⫹03


(continued )


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Appendix 5 Atom Exp. # Symm. Exponents 3 4 5 6 7 8 9 10 11 12 13

0.2685630313E⫹03 0.1129677844E⫹03 0.5135047750E⫹02 0.2468716424E⫹02 0.1228542317E⫹02 0.6193791457E⫹01 0.3096178075E⫹01 0.1501943262E⫹01 0.6919818702E⫹00 0.2963499435E⫹00 0.1154620528E⫹00

Atom Exp. # Symm. Exponents Ru

Symm. Exponents

217 Symm. Exponents

0.2685630313E⫹03 0.1129677844E⫹03 0.5135047750E⫹02 0.2468716424E⫹02 0.1228542317E⫹02 0.6193791457E⫹01 0.3096178075E⫹01 0.1501943262E⫹01 0.6919818702E⫹00 0.2963499435E⫹00 0.1154620528E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7

D⫺

0.1914069549E⫹04 D⫹ 0.6473621501E⫹03 0.2469160509E⫹03 0.1039034494E⫹03 0.4719059217E⫹02 0.2263038290E⫹02 0.1120999929E⫹02

0.1914069549E⫹04 0.6473621501E⫹03 0.2469160509E⫹03 0.1039034494E⫹03 0.4719059217E⫹02 0.2263038290E⫹02 0.1120999929E⫹02


(continued )


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218

Appendix 5


Symm. Exponents

8 9 10 11 12 13

0.5611289438E⫹01 0.2776702514E⫹01 0.1328837355E⫹01 0.6016671342E⫹00 0.2521441700E⫹00 0.9567885039E⫺01

Atom Exp. # Symm. Exponents Rh

Symm. Exponents

0.5611289438E⫹01 0.2776702514E⫹01 0.1328837355E⫹01 0.6016671342E⫹00 0.2521441700E⫹00 0.9567885039E⫺01

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.2150541641E⫹04 D⫹ 0.7278304844E⫹03 0.2782111434E⫹03 0.1175273656E⫹03 0.5368858880E⫹02 0.2595145965E⫹02 0.1298783347E⫹02 0.6585140718E⫹01 0.3309825836E⫹01 0.1613672001E⫹01 0.7467130337E⫹00 0.3209067509E⫹00 0.1253282382E⫹00

0.2150541641E⫹03 0.7278304844E⫹03 0.2782111434E⫹03 0.1175273656E⫹03 0.5368858880E⫹02 0.2595145965E⫹02 0.1298783347E⫹02 0.6585140718E⫹01 0.3309825836E⫹01 0.1613672001E⫹01 0.7467130337E⫹00 0.3209067509E⫹00 0.1253282382E⫹00



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Appendix 5 Atom Exp. # Symm. Exponents Rh

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.1974411246E⫹04 D⫹ 0.6692546256E⫹03 0.2562224176E⫹03 0.1083700640E⫹03 0.4952933321E⫹02 0.2392605319E⫹02 0.1194894499E⫹02 0.6034377274E⫹01 0.3014215098E⫹01 0.1456630473E⫹01 0.6661198797E⫹00 0.2819536598E⫹00 0.1080487879E⫹00

0.1974411246E⫹04 0.6692546256E⫹03 0.2562224176E⫹03 0.1083700640E⫹03 0.4952933321E⫹02 0.2392605319E⫹02 0.1194894499E⫹02 0.6034377274E⫹01 0.3014215098E⫹01 0.1456630473E⫹01 0.6661198797E⫹00 0.2819536598E⫹00 0.1080487879E⫹00

Atom Exp. # Symm. Exponents Pd

Symm. Exponents

219

1 2 3 4

S⫹

Symm. Exponents

0.1594112295E⫹09 P⫺ 0.3653200184E⫹08 0.9290400255E⫹07 0.2601885155E⫹07


Symm. Exponents

0.9621519759E⫹06 P⫹ 0.1927135005E⫹06 0.4595249932E⫹05 0.1281404505E⫹05

0.9621519759E⫹06 0.1927135005E⫹06 0.4595249932E⫹05 0.1281404505E⫹05 (continued )


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Appendix 5


Symm. Exponents

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.7963810466E⫹06 0.2643742490E⫹06 0.9446461946E⫹05 0.3605432342E⫹05 0.1458713536E⫹05 0.6208597661E⫹04 0.2758761853E⫹04 0.1270043066E⫹04 0.6011634570E⫹03 0.2903507408E⫹03 0.1420024680E⫹03 0.6979073714E⫹02 0.3420699832E⫹02 0.1659333896E⫹02 0.7905704745E⫹01 0.3671315714E⫹01 0.1649162921E⫹01 0.7111356099E⫹00 0.2921290027E⫹00 0.1134531767E⫹00 0.4133945335E⫺01

0.4104828775E⫹04 0.1483840139E⫹04 0.5945851134E⫹03 0.2594342886E⫹03 0.1210817595E⫹03 0.5937719198E⫹02 0.3005396319E⫹02 0.1542327442E⫹02 0.7883076506E⫹01 0.3941947133E⫹01 0.1894407161E⫹01 0.8594782949E⫹00 0.3616155811E⫹00

0.2425662023E⫹04 D⫹ 0.8068110627E⫹03 0.3046810970E⫹03 0.1277433595E⫹03 0.5814834501E⫹02 0.2810150844E⫹02 0.1409945097E⫹02 0.7181945609E⫹01 0.3631921130E⫹01 0.1783085381E⫹01 0.8310675963E⫹00 0.3595970206E⫹00 0.1412532314E⫹00

0.2425662023E⫹04 0.8068110627E⫹03 0.3046810970E⫹03 0.1277433595E⫹03 0.5814834501E⫹02 0.2810150844E⫹02 0.1409945097E⫹02 0.7181945609E⫹01 0.3631921130E⫹01 0.1783085381E⫹01 0.8310675963E⫹00 0.3595970206E⫹00 0.1412532314E⫹00


1 2 3 4 5 6 7 8

S⫹

Symm. Exponents

0.1568006564E⫹09 P⫺ 0.3549218306E⫹08 0.8924411199E⫹07 0.2473743195E⫹07 0.7501083322E⫹06 0.2469186314E⫹06 0.8756093370E⫹05 0.3319400960E⫹05


Symm. Exponents

0.9477639491E⫹06 P⫹ 0.1949596596E⫹06 0.4736098156E⫹05 0.1335892163E⫹05 0.4301701665E⫹04 0.1554786489E⫹04 0.6201635267E⫹03 0.2684046792E⫹03

0.9477639491E⫹06 0.1949596596E⫹06 0.4736098156E⫹05 0.1335892163E⫹05 0.4301701665E⫹04 0.1554786489E⫹04 0.6201635267E⫹03 0.2684046792E⫹03 (continued )


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Appendix 5 Atom Exp. # Symm. Exponents 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺


0.1239269472E⫹03 0.6001732481E⫹02 0.2997547931E⫹02 0.1518020606E⫹02 0.7663984761E⫹01 0.3792629687E⫹01 0.1808749804E⫹01 0.8173577981E⫹00 0.3441002492E⫹00

0.2205627338E⫹04 D⫹ 0.7535045759E⫹03 0.2897464749E⫹03 0.1227234754E⫹03 0.5602902388E⫹02 0.2698193567E⫹02 0.1341239986E⫹02 0.6734602003E⫹01 0.3342628712E⫹01 0.1604847661E⫹01 0.7293696146E⫹00 0.3070636497E⫹00 0.1171857341E⫹00

0.2205627338E⫹04 0.7535045759E⫹03 0.2897464749E⫹03 0.1227234754E⫹03 0.5602902388E⫹02 0.2698193567E⫹02 0.1341239986E⫹02 0.6734602003E⫹01 0.3342628712E⫹01 0.1604847661E⫹01 0.7293696146E⫹00 0.3070636497E⫹00 0.1171857341E⫹00


1 2 3 4 5 6 7 8 9 10 11 12

S⫹

Symm. Exponents

Symm. Exponents

0.1718569821E⫹09 P⫺ 0.3869939731E⫹08 0.9756676216E⫹07 0.2732309697E⫹07 0.8432531311E⫹06 0.2845480924E⫹06 0.1041580259E⫹06 0.4103352404E⫹05 0.1726090556E⫹05 0.7691954179E⫹04 0.3602697694E⫹04 0.1759570387E⫹04

221 Symm. Exponents 0.1239269472E⫹03 0.6001732481E⫹02 0.2997547931E⫹02 0.1518020606E⫹02 0.7663984761E⫹01 0.3792629687E⫹01 0.1808749804E⫹01 0.8173577981E⫹00 0.3441002492E⫹00

Symm. Exponents

0.9959637868E⫹06 P⫹ 0.1977549193E⫹06 0.4685017601E⫹05 0.1300314843E⫹05 0.4151378712E⫹04 0.1496910750E⫹04 0.5985664330E⫹03 0.2606122863E⫹03 0.1213100187E⫹03 0.5927490775E⫹02 0.2985181329E⫹02 0.1521420760E⫹02

0.9959637868E⫹05 0.1977549193E⫹06 0.4685017601E⫹05 0.1300314843E⫹05 0.4151378712E⫹04 0.1496910750E⫹04 0.5985664330E⫹03 0.2606122863E⫹03 0.1213100187E⫹03 0.5927490775E⫹02 0.2985181329E⫹02 0.1521420760E⫹02 (continued )


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Appendix 5


Symm. Exponents

13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.8890840636E⫹03 0.4611115861E⫹03 0.2435375539E⫹03 0.1299547654E⫹03 0.6951115636E⫹02 0.3697622064E⫹02 0.1940733476E⫹02 0.9971358139E⫹01 0.4975737283E⫹01 0.2392461436E⫹01 0.1099731075E⫹01 0.4794593181E⫹00 0.1967027419E⫹00

0.7704774438E⫹01 0.3806761504E⫹01 0.1801730108E⫹01 0.8020756075E⫹00 0.3297505788E⫹00

0.1859683876E⫹04 D⫹ 0.6372664379E⫹03 0.2458257609E⫹03 0.1043875042E⫹03 0.4771717948E⫹02 0.2296130126E⫹02 0.1137376005E⫹02 0.5671367785E⫹01 0.2783800444E⫹01 0.1315361957E⫹01 0.5850590431E⫹00 0.2395475371E⫹00 0.8829003108E⫺01

0.1859683876E⫹04 0.6372664379E⫹03 0.2458257609E⫹03 0.1043875042E⫹03 0.4771717948E⫹02 0.2296130126E⫹02 0.1137376005E⫹02 0.5671367785E⫹01 0.2783800444E⫹01 0.1315361957E⫹01 0.5850590431E⫹00 0.2395475371E⫹00 0.8829003108E⫺01

Atom Exp. # Symm. Exponents Ag

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S⫹

Symm. Exponents

0.1510571056E⫹09 P⫺ 0.3451946817E⫹08 0.8767365077E⫹07 0.2455766579E⫹07 0.7527469722E⫹06 0.2505450514E⫹06 0.8985214744E⫹05 0.3445147760E⫹05 0.1401376873E⫹05 0.6000682938E⫹04 0.2683959734E⫹04 0.1244261989E⫹04 0.5932518955E⫹03 0.2886614827E⫹03 0.1422302211E⫹03 0.7041727521E⫹02

Symm. Exponents 0.7704774438E⫹01 0.3806761504E⫹01 0.1801730108E⫹01 0.8020756075E⫹00 0.3297505788E⫹00

Symm. Exponents

0.1067066851E⫹07 P⫹ 0.2125277830E⫹06 0.5042778983E⫹05 0.1400194242E⫹05 0.4468942472E⫹04 0.1610472661E⫹04 0.6436770982E⫹03 0.2802740303E⫹03 0.1305965646E⫹03 0.6396593208E⫹02 0.3234951620E⫹02 0.1659292375E⫹02 0.8479067793E⫹01 0.4240112237E⫹01 0.2038189899E⫹01 0.9250899798E⫹00

0.1067066851E⫹07 0.2125277830E⫹06 0.5042778983E⫹05 0.1400194242E⫹05 0.4468942472E⫹04 0.1610472661E⫹04 0.6436770982E⫹03 0.2802740303E⫹03 0.1305965646E⫹03 0.6396593208E⫹02 0.3234951620E⫹02 0.1659292375E⫹02 0.8479067793E⫹01 0.4240112237E⫹01 0.2038189899E⫹01 0.9250899798E⫹00 (continued )


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Appendix 5 Atom Exp. # Symm. Exponents 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.3476016037E⫹02 0.1697580945E⫹02 0.8138723928E⫹01 0.3800932773E⫹01 0.1715785707E⫹01 0.7428580430E⫹00 0.3060902394E⫹00 0.1191036140E⫹00 0.4342733176E⫺01

0.3894299429E⫹00

0.2412406479E⫹04 D⫹ 0.8106619270E⫹03 0.3091649991E⫹03 0.1308443228E⫹03 0.6008778966E⫹02 0.2927771106E⫹02 0.1479995256E⫹02 0.7589420242E⫹01 0.3860421439E⫹01 0.1904545884E⫹01 0.8911104352E⫹00 0.3866408626E⫹00 0.1521152898E⫹00

0.2412406479E⫹04 0.8106619270E⫹03 0.3091649991E⫹03 0.1308443228E⫹03 0.6008778966E⫹02 0.2927771106E⫹02 0.1479995256E⫹02 0.7589420242E⫹01 0.3860421439E⫹01 0.1904545884E⫹01 0.8911104352E⫹00 0.3866408626E⫹00 0.1521152898E⫹00

Atom Exp. # Symm. Exponents Ag

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S⫹

Symm. Exponents

Symm. Exponents

0.1559462084E⫹09 P⫺ 0.3582872462E⫹08 0.9133497564E⫹07 0.2563655536E⫹07 0.7862606781E⫹06 0.2614720790E⫹06 0.9356286402E⫹05 0.3574939241E⫹05 0.1447399236E⫹05 0.6162118925E⫹04 0.2737545972E⫹04 0.1259360077E⫹04 0.5953373404E⫹03 0.2869911945E⫹03 0.1400018628E⫹03 0.6858456622E⫹02 0.3348219058E⫹02 0.1616455540E⫹02 0.7658498685E⫹01 0.3533629767E⫹01

223 Symm. Exponents 0.3894299429E⫹00

Symm. Exponents



(continued )


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224

Appendix 5


Symm. Exponents

21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

0.1575663656E⫹01 0.6738117933E⫹00 0.2742293036E⫹00 0.1054039346E⫹00 0.3796951417E⫺01 D⫺

0.2346979016E⫹04 D⫹ 0.7895659183E⫹03 0.3002116891E⫹03 0.1262181656E⫹03 0.5740718302E⫹02 0.2763476398E⫹02 0.1377480561E⫹02 0.6955853675E⫹01 0.3481333914E⫹01 0.1689531345E⫹01 0.7778703174E⫹00 0.3324019509E⫹00 0.1289823814E⫹00

Atom Exp. # Symm. Exponents Cd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Symm. Exponents

S⫹

0.2346979016E⫹04 0.7895659183E⫹03 0.3002116891E⫹03 0.1262181656E⫹03 0.5740718302E⫹02 0.2763476398E⫹02 0.1377480561E⫹02 0.6955853675E⫹01 0.3481333914E⫹01 0.1689531345E⫹01 0.7778703174E⫹00 0.3324019509E⫹00 0.1289823814E⫹00

Symm. Exponents

0.1439742140E⫹09 P⫺ 0.3348673768E⫹08 0.8625844795E⫹07 0.2442528635E⫹07 0.7546671786E⫹06 0.2525326780E⫹06 0.9084365075E⫹05 0.3487024475E⫹05 0.1417644651E⫹05 0.6058990788E⫹04 0.2702229748E⫹04 0.1248250844E⫹04 0.5927992346E⫹03 0.2872818032E⫹03 0.1410171291E⫹03 0.6959330087E⫹02 0.3427388764E⫹02 0.1671968982E⫹02 0.8019190456E⫹01 0.3753521029E⫹01 0.1701851812E⫹01 0.7419040491E⫹00 0.3086635897E⫹00

Symm. Exponents



(continued )


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Appendix 5 Atom Exp. # Symm. Exponents 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

Symm. Exponents

0.1216475242E⫹00 0.4507854673E⫺01 D⫺

0.2912502614E⫹04 D⫹ 0.9458028609E⫹03 0.3518585675E⫹03 0.1464439958E⫹03 0.6659071375E⫹02 0.3230703890E⫹02 0.1633145019E⫹02 0.8400384867E⫹01 0.4293615799E⫹01 0.2129607002E⫹01 0.1000992370E⫹01 0.4354314656E⫹00 0.1711864236E⫹00

Atom Exp. # Symm. Exponents In

Symm. Exponents

225

0.2912502614E⫹04 0.9458028609E⫹03 0.3518585675E⫹03 0.1464439958E⫹03 0.6659071375E⫹02 0.3230703890E⫹02 0.1633145019E⫹02 0.8400384867E⫹01 0.4293615799E⫹01 0.2129607002E⫹01 0.1000992370E⫹01 0.4354314656E⫹00 0.1711864236E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹


0.2535808680E⫹07 P⫹ 0.5497144499E⫹06 0.1370375986E⫹06 0.3880580142E⫹05 0.1233048954E⫹05 0.4342733112E⫹04 0.1674624856E⫹04 0.6984187854E⫹03 0.3111936243E⫹03 0.1463303066E⫹03 0.7172971436E⫹02 0.3620741844E⫹02 0.1859097107E⫹02 0.9591467450E⫹01 0.4911553978E⫹01 0.2465903522E⫹01 0.1199026964E⫹01 0.5577633085E⫹00 0.2451949210E⫹00 0.1006203790E⫹00 0.3807551547E⫺01

1 2

D⫺

0.2919907535E⫹04 D⫹ 0.1010715927E⫹04

0.2919907535E⫹04 0.1010715927E⫹04


(continued )


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Appendix 5


Symm. Exponents

3 4 5 6 7 8 9 10 11 12 13

0.3905601419E⫹03 0.1653818952E⫹03 0.7533064367E⫹02 0.3623109802E⫹02 0.1806175106E⫹02 0.9161132870E⫹01 0.4640790035E⫹01 0.2304785435E⫹01 0.1101557975E⫹01 0.4973532155E⫹00 0.2082311263E⫹00

Atom Exp. # Symm. Exponents Sn

Symm. Exponents

0.3905601419E⫹03 0.1653818952E⫹03 0.7533064367E⫹02 0.3623109802E⫹02 0.1806175106E⫹02 0.9161132870E⫹01 0.4640790035E⫹01 0.2304785435E⫹01 0.1101557975E⫹01 0.4973532155E⫹00 0.2082311263E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7

D⫺

0.3365477181E⫹04 D⫹ 0.1134031272E⫹04 0.4301877047E⫹03 0.1801513356E⫹03 0.8166866523E⫹02 0.3930102306E⫹02 0.1968677733E⫹02

0.3365477181E⫹04 0.1134031272E⫹04 0.4301877047E⫹03 0.1801513356E⫹03 0.8166866523E⫹02 0.3930102306E⫹02 0.1968677733E⫹02


(continued )


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Appendix 5 Atom Exp. # Symm. Exponents 8 9 10 11 12 13

0.1006606490E⫹02 0.5151706289E⫹01 0.2587859440E⫹01 0.1251181866E⫹01 0.5709292793E⫹00 0.2411116835E⫹00

Atom Exp. # Symm. Exponents Sb

Symm. Exponents

227 Symm. Exponents

0.1006606490E⫹02 0.5151706289E⫹01 0.2587859440E⫹01 0.1251181866E⫹01 0.5709292793E⫹00 0.2411116835E⫹00

Symm. Exponents

Symm. Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.3620812446E⫹04 D⫹ 0.1205684687E⫹04 0.4540956270E⫹03 0.1895993133E⫹03 0.8601854148E⫹02 0.4156265385E⫹02 0.2096333407E⫹02 0.1081815842E⫹02 0.5598506971E⫹01 0.2847783843E⫹01 0.1395557516E⫹01 0.6457796952E⫹00 0.2765712079E⫹00

0.3620812446E⫹04 0.1205684687E⫹04 0.4540956270E⫹03 0.1895993133E⫹03 0.8601854148E⫹02 0.4156265385E⫹02 0.2096333407E⫹02 0.1081815842E⫹02 0.5598506971E⫹01 0.2847783843E⫹01 0.1395557516E⫹01 0.6457796952E⫹00 0.2765712079E⫹00



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228

Appendix 5


Symm. Exponents

Te

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

S⫹



1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.4196390059E⫹04 D⫹ 0.1371511947E⫹04 0.5093213667E⫹03 0.2105673979E⫹03 0.9495890036E⫹02 0.4576820850E⫹02 0.2310011903E⫹02 0.1196258312E⫹02 0.6227798369E⫹01 0.3193605892E⫹01 0.1580533404E⫹01 0.7396733910E⫹00 0.3207219245E⫹00

0.4196390059E⫹04 0.1371511947E⫹04 0.5093213667E⫹03 0.2105673979E⫹03 0.9495890036E⫹02 0.4576820850E⫹02 0.2310011903E⫹02 0.1196258312E⫹02 0.6227798369E⫹01 0.3193605892E⫹01 0.1580533404E⫹01 0.7396733910E⫹00 0.3207219245E⫹00

Atom Exp. # Symm. Exponents I

Symm. Exponents

1 2 3 4

S⫹

Symm. Exponents

0.1235253074E⫹09 P⫺ 0.2763027891E⫹08 0.6959159879E⫹07 0.1956557522E⫹07


Symm. Exponents

0.5058911310E⫹07 P⫹ 0.9900977547E⫹06 0.2268404533E⫹06 0.6002554985E⫹05

0.5058911310E⫹07 0.9900977547E⫹06 0.2268404533E⫹06 0.6002554985E⫹05 (continued )


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Appendix 5 Atom Exp. # Symm. Exponents 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺



0.4246986262E⫹04 D⫹ 0.1380058294E⫹04 0.5116191497E⫹03 0.2118722221E⫹03 0.9596750538E⫹02 0.4655243786E⫹02 0.2367956806E⫹02 0.1236697069E⫹02 0.6493161128E⫹01 0.3355810328E⫹01 0.1671598971E⫹01 0.7857878892E⫹00 0.3413203629E⫹00

0.4246986262E⫹04 0.1380058294E⫹04 0.5116191497E⫹03 0.2118722221E⫹03 0.9596750538E⫹02 0.4655243786E⫹02 0.2367956806E⫹02 0.1236697069E⫹02 0.6493161128E⫹01 0.3355810328E⫹01 0.1671598971E⫹01 0.7857878892E⫹00 0.3413203629E⫹00

Atom Exp. # Symm. Exponents Xe

1 2 3 4 5 6 7 8

S⫹

Symm. Exponents

Symm. Exponents

0.1188842243E⫹09 P⫺ 0.2635274948E⫹08 0.6603811044E⫹07 0.1853933660E⫹07 0.5778144966E⫹06 0.1981258838E⫹06 0.7406557287E⫹05 0.2991422722E⫹05

229 Symm. Exponents 0.1809992545E⫹05 0.6136125205E⫹04 0.2307492060E⫹04 0.9496565834E⫹03 0.4220122962E⫹03 0.1997871584E⫹03 0.9941359212E⫹02 0.5129930526E⫹02 0.2708423105E⫹02 0.1443486195E⫹02 0.7662167589E⫹01 0.3996550773E⫹01 0.2020996156E⫹01 0.9775610607E⫹00 0.4462439519E⫹00 0.1896717379E⫹00 0.7406048032E⫺01

Symm. Exponents

0.6350064231E⫹07 P⫹ 0.1212927247E⫹07 0.2721059374E⫹06 0.7072093605E⫹05 0.2100510138E⫹05 0.7032802908E⫹04 0.2618298024E⫹04 0.1069194607E⫹04

0.6350064231E⫹06 0.1212927247E⫹07 0.2721059374E⫹06 0.7072093605E⫹05 0.2100510138E⫹05 0.7032802908E⫹04 0.2618298024E⫹04 0.1069194607E⫹04 (continued )


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230

Appendix 5


Symm. Exponents

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺


0.4723909569E⫹03 0.2227478948E⫹03 0.1105739801E⫹03 0.5700063067E⫹02 0.3009912342E⫹02 0.1605963189E⫹02 0.8540532713E⫹01 0.4465417330E⫹01 0.2264263413E⫹01 0.1098347412E⫹01 0.5027597278E⫹00 0.2142147085E⫹00 0.8380428577E⫺01

0.4354113781E⫹04 D⫹ 0.1415449281E⫹04 0.5256557401E⫹03 0.2182902913E⫹03 0.9922170579E⫹02 0.4832051629E⫹02 0.2467870693E⫹02 0.1293875656E⫹02 0.6816404260E⫹01 0.3532028031E⫹01 0.1762025943E⫹01 0.8283867782E⫹00 0.3592527253E⫹00

0.4354113781E⫹04 0.1415449281E⫹04 0.5256557401E⫹03 0.2182902913E⫹03 0.9922170579E⫹02 0.4832051629E⫹02 0.2467870693E⫹02 0.1293875656E⫹02 0.6816404260E⫹01 0.3532028031E⫹01 0.1762025943E⫹01 0.8283867782E⫹00 0.3592527253E⫹00

Atom Exp. # Symm. Exponents Cs

1 2 3 4 5 6 7 8 9 10 11 12

S⫹

Symm. Exponents

0.1532069592E⫹09 P⫺ 0.3868527134E⫹08 0.1082603162E⫹08 0.3335030815E⫹07 0.1123279296E⫹07 0.4108501282E⫹06 0.1620829365E⫹06 0.6850145392E⫹05 0.3080500847E⫹05 0.1464038012E⫹05 0.7303705903E⫹04 0.3798783406E⫹04

Symm. Exponents 0.4723909569E⫹03 0.2227478948E⫹03 0.1105739801E⫹03 0.5700063067E⫹02 0.3009912342E⫹02 0.1605963189E⫹02 0.8540532713E⫹01 0.4465417330E⫹01 0.2264263413E⫹01 0.1098347412E⫹01 0.5027597278E⫹00 0.2142147085E⫹00 0.8380428577E⫺01

Symm. Exponents

0.6627847402E⫹07 P⫹ 0.1317373589E⫹07 0.3041650948E⫹06 0.8057516929E⫹05 0.2418866898E⫹05 0.8127726277E⫹04 0.3019256302E⫹04 0.1224706913E⫹04 0.5357881369E⫹03 0.2496952125E⫹03 0.1224365254E⫹03 0.6239113661E⫹02



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Appendix 5 Atom Exp. # Symm. Exponents 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.2046004312E⫹04 0.1133390953E⫹04 0.6413787245E⫹03 0.3682658058E⫹03 0.2130937472E⫹03 0.1234220935E⫹03 0.7106870953E⫹02 0.4040896208E⫹02 0.2253414755E⫹02 0.1224106405E⫹02 0.6433721139E⫹01 0.3249531476E⫹01 0.1566553179E⫹01 0.7159551918E⫹00 0.3081011193E⫹00 0.1239989104E⫹00 0.4635643704E⫺01 0.1598896094E⫺01

0.3263423071E⫹02 0.1730572921E⫹02 0.9189670325E⫹01 0.4826484589E⫹01 0.2476340393E⫹01 0.1225929925E⫹01 0.5783946922E⫹00 0.2568702217E⫹00 0.1060624103E⫹00

0.5231139634E⫹04 D⫹ 0.1665987031E⫹04 0.6074744909E⫹03 0.2483995491E⫹03 0.1115642604E⫹03 0.5390587636E⫹02 0.2744536845E⫹02 0.1442145758E⫹02 0.7660226080E⫹01 0.4028575979E⫹01 0.2054585760E⫹01 0.9952792138E⫹00 0.4485374834E⫹00

0.5231139634E⫹04 0.1665987031E⫹04 0.6074744909E⫹03 0.2483995491E⫹03 0.1115642604E⫹03 0.5390587636E⫹02 0.2744536845E⫹02 0.1442145758E⫹02 0.7660226080E⫹01 0.4028575979E⫹01 0.2054585760E⫹01 0.9952792138E⫹00 0.4485374834E⫹00

AtomExp. #

Symm. Exponents

Ba

S⫹

1 2 3 4 5 6 7 8 9 10 11

Symm. Exponents

Symm. Exponents

0.1585253443E⫹09 P⫺ 0.3898929057E⫹08 0.1065151567E⫹08 0.3210403876E⫹07 0.1060358809E⫹07 0.3812006891E⫹06 0.1481579261E⫹06 0.6183426772E⫹05 0.2752513173E⫹05 0.1298040301E⫹05 0.6441221986E⫹04

231 Symm. Exponents 0.3263423071E⫹02 0.1730572921E⫹02 0.9189670325E⫹01 0.4826484589E⫹01 0.2476340393E⫹01 0.1225929925E⫹01 0.5783946922E⫹00 0.2568702217E⫹00 0.1060624103E⫹00

Symm. Exponents

0.8077989360E⫹07 P⫹ 0.1530396258E⫹07 0.3403734417E⫹06 0.8768886235E⫹05 0.2581998885E⫹05 0.8573859375E⫹04 0.3168043960E⫹04 0.1285246488E⫹04 0.5648691831E⫹03 0.2653758598E⫹03 0.1314960988E⫹03



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232

Appendix 5


Symm. Exponents

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3 4 5 6 7 8 9 10 11 12 13

D⫺

0.3340661030E⫹04 0.1798636254E⫹04 0.9985361597E⫹03 0.5677491667E⫹03 0.3283856749E⫹03 0.1919153114E⫹03 0.1125629596E⫹03 0.6581192169E⫹02 0.3809777091E⫹02 0.2168918424E⫹02 0.1206141505E⫹02 0.6507698633E⫹01 0.3383712559E⫹01 0.1684065921E⫹01 0.7968691970E⫹00 0.3560738275E⫹00 0.1492384953E⫹00 0.5827364709E⫺01 0.2105604501E⫺01

0.6780911725E⫹02 0.3590648787E⫹02 0.1926429964E⫹02 0.1033272400E⫹02 0.5466930916E⫹01 0.2815304289E⫹01 0.1392342063E⫹01 0.6525161067E⫹00 0.2859216667E⫹00 0.1155844155E⫹00

0.5990803736E⫹04 D⫹ 0.1849760030E⫹04 0.6599296840E⫹03 0.2660878748E⫹03 0.1186016770E⫹03 0.5715945426E⫹02 0.2913466248E⫹02 0.1536207064E⫹02 0.8195973556E⫹01 0.4327683964E⫹01 0.2212116027E⫹01 0.1070658121E⫹01 0.4799295081E⫹00

0.5990803736E⫹04 0.1849760030E⫹04 0.6599296840E⫹03 0.2660878748E⫹03 0.1186016770E⫹03 0.5715945426E⫹02 0.2913466248E⫹02 0.1536207064E⫹02 0.8195973556E⫹01 0.4327683964E⫹01 0.2212116027E⫹01 0.1070658121E⫹01 0.4799295081E⫹00

Symm. Exponents 0.6780911725E⫹02 0.3590648787E⫹02 0.1926429964E⫹02 0.1033272400E⫹02 0.5466930916E⫹01 0.2815304289E⫹01 0.1392342063E⫹01 0.6525161067E⫹00 0.2859216667E⫹00 0.1155844155E⫹00

Symm.

Exponents

Symm.

Exponents

La

S⫹

0.1690768754E⫹09 0.4063743075E⫹08 0.1088808219E⫹08 0.3229452276E⫹07 0.1052998849E⫹07 0.3748156190E⫹06 0.1446330134E⫹06 0.6008227561E⫹05 0.2668232738E⫹05 0.1257965277E⫹05 0.6252452794E⫹04 0.3253405714E⫹04 0.1759953840E⫹04 0.9828987685E⫹03 0.5627687573E⫹03 0.3280454774E⫹03 0.1933259846E⫹03 0.1143846427E⫹03 0.6747388796E⫹02 0.3940610321E⫹02 0.2262664678E⫹02 0.1268455013E⫹02 0.6894413374E⫹01 0.3607913860E⫹01 0.1805181212E⫹01 0.8575520045E⫹00 0.3841000447E⫹00 0.1610800263E⫹00 0.6280897143E⫺01 0.2261275824E⫺01

P⫺


P⫹


D⫺

0.5658367057E⫹04 0.1859304540E⫹04 0.6958393192E⫹03 0.2906824142E⫹03 0.1328410331E⫹03 0.6508812458E⫹02 0.3351040178E⫹02 0.1776716805E⫹02 0.9507576281E⫹01 0.5032534902E⫹01 0.2582387889E⫹01 0.1259001350E⫹01 0.5715487637E⫹00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

233

(continued )

Page 233

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Atom Exp. #

Symm.

Exponents

Symm.

1 2 3 4 5 6 7 8 9 10 11 12 13

D⫹

0.5658367057E⫹04 0.1859304540E⫹04 0.6958393192E⫹03 0.2906824142E⫹03 0.1328410331E⫹03 0.6508812458E⫹02 0.3351040178E⫹02 0.1776716805E⫹02 0.9507576281E⫹01 0.5032534902E⫹01 0.2582387889E⫹01 0.1259001350E⫹01 0.5715487637E⫹00

F⫺

0.2092088640E⫹03 0.8623295998E⫹02 0.3729218805E⫹02 0.1673248900E⫹02 0.7702790609E⫹01 0.3597719080E⫹01 0.1685950311E⫹01 0.7838760711E⫹00 0.3575871552E⫹00 0.1582686985E⫹00 0.6720994200E⫺01

F⫹

0.2092088640E⫹03 0.8623295998E⫹02 0.3729218805E⫹02 0.1673248900E⫹02 0.7702790609E⫹01 0.3597719080E⫹01 0.1685950311E⫹01 0.7838760711E⫹00 0.3575871552E⫹00 0.1582686985E⫹00 0.6720994200E⫺01

Exponents

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

La

1 2 3 4 5 6 7 8 9 10 11 12 13

S⫹

0.1693237518E⫹09 0.4026346052E⫹08 0.1068651416E⫹08 0.3143715721E⫹07 0.1017848604E⫹07 0.3601689774E⫹06 0.1383130323E⫹06 0.5724058554E⫹05 0.2535010020E⫹05 0.1192999944E⫹05 0.5924296407E⫹04 0.3082614600E⫹04 0.1668930537E⫹04

P⫺

0.1006323333E⫹08 0.1896305287E⫹07 0.4193792744E⫹06 0.1074162164E⫹06 0.3144366069E⫹05 0.1038084959E⫹05 0.3814217268E⫹04 0.1539171004E⫹04 0.6731524308E⫹03 0.3148627237E⫹03 0.1554341871E⫹03 0.7991448358E⫹02 0.4222739449E⫹02

P⫹

0.1006323333E⫹08 0.1896305287E⫹07 0.4193792744E⫹06 0.1074162164E⫹06 0.3144366069E⫹05 0.1038084959E⫹05 0.3814217268E⫹04 0.1539171004E⫹04 0.6731524308E⫹03 0.3148627237E⫹03 0.1554341871E⫹03 0.7991448358E⫹02 0.4222739449E⫹02

Page 234

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Exp. #

234

Atom

0.2263023662E⫹02 0.1213797140E⫹02 0.6429859964E⫹01 0.3319643854E⫹01 0.1648360619E⫹01 0.7768191256E⫹00 0.3428709697E⫹00 0.1398687948E⫹00

5/19/2007 21:38

D⫹

0.4775657703E⫹04 0.1694015057E⫹04 0.6625280324E⫹03 0.2813420277E⫹03 0.1277464308E⫹03 0.6107833490E⫹02 0.3028240056E⫹02 0.1533199089E⫹02 0.7806428607E⫹01 0.3936336921E⫹01 0.1935791648E⫹01 0.9143066559E⫹00 0.4084439970E⫹00 0.1699499202E⫹00 0.6486309520E⫺01

Page 235



D⫺

0.2263023662E⫹02 0.1213797140E⫹02 0.6429859964E⫹01 0.3319643854E⫹01 0.1648360619E⫹01 0.7768191256E⫹00 0.3428709697E⫹00 0.1398687948E⫹00

235

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.9335645511E⫹03 0.5357818380E⫹03 0.3132706077E⫹03 0.1853061447E⫹03 0.1101156340E⫹03 0.6527511134E⫹02 0.3832977027E⫹02 0.2213936965E⫹02 0.1249065490E⫹02 0.6835126755E⫹01 0.3602465450E⫹01 0.1815914711E⫹01 0.8693285102E⫹00 0.3924783772E⫹00 0.1659362864E⫹00 0.6523960148E⫺01 0.2368514331E⫺01

Appendix 5

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Exponents

Symm.

Exponents

Ce

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺

0.6053135723E⫹04 0.2035709607E⫹04 0.7709593956E⫹03 0.3230595562E⫹03 0.1471716735E⫹03 0.7161624660E⫹02 0.3657629553E⫹02 0.1926388406E⫹02 0.1028014197E⫹02 0.5461609303E⫹01 0.2838334329E⫹01 0.1417695678E⫹01 0.6687043810E⫹00

Page 236

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Exp. #

236

Atom


D⫹

0.6053135723E⫹04 0.2035709607E⫹04 0.7709593956E⫹03 0.3230595562E⫹03 0.1471716735E⫹03 0.7161624660E⫹02 0.3657629553E⫹02 0.1926388406E⫹02 0.1028014197E⫹02 0.5461609303E⫹01 0.2838334329E⫹01 0.1417695678E⫹01 0.6687043810E⫹00

F⫺

0.2410059399E⫹03 0.9641818099E⫹02 0.4071502271E⫹02 0.1793716347E⫹02 0.8148824502E⫹01 0.3773262804E⫹01 0.1760187013E⫹01 0.8176341370E⫹00 0.3738147558E⫹00 0.1662607996E⫹00 0.7110466020E⫺01

F⫹

0.2410059399E⫹03 0.9641818099E⫹02 0.4071502271E⫹02 0.1793716347E⫹02 0.8148824502E⫹01 0.3773262804E⫹01 0.1760187013E⫹01 0.8176341370E⫹00 0.3738147558E⫹00 0.1662607996E⫹00 0.7110466020E⫺01

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Ce

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S⫹

0.1717256364E⫹09 0.4172259351E⫹08 0.1128832905E⫹08 0.3377585130E⫹07 0.1109934383E⫹07 0.3978311103E⫹06 0.1544566788E⫹06 0.6450854322E⫹05 0.2878236301E⫹05 0.1362480847E⫹05 0.6795559058E⫹04 0.3546556477E⫹04 0.1923408588E⫹04 0.1076503092E⫹04 0.6174960017E⫹03 0.3605157428E⫹03

P⫺


P⫹


D⫺


5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

21:38 Page 237

Appendix 5 237

(continued )

Symm.

0.2127558998E⫹03 0.1260382492E⫹03 0.7443592018E⫹02 0.4352294647E⫹02 0.2502105528E⫹02 0.1404561111E⫹02 0.7645738127E⫹01 0.4008095291E⫹01 0.2009521121E⫹01 0.9569276604E⫹00 0.4298270224E⫹00 0.1808558138E⫹00 0.7079318686E⫺01 0.2560157950E⫺01

0.3320368909E⫹01 0.1642656911E⫹01 0.7698792089E⫹00 0.3373125997E⫹00 0.1363313794E⫹00

Exponents

Symm.

Exponents

0.3320368909E⫹01 0.1642656911E⫹01 0.7698792089E⫹00 0.3373125997E⫹00 0.1363313794E⫹00

F⫺

0.2480995597E⫹03 0.1037479793E⫹03 0.4578668838E⫹02 0.2109137991E⫹02 0.1002945121E⫹02 0.4869181916E⫹01 0.2386941227E⫹01 0.1168515792E⫹01 0.5649822740E⫹00 0.2668349263E⫹00 0.1217470319E⫹00

F⫹

0.2480995597E⫹03 0.1037479793E⫹03 0.4578668838E⫹02 0.2109137991E⫹02 0.1002945121E⫹02 0.4869181916E⫹01 0.2386941227E⫹01 0.1168515792E⫹01 0.5649822740E⫹00 0.2668349263E⫹00 0.1217470319E⫹00

Page 238


Symm.

21:38

D⫹

Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Symm.

Appendix 5

17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

238

Atom

Exponents

Symm.

Exponents

Symm.

Exponents

Pr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺

0.7298437961E⫹04 0.2377619340E⫹04 0.8745546960E⫹03 0.3568584343E⫹03 0.1587089376E⫹03 0.7558494880E⫹02 0.3787301441E⫹02 0.1961630137E⫹02 0.1031879941E⫹02 0.5416242480E⫹01 0.2787122151E⫹01 0.1381449370E⫹01 0.6479882030E⫹00

239

(continued )

Page 239

Symm.

21:38

Exponents

5/19/2007

Symm.


Exp. #

Appendix 5

Atom

Exponents

Symm.

Exponents

Appendix 5

0.9842830204E⫹07 0.1897373842E⫹07 0.4278989462E⫹06 0.1114286768E⫹06 0.3306985063E⫹05 0.1103974528E⫹05 0.4091565311E⫹04 0.1661634856E⫹04 0.7298093556E⫹03 0.3421551186E⫹03 0.1690006965E⫹03 0.8679962345E⫹02 0.4575337522E⫹02 0.2442960466E⫹02

D⫺

0.7063732040E⫹04 0.2307956999E⫹04 0.8509658031E⫹03 0.3479136392E⫹03 0.1549850498E⫹03 0.7391817219E⫹02 0.3708854065E⫹02 0.1923709273E⫹02 0.1013524738E⫹02 0.5329770926E⫹01 0.2748818486E⫹01 0.1366252487E⫹01 0.6430537944E⫹00

Symm.

Exponents

Symm.

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13

D⫹

0.7298437961E⫹04 0.2377619340E⫹04 0.8745546960E⫹03 0.3568584343E⫹03 0.1587089376E⫹03 0.7558494880E⫹02 0.3787301441E⫹02 0.1961630137E⫹02 0.1031879941E⫹02 0.5416242480E⫹01 0.2787122151E⫹01 0.1381449370E⫹01 0.6479882030E⫹00

F⫺

0.2575287643E⫹03 0.1049843308E⫹03 0.4502499952E⫹02 0.2008110975E⫹02 0.9206630858E⫹01 0.4289108633E⫹01 0.2007062628E⫹01 0.9325167316E⫹00 0.4252344477E⫹00 0.1881267748E⫹00 0.7981737008E⫺01

F⫹

0.2575287643E⫹03 0.1049843308E⫹03 0.4502499952E⫹02 0.2008110975E⫹02 0.9206630858E⫹01 0.4289108633E⫹01 0.2007062628E⫹01 0.9325167316E⫹00 0.4252344477E⫹00 0.1881267748E⫹00 0.7981737008E⫺01

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Nd

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.1691153375E⫹09 0.4122575013E⫹08 0.1118983006E⫹08 0.3358376339E⫹07 0.1106796485E⫹07 0.3977592719E⫹06 0.1547991518E⫹06 0.6478793794E⫹05 0.2895867233E⫹05 0.1372788516E⫹05 0.6854104494E⫹04 0.3579327942E⫹04 0.1941500643E⫹04 0.1086276682E⫹04

P⫺

0.9842830204E⫹07 0.1897373842E⫹07 0.4278989462E⫹06 0.1114286768E⫹06 0.3306985063E⫹05 0.1103974528E⫹05 0.4091565311E⫹04 0.1661634856E⫹04 0.7298093556E⫹03 0.3421551186E⫹03 0.1690006965E⫹03 0.8679962345E⫹02 0.4575337522E⫹02 0.2442960466E⫹02

P⫹

Page 240

Exponents

21:38

Symm.

5/19/2007

Exponents

Exp. #


Symm.

240

Atom

21:38

F⫺

0.2566803987E⫹03 0.1065449337E⫹03 0.4642639788E⫹02 0.2099296682E⫹02 0.9737418842E⫹01 0.4579938447E⫹01 0.2159264015E⫹01 0.1008710927E⫹01 0.4615589043E⫹00 0.2044897427E⫹00 0.8671305100E⫺01

F⫹

0.2566803987E⫹02 0.1065449337E⫹03 0.4642639788E⫹02 0.2099296682E⫹02 0.9737418842E⫹01 0.4579938447E⫹01 0.2159264015E⫹01 0.1008710927E⫹01 0.4615589043E⫹00 0.2044897427E⫹00 0.8671305100E⫺01

Page 241

0.7063732040E⫹04 0.2307956999E⫹04 0.8509658031E⫹03 0.3479136392E⫹03 0.1549850498E⫹03 0.7391817219E⫹02 0.3708854065E⫹02 0.1923709273E⫹02 0.1013524738E⫹02 0.5329770926E⫹01 0.2748818486E⫹01 0.1366252487E⫹01 0.6430537944E⫹00

0.1304096716E⫹02 0.6869344233E⫹01 0.3524073285E⫹01 0.1737844086E⫹01 0.8130644512E⫹00 0.3562045605E⫹00 0.1442269345E⫹00

5/19/2007

D⫹

0.1304096716E⫹02 0.6869344233E⫹01 0.3524073285E⫹01 0.1737844086E⫹01 0.8130644512E⫹00 0.3562045605E⫹00 0.1442269345E⫹00


1 2 3 4 5 6 7 8 9 10 11 12 13


Appendix 5

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

241

Symm.

Exponents

Symm.

Exponents

Pm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺

0.6994773025E⫹04 0.2290822475E⫹04 0.8475971160E⫹03 0.3480874383E⫹03 0.1558865680E⫹03 0.7479475150E⫹02 0.3777428508E⫹02 0.1972898328E⫹02 0.1046928676E⫹02 0.5545670285E⫹01 0.2880952996E⫹01 0.1442062543E⫹01 0.6833110374E⫹00

Page 242

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Exp. #

242

Atom


D⫹

0.6994773025E⫹04 0.2290822475E⫹04 0.8475971160E⫹03 0.3480874383E⫹03 0.1558865680E⫹03 0.7479475150E⫹02 0.3777428508E⫹02 0.1972898328E⫹02 0.1046928676E⫹02 0.5545670285E⫹01 0.2880952996E⫹01 0.1442062543E⫹01 0.6833110374E⫹00

F⫺

0.2725688394E⫹03 0.1113164159E⫹03 0.4814403074E⫹02 0.2175928159E⫹02 0.1014107927E⫹02 0.4809278991E⫹01 0.2290079626E⫹01 0.1080474845E⫹01 0.4984147102E⫹00 0.2218185213E⫹00 0.9398405338E⫺01

F⫹

0.2725688394E⫹03 0.1113164159E⫹03 0.4814403074E⫹02 0.2175928159E⫹02 0.1014107927E⫹02 0.4809278991E⫹01 0.2290079626E⫹01 0.1080474845E⫹01 0.4984147102E⫹00 0.2218185213E⫹00 0.9398405338E⫺01

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Sm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S⫹


P⫺


P⫹


D⫺

0.7706916812E⫹04 0.2497594654E⫹04 0.9157489831E⫹03 0.3731772534E⫹03 0.1660386290E⫹03 0.7923725004E⫹02 0.3984253457E⫹02 0.2073637579E⫹02 0.1097382701E⫹02 0.5800878238E⫹01 0.3008921100E⫹01 0.1504458755E⫹01 0.7123168769E⫹00

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

21:38 Page 243

Appendix 5 243

(continued )

Symm.

0.2500577636E⫹03 0.1433609028E⫹03 0.8189813353E⫹02 0.4633383427E⫹02 0.2580070094E⫹02 0.1405406194E⫹02 0.7442817731E⫹01 0.3808606755E⫹01 0.1871613546E⫹01 0.8778404535E⫹00 0.3905642005E⫹00 0.1638229621E⫹00 0.6438588683E⫺01 0.2356504229E⫺01

0.4013539806E⫹01 0.1951809233E⫹01 0.9002464566E⫹00 0.3887242394E⫹00 0.1551028807E⫹00

Exponents

Symm.

Exponents

0.4013539806E⫹01 0.1951809233E⫹01 0.9002464566E⫹00 0.3887242394E⫹00 0.1551028807E⫹00

F⫺

0.3126018494E⫹03 0.1252448045E⫹03 0.5367478840E⫹02 0.2422282759E⫹02 0.1133247160E⫹02 0.5410920756E⫹01 0.2595767977E⫹01 0.1231714796E⫹01 0.5691221221E⫹00 0.2520886111E⫹00 0.1053791915E⫹00

F⫹

0.3126018494E⫹03 0.1252448045E⫹03 0.5367478840E⫹02 0.2422282759E⫹02 0.1133247160E⫹02 0.5410920756E⫹01 0.2595767977E⫹01 0.1231714796E⫹01 0.5691221221E⫹00 0.2520886111E⫹00 0.1053791915E⫹00

Page 244

0.7706916812E⫹04 0.2497594654E⫹04 0.9157489831E⫹03 0.3731772534E⫹03 0.1660386290E⫹03 0.7923725004E⫹02 0.3984253457E⫹02 0.2073637579E⫹02 0.1097382701E⫹02 0.5800878238E⫹01 0.3008921100E⫹01 0.1504458755E⫹01 0.7123168769E⫹00

Symm.

21:38

D⫹

Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

244

Atom

Exponents

Symm.

Exponents

Symm.

Exponents

Eu

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺

0.7808850447E⫹04 0.2635076640E⫹04 0.9949009856E⫹03 0.4133760494E⫹03 0.1859036743E⫹03 0.8900332368E⫹02 0.4461680211E⫹02 0.2303366206E⫹02 0.1204476287E⫹02 0.6274839246E⫹01 0.3203128894E⫹01 0.1575839788E⫹01 0.7348757528E⫹00

245

(continued )

Page 245

Symm.

21:38

Exponents

5/19/2007

Symm.


Exp. #

Appendix 5

Atom

Exponents

Symm.

Exponents

Appendix 5

0.1296942432E⫹08 0.2565070216E⫹07 0.5885442607E⫹06 0.1547472734E⫹06 0.4605678111E⫹05 0.1532687274E⫹05 0.5633335375E⫹04 0.2258876272E⫹04 0.9761054853E⫹03 0.4489947166E⫹03 0.2171643682E⫹03 0.1090941497E⫹03 0.5622670093E⫹02 0.2936802118E⫹02

D⫺

0.8963245240E⫹04 0.2903473411E⫹04 0.1070051183E⫹04 0.4401302502E⫹03 0.1982000230E⫹03 0.9585788490E⫹02 0.4884380500E⫹02 0.2572198531E⫹02 0.1373310304E⫹02 0.7292188930E⫹01 0.3777704920E⫹01 0.1872987518E⫹01 0.8718350225E⫹00

Symm.

Exponents

Symm.

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13

D⫹

0.7808850447E⫹04 0.2635076640E⫹04 0.9949009856E⫹03 0.4133760494E⫹03 0.1859036743E⫹03 0.8900332368E⫹02 0.4461680211E⫹02 0.2303366206E⫹02 0.1204476287E⫹02 0.6274839246E⫹01 0.3203128894E⫹01 0.1575839788E⫹01 0.7348757528E⫹00

F⫺

0.3141936420E⫹03 0.1292530954E⫹03 0.5668323094E⫹02 0.2610001359E⫹02 0.1242801501E⫹02 0.6027540140E⫹01 0.2932636039E⫹01 0.1409802721E⫹01 0.6595429015E⫹00 0.2957431487E⫹00 0.1251917406E⫹00

F⫹

0.3141936420E⫹03 0.1292530954E⫹03 0.5668323094E⫹02 0.2610001359E⫹02 0.1242801501E⫹02 0.6027540140E⫹01 0.2932636039E⫹01 0.1409802721E⫹01 0.6595429015E⫹00 0.2957431487E⫹00 0.1251917406E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Gd

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.1691090010E⫹09 0.4478131255E⫹08 0.1305765757E⫹08 0.4166019649E⫹07 0.1445154089E⫹07 0.5416179157E⫹06 0.2179250386E⫹06 0.9354179661E⫹05 0.4256356225E⫹05 0.2040107891E⫹05 0.1023529656E⫹05 0.5341079879E⫹04 0.2880635340E⫹04 0.1595611968E⫹04

P⫺

0.1296942432E⫹08 0.2565070216E⫹07 0.5885442607E⫹06 0.1547472734E⫹06 0.4605678111E⫹05 0.1532687274E⫹05 0.5633335375E⫹04 0.2258876272E⫹04 0.9761054853E⫹03 0.4489947166E⫹03 0.2171643682E⫹03 0.1090941497E⫹03 0.5622670093E⫹02 0.2936802118E⫹02

P⫹

Page 246

Exponents

21:38

Symm.

5/19/2007

Exponents

Exp. #


Symm.

246

Atom

21:38

F⫺

0.3870580284E⫹03 0.1591000454E⫹03 0.7054733286E⫹02 0.3313412018E⫹02 0.1618539532E⫹02 0.8074073155E⫹01 0.4038797006E⫹01 0.1989160489E⫹01 0.9471408152E⫹00 0.4281093228E⫹00 0.1803676722E⫹00

F⫹

0.3870580284E⫹03 0.1591000454E⫹03 0.7054733286E⫹02 0.3313412018E⫹02 0.1618539532E⫹02 0.8074073155E⫹01 0.4038797006E⫹01 0.1989160489E⫹01 0.9471408152E⫹00 0.4281093228E⫹00 0.1803676722E⫹00

Page 247

0.8963245240E⫹04 0.2903473411E⫹04 0.1070051183E⫹04 0.4401302502E⫹03 0.1982000230E⫹03 0.9585788490E⫹02 0.4884380500E⫹02 0.2572198531E⫹02 0.1373310304E⫹02 0.7292188930E⫹01 0.3777704920E⫹01 0.1872987518E⫹01 0.8718350225E⫹00

0.1535537413E⫹02 0.7938938707E⫹01 0.4009068880E⫹01 0.1953288094E⫹01 0.9069712012E⫹00 0.3964494273E⫹00 0.1611431649E⫹00

5/19/2007

D⫹

0.1535537413E⫹02 0.7938938707E⫹01 0.4009068880E⫹01 0.1953288094E⫹01 0.9069712012E⫹00 0.3964494273E⫹00 0.1611431649E⫹00


1 2 3 4 5 6 7 8 9 10 11 12 13


Appendix 5

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

247

Symm.

Exponents

Symm.

Exponents

Gd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺


Page 248

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Exp. #

248

Atom


D⫹


F⫺

0.4313300740E⫹03 0.1840308903E⫹03 0.8264134856E⫹02 0.3863836863E⫹02 0.1860561036E⫹02 0.9127710384E⫹01 0.4512961290E⫹01 0.2224499066E⫹01 0.1081341190E⫹01 0.5127931321E⫹00 0.2346711088E⫹00

F⫹

0.4313300740E⫹03 0.1840308903E⫹03 0.8264134856E⫹02 0.3863836863E⫹02 0.1860561036E⫹02 0.9127710384E⫹01 0.4512961290E⫹01 0.2224499066E⫹01 0.1081341190E⫹01 0.5127931321E⫹00 0.2346711088E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Tb

1 2 3 4 5 6 7 8 9 10 11 12 13

S⫹

0.1690551753E⫹09 0.4398733482E⫹08 0.1263577992E⫹08 0.3981411106E⫹07 0.1367155460E⫹07 0.5083116069E⫹06 0.2033100001E⫹06 0.8691392925E⫹05 0.3945547697E⫹05 0.1889721580E⫹05 0.9487380070E⫹04 0.4960644079E⫹04 0.2683851426E⫹04

P⫺

0.1492826638E⫹08 0.2936009189E⫹07 0.6702983935E⫹06 0.1754670133E⫹06 0.5202272888E⫹05 0.1725499825E⫹05 0.6324331952E⫹04 0.2530149282E⫹04 0.1091347830E⫹04 0.5013252636E⫹03 0.2422529945E⫹03 0.1216369950E⫹03 0.6268490611E⫹02

P⫹

0.1492826638E⫹08 0.2936009189E⫹07 0.6702983935E⫹06 0.1754670133E⫹06 0.5202272888E⫹05 0.1725499825E⫹05 0.6324331952E⫹04 0.2530149282E⫹04 0.1091347830E⫹04 0.5013252636E⫹03 0.2422529945E⫹03 0.1216369950E⫹03 0.6268490611E⫹02

D⫺

0.9775593468E⫹04 0.3124141483E⫹04 0.1138852299E⫹04 0.4643215893E⫹03 0.2076111782E⫹03 0.9982253313E⫹02 0.5060786944E⫹02 0.2652679325E⫹02 0.1409594829E⫹02 0.7445816119E⫹01 0.3833578807E⫹01 0.1886410393E⫹01 0.8699079071E⫹00

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

21:38 Page 249

Appendix 5 249

(continued )

Symm.

0.9775593468E⫹04 0.3124141483E⫹04 0.1138852299E⫹04 0.4643215893E⫹03 0.2076111782E⫹03 0.9982253313E⫹02 0.5060786944E⫹02 0.2652679325E⫹02 0.1409594829E⫹02 0.7445816119E⫹01 0.3833578807E⫹01 0.1886410393E⫹01 0.8699079071E⫹00

0.3275026731E⫹02 0.1713464433E⫹02 0.8867424887E⫹01 0.4483693586E⫹01 0.2187983883E⫹01 0.1017831533E⫹01 0.4458462087E⫹00 0.1816458378E⫹00

Exponents

Symm.

Exponents

0.3275026731E⫹02 0.1713464433E⫹02 0.8867424887E⫹01 0.4483693586E⫹01 0.2187983883E⫹01 0.1017831533E⫹01 0.4458462087E⫹00 0.1816458378E⫹00

F⫺

0.5092038278E⫹03 0.2029714741E⫹03 0.8760969582E⫹02 0.4019878407E⫹02 0.1924806142E⫹02 0.9441568908E⫹01 0.4657514199E⫹01 0.2268228560E⫹01 0.1070561954E⫹01 0.4807274446E⫹00 0.2016127094E⫹00

F⫹

0.5092038278E⫹03 0.2029714741E⫹03 0.8760969582E⫹02 0.4019878407E⫹02 0.1924806142E⫹02 0.9441568908E⫹01 0.4657514199E⫹01 0.2268228560E⫹01 0.1070561954E⫹01 0.4807274446E⫹00 0.2016127094E⫹00

Page 250

D⫹

Symm.

21:38

0.1492769821E⫹04 0.8480599182E⫹03 0.4889279721E⫹03 0.2842059904E⫹03 0.1654919632E⫹03 0.9590942032E⫹02 0.5496315359E⫹02 0.3094520610E⫹02 0.1700637754E⫹02 0.9063841322E⫹01 0.4654576295E⫹01 0.2288235587E⫹01 0.1069939894E⫹01 0.4727607691E⫹00 0.1961246696E⫹00 0.7589568851E⫺01 0.2721957950E⫺01

Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

250

Atom

Exponents

Symm.

Exponents

Symm.

Exponents

Dy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺

0.1065705362E⫹05 0.3336605330E⫹04 0.1195738603E⫹04 0.4808082674E⫹03 0.2126436462E⫹03 0.1013953980E⫹03 0.5109871400E⫹02 0.2667891300E⫹02 0.1414598744E⫹02 0.7466996098E⫹01 0.3846330884E⫹01 0.1895291276E⫹01 0.8757384756E⫹00

251

(continued )

Page 251

Symm.

21:38

Exponents

5/19/2007

Symm.


Exp. #

Appendix 5

Atom

Exponents

Symm.

Exponents

Appendix 5

0.1367774085E⫹08 0.2718206220E⫹07 0.6263434920E⫹06 0.1653043868E⫹06 0.4936024839E⫹05 0.1647293359E⫹05 0.6069385437E⫹04 0.2438811049E⫹04 0.1055724889E⫹04 0.4863422990E⫹03 0.2355220152E⫹03 0.1184400718E⫹03 0.6109743931E⫹02 0.3193625997E⫹02

D⫺

0.1076046342E⫹05 0.3416808552E⫹04 0.1236196284E⫹04 0.4998072270E⫹03 0.2214813339E⫹03 0.1055024201E⫹03 0.5298459232E⫹02 0.2751502095E⫹02 0.1449086683E⫹02 0.7590918723E⫹01 0.3879189247E⫹01 0.1896729655E⫹01 0.8702798618E⫹00

Symm.

Exponents

Symm.

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13

D⫹

0.1065705362E⫹05 0.3336605330E⫹04 0.1195738603E⫹04 0.4808082674E⫹03 0.2126436462E⫹03 0.1013953980E⫹03 0.5109871400E⫹02 0.2667891300E⫹02 0.1414598744E⫹02 0.7466996098E⫹01 0.3846330884E⫹01 0.1895291276E⫹01 0.8757384756E⫹00

F⫺

0.4205025022E⫹03 0.1706711037E⫹03 0.7476953295E⫹02 0.3471998835E⫹02 0.1678192126E⫹02 0.8291401888E⫹01 0.4112021771E⫹01 0.2010207876E⫹01 0.9512652543E⫹00 0.4279119062E⫹00 0.1796867910E⫹00

F⫹

0.4205025022E⫹03 0.1706711037E⫹03 0.7476953295E⫹02 0.3471998835E⫹02 0.1678192126E⫹02 0.8291401888E⫹01 0.4112021771E⫹01 0.2010207876E⫹01 0.9512652543E⫹00 0.4279119062E⫹00 0.1796867910E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Ho

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.1694621245E⫹09 0.4340862049E⫹08 0.1230174031E⫹08 0.3831645861E⫹07 0.1303087002E⫹07 0.4806973301E⫹06 0.1910826514E⫹06 0.8131357664E⫹05 0.3679917912E⫹05 0.1759492677E⫹05 0.8829824411E⫹04 0.4620329479E⫹04 0.2504327405E⫹04 0.1396842855E⫹04

P⫺

0.1367774085E⫹08 0.2718206220E⫹07 0.6263434920E⫹06 0.1653043868E⫹06 0.4936024839E⫹05 0.1647293359E⫹05 0.6069385437E⫹04 0.2438811049E⫹04 0.1055724889E⫹04 0.4863422990E⫹03 0.2355220152E⫹03 0.1184400718E⫹03 0.6109743931E⫹02 0.3193625997E⫹02

P⫹

Page 252

Exponents

21:38

Symm.

5/19/2007

Exponents

Exp. #


Symm.

252

Atom

21:38

F⫺

0.4260316537E⫹03 0.1746420740E⫹03 0.7708270062E⫹02 0.3598343492E⫹02 0.1745109141E⫹02 0.8636812138E⫹01 0.4284813092E⫹01 0.2093121533E⫹01 0.9889578632E⫹00 0.4439341292E⫹00 0.1859744616E⫹00

F⫹

0.4260316537E⫹03 0.1746420740E⫹03 0.7708270062E⫹02 0.3598343492E⫹02 0.1745109141E⫹02 0.8636812138E⫹01 0.4284813092E⫹01 0.2093121533E⫹01 0.9889578632E⫹00 0.4439341292E⫹00 0.1859744616E⫹00

Page 253

0.1076046342E⫹05 0.3416808552E⫹04 0.1236196284E⫹04 0.4998072270E⫹03 0.2214813339E⫹03 0.1055024201E⫹03 0.5298459232E⫹02 0.2751502095E⫹02 0.1449086683E⫹02 0.7590918723E⫹01 0.3879189247E⫹01 0.1896729655E⫹01 0.8702798618E⫹00

0.1670941734E⫹02 0.8644388328E⫹01 0.4367998162E⫹01 0.2129537773E⫹01 0.9895152154E⫹00 0.4328857992E⫹00 0.1761235542E⫹00

5/19/2007

D⫹

0.1670941734E⫹02 0.8644388328E⫹01 0.4367998162E⫹01 0.2129537773E⫹01 0.9895152154E⫹00 0.4328857992E⫹00 0.1761235542E⫹00


1 2 3 4 5 6 7 8 9 10 11 12 13


Appendix 5

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

253

Symm.

Exponents

Symm.

Exponents

Er

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺

0.1022804216E⫹05 0.3251031433E⫹04 0.1180358384E⫹04 0.4798481771E⫹03 0.2141037275E⫹03 0.1027799033E⫹03 0.5203420901E⫹02 0.2723326200E⫹02 0.1444353522E⫹02 0.7609281614E⫹01 0.3903402183E⫹01 0.1911197698E⫹01 0.8755153699E⫹00

Page 254

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Exp. #

254

Atom


D⫹

0.1022804216E⫹05 0.3251031433E⫹04 0.1180358384E⫹04 0.4798481771E⫹03 0.2141037275E⫹03 0.1027799033E⫹03 0.5203420901E⫹02 0.2723326200E⫹02 0.1444353522E⫹02 0.7609281614E⫹01 0.3903402183E⫹01 0.1911197698E⫹01 0.8755153699E⫹00

F⫺

0.4810157991E⫹03 0.1961000236E⫹03 0.8616533139E⫹02 0.4006951590E⫹02 0.1936476578E⫹02 0.9550300247E⫹01 0.4719743861E⫹01 0.2295125653E⫹01 0.1078378299E⫹01 0.4807313585E⫹00 0.1996596975E⫹00

F⫹

0.4810157991E⫹03 0.1961000236E⫹03 0.8616533139E⫹02 0.4006951590E⫹02 0.1936476578E⫹02 0.9550300247E⫹01 0.4719743861E⫹01 0.2295125653E⫹01 0.1078378299E⫹01 0.4807313585E⫹00 0.1996596975E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Tm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

S⫹

0.1690532079E⫹09 0.4299514555E⫹08 0.1210227324E⫹08 0.3745561724E⫹07 0.1266252709E⫹07 0.4645454298E⫹06 0.1837347583E⫹06 0.7783245513E⫹05 0.3508224402E⫹05 0.1671560802E⫹05 0.8364040382E⫹04 0.4366366067E⫹04 0.2362572780E⫹04 0.1316322274E⫹04 0.7502434218E⫹03

P⫺


P⫹


D⫺

0.1121380252E⫹05 0.3471149164E⫹04 0.1232828837E⫹04 0.4923200683E⫹03 0.2166275329E⫹03 0.1029217687E⫹03 0.5174095177E⫹02 0.2697126298E⫹02 0.1428610916E⫹02 0.7534918659E⫹01 0.3877941098E⫹01 0.1908481687E⫹01 0.8801263971E⫹00

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

21:38 Page 255

Appendix 5 255

(continued )

Symm.

0.1121380252E⫹05 0.3471149164E⫹04 0.1232828837E⫹04 0.4923200683E⫹03 0.2166275329E⫹03 0.1029217687E⫹03 0.5174095177E⫹02 0.2697126298E⫹02 0.1428610916E⫹02 0.7534918659E⫹01 0.3877941098E⫹01 0.1908481687E⫹01 0.8801263971E⫹00

0.9168311562E⫹01 0.4644512955E⫹01 0.2269196098E⫹01 0.1056218097E⫹01 0.4626511008E⫹00 0.1883825657E⫹00

Exponents

Symm.

Exponents

0.9168311562E⫹01 0.4644512955E⫹01 0.2269196098E⫹01 0.1056218097E⫹01 0.4626511008E⫹00 0.1883825657E⫹00

F⫺

0.4933650201E⫹03 0.2036727886E⫹03 0.9033005444E⫹02 0.4224827530E⫹02 0.2045530645E⫹02 0.1006389581E⫹02 0.4938922968E⫹01 0.2373270250E⫹01 0.1096109149E⫹01 0.4776336522E⫹00 0.1927584316E⫹00

F⫹

0.4933650201E⫹03 0.2036727886E⫹03 0.9033005444E⫹02 0.4224827530E⫹02 0.2045530645E⫹02 0.1006389581E⫹02 0.4938922968E⫹01 0.2373270250E⫹01 0.1096109149E⫹01 0.4776336522E⫹00 0.1927584316E⫹00

Page 256

D⫹

Symm.

21:38

0.4345663006E⫹03 0.2541409958E⫹03 0.1490765020E⫹03 0.8713856328E⫹02 0.5042306978E⫹02 0.2869569883E⫹02 0.1595599591E⫹02 0.8611951288E⫹01 0.4482297742E⫹01 0.2234972985E⫹01 0.1060638151E⫹01 0.4759228445E⫹00 0.2006001024E⫹00 0.7890453982E⫺01 0.2877398583E⫺01

Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

256

Atom

Symm.

Exponents

Symm.

Exponents

Yb

S⫹


P⫺


P⫹


D⫺

0.1189615062E⫹05 0.3656012412E⫹04 0.1290997690E⫹04 0.5131039450E⫹03 0.2248499606E⫹03 0.1064224713E⫹03 0.5329327561E⫹02 0.2766021351E⫹02 0.1457565799E⫹02 0.7638987972E⫹01 0.3900533280E⫹01 0.1900807524E⫹01 0.8660117377E⫹00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

257

(continued )

Page 257

Exponents

21:38

Symm.

5/19/2007

Exponents


Symm.

Appendix 5

Atom Exp. #

Exponents

Symm.

Exponents

Appendix 5

0.2029389073E⫹08 0.4122821908E⫹07 0.9668915279E⫹06 0.2588291998E⫹06 0.7819869987E⫹05 0.2636543593E⫹05 0.9808846862E⫹04 0.3981502008E⫹04 0.1743493464E⫹04 0.8143976677E⫹03 0.4012305051E⫹03 0.2061536396E⫹03 0.1092260228E⫹03 0.5900620702E⫹02

D⫺

0.1464842357E⫹05 0.5343492749E⫹04 0.2130929377E⫹04 0.9175548485E⫹03 0.4213313805E⫹03 0.2037764634E⫹03 0.1025256015E⫹03 0.5299908481E⫹02 0.2780172878E⫹02 0.1461679885E⫹02 0.7607105210E⫹01 0.3870641138E⫹01 0.1901745400E⫹01 0.8911226256E⫹00

Symm.

Exponents

Symm.

Exponents

1 2 3 4 5 6 7 8 9 10 11 12 13

D⫹

0.1189615062E⫹05 0.3656012412E⫹04 0.1290997690E⫹04 0.5131039450E⫹03 0.2248499606E⫹03 0.1064224713E⫹03 0.5329327561E⫹02 0.2766021351E⫹02 0.1457565799E⫹02 0.7638987972E⫹01 0.3900533280E⫹01 0.1900807524E⫹01 0.8660117377E⫹00

F⫺

0.4710253235E⫹03 0.1929199564E⫹03 0.8455516396E⫹02 0.3898370751E⫹02 0.1858472365E⫹02 0.9005523830E⫹01 0.4360052285E⫹01 0.2073262062E⫹01 0.9518016014E⫹00 0.4146850026E⫹00 0.1685465376E⫹00

F⫹

0.4710253235E⫹03 0.1929199564E⫹03 0.8455516396E⫹02 0.3898370751E⫹02 0.1858472365E⫹02 0.9005523830E⫹01 0.4360052285E⫹01 0.2073262062E⫹01 0.9518016014E⫹00 0.4146850026E⫹00 0.1685465376E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Lu

1 2 3 4 5 6 7 8 9 10 11 12 13 14

S⫹

0.4085370609E⫹09 0.9995341697E⫹08 0.2715733446E⫹08 0.8138323089E⫹07 0.2671627988E⫹07 0.9542129206E⫹06 0.3682797442E⫹06 0.1525487258E⫹06 0.6735541682E⫹05 0.3148506861E⫹05 0.1547532977E⫹05 0.7943525860E⫹04 0.4229215625E⫹04 0.2319607133E⫹04

P⫺

0.2029389073E⫹08 0.4122821908E⫹07 0.9668915279E⫹06 0.2588291998E⫹06 0.7819869987E⫹05 0.2636543593E⫹05 0.9808846862E⫹04 0.3981502008E⫹04 0.1743493464E⫹04 0.8143976677E⫹03 0.4012305051E⫹03 0.2061536396E⫹03 0.1092260228E⫹03 0.5900620702E⫹02

P⫹

Page 258

Exponents

21:38

Symm.

5/19/2007

Exponents

Exp. #


Symm.

258

Atom

0.3933216429E⫹00 0.1615076675E⫹00 0.6093727369E⫺01

5/19/2007 21:38

F⫺

0.6250352087E⫹03 0.2477873290E⫹03 0.1069567716E⫹03 0.4936006082E⫹02 0.2391469939E⫹02 0.1194427065E⫹02 0.6038699865E⫹01 0.3034589732E⫹01 0.1488376770E⫹01 0.6996257291E⫹00 0.3094869081E⫹00

F⫹

0.6250352087E⫹03 0.2477873290E⫹03 0.1069567716E⫹03 0.4936006082E⫹02 0.2391469939E⫹02 0.1194427065E⫹02 0.6038699865E⫹01 0.3034589732E⫹01 0.1488376770E⫹01 0.6996257291E⫹00 0.3094869081E⫹00

Page 259


0.3213691282E⫹02 0.1744792963E⫹02 0.9337171830E⫹01 0.4869864950E⫹01 0.2447637022E⫹01 0.1172206302E⫹01 0.5289156840E⫹00 0.2223272045E⫹00


D⫹

0.3213691282E⫹02 0.1744792963E⫹02 0.9337171830E⫹01 0.4869864950E⫹01 0.2447637022E⫹01 0.1172206302E⫹01 0.5289156840E⫹00 0.2223272045E⫹00

259

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17


Appendix 5

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

260

Hf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹

0.4035098028E⫹09 0.1005172537E⫹09 0.2776336215E⫹08 0.8445229483E⫹07 0.2810100424E⫹07 0.1015935763E⫹07 0.3963763322E⫹06 0.1657712732E⫹06 0.7381318722E⫹05 0.3475720182E⫹05 0.1719114682E⫹05 0.8871092838E⫹04 0.4743781370E⫹04 0.2611021853E⫹04 0.1469256713E⫹04 0.8395551510E⫹03 0.4838694269E⫹03 0.2793814542E⫹03 0.1605167799E⫹03 0.9115065910E⫹02 0.5081345571E⫹02 0.2762105277E⫹02 0.1454143737E⫹02 0.7364497078E⫹01 0.3563777108E⫹01 0.1636713598E⫹01 0.7085860520E⫹00 0.2872323124E⫹00 0.1082823846E⫹00 0.3770762539E⫺01

P⫺

0.2389384188E⫹08 0.4933321907E⫹07 0.1172481266E⫹07 0.3172067955E⫹06 0.9660648734E⫹05 0.3275325476E⫹05 0.1222486646E⫹05 0.4967437347E⫹04 0.2173083486E⫹04 0.1012122987E⫹04 0.4963181475E⫹03 0.2534043910E⫹03 0.1332147630E⫹03 0.7130696327E⫹02 0.3843350715E⫹02 0.2062736279E⫹02 0.1090159735E⫹02 0.5610562865E⫹01 0.2780671715E⫹01 0.1312432938E⫹01 0.5833722465E⫹00 0.2414975744E⫹00

P⫹


D⫺


Appendix 5

5/19/2007

Exp. #


Atom

21:38 Page 260


D⫹


F⫺

0.8060066282E⫹03 0.3095669662E⫹03 0.1325338695E⫹03 0.6171960444E⫹02 0.3050781470E⫹02 0.1561918956E⫹02 0.8082264177E⫹01 0.4124795846E⫹01 0.2025976790E⫹01 0.9345376106E⫹00 0.3950546102E⫹00

F⫹

0.8060066282E⫹03 0.3095669662E⫹03 0.1325338695E⫹03 0.6171960444E⫹02 0.3050781470E⫹02 0.1561918956E⫹02 0.8082264177E⫹01 0.4124795846E⫹01 0.2025976790E⫹01 0.9345376106E⫹00 0.3950546102E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Hf

1 2 3 4 5 6 7 8 9 10 11

S⫹

0.4064202208E⫹09 0.1060274894E⫹09 0.3049103105E⫹08 0.9603445484E⫹07 0.3291351349E⫹07 0.1219563707E⫹07 0.4854099632E⫹06 0.2061947960E⫹06 0.9287576540E⫹05 0.4407310767E⫹05 0.2189184566E⫹05

P⫺

0.2220483811E⫹08 0.4548864585E⫹07 0.1074023790E⫹07 0.2890040097E⫹06 0.8763881315E⫹05 0.2961524552E⫹05 0.1102767596E⫹05 0.4474309711E⫹04 0.1955982904E⫹04 0.9110130761E⫹03 0.4470213739E⫹03

P⫹

0.2220483811E⫹08 0.4548864585E⫹07 0.1074023790E⫹07 0.2890040097E⫹06 0.8763881315E⫹05 0.2961524552E⫹05 0.1102767596E⫹05 0.4474309711E⫹04 0.1955982904E⫹04 0.9110130761E⫹03 0.4470213739E⫹03

D⫺

0.1321226722E⫹05 0.4610581956E⫹04 0.1770226707E⫹04 0.7382905562E⫹03 0.3302034130E⫹03 0.1563587393E⫹03 0.7738907525E⫹02 0.3952615659E⫹02 0.2056683608E⫹02 0.1076363332E⫹02 0.5593573472E⫹01

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

21:38 Page 261

Appendix 5 261

(continued )

Symm.

0.1321226722E⫹04 0.4610581956E⫹04 0.1770226707E⫹04 0.7382905562E⫹03 0.3302034130E⫹03 0.1563587393E⫹03 0.7738907525E⫹02 0.3952615659E⫹02 0.2056683608E⫹02 0.1076363332E⫹02 0.5593573472E⫹01 0.2849635958E⫹01

0.2285070840E⫹03 0.1203267888E⫹03 0.6454161707E⫹02 0.3487030884E⫹02 0.1876436013E⫹02 0.9944828949E⫹01 0.5132985609E⫹01 0.2551382221E⫹01 0.1207639558E⫹01 0.5382438076E⫹00 0.2233696881E⫹00

Exponents 0.2285070840E⫹03 0.1203267888E⫹03 0.6454161707E⫹02 0.3487030884E⫹02 0.1876436013E⫹02 0.9944828949E⫹01 0.5132985609E⫹01 0.2551382221E⫹01 0.1207639558E⫹01 0.5382438076E⫹00 0.2233696881E⫹00

Symm.

Exponents 0.2849635958E⫹01 0.1405042749E⫹01 0.6619426306E⫹00 0.2941794695E⫹00 0.1217576065E⫹00 0.4633424752E⫺01

F⫺

0.6281362603E⫹02 0.2486459196E⫹03 0.1089501632E⫹03 0.5164150049E⫹02 0.2587614535E⫹02 0.1339477478E⫹02 0.7000220117E⫹01 0.3609389634E⫹01 0.1794349217E⫹01 0.8404979281E⫹00 0.3625168040E⫹00

F⫹

0.6281362603E⫹02 0.2486459196E⫹03 0.1089501632E⫹03 0.5164150049E⫹02 0.2587614535E⫹02 0.1339477478E⫹02 0.7000220117E⫹01 0.3609389634E⫹01 0.1794349217E⫹01 0.8404979281E⫹00 0.3625168040E⫹00

Page 262

D⫹

Symm.

21:38

0.1130887457E⫹05 0.6036366148E⫹04 0.3307819569E⫹04 0.1848881048E⫹04 0.1047292328E⫹04 0.5973234138E⫹03 0.3408197008E⫹03 0.1932877147E⫹03 0.1082527882E⫹03 0.5948672809E⫹02 0.3186677371E⫹02 0.1653423116E⫹02 0.8255600713E⫹01 0.3941150256E⫹01 0.1787301479E⫹01 0.7650045499E⫹00 0.3070523845E⫹00 0.1148243657E⫹00 0.3974839818E⫺01

Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12

Symm.

Appendix 5

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

262

Atom


13 14 15 16 17

0.1405042749E⫹01 0.6619426306E⫹00 0.2941794695E⫹00 0.1217576065E⫹00 0.4633424752E⫺01

Exponents

Symm.

Exponents

Symm.

Exponents

Ta

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

S⫹

0.4029653741E⫹09 0.9938239359E⫹08 0.2719682560E⫹08 0.8202873159E⫹07 0.2708474590E⫹07 0.9724482495E⫹06 0.3771057637E⫹06 0.1568869260E⫹06 0.6955193604E⫹05 0.3263642301E⫹05 0.1610047609E⫹05 0.8294490067E⫹04 0.4432282645E⫹04 0.2440191782E⫹04 0.1374836976E⫹04 0.7873741509E⫹03 0.4552875570E⫹03 0.2640201435E⫹03 0.1525130379E⫹03 0.8717008760E⫹02 0.4896554231E⫹02 0.2685024205E⫹02 0.1427619151E⫹02 0.7310633692E⫹01

P⫺


P⫹


D⫺


(continued )

Page 263

Symm.

21:38

Exponents

5/19/2007

Symm.

263

Exp. #

Appendix 5

Atom

Symm.

Symm.

Exponents

Symm.

Exponents

0.3581361447E⫹01 0.1667107983E⫹01 0.7324428239E⫹00 0.3016825014E⫹00 0.1157083981E⫹00 0.4104780935E⫺01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

D⫹


F⫺

0.9868407748E⫹03 0.3859490338E⫹03 0.1662537720E⫹03 0.7718107692E⫹02 0.3778218218E⫹02 0.1908267347E⫹02 0.9729857447E⫹01 0.4900359179E⫹01 0.2385297907E⫹01 0.1097968090E⫹01 0.4676363776E⫹00

F⫹

0.9868407748E⫹03 0.3859490338E⫹03 0.1662537720E⫹03 0.7718107692E⫹02 0.3778218218E⫹02 0.1908267347E⫹02 0.9729857447E⫹01 0.4900359179E⫹01 0.2385297907E⫹01 0.1097968090E⫹01 0.4676363776E⫹00

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

S⫹

0.4055824106E⫹09 0.1007419648E⫹09 0.2774895370E⫹08 0.8418968500E⫹07

P⫺

0.2257809132E⫹08 0.4780172024E⫹07 0.1161882792E⫹07 0.3206565157E⫹06

P⫹

0.2257809132E⫹08 0.4780172024E⫹07 0.1161882792E⫹07 0.3206565157E⫹06

D⫺

0.1587092359E⫹05 0.5671685668E⫹04 0.2222860718E⫹04 0.9434583323E⫹03

Page 264

1 2 3 4

21:38

Ta

Exponents

Appendix 5

Atom

Symm.

5/19/2007

25 26 27 28 29 30

Exponents


Exp. #

264

Atom

F⫹

0.7007314780E⫹03 0.2755704966E⫹03 0.1198161085E⫹03 0.5638900302E⫹02 0.2812325934E⫹02 0.1455205179E⫹02 (continued )

Page 265

0.7007314780E⫹03 0.2755704966E⫹03 0.1198161085E⫹03 0.5638900302E⫹02 0.2812325934E⫹02 0.1455205179E⫹02

21:38

F⫺

0.4282169015E⫹03 0.2052370309E⫹03 0.1025696116E⫹03 0.5278043921E⫹02 0.2761460405E⫹02 0.1450562361E⫹02 0.7554156582E⫹01 0.3851303874E⫹01 0.1898112928E⫹01 0.8929949360E⫹00 0.3960124725E⫹00 0.1634639101E⫹00 0.6201664725E⫺01

5/19/2007

0.1587092359E⫹05 0.5671685668E⫹04 0.2222860718E⫹04 0.9434583323E⫹03 0.4282169015E⫹03 0.2052370309E⫹03

0.9937410617E⫹05 0.3420253378E⫹05 0.1292979580E⫹05 0.5309698411E⫹04 0.2342552356E⫹04 0.1098111612E⫹04 0.5409259912E⫹03 0.2769235336E⫹03 0.1457165871E⫹03 0.7794382063E⫹02 0.4191547809E⫹02 0.2241215692E⫹02 0.1178435572E⫹02 0.6026115515E⫹01 0.2963977778E⫹01 0.1386803017E⫹01 0.6104546660E⫹00 0.2500272388E⫹00


D⫹


265

1 2 3 4 5 6

0.2794592056E⫹07 0.1008083266E⫹07 0.3925220217E⫹06 0.1638673653E⫹06 0.7285391988E⫹05 0.3426230461E⫹05 0.1692990814E⫹05 0.8730481565E⫹04 0.4667008071E⫹04 0.2568779609E⫹04 0.1446019886E⫹04 0.8268981104E⫹03 0.4771239679E⫹03 0.2759199826E⫹03 0.1588469744E⫹03 0.9042526695E⫹02 0.5055765215E⫹02 0.2757664494E⫹02 0.1457553738E⫹02 0.7414938079E⫹01 0.3606296852E⫹01 0.1665549692E⫹01 0.7255495547E⫹00 0.2961154828E⫹00 0.1124633169E⫹00 0.3948094905E⫺01

Appendix 5

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

0.1025696116E⫹03 0.5278043921E⫹02 0.2761460405E⫹02 0.1450562361E⫹02 0.7554156582E⫹01 0.3851303874E⫹01 0.1898112928E⫹01 0.8929949360E⫹00 0.3960124725E⫹00 0.1634639101E⫹00 0.6201664725E⫺01

Exponents

Symm.

0.7648309810E⫹01 0.3997465147E⫹01 0.2034122904E⫹01 0.9865936107E⫹00 0.4465430423E⫹00

Exponents

Symm.

Exponents

0.7648309810E⫹01 0.3997465147E⫹01 0.2034122904E⫹01 0.9865936107E⫹00 0.4465430423E⫹00

5/19/2007 21:38

7 8 9 10 11 12 13 14 15 16 17

Exponents


Symm.

Appendix 5

Exp. #

266

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S⫹


P⫺


P⫹


D⫺


Page 266

Atom

0.8151632680E⫺01

21:38

F⫺

0.8966095057E⫹03 0.3633872962E⫹03 0.1616795926E⫹03 0.7733277626E⫹02 0.3894025525E⫹02 0.2021457660E⫹02 0.1059412357E⫹02 0.5489133782E⫹01 0.2753498596E⫹01 0.1309519688E⫹01 0.5782147957E⫹00

F⫹

0.8966095057E⫹03 0.3633872962E⫹03 0.1616795926E⫹03 0.7733277626E⫹02 0.3894025525E⫹02 0.2021457660E⫹02 0.1059412357E⫹02 0.5489133782E⫹01 0.2753498596E⫹01 0.1309519688E⫹01 0.5782147957E⫹00

Page 267


0.1180945384E⫹02 0.6156594684E⫹01 0.3090795564E⫹01 0.1477451244E⫹01 0.6649108044E⫹00 0.2785577010E⫹00

5/19/2007

D⫹

0.1180945384E⫹02 0.6156594684E⫹01 0.3090795564E⫹01 0.1477451244E⫹01 0.6649108044E⫹00 0.2785577010E⫹00

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.5303399858E⫹03 0.3077211146E⫹03 0.1776120179E⫹03 0.1012853363E⫹03 0.5667963214E⫹02 0.3091439421E⫹02 0.1632280434E⫹02 0.8286596137E⫹01 0.4017463898E⫹01 0.1847435357E⫹01 0.8003409255E⫹00 0.3244265233E⫹00 0.1222194375E⫹00 0.4250048208E⫺01


17 18 19 20 21 22 23 24 25 26 27 28 29 30

267

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

268

W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺


Appendix 5

5/19/2007

Exp. #


Atom

21:38 Page 268



F⫺

0.7803787229E⫹03 0.3016865533E⫹03 0.1292397088E⫹03 0.6004825531E⫹02 0.2961720982E⫹02 0.1517756626E⫹02 0.7909504920E⫹01 0.4102613005E⫹01 0.2073054949E⫹01 0.9987918860E⫹00 0.4490844825E⫹00

F⫹

0.7803787229E⫹03 0.3016865533E⫹03 0.1292397088E⫹03 0.6004825531E⫹02 0.2961720982E⫹02 0.1517756626E⫹02 0.7909504920E⫹01 0.4102613005E⫹01 0.2073054949E⫹01 0.9987918860E⫹00 0.4490844825E⫹00

AtomExp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Re

S⫹

0.4069415178E⫹09 0.1043096748E⫹09 0.2958259343E⫹08 0.9220427258E⫹07 0.3137278546E⫹07 0.1157514475E⫹07 0.4599971955E⫹06 0.1955795417E⫹06 0.8837211241E⫹05 0.4215166220E⫹05 0.2108171392E⫹05 0.1098178539E⫹05

P⫺

0.2512951096E⫹08 0.5195680795E⫹07 0.1237379184E⫹07 0.3356686553E⫹06 0.1025680576E⫹06 0.3491021737E⫹05 0.1308813524E⫹05 0.5344825296E⫹04 0.2351069642E⫹04 0.1101588599E⫹04 0.5436771125E⫹03 0.2794963032E⫹03

P⫹

0.2512951096E⫹08 0.5195680795E⫹07 0.1237379184E⫹07 0.3356686553E⫹06 0.1025680576E⫹06 0.3491021737E⫹05 0.1308813524E⫹05 0.5344825296E⫹04 0.2351069642E⫹04 0.1101588599E⫹04 0.5436771125E⫹03 0.2794963032E⫹03

D⫺

0.1907989292E⫹05 0.6303795739E⫹04 0.2342123823E⫹04 0.9636372504E⫹03 0.4323423773E⫹03 0.2082893287E⫹03 0.1061074093E⫹03 0.5628314635E⫹02 0.3061113828E⫹02 0.1680983828E⫹02 0.9177936003E⫹01 0.4906123063E⫹01

21:38 Page 269

(continued )

269

1 2 3 4 5 6 7 8 9 10 11 12

5/19/2007

D⫹

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Symm.

0.1907989292E⫹05 0.6303795739E⫹04 0.2342123823E⫹04 0.9636372504E⫹03 0.4323423773E⫹03 0.2082893287E⫹03 0.1061074093E⫹03 0.5628314635E⫹02 0.3061113828E⫹02 0.1680983828E⫹02 0.9177936003E⫹01 0.4906123063E⫹01 0.2528472662E⫹01

0.1480028930E⫹03 0.7983056089E⫹02 0.4337292546E⫹02 0.2347278012E⫹02 0.1251273781E⫹02 0.6497224565E⫹01 0.3249651960E⫹01 0.1548192582E⫹01 0.6947650982E⫹00 0.2904167118E⫹00

Exponents 0.1480028930E⫹03 0.7983056089E⫹02 0.4337292546E⫹02 0.2347278012E⫹02 0.1251273781E⫹02 0.6497224565E⫹01 0.3249651960E⫹01 0.1548192582E⫹01 0.6947650982E⫹00 0.2904167118E⫹00

Symm.

Exponents 0.2528472662E⫹01 0.1237140397E⫹01 0.5658940202E⫹00 0.2382983861E⫹00 0.9096859063E⫺01

F⫺

0.1033779018E⫹04 0.4089086405E⫹03 0.1788546262E⫹03 0.8463618080E⫹02 0.4239362951E⫹02 0.2199077213E⫹02 0.1155799138E⫹02 0.6021891640E⫹01 0.3042978350E⫹01 0.1459105914E⫹01 0.6495363727E⫹00

F⫹

0.1033779018E⫹04 0.4089086405E⫹03 0.1788546262E⫹03 0.8463618080E⫹02 0.4239362951E⫹02 0.2199077213E⫹02 0.1155799138E⫹02 0.6021891640E⫹01 0.3042978350E⫹01 0.1459105914E⫹01 0.6495363727E⫹00

Page 270

D⫹

Symm.

21:38

0.5918339855E⫹04 0.3277712404E⫹04 0.1852976322E⫹04 0.1062138278E⫹04 0.6131807356E⫹03 0.3541410569E⫹03 0.2032490275E⫹03 0.1151407830E⫹03 0.6395322813E⫹02 0.3459492856E⫹02 0.1810352579E⫹02 0.9103297509E⫹01 0.4369213199E⫹01 0.1988203124E⫹01 0.8520309401E⫹00 0.3415632707E⫹00 0.1272307918E⫹00 0.4374240427E⫺01

Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

270

Atom


14 15 16 17

0.1237140397E⫹01 0.5658940202E⫹00 0.2382983861E⫹00 0.9096859063E⫺01

Exponents

Symm.

Exponents

Symm.

Exponents

Re

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

S⫹


P⫺


P⫹


D⫺


(continued )

Page 271

Symm.

21:38

Exponents

5/19/2007

Symm.

271

Exp. #

Appendix 5

Atom

Symm.

Symm.

Exponents

Symm.

Exponents

0.4203380012E⫹01 0.1950572527E⫹01 0.8534734812E⫹00 0.3497369890E⫹00 0.1333140235E⫹00 0.4695176964E⫺01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

D⫹


F⫺

0.1084875180E⫹04 0.4233741321E⫹03 0.1824656580E⫹03 0.8504412555E⫹02 0.4197666760E⫹02 0.2148652309E⫹02 0.1116896604E⫹02 0.5773544636E⫹01 0.2906356660E⫹01 0.1395167031E⫹01 0.6254153098E⫹00

F⫹

0.1084875180E⫹04 0.4233741321E⫹03 0.1824656580E⫹03 0.8504412555E⫹02 0.4197666760E⫹02 0.2148652309E⫹02 0.1116896604E⫹02 0.5773544636E⫹01 0.2906356660E⫹01 0.1395167031E⫹01 0.6254153098E⫹00

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

S⫹

0.4086310495E⫹09 0.1011921401E⫹09 0.2783375253E⫹08

P⫺

0.2779355678E⫹08 0.5713911184E⫹07 0.1355101552E⫹07

P⫹

0.2779355678E⫹08 0.5713911184E⫹07 0.1355101552E⫹07

D⫺

0.2949572348E⫹05 0.9060376450E⫹04 0.3164681096E⫹04

Page 272

1 2 3

21:38

Os

Exponents

Appendix 5

Atom

Symm.

5/19/2007

25 26 27 28 29 30

Exponents


Exp. #

272

Atom

F⫹

0.1080012686E⫹04 0.4039400774E⫹03 0.1702401381E⫹03 0.7888949721E⫹02 0.3922327892E⫹02 (continued )

Page 273

0.1080012686E⫹04 0.4039400774E⫹03 0.1702401381E⫹03 0.7888949721E⫹02 0.3922327892E⫹02

21:38

F⫺

0.1236689001E⫹04 0.5319712894E⫹03 0.2478354083E⫹03 0.1230371588E⫹03 0.6404088535E⫹02 0.3438561287E⫹02 0.1873893629E⫹02 0.1019793261E⫹02 0.5452914563E⫹01 0.2818674430E⫹01 0.1385833706E⫹01 0.6376430357E⫹00 0.2701436306E⫹00 0.1036842450E⫹00

5/19/2007

0.2949572348E⫹05 0.9060376450E⫹04 0.3164681096E⫹04 0.1236689001E⫹04 0.5319712894E⫹03

0.3665684720E⫹06 0.1118353739E⫹06 0.3804871617E⫹05 0.1427359791E⫹05 0.5837883945E⫹04 0.2573960706E⫹04 0.1209675083E⫹04 0.5991724462E⫹03 0.3092766395E⫹03 0.1644941652E⫹03 0.8913697318E⫹02 0.4865922440E⫹02 0.2645868806E⫹02 0.1416978184E⫹02 0.7390020876E⫹01 0.3711177441E⫹01 0.1774423321E⫹01 0.7986897779E⫹00 0.3346341793E⫹00


D⫹


273

1 2 3 4 5

0.8445672772E⫹07 0.2807774293E⫹07 0.1015742293E⫹07 0.3971240807E⫹06 0.1666549431E⫹06 0.7455682618E⫹05 0.3531521105E⫹05 0.1759014123E⫹05 0.9150354480E⫹04 0.4937368348E⫹04 0.2744543391E⫹04 0.1560952123E⫹04 0.9021565730E⫹03 0.5262293321E⫹03 0.3076783453E⫹03 0.1790917172E⫹03 0.1030715260E⫹03 0.5825248974E⫹02 0.3210929757E⫹02 0.1714413548E⫹02 0.8806367243E⫹01 0.4322175109E⫹01 0.2013076282E⫹01 0.8836867838E⫹00 0.3631154159E⫹00 0.1387161923E⫹00 0.4892983167E⫺01

Appendix 5

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Symm.

Symm.

0.2041697340E⫹02 0.1085717565E⫹02 0.5755402161E⫹01 0.2967721023E⫹01 0.1452493953E⫹01 0.6584226020E⫹00

Exponents

Symm.

Exponents

0.2041697340E⫹02 0.1085717565E⫹02 0.5755402161E⫹01 0.2967721023E⫹01 0.1452493953E⫹01 0.6584226020E⫹00

21:38

0.2478354083E⫹03 0.1230371588E⫹03 0.6404088535E⫹02 0.3438561287E⫹02 0.1873893629E⫹02 0.1019793261E⫹02 0.5452914563E⫹01 0.2818674430E⫹01 0.1385833706E⫹01 0.6376430357E⫹00 0.2701436306E⫹00 0.1036842450E⫹00

Exponents

5/19/2007

6 7 8 9 10 11 12 13 14 15 16 17

Exponents


Exp. #

274

Atom

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Os

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

S⫹


P⫺


P⫹


D⫺


Page 274

Exp. #

Appendix 5

Atom

0.2261518115E⫹00 0.8545389502E⫺01

21:38

0.1116959292E⫹04 0.4209006657E⫹03 0.1779745014E⫹03 0.8245508897E⫹02 0.4086996385E⫹02 0.2116234324E⫹02 0.1117741995E⫹02 0.5880080863E⫹01 0.3008392843E⫹01 0.1461641668E⫹01 0.6584882141E⫹00

F⫹

0.1116959292E⫹04 0.4209006657E⫹03 0.1779745014E⫹03 0.8245508897E⫹02 0.4086996385E⫹02 0.2116234324E⫹02 0.1117741995E⫹02 0.5880080863E⫹01 0.3008392843E⫹01 0.1461641668E⫹01 0.6584882141E⫹00

Page 275

F⫺

275


0.2524569093E⫹02 0.1352828995E⫹02 0.7056080244E⫹01 0.3542174996E⫹01 0.1692323311E⫹01 0.7608944381E⫹00 0.3183568212E⫹00

5/19/2007

D⫹

0.2524569093E⫹02 0.1352828995E⫹02 0.7056080244E⫹01 0.3542174996E⫹01 0.1692323311E⫹01 0.7608944381E⫹00 0.3183568212E⫹00

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17



16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

276

Ir

S⫹


P⫺


P⫹


D⫺


Appendix 5

21:38 Page 276

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5/19/2007

Symm.


AtomExp. #



F⫺

0.9586109805E⫹03 0.3683769739E⫹03 0.1591151033E⫹03 0.7535830800E⫹02 0.3817540295E⫹02 0.2017905530E⫹02 0.1085712138E⫹02 0.5800389666E⫹01 0.3001651125E⫹01 0.1467762463E⫹01 0.6615700121E⫹00

F⫹

0.9586109805E⫹03 0.3683769739E⫹03 0.1591151033E⫹03 0.7535830800E⫹02 0.3817540295E⫹02 0.2017905530E⫹02 0.1085712138E⫹02 0.5800389666E⫹01 0.3001651125E⫹01 0.1467762463E⫹01 0.6615700121E⫹00

Atom Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Ir

S⫹

0.4048743529E⫹09 0.1003387811E⫹09 0.2761218824E⫹08 0.8380151754E⫹07 0.2785849158E⫹07 0.1007519757E⫹07 0.3937090588E⫹06 0.1651044236E⫹06 0.7379706329E⫹05 0.3491822199E⫹05 0.1737134471E⫹05 0.9024406146E⫹04

P⫺

0.2420551978E⫹08 0.5052664329E⫹07 0.1213878071E⫹07 0.3319289005E⫹06 0.1021638496E⫹06 0.3500244719E⫹05 0.1320125879E⫹05 0.5420194294E⫹04 0.2395872684E⫹04 0.1127531768E⫹04 0.5586975987E⫹03 0.2882534990E⫹03

P⫹

0.2420551978E⫹08 0.5052664329E⫹07 0.1213878071E⫹07 0.3319289005E⫹06 0.1021638496E⫹06 0.3500244719E⫹05 0.1320125879E⫹05 0.5420194294E⫹04 0.2395872684E⫹04 0.1127531768E⫹04 0.5586975987E⫹03 0.2882534990E⫹03

D⫺

0.2353755986E⫹05 0.7622930683E⫹04 0.2787338430E⫹04 0.1132588257E⫹04 0.5033573880E⫹03 0.2408298429E⫹03 0.1220902662E⫹03 0.6454997574E⫹02 0.3503183217E⫹02 0.1920829277E⫹02 0.1047324467E⫹02 0.5589175167E⫹01

21:38 Page 277

(continued )

277

1 2 3 4 5 6 7 8 9 10 11 12

5/19/2007

D⫹

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.2353755986E⫹05 0.7622930683E⫹04 0.2787338430E⫹04 0.1132588257E⫹04 0.5033573880E⫹03 0.2408298429E⫹03 0.1220902662E⫹03 0.6454997574E⫹02 0.3503183217E⫹02 0.1920829277E⫹02 0.1047324467E⫹02 0.5589175167E⫹01 0.2873396792E⫹01

0.1531401026E⫹03 0.8284887970E⫹02 0.4513718859E⫹02 0.2449059443E⫹02 0.1308723503E⫹02 0.6811566940E⫹01 0.3414786108E⫹01 0.1630662674E⫹01 0.7335263175E⫹00 0.3073866579E⫹00


Symm.

Exponents 0.2873396792E⫹01 0.1400658675E⫹01 0.6371846141E⫹00 0.2662575424E⫹00 0.1005888058E⫹00

F⫺

0.1107522476E⫹04 0.4132858342E⫹03 0.1734823240E⫹03 0.7999551998E⫹02 0.3957134435E⫹02 0.2050688191E⫹02 0.1087229243E⫹02 0.5758962760E⫹01 0.2976240679E⫹01 0.1465518960E⫹01 0.6714496186E⫹00

F⫹

0.1107522476E⫹04 0.4132858342E⫹03 0.1734823240E⫹03 0.7999551998E⫹02 0.3957134435E⫹02 0.2050688191E⫹02 0.1087229243E⫹02 0.5758962760E⫹01 0.2976240679E⫹01 0.1465518960E⫹01 0.6714496186E⫹00

Page 278

D⫹

Symm.

21:38


Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Symm.

278

Atom Exp. #


14 15 16 17

0.1400658675E⫹01 0.6371846141E⫹00 0.2662575424E⫹00 0.1005888058E⫹00

Symm.

Exponents

Symm.

Exponents

Pt

S⫹


P⫺


P⫹


D⫺


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

(continued )

Page 279

Exponents

21:38

Symm.

5/19/2007

Exponents

279

Symm.

Appendix 5

Atom Exp. #

Symm.

Exponents


F⫺

0.1265788055E⫹04 0.4549376132E⫹03 0.1863037997E⫹03 0.8466684842E⫹02 0.4158815825E⫹02 0.2150470015E⫹02 0.1140111439E⫹02 0.6036069343E⫹01 0.3108114015E⫹01 0.1516066547E⫹01 0.6822774033E⫹00

F⫹

0.1265788055E⫹04 0.4549376132E⫹03 0.1863037997E⫹03 0.8466684842E⫹02 0.4158815825E⫹02 0.2150470015E⫹02 0.1140111439E⫹02 0.6036069343E⫹01 0.3108114015E⫹01 0.1516066547E⫹01 0.6822774033E⫹00

Symm.

Exponents

0.4820361727E⫹01 0.2248299709E⫹01 0.9881479901E⫹00 0.4064578138E⫹00 0.1554057721E⫹00 0.5485414903E⫺01 D⫹

21:38

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Pt

1 2 3

S⫹

0.4025108442E⫹09 0.1003914263E⫹09 0.2778631465E⫹08

P⫺

0.2521709505E⫹08 0.5257627178E⫹07 0.1261105378E⫹07

P⫹

0.2521709505E⫹08 0.5257627178E⫹07 0.1261105378E⫹07

D⫺

0.2983846478E⫹05 0.8927434537E⫹04 0.3061904062E⫹04

Page 280

Exponents

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Symm.

5/19/2007

25 26 27 28 29 30

Exponents


Symm.

280

Atom Exp. #

F⫹

0.1006934451E⫹04 0.3841434327E⫹03 0.1645807917E⫹03 0.7724147202E⫹02 0.3873465887E⫹02 (continued )

Page 281

0.1006934451E⫹04 0.3841434327E⫹03 0.1645807917E⫹03 0.7724147202E⫹02 0.3873465887E⫹02

21:38

F⫺

0.1183435768E⫹04 0.5067100459E⫹03 0.2362705279E⫹03 0.1179419504E⫹03 0.6195969614E⫹02 0.3367487371E⫹02 0.1861366519E⫹02 0.1028633227E⫹02 0.5586829682E⫹01 0.2931703526E⫹01 0.1461160438E⫹01 0.6799429902E⫹00 0.2904138609E⫹00 0.1119195428E⫹00

5/19/2007

0.2983846478E⫹05 0.8927434537E⫹04 0.3061904062E⫹04 0.1183435768E⫹04 0.5067100459E⫹03



D⫹


281

1 2 3 4 5


Appendix 5

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Symm.

Symm.

0.2024505065E⫹02 0.1075722064E⫹02 0.5668073872E⫹01 0.2888798712E⫹01 0.1389112936E⫹01 0.6147357389E⫹00

Exponents

Symm.

Exponents

0.2024505065E⫹02 0.1075722064E⫹02 0.5668073872E⫹01 0.2888798712E⫹01 0.1389112936E⫹01 0.6147357389E⫹00

21:38

0.2362705279E⫹03 0.1179419504E⫹03 0.6195969614E⫹02 0.3367487371E⫹02 0.1861366519E⫹02 0.1028633227E⫹02 0.5586829682E⫹01 0.2931703526E⫹01 0.1461160438E⫹01 0.6799429902E⫹00 0.2904138609E⫹00 0.1119195428E⫹00

Exponents

5/19/2007

6 7 8 9 10 11 12 13 14 15 16 17

Exponents


Exp. #

282

Atom

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Au

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

S⫹


P⫺


P⫹


D⫺


Page 282

Exp. #

Appendix 5

Atom

0.3559232184E⫹00 0.1369509868E⫹00

21:38

0.9710235157E⫹03 0.3731582036E⫹03 0.1612172712E⫹03 0.7636023630E⫹02 0.3866712529E⫹02 0.2041348862E⫹02 0.1095658214E⫹02 0.5830385732E⫹01 0.2999615562E⫹01 0.1454996094E⫹01 0.6488850608E⫹00

F⫹

0.9710235157E⫹03 0.3731582036E⫹03 0.1612172712E⫹03 0.7636023630E⫹02 0.3866712529E⫹02 0.2041348862E⫹02 0.1095658214E⫹02 0.5830385732E⫹01 0.2999615562E⫹01 0.1454996094E⫹01 0.6488850608E⫹00

Page 283

F⫺

283


0.2735249698E⫹02 0.1484375151E⫹02 0.7855797148E⫹01 0.4008978001E⫹01 0.1950611822E⫹01 0.8947438792E⫹00 0.3825729542E⫹00

5/19/2007

D⫹

0.2735249698E⫹02 0.1484375151E⫹02 0.7855797148E⫹01 0.4008978001E⫹01 0.1950611822E⫹01 0.8947438792E⫹00 0.3825729542E⫹00

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17



16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

284

Au

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹


P⫺


P⫹


D⫺


Appendix 5

5/19/2007

Exp. #


Atom

21:38 Page 284


D⫹


F⫺

0.8572354967E⫹03 0.3339664579E⫹03 0.1459968492E⫹03 0.6984674979E⫹02 0.3566441980E⫹02 0.1895546858E⫹02 0.1022747540E⫹02 0.5463375835E⫹01 0.2817971411E⫹01 0.1368734195E⫹01 0.6105663114E⫹00

F⫹

0.8572354967E⫹03 0.3339664579E⫹03 0.1459968492E⫹03 0.6984674979E⫹02 0.3566441980E⫹02 0.1895546858E⫹02 0.1022747540E⫹02 0.5463375835E⫹01 0.2817971411E⫹01 0.1368734195E⫹01 0.6105663114E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Hg

1 2 3 4 5 6 7 8 9 10 11 12

S⫹

0.4082595892E⫹09 0.1020218335E⫹09 0.2830329863E⫹08 0.8657800273E⫹07 0.2900307411E⫹07 0.1056783060E⫹07 0.4159799276E⫹06 0.1756883683E⫹06 0.7907462234E⫹05 0.3766986873E⫹05 0.1886484891E⫹05 0.9864025359E⫹04

P⫺

0.2519545305E⫹08 0.5216082410E⫹07 0.1245938840E⫹07 0.3395209340E⫹06 0.1043617821E⫹06 0.3577742321E⫹05 0.1352560863E⫹05 0.5575352189E⫹04 0.2477667776E⫹04 0.1173698798E⫹04 0.5860040434E⫹03 0.3049035466E⫹03

P⫹

0.2519545305E⫹08 0.5216082410E⫹07 0.1245938840E⫹07 0.3395209340E⫹06 0.1043617821E⫹06 0.3577742321E⫹05 0.1352560863E⫹05 0.5575352189E⫹04 0.2477667776E⫹04 0.1173698798E⫹04 0.5860040434E⫹03 0.3049035466E⫹03

D⫺


5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

21:38 Page 285

Appendix 5 285

Symm.

0.3096414137E⫹05 0.9372516661E⫹04 0.3255952821E⫹04 0.1275788592E⫹04 0.5541305326E⫹03 0.2621997227E⫹03 0.1328290857E⫹03 0.7080256192E⫹02 0.3902596240E⫹02 0.2186055368E⫹02 0.1222997603E⫹02 0.6715850084E⫹01 0.3557470810E⫹01

0.1634668880E⫹03 0.8928738704E⫹02 0.4912821414E⫹02 0.2692402162E⫹02 0.1453130165E⫹02 0.7636815263E⫹01 0.3864121519E⫹01 0.1861262104E⫹01 0.8438594296E⫹00 0.3560623375E⫹00


Symm.

Exponents 0.3557470810E⫹01 0.1786490409E⫹01 0.8358590525E⫹00 0.3580902163E⫹00 0.1380488938E⫹00

F⫺

0.1206283892E⫹04 0.4627868654E⫹03 0.1971486389E⫹03 0.9116005461E⫹02 0.4472292273E⫹02 0.2275555116E⫹02 0.1173796723E⫹02 0.6000169944E⫹01 0.2971092049E⫹01 0.1393052016E⫹01 0.6045524717E⫹00

F⫹

0.1206283892E⫹04 0.4627868654E⫹03 0.1971486389E⫹03 0.9116005461E⫹02 0.4472292273E⫹02 0.2275555116E⫹02 0.1173796723E⫹02 0.6000169944E⫹01 0.2971092049E⫹01 0.1393052016E⫹01 0.6045524717E⫹00

Page 286

D⫹

Symm.

21:38


Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

286

Atom


14 15 16 17

0.1786490409E⫹01 0.8358590525E⫹00 0.3580902163E⫹00 0.1380488938E⫹00

Symm.

Exponents

Symm.

Exponents

Tl

S⫹


P⫺


P⫹


D⫺


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

(continued )

Page 287

Exponents

21:38

Symm.

5/19/2007

Exponents

287

Symm.

Appendix 5

AtomExp. #

0.5634668059E⫹01 0.2739203885E⫹01 0.1250958459E⫹01 0.5326674541E⫹00 0.2098922582E⫹00 0.7596191414E⫺01

Exponents

Symm.

0.2170325799E⫹00 0.8549975957E⫺01 0.3125305367E⫺01

Exponents

Symm.

Exponents

0.2170325799E⫹00 0.8549975957E⫺01 0.3125305367E⫺01


F⫺

0.1317339294E⫹04 0.5111787958E⫹03 0.2213273832E⫹03 0.1044616537E⫹03 0.5250617311E⫹02 0.2745793368E⫹02 0.1459491761E⫹02 0.7703428352E⫹01 0.3944458086E⫹01 0.1914184678E⫹01 0.8600937886E⫹00

F⫹

0.1317339294E⫹04 0.5111787958E⫹03 0.2213273832E⫹03 0.1044616537E⫹03 0.5250617311E⫹02 0.2745793368E⫹02 0.1459491761E⫹02 0.7703428352E⫹01 0.3944458086E⫹01 0.1914184678E⫹01 0.8600937886E⫹00

Atom Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Pb

S⫹

0.4397170675E⫹09 0.9997473321E⫹08 0.2559639542E⫹08

P⫺

0.1019329950E⫹09 0.2424307453E⫹08 0.6429845967E⫹07

P⫹

0.1019329950E⫹09 0.2424307453E⫹08 0.6429845967E⫹07

D⫺

0.3909520062E⫹05 0.1183891730E⫹05 0.4096297745E⫹04

Page 288

1 2 3

21:38

D⫹

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Symm.

5/19/2007

25 26 27 28 29 30

Exponents


Symm.

288

Atom Exp. #

F⫹

0.1266249494E⫹04 0.4951314662E⫹03 0.2162558019E⫹03 0.1030231291E⫹03 0.5227513482E⫹02 (continued )

Page 289

0.1266249494E⫹04 0.4951314662E⫹03 0.2162558019E⫹03 0.1030231291E⫹03 0.5227513482E⫹02

21:38

F⫺

0.1593069033E⫹04 0.6850366165E⫹03 0.3204067440E⫹03 0.1603506574E⫹03 0.8446827707E⫹02 0.4607261713E⫹02 0.2559711662E⫹02 0.1424986303E⫹02 0.7819416972E⫹01 0.4160587464E⫹01 0.2111660528E⫹01 0.1005669482E⫹01 0.4421002143E⫹00 0.1764789964E⫹00

5/19/2007

0.3909520062E⫹05 0.1183891730E⫹05 0.4096297745E⫹04 0.1593069033E⫹04 0.6850366165E⫹03

0.1887308537E⫹07 0.6084210114E⫹06 0.2137833946E⫹06 0.8125342962E⫹05 0.3315102053E⫹05 0.1440884183E⫹05 0.6621062958E⫹04 0.3192142972E⫹04 0.1602442396E⫹04 0.8312228924E⫹03 0.4421575472E⫹03 0.2393594449E⫹03 0.1308662256E⫹03 0.7171294663E⫹02 0.3908859342E⫹02 0.2103170609E⫹02 0.1108563187E⫹02 0.5680641606E⫹01 0.2808502525E⫹01 0.1329481319E⫹01 0.5980109735E⫹00 0.2536559552E⫹00 0.1006884068E⫹00 0.3711947918E⫺01


D⫹

0.1887308537E⫹07 0.6084210114E⫹06 0.2137833946E⫹06 0.8125342962E⫹05 0.3315102053E⫹05 0.1440884183E⫹05 0.6621062958E⫹04 0.3192142972E⫹04 0.1602442396E⫹04 0.8312228924E⫹03 0.4421575472E⫹03 0.2393594449E⫹03 0.1308662256E⫹03 0.7171294663E⫹02 0.3908859342E⫹02 0.2103170609E⫹02 0.1108563187E⫹02 0.5680641606E⫹01 0.2808502525E⫹01 0.1329481319E⫹01 0.5980109735E⫹00 0.2536559552E⫹00 0.1006884068E⫹00 0.3711947918E⫺01

289

1 2 3 4 5


Appendix 5

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Symm.

Symm.

0.2758816231E⫹02 0.1478738220E⫹02 0.7860939748E⫹01 0.4047119330E⫹01 0.1970514586E⫹01 0.8860314326E⫹00

Exponents

Symm.

Exponents

0.2758816231E⫹02 0.1478738220E⫹02 0.7860939748E⫹01 0.4047119330E⫹01 0.1970514586E⫹01 0.8860314326E⫹00

21:38

0.3204067440E⫹03 0.1603506574E⫹03 0.8446827707E⫹02 0.4607261713E⫹02 0.2559711662E⫹02 0.1424986303E⫹02 0.7819416972E⫹01 0.4160587464E⫹01 0.2111660528E⫹01 0.1005669482E⫹01 0.4421002143E⫹00 0.1764789964E⫹00

Exponents

5/19/2007

6 7 8 9 10 11 12 13 14 15 16 17

Exponents


Exp. #

290

Atom

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Bi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

S⫹


P⫺


P⫹


D⫺


Page 290

Exp. #

Appendix 5

Atom

0.4358389653E⫹00 0.1739192931E⫹00

21:38

0.1331393026E⫹04 0.5236688724E⫹03 0.2298306427E⫹03 0.1099069495E⫹03 0.5592085304E⫹02 0.2956100488E⫹02 0.1585355746E⫹02 0.8422902279E⫹01 0.4329020543E⫹01 0.2101724301E⫹01 0.9412075872E⫹00

F⫹

0.1331393026E⫹04 0.5236688724E⫹03 0.2298306427E⫹03 0.1099069495E⫹03 0.5592085304E⫹02 0.2956100488E⫹02 0.1585355746E⫹02 0.8422902279E⫹01 0.4329020543E⫹01 0.2101724301E⫹01 0.9412075872E⫹00

Page 291

F⫺

291


0.1444729247E⫹03 0.7980341532E⫹02 0.4383263997E⫹02 0.2375581289E⫹02 0.1260646323E⫹02 0.6500139924E⫹01 0.3231567327E⫹01 0.1537162395E⫹01 0.6942203083E⫹00 0.2953934613E⫹00 0.1175128672E⫹00 0.4337164849E⫺01

5/19/2007

D⫹

0.1444729247E⫹03 0.7980341532E⫹02 0.4383263997E⫹02 0.2375581289E⫹02 0.1260646323E⫹02 0.6500139924E⫹01 0.3231567327E⫹01 0.1537162395E⫹01 0.6942203083E⫹00 0.2953934613E⫹00 0.1175128672E⫹00 0.4337164849E⫺01

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17



16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

292

Po

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S⫹

0.4793529497E⫹09 0.1150969767E⫹09 0.3096165075E⫹08 0.9262636091E⫹07 0.3059102459E⫹07 0.1107136056E⫹07 0.4358671517E⫹06 0.1852900438E⫹06 0.8442935065E⫹05 0.4093330951E⫹05 0.2096046622E⫹05 0.1125289612E⫹05 0.6287320476E⫹04 0.3629137519E⫹04 0.2148208430E⫹04 0.1294443786E⫹04 0.7881745077E⫹03 0.4813850605E⫹03 0.2927470249E⫹03 0.1759627150E⫹03 0.1037709336E⫹03 0.5960138103E⫹02 0.3309490013E⫹02 0.1763553355E⫹02 0.8952359884E⫹01 0.4297406985E⫹01 0.1936395595E⫹01 0.8130157483E⫹00 0.3157328175E⫹00 0.1125783841E⫹00

P⫺


P⫹


D⫺


Appendix 5

5/19/2007

Exp. #


Atom

21:38 Page 292


D⫹


F⫺

0.1367160272E⫹04 0.5474848526E⫹03 0.2428772741E⫹03 0.1166999165E⫹03 0.5937859308E⫹02 0.3128036071E⫹02 0.1668026294E⫹02 0.8802988249E⫹01 0.4495316152E⫹01 0.2171701080E⫹01 0.9704124086E⫹00

F⫹

0.1367160272E⫹04 0.5474848526E⫹03 0.2428772741E⫹03 0.1166999165E⫹03 0.5937859308E⫹02 0.3128036071E⫹02 0.1668026294E⫹02 0.8802988249E⫹01 0.4495316152E⫹01 0.2171701080E⫹01 0.9704124086E⫹00

Atom

Exp. #

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

Symm.

Exponents

At

1 2 3 4 5 6 7 8 9 10 11 12

S⫹

0.4995639590E⫹09 0.1214680951E⫹09 0.3304753792E⫹08 0.9987133025E⫹07 0.3328024118E⫹07 0.1213933985E⫹07 0.4811557931E⫹06 0.2057204576E⫹06 0.9418666659E⫹05 0.4583957940E⫹05 0.2354236793E⫹05 0.1266592909E⫹05

P⫺

0.1024562876E⫹09 0.2560347544E⫹08 0.7111669722E⫹07 0.2179141879E⫹07 0.7310879445E⫹06 0.2665341384E⫹06 0.1048006761E⫹06 0.4410951325E⫹05 0.1972360530E⫹05 0.9299403976E⫹04 0.4588467045E⫹04 0.2351541934E⫹04

P⫹

0.1024562876E⫹09 0.2560347544E⫹08 0.7111669722E⫹07 0.2179141879E⫹07 0.7310879445E⫹06 0.2665341384E⫹06 0.1048006761E⫹06 0.4410951325E⫹05 0.1972360530E⫹05 0.9299403976E⫹04 0.4588467045E⫹04 0.2351541934E⫹04

D⫺

0.4736452685E⫹05 0.1360661431E⫹05 0.4527344172E⫹04 0.1714644286E⫹04 0.7264106021E⫹03 0.3383050242E⫹03 0.1702132563E⫹03 0.9092378277E⫹02 0.5067588213E⫹02 0.2896045419E⫹02 0.1667747020E⫹02 0.9510782598E⫹01

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

21:38 Page 293

Appendix 5 293

(continued )

Symm.

0.4736452685E⫹05 0.1360661431E⫹05 0.4527344172E⫹04 0.1714644286E⫹04 0.7264106021E⫹03 0.3383050242E⫹03 0.1702132563E⫹03 0.9092378277E⫹02 0.5067588213E⫹02 0.2896045419E⫹02 0.1667747020E⫹02 0.9510782598E⫹01 0.5278422335E⫹01


F⫺

0.1335057098E⫹04 0.5475634642E⫹03 0.2473267456E⫹03 0.1203568356E⫹03 0.6172975537E⫹02 0.3264401190E⫹02 0.1741240401E⫹02 0.9164763005E⫹01 0.4656419483E⫹01 0.2234150222E⫹01 0.9902916549E⫹00

Exponents 0.1242336842E⫹04 0.6715161226E⫹03 0.3685809494E⫹03 0.2038907869E⫹03 0.1128182905E⫹03 0.6197371632E⫹02 0.3354367136E⫹02 0.1775487406E⫹02 0.9121315159E⫹01 0.4513974579E⫹01 0.2135755114E⫹01 0.9588781384E⫹00 0.4054374977E⫹00 0.1602366509E⫹00 0.5874977040E⫺01

F⫹

0.1335057098E⫹04 0.5475634642E⫹03 0.2473267456E⫹03 0.1203568356E⫹03 0.6172975537E⫹02 0.3264401190E⫹02 0.1741240401E⫹02 0.9164763005E⫹01 0.4656419483E⫹01 0.2234150222E⫹01 0.9902916549E⫹00

Symm.

Exponents 0.5278422335E⫹01 0.2801782000E⫹01 0.1397807586E⫹01 0.6441452608E⫹00 0.2694536884E⫹00

Page 294

D⫹

Symm.

21:38


Exponents

5/19/2007

1 2 3 4 5 6 7 8 9 10 11 12 13

Symm.

Appendix 5

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Exponents


Exp. #

294

Atom


14 15 16 17

0.2801782000E⫹01 0.1397807586E⫹01 0.6441452608E⫹00 0.2694536884E⫹00

Exponents

Symm.

Exponents

Symm.

Exponents

Rn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

S⫹


P⫺


P⫹


D⫺


(continued )

Page 295

Symm.

21:38

Exponents

5/19/2007

Symm.

295

Exp. #

Appendix 5

Atom

Symm.

0.1089763045E⫹02 0.5213284369E⫹01 0.2340431945E⫹01 0.9788292754E⫹00 0.3785866368E⫹00 0.1344287237E⫹00 D⫹

Symm.

0.4357561881E⫹00 0.1741562989E⫹00 0.6458159887E⫺01

F⫺

Symm.

Exponents

0.4357561881E⫹00 0.1741562989E⫹00 0.6458159887E⫺01

F⫹

0.1329749051E⫹04 0.5507367179E⫹03 0.2505258923E⫹03 0.1224792061E⫹03 0.6297094106E⫹02 0.3331601106E⫹02 0.1774874793E⫹02 0.9316467477E⫹01 0.4714879160E⫹01 0.2251088251E⫹01 0.9921653440E⫹00

Page 296

0.1329749051E⫹04 0.5507367179E⫹03 0.2505258923E⫹03 0.1224792061E⫹03 0.6297094106E⫹02 0.3331601106E⫹02 0.1774874793E⫹02 0.9316467477E⫹01 0.4714879160E⫹01 0.2251088251E⫹01 0.9921653440E⫹00

Exponents

21:38


Exponents

Appendix 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Symm.

5/19/2007

25 26 27 28 29 30

Exponents


Exp. #

296

Atom


5/19/2007

21:38

Page 297

Appendix 5

References 1. R. L. A. Haiduke, and A. B. F. da Silva, J. Comput. Chem., 2006, 27, 61. 2. R. L. A. Haiduke, and A. B. F. da Silva, J. Comput. Chem., 2006, 27, 1970.

297


Else_EAMC-TRSIC_index.qxd

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Page 299

Subject Index

1s and 2s orbital energies, 114 1s generator functions, 62 2s HF weight function, 156 4d weight function, 62, 65 ab initio calculations, 7 accuracy, 1, 37, 50–52, 72–73, 75, 87, 89–90, 93–94, 110, 120, 131, 136 accurate Adapted Gaussian Basis Sets, 119 accurate basis set, 50, 94 accurate GTF basis sets, 50, 52 accurate numerical integration, 22 accurate relativistic energy, 119 adapted basis set, 67, 87, 135 advanced numerical integration techniques, 34 algebraic approximation, 108 algebraic structure, 20 alkaline atoms, 73–74 alternative paths, 14 analytical formulas, 46 analytical integration, 44 analytical mathematical case, 11 analytical solutions, 2, 5, 7, 9, 31 angular functions, 81 angular momentum, 81, 117 anions, 67, 72 antisymmetry, 20 approximate eigenvalues, 16 approximate linear dependence, 41 approximate relativistic many-electron Hamiltonian, 108 approximate schemes, 2, 9 arbitrary discretization parameters, 23 atom-adapted basis sets, 67 atom-adapted STF or GTF basis sets, 47, 49–52, 74

atomic and molecular relativistic calculations, 2, 93 atomic GTF exponents, 49 atomic mass number, 86–87, 93, 110, 114, 119 atomic natural orbitals, 156 atomic nuclear charge, 67 atomic nucleus, 20, 89, 100–101, 117, 132–133, 148 atomic number, 89, 94, 100, 115, 117, 120, 135–136, 146 atomic orbital coefficients, 94 atomic orbital symmetry, 48–51, 113–114, 119, 135 atomic orbital, 48–51, 67, 73, 76, 93–94, 100, 107, 113–114, 119–120, 135 atomic region, 132 atomic relativistic energy, 87 atomic symmetry, 76 atomic system, 6, 20, 49, 72–73, 87, 89, 107, 110, 114, 117, 133, 148, 157 atomic units, 9, 15, 27, 80 atom-optimized results, 67 atoms, 1, 3, 5, 12, 19, 23, 25, 46, 49–52, 55–58, 61, 63–65, 67–71, 73–75, 79, 87–88, 93–94, 100–101, 107, 110, 114, 117–120, 131–133, 135–136, 145–146, 151–153 attractive interactions, 20 augmentation, 145–148 average energy, 93, 120 basis components, 155 basis functions, 7, 27–28, 46, 56, 67, 72, 79, 83, 86–90, 111, 117 basis set design, 2, 26


300

6/6/2007

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Page 300

Subject Index

basis set error, 93–95, 107, 120, 128, 135–136, 143 basis set exponents, 50, 52, 56, 74, 83, 108, 114 basis set functions, 55, 89, 94, 107, 117 basis set segmentation, 52 basis set selection, 156 basis set size reduction, 117–118 basis set size, 51–52, 67, 72, 75–76, 94–96, 98, 107, 113–114, 117–120, 131, 135 basis set truncation process, 94 basis set, 2, 23, 26, 46–47, 49–53, 55–57, 59, 61, 63–67, 69, 71–77, 79, 83, 86–96, 98, 100–101, 107–108, 110–120, 128, 130–132, 135–136, 143–145, 147–148, 152–153, 155–156 basis, 2, 7, 23, 26–28, 32, 35, 39, 46–47, 49–53, 55–57, 59, 61, 63–67, 69, 71–77, 79, 83, 86–96, 98, 100–101, 107–108, 110–120, 128, 130–132, 135–136, 143–145, 147–148, 152–153, 155–156 Be-like species, 114, 117 biorthogonal natural state expansion, 35 block-diagonal form, 84 Bohr model, 19 bound state, 4–5 boundary conditions, 33, 79, 85, 87, 90, 111 Breit interaction matrices, 109 Breit interaction, 108–113, 117 Breit operator, 108 brute force optimization, 44

close-shell, 80 collective aspects of nuclei, 3 common ID interval, 94 complete set, 46, 50 completeness, 32 computational cost of molecular calculations, 50 computational demanding process, 114 computational effort, 3, 120 computational time, 42, 45 computer codes, 156 configuration interaction, 1, 3, 151, 153 continuous character, 83, 155–156 continuous natural weight function, 153 continuous representation, 22, 26, 39, 48 continuous superpositions, 82 continuous weight function, 158 contracted basis, 155 contraction, 67, 89 convergence tests, 101, 107 convergence, 22, 40–41, 44, 90, 100–107, 111–112, 132–134, 145–148 coordinate space, 22, 48, 83, 88, 93 correlated atomic and molecular energies, 156 correlated functions, 6 Coulomb and Breit interactions, 108, 110 Coulomb and exchange integrals, 85 Coulomb and exchange kernels, 48 Coulomb and exchange operators, 20 Coulomb and exchange potentials, 26 covariance, 108–109 covariant contributions, 116 cusp, 46 cut-off value, 41

Canonical orthonormalisation, 39 cations, 67, 72 central field potential, 81 charge distribution, 87, 89, 93–94, 101, 110, 114, 119, 133, 145, 147–148 charged fluorine atoms, 74 CI calculation, 153, 158–159 Clebsch-Gordon coefficients, 81 closed-shell DF equations, 108

d orbital, 64, 72–73, 75, 77, 92, 116 DC Hamiltonian, 108–109 definite integral formulae, 13 degree of the polynomial, 50–51, 114 delta distribution, 42 density functional theory, 1, 3, 151, 153, 155–157, 159 density matrices, 85 density of discretization points, 117


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Page 301

Subject Index determinant, 20–21, 36, 47, 151 determinants, 158 DF environment, 80 DF equations, 79–80, 82, 87, 90, 93, 108, 110, 114 DF formalism, 80 DF matrix, 84, 108 DF orbital energies, 90 DF theory, 119 DFB calculations, 108, 110, 113, 116, 119, 148 DFB energies, 110–111 DFB equations, 108 DFB matrix, 109 DFB orbital energies, 112–113 DFB-SCF calculations, 110 DFC calculations, 87, 89, 93–94, 115, 119 DFC energies, 87–89, 94–95, 111, 114, 128, 143 DFC energy, 79, 87, 89–92, 100–101, 107, 114, 120, 131–132, 136, 145 DFC matrix, 109 DFC orbital energies, 90–92, 118–119 DFC total energies, 94, 100–101, 107, 120, 135 DFC total energy convergence, 148 DFC total energy reduction, 133 DFC total energy, 94, 101, 107, 132–133, 145, 148 diagonal form, 84, 153 diagonal matrix, 35 diagonalization, 24, 35, 42, 45 diatomic molecules, 46 diffuse basis set function, 117, 120 diffuse orbitals, 74–75, 154–155 diffuseness, 94 dimensionality, 36 Dirac equation, 47 Dirac Hamiltonian, 80 Dirac matrices, 80–81 Dirac distribution, 76 direct solution, 11 discrete basis, 39 discrete character, 19

301

discrete weight function, 41 discretization parameters, 23, 25, 57, 65–67, 73, 76, 93, 100, 107, 114, 117, 119, 135, 145 discretization, 2, 5, 7, 9, 11, 19, 22–23, 25, 31, 33–35, 37–39, 41, 43–49, 51, 56–57, 65–67, 73, 76, 82–83, 90, 93, 100, 107–108, 110–111, 113–114, 117, 119–120, 132, 135, 145, 152 discretized eigenvalue problem, 35 discretized integration grid, 156 discretized weight functions, 35, 46 distribution, 6, 42, 75–76, 87, 89, 93–94, 101, 110, 114, 119, 133, 145, 147–148 Dyall’s basis sets, 131 effective discretization process, 117 Eigenfunction, 4 eigenfunctions, 32, 35 eigenvalue problem, 34–35, 37 eigenvalues, 16, 21, 31, 35–37, 39, 41, 47 eigenvector matrix, 35 electron charge, 21 electronic cloud, 94, 133, 136 electronic configuration, 94, 120, 136, 145 electronic structure, 46, 151, 158 energy expectation value, 47, 82 energy functional, 4 equally spaced mesh, 23, 44 even-tempered scheme, 114 exact ground state energy, 156 exact solution, 2, 4, 11–12, 15, 41 exchange term, 157 exchange-correlation term, 156–157 excited state, 16, 42, 44–45, 56, 58–59, 61, 65, 67, 69, 71, 156, 158–159 exclusion principle, 20 expansion coefficients, 83 fine-structure constant, 108 finite basis set expansion method, 46 finite body, 89 finite nuclear boundary conditions, 79


302

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Page 302

Subject Index

finite nuclear boundary results, 86 finite nuclear model, 79, 85–87, 119 finite nucleus approximation, 47 finite nucleus model of uniform proton-charge distribution, 87, 93, 110, 114, 119 finite nucleus of Gaussian proton-charge distribution, 94, 119 finite-difference procedure, 89 first-order perturbative approach, 115–116 first-order reduced density matrix, 152 fission problem, 3 Fock kernels, 21 force constant, 33 Fredhom equation, 34 frontier atomic orbital, 67 function exponents, 56, 79, 87, 90, 93–94, 110, 120, 132 functional form, 3, 14 Gaussian 2p weight functions, 74 Gaussian 2s weight functions, 73–74 Gaussian basis functions, 87–89 Gaussian distributions, 6 Gaussian overlap approximation, 5, 9, 31 Gaussian trial function, 11 Gaussians weight functions, 73 Gaussian-type functions, 1, 55, 79 Gaussian-type orbitals, 44 GC method, 52, 61, 67, 163, 164 GCDF method, 80, 82–83, 87–90, 93–94, 100, 107–108, 110–111, 113–114, 117–120, 132, 135–136, 148 GCDFB equations, 108 GCDFB formalism, 108, 110 GCDFC formalism, 85, 87, 108–110 GCHF method, 38, 47–53, 55–56, 65, 73–77 generating function, 38 generator coordinate ansatz, 10, 32, 47, 82 generator coordinate integral view, 158 generator coordinate space, 22, 83, 88, 93

generator coordinate, 1–6, 9–10, 19, 21–23, 25, 27, 31–34, 42, 47, 49, 55, 75–76, 79–80, 82–88, 93, 100, 108–109, 113, 119, 148, 151–152, 154–155, 157–158 generator function, 25–27, 38, 42, 44, 47, 62, 75, 82, 153, 155 geometrical Gaussian basis sets, 87 GHW equation, 2, 5, 10–13, 15, 19, 27, 32, 34–35, 41, 44 GHWHF equations, 21, 23, 47–49, 53, 82 Gram determinant, 36 Gram-matrix, 36 ground-state hydrogen atom function, 11 Hamiltonian, 2, 4, 9–11, 33, 35, 80, 108–109, 156–157 harmonic oscillator problem, 32, 34, 36, 38, 43 Hartree-Fock theory, 2, 3 He- and Be-like species, 114 He atom, 20, 22, 25–26, 49–50, 52, 55–56, 58, 63, 65, 67–68, 74, 86–89, 93–94, 100–101, 107, 110, 114–115, 117, 119–120, 132–133, 135–136, 146, 148, 151–153, 157–158 heavy atoms, 49, 52, 146 Hermitian, 11, 14, 35 HF energies, 51–52, 67, 72 HF energy, 50–52, 73–74 HF equations, 21, 23, 47–49, 53, 82 HF limit, 25, 72 HF orbitals, 152, 156 HF weight function, 155–156 HFS equations, 157 Hohemberg and Kohn expansion, 156 hydrogen atom, 11, 13, 15–16, 19, 44–45 ID technique, 56, 61, 65, 73, 83, 88, 90, 93, 107–108, 110, 119 independent particle model, 47, 76, 82 inner electrons, 80 inner orbitals, 64, 100 inner-core electronic penetration, 145–146


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Page 303

Subject Index innermost atomic orbitals, 94 instantaneous Coulomb operator, 108 integral character, 48, 107–108, 111, 153 integral Hellmann-Feynman formulation, 11 integral transform method, 6 integration by parts, 14 integration technique, 34, 44 ions, 19, 55–56, 65–71, 73–74, 115–117 isoelectronic species, 73 iterative procedure, 22 kinetic balance condition, 87, 90, 93, 110–111, 114, 117, 119 Koga’s basis set, 72 Kohn and Sham scheme, 156 Kohn-Sham orbitals, 156 Kronecker delta, 81 Lagrange multipliers, 21 Laplace transform, 6, 12 large and small components, 81, 82, 84, 87, 117 large and small exponents, 117 large component, 117 lighter atomic systems, 72, 107 linear combination coefficients, 1 linear dependence, 36, 41 linear transformation, 156 linearly independent variations, 5 liner combination of functions, 15 lowest positive parity solutions, 38 low-lying excited states, 56, 58, 65, 67 low-lying states, 37 many body perturbation theory, 1, 3 many-electron system, 20, 156 many-electron wave function, 20 mass number, 86–87, 93, 110, 114, 119 matrices of two-electron Coulomb and exchange interactions, 84 matrix DF equations, 87 Matsuoka’s method, 93, 120 mesh of ID points, 83 model problems, 9, 31, 61

303

molecular nonrelativistic calculations, 55, 76 molecular nuclei, 20 molecular relativistic calculations, 2, 93 momentum operator, 81 multicentre integrals, 46 natural orbital, 2, 151, 153, 155–157, 159 natural weight function, 153–156 negative ions, 55–56, 65–67, 70–71, 73–74 negative parity solutions, 33 negative parity states, 38 neutral atoms, 55–57, 65, 67, 73, 117 NHF energies, 51 noble gases, 61–64 nonlinear parameters, 22, 42, 44 nonorhtogonal representation, 34 nonrelativistic atomic and molecular calculations, 77 nonrelativistic limit, 82, 85, 87 normalisation kernels, 33 normalization constants, 86 normalized spherical harmonics, 81 normalized total wave function, 81 nuclear charge density, 89 nuclear charge, 9, 25–26, 67, 80, 89 nuclear liquid drop model, 3 nuclear models, 93–94, 101, 107, 119–120, 132–133 Nuclear Physics, 1–5, 9, 19, 31 nuclear proton-charge distribution, 145 nuclear radius, 86–87, 93, 110, 114, 119, 148 numerical accuracy, 89–90, 110 numerical approach, 32, 34 numerical approaches, 164 numerical DFC calculations, 94 numerical DFC energy results, 120 numerical errors, 36, 41 numerical HF functions, 65 numerical integration procedure, 108, 110 numerical techniques, 164 numerical-finite-difference DF calculations, 88, 90


304

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Page 304

Subject Index

one-electron density, 157 one-electron eigenvalue, 20 one-electron functions, 19–21, 47, 157–158 one-electron Hamiltonian, 108, 156 one-electron kernel, 48 one-electron kinetic energy operator, 20 one-electron matrix elements, 84 one-electron spin-functions, 21 optimized discretization parameters, 117 optimized pGCDF parameters, 135 orbital angular momentum, 81 orbital energies, 90–92, 101, 107, 112–114, 116–119 orbital exponents, 6, 57, 65, 79, 83, 108 orbital reorganization, 110 orbital symmetry, 48–51, 87, 113–114, 119–120, 132–133, 135 orbital wave function, 62, 89 orthogonal bases, 76 orthogonality, 32, 76, 155 orthonormal basis, 39 orthonormality, 40, 81 oscillator Hamiltonian, 33 oscillator wave functions, 33 outer shell electrons, 50 overlap and two-electron integrals, 131 overlap matrix, 31, 37, 84 parabolic-cylinder functions, 13 para-helium independent particle case, 9 Pauli matrices, 81 pGCDF basis sets, 114–117 pGCDF discretization parameters, 119, 135, 145 pGCDF method, 113–114, 117, 119–120, 132, 135–136, 148 point nucleus model, 46 polyatomic calculations, 46 polynomial degree, 114 polynomial expansion, 49–53, 113, 120 positive ions, 65–69 post-HF calculations, 156 potential energy, 81 prolapse analysis, 100, 145, 148

prolapse problem, 100, 120, 132, 135 prolapse-free RUGBS, 135, 146 proton-charge density, 148 proton-charge distribution, 87, 93–94, 101, 110, 114, 119, 133, 145, 147–148 quadrature method, 35 quantum electrodynamics, 108 quantum lattice dynamics, 7 radial large and small components, 82, 117 radial large-component basis set, 86 radial small-component basis set, 86 radial wave functions, 81 RAGBS augmented, 132, 145 RAGBS error, 120, 130–131, 135–136, 144 RAGBS exponents, 135 Rayleigh-Ritz variational method, 1 relativistic adapted Gaussian basis sets, 119, 135 relativistic and nonrelativistic calculations, 163, 164 relativistic angular coefficients, 110 relativistic atomic orbital symmetries, 93, 100, 107, 120 relativistic atomic orbital symmetry, 135 relativistic boundary conditions, 85 relativistic calculation, 2, 47, 52, 55, 76, 79, 81, 83, 85, 87–91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147 relativistic closed-shell atoms, 87, 110 relativistic corrections, 108 relativistic energies, 114 relativistic Gaussian basis sets, 77, 113–114, 119, 148 relativistic generator coordinate formalism, 148 relativistic GTF basis sets, 52 relativistic Hartree-Fock limit, 79 relativistic kinematics, 87, 89 relativistic triple-zeta basis sets, 131


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Page 305

Subject Index relativistic universal Gaussian basis set, 87–88, 91–93, 100, 111 restricted kinetic balance condition, 87, 93, 110, 114, 119 Roothaan equations, 22 Roothaan expansions, 1 RUGBS functions, 94 s, p and d orbitals, 72 saddle critical point, 15 saddle-like surface, 15 scaling parameter, 48–49, 88, 110, 114 scattering problem, 5 SCF Breit energies, 115 SCF energy, 110 SCF equations, 93, 109, 120 SCF procedure, 83, 108–110, 115 Schrödinger equation, 4, 19, 156 screened non interacting two-electron system, 10 secular equation, 15, 38 segmentation, 50, 52 segmentations, 50–51 self-consistent random phase approximation, 7 shape parameter, 3 SIMPLEX algorithm, 114, 120, 135 Simpson rule, 41 singularities, 31 Slater- and Gaussian-function exponents, 163, 164 Slater determinant, 21, 47 Slater determinants, 163 Slater orbitals, 163 Slater universal basis set, 55 Slater’s rules, 163 Slater’s transition state method, 156 Slater’s X potential, 157 small component, 117 spatial one-electron functions, 21 spectral problem, 37 speed of light, 79, 85, 87, 93, 110, 114, 119 square-integrable, 38–40 square-well problem, 32, 38

305

square-well solutions, 32 step function, 33 STF and GTF HF energies, 67 STF and GTF negative ions, 55, 65–67, 70, 73, 74 STF and GTF positive ions, 67 STF weight functions, 62, 74 symmetrical character, 42 test function, 76 tight d or f functions, 101, 107, 148 tight function, 100–101, 107, 119, 132, 145–147 tight p function, 101, 132–133, 145–146 tight s function, 101, 107, 133, 146 total electronic energy, 55–56 total wave function, 81 translated Gaussian, 42 trial function, 1–4, 6, 10–11, 21, 51, 157 triangular inequality, 40 truncation, 34, 37–38, 41, 51, 74, 94, 114 two-body interactions, 108 two-component Pauli spinors, 81 two-electron interaction, 108 two-electron kernels, 48 two-electron matrices, 85 two-parameter Hulthen function, 6 uncertainty principle, 20 unequally spaced numerical mesh, 50 uniform charge approximation, 85 uniform density, 147 uniform proton charge distribution, 89 uniform sphere and Gaussian nucleus models, 119, 133, 148 unitary transformation, 153 universal Gaussian basis set, 56–57, 79, 87–88, 91–94, 100, 111–112 universal mesh, 56 universal Slater basis set, 57 unperturbed DC Hamiltonian, 80, 108–109 valence region, 94, 119 variational approach, 10


306

6/6/2007

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Subject Index

variational Breit interaction energy, 110, 112 variational Breit interaction, 110–112 variational discretization, 19, 38 variational exponent optimization, 90, 93 variational principle, 21, 47 variational prolapse analysis, 100, 145 variational prolapse problem, 120, 132

weight function behavior, 94 weight function, 2, 4, 6–7, 9, 12, 15, 19, 21–23, 26–28, 31, 35, 41–42, 44–48, 55–57, 59, 61–65, 67, 69, 71–76, 82, 85, 94, 152–156, 158–159 well-tempered scheme, 79 X method, 163

wave function, 5, 12, 20, 32–33, 42, 46, 62–63, 65, 79, 81, 89, 152, 158

zero-order Hamiltonian, 108

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