Quantal Density Functional Theory II: Approximation Methods and Applications

Quantal Density Functional Theory II

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Viraht Sahni

Quantal Density Functional Theory II Approximation Methods and Applications

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Professor Dr. Viraht Sahni Brooklyn College and the Graduate School of the City University of New York 2900 Bedford Avenue Brooklyn, NY 11210, USA E-mail: [email protected]

ISBN 978-3-540-92228-5 DOI 10.1007/978-3-540-92229-2

e-ISBN 978-3-540-92229-2

Springer Heidelberg Dordrecht London New york Library of Congress Control Number: 2009060243 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar Steinen Printed on acid-free paper Springer is part of Springer Science+Business + Media (( (www.springer.com) )

“The best thing for being sad”, replied Merlyn. . . “is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world around you devastated by evil lunatics, or know your honor trampled in the sewers of baser minds. There is only one thing for it then – to learn.” – Merlyn, advising the young Arthur from The Once and Future King by T.H. White.

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To my wife, Catherine, the love of my life

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Preface

In my original proposal to Springer for a book on Quantal Density Functional Theory, I had envisaged one that was as complete in its presentation as possible, describing the basic theory as well as the approximation methods and a host of applications. However, after working on the book for about five years, I realized that the goal was too ambitious, and that I would be writing for another five years for it to be achieved. Fortunately, there was a natural break in the material, and I proposed to my editor, Dr. Claus Ascheron, that we split the book into two components: the first on the basic theoretical framework, and the second on approximation methods and applications. Dr. Ascheron consented, and I am thankful to him for agreeing to do so. Hence, we published Quantal Density Functional Theory in 2004, and are now publishing Quantal Density Functional Theory II: Approximation Methods and Applications. One significant advantage of this, as it turns out, is that I have been able to incorporate in each volume the most recent understandings available. This volume, like the earlier one, is aimed at advanced undergraduates in physics and chemistry, graduate students and researchers in the field. It is written in the same pedagogical style with details of all proofs and numerous figures provided to explain the physics. The book is independent of the first volume and stands on its own. However, proofs given in the first volume are not repeated here. I wish to acknowledge my graduate students Dr. Cheng Quinn Ma, Dr. Abdel Raouf Mohammed, Professor Manoj Kumar Harbola, Professor Marlina Slamet, Dr. Alexander Solomatin, Professor Zhixin Qian, and Professor Xiao Yin Pan whose creativity is being reported. They have each contributed in their own special way to this painting. Again, I owe a debt of gratitude to my friend and colleague Professor Lou Massa for his continued enthusiasm for this work, and for his generosity in taking the time to critique many sections of the book. I would also like to express my appreciation to Professor Marlina Slamet for her considerable assistance with the graphs. I thank Brooklyn College for its support of my research. It is the freedom afforded, a hallmark of the institution, that has allowed the pursuit of ideas different from the norm.

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Finally, my many thanks to my wife, Catherine, for undertaking the arduous and formidable task of typing the book, most of which was hand written. Without her, it would probably have taken another five years to complete. Brooklyn, New York, September 2009

Viraht Sahni

Contents

1

Introduction .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

1

2

Schr¨odinger Theory from a “Newtonian” Perspective . . . . . . . .. . . . . . . . . . . 2.1 Time-Independent Schr¨odinger Theory . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2 Schr¨odinger Theory from a “Newtonian” Perspective: The Pure State Differential Virial Theorem . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3 Definitions of Quantal Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3.1 Electron Density .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3.2 Spinless Single-Particle Density Matrix .rr 0 / . .. . . . . . . . . . . 2.3.3 Pair-Correlation Density g.rr 0 / and Fermi–Coulomb Hole xc .rr 0 / . . . . . . . . . . . . . . .. . . . . . . . . . . 2.4 Definitions of “Classical” Fields . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.4.1 Electron-Interaction Field E ee .r/ . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.4.2 Differential Density Field D.r/ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.4.3 Kinetic Field Z.r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.5 Energy Components in Terms of Quantal Sources and Fields.. . . . . . 2.5.1 Electron-Interaction Potential Energy Eee . . . . . . . .. . . . . . . . . . . 2.5.2 Kinetic Energy T .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.5.3 External Potential Energy Eext . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.6 Integral Virial, Force, and Torque Sum Rules . . . . . . . . . . . . .. . . . . . . . . . . 2.7 Coalescence Constraints.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

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Quantal Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.1 Quantal Density Functional Theory from a “Newtonian” Perspective .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2 Definitions of Quantal Sources Within Quantal Density Functional Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.1 Electron Density .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.2 Dirac Spinless Single-Particle Density Matrix s .rr 0 / . . . . . 3.2.3 Pair-Correlation Density gs .rr 0 /; Fermi x .rr 0 / and Coulomb c .rr 0 / Holes . . . . . .. . . . . . . . . . .

17 18 18 18 20 22 22 23 23 24 24 25 25 26 29 35 36 37 37 38 38

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3.3

3.4

3.5 3.6 3.7 3.8 4

5

Definitions of “Classical” Fields Within Quantal Density Functional Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3.1 Electron-Interaction Field E ee .r/, and Its Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ Components .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3.2 Differential Density Field D.r/ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3.3 Kinetic Z s .r/ and Correlation–Kinetic Z tc .r/ Fields . . . . . Total Energy and Its Components in Terms of Quantal Sources and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4.1 Electron-Interaction Potential Energy Eee , and Its Hartree EH , Pauli Ex , and Coulomb Ec Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4.2 Kinetic Ts , and Correlation-Kinetic Tc Energies .. . . . . . . . . . . 3.4.3 External Potential Energy Eext . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4.4 Total Energy E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Effective Field F eff .r/ and Electron-Interaction Potential Energy vee .r/.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Integral Virial, Force, and Torque Sum Rules . . . . . . . . . . . . .. . . . . . . . . . . Highest Occupied Eigenvalue m . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Quantal Density Functional Theory of Degenerate States .. . . . . . . . . .

New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.1 The Hohenberg–Kohn Theorems and Corollary . . . . . . . . . .. . . . . . . . . . . 4.2 Kohn–Sham Density Functional Theory . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2.1 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Generalization of the Fundamental Theorem 4.3 of Hohenberg–Kohn .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.3.1 The Unitary Transformation .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.3.2 New Insights as a Consequence of the Generalization . . . . . . Nonuniqueness of the Effective Potential Energy and Wave Function in Quantal Density Functional Theory . . . . . . .. . . . . . . . . . . 5.1 The Interacting System: Hooke’s Atom in a Ground State . . . . . . . . . . 5.2 Mapping to the S system in Its 11 S Ground State . . . . . . . .. . . . . . . . . . . 5.3 Mapping to an S system in Its 21 S Singlet Excited State . . . . . . . . . . . 5.4 Nonuniqueness of the Wave Function of the S system in an Excited State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.4.1 The Single Slater Determinant Case . . . . . . . . . . . . . .. . . . . . . . . . . 5.4.2 The Linear Combination of Slater Determinants Case . . . . . . 5.5 Proof that Nonuniqueness of Effective Potential Energy Is Solely Due to Correlation-Kinetic Effects . . . . .. . . . . . . . . . .

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Ad Hoc Approximations Within Quantal Density Functional Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 99 6.1 The Q-DFT of Hartree Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .103 6.1.1 The Q-DFT Hartree Uncorrelated Approximation . . . . . . . . . .106 6.1.2 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .107 6.2 The Q-DFT of Hartree–Fock Theory .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .107 6.2.1 The Q-DFT Pauli Approximation . . . . . . . . . . . . . . . . .. . . . . . . . . . .110 6.2.2 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .112 Time-independent Quantal-Density Functional Theory . .. . . . . . . . . . .113 6.3 6.3.1 The Q-DFT Pauli–Coulomb Approximation . . . . .. . . . . . . . . . .114 6.3.2 The Q-DFT Fully Correlated Approximation.. . . .. . . . . . . . . . .117 6.4 The Case of Nonconservative Fields. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .119 6.4.1 The Central Field Approximation . . . . . . . . . . . . . . . . .. . . . . . . . . . .119 6.4.2 The Irrotational Component Approximation.. . . . .. . . . . . . . . . .121

7

Analytical Asymptotic Structure in the Classically Forbidden Region of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .125 7.1 The Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .127 7.2 The Single-Particle Density Matrix and Density . . . . . . . . . .. . . . . . . . . . .130 7.3 The Pair-Correlation Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .132 7.4 The Work Done in the Electron-Interaction Field. . . . . . . . .. . . . . . . . . . .133 7.4.1 The Hartree, Pauli, and Coulomb Potential Energies.. . . . . . .135 7.5 The Correlation-Kinetic Potential Energy .. . . . . . . . . . . . . . . .. . . . . . . . . . .137 7.6 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .138

8

Analytical Asymptotic Structure At and Near the Nucleus of Atoms . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .141 8.1 Proof of Finiteness of Potential Energies vee .r/ and vB ee .r/ at the Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .143 8.2 Criticality of the Electron–Nucleus Coalescence Condition to Local Effective Potential Energy Theories... . . . . . . . . . .145 8.3 General Structure of vee .r/ Near the Nucleus of Spherically Symmetric and Sphericalized Systems . . . .. . . . . . . . . . .148 8.4 Exact Structure of vee .r/ Near the Nucleus of Spherically Symmetric and Sphericalized Systems . . . . . . .. . . . . . . . . . .153 8.4.1 Near Nucleus Structure of the Wave Functions and the Density .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .154 8.4.2 Electron-Interaction Field E ee .r/ at the Nucleus .. . . . . . . . . . .159 8.4.3 Kinetic “Force” z.rI  / Near the Nucleus . . . . . . . .. . . . . . . . . . .160 8.4.4 Kinetic “Force” zs .rI s / Near the Nucleus . . . . . .. . . . . . . . . . .163 8.4.5 Correlation-Kinetic Field Z tc .r/ Near the Nucleus . . . . . . . .164 8.4.6 Structure of Potential Energy vee .r/ Near the Nucleus . . . . .165 8.5 Endnote . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .165

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Application of the Q-DFT Hartree Uncorrelated Approximation to Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .167 9.1 Electronic Structure of the Neon Atom .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . .168 9.2 Atomic Shell Structure and Core–Valence Separation .. . .. . . . . . . . . . .175 9.2.1 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .178 9.3 Total Ground State Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179 Highest Occupied Eigenvalues .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .179 9.4 9.4.1 Satisfaction of the Aufbau Principle .. . . . . . . . . . . . . .. . . . . . . . . . .184

10 Application of the Q-DFT Pauli Correlated Approximation to Atoms and Negative Ions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .187 10.1 Ground State Properties of Atoms . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .188 10.1.1 Electronic Structure of the Argon Atom . . . . . . . . . .. . . . . . . . . . .188 10.1.2 Atomic Shell Structure and Core–Valence Separation . . . . . .196 10.1.3 Total Ground State Energies .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .199 10.1.4 Highest Occupied Eigenvalues . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .202 10.1.5 Satisfaction of the Aufbau Principle .. . . . . . . . . . . . . .. . . . . . . . . . .207 10.1.6 Single-Particle Expectation Values .. . . . . . . . . . . . . . .. . . . . . . . . . .209 10.2 Ground State Properties of Mononegative Ions . . . . . . . . . . .. . . . . . . . . . .214 10.2.1 Total Ground State Energies .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .215 10.2.2 Highest Occupied Eigenvalues . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .216 10.3 Static Polarizabilities of the Neon Isoelectronic Sequence . . . . . . . . . .217 11 Quantal Density Functional Theory of the Density Amplitude: Application to Atoms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .221 11.1 Quantal Density Functional Theory of the Density Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .222 11.2 Application to Atoms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .226 11.3 Conclusions and Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231 11.4 Consequences for Traditional Density Functional Theory . . . . . . . . . .232 12 Application of the Irrotational Component Approximation to Nonspherical Density Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .235 12.1 Scalar Effective Fermi Hole Source xeff .r/ . . . . . . . . . . . . . . .. . . . . . . . . . .236 12.1.1 Spherically Symmetric Density Atoms . . . . . . . . . . .. . . . . . . . . . .237 12.1.2 Nonspherical Density Atoms . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .237 12.2 Vector Vortex Fermi Hole Source J x .r/ . . . . . . . . . . . . . . . . .. . . . . . . . . . .239 12.3 Irrotational E Ix .r/ and Solenoidal E Sx .r/ Components of the Pauli Field E x .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .241 12.4 Path-Independent Pauli Potential Energy WxI .r/ . . . . . . . . .. . . . . . . . . . .245 12.5 Endnotes on the Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .246 13 Application of Q-DFT to Atoms in Excited States . . . . . . . . . . . . .. . . . . . . . . . .249 13.1 The Triplet 2 3 S State Isoelectronic Sequence of He . . . . .. . . . . . . . . . .251 13.2 One-electron Excited States of Li.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .252 13.3 One-electron Excited States of Na .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .254

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Multiplet Structure of C and Si . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .256 Doubly Excited Autoionizing States of He . . . . . . . . . . . . . . . .. . . . . . . . . . .258 Endnote . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .261

14 Application of the Multi-Component Q-DFT Pauli Approximation to the Anion–Positron Complex: Energies, Positron and Positronium Affinities . . . . .. . . . . . . . . . .263 14.1 Equations of the Multi-Component Q-DFT Pauli Approximation . .264 14.2 Brief Remarks on Hartree–Fock Theory of Positron Binding to Anions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .267 14.3 Total Energy of the Anion–Positron Complex and Positron Affinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .268 14.4 Positronium Affinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .272 15 Application of the Q-DFT Fully Correlated Approximation to the Helium Atom .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .275 15.1 The Interacting System: Helium Atom in Its Ground State . . . . . . . . .276 15.2 Mapping to an S System in Its 11 S Ground State . . . . . . . .. . . . . . . . . . .277 15.2.1 Coulomb Hole Charge Distribution c .rr 0 / . . . . . .. . . . . . . . . . .277 15.2.2 Pauli–Coulomb E xc .r/, Pauli E x .r/, and Coulomb E c .r/ Fields, and the Pauli–Coulomb Exc , Pauli Ex , and Coulomb Ec Energies . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .281 15.2.3 Pauli–Coulomb Wxc .r/, Pauli Wx .r/, and Coulomb Wc .r/ Potential Energies.. . . . . . . . . .. . . . . . . . . . .283 15.2.4 Correlation-Kinetic Field Z tc .r/, Potential Energy Wtc .r/, and Energy Tc . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .284 15.2.5 Total Energy and Ionization Potential .. . . . . . . . . . . .. . . . . . . . . . .286 15.3 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .287 16 Application of the Q-DFT Fully Correlated Approximation to the Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .289 16.1 The Interacting System: Hydrogen Molecule in Its Ground State .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .289 16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .290 16.2.1 Fermi–Coulomb xc .rr 0 /, Fermi x .rr 0 /, and Coulomb c .rr 0 / Hole Charge Distributions . . . . . . . . . . .291 16.2.2 Electron Interaction E ee .r/ and Correlation-Kinetic Z tc .r/ Fields. . . . . . . . . . . .. . . . . . . . . . .296 16.2.3 Electron-Interaction Potential Energy vee .r/ . . . . .. . . . . . . . . . .298 16.2.4 Total Energy and Ionization Potential .. . . . . . . . . . . .. . . . . . . . . . .300 16.3 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .301

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17 Application of Q-DFT to the Metal–Vacuum Interface . . . . . . .. . . . . . . . . . .303 17.1 Jellium Model of a Metal Surface . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .306 17.2 Surface Model Effective Potential Energies and Orbitals . . . . . . . . . . .311 17.2.1 The Finite Linear Potential Model . . . . . . . . . . . . . . . .. . . . . . . . . . .311 17.2.2 The Linear Potential Model.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .312 17.3 Accuracy of the Model Potentials . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .314 17.4 Structure of the Fermi Hole at a Metal Surface . . . . . . . . . . .. . . . . . . . . . .316 17.4.1 General Expression for the Planar Averaged Fermi Hole x .xx 0 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .317 17.4.2 Structure of the Planar Averaged Fermi Hole x .xx 0 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .321 17.4.3 Structure of Fermi Hole in Planes Parallel to the Surface .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .325 17.5 General Expression for the Pauli Field E x .x/ and Potential Energy Wx .x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .328 17.6 Structure of the Pauli Field E x .x/ and Potential Energy Wx .x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .332 17.7 Analytical Structure of the Pauli Potential Energy Wx .x/. . . . . . . . . . .335 17.8 Analytical Structure of the Lowest Order Correlation.1/ Kinetic Potential Energy Wtc .x/ in the Classically Forbidden Region .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .338 17.8.1 Analytic Asymptotic Structure of the Slater Function VxS .x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .339 17.8.2 Analytical Asymptotic Structure of the Kohn–Sham “Exchange” Potential Energy vx .r/ . . . . . . . . . . .340 17.8.3 Analytical Asymptotic Structure of the Lowest-Order Correlation-Kinetic Potential Energy Wt1c .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .343 17.9 Analytical Structure of the Coulomb Wc .x/ and Second-and Higher-Order Correlation Kinetic Wt2c .x/, Wt3c .x/ : : : Potential Energies in the Classically Forbidden Region .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .344 17.9.1 New Expression for Kohn–Sham “ExchangeCorrelation” vxc .r/ Potential Energy in Classically Forbidden Region.. . . . . . . . . . . . . . . . . .. . . . . . . . . . .345 17.9.2 Analytical Asymptotic Structure of the Orbital k .x/, Dirac Density Matrix s .xx 0 /, and Density .x/ . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .347 17.9.3 Analytical Asymptotic Structure of the Kohn–Sham “Correlation” Potential Energy vc .x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .348 17.10 Analytical Asymptotic Structure of the Effective Potential Energy vs .x/ in the Classically Forbidden Region .. . . . . . .350 17.11 Endnote on Image-Potential-Bound Surface States . . . . . . .. . . . . . . . . . .353

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18 Many-Body and Pseudo Møller-Plesset Perturbation Theory within Quantal Density Functional Theory . . . . . . . . . . .. . . . . . . . . . .355 18.1 Many-Body Perturbation Theory within Q-DFT .. . . . . . . . .. . . . . . . . . . .356 18.1.1 Quantal Sources in Terms of Green’s Functions ... . . . . . . . . . .356 18.1.2 Perturbation Series for the Electron-Interaction Field E ee .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .360 18.1.3 Perturbation Series for the Correlation-Kinetic Field Z tc .r/ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .362 18.1.4 Approximations within the Perturbation Theory .. . . . . . . . . . .365 18.1.5 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .366 18.2 Pseudo Møller–Plesset Perturbation Theory Within Q-DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .367 18.2.1 Pseudo Møller–Plesset Q-DFT Perturbation Theory . . . . . . . .367 18.2.2 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .371 19 Epilogue . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .373 A

Quantal Density Functional Theory of Degenerate States . . . .. . . . . . . . . . .375

B

Generalization of the Runge–Gross Theorem of Time-Dependent Density Functional Theory . . . . . . . . . . . . . . . .. . . . . . . . . . .383

C

Analytical Asymptotic Structure of the CorrelationKinetic Potential Energy in the Classically Forbidden Region of Atoms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .387

D

The Pauli Field E x .r/ and Potential Energy Wx .r/ in the Central Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .393

E

Equations of the Irrotational Component Approximation as Applied to the Carbon Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .397 E.1 Electron Density .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .397 E.2 Fermi Hole x .rr 0 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .398 E.3 Gradient of Fermi Hole rx .rr 0 / . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .398 E.4 Pauli Field E x .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .399 E.5 Scalar Effective Fermi Hole xeff .r/ . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .399 E.6 Vector Vortex Fermi Hole J x .r/ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .399 E.7 Irrotational Component E Ix .r/ of the Pauli Field E x .r/ . .. . . . . . . . . . .400 E.8 Solenoidal Component E Sx .r/ of the Pauli Field E x .r/ . .. . . . . . . . . . .400 E.9 The Potential Energy WxI .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .401

F

Ground State Properties of the Helium Atom as Determined by the Kinoshita Wave Function . . . . . . . . . . . . . . .. . . . . . . . . . .403 F.1 Wave Function .r 1 r 2 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .403 F.2 Electron Density .r/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .404 Coulomb Hole c .rr 0 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .405 F.3

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F.4 F.5 G

Coulomb Field E c .r/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .405 Coulomb Potential Energy Wc .r/ . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .406

Approximate Wave Function for the Hydrogen Molecule . . . .. . . . . . . . . . .407

References .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .409 Index . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .423

Chapter 1

Introduction

In a recent book [1] entitled Quantal Density Functional Theory, referred to from now on in abbreviated form as QDFT, I described a new theory of the electronic structure of matter. Quantal density functional theory (Q-DFT) is the description [1–8] of a quantum-mechanical system of electrons in terms of “classical” fields and their quantal sources within the framework of local effective potential energy theory. The theory is based on a similar description of Schr¨odinger theory in terms of quantal sources and fields [1, 9, 10]. This “classical” description of a quantummechanical system is based on the integral and differential virial theorems of quantum mechanics [1–5,11]. The formal ideas underlying both time-dependent and time-independent Q-DFT as explicated in QDFT are within the Born–Oppenheimer Approximation [12]. As with Schr¨odinger theory, time-independent Q-DFT is a special case of the time-dependent theory. Q-DFT additionally provides insights into other traditional local effective potential energy theories. Thus, an understanding of Slater theory [13], and a rigorous physical interpretation of the Optimized Potential Method [14, 15] and of Hohenberg–Kohn–Sham density functional theory [16, 17] are also provided in QDFT. (For the rigorous physical interpretation of time-dependent Runge–Gross [18] density functional theory via Q-DFT, the reader is referred to the literature [4–6].) In QDFT, further understandings in terms of electron correlations of the Local Density Approximation of Hohenberg–Kohn– Sham density functional theory and of the discontinuity issue [19] within local effective potential energy theory are also described. Hence, QDFT is comprised of the theoretical foundations of the theory, the further elucidation of the theory by application to the ground and excited states of an exactly solvable interacting model system, its relationship to and physical interpretation of other local effective potential energy theories, and of many fundamental insights arrived at via Q-DFT of the local effective potential energy approach to electronic structure. This volume is on various approximation methods within and selected applications of time-independent Q-DFT. To understand time-independent Q-DFT, one must first describe Schr¨odinger theory from the new perspective of “classical” fields and quantal sources. This new perspective on time-independent Schr¨odinger theory is described in Chap. 2. Thus, in addition to approximation schemes and applications, there are formal in principle components to the book: the description of Schr¨odinger theory from this “Newtonian” perspective; the extension of

1

2

1 Introduction

nondegenerate state Q-DFT to degenerate states [20]; and the generalization [21] of the fundamental theorem of both time-independent density functional theory of Hohenberg–Kohn [16] and of its extension to time-dependent phenomenon due to Runge–Gross [18]. The book is written to be as independent of QDFT as possible: all the requisite physics, and the corresponding equations and sum rules relevant to time-independent Q-DFT are given in Chap. 3. For the description of time-dependent Q-DFT, and proofs of the formal framework of the theory, however, the reader is referred to QDFT. Quantal density functional theory (Q-DFT) is a theory of both ground and excited states of a many-electron system. Nondegenerate state Q-DFT is described in Chap. 3, and degenerate state Q-DFT in Appendix A. Q-DFT provides the most general definition of local effective potential energy theory [22]. Consider a system of N electrons in the presence of an arbitrary time-independent external field F ext .r/ such that F ext .r/ D r v.r/, and in an arbitrary, nondegenerate ground or excited state as described by Schr¨odinger theory. Q-DFT is the direct mapping from this interacting system of electrons with electron density .r/, as determined by solution of the time-independent Schr¨odinger equation, to one of N noninteracting fermions with equivalent density .r/. The model system of noninteracting fermions with equivalent density .r/ is referred to as the S system. The existence of such S systems is an assumption. (The mapping to a model system of N noninteracting bosons with equivalent density .r/ is discussed later.) In the mapping from the interacting electronic to the noninteracting fermion model system the state of the latter is arbitrary. Thus, for example, it is possible via Q-DFT to map an interacting system in its ground state to an S system, which is also in a ground state (see Fig. 1.1). However, the mapping could be to an S system in an excited state, again with a density which is the same as that of the interacting system in its ground state (see Fig. 1.1). Similarly, a system of electrons in an excited state can be mapped via Q-DFT to an S system that is either in its ground state or an excited state of the same configuration as that of the electrons or any other excited state with a different configuration (see Fig. 1.2). From any of these model systems, whether in a ground or excited state, the corresponding total energy E and ionization potential I (or electron affinity A) of the interacting system are also thereby obtained. As the model fermions of the S system are noninteracting, the effective potential energy of each fermion in the presence of the external field F ext .r/ is the same. As a consequence, this potential energy can be represented in the corresponding Schr¨odinger equation – the S system differential equation for the model-fermion single-particle spin-orbitals – by a local (multiplicative) potential energy operator vs .r/. The resulting wave function of the S system is then a Slater determinant of these spin-orbitals. This Slater determinantal wave function then leads to the same density .r/ as that of the interacting system, and thereby to all expectations of nondifferential Hermitian single-particle operators. Note that since the state of the S system is arbitrary, and one may construct an S system in a ground or excited state, there exist in principle an infinite number of local effective potential energy functions vs .r/ that will generate the density .r/ of the interacting system in question. Additionally, in the mapping to a model system in an excited state, there is also a

1 Introduction

3

SYSTEM OF ELECTRONS

MODEL SYSTEMS OF NONINTERACTING FERMIONS

EXCITED State Configurations

Mapping via Q-DFT

GROUND STATE

GROUND State Configuration Mapping via Q-DFT & HKS-DFT

Fig. 1.1 A pictorial representation of the Q-DFT mapping from the ground state of the interacting system of electrons to model systems of noninteracting fermions with equivalent density. This mapping can be used to model systems in either their ground or excited states. In HKS-DFT, the mapping is from the ground state of the interacting system to a model system only in its ground state

SYSTEM OF ELECTRONS

Mapping via Q-DFT

MODEL SYSTEMS OF NONINTERACTING FERMIONS

Mapping via Q-DFT & HKS-DFT To Model System With Same Configuration

EXCITED State Configurations

ANY EXCITED STATE

Mapping via Q-DFT GROUND State Configuration

Fig. 1.2 A pictorial representation of the Q-DFT mapping from any excited state of the interacting electrons to model systems of noninteracting fermions with equivalent density. This mapping can be used to model systems in either their ground or excited states. The HKS-DFT mapping is from an excited state of the interacting system to a model system in an excited state of the same configuration

4

1 Introduction

nonuniqueness of the model system wave function. Different wave functions lead to the same density, each thereby satisfying the sole requirement of reproducing the interacting system density. As noted above, Q-DFT does not require the model system to be constructed in the same configuration or be an image of the true interacting system. Thus, these wave functions are not constrained to be eigenfunctions of various spin-symmetry operators. The nonuniqueness of the effective potential energy vs .r/ of the S system and of its wave function is demonstrated [22] in Chap. 5 for an exactly solvable atomic model of an interacting electronic system. In the interacting system as described by Schr¨odinger theory, the electrons are correlated as a result of the Pauli exclusion principle and Coulomb repulsion. The Pauli principle is manifested by the requirement that the wave function be antisymmetric in an interchange of the coordinates of the electrons including the spin coordinate. The analytical dependence of Coulomb repulsion between the electrons in the solution to the Schr¨odinger equation, however, is unknown at present. In the mapping to the model S system with equivalent density .r/, the correlations due to the Pauli principle and Coulomb repulsion must be accounted for. The effective potential energy of the model fermions vs .r/ is thus comprised of the sum of their potential energy v.r/ in the external field and an electron-interaction potential energy component vee .r/ representative of these correlations. The S system must, however, also account for the difference between the kinetic energy of the interacting and noninteracting systems, both with the same density .r/. It is evident from the Heisenberg’s uncertainty principle that these kinetic energies cannot be the same because the effective volume available to the interacting electrons is different from that for the model noninteracting fermions. The difference is the correlation contribution to the kinetic energy, and it is referred to as the Correlation-Kinetic effect. Thus, the effective electron-interaction potential energy vee .r/ is representative of correlations due to the Pauli principle, Coulomb repulsion, and the CorrelationKinetic effect. Similarly, the expression for the total energy E involves, in addition to the external potential energy, terms representative of each of these three types of correlations. (In the time-dependent case as explained in QDFT, the differences between the current densities of the interacting and model systems must also be accounted for by the S system. This difference is the Correlation-Current-Density effect.) Within Q-DFT, we refer to correlations due to the Pauli exclusion principle as Pauli correlations. Since the S system wave function is a Slater determinant of spin orbitals, it is evident that these correlations exist between model fermions of parallel spin. Correlations due to Coulomb repulsion are referred to as Coulomb correlations. These correlations are representative of the repulsion between electrons of both parallel and antiparallel spin. The contribution of correlations due to the Pauli principle and Coulomb repulsion to the kinetic energy are the Correlation-Kinetic effects. Within Q-DFT, it is possible to delineate the three different electron correlations – Pauli correlations, Coulomb correlations, and Correlation-Kinetic effects – and their separate contributions to properties such as the effective potential energy vee .r/ of the model fermions, the total energy E, and other properties. Within the

1 Introduction

5

context of Q-DFT, we thus refer to Pauli correlations, Coulomb correlations, and Correlation-Kinetic effects. In Q-DFT, the description of the S system of model fermions is in terms of a “classical” conservative effective field F eff .r/, which is representative of all the different electron correlations that must be accounted for. The field is “classical” in the sense that it pervades all space. The effective field F eff .r/ may be written as a sum of its components, with each component separately representing a particular electron correlation. The assignation of the component fields as being representative of Pauli correlations, Coulomb correlations or Correlation-Kinetic effects is via their respective quantal sources. The sources of the component fields are quantal in that they are quantum-mechanical expectations of Hermitian operators or the complex sum of Hermitian operators taken with respect to the interacting and S system wave functions. Within Q-DFT then, the effective electron-interaction potential energy vee .r/ of the model fermions is the work done to move such a fermion in the force of the conservative field F eff .r/. The total energy E and its components can also be expressed in integral virial form in terms of these component fields. The highest occupied eigenvalue of the S system, whether in a ground or excited state, is the negative of the ionization potential I (or the electron affinity A). Hence, the description of the interacting quantum-mechanical system via Q-DFT is in terms of fields and their quantal sources. As in classical physics, each component field contributes to both the potential energy of the model fermions as well as to the corresponding component of the total energy. As explained in QDFT, it is also possible within time-independent Q-DFT to map a system of electrons in any state, ground or excited, as described by Schr¨odinger theory to one of noninteracting bosons in their ground state such that the corresponding interacting system density .r/, energy E, and ionization potential I are obtained [1]. Such a model system is referred to as a B system. Again, as the bosons are noninteracting, the effective potential energy vB .r/ of each boson is the same. The solution p to the corresponding B-system differential equation is the density amplitude .r/, and its eigenvalue is the negative of the ionization potential I . As was the case for the S system, the potential energy vB .r/ of the bosons and the total energy E are expressed in terms of “classical” fields whose sources are quantal. The potential energy vB .r/ is the sum of the potential energy v.r/ due to the external field and an electron-interaction potential energy vB ee .r/ that represents all the correlations the B system must account for. These are the correlations due to the Pauli exclusion principle and Coulomb repulsion, as well as the Correlation-Kinetic effects due to the difference in kinetic energy between the interacting electronic and noninteracting bosonic systems of the same density .r/. The potential energy vB ee .r/ of the model bosons is the work done to move such a boson in the force of a conservative effective field F eff B .r/. The total energy E can be expressed in integral virial form in terms of the components of this field which are each representative of a different electron correlation. The single eigenvalue of the B system is the negative of the ionization potential I . From the proofs of these definitions given in QDFT, it becomes evident that the model B system is a special case of the S system.

6

1 Introduction

The rationale for the mapping from the interacting system to model systems with equivalent density .r/ stems historically from Hohenberg–Kohn [16] and Kohn–Sham [17] density functional theory, a ground state theory. Hohenberg– Kohn–Sham density functional theory and new perspectives on it are described in Chap. 4. The theory is founded on the two theorems of Hohenberg and Kohn. According to the first or fundamental theorem, the stationary-state solutions to the Schr¨odinger equation for the ground or excited states are unique functionals of the ground state density .r/ W D Œ.r/. The theorem is proved for N electrons in a nondegenerate ground state in the presence of an arbitrary external field F ext .r/ D r v.r/, and thus for arbitrary scalar external potential energy operator v.r/. The theorem proves the bijectivity between the operator v.r/ and the ground state density .r/ to which it leads on solution of the Schr¨odinger equation. Knowledge of the ground state density thus uniquely determines the external potential energy operator v.r/ of the electrons to within an arbitrary constant C . Since the operators representing the kinetic energy and the electron-interaction potential energy are assumed known, the ground state density uniquely determines the Hamiltonian HO of the system. Solution of the Schr¨odinger equation for this Hamiltonian then leads to the system ground and excited state wave functions . Thus, the stationary state is a functional of the ground state density .r/. As such the expectations of all operators, and thus of the Hamiltonian, are unique functionals of the ground state density .r/. The second theorem of Hohenberg and Kohn states that the ground state energy E, a functional of the ground state density .r/ W E D EŒ, can be obtained by a variational principle involving the density. The variational densities .r/Cı.r/ are determined from similar interacting system Hamiltonians but for different external potential energy operators v.r/. These densities preserve the R number N of electrons so that ı.r/d r D 0. For arbitrary variations of such densities, the ground state energy EŒ is obtained for the true ground state density .r/. In a recent work [21–23], further understandings of the fundamental theorem of Hohenberg–Kohn have been achieved. These new insights are described in Chap. 4. For one, we now understand the conditions under which the fundamental theorem of bijectivity between the ground state density .r/ and Hamiltonian HO is not applicable. A Corollary [23] stating these conditions together with an example is provided. It is possible to construct an infinite number of degenerate Hamiltonians fHO g, corresponding to different physical systems, all with the same density .r/. In such a case, there is no bijectivity because the density .r/ then cannot distinguish between the different Hamiltonians fHO g. (For the corresponding Corollary to the fundamental theorem of Runge–Gross time dependent density functional theory, see [23] or QDFT.) A second understanding [21] is arrived at by a generalization of the fundamental theorem achieved via a unitary transformation that preserves the density .r/. Hence, it is proved that the theorem is valid not only for Hamiltonians with a scalar external potential energy operator v.r/, but also for those O that additionally include the momentum pO and curl-free vector potential energy A operators. The generalization of the time dependent theorem of Runge–Gross is described in Appendix B. As a consequence of the generalization, the fundamental theorems of Hohenberg–Kohn and Runge–Gross then each constitute a special

1 Introduction

7

case. The generalizations hence expand the realm of the application of these theorems. Another fundamental understanding achieved by the generalization is that in the most general case, the time-independent Schr¨odinger theory wave function is a functional of the ground state density .r/ and a gauge function ˛.R/, where R D r 1 ; : : : ; r N I D Œ˛.R/I .r/. (In the time-dependent case (see Appendix B), the density and gauge function also depend upon time.) The choice of gauge function ˛.R/, however, is arbitrary, so that the choice of ˛.R/ D 0 is a valid one. This then provides a deeper understanding of the Hohenberg–Kohn statement above that the wave function is solely a functional of the density .r/. Kohn–Sham density functional theory (KS-DFT) [17] is a manifestation of the Hohenberg–Kohn theorems within the context of local effective potential energy theory. Thus, it is also a ground state theory: the nondegenerate ground state of the interacting system is mapped to model S and B systems also in their ground state. The existence of such model systems is an assumption. In KS-DFT, the first Hohenberg–Kohn theorem is explicitly employed to express the ground state energy E of the system of noninteracting fermions or bosons as a functional of the ground state density .r/ W E D EŒ. The many-body effects – Pauli and Coulomb correlations and Correlation-Kinetic effects – are embedded in the KS electron-interaction KS energy functional component of the total energy EŒ W Eee Œ for the S system; B Eee Œ for the B system. As a consequence of the second Hohenberg–Kohn theorem, the corresponding electron-interaction potential energies of the noninteracting fermions and bosons are then defined, respectively, as the functional derivatives KS B vee .r/ D ıEee Œ=ı.r/ and vB ee .r/ D ıEee Œ=ı.r/. The manner in which the various electron correlations are incorporated in the respective electron-interaction energy functionals is, however, not described by KS-DFT. Thus, these functionals and their derivatives are unknown. Hence, in any application of the theory, all the physics of the various electron correlations must be extrinsically incorporated into constructing these energy functionals and their functional derivatives. One consequence of the use of such approximate energy functionals is that the variational principle aspect of the second Hohenberg–Kohn theorem is then no longer valid. Thus, there is no bound on the total energy obtained by these functionals. From the first Hohenberg–Kohn theorem, it follows that knowledge of the ground state density .r/ then uniquely determines the KS-DFT S system electroninteraction potential energy or functional derivative vee .r/. The potential energy is unique for the mapping from the ground state of the interacting system to the S system in its ground state. Thus, within the context of KS-DFT, there is only one S system local effective potential energy function that generates the ground state density .r/ (see Fig. 1.1). The same potential energy is obtained via Q-DFT. However, as noted above, within Q-DFT, the state of the S system is arbitrary in that it may be in a ground or excited state. In Q-DFT, the S system is not restricted to being solely in its ground state. This generality is possible because, Q-DFT is not based on the theorems of Hohenberg–Kohn but rather on the integral and differential virial theorems of quantum mechanics. Thus, we observe that it is not possible to learn from ground state KS-DFT that there exist other local effective potential energy functions

8

1 Introduction

that can generate the ground state density of the interacting system. In this context then, KS-DFT constitutes a special case of Q-DFT. There is no equivalent to the first Hohenberg–Kohn theorem for excited states [24–26]. In other words, knowledge of the excited state density .r/ does not uniquely determine the external potential energy operator v.r/. Thus, there is no one-to-one correspondence between the excited state density and the Hamiltonian of the interacting system. For the model S system of noninteracting fermions, the implication of the lack of the first Hohenberg–Kohn theorem for excited states means that there is no unique local effective potential energy function vs .r/ that would generate orbitals leading to the excited state density .r/. In excited state KS-DFT [27], for a specific excited state k of density k .r/, there exists a bidensity energy functional Ek Œ; g , where g .r/ is the exact ground state density, whose value at  D k is the energy Ek of that state. For the S system, this means that KS there exists a bidensity electron-interaction energy functional Ek;ee Œ; g , whose functional derivative evaluated at the excited state density k is the local electroninteraction potential energy vee .r/ that generates orbitals which reproduce the KS excited state density: vee .r/ D ıEk;ee Œ; g =ı.r/jDk . In KS-DFT, one is mapping to a model system with the same excited state configuration as that of the interacting system. In this manner, one local effective potential energy function that generates the excited state density is selected (see Fig. 1.2). In contrast, as noted previously, within Q-DFT it is possible to map an excited state of the interacting system to model S systems in their ground or any excited state such that the specific interacting system excited state density is reproduced [22]. As there exists only KS one bidensity energy functional Ek;ee Œ; g , KS-DFT of excited states thus also constitutes a special case of Q-DFT. In Chap. 6 ad hoc approximation schemes within Q-DFT are described. In a manner similar to that of Schr¨odinger theory, the approximations are based on the systematic but ad hoc inclusion of the different electron correlations. As described briefly below, at the lowest order approximation of Q-DFT, there are no electron correlations. At the next and higher order levels of approximation, one introduces, respectively, Pauli correlations, Coulomb correlations, and CorrelationKinetic effects. Approximations within Schr¨odinger theory are made on the basis of the electron correlations assumed in the approximate wave function. Thus, in the simplest Hartree product of spin-orbitals–type wave function, correlations due to the Pauli exclusion principle are not accounted for because such a wave function is not antisymmetric in an interchange of the coordinates of the particles including those of the spin coordinate. (In calculations within Hartree theory, the occupation of states to satisfy the consequence of the Pauli exclusion principle that no two electrons occupy the same state is thus ad hoc.) Nor are correlations due to Coulomb repulsion explicitly accounted for in this approximate wave function. The best spin-orbitals, from the perspective of the total energy as derived via the variational principle [28], are those obtained via Hartree theory [29]. Although the Hartree theory wave function is a product of single-particle orbitals, the theory is not rigorously an independent particle theory. This is because the potential energy of each electron in this theory

1 Introduction

9

depends upon the charge density of all the other electrons. Thus, the potential energy of each electron is different. Hartree theory is therefore an orbital-dependent theory. It is, however, possible to map the interacting system as described by Hartree theory to one of noninteracting particles whereby the density and energy equivalent to that of Hartree theory is obtained. This is the Q-DFT of Hartree theory (see Chap. 6 and QDFT) where both an S and a B system may be constructed. If the corresponding Correlation-Kinetic effects are ignored, then the model particles are rigorously independent, and one obtains the Q-DFT Hartree Uncorrelated Approximation. The difference between the results of this approximation and those of Hartree theory are then an estimate of the Correlation-Kinetic effect contributions. If, within Schr¨odinger theory, the approximate wave function is assumed to be a Slater determinant of spin-orbitals, then correlations due to the Pauli exclusion principle are accounted for because the wave function is then antisymmetric. Once again Coulomb correlations are not represented explicitly in this single-particle type wave function. From the perspective of the total energy, the best spin-orbitals are then obtained by application of the variational principle via Hartree–Fock theory [30,31]. (As explained in QDFT, according to Bardeen [32] and Slater [13], Hartree–Fock theory may also be reinterpreted as an orbital-dependent theory, with the potential energy of each electron being different.) Within Hartree–Fock theory, the correlations due to the Pauli principle which keep electrons of parallel spin apart are referred to as exchange effects. Furthermore, there is an inherent Correlation-Kinetic contribution to the kinetic energy because of these correlations between electrons of parallel spin. This contribution to the kinetic energy is not determined by Hartree– Fock theory. Once again, it is possible to map the interacting system as described by Hartree–Fock theory to one of noninteracting fermions or bosons whereby the equivalent density and energy are obtained. This is the Q-DFT of Hartree–Fock theory (see Chap. 6 and QDFT). In this Q-DFT or local effective potential energy theory representation of Hartree–Fock theory, it is once again possible to separate out the contribution to the kinetic energy as a result of the model fermions of parallel spin being kept apart – the Correlation-Kinetic contribution. Thus, one distinguishes between the Pauli correlation and Correlation-Kinetic contributions to the effective potential energy and total energy of the model fermions. (The reason why in Q-DFT the correlations due to the Pauli principle are not referred to as exchange effects is twofold. In Hartree–Fock theory, exchange is represented by a nonlocal (integral) operator. In Q-DFT, these effects are represented by a local (multiplicative) operator. Furthermore, as noted above, it is possible to separate out the contribution of the Correlation-Kinetic contribution to the kinetic energy.) If in the Q-DFT of Hartree–Fock theory, the Correlation-Kinetic effects are ignored, one obtains the Q-DFT Pauli Approximation. The difference between the results of this approximation and those of Hartree–Fock theory are then an estimate of these Correlation-Kinetic effects. Within the rubric of ad hoc approximation methods within Q-DFT, Coulomb correlations and Correlation-Kinetic effects are incorporated by assuming a correlatedtype wave function or a wave function functional (Chap. 6). For example, the wave function functional may be of the form of a Slater determinant times a

10

1 Introduction

correlation factor term that depends upon a function: the orbitals of the Slater determinant are obtained by self-consistent solution of the Q-DFT differential equation while simultaneously the function on which the correlation functional depends upon is determined by satisfaction of a physical constraint. In the Q-DFT Pauli–Coulomb Approximation, only Coulomb correlations are included additionally. Finally, with Correlation-Kinetic effects also included, one obtains the Q-DFT Fully Correlated Approximation. In Chap. 6, two additional approximation methods within Q-DFT are discussed: The Central Field Approximation and The Irrotational Component Approximation. These methods address cases for which the components of the effective field F eff .r/, and hence the effective field F eff .r/, may not be conservative. In Chap. 18, we develop a Many-Body Perturbation Theory within Q-DFT. In Q-DFT, as noted previously, the correlations due to the Pauli principle and Coulomb repulsion – the electron-interaction component – are separated from those of the Correlation-Kinetic effects. Thus, a separate perturbation series is developed for the electron-interaction and Correlation-Kinetic components. At lowest order of the perturbation theory, only Pauli correlations are accounted for. As in standard many-body perturbation theory [33], the bound on the total energy at this lowest order is rigorous. Coulomb correlations and Correlation-Kinetic effects may be incorporated separately to the higher order desired from the corresponding perturbation series. The many-body perturbation theory within Q-DFT also provides a formal justification for the approximation methods described above based on the ad hoc but systematic inclusion of the different electron correlations. In Chap. 18, we also describe the Pseudo-Møller—Plesset Perturbation Theory within Q-DFT. In this theory, the key attributes of Møller—Plesset Perturbation Theory [34] are employed in self-consistent conjunction with Q-DFT to construct an accurate wave function. The wave function is superior to that of Møller—Plesset Perturbation Theory because its orbitals incorporate all three types of electron correlations – Pauli, Coulomb, and Correlation-Kinetic. Furthermore, the bound on the total energy as determined via the equations of Q-DFT or equivalently from the wave function, is rigorous. (This chapter assumes knowledge of the more advanced Many-Body and Møller-Plesset Perturbation theories, and for this reason is placed towards the end of the book.) The applications of Q-DFT that have been selected are described in Chaps. 7–17. Many of these applications involve calculations that incorporate all the electron correlations, viz. those due to the Pauli exclusion principle, Coulomb repulsion and Correlation-Kinetic effects. Others involve the inclusion of only specific electron correlations. One of the purposes of the book is to study the evolution of a property of a system, as each higher order of electron correlation is introduced. Thus, beginning with the case of rigorously independent particles with none of the above electron correlations present, one then systematically introduces Pauli correlations, Coulomb correlations, and Correlation-Kinetic effects. As a consequence of the fact that the contributions of the different electron correlations are defined explicitly within Q-DFT, it is possible to obtain many results analytically. These results are valid for fully self-consistent orbitals when all the electron correlations are present. (By fully self-consistent is meant the equivalent of employing an

1 Introduction

11

infinite basis set.) The detailed derivations of these results are provided. The majority of numerical results obtained are those determined in a fully self-consistent manner. Other numerical results are obtained employing accurate analytical orbitals and wave functions. Comparisons with the results of other theories and with experiment are made throughout. The emphasis in each chapter is to elucidate understandings in terms of the different electron correlations, and thereby to demonstrate the rigor of the physics achieved via Q-DFT. The analytical asymptotic structure of the S system local electron-interaction potential energy vee .r/ in the classically forbidden region of atoms is derived in Chap. 7 and Appendix C. The explicit separate contributions to this structure due to Pauli correlations, Coulomb correlations, and Correlation-Kinetic effects are obtained. It is shown that the asymptotic structure is solely due to Pauli correlations, with the Coulomb correlation and Correlation-Kinetic contributions decaying more rapidly. The analytical asymptotic structure of the S system electron-interaction potential energy vee .r/ at and near the nucleus of atoms is derived in Chap. 8. Again, the explicit separate contributions to this structure of the different electron correlations are provided. It is also proved here that the potential energy function vee .r/, and the corresponding B system potential energy vB ee .r/, are finite at the nucleus of atoms, molecules, and solids, and that this result is valid for arbitrary state and symmetry. Furthermore, this finiteness of the potential energy functions at the nucleus is shown to be a direct consequence of the electron–nucleus coalescence constraint [35] on the Schr¨odinger theory wave function. In Chap. 9, the Q-DFT Hartree Uncorrelated Approximation in which the model particles are rigorously independent is applied to atoms in their ground state. In this chapter, The Central Field Approximation is invoked for open-shell atoms by spherically averaging the orbitals. In Chap. 10 and affiliated Appendix D, the Q-DFT Pauli Approximation in which Pauli correlations are introduced is applied to atoms and mononegative ions in their ground state. In this chapter The Central Field Approximation is invoked for open-shell atoms by spherically averaging the fields. In each chapter, a description of the physics from a Q-DFT perspective for a specific atom is first provided. The approximations are then applied to the elements of the Periodic Table to obtain properties such as the shell structure and core–valence separation, total ground state energy, highest occupied eigenvalue (which is related to the ionization potential or electron affinity), the satisfaction of Madelung’s aufbau principle, single-particle expectations, and static polarizabilities. As expected, there is an improvement of the results on the introduction of Pauli correlations into the wave function. However, it is interesting to note that even at the entirely uncorrelated level of approximation, with the Pauli exclusion principle invoked in an ad hoc manner, properties such as atomic shell structure and core–valence separations are obtained accurately. In Chap. 11 we apply the Q-DFT of the Density Amplitude to atoms, and compare the mapping to the system of noninteracting bosons with that of the mapping to the system of noninteracting fermions. In these mappings, the Pauli and Coulomb correlation contributions are the same, with the differences arising solely due to Correlation-Kinetic effects. It turns out that these latter effects are particularly

12

1 Introduction

significant in the mapping to the system of noninteracting bosons. In Chap. 12 and Appendix E, we demonstrate the Q-DFT Irrotational Component Approximation as applied to a model open-shell nonspherical density atom for which the effective field F eff .r/ is not conservative. A key understanding achieved by this study is that, in essence all the many-body correlations are represented by the irrotational component of the field F eff .r/, the solenoidal component being many orders of magnitude smaller. For atoms in their excited states, it is expected that Coulomb correlations and Correlation-Kinetic effects are smaller than when the atoms are in their ground state. As such in Chap. 13, the Q-DFT Pauli Approximation is applied to certain atoms to study one-electron excited states, multiplet structure, and doubly-excited autoionizing states. In these calculations, the mapping from the interacting system is to the S systems of the same configuration. In Chap. 14, Multi-Component Q-DFT in its Pauli Approximation is applied to the anion-positron complex with the positron in its ground and excited states to determine the energy of the complex as well as positron affinities. Calculations for the determination of positronium affinities at this level of electron correlation are also described, and it is shown that the anion-positron complex is stable against the dissociation into a positronium and a neutral atom. The role of Coulomb correlations and Correlation-Kinetic effects is shown to be significant. In Chap. 15, in addition to Pauli correlations we now include Coulomb correlations and Correlation-Kinetic effects by application of the Q-DFT Fully Correlated Approximation to the ground state of the Helium atom. These calculations are performed by employing an accurate wave function given in Appendix F. As such, the Q-DFT properties obtained are “exact.” The mapping is to an S system also in its ground state. (This mapping of the ground state two-electron system to two model fermions in their ground state is equivalent to the mapping to two model bosons in their ground state: the S and B systems are equivalent in this case.) In this manner, a complete description of the Helium atom within the framework of Q-DFT is provided: the structure of all the quantal sources, fields, potential energies, the components of the total energy, and the ionization potential are all given. As a result of incorporating additionally the Coulomb correlations and CorrelationKinetic effects, there is an improvement in the results for both the total energy as well as for the highest occupied eigenvalue over the purely Pauli correlated results. (The highest occupied eigenvalue is the negative of the ionization potential.) A significant fact gleaned from these calculations is that there is substantial cancelation of the Coulomb correlation and Correlation-Kinetic potential energies of the model fermions. As expected, the Coulomb correlation potential energy is negative, and lowers the total energy. The Correlation-Kinetic potential energy, on the other hand, is positive, and raises the total energy. The magnitude of the Coulomb correlation component of the total energy, however, is nearly twice as large, thereby leading to a lowering of the total energy to its exact value. As an example of the application of Q-DFT to molecular systems, in Chap. 16 we apply the Q-DFT Fully Correlated Approximation to the ground state of the Hydrogen molecule. As in the case of the application of this approximation to the

1 Introduction

13

Helium atom, these calculations too are performed employing an accurate wave function given in Appendix G, so that all the Q-DFT properties obtained are “exact.” The mapping is once again to an S (or equivalently a B) system also in its ground state. A complete description in terms of quantal sources, fields, potential energies, total energy components, and the ionization potential is given. The description of the Hydrogen molecule from the perspective of Q-DFT is distinct from the conventional description given in textbooks. Consequently, new physics is gleaned as a result of this different perspective. The final application of Q-DFT is to the inhomogeneous electron gas at a metal–vacuum interface as described in Chap. 17. The metal is represented by the semi-infinite jellium model, with vacuum as the other half-space [32]. The model provides the essential physics of the metal–vacuum interface. This nonuniform electron gas system differs from the few-electron, finite, discrete energy spectrum systems studied thus far in that it is a many-electron, extended, continuum energy spectrum system. Thus, the physics of the metal–vacuum interface is significantly different. For example, whereas in finite systems the quantal sources are principally localized about the nucleus with an extension into the classically forbidden region, in the semi-infinite-metal surface problem, these sources are not localized to the metal surface but delocalized and spread principally throughout the crystal with again a fraction of the quantal source extending into the classically forbidden vacuum region. A principal advantage of the jellium model of the metal surface is that within the framework of local effective potential energy theory, accurate analytical model effective potential energy functions may be constructed such that the delocalized orbitals of the noninteracting fermions may be written analytically. This allows for many results to be obtained in closed analytical form or semi-analytically, thereby providing for a considerable simplification of the numerical calculations. As in prior applications, since the contribution of Pauli and Coulomb correlations and Correlation-Kinetic effects are defined and delineated within Q-DFT, it is possible to study metal surface properties as a function of the different electron correlations. As a further consequence, analytical results are derived that are valid for the exact fully self-consistent orbitals. One such result concerns the asymptotic structure of the effective potential energy function in the vacuum region. In classical physics, the structure of the potential energy of a test charge in front of a metal surface is the image potential, and this structure is the same for all metals. Here it is shown that the quantum-mechanical structure of the potential energy function of the asymptotic model fermion is image-potential-like but not the image potential, with a coefficient that is approximately twice as large as the classical result and one that is dependent on the metal density. In contrast to finite systems, all the correlations – Pauli, Coulomb, and Correlation-Kinetic – contribute to this asymptotic structure. Key results within KS-DFT are also derived. Metal surface physics is a field unto itself, and hence it is only selected aspects of the subject that are discussed. However, the chapter is self-contained and written so as to be understood without any prior knowledge of the subject: all the fundamental properties of interest are explained and defined, and the details of all derivations are provided. The Epilogue is Chap. 19.

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Chapter 2

Schr¨odinger Theory from a “Newtonian” Perspective

Time-independent quantal density functional theory (Q-DFT) [1] is a description of the mapping from a system of electrons in an external field in their ground or excited state as described by Schr¨odinger theory, to a model system of noninteracting fermions – the S system – whereby the equivalent density .r/, the energy E, and the ionization potential I are obtained. The reason for the mapping to the model S system is that for a system of N electrons, it is easier to solve the corresponding N model-fermion single-particle differential equations than it is to solve the single N -electron Schr¨odinger equation. The model system is described by Q-DFT via a “Newtonian” perspective. This perspective is in terms of fields that are “classical” in nature and which pervade all space, but whose sources are quantal in that they are quantum-mechanical expectations of Hermitian operators or of the complex sum of Hermitian operators. As the model system is in effect a representation of the interacting system, it is best to first describe Schr¨odinger theory [2] from the same “Newtonian” perspective of “classical” fields and quantal sources [3, 4]. This perspective of Schr¨odinger theory is new. To quote from Einstein and Infeld [5]: “A new concept appeared in physics, the most important invention since Newton’s time: the field. It needed great scientific imagination to realize that it is not the charges nor the particles but the field in the space between the charges and particles that is essential for the description of the physical phenomenon.” These remarks were made with reference to the classical physics of Faraday and Maxwell. Einstein and Infeld may not have imagined then that nonrelativistic Quantum Mechanics/Schr¨odinger theory too could be similarly described in terms of fields that are “classical” in nature but which arise from sources that are quantal.

2.1 Time-Independent Schr¨odinger Theory Consider a system of N electrons in the presence of a time-independent external field F ext .r/ such that F ext .r/ D r v.r/, where v.r/ is the potential energy of an electron. The Hamiltonian operator HO of this system is the sum of the kinetic energy TO , external potential energy VO , and the electron-interaction potential energy UO operators:

15

16

2 Schr¨odinger Theory from a “Newtonian” Perspective

HO D TO C VO C UO ;

(2.1)

where 1 TO D  2

N X

ri2 ;

(2.2)

v.r i /;

(2.3)

N 1 1 X0 : UO D 2 jr i  r j j

(2.4)

VO D

i D1

N X i D1

i;j D1

The time-independent Schr¨odinger equation [2] in the Born–Oppenheimer [6] Approximation is (in atomic units: e D „ D m D 1) HO .X / D E .X /;

(2.5)

where the wave functions .X / are the eigenfunctions of HO , and E the eigenvalues of the energy. (Note that no subscripts are employed for the wave function and eigenvalue to distinguish between ground and excited states.) The symbol X designates the coordinates of the electrons: X D x 1 ; x 2 ; : : : ; x N , x D r, where r and  are the spatial and spin coordinates. The energy E of the system in a particular state is the expectation value of the Hamiltonian: E D h .X / j HO j

.X /i:

(2.6)

This energy may then be written in terms of its kinetic T , external potential Eext , and electron-interaction potential Eee energy components as E D T C Eext C Eee ; where

T D h .X / j TO j

.X /i;

(2.7)

(2.8)

Eext D h .X / j VO j

.X /i;

(2.9)

Eee D h .X / j UO j

.X /i:

(2.10)

The quantum-mechanical system of the N electrons in the presence of the external field F ext .r/ as described by time-independent Schr¨odinger theory can alternatively be afforded a rigorous “Newtonian” interpretation. This description is in terms of fields that are “classical,” but whose sources are quantum-mechanical in that they are expectations of Hermitian operators or of the complex sum of Hermitian operators. These quantal sources, and therefore the fields, are separately representative of the kinetic, external, and electron-interaction components of the physical system. We next describe the quantal system from this “Newtonian” perspective [3, 4].

2.2 Schr¨odinger Theory from a “Newtonian” Perspective

17

2.2 Schr¨odinger Theory from a “Newtonian” Perspective: The Pure State Differential Virial Theorem According to Newton’s first law of motion [7], the law of equilibrium, for a system of N particles that obey Newton’s third law of action and reaction, exert forces on each other that are equal and opposite, and lie along the line joining them, the sum of all the forces both external and internal acting on a particle vanish. The analogous “Newtonian” equation of motion for the sum of the forces acting on an electron – the “Quantal Newtonian” first law – is F ext .r/ C F int .r/ D 0;

(2.11)

where F int .r/ is the internal field of the electrons. Since F ext .r/ D r v.r/, the potential energy v.r/ of an electron due to the external field is directly related to the internal field F int .r/. Thus Z

r

v.r/ D 1

0

0

r v.r /  d` D

Z

r 1

F int .r 0 /  d` 0

(2.12)

where we have assumed v.1/ D 0. Therefore, v.r/, the external potential energy is the work done to move an electron from some reference point at infinity to its position at r in the force of the internal field. The potential energy v.r/ of an electron due to the external field F ext .r/ is thereby intrinsically linked to the internal field F int .r/ experienced by the electron. From (2.11) and the definition of F ext .r/, it follows that r  F int .r/ D 0;

(2.13)

so that the work done v.r/ is path-independent as originally assumed. The internal field F int .r/ is comprised of the sum of three components: an electron-interaction field E ee .r/, a differential density field D.r/, and a kinetic field Z.r/: F int .r/ D E ee .r/  D.r/  Z.r/:

(2.14)

These fields and the quantal sources which give rise to them are described in the following sections. The “Quantal Newtonian” first law (2.11) is the pure state time-independent differential virial theorem of quantum mechanics [1, 8, 9]. The quantal analogue to Newton’s second law of motion for an individual electron – the “Quantal Newtonian” second law – is the pure state time-dependent differential virial theorem [10–12]. For a description of the “Quantal Newtonian” second law, see QDFT. As in classical physics, the “Quantal Newton’s” first law is a special case of the “Quantal Newton’s” second law [4]. For a proof of the time-dependent differential virial theorem see Appendix A of QDFT.

18

2 Schr¨odinger Theory from a “Newtonian” Perspective

2.3 Definitions of Quantal Sources The various quantal sources are defined in terms of their probabilistic interpretations as well as expectations of Hermitian operators. The quantal sources are the electronic density .r/, the spinless single-particle density matrix .rr 0 /, the paircorrelation density g.rr 0 /, and from it the Fermi–Coulomb hole charge distribution xc .rr 0 /.

2.3.1 Electron Density .r/ The electron density .r/ is N times the probability of an electron being at r: .r/ D N

XZ

?

.r; X N 1 / .r; X N 1 /dX N 1 ;

(2.15)



R P R where X N 1 D x 2 ; x 3 ; : : : ; x N ; dX N 1 D dx 2 ; : : : ; dx N , and dx D  dr. The density may also be expressed as the expectation value of the Hermitian density operator X .r/ O D ı.r  r i /; (2.16) i

so that .r/ D h .X / j .r/ O j

.X /i:

(2.17)

Integration of the electronic density over all space then gives the total number of electrons: Z .r/dr D N: (2.18) The electron density is a static or local charge distribution. Although its value is different for each electron position r, the structure of this charge remains unchanged.

2.3.2 Spinless Single-Particle Density Matrix .rr 0 / The spinless single-particle density matrix .rr 0 / is defined as .rr 0 / D N

XZ

?

.r; X N 1 / .r 0 ; X N 1 /dX N 1 :

(2.19)



Note that in general, .rr 0 / is complex. Thus, there exists no Hermitian operator whose expectation value yields .rr 0 /. Consider, however, the Hermitian operator AO defined as

2.3 Definitions of Quantal Sources

19

  1X 0 O ı.r j  r/Tj .a/ C ı.r j  r /Tj .a/ ; AD 2

(2.20)

j

where Tj .a/ is a translation operator such that Tj .a/ .r 1 ; : : : ; r j ; : : : ; r N / D .r 1 ; : : : ; r j C a; : : : ; r N /, and a D r 0  r. Then the expectation value h .X / j AO j and since we have

  1 0 0 O .rr / C .r r/ ; .X /i  hAi D 2

(2.21)

.r 0 r/ D  ? .rr 0 /;

(2.22)

O D <.rr 0 /: hAi

(2.23)

Similarly, the expectation value of the Hermitian operator BO defined as i BO D  2

 X ı.r j  r/Tj .a/  ı.r j  r 0 /Tj .a/ ;

(2.24)

j

is   i 0 0 O hBi D  .rr /  .r r/ 2 D =.rr 0 /:

(2.25) (2.26)

Thus, defining the density matrix operator [13, 14] O O .rr 0 / D AO C i B;

(2.27)

we can write the single-particle density matrix as the expectation .rr 0 / D h .X / j O .rr 0 / j

.X /i;

(2.28)

which in turn is the expectation value of the complex sum of two Hermitian operators. The single-particle density matrix .rr 0 / differs from the Dirac density matrix s .rr 0 / to be defined later in that it is not idempotent but satisfies the inequality Z

.rr 00 /.r 00 r 0 /dr 00 < .rr 0 /:

(2.29)

The diagonal matrix element of .rr 0 / is the density: .rr/ D .r/;

(2.30)

20

2 Schr¨odinger Theory from a “Newtonian” Perspective

and this property is the same as the diagonal element of the Dirac density matrix s .rr/ D .r/.

2.3.3 Pair-Correlation Density g.rr 0 / and Fermi–Coulomb Hole xc .rr 0 / The pair-correlation density g.rr 0 / is a property representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion. It is the conditional density at r 0 of the other electrons, given that one electron is at r. It is defined as the ratio of the expectations of two Hermitian operators: g.rr 0 / D

P .rr 0 / ; .r/

(2.31)

with the pair function P .rr 0 / being the expectation P .rr 0 / D h .X / j PO .rr 0 / j

.X /i;

(2.32)

where PO .rr 0 / is the Hermitian pair-correlation operator PO .rr 0 / D

X ı.r i  r/ı.r j  r 0 /:

(2.33)

i ¤j

The pair function P .rr 0 / is the probability of simultaneously finding electrons at r and r 0 . The total charge of the pair-correlation density for each electron position r is Z g.rr 0 /dr 0 D N  1: (2.34) To prove the sum rule of (2.34), we rewrite the expectation of P .rr 0 / as ˇ   ˇX ˇ ˇ 0 ˇ ˇ P .rr / D ˇ ı.r i  r/ı.r j  r /ˇ 0

i;j

ˇ   ˇX ˇ ˇ 0 ˇ ˇ  ˇ ı.r i  r/ı.r i  r /ˇ

(2.35)

i

ˇ   ˇX X ˇ ˇ 0 ˇ ˇ D ı.r i  r/ ı.r j  r /ˇ  ı.r  r 0 /.r/: ˇ i

j

(2.36)

2.3 Definitions of Quantal Sources

21

Thus, Z

ˇ   ˇX XZ ˇ ˇ 0 0ˇ ˇ ı.r j  r /dr ˇ ı.r i  r/ P .rr /dr D ˇ 0

0

i

Z

.r/

j

ı.r  r 0 /dr 0

D N.r/  .r/; so that

1 .r/

Z

P .rr 0 /dr 0 D N  1:

(2.37) (2.38)

(2.39)

We refer to the pair-correlation density as a dynamic or nonlocal charge distribution, because for nonuniform electron density systems it depends on both the coordinates r and r 0 , and as such its structure changes as a function of electron position r. (For the uniform electron gas, g.rr 0 / depends upon j r  r 0 j.) If there were no electron correlations, the density at r 0 would simply be .r 0 /. However, due to the electron correlations – the keeping apart of the electrons – there is a reduction in the density at r 0 . This change in the pair-correlation density relative to the density, which occurs as a consequence of the Pauli exclusion principle and Coulomb repulsion, is called the Fermi–Coulomb hole charge distribution xc .rr 0 /. Thus, we may also write the pair-correlation density as g.rr 0 / D .r 0 / C xc .rr 0 /;

(2.40)

and thereby express it as the sum of its local and nonlocal components. As a consequence of the sum rules (2.18) and (2.34), the total charge of the Fermi–Coulomb hole for each electron position r is Z

xc .rr 0 /dr 0 D 1:

(2.41)

Note further that due to the definition of the pair-correlation operator (2.33), there is no self-interaction in the pair-correlation density. A property related to the pair-correlation density is the pair-correlation function h.rr 0 / defined as g.rr 0 / ; (2.42) h.rr 0 / D .r 0 / which is symmetrical in an interchange of r and r 0 : h.rr 0 / D h.r 0 r/:

(2.43)

It is due to this symmetry that the pair-correlation function h.rr 0 / is employed to describe the electron correlations in a uniform electron gas. The quantal sources defined above then give rise to “classical” fields that pervade all space. The fields are defined next.

22

2 Schr¨odinger Theory from a “Newtonian” Perspective

2.4 Definitions of “Classical” Fields The “classical” fields that fully characterize a quantum system are the electroninteraction E ee .r/ field which is the sum of its Hartree E H .r/ and Pauli–Coulomb E xc .r/ field components, the differential density D.r/ field, and the kinetic Z.r/ field.

2.4.1 Electron-Interaction Field E ee .r/ The quantal source of the electron-interaction field E ee .r/ is the pair-correlation density g.rr 0 /. Hence, this field is representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion. It is obtained from its quantal source via Coulomb’s law as Z g.rr 0 /.r  r 0 / 0 E ee .r/ D dr : (2.44) j r  r 0 j3 The field E ee .r/ may be rewritten in terms of an electron-interaction “force” e ee .r/ and the electron density .r/ as E ee .r/ D where

Z e ee .r/ D

e ee .r/ .r/

P .rr 0 /.r  r 0 / 0 dr : j r  r 0 j3

(2.45)

(2.46)

The electron-interaction field E ee .r/ may be written in terms of its Hartree E H .r/ and Pauli–Coulomb E xc .r/ field components by substituting the expression for g.rr 0 / of (2.40) into (2.44). Thus, E ee .r/ D E H .r/ C E xc .r/; where

Z E H .r/ D

and

Z E xc .r/ D

(2.47)

.r 0 /.r  r 0 / 0 dr ; j r  r 0 j3

(2.48)

xc .rr 0 /.r  r 0 / 0 dr : j r  r 0 j3

(2.49)

The Hartree field E H .r/ is conservative: r  E H .r/ D 0. This is because it arises due to a static charge distribution .r/. In general, quantal fields due to nonlocal sources such as the pair-correlation density g.rr 0 /, the Fermi–Coulomb hole charge distribution xc .rr 0 /, or the single-particle density matrix .rr 0 / are not

2.4 Definitions of “Classical” Fields

23

conservative. Thus, in general, r  E ee .r/ ¤ 0 and r  E xc .r/ ¤ 0. However, for systems of special symmetry such as closed shell atoms, open shell atoms in the Central Field Approximation, jellium metal surfaces and clusters etc., the fields arising from nonlocal sources are conservative so that the curl of these fields vanish.

2.4.2 Differential Density Field D.r/ The differential density field D.r/ is defined as D.r/ D

d.r/ ; .r/

(2.50)

where the corresponding “force” is 1 d.r/ D  r r 2 .r/: 4

(2.51)

As the source of this field is the density .r/, a local source, the field is conservative: r  D.r/ D 0. The vanishing of the curl is because the curl of the gradient of a scalar function vanishes.

2.4.3 Kinetic Field Z.r/ It is possible to describe kinetic effects in a quantum system in terms of a field Z.r/ whose quantal source is the single-particle density matrix .rr 0 /. It is defined as Z.r/ D

z.rI Œ/ ; .r/

(2.52)

where the kinetic “force” z.rI Œ / is defined by its component z˛ .r/ in terms of the kinetic-energy-density tensor t˛ˇ .r/ as z˛ .r/ D 2

X @ t˛ˇ .r/; @rˇ

(2.53)

ˇ

where 1 t˛ˇ .r/ D 4

"

# ˇ ˇ @2 @2 0 00 ˇ .r C r / 00 0 ˇ 0 00 : 0 00 @r˛ @rˇ @rˇ @r˛ r Dr Dr

(2.54)

As the source of the field Z.r/ is nonlocal, the field in general is not conservative: r Z.r/ ¤ 0. Again, for systems of special symmetry, this field too is conservative.

24

2 Schr¨odinger Theory from a “Newtonian” Perspective

2.5 Energy Components in Terms of Quantal Sources and Fields The components of the total energy E – the kinetic T , external potential Eext , and electron-interaction potential Eee energies – as defined by the expectations of (2.8)–(2.10), may be expressed in terms of the quantal sources, and in integral virial form in terms of the respective fields described in the previous sections.

2.5.1 Electron-Interaction Potential Energy Eee The electron-interaction potential energy Eee of (2.10) can be written such that it may be interpreted as the energy of the interaction between the density .r/ and the pair-correlation density g.rr 0 /: Eee D

1 2

Z Z

.r/g.rr 0 / drdr 0 : j r  r0 j

(2.55)

Employing the decomposition of g.rr 0 / as in (2.40), we may write Eee D EH C Exc ;

(2.56)

where EH is the Hartree or Coulomb self-energy: EH D

1 2

Z Z

.r/.r 0 / drdr 0 ; j r  r0 j

(2.57)

and Exc the quantum-mechanical exchange-correlation or Pauli-Coulomb energy: Exc

1 D 2

Z Z

.r/xc .rr 0 / drdr 0 : jr  r 0 j

(2.58)

The Pauli–Coulomb energy is thus the energy of interaction between the density .r/ and the Fermi–Coulomb hole charge xc .rr 0 /. The electron-interaction potential energy Eee , and its Hartree EH and Pauli– Coulomb Exc components, may also be expressed in integral virial form in terms of the corresponding electron-interaction E ee .r/, Hartree E H .r/ and Pauli–Coulomb E xc .r/ fields as [1, 9] Z Eee D

.r/r  E ee .r/dr;

(2.59)

.r/r  E H .r/dr;

(2.60)

Z EH D

2.5 Energy Components in Terms of Quantal Sources and Fields

25

Z

and Exc D

.r/r  E xc .r/dr:

(2.61)

The derivation of (2.59)–(2.61) requires writing Eee of (2.55) in terms of the paircorrelation function h.rr 0 / of (2.42) and employing its symmetry property of (2.43). For details, see QDFT. The derivation, however, makes clear that the expressions for the energies in terms of the fields are independent of whether or not the fields are conservative. (The above definitions of the Fermi–Coulomb hole charge xc .rr 0 / and energy Exc are written within the framework of Schr¨odinger theory. They differ from the “exchange-correlation” hole and energy of traditional Kohn–Sham density functional theory [15] (see QDFT).)

2.5.2 Kinetic Energy T The kinetic energy T may be written in terms of its quantal source, the singleparticle density matrix, as Z T D t.r/dr; (2.62) where the kinetic-energy-density t.r/ is the trace of the kinetic-energy-density tensor t˛ˇ .r/ of (2.54): t.r/ D

X ˛

ˇ ˇ 1 0 00 ˇ t˛˛ .r/ D r r 0  r r 00 .r r /ˇ : 2 r 0 Dr 00 Dr

(2.63)

The kinetic energy T may also be expressed in virial form in terms of the kinetic field Z.r/ as [1, 9] Z 1 T D .r/r  Z.r/dr: (2.64) 2 The equivalence of the expressions (2.62) and (2.64) is proved by partial integration and by employing the vanishing of the wave function, and hence of the singleparticle density matrix, at infinity. Once again, the expression for the kinetic energy in terms of the field Z.r/ is independent of whether or not the field is conservative.

2.5.3 External Potential Energy Eext The external potential energy Eext of (2.9) may be expressed in terms of the density .r/ and the potential energy v.r/ of an electron in the external field F ext .r/ as Z Eext D

.r/v.r/dr:

(2.65)

26

2 Schr¨odinger Theory from a “Newtonian” Perspective

From the “Quantal Newtonian” equation (2.11) or (2.12) for v.r/, it is evident that Eext depends explicitly on the internal field F int .r/ of the system. Thus, on substituting (2.12) into (2.65) we obtain Z Eext D

Z

r

dr.r/ 1

F int .r 0 /  d` 0 ;

(2.66)

with F int .r/ defined in terms of its components by (2.14). Hence, in contrast to the electron-interaction Eee and kinetic T energies, the external potential energy Eext depends on all the fields present in the quantal system.

2.6 Integral Virial, Force, and Torque Sum Rules The integral virial theorem of quantum mechanics can be derived by operating on R the “Quantal Newtonian” first law (2.11) by dr.r/r to obtain Z

  dr.r/r  F ext .r/ C E ee .r/  D.r/  Z.r/ D 0:

(2.67)

The term involving the differential density field D.r/ can be shown to vanish by transforming it into a surface integral and by employing the vanishing of the density there at infinity. Using the expressions for the energies Eee and T in terms of the fields E ee .r/ and Z.r/ of (2.59) and (2.64), (2.67) reduces to Z dr.r/r  F ext .r/ C Eee C 2T D 0;

(2.68)

which is the virial theorem in integral form. In texts on quantum mechanics [16], the theorem is usually stated as X N

  r i  r i VO .r 1 ; : : : ; r N / C UO .r 1 ; : : : ; r N / D 2T:

(2.69)

i D1

The equivalence of (2.68) and (2.69) follows from the fact that for Coulombic interaction the UO .r 1 ; : : : ; r N / is a homogeneous function of degree 1. From Euler’s theorem on homogeneous functions, if f .x1 ; : : : ; xj / is homogeneous of degree n, P then jkD1 xk .@f =@xk / D nf . Thus, X N i D1

 ˝ ˛ O r i  r i U .r 1 ; : : : ; r N / D  UO D Eee :

(2.70)

2.6 Integral Virial, Force, and Torque Sum Rules

27

Furthermore, X N

 Z O r i  r i V .r 1 ; : : : ; r N / D .r/r  F ext .r/;

(2.71)

i D1

and the equivalence follows. Now if VO .r 1 ; : : : ; r N / is also Coulombic, then it too is a homogeneous function of degree 1, so that X N

 ˝ ˛ r i  r i VO .r 1 ; : : : ; r N / D  VO D Eext :

(2.72)

i D1

Thus, for Coulombic systems, the integral virial theorem may be expressed as Eext C Eee C 2T D 0:

(2.73)

The integral virial theorem for Coulombic systems (2.73) may also be derived through scaling arguments. Thus, if .r 1 ; : : : ; r N / is a normalized eigenstate of the Hamiltonian HO of (2.1), then the scaled function defined as ˛ .r 1 ; : : : ; r N /

D ˛ 3N=2 .˛r 1 ; : : : ; ˛r N /

(2.74)

is also normalized to unity. The kinetic, external, and electron-interaction energies scale as h h h

˛ ˛ ˛

j TO j j VO j j UO j

˛i

D ˛2 h

˛ i D ˛h ˛ i D ˛h

j TO j i; j VO j i; j UO j i:

(2.75) (2.76) (2.77)

Now, the variational principle for the energy ensures that d h d˛

˛

j HO j

˛ i j˛D1 D

0:

(2.78)

Application of (2.78) leads to the virial theorem (2.73). What this proof shows is that if an approximate wave function scales according to (2.74), then it will satisfy the integral virial theorem for Coulombic systems. This is an important fact since approximate wave functions that lead to poor results but which scale correctly still satisfy the virial theorem. The virial theorem is also satisfied for calculations that are fully self-consistent. For an external potential energy that is harmonic: v .r/ D 12 kr 2 , the function VO .r 1 ; : : : ; r N / is a homogeneous function of degree 2. Thus, on application of Euler’s theorem,

28

2 Schr¨odinger Theory from a “Newtonian” Perspective

X N

 ˝ ˛ O r i  r i V .r 1 ; : : : ; r N / D 2 VO D 2Eext :

(2.79)

i D1

For the harmonic external potential energy, the integral virial theorem is then 2Eext C Eee C 2T D 0:

(2.80)

(The reader is referred to Table 2.1 of QDFT, where the numerical values of T , Eext , and Eee are quoted for both a ground and first excited singlet state of the Hooke’s atom [17–19]. The satisfaction of the virial theorem of (2.80) is evident there.) In classical physics, the total internal force between the N interacting particles and their torque vanishes as a consequence of Newton’s third law. In a similar manner, the quantal average of the internal field F int .r/ and the quantal average of the torque of this field also vanish. Thus, Z

and

.r/F int .r/dr D 0;

(2.81)

.r/r  F int .r/dr D 0:

(2.82)

Z

This is the case because the quantal average of each component of the internal field F int .r/ as well as the quantal average of the torque of each component vanishes: Z .r/E ee .r/dr D 0;

(2.83)

.r/D.r/dr D 0;

(2.84)

.r/Z.r/dr D 0;

(2.85)

.r/r  E ee .r/dr D 0;

(2.86)

.r/r  D.r/dr D 0;

(2.87)

.r/r  Z.r/dr D 0:

(2.88)

Z Z Z Z Z

The vanishing of the averaged electron interaction field E ee .r/ (2.83) and that of its averaged torque (2.86) can be attributed to Newton’s third law. This is because, Coulomb’s law and hence the field E ee .r/ obey the third law. The proof of these sum rules clearly demonstrates this. On the other hand, the vanishing of the average of the other components of the internal field and that of their averaged torque is not directly a consequence of the third law. The reader is referred to QDFT for the proof of the above sum rules.

2.7 Coalescence Constraints

29

Thus, each electron in a sea of electrons experiences an external field F ext .r/ and an internal field F int .r/ so as to satisfy the “Quantal Newtonian” first law. The internal field accounts for the motion of the electrons, and the fact that they are kept apart as a result of Coulomb repulsion and the Pauli exclusion principle. And, in a manner similar to that of classical physics, the quantal average of the internal field and of its torque vanishes.

2.7 Coalescence Constraints The Hamiltonian HO of (2.1) with Coulombic interaction is singular when either two electrons coalesce or when an electron coalesces with the nucleus. For the wave function .X / to satisfy the Schr¨odinger equation (2.5) and remain bounded, it therefore must satisfy coalescence constraints. The coalescence constraints of Schr¨odinger theory play an important role in the local effective potential energy theories: properties of this potential energy are dictated by these constraints as shown later. Hence, we derive [20] here the integral form of these coalescence constraints for dimensionality D  2. This form of constraints is more general than the originally derived and commonly employed differential form [21–25] because it retains the angular dependence of the wave function at coalescence. Further, the differential form is then readily obtained by spherically averaging and differentiating. The generalization to dimensions other than D D 3 is done because there is considerable interest in both lower dimensional systems such as the two-dimensional (D D 2) electron gas [26, 27], the two-dimensional gas in the presence of a magnetic field (the Quantum Hall effect [28]), and in their high-dimensional .D  4/ generalizations [29, 30]. Employing the integral coalescence constraints, the integral and differential forms of the pair function P .rr 0 / of (2.32) at the coalescence of two identical particles in D  2 dimensions are also derived [20]. Consider a nonrelativistic system of N charged particles in D. 2/ dimension space with the Hamiltonian .„ D e D 1/ HO D 

N N X X 1 2 Zi Zj r C ; 2mi i rij i D1

(2.89)

j >i D1

where mi and Zi are the mass and charge of the i th particle, and rij Dj r i  r j j. q PD 2 In D-dimension space, r D .x1 ; x2 ; : : : ; xD /, r D D kD1 xk xk and r PD kD1 .@=@xk /.@=@xk /. Due to the Coulomb potential energy term, the Hamiltonian is singular when two particles i and j coalesce .rij ! 0/. For the wave function .r 1 ; r 2 ; : : : ; r N / which satisfies the Schr¨odinger equation HO .r 1 ; r 2 ; : : : ; r N / D E .r 1 ; r 2 ; : : : ; r N /;

(2.90)

30

2 Schr¨odinger Theory from a “Newtonian” Perspective

to be bounded and remain finite at the singularities, it must satisfy a cusp coalescence condition. If the wave function vanishes at the singularity, it must satisfy a node coalescence condition. (Here we suppress the spin index.) We are interested in the form of the wave function when two particles approach each other, i.e., when rij is very small. Following Pack and Byers Brown [22] we focus our attention on two particles 1 and 2, and transform their coordinates r 1 and r 2 to the center-of-mass R 12 and relative coordinates r 12 as R 12 D

m1 r 1 C m2 r 2 ; m1 C m2

(2.91)

r 12 D r 1  r 2 :

(2.92)

The Hamiltonian of (2.89) may then be rewritten as Z1 Z2 1 1 r2 rr212 C  212 r12 2.m1 C m2 / R12   X N  N N X X Z2 1 2 Zi Zj Z1 Zi C C C rr i C ; r1i r2i 2mi rij

HO D 

i D3

i D3

(2.93)

j >i D3

where 12 D m1 m2 =.m1 C m2 / is the reduced mass of particles 1 and 2. When particles 1 and 2 are within a small distance of each other .0 < r12 < /, and all other particles are well separated, there is only one singularity in the Hamiltonian. Retaining only terms of lower order in r12 , (2.90) reduces to   1 Z1 Z2 2 0  r C C O. / .r 1 ; r 2 ; : : : ; r N / D 0; 212 r 12 r12

(2.94)

where O. 0 / refers to terms of order zero (constant) and higher order in r12 and the vector components .r 12 /k , k D 1; 2; : : : ; D. Equation (2.94) is a one-electron-atom equation in D-dimension space. Furthermore, it is not an eigenvalue equation. We are interested in the solution of the equation which is finite and continuous at the singularities. Thus, in the limit as r 1 ! r 2 , we could write the wave function in two parts as .r 1 ; r 2 ; : : : ; r N / D .r 2 ; r 2 ; r 3 ; : : : ; r N / C ı .r 1 ; r 2 ; : : : ; r N /;

(2.95)

where the term ı .r 1 ; r 2 ; : : : ; r N / must vanish at the singularity r 1 D r 2 . From the above, it follows that we need to consider only terms of first order in r12 in ı when r 1 is near r 2 . Dropping the subscript 12 and writingPr12 as r, we see that ı could have terms of the form rD B.r 2 ; r 3 ; : : : ; r N / or l rl B` .r 2 ; : : : ; r N /, where l denotes the 2 to .D1/-dimensional subspace and r` the distance in the subspace, or r  C .r 2 ; : : : ; r N /. Here, rD D r is just the conventional distance in D-dimensional space. As an example of the second kind of terms, consider the case of D D 4. There

2.7 Coalescence Constraints

31

are four three-dimensional subspaces, and six two-dimensional subspaces. The disq

tances in the three-dimensional subspaces constitute the terms x12 C x22 C x32 , q q q x12 C x22 C x42 , x22 C x32 C x42 , and x12 C x42 C x32 . The distances in the twoq dimensional subspaces are xi2 C xj2 , where i ¤ j and i; j D 1; 2; 3; 4. In the limit r ! 0, such terms do not effect the O.D/ symmetry of the first two terms on the left-hand side of (2.94). However, the Laplacian of these terms is singular since r 2 rd D .d  1/=r, where rd is the distance in the d -dimensional subspace. These singularities cannot be canceled by the Coulomb potential energy singularity term of (2.94). Thus, their coefficients must vanish. Therefore (recovering the subscript 12), (2.95) must be written as .r 1 ; r 2 ; : : : ; r N / D

.r 2 ; r 2 ; r 3 ; : : : ; r N /

C r12 B.r 2 ; r 3 ; : : : ; r N / C r 12  C .r 2 ; r 3 ; : : : ; r N / C O. 2 /:

(2.96)

Next substitute (2.96) for .r 1 ; r 2 ; : : : ; r N / into (2.94) and employ r 2 r D 0. For the Coulomb potential energy singularity to be canceled, we have 

1 D1 Z1 Z2 B.r 2 ; r 3 ; : : : ; r N / C .r 2 ; r 2 ; r 3 ; : : : ; r N / D 0: (2.97) 212 r12 r12

At the point of coalescence, the wave function is .r 2 ; r 2 ; r 3 ; : : : ; r N /. Since we require the wave function to be finite at this point, we have B.r 2 ; r 3 ; : : : ; r N / D

2Z1 Z2 12 .r 2 ; r 2 ; r 3 ; : : : ; r N /: D1

(2.98)

Thus, in the limit as r12 ! 0, we may write the wave function as   2Z1 Z2 12 r12 .r 1 ; r 2 ; : : : ; r N / D .r 2 ; r 2 ; r 3 ; : : : ; r N / 1 C D1 (2.99) C r 12  C .r 2 ; r 3 ; : : : ; r N /: This is the general form of the integral cusp coalescence condition in D-dimension space. Note that this expression is equally valid even if the wave function vanishes at the point of coalescence, i.e., if .r 2 ; r 2 ; : : : ; r N / D 0. The latter is referred to as a node coalescence condition. By taking the spherical average of (2.99) about the point of coalescence in D-dimension space, we obtain the differential form of the cusp condition: 

@N @r12

ˇ ˇ ˇ ˇ

D r12 !0

2Z1 Z2 12 .r12 D 0/; D1

(2.100)

32

2 Schr¨odinger Theory from a “Newtonian” Perspective

where N is the spherically averaged wave function. For the electron-nucleus coalescence, Z1 D 1; Z2 D Z the nuclear charge, and 12  me the mass of the electron. For the electron-electron coalescence, Z1 D 1; Z2 D 1; 12 D 12 me . In D D 3 dimensions, the traditional integral and differential cusp conditions are recovered [21–25]. It follows from the definition of the density .r 1 /, which is Z .r 1 / D N

?

.r 1 ; r 2 ; : : : ; r N / .r 1 ; r 2 ; : : : ; r N /dr 2 ; : : : ; dr N ;

(2.101)

that the differential form of the electron–nucleus cusp condition may be expressed in terms of the density at and about the nucleus. In this case with the nuclear positions fixed, the wave function does not depend upon the nuclear coordinates. Then when the electron at r 1 approaches a nucleus, say fixed at the origin, the integral for the density which is over all the remaining N  1 electrons can be performed. The result is 4Zme d.r/ lim D .r D 0/: (2.102) r!0 dr D1 For the proof see QDFT. It is evident from the integral cusp coalescence condition (2.99) and the above definition of the density, that there can be no such differential form of the cusp condition for electron-electron coalescence. It is, however, possible to derive the integral and differential forms of the cusp coalescence condition in D-dimensions for the pair function P .rr 0 / of (2.32) rewritten here as P .r 1 r 2 /: Z P .r 1 r 2 / D N.N  1/

?

.r 1 ; r 2 ; : : : ; r N /

 .r 1 ; r 2 ; : : : ; r N /dr 3 ; : : : ; dr N :

(2.103)

In the limit as r 1 ! r 2 and employing the integral cusp expression of (2.99) we have that to first order in r12 , 2  2Z1 Z2 12 r12 P .r 1 r 2 / D P .r 2 r 2 / 1 C D1 Z CN.N  1/ .r 12  C .r 2 ; : : : ; r N //2 dr 3 : : : dr N   2Z1 Z2 12 r12 C2N.N  1/ 1 C D1 Z  .r 12  C .r 2 ; : : : ; r N // .r 2 ; r 2 ; : : : ; r N / dr 3 : : : dr N C O. 3 /:

(2.104)

2.7 Coalescence Constraints

33

This is the integral form of the cusp coalescence condition for the pair function. The differential form of the cusp condition for the pair function is !ˇ 4Z1 Z2 12 N @PN .r 1 r 2 / ˇˇ D P .r 2 r 2 /; ˇ @r12 D1 r12 !0

(2.105)

where PN is the spherical average of the correlation function. The differential cusp condition derived for the uniform electron gas in dimensions D D 3 and 2 [31–35] constitute a special case of this general form of the differential cusp condition. Note that the results derived are valid for the case Pof a more generalized Hamiltonian which includes additional terms of the form i f .r i /, where f .r i / is a local potential energy. This is because, such terms do not contribute any singularities at the coalescence of two charged particles.

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Chapter 3

Quantal Density Functional Theory

Quantal density functional theory (Q-DFT) [1–18] is a local effective potential energy theory in which the interacting system as described by the Schr¨odinger equation is mapped into one of noninteracting fermions with equivalent density .r/. The existence of such a model system is an assumption. From the model system, it is then possible to obtain the energy E of the interacting system as well as its first ionization potential I (or electron affinity A). As the fermions of the model system are noninteracting, their effective potential energy vs .r/ is the same. Thus, in the corresponding Hamiltonian, the potential energy operator is multiplicative or local. The model system is referred to as the S system, S being a mnemonic for “single Slater determinant.” For the mapping from any ground or bound excited state, nondegenerate or degenerate pure state of the interacting system, the corresponding state of the S system is arbitrary in that it could be in a ground or excited state. Irrespective of the state of the S system, the energy E, and ionization potential I (or electron affinity A) of the interacting system are once again obtained. Within the framework of Q-DFT, it is also possible to transform any state of the interacting system to one of noninteracting bosons in a ground state such that the density, energy, and ionization potential are obtained. Examples of such mappings as applied to atoms are described in Chap. 11. Consider a system of N noninteracting fermions in an external field F ext .r/ D r v.r/. The Hamiltonian of this system may be written as HO s D

X

hO s .r i /;

(3.1)

i

1 hO s .r/ D  r 2 C vs .r/; (3.2) 2 where vs .r/ is their effective potential energy. The corresponding S system Schr¨odinger equation for each of these fermions is then 

 1  r 2 C vs .r/ i .x/ D i i .x/I 2

i D 1; : : : ; N:

(3.3)

35

36

3 Quantal Density Functional Theory

The wave function of the S system is a single Slater determinant ˆfi g of the single particle orbitals i .x/. However, for S systems in certain excited states, it is also possible to represent the wave function as a linear combination of Slater determinants. Both wave functions lead to the same density .r/, but the latter may represent the spin-symmetry of the system more accurately. The effective potential energy vs .r/ of the model fermions is written as the sum of the external potential energy v.r/ and an effective “electron-interaction” potential energy vee .r/, which is representative of all the electron correlations the S system must account for in order that its density .r/ be the same as that of the interacting system: vs .r/ D v.r/ C vee .r/: (3.4) The electron correlations that vee .r/ must account for are those due to the Pauli exclusion principle, Coulomb repulsion, and Correlation-Kinetic effects. The last of these arises because of the difference between the kinetic energy of the interacting and noninteracting systems, and represents the correlation contribution to the kinetic energy. The satisfaction of the Pauli principle is explicitly achieved as the Slater determinant ˆfi g is anti-symmetric in an interchange of the coordinates of the fermions including their spins. How the various electron correlations, which are intrinsically nonlocal in the sense that they depend on the coordinates of two particles, are incorporated into the local electron-interaction potential energy operator vee .r/ is described later.

3.1 Quantal Density Functional Theory from a “Newtonian” Perspective We have seen that Schr¨odinger theory can be described in a “Newtonian” context. As the S system represents the transformation to one of noninteracting fermions with equivalent density .r/, then it also can be described in such a manner. The S system “Quantal Newtonian” first law for the sum of the forces acting on each model fermion is F ext .r/ C F int (3.5) s .r/ D 0; ext .r/ D where F int s .r/ is the internal field of these fermions. Again, since F r v.r/, the external potential energy v.r/, is the work done to move the model fermion from some reference point at infinity to its position at r in the force of the internal field F int s .r/: Z

v.r/ D

r

1

0 0 F int s .r /  d` ;

(3.6)

where v.1/ D 0 is assumed. It also follows from (3.5) that r  F int s .r/ D 0; so that the work done v.r/ is path-independent as originally assumed.

(3.7)

3.2 Definitions of Quantal Sources Within Quantal Density Functional Theory

37

The internal field F int s .r/ of the model fermions is also comprised of three components: the S system effective field F eff .r/, the differential density field D.r/, and the S system kinetic field Z s .r/: eff F int s .r/ D F .r/  D.r/  Z s .r/:

(3.8)

The effective field F eff .r/ is a consequence of the assumption of existence of the S system or equivalently that of the electron-interaction potential energy vee .r/. Thus, F eff .r/ D r vee .r/:

(3.9)

The fields D.r/ and Z s .r/, and their quantal sources are defined in the following sections. The “Quantal Newtonian” first law of (3.5) is the pure state time-independent differential virial theorem for the S system [1, 13]. For time-dependent external fields F ext .rt/, there also exists a pure state differential virial theorem for the S system [1, 14, 15]. This corresponds to the S system “Quantal Newtonian” second law. Once again, for the S system, the first law is a special case of the second law. The derivations of these laws are given in Appendix E of QDFT.

3.2 Definitions of Quantal Sources Within Quantal Density Functional Theory The quantal sources for the S system are the density .r/, the Dirac spinless singleparticle density matrix s .rr 0 /, and the pair-correlation density gs .rr 0 / from which one then defines the Fermi hole charge distribution x .rr 0 /. The Fermi hole represents the reduction in density at r 0 for an electron at r due to the Pauli exclusion principle. As the wave function of the S system is a Slater determinant ˆfi g of the orbitals i .x/ of (3.3), and hence is antisymmetric, the Fermi hole can be defined explicitly in terms of these orbitals. With the Fermi hole thus defined, it is then possible to define the Coulomb hole charge c .rr 0 / for the S system. The quantal sources are once again expectations of Hermitian operators but now taken with respect to the Slater determinant ˆfi g. As such, these sources are expressed explicitly in terms of the orbitals i .x/.

3.2.1 Electron Density .r/ The electronic density .r/ is N times the probability of an electron being at r, or equivalently the expectation of the density operator .r/ O of (2.16): ˇ ˇ ˇ ˝ ˛ XXˇ ˇi .x/ˇ2 : .r/ D ˆfi gˇ.r/ O ˇˆfi g D 

i

(3.10)

38

3 Quantal Density Functional Theory

As the Slater determinant ˆfi g is normalized, it satisfies the charge conservation constraint: Z .r/dr D N: (3.11) Note that the density .r/ of (3.10) is the same as that for the interacting system.

3.2.2 Dirac Spinless Single-Particle Density Matrix s .rr 0 / The Dirac spinless single-particle density matrix s .rr 0 / is the expectation of the density matrix operator O .rr 0 / of (2.27): ˇ ˇ ˝ ˛ XX ? i .r/i .r 0 /: s .rr 0 / D ˆfi gˇO .rr 0 /ˇˆfi g D 

(3.12)

i

The properties of this density matrix are that its diagonal matrix element is the density: s .rr/ D .r/; (3.13) that its complex conjugate corresponds to an interchange of the coordinates: s? .rr 0 / D s .r 0 r/;

(3.14)

and that it is idempotent: Z

s .rr 00 /s .r 00 r 0 /dr 00 D s .rr 0 /:

(3.15)

It is important to note that the S system Dirac density matrix s .rr 0 / and the interacting system density matrix .rr 0 / of (2.28) are inequivalent. It is only their diagonal matrix elements that are equal.

3.2.3 Pair-Correlation Density gs .rr 0 /; Fermi x .rr 0 / and Coulomb c .rr 0 / Holes As was the case for the interacting system, the pair-correlation density of the S system gs .rr 0 / represents the conditional density at r 0 for a model fermion at r as obtained via the Slater determinant ˆfi g. As such, it is the ratio of the expectations of the pair-correlation and density operators of (2.33) and (2.16), respectively, gs .rr 0 / D

hˆfi gjPO .rr 0 /jˆfi gi ; .r/

(3.16)

3.2 Definitions of Quantal Sources Within Quantal Density Functional Theory

39

and thus also satisfies the constraint Z

gs .rr 0 /dr 0 D N  1;

(3.17)

for each electron position r. The pair density gs .rr 0 / may also be expressed as a sum of its local and nonlocal components as gs .rr 0 / D .r 0 / C x .rr 0 /;

(3.18)

where the nonlocal component is the Fermi hole charge x .rr 0 /. It follows from (3.11) and (3.17) that the Fermi hole satisfies the charge conservation sum rule: Z x .rr 0 /dr 0 D 1; (3.19) for each electron position r. It also satisfies the constraint of negativity: x .rr 0 /  0:

(3.20)

Comment: In local effective potential energy theories such as Q-DFT, the Fermi hole is defined through (3.16) and (3.18). In general, for ground state S systems with two model fermions of opposite spin occupying each discrete state, as for example, in a closed shell system or the continuum states of a metal, the Fermi hole is the reduction in density at r 0 in the distribution of electrons with spin parallel to that of the electron at r. The Fermi hole may then be expressed as js .rr 0 /j2 x .rr 0 / D  : (3.21) 2.r/ The constraint on the Fermi hole at the electron position r 0 D r is then x .rr/ D .r/=2:

(3.22)

Next, consider a two model-fermion S system in its ground state. This is a singlet state: The two model fermions with opposite spin occupy the lowest energy state. From (3.18) it follows (see QDFT) that the Fermi hole x .rr 0 / D .r 0 /=2;

(3.23)

which also satisfies the sum rule of (3.22). Thus, in Q-DFT and other local effective potential energy theories, one defines a Fermi hole even for singlet states. For excited singlet states, such as for the singlet 21 S state of the two model-fermion

40

3 Quantal Density Functional Theory

S system, the sum rules of charge conservation and negativity are also satisfied. However, the constraint at the electron position will differ from that of (3.22). The Fermi hole is, of course, also defined through (3.18) for excited triplet states, but once again the condition at the electron position will differ from that of (3.22). For example, for the two model-fermion first excited triplet 23 S state of the S system, it readily follows that x .rr/ D .r/:

(3.24)

These examples are discussed in detail in Chap. 5. Note that in contrast, within Hartree–Fock Theory [19, 20], the Fermi hole is defined only for electrons of parallel spin, and there is, for example, no Fermi hole for the two-electron atom in its ground or first excited singlet 21 S states. In a manner similar to Schr¨odinger theory, one defines the pair-correlation function hs .rr 0 / for the S system as hs .rr 0 / D

gs .rr 0 / ; .r 0 /

(3.25)

which is also symmetrical in an interchange of r and r 0 : hs .rr 0 / D hs .r 0 r/:

(3.26)

Since the wave function of the S system is a Slater determinant, electron correlations due to the Pauli principle are explicitly represented by the Fermi hole x .rr 0 /. The correlations due to Coulomb repulsion can then be represented by a Coulomb hole charge c .rr 0 /, which is also a nonlocal charge distribution. The Coulomb hole at r 0 for an electron at r is defined as c .rr 0 / D g.rr 0 /  gs .rr 0 / 0

(3.27) 0

D xc .rr /  x .rr /;

(3.28)

where g.rr 0 / and xc .rr 0 / are the Schr¨odinger theory pair-correlation density and Fermi–Coulomb hole charge, respectively. As the Fermi–Coulomb and Fermi holes both satisfy the same charge conservation sum rule (see (2.41) and (3.19)), the total charge of the Coulomb hole is zero: Z

c .rr 0 /dr 0 D 0;

for each electron position r. This is the Coulomb hole sum rule.

(3.29)

3.3 Definitions of “Classical” Fields Within Quantal Density Functional Theory

41

3.3 Definitions of “Classical” Fields Within Quantal Density Functional Theory The assumption of existence of the S system, or equivalently of the local electroninteraction potential energy vee .r/, implies that there exists a conservative effective field F eff .r/ as defined by (3.9). This field must account for electron correlations arising from the Pauli principle and Coulomb repulsion. Furthermore, as the kinetic energies of the interacting and noninteracting systems of the same density .r/ are different, the field F eff .r/ must also account for the correlation contribution to the kinetic energy. This contribution is referred to as the Correlation–Kinetic effect. It is evident that for F eff .r/ to represent all these correlations, it must be defined in terms of the properties of both the interacting and noninteracting systems. It turns out that Pauli and Coulomb correlations are represented by the electron-interaction field E ee .r/ of (2.44), and its Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ field components. The kinetic energy and the correlation contributions to the kinetic energy are represented by the kinetic Z s .r/ and Correlation–Kinetic Z tc .r/ fields, respectively. There is also a corresponding differential density field D.r/. These individual fields are defined below.

3.3.1 Electron-Interaction Field E ee .r/, and Its Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ Components The electron-interaction field E ee .r/ of the interacting system, which constitutes a component of the effective field F eff .r/, can be expressed in terms of its Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ components. The Hartree field E H .r/ is defined by (2.48). The definition of the Pauli field E x .r/ derives from the S system electron-interaction field E ee;s .r/. The latter is obtained via Coulomb’s law from its quantal source, the S system pair-correlation density gs .rr 0 / as Z E ee;s .r/ D

gs .rr 0 /.r  r 0 / 0 dr ; jr  r 0 j3

(3.30)

On employing the decomposition (3.18), the field E ee;s .r/ is the sum of the Hartree E H .r/ and Pauli E x .r/ fields: E ee;s .r/ D E H .r/ C E x .r/;

(3.31)

where E x .r/ arises from its quantal source the Fermi hole charge x .rr 0 / via Coulomb’s law as Z x .rr 0 /.r  r 0 / 0 E x .r/ D dr : (3.32) jr  r 0 j3

42

3 Quantal Density Functional Theory

The Coulomb field E c .r/, due to the Coulomb hole charge c .rr 0 / defined by (3.27), is also obtained via Coulomb’s law as Z E c .r/ D

c .rr 0 /.r  r 0 / 0 dr : jr  r 0 j3

(3.33)

Thus, the electron-interaction field E ee .r/ of the interacting system may be written in terms of its components as E ee .r/ D E H .r/ C E x .r/ C E c .r/:

(3.34)

The Pauli field E x .r/ represents correlations due to the Pauli exclusion principle, and the Coulomb field E c .r/ due to Coulomb repulsion beyond those incorporated in the Hartree field E H .r/. As both the Pauli E x .r/ and Coulomb E c .r/ fields are due to nonlocal sources, they are in general not conservative: r  E x .r/ ¤ 0; r  E c .r/ ¤ 0:

3.3.2 Differential Density Field D.r/ The differential density field D.r/ of the S system is defined in the same manner as that of the interacting system given in Sect. 2.4.2, the quantal source being the density .r/. Since the densities of the two systems are the same, the fields are equivalent: d.r/ D.r/ D ; (3.35) .r/ where the “force” is

1 (3.36) d.r/ D  r r 2 .r/: 4 The field D.r/ is conservative: r  D.r/ D 0. This is because, the curl of the gradient of a scalar function vanishes.

3.3.3 Kinetic Z s .r/ and Correlation–Kinetic Z tc .r/ Fields The kinetic effects of the noninteracting fermions are described by the kinetic field Z s .r/, whose quantal source is the Dirac density matric s .rr 0 /. Thus, Z s .r/ D

zs .rI Œs / ; .r/

where the S system kinetic “force” is defined by its component zs;˛ .r/ as

(3.37)

3.4 Total Energy and Its Components in Terms of Quantal Sources and Fields

zs;˛ .r/ D 2

X @ ts;˛ˇ .r/; @rˇ

43

(3.38)

ˇ

and where ts;˛ˇ .r/ is the S system kinetic-energy-density tensor defined in turn as 1 ts;˛ˇ .r/ D 4

"

# ˇ @2 @2 C 0 00 s .r 0 r 00 /ˇr 0 Dr 00 Dr : @r˛0 @rˇ00 @rˇ @r˛

(3.39)

From (3.7)–(3.9), it follows that the field Z s .r/ is conservative: r  Z s .r/ D 0. The Correlation-Kinetic field Z tc .r/ is defined as the difference between the interacting and noninteracting system kinetic fields: Z tc .r/ D Z s .r/  Z.r/:

(3.40)

Thus, Z tc .r/ represents the correlation component of the interacting system kinetic field Z.r/. In general, the field Z tc .r/ is not conservative: r  Z tc .r/ ¤ 0. As noted earlier, the fields E ee .r/, E x .r/, E c .r/, Z.r/ and Z tc .r/ are in general not conservative. However, as shown in Sect. 3.5, the sum ŒE ee .r/ C Z tc .r/ or ŒE x .r/ C E c .r/ C Z tc .r/ is always conservative so that r  ŒE ee .r/ C Z tc .r/ D 0;

(3.41)

r  ŒE x .r/ C E c .r/ C Z tc .r/ D 0:

(3.42)

For systems with a certain symmetry, or when such symmetry is imposed, the individual fields are then separately conservative in which case r  E ee D 0, r  Z tc .r/ D 0, r  E x .r/ D 0, and r  E c .r/ D 0.

3.4 Total Energy and Its Components in Terms of Quantal Sources and Fields Within Q-DFT, the total energy E and its components may also be expressed in terms of the quantal sources, as well as in integral virial form in terms of the corresponding fields. The virial expressions are independent of whether or not the individual fields are conservative.

3.4.1 Electron-Interaction Potential Energy Eee , and Its Hartree EH , Pauli Ex , and Coulomb Ec Components We wish to write the electron-interaction energy Eee of the interacting system in terms of its components. To do so, note that the S system electron-interaction energy

44

3 Quantal Density Functional Theory

Eee;s is the energy of interaction between the density .r/ and the pair-correlation density gs .rr 0 /: Z Z .r/gs .rr 0 / 1 drdr 0 : (3.43) Eee;s D 2 jr  r 0 j On employing the decomposition (3.18), Eee;s may be written as Eee;s D EH C Ex ;

(3.44)

where the Hartree energy EH is defined by (2.57), and the Pauli energy Ex is the energy of the interaction between the density .r/ and the Fermi hole charge x .rr 0 /: Z Z .r/x .rr 0 / 1 drdr 0 : (3.45) Ex D 2 jr  r 0 j By employing the symmetry property of the S system pair-correlation function hs .rr 0 / of (3.25), these energies may be written in integral virial form in terms of the respective fields as Z Eee;s D

.r/r  E ee;s .r/dr;

(3.46)

.r/r  E x .r/dr;

(3.47)

Z Ex D

with EH given in terms of E H .r/ as in (2.60). Then, by the definition of the Coulomb hole c .rr 0 / of (3.27) or (3.28), the interacting system electron-interaction energy Eee may be expressed in terms of its Hartree, Pauli, and Coulomb components as Eee D EH C Ex C Ec ;

(3.48)

where the Coulomb energy Ec is the energy of interaction between the density .r/ and the Coulomb hole charge c .rr 0 /: 1 Ec D 2

Z Z

.r/c .rr 0 / drdr 0 ; jr  r 0 j

(3.49)

or equivalently in integral virial form in terms of the Coulomb field E c .r/ as Z Ec D

.r/r  E c .r/dr:

(3.50)

3.4.2 Kinetic Ts , and Correlation-Kinetic Tc Energies The kinetic energy Ts of the S system may be expressed in terms of its quantal source, the Dirac density matrix s .rr 0 /, as

3.4 Total Energy and Its Components in Terms of Quantal Sources and Fields

45

Z Ts D

ts .r/dr;

(3.51)

where the kinetic-energy-density ts .r/ is the trace of the kinetic-energy-density tensor ts;˛ˇ .r/: ts .r/ D

X

ts;˛˛ .r/ D

˛

ˇ 1 r r 0  r r 00 s .r 0 r 00 /ˇr 0 Dr 00 Dr : 2

(3.52)

In terms of the kinetic field Z s .r/, the kinetic energy is Ts D 

1 2

Z .r/r  Z s .r/dr:

(3.53)

The kinetic energy may also be obtained directly from the S system orbitals i .x/ as the expectation Ts D hˆfi gjTO jˆfi gi D

XX 1 hi .r/j  r 2 ji .r/i: 2 

(3.54)

i

The Correlation-Kinetic energy Tc is defined as the difference between the interacting T and the noninteracting Ts system kinetic energies: Tc D T  Ts :

(3.55)

Employing (2.64) and (3.53), Tc may be expressed in terms of the CorrelationKinetic field Z tc .r/ as Tc D

1 2

Z .r/r  Z tc .r/dr:

(3.56)

For the transformation from a ground state of the interacting system to an S system in its ground state, Tc > 0. (See QDFT for an example.) However, it is possible for Tc < 0 as when mapping from an excited state of the interacting system to an S system in an excited state, or on mapping a ground state of the interacting system to an S system in an excited state. Such examples are described in Chap. 5.

3.4.3 External Potential Energy Eext As it is assumed that the external potential energy v.r/ of the model noninteracting fermions is the same as that of the interacting electrons (see (3.4)), and since their densities .r/ too are the same, the external energy Eext is also the same: Z Eext D

.r/v.r/dr:

(3.57)

46

3 Quantal Density Functional Theory

However, the external potential energy v.r/ may be expressed via the S system “Quantal Newtonian” first law (3.5, 3.6) in terms of the internal field F int s .r/ of the S system. Thus, we may express Eext in terms of the S system properties as Z Eext D

Z

r

dr.r/ 1

0 0 F int s .r /  d` ;

(3.58)

where the various components of F int s .r/ are defined via (3.8).

3.4.4 Total Energy E The total energy E may then be written as E D Ts C Eext C Eee C Tc ;

(3.59)

or by employing the decomposition (3.48) as Z E D Ts C

.r/v.r/dr C EH C Ex C Ec C Tc :

(3.60)

In this manner, the separate contributions of the various electron correlations to the total energy are delineated. As shown later, the potential energy vee .r/ of (3.4) can also be written explicitly in terms of the various electron correlations i.e., its Hartree, Pauli, Coulomb, and Correlation-Kinetic components. Hence, within Q-DFT it is possible to study a system as a function of the different electron correlations. One simply truncates the expressions for E and vee .r/ at the particular level of correlation of interest. This also constitutes one approach to approximation methods within Q-DFT (see Chap. 6). The total energy may also be expressed in terms of the eigenvalues i of the S system differential equation (3.3). Multiplying (3.3) by i? .r/, summing over all the model fermions, and integrating over spatial and spin coordinates leads to Ts D

X

Z i 

Z .r/v.r/dr 

.r/vee .r/dr;

(3.61)

i

which on substitution into (3.60) for the total energy E gives ED

X

Z i 

.r/vee .r/dr C Eee C Tc :

(3.62)

i

P Note that although the model fermions are noninteracting, we have E ¤ i i . This is because, the S system explicitly accounts for correlations due to the Pauli principle, Coulomb repulsion, and Correlation-Kinetic effects. The expression (3.62) is analogous to the corresponding expressions for the energy E in Hartree [21] and Hartree–Fock [19, 20] theories (see also QDFT).

3.5 Effective Field F eff .r/ and Electron-Interaction Potential Energy vee .r/

47

3.5 Effective Field F eff .r/ and Electron-Interaction Potential Energy vee .r/ The assumption of existence of the S system implies that there exists an effective field F eff .r/, which is representative of all the electron correlations of the interacting system. The corresponding local electron-interaction potential energy of each model fermion in this field is vee .r/. The effective field F eff .r/ is derived by equating the “Quantal Newtonian” first law of the interacting and model systems (see (2.11) and (3.5)). As the external field F ext .r/ is the same for both systems, we have int F int .r/ (3.63) s .r/ D F or from (2.14) and (3.8) F eff .r/  D.r/  Z s .r/ D E ee .r/  D.r/  Z.r/;

(3.64)

F eff .r/ D E ee .r/ C Z tc .r/:

(3.65)

so that As the curl of the gradient of a scalar function vanishes, it follows from (3.9) that the field F eff .r/ is conservative: r  F eff .r/ D 0:

(3.66)

Equation (3.41) then follows from (3.66). It also follows from (3.9) and (3.66) that the electron-interaction potential energy vee .r/ is the work done to move a model fermion from its reference point at infinity to its position at r in the force of the conservative effective field F eff .r/: Z r vee .r/ D  F eff .r 0 /  d` 0 ;

(3.67)

1

where it is assumed that vee .1/ D 0. This work done, of course, is path independent. Employing the decomposition (3.34) of the electron-interaction field E ee .r/ into its Hartree E H .r/, Pauli E x .r/ and Coulomb E c .r/ components, the effective field F eff .r/ of (3.65) may be written as F eff .r/ D E H .r/ C E x .r/ C E c .r/ C Z tc .r/:

(3.68)

Thus, as for the total energy, the separate contributions of the different correlations to the potential energy vee .r/ are delineated. As noted previously, the Hartree field E H .r/ defined by (2.48) is conservative because it arises from a local charge distribution: the density .r/. Hence, the field may be written as E H .r/ D r WH .r/; (3.69)

48

3 Quantal Density Functional Theory

where WH .r/ is the Hartree potential energy. The Hartree potential energy, which is the work done in the field E H .r/: Z r WH .r/ D  E H .r 0 /  d` 0 ;

(3.70)

1

may then be expressed as Z WH .r/ D

.r 0 / dr 0 : jr  r 0 j

(3.71)

Hence, the potential energy vee .r/ for an S system of arbitrary symmetry may be written as   Z r vee .r/ D WH .r/ C  (3.72) ŒE x .r 0 / C E c .r 0 / C Z tc .r 0 /  d` 0 : 1

Note that (3.42) follows from (3.66) and the fact that the Hartree field E H .r/ is conservative. Thus, the work done in (3.72) is path-independent. For systems with symmetry such that the individual fields E x .r/, E c .r/, and Z tc .r/ are conservative, the potential energy vee .r/ may be written as the sum of the separate work done in the force of these fields:

where

vee .r/ D WH .r/ C Wx .r/ C Wc .r/ C Wtc .r/;

(3.73)

Z r Wx .r/ D  E x .r 0 /  d` 0 ;

(3.74)

E c .r 0 /  d` 0 ;

(3.75)

Z tc .r 0 /  d` 0 ;

(3.76)

1 Z r

Wc .r/ D 

1 Z r

Wtc .r/ D 

1

are the Pauli Wx .r/, Coulomb Wc .r/ and Correlation-Kinetic Wtc .r/ potential energies. Note, each work done is path independent.

3.6 Integral Virial, Force, and Torque Sum Rules For the interacting system (see Sect. 2.6), the integral virial theorem (2.68) is expressed in terms of the external field F ext .r/. The force and torque sum rules (2.81, 2.82), of course, are concerned only with the internal field F int .r/. However, for the S system, it is best to write the corresponding sum rules in terms of the effective field F eff .r/ which is key to the mapping from the interacting to the noninteracting system.

3.7 Highest Occupied Eigenvalue m

49

R

Thus, operating by d r.r/r on (3.65) and employing (2.59) and (3.56) for the energies Eee and Tc , respectively, leads to the integral virial theorem for the S system: Z .r/r  F eff .r/dr D Eee C 2Tc :

(3.77)

R R Similarly, by operating on (3.65) by dr.r/ and dr.r/r, we have the S system force and torque sum rules which state that the averaged and averaged torque of the effective field F eff .r/ vanishes: Z .r/F eff .r/dr D 0;

(3.78)

.r/r  F eff .r/dr D 0:

(3.79)

Z

These sum rules follow from (2.83) to (2.88), and the fact that Z .r/Z s .r/dr D 0;

(3.80)

Z .r/  Z s .r/dr D 0;

(3.81)

which can be proved in a manner similar to that of (2.85) and (2.88). Note that it is only the vanishing of the averaged and averaged torque of the electron-interaction field E ee .r/ component of F eff .r/ that is attributable to Newton’s third law. That of its Correlation-Kinetic field Z tc .r/ component is not.

3.7 Highest Occupied Eigenvalue m One of the principle attributes of the mapping from the interacting to the noninteracting system, and a key property of the latter, is that the highest occupied eigenvalue of the S system is the negative of the ionization potential I [1, 22–24]. (The remaining eigenvalues, both occupied and unoccupied, have no rigorous physical interpretation.) Since the effective potential energy vs .r/ of each model fermion is the same, the asymptotic decay of each orbital i .x/ in the classically forbidden region depends on its corresponding eigenvalue. Thus, the asymptotic structure of the density for finite systems, for which the eigenvalues are discrete, is due entirely to the highest occupied state m .x/. The asymptotic structure of the density as determined via the S system is then h p i lim .r/ D jm .x/j2 exp 2 2m r :

r!1

This expression is valid whether the S system is in a ground or excited state.

(3.82)

50

3 Quantal Density Functional Theory

The asymptotic structure of the density may also be obtained for the interacting system by solution of the Schr¨odinger equation in the classically forbidden region. With the asymptotic structure of the wave function thus known (see QDFT for the derivation), the decay of the density is determined as

p  lim .r/ exp 2 2I r ;

r!1

(3.83)

where the ionization potential I is I D E .N 1/  E;

(3.84)

and E .N 1/ the energy of the .N  1/–electron ion. Equivalently, the ionization potential is the energy difference between the highest occupied eigenvalue and the reference vacuum level. The expression for the asymptotic density is valid for both ground and excited states. A comparison of (3.82) and (3.84) then leads to m D I:

(3.85)

Thus, the highest occupied eigenvalue is the negative of the ionization potential. For finite systems, highly accurate ionization potentials can be obtained in the Q-DFT approximation in which only Pauli correlations are considered. To see why this is the case, consider for example a spherically symmetric atom by which we mean an atom that has a spherical electron density. The S system effective potential energy vs .r/ may then be written as the sum of the external v.r/, Hartree WH .r/, Pauli Wx .r/, Coulomb Wc .r/, and Correlation-Kinetic Wtc .r/ potential energies: vs .r/ D v.r/ C WH .r/ C Wx .r/ C Wc .r/ C Wtc .r/:

(3.86)

Asymptotically, as r ! 1, the contribution of v.r/ C WH .r/ vanishes since v.r/ D Z=r, WH .r ! 1/ D N=r, and N D Z. Thus, asymptotically vs .r ! 1/ D Wx .r ! 1/ C Wc .r ! 1/ C Wtc .r ! 1/:

(3.87)

Now it can be proved (see Chap. 7 for the derivation) that Wx .r ! 1/ D 1=r, Wc .r ! 1/ O.1=r 4 / and Wtc .r ! 1/ O.1=r 5/. Hence, the asymptotic structure of vs .r/, on which depends the highest occupied eigenvalue m , is governed solely by the Pauli potential energy Wx .r/ in this region. Now in the Pauli Correlated Approximation of Q-DFT, the effective potential energy is assumed to be vs .r/ D v.r/ C WH .r/ C Wx .r/. Thus, the S system differential equation is the same as that of the Pauli Correlated Approximation in the classically forbidden region. Solving the differential equation in the latter approximation then leads to highest occupied eigenvalues that are good approximations to the experimental ionization potential. This will be demonstrated by application of the Q-DFT Pauli Approximation to atoms in Chap. 10.

3.8 Quantal Density Functional Theory of Degenerate States

51

At the metal–vacuum interface, the asymptotic decay of the highest occupied S system orbital, and hence that of the density, in the classically forbidden vacuum region depends upon the Fermi energy. (The analytical asymptotic structure of the orbitals at a metal surface and of the density is derived in Chap. 17.) The difference between the Fermi-level energy and the reference vacuum level is the Work Function of the metal. At the metal surface, in contrast to the atomic case discussed above, Pauli and Coulomb correlations as well as Correlation-Kinetic effects contribute to the asymptotic structure of the effective potential energy vs .r/. These results are derived analytically in Chap. 17.

3.8 Quantal Density Functional Theory of Degenerate States The Quantal density functional theory of the mapping from both a degenerate ground and excited state of the interacting system to one of noninteracting fermions such that the equivalent density and energy are obtained [9] is described in Appendix A. The cases of both pure state and ensemble v-representable densities are considered. Examples of such mappings within Q-DFT are also presented.

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Chapter 4

New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

For completeness and comparison with Q-DFT, we briefly describe in this chapter the principal tenets of Hohenberg–Kohn [1] and Kohn–Sham density functional theory [2]. A corollary [3] (see also Chap. 4 of QDFT [4]) on the first or fundamental theorem of Hohenberg and Kohn [1] stating the conditions under which the theorem is not valid is described and further explained diagrammatically. We then generalize [5] this fundamental theorem of Hohenberg and Kohn [1] to external potential energy operators, that in addition to the standard scalar potential energy operator also include the momentum and curl-free vector potential energy operators. The theorem as originally formulated by Hohenberg and Kohn then constitutes a special case of this generalization. We provide here the detailed proof of this generalization, and discuss its consequences. For details of the various proofs of Hohenberg–Kohn–Sham theory as derived originally, we refer the reader to the literature [1, 2] or to Chap. 4 of QDFT [4]. There are, in addition, numerous excellent texts on the subject [6–9] with different emphases in which the precursory material is also described. There are also many books with reviews and articles on the broader aspects of the theory and its applications [10–19]. The extension of the theory to phenomenon with time-dependent scalar external potential energy operators is due to Runge and Gross [20]. Once again, we refer the reader to the original paper, and to the review articles on time-dependent DFT given in [10–19] for details and for further developments. A brief description of time-dependent theory as well as a corollary [3] to the fundamental Runge– Gross theorem is also given in Chap. 4 of QDFT. The generalization [5] of this fundamental time-dependent theorem to additionally include the momentum and curl-free vector potential energy operators is given in Appendix B. The fundamental Runge–Gross and Hohenberg–Kohn theorems, each constitute a special case of the generalized time-dependent theorem. Hohenberg–Kohn–Sham DFT is a ground state theory. The theory is based on the two profound theorems of Hohenberg and Kohn [1]. The Kohn–Sham version of the theory [2] (KS-DFT) is based on these theorems. KS-DFT provides an alternate method whereby a system of electrons in a ground state as described by the Schr¨odinger equation is mapped to one of noninteracting fermions, also in their ground state, but with the same density as that of the interacting electrons. As

53

54

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

opposed to Q-DFT [21], there is, therefore, only one model S system that can be constructed within KS-DFT. The generalization of the fundamental theorem of Hohenberg–Kohn/Runge– Gross is arrived via a unitary transformation that preserves the density. In a unitary transformation, the number of degrees of freedom is conserved. Equivalently, the generalization is achieved by a gauge transformation. Unitary transformations lead to new physical insights. An instructive example of such a transformation is the homogeneous electron gas in the field of a uniform positive jellium charge distribution. This physical system may be represented either by the corresponding time-independent Schr¨odinger equation or equivalently by a unitary transformation [22] as one comprised of (fermionic) quasi-particles with short-ranged interaction, the long-ranged component of the Coulomb interaction being described by (bosonic) plasmons. The new physics, based on this unitary transformation, then constitutes a principal justification for the use of the independent particle model in condensed matter physics. In a similar manner, the unitary transformation to be described leads to new understandings of the fundamental theorem of Hohenberg–Kohn/Runge– Gross. In particular, as a consequence of the generalization, we now understand that the Schr¨odinger theory wave function in general is a functional of the density as well as of a gauge function. We begin with the statements and implications of the two Hohenberg–Kohn theorems, and the corollary to the first theorem.

4.1 The Hohenberg–Kohn Theorems and Corollary Consider a system of N electrons in the presence of an arbitrary external field F ext .r/ D r v.r/ as described by the Hamiltonian of (2.1). The external potential energy of the electrons is represented by the local operator v.r/. This potential energy is not restricted to being Coulombic: it could be Harmonic, Yukawa, oscillatory, etc. In the Hamiltonian of (2.1), the interaction between the electrons is, of course, Coulombic. However, as is the case for the external potential energy, the interaction between the electrons is also not restricted to being Coulombic. For this system of electrons in a nondegenerate ground state as described by the Schr¨odinger equation (2.5), the statement of the first theorem is as follows: Theorem 4.1. The ground state density .r/ determines the external potential energy v.r/ to within a trivial additive constant C. The following are consequences of Theorem 4.1: 1. Knowledge of the ground state density .r/ uniquely determines the Hamiltonian HO of the system. The ground state density, therefore, identifies the physical system. To see this, consider external potential energies v.r/ of the form of the Coulomb or Yukawa interaction, which are singular for electron-nucleus coalescence, and for which the wave function therefore exhibits a cusp at each nucleus.

4.1 The Hohenberg–Kohn Theorems and Corollary

55

Integration of the density (see (2.18)) gives the number N of electrons. With the form of the kinetic energy operator known for each electron, the kinetic energy operator TO is thus known. Having assumed a form for the interaction among the electrons, the electron-interaction operator UO is also now known. The electron density exhibits cusps at each nucleus, thereby identifying their position. The electron density also satisfies the electron–nucleus coalescence condition (see Sect. 2.7) through which the charge Z of each nuclei is then obtained. Since, by Theorem 1, the ground state density determines the form of the external potential energy v.r/, the external potential energy operator VO is now known. Hence, the time-independent Hamiltonian HO of the system is fully defined by knowledge of the ground state density .r/. As an example, consider the Coulomb species [3, 4] comprised of two electrons and an arbitrary number N of nuclei as shown in Fig. 4.1. The elements of this species are the Helium atom (N D 1; atomic number Z D 2), the Hydrogen molecule (N D 2; atomic number of each nuclei Z D 1), and the positive molecular ions

Coulomb Species Number of Electrons N = 2 Number of Nuclei arbitrary = Coulomb Interaction =2

=1 e–

e–

e–

z=2 Heilum Atom (a)

e–

z=1 z=1 Hydrogen Molecule (b) =3,..... e–

e–

........ z=1

z=1

z = 1......

Positive Molecular Ions (c),(d),.......

Fig. 4.1 The Coulomb species comprises of two electrons and an arbitrary number N of nuclei, the interaction between the electrons and between the electrons and nuclei being Coulombic: (a) Helium atom; (b) Hydrogen molecule; (c), (d), . . . , Positive molecular ions. Here N is the number of nuclei, Z the nuclear charge, e  the electronic charge. Note that each element of the species corresponds to a different physical system

56

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

(N > 2; atomic number of each nuclei Z D 1). The external potential energy operator VON of this species is 2 X VON D vN .r i /; (4.1) i D1

where vN .r/ D

N X

fC .r  R j /

(4.2)

j D1

with fC .r  R j / D 

1 : jr  R j j

(4.3)

Here r 1 and r 2 are the positions of the electrons, R j .j D 1; : : : ; N / the positions of the nuclei, and fC .r  R j / the Coulomb external potential energy function. Note that each element of the Coulomb species represents a different physical system. Now, suppose the ground state density .r/ of the Hydrogen molecule were known. Then, according to the first Hohenberg–Kohn theorem, this density uniquely determines the external potential energy operator to within an additive constant C: VON D2 D 

1 1 1 1    C C: jr 1  R 1 j jr 1  R 2 j jr 2  R 1 j jr 2  R 2 j

(4.4)

Thus, as described above, the Hamiltonian of the Hydrogen molecule is known. Note that in addition to the functional form of the external potential energy, the ground state density also explicitly defines the positions R 1 and R 2 of the nuclei. Thus, the Hydrogen molecule, and each element of the Coulomb species, is explicitly identified through knowledge of its ground state density. It is, however, possible to construct [3, 4] an infinite number of degenerate Hamiltonians fHO g that represent different physical systems, but which have the same ground state density .r/. In such a case, it is not possible on the basis of the Hohenberg–Kohn theorem to distinguish between the different systems. As an example, consider the Hooke’s species which is comprised of two electrons coupled harmonically to a variable number N of nuclei (as shown in Fig. 4.2). The elements of the species are the Hooke’s atom [23] (N D 1; atomic number Z D 2, spring constant k), the Hooke’s molecule (N D 2; atomic number of each nuclei Z D 1, spring constants k1 and k2 ), and the Hooke’s positive molecular ions (N > 2; atomic number of each nuclei Z D 1, spring constants k1 ; k2 ; k3 ; : : : ; kN ). The Hamiltonian of this species is the same as that of the Coulomb species except that the external potential energy function is fH .r  R j / D .1=2/kj .r  R j /2 :

(4.5)

As was the case for the Coulomb species, each element of the Hooke’s species represents a different physical system. Now it can be shown (see Chap. 4 of QDFT) that the Hamiltonians of the Hooke’s species are those of the Hooke’s atom to within

4.1 The Hohenberg–Kohn Theorems and Corollary

57

Hooke’s Species Number of Electrons N = 2 Number of Nuclei arbitrary = Coulomb Interaction = Harmonic Interaction e–

=1

e–

=2

e–

k1

k2 k1

k

k

e–

k2

z=2 Hooke’s Atom (a)

z=1 z=1 Hooke’s Molecule (b) =3,..... e–

e– k2

k1

k2 k1

z=1

k3

k3 z=1

........ z = 1......

Hooke’s Positive Molecular Ions (c),(d),.......

Fig. 4.2 The Hooke’s species comprises of two electrons and an arbitrary number N of nuclei, the interaction between the electrons is Coulombic, and that between the electrons and nuclei is harmonic with spring constant k; k1 ; : : : kN : (a) Hooke’s atom; (b) Hooke’s molecule; (c), (d), . . . Hooke’s positive molecular ions. Here, N is the number of nuclei, Z the nuclear charge, e  the electronic charge. Note that each element of the species corresponds to a different physical system

a constant C that depends upon the spring constants fkg, the positions of the nuclei fRg, and the number N of nuclei. It is the constant C that differentiates between the different elements of the species. Since the density of each element of the Hooke’s species is that of the Hooke’s atom, it can only recognize the Hamiltonian of a Hooke’s atom and not the constant C.fkg; fRg; N /. Thus, it cannot determine the Hamiltonians for N > 1. It is also possible to construct [3, 4] a Hooke’s species for which the ground state densities of all the elements are the same. Once again, the density cannot, on the basis of the first Hohenberg–Kohn Theorem distinguish between these elements of the species. This is a reflection of the fact that the wave function and density of the elements of the Hooke’s species do not exhibit a cusp at the positions of the nuclei. As such one arrives at the following corollary [3, 4] to the first Hohenberg–Kohn theorem.

58

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

Corollary 4.1. Degenerate time-independent Hamiltonians fHO g that represent different physical systems, but which differ by a constant C and yet possess the same density .r/, cannot be distinguished on the basis of the Hohenberg–Kohn theorem. A similar corollary may be derived [3, 4] for the Runge–Gross theorem [20] for time-dependent phenomenon (see Chap. 4 of QDFT). We note that the Hooke’s species for time-independent and time-dependent theories do not constitute a counter example to the Hohenberg–Kohn and Runge–Gross theorems. 2. As described earlier, knowledge of the ground state density uniquely determines the Hamiltonian of the system. With the Hamiltonian known, the Schr¨odinger equation (2.5) can be solved to determine the wave function and eigen energy for both ground and excited states. Thus, knowledge of the ground state density is equivalent to the knowledge of all the properties of the system. 3. Since the wave function .X /, whether for a ground or excited state, is determined through the ground state density .r/, the wave function is a functional of the ground state density: D Œ. Hence, the expectation value of any operator OO is a unique functional of the ground state density: O D OŒ D h ŒjOj O Œi: hOi

(4.6)

The eigen energy E of the Schr¨odinger equation for any state, ground or excited, which is the expectation value of the Hamiltonian HO , is consequently also a functional of the ground state density: E D EŒ. Although Theorem 1 establishes the fact that the wave function is a functional of the ground state density, it does not provide the explicit dependence of on .r/. Hence, all the unique expectation value functionals are unknown. It is also unknown whether the functional dependence of on .r/ is the same for all states, or whether the functional Œ for the ground state differs from that for an excited state. 4. By separating out the external potential energy component of (2.65), the energy functional EŒ may be written as Z EŒ D where

.r/v.r/dr C FHK Œ;

FHK Œ D h ŒjTO C UO j Œi;

(4.7)

(4.8)

is the Hohenberg–Kohn (HK) functional. Note that this functional depends only on the kinetic TO and electron-interaction UO operators, and is independent of the external potential energy operator. The functional FHK Œ is universal in that it is the same functional for all Coulombic systems such as atoms, molecules, and solids, or a set of systems with the same assumed interaction between the particles. Since the functional dependence of on .r/ is unknown, the functional FHK Œ is unknown. The statement of the second theorem of Hohenberg and Kohn is the following: Theorem 4.2. The ground state density .r/ can be determined by application of the variational principle for the energy by variation only of the density.

4.2 Kohn–Sham Density Functional Theory

59

As noted above, the ground state energy E is a functional of the ground state density .r/: E D EŒ D h ŒjHO j Œi: (4.9) According to Theorem 2, the ground state density can be obtained by variational minimization of the energy functional EŒ for arbitrary variations ı.r/ of the density. For the ground state density, the energy is that of the ground state. For densities .r/ Q that differ from the true ground state density, the resulting energy is an upper bound to the ground state energy: EŒ.r/ Q >E

for

.r/ Q ¤ .r/:

(4.10)

Application of the variational principle for the energy, together with the introduction of a Lagrange multiplier  to ensure charge conservation of (2.18), leads to the Euler–Lagrange equation ıEŒ=ı.r/ D ; (4.11) from which the ground state density may be determined. It can be proved that the Lagrange multiplier  has the physical interpretation of being the chemical potential (see QDFT). Note that the upper bound on the energy obtained for an approximate ground state density .r/ Q is rigorous only for the true ground state energy functional EŒ. If the energy functional itself is approximated, the bound is no longer rigorous. Solution of the Euler–Lagrange equation (4.11) with an approximate ground state energy functional will lead to an approximate density .r/. Q The corresponding ground state energy determined from this approximate functional is not a rigorous upper bound to the true value. (From the proof of Theorem 4.1, it becomes evident that the energy functional EŒ and other functionals OŒ are functionals of v-representable densities. A density is said to be v-representable if it is obtained from an antisymmetric ground state wave function of the time-independent Schr¨odinger equation (2.5) for arbitrary external potential energy v.r/. Consequently, the densities in the above variational procedure must be v-representable. It turns out, however, that a weaker constraint, namely that of N-representability in fact suffices. A density is N -representable if it is obtained from any antisymmetric function that satisfies the physical constraints of charge conservation, positivity, and continuity. For a more complete discussion of these points, the reader is referred to QDFT.)

4.2 Kohn–Sham Density Functional Theory Kohn–Sham density functional theory (KS-DFT) is an alternate mapping from a nondegenerate ground state of a system of N electrons in an external field F ext .r/ D r v.r/ as described by the Schr¨odinger equation (2.5), to one of noninteracting fermions also in their ground state but whose density .r/ is the same

60

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

as that of the interacting system. The theory, therefore, is limited to the construction of only one model S system. In contrast to Q-DFT [21], other model S systems with the same density cannot be constructed within the framework of KS-DFT. As in Q-DFT, the existence of such an S system is an assumption. This assumption is referred to as noninteracting v-representability. The terminology emphasizes that the interacting system v-representable densities .r/ can also be obtained from the model noninteracting fermion system. However, as is the case for the (interacting) electrons, the weaker constraint of N-representability suffices. The S system determined by KS-DFT is the same as that obtained by Q-DFT in that the ground state density .r/, energy E, and ionization potential I are the same. It is the expressions for the ground state energy E and the local electron-interaction potential energy vee .r/ that differ in the two theories. The assumption of existence of the S system allows one to directly write down the Hamiltonian HO s and Schr¨odinger equation for the noninteracting fermions. Thus, the Hamiltonian is X HO s D TO C VOs D (4.12) hO s .r i /; i

1X 2 TO D  ri ; 2 i X vs .r i /; VOs D

(4.13) (4.14)

i

1 hO s .r/ D  r 2 C vs .r/; 2 vs .r/ D v.r/ C vee .r/;

(4.15) (4.16)

and the corresponding Schr¨odinger equation for each model fermion is hO s .r/i .x/ D i i .x/I

i D 1; : : : ; N:

(4.17)

The wave function is a Slater determinant ˆfi g of the N lowest lying orbitals i .x/. The ground state density .r/, equivalent to that of the interacting system, is O .r/ D hˆfi gj.r/jˆf i gi D

XX 

ji .r/j2 :

(4.18)

i

From the first Hohenberg–Kohn theorem, it follows that the density .r/ uniquely determines the “external” potential energy vs .r/, and therefore the electroninteraction potential energy vee .r/. (It is important to note that the potential energies vs .r/; vee .r/ are unique for the S system in its ground state.) It is, however, possible to construct [21, 24] via Q-DFT S systems in an excited state with different local potential energies vs .r/; vee .r/ that lead to the same ground state density .r/. Hence, the potential energies vs .r/; vee .r/ are not unique in the rigorous sense

4.2 Kohn–Sham Density Functional Theory

61

of the word. There are many local potential energies vs .r/; vee .r/ that generate the ground state density .r/ via the Schr¨odinger equation (4.17). For the S system in its ground state, the potential energy vs .r/ is known through the density .r/. Since the kinetic energy operator TO is known, the Hamiltonian HO s is fully defined. The corresponding wave function ˆfi g and hence the orbitals i .x/ are functionals of the ground state density .r/ W i .x/  i Œ. Thus, the kinetic energy Ts of the noninteracting fermions is a functional of the density .r/: Ts D Ts Œ D

XX 

i

1 hi .rI Œ/j  r 2 ji .rI Œ/i: 2

(4.19)

(Since the kinetic energy is a functional of the ground state density: T D T Œ, it follows as a consequence of (4.19) that the Correlation-Kinetic energy Tc defined in Chap. 3 is also a functional of the density: Tc D Tc Œ). By adding and subtracting Ts Œ from the ground state energy expression of (4.7), we obtain Z KS EŒ D Ts Œ C .r/v.r/dr C Eee Œ; (4.20) where KS Eee Œ D FHK Œ  Ts Œ;

(4.21)

KS Œ. which then defines the KS-DFT electron interaction energy functional Eee As in Chap. 3, the energy EŒ may also be expressed in terms of the eigenvalues i of the S system. Thus, with Ts Œ obtained as in (3.61), we have

EΠD

X

Z i 

KS .r/vee .r/dr C Eee Œ:

(4.22)

i

All that remains to complete the set of self-consistent equations, is the definition of the electron-interaction potential energy vee .r/. This is arrived by the second theorem of Hohenberg and Kohn by application of the variational principle in terms of the density to the ground state energy functional EΠof (4.20). Thus, at the vanishing of the first-order variation of the energy, we have Z

ıEŒ ı.r/dr ı.r/ Z D ıTs Œ C Œv.r/ C vee .r/ı.r/dr

ıE D

D 0;

(4.23)

where the electron-interaction potential energy vee .r/ is the functional derivative of KS the functional Eee Œ: KS ıEee Œ : (4.24) vee .r/ D ı.r/

62

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

By employing the definition (4.19) for Ts Œ, the S system differential equation (4.17), and the normalization of the orbitals i .x/, we obtain the first-order variation of Ts Œ to be Z ıTs Œ D 

vs .r/ı.r/dr:

Substitution of (4.25) into (4.23) leads to Z Œvs .r/ C v.r/ C vee .r/ı.r/dr D 0:

(4.25)

(4.26)

Since the variations ı.r/ within the realm of N -representable densities are arbitrary, we recover (4.16) with the electron-interaction potential energy vee .r/ defined by the functional derivative of (4.24). With this definition for vee .r/, (4.17) and (4.20) are then solved self-consistently. KS The functional Eee Œ is unknown, and hence so is its functional derivative vee .r/. However, the Hartree or Coulomb self-energy EH Œ functional of the density and its functional derivative are known: 1 EH Œ D 2 vH .r/ D

Z Z

.r/.r 0 / drdr 0 ; jr  r 0 j

ıEH Œ D ı.r/

Z

.r 0 / dr 0 : jr  r 0 j

(4.27)

(4.28)

KS Thus, the functional Eee Œ is customarily partitioned as KS KS Œ D EH Œ C Exc Œ; Eee

(4.29)

KS which then defines the KS “exchange-correlation” energy functional Exc Œ. From (4.24) and (4.28), the potential energy vee .r/ within KS-DFT is written as

vee .r/ D vH .r/ C vxc .r/;

(4.30)

where the functional derivative vxc .r/ D

KS ıExc Œ ; ı.r/

(4.31)

is the KS “exchange-correlation” potential energy. Neither the KS “exchangeKS correlation” energy functional Exc Œ nor its derivative vxc .r/ are known. Note that since Ts Œ is the kinetic energy of a system of noninteracting fermions KS KS of density .r/, the energy functionals Eee Œ and Exc Œ and their functional derivatives vee .r/ and vxc .r/ are representative of electron correlations due to the Pauli principle, Coulomb repulsion, and Correlation-Kinetic effects. KS The functional Exc Œ is usually further partitioned into its KS “exchange” KS Ex Œ and KS “correlation” EcKS Œ energy functional components. Thus,

4.2 Kohn–Sham Density Functional Theory

63

KS Exc Œ D ExKS Œ C EcKS Œ;

(4.32)

so that the KS “exchange-correlation” potential energy vxc .r/ is vxc .r/ D vx .r/ C vc .r/;

(4.33)

where the KS “exchange” potential energy vx .r/ is defined as vx .r/ D

ıExKS Œ ; ı.r/

(4.34)

and the KS “correlation” potential energy vc .r/ as vc .r/ D

ıEcKS Œ : ı.r/

(4.35)

KS Œ into its “exchange” ExKS Œ and “correlation” EcKS Œ The partitioning of Exc energy components of (4.32) is based on the ad hoc choice that ExKS Œ is given by the Hartree-Fock theory [25,26] expression for the exchange energy, but with the S system orbitals i .x/ employed instead. Thus,

ExKS ΠD

1 2

Z Z

.r/x .rr 0 / drdr 0 ; jr  r 0 j

(4.36)

where x .rr 0 / is the S system Fermi hole (see also 3.45). The KS “exchange” energy thus defined is a functional of the ground state density .r/ because the orbitals i .x/ are such functionals. The partitioning (4.32) then defines the KS “correlation” energy functional EcKS Œ. Since ExKS Œ is defined in terms of the orbitals i .x/, and the functional dependence of these orbitals on the density in unknown, the “exchange” potential energy vx .r/ of (4.34) cannot be obtained as a functional derivative from (4.36). It can, however, be determined by the optimized potential method [27, 28] as described in QDFT. For the S system of KS-DFT, i.e., for the mapping from a ground state of the interacting system to one of noninteracting fermions in their ground state, the KS KS energy functionals Eee Œ; Exc Œ; EH Œ; ExKS Œ; EcKS Œ, and their respective functional derivatives vee .r/; vxc .r/; vH .r/; vx .r/; vc .r/ satisfy [29] the following integral virial theorems: Z KS Eee Œ C

.r/r  r vee .r/dr D Tc Π 0;

(4.37)

.r/r  r vxc .r/dr D Tc Π 0;

(4.38)

Z KS Exc ΠC

Z EH ΠC

.r/r  r vH .r/dr D 0;

(4.39)

64

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

Z ExKS ΠC

.r/r  r vx .r/dr D 0;

(4.40)

.r/r  r vc .r/dr D Tc Π 0:

(4.41)

Z EcKS Œ

C

Note that for this ground state S system we have Tc  0. However, for other S systems not in their ground state, Tc < 0. An example of such an S system is given in Chapter 5. Finally, the functional Ts Πsatisfies the sum rule Z 2Ts ΠD

.r/r  r vs .r/dr;

(4.42)

where vs .r/ is the effective potential energy of the noninteracting fermions as defined by (4.15).

4.2.1 Endnote The following are remarks relevant to KS-DFT: 1. A comparison of the expression for the functional derivative vH .r/ of (4.28) with that of the potential energy WH .r/ of (3.71) shows them to be equivalent. Thus, the physical interpretation [4, 30] of the functional derivative vH .r/ is that it is the work done in the Hartree field E H .r/ of the electron density .r/ as expressed by (3.70). KS KS 2. Although it is known that the functionals Eee Œ and Exc Œ, and their functional derivatives vee .r/ and vxc .r/, are representative of Pauli and Coulomb correlations and Correlation-Kinetic effects, KS-DFT does not describe how these correlations are incorporated in these functionals or their derivatives. A rigorous physical interpretation of these functionals and derivatives in terms of the various electron correlations is afforded [31, 32] by Q-DFT. We refer the reader to Chap. 5 of QDFT for details of this description. 3. Since the Hartree–Fock theory exchange energy expression is employed for the “exchange” energy functional ExKS Œ, and the fact that the functional and its functional derivative vx .r/ satisfy the sum rule (4.40) (with Tc absent), it could be erroneously construed that ExKS Œ and vx .r/ are strictly representative of Pauli correlations. Furthermore, as a consequence, that the “correlation” energy functional EcKS Œ and its derivative vc .r/ are therefore representative of Coulomb correlations and Correlation-Kinetic effects. This, however, is not the case. In Chap. 5 of QDFT it is proved that ExKS Œ and vx .r/ are representative not only of correlations due to the Pauli principle, but also of lowest-order Correlation-Kinetic effects. And that EcKS Œ and vc .r/ are, therefore, representative of Coulomb correlations and higherorder Correlation-Kinetic effects. 4. The definition of the correlation energy EcKS Œ within KS-DFT (4.32) is a consequence of the ad hoc choice of the Hartree–Fock theory exchange energy expression for the KS “exchange” energy ExKS Œ (see remarks prior to (4.36). This

4.2 Kohn–Sham Density Functional Theory

65

definition differs from the quantum chemistry definition of the correlation energy EcHF Œ, which is the difference between the ground state energy EŒ and the Hartree–Fock theory energy EHF Œ:

with

EcHF Œ D EŒ  EHF Œ;

(4.43)

EHF Œ D hˆHF jTO C UO C VO jˆHF i;

(4.44)

and where ˆHF , the Hartree–Fock theory wave function, is that single Slater determinant that minimizes the expectation (4.44) of the Hamiltonian (2.1) with no further restrictions. There is yet another definition [33, 34] of the correlation energy Ec0 Œ within the context of local effective potential energy theories. This is the difference between the ground state energy EŒ, and the energy Exo Œ obtained within an “exchangeonly” (xo) local effective potential energy calculation: Ec0 Œ D EŒ  Exo Œ:

(4.45)

In (4.45), Exo Œ is the exact “exchange-only” total energy [33] for external potential energy operator VO . That is Exo Œ D hˆ0 jTO C UO C VO jˆ0 i;

(4.46)

to be a ground state where ˆ0 is that single Slater determinant which is constrained P of some noninteracting Hamiltonian of the form T C i v0s .r i /, and which simultaneously minimizes the expectation (4.46) of the Hamiltonian of (2.1). (This is equivalent to the xo optimized potential method [4, 27, 28] in which the Hartree– Fock theory expression for the total energy is minimized with respect to arbitrary variations of a local effective potential energy function v0s .r/.) Note that the S system Slater determinantal wave function ˆfi g, the Hartree-Fock theory wave function ˆHF , and the Slater determinant ˆ0 are different and lead to different densities. The following rigorous bounds on the corresponding correlation energies can be proved [34]: EcKS Œ < Ec0 Œ < EcHF Œ: (4.47) KS KS Œ, Exc Œ, ExKS Œ, EcKS Œ representative 5. Since the energy functionals Eee of the electron correlations are unknown, they must be approximated in any application of KS-DFT. However, approximating these functionals is akin to approximating the electron-interaction potential energy operator UO , and thus the Hamiltonian HO of (2.1). As a consequence, the physical system itself is approximated. As a further consequence, the rigor of the Hohenberg–Kohn theorems is lost, and the bound on the ground state energy obtained via the approximate functional is no longer rigorous. The results could lie below the true nonrelativistic value [35]. 6. In the construction of approximate “exchange-correlation” energy functionals, the physics of the different electron correlations is an extrinsic input. As such

66

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

all approximate “exchange-correlation” energy functionals are ad hoc. The manner in which approximate functionals are usually constructed is by requiring these functionals to satisfy various scaling laws [36, 37] and charge conservation sum rules [38], and by the fitting of the corresponding functional derivative to the known asymptotic structure of the potential energy determined independently either through rigorous analytical work or via quantum chemical calculations. Typically, the attempt is not to approximate accurately the electron correlations themselves via the structure of the approximate Fermi or Coulomb holes. Rather, the approach is to first determine an approximate hole, for example from a gradient expansion approximation, and then to enforce the various charge conservation rules on these holes. Generally, the holes thus constructed, do not bear much resemblance to the structure of the true holes [39, 40]. Nevertheless, over the past 4 decades, approximate “exchange-correlation” energy functionals have been developed that lead to highly accurate results [41]. KS KS 7. Since the KS-DFT energy functionals Eee Œ and Exc Œ are unknown, the corresponding potential energies vee .r/ and vxc .r/ cannot be obtained as their respective functional derivatives. Thus, even for model interacting systems for which the exact wave function .X / and ground state density .r/ are known, it is not possible to determine these potential energies directly via the KS formalism. Instead, they are usually determined indirectly with the known density as the starting point. A local effective potential energy vs .r/ is determined self-consistently such that the orbitals generated by it reproduce the density. Then the potential energy vee .r/ D vs .r/  v.r/, with v.r/ the interacting system external potential energy, and vxc .r/ D vee .r/  vH .r/. Numerical schemes whereby the potential energy vs .r/ is determined from the density have been developed [42–44]. Such schemes, whether employed for ground or excited states, are entirely independent of Kohn–Sham theory. 8. In the construction of an approximate KS “exchange-correlation” energy funcKSapprox tional Exc Œ, such as for example that of the local density approximation (LDA), the correlations between the electrons are approximated. Now it is commonly assumed that the electron correlations of the system are those represented by this approximate energy functional. Thus, the approximate potential energy or KSapprox functional derivative vapprox .r/ D ıExc Œ=ı.r/ is also assumed to be xc representative of these approximate correlations. The analysis of results thereby obtained is also based on this assumption. It turns out that the assumption may not be correct. There could exist additional electron correlations that the approximate functional derivative is representative of, but which do not contribute explicitly to the approximate energy functional. There is, of course, no mechanism within KS theory whereby these additional correlations can be determined. However, it is possible to obtain these additional correlations via Q-DFT because within Q-DFT a component of the total energy and the corresponding potential energy are both determined from the same source and therefore are representative of the same electron correlations. In QDFT, it is shown that the electron correlations in the LDA are not only those of the uniform electron gas corresponding to the density at each electron position, but that there are also additional correlations inherently present that are proportional to

4.3 Generalization of the Fundamental Theorem of Hohenberg–Kohn

67

the gradient of the density at each point in space. It is these additional correlations that give rise to the potential energy in the LDA. However, these correlations, while contributing to the potential energy explicitly, do not contribute explicitly to the energy in the LDA. The contribution of these correlations to the energy in the LDA is indirect via the orbitals generated by the potential energy function. The reader is referred to QDFT and the original literature [45–48] for further details.

4.3 Generalization of the Fundamental Theorem of Hohenberg–Kohn The fundamental theorem of Hohenberg and Kohn (Theorem 1), of the bijectivity, between the density .r/ and the Hamiltonian HO .R/ to within a constant C i.e., .r/ $ HO .R/ C C , is proved for the Hamiltonian HO .R/ of (2.1)–(2.4) where R D r1; : : : ; rN . P In this Hamiltonian, the external potential energy operator VO D i v.r i / is a scalar. Furthermore, in the proof of the theorem it is assumed that the kinetic energy TO and electron-interaction potential energy WO operators are known. (The symbol UO of (2.4) is replaced here by WO .) Thus, in the proof, these operators are kept fixed. The theorem is then proved by considering different external potential energy operators VO . The density .r/ is the expectation of the density operator .r/ O of (2.16) taken with respect to the wave function .X /, where X D x 1 ; : : : ; r N ; x D r. We generalize the theorem of bijectivity by a unitary transformation of the Hamiltonian HO .R/ to Hamiltonians HO 0 .R/ which in addition to the scalar potential energy v.r/ operator also include the momentum pO and a curl-free vector potential O energy A.r/ operator.

4.3.1 The Unitary Transformation To generalize the fundamental theorem, we perform a unitary transformation of the Hamiltonian HO .R/. The unitary operator UO we employ is UO D ei˛.R/ ; so that the transformed wave function 0

0

(4.48)

.X / is

.X / D UO  .X /;

(4.49)

and the transformed density 0 .r/ is 0 .r/ D h

0

.X /j.r/j O

0

.X /i D .r/:

The unitary transformation thus preserves the density.

(4.50)

68

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

The transformed Hamiltonian HO 0 .R/ is HO 0 .R/ D UO  HO .R/UO ;

(4.51)

so that the transformed time-independent Schr¨odinger equation is HO 0 .R/

0

.X / D E 0

0

.X /;

(4.52)

with E 0 D E of (2.5). In a unitary transformation, the eigen energies remain unchanged. (That E 0 D E also follows from the fact that the eigen energies E are unique functionals of the ground state density .r/. As the density .r/ is invariant in this unitary transformation, the eigen energies of the Hamiltonian HO .R/ and HO 0 .R/ are the same.) We next obtain the transformed Hamiltonian HO 0 .R/. From (4.51) HO 0 .R/ D ei˛.R/

X 1   ri2 ei˛.R/ C VO C WO : 2

(4.53)

i

Since



r 2 ; ei˛ D r 2 ei˛  ei˛ r 2 ;

(4.54)

the Hamiltonian HO 0 .R/ is 1 HO 0 .R/ D  2 or

Xn

o ei˛.R/ Œri2 ; ei˛.R/  C ri2 C VO C WO

(4.55)

i

1 X n i˛.R/ 2 i˛.R/ o e Œri ; e  : HO 0 .R/ D HO .R/  2

(4.56)

i

Next, we determine the commutator of (4.56). Employing the commutator relationship

2 (4.57) r ; f .r/ D r 2 f .r/ C 2r f .r/  r ; we have With r e

i˛

r 2 ; ei˛ D r 2 ei˛ C 2r ei˛  r :

(4.58)

D ie r ˛, then i˛

r 2 ei˛ D r  r ei˛ D ei˛ .r ˛/2 C iei˛ r 2 ˛:

(4.59)

Thus, the commutator

r 2 ; ei˛ D ei˛ .r ˛/2 C iei˛ r 2 ˛ C 2iei˛ r ˛  r ;

(4.60)

4.3 Generalization of the Fundamental Theorem of Hohenberg–Kohn

69

and therefore

ei˛ r 2 ; ei˛ D .r ˛/2 C ir 2 ˛ C 2ir ˛  r :

(4.61)

Employing the vector identity r  .C / D r   C C .r  C /;

(4.62)

r  .r ˛/ D r ˛  r C r 2 ˛;

(4.63)

r 2˛ D r  r ˛  r ˛  r :

(4.64)

we have so that Therefore, on substituting (4.64) into (4.61), we have

ei˛ r 2 ; ei˛ D .r ˛/2 C ir  r ˛ C ir ˛  r :

(4.65)

Hence, the transformed Hamiltonian HO 0 .R/ of (4.56) may be expressed as  1 X

Oi CA O i  pO i C A O 2i ; pO i  A HO 0 .R/ D HO .R/ C 2

(4.66)

i

where pO i D i r i is the momentum operator, and where the vector potential energy O i D r i ˛.R/ so that r  Ai D 0. (It is implicit that for the operator is defined as A transformed system, the boundary conditions too are transformed.) Note that, by writing the transformed Hamiltonian HO 0 .R/ as in (4.66), we emphasize the fact that the operators TO and WO are the same as those of the untransformed Hamiltonian HO .R/ of (2.1)-(2.4). Thus, we preserve the Hohenberg–Kohn assumption that the operators TO and WO are fixed. It is evident that HO 0 .R/ may also be written as 2 1 X

O i C VO C WO : pO i C A HO 0 .R/ D 2

(4.67)

i

Note that, as is the case for the Hamiltonian HO .R/, there is no magnetic field in O i as the transformed Hamiltonian HO 0 .R/. The vector potential energy operator A defined above is curl-free. As we have performed a unitary transformation, the physical system described by HO 0 .R/ and HO .R/ is the same. That HO 0 .R/ and HO .R/ represent the same physical system may also be seen by performing the following gauge transformation of HO .R/ to obtain HO 0 .R/. Rewriting HO .R/ as 2 ˇˇ 1 X

O O pO i C Ai ˇˇ H .R/ D C VO C WO ; 2 O i D0 A i

(4.68)

70

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

Oi ! A O0 D A O i C r i ˛.R/ such that B D r  Ai D 0, we make the transformation A i 0 0 O O with Ai D 0 so that B D r  Ai D 0. One then reobtains the Hamiltonian HO 0 .R/ as written in (4.67). It is well known in quantum mechanics [49] that the above gauge transformation for a Hamiltonian with nonzero but finite magnetic field B leaves the Schr¨odinger equation invariant provided the wave functions are related by the gauge transformation ˛.R/ as in (4.49). The generalization of the time-dependent Runge–Gross theorem via a unitary or gauge transformation is given in Appendix B.

4.3.2 New Insights as a Consequence of the Generalization As a consequence of the unitary transformation, there are several new insights that are achieved with regard to the theorem of bijectivity between the ground state density .r/ and the Hamiltonian HO .R/ of a system: .r/ $ HO .R/. We describe here these insights together with other clarifactory remarks. 1. The Hamiltonian HO 0 .R/ of (4.66), (4.67) obtained from the gauge function ˛.R/ is the most general form of the Hamiltonian for which the Hohenberg–Kohn theorem is valid. This Hamiltonian includes not only a scalar potential energy operator v.r i / but also the momentum operator pO i D i r i and a curl-free vector O i D r i ˛.R/. The bijectivity of the fundamental thepotential energy operator A orem in its general form is represented pictorially in Fig. 4.3. The figure shows that the bijectivity is .r/ $ HO .R/ with HO .R/ of (2.1)–(2.4), or equivalently

Fig. 4.3 The generalization of the fundamental theorem of Hohenberg–Kohn demonstrating the bijectivity between the density .r/ and the Hamiltonians H.R/ and Hj0 .R/ representing that physical system. The figure is drawn for the most general form of the time-independent theorem for which the gauge function is ˛j .R/. The theorem as originally enunciated is recovered when ˛.R/ D ˛, a constant

4.3 Generalization of the Fundamental Theorem of Hohenberg–Kohn

71

.r/ $ HO j0 .R/ with HO j0 .R/ of (4.66), (4.67), depending on the choice of the gauge function ˛j .R/. It is emphasized that the Hamiltonian HO .R/ and Hamiltonians HO j0 .R/ all correspond to the same physical system. 2. The Hohenberg–Kohn theorem as originally enunciated is recovered as a special case when ˛.R/ D ˛, a constant (see (4.66) and Fig. 4.3). (As an aside we point out that the more general statement of the bijectivity between the density .r/ and the wave function .X /, as proved and then employed in the proof of the fundamental theorem, is that the latter is known to within a phase factor ˛.) Note, that for the special case ˛.R/ D ˛, there is no constant C present in (4.66). Of course, this must be so because in this case HO j0 .R/ D HO .R/, and the energies E 0 and E are equivalent. Therefore the constant C of the Hohenberg–Kohn theorem is arbitrary and extrinsically additive. This has also been the understanding since the advent of the theorem. Put another way, the bijectivity .r/ $ HO .R/ or .r/ $ HO .R/ C C is for the same physical system since the constant C simply adjusts the energy reference level. (Note that as discussed above for the Hooke’s species (see Fig. 4.2 and also QDFT), it is possible to construct an infinite number of degenerate Hamiltonians fH g that differ by an intrinsic constant C , represent different physical systems, and which all possess the same density .r/. In this case, the density .r/ cannot distinguish between the different physical systems, and consequently the theorem of bijectivity is no longer valid (see corollary above).) 3. It becomes evident from the above unitary or gauge transformation that in the general case the wave function .X / must be a functional of both the density .r/ as well as the gauge function ˛.R/ i.e., .X / D Œ.r/I ˛.R/. If the wave function .X / was solely a functional of the density .r/, then that wave function as a functional of the density would be gauge invariant because the density is gauge invariant. However, it is well known in quantum mechanics [49] that the Hamiltonian HO and wave function .X / are gauge variant. It is the functional dependence of the wave function functional on the gauge function ˛.R/ that ensures it is gauge variant. 4. Because the bijectivity is between the density .r/ and the Hamiltonian representation of the physical system HO .R/, HO .R/ C C , or HO j0 .R/ (see Fig. 4.3), the choice of gauge function is arbitrary. Thus the choice ˛.R/ D 0 is equally valid. This provides a deeper understanding of the fundamental theorem of Hohenberg– Kohn. In their original paper [1] they state: “Thus, v.r/ is (to within a constant) a unique functional of .r/; since, in turn, v.r/ fixes H we see that the full manyparticle ground state is a unique functional of .r/.” (our emphases). The statement implies that the many-particle ground state functional is gauge invariant. However, we now understand that their statement is consistent with the fact that the choice of gauge function ˛.R/ D 0 is valid. 5. As a point of information, we note that the two Hohenberg–Kohn theorems can be derived employing the original reductio ad absurdum argument for a general form of the Hamiltonian HO D HO 0 C VO , where VO is a local potential energy operator, and HO 0 any Hermitian operator defined on the Hilbert space of quadratically integrable functions. The only requirement that HO 0 must have is that it be bounded from below and have normalizable eigen functions. The Hamiltonian HO 0

72

4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory

could contain a magnetic field or a vector potential with vanishing or nonvanishing curl. This form of the generalization of the theorem differs from the generalized form derived via the unitary transformation in a fundamental way. For different Hermitian operators HO 0 , the Hamiltonian HO corresponds to different physical systems, and therefore to different ground state densities. In the generalization derived via the unitary transformation, the physical system is unchanged and therefore the density is preserved. 6. As noted previously, the Hohenberg–Kohn theorems can be proved for different Hamiltonians HO as for example when different potential energy operators WO such as the Coulomb or Yukawa interactions are employed. Thus, one can state that the wave function .X / is a functional of the operator WO . The physical systems corresponding to different WO are different, and hence the density for these different Hamiltonians will be different. However, it is important to note that in proving the Hohenberg–Kohn theorems, the operator WO is assumed known and kept fixed throughout the proof. Hence the statement that the wave function .X / is a functional of both the ground state density .r/ and the gauge function ˛.R/ is valid for each Hamiltonian HO with a fixed electron-interaction operator WO . In conclusion it is reiterated that in the most general case when the gauge function is ˛.R/, the functional dependence of the wave function .X / on the gauge function is important because the corresponding Hamiltonian HO 0 .R/ of (4.66) explicitly involves the gauge function via the momentum pO i and curl-free vector potential O i operators. This functional dependence hence also enhances the signifienergy A cance of the phase factor in density functional theory in a manner similar to that of quantum mechanics (see also Appendix B). The understanding that the wave function .X / is a functional of both the density .r/ and the gauge function ˛.R/ is fundamental.

Chapter 5

Nonuniqueness of the Effective Potential Energy and Wave Function in Quantal Density Functional Theory

Since its advent, a key precept of Kohn–Sham density functional theory (KS-DFT) [1, 2] has been the uniqueness of the local effective potential energy function vs .r/, or equivalently of the electron-interaction potential energy function vee .r/ of the model S system of noninteracting fermions. Nondegenerate ground state KS-DFT maps an interacting system in its ground state to an S system that is also in its ground state. As the density .r/ of the interacting and noninteracting fermions is the same, it then follows from the first Hohenberg–Kohn theorem, as explained in Chap. 4, that the potential energy vee .r/ is unique. In Kohn–Sham terms, there is only one such potential energy function because vee .r/ is the functional derivative KS KS ıEee Œ=ı.r/ taken at the ground state density, where Eee Œ is the unique KS ground state electron-interaction energy functional. It is evident, however, from quantal density functional theory (Q-DFT) [3], that one is not limited to constructing S systems only in a ground state. One can equally well construct S systems in an excited state such that the ground state density .r/, energy E, and ionization potential I of the interacting system are reproduced [3, 4]. The state of the model S system is arbitrary. This means that there exist, in principle, an infinite number of local effective potential energy functions that generate the density, energy and ionization potential of the interacting system in its ground state. Furthermore, it is solely the Correlation-Kinetic components of these potential energy functions that differ. To understand this, consider the Q-DFT interpretation of the electron-interaction potential energy vee .r/ as the work done in the conservative field F eff .r/ (see 3.67): Z vee .r/ D 

r

1

F eff .r 0 /  d` 0 ;

(5.1)

where F eff .r/ is the sum of the electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ fields: F eff .r/ D E ee .r/ C Z tc .r/; (5.2) and Z tc .r/ D Z s .r/  Z.r/;

(5.3)

73

74

5 Nonuniqueness of the Effective Potential Energy and Wave Function

is the difference between the noninteracting Z s .r/ and interacting Z.r/ kinetic fields. It is only the S system kinetic field Z s .rI Œs / through its source, the Dirac density matrix s .rr 0 /, and therefore the Correlation-Kinetic field Z tc .r/ that depends on the orbitals i .x/ of the model fermions (see (2.52) and (3.37)). The state of the S system may be arbitrarily chosen such that the orbitals i .x/ occupy excited states. In doing so, it is only the Correlation-Kinetic field Z tc .r/ that is modified. The electron-interaction field E ee .r/ remains the same whether the S system is in a ground or excited state. However, although the field E ee .r/ remains unchanged, its Pauli E x .r/ and Coulomb E c .r/ components will change as a function of the state the S system is in. Thus, although the contributions of Pauli and Coulomb correlations taken together remain unchanged, the separate Pauli and Coulomb components are different. Proof that the nonuniqueness of the effective potential energy of the S system is due solely to Correlation-Kinetic effects is given in the final section of the chapter. As the equations of Q-DFT encompass both ground and excited states, the above remarks on the state arbitrariness of the S system are equally valid for a nondegenerate excited state [5–7]. Thus, it is possible to obtain the density of an excited state of the interacting system via an infinite number of S systems in different states. In other words, the excited state density can be obtained from many local potential energy functions. Once again, the difference between the various potential energy functions is in their Correlation-Kinetic component, the sum of the Pauli and Coulomb components remain unchanged. The proof of this is also given in the final section of the chapter. The fact that there exist an infinite number of local potential energy functions that can generate the density of an excited state of the interacting system, has the following implication with regard to traditional density functional theory. Although the excited state energy is a unique functional of the ground state density, as explained in Chap. 4, it is not a unique functional of the excited state density. Hence, there is no first Hohenberg–Kohn theorem for excited states. It is well known that there is no such theorem for excited states [8–10]. However, Q-DFT via its equations, confirms in a rigorous mathematical manner that there can be no theorems for excited states similar to those of the ground state Hohenberg–Kohn theorems. Additionally, what Q-DFT shows is that this is a direct consequence of Correlation-Kinetic effects. The concept of state arbitrariness of the S system is also valid with regard to the Q-DFT of degenerate ground and excited states [11] as described in Appendix A. In this chapter, the state arbitrariness of the S system or equivalently the nonuniqueness of the effective potential energy function is demonstrated by application to a ground state of the exactly solvable Hooke’s atom [12–14]. This ground state is mapped via Q-DFT to two S systems: one in its 11 S ground state [15], and the other in an excited 21 S singlet state [4]. For the mapping from an excited 21 S singlet state of the Hooke’s atom to two S systems, one in its ground 11 S state, and the other in an excited 21 S singlet state, the reader is referred to [3, 5–7]. (For other work along similar lines where an approximate density is the starting point, and the S system constructed indirectly by density-based methods, see [16].)

5 Nonuniqueness of the Effective Potential Energy and Wave Function

75

There is yet another arbitrariness to be noted. This arbitrariness has to do with the wave function of the S system in an excited state [17, 18]. Consider the case of the mapping from a ground state of the Hooke’s two-electron atom to the excited 21 S singlet (anti-parallel spin) state of the S system to be described in Sect. 5.3. (By the 21 S singlet state of the S system we mean that one electron is in the ground state, and the other of opposite spin in the first excited state.) It turns out that the same density .r/ may be obtained from two different single Slater determinants of the orbitals of the S system, as well as from a linear combination of these Slater determinants. However, although the single Slater determinant wave functions are an eigenfunction of SOz , the z component of the total spin operator, they are not an eigenfunction of SO 2 , the square of the total spin operator. The wave function formed by the linear combination of Slater determinants, on the other hand, is an eigenfunction of both the SO 2 and SOz operators. Is the latter then more appropriate choice for the wave function of the S system in this excited singlet state? The answer to the question is that, it is not any more or less appropriate than the single determinant wave function. The reason for this is because all that is demanded from the model system of noninteracting fermions is that it reproduces the interacting system density .r/. It is irrelevant from wave function that this density is obtained. However, based on the choice of the wave function, the structure of the corresponding Fermi and Coulomb holes and therefore the values of the resulting Pauli and Coulomb correlation energies will differ. Their sum, the Fermi–Coulomb holes, and the corresponding Pauli–Coulomb energy, remains unchanged. The remarks of the previous paragraph are equally valid for the mapping from a nondegenerate excited state of the interacting system to one of noninteracting fermions in the same configuration [17, 18]. For example, for the mapping from the excited 21 S singlet state of the Hooke’s atom to an S system with the same configuration, there are once again two wave functions, a single Slater determinant and a linear combination of Slater determinants, both of which lead to the same excited state density .r/ as that of the interacting atom. Both wave functions are equally appropriate. Once again, the Fermi and Coulomb holes, and the Pauli and Coulomb energies will differ based on the choice of wave function employed. It has also been shown [19, 20] that an excited state density can be reproduced by different local potential energy functions with the S system in a fixed excited state configuration. The multiplicity of these potentials has been related [19] to the eigenvalues of the linear nonlocal susceptibility of the system in its excited state. These different potential energy functions can also be obtained [20] via a density functional theory constrained search approach [21]. Once again, an insight that can be provided via Q-DFT, is that the difference between these functions arises solely due to the difference in their Correlation-Kinetic fields. Furthermore, the state of the S system is arbitrary. Expressions in terms of fields relating the different potential energy functions of the S system in a fixed configuration that lead to the same excited state density are given in the final section of the chapter.

76

5 Nonuniqueness of the Effective Potential Energy and Wave Function

5.1 The Interacting System: Hooke’s Atom in a Ground State The interacting system we consider is the Hooke’s atom [14] in a ground state. The Hooke’s atom is composed of two electrons whose external potential energy is harmonic. The Hamiltonian is therefore 1 1 1 1 1 HO D  rr21  rr22 C kr12 C kr22 C ; 2 2 2 2 jr 1  r 2 j

(5.4)

where r 1 and r 2 are the coordinates of the electrons. For the atom in the ground state with electrons of opposite spin corresponding to k D 1=4, the solution of the Schr¨odinger equation (2.5) is 00 .r 1 r 2 /

 0 .R/ D

D 0 .R/0 .r/; 2!

3=4

2

e!R ;

(5.5)

(5.6)

2

0 .r/ D a00 e!r .1 C !r/; (5.7) p p 5=4 where R D .rp1 C r 2 /=2; r D r 1  r 2 ; ! D k D 0:5; a00 D ! .3 =2 C p 8 ! C 2 2 !/1=2 . Note that the atom in its ground state is spherically symmetric. A study of the atom in this state from the “Newtonian” perspective of Schr¨odinger theory, as described in Chap. 2, is given in QDFT [3]. The total energy of this ground state is E D 2:000000 a.u., and its ionization potential I00 D 1:250000 a.u. We next map the interacting system in its ground state via Q-DFT to two different S systems. The first is an S system in its singlet 1s 2 .11 S / ground state [15]. The second is the first excited singlet 1s2s.21 S / state [4].

5.2 Mapping to the S system in Its 11 S Ground State The mapping to an S system in its ground state is described in detail in QDFT. Here, just the basic equations and results are given. In the S system ground state, both the model fermions occupy the same 1s orbital and have opposite spins. Thus, the two one-particle spin-orbitals of the S system differential equation (3.3) are 1 .x/ D

1s 2 .r/˛./;

2 .x/ D

1s 2 .r/ˇ./;

(5.8)

where the normalized 1s 2 .r/ is the spatial part of the spin orbital, and ˛./; ˇ./ the spin functions. The spin coordinate ˛ can have only two values ˙1.The spin

5.2 Mapping to the S system in Its 11 S Ground State

77

functions have only two values 0 and 1, so that ˛.1/ D 1; ˛.1/ D 0; ˇ.1/ D 0; ˇ.1/ D 1. The normalized S system wave function is the Slater determinant. ˇ ˇ 1 ˇˇ 1 .x 1 / 1 .x 2 / ˇˇ ˆ.x 1 x 2 / D p ˇ 2 2 .x 1 / 2 .x 2 / ˇ 1 D p Œ 1s 2 .r 1 / 1s 2 .r 2 /Œ˛.1 /ˇ.2 /  ˛.2 /ˇ.1 / : (5.9) 2 As the electrons have opposite spin, the density .r/ D hˆj.r/jˆi O  D 2 1s 2 .r/ 1s 2 .r/: Thus, the S system orbitals

1s 2 .r/

(5.10)

are known in terms of the density .r/ as r

1s 2 .r/

D

.r/ : 2

(5.11)

As the density .r/ is known via the wave function so is the spatial orbital 1s 2 .r/. This orbital, and the radial probability density r 2 .r/ indicated as r 2 1s 2 .r/ are plotted in Figs. 5.1 and 5.2, respectively. The S system pair-correlation density gs .rr 0 / (see Sect. 3.2.3 for definitions) is gs .rr 0 / D

.r 0 / ; 2

(5.12)

so that the Fermi hole

.r 0 / ; (5.13) 2 which is a local charge distribution independent of electron position r. As is evident, the Fermi hole satisfies the charge conservation (3.19), negativity (3.20), and value at the electron position (3.22) sum rules. The Coulomb hole c .rr 0 / is obtained from the Fermi–Coulomb xc .rr 0 / and Fermi x .rr 0 / hole via its definition (3.28). Thus, x .rr 0 / D 

c .rr 0 / D xc .rr 0 / 

.r 0 / ; 2

(5.14)

where the nonlocal Fermi–Coulomb holes xc .rr 0 / as a function of electron position r are plotted in Fig. 5.3. (The electron position is on the z axis corresponding to  D 0ı . The cross- section through the Coulomb hole plotted corresponds to  0 D 0ı with respect to the electron–nucleus direction. The graph for r 0 < 0 corresponds to the structure for  0 D and r 0 > 0.) The electron positions are at r D 0; 0:5; 1:0; 2:0; 7:0 a.u. The Fermi–Coulomb holes xc .rr 0 / remain unchanged irrespective of the state the S system is in. For the structure of the Fermi x .rr 0 /

78

5 Nonuniqueness of the Effective Potential Energy and Wave Function

Fig. 5.1 The orbitals of the S system in its singlet ground state state 1s .r/ and 2s .r/

1s 2 .r/,

and its singlet first excited

and the Coulomb c .rr 0 / holes for the S system in its ground state, the reader is referred to Figs. 3.1 and 3.4 of QDFT. The resulting electron-interaction field E ee .r/ and its Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ field components are plotted in Fig. 5.4. The corresponding Pauli–Coulomb potential energy Wee .r/ obtained as the work done in the field E ee .r/, and its Hartree WH .r/, Pauli Wx .r/, and Coulomb Wc .r/ components are plotted in Fig. 5.5. (Since the interacting system is spherically symmetric, all fields are curl free, and all the work done are path-independent.) The contribution of these potential energies to the electron-interaction potential energy vee .r/ is the same irrespective of the state of the S system. The electron-interaction energy Eee , and its Hartree EH , Pauli Ex , and Coulomb Ec energy component values also remain the same. Those energy values are given in Table 5.1

5.2 Mapping to the S system in Its 11 S Ground State

79

Fermi-Coulomb Hole ρxc (r r') (a.u.)

Fig. 5.2 The radial probability densities r 2 1s 2 .r/, and r 2 1s .r/, r 2 2s .r/

–0.02

r = 7 a.u.

–0.04

r = 2 a.u. r = 1 a.u. –0.06 r = 0.5 a.u. r=0 –0.08 –4

–3

–2

–1

0

1

2

3

4

5

r' (a.u.)

Fig. 5.3 Cross-sections through the Fermi–Coulomb hole charge xc .rr 0 / as a function of electron positions at r D 0; 0:5; 1; 2; 7 a.u. The electron is on the z-axis corresponding to  D 0. The cross-sections plotted correspond to  0 D 0ı with respect to the nucleus–electron direction. The graph for r 0 < 0 is the structure for  0 D and r 0 > 0

80

5 Nonuniqueness of the Effective Potential Energy and Wave Function

Fig. 5.4 The electron-interaction field E ee .r/, and its Hartree E H .r/, Pauli Ex .r/, and Coulomb E c .r/ field components

The interacting system kinetic “force” z.r/ and the corresponding ground-state S system kinetic “force” zs .r/j1s 2 are plotted in Fig. 5.6a. The resulting CorrelationKinetic field Z tc .r/j1s 2 which is curl free, and the path-independent CorrelationKinetic potential energy Wtc .r/j1s 2 are plotted in Figs. 5.7(a) and 5.8, respectively. The S system kinetic energy Ts j1s 2 and the Correlation-Kinetic energy Tc j1s 2 are given in Table 5.1. The path-independent electron-interaction potential energy vee .r/ of the S system in its ground state, as determined via (5.1) or equivalently as the sum of the work Wee .r/ and Wtc .r/, is plotted in Fig. 5.9.

5.2 Mapping to the S system in Its 11 S Ground State

81

Fig. 5.5 The potential energy Wee .r/ and its Hartree WH .r/, Pauli Wx .r/, and Coulomb Wc .r/ potential energy components

The single eigenvalue 1s 2 may be determined from the S system differential equation (3.3) via 1s 2

p 1 r 2 .r/ D p C v.r/ C vee .r/; 2 .r/

(5.15)

which is an expression valid for arbitrary r, and where vee .r/ is as given in Fig. 5.9. The eigenvalue 1s 2 may also be determined by substituting for vee .r/ into the differential equation (3.3), and solving numerically for the single zero node orbital and single eigenvalue. The orbital leads to the density .r/, and the eigenvalue 1s 2 given in Table 5.1 is the negative of the ionization potential.

82

5 Nonuniqueness of the Effective Potential Energy and Wave Function

Table 5.1 The components of the total energy and eigenvalues for the mapping from a ground state of Hooke’s atom to S systems in the singlet ground 11 S and first excited singlet 21 S states. For the latter, the Pauli energies, as determined from a single Slater determinant (SD) and from a linear combination of Slater determinants (LCD), and the corresponding Coulomb energies are also given Property S systems (a.u.)

Ground State 11 S .a/

Eext Eee EH Ex

0:888141 0:447443 1:030250 0:515125

Ec

0:067682

Ts Tc E

0:635245 0:029173 2:000000 1s 2 D 1:250

Eigenvalues a:

Excited State 21 S .b/

0:888141 0:447443 1:030250 0:526185.SD/ 0:487669.LCD/ 0:056622.SD/ 0:095138.LCD/ 2:344949 1:680531 2:000000 1s D 1:799 2s D 1:250

From [15]; b: From [4]

5.3 Mapping to an S system in Its 21 S Singlet Excited State In the 21 S singlet excited state, the 1s and 2s eigenstates are occupied by the model fermions of opposite spin. With this occupation, the S system differential equation (3.3) is then solved numerically till self-consistency is achieved. The orbitals 1s .r/ and 2s .r/ are plotted in Fig. 5.1. The density .r/ determined from these orbitals is the same as that obtained from the ground state S system orbital 1s 2 .r/ of Sect. 5.2. The radial probability densities r 2 1s .r/ and r 2 2s .r/ are plotted in Fig. 5.2. The sum of the excited state S system radial probability densities r 2 Œ1s .r/ C 2s .r/ is equivalent to the ground-state S system radial probability density r 2 1s 2 .r/. Each is equivalent to the radial probability density r 2 .r/ of the interacting system. Observe that the asymptotic structure of r 2 2s .r/ and that of r 2 1s 2 .r/ are the same. This is why the eigenvalues 2s and 1s 2 are equivalent. As noted earlier, the electron-interaction E ee .r/ component of the effective field F eff .r/ (see 5.2) for the S system in its 21 S state is the same as that for the S system in its ground state. The corresponding Pauli–Coulomb Wee .r/ potential energy component of vee .r/ is also the same (see Figs. 5.4 and 5.5). As the density of the S system in their ground and excited states is the same, so are Hartree field E H .r/ and potential energy WH .r/. The resulting electron-interaction Eee and Hartree EH energies are consequently also the same (see Table 5.1). It is the kinetic field Z s .r/j1s2s , and therefore the Correlation-Kinetic field Z tc .r/j1s2s , of the S system in its excited state that are different. In Fig. 5.6b, the kinetic “force” zs .r/j1s2s is plotted together with the interacting system “force”

5.3 Mapping to an S system in Its 21 S Singlet Excited State

83

Fig. 5.6 (a) The kinetic “forces” z.r/ of the interacting system, and zs .r/j1s 2 of the S system in the ground 1s 2 state. (b) The kinetic “forces” z.r/ of the interacting system, and zs .r/j1s2s of the S system in the excited 1s2s state

z.r/. Observe that zs .r/j1s2s is an order of magnitude greater than both z.r/ as well as zs .r/j1s 2 (see Fig. 5.6a). These differences are reflected and enhanced in the corresponding Correlation-Kinetic field Z tc .r/j1s2s plotted in Fig. 5.7b. This field is 2 orders of magnitude greater than the ground state Z tc .r/j1s 2 . The Correlation-Kinetic potential energies Tc j1s2s and Tc j1s 2 thus also differ by 2 orders of magnitude (see Table 5.1). The Correlation-Kinetic potential energy Wtc .r/j1s2s is plotted in Fig. 5.8, and as expected is 2 orders-of-magnitude greater than Wtc .r/j1s 2 . Observe, however, that the asymptotic structure of these potential energies is the same. As a consequence, the S system electron-interaction potential energies vee .r/j1s2s and vee .r/j1s 2 plotted in Fig. 5.9 also have an equivalent asymptotic structure. This ensures that the corresponding highest occupied eigenvalues 2s and 1s 2 are the same (see Table 5.1).

84

5 Nonuniqueness of the Effective Potential Energy and Wave Function

Fig. 5.7 (a) The Correlation-Kinetic field Z tc for the S system in the ground .1s 2 / state. (b) The Correlation-Kinetic field Z tc for the S system in the excited .1s2s/ state

Note that the two potential energy functions vee .r/j1s 2 and vee .r/j1s2s of Fig. 5.9 both generate the same density .r/ as that of the interacting system. The total energy E determined from (3.59), and the ionization potential I as obtained from the highest occupied eigenvalues 1s 2 and 2s are also equivalent. The noninteracting kinetic Ts and Correlation-Kinetic Tc energies of the S system in their ground and excited states differ significantly. In each case, however, their sum is equivalent to the kinetic energy of the interacting system T D 0:664418 a.u. For the S system in its ground state .1s 2 /; Ts Tc and Tc > 0. For the S system in the 1s2s excited state, once again Ts > Tc , but in this case Tc < 0: (Similar results for the sign of Tc are obtained for the ground state of the He atom when an S system in the 1s2s state with equivalent density is constructed [16]. In this calculation, an approximate He atom density is employed, and the S system constructed indirectly by density-based methods.) As a point of information, we note that in the

5.4 Nonuniqueness of the Wave Function of the S system in an Excited State

85

Fig. 5.8 The Correlation-Kinetic potential energies Wtc .r/ for the S systems in the ground .1s 2 / and excited .1s2s/ states

mapping [3, 5–7] from an excited state of the Hooke’s atom to an S system in a ground state .1s 2 /Tc > 0, whereas for the mapping to an S system in the excited 1s2s state Tc < 0:

5.4 Nonuniqueness of the Wave Function of the S system in an Excited State Just as there is an arbitrariness of the state of the S system that reproduces the density .r/ of the interacting system, there is a nonuniqueness of the wave function for an S system in an excited state [17, 18]. Consider the mapping from the ground state of a two-electron atom to an S system in its excited 21 S singlet state.

86

5 Nonuniqueness of the Effective Potential Energy and Wave Function

Fig. 5.9 The electron-interaction potential energies vee .r/ for the S systems in the ground .1s 2 / and excited .1s2s/ states

The two single Slater determinants ˇ 1 ˇˇ ˆ1 .x 1 x 2 / D p ˇ 2 1 Dp Œ 2 and

1s .r 1 / 2s .r 2 /˛.1 /ˇ.2 / 

ˇ 1 ˇ ˆ2 .x 1 x 2 / D p ˇˇ 2 1 Dp Œ 2

ˇ ˇ ˇ 2s .r 1 /ˇ.1 / 2s .r 2 /ˇ.2 / 1s .r 1 /˛.1 /

1s .r 2 /˛.2 / ˇ

1s .r 2 / 2s .r 1 /˛.2 /ˇ.1 /;

(5.16)

ˇ ˇ ˇ 2s .r 1 /˛.1 / 2s .r 2 /˛.2 / 1s .r 1 /ˇ.1 /

1s .r 1 / 2s .r 2 /ˇ.1 /˛.2 /



1s .r 2 /ˇ.2 / ˇ

1s .r 2 / 2s .r 1 /ˇ.2 /˛.1 /;

(5.17)

5.4 Nonuniqueness of the Wave Function of the S system in an Excited State

87

and the wave function constructed by the linear combination of these Slater determinants 1 ˆs .x 1 x 2 / D p .ˆ1  ˆ2 / (5.18) 2 1 D Œ 1s .r 1 / 2s .r 2 / C 1s .r 2 / 2s .r 1 /Œ˛.1 /ˇ.2 /  ˛.2 /ˇ.1 /; (5.19) 2 all lead to the same density .r/. Hence, from the perspective of constructing model systems of noninteracting fermions that lead to the density .r/, each of these wave functions is equally valid. The two single Slater determinants ˆ1 .x 1 x 2 / and ˆ2 .x 1 x 2 / are eigenfunctions of SOz , the z-component of the total spin operator SO , but not of the operator SO 2 . Their linear combination ˆs .x 1 x 2 / is an eigenfunction of both operators. Furthermore, ˆs .x 1 x 2 / is a product of a symmetric spatial part and an antisymmetric spin part. In quantum mechanics, it is this wave function that defines the 21 S singlet state. However, in local effective potential energy theories such as Q-DFT or KS–DFT, there are no constraints on the S system wave function other than that it reproduce the density .r/. Therefore, from the perspective of these theories, all three wave functions are appropriate representation of the singlet 21 S state of the S system. The same reasoning applies if one were to map [18] the excited 21 S state of the interacting two-electron atom to an S system also in the excited 21 S state. Once again, there are three wave functions that reproduce the density .r/ of the interacting system. And again, there are no constraints on the S system that require it to mimic the interacting system other than reproduce its density. There are no restrictions that the S system must be in the same configuration. Nor are there any constraints that since the wave function of the interacting system is an eigenfunction of SOz and SO 2 , that the corresponding S system wave function also be such an eigenfunction. The same arguments are equally applicable to a mapping to an S system in the triplet 23 S state. In this case, the two model fermions in the 1s and 2s states have parallel spin. The two single Slater determinants ˇ 1 ˇ ˆ3 .x 1 x 2 / D p ˇˇ 2 1 Dp Œ 2

ˇ ˇ ˇ 2s .r 1 /˛.1 / 2s .r 2 /˛.2 / 1s .r 1 /˛.1 /

1s .r 1 / 2s .r 2 /



1s .r 2 /˛.2 / ˇ

1s .r 2 / 2s .r 1 /˛.1 /˛.2 /;

(5.20)

and ˇ 1 ˇˇ ˆ4 .x 1 x 2 / D p ˇ 2 1 Dp Œ 2

1s .r 1 / 2s .r 2 /

ˇ ˇ ˇ .r /ˇ. / .r /ˇ. / 2s 1 1 2s 2 2 1s .r 1 /ˇ.1 /



1s .r 2 /ˇ.2 / ˇ

1s .r 2 / 2s .r 1 /ˇ.1 /ˇ.2 /;

(5.21)

88

5 Nonuniqueness of the Effective Potential Energy and Wave Function

and the linear contributions of the Slater determinants ˆ1 .x 1 x 2 / and ˆ2 .x 1 x 2 /: 1 ˆt .x 1 x 2 / D p Œˆ1 .x 1 x 2 / C ˆ2 .x 1 x 2 / 2

(5.22)

1 Œ 1s .r 1 / 2s .r 2 /  1s .r 2 / 2s .r 1 /Œ˛.1 /ˇ.2 / C ˛.2 /ˇ.1 /; (5.23) 2 all lead to the same density .r/. All three wave functions are eigenfunctions of both the SOz and SO 2 operators. They are all written as a product of an antisymmetric spatial function and a symmetric spin function, the spatial function being the same in each case. Hence, from the perspective of both quantum mechanics and local effective potential energy theory, all three wave functions are appropriate representations of the triplet state. For the triplet state 23 S , of the S system, the physical meaning of the Fermi hole is the same as that in Hartree-Fock theory because the two electrons have parallel spin. The three wave functions all lead to the same expression for the pair-correlation density gs .rr 0 / of (3.16) which is D

gs .rr 0 / D Œ

1s .r/ 2s .r

0

/

1s .r

0

/

2 2s .r/ =.r/;

(5.24)

and hence to the same Fermi hole from (3.18) as x .rr 0 / D 

Œ

1s .r/ 1s .r

0

/ C 2s .r/ .r/

2s .r

0

/2

:

(5.25)

This Fermi hole satisfies the sum rules of charge conservation (3.19) and negativity (3.20). Its value at the electron position, as must be the case, is x .rr/ D .r/:

(5.26)

For the mapping to the excited 21 S singlet state, the two single Slater determinants ˆ1 .x 1 x 2 / and ˆ2 .x 1 x 2 /, and their linear combination ˆs .x 1 x 2 / lead to different Fermi and Coulomb holes. Consequently, the corresponding Pauli and Coulomb potentials and energies also differ. These differences are discussed in the subsections below.

5.4.1 The Single Slater Determinant Case 5.4.1.1 Fermi and Coulomb Holes In the case of the mapping to the excited 21 S singlet state, the two Slater determinants ˆ1 .x 1 x 2 / and ˆ2 .x 1 x 2 / lead to the same expression for the Fermi Hole x .rr 0 /. With either of these wave functions, the S system pair-correlation density

5.4 Nonuniqueness of the Wave Function of the S system in an Excited State

89

gs .rr 0 / of (3.16) is gs .rr 0 / D

2 X

0

  0 0 i .r/ i .r/ j .r / j .r /=.r/;

(5.27)

i;j D1

so that the corresponding Fermi hole from (3.18) is xSD .rr 0 / D 

2 2 0 1s .r/ 1s .r /

C .r/

2 2 0 2s .r/ 2s .r /

;

(5.28)

where the superscript SD stands for single determinant. Note that in contrast to the mapping to the S system in a ground state, the Fermi hole in this case depends explicitly on the electron position. The Fermi hole also satisfies the charge conservation (3.19) and negativity (3.20) sum rules. Its value at the electron position, however, is 4 4 .r/ C 2s .r/ xSD .rr/ D  1s : (5.29) .r/ This value of the Fermi hole has no physical meaning. The Fermi hole xSD .rr 0 / is plotted in Fig. 5.10 for the electron positions at r D 0; 0:5; 1:0; 2:0; 7:0 a.u. Observe that the Fermi holes are spherically symmetric about the nucleus for all electron positions. This is because the orbitals 1s .r/ and 2s .r/ are spherically symmetric. The magnitude of the Fermi hole diminishes as the electron position is moved further away from the nucleus to a position about 1.6 a.u. For positions beyond this point, its magnitude increases, approaching its stabilized value for asymptotic positions of the electron. Note that the position r D 1:6 a.u. corresponds to the point where the 1s .r/ orbital is a maximum, and the 2s .r/ orbital is a minimum (see Fig. 5.2). The Coulomb holes cSD .rr 0 /, as determined from equation (3.28) and the Fermi–Coulomb holes of Fig. 5.3, are plotted in Fig. 5.11 for the different electron positions. With the exception of the electron position at the nucleus, the Coulomb holes are not spherically symmetric about the nucleus. The holes are both positive and negative as they satisfy the Coulomb hole sum rule (3.29) of zero total charge. Observe the cusp in the Coulomb holes at the electron position that is evident in the figure for electron positions near the nucleus.

5.4.1.2 Pauli and Coulomb Fields, Potentials and Energies The Pauli E x .r/ and Coulomb E c .r/ fields determined via (3.32) and (3.33) from the Fermi and Coulomb holes of Sect. 5.3 are plotted in Fig. 5.12. For comparison, the electron-interaction E ee .r/ and its Hartree E H .r/ component are also plotted. The sum of E H .r/; E x .r/; E c .r/ is, of course, equivalent to E ee .r/. The asymptotic structure of these fields is also indicated in the figure. As expected, because of the

90

5 Nonuniqueness of the Effective Potential Energy and Wave Function

Fermi Hole ρxSD (r r′) (a.u.)

r = 2 a.u. –0.01

–0.02

–0.03 r = 0.5 a.u.

r = 7 a.u. –0.04 r = 1 a.u.

–4

–3

–2

r=0 –1

0

1

2

3

4

5

r′(a.u.)

Fig. 5.10 Cross-section of the Fermi holes xSD .rr 0 / as a function of electron positions at r D 0, 0:5; 1:0; 2:0; 7:0 a.u., as determined from a single Slater determinant (SD)

0.06

Coulomb Hole ρcSD (r r′) (a.u.)

r = 7 a.u. 0.04

r = 0.5 a.u.

r=0

0.02

0.00

r = 2 a.u.

–0.02

r = 1 a.u. –4

–3

–2

–1

0

1

2

3

4

5

r′ (a.u.)

Fig. 5.11 Cross-section of the Coulomb holes cSD .rr 0 / for electron positions at r D 0; 0:5; 1:0; 2:0; 7:0 a.u., as determined from the Fermi–Coulomb holes of Fig. 5.3 and Fermi holes of Fig. 5.10

5.4 Nonuniqueness of the Wave Function of the S system in an Excited State

91

0.3

eH 0.2

Fields (a.u.)

eee 0.1

1/r2

2/r2

eC

0.0

–1/r2

O(–1/r4)

–0.1

eX –0.2

0

1

2

3

4

5

6

7

8

9

10

r (a.u.)

Fig. 5.12 The electron-interaction Eee .r/ field, and its Hartree EH .r/, Pauli Ex .r/, and Coulomb E c .r/ field components. The E x .r/ and E c .r/ are a consequence of the Fermi hole xSD .rr 0 / determined via a single Slater determinant

negativity of the Fermi hole, the Pauli field E x .r/ is negative. Further, it exhibits shell structure. The Coulomb field E c .r/, as expected is both positive and negative, and it too exhibits shell-like structure. The Pauli Wx .r/ and Coulomb Wc .r/ potential energies determined via (3.74) and (3.75) together with the Pauli–Coulomb Wee .r/ and its Hartree WH .r/ potential energy component are plotted in Fig 5.13. Because the fields E x .r/ and E c .r/ vanish at the nucleus, the potential energies Wx .r/ and Wc .r/ have zero slope there. The asymptotic structure of the potential energies are also indicated in the figure. The two shells are weakly evident in the Wx .r/ and Wc .r/ curves. The Pauli Ex and Coulomb Ec energies (see (3.45), (3.47), (3.49), and (3.50)) are given in Table 5.1 indicated by SD in parenthesis. For the single determinant case,

92

5 Nonuniqueness of the Effective Potential Energy and Wave Function 1.5

WH

Potential Energies (a.u.)

1.0

Wee

0.5

1/r

2/r

0.0 Wc –1/r

O(–1/r3)

–0.5 Wx

0

1

2

3

4

5 6 r (a.u.)

7

8

9

10

Fig. 5.13 The Pauli–Coulomb Wee .r/ potential energy, and its Hartree WH .r/, Pauli Wx .r/, and Coulomb Wc .r/ components determined as the work done in the corresponding fields of Fig 5.12

the energies Ex and Ec are very close to those of the mapping to the ground state which also involves a single determinant (see Table 5.1). Note that the Coulomb energy is an order of magnitude smaller than the Pauli energy.

5.4.2 The Linear Combination of Slater Determinants Case 5.4.2.1 Fermi and Coulomb Hole For the wave function ˆs .x 1 x 2 / of (5.18) formed by the linear combination of the Slater determinants ˆ1 .x 1 x 2 / and ˆ2 .x 1 x 2 /, the S system pair-correlation density

Fermi Hole ρxLCD (r r′) (a.u.)

5.4 Nonuniqueness of the Wave Function of the S system in an Excited State

93

r = 1 a.u.

–0.02

–0.04 r = 2 a.u. –0.06 r = 0.5 a.u. r=0 –0.08

r = 7 a.u. –4

–3

–2

–1

0

1

2

3

4

5

r′ (a.u.)

Fig. 5.14 Cross-sections of the Fermi holes xLCD .rr 0 / for electron positions at r D 0, 0:5; 1:0; 2:0; 7:0 a.u. as determined by the linear combination of Slater determinants (LCD)

gs .rr 0 / of (3.16) differs from that of (5.24) and is gs .rr 0 / D Œ

1s .r/ 2s .r/

C

1s .r

0

/

2s .r

0

/2 =.r/:

(5.30)

The Fermi hole from (3.18) thus also differs from (5.25) and is xLCD .rr 0 / D 

Œ

1s .r/ 1s .r

0

/  2s .r/ .r/

2s .r

0

/2

;

(5.31)

where the superscript LCD stands for linear combination of determinants. This Fermi hole also depends explicitly on the electron position, and satisfies the constraints of charge conservation (3.19) and negativity (3.20). Its value at the electron position which is different from (5.26) is xLCD .rr/ D 

Œ

2

2  2s .r/ : .r/

2 1s .r/

(5.32)

In this case too, this value of the Fermi hole cannot be interpreted physically. The Fermi hole xLCD .rr 0 / for different electron positions is plotted in Fig. 5.14. Although these holes differ from those of the single determinant hole xSD .rr 0 / of Fig. 5.10, their general structure such as symmetry about the nucleus etc., are the same as described previously for xSD .rr 0 /. The corresponding Coulomb holes cLCD .rr 0 / are given in Fig. 5.15. Again, these holes differ from those of cSD .rr 0 /, but their general features are similar, with the cusp at the electron position clearly evident for positions near the nucleus.

94

5 Nonuniqueness of the Effective Potential Energy and Wave Function r = 7 a.u. Coulomb Hole ρcLCD (r r′) (a.u.)

0.05

0.03

r = 2 a.u.

r=0 0.01

–0.01

–0.03 r = 1 a.u. r = 0.5 a.u.

–0.05 –4

–3

–2

–1

0 r′ (a.u.)

1

2

3

4

5

Fig. 5.15 Cross-sections of the Coulomb holes cLCD .rr 0 / for electron positions at r D 0, 0:5; 1:0; 2:0; 7:0 a.u. as determined from the Fermi–Coulomb holes of Fig. 5.3 and Fermi holes of Fig. 5.14

5.4.2.2 Pauli and Coulomb Fields, Potentials and Energies The Pauli field E x .r/ determined by the linear combination of determinants wave function ˆs .x 1 x 2 /, and the corresponding Coulomb field E c .r/ are plotted in Fig. 5.16. In this case the shell structure is far more dramatic than in the single determinant case of Fig. 5.12. Similarly, in the resulting graphs of the potential energies Wx .r/ and Wc .r/ of Fig. 5.17, shell structure is more clearly exhibited. The asymptotic structure of the fields and potential energies is, of course, the same as in the single determinant case. The Pauli Ex and Coulomb Ec energies, indicated by (LCD), are given in Table 5.1. These energies differ from their single determinant counterparts, although once again the Coulomb energy is an order of magnitude less than the Pauli energy. However, the Coulomb energy is twice as large as in the single determinant case. Note that the sum of Ex and Ec is the same as in the single determinant example, as must be the case.

5.5 Proof that Nonuniqueness of Effective Potential Energy Is Solely Due to Correlation-Kinetic Effects In the construction of S systems that reproduce the ground or excited state density of the interacting system, it is assumed that the external field F ext .r/ D r v.r/ is the same for both the interacting and model fermions. This in turn leads to

5.5 Proof: Nonuniqueness of Potential due to Correlation-Kinetic Effects

95

0.3

eH 0.2

eee 1/r2

Fields (a.u.)

0.1

2/r2

eC 0.0

–1/r2

O(–1/r4)

–0.1

–0.2

eX 0

1

2

3

4

5

6

7

8

9

10

r (a.u.)

Fig. 5.16 The electron-interaction Eee .r/ field, and its Hartree EH .r/, Pauli Ex .r/, and Coulomb E c .r/ field components. The E x .r/ and E c .r/ fields are a consequence of the Fermi hole xLCD .rr 0 / determined via a linear combination of Slater determinants

the interpretation (5.1) for the corresponding electron-interaction potential energy vee .r/ of the S system. Here, we prove that the vee .r/ of the different S systems, whether they correspond to S systems in different states or whether they are different S systems corresponding to the same excited state configuration [19, 20], differ solely in their Correlation-Kinetic component. The component due to the Pauli exclusion principle and Coulomb repulsion remains the same. Consider the mapping from a ground or excited state of the interacting system with density .r/. Next, consider two noninteracting fermion systems S and S 0 that in the presence of the same external field F ext .r/ D r v.r/, reproduce the same density .r/. For the S system, the differential equation and the corresponding local effective potential energy vs .r/ are defined by (3.3) and (3.4), respectively. The electron-interaction potential energy vee .r/ is the work done as given by (5.1).

96

5 Nonuniqueness of the Effective Potential Energy and Wave Function 1.5

WH

Potential Energies (a.u.)

1.0

Wee 1/r

0.5

2/r

0.0 Wc

–1/r

O(–1/r3) –0.5

Wx

0

1

2

3

4

5 r (a.u.)

6

7

8

9

10

Fig. 5.17 The Pauli–Coulomb Wee .r/ potential energy, and its Hartree WH .r/, Pauli Wx .r/, and Coulomb Wc .r/ components determined as the work done in the corresponding fields of Fig. 5.16

For the S 0 system, the differential equation is   1 2 0  r C vs .r/ i0 .x/ D i0 i0 .x/; 2

(5.33)

where the corresponding local effective potential energy v0s .r/ is v0s .r/ D v.r/ C v0ee .r/;

(5.34)

with v0ee .r/ being the electron-interaction potential energy. The resulting “Quantal Newtonian” first law is F ext .r/ C F 0int (5.35) s .r/ D 0;

5.5 Proof: Nonuniqueness of Potential due to Correlation-Kinetic Effects

97

0 where F 0int s .r/ is the internal field of the S model fermions: 0 0 F 0int s .r/ D r vee .r/  D.r/  Z s .r/;

(5.36)

with the definitions of the fields D.r/ and Z 0s .r/ being the same as in Chap. 3. A comparison of (5.35) with the interacting system first law of (2.11) then yields v0ee .r/

Z

r

D 1

ŒE ee .r 0 / C Z 0tc .r 0 /  d` 0 ;

(5.37)

where the Correlation-Kinetic field Z 0tc .r/ is Z 0tc .r/ D Z 0s .r/  Z.r/:

(5.38)

Here E ee .r/ and Z.r/ are the electron-interaction and kinetic fields of the interacting system as defined in Chap. 2. The difference between vee .r/ and v0ee .r/ of the S and S 1 systems is then vee .r/  v0ee .r/ D 

Z

r 1

ŒZ tc .r 0 /  Z 0tc .r 0 /  d` 0 ;

(5.39)

or equivalently vee .r/ 

v0ee .r/

Z D

r 1

ŒZ s .r 0 /  Z 0s .r 0 /  d` 0 :

(5.40)

Note that both (5.39) and (5.40) are independent of the electron-interaction field E ee .r/. As such the contribution of E ee .r/ to vee .r/ and v0ee .r/ is the same. Thus, the difference between the electron-interaction potential energies arises solely due to the difference in their Correlation-Kinetic or equivalently their kinetic fields. This completes the proof.

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Chapter 6

Ad Hoc Approximations Within Quantal Density Functional Theory

In quantum mechanics, one chooses the level of approximation in terms of electron correlations by the ad hoc choice of approximate wave function. For example, in the Hartree approximation [1] (see also QDFT), one chooses a wave function that is a product of single-particle spin-orbitals. The Hartree wave function does not obey the Pauli exclusion principle, as it is not antisymmetric in an interchange of the coordinates of the electrons including its spin coordinate. Thus, electron correlations due to the Pauli principle are ignored in this choice of wave function. (In calculations performed within the Hartree approximation [2], one incorporates the Pauli exclusion principle in an ad hoc manner by ensuring that no two electrons occupy the same state.) The Hartree wave function also ignores Coulomb correlations between the electrons: there are no terms, for example, involving the inter-electronic separation in the wave function. Hence, Hartree theory is said to be an independent particle approximation. The best orbitals for this product wave function in terms of the energy, are then obtained by application of the variational principle for the energy [3] leading to the Hartree equations. The energy obtained is then a rigorous upper bound to the true nonrelativistic value. It is interesting to note that this independent particle model leads to various properties that are obtained accurately. For example, atomic shell structure throughout the Periodic Table is exhibited [4] via Hartree theory, with highly accurate core-valence separations. The Sommerfeld–Hartree model of a simple metal [5], which is an example of a noninteracting uniform electron gas, also reproduces accurately experimental properties such as the electronic specific heat at low temperatures. On the other hand, as no electron correlations are accounted for in the theory, the Sommerfeld–Hartree model does not allow for cohesion in the jellium approximation of metals. At the next level of approximation, one includes electron correlations due to the Pauli exclusion principle. One way to do this is to assume the wave function to be a Slater determinant of spin-orbitals, as in Hartree–Fock theory [6, 7] (see also QDFT). As was the case in Hartree theory, Hartree–Fock theory too does not account for electron correlations due to Coulomb repulsion. Once again, from the perspective of the energy, the best spin-orbitals for use in the Slater determinant are obtained by application of the variational principle [3] leading to the Hartree– Fock equations. As the physics is improved by explicitly accounting for Pauli

99

100

6 Ad Hoc Approximations Within Quantal Density Functional Theory

correlations, the upper bound to the energy [8] is superior to that of Hartree theory. Hence, it is possible to have cohesion of metals within Hartree–Fock theory. Again, highly accurate shell structure is obtained [9] for atoms throughout the periodic table. What is interesting, however, is that in spite of accounting for the Pauli correlations, other properties turn out not to be better. For example, the band width of metals, as obtained in Hartree–Fock theory, is significantly in error [5]. Furthermore, within this theory, the density of states at the Fermi level in metals vanishes [5], so that the ratio of the electronic specific heat to temperature vanishes as the temperature approaches zero, instead of being finite as observed experimentally. As another example of where the Pauli correlations prove inadequate is the Florine molecule which is unbound [10] in Hartree–Fock theory. At the final level of approximation, one incorporates both Pauli and Coulomb correlations in an approximate wave function. Such correlated wave functions are antisymmetric in an interchange of the coordinates of the electrons, and include explicit factors involving the inter-electronic separation. Examples of such wave functions are the correlated-determinantal or Jastrow type wave functions. [11–14] One may also include Coulomb correlations via the use of configuration-interaction type wave functions [15]. These wave functions are linear combinations of Slater determinants of spin-orbitals corresponding to different electronic configurations. Variation of the coefficients then leads to the secular equation for the eigen energies of the system, the lowest eigenvalue being a rigorous upper bound to the true nonrelativistic ground state energy. (Note that the set of all possible configuration functions constitutes a complete set. Hence, the exact wave function can be expressed as a linear combination of this complete set of configuration functions.) With the inclusion of Couloumb correlations, the bound on the energy, as expected, is superior to that of Hartree–Fock theory. Another way to incorporate Coulomb correlations is via many-body perturbation theory [16], the lowest order term representing correlations due to the Pauli principle. Correlated wave functions, of course, lead to accurate atomic shell structure [17,18]. However, the effects of Coulomb correlations on shell structure and core-valence separation are minimal when compared to those of Hartree–Fock theory. Although correlated wave functions are employed in condensed matter theory (see e.g. [[19–21]] for applications to metal surfaces), it has proved to be easier to describe the many-electron system in both solids and large molecules within the context of approximate Kohn–Sham density functional theory instead. The application to solids was the original motivation for the development of Kohn–Sham theory. At each of the above levels of approximation for the wave function based on electron correlations, there exists a corresponding level within quantal density functional theory (Q-DFT). Thus, for example, it is possible to construct model noninteracting-fermion S systems such that the same density .r/ and energy E as that determined within Hartree and Hartree–Fock theories is obtained. We refer to these as the Q-DFT of Hartree Theory and the Q-DFT of Hartree–Fock Theory, and these are described in Sect. 6.1 and 6.2, respectively (See also QDFT). The Q-DFT of Schr¨odinger Theory, whereby the density and energy when all electron correlations are present, is described in Chap. 3. The reason why a Q-DFT exists for

6 Ad Hoc Approximations Within Quantal Density Functional Theory

101

Hartree, Hartree–Fock, and Schr¨odinger theories is that for each theory, there is a “Quantal Newtonian” first law equation or differential virial theorem for the corresponding interacting and model noninteracting systems. Equating the internal fields of the respective interacting and noninteracting systems (see 3.63), then leads to the corresponding Q-DFT equations of each theory. We note that as was the case with regard to Schr¨odinger theory, there exists a Hohenberg–Kohn theorem [22,23] for Hartree and Hartree–Fock theories. However, once again, the theorem is valid only for the ground state. Thus, in the corresponding Kohn–Sham manifestations, only the mapping from a ground state of the interacting system to an S system in a ground state can be achieved. On the other hand, the “Quantal Newtonian” first law equation within Hartree and Hartree–Fock theories is equally valid for ground and excited states. Hence, the Q-DFT of both Hartree and Hartree–Fock theories is more general and applicable for arbitrary state of the interacting system. In quantum mechanics, the kinetic energy T , at each level of approximation, is that of the interacting system for the assumed level of electron correlation. For example, in Hartree–Fock theory, the kinetic energy is that of a system of electrons for which only those correlations that arise due to the Pauli principle are considered. In Q-DFT, at each level of approximation, the kinetic energy Ts is that of a system of noninteracting fermions with the same density as that of the corresponding interacting system. It follows from the Heisenberg uncertainty principle that Ts ¤ T . Hence, to obtain the correct kinetic and therefore total energy within QDFT, one must add to Ts the contribution to the kinetic energy due to the electron correlations, viz. the corresponding Correlation-Kinetic energy Tc . Thus, as opposed to traditional quantum mechanics, an advantage of Q-DFT is that it is possible to determine the component Tc of the kinetic energy that is due to electron correlations. (For this, the mapping from the interacting system must be to an S system with the same configuration.) At each level of electron correlation assumed, the corresponding Q-DFT description: the Q-DFT of Hartree, Hartree–Fock, and Schr¨odinger theories, is the same. There is an electron-interaction field E ee .r/ component representative of the electron correlations, and a Correlation-Kinetic field Z tc .r/ component that accounts for the correlation contribution to the kinetic energy. At each level of approximation, the field E ee .r/ may be written as a sum of its Hartree E H .r/ component due to the local part of the pair-correlation density g.rr 0 / quantal source, and a second component, (the self-interaction-correction field E SIC .r/, the Pauli field E x .r/, and the Pauli–Coulomb field E xc .r/, respectively) due to the nonlocal part of this source. The local electron-interaction potential energy vee .r/ of the model fermions is then (see (3.67)) the work done in the conservative effective field F eff .r/ D E ee .r/ C Z tc .r/. In turn, the electron-interaction Eee and CorrelationKinetic Tc energies are expressed in integral virial form in terms of the fields E ee .r/ and Z tc .r/, respectively. In each ad hoc approximation within Q-DFT based on the choice of wave function, a further approximation can be made. This second approximation in the Q-DFT of Hartree, Hartree–Fock, and Schr¨odinger theories is based on the

102

6 Ad Hoc Approximations Within Quantal Density Functional Theory

ad hoc neglect of the Correlation-Kinetic effects in each of them. We thus have the Q-DFT Hartree Uncorrelated Approximation, the Q-DFT Pauli Approximation, and the Q-DFT Pauli–Coulomb Approximation. The energies obtained in these approximations are rigorous upper bounds to the Hartree, Hartree–Fock, and the exact Schr¨odinger theory values, respectively. The approximation of neglecting Correlation-Kinetic effects is a particularly good when the mapping from the interacting system is to an S system with the same configuration. This is because, as will be demonstrated in future chapters, the Correlation-Kinetic contributions are then negligible. (Working within these approximations then allows for an estimation of these Correlation-Kinetic contributions.) Finally, with an approximate correlated wave function, Correlation-Kinetic effects may also be included, in which case we have the Q-DFT Fully Correlated Approximation. With all the correlations now included in this approximation, the energy determined thereby will be a superior upper bound to the exact value. In the application of an approximation within Q-DFT, there may exist systems of symmetry for which the resulting effective field F eff .r/ of (3.65) is not conservative. Consider for example an approximation in which the Correlation-Kinetic effects are ignored. The effective field is then equivalent to the electron-interaction field: F eff .r/  E ee .r/. The effective field is conservative only if the electroninteraction field is conservative. In many of the applications to be described, the symmetry of the system will be such that this is the case. For systems, for which the field E ee .r/ is not conservative, there are two ways to obtain a local electroninteraction potential energy vee .r/ from this field such that it is path-independent. The first is to take an appropriate average of the field E ee .r/ or of the orbitals i .x/, so that the curl of the averaged field E ee .r/ vanishes. For example, in the case of open shell atoms, one works within the Central Field Approximation in which the spherical average of the field E ee .r/ is employed [24]. One could equally, well-spherically average the orbitals i .x/ prior to determining the field. (The Hartree–Fock theory calculations of atoms [8] are also performed within the central field approximation. In these calculations, it is the orbitals that are spherically averaged.) The second way to obtain a path-independent potential energy vee .r/ is via the Irrotational Component Approximation [25, 26]. In this approximation, one employs the irrotational component of the nonconservative field E ee .r/. As will be explained, this is equivalent to determining the potential energy from an effective charge distribution that is static or local. (Recall that the Hartree potential energy WH .r/ is also obtained from such a source, the electron density .r/.) A study [27] of a nonspherically symmetric system shows that the field E ee .r/ and its irrotational component are essentially equivalent, and that the solenoidal component of E ee .r/ is 2 orders of magnitude smaller. As the Hartree field E H .r/ component of E ee .r/ is conservative, it is the irrotational component of the fields E SIC .r/, E x .r/, and E xc .r/ of the various approximations described above that must be employed. The various ad hoc approximations within Q-DFT are described in the following sections. A many-body perturbation theory and a pseudo Møller-Plesset perturbation theory within the context of Q-DFT are described in Chap. 18.

6.1 The Q-DFT of Hartree Theory

103

6.1 The Q-DFT of Hartree Theory In Hartree theory, the wave function ‰ H .X / is a product of spin-orbitals iH .x/. The potential energy of each electron is the sum of the external potential energy v.r/ and the potential energy due to the density of all the other electrons. The latter is equivalent to the Hartree potential energy WH .r/ due to the charge density of all the electrons minus vSIC .r/ due to the self-interaction correction charge of the i electron in question (see QDFT). Hence, the potential energy of each electron is different, and therefore, Hartree theory is an orbital-dependent theory. In the QDFT of Hartree theory, whereby the same density .r/ and energy E H as that of Hartree theory are obtained, the wave function ‰.X / is also assumed to be a product of spin-orbitals: N Y ‰.X / D i .x/; (6.1) i D1

where i .x/ D i .r/ i ./. The potential energy of each model noninteracting fermion, which is the same, is the sum of the external component v.r/ and the local effective electron-interaction potential energy vH ee .r/. The S system differential equation is then 

 1  r 2 C v.r/ C vH .r/ i .x/ D i i .x/I ee 2

i D 1; : : : ; N:

(6.2)

The orbitals i .x/ differ from those of Hartree theory, so that the corresponding Dirac density matrices differ: s .rr 0 / D

XX 

¤

i

XX 

i .r/i .r 0 / iH .r/iH .r 0 / D  H .rr 0 /:

(6.3)

i

However, the diagonal matrix elements of these matrices are the same: s .rr/ D  H .rr/ D .r/. The potential energy vH ee .r/ is the work done to move the model fermion in the force of the conservative field F H .r/: Z vH ee .r/ D 

r 1

F H .r 0 /  d` 0 ;

(6.4)

where F H .r/, the effective Hartree field, is the sum of the Hartree electronH interaction E H ee .r/ and Correlation-Kinetic Z tc .r/ fields: H F H .r/ D E H ee .r/ C Z tc .r/: H Note that the work done vH ee .r/ is path-independent since r  F .r/ D 0.

(6.5)

104

6 Ad Hoc Approximations Within Quantal Density Functional Theory

The field E H ee .r/ is obtained from the Hartree theory pair-correlation density g H .rr 0 / via Coulomb’s law. Thus Z

g H .rr 0 /.r  r 0 / 0 dr ; jr  r 0 j3

EH ee .r/ D

(6.6)

with gH .rr 0 / D h‰ H .X /jPO .rr 0 /j‰ H .X /i=.r/; D .r 0 / C SIC .rr 0 /;

(6.7) (6.8)

where PO .rr 0 / is the pair-correlation operator (2.33), and SIC .rr 0 / the nonlocal self-interaction correction .SIC/ component of gH .rr 0 /: SIC .rr 0 / D 

XX 

qi .r/qi .r 0 /=.r/

(6.9)

i

with qi .r/ D i .r/i .r/. Employing (6.8) in (6.6), we can write SIC EH ee .r/ D E H .r/ C E H .r/;

(6.10)

where the Hartree field E H .r/ is Z E H .r/ D

.r 0 /.r  r 0 / 0 dr ; jr  r 0 j3

(6.11)

and the SIC field E SIC ee .r/ is Z E SIC H .r/

D

SIC .rr 0 /.r  r 0 / 0 dr : jr  r 0 j3

(6.12)

The Correlation-Kinetic field Z H tc .r/ is the difference between the S system and Hartree theory kinetic fields: H ZH tc .r/ D Z s .r/  Z .r/;

where Z s .r/ D

zs .rI Œs / .r/

and Z H .r/ D

(6.13) zH .rI ΠH / ; .r/

(6.14)

with zs .rI Œs / and zH .rI Œ H / the corresponding kinetic “forces.” These kinetic “forces” in turn are derived from the noninteracting and Hartree theory kineticenergy-density tensors defined in terms of the density matrices s .rr 0 / and  H .rr 0 /, respectively (see (2.54)).

6.1 The Q-DFT of Hartree Theory

105

Since the Hartree field E H .r/ is conservative, the potential energy vH ee .r/ may be written as   Z r  0 SIC 0 H 0 vH E ; (6.15) .r/ D W .r/ C  .r / C Z .r /  d` H ee H tc 1

Z

where

.r 0 / dr 0 : jr  r 0 j

WH .r/ D

(6.16)

For systems with symmetry such that the fields E SIC .r/ and Z H tc .r/ are each conservative, we then can write vee .r/ D WH .r/ C WHSIC .r/ C WtH .r/; c

(6.17)

where WHSIC .r/ and WtHc .r/ are, respectively, the separate work done in the fields H E SIC H .r/ and Z tc .r/: Z WHSIC .r/

1

Z

and .r/ WtH c

r

D

r

D 1

0 0 E SIC H .r /  d`

0 0 ZH tc .r /  d` :

(6.18)

(6.19)

The total energy of Hartree theory E H may be written in terms of the fields as Z E H D Ts C Z D Ts C

H .r/v.r/dr C Eee C TcH

(6.20)

SIC .r/v.r/dr C EH C EH C TcH ;

(6.21)

where Ts is the S system kinetic energy: Ts D

XX 1 hi .r/j  r 2 ji .r/i; 2 

(6.22)

i

the second term is the external potential energy, and where in integral virial form Z H Eee D

.r/r  E H ee .r/dr;

(6.23)

EH D .r/r  E H .r/dr; Z SIC D .r/r  E SIC EH H .r/dr; Z 1 .r/r  Z H TcH D tc .r/dr: 2

(6.24)

Z

(6.25) (6.26)

106

6 Ad Hoc Approximations Within Quantal Density Functional Theory

The expressions for the individual components of the energy in terms of the individual fields are independent of whether or not the fields are conservative. Note that it is the Correlation-Kinetic component of both the electron-interaction H potential energy vH ee .r/ of (6.2) and the total energy E expression of (6.20) that ensures the density and total energy of Hartree theory are obtained. It is also important to understand that the S system expression for the Hartree energy E H of (6.20) is not the expectation value of the Hamiltonian HO of (2.1) taken with respect to the Q-DFT of Hartree theory wave function ‰.X / of (6.1).

6.1.1 The Q-DFT Hartree Uncorrelated Approximation The Q-DFT Hartree Uncorrelated (HU) Approximation is derived from the above described Q-DFT of Hartree theory by neglecting the Correlation-Kinetic effects. HU That is, we assume that the field Z H .X / is again tc .r/ D 0. The wave function ‰ assumed to be a Hartree type product of spin orbitals i .x/ as in (6.1). Let us further assume that the remaining electron-interaction field E HU ee .r/ is conservative, so that the resulting effective Q-DFT Hartree Approximation field F HU .r/ is also conservative. (The case of a nonconservative field is discussed in a general manner in Section 6.4). That is, now F HU .r/ D E HU ee .r/;

(6.27)

HU .r/ D 0. The resulting S system differential equation and r  E HU ee .r/ D r  F is then   1 2 HU  r C v.r/ C vee .r/ i .x/ D i i .x/I i D 1; : : : ; N; (6.28) 2 HU .r/: where vHU ee .r/ is the work done in the field F

Z vHU ee .r/

r

D 1

F HU .r 0 /  d` 0 :

(6.29)

Employing (6.27), and writing E HU ee .r/ in terms of its Hartree E H .r/ and SIC E SIC .r/ components as in (6.10), we have that H SIC vHU ee .r/ D WH .r/ C WH .r/;

(6.30)

with WH .r/ given by (6.16), and WHSIC .r/ by (6.18). With the neglect of the Correlation-Kinetic energy TcH (since Z H tc .r/ D 0), the total energy within the Q-DFT Hartree Uncorrelated Approximation is then Z EHU D Ts C Z D Ts C

.r/v.r/dr C EH ee ;

(6.31)

.r/v.r/dr C EH C ESIC H ;

(6.32)

6.2 The Q-DFT of Hartree–Fock Theory

107

SIC where Ts , EH ee , EH , and EH , are defined as in (6.22)–(6.25) but for the orbitals and fields of this approximation. This expression is a rigorous upper bound to the Hartree theory energy EH of (6.21). The reason for this is that the expression (6.31) in terms of the fields is equivalent to the expectation value of the Hamiltonian HO of (2.1) taken with respect to the Q-DFT Hartree Uncorrelated Approximation wave function ‰ HU .X /. Since ‰ HU .X / is not the same as the Hartree theory wave function ‰ H .X /, one obtains a rigorous upper bound to the Hartree theory value. (Note that the expectation value of the Hamiltonian HO of (2.1) taken with respect to the QDFT of Hartree theory wave function ‰.X / of (6.1), would also be an upper bound to the true Hartree theory value EH because once again ‰.X / differs from ‰ H .X /.) For the application of the Q-DFT Hartree Uncorrelated Approximation to atoms, see Chap. 9.

6.1.2 Endnote As noted in the introduction to this chapter, Hartree theory is thought of as an independent particle theory because the wave function is a product of spin orbitals, and neither Pauli nor Coulomb correlations are considered. However, via the Q-DFT of Hartree Theory, it is possible to construct a model of noninteracting fermions whose density .r/ is the same as that of Hartree theory. (The focus on ensuring the equivalence of the density is, of course, because of the fundamental significance of this property.) The corresponding Hartree theory energy E H as determined via this model system, is then obtained via the expression of (6.20). The expression shows that there is a Correlation-Kinetic component TcH to the total energy E H . This implies that in Hartree theory there exist correlation contributions to the kinetic energy, and hence that the particles within this theory are in fact correlated and not truly independent. Recall, that in Hartree theory, the potential energy of each electron is different and not the same. Now in the Q-DFT Hartree Uncorrelated Approximation described above, these Correlation-Kinetic effects are ignored. Furthermore, the potential energy of each model fermion is also the same. Thus, the fermions in this approximation are independent in the rigorous sense of the word. As such, the Q-DFT Hartree Uncorrelated Approximation is the truly independent particle model.

6.2 The Q-DFT of Hartree–Fock Theory In Hartree–Fock theory, the wave function ‰ HF .X / is assumed to be a Slater determinant ˆfiHF .x/g of the spin orbitals iHF .x/. In the Slater-Bardeen [28, 29] interpretation of Hartree–Fock theory (see QDFT), the potential energy of each electron is the sum of the external potential energy v.r/, the Hartree potential energy WH .r/ due to the charge density of all the electrons, and the potential energy vx;i .r/

108

6 Ad Hoc Approximations Within Quantal Density Functional Theory

due to the charge of the orbital-dependent Fermi hole. Hence, as in Hartree theory, the potential energy of each electron is different, and as such, Hartree–Fock theory too can be thought of as an orbital-dependent theory. In the Q-DFT of Hartree–Fock theory, whereby the same density .r/ and energy E HF as that of Hartree–Fock theory is obtained, the wave function is also assumed to be a Slater determinant ˆf.x/g of spin-orbitals: 1 ‰.X / D ˆf.x/g D p deti .r j j /; N

(6.33)

where i .r/ D i .x/ D i .r/ i ./. The potential energy of each model fermion is the same, and is the sum of the external component v.r/ and the local effective electron-interaction potential energy vHF ee .r/. The S system differential equation is   1 2 HF  r C v.r/ C vee .r/ i .x/ D i i .x/I 2

i D 1; : : : ; N:

(6.34)

These orbitals differ from those of Hartree–Fock theory, so that the corresponding Dirac density matrices differ: s .rr 0 / D

XX 

¤

i .r/i .r 0 /

i

XX 

iHF .r/iHF .r 0 / D  HF .rr 0 /:

(6.35)

i

However, the diagonal matrix elements of these matrices are the same: s .rr/ D  HF .rr/ D .r/. HF The potential energy vHF .r/: ee .r/ is the work done in the conservative field F Z vHF ee .r/

r

D 1

F HF .r 0 /  d` 0 ;

(6.36)

where the effective Hartree–Fock field F HF .r/ is the sum of the electron-interaction HF E HF ee .r/ and the Correlation-Kinetic Z tc .r/ fields: HF F HF .r/ D E HF ee .r/ C Z tc .r/:

(6.37)

HF The work done vHF .r/ D 0. The field ee .r/ is path-independent since r  F HF E ee .r/ is obtained via Coulomb’s law from its quantal source, the Hartree–Fock theory pair-correlation density gHF .rr 0 /. Thus,

Z E HF ee .r/

D

g HF .rr 0 /.r  r 0 / 0 dr ; jr  r 0 j3

(6.38)

6.2 The Q-DFT of Hartree–Fock Theory

109

where g HF .rr 0 / D h‰ HF .X /jPO .rr 0 /j‰ HF .X /i=.r/ D .r 0 / C xHF .rr 0 /;

(6.39) (6.40)

and PO .rr 0 / the pair-correlation operator (2.33). Here xHF .rr 0 /, the nonlocal component of g HF .rr 0 / is the Hartree–Fock theory Fermi hole: xHF .rr 0 / D 

j HF .rr 0 /j2 : 2.r/

(6.41)

Employing (6.40) via (6.38), we can write HF E HF ee .r/ D E H .r/ C E x .r/;

(6.42)

where the Hartree field E H .r/ expression is the same as in (6.11), and where the HF 0 Pauli field E HF x .r/ due to the Fermi hole x .rr / is Z

xHF .rr 0 /.r  r 0 / 0 dr : jr  r 0 j3

E HF x .r/ D

(6.43)

The Correlation-Kinetic field Z HF tc .r/ is the difference between the S system and Hartree–Fock theory kinetic fields: HF Z HF .r/; tc .r/ D Z s .r/  Z

where Z s .r/ D

zs .rI Œs / .r/

and Z HF .r/ D

zHF .rI ΠHF / ; .r/

(6.44)

(6.45)

with zs .rI Œs / and zHF .rI Œ HF / the corresponding kinetic “forces.” These kinetic “forces” are derived from the noninteracting and Hartree–Fock theory kineticenergy-density tensors defined respectively, in terms of the density matrices s .rr 0 / and  HF .rr 0 / (see (2.54)). Once again, as the Hartree field E H .r/ component is stationary, the potential energy vHF ee .r/ may be written as  Z vHF .r/ D W .r/ C  H ee

r 1





0 HF 0 0 ; E HF .r / C Z .r /  d` x tc

(6.46)

where the Hartree potential energy WH .r/ is defined as in (6.16). For systems of HF symmetry such that the fields E HF x .r/ and Z tc .r/ are separately conservative, we can write HF HF vHF (6.47) ee .r/ D WH .r/ C Wx .r/ C Wtc .r/;

110

6 Ad Hoc Approximations Within Quantal Density Functional Theory

where WxHF .r/ and WtHF .r/ are, respectively, the work done in the force of the c HF HF fields E x .r/ and Z tc .r/: Z WxHF .r/ D 

1

Z

and .r/ D  WtHF c

r

r 1

0 0 E HF x .r /  d` ;

(6.48)

0 0 Z HF tc .r /  d` :

(6.49)

In terms of the fields, the total energy E HF in Hartree–Fock theory may be written as Z HF E HF D Ts C .r/v.r/dr C Eee C TcHF (6.50) Z (6.51) D Ts C .r/v.r/dr C EH C ExHF C TcHF ; where the expressions for Ts , the kinetic energy of the S system, and EH , the Hartree energy, are the same as in (6.22) and (6.24), respectively, and where in integral virial form Z HF Eee D .r/r  E HF (6.52) ee .r/dr; Z (6.53) ExHF D .r/r  E HF x .r/dr; Z 1 .r/r  Z HF (6.54) TcHF D tc .r/dr: 2 Again, the expressions for the individual components of the energy in terms of the corresponding fields is independent of whether or not the fields are conservative. Note that it is the presence of the Correlation-Kinetic component to both the HF electron–interaction potential energy vHF that ensures ee .r/ and the total energy E the equivalence to the Hartree–Fock theory density and energy. Also note that the S system expression for the Hartree–Fock theory energy E HF of (6.50) is not the expectation value of the Hamiltonian HO of (2.1) taken with respect to the Q-DFT of Hartree–Fock theory wave function ‰.X/ of (6.33). As this wave function differs from the Hartree–Fock theory wave function ‰ HF .X /, the expectation value of HO of (2.1) taken with respect to ‰.X / would be an upper bound to the Hartree–Fock theory value E HF .

6.2.1 The Q-DFT Pauli Approximation The Q-DFT Pauli (P ) Approximation is obtained from the Q-DFT of Hartree–Fock theory by neglecting the Correlation-Kinetic effects. That is, we assume the field

6.2 The Q-DFT of Hartree–Fock Theory

111

P Z HF tc .r/ D 0. The wave function ‰ .X / is again assumed to be a Slater deterP minant ˆ fi g of spin orbitals i .x/ as in (6.33). Let us further assume that the remaining electron-interaction field E P ee .r/ in this approximation is conservative, so that the effective field F P .r/ is also conservative. (The case of a nonconservative field is discussed in Section 6.4). Thus,

F P .r/ D E P ee .r/;

(6.55)

P with r  E P ee .r/ D 0 and r  F .r/ D 0. The resulting S system differential equation is then

  1  r 2 C v.r/ C vP .r/ i .x/ D i i .x/I ee 2

i D 1; : : : ; N

(6.56)

P where vP ee .r/ is the work done in the force of the field F .r/:

Z vP ee .r/ D 

r 1

F P .r 0 /  d` 0 :

(6.57)

P Employing (6.55), and writing E P ee .r/ in terms of its Hartree E H .r/ and Pauli E x .r/ components as in (6.42), we have P vP ee .r/ D WH .r/ C Wx .r/;

(6.58)

where WH .r/ is given by (6.16), and where WxP .r/ is the work done in the field EP x .r/: Z r 0 0 EP (6.59) WxP .r/ D  x .r /  d` : 1

Since the Correlation-Kinetic energy Q-DFT Pauli Approximation is

TcHF

is neglected, the total energy E P in the

Z E P D Ts C Z D Ts C

P .r/v.r/dr C Eee

(6.60)

.r/v.r/dr C EH C ExP ;

(6.61)

where Ts and EH are defined as in (6.22) and (6.24), respectively, and where the P electron-interaction Eee and Pauli ExP energies are Z P D Eee

.r/r  E P ee .r/dr;

(6.62)

.r/r  E P x .r/dr:

(6.63)

Z ExP D

112

6 Ad Hoc Approximations Within Quantal Density Functional Theory

These expressions for the energy components are again independent of whether or not the individual fields are conservative. The energy E P of the Q-DFT Pauli Approximation constitutes a rigorous upper bound to the Hartree–Fock theory value. The reason for this is that the expression for E P of (6.60) in terms of fields is equivalent to the expectation value of the Hamiltonian HO of (2.1) taken with respect to the wave function ‰ P .X / D ˆP fi g of this approximation. As ‰ P .X / differs from the Hartree–Fock theory wave function ‰ HF .X /, an upper bound is obtained.

6.2.2 Endnote In Hartree–Fock theory, only those electron correlations that arise due to the Pauli exclusion principle are considered. These correlations are manifested by the resulting exchange operator of the Hartree–Fock theory equations being an integral or nonlocal operator. Equivalently, from the Slater-Bardeen perspective, the potential energy of each electron is different. The kinetic energy T HF in Hartree–Fock theory, therefore, is that of a system of electrons correlated in this manner. Now, via the Q-DFT of Hartree–Fock theory, it is possible to construct a system of noninteracting fermions such that the equivalent density and energy are obtained. The kinetic energy Ts of these model fermions differs from T HF , the difference being the contribution of the Pauli-correlations. The Pauli-correlation contribution to the kinetic energy within Hartree–Fock theory can, therefore, be explicitly determined via the Q-DFT of Hartree–Fock theory: it is the Correlation-Kinetic energy TcHF of (6.54). In the Q-DFT Pauli Approximation, the correlation contribution to the kinetic energy is neglected. Thus, in this approximation, the resulting local effective electron-interaction potential energy function vP ee .r/ of (6.57) is representative solely of correlations due to the Pauli principle. In contrast, in the Kohn–Sham theory “exchange-only” approximation [30, 31], or equivalently the Optimized Potential Method (OPM) [32,33], (wherein one also assumes a local effective potential energy framework and a corresponding Slater determinantal wave function), the resulting electron-interaction potential energy function is representative not only of Pauli correlations but also of the lowest-order Correlation-Kinetic effects. (For a proof of this statement, see QDFT). This potential energy function, therefore, is different [34] from that of the potential energy vP ee .r/ of the Q-DFT Pauli Approximation. As a consequence of the additional correlations present, the upper bound on the energy obtained via the expectation of the Hamiltonian of (2.1) taken with respect to the OPM Slater determinant, is different and superior to that of the QDFT Pauli Approximation. The OPM electron-interaction potential energy can be expressed [35] as the work done in the sum of a Pauli and Correlation-Kinetic field (see QDFT).

6.3 Time-independent Quantal-Density Functional Theory

113

6.3 Time-independent Quantal-Density Functional Theory The in principle exact Q-DFT of when Pauli and Coulomb correlations, and Correlation-Kinetic effects, are all considered, is described in Chap. 3. The key equations are those for the S system local electron-interaction potential energy vee .r/, and for the total energy E, in which all these electron correlations are incorporated. To summarize, the S system differential equation that leads to the same density .r/ as that of the interacting system is   1 2  r C v.r/ C vee .r/ i .x/ D i i .x/I 2

i D 1; : : : ; N:

(6.64)

The potential energy vee .r/ is the work done in the force of the effective field F eff .r/ (see (3.67)): Z r vee .r/ D  F eff .r 0 /  d` 0 ; (6.65) 1

where F .r/ is the sum of its electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ components: F eff .r/ D E ee .r/ C Z tc .r/: (6.66) eff

Equivalently, in terms of the Hartree E H .r/, Pauli–Coulomb E xc .r/, Pauli E x .r/, and Coulomb E c .r/ components of E ee .r/, the effective field is the sum F eff .r/ D E H .r/ C E xc .r/ C Z tc .r/ D E H .r/ C E x .r/ C E c .r/ C Z tc .r/:

(6.67) (6.68)

The work done vee .r/ is path-independent. For systems of symmetry such that the fields E ee .r/ and Z tc .r/ are separately conservative, the potential energy vee .r/ may be written as the sum (see (3.73)) vee .r/ D WH .r/ C Wxc .r/ C Wtc .r/ D WH .r/ C Wx .r/ C Wc .r/ C Wtc .r/;

(6.69) (6.70)

where the Hartree WH .r/, Pauli–Coulomb Wxc .r/, Pauli Wx .r/, Coulomb Wc .r/, and Correlation-Kinetic Wtc .r/ components of the potential energy are each the work done in the corresponding fields. Each work done is path-independent. The total energy E is (see (3.60)) Z E D Ts C

.r/v.r/dr C Eee C Tc

(6.71)

.r/v.r/dr C EH C Exc C Tc

(6.72)

.r/v.r/dr C EH C Ex C Ec C Tc ;

(6.73)

Z D Ts C Z D Ts C

114

6 Ad Hoc Approximations Within Quantal Density Functional Theory

where Ts is the kinetic energy of the noninteracting fermions, the second term is the external potential energy, and Eee, EH , Exc , Ex , Ec , Tc are, respectively, the electron-interaction, Hartree, Pauli–Coulomb, Pauli, Coulomb, and CorrelationKinetic contributions. For the definitions of these components of the energy written in virial form in terms of their respective fields, the reader is referred to Chap. 3.

6.3.1 The Q-DFT Pauli–Coulomb Approximation The Q-DFT Pauli–Coulomb (PC) Approximation is obtained from the above description by neglecting the Correlation-Kinetic effects only in the electron-interaction potential energy vee .r/, i.e. by assuming the field Z tc .r/ D 0 in (6.65 - 6.66), but not in the total energy expression (6.71) so that Tc ¤ 0. The reason for the latter, as further explained below, is that the approximate energy obtained must constitute a rigorous upper bound to the true value. Thus, in this approximation, only Pauli and Coulomb correlations are accounted for in the local electron-interaction potential energy of the model fermions. However, the contribution of these correlations to the kinetic energy of the model fermions, viz. the Correlation-Kinetic effects, is explicitly considered. The approximate trial wave function ‰.X /, from which the pair-correlation C density g PC .rr 0 / and electron-interaction field E P ee .r/ are determined, may be a parameterized correlated wave function or a few-parameter correlated wave function functional of functions [36–38]. The functions of the wave function functional are determined by satisfaction of a physical constraint. One of the benefits of employing a wave function functional is that fewer parameters are required to achieve a given accuracy. If the wave function or wave function functional ‰.X/ is of a correlateddeterminantal type, then the orbitals of the Slater determinant can be determined self-consistently via the S system differential equation within this approximation. A correlated-determinantal wave function is a Slater determinant times one minus a correlated function. This wave function or functional is of the form Y ‰.X / D ˆfi g f1  f .r i r j /g; (6.74) i¤j

where ˆfi g is the Slater determinant of the orbitals i .x/ of the S system differential equation in this approximation, and f .ri rj / a spinless few parameter correlation function that depends explicitly on the electronic coordinates, and which is symmetric to a permutation of the coordinates of the particle pairs. Thus, the trial wave function ‰.X/ is antisymmetric in an interchange of the coordinates including spin of any two electrons. One also constructs the function f .ri rj / in such a manner that the coalescence conditions are satisfied. C Once again, let us assume that the field E P ee .r/ is conservative, so that the effecPC tive field F .r/ is conservative. (The case when these fields are not conservative is discussed in Sect. 6.4.) Thus, F PC .r/ D E PC ee .r/;

(6.75)

6.3 Time-independent Quantal-Density Functional Theory

115

with r  F PC .r/ D 0 and r  E PC ee .r/ D 0. The S system differential equation is 

 1 2 PC  r C v.r/ C vee .r/ i .x/ D i i .x/I 2

i D 1; : : : ; N;

(6.76)

PC .r/: where vPC ee .r/ is the work done in the force of the field F

Z r vPC .r/ D  F PC .r 0 /  d` 0 : ee

(6.77)

1

Employing (6.75) and writing the electron-interaction field E PC ee .r/ in terms of its Hartree E H .r/ and Pauli–Coulomb E PC xc .r/ components, we may write the potential energy vPC ee .r/ as PC vPC (6.78) ee .r/ D WH .r/ C Wxc .r/; where WH .r/, the Hartree potential energy, is given by the expression of (6.16), and PC where Wxc .r/ is the work done in the field E PC xc .r/: PC Wxc .r/

Z r 0 0 D  E PC xc .r /  d` :

(6.79)

1

To study the separate contributions of the Pauli and Coulomb correlations, the PC Fermi–Coulomb hole xc .rr 0 / of this approximation may be written as the sum PC 0 of its Fermi x .rr / and Coulomb cPC .rr 0 / hole components (see 3.28). These PC holes then give rise to the Pauli E PC x .r/ and Coulomb E c .r/ fields. Thus, one may write PC Wxc .r/ D WxPC .r/ C WcPC .r/; (6.80) where WxPC .r/; WcPC .r/ are, respectively, the work done in the fields E PC x .r/ and E PC .r/: c WxPC .r/

Z r 0 0 D  E PC x .r /  d` 1

and

WcPC .r/

Z r 0 0 D  E PC c .r /  d` : (6.81) 1

Since in this approximation the trial wave function ‰.X/ is correlated – possibly a correlated-determinantal wave function or wave function functional – there are Pauli and Coulomb correlation contributions to the kinetic energy. Hence, in determining the total energy of the model S system fermions, one must include the corresponding Correlation-Kinetic energy TcPC . The total energy in the Q-DFT Pauli–Coulomb Approximation is thus Z E

PC

D Ts C Z D Ts C Z D Ts C

PC .r/v.r/dr C Eee C TcPC

(6.82)

PC .r/v.r/dr C EH C Exc C TcPC

(6.83)

.r/v.r/dr C EH C ExPC C EcPC C TcPC ;

(6.84)

116

6 Ad Hoc Approximations Within Quantal Density Functional Theory

where Ts and EH are defined as in (6.22) and (6.24), respectively, and where in terms of the individual fields Z PC Eee D .r/r  E PC (6.85) ee .r/dr Z PC Exc

D

.r/r  E PC xc .r/dr

(6.86)

.r/r  E PC x .r/dr

(6.87)

.r/r  E PC c .r/dr

(6.88)

.r/r  Z PC tc .r/dr;

(6.89)

Z ExPC D Z EcPC

D

and TcPC D

1 2

Z

with Z PC tc .r/ the Correlation-Kinetic field in this approximation: PC PC .r/: Z PC tc .r/ D Z s .r/  Z

(6.90)

PC 0 Here the kinetic field Z PC s .r/ is determined from the Dirac density matrix s .rr / PC formed by the orbitals of (6.76) (see 3.39), and the kinetic field Z .r/ from the density matrix  PC .rr 0 / formed from the trial correlated wave function ‰.X / (see 2.54). The procedure whereby the properties are obtained via the S system is as follows. Consider, for example, that the trial wave function is of the correlated-determinantal form. For an initial choice of parameters in the trial wave function ‰.X /, together with an initial set of orbitals i .x/ of the Slater determinant ˆfi g, the field PC PC PC PC E PC ee .r/, its components E H .r/, E xc .r/, E x .r/, E c .r/, and the field Z tc .r/ PC PC are first determined. (Note that Z .r/, Z s .r/ are obtained from ‰.X/, ˆfi g, respectively.) From these fields the corresponding total energy E PC of (6.82) is obtained. The electron interaction potential energy vPC ee .r/ is determined next via (6.75) and (6.77). The S system differential equation (6.76) is then solved to obtain self-consistently a new set of orbitals i .x/, leading to a new Slater determinant ˆfi g, and hence a new wave function ‰.X /. This then leads to a new set of fields and a new total energy. The parameters of the wave function are then varied, and the self-consistent procedure repeated, and this process is continued till the total energy is minimized. The calculational procedure is thus a variational self-consistent one. As all the terms in the total energy expression of (6.82) are considered, which is equivalent to the expectation of the Hamiltonian HO of (2.1) taken with the wave function ‰.X/, the upper bound on the energy is rigorous.

6.3 Time-independent Quantal-Density Functional Theory

117

6.3.2 The Q-DFT Fully Correlated Approximation In contrast to the Q-DFT Pauli–Coulomb Approximation, in the Q-DFT Fully Correlated Approximation, Pauli and Coulomb correlations, and Correlation-kinetic effects, are all considered, albeit approximately, in both the corresponding S system electron-interaction potential energy vee .r/ as well as in the expression for the total energy E. Thus, this approximation corresponds to the Quantal Density Functional Theory description of Sect. 6.3, but with an approximate trial correlated wave function or wave function functional ‰.X / employed instead. The equations of this approximation are the same as those of Sect. 6.3, and hence will not be repeated with any superscript. To understand how this approximation may be invoked in practice, say for an atomic system, let us consider the trial wave function ‰.X / to be a parametrized correlated-determinantal wave function functional ‰Œ  of the form of (6.74) with the dependence on the functions being incorporated into the correlation factor f .r i r j I /. For such a choice of approximate wave function functional, and for specific values of the variational parameters, the function is determined [36–38] such that ‰Œ  satisfies a constraint such as normalization, or reproduces the experimentally determined value of an observable such as the diamagnetic susceptibility, etc. Next the quantal sources – the pair-correlation density g.rr 0 /, the density matrices .rr 0 /, s .rr 0 /, and the density .r/ – are determined, from which are obtained the corresponding electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ fields, and thereby the potential energy vee .r/ and the total energy E. The S system differential equation is then solved self-consistently to obtain the orbitals i .x/ and the highest occupied eigenvalue m . (The exact highest occupied eigenvalue m is the negative of the ionization potential (see Sect. 3.7).) This leads to a new Slater determinant ˆfi g, and a new wave function functional ‰Œ , and the procedure repeated for a new set of parameters till the energy E is minimized. Since all terms in the energy expression (6.71) are considered, and this is equivalent to the expectation of the Hamiltonian HO of (2.1) taken with the wave function ‰Œ , the bound on the true energy is rigorous. A flow chart of this variational-self-consistent procedure is detailed in Fig. 6.1. The calculational procedure described earlier is the same for both ground and excited states. Orthogonality of the approximate excited and ground state wave functions can be insured. For those cases where the excited state cannot be defined via a single determinant S system, Slater’s diagonal sum rule [39] may be applied. For excited states, the self-consistent procedure may also be employed in conjunction with a theorem due to Theophilou [40]. According to the theorem, if 1 ; 2 ; : : : m , are the orthonormal trial functions for the m lowest eigen states of the Hamiltonian HO of (2.1), having exact eigenvalues E1 ; E2 ; : : : Em , then m X h i D1

O

i jH j

ii



m X i D1

Ei :

(6.91)

118

6 Ad Hoc Approximations Within Quantal Density Functional Theory

Φ{φi (x)}

Ψ[χ] Determine χ:

ò ρc (rr¢) dr ¢ = 0 or ò ρ (r) dr = N

g(rr¢)

Eee

ee

γ(rr¢) ρ(r)

γs(rr¢) Z tc

Tc

υee(r) Eee + Tc

φi (r), εm E Fig. 6.1 Flow chart for the variational-self-consistent procedure in the Q-DFT Fully Correlated Approximation

In this way, a rigorous upper bound to the sum of the ground and excited states is achieved. With the ground state energy known, a rigorous upper bound to the excited state energy is then determined, while simultaneously a physical constraint or sum rule is satisfied or an observable obtained exactly. There are three additional points worth noting about the above variational-selfconsistent procedure. The first is that all the electron correlations – Pauli, Coulomb, and Correlation-Kinetic – are implicitly incorporated in the orbitals i .x/ of the self-consistently determined Slater determinant ˆfi g. This is because the local electron-interaction potential energy vee .r/ that generates these orbitals is representative of these correlations. Thus, although the total energy obtained via the Slater determinant ˆfi g by itself will not be superior to that of Hartree–Fock theory, the energy obtained via the resulting wave function functional ‰Œ  will be superior. Second, the densities obtained via the Slater determinant ˆfi g and the wave function functional ‰Œ  are the same. Finally, the wave function functional ‰Œ  thus determined will be accurate throughout space. For example, it is accurate in the classically forbidden region of an atom because, as proved in Chap. 7 (see also Sect. 3.7), the asymptotic structure of the electron-interaction potential energy vee .r/ is known there exactly. (As a consequence, the highest occupied eigenvalue m , and hence

6.4 The Case of Nonconservative Fields

119

the ionization potential is determined accurately.) The functional ‰Œ  is accurate in the interior regions of the atom because that is where the principal contributions to the energy arise, and the variational principle is being applied. These accuracies, of course, all feed back on each other in the variational-self-consistent formulation.

6.4 The Case of Nonconservative Fields In the in principle exact formulation of Q-DFT with all the electron correlations present, the effective field F eff .r/ of (6.66) is conservative. This is because the sum of its electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ components is conservative. As such the electron-interaction potential energy vee .r/ of (6.65) is path-independent. This is the case irrespective of the symmetry of the system. There are, of course, systems of symmetry such that the individual electron-interaction and Correlation Kinetic fields are separately conservative. In such cases, the different components of the potential energy vee .r/ are individually path-independent. When an approximate wave function is employed, the appropriate system symmetries ought to be incorporated in it such that the corresponding effective field is also conservative. For systems of symmetry such that the fields E ee .r/ and Z tc .r/ are not separately conservative, the neglect of say the latter in an approximation such as the Q-DFT Pauli–Coulomb Approximation, then leads to an effective field F eff .r/  E ee .r/ that is not conservative. Hence, the resulting work done vee .r/ in this field is not path-independent. As such this work done can no longer be interpreted as a potential energy. In such an approximation, how then does one ensure that the work done vee .r/ employed in the S system differential equation is path-independent, and therefore a potential energy in the rigorous sense? The following Sects. 6.4.1 and 6.4.2 describe two ways by which this may be achieved.

6.4.1 The Central Field Approximation For purposes of explanation, let us consider the Q-DFT Pauli Approximation of Sect. 6.2.1 applied to atoms as an example. In this approximation, Coulomb correlation and Correlation-Kinetic effects are ignored. Hence, the effective field F P .r/, which is then equivalent to the corresponding electron-interaction field E ee .r/ (see 6.55), is the sum of its Hartree E H .r/ and Pauli E P x .r/ components. As the Hartree field is conservative, and the work done WH .r/ in this field path-independent, it is the Pauli component E P x .r/ that must be modified such that the resulting field is conservative. In the central field model of atoms, as for example in the Hartree–Fock theory calculations [8], the electrons are assumed to move in a central potential. Thus, the electronic wave functions may be written as

120

6 Ad Hoc Approximations Within Quantal Density Functional Theory

‰nlm .r/ D Rnl .r/Ylm ./;

(6.92)

where Rnl .r/ is the radial part of the wave function, and Ylm ./, the angular part, is the spherical harmonic of order (lm). One way to achieve the Central Field Approximation within the Q-DFT Pauli Approximation is by spherically averaging the corresponding Dirac density matrix s .rr 0 / over the coordinates of the electrons of a given orbital-angular-momentum quantum number. For open-shell atoms, this is equivalent to spherically averaging the radial component of the Pauli field E P x .r/ due to the Fermi hole charge x .rr 0 /. In this manner, the resulting field E x;r .r/, and corresponding work done Wx .r/ in this field, are spherically symmetric. For closed-subshell atoms, this is automatically the case. The spherical average of the radial component of the Pauli field E P x .r/ is EP x;r .r/

1 D 4

Z

x .rr 0 /

@ 1 dr 0 d r : @r jr  r 0 j

(6.93)

Since the curl of this field vanishes, the spherically symmetric work done Wx .r/ is given by the integral Z r

WxP .r/ D 

1

0 0 EP x;r .r /dr :

(6.94)

This work done is path-independent, and can thus be rigorously interpreted as the potential energy of the model fermions due to the Pauli exclusion principle. In the application of the Q-DFT Pauli Approximation to atoms [24] described in Chap. 10, it is the central field model achieved by spherically averaging the Pauli field that is employed. In the application of Hartree and Hartree–Fock theories to atoms [1, 2, 4, 8, 9], the central field model is accomplished by spherically averaging the corresponding single-particle orbitals. As such, a second way of accomplishing the central field model within Q-DFT is by averaging the orbitals i .x/ of the S system differential equation. The corresponding Pauli field E x .r/, for example, then only has a radial component Ex i r , and this component then only depends on the radial coordinate r W E x .r/ D Ex .r/i r . This field is spherically symmetric, and hence curl free. The corresponding work done in this field is also spherically symmetric and hence pathindependent. As such the work done can be rigorously interpreted as a potential energy. In the application of the Q-DFT Hartree Uncorrelated Approximation to atoms [4] described in Chap. 9, it is the central field model constructed by spherically averaging the orbitals i .x/ that is employed. The idea of averaging the three-dimensional fields or orbitals such that the resulting fields are conservative may also be applied to systems of a different symmetry. Consider, for example, the case of the nonuniform electron gas at the metal–vacuum interface. Let us further assume that one is interested in properties at this interface in the direction perpendicular to the surface. In the jellium model of a metal surface [29], there is translational symmetry in the plane parallel to the surface. The com-

6.4 The Case of Nonconservative Fields

121

ponents of the orbitals in this plane are plane waves. As such, all the fields at the interface depend only upon the coordinate in the perpendicular direction. The fields are thus curl free, and therefore conservative. The work done in these fields is consequently path-independent [41, 42]. The application of Q-DFT to the metal–vacuum interface is described in Chap. 17. On the other hand, consider a more realistic surface that is corrugated, and represented by a periodic potential in the plane parallel to the surface. Then a conservative field in the direction perpendicular to the surface can be achieved by taking the planar average of either the three-dimensional orbitals or the sources or the fields. In this manner, once again, the work done in these fields in the plane perpendicular to the surface is path-independent, and thus the interpretation of this work as a potential energy is rigorous.

6.4.2 The Irrotational Component Approximation To explain this approximation, let us consider the Q-DFT Pauli–Coulomb Approximation of Sect. 6.3.1 in which the Correlation-Kinetic effects are ignored in the effective electron-interaction potential energy vee .r/ of the S system. (In the discussion and expressions given later, the superscript PC to indicate the level of electron correlation assumed, is dropped.) The effective field (see 6.65, 6.75) is then the electron-interaction field: F eff .r/  E ee .r/. We consider a symmetry for which the latter is not conservative. As the Hartree component E H .r/ of E ee .r/ is conservative, it is the Pauli–Coulomb component E xc .r/ that is then not conservative. The Irrotational Component Approximation is based on Helmholtz’s theorem [43]. According to the first part of the theorem, the most general vector field has both a nonzero divergence and a nonzero curl, and can be derived from the negative gradient of a scalar potential and the curl of a vector potential. We can, therefore, write the Pauli–Coulomb field E xc .r/, due to the Fermi–Coulomb hole charge xc .rr 0 / of a system for which the curl of the field does not vanish, as the sum of its irrotational E Ixc .r/ and solenoidal E Sxc .r/ components: E xc .r/ D E Ixc .r/ C E Sxc .r/ I .r/ C r  A Sxc .r/; D r Wxc

(6.95) (6.96)

where I .r/; E Ixc .r/ D r Wxc

(6.97)

Dr

(6.98)

E Sxc .r/

A Sxc .r/;

I .r/, A Sxc .r/ are, respectively, the Pauli–Coulomb scalar and vector and where Wxc potentials.

122

6 Ad Hoc Approximations Within Quantal Density Functional Theory

The second part of the mathematical statement of Helmholtz’s theorem in this case is that the field E xc .r/ may also be written as Z E xc .r/ D r

r 0  E xc .r 0 / 0 dr C r  4 jr  r 0 j

Z

r 0  E xc .r 0 / 0 dr ; 4 jr  r 0 j

(6.99)

where we have assumed that the surface of the system is at infinity, and that the field E xc .r/ vanishes there. Hence, there are no surface source contributions. Of course, if the system is such that the surface is not at infinity, sources will occur on the surface, and must be accounted for. A comparison of the two statements (6.96) and (6.99) of Helmholtz’s theI orem shows that the scalar Pauli–Coulomb potential energy Wxc .r/, which is path-independent, is Z eff 0 xc .r / 0 I Wxc .r/ D dr ; (6.100) jr  r 0 j eff and arises from a static scalar effective Fermi–Coulomb hole charge xc .r/ given by eff xc .r/ D

1 r  E xc .r/: 4

(6.101)

eff .r/ can in turn be expressed solely in terms of the The effective hole charge xc Fermi–Coulomb hole charge xc .rr 0 / as follows. The Pauli–Coulomb field E xc .r/ due to the charge xc .rr 0 / is derived via Coulomb’s law as (see 2.49)

Z

xc .rr 0 /.r  r 0 / 0 dr ; jr  r 0 j3

E xc .r/ D and may be rewritten as

Z

xc .rr 0 /r

E xc .r/ D 

(6.102)

1 dr 0 ; jr  r 0 j

(6.103)

since .r  r 0 /=jr  r 0 j3 D r .1=jr  r 0 j/. Thus Z

xc .rr 0 /r 2

r  E xc .r/ D 

1 dr 0 C jr  r 0 j

Z

r xc .rr 0 / 

.r  r 0 / 0 dr : (6.104) jr  r 0 j3

r xc .rr 0 / 

.r  r 0 / 0 dr ; (6.105) jr  r 0 j3

Using r 2 .1=jr  r 0 j/ D 4 ı.r  r 0 /, we have Z r  E xc .r/ D 4

xc .rr 0 /ı.r  r 0 /dr 0 C

Z

eff of (6.101) may be expressed in terms of xc .rr 0 / as so that the effective charge xc eff xc .r/ D xc .rr/ C

1 4

Z

r xc .rr 0 / 

.r  r 0 / 0 dr : jr  r 0 j3

(6.106)

6.4 The Case of Nonconservative Fields

123

As the total charge of the Fermi–Coulomb hole is negative unity (see (2.41)), so is eff the R total chargeHof the effective Fermi–Coulomb hole xc .r/. Applying Gauss’s law ( r  C dr D S C  dS ) to (6.101) we have Z 1 r  E xc .r/dr 4

Z 1 E xc .r/  dS : D 4

Z

eff xc .r/dr D

(6.107) (6.108)

For finite systems such as atoms and molecules, the structure of the field E xc .r/  for asymptotic positions of the electron is E xc .r/ r!1  1=r 2 (see QDFT). Substituting this into (6.108) gives Z

1 4

D 1:

eff xc .r/dr D 

Z

 0

Z

2 0

1  r 2 sin dd r2

(6.109) (6.110)

The irrotational component E Ixc .r/ of (6.95) can also be obtained directly from the eff effective charge xc .r/ via Coulomb’s law: Z E Ixc .r/ D

eff xc .r/.r  r 0 / 0 dr : .r  r 0 /3

(6.111)

As a consequence of the sum rule of (6.110), and the fact that the effective charge eff xc .r/ is static, it is evident that the asymptotic structure of E Ixc .r/ for finite sysI tems is also 1=r 2 . As such, the asymptotic structure of the potential energy Wxc .r/ must be 1=r. In the Irrotational Component Approximation, one employs the irrotational component E Ixc .r/ of the nonconservative field E xc .r/ to determine the pathI independent potential energy Wxc .r/. It is this potential energy that is then employed in the corresponding S system differential equation which is   1 2 I  r C v.r/ C WH .r/ C Wxc .r/ i .x/ D i i .x/; 2

(6.112)

I .r/ are determined self-consistently by solution of the differwhere WH .r/ and Wxc ential equation (6.112). The total energy E can then be determined as described in Sect. 6.3.1. The value obtained will be a rigorous upper bound. Let us next discuss the solenoidal component E Sxc .r/ of the non conservative field E xc .r/. According to (6.98), it is the curl of the Pauli–Coulomb vector potential A Sxc .r/. A comparison of (6.96) and (6.99) shows that this vector potential is

Z A Sxc .r/

D

J xc .r 0 / 0 dr ; jr  r 0 j

(6.113)

124

6 Ad Hoc Approximations Within Quantal Density Functional Theory

and due to a Pauli–Coulomb vector vortex source J xc .r/ given as J xc .r/ D

1 r  E xc .r/: 4

(6.114)

The vector vortex source J xc .r/ may also be expressed solely in terms of the Fermi–Coulomb hole charge xc .rr 0 /. Using the vector identities r   C D r  C C r  C ; r  r  D 0, and the expressions (6.102) and (6.103) for E xc .r/, we have Z 1 .r  r 0 / 0 Œr xc .rr 0 /  J xc .r/ D dr : (6.115) 4

jr  r 0 j3 Again, employing the above vector identities and expressions for E xc .r/, in (6.98), the solenoidal component E Sxc .r/ can be obtained directly from the vector vortex source as Z .r  r 0 / 0 E Sxc .r/ D J xc .r 0 /  dr : (6.116) jr  r 0 j3 Although within this approximation, it is only the irrotational component E Ixc .r/ that plays a role, nonetheless, it is important to also determine the solenoidal component E Sxc .r/ for the following reasons. First, a comparison between the irrotational and solenoidal field components allows for an estimation of the accuracy of the approximation. For example, if the solenoidal component is negligible in comparison to the irrotational component, then the many-body effects are esseneff tially all accounted for by the effective charge xc .r/. As such the results obtained by employing just the irrotational component will be accurate. Preliminary studies described in Chap. 12 indicate the solenoidal component to be many orders of magnitude smaller, thereby providing a justification for the Irrotational Component Approximation. Secondly, there are symmetry directions and regions of space in which the vector vortex source J xc .r/ may vanish. In these directions and regions I then, the potential energy Wxc .r/ is the same as Wxc .r/, the work done in the field E xc .r/ due to the Fermi–Coulomb hole charge xc .rr 0 /. The work Wxc .r/ is consequently path-independent.

Chapter 7

Analytical Asymptotic Structure in the Classically Forbidden Region of Atoms

In the application of a theory, it is always best to perform analytically as much of the calculation as possible. This is because (a) one obtains a result that is rigorous; (b) the derivation can lead to physical insights; (c) it obviates the need for numerical work in those regions of space where the analytical results are valid; (d) the analytical results help in confirming the correctness of the numerical component of the calculation by the requirement that it match smoothly with the analytical expressions, and that they in fact reproduce the results of these expressions; and (e) the analytical expressions help in determining the accuracy of the numerical work, which by its nature is always approximate. In this chapter, we apply Q-DFT to atoms to derive [1–3] the analytical structure of the local electron-interaction potential energy vee .r/ of the model S system (see (3.3), (3.4), (3.67)), and of its Hartree, Pauli, Coulomb, and Correlation-Kinetic components, in the asymptotic classically forbidden region. To do so, we derive the asymptotic structure of the interacting system wave function to terms including the quadrupole moment contribution, and thereby that of the corresponding single particle density matrix, the density, and pair-correlation density to this order. For specificity, let us consider systems for which the N -electron atom may be orbitally nondegenerate or degenerate, but the .N  1/-electron ion is always orbitally nondegenerate except for the twofold spin degeneracy. Examples of such systems are the B and Mg atoms and their ions. For the case when both the N - and .N  1/-electron systems are nondegenerate, the atom and its ion are spherically symmetric. For such systems, the electron-interaction potential energy vee within Q-DFT is then (see (3.73)) vee .r/ D WH .r/ C Wx .r/ C Wc .r/ C Wtc .r/;

(7.1)

where WH .r/, Wx .r/, Wc .r/, Wtc .r/, are its Hartree, Pauli, Coulomb, and Correlation-Kinetic potential energy components, respectively. In the asymptotic limit as r!1, it is proved that N ; r 1 Wx .r!1/ D  ; r

WH .r!1/ D

(7.2) (7.3) 125

126

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

˛ ; 2r 4 80 Wtc .r!1/ D ; 5r 5 Wc .r!1/ D 

(7.4) (7.5)

where ˛ is the ground-state polarizability of the .N  1/-electron ion, 02 =2 the ionization potential energy, and an expectation of the .N  1/-electron ion. Thus, the asymptotic structure of the potential energy vee .r/ is vee .r!1/ D

1 ˛ N 80   4C : r r 2r 5r 5

(7.6)

The derivations of the above analytical results via Q-DFT lead to the following understandings: 1. The asymptotic structure of N=r of the Hartree potential energy WH .r/ arises from the static electron density source .r/, and is a consequence of charge conservation, i.e. the electron density integrates to N , the number of electrons. 2. The Pauli potential energy Wx .r/ asymptotic structure of .1=r/ is solely due to Pauli correlations, and obtained directly from the Pauli field of the S system Fermi hole charge x .rr 0 /. This structure may also be understood from a physical perspective [4, 5] by the fact that the total charge of the Fermi hole is negative unity, and that for asymptotic positions of the electron, the Fermi hole becomes an essentially static charge distribution (see QDFT). 3. The O.1=r 4/ term obtained through Wc .r/ is solely due to quantummechanical Coulomb correlations, and results directly from the field of the Coulomb hole charge c .rr 0 /. For asymptotic positions of the electron, the Coulomb hole is also an essentially static charge distribution (see QDFT). However, the total charge of the Coulomb hole is zero. This explains why the Coulomb potential energy Wc .r/ decays more rapidly than the Pauli potential energy Wx .r/. 4. There are no O.1=r 5 / contributions due to quantum-mechanical Coulomb correlations. 5. The O.1=r 5 / contribution is solely due to Correlation-Kinetic effects as determined through Wtc .r/. In atoms, therefore, Correlation-Kinetic effects are very short-ranged. 6. As opposed to the Pauli and Coulomb potential energies that decay asymptotically as negative functions, the Correlation-Kinetic potential energy decays as a positive function. 7. The far asymptotic structure of vee .r/ is then .N  1/=r, and governed solely by the Hartree electrostatic term and Pauli correlations. 8. The highest occupied eigenvalue m of the S system differential equation is governed principally by the asymptotic structure of vee .r/. In the far asymptotic region as noted in (7), vee .r/ depends only on the sum of WH .r/ and Wx .r/. Now in the Q-DFT Pauli Approximation of Sect. 6.2.1, the electron-interaction potential energy is vP ee .r/ D WH .r/ C Wx .r/. In this approximation then, the asymptotic structure of vP ee .r/ is the same as vee .r/ of the fully correlated system. Thus,

7.1 The Wave Function

127

accurate values for the highest occupied eigenvalue, and hence of the ionization potential, are obtained in the Q-DFT Pauli Approximation (see Chap. 10). For the case when the N -electron atom is orbitally degenerate, there are higher order contributions to both the Hartree WH .r/ and Pauli Wx .r/ potential energies that are the same but with opposite signs. These higher order contributions are WH ; Wx .r!1/ D    ˙

Q R ˙ 5; r3 r

(7.7)

where the .˙/ signs are for the Hartree and Pauli terms, respectively, and where Q and R are multipole moments of the density. Thus, the asymptotic structure of vee .r!1/ to O.1=r 5 / remains the same as in (7.6). The physics of the other terms to this order remain unchanged. In the following sections we derive the classically forbidden region asymptotic structure of the interacting system ground state wave function .X /, the single-particle density matrix .rr 0 /, the density .r/, the pair-correlation density g.rr 0 /, and thereby that of the Hartree WH .r/, Pauli Wx .r/, Coulomb Wc .r/, and Correlation-Kinetic Wtc .r/ potential energy components of the S system electroninteraction potential energy vee .r/. In Sect. 7.6, we discuss how the above results derived via Q-DFT lead to insights into the work of others done within the framework of traditional density functional theory.

7.1 The Wave Function The N -electron interacting system Hamiltonian of (2.1) may be rewritten as 1 HO D  r 2 C v.r/ C 2

N X i D2

1 C HO .N 1/ ; jr  r i j

(7.8)

where the .N  1/-electron Hamiltonian HO .N 1/ is 1 HO .N 1/ D  2

N X

ri2 C

i D2

N X

v.r i / C

i D2

N 1 X 1 : 2 jr i  r j j

(7.9)

i ¤j ¤1

The complete set of eigenfunctions and eigen energies of the .N  1/-electron system are defined by the Schr¨odinger equation HO .N 1/

.N 1/ .X N 1 / s

D Es.N 1/

.N 1/ .X N 1 /: s

We next expand the ground state wave function of the N -electron system (2.5) in terms of the eigenfunctions s.N 1/ .X N 1 /: .r; X N 1 / D

X s

Cs .r/

.N 1/ .X N 1 /: s

(7.10) .X / of

(7.11)

128

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

The Schr¨odinger equation (2.5) for the ground state may then be rewritten as 

X 1 1X 2 1  ri  r 2 C v.r/ C 2 jr  r i j 2 N

N

i D2

N X

C

v.r i / C

i D2

1 2

N X i ¤j ¤1

i D2

1 jr i  r j j

X

Cs .r/

.N 1/ .X N 1 / s

s

X D E0 Cs .r/

.N 1/ .X N 1 /; s

(7.12)

s

where E0 is the N -electron ground state energy. For asymptotic positions of the electron, we have by Taylor expansion 1 ri  r 1X 1 @2 1 D C 3 C C : ri ˛ riˇ jr  r i j r r 2 @r˛ @rˇ r

(7.13)

˛;ˇ

On substituting this expansion into (7.12) we have 

 N  1 N  1 X ri  r 1X @2 1 C  r 2 C v.r/ C C r r i ˛ iˇ 2 r r3 2 @r˛ @rˇ r i D2 ˛;ˇ X X X  Cs .r/ s.N 1/ C HO N 1 Cs .r/ s.N 1/ D E0 Cs s.N 1/ ;(7.14) s

s

s

which reduces further to   1 N 1 X  r 2 C v.r/ C Cs .r/ s.N 1/ 2 r s   N X ri  r 1X @2 1 C C r r i ˛ iˇ r3 2 @r˛ @rˇ r i D2 ˛;ˇ X  Cs .r/ s.N 1/ D

X

s

ŒE0  EsN 1 Cs .r/

.N 1/ : s

(7.15)

s 1/? Multiplying (7.15) by s.N .X N 1 / from the left, integrating over 0 and employing the orthonormality condition

h

.N 1/ j s0

.N 1/ i s

D ıss 0 ;

R

dX N 1 ,

(7.16)

7.1 The Wave Function

129

we have 

 1 N 1 r X Cs .r/ C 3   r 2 C v.r/ C Cs 0  .r/P ss 0 2 r r 0 s

C

1X 2

˛;ˇ

where P ss 0 D

2

1X

@ @r˛ @rˇ r N Z X

Cs 0  .r/.Qss 0 /˛ˇ D ŒE0  EsN 1 Cs .r/; (7.17)

s0

.N 1/? .X N 1 /r i s

.N 1/ .X N 1 /dX N 1 s0

(7.18)

i D2

and .Qss 0 /˛ˇ D

N Z X

.N 1/?

.X N 1 /ri ˛ riˇ

.N 1/ .X N 1 /dX N 1 : s0

(7.19)

i D2

Here P ss 0 is the dipole moment and .Qss 0 /˛ˇ is the quadrupole moment tensor of the .N  1/-electron system. With the definitions Dss 0 .r/ O D rO  P ss 0 ; and O D Qss 0 .r/

(7.20)

1X .3rO˛ rOˇ  ı˛ˇ /.Qss 0 /˛ˇ ; 2

(7.21)

˛;ˇ

with rO D r=r the unit vector, (7.17) becomes  1 2 N 1 Cs .r/  r C v.r/ C 2 r  X  1X 1 C 2 Dss 0 .r/ O C 3 Qss 0 .r/ O Cs 0  .r/ D s Cs .r/; r 0 r 0 

s

where

(7.22)

s

s D E0  Es.N 1/

(7.23)

is the negative of the ionization potential Is from the ground state of the N -electron atom to various states of the .N  1/-electron ion. In the asymptotic region as r ! 1, (7.22) reduces to    X  1 2 1X 1 0 0  r  s / Cs .r/ D  2 Dss .r/ O C 3 Qss .r/ O Cs 0  .r/: (7.24) 2 r 0 r 0 s

s

130

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

Now, asymptotically, the ground state wave function 0.N 1/ .X N 1 / of the .N 1/p electron system, decays as exp.0 r/ where 0pD 20 , and the states s.N 1/ .X N 1 / for s ¤ 0 as exp.s r/ where s D 2s (see QDFT). Also s > 0 for s ¤ 0. Therefore, the decay of all the s.N 1/ .X N 1 / will be faster than that of .N 1/ .X N 1 /. Hence, the only term contributing to the right- hand side of (7.24) 0 is that for s 0 D 0, and the decay of all the Cs .r/ is governed by C0 .r/. Thus, in the asymptotic region, (7.24) reduces to   1  r 2  s Cs .r/ .0  s /Cs .r/ 2   1 1 D  2 Ds0 .r/ O C 3 Qs0 .r/ O Cs .r/; r r

(7.25)

and the asymptotic structure of the coefficients Cs .r/ for s ¤ 0 is then   1 1 C0 .r/ Cs .r/ D  2 Ds0 .r/ O C 3 Qs0 .r/ O ; r r ws .N 1/

(7.26)

.N 1/

where ws D 0  s D Es  E0 is an excitation of the .N  1/-electron system. The asymptotic structure of the wave function to 0.1=r 3/ from (7.11) and (7.26) is therefore  X  1 1 C0 .r/ N 1 .N 1/ .r; X  2 Ds0 .r/ /D O C 3 Qs0 .r/ O .X n1 /: s r r w s s (7.27)

7.2 The Single-Particle Density Matrix and Density On substituting (7.11) and (7.27) into (2.19), and employing the orthonormality condition of (7.16), we obtain the asymptotic structure of the single-particle density matrix .rr 0 / as .rr 0 / D N

XX 

DN

X

? Cs .r/Cs .r 0 /

s



? C0 .r/C0 .r 0 /

 D ? .r/ O C s02 r

 ? 0 X 1 Ds0 .r/ O Ds0 .rO 0 / 1C w2s r2 r 02 s

Q? .r/ Q? .r/ Qs0 .rO 0 / O Ds0 .rO 0 / O Qs0 .rO 0 / C s03 C s03 3 2 r r r0 r0 r 03

 : (7.28)

(Here

P0

s

D

P

s¤0 .)

7.2 The Single-Particle Density Matrix and Density

131

The dipole-quadrupole cross terms of (7.28) can be shown to vanish by rewriting each term as 0 ? X .rO 0 / 1 1 X 0 0 1 Ds0 .r/ O Qs0 D  rO˛ rOˇ r w2s r 2 r 2 r 03 r 03 s ˛ˇ ˇ ˇ  ˇ ˇ 1  PO .N 1/ ˇ O d ˇ 0 ˇqO˛ˇ O .N 1/ 2  ˇ ŒH  E 

.N 1/ 0

 ;(7.29)

0

R R where d D rı .r/dr, O O ı .r/ O D .r/ O h 0.N 1/ j.r/j O qO ˛ˇ D r˛ rˇ ı .r/dr, P .N 1/ .N 1/ .N 1/ O i, .r/ O D i ı.r  r i /; and P D j 0 ih 0 j is the projector onto 0 the .N  1/-electron ground state. (Note that the dipole d and quadrupole qO˛ˇ moment operators are the same as (7.18) and (7.19) since the second term of the operator ı .r/ O does not contribute on account of the fact that the .N  1/-electron system is spherically symmetric.) Now the operators PO ; HO , and qO˛ˇ are invariant under the inversion operator IO, but dO changes sign. Thus, the integral of (7.29), and hence the dipole-quadrupole terms of (7.28) vanish. The asymptotic structure of the density matrix is then .rr 0 /

 r;r 0 !1

N

X ? C0 .r/C0 .r 0 / 

 ?   0 ? X Qs0 1 Ds0 .r/ O Ds0 .rO 0 / .r/ O Qs0 .rO 0 / :  C   1C w2s r2 r3 r 02 r 03 s

(7.30)

The asymptotic structure of the density .r/ which is given by the diagonal matrix element .rr/ is then derived to O.1=r 6 / as

 .r/ r!1

   0 X X jQs0 .r/j 1 jDs0 .r/j O 2 O 2 2 : N jC0 .r/j 1 C C w2s r4 r6  s

(7.31)

For the systems considered, the leading term of .rr 0 / as r; r 0 ! 1 is obtained from (7.30) and (7.31) as .rr 0 /

 r;r 0 !1

p p .r/ .r 0 /;

(7.32)

which is a well-known result [6–10]. Here, we have derived the higher order contributions.

132

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

7.3 The Pair-Correlation Density The pair function P .rr 0 / of (2.32) may be written as XZ ? P .rr 0 / D N.N  1/ .r; r 0  0 ; X N 2 / .r; r 0  0 ; X N 2 /dX N 2 ;  0

(7.33) R

R

where X N 2 D x 3 ; : : : ; x N and dX N 2 D dx 3 ; : : : ; dx N . On substituting (7.11) into (7.33) we obtain XZ  0 ? P .rr / D N.N  1/ C0 .r/ 0.N 1/? .r 0  0 ; X N 2 /  0

C

0 X

? Cs .r/ s.N 1/? .r 0  0 ; X N 2 /



s

  C0 .r/

.N 1/ 0 0 .r  ; X N 2 / 0

C

0 X

Cs 0  .r/

.N 1/ 0 0 .r  ; X N 2 / s0

 dX N 2

s0

(7.34)

0 X X .N 1/ 0 ? jC0 .r/j2 .N 1/ .r 0 / C DN C0 .r/Cs .r/0s .r / 

C

s

0 X

.N 1/ 0 ? Cs .r/C0 .r/s0 .r /

C

0 X

 ;(7.35)

.N 1/ 0 ? Cs .r/Cs 0  .r/ss .r / 0

ss 0

s

where .N 1/ 0 ss .r / 0

D .N  1/

XZ

1/ 0 0 .N 1/? 0 0 .r  ; X N 2 / s.N .r  ; X N 2 /dX N 2 : 0 s

0

(7.36) The asymptotic structure of the pair-correlation density g.rr 0 / of (2.31) is then obtained to 0.1=r 6/ by substituting (7.26) for Cs .r/ into (7.35): g.rr 0 / D D

P .rr 0 / .r/ N X .r/



 jC0 .r/j2 .N 1/ .r 0 /

  0 X Qs0 .r/ 1 Ds0 .r/ O O .N 1/ 0 s0 C .r /  2< 2 3 w r r s s  ?  0 ? X Qs0 O s 0 0 .r/ O .r/Q O s 0 0 .r/ O Ds0 .r/D 1 .N 1/ 0  C .r / : (7.37) C 0 ss ws ws 0 r4 r6 0 ss

7.4 The Work Done in the Electron-Interaction Field

133

Employing (7.31), it is evident that the leading term of (7.37) is .N 1/ .r 0 /. This also is a well-known result [6–11]. Here we have provided the higher order corrections.

7.4 The Work Done in the Electron-Interaction Field With the asymptotic structure of the pair-correlation density g.rr 0 / given by (7.37), we next determine the asymptotic structure of the electron-interaction field E ee .r/ of (2.44). Recall that this field due to the pair-correlation density is representative of Pauli and Coulomb correlations. Substituting (7.37) with up to only the dipole moment terms into (2.44), we obtain E ee .r/ D

 Z .N 1/ 0  N X .r /.r  r 0 / 0 jC0 .r/j2 dr .r/  jr  r 0 j3 Z .N 1/ 0 0 X s0 Ds0 .r/ O .r /.r  r 0 / 0 2< dr ws r 2 jr  r 0 j3 s  Z .N 1/ 0 0 ? X ss 0 .r /.r  r 0 / 0 1 Ds0 .r/D O s0 .r/ O : (7.38) C dr ws w0s r4 jr  r 0 j3 0 ss

Since r .1=jr  r 0 j/ D .r  r 0 /=jr  r 0 j3 , we rewrite (7.38) as  Z .N 1/ 0  N X .r / 0 dr jC0 .r/j2 r .r/  jr  r 0 j Z .N 1/ 0 0 X s0 Ds0 .r/ O .r / 0 dr  2< r 2 ws r jr  r 0 j s  Z .N 1/ 0 0 ? X ss 0 .r / 0 1 Ds0 .r/D O s0 .r/ O dr : C r ws w0s r4 jr  r 0 j 0

E ee .r/ D 

(7.39)

ss

In the limit r ! 1, the first term in the square parentheses of (7.39) is Z r

.N 1/ .r 0 / 0 dr jr  r 0 j



r!1

    Z N 1 1 .N 1/ 0 0  : r .r /dr D r r r

(7.40)

In the same limit, the second term of (7.39) employing the expansion (7.13) may be written as 2<

 Z  Z 0 X 1 1 Ds0 .r/ O 1 .N 1/ 0 .N 1/ 0 0 0 0   r .r /dr C .r / r O  r dr C : : : s0 s0 ws r 2 r r2 s (7.41)

134

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

D 2<

 Z  0 X Ds0 .r/ 1 Ds0 .r/ O O 1 .N 1/ 0 0  r .r /dr C C : : : : s0 ws r 2 r r2 s .N 1/

Now, from the definition of ss 0 Z

(7.42)

.r 0 / of (7.36) we have

.N 1/ 0 ss .r /dr 0 D .N  1/ıss 0 : 0

(7.43)

Thus, there is no contribution from the monopole term of (7.42) which then reduces to 2<

  0 0 ? X X 1 Ds0 .r/ O O Ds0 .r/D O s0 .r/ O Ds0 .r/ D r r : 2 2 4 w r r w r s s s s

(7.44)

In the asymptotic limit, the last term of (7.39) is 0 X ss 0

  Z ? 1 Ds0 .r/D O s0 .r/ O 1 .N 1/ 0 0 ss0 .r /dr : r ws ws 0 r4 r

(7.45)

Again, using (7.43) we see that the only contribution to the integral is for s D s 0 ; s; s 0 ¤ 0. Furthermore, this last term is of 0.1=r 6/. Hence, we have proved that asymptotically there are no 0.1=r 5/ contributions to the field due to Pauli and Coulomb correlations. Employing the asymptotic structure of .r/ given by (7.31) together with (7.40) and (7.44), we have the asymptotic structure of the field E ee .r/ to the accuracy of 0.1=r 6/ to be   E ee .r/ r!1 r

 0 ? 1 .r/D O s0 .r/ O N  1 X Ds0   4 : r ws r s

(7.46)

Thus, asymptotically, the work done in the field is  Wee .r/ r!1

where ˛D2

˛ N 1  4; r 2r

0 ? X Ds0 .r/D O s0 .r/ O ; ws s

(7.47)

(7.48)

is the ground-state polarizability of the .N  1/-electron system. Note that the result for Wee .r/ is to the accuracy of 0.1=r 5/. In other words, there are no 0.1=r 5 / contributions to vee .r/ due to Pauli and Coulomb correlations.

7.4 The Work Done in the Electron-Interaction Field

135

The asymptotic structure of E ee .r/ and Wee .r/ given by (7.46) and (7.47) are valid for when the .N  1/-electron system is either orbitally degenerate or nondegenerate.

7.4.1 The Hartree, Pauli, and Coulomb Potential Energies The asymptotic structure of the Hartree field E H .r/ of (2.48) is readily obtained as Z E H .r/ D r 

r!1

.r 0 / dr 0 jr  r 0 j     Z N 1 0 0 .r /dr D r : r r r

(7.49)

The work done in this field is then  WH .r/ r!1

N : r

(7.50)

(This asymptotic structure may also be obtained directly from the definitions (3.71) of WH .r/.) Thus, the leading .N=r/ term of Wee .r/ of (7.47), and hence of the leading term of vee .r/ of (7.6), arises solely from the density .r/, and constitutes the Hartree contribution to the asymptotic structure. The Pauli field E x .r/ of (3.32) is due to the Fermi hole charge x .rr 0 / of (3.21), the latter being defined in terms of the S system Dirac density matrix s .rr 0 / of (3.12). Now for the S system, it is only the highest occupied orbital N .x/ that contributes to the asymptotic structure. To exponential accuracy then  s .rr 0 / r!1

X

? N .r/N .r 0 /;

(7.51)



and  .r/ r!1

X

jN .r/j2 :

(7.52)



Thus, the Pauli field



r!1

Z

js .rr 0 /j2 .r  r 0 / 0 dr j.r  r 0 /j3 Z 1 4 jN .r/j2 r jN .r 0 /j2 dr 0 2.r/ r

1 E x .r/ D  2.r/

D

r : r3

(7.53) (7.54) (7.55)

136

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

The asymptotic structure of the Pauli potential energy Wx .r/ of (3.74), which is the work done in the field E x .r/ is then 1   ; Wx .r/ r!1 r

(7.56)

to exponential accuracy. Thus, we have proved that the asymptotic .1=r/ structure of the potential energies Wee .r/ and vee .r/ of (7.47) and (7.6), respectively, is due solely to Pauli correlations arising from the Fermi hole charge x .rr 0 /. From the definitions of the Coulomb hole charge c .rr 0 / of (3.27) and (3.28), of the corresponding Coulomb field E c .r/ of (3.33) and potential energy Wc .r/ of (3.75), it follows from the expressions for Wee .r/ of (7.47) and Wx .r/ of (7.56) that the asymptotic structure of Wc .r/ is  Wc .r/ r!1 

˛ : 2r 4

(7.57)

Thus, we have proved that the 0.1=r 4/ term of the potential energy vee .r/ is solely due to Coulomb correlations arising from the Coulomb hole charge c .rr 0 /. Finally, when the N -electron system is orbitally degenerate, the electron density .r/ is no longer spherically symmetric. Thus, the Hartree WH .r/ and Pauli Wx .r/ potential energies will have higher order contributions. To see this, consider the Pauli field E x .r/ of (7.53). Then, asymptotically,  Ex;’ .r/ r!1

4 @ jN .r/j2 2.r/ @r˛

Z

jN .r 0 /j2

1 dr 0 : j.r  r 0 /j

(7.58)

Expanding 1=j.r  r 0 /j in Legendre polynomials P` .x/, we have Ex;’ .r/



r!1

@ @r˛

Z

0

jN .r /j

2

X 1 `D0

Z

`

r0

r `C1



P` .cos  / dr 0



0

(7.59)

2

r0 r0 1 C 2 cos  0 C 3 P2 .cos  0 / r r r 03 04 r r 0 0 C 4 P3 .cos  / C 5 P4 .cos  / C    dr 0 (7.60) r r     r˛ @ @ 1 1 D 3 C QC R C (7.61) r @r˛ r 3 @r˛ r 5 D

@ @r˛

jN .r 0 /j2

where Z QD Z RD

2

jN .r 0 /j2 r 0 P2 .cos  0 /dr 0 ; 4

jN .r 0 /j2 r 0 P4 .cos  0 /dr 0 :

(7.62) (7.63)

7.5 The Correlation-Kinetic Potential Energy

Thus,  Ex;’ .r/ r!1 

137

r˛ 3r˛ 5r˛  5 Q  7 R C ; r3 r r

(7.64)

so that

r 3r 5r  5 Q  7 R C ; r3 r r and consequently, the Pauli potential energy is E x .r/ D 

  Wx .r/ r!1

1 Q R  3  5 C : r r r

(7.65)

(7.66)

The same procedure for the Hartree potential energy WH .r/ leads to  WH .r/ r!1

Q R N C 3 C 5 C : r r r

(7.67)

Thus, for both the orbitally degenerate and nondegenerate N -electron systems, the asymptotic structure of the electron-interaction potential energy vee .r/ is given by (7.6).

7.5 The Correlation-Kinetic Potential Energy To determine the asymptotic structure of the Correlation-Kinetic potential energy Wtc .r/ component of vee .r/, one needs to first obtain the corresponding structure of the Correlation-Kinetic field Z tc .r/ of (3.40). This in turn requires the determination of the interacting and noninteracting system kinetic fields Z.r/ and Z s .r/ that are derived from the corresponding system tensors t˛ˇ .rI Œ/ and ts;˛ˇ .rI Œs / (see (2.54) and (3.39)). The tensors depend, respectively, on the density matrix .rr 0 / and the Dirac density matrix s .rr 0 /. The asymptotic structure of the density matrix .rr 0 / is given by (7.30), and consequently the structure of the tensor t˛ˇ .rI Œ/ and that of the interacting system kinetic field Z.r/, can be determined. Now we know that the asymptotic structure of the density .r/ and Dirac density matrix s .rr 0 / are governed by the highest occupied S system orbital N .x/ (see (7.51) and (7.52)). Then, from the expression for the density .r/ of (7.31) and (7.52), the asymptotic structure of N .r/ is N .r/ D

   0 X p X jQs0 .r/j 1 jDs0 .r/j O 2 O 2 1=2 N jC0 .r/j 1 C C : (7.68) w2s r4 r6  s

The asymptotic structure of the matrix s .rr 0 / is thus known. The corresponding S system tensor ts;˛ˇ .rI Œs / and kinetic field Z s .r/ can then be determined. With the kinetic fields Z.r/ and Z s .r/ known, the asymptotic structure of the

138

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

Correlation-Kinetic field is derived to be  Z tc .r/ r!1 80

r ; r7

(7.69)

where, as previously defined, 02 =2 D E0.N 1/  E0 is the ionization potential, and is an expectation of the spherically symmetric .N  1/-electron system defined by ˛ˇ D ı˛ˇ ; (7.70) with

 ˛ˇ D

ˇ ˇ

.N 1/ ˇ 0 ˇdˇ

ˇ ˇ d˛ ˇˇ .N 1/ 2 ŒHO  E0  1  PO

.N 1/ 0

 :

(7.71)

(The operators d and PO are defined in Sect. 7.2). The work done in the field Z tc .r/ of (7.69) then yields the asymptotic structure of the Correlation-Kinetic potential energy Wtc .r/ as  Wtc .r/ r!1

80 : 5r 5

(7.72)

(In Appendix C, we derive the above structure of Wtc .r/ using the concept of quasiparticle amplitudes as described by Almbladh and von Barth [12]. Quasi-particle amplitudes are the interacting system counterparts of the single-particle orbitals of the S system. The reason for performing the calculation in this framework is to demonstrate the mathematical equivalence of the quasi-particle amplitude approach and that of the expansion of the wave function within Q-DFT. There is, however, an important difference. In contrast to Q-DFT, the rigorous assignation of each term of the asymptotic structure of vee .r/ to a specific electron correlation cannot be made via the approach of quasi-particle amplitudes (see Endnotes below).)

7.6 Endnotes We conclude with a few remarks about the asymptotic structure of the electroninteraction potential energy vee .r/ as obtained within Kohn–Sham density functional theory, and the work of others within this framework. 1. In Chap. 5 of QDFT, a rigorous physical interpretation of the Kohn–Sham KS theory “exchange-correlation” Exc Œ, “exchange” ExKS Œ, and “correlation” EcKS Œ energy functionals, and of their respective functional derivatives vxc .r/, vx .r/, and vc .r/ was derived via Q-DFT and adiabatic coupling constant perturbation theory. There, for example, it was proved that the KS “exchange” potential energy, the lowest-order term of the perturbation theory, was the sum of the work Wx .r/ in the field of the S system Fermi hole charge, and the lowest-order Correlation-Kinetic contribution Wtc ;1 .r/: vx .r/ D Wx .r/  Wtc ;1 .r/:

(7.73)

7.6 Endnotes

139

Thus, the KS “correlation” potential energy vc .r/, which commences in secondorder, is comprised of Coulomb correlations and second and higher order Correlation-Kinetic effects. Many rigorous results for the asymptotic structure of the various KS potential energies have thereby been proved [3] via Q-DFT. For example, it is proved [3] that there is no contribution of Wtc ;1 .r/ in the asymptotic classically forbidden region. Thus, the asymptotic structure of vx .r/ is that of Wx .r/, which as we have shown decays as 1=r. Therefore, via Q-DFT, the .1=r/ structure of vx .r/ can be attributed solely to Pauli correlations. It is also proved [3] that the lowest order contribution to vc .r/, which is of second order, is due to Correlation-Kinetic effects with a term of O.1=r 5 /. The third-order contribution to vc .r/ is a strictly Coulomb correlation term of O.1=r 4 /. The reader is referred to [3] for further details. These rigorous results are also of value from the perspective that they ought to be incorporated into the construction of approximate KS energy functionals and their derivatives. 2. The first three terms of (7.6) have also been derived by Almbladh and von Barth [12] using the approach of quasi-particle amplitudes. A brief description of this method is given in Appendix C, where its mathematical equivalence to the expansion of the wave function approach is shown. The two approaches, however, differ in a very fundamental way. It is not possible within the quasi-particle method to rigorously ascribe the .1=r/ term to Pauli correlations because the Fermi hole charge or the Pauli exclusion principle appears nowhere in the derivation. Furthermore, the assignation of the .˛=2r 4/ term to Coulomb correlations is based on a comparison with a classical calculation of an ion and an asymptotic test charge. Once again, there is no direct relationship in the quasi-particle approach between this term and the Coulomb hole charge or other representation of Coulomb correlations. However, in contrast, within Q-DFT, the dependence of this structure on the Fermi and Coulomb holes is explicit. Of course, when the quasi-particle amplitude derivation is extended to higher order, one obtains the O.1=r 5 / term. But again, there is no way within the quasi-particle approach to attribute this term to Correlation-Kinetic effects. That is accomplished rigorously via Q-DFT. Note that in the classical calculation mentioned above, the higher order contributions [13] are of O.1=r 6 ; 1=r 8 /, etc., and thus of even order. There are no terms of O.1=r 5 / in the classical calculation. Such a term is strictly a consequence of quantum mechanics, and is a result of the mapping from the interacting to the noninteracting model system. Of course, in the classical physics calculation, even the .1=r/ term is attributed to Coulomb correlations. 3. The term .1=r/ of (7.6) was also derived by Sham [14]. In the asymptotic limit, the integral equation [15,16] relating the KS “exchange-correlation” potential energy vxc .r/ to the nonlocal exchange-correlation component †xc .rr 0 I !/ of the self-energy †.rr 0 I !/ reduces to vxc .r/ D

Z 1 dr 0 †xc .rr 0 I /i .r 0 / 2i .r/ Z 1 dr 0 †xc .rr 0 I /i? .r 0 /; C ? 2i .r/

(7.74)

140

7 Asymptotic Structure in the Classically Forbidden Region of Atoms

where i .r/ is the S system orbitals,  is the chemical potential, and the electron is in the highest occupied orbital. By considering the leading exchange term in a diagrammatical analysis, so that (7.74) reduces to that for vx .r/ and †x .rr 0 /, Sham obtained the asymptotic .1=r/ term . Thus, within traditional density functional theory, this structure is attributed to Kohn–Sham “exchange.” However, as indicated by (7.73), the potential energy vx .r/ also has a Correlation-Kinetic component. Thus, within Kohn–Sham theory, one cannot attribute the .1=r/ structure solely to Pauli correlations because there is no way to prove in this framework that the lowest-order Correlation-Kinetic contribution to vx .r/ vanishes asymptotically. For a discussion of the lack of validity of (7.74) for the KS “correlation” potential energy vc .r/, see [17]. 4. In the Kohn–Sham “exchange-only” approximation [18,19], the integral equation [14] relating the KS potential energy vx .r/ to the self-energy †x .rr 0 / is the same as that of the Optimized Potential method [20, 21] (see QDFT). The .1=r/ asymptotic structure of vx .r/ has also been obtained [21] directly from the integral equation of the latter approximation. However, again, one cannot rigorously attribute this structure to Pauli correlations as the Correlation-Kinetic contribution (see (7.73)) to vx .r/ is not explicitly present in the integral equation. It is only when one proves, as in Q-DFT, that the Correlation-Kinetic term Wtc ;1 .r/ vanishes asymptotically, that one can assign the .1=r/ structure of vx .r/ solely to Pauli correlations.

Chapter 8

Analytical Asymptotic Structure At and Near the Nucleus of Atoms

In the previous chapter we derived the analytical structure of the electron-interaction potential energy vee .r/ (see (3.3) and (3.4)) of the model S system of noninteracting fermions in the asymptotic classically forbidden region of atoms. In this chapter, we derive the analytical structure of vee .r/ in the asymptotic region at and near the nucleus of atoms [[1–3]]. We begin Sect. 8.1 by first proving that the potential energy vee .r/ is finite at the nucleus. This finiteness is a direct consequence of the Kato electron–nucleus coalescence constraint on the system wave function .X / as described in Sect. 2.7, and is proved by employing the integral version of the constraint. Now, as explained in QDFT, it is also possible to map the interacting electronic system described by the Schr¨odinger equation to one of noninteracting bosons with equivalent density referred to as the B system. As in the S system, there is a corresponding effective electron-interaction potential energy vB ee .r/ of the B system. This potential energy accounts for Pauli and Coulomb correlation, and Correlation– Kinetic effects due to the difference in kinetic energy of the interacting electron and noninteracting boson systems. We prove that the potential energy vB ee .r/ too is finite at the nucleus. The results of finiteness of vee .r/ and vB ee .r/ at the nucleus are general, and valid for arbitrary system whether it be atomic, molecular, or solid state, and for arbitrary state and symmetry. That the finiteness of the potential energies vee .r/ and vB ee .r/ at the nucleus of the model systems are a reflection of the electron–nucleus coalescence constraint on the wave function, then constitutes a rigorous condition on approximate calculations performed within local effective potential energy theories. If electronic densities .r/ obtained from accurate approximate wave functions are employed in such calculations, the corresponding approximate potential energies vee .r/ and vB ee .r/ will be finite at the nucleus only if these wave functions satisfy the electron–nucleus coalescence constraint. If the approximate wave functions do not satisfy the constraint, then the potential energies vee .r/ and vB ee .r/ will be singular at the nucleus. In Sect. 8.2 we demonstrate this criticality of the electron–nucleus coalescence condition to local effective potential energy theories by application to the Hydrogen molecule.

141

142

8 Asymptotic Structure At and Near the Nucleus

Next, in Sect. 8.3, we derive the analytical asymptotic structure of the potential energy vee .r/ near the nucleus of spherically symmetric atoms, open-shell atoms in the central field approximation, and of their corresponding isoelectronic series. We initially prove, by employing either the integral or differential form of the Kato electron–nucleus coalescence constraint, that in the asymptotic r ! 0 limit, the structure of vee .r/ is of the form vee .r/ D ˛ C ˇr C  r 2 ;

(8.1)

where ˛; ˇ; and  are constants. It is evident from the structure of the electron interaction E ee .r/ and CorrelationKinetic Z tc .r/ fields for spherical or sphericalized systems, (see QDFT for the exactly solvable Hooke’s atom example), that Pauli, Coulomb, and CorrelationKinetic effects all contribute to the coefficient ˛ D vee .0/. By employing Q-DFT, we prove [3] in Sect. 8.4 the following for spherically symmetric and sphericalized systems: (1) correlations due to the Pauli principle and Coulomb repulsion do not contribute to the linear structure; (2) these Pauli and Coulomb correlations contribute quadratically; (3) the linear structure is solely due to Correlation-Kinetic effects, and this contribution is determined analytically to be  ˇD

4Z 3.0/

  t.0/  ts .0/ ;

(8.2)

where t.r/ and ts .r/ are the kinetic energy densities of the interacting and noninteracting S systems, respectively: (4) all these conclusions are equally valid for systems of arbitrary symmetry, provided spherical averages of the properties are employed. The above results resolve a long standing controversy about the structure of vee .r/ at and near the nucleus of atoms. Here are a few historical remarks to put the work of this chapter into context. The traditional (non Q-DFT) way of determining the exact structure of vee .r/ employs methods [4, 5] that assume knowledge of the exact density .r/. The densities in turn are obtained from correlated or configuration-interaction type wave functions that are highly accurate from the perspective of the total energy. Work on the He atom by Smith et al. [6], Davidson [7], and Umrigar and Gonze [8], show vee .r/ to be finite at the nucleus. Almbladh and Pedroza [9, 10], on the other hand, showed it to be singular there. Additional work on light atoms show it to be either finite [[8,11–13]] or to diverge at the nucleus [14]. For the determination of its structure for few electron molecular systems [15] such as H2 and LiH, the potential energy vee .r/ is assumed to be finite at each nucleus. Expressions for vee .r/ at a nucleus have also been derived [16], but once again they are based on the assumption that it is finite there. In various approximations within Kohn–Sham DFT [[17–21]], the potential energy vee .r/ also diverges at the nucleus. Hence, the structure of vee .r/ at and near the nucleus remained controversial until the work of Pan and Sahni [1] and of Qian and Sahni [2,3]. Furthermore, these authors were the first to show that this asymptotic structure of vee .r/ is a

8.1 Proof of Finiteness of Potential Energies at the Nucleus

143

direct consequence of the electron–nucleus coalescence constraint on the system wave function. That vee .r/ approaches the nucleus linearly, and that this structure is solely due to Correlation-Kinetic effects is then proved via Q-DFT.

8.1 Proof of Finiteness of Potential Energies vee .r/ and vB ee .r/ at the Nucleus For three dimensions .D D 3/, the D-dimensional expression for the electron– nucleus coalescence constraint of (2.99) reduces to .r; r 2 ; : : : ; r N / D

.0; r 2 ; : : : ; r N /.1  Zr/ Cr  a.r 2 ; : : : ; r N / C O.r 2 /;

(8.3)

where the vectors r i now represent the position of the electrons from a particular nucleus. The vector a.r 2 ; : : : ; r N / is undetermined. The derivation of the integrated form of the cusp condition does not involve any boundary conditions far from the nucleus. Hence, it is valid at the nuclei of atoms, molecules, and periodic, and aperiodic solids. Substitution of (8.3) into the definition (2.15) or (2.17) for the density leads to  3 X .r/ D .0/ .1  Zr/2 C 2.1  Zr/ Bk rk C

3 X



kD1

rl rm Alm C    ;

(8.4)

lD1;mD1

R R where Bk D ak .0; r 2 ; : : : ; r N /1 dr 2 : : : dr N and Alm D al am .0; r 2 ; : : : ; r N /2 dr 2 : : : dr N are constants, and rk ; ak , etc., components of the vector r and a. It follows then that p

.r/ D

p

C

 3 X 2 .0/ .1  Zr/ C 2.1  Zr/ Bk rk 3 X

rl rm Alm C   

 12

kD1

lD1;mD1

D

p

.0/ 1  Zr C B  r C O.r 2 / ;

where in the second step we have retained only the terms of O.r/.

(8.5)

144

8 Asymptotic Structure At and Near the Nucleus

We first consider the electron-interaction potential energy vee .r/ of the S system. In the derivation, the spin coordinate is suppressed. The S system differential equation (see (3.3), (3.4)) is then

1  r 2 C v.r/ C vee .r/ i .r/ D i i .r/I i D 1; : : : ; N; (8.6) 2 P where v.r/ D ˛ Z=jr  R ˛ j is the potential energy of the electrons due to the external charge Z of the nuclei at R ˛ . Inverting this differential equation, we obtain for any occupied orbital i .r/ the expression for the electron-interaction potential energy vee .r/ as r 2 i .r/  v.r/: (8.7) vee .r/ D i C 2i .r/ Next, we rewrite the orbitals i .r/ as i .r/ D

p

i D 1; 2; : : : ; N;

.r/ci .r/;

(8.8)

where the coefficients ci .r/ satisfy N X

ci .r/2 D 1:

(8.9)

i D1

This definition of the i .r/ is consistent with the density .r/ written in terms of the orbitals (see(3.10)): Xˇ ˇ ˇi .r/ˇ2 : .r/ D (8.10) i

Expanding the coefficient ci .r/ about the nucleus we obtain

ci .r/ D ci .0/ C r ci .0/  r C O.r 2 / D ci .0/ 1 C D  r C O.r 2 / ;

(8.11)

where D D r ci .0/=ci .0/ is some constant vector. Inserting (8.5) and (8.11) into (8.8), we obtain the expression for the orbitals i .r/ near the nucleus as i .r/ D

p



.0/ci .0/ 1  Zr C .B C D/  r C O.r 2 / :

(8.12)

Now the expression for the external potential energy near the nucleus is 0

v.r/ D 

Z X Z  ; r jr  R˛ j ˛

(8.13)

where the sum is over all the other nuclei. At the nucleus, the term .Z=r/ is singular, whereas the other terms are constants. From

(8.7) it is evident that this singularity must be canceled by the r 2 i .r/=2i .r/ term.

8.2 Criticality of the Electron–Nucleus Coalescence Condition



145

Consider the term r 2 i .r/=2i .r/ near the nucleus with i .r/ given by (8.12). We have r 2 r D 2=r and r 2 .B C D/  r D 0. After acting by r 2 and taking the limit as r ! 0, terms of O.r 2 / lead higher-order to constants while

terms vanish. Thus, near the nucleus, the term r 2 i .r/=2i .r/ is .Z=r/ plus some constant, and therefore in this limit the singularity of the external potential energy is canceled. Therefore, vee .0/ is finite. The B system differential equation (see Chap. 6 of QDFT) is  p p 1 2 B  r C v.r/ C vee .r/ .r/ D  .r/; 2

(8.14)

where vB ee .r/ is the corresponding effective electron-interaction potential energy, and  is the chemical potential or the negative of the ionization energy. Inverting the B system differential equation we have the potential energy vB ee .r/ to be vB ee .r/

p r .r/ DC p  v.r/: 2 .r/

(8.15)

The proof of the finiteness of the B system potential energy vB ee .r/ at the nucleus is along the same lines as above. Substitution of (8.5) into the expression for vB ee .r/ of (8.15) leads to the result that vB may also be arrived at as a ee .0/ is finite. This result p special case of the S system proof for which ci D 1= N .

8.2 Criticality of the Electron–Nucleus Coalescence Condition to Local Effective Potential Energy Theories In this section we demonstrate the criticality of the electron–nucleus coalescence to the finiteness of the potential energies vee .r/ and vB ee .r/ at a nucleus by the application to the Hydrogen molecule. It becomes evident thereby that densities derived from wave functions that do not satisfy the cusp condition lead to potential energies that are singular at a nucleus, irrespective of how accurate the wave function may be from an energy standpoint. For two-electron systems in the ground state such as the Helium atom, Hooke’s atom, and Hydrogen molecule, the S p and B systems are equivalent. This is because the S system orbital is then i .r/ D .r/=2, i D 1; 2. Hence, the demonstration of the significance of the coalescence condition as applied to the Hydrogen molecule is equally valid for both systems. Equation (8.7) for vee .r/ of the S system is then the same as (8.15) for vB ee .r/ of the B system: vee .r/ D vB ee .r/:

(8.16)

146

8 Asymptotic Structure At and Near the Nucleus

(In this example, the S system differential equation has only one eigenvalue.) It is evident,ptherefore, pthat the singularity in v.r/ at each nucleus must be canceled by the Œr 2 .r/=2 .r/ term for vee .r/ or vB ee .r/ to be finite there. In the calculation we employ the accurate Gaussian geminal wave function of Komasa and Thakkar [22, 23] for the Hydrogen molecule which is spin free and is of the form 150 X 1 .r 1 r 2 / D .1 C PO12 /.1 C POab / ck k ; (8.17) 4 kD1

in which k D exp.˛k jr 1  R a j2  ˇk jr 1  R b j2  k jr 2  R a j2 k jr 2  R a j2  k jr 1  r 2 j2 /;

(8.18)

where r j for j 2 f1; 2g are the position vectors of the electrons, R j for j 2 fa; bg are the position vectors of the nuclei, PO12 and POab are permutation operators that interchange the electronic and nuclear coordinates, respectively, and ck ; ˛k ; ˇk ; k ; k ; k are variationally determined parameters subject to the squareintegrability constraint .˛k C ˇk /.k C k / C k .˛k C ˇk C k C k / > 0;

(8.19)

for each k. The individual exponential parameters are allowed to become negative as long as the square integrability is satisfied. The ground-state energy obtained with this wave function is E D 1:174475313 a.u. and that of the most accurate correlated wave function [24] is E D 1:174475668 a.u. Thus the energy obtained by the Gaussian wave function is accurate to the sixth decimal place. It is well known that such Gaussian geminal or orbital wave functions do not satisfy the electron–nucleus cusp condition, and it is for this reason we employ this wave function. Additionally, in contrast to wave functions such as the Kolos–Roothan type wave function [[24, 25]] the calculations are analytical. In Fig. 8.1, we plot the density along the nuclear bond axis with the two nuclei on the z axis at R D ˙0:7 a.u. It is evident that the density is very accurate right up to the nucleus, and on the scale of the figure appears to possess a cusp at each nuclear position. However, in magnifying the scale as in Fig. 8.2, we see that there is no cusp as expected, and that the density is smooth across the nucleus. p p In Fig. 8.3 we plot v.r/ and Œr 2 .r/=2 .r/ along axis about one p the z p nucleus. Whereas, v.r/ is singular as expected, the Œr 2 .r/=2 .r/ term is finite at the nucleus. Therefore, the singularity in v.r/ is not canceled. Hence, although the wave function is very accurate from the perspective of the ground-state energy, the fact that it does not satisfy the electron–nucleus cusp condition leads to the potential energy vee .r/ [or vB ee .r/] being singular at each nucleus. In determining these potential energies from accurate densities, it is therefore imperative that the densities are obtained from wave functions that satisfy the electron–nucleus cusp condition.

8.2 Criticality of the Electron–Nucleus Coalescence Condition

147

Fig. 8.1 The electron density .r/ of the hydrogen molecule along the nuclear bond axis in atomic units (a.u.). The nuclei are on this axis at ˙0:7 a.u. The density is determined by the wave function of (8.17)

Fig. 8.2 The electron density .r/ of the hydrogen molecule near a nucleus as determined by the wave function of (8.17) in atomic units (a.u.). The nucleus is indicated by a large dot on the axis

148

8 Asymptotic Structure At and Near the Nucleus

p p Fig. 8.3 The external potential energy v.r/ and the function Œr 2 .r/=2 .r/ about a nucleus of the hydrogen molecule in atomic units (a.u.). The density .r/ is determined by the wave functionp of (8.17).p The singularity of the external potential energy at the nucleus is not canceled by the Œr 2 .r/=2 .r / function

8.3 General Structure of vee .r/ Near the Nucleus of Spherically Symmetric and Sphericalized Systems The effective potential energy vs .r/ for a spherically symmetric or sphericalized system of noninteracting fermions in an external potential v.r/ D Z=r is vs .r/ D 

Z C vee .r/; r

(8.20)

where vee .r/, the local electron-interaction potential energy, is spherically symmetric. The corresponding S system differential equation is then 

 1 2  r C vs .r/ nlm .r/ D nl nlm .r/; 2

(8.21)

with orbitals nlm .r/ separable as nlm .r/ D Rnl .r/Ylm ./;

(8.22)

8.3 General Structure of vee .r/ Near the Nucleus

149

and where Rnl .r/ is the radial part and Ylm ./ the spherical harmonic of order .l m/ being the angular part. The radial function Rnl .r/ is then a solution of the equation. 



l .l C 1/ 2 @ @2  Rnl .r/ D 2 nl  vs .r/ Rnl .r/: C 2 2 @r r @r r

(8.23)

In the asymptotic limit r ! 0, we assume the most general form for the structure of the radial function Rnl .r/ to be (with .i D nl / Ri D ai C bi r C ci r 2 C di r 3 C fi r 4 C    ;

(8.24)

and where ai ; bi , etc. are coefficients. The above form of solution is dictated by the fact that in the asymptotic r ! 0 limit, the structure of the bounded solution of the Schr¨odinger equation (2.5) as obtained by the Frobenius method [26] is the same (see [27, 28]). This assumption guarantees that the densities of the interacting Schr¨odinger and noninteracting S systems are equivalent in this limit. Substitution of (8.24) into the differential equation leads to 2 6ci C bi C 12di r r  

bi ai C ci C di r  2i ai C bi r C ci r 2 C di r 3 D l .l C 1/ 2 C r r  

ai  2Z C bi C ci r C di r 2 C 2vee.r/ ai C bi r C ci r 2 C di r 3 : r

(8.25)

Now, we first prove that for l ¤ 0, all ai D 0. (This, of course, is the case for Coulombic effective potentials.) To do so let us assume that there is at least one ai ¤ 0 for l ¤ 0. From (8.25) the implication of this assumption is that there is a 1=r 2 term in vee .r/ divergent at the nucleus. Writing this term of vee .r/ as ı=r 2 , we have from (8.25) that l .l C 1/ai D 2ıai ; (8.26) for any principal quantum number n. We next show that this then leads to a null solution with ai D bi D ci D    D 0 for all l D 0. For ı ¤ 0, (8.26) leads to ai D 0 for all orbitals with l D 0. For l D 0 (8.25) then reduces to

2 6ci C bi C 12di r D 2i bi r C ci r 2 C di r 3 r   2 2Z bi C ci r C di r

C2vee .r/ bi r C ci r 2 C di r 3 C    ;

(8.27)

150

8 Asymptotic Structure At and Near the Nucleus

which leads to

2 2 bi D ıbi : (8.28) r r Then, since ı is finite, but negative definite as indicated by (8.26), we have bi D 0. Using this fact, (8.27) reduces to

6ci C 12di r C 20fi r 2 D 2i ci r 2 C di r 3 C fi r 4

2Z ci r C di r 2 C fi r 3

C2vee .r/ ci r 2 C di r 3 C fi r 4 C    ;

(8.29)

which leads to 6ci D 2ıci ;

(8.30)

so that again, ci D 0. Similarly, one obtains 12di r D 2ıdi r;

(8.31)

so that di D 0, and so on. Thus, the assumption that there is at least one ai ¤ 0 for l ¤ 0 leads to the null solution. We have therefore proved that ai D 0 for l ¤ 0. The radial equation (8.25) for l D 0 is

2 6ci C bi C 12di r D 2i ai C bi r C ci r 2 C di r 3 r   ai 2 C bi C ci r C di r 2Z r

C2vee .r/ ai C bi r C ci r 2 C di r 3 :

(8.32)

Rewriting this equation by collecting terms of equivalent powers of r, we have





2 bi C Zai C 6ci C 2i ai C 2Zbi C 12di C 2i bi C 2Zci r r

C    D 2vee.r/ ai C bi r C    : (8.33) There is a similar equation corresponding to the complex conjugate of (8.33). If (8.33) is multiplied by ai? and the complex-conjugate equation by ai , and the two resulting equations summed over i and added, we obtain X

2X .3ci C i ai C Zbi /ai? .bi ai? C bi? ai C 2Zai ai? / C 4< r i i X

C4r< .6di C i bi C Zci /ai? i

D 2vee .r/

X i

2ai? ai C .ai? bi C ai bi? /r C    :

(8.34)

8.3 General Structure of vee .r/ Near the Nucleus

151

Now the equivalent electronic density .r/ is obtained from the noninteracting system as X .r/ D jnlm .r/j2 ; (8.35) nlm

and has the asymptotic r ! 0 structure of X X ai? ai C .ai? bi C ai bi? /r C    : .r/ D i

(8.36)

i

[Note that the summations in (8.34) and (8.36) are for all l , but in fact only the l D 0 terms contribute since ai D 0 for l ¤ 0.] Now either from (8.4) which is obtained from the integral form of Kato’s electron–nucleus coalescence condition, or from its differential form (see 2.102) which for three dimensions .D D 3/ is ˇ d.r/ ˇˇ D 2Z.0/; (8.37) dr ˇrD0 it follows that for r ! 0, .r/ D .0/Œ1  2Zr: A comparison of (8.36) and (8.38) leads to X .0/ D ai? ai ;

(8.38)

(8.39)

i

and 2Z.0/ D 

 X ai? bi C ai bi? :

(8.40)

i

Substituting (8.39) and (8.40) into (8.34) leads to X

.3ci C i ai C Zbi /ai? vee .r/Œ2.0/  2Z.0/r D 2< i

C 2r<

X

.6di C i bi C Zci /ai? C    : (8.41)

i

Therefore,  X

1 Œ1 C Zr C     < .3ci C i ai C Zbi /ai? vee .r/ D .0/ i X

.6di C i bi C Zci /ai? C    ; Cr< i

(8.42)

152

8 Asymptotic Structure At and Near the Nucleus

or vee .r/ D

X 1 < .3ci C i ai C Zbi /ai? C .6di C i bi .0/ i 2 ? C4Zci C i ai Z C Z bi /ai r C    :

(8.43)

Since the leading term is a constant, we have proved that the electron interaction potential energy vee .r/ is finite at the nucleus. Thus, in the asymptotic r ! 0 limit, the structure of vee .r/ is vee .r/ D ˛ C ˇr C  r 2 ;

(8.44)

where ˛, ˇ, and  are constants. The asymptotic structure of (8.44) can also be arrived at for a more general form of the potential energy vee .r/: vee .r/ D

X

˛p r p :

(8.45)

p

It can be proved that ˛p D 0 for p < 0 as follows. Substitution of (8.45) into (8.25) leads to 2ai ˛p D 0 for

p < 2:

(8.46)

Since ai ¤ 0 for all i, we have ˛p D 0 for p < 2. The constant ˛2 for p D 2 is just the ı in the above proof where it is shown to be zero. For the ˛1 =r; .p D 1/ term, the equation one obtains from (8.32) for l D 0 is 4˛1 ai? ai 2 ? a? ai .ai bi C ai bi? / D 4Z i C : r r r

(8.47)

Summing over all i and using (8.39) and (8.40) then leads to ˛1 D 0. Similar arguments as above apply for 2 < p < 1 and 1 < p < 0. Thus, ˛p D 0 for p < 0. A similar approach as above also proves that any term with a lnr dependence in vee .r/ is also zero. Thus, the general form of the asymptotic structure of vee .r/ near the nucleus is given by (8.44). Finally, on substituting (8.44) into (8.25) we have for any i .l D 0 and l ¤ 0/   2 bi ai 6ci C bi C 12di r D l .l C 1/ 2 C C ci C di r r r r

2i ai C bi r C ci r 2 C di r 3   ai C bi C ci r C di r 2 2Z r C2.˛ C ˇr C  r 2 /Œai C bi r C ci r 2 C di r 3 :

(8.48)

8.4 Near Nucleus Structure of vee .r/

153

Thus, for l D 1: ai D 0; for l D 2: ai D 0; bi D 0; for l D 3: ai D 0; bi D 0, ci D 0; for l D 4: ai D 0; bi D 0; ci D 0; di D 0; and so on. However, we only need these results as we are interested in vee .r/ only to 0.r 2 / as in (8.44).

8.4 Exact Structure of vee .r/ Near the Nucleus of Spherically Symmetric and Sphericalized Systems In this section we employ Q–DFT to prove that for spherically symmetric and sphericalized systems, the electron-interaction potential energy vee .r/ approaches the nucleus linearly, with the coefficient ˇ of (8.1) being due solely to CorrelationKinetic effects as given by (8.2). It is further shown that correlations due to the Pauli principle and Coulomb repulsion do not contribute to this linear term but do contribute to the quadratic term. The outline of the proof is as follows. We know from (3.67) that vee .r/ is the work done in the conservative field F eff .r/: Z r vee .r/ D  F eff .r/  d` 0 ; (8.49) 1

where F eff .r/ is the sum of the electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ fields (see 3.65): F eff .r/ D E ee .r/ C Z tc .r/:

(8.50)

Step 1: As the density .r/ appears in the expressions for the fields, we first determine in Sect. 8.4.1 the near nucleus structure of the wave function and density for the interacting and noninteracting systems, ensuring that the densities are equivalent. Step 2: We next prove in Sect. 8.4.2 that the field E ee .r/, and its Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ components all vanish at the nucleus: ŒE ee .r/; E H .r/; E x .r/; E c .r/ D 0 for r ! 0:

(8.51)

(Now for spherically symmetric systems, the fields E ee .r/ and Z tc .r/ are separately conservative. Thus, each field is the negative gradient of a scalar function: E ee .r/ D r Wee .r/ and Z tc .r/ D r Wtc .r/. Thus, vee .r/ D Wee .r/ C Wtc .r/ (see Sect. 3.5)). As a consequence of the vanishing of the field E ee .r/ at the nucleus, and the fact that it is the negative gradient of the potential energy Wee .r/, we see that there can be no linear term contribution to vee .r/ due to E ee .r/. Equivalently, correlations due to the Pauli exclusion principle and Coulomb repulsion do not contribute to the coefficient ˇ in (8.1).

154

8 Asymptotic Structure At and Near the Nucleus

Step 3: What remains is the determination of the structure of the Correlation-Kinetic field Z tc .r/ as r ! 0. The field Z tc .r/ is defined as (see 3.40) Z tc .r/ D Z s .r/  Z.r/;

(8.52)

where Z s .r/, Z.r/ are the S and interacting system kinetic fields. These fields in turn are defined in terms of the respective kinetic “forces” zs .r/ and z.r/ as Z s .r/ D

zs .rI s / .r/

and Z.r/ D

z.rI  / : .r/

(8.53)

Hence, one must determine the structure of the “forces” z.rI  / and zs .rI s / as r ! 0 to obtain that of the field Z tc .r/ in this limit. What we finally prove is that Z tc .r/ is finite at the nucleus. Thus, the Correlation-Kinetic potential energy Wtc .r/ contributes to vee .r/ linearly.

8.4.1 Near Nucleus Structure of the Wave Functions and the Density We first determine the asymptotic r ! 0 structure of the wave function as obtained from the Schr¨odinger equation (2.5). The equation may be rewritten as 

N N N X 1 1X 2 X 1 1 C  rr2  ri C 2 2 jr  r i j jr i  r j j i D2

 4

Z  r

N X i D2

Z ri

i D2



i ¤j ¤1

.rX / D E .rX /;

(8.54)

4

where r 1 D r; 1 D , and X D ; x 2 ; : : : xN . (Note that in contrast to Chap. 2, the symbol X now excludes the spatial coordinate r 1 or r.) For small r D jrj, we employ the expansion 1 1 1 X rl D Pl .cos i1 /; jr  r i j ri rl lD0 i

(8.55)

where Pl .x/ is the Legendre polynomial of order l , and i1 is the angle between r 1 (or r/ and r i . Thus for small r, N X i D2

X 1 X 1 1 D Cr P1 .cos i1 /: jr  r i j ri ri2 N

N

i D2

i D2

(8.56)

8.4 Near Nucleus Structure of vee .r/

155

Employing the addition theorem for spherical harmonics which is Pl .cos  / D

l 4 X ? 0 0 Y .  /Ylm ./; 2l C 1 mDl lm

(8.57)

where  is the angle between r.r/ and r 0 .r 0  0  0 /, we can rewrite (8.56) as N X i D2

N 1 X X 1 1 D Cr gm .X /Y1m ./; jr  r i j ri mD1

(8.58)

i D2

where gm .X / D

N 4 X ? Y1m .i i /: 3

(8.59)

i D2

The Schr¨odinger equation (8.54) may then be written as 

 1 X 1 Z  rr2  Cr gm .X /Y1m ./ .rX / 2 r mD1   N N N X X 1X 2 1 Z1 D EC C .rX /: ri  2 jr i  r j j ri i D2

i ¤j ¤1

(8.60)

i D2

P For small r, and of course r  ri , the term r 1mD1 gm .X /Y1m ./ in (8.60) 0 may be neglected if we consider accuracy to 0.r /. Thus, the Schr¨odinger equation (8.60) reduces to 1 2 Z

 rr  C 0. 0 / .rX / D 0; (8.61) 2 r where 0. 0 / implies that terms of 0.r 0 / remain, and terms of 0.r/ and higher than r are neglected. The ground state many body wave function for small r can then be written as [27, 28] (see also (8.3)) .rX / D C

1 X

.0X / C a.X /r C b.X /r 2 C   

a1m .X /r C b1m .X /r 2 Y1m ./ C   

mD1

C

2 X

b2m r 2 Y2m ./ C    :

(8.62)

mD2

Substitution of the .rX / of (8.62) into the differential equation (8.61) and considering only terms up to 0.r 2 / and Y1m ./, leads to

156

8 Asymptotic Structure At and Near the Nucleus

2 a.X / C Z .0X / C 2Za.X / C 6b.X/ C 2Zb.X/r r   1 X C 2Za1m .X / C 4b1m .X / Y1m ./ mD1

C2Z

1 X

b1m .X /rY1m ./ C    D 0:

(8.63)

mD1

By equating the coefficients of the r 1 and r 0 Y1m ./ terms, we obtain the following relations: a.X / C Z .0X / D 0; Za1m .X / C 2b1m .X / D 0:

(8.64) (8.65)

The relations (8.64) and (8.65) can also be derived [27, 28] by substituting .rX / of (8.62) into the differential form of the electron–nucleus coalescence condition (2.100). Employing these relations, the function .rX / becomes .rX / D

.0X /.1  Zr/ C b.X /r 2  C1  X Zr 2 Y1m ./ C    : a1m .X /r  a1m .X / C 2 mD1

(8.66)

The density is defined as Z .r/ D N

dX j .rX /j2 ;

(8.67)

R P R where dX  dx 2 ; : : : ; dx N . Note that we are considering spherically  symmetric or sphericalized systems, so that for terms involving the spherical harmonics, an integral over the solid angle has to be performed. Then on employing the orthonormality condition Z

Z

2



d 0

0

? d sin Ylm ./Yl 0 m 0 ./ D ıll 0 ımm 0 ;

(8.68)

the density is obtained as Z .r/ D N Cr 2

 dX j .0X /j2 .1  Zr/2 C 2r 2 <Π1 ja1m .X /j2 C 0.r 3 /: 4

mD1 1 X

?

.0X/b.X / (8.69)

8.4 Near Nucleus Structure of vee .r/

157

Z

Since .0/ D N

dX j .0X /j2 ;

(8.70)

we have the density of the interacting system to be Z .r/ D .1  Zr/2 .0/ C r 2 2N Z Cr N 2

dX

dX <Œ

?

.0X /b.X/

1 X

1 ja1m .X /j2 C 0.r 3 /: 4

mD1

(8.71)

We next obtain the expression for the density .r/ of the S system. Then by comparing the coefficients of this expression with those of the interacting system density of (8.71), we ensure the two densities are the same. We rewrite the S system orbitals nlm .r/ of (8.22) as nlm .r/ D Rnl .r/Ylm ./ D r l ŒAnl C Bnl r C Cnl r 2 C    Ylm ./: (8.72) Defining Rnl .r/ D nl .r/=r, we have nl .r/ D r lC1 ŒAnl C Bnl r C Cnl r 2 C    ;

(8.73)

and the corresponding radial equation for the nl .r/ is (see 8.23) l .l C 1/ d 2 nl nl .r/ D 0: C 2 nl  vs .r/  2 dr 2r 2

(8.74)

On substituting for nl .r/ from (8.73) into (8.74) and noting that vs .r/ D v.r/ C vee .r/ Z D  C ˛ C ˇr C  r 2 ; r

(8.75) (8.76)

the differential equation (8.74) reduces to Anl l .l C 1/r l1 C Bnl .l C 1/.l C 2/r l C Cnl .l C 2/.l C 3/r lC1 C    Z l .l C 1/  ˛  ˇr   r 2  D 2 nl C r r2

lC1 lC2 lC3 C Bnl r C Cnl r C : (8.77)  Anl r For l D 0, this equation is

Z  ˛  ˇr   r 2 2Bn0 C 6Cn0 r C    D  2 n0 C r

 An0 r C Bn0 r 2 C Cn0 r 3 C    :

(8.78)

158

8 Asymptotic Structure At and Near the Nucleus

For l D 1, (8.77) is

Z  ˛  ˇr   r 2 2An1 C 6Bn1 r C 12Cn1 r 2 C    D 2 n1 C r

 r 2 An1 C r 3 Bn1 C r 4 Cn1 C    : (8.79) On comparing coefficients of equal powers of r in (8.78) and (8.79) leads to the relations Bn0 C ZAn0 D 0;

(8.80)

2Bn1 C ZAn1 D 0:

(8.81)

The S system density .r/ which is defined as .r/ D

Xˇ ˇ ˇnlm .r/ˇ2 ;

(8.82)

nlm

is then .r/ D D

X

ˇ2

2 ˇ r 2l Anl C Bnl r C Cnl r 2 C    ˇYlm ./ˇ

nlm

 r 2l jAnl j2 C 2<.A?nl Bnl /r C r 2 jBnl j2

X

(8.83)

nlm

 ˇ ˇ2 C r 2 2<.A?nl Cnl / C    ˇYlm ./ˇ (8.84)  X D jAn0 j2 C 2<.A?n0 Bn0 /r C r 2 jBn0 j2 C r 2 2<.A?n0 Cn0 / C    jY00 j2 n

Cr

2

 1 X

jAn1 j2 C 2<.A?n1 Bn1 /r C r 2 jBn1 j2

n;mD1

Cr

2

2<.A?n1 Cn1 /

 C    jY1m ./j2 :

(8.85)

Comparing the expressions (8.71) and (8.85) for the density as obtained respectively from the interacting and S system differential equations, we have for the p .l D 0; m D 0I r 0 I rI r 2 / terms with Y00 D 1= 4 that X <

X n

jAn0 j2 D 4 .0/;

(8.86)

A?n0 Bn0 D 4 Z.0/:

(8.87)

n

Employing these equivalences, together with the relation of (8.80), and that P1 jY ./j2 D 3=4 , we have the S system density to be 1m mD1

8.4 Near Nucleus Structure of vee .r/

159

 X  X 2 2 1 ? 2 2 .r/ D .1Zr/ .0/Cr <.An0 Cn0 /C3 jAn1 j C0.r 3 /: (8.88) 4

n n Thus, to the leading term, the expressions for the density of the interacting and S systems are equivalent. The equivalence of the other terms can be achieved by considering the higher-order terms in the various expansions in a manner similar to that described above.

8.4.2 Electron-Interaction Field E ee .r/ at the Nucleus In this section, we prove that the electron-interaction field E ee .r/ vanishes at the nucleus. This can be understood on physical grounds. The pair-correlation density g.rr 0 /, the quantal source of the field E ee .r/ (see (2.44)), is in general not spherically symmetric about the electron position at r. However, for an electron at the nucleus, this charge distribution is spherically symmetric. Hence, the field due to this charge as obtained by Coulomb’s law, must vanish at the nucleus. This result can also be proved mathematically. The electron-interaction field E ee .r/ may be expressed as Z 1 dr 0 : E ee .r/ D  g.rr 0 /r (8.89) jr  r 0 j For spherically symmetric systems, the field E ee .r/ can only have a radial component, and can only depend on the radial coordinate r, i.e., E ee .r/ D rE O ee;r .r/. Thus, by employing the identity rO  r D @=@r;

(8.90)

we have Z Eee;r .r/ D rO  E ee .r/ D 

g.rr 0 /

@ 1 dr 0 : @r jr  r 0 j

(8.91)

Since for small r, (see (8.55) and (8.57)), 1 1 1 r 4 X ? 0 0 D C  Y1m .  /Y1m ./; jr  r 0 j r0 r 02 3 mD1

(8.92)

and the fact that for an electron at the nucleus, the pair-correlation density is spherically symmetric, i.e., g.0r 0 / D g.0r 0 /, we have ˇ ˇ Eee;r .r/ˇˇ

rD0

D

4

3

D 0:

Z

dr 0 g.0r 0 /

1 r 02



1 X

? Y1m . 0  0 /Y1m ./ (8.93)

mD1

(8.94)

160

8 Asymptotic Structure At and Near the Nucleus

Here, we have again employed the vanishing of the solid angle integral of the spherical harmonics Y1m ./. (For open shell atoms, the central field approximation is invoked which means taking the spherical average of the field E ee .r/. Thus, in this approximation, the field is again radial and depends only on the radial coordinate. See Chap. 10 where this approximation is employed.) Arguments similar to those above can also be made to prove that the Hartree E H .r/, Pauli E x .r/, and Coulomb E c .r/ components of E ee .r/ vanish at the nucleus. The density .r/ is spherically symmetric about the nucleus, and hence the field E H .r/ vanishes there. Again, for an electron at the nucleus, the Fermi x .rr 0 / and Coulomb c .rr 0 / holes are spherically symmetric about the electron. Hence, the corresponding fields E x .r/ and E c .r/ vanish at the nucleus. The vanishing of the field E ee .r/ and its components E H .r/, E x .r/, and E c .r/ at the nucleus means that correlations due to the Pauli principle and Coulomb repulsion contribute quadratically to the near nucleus structure of vee .r/. These correlations do not contribute to the linear structure of vee .r/ near the nucleus.

8.4.3 Kinetic “Force” z.rI / Near the Nucleus We next determine the near nucleus structure of the kinetic “force” z.rI  / of the interacting system. (In this and the following sections, the summations over the dummy cartesian indices are assumed.) To make the derivation more transparent, we rewrite the wave function (8.66) as .rX / D

.0X /.1  Zr/ C ˛.X /  r.1  Zr=2/ C ˇij .X /ri rj :

(8.95)

The relations between ˛.X / and a1m .X /, and ˇij .X / and b.X /, b2m .X / are those between cartesian and spherical tensors as listed below: r

4

˛z .X /; 3 r 4 1 p Œ˛x .X /  i˛y .X /; a11 .X / D  3 2 r 4 1 p Œ˛x .X / C i˛y .X /; a11 .X / D 3 2 a10 .X / D

(8.96) (8.97) (8.98)

and b.X/ D ˇi i .X /=3; r 2

Œ2ˇzz .X /  ˇxx .X /  ˇyy .X /; b20 .X / D 3 5

(8.99) (8.100)

8.4 Near Nucleus Structure of vee .r/

r b2˙1 .X / D r b2˙2 .X / D

161

2

Œˇxz .X / C ˇzx .X / i.ˇyz .X / C ˇzy .X //; 15

(8.101)

2

Œˇxx .X / C ˇyy .X / i.ˇxy .X / C ˇyx .X //: 15

(8.102)

By substituting (8.95) into (2.19), we have the structure of .rr 0 / for small r; r 0 as .rr 0 / D .0/.1  Zr/.1  Zr 0 / C .1  Zr/.1  Zr 0 =2/A?  r 0 C .1  Zr 0 /.1  Zr=2/A  r C Eij? .1  Zr=2/.1  Zr 0 =2/ri rj0 ? C ŒDij? .1  Zr/ C Flij rl ri0 rj0 C ŒDij .1  Zr 0 / C Flij rl0 ri rj ; (8.103)

where Z Ai D N Z Eij D N Z Dij D N Z Fijl D N

dX .0X /˛i? .X /;

(8.104)

dX ˛i .X /˛j? .X /;

(8.105)

dX .0X /ˇij? .X /;

(8.106)

? dX ˛i .X /ˇjl .X /:

(8.107)

Note that the above expression for .rr 0 / is not restricted to spherically symmetric or sphericalized systems, but is valid in general. For systems with central-reflection symmetry, (8.103) reduces to .rr 0 / D .0/.1  Zr/.1  Zr 0 / CEij? .1  Zr=2/.1  Zr 0 =2/ri rj0 CDij? .1  Zr/ri0 rj0 C Dij .1  Zr 0 /ri rj :

(8.108)

(In the literature [29,30], this expansion is given only up to the first term.) Also note that Z Di i D 3N dX ? .0X /b.X/; (8.109) Z 1 ja1m .X /j2 : (8.110) Ei i D 3N dX 4

Thus, the expression for the density .r/ of (8.71) may be rewritten as 1 .r/ D .0/.1  Zr/2 C .Ei i C 2Di i /r 2 C O.r 3 /: 3

(8.111)

162

8 Asymptotic Structure At and Near the Nucleus

The kinetic-energy-density tensor tij which is defined as tij .r/ D

ˇ ˇ 1 @2 @2 0 ˇ .rr C / ˇ0 ; 0 0 4 @ri @rj @rj @ri r Dr

(8.112)

is determined next. Though tedious, the derivation of the tensor is straightforward. After performing the various derivatives, we arrive at tij .r/ D

 1 ri rj Z < Z 2 ..0/ C A  r/ 2  .2  Zr/.ri Aj C rj Ai / 2 r 2r Zrl Œri .Elj C 4Dlj / C rj .Eli C 4Dli / C Eij .1  Zr/  2r  C 2.Fijl C Fj i l /rl C O.ri2 /:

(8.113)

The corresponding kinetic energy density is t.r/ D ti i .r/ D

1 2 2Z ŒZ .0/ C Ei i   rŒDi i C Ei i  C O.r 2 /: 2 3

(8.114)

The kinetic “force” z.rI  / which is defined in terms of tij .r/ as zi .r/ D 2 is then z.r/ D 2Z 2 .0/

@ tij @rj

r 4 r  Z.3Di i C Ei i / C O.r/: r2 3 r

(8.115)

(8.116)

From (8.114) it follows that t.0/ D

1 2 ŒZ .0/ C Ei i ; 2

(8.117)

so that finally, z.r/ D 2Z 2 .0/

r 4 r .1 C 2Zr=3/  ZŒ2t.0/ C 3Di i  C O.r/: (8.118) r2 3 r

Using (8.117), the expression (8.111) for the density may be written as 2 2 2 2 .r/ D .0/ 1  2Zr C ŒZ .0/ C t.0/r C r 2 Di i C O.r 3 /: (8.119) 3.0/ 3 

This expression for the density is employed later.

8.4 Near Nucleus Structure of vee .r/

163

8.4.4 Kinetic “Force” zs .rI s / Near the Nucleus The kinetic-energy-density tensor ts;ij .rI s / of the S system is defined as (see(3.39)) ts;ij .r/ D

ˇ   ˇ 1 @2 @2 0 00 ˇ  C .r r / s 0 00 0 00 ˇ 0 00 ; 4 @ri @rj @rj @ri r Dr Dr

(8.120)

where s .rr 0 / is the Dirac density matrix (see (3.12)). In terms of the S system orbitals nlm .r/ the tensor is    1 X @ ? @ ts;ij .r/ D <  .r/ nlm .r/ : 2 nlm @ri nlm @rj

(8.121)

Substituting (8.72) into the above expression, and employing (8.80) and (8.81), one obtains ts;ij .r/ D

1 2 Z .0/ri rj =r 2 2 ˇ2 3 X ˇˇ An1 ˇ .ıij  Zrıij  Zri rj =r/ C 8 n X 1 2Z <.A?n0 Cn0 /ri rj =r C O.ri2 /: 4

n

(8.122)

The kinetic-energy-density ts .r/ of the S system is then X 1 1 2 Z .0/  2Zr
n X 3 ˇ ˇ 1 ˇAn1 ˇ2 C O.r 2 /: C .3  4Zr/ 2 4

n

ts .r/ D ts;i i .r/ D

(8.123)

The kinetic “force” zs .rI s / which is defined as zs;i .r/ D 2

@ ts;ij .r/; @rj

(8.124)

is then obtained as X 1 r r <.A?n0 Cn0 /  12Z r2 4

r n X 3 ˇ ˇ ˇAn1 ˇ2 r C O.r/: 4Z 4

r n

zs .r/ D 2Z 2 .0/

(8.125)

164

8 Asymptotic Structure At and Near the Nucleus

From (8.123) it follows that ts .0/ D

ˇ2 1 2 9 X ˇˇ Z .0/ C An1 ˇ ; 2 4 n

(8.126)

so that the kinetic “force” is r zs .r/ D 2Z 2 .0/ 2 .1 C 2Zr=3/ r   X 1 r 2 ? ts .0/ C 3 <.An0 Cn0 / C O.r/: 4Z r 3 4

n

(8.127)

Employing (8.126), the S system density given by (8.88) may be written as  .r/ D .0/ 1  2Zr C C2r 2

2 ŒZ 2 .0/ C ts .0/r 2 3.0/



X 1 <.A?n0 Cn0 / C O.r 3 /: 4

n

(8.128)

This expression for the density is employed in Sect. 8.4.5.

8.4.5 Correlation-Kinetic Field Z tc .r/ Near the Nucleus The Correlation-Kinetic field Z tc .r/ near the nucleus is then obtained by substituting (8.118) and (8.127) into the definition (8.52), (8.53) of Z tc .r/. Thus,   X 1 4Z r 2 ? .t.0/  ts .0// C Di i  3 <.An0 Cn0 / C O.r/: (8.129) Z tc .r/ D .r/ r 3 4

n Since the densities of the interacting and S systems are the same, comparison of (8.119) and (8.128) leads to 1 X 2 <.A?n0 Cn0 / D Œt.0/  ts .0/ C Di i : 2 n 3

(8.130)

Making use of this relation, one obtains from (8.129) that Z tc .r/ D

4Z 1 r Œts .0/  t.0/ C O.r/: 3 .0/ r

(8.131)

Thus, we observe, that in contrast to the electron-interaction field E ee .r/, the Correlation-Kinetic field Z tc .r/ is finite at the nucleus of spherically symmetric or sphericalized systems.

8.5 Endnote

165

8.4.6 Structure of Potential Energy vee .r/ Near the Nucleus Finally, on combining (8.94) and (8.128) in (8.49), one obtains that near the nucleus r vee .r/ D

4Z r Œt.0/  ts .0/ C O.r/; 3.0/ r

(8.132)

so that vee .r/ approaches the nucleus linearly. This linearity is solely a consequence of Correlation-Kinetic effects. For completeness, we note that since the Hartree potential energy WH .r/ has zero slope at the nucleus, the HKS-DFT “exchange-correlation” potential energy vxc .r/ near the nucleus behaves as vxc .r/ D vxc .0/ C

4Z Œt.0/  ts .0/r C O.r 2 /: 3.0/

(8.133)

The linear term of vxc .r/ can thus be identified as being due to Correlation-Kinetic effects. The expression for vxc .r/ can also be derived [31] without employing the Q-DFT definitions of vee .r/. However, the advantage of working within the framework of Q-DFT is that the near nucleus structure of vee .r/ can be understood as a function of the separate electron correlations that contribute to it.

8.5 Endnote To summarize, in this chapter we have proved that the electron-interaction potential energy vee .r/ of the noninteracting fermions of the model S system is finite at the nucleus of atoms, molecules and solids irrespective of their state or their symmetry. Further, that correlations due to the Pauli principle and Coulomb repulsion, as well as Correlation-Kinetic effects contribute to this value. We have also proved via Q-DFT that for spherically symmetric or sphericalized systems, this potential energy vee .r/ approaches the nucleus linearly. This nearnucleus linearity of vee .r/ is solely due to Correlation-Kinetic effects. Correlations due to the Pauli principle and Coulomb repulsion do not contribute to this linear structure. All the various electron correlations also contribute to the structure quadratically. The conclusions of the near-nucleus structure were arrived at for spherically symmetric or sphericalized systems. However, these conclusions are equally valid for systems of arbitrary symmetry, provided the spherical average of the various properties is employed instead. We have further proved that the potential energy vB ee .r/ of the non interacting bosons of the model B system is also finite at the nucleus. As shown in QDFT,

166

8 Asymptotic Structure At and Near the Nucleus

Chap. 6, the B system is a special case of the S system. From the examples to be discussed later, it also becomes evident that Pauli and Coulomb correlations, as well as the corresponding bosonic Correlation-Kinetic effects all contribute to the value at the nucleus. Furthermore, for spherically symmetric B systems, the asymptotic near-nucleus structure of vB ee .r/ is also linear, but with a coefficient that differs from the one derived for the S system.

Chapter 9

Application of the Q-DFT Hartree Uncorrelated Approximation to Atoms

The Q-DFT of Hartree theory (Sect. 6.1) [1] maps the interacting system as described by Hartree theory [2, 3] to one of noninteracting fermions such that the equivalent density .r/ and energy E are obtained. For this mapping, the corresponding expressions for the local electron-interaction potential energy vH ee .r/ of the model fermions, and the energy E, contain Correlation-Kinetic components. If the model system is constructed with the Correlation-Kinetic contribution to these properties neglected, then one obtains the Q-DFT Hartree Uncorrelated Approximation (Sect. 6.1.1). As the wave function of these fermions is assumed to be a product of spin-orbitals, the neglect of the correlation contributions to the potential and kinetic energy, ensures that the model fermions are noninteracting in the rigorous sense of the word (see Sect. 9.2.1). This is thus the lowest level of approximation within Q-DFT and in quantum mechanics. In this chapter we apply the Q-DFT Hartree Uncorrelated Approximation to determine the ground state electronic structure of atoms [4, 5]. The self-consistently determined results are thus for atoms constructed of uncorrelated particles with external potential energy of v.r/ D Z=r: (Note that the local effective potential energy vs .r/ of these particles is not Z=r, in which case the solutions would be Hydrogenic.) What is remarkable is that even at this lowest level of approximation where the wave function satisfies neither the Pauli exclusion principle requirement of antisymmetry, nor contains any Coulomb correlations, certain atomic properties are determined very accurately. For example, atomic shell structure is observed throughout the Periodic Table, with highly accurate core–valence separations thereby obtained [4]. Thus, the requirement that the wave function be antisymmetric is not fundamental to obtaining the shell structure of atoms. The results of other properties determined are also compared with those of Hartree theory [5], thereby demonstrating the insignificance of Correlation-Kinetic effects for ground state properties so obtained. The calculations further demonstrate the commonality of the theoretical framework of Q-DFT at each level of approximation: local and nonlocal quantal sources give rise to fields which in turn lead to potential energies and components of the total energy, with the highest occupied eigenvalues compared with experimental ionization potentials. In the chapters to follow, it will be demonstrated how the systematic

167

168

9 Q-DFT Hartree Uncorrelated Approximation

inclusion of higher-order correlations viz. those due to the Pauli exclusion principle, Coulomb repulsion and Correlation-Kinetic effects, improve on these results. In both the Q-DFT Hartree Uncorrelated (see Sect. 6.1.1 for the equations) and Hartree theory approximations (see QDFT), for open-shell atoms, the orbital densities are spherically averaged. (The other choice for the former approximation would have been to spherically average the electron-interaction field of (6.6).) Thus, in both theories, the atoms are spherically symmetric. This is the central field model. Thus, all the fields are conservative, and the corresponding work done in these fields is path-independent. The results for the Q-DFT Hartree Uncorrelated Approximation are obtained by modification of the Q-DFT Pauli Approximation code [6]. The Hartree theory calculations are done by modification of the Froese–Fischer Hartree–Fock theory code [7]. The calculations are performed for the elements of the Periodic Table up to Ac (Z D 89). Atomic shell structure is thus exhibited up to this atomic number. The results for the energy E and highest occupied eigenvalue m are given for He (Z D 2) to Kr (Z D 36). To explain the physics underlying the mapping via Q-DFT, we begin by first discussing in the following section, the details of the electronic structure of a typical atom such as the Ne atom as determined via the Q-DFT Hartree Uncorrelated Approximation. Thus, for this atom, we show the structure of the self-consistently determined local and nonlocal quantal sources, the corresponding fields, and the resulting potential energies. In the sections to follow, we present for the other atoms, the results for the shell structure, core–valence separations, the energy, the highest occupied eigenvalue, and discuss the degree to which the aufbau principle – Madelung’s rule [8] for the order in which electron shells are filled – is satisfied.

9.1 Electronic Structure of the Neon Atom The electronic density .r/ of the Ne atom .1s 2 2s 2 2p 6 / – the local quantal source – is spherically symmetric about the nucleus, and is plotted in Fig. 9.1. The radial probability density 4 r 2 .r/ of the atom is plotted in Fig. 9.2. The two atomic shells are clearly evident. The maximum of the inner shell is at r D 0:105 a.u., and that of the outer shell at r D 0:649 a.u. The minimum at the intershell region is at r D 0:312 a.u. The self-interaction-correction (SIC) charge distribution SIC .rr 0 / – the nonlocal quantal source – is plotted for different electron positions r in Figs. 9.3–9.7. In these figures, the electron position is along the z-axis corresponding to  D 0ı . The cross-sections plotted in Figs. 9.3 and 9.4 correspond to  0 D 0ı with respect to the nucleus–electron direction. In Figs. 9.5–9.7, the cross-sections plotted are for  0 D 0ı , 45ı , 90ı . The graph for r 0 < 0 is the structure for  0 D and r 0 > 0. The dynamic nature of these charge distributions as a function of electron position is evident from these figures. Observe that for all electron positions, the charge SIC .rr 0 / is negative.

9.1 Electronic Structure of the Neon Atom

169

600 Ne Atom 500

ρ (r) (a.u.)

400

300

200

100

0 0.0

0.1

0.2

0.3

r (a.u.)

Fig. 9.1 The electron density .r/ of the Ne atom as determined in the Q-DFT Hartree Uncorrelated Approximation

10

4 π r2 ρ (r) (a.u.)

8

6

4

2

0 0.0

0.5

1.0

1.5

2.0

2.5

r (a.u.) 2

Fig. 9.2 The radial probability density 4 r .r/ of the Ne atom as determined in the Q-DFT Hartree Uncorrelated Approximation

170

9 Q-DFT Hartree Uncorrelated Approximation Ne Atom

ρSIC (r r’) (a.u.)

0

–100 r=0

–200

–300 –0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

r′ (a.u.)

Fig. 9.3 Cross-section through the SIC hole  .rr 0 / for a model fermion at the nucleus r D 0 a.u. The fermion is on the z-axis corresponding to  D 0. The cross-section plotted corresponds to  0 D 0ı . The graph for r 0 < 0 corresponds to the structure for  0 D and r 0 > 0. For this electron position, the hole is spherically symmetric SIC

Ne Atom 0

ρSIC (r r’) (a.u.)

r = 0.312 a.u. –100 r = 0.105 a.u.

–200

–300 –0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

r′ (a.u.)

Fig. 9.4 Same as in Fig. 9.3 except that the model fermion is at r D 0:105 and 0:312 a.u. The cross-sections for  0 D 45ı , 90ı are different, but indistinguishable on this scale

9.1 Electronic Structure of the Neon Atom

171 Ne Atom

Electron at r = 0.649 a.u.

θ’ = 45°

–1 ρSIC (r r’) (a.u.)

θ’ = 90°

θ’ = 0°

–2

–3

–4

–5 –1.5

–1.0

–0.5

0.0 r′ (a.u.)

0.5

1.0

1.5

Fig. 9.5 Same as in Fig. 9.3 except that the model fermion is at r D 0:649 a.u. The cross-sections for  0 D 0ı ; 45ı ; 90ı are plotted Ne Atom θ’ = 90°

Electron at r = 2 a.u. 0.0

θ’ = 45°

ρSIC (r r’) (a.u.)

–0.5

θ’ = 0° –1.0

–1.5

–2.0

–1.5

–1.0

–0.5

0.0

0.5

r′ (a.u.)

Fig. 9.6 Same as in Fig. 9.5 except that the model fermion is at r D 2 a.u.

1.0

1.5

172

9 Q-DFT Hartree Uncorrelated Approximation Ne Atom Electron at r = 10 a.u.

θ’ = 90°

ρ SIC (r r’) (a.u.)

0.0

–0.5 θ’ = 45°

θ’ = 0°

–1.0

–1.5 –1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

r′ (a.u.)

Fig. 9.7 Same as in Fig. 9.5 except that the model fermion is at r D 10 a.u.

In Fig. 9.3, the electron is at the nucleus r D 0 a.u. For this and all electron positions, the SIC .rr 0 / charge is spherically symmetric about the nucleus. Each cross-section of SIC .rr 0 / corresponding to different  0 is also symmetrical about it. In Fig. 9.4, the electron positions considered are at r D 0:105 a.u. and r D 0:312 a.u. On the scale of this figure, the cross sections for  0 D 0ı , 45ı , 90ı are indistinguishable. The differences in the structure for different  0 become evident in Figs. 9.5–9.7 for r D 0:649, 2.0, and 10.0 a.u. By about r D 10 a.u. in the classically forbidden region, SIC .rr 0 / is essentially a static charge whose structure no longer changes with electron position. In Fig. 9.8, the electron-interaction field E ee .r/ due to the pair-correlation density g H .rr 0 / (see (6.6)), and its Hartree E H .r/ and SIC E SIC .r/ components are plotted. The Hartree field E H .r/ is positive because its source charge, the density .r/, is positive. Since the density .r/ is spherically symmetric about the nucleus, the Hartree field E H .r/ (see (6.11)) vanishes there. (This is not shown in the graph because the scale is logarithmic.) Although the first shell is evident, a careful examination of the graph for E H .r/ indicates the existence of the second shell. As the density is a static charge, and the total number of electrons in this atom is N D 10, the field decays asymptotically in the classically forbidden region as E H .r/ 10=r 2. The field E SIC .r/ is due to the dynamic source charge SIC .rr 0 / (see (6.12)). It is negative because SIC .rr 0 / is negative for all electron positions. E SIC .r/ also vanishes at the nucleus. (Again this is not shown because of the logarithmic scale.) The

9.1 Electronic Structure of the Neon Atom

60

173

Ne Atom

eH

Force Fields (a.u.)

40

eee eH~10/r2 eee~9/r2

20

0

eSIC –1/r2 –20

0.001

0.01

0.1

1

10

r (a.u.)

Fig. 9.8 The electron-interaction Eee .r/ field, and its Hartree EH .r/ and SIC E SIC .r/ components

reason why the field vanishes there is that the source charge SIC .rr 0 / is spherically symmetric about the nucleus for this electron position (see Fig. 9.3). The two-shell structure is clearly evident in the plot of E SIC .r/. Asymptotically, in the classically forbidden region, the field decays as E SIC .r/  1=r 2 . This is because the total charge of the source SIC .rr 0 / is 1, and for these asymptotic positions of the electron, the source charge becomes static as noted above. The electron-interaction field E ee .r/, which is the sum of E H .r/ and E SIC .r/, is positive, vanishes at the nucleus, clearly exhibits shell structure, and decays asymptotically as E ee .r/ 9=r 2 . The structure of the field E ee .r/ may also be understood from the perspective of its dynamic source, the pair-correlation density g H .rr 0 /. The field E ee .r/ is positive because the pair-correlation density g.rr 0 / is positive for all electron positions. The field E ee .r/ vanishes at the nucleus because the pair density g H .rr 0 / is spherically symmetric about the nucleus for an electron at that position. The total charge of this source is N  1 D 9, and for asymptotic positions of the electron, the charge also becomes static. Hence, the asymptotic decay of E ee .r/ given earlier.

174

9 Q-DFT Hartree Uncorrelated Approximation

Finally, the corresponding work done in the above fields is plotted in Fig. 9.9, i.e. the electron-interaction Wee .r/, the Hartree WH .r/, and SIC W SIC .r/ potential energies. The work done Wee .r/ and WH .r/ are positive and finite at the nucleus because the fields E ee .r/ and E H .r/ are positive, and to determine the work, it is the area under these curves that is calculated. The potential energies decay asymptotically as Wee .r/ 9=r and WH .r/ 10=r because of how the corresponding fields decay in this region as discussed above. The structure of the work W SIC .r/ may similarly be understood from that of the field E SIC .r/. The potential energy W SIC .r/ is negative because the field E SIC .r/ is negative, is finite at the nucleus,

Ne Atom 30 WH

Wee

Potentials (a.u.)

20

WH ~ 10/r 10 Wee ~ 9/r

0

–1/r

W SIC –10

0.001

0.01

0.1

1

10

r (a.u.)

Fig. 9.9 The electron-interaction Wee .r/ potential energy, and its Hartree WH .r/, and SIC W SIC .r/ components

9.2 Atomic Shell Structure and Core–Valence Separation

175

and decays as W SIC .r/ 1=r as a consequence of the decay of E SIC .r/. The two shells are particularly evident in the structure of W SIC .r/. The results for the total energy E as obtained from the fields E H .r/ and E SIC .r/ and (6.32), and the highest occupied eigen value m of the differential equation (6.28) are given together with the results for the other atoms in the Tables 9.2 and 9.3.

9.2 Atomic Shell Structure and Core–Valence Separation There is considerable history on the subject matter of atomic shell structure [9]. Until fairly recently [10], it was thought that for atoms of atomic number Z > 18, the correct number of atomic shells was not exhibited by Hartree–Fock theory in which only correlations due to the Pauli exclusion principle are considered in the wave function which is a single Slater determinant. As such highly inaccurate core– valence separations were obtained. Finally, it was shown [10] that Hartree–Fock theory did in fact lead to accurate atomic shell structure. More recent work [11, 12] has focused on studying the effect of Coulomb correlations on shell structure via manybody perturbation theory and the use of configuration-interaction wave functions. The principle conclusion of these papers is that Coulomb correlations have negligible effect on shell structure and core–valence separations. The question then arose as to whether atomic shell structure is a consequence of and thereby attributable to Pauli correlations? To answer this question, calculations were performed [4] within both Hartree theory and the Q-DFT Hartree Uncorrelated Approximation. In Figs. 9.10–9.13 we plot the radial probability density D.r/ D 4 r 2 .r/ as determined by the Q-DFT HartreeUncorrelated Approximation for K, Ca, Sc, Kr;

Fig. 9.10 The radial probability density D.r/ D 4 r 2 .r/ for the fourth period atoms K, Ca, Sc, and Kr as determined in the Q-DFT Hartree Uncorrelated Approximation

176

Fig. 9.11 11 The radial probability density D.r/ D 4 r 2 .r/ for the fifth period atoms Rb, Sr, Y, Xe as determined in the Q-DFT Hartree Uncorrelated Approximation

Fig. 9.12 The radial probability density D.r/ D 4 r 2 .r/ for the sixth period atoms Cs, Ba, La, Rn as determined in the Q-DFT Hartree Uncorrelated Approximation

Fig. 9.13 The radial probability density D.r/ D 4 r 2 .r/ for the seventh period atoms Fr, Ra, Ac as determined in the QDFT Hartree Uncorrelated Approximation

9 Q-DFT Hartree Uncorrelated Approximation

9.2 Atomic Shell Structure and Core–Valence Separation

177

Rb, Sr, Y, Xe; Cs, Ba, La, Rn; and Fr, Ra, and Ac atoms, respectively. Note that these figures are plotted on a logarithmic scale. It is evident that the correct number of shells is exhibited for each period beginning at the corresponding alkali metal, the outermost shoulder being indicative of the valence shell for each atom. The corresponding D.r/ of Hartree theory is indistinguishable on the scale of these figures. The difference occurs primarily in the asymptotic classically forbidden region [4]. For Hartree theory shell structure up to Z D 18, see [13]. For earlier precomputer era calculations of D.r/ using crude Hartree functions, see [14]. From the above shell structure, the core–valence separation is obtained by considering the midpoint ri of the two outermost points of inflection in D.r/. The core radius ri and the core charge q are defined as Z

ri

qD

D.r/dr

(9.1)

0

and the results thus obtained via the Q-DFT Hartree Uncorrelated and Hartree theory approximations are given in Table 9.1. The standard core radius rs determined from the D.r/ of each theory such that the standard core charge q based on the electronic configuration of atoms in their ground state is also given. It is evident that the core– valence separations as obtained by both theories are accurate when compared to the standard values. Thus, we see that accurate atomic shell structure is exhibited by a completely uncorrelated wave function, i.e. one that does not incorporate correlations due to

Table 9.1 The core charge q and radius ri as determined by Hartree theory and the Q-DFT Hartree Uncorrelated Approximation. The radius rs corresponding to the standard ground state electronic configuration is also given. All values are in (a.u.). The atomic configurations are based on Madelung’s law [8] Atom Hartree theory Approximate Q-DFT ri q rs q ri q rs q K(4s 1 ) 3.52 18.04 3.39 18 3.42 18.10 3.19 18 Ca(4s 2 ) 2.79 18.10 2.64 18 2.68 18.13 2.53 18 Sc(4s 2 3d 1 ) 2.43 18.63 2.83 19 2.33 18.64 2.65 19 1.09 28.01 1.09 28 1.07 27.95 1.07 28 Kr(4s 2 4p 6 ) Rb(5s 1 ) Sr(5s 2 ) Y(5s 2 4d 1 ) Xe(5s 2 5p 6 )

4.05 3.09 2.77 1.40

36.07 36.02 36.39 45.60

3.84 3.07 3.38 1.46

36 36 37 46

3.71 2.99 2.75 1.37

36.04 36.04 36.55 45.52

3.62 2.95 3.16 1.44

36 36 37 46

Cs(6s 1 ) Ba(6s 2 ) La(6s 2 5d 1 ) Rn(6s 2 6p 6 )

4.82 3.75 3.29 1.68

54.11 54.09 54.46 78.04

4.45 3.62 3.85 1.68

54 54 55 78

4.56 3.64 3.22 1.65

54.13 54.13 54.53 77.98

4.21 3.49 3.65 1.65

54 54 55 78

Fr(7s 1 ) Ra(7s 2 ) Ac(7s 2 6d 1 )

4.78 4.12 3.72

86.00 86.14 86.60

4.78 3.92 4.17

86 86 87

4.65 3.96 3.57

86.05 86.15 86.59

4.53 3.78 3.96

86 86 87

178

9 Q-DFT Hartree Uncorrelated Approximation

the Pauli exclusion principle, Coulomb repulsion, and the correlation contribution to the kinetic energy. Accurate atomic shell structure is also obtained from the Hartree theory wave function which also does not incorporate the Pauli or Coulomb correlations, but whose orbitals do account for the Correlation-Kinetic effects. Additionally, accurate atomic shell structure is obtained from purely electrostatic fields whose sources are quantum-mechanical and derived from uncorrelated wave functions. In Hartree theory, the orbital-dependent potential energy of each electron is the work done in the electric field of the static charge distribution of all the other electrons. In the Q-DFT Hartree Uncorrelated Approximation, the potential energy of the model fermions is the work done in the electric field of the dynamic paircorrelation density.

9.2.1 Endnote The Pauli exclusion principle, a fundamental postulate of quantum mechanics, is a consequence of the indistinguishability and spin quantum number of the electrons. To quote Pauli in his Nobel lecture [15]: “A half-integer value of the spin quantum number is always connected with antisymmetrical states (exclusion principle), an integer spin with symmetrical states.” (The emphasis is by Pauli.) The state here means the many-body wave function. The principle is also described in the words of Coulson [16]: “The statement that every final wave function for an atom or molecule must be antisymmetrical in every pair of electrons is one of the most fundamental statements of the Pauli exclusion principle.” (For the historical evolution of the Pauli principle in quantum mechanics, see Slater’s book [17]. On page 191, Slater states that, “Pauli’s principle was formulated as a result of the study of the periodic table of the elements, principally by Bohr, during the years from 1920 to 1925.” Then on page 285, “These facts, and others like them, led Heisenberg and Dirac to the fundamental postulate that only the antisymmetric solution existed in nature and that the symmetric solution represented a solution which we do not find in the ordinary universe.”) A consequence of the quantum mechanical principle is that no more than one electron can occupy each one-orbital state. The one-orbital states are approximate, except when there are no interactions assumed in the Hamiltonian. The statement of antisymmetry of the many-body wave function is therefore more fundamental than the statement that only one electron can occupy each one-electron state. In the central field model, the fact that no more than one electron can occupy each orbital state means “that no two electrons in the same atom can have the same values of all four quantum numbers n; l; ml ; ms :” [17]. In theoretical calculations, such as those of Hartree theory and the Q-DFT Hartree Uncorrelated Approximation of Sect. 9.1, the occupation of the states in this manner, devoid of the requirement of antisymmetry of the wave function, is therefore ad hoc. In the past, it has been assumed that such an occupation of the states, and hence of shell structure in atoms, was possible only with an antisymmetric wave function,

9.4 Highest Occupied Eigenvalues

179

or equivalently one that obeyed the Pauli principle in the sense of Pauli and Coulson, above. The results of the previous section show that for the ad hoc occupation of states, accurate atomic shell structure and core–valence separations are possible even for Hartree product-type wave functions that are not antisymmetrical and which therefore violate the Pauli exclusion principle in its quantum mechanical formulation.

9.3 Total Ground State Energies The total ground state energy of atoms from He.Z D 2/ to Kr.Z D 36/ as determined within Hartree theory and the Q-DFT Hartree Uncorrelated Approximation is given in Table 9.2. The negative values in atomic units are given. As the wave function of the latter approximation is not the same as that of Hartree theory, the corresponding energies are upper bounds to the Hartree theory values. The two sets of results, however are essentially equivalent. The differences between the two sets of energies in parts per million .ppm/ are also given. For He .Z D 2/, the energies of the two theories are exactly equivalent. (The same value is also obtained within Hartree–Fock theory and in the Q-DFT Pauli Approximation (see Chap. 10). The reason for this is that the differential equation of all four theories for this atom is the same.) For Li.Z D 3/, the difference is 13 ppm; for B.Z D 5/ to Al.Z D 13/, the difference lies between 151 and 102 ppm; for Si.Z D 14/ to Cl.Z D 17/ the difference lies between 84 and 48 ppm; and for Ar.Z D 18/ to Kr.Z D 36/, the difference is 41 ppm and less. A plot of these differences is given in Fig. 9.14. The minimal difference between the two sets of energies is an approximate measure of the Correlation-Kinetic energy. Thus, for the ground state of atoms, at this Hartree level of approximation, the correlation contribution to the kinetic energy is negligible. (A similar conclusion will be arrived at when Pauli correlations are incorporated (see Chap. 10)). (The Q-DFT Hartree Uncorrelated Approximation ground state energy values for Be.Z D 4/ and Cr.Z D 24/ lie below those of Hartree theory in spite of considerable refinement of the two respective codes. There appears no reasonable explanation for this anomalous result.)

9.4 Highest Occupied Eigenvalues The highest occupied eigenvalues of the Q-DFT Hartree Uncorrelated Approximation differential equation for the atoms 2 He–36 Kr, with the exception of the first transition group (21 Sc–28 Ni/ and the noble metal 29 Cu, are given in Table 9.3. The values for the latter atoms are given in Table 9.4. The configuration of these atoms according to Madelung’s law [8] is also given. The exception is in Table 9.3 for 30 Zn whose structure as obtained is noted.

180

9 Q-DFT Hartree Uncorrelated Approximation

Table 9.2 Total ground state energy of atoms in the central field model as obtained within Hartree theory and the Q-DFT Hartree Uncorrelated Approximation. The negative values of the energies in a.u. are given. The difference between the results in ppm are also given Atomic Atom Hartree Q-DFT Hartree Difference Number Theory Uncorrelated (ppm) Z Approximation 2

He

2:86168

2:86168

0

3

Li

7:412379

7:412284

13

4

Be?

14:50897

14:50995

5

B

24:33916

24:33660

105

6

C

37:26941

37:26392

147

7

N

53:62095

53:61288

151

8

O

73:71391

73:70358

140

9

F

97:86820

97:85586

126

10

Ne

126:4036

126:3895

112

11

Na

159:2532

159:2330

127

12

Mg

196:5260

196:5043

110

13

Al

238:2409

238:2165

102

14

Si

284:5890

284:5652

84

15

P

335:7290

335:7060

69

16

S

391:8153

391:7929

57

17

Cl

453:0009

452:9790

48

18

Ar

519:4385

519:4171

41

19

K

590:9995

590:9758

40

20

Ca

667:7956

667:7721

35

21

Sc

749:8229

749:7937

39

22

Ti

837:4135

837:3811

39

23

V

930:7207

930:6854

38

24

Cr?

1029:741

1029:846

25

Mn

1135:039

1134:998

36

26

Fe

1246:322

1246:279

35

27

Co

1363:867

1363:822

33

28

Ni

1487:809

1487:762

32

29

Cu

1618:281

1618:232

30

30

Zn

1755:418

1755:367

29

31

Ga

1899:229

1899:177

27

32

Ge

2049:676

2049:625

25

33

As

2206:837

2206:788

22

34

Se

2370:791

2370:743

20

35

Br

2541:618

2541:573

18

36

Kr

2719:403

2719:359

16

?

See text

9.4 Highest Occupied Eigenvalues

181

N

Energy Difference (ppm)

150

C Na

B

100

Ne

Al

50 Ar

Sc Ca

Ni

Kr

Li 0

He 0

10

20

30

Atomic Number Z Fig. 9.14 The difference between the ground state energy obtained via the Q-DFT Hartree Uncorrelated Approximation and Hartree theory, in parts per million (ppm). This difference represents an accurate approximation to the Correlation-Kinetic contribution to the total energy of Hartree theory

Now, as proved in Chap. 7 (see also Sect. 3.7), the asymptotic structure of the effective potential energy vs .r/ in the classically forbidden region for the fully correlated model S system is vs .r/ 1=r. (In the fully correlated case, the model S system accounts for all the correlations: Pauli, Coulomb, and Correlation-Kinetic effects. The highest occupied eigenvalue of the differential equation is the negative of the ionization potential.) In the Q-DFT Hartree Uncorrelated Approximation, the asymptotic structure of the corresponding effective potential energy function is also vs .r/ 1=r. This follows from the fact that in this approximation (see (6.28)–(6.30)): vs D v.r/ C WH .r/ C WHSIC .r/: (9.2) Since v.r/ D Z=r, and in the asymptotic r ! 1 limit, WH .r/ N=r with N D Z, the contributions of v.r/ and WH .r/ in this region cancel. Thus, vs .r ! 1/ D WHSIC .r ! 1/. In the asymptotic region, the dynamic quantal source SIC .rr 0 / of the potential energy WHSIC .r/ becomes a static charge (see Sect. 9.1). The total charge of this source is 1, so that WHSIC .r ! 1/ 1=r. The highest occupied eigenvalue is governed principally by this asymptotic structure of the effective potential energy vs .r/. Thus, it is meaningful to compare the highest occupied eigenvalue as obtained within this approximation to the experimental ionization potential. The experimental ionization potentials [18] are also given in Tables 9.3 and 9.4.

182

9 Q-DFT Hartree Uncorrelated Approximation

Table 9.3 Highest occupied eigenvalues of atoms as determined within Hartree theory and the Q-DFT Hartree Uncorrelated Approximation. The “exact” theoretical values for the highest occupied eigenvalue of the fully interacting model system differential equation, and the experimental ionization potentials are also given. The negative values of the eigen energies in Rydbergs are given Atom Hartree Q-DFT – Hartree “Exact” Experimentb Theory Uncorrelated Theorya Approximation 2 4

Atoms with last closed shell a s subshell 1.836 1.836 1.808

He:1s 2 Be:[He]2s

2

1.807

0.576

0.627

0.676

0.685

4Mg:[Ne]3s 2

0.464

0.506

0.518

0.562

4Ca:[Ar]4s 2 30 Zn:[Ar]3d 10 4s 2

0.357 0.540

0.387 0.623

10

1.251

12 20

Ne:[He]2s 2 2p 6

Ar:[Ne]3s 2 3p 6 36 Kr:[Ar]3d 10 4s 2 4p 6

Noble gas atoms 1.259

18

0.834 0.731

0.870 0.763

3

0.352

Alkali metals 0.406

11

0.328 0.265

0.375 0.301

9

1.081

Halogens 1.092

0.718 0.642

0.755 0.675

Li:[He]2s 1

Na:[Ne]3s 1 19 K:[Ar]4s 1

F:[He]2s 2 2p 5

17

Cl:[Ne]3s 2 3p 5 35 Br:[Ar]3d 10 4s 2 4p 5

5

B:[He]2s 2 2p 1

6

C:[He]2s2 2p 2 2

0.449 0.690

1.594

1.585

1.094

1.158 1.029

0.400

0.396

0.364

0.378 0.319

1.368

1.281

0.982

0.953 0.868

Atoms with less than half-filled p subshells 0.456 0.497 0.598

0.610

0.609

0.634

0.820

0.828

13

Al:[Ne]3s 3p

1

0.282

0.347

0.428

0.440

14

Si:[Ne]3s2 3p 2

0.392

0.442

0.714

0.599

0.284 0.380

0.351 0.428

31

Ga:[Ar]3d10 4s 2 4p 1 32 Ge:[Ar]3d10 4s 2 4p 2

7 8

0.441 0.581

Atoms with half- and two thirds-filled p subshells N:[He]2s 2 2p 3 0.761 0.779 1.056

1.068

O:[He]2s 2 2p 4

0.932

1.172

1.001

15

P:[Ne]3s 3p

3

0.498

0.541

0.748

0.771

16

S:[Ne]3s 2 3p 4

0.606

0.646

0.832

0.761

3

0.468

0.508

0.721

Se:[Ar]3d 10 4s 2 4p 4 a See [20]; b See [18]

0.555

0.591

0.717

33 34

2

0.918

10

2

As:[Ar]3d 4s 4p

9.4 Highest Occupied Eigenvalues

183

Table 9.4 Highest occupied eigenvalues for the first transition group of atoms (21 Sc–28 Ni) and the noble metal atom 29 Cu as determined within Hartree theory and the Q-DFT Hartree Uncorrelated Approximation. The 4s and 3d eigenvalues are given for those atoms which obey Madelung’s law for which the 4s subshell fills before the 3d subshell, so that the highest occupied state is 3d . For the other atoms, the highest occupied eigenvalue, which is 4s, is given. The experimental values are also given. The negative values of the eigen energies in Rydbergs are given Atom

Hartree Theory

21

Sc:[Ar]4s 2 3d n

nD1

22

Ti

nD2

23

V

nD3

24

Cr

nD4

25

Mn

nD5

27

Fe Co 28 Ni

nD6 nD7 nD8

29

nD9

26

Cu a See [18]

Q-DFT - Hartree Uncorrelated Approximation

First transition group 0.398(4s) 0.435(4s) 0.324(3d ) 0.284(3d ) 0.417(4s) 0.465(4s) 0.356(3d ) 0.434(4s) 0.489(4s) 0.419(3d ) 0.401(4s) 0.511(4s) 0.475(3d ) 0.467(4s) 0.532(4s) 0.527(3d ) 0.483(4s) 0.551(4s) 0.498(4s) 0.570(4s) 0.512(4s) 0.588(4s) Noble metal 0.526(4s)

0.606(4s)

Experimenta

0.481 0.501 0.495 0.497 0.546 0.578 0.578 0.561

0.568

The highest occupied eigenvalues as obtained via Hartree theory are also given in the tables. These eigenvalues can be interpreted as removal energies only in the context of Koopmans’s theorem [19]. In other words, this interpretation is rigorous provided that when an electron is removed to infinity, the remaining orbitals do not relax but retain their original neutral atom structure. For purposes of comparison, the “exact” theoretical highest occupied eigenvalue for the fully correlated model S system for the atoms 2 He18 Ar are also given [20] in Table 9.3. These “exact” values are obtained indirectly by numerical methods [21–23] whereby the S system orbitals and eigenvalues are determined by working backwards from knowledge of the “exact” densities. The densities are those determined from accurate configuration-interaction type wave functions. exact HU Let us first consider the results of Table 9.3. With the designations m , m , H and m for the highest occupied eigenvalue of the “exact,” Q-DFT Hartree Uncorrelated Approximation, and Hartree theory, respectively, the following remarks can HU be made. For all the atoms of the table, the m results are consistently superior to H exact those of m when compared to the “exact” values m . This is graphically demonexact strated in Fig. 9.15, where the magnitude of the difference relative to m is plotted

184

9 Q-DFT Hartree Uncorrelated Approximation Ne N Hartree Theory

Si

F

0.2

Cl Ar

P

O

S

C

Exact

|∈m - ∈m | (Ryd)

0.3

B

Al

0.1

Q-DFT Hartree Uncorrelated Approximation

Be He

Li

Na

Mg

0.0 0

5

10

15

Atomic Number Z

Fig. 9.15 The magnitude of the difference between the “exact” highest occupied eigenvalue exact m , and the corresponding eigenvalues m of the Q-DFT Hartree Uncorrelated Approximation and Hartree theory, as a function of the atomic number Z

as a function of the atomic number Z. The reason for this superiority is that in the Q-DFT Hartree Uncorrelated Approximation, the asymptotic structure of the local effective potential energy is vs .r/ 1=r, which is the same as that for the fully HU correlated model S system as explained above. In particular, the results of m for 2 3 4 11 12 the atoms He, Li, Be, Na, and Mg, for which the valence shell is an s subshell, are very accurate (see Fig. 9.15). This accuracy, even at the purely uncorrelated level, arises because exchange effects are not significant for these atoms. There are no exchange effects in the s shells. For 11 Na, and 12 Mg, the valence subshell is separated both in energy and distance from the core s and p subshells. Thus, for these atoms, intershell exchange effects are also negligible. HU H The results of Table 9.3 also show the m values to be superior to those of m when compared to the experimental ionization potentials. The results of Table 9.3 also clearly demonstrate the lack of rigor of the Koopmans’s theorem interpretation of the highest occupied eigenvalue of Hartree theory as the removal energy when applied to finite systems.

9.4.1 Satisfaction of the Aufbau Principle The filling up of the electron shells in atoms, the aufbau or building-up principle, was discovered empirically by Madelung [8]. The rule, in two parts, states:

9.4 Highest Occupied Eigenvalues

185

1. When consecutive atoms are considered, the electron shells are filled in the order of the sum of two quantum numbers n and l, that is .n C l/. 2. Shells with equal .nCl/ numbers are filled in the order of the quantum number n. (n; l are the principal and orbital angular momentum quantum numbers.) This rule, depicted in Fig. 9.16, predicts that for the first transition group atoms, the 4s shell will be filled before the 3d shell. The self-consistently determined configuration of the atoms listed in Table 9.3 as determined via both the Q-DFT Hartree Uncorrelated Approximation and Hartree theory are consistent with the aufbau principle. The Q-DFT Hartree Uncorrelated Approximation configuration of the first transition group atoms 21 Sc, 22 Ti, 23 V, 24 Cr, 25 Mn are also consistent with the aufbau principle (see Table 9.4). In other words for these atoms, the 4s subshell fills before the 3d subshell: the 4s eigenvalue is more negative than the 3d eigenvalue. Thus, it is the 3d eigen state that is the highest occupied. For the other atoms of the group n

10

O

X

O

X

9

X

O

X

O

X

8 O

X

O

X

O

X

O

X

O

X

O

6 O

X

O

X

O

X

X

O

X

O

X

4 O

X

O

X

X

O

X

2 O

X

7

5

3

1

O (n + ) = EVEN X (n + ) = ODD

X

0

1

2

3

4

5

Fig. 9.16 The aufbau principle for neutral atoms according to Madelung [8]. (n; l are the principal and orbital angular momentum quantum numbers)

186

9 Q-DFT Hartree Uncorrelated Approximation

viz. 26 Fe, 27 Co, 28 Ni, and the noble metal 29 Cu, the highest occupied state is 4s. It is only for these four atoms that the aufbau principle is not obeyed by the approximation. In contrast, for the first transition group atoms and 29 Cu, it is only for 21 Sc that Hartree theory obeys the principle. It is interesting that the noninteracting model system so accurately reflects the aufbau principle. In comparison to experimental ionization potentials (see Table 9.4), for 21 Sc, H HU the highest occupied eigenvalue m .3d / is superior to m .3d /. For 22 Ti and 23 V, H HU HU m .4s/ is also superior to m .3d /. However, for 24 Cr and 25 Mn, m .3d / is supeH rior to m .4s/, and in contrast to the latter, are close to the experimental values, as HU are the results of m .4s/ for 26 Fe, 27 Co and 28 Ni. Hence, as a consequence of the correct asymptotic structure, the Q-DFT Hartree Uncorrelated Approximation leads to reasonably accurate highest occupied eigenvalues when compared to experimental ionization potentials.

Chapter 10

Application of the Q-DFT Pauli Correlated Approximation to Atoms and Negative Ions

In Chap. 9, we determined the ground state electronic structure of atoms within Q-DFT via a system of rigorously uncorrelated fermions. Neither Pauli nor Coulomb correlations, nor the correlation contribution to the kinetic energy were considered, the approximate wave function being a product of spin-orbitals. However, in these calculations, the incorporation of the Pauli exclusion principle, and thereby the occupation of states, was ad hoc (see Sect. 9.2.1). In this chapter, we determine the ground state electronic properties of atoms and negative ions via the Q-DFT Pauli Approximation (see Sect. 6.2.1). In this approximation, one considers a system of noninteracting fermions, with the approximate wave function being a single Slater determinant of spin-orbitals. Thus, correlations between the model fermions of parallel spin – Pauli correlations – are now explicitly incorporated via the wave function. However, Coulomb correlations and Correlation-Kinetic effects are once again ignored. The fact that Coulomb correlations are not considered is of course evident as the choice of the approximate wave function is a Slater determinant. Note, however, that the contributions of Correlation-Kinetic effects to (1) the kinetic energy, (2) the local electroninteraction potential energy that generates the spin-orbitals, and (3) the electroninteraction energy component of the total energy, are also not considered. Thus, within the Q-DFT Pauli Approximation, the local electron-interaction potential energy of the model fermions vP ee .r/ (6.57) and the electron-interaction energy P Eee .r/ (6.62) are solely due to Pauli correlations. In contrast, within Hartree– Fock theory, the fact that electrons of parallel spin are correlated, is explicitly accounted for in the kinetic energy. These correlations are then reflected via the selfconsistently determined spin-orbitals in the nonlocal exchange operator and in the Hartree and exchange energy components of the electron-interaction energy. Thus, differences between the results of the Q-DFT Pauli Approximation and Hartree– Fock theory can be attributed to, and are an estimate of, the contributions to the kinetic energy due to correlations arising from the Pauli exclusion principle. As in the previous Chap. 9, the calculations are performed in the central field model. There, for open-shell systems, the central field model was achieved by spherically averaging the spin-orbitals. In this chapter, the central field model for such systems is accomplished [1, 2] by spherically averaging the Pauli field E P x .r/ due to the corresponding Fermi hole charge (see Sect. 6.2.1). The analytical expressions

187

188

10 Q-DFT Pauli Correlated Approximation

for the spherical average of the Pauli field E P x .r/, and the resulting Pauli potential energy WxP .r/ are derived in Appendix D. In the central field model, the field EP x .r/ is therefore spherically symmetric. Thus, the field is conservative, and the work done to move the model fermion in this field path-independent. (In the remainder of this chapter and in Appendix D, the superscript P will be dropped, it being understood that the calculations are performed within the Pauli approximation.) To demonstrate the underlying physics, we begin in the following section by describing in detail the structure of a typical atom – the three-shell Argon atom – as obtained via the Q-DFT Pauli Approximation. Here we show the self-consistently determined structure of the local and nonlocal source charge distributions, the corresponding fields and potential energies. In the sections to follow, we present for the atoms He.Z D 2/ to Ac.Z D 89/ in their ground state, results for the shell structure, core–valence separation, total ground state energies, highest occupied eigenvalues, various single-particle expectations, and discuss the degree to which the aufbau principle is satisfied by this approximation. The results for the ground state energy and highest occupied eigenvalue of various negative ions, as well as the static dipole and quadrupole polarizabilities of the Ne isoelectronic series are also given. The results of Hartree–Fock theory and of experiment are also noted for purposes of comparison and to provide accurate estimates of the Pauli correlation contribution to the kinetic energy.

10.1 Ground State Properties of Atoms In the subsections to follow, the ground state properties of atoms 2 He to determined within the Q-DFT Pauli Approximation are given.

89

Ac as

10.1.1 Electronic Structure of the Argon Atom The electronic configuration of the Ar atom is 1s 2 2s 2 2p 6 3s 2 3p 6 . The radial probability density 4 r 2 .r/, with the density .r/ the spherically symmetric local quantal source as obtained self-consistently via the Q-DFT Pauli Approximation [1, 2], is plotted on a semi-log scale in Fig. 10.1. The three maxima occur at r D 0:059, 0:291, and 1.236 a.u., and the two minima at r D 0:142, 0.791 a.u. (The structure of the density .r/ is similar to that of the Ne atom of Fig. 9.1.) The nonlocal quantal source – the Fermi hole charge x .rr 0 / – for different electron positions is plotted in Figs. 10.2–10.6. The electron positions are indicated by an arrow. These electron positions correspond to the nucleus, the various maxima and minima of the radial probability density, and in the classically forbidden region. In these figures, the electron is along the z-axis corresponding to  D 0ı . The cross-sections plotted in each figure correspond to  0 D 0ı with respect to the

10.1 Ground State Properties of Atoms

189

Argon Atom 20

4πr2ρ (a.u.)

15

10

5

0 0.001

0.01

0.1 r (a.u.)

1

10

Fig. 10.1 The radial probability density 4 r 2 .r/ of the Ar atom as determined in the Q-DFT Pauli Approximation

Argon Atom

Fermi Hole ρx (r r’) (a.u.)

0

–500 r = 0.142 a.u.

–1000 r = 0.059 a.u. –1500

–0.2

r=0

–0.1

0.0 r’ (a.u)

0.1

0.2

Fig. 10.2 Cross-sections of the Fermi hole x .rr 0 / for a model fermion at the nucleus r D 0 a.u., at the first maximum r D 0:059 a.u., and at the first minimum r D 0:142 a.u. of the radial probability density

190

10 Q-DFT Pauli Correlated Approximation Argon Atom

r = 0.291 a.u.

Fermi Hole ρx (r r’) (a.u.)

0

–5

–10

–15

–1.0

–0.5

0.0

0.5

1.0

r’ (a.u)

Fig. 10.3 Cross-section of the Fermi hole x .rr 0 / for a model fermion at the second maximum r D 0:291 a.u. of the radial probability density

Argon Atom

r = 0.791 a.u.

Fermi Hole ρx (r r’) (a.u.)

0

–2

–4

–6 –1.0

–0.5

0.0 r’ (a.u)

0.5

1.0

Fig. 10.4 Cross-section of the Fermi hole x .rr 0 / for a model fermion at the second minimum r D 0:791 a.u. of the radial probability density

10.1 Ground State Properties of Atoms

191

Argon Atom

r = 1.236 a.u.

Fermi Hole ρx (r r’) (a.u.)

0.0

–0.2

–0.4

–0.6

–0.8

–1.0 –3

–2

–1

0

1

2

3

r’ (a.u)

Fig. 10.5 Cross-section of the Fermi hole x .rr 0 / for a model fermion at the third maximum r D 1:236 a.u. of the radial probability density 0.2 Argon Atom

r = 5 a.u.

Fermi Hole ρx (r r’) (a.u.)

0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2

–2

–1

0 r’ (a.u)

1

2

Fig. 10.6 Cross-section of the Fermi hole x .rr 0 / for a model fermion in the classically forbidden region at r D 5 a.u.

192

10 Q-DFT Pauli Correlated Approximation

nucleus-electron direction. The graph for r 0 < 0 is the structure for  0 D and r 0 > 0. The dynamic nature of this quantal source as a function of electron position is evident from these figures. For all electron positions, the Fermi hole charge x .rr 0 / is negative. For an electron at the origin r D 0 a.u. (Fig. 10.2), the Fermi hole x .rr 0 / is spherically symmetric about it. For all other electron positions (Figs. 10.2–10.6), the Fermi hole is not symmetric about either the nucleus or the electron. (It is only the  0 D 90ı cross-section of the hole which is not shown that is symmetric about the nucleus.) For electron positions in the far classically forbidden region (Fig. 10.6), the Fermi hole becomes an essentially static charge distribution that does not change with electron position. In Fig. 10.7, the electron-interaction field E ee .r/ and its Hartree E H .r/ and Pauli E x .r/ components are plotted. The Hartree field E H .r/ is positive because its source charge, the density .r/, is positive. Since the density .r/ is spherically

250 Argon Atom

eH(r)

200

150

Fields (a.u)

100

eee(r)

eH~18/r2 eee~17/r2

50

0

eX ~ –1/r2

eX(r)

–50

–100 0.001

0.01

0.1

1

10

r (a.u)

Fig. 10.7 The electron-interaction field Eee .r/, and its Hartree E H .r/ and Pauli Ex .r/ components

10.1 Ground State Properties of Atoms

193

symmetric about the nucleus, the Hartree field E H .r/ vanishes there. (This is not shown in the graph because the scale is logarithmic.) A careful examination of the graph for E H .r/ indicates the existence of all three shells. As the density is a static charge, and the total number of electrons in this atom is N D 18, the field decays asymptotically in the classically forbidden region as E H .r/ 18=r 2. The Pauli field E x .r/ due to the dynamic Fermi hole x .rr 0 / is negative throughout space because its source charge is negative. Since for an electron at the nucleus, the Fermi hole is spherically symmetric (see Fig. 10.2), the field E x .r/ vanishes there. (Once again, this is not shown because of the logarithmic scale.) Although the third and outermost shell is not evident in Fig. 10.7, it is clearly so when the scale is expanded as in Fig. 10.8. As the total charge of the Fermi hole x .rr 0 / is 1, and since the Fermi hole becomes an essentially static charge for electron positions in the classically forbidden region, the asymptotic structure of E x .r/ 1=r 2 . The electron-interaction field E ee .r/, which is the sum of its Hartree E H .r/ and Pauli E x .r/ components, is positive, vanishes at the nucleus, clearly exhibits shell structure, and decays asymptotically as E ee .r/ 17=r 2. The structure of the field

Argon Atom 0

Third Shell

Pauli Field e x(r) (a.u)

–5

–10

Second Shell

–15

–20 0.1

1

10

r (a.u)

Fig. 10.8 The second and third shells of the Pauli field Ex .r/ exhibited on an expanded scale

194

10 Q-DFT Pauli Correlated Approximation

E ee .r/ may also be understood from the perspective of its dynamic source, the paircorrelation density g.rr 0 /. The field E ee .r/ is positive because the pair-correlation density g.rr 0 / is positive for all electron positions. The field E ee .r/ vanishes at the nucleus because the pair-correlation density g.rr 0 / is spherically symmetric about the nucleus for an electron at that position. The total charge of this quantal source is N  1 D 17, and for asymptotic positions of the electron, the charge also becomes static. Hence, the asymptotic decay of E ee .r/ given above. The electron-interaction Wee .r/ potential energy, and its Hartree WH .r/ and Pauli Wx .r/ components are plotted in Fig. 10.9. These potential energies are the work done in the fields E ee .r/, E H .r/ and E x .r/, respectively. In other words,

Argon Atom 60 WH (r)

Wee (r)

Potentials (a.u.)

40

WH ~ 18/r

20

Wee ~ 17/r

0

Wx ~ –1/r Wx (r)

0.001

0.01

0.1 r (a.u.)

1

10

Fig. 10.9 The electron-interaction Wee .r/ potential energy, and its Hartree WH .r/ and Pauli Wx .r/ components

10.1 Ground State Properties of Atoms

195

they are the areas under the field curves. Hence, it follows from the structure of the fields given in Fig. 10.7, that Wee .r/ and WH .r/ are positive, and Wx .r/ negative, all being finite at the nucleus. As all the fields also vanish at the nucleus, the slope of the potential energies there is zero (see also Sect. 8.4). The asymptotic decay of these potential energies are respectively, Wee .r/ 17=r, WH .r/ 18=r, and Wx .r/ 1=r, and this follows from that of the asymptotic structure of the corresponding fields (see also Sect. 7.4). The three shells are clearly delineated in the structure of the Pauli potential energy Wx .r/ plotted separately in Fig. 10.10. Note that Wx .r/, which is representative purely of Pauli correlations, is monotonic throughout space, with a dramatic change in the slope of the function in the inter shell regions. (As a point of contrast,

Argon Atom 0

Pauli Potential Wx (a.u.)

–1/r

–5

Wx(r)

–10

–15 0.001

0.01

0.1

1 r (a.u.)

Fig. 10.10 The Pauli potential energy Wx .r/

10

196

10 Q-DFT Pauli Correlated Approximation

the Kohn–Sham “exchange” potential energy vx .r/ is comprised of a Pauli component Wx .r/ and the lowest-order Correlation-Kinetic contribution Wt.1/ .r/ (see c Chap. 5 of QDFT.) In atoms, this Correlation-Kinetic contribution exhibits itself as bumps in vx .r/ in the inter shell region. Thus, vx .r/ is not a monotonic function. For .1/ a study of the lowest-order Correlation-Kinetic potential energy Wtc .r/ in atoms and at metal surfaces, the reader is referred to [3].) The results for the total energy E and the highest occupied eigenvalue m as obtained self consistently within this Q-DFT Pauli Approximation are given in tables in the following sections.

10.1.2 Atomic Shell Structure and Core–Valence Separation For a brief history of the literature on atomic shell structure and the determination of core–valence separations for atoms with Z > 18, the reader is referred to Sect. 9.2. In the present section, the results for this shell structure as determined within the Q-DFT Pauli Approximation are presented [4]. In Figs. 10.11–10.14, the radial probability density D.r/ D 4 r 2 .r/ for K, Ca, Sc, Kr; Rb, Sr, Y, Xe; Cs, Ba, La, Rn; and Fr, Ra, and Ac atoms are plotted. These figures are plotted on a log-log scale. Throughout the fourth, fifth, sixth, and seventh periods, the appropriate number of shells is exhibited. For each period, the valence shell begins at the corresponding alkali metal atom. On the scale of these figures, the minuscule difference between these plots and those of Hartree–Fock theory [5] are not discernable.

Fig. 10.11 The radial probability density D.r/ D 4 r 2 .r/ for the fourth period atoms K, Ca, Sc, and Kr as determined in the Q-DFT Pauli Approximation

10.1 Ground State Properties of Atoms

197

Fig. 10.12 The radial probability density D.r/ D 4 r 2 .r/ for the fifth period atoms Rb, Sr, Y, Xe as determined in the Q-DFT Pauli Approximation

Fig. 10.13 The radial probability density D.r/ D 4 r 2 .r/ for the sixth period atoms Cs, Ba, La, Rn as determined in the Q-DFT Pauli Approximation

In Table 10.1, the results for the core–valence separations are presented. These core–valence separations are obtained from the above shell structure by considering the midpoint ri of the two outermost points of inflection in D.r/. The core radius ri and the core charge q are defined as Z

ri

qD

D.r/dr:

(10.1)

0

For purposes of comparison, the Hartree–Fock theory results [5] obtained in a similar manner are considered. For both theories, the standard core radius rs determined from the D.r/ such that the standard core charge q based on the electronic configuration of atoms in their ground state are also considered. It is evident that

198

10 Q-DFT Pauli Correlated Approximation

Fig. 10.14 The radial probability density D.r/ D 4 r 2 .r/ for the seventh period atoms Fr, Ra, Ac as determined in the Q-DFT Pauli Approximation

Table 10.1 The core charge q and radius ri as determined by Hartree–Fock theory [4] and the Q-DFT Pauli Approximation. The radius rs corresponding to the standard ground state electronic configuration is also given. All values are in a.u. The atomic configurations are based on Madelung’s law [7] (see Fig. 9.16) Atom Hartree–Fock Q-DFT Pauli Theorya Approximationb 19

K(4s 1 )

20

Ca(4s 2 ) 1

2

ri

q

rs

q

ri

q

rs

q

2.91

17.94

3.06

18

2.97

18.00

2.98

18

2.33

17.89

2.45

18

2.36

17.94

2.42

18

21

Sc(3d 4s )

2.26

18.78

2.47

19

2.25

18.77

2.45

19

36

Kr(4s 2 4p 6 )

0.97

27.49

1.04

28

0.96

27.48

1.03

28

37

Rb(5s 1 )

3.34

35.95

3.46

36

3.43

36.03

3.37

36

38

Sr(5s 2 )

2.76

35.93

2.83

36

2.78

36.00

2.79

36

39

Y(4d 1 5s 2 )

2.70

36.73

2.94

37

3.02

37.10

2.93

37

54

Xe(5s 2 5p 6 )

1.34

45.61

1.39

46

1.34

45.62

1.39

46

55

Cs(6s 1 )

3.90

53.95

4.01

54

4.19

54.10

3.93

54

56

2

Ba(6s )

3.27

53.93

3.34

54

3.31

54.01

3.31

54

57

La(5d 1 6s 2 )

3.41

54.98

3.42

55

3.31

54.88

3.44

55

1.54

77.61

1.59

78

1.55

77.85

1.58

78

86

Rn(6s 6p )

87

1

Fr(7s )

4.17

85.90

4.55

86

4.46

86.11

4.20

86

88

Ra(7s 2 )

3.52

85.90

3.65

86

3.58

86.00

3.55

86

89

Ac(6d 1 7s 2 )

4.15

87.30

3.75

87

3.57

86.86

3.72

87

a

2

6

See [5]; b see [4]

10.1 Ground State Properties of Atoms

199

the values of ri and q as obtained from the two theories are accurate when compared with the standard values, and essentially equivalent. The latter is a reflection of the fact that in atoms, the Correlation-Kinetic effects within Hartree–Fock theory (see QDFT) are very small. This fact is reaffirmed in the comparison of the total energy of atoms as obtained by Hartree–Fock theory [6] and the Q-DFT Pauli Approximation discussed in Sect. 10.1.3. Thus, accurate atomic shell structure is obtained when only Pauli correlations are assumed for the model system of noninteracting fermions, the Coulomb correlations and Correlation-Kinetic effects being ignored.

10.1.3 Total Ground State Energies The total ground state energies of atoms from He.Z D 2/ to Rn.Z D 86/ as obtained [1, 2] via the Q-DFT Pauli Approximation in the central field model (see Appendix D) are given in Table 10.2. For purposes of comparison, the Hartree–Fock theory results [6], also performed in the central field model of an atom, are included. The negative values of the energies in atomic units are given. In Table 10.3, the difference in parts per million between these sets of energies are given. These differences represent a very accurate estimate of the Correlation-Kinetic contribution TcHF within Hartree–Fock theory (see Sect. 6.2). (It is an accurate estimate because although the densities of the two theories differ in principle, in practice they are essentially equivalent.) Because of the absence of these Correlation-Kinetic contributions, the results of the Q-DFT Pauli Approximation are an upper bound to those of Hartree–Fock theory. Another way to understand this is that the Q-DFT Pauli Approximation wave function differs from that of Hartree–Fock theory, and thus the energy as obtained by the former must be an upper bound to that of the latter. It is interesting that as the number of electrons in the atoms increase, the CorrelationKinetic energy TcHF diminishes. For the 2 He atom, the energies of the two theories are the same. As such there is no Correlation-Kinetic energy TcHF . (This may also be understood from the fact that the two electrons are of opposite spin, and as such there is no Pauli correlation between them.) For the lighter atoms 3 Li–8 O, the differences in the energies lie between 157 and 53 ppm. By 9 F these differences are down to 50 ppm, by 35 Br they are down to 10 ppm, and by 72 Hf they are less than 5 ppm. For 86 Rn, the energies differ by 2 ppm. Thus, excluding the He atom, the contribution to the kinetic energy due to correlations resulting from the Pauli exclusion principle decreases as the atomic number increases. In a comparison with the results given in Table 9.2 of the Q-DFT Hartree Uncorrelated Approximation in which all electron correlations are neglected, the inclusion of Pauli correlations as in the Q-DFT Pauli Approximation lowers the total energy as it must because now the fermions of parallel spin are being kept apart. The difference between these sets of energy values is then a measure of the contribution to the total energy due to correlations arising from the Pauli exclusion principle. As noted in the previous paragraph, the difference between the Q-DFT Pauli Approximation values and those of Hartree–Fock theory is in turn a measure of the contributions of Pauli correlations to the kinetic energy.

200

10 Q-DFT Pauli Correlated Approximation

Table 10.2 The total ground state energy of atoms in the central field model as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory. The negative values of the energies in atomic units are given Atomic Atom Hartree–Fock Q-DFT Pauli Number Theorya Approximationb 2 He 2:861680 2:861680 3 Li 7:432730 7:431573 4 Be 14:57302 14:57144 5 B 24:52906 24:52623 6 C 37:68862 37:68472 7 N 54:40093 54:39611 8 O 74:80940 74:80505 9 F 99:40935 99:40467 10 Ne 128:5471 128:5419 11 Na 161:8589 161:8508 12 Mg 199:6146 199:6063 13 Al 241:8767 241:8676 15 Si 288:8544 288:8440 15 P 340:7188 340:7070 16 S 397:5049 397:4930 17 Cl 459:4821 459:4697 18 Ar 526:8175 526:8043 19 K 599:1648 599:1491 20 Ca 676:7582 676:7426 21 Sc 759:7357 759:7166 22 Ti 848:4060 848:3825 23 V 942:8843 942:8575 24 Cr 1043:310 1043:280 25 Mn 1149:866 1149:833 26 Fe 1262:444 1262:414 27 Co 1381:415 1381:386 28 Ni 1506:871 1506:843 29 Cu 1638:950 1638:922 30 Zn 1777:848 1777:820 31 Ga 1923:261 1923:235 32 Ge 2075:360 2075:334 33 As 2234:239 2234:212 34 Se 2399:868 2399:842 35 Br 2572:441 2572:416 36 Kr 2752:055 2752:030 37 Rb 2938:357 2938:330 38 Sr 3131:546 3131:519 39 Y 3331:684 3331:656 40 Zr 3538:995 3538:963 41 Nb 3753:552 3753:517 42 Mo 3975:443 3975:405 43 Tc 4204:789 4204:747 (continued)

10.1 Ground State Properties of Atoms Table 10.2 (continued) Atomic Atom Number 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 55 Cs 56 Ba 57 La 58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn a See [6]; b see [1]

201

Hartree–Fock Theorya 4441:487 4685:801 4937:783 5197:518 5465:133 5740:169 6022:932 6313:485 6611:784 6917:981 7232:138 7553:934 7883:544 8221:064 8566:920 8921:181 9283:883 9655:099 10034:953 10423:543 10820:617 11226:568 11641:453 12065:290 12498:153 12940:174 13391:456 13851:808 14321:250 14799:813 15287:546 15784:533 16290:649 16806:113 17330:949 17865:212 18408:991 18961:825 19524:008 20095:586 20676:501 21266:882 21866:772

Q-DFT Pauli Approximationb 4441:448 4685:762 4937:744 5197:479 5465:093 5740:130 6022:893 6313:447 6611:746 6917:944 7232:101 7553:895 7883:506 8221:021 8566:871 8921:125 9283:822 9655:034 10034:883 10423:466 10820:546 11226:500 11641:385 12065:223 12498:087 12940:109 13391:389 13851:745 14321:187 14799:750 15287:484 15784:469 16290:588 16806:054 17330:891 17865:153 18408:932 18961:768 19523:953 20095:532 20676:448 21266:830 21866:721

202

10 Q-DFT Pauli Correlated Approximation

Table 10.3 Total ground state energy differences between the Q-DFT Pauli Approximation values and those of Hartree–Fock theory. These differences are an accurate measure of the CorrelationKinetic energy TcHF arising due to Pauli correlations Atoms Differences (ppm) 2

He Li, 4 Be 5 4B 6 C 7 N 8 O 9 Fe–34 Se 35 Br–71 Lu 72 Hf–86 Rn 3

0 157, 108 115 103 87 58 50–10 10–5 <5

10.1.4 Highest Occupied Eigenvalues As proved in QDFT and discussed in Sect. 3.7, the highest occupied eigenvalue m of the S system differential equation (3.3) with all the electron correlations accounted for, is the negative of the ionization potential I . Furthermore, as proved in Chap. 7, the asymptotic structure of the effective potential energy vs .r/ in the classically forbidden region is 1=r, and that this structure is solely due to the Pauli potential energy Wx .r/. The Coulomb Wc .r/ and Correlation-Kinetic Wtc .r/ potential energies decay more rapidly. Thus, in this region the fully correlated S system differential equation reduces to being that of the Q-DFT Pauli Approximation. Since the highest occupied eigenvalue is governed principally by the asymptotic structure of the local effective potential energy vs .r/, it is meaningful to compare the highest occupied eigenvalues m of the Q-DFT Pauli Approximation with those of experimental ionization potentials. In Table 10.4, the m of the Q-DFT Pauli Approximation [1] and experimental ionization potentials [8] are given. The negative values of the eigen energies in Rydbergs are given. The atoms are grouped together on the basis of the commonality of properties. (The configuration of each atom as determined via the self consistent calculations is also given. In those cases, such as the first element of each transition group, where the configuration satisfies Madelung’s law [7] as discussed later, both the highest and next highest eigenvalues are given.) The highest occupied eigenvalues as obtained [6] within Hartree–Fock theory are also given. Note that these latter eigenvalues have meaning as removal energies only within the strict constraint of Koopmans’ theorem [9]. The constraint is that no relaxation of the remaining orbitals is allowed when a valence electron is removed to infinity. The results for the atoms .2 He–86 Rn/ are summarized in Figs. 10.15 and 10.16 where the magnitude of the difference between the experimental ionization potential and the highest occupied eigenvalues as obtained via the Q-DFT Pauli Approximation and Hartree–Fock theory are plotted. Panels (a) of these figures correspond to those atoms for which the Q-DFT Pauli Approximation results are closer to the experimental ionization potentials than those due to Hartree–Fock theory, and

10.1 Ground State Properties of Atoms

203

Table 10.4 Highest occupied eigenvalues of atoms (2 He–86 Rn) in the central field model as obtained by the Q-DFT Pauli Approximation. The atoms, 21 Sc, 39 Y, 40 Zr, 57 La, 71 Lu, 72 Hf, for which the highest two eigenvalues are given obey Madelung’s aufbau principle [7]. The Hartree– Fock theory and the experimental ionization potentials for all the atoms, as well as the “exact” theoretical values of the highest occupied eigenvalues for the fully correlated S system for the atoms (2 He–18 Ar) are also given. The magnitude of the difference between the Q-DFT Pauli Approximation and the “exact” results for these atoms are noted in the last column. The negative values of the eigen energies in Rydbergs are given Atom Hartree–Fock Q-DFT Pauli Expt.c “Exact” jDifferencej Approxb Theoryd Theorya 2

He 4 Be 12 Mg 20 Ca 30 Zn 38 Sr 48 Cd 56 Ba 70 Yb 80 Hg

1.836 0.619 0.506 0.391 0.585 0.357 0.530 0.315 0.365 0.522

Atoms with last closed subshell a s subshell 1.836 1.807 1.808 0.626 0.685 0.676 0.521 0.562 0.518 0.402 0.449 0.646 0.690 0.369 0.419 0.583 0.661 0.325 0.383 0.383 0.460 0.580 0.767

10

Ne 18 Ar 36 Kr 54 Xe 86 Rn

1.701 1.182 1.048 0.915 0.856

Noble gas atoms 1.713 1.585 1.178 1.158 1.035 1.029 0.899 0.892 0.838 0.790

21

Sc

0.420

Ti V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni

0.442 0.461 0.479 0.496 0.516 0.535 0.553

39

Y

0.392

40

Zr

0.415

Nb Mo 43 Tc 44 Ru 45 Rh 46 Pd

0.433 0.449 0.463 0.480 0.494 0.507

22 23

41 42

1.594 1.094

0.028 0.050 0.003

0.119 0.084

First transition group 0.446 .4s/ 0.481 0.415 .3d / 0.478 0.501 0.505 0.495 0.529 0.497 0.552 0.546 0.571 0.578 0.591 0.578 0.610 0.561 Second transition group 0.423 .5s/ 0.469 0.337 .4d / 0.458 .5s/ 0.503 0.438 .4d / 0.484 0.506 0.504 0.522 0.523 0.535 0.536 0.542 0.548 0.548 0.560 0.613 (continued)

204

10 Q-DFT Pauli Correlated Approximation

Table 10.4 (continued) Atom Hartree–Fock Theorya

Q-DFT Pauli Approxb

Exptc

Hf

0.418

Ta W 75 Re 76 Os 77 Ir 78 Pt

0.435 0.450 0.461 0.478 0.491 0.503

Third transition group 0.470 .6s/ 0.515 0.426 .5d / 0.494 0.580 0.513 0.587 0.531 0.579 0.540 0.639 0.551 0.669 0.561 0.662

58

Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb

0.324 0.328 0.332 0.335 0.339 0.342 0.346 0.349 0.352 0.356 0.359 0.362 0.365

Rare earths 0.338 0.402 0.342 0.398 0.347 0.404 0.351 0.408 0.355 0.414 0.359 0.417 0.362 0.451 0.366 0.430 0.369 0.436 0.373 0.442 0.376 0.448 0.380 0.454 0.383 0.460

57

La

0.320

71

Lu

0.398

3

Li 11 Na 19 K 37 Rb 55 Cs

0.393 0.364 0.295 0.276 0.247

Alkali metals 0.405 0.396 0.390 0.378 0.317 0.319 0.299 0.307 0.268 0.286

9

1.460 1.013 0.914 0.806 0.760

Halogens 1.464 1.281 1.006 0.953 0.900 0.868 0.791 0.768 0.743 0.708

72

73 74

F 17 Cl 35 Br 53 I 85 At

0.332 .6s/ 0.267 .4f / 0.435 .6s/ 0.330 .5d /

“Exact” Theoryd

jDifferencej

0.410 0.399

0.400 0.364

0.005 0.026

1.368 0.982

0.096 0.024

(continued)

10.1 Ground State Properties of Atoms Table 10.4 (continued) Atom Hartree–Fock Theorya 5

B 6 C 13 Al 14 Si 31 Ga 32 Ge 49 In 50 Sn 81 Tl 82 Pb

7

N 8 O 15 P 16 S 33 As 34 Se 51 Sb 52 Te 83 Bi 84 Po

29

0.620 0.867 0.420 0.594 0.417 0.575 0.395 0.530 0.385 0.511

Q-DFT Pauli Approxb

205

Exptc

“Exact” Theoryd

Atoms with less than half-filled p subshells 0.581 0.610 0.598 0.818 0.828 0.820 0.406 0.440 0.428 0.571 0.599 0.714 0.410 0.441 0.556 0.581 0.387 0.425 0.511 0.540 0.378 0.449 0.494 0.545

jDifferencej

0.017 0.002 0.022 0.143

Atoms with less than half and two-thirds filled p subshells 1.135 1.078 1.068 1.056 0.022 1.264 1.249 1.001 1.172 0.077 0.783 0.754 0.771 0.748 0.006 0.875 0.861 0.761 0.832 0.029 0.739 0.712 0.721 0.806 0.789 0.717 0.669 0.644 0.635 0.720 0.702 0.662 0.640 0.616 0.536 0.683 0.665 0.619

Noble metal atoms 0.569 0.628 0.568 0.519 0.572 0.557 0.513 0.570 0.678 a See [6]; b see [1]; c see [8]; d see [10]

Cu 47 Ag 79 Au

panels (b) for the cases where the reverse is true. Note that whereas the scales of Figs. 10.15a and 10.16a are the same, those of Figs. 10.15b and 10.16b are different, being smaller by a factor of two and an order of magnitude, respectively. It is evident from these figures that for the majority of atoms considered, the Q-DFT Pauli Approximation results lie closer to experiment. Furthermore, for those atoms for which the Hartree–Fock theory results are closer, the difference between the two theories (see Figs. 10.15b and 10.16b) are of the order of a few hundreds of a Rydberg. For atoms with Z < 55, the Q-DFT Pauli Approximation highest occupied eigenvalues lie within 0.05 ryd of the experimental ionization potentials, and for the heavier atoms within 0.1 ryd. These differences, of course, are a consequence of the neglect of Coulomb correlations, Correlation-Kinetic effects due to both Pauli and Coulomb correlations, and relativistic effects, in the calculations.

206 Fig. 10.15 The magnitude of the difference (in ryd) between the experimental ionization potential I and the highest occupied eigenvalues m as obtained by the QDFT Pauli Approximation and Hartree–Fock theory as a function of the atomic number Z for Z D 2  42. Panel (a) corresponds to atoms for which the Q-DFT Pauli Approximation results are closer to experiment than those of Hartree–Fock theory, and panel (b) to atoms for which the reverse is true. (Note the difference in scales between panels (a) and (b))

Fig. 10.16 The magnitude of the difference (in ryd) between the experimental ionization potential I and the highest occupied eigenvalues m as obtained by the QDFT Pauli Approximation and Hartree–Fock theory as a function of the atomic number Z ranging from Z D 42– 86. The legend for panels (a) and (b) are the same as in Fig. 10.15 (Note the difference in scales between panels (a) and (b))

10 Q-DFT Pauli Correlated Approximation

10.1 Ground State Properties of Atoms

207

Finally, the “exact” theoretical nonrelativistic values of the highest occupied eigenvalues of the fully interacting S system for the atoms 2 He–18 Ar are also given. These results are obtained [10] by constructing [11–13] the fully correlated S system numerically by working backwards from the “exact” density as determined from accurate correlated wave functions. The magnitude of the difference between these values and the Q-DFT Pauli Approximation results are given in the last column. Observe that, with the exception of 10 Ne and 14 Si for which the differences are less than two tenths of a Rydberg, the eigenvalues of the remaining atoms differ by hundredths or less of a Rydberg. These differences thus constitute the Coulomb correlation and Correlation-Kinetic contributions to the highest occupied eigenvalues. It is evident from the results of Table 10.4 that reasonably accurate approximations to the experimental ionization potentials can be obtained at the Pauli level of approximation within Q-DFT. It is also noteworthy that the highest occupied eigenvalues of the Q-DFT Pauli Approximation are superior to those of the Q-DFT Hartree Uncorrelated Approximation (see Table 9.3) when compared to experimental ionization potentials. Thus, the inclusion of Pauli correlations to the uncorrelated model particles improves both the total energy as well as the highest occupied eigenvalue for each atom.

10.1.5 Satisfaction of the Aufbau Principle Madelung’s law for the filling up of the electronic shells, the aufbau principle, is stated in Sect. 9.4.1 (see Fig. 9.16). According to this law, for the transition elements, the 4s shell fills before the 3d shell, the 5s before the 4d , and 6s before the 5d . The Q-DFT Pauli Approximation results (see Table 10.4) show that the first element of the first, second, and third transition group of elements obeys Madelung’s law. Thus the highest occupied orbitals for 21 Sc, 39 Y, and 72 Hf are the 3d , 4d , and 5d eigen states. (For 40 Zr, the 4d eigenvalue is also higher than the 5s eigenvalue.) For the remaining elements of each transition group, the highest occupied orbital is the s orbital. For the rare earth element 57 La, the 6s shell fills before the 4f shell, in accordance with Madelung’s law. For the rare earth element 71 Lu, the 6s shell fills before the 5d shell, again in satisfaction of the aufbau principle. In contrast, in Hartree–Fock theory, the highest occupied orbitals for 21 Sc, 39 Y, are the 4s, and 5s eigen states, whereas for 57 La, 71 Lu, and 72 Hf, it is the 6s eigen state. To understand why the results of the Q-DFT Pauli Approximation satisfy the aufbau principle whereas those of Hartree–Fock theory do not, the ratio of the magnitude of the 4d and 5s radial orbitals of the two theories for 39 Y in the far classically forbidden region is plotted in Fig. 10.17. It is evident from this graph that asymptotically, the Q-DFT 4d orbital spreads out much further than the 5s orbital, such that the ratio jR4d =R5s j > 1, and becomes more singular with distance from

208

10 Q-DFT Pauli Correlated Approximation

Fig. 10.17 The ratio of the magnitude of the 4d to 5s radial orbitals jR4d =R5s j for 39 Y in the far classically forbidden region as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory

20 a.u. onwards. In Hartree–Fock theory, the 4d orbital decays faster than the 5s orbital, so that the ratio jR4d =R5s j < 1, and decreases to zero asymptotically. From the Q-DFT Pauli Approximation core–valence separation results of Table 10.1, we have seen that the number of valence electrons for 21 Sc, 39 Y, and 57 La is essentially 2. However, these valence electrons occupy the 5s orbital, the occupation of the 4d orbital still being one electron, in spite of the 4d eigenvalues 2 2 being higher. To understand this, the radial probability densities r 2 R4d and r 2 .2R5s / 39 for Y are plotted in Fig. 10.18. In the valence shell region from 3–10 a.u., the latter dominates. Thus the 5s orbitals contain the two valence electrons, with the 4d orbital being filled last, and having only one electron. (Recall, that the 4d orbital dominates the 5s orbital only in the far classically forbidden region beyond 20 a.u. (see Fig. 10.17)). These conclusions are consistent with the fact that if the selfconsistent calculations were performed with two electrons in the 4d orbital, the value for the ground state energy obtained is higher.

10.1 Ground State Properties of Atoms

2 2 2 2 Fig. 10.18 The radial probability densities r 2 .2R4s /, r 2 .6R4p /, r 2 R4d , r 2 .2R5s / for obtained in the Q-DFT Pauli Approximation

209

39

Y as

10.1.6 Single-Particle Expectation Values A measure of the accuracy of the density .r/ within an approximation is the expectation of various single particle operators. The density is such an expectation value. The expectation value of the operators P Furthermore, it is an observable. P W D i rin , n D 1, 2, 1, 2, and W D i ı.r i ) for atoms in the central-field model as obtained by the Q-DFT Pauli Approximation [1] are given in atomic units in Table 10.5. In terms of the density .r/, the expectations of the operators W may be written as Z hW i D Œr n ; ı.r/.r/dr: (10.2) These expectation values sample both the exterior and interior of atoms as well as determine its size. They are also related to various atomic properties [14] such as the diamagnetic susceptibility .hr 2 i/, nuclear magnetic shielding .hr 1 i/, and the Fermi contact term .hı.r/i/. The Hartree–Fock theory expectations [6] are also given for purposes of comparison. According to Brillouin’s theorem [15–17], single-particle expectation values taken with respect to the Hartree–Fock theory wave function are

210

10 Q-DFT Pauli Correlated Approximation

Table 10.5 The expectation value of the single particle operators r, r 2 , r 1 , r 2 , and ı.r / for atoms in the central-field model as obtained by the Q-DFT Pauli Approximation [1] and Hartree– Fock theory [6]. The expectations given are in atomic units Z Atom Theory hri hr 2 i hr 1 i hr 2 i hı.r /i 2 He QDFT 0.9273 1.1848 1.6873 5.9955 3.5959 HF 0.9273 1.1848 1.6873 5.9955 3.5959 3

Li

QDFT HF

1.6320 1.6733

5.8541 6.2107

1.9073 1.9052

10.0624 10.0706

13.8014 13.8148

4

Be

QDFT HF

1.5140 1.5322

4.2211 4.3297

2.1040 2.1022

14.3918 14.4045

35.3514 35.3877

5

B

QDFT HF

1.3525 1.3621

3.1391 3.1702

2.2779 2.2760

18.6997 18.7310

71.7560 71.9214

6

C

QDFT HF

1.1854 1.1908

2.2892 2.2987

2.4502 2.4482

23.0792 23.1273

127.0741 127.4580

7

N

QDFT HF

1.0468 1.0500

1.7234 1.7261

2.6212 2.6194

27.5369 27.6000

205.2651 205.9683

8

O

QDFT HF

0.9466 0.9512

1.3851 1.3961

2.7847 2.7824

32.0602 32.1529

310.2906 311.6609

9

F

QDFT HF

0.8597 0.8642

1.1260 1.1371

2.9490 2.9465

36.6658 36.7818

446.1298 448.3222

10

Ne

QDFT HF

0.7852 0.7891

0.9280 0.9372

3.1138 3.1113

41.3537 41.4890

616.7369 619.9221

11

Na

QDFT HF

0.9594 0.9858

2.2373 2.4687

3.2241 3.2209

46.1839 46.3181

829.9235 833.7575

12

Mg

QDFT HF

1.0051 1.0215

2.3549 2.4676

3.3294 3.3267

51.1017 51.2379

1089.0796 1093.7178

13

Al

QDFT HF

1.0444 1.0551

2.5191 2.5751

3.4255 3.4231

56.0418 56.1829

1397.2025 1402.8456

14

Si

QDFT HF

1.0265 1.0343

2.2729 2.3043

3.5197 3.5174

61.0165 61.1619

1758.8655 1765.6069

15

P

QDFT HF

0.9920 0.9982

1.9968 2.0183

3.6120 3.6099

66.0241 66.1736

2178.3614 2186.3141

16

S

QDFT HF

0.9616 0.9673

1.8021 1.8227

3.7009 3.6988

71.0606 71.2166

2660.0332 2669.4696

17

Cl

QDFT HF

0.9257 0.9307

1.6081 1.6255

3.7887 3.7869

76.1275 76.2892

3208.1531 3219.1902

18

Ar

QDFT HF

0.8885 0.8928

1.4322 1.4464

3.8755 3.8736

81.2239 81.3908

3827.0091 3839.7817

19

K

QDFT HF

1.0028 1.0237

2.4780 2.6964

3.9439 3.9417

86.3706 86.5382

4524.3597 4538.6539

20

Ca

QDFT HF

1.0484 1.0623

2.7114 2.8283

4.0100 4.0080

91.5533 91.7232

5303.5527 5319.6070 (continued)

10.1 Ground State Properties of Atoms Table 10.5 (continued) Z Atom Theory

211

hri

hr 2 i

hr 1 i

hr 2 i

hı.r/i

21

Sc

QDFT HF

1.0099 1.0227

2.4179 2.5317

4.0831 4.0814

96.7001 96.8718

6164.4231 6182.3042

22

Ti

QDFT HF

0.9686 0.9815

2.1585 2.2798

4.1569 4.1555

101.8553 102.0276

7113.4939 7133.2301

23

V

QDFT HF

0.9295 0.9426

1.9439 2.0659

4.2306 4.2293

107.0212 107.1951

8155.1872 8176.9552

24

Cr

QDFT HF

0.8929 0.9061

1.7627 1.8820

4.3041 4.3028

112.1992 112.3751

9293.8828 9317.8305

25

Mn

QDFT HF

0.8582 0.8715

1.6042 1.7229

4.3776 4.3764

117.3902 117.5669

10533.938 10560.077

26

Fe

QDFT HF

0.8284 0.8407

1.4793 1.5822

4.4497 4.4483

122.5926 122.7760

11879.715 11908.704

27

Co

QDFT HF

0.7997 0.8114

1.3656 1.4588

4.5218 4.5203

127.8085 127.9967

13335.544 13367.382

28

Ni

QDFT HF

0.7727 0.7839

1.2645 1.3497

4.5938 4.5921

133.0376 133.2301

14905.764 14940.568

29

Cu

QDFT HF

0.7473 0.7579

1.1742 1.2526

4.6655 4.6638

138.2798 138.4765

16594.700 16632.599

30

Zn

QDFT HF

0.7230 0.7334

1.0920 1.1660

4.7372 4.7355

143.5356 143.7352

18406.677 18447.675

31

Ga

QDFT HF

0.7467 0.7548

1.2722 1.3219

4.7969 4.7952

148.8447 149.0477

20352.768 20397.202

32

Ge

QDFT HF

0.7502 0.7563

1.2703 1.3005

4.8557 4.8540

154.1725 154.3779

22432.300 22480.118

33

As

QDFT HF

0.7461 0.7509

1.2235 1.2444

4.9136 4.9119

159.5176 159.7251

24649.641 24700.939

34

Se

QDFT HF

0.7430 0.7472

1.1940 1.2110

4.9699 4.9683

164.8775 165.0882

27009.198 27064.393

35

Br

QDFT HF

0.7356 0.7392

1.1443 1.1577

5.0256 5.0240

170.2531 170.4667

29515.406 29574.567

36

Kr

QDFT HF

0.7258 0.7289

1.0875 1.0981

5.0807 5.0792

175.6439 175.8599

32172.660 32235.890

37

Rb

QDFT HF

0.7927 0.8055

1.7051 1.8447

5.1281 5.1264

181.0577 181.2747

34990.627 35057.626

38

Sr

QDFT HF

0.8279 0.8372

1.9179 2.0016

5.1745 5.1729

186.4894 186.7082

37971.637 38042.812

39

Y

QDFT HF

0.8229 0.8298

1.8176 1.8818

5.2226 5.2212

191.9137 192.1329

41114.120 41189.282 (continued)

212

10 Q-DFT Pauli Correlated Approximation

Table 10.5 (continued) Z Atom Theory

hri

hr 2 i

hr 1 i

hr 2 i

hı.r/i

40

Zr

QDFT HF

0.8102 0.8173

1.6916 1.7611

5.2710 5.2698

197.3434 197.5623

44425.160 44504.257

41

Nb

QDFT HF

0.7962 0.8036

1.5807 1.6536

5.3193 5.3182

202.7793 202.9982

47909.362 47992.582

42

Mo

QDFT HF

0.7818 0.7895

1.4828 1.5569

5.3674 5.3663

208.2219 208.4409

51571.257 51658.750

43

Tc

QDFT HF

0.7668 0.7750

1.3921 1.4693

5.4154 5.4143

213.6716 213.8903

55415.360 55507.100

44

Ru

QDFT HF

0.7542 0.7616

1.3239 1.3903

5.4627 5.4616

219.1275 219.3482

59446.034 59542.829

45

Rh

QDFT HF

0.7409 0.7480

1.2570 1.3177

5.5098 5.5087

224.5906 224.8126

63667.865 63769.628

46

Pd

QDFT HF

0.7277 0.7345

1.1946 1.2509

5.5568 5.5557

230.0607 230.2839

68085.261 68192.073

47

Ag

QDFT HF

0.7147 0.7212

1.1365 1.1893

5.6035 5.6024

235.5378 235.7619

72702.656 72814.609

48

Cd

QDFT HF

0.7016 0.7081

1.0813 1.1325

5.6502 5.6490

241.0218 241.2466

77524.497 77641.557

49

In

QDFT HF

0.7195 0.7248

1.2188 1.2552

5.6909 5.6898

246.5312 246.7584

82563.142 82686.163

50

Sn

QDFT HF

0.7253 0.7294

1.2374 1.2609

5.7313 5.7301

252.0496 252.2786

87816.418 87945.278

51

Sb

QDFT HF

0.7259 0.7294

1.2197 1.2366

5.7713 5.7701

257.5767 257.8074

93288.803 93423.567

52

Te

QDFT HF

0.7271 0.7301

1.2129 1.2261

5.8104 5.8093

263.1115 263.3444

98984.739 99125.883

53

I

QDFT HF

0.7252 0.7277

1.1872 1.1973

5.8492 5.8481

268.6545 268.8893

104908.79 105056.34

54

Xe

QDFT HF

0.7212 0.7233

1.1524 1.1602

5.8876 5.8866

274.2056 274.4421

111065.41 111219.44

55

Cs

QDFT HF

0.7721 0.7816

1.6646 1.7796

5.9220 5.9208

279.7693 280.0067

117465.26 117625.45

56

Ba

QDFT HF

0.8015 0.8085

1.8685 1.9399

5.9558 5.9547

285.3428 285.5817

124109.98 124276.86

57

La

QDFT HF

0.7891 0.7956

1.7915 1.8627

5.9952 5.9943

290.8814 291.1199

130987.12 131160.11 (continued)

10.1 Ground State Properties of Atoms Table 10.5 (continued) Z Atom Theory

213

hri

hr 2 i

hr 1 i

hr 2 i

hı.r/i

58

Ce

QDFT HF

0.7762 0.7825

1.7195 1.7913

6.0350 6.0343

296.4212 296.6589

138113.04 138292.14

59

Pr

QDFT HF

0.7634 0.7696

1.6526 1.7248

6.0749 6.0744

301.9633 302.2001

145492.49 145677.79

60

Nd

QDFT HF

0.7508 0.7569

1.5906 1.6622

6.1149 6.1145

307.5081 307.7447

153130.05 153321.83

61

Pm

QDFT HF

0.7386 0.7446

1.5327 1.6034

6.1548 6.1545

313.0560 313.2926

161030.21 161228.74

62

Sm

QDFT HF

0.7266 0.7326

1.4782 1.5481

6.1948 6.1945

318.6072 318.8437

169197.52 169402.77

63

Eu

QDFT HF

0.7149 0.7209

1.4265 1.4960

6.2347 6.2346

324.1620 324.3977

177636.42 177848.32

64

Gd

QDFT HF

0.7039 0.7098

1.3793 1.4460

6.2744 6.2742

329.7198 329.9579

186351.40 186571.24

65

Tb

QDFT HF

0.6931 0.6989

1.3342 1.3988

6.3141 6.3138

335.2813 335.5210

195346.93 195574.58

66

Dy

QDFT HF

0.6826 0.6883

1.2912 1.3541

6.3537 6.3535

340.8466 341.0874

204627.47 204862.90

67

Ho

QDFT HF

0.6723 0.6779

1.2505 1.3117

6.3933 6.3931

346.4155 346.6576

214197.48 214440.88

68

Er

QDFT HF

0.6624 0.6679

1.2119 1.2713

6.4329 6.4326

351.9881 352.2316

224061.42 224312.96

69

Tm

QDFT HF

0.6527 0.6581

1.1750 1.2330

6.4724 6.4721

357.5645 357.8089

234223.74 234483.40

70

Yb

QDFT HF

0.6432 0.6485

1.1396 1.1965

6.5119 6.5116

363.1448 363.3897

244688.89 244956.57

71

Lu

QDFT HF

0.6427 0.6468

1.1189 1.1609

6.5470 6.5466

368.7549 369.0013

255475.02 255751.36

72

Hf

QDFT HF

0.6384 0.6426

1.0756 1.1186

6.5820 6.5815

374.3710 374.6179

266574.60 266859.03

73

Ta

QDFT HF

0.6335 0.6378

1.0349 1.0790

6.6167 6.6162

379.9924 380.2400

277991.90 278284.59

74

W

QDFT HF

0.6281 0.6326

0.9971 1.0416

6.6511 6.6505

385.6192 385.8672

289731.46 290032.49

75

Re

QDFT HF

0.6223 0.6271

0.9597 1.0061

6.6853 6.6847

391.2511 391.4992

301797.83 302107.03

76

Os

QDFT HF

0.6178 0.6221

0.9340 0.9738

6.7191 6.7184

396.8873 397.1367

314195.08 314513.80 (continued)

214

10 Q-DFT Pauli Correlated Approximation

Table 10.5 (continued) Z Atom Theory

hri

hr 2 i

hr 1 i

hr 2 i

hı.r/i

77

Ir

QDFT HF

0.6127 0.6168

0.9064 0.9426

6.7526 6.7520

402.5284 402.7788

326928.18 327256.11

78

Pt

QDFT HF

0.6075 0.6113

0.8794 0.9130

6.7860 6.7853

408.1742 408.4253

340001.48 340338.64

79

Au

QDFT HF

0.6021 0.6057

0.8532 0.8847

6.8193 6.8186

413.8246 414.0762

353419.47 353765.87

80

Hg

QDFT HF

0.5964 0.6001

0.8270 0.8578

6.8524 6.8516

419.4795 419.7314

367186.65 367542.13

81

Tl

QDFT HF

0.6093 0.6126

0.9241 0.9480

6.8824 6.8816

425.1479 425.4016

381318.41 381684.67

82

Pb

QDFT HF

0.6153 0.6180

0.9496 0.9658

6.9122 6.9114

430.8217 431.0767

395809.60 396186.26

83

Bi

QDFT HF

0.6182 0.6204

0.9508 0.9628

6.9418 6.9410

436.5006 436.7569

410664.66 411051.72

84

Po

QDFT HF

0.6214 0.6234

0.9586 0.9680

6.9709 6.9701

442.1842 442.4420

425888.02 426286.07

85

At

QDFT HF

0.6226 0.6243

0.9525 0.9597

6.9998 6.9991

447.8727 448.1319

441484.32 441893.32

86

Rn

QDFT HF

0.6224 0.6239

0.9391 0.9447

7.0286 7.0278

453.5661 453.8265

457458.02 457878.01

correct to second order in the error of the wave function. Thus, the Hartree–Fock theory density and the various other single-particle expectations are accurate. It is evident from a study of Table 10.5 that the Q-DFT Pauli Approximation and Hartree–Fock theory densities are essentially equivalent. The difference between the two sets of results can thus be attributed to Correlation-Kinetic effects due to Pauli correlations. These effects that are present in Hartree–Fock theory are absent from the Q-DFT Pauli Approximation.

10.2 Ground State Properties of Mononegative Ions In the subsections to follow, the results of spin-unrestricted calculations [18, 19] for mononegative ions performed within the Q-DFT Pauli Approximation and Hartree– Fock theory in the central field model are presented. In spin-unrestricted calculations [20], the spatial factor of each spin-orbital of the doubly occupied state is allowed to be different. The reason for performing spin-unrestricted calculations is because some of these systems under consideration are spin-polarized.

10.2 Ground State Properties of Mononegative Ions

215

The spin-polarized Q-DFT Pauli potential energy Wx .r/ for spin  to be employed in the S system differential equation is the work done in the Pauli field E x .r/: Z r

Wx .r/ D  Z

where E x .r/ D

1

E x .r 0 /  d` 0 ;

x .rr 0 /.r  r 0 / 0 dr : jr  r 0 j3

(10.3)

(10.4)

The Fermi hole charge x .rr 0 / of spin  is x .rr 0 / D 

j .rr 0 /j2 ;  .r/

(10.5)

with the single-particle density matrix  .rr 0 / defined as  .rr 0 / D

X

fi 

i  .r/ i  .r

0

/

(10.6)

i

for real f i  g, where fi  is the fractional occupancy of the i  orbital. As was the case for the neutral atoms, the central field model is achieved by spherically averaging the field E x .r).

10.2.1 Total Ground State Energies The total ground state energies of mononegative ions of various groups: Hydrogen, the Alkalis, the Halogens, and Ions with incomplete p shells, as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory in Rydbergs are given in Table 10.6. For H , the two results are equivalent as expected, because for two electrons in the ground 1s 2 state, the two theories are equivalent. (Recall, this was also the case for the ground state of the He atom: see Table 10.2.) As expected, for the other mononegative ions, the Q-DFT Pauli Approximation values are an upper bound to the Hartree–Fock theory results, because the two wave functions are different. The difference between these sets of results in parts per million (ppm) are given in the last column. Within each group, as the atomic number increases, the difference decreases. As the densities of the two theories are very close, the difference between the ground state energies is an accurate representation of the contribution TcHF to the kinetic energy due to correlations arising from the Pauli principle. It is evident from Table 10.6 that this Correlation-Kinetic contribution to the total energy is negligible. For H there are no such Correlation-Kinetic effects because the two electrons have opposite spin.

216

10 Q-DFT Pauli Correlated Approximation

Table 10.6 Ground state total energies of negative ions as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory. The difference between the two approximations in parts per million (ppm) are also given. The negative values of the energy in Rydbergs are given Z

Ion

State

1

H

1s 2 1S

3 11 19 37

Li Na K Rb

2s 2 1S 3s 2 1S 4s 2 1S 5s 2 1S

Alkalis 14:8565 323:7103 1198:3238 5876:7098

14:8540 323:6935 1198:2918 5876:6546

168 52 27 9

9 17 35 53

F Cl Br I

2p 6 1S 3p 6 1S 4p 6 1S 5p 6 1S

Halogens 198:9189 919:1539 5145:1254 13836:1518

198:9085 919:1278 5145:0200 13836:0753

52 28 20 6

Ions with incomplete p shells 49:0386 49:0326 75:4206 75:4140 149:5862 149:5764 483:7575 483:7383 577:7801 577:7589 795:0789 795:0545

122 88 66 40 37 31

5 B 6 C 8 O 13 Al 14 Si 16 S a See [18, 19]

2p 2 3P 2p 3 4S 2p 5 2P 3p 2 3P 3p 3 4S 3p 5 2P

Hartree–Fock Theorya

Q-DFT Pauli Approximationa

Hydrogen 0:9759

0:9759

Difference (ppm) 0

10.2.2 Highest Occupied Eigenvalues The negative of the highest occupied eigenvalues m of the various mononegative ions as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory in Rydbergs are given in Table 10.7. The A experimental electron affinities A of these ions [21] are also given. For H , the eigenvalues of the two theories are equivalent, as expected. Note that in both the Q-DFT Pauli Approximation and Hartree–Fock theory, the ion H is not stable. For a negative ion to be stable, the electron affinity A must be positive. The electron affinity is defined as A D E(neutral atom) – E(negative ion). Thus, for H , the electron affinity obtained by these theories is A D E.H /  E.H  / D 1:0000 .0:9759/ D 0:0241 ryd. (It is evident that if a stable H  is to be obtained within Q-DFT, Coulomb correlations and the Correlation-Kinetic contributions to the total energy must be included.) Irrespective of whether the experimental value of A lies below (the Alkalis) or above (the Halogens and Ions with incomplete p shells) the Hartree–Fock theory values, the Q-DFT Pauli Approximation results are superior. (The only exception is

10.3 Static Polarizabilities of the Neon Isoelectronic Sequence

217

Table 10.7 Highest occupied eigenvalues of negative ions as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory. The experimental electron affinities are also given. The negative values of the eigen energies in Rydbergs are given Z

Ion

State

1

H

1s 2 1S

3 11 19 37

Li Na K Rb

2s 2 3s 2 4s 2 5s 2

9 17 35 53

F Cl Br I

2p 6 1S 3p 6 1S 4p 6 1S 5p 6 1S

1

S S 1 S 1 S 1

Hartree–Fock Theorya

Q-DFT Pauli Approximationa

Hydrogen 0.09244 0.09244 Alkalis 0.02908 0.03017 0.02670 0.02918 0.02064 0.02280 0.01906 0.02136

Experimental Electron Affinityb 0.0554 0.04542 0.04027 0.03686 0.03571

Halogens

2p 2 3P 5 B  6 C 2p 3 4S  8 O 2p 5 2P  13 Al 3p 2 3P  14 Si 3p 3 4S  16 S 3p 5 2P a b See [18, 19]; see [21]

0.3620 0.3006 0.2787 0.2583

0.3576 0.2884 0.2612 0.2401

Ions with incomplete p shells 0.05314 0.04933 0.1562 0.1511 0.1434 0.1414 0.04124 0.03259 0.1241 0.1099 0.1551 0.1457

0.2498 0.2658 0.2473 0.2248

0.0204 0.0928 0.1074 0.0324 0.1018 0.1527

the case of S ). These negative ions are stable in both theories. The superiority of the Q-DFT Pauli Approximation results for these negative ions is similar to those for neutral atoms, and occurs for the same reasons (see Sect. 10.1.4). It is reiterated that the comparison of the Hartree–Fock theory m with the experimental electron affinities is appropriate only in the strict context of Koopmans’ theorem as also explained previously.

10.3 Static Polarizabilities of the Neon Isoelectronic Sequence As a final example of ground state properties determined within the Q-DFT Pauli Approximation, the results [22, 23] of the static dipole ˛d and quadrupole ˛q polarizabilities of the Neon isoelectronic series F , Ne, NaC , Mg2C , Al3C , Si4C , are presented. The method employed for the determination of these polarizabilities is that due to Sternheimer [24, 25]. When an electric field is applied to an atom or ion,

218

10 Q-DFT Pauli Correlated Approximation

the field distorts the wave function from its unperturbed state, thereby distorting the electronic charge density. This distortion of the density gives rise to induced electric dipole and higher order moments. Treating the field as a perturbation, the polarizabilities are obtained by determining the first-order correction to the unperturbed Q-DFT Pauli Approximation wave function. In the case of the dipole polarizability ˛d , the perturbation is proportional to the field, whereas for the quadrupole polarizability ˛q , the perturbation is proportional to the gradient of the field. In each case, the first-order perturbation to the density is then determined, from which the induced dipole and quadrupole moments are obtained. The dipole polarizability ˛d is the ratio of the induced dipole moment to the field, and the quadrupole polarizability ˛q , the ratio of the induced quadrupole moment to the gradient of the field. The unperturbed wave function is obtained via the Q-DFT Pauli Approximation in the central field model. With the designation u0 .nlm/ for the unperturbed wave function times r, where nlm are the principal, azimuthal, and magnetic quantum numbers, the radial part of the wave function times r designated as u00 .nl/, is obtained by solution of the differential equation 

 1 2  r C vs .r/ u00 .nl / D nl u00 .nl /; 2

(10.7)

where vs .r/ D v.r/ C WH .r/ C Wx .r/, and nl the corresponding eigenvalues. The first-order part of the perturbed wave function times r, designated as u1 .nlm/, is obtained by solution of the inhomogeneous Schr¨odinger equation for u1 .nlm/. (The reader is referred to [24] for details.) For the dipole case, the radial component times r; designated as u01 .nl ! l 0 ) is then obtained by solution of the corresponding radial differential equation  

 d2 l 0 .l 0 C 1/ u01 .nl ! l 0 / D u00 .nl/r: C C v .r/   s nl dr 2 r2

(10.8)

The notation u01 .nl ! l 0 ) denotes the radial function for the excitation of a state 0 with azimuthal quantum number l into R 1states with azimuthal quantum number l . In (10.8), the normalization condition 0 Œu00 .nl /2 dr D 1 holds. Equation (10.8) is solved for all perturbations defined by jl  l 0 j D ˙1. P The first-order perturbation of the density which is nlm Œu0 .nlm/ C u1 .nlm/2 is given by 2u0 .nlm/Cu1 .nlm). From this perturbation term, one then obtains the induced dipole moment. Dividing the induced dipole moment by the field leads to the dipole polarizability. The dipole polarizability ˛d is obtained from the expression V 3 / D 0:148 ˛d .A

X nl

0

Z

C.nl ! l / 0

1

u00 .nl /ru01 .nl ! l 0 /dr:

(10.9)

10.3 Static Polarizabilities of the Neon Isoelectronic Sequence

219

Table 10.8 The static electronic dipole polarizability ˛d of the Ne isoelectronic series as obtained by the Q-DFT Pauli Approximation and Hartree–Fock theory together with the exper˚3 imental values in A Atom Hartree–Fock Q-DFT Pauli Experimentb a a and Ions Theory Approximation F 1.788 Ne 0.389 NaC 0.142 Mg2C 0.069 Al3C 0.039 Si4C 0.022 a See [22, 23]; b see[26, 27]

1.824 0.384 0.147 0.070 0.037 0.023

1:38 0:395 0:158 0:0784 0:038

The angular factors C.nl ! l 0 ) are tabulated in [24]. The factor 0:148 D 0:5293 ˚ 3. ensures ˛d is in units of A The results for the dipole polarizability ˛d , together with those in which the Hartree–Fock theory wave functions were employed as the unperturbed wave function, and the experimental values [26, 27], are given in Table 10.8. The two sets of theoretical results closely approximate each other. As it is the effective potential energy vs .r/ and eigenvalues nl of the unperturbed system that appear in the differential equation (10.8) for the perturbed part u01 .nl ! l 0 /, the difference between the two sets of theoretical results can be attributed to Pauli CorrelationKinetic effects. Even in the presence of an electric field, these effects are very small. The difference between the Q-DFT Pauli Approximation results and those of experiment are an estimate of the contributions of the Couloumb correlations plus the Pauli and Coulomb Correlation-Kinetic effects. For the quadrupole case, the first-order perturbed wave function is obtained by solution of the inhomogeneous differential equation  

 d2 l 0 .l 0 C 1/ 0 0 C C v .r/   s nl u1 .nl ! l / dr 2 r2   D u00 .nl/ r 2  hr 2 inl ıll 0 ;

(10.10)

where hr 2 i is the expectation of r 2 taken with respect to the unperturbed u00 .nl/. Equation (10.10) is solved for the perturbations defined by jl  l 0 j D 0; ˙2. The induced density is again obtained and from it the induced quadrupole moment is determined. Division by the gradient of the field then leads to the quadrupole polarizability. The quadrupole polarizability ˛q is obtained from the expression V 5 / D 0:0415 ˛q .A

X nl

C.nl ! l 0 /

Z 0

1

u00 .nl /r 2 u01 .nl ! l 0 /dr:

(10.11)

220

10 Q-DFT Pauli Correlated Approximation

Table 10.9 The quadrupole electronic polarizability ˛q of the Ne isoelectronic series as obtained ˚5 by the Q-DFT Pauli Approximation and Hartree–Fock theory in A Atom and Ions

Hartree–Fock Theorya

F Ne NaC Mg2C Al3C Si4C a See [22, 23]

2.809 0.272 0.064 0.022 0.009 0.004

Q-DFT Pauli Approximationa 2.950 0.258 0.062 0.022 0.009 0.004

The angular coefficients C.nl ! l 0 ) are listed in [24], and the factor of 0:0415 D V 5. 0:5295 ensures ˛q is in units of A The Q-DFT Pauli Approximation and Hartree–Fock theory results for ˛q are given in Table 10.9. Their differences are once again small, the two sets of results being equivalent for Mg2C , Al3C , and Si4C . Again, the differences are an estimate of the Pauli Correlation-Kinetic effects.

Chapter 11

Quantal Density Functional Theory of the Density Amplitude: Application to Atoms

The Q-DFT of Chap. 3 is a description of the mapping from the interacting electronic system as defined by the Schr¨odinger equation of (2.1)–(2.5) to one of a model system of noninteracting fermions – the S system – with equivalent density .r/. Thus, via Q-DFT, an interacting system in a ground or excited state can be mapped into a model S system whose density is the same. The state of the S system is arbitrary, in that it could be in a ground or excited state (see Chap. 5). For an interacting system of N electrons, if the mapping is to an S system with the same electronic configuration, the S system differential equation with the same self-consistently determined local effective potential energy function vs .r/ must be solved N times. The resulting self-consistently determined single-particle spinorbitals lead to the equivalent density. A significant observation of the application of this Q-DFT to atoms (see QDFT and Chaps. 5, 10, 13, and 15) is that if the mapping is to an S system with the same configuration as that of the interacting system, the Correlation-Kinetic energy contributions are a very small fraction of the interacting system kinetic energy, and hence, from the virial theorem of the total energy. Thus, Correlation-Kinetic effects play a relatively insignificant role in such calculations, the contributions to the various atomic properties as obtained from the S system equations arising principally from correlations due to the Pauli principle and Coulomb repulsion. As explained in Chap. 6 of QDFT, there is a Q-DFT whereby the N -electron interacting system in a ground or excited state as described by the Schr¨odinger equation of (2.1)–(2.5), can be mapped to a model system of noninteracting bosons in its ground state – the B system – with equivalent density .r/. The corresponding B system differential equation, also one with a local effective p potential energy function vB .r/, is for a single orbital – the density amplitude .r/. This effective potential energy function once again accounts for electron correlations due to the Pauli principle and Coulomb repulsion, as well as Correlation-Kinetic effects due to the difference in the kinetic energies of the interacting electronic and noninteracting bosonic systems. The model B system is a special case of the S system [1]. The B system equations can be derived in three different ways: via traditional Hohenberg– Kohn [2] density functional theory [3] employing the von Weizs¨acker kinetic energy functional of the density [4]; directly from the interacting system Schr¨odinger equation [5]; and via Q-DFT. The reader is referred to QDFT for the details of

221

222

11 Quantal Density Functional Theory of the Density Amplitude

these derivations. Within traditional DFT, the corresponding local effective potential energy function vB .r/ of the B system is in terms of the functional derivative of an energy functional of the density in which all the correlations are embedded. In the derivation from the interacting system Schr¨odinger equation, the local potential energy function is defined in terms of the conditional probability amplitude [6] associated with (N  1) electrons when the position of one electron is known . Within Q-DFT, this potential energy is the work done in a conservative field, with each component of the field being representative of a particular electron correlation. In general, the local effective potential energy functions of the S and B systems, vs .r/ and vB .r/ are different. The former function generates the density via N spin-orbitals, whereas the latter accomplishes this via a single orbital. However, for two-electron systems such as the Hooke’s atom [7], the Helium atom, and the Hydrogen molecule, in their ground or excited states, the corresponding S and B systems are equivalent. It has also been proved [5] that in the classically forbidden region of atoms, vs .1/ D vB .1/. As a final point of comparison between the two mappings, recall that irrespective of the whether the mapping is to an S system in a ground or excited state, the highest occupied eigenvalue m of the S system differential equation is equivalent to the negative of the ionization potential I [8]. For the B system, the single eigenvalue  of the corresponding differential equation is the negative of the ionization potential I [5]. p The equations governing the Q-DFT of the density amplitude .r/ are given in Sect. 11.1. As a preliminary application, the theory is applied [9] in Sect. 11.2 to the Be and Mg atoms to map to a B system with a density and energy that is equivalent to that of Hartree–Fock theory [10, 11]. What is learned from this application, and its implications for traditional density functional theory of the density amplitude, are discussed in the concluding Sects. 11.3 and 11.4.

11.1 Quantal Density Functional Theory of the Density Amplitude In the mapping from the time-independent interacting system as defined by (2.1)–(2.5) to one of noninteracting bosons in their ground state, electron correlations due to the Pauli principle and Coulomb repulsion must be accounted for. In addition, the kinetic energy of the electrons is different from that of the noninteracting bosons having the same density. Thus, the model B system must also account for this difference i.e., for Correlation-Kinetic effects. With the assumption that the bosons experience the same external field F ext .r/ D r v.r/, the corresponding B system differential equation is (see QDFT) 

p p 1  r 2 C vB .r/ .r/ D  .r/; 2

(11.1)

11.1 Quantal Density Functional Theoryof the Density Amplitude

223

where the effective potential energy vB .r/ of the bosons is vB .r/ D v.r/ C vB ee .r/;

(11.2)

with vB ee .r/ being the “electron-interaction” potential energy function that accounts for the Pauli and Coulomb correlations, and the Correlation-Kinetic effects. The single eigenvalue  is the negative of the ionization potential I . Within Q-DFT, the potential energy vB ee .r/ is the work done in a conservative field F eff .r/: B Z vB .r/ D 

r 1

0 0 F eff B .r /  d` :

(11.3)

This work done is path-independent since r  F eff B .r/ D 0. The effective field F eff B .r/ is the sum of the interacting system electroninteraction field E ee and a Correlation-Kinetic field Z B tc .r/: B F eff B .r/ D E ee .r/ C Z tc .r/:

(11.4)

Note that in general, the fields E ee .r/ and Z B tc .r/ are not separately conservative. Their sum always is. The field E ee .r/, representative of electron correlations due to the Pauli principle and Coulomb repulsion, is obtained via Coulomb’s law from the pair-correlation density g.rr 0 / which constitutes its source: Z

g.rr 0 /.r  r 0 / 0 dr ; jr  r 0 j3

E ee .r/ D

(11.5)

where g.rr 0 / D h .X /jPO .rr 0 /j .X /i=.r/ with .X / the solution of the Schr¨odinger equation (2.5) and PO .rr 0 / is the pair-correlation operator (2.33). The field Z B tc .r/, representative of the Correlation-Kinetic effects, is the difference between the kinetic fields Z B .r/ and Z.r/ of the bosonic and electronic systems, respectively: ZB tc .r/ D Z B .r/  Z.r/; where Z B .r/ D

zB .rI ŒB / .r/

and

Z.r/ D

(11.6) z.rI Œ/ : .r/

(11.7)

The B system kinetic “force” zB .rI ŒB / is defined in terms of the bosonic kineticenergy-density tensor tB;˛ˇ .r/ as zB;˛ .r/ D 2

X @ tB;˛ˇ .rI ŒB /; @rˇ ˇ

(11.8)

224

with

11 Quantal Density Functional Theory of the Density Amplitude

ˇ   ˇ 1 @2 @2 0 00 ˇ  tB;˛ˇ .r/ D C .r r / B ˇ 0 00 ; 00 0 4 @r˛0 @rˇ @rˇ @r˛00 r Dr Dr

(11.9)

and where the bosonic density matrix B .rr 0 / quantal source is B .rr 0 / D

p

p .r/ .r 0 /:

(11.10)

Note that r  Z B .r/ D 0, so that the field Z B .r/ is conservative. The interacting system kinetic “force” z.rI Œ / is as defined by (2.53) and (2.54) in terms of its quantal source, the single-particle density matrix .rr 0 / of (2.28). The field Z.r/ is in general not conservative. The interacting system energy E written in terms of its components and respective fields is Z E D TB C .r/v.r/dr C Eee C TcB ; (11.11) where TB is the kinetic energy of the bosons: 1 TB D  2

Z .r/r  Z B .r/dr;

(11.12)

the second term is the external energy Eext ; Eee is the electron-interaction energy: Z Eee D

.r/r  E ee .r/dr;

(11.13)

and TcB the Correlation-Kinetic energy: TcB D

1 2

Z .r/r  Z B tc .r/dr:

(11.14)

The kinetic energy of the bosons may also be written in terms of the density amplitude as p  Z p 1 .r/  r 2 .r/dr: (11.15) TB D 2 The expressions for Eee and TcB are independent of whether the fields E ee .r/ and ZB tc .r/ are conservative or not. For systems of symmetry such that r  E ee .r/ D 0 B and r  Z B tc .r/ D 0, the potential energy vee .r/ may be written as B vB ee .r/ D Wee .r/ C Wtc .r/;

(11.16)

where Wee .r/ and WtBc .r/ are, respectively, the work done in the fields E ee .r/ and ZB tc .r/: Z r Wee .r/ D  E ee .r 0 /  d` 0 ; (11.17) 1

11.1 Quantal Density Functional Theoryof the Density Amplitude

Z

and WtBc .r/ D 

r 1

0 0 ZB tc .r /  d` :

225

(11.18)

Each work done is path-independent. The field E ee .r/ component of F eff B .r/ is the same as that in the mapping to the S system of noninteracting fermions. Thus, as in that mapping, the field E ee .r/ may be expressed as a sum of its Hartree E H .r/ and Pauli-Coulomb E xc .r/ components (see Sect. 2.4.1). Therefore, the effective field F eff B .r/ may be written as B F eff B .r/ D E H .r/ C E xc .r/ C Z tc .r/:

(11.19)

In turn, for systems for which the field E ee .r/ is conservative, the potential energy vB ee .r/ is B vB (11.20) ee .r/ D WH .r/ C Wxc .r/ C Wtc .r/; where WH .r/, the Hartree potential energy, is the work done in the field E H .r/ of the density .r/, or equivalently Z WH .r/ D

.r 0 / dr 0 ; jr  r 0 j

(11.21)

and Wxc .r/, the work done in the field E xc .r/ of the Fermi–Coulomb hole charge xc .rr 0 /: Z r

Wxc .r/ D 

1

E xc .r 0 /  d` 0 :

(11.22)

The above equation represents the Q-DFT mapping to a system of noninteracting bosons with density, energy, and ionization potential equivalent to that of the interacting system of electrons. It is also possible to map to B systems with densities and energies equivalent to those of Hartree–Fock and Hartree theories. The equations for these mappings are the same but with the electron-interaction E ee .r/ and kinetic Z.r/ fields of the fully interacting system replaced by their Hartree–Fock and Hartree theory counterparts. As the B system is a special case of the S system [1], it follows from the derivations of Chap. 8 that the potential energy vB ee .r/ of (11.2) is finite at the nucleus. (This is borne out by the calculations in Sect. 11.2.) However, its asymptotic structure near the nucleus differs from that of vee .r/ of the S system (see 8.132) due to the fact that the Correlation-Kinetic field Z B tc .r/ differs from Z tc .r/. The analytB ical asymptotic structure of Z tc .r/, and hence of vB ee .r/, near the nucleus has not been derived yet. The analytical asymptotic structure of Z B tc .r/ in the classically forbidden region of atoms has also not yet been derived. However, the calculations in Sect. 11.2 indicate that in the classically forbidden region of atoms, the 2 field Z B tc .r/ decays more rapidly than E ee .r/ r!1 .N  1/=r . Thus, one may conclude that the asymptotic structure of the effective potential energy vB .r/ in 1=r. This is consistent with the earlier remark that this region is vB .r/ r!1 vs .1/ D vB .1/.

226

11 Quantal Density Functional Theory of the Density Amplitude

11.2 Application to Atoms For the application of the Q-DFT of the density amplitude to the exactly solvable two electron Hooke’s atom [7] from both a ground and excited state to model B systems in their ground state, the reader is referred to QDFT. These model B systems are exact in that they reproduce the interacting system ground and excited state densities, energies, and ionization potentials. If the only correlations considered in the choice of the approximate wave function are those due to the Pauli exclusion principle, then the exact solution of the interacting system is that of Hartree–Fock theory. Exact, fully self-consistent calculations for atoms in the Hartree–Fock approximation exist [12]. Hence, via the Q-DFT of the density amplitude, it is possible to map to the exact B systems that would reproduce the Hartree–Fock theory density and energy. In this section, we demonstrate this mapping for the spherically symmetric Be and Mg atoms. Rather than use the self-consistent numerical orbitals of [12] in the demonstration, we employ instead the analytical Hartree–Fock theory wave functions of Clementi and Roetti [13]. From the fact that the B system is a special case of the S system, it follows from the explanation in Chap. 8, that the potential energy vB ee .r/ must be finite at the nucleus. This, of course, is a consequence of the cusp in the wave function of the interacting system at the nucleus. The fully self-consistent Hartree–Fock theory orbitals satisfy the electron–nucleus coalescence condition exactly. According to (8.4), this condition to leading order in terms of the density is .r/r!1 .0/.1  2Zr/;

(11.23)

where Z is the atomic number. For analytical Hartree–Fock wave functions, the coalescence condition is satisfied essentially exactly. The corresponding relations for Be and Mg using these wave functions are Be W Mg W

.r/r!1 35:4277.1  8:0389r/;

(11.24)

.r/r!1 1093:7310.1  24:0228r/;

(11.25)

With the analytical Hartree–Fock theory wave functions, the quantal sources – the Hartree–Fock theory density HF .r/, pair-correlation density gsHF .rr 0 /, Dirac density matrix sHF .rr 0 /, and the bosonic system density matrix BHF .rr 0 / are first determined. The corresponding radial probability density of the Be atom is plotted in Fig. 11.1 with the two shells clearly exhibited. (The superscript HF for the various properties is dropped in the remainder of the chapter.) The separately conservative electron-interaction E ee .r/ and Correlation-Kinetic ZB tc .r/ fields for Be are plotted in Fig. 11.2. Observe that both the fields exhibit the two-shell structure of the atom. The field E ee .r/ is positive and has a larger magnitude in the K shell. This field decays as 3=r 2 asymptotically. This is because, the quantal source gs .rr 0 / of the field, which has a total charge of 3 a.u., becomes an essentially static charge distribution for asymptotic positions of the electron.

11.2 Application to Atoms

227

Fig. 11.1 The Hartree–Fock theory radial probability density of the Be atom

Fig. 11.2 The electron-interaction Eee .r/ and Correlation-Kinetic Z Btc .r/ fields of the B system for the Be atom

228

11 Quantal Density Functional Theory of the Density Amplitude

Fig. 11.3 The potential energies Wee .r/, WtBc .r/, and vBee .r/ of the B system for the Be atom

The Correlation-Kinetic field Z B tc .r/ (see Fig. 11.2) is interesting in that the magnitude of the maximum is greater than the electron-interaction field E ee .r/ in both the K and L shells. However, it decays more rapidly in the asymptotic classically forbidden region than the electron-interaction field E ee .r/ which decays as 3=r 2 . The work done Wee .r/ and WtBc .r/ in the fields E ee .r/ and Z B tc .r/, respectively, eff B and the work vee .r/ in the effective field F B .r/ are plotted in Fig. 11.3. The twoshell structure is evident in the potential energy WtBc .r/. Asymptotically, this work decays more rapidly than Wee .r/ which in turn decays as 3=r. On the scale of the figure, the shell structure exhibited by Wee .r/ is not that clearly visible. However, the shell structure is clearly exhibited by the structure of the potential energy vB ee .r/. Asymptotically, the curves of Wee .r/ and vB .r/ merge, both decaying as 3=r. Thus, ee the effective potential energy vB .r/ decays asymptotically as 1=r. The B system properties for the Be atom viz. the kinetic energy TB , the Correlation-Kinetic energy TcB , the external potential energy Eext , the electron interaction energy Eee , and the total energy E, as determined from the corresponding fields, as well as the eigenvalue , and the experimental ionization potential I are given in a.u. in Table 11.1. The results indicate the following. The CorrelationKinetic energy TcB component is 6:3% of the total energy E, or equivalently of the (Hartree–Fock theory) kinetic energy T of the interacting system. It comprises 20% of the electron-interaction energy Eee . Thus, in the mapping to the B system, the Correlation-Kinetic contribution is significant. (That this is the case is more

11.2 Application to Atoms

229

Table 11.1 Properties of the B system for the Be and Mg atoms in a.u. Atom Property

Be

Mg

TB 13:6587 132:5942 TcB 0:9143 67:0204 Eext 33:6647 479:0672 Eee 4:5186 79:8423 E 14:5730 199:6146  0:3077 0:3968 I(experiment) 0:3425 0:2810

Fig. 11.4 The Hartree–Fock theory radial probability density of the Mg atom

dramatically demonstrated in the results for the Mg atom to be discussed later.) The eigenvalue  differs by 10% of the experimental result. The B system results for Mg are presented as follows: the radial probability density is plotted in Fig. 11.4; the electron-interaction E ee .r/ and Correlation-Kinetic B ZB tc .r/ fields in Figs. 11.5 and 11.6; the work done Wee .r/ and Wtc .r/ in the fields B eff E ee .r/ and Z B tc .r/, respectively, and the work vee .r/ in the effective field F B .r/ B are plotted in Fig. 11.7; the values of the properties TB , Tc , Eext , Eee , E, , and the experimental I in Table 11.1. The three-shell structure of the Mg atom (see Fig. 11.4 for the radial probability density) is clearly evident in the plots for the Correlation-Kinetic field Z B tc .r/ (see Figs. 11.5 and 11.6). Observe that the magnitude of the field Z B .r/ is substantially tc larger than the field E ee .r/ (see Fig. 11.5), but again decays asymptotically much faster than E ee .r/ 11=r 2. On the scale of Fig. 11.7, the shell structure is also

230

11 Quantal Density Functional Theory of the Density Amplitude

Fig. 11.5 The electron-interaction Eee .r/ and Correlation-Kinetic Z Btc .r/ fields of the B system for the Mg atom

Fig. 11.6 The electron-interaction Eee .r/ and Correlation-Kinetic Z Btc .r/ fields of the B system for the Mg atom in the classically forbidden region

11.3 Conclusions and Endnotes

231

Fig. 11.7 The potential energies Wee .r/, WtBc .r/, and vBee .r/ of the B system for the Mg atom

evident in that of the potential energy WtBc .r/, and as expected, it decays asymptotically faster than Wee .r/ 11=r. The potential energy vB ee .r/ also exhibits the shell structure, and decays as the potential energy Wee .r/ (see Fig. 11.7). Thus, the B system effective potential energy vB .r/ for Mg again decays asymptotically as 1=r. The trends observed for the bosonic system properties for the Be atom are magnified for those of Mg (see Table 11.1) i.e. with an increase in the number of electrons. For Mg, the Correlation-Kinetic energy TcB constitutes 34% of the total energy E, and is 84% of the electron-interaction energy Eee . Furthermore, the eigenvalue  differs from the experimental ionization potential by 41%. The B systems described earlier, reproduce the Hartree–Fock theory density and energy of the Be and Mg atoms: the potential p energy vB .r/ obtained from the various fields generates the density amplitude .r/ via (11.1), and the energy E and its components are obtained from these fields via (11.11).

11.3 Conclusions and Endnotes As noted in the introduction section of the chapter, the mapping to the p B system requires the solution of a differential equation for a single orbital .r/, whereas for the mapping to an S system, the corresponding differential equation has to be solved for N orbitals, N being the number of electrons. The calculations of Sect. 11.2, although performed at the Hartree–Fock theory level of correlation,

232

11 Quantal Density Functional Theory of the Density Amplitude

indicate yet another significant difference between the two mappings. The corresponding Correlation-Kinetic contributions, Tc of the S system and TcB of the B system, differ substantially. In Chap. 10, S system calculations at the Pauli level of correlation were performed. The difference between the resulting energies and those of Hartree–Fock theory are an accurate estimate of the Correlation-Kinetic energy Tc . From Tables 10.2 and 10.3, these differences for Be and Mg are 108 and 42 ppm, respectively. As noted earlier, the corresponding B system Correlation-Kinetic energies TcB for Be and Mg are 6% and 34% of the Hartree-Fock theory energy. (For the He atom, Hartree–Fock theory, the Q-DFT Pauli Correlated Approximation, and the Q-DFT of the density amplitude at the Hartree–Fock theory level, are all equivalent. It follows that Tc D TcB D 0 for this atom.) Obviously, requiring the model noninteracting particles – bosons – to be in their ground state occupying a single state, rather than allowing the model noninteracting particles – fermions – to be in their ground state occupying states according to the Pauli exclusion principle, is reflected in the Correlation-Kinetic component of the former being very much greater than that of the latter. It is reasonable to assume that this will also be the case when all the remaining correlations – those due to Coulomb repulsion and the remaining Correlation-Kinetic contributions – are included in the mapping. Note also, that whereas for the S system at the Hartree–Fock level of correlation, with increasing number of electrons, the Correlation-Kinetic energy Tc diminishes (see Tables 10.2 and 10.3), the trend for the B system is the opposite. As the number of electrons increases, the Correlation-Kinetic piece TcB increases, thus becoming a greater fraction of the electron-interaction energy component Eee . Yet another comparison that can be made from the results obtained above is that between the highest occupied eigenvalue m of the S system and the eigenvalue  of the B system. For the mappings when all the electron correlations are considered, both m and  are equivalent to the negative of the ionization potential I . However, at the approximate Hartree–Fock theory level of correlation, they are about the same for Be: m and  differ from the experimental values of I by 9% and 10%, respectively, but differ significantly for Mg: in this case m and  differ from the experimental values of I by 7% and 41%, respectively (see Tables 10.4 and 11.1). Thus, with an increase in the number of electrons, the single eigenvalue  is far less accurate. This is in spite of the fact that both the S and B system effective potential energy functions vs .r/ and vB .r/ decay asymptotically as 1=r. With the incorporation of the Coulomb correlations and the remaining Correlation-Kinetic effects, the two eigenvalues m and  of course become equivalent.

11.4 Consequences for Traditional Density Functional Theory The consequences of the above results and conclusions for traditional KS-DFT of the density amplitude (see QDFT) are as follows. In traditional theory, the mapping is from an interacting system in its ground state to one of noninteracting bosons in their ground state. The many body effects of the Pauli principle, Coulomb

11.4 Consequences for Traditional Density Functional Theory

233

repulsion and the Correlation-Kinetic effects of the bosons are all incorporated in B the ground state electron-interaction energy functional Eee Œ, with the potential B B energy vB .r/ being its functional derivative: v .r/ D ıE ee ee ee Œ=ı.r/. Since the B functional Eee Œ is unknown, approximations to this functional are constructed with the physics of the various electron correlations being incorporated extrinsically. Now the Q-DFT derivations of the B and S systems indicates that the component B due to Pauli and Coulomb correlations of the functional Eee Œ and of the Kohn– KS Sham electron-interaction energy functional Eee Œ of the S system is the same. B Thus, the approximate incorporation of these correlations into Eee Œ can be done in a manner similar to that presently being employed for the construction of S system KS approximate energy functionals Eee Œ. However, due to the large magnitude of the Correlation-Kinetic effects for the bosonic system, the incorporation of these effects B into the functional Eee Œ poses a difficult challenge. Unless these effects, which for bosonic systems are large, can be reasonably well accounted for in the functional B Eee Œ, the functional and its derivative will be inaccurate, and the KS-DFT of the density amplitude will then not lead to meaningful results.

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Chapter 12

Application of the Irrotational Component Approximation to Nonspherical Density Atoms

The electronic density .r/ of majority of atoms in the Periodic Table are nonspherically symmetric. These are open shell atoms with partially filled degenerate sublevels. However, in most calculations of atomic structure, such atoms are treated in the central field model wherein their densities are sphericalized by ensemble averaging the different orientations (see e.g. Appendix D). Thus, in order that these atoms be represented in a more physically correct manner, and thereby obtain their properties more accurately, the nonsphericity of the density must be taken into account [1, 2]. For nonspherical density atoms as treated within local effective potential energy theory, this means that the electron-interaction potential energy function vee .r/ is nonspherical and angle dependent, the external potential energy v.r/ being central (see (3.3) and (3.4)). In Q-DFT, the potential energy vee .r/ is the work done in the conservative effective field F eff .r/ which is the sum of its electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ components. In turn, the field E ee .r/ may be expressed as the sum of its Hartree E H .r/, Pauli E x .r/ and Coulomb E c .r/ components. Thus, F eff .r/ D E ee .r/ C Z tc .r/ D E H .r/ C E x .r/ C E c .r/ C Z tc .r/ (see (3.65)–(3.68)). Irrespective of the symmetry of a system, the effective field F eff .r/ is conservative, and therefore the potential energy vee .r/ path-independent. However, depending on the symmetry of a system, the individual components of F eff .r/, with the exception of E H .r/, may not be separately conservative. Thus, in such a case, if a particular electron correlation, say for example CorrelationKinetic effects are ignored so that Z tc .r/ D 0, the resulting approximate effective field F eff .r/ will not be conservative. The approximate electron-interaction potential energy vee .r/ determined from this field would then not be path-independent. As demonstrated in Chaps. 9 and 10, one approach to obtaining approximate pathindependent vee .r/ for nonspherical density atoms within Q-DFT is to sphericalize these atoms in some manner, whether by spherically averaging the orbitals or by taking the spherical average of the approximate F eff .r/. Another approach to obtaining path-independent vee .r/ when the approximate F eff .r/ is not conservative is to employ the irrotational component of the field. In this chapter, we demonstrate the Irrotational Component Approximation [3, 4] as described in Sect. 6.4.2 by application to the ground state of the carbon atom in its degenerate 1s 2 2s 2 2pz2 state. To make the application of the approximation

235

236

12 Application of the Irrotational Component Approximation

accessible, and to make the calculations as analytical as possible, we employ Hydrogenic orbitals. With this choice of orbitals, the nonsphericity of the atom in this degenerate state is exaggerated. Furthermore, we neglect Coulomb correlations and Correlation-Kinetic effects, and consider only correlations due to the Pauli exclusion principle. Brief details of the derivation and the specific analytical and integral expressions for the various properties are given in Appendix E. A key result of these calculations amongst many others, is that the solenoidal component E Sx .r/ of the Pauli field E x .r/ (see (3.32)) due to the Fermi hole x .rr 0 / is negligible in comparison to its irrotational component E Ix .r/. This implies that essentially all the many-body effects are accounted for by the irrotational component. For self-consistently determined orbitals for which the distortion of the electron density from sphericity is far less pronounced, the solenoidal component will be still smaller. Another key observation is that the path-independent Pauli potential energy function obtained from the irrotational component of the Pauli field E x .r/ is different in the different directions primarily in the intershell region of the atom. These two observations thus explain why the central field model of atoms leads to accurate results. The application of the approximation to the nonspherical carbon atom in its degenerate state is described in Sect. 12.1 to 12.4. Conclusions regarding the accuracy of the approximation are arrived at in Sect. 12.5.

12.1 Scalar Effective Fermi Hole Source eff x .r/ For nonspherically symmetric systems treated within the Irrotational Component Approximation, the path-independent Pauli potential energy WxI .r/ is obtained from the irrotational component E Ix .r/ of the Pauli field E x .r/. Equivalently, the potential energy WxI .r/ is determined from a static scalar effective Fermi hole source charge xeff .r/ defined as (see (6.100), (6.101), (6.106)) as 1 r  E x .r/ 4

Z 1 .r  r 0 / 0 r x .rr 0 /  dr ; D x .rr 0 / C 4

jr  r 0 j3

xeff .r/ D

(12.1) (12.2)

where E x .r/ is the Pauli field due to the Fermi hole x .rr 0 / obtained via Coulomb’s law: Z x .rr 0 /.r  r 0 / 0 E x .r/ D dr (12.3) jr  r 0 j3 with the Fermi hole defined by (3.21). (In this chapter, we treat the Fermi hole as being a positive charge distribution so that x .rr 0 / D Cjs .rr 0 /j2 =2.r/.) The charge conservation sum rule for the effective Fermi hole is then

12.1 Scalar Effective Fermi Hole Source xeff .r/

237

Z

xeff .r/dr D 1:

(12.4)

For spherically symmetric systems, the Pauli field E x .r/ and its irrotational component E Ix .r/ are, of course, equivalent. As noted earlier, the Pauli field E x .r/ is determined from the dynamic Fermi hole charge x .rr 0 / via Coulomb’s law. In turn, the irrotational component E Ix .r/ is also obtained via Coulomb’s law from the static effective Fermi hole charge xeff .r/ (see (6.111)). Thus, for spherically symmetric systems, the Pauli potential energy Wx .r/ which is the work done in the field E x .r/, may also be obtained directly from the effective Fermi hole xeff .r/. As xeff .r/ is a static charge distribution, the potential energy Wx .r/ may then be determined in a manner similar to that of the Hartree potential energy WH .r/ which is obtained from the static electron density charge .r/. Therefore, for spherically symmetric or sphericalized atoms, Z Wx .r/ D WxI .r/ D

xeff .r 0 / 0 dr : jr  r 0 j

(12.5)

Let us, therefore, first study the structure of the static effective Fermi hole charge xeff .r/ for such systems.

12.1.1 Spherically Symmetric Density Atoms Consider the Ar atom. For this atom, we determine the effective Fermi hole charge distribution xeff .r/ employing analytical Hartree-Fock theory wave functions [5]. In Fig. 12.1 the quantity 4 r 2 xeff .r/ is plotted together with the radial probability density 4 r 2 .r/. Observe that although the effective charge distribution is both positive and negative, it closely resembles the radial probability density in exhibiting the shell structure of the atom. Thus, the effective Fermi hole charge for spherically symmetric atoms has a structure similar to that of the radial probability density. This static charge distribution then gives rise to the same potential energy Wx .r/ via (12.5) as the work done in the field E x .r/ due to the dynamic Fermi hole x .rr 0 /. The plot of Wx .r/ as obtained with these Hartree–Fock theory wave functions is essentially the same as that of Fig. 10.10 as determined from the Q-DFT Pauli Correlated Approximation.

12.1.2 Nonspherical Density Atoms Next, let us consider the open-shell carbon atom in its degenerate 1s 2 2s 2 2pz2 state. We assume the single particle orbitals of the model system of noninteracting fermions to be Hydrogenic so that

238

12 Application of the Irrotational Component Approximation

Radial Charge Densities (a.u.)

24 Argon Atom 16 4pr2rxeff(r) 4pr2r (r)

8

0

–8 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

r (a.u.)

Fig. 12.1 Variation of the effective radial Fermi hole charge 4 r 2 xeff .r/, and the radial probability density 4 r 2 .r/ for the Ar atom

1 D p Z 3=2 e Zr

  Zr Zr=2 1 3=2 e 1 Z 2s .r/ D p 2 2 2

1 Z 5=2 r cos e Zr=2 2pz .r/ D p 4 2

1s .r/

(12.6)

with Z D 6. (A way to think of these orbitals in the present context is that there exists some hypothetical external potential energy v.r/ such that the resulting effective potential energy vs .r/ generates these model fermion orbitals self-consistently.) For this choice of orbitals, the density .r/ has azimuthal symmetry and depends only on the coordinates .r/  q. The highly nonspherically symmetric structure of the density .r/ is evident in Fig. 12.2 where the radial probability density r 2 .q/ is plotted for different directions corresponding to  D 0ı , 30ı , 60ı , and 90ı . The Fermi hole x .rr 0 /  x .qq 0 / is also independent of the azimuthal angles  and  0 . In the panels of Fig. 12.2, we also plot the effective Fermi hole charge xeff .r/ via the quantity r 2 xeff .r/ for the same angles . Observe again how the effective Fermi hole resembles the probability density. For example, as the probability density in the intershell region diminishes for increasing , so does the effective Fermi hole distribution. To further demonstrate the similarity in structure between the effective Fermi hole and the probability density, we plot in the panels of Fig. 12.3 these charge distributions as a function of the angle  for specific electron positions. These positions correspond to the maxima and intershell minimum of the radial probability distribution as well as a point in the classically forbidden region for  D 0ı . Again observe that the effective Fermi hole charge essentially follows the probability density distribution. Thus, the structure of the effective Fermi hole charge for nonsymmetrical

12.2 Vector Vortex Fermi Hole Source J x .r/

239

0.8 q = 0°

q = 30°

q = 60°

q = 90°

0.4

Charge Densities (a.u.)

0.0

0.6 0.4

0.0

r2 rxeff(r,q) –0.4

r2 r (r,q)

–0.8 0.0

1.0

2.0

0.0

1.0

2.0

r (a.u.)

Fig. 12.2 Variation of the effective Fermi hole charge density r 2 xeff .r / and the radial probability density r 2 .r / as a function of the radial distance r for different angles  for the degenerate state of the C atom

density atoms is also similar to that of the probability density distribution. Irrespective of the fact that the density distribution is nonspherical, the effective Fermi hole charge satisfies the conservation sum rule of (12.4).

12.2 Vector Vortex Fermi Hole Source J x .r/ The vector vortex Fermi hole source J x .r/ is defined as (see (6.114), (6.115)) J x .r/ D D

1 4

Z

1 r  E x .r/ 4

Œr x .rr 0 / 

.r  r 0 / 0 dr jr  r 0 j3

with E x .r/ the Pauli field due to the Fermi hole x .rr 0 /.

(12.7)

(12.8)

240

12 Application of the Irrotational Component Approximation 0.6 0.6 0.4 r = 0.18 r = 0.35

0.2

0.4

Charge Densities (a.u.)

0.0 0.2

0.0

–0.2

0

60

0.6

120

–0.4 180 0 0.4

60

120

180

r = 1.25

r = 0.71

0.4

0.3

0.2

0.2

0.0 r2rxeff(r,q)

–0.2 –0.4

0.1

r2r (r,q) 0

60

120 q°

180

0.0

0

60

120

180

q°

Fig. 12.3 Variation of the effective Fermi hole charge density r 2 xeff .r / and the radial probability density r 2 .r / as a function of the angle  for different positions r of the electron

For the carbon atom in the degenerate state specified above with a nonspherical density, the curl of the Pauli field E x .r/ does not vanish: r  E x .r/ ¤ 0. As the source x .rr 0 / is independent of the azimuthal angles  and  0 , E x .r/ is independent of the angle  and does not have an azimuthal component. (This is also the case for the irrotational E Ix .r/ and solenoidal E Sx .r/ components of the field.) The Pauli field is thus E x .r/ D i r Ex;r .q/ C i  Ex; .q/; (12.9) so that its curl is also independent of , and only has an azimuthal component: r  E x .r/ D i  Œr  E x .q/ . In Fig. 12.4 the vector vortex Fermi hole source 4 J x .r/, or equivalently the curl of the Pauli field E x .r/, is plotted as a function of the radial distance r from the nucleus for different angles . The source vanishes for  D 0ı , but its magnitude increases (Fig. 12.4a) with increasing values of  reaching a maximum for  D 60ı . With a further increase in  (Fig. 12.4b) the magnitude of the source decreases vanishing once again for  D 90ı . As expected, the vortex source is finite within the atom, and vanishes in all directions for r > 2:0 a.u. Thus, in the classically

12.3 Irrotational E Ix .r/ and Solenoidal ESx .r/ Components

241

Fig. 12.4 Variation of the vector vortex Fermi hole source 4 J x .r/ D r  Ex .r/ D i  Œr  E x .q/ as a function of the model fermion position r from the nucleus for different angles 

forbidden region and right up to the surface of the atom, the vortex source and thus the solenoidal component of the Pauli field are zero. The Pauli field E x .r/ in these regions is thus equivalent to its irrotational component E Ix .r/. Therefore, in the classically forbidden region as well as at the origin and in the directions  D 0ı and 90ı , the potential energy WxI .r/ is equal to the potential energy Wx .r/ due to the Fermi hole itself. What the study of the vortex source J x .r/ shows is that even for nonsymmetrical density atoms, the work Wx .r/ is path-independent over substantial regions of configuration space. Further, with a knowledge of the symmetry of the system, specific directions for which this is also the case can be determined. In addition, the self-consistent determination of the structure of these atoms is facilitated since the local many-body potential energy Wx .r/ in these regions and directions is then obtained directly from the Fermi hole x .rr 0 / itself.

12.3 Irrotational E Ix .r/ and Solenoidal E Sx .r/ Components of the Pauli Field E x .r/ It is evident from Figs. 12.2 and 12.4, that both the scalar effective xeff .r/ and the vector vortex J x .r/ sources are confined to within the atom. This, of course, must be the case as both these sources are derived from the Fermi hole charge x .rr 0 /

242

12 Application of the Irrotational Component Approximation

that is localized about the nucleus. The significant question that next needs to be answered is how the irrotational E Ix .r/ and solenoidal E Sx .r/ components of the Pauli Field E x .r/ compare to each other. This will answer what fraction of the many-body effects is incorporated in the effective source xeff .r/, and therefore how accurate the path-independent Pauli potential energy WxI .r/ obtained from this source is. The irrotational E Ix .r/ and solenoidal E Sx .r/ components of the Pauli field E x .r/ are defined in terms of their sources, respectively, as Z E Ix .r/ D Z

and E Sx .r/ D

xeff .r 0 /.r  r 0 / 0 dr jr  r 0 j3

J x .r 0 / 

.r  r 0 / 0 dr : jr  r 0 j3

(12.10)

(12.11)

In Figs. 12.5–12.7, we compare the magnitudes jE Ix .q/j and jE Sx .q/j of the irrotational and solenoidal components of the Pauli field E x .r/ as a function of the radial distance r for the different angles  D 0ı , 30ı , and 60ı , respectively. In Fig. 12.5, which corresponds to  D 0ı , there is only the irrotational component jE Ix .q/j since in this direction the vortex source vanishes. This force field increases from a small value in the deep interior to a maximum within the K shell of the atom. The structure of the field clearly delineates between the atomic shells, with its value in the L shell being much smaller. In Fig. 12.5, the function r 2 jE Ix .q/j is also plotted as the dashed

Fig. 12.5 The magnitudes jE Ix .q/j and jESx .q/j of the irrotational and solenoidal components of the Pauli field Ex .r/ as a function of the radial distance r for  D 0ı . The function r 2 jE Ix .q/j is also plotted as the dashed line

12.3 Irrotational E Ix .r/ and Solenoidal ESx .r/ Components

243

Fig. 12.6 The magnitudes jE Ix .q/j and jESx .q/j of the irrotational and solenoidal components of the Pauli field Ex .r/ as a function of the radial distance r for  D 30ı

Fig. 12.7 The figure caption is the same as that for Fig. 12.5 with the exception that this figure is plotted for  D 60ı

244

12 Application of the Irrotational Component Approximation

line. From this graph it is evident that the force field E Sx .q/ decays asymptotically as 1=r 2 in the  D 0ı direction. Consequently, the asymptotic structure of the potential energy WxI .r/ in this direction must be 1=r. Figs. 12.6 and 12.7 correspond to  D 30ı and 60ı for which the vortex source is finite (see Fig. 12.4), so that there is a solenoidal component jE Sx .q/j in these directions. Observe that the solenoidal component is negligible in the deep interior of the atom. It reaches a maximum at approximately the same electron position as that of the irrotational component. However, this maximum in jE Sx .q/j is 2 orders of magnitude smaller than that of jE Ix .q/j. The solenoidal component vanishes for r > 2:0 a.u., as expected since the vortex source vanishes in this region. The fact that the solenoidal component is negligible implies that essentially all the many-body effects are incorporated in the effective charge xeff .r/, and that as a consequence the potential energy WxI .r/ is accurate throughout space. The structure of the irrotational component in the  D 30ı and 60ı directions is similar to that for  D 0ı , once again delineating between the shells and decaying asymptotically as 1=r 2 . Thus, as is the case for spherically symmetric atoms, the potential energy WxI .r/ for open shell atoms decays asymptotically as 1=r in all directions. This is because asymptotically the structure of WxI .r/ is due to a unit static (negative) charge contained in xeff .r/. Equivalently, since the vortex source vanishes in the classically forbidden region, the asymptotic structure of WxI .r/ is the same as that of Wx .r/ which in turn arises from the unit (negative) charge of the Fermi hole x .rr 0 / which is also essentially static for these electron positions. It is also evident from the plots of r 2 jE Ix .q/j in Figs. 12.5–12.7 that the Pauli field E x .r/ approaches the 1=r 2 asymptotic limit at different rates in different directions. This is to be expected due to the nonspherical nature of the density. However, the different asymptotic decay rates of the Pauli field, and consequently of the potential energy Wx .r/, can be explained more rigorously by a study of the center of mass hr 0 i of the Fermi hole for these positions. Since for the orbitals assumed, the Fermi hole has azimuthal symmetry, the center of mass always lies along the z0 . 0 D 0ı / axis irrespective of the coordinates .r/ of the electron. In Fig. 12.8 we plot hr 0 i as a function of the radial distance r for  D 0ı , 30ı , 60ı , and 90ı . For asymptotic positions of the electron, the center of mass approaches a constant value which lies between 0 and 0.5 a.u. for 90ı    0ı since the limr!1 hr 0 i cos =Œ1Ccos2 . That the center of mass approaches a different constant in different directions can also be seen by plotting the Fermi hole x .rr 0 / for asymptotic positions of the electron. (Recall that for spherical density atoms, the Fermi hole charge is essentially static and spherically symmetric about the nucleus for asymptotic positions of the electron. Thus, asymptotically, the center of mass is at the nucleus.) For the nonspherical density atom, the Fermi hole charge is again essentially static but not spherically symmetric about the nucleus [4]. Thus, for asymptotic positions of the electron, the center of mass in different directions is at different positions. Therefore, together with the fact that the total charge of the Fermi hole is unity, the different asymptotic values of its center of mass in different directions explains why the Pauli field decays toward 1=r 2 with different decay rates in these directions. For orbitals more accurate than the Hydrogenic functions such as those determined self-consistently, the center of mass for asymptotic positions of the electron would

12.4 Path-Independent Pauli Potential Energy WxI .r/

245

Fig. 12.8 The center of mass hr 0 i of the Fermi hole x .rr 0 / as a function of the radial electron distance r for different directions  D 0ı ; 30ı ; 60ı , and 90ı . Due to the azimuthal symmetry of the Fermi hole, its center of mass lies along the z0 axis irrespective of the coordinates .r / of the electron

be closer to the nucleus. Consequently, the Pauli field and potential energy would achieve their asymptotic structures closer to the surface of the atom.

12.4 Path-Independent Pauli Potential Energy WxI .r/ The path-independent Pauli potential energy WxI .r/ is obtained either from the irrotational component E Ix .r/ of the field E x .r/ or from the effective Fermi hole charge xeff .r/: Z WxI .r/ D  Z D

r

1

E Ix .r 0 /  d` 0

xeff .r 0 / 0 dr : jr  r 0 j

(12.12) (12.13)

The potential energy WxI .r/ is plotted in Fig. 12.9 as a function of the radial distance r for different values of . As for spherical density atoms, the potential energies in the different directions WxI .r/ increase monotonically throughout space so that the work done to remove the model fermion in any direction is always positive. The potential energies in each direction also clearly delineate between the K and L shells, there being a distinct change in the structure primarily in the intershell region. As expected from our discussion above of the asymptotic structure of the field E Ix .r/, the potential energy WxI .r/ decays as 1=r at different rates in the different directions. By r D 10 a.u. these functions are all equivalent to two decimal places. It is also evident from the figure that in the deep interior of the atom, the differences between the potential energies in the different directions is negligible. At the nucleus, all the potential energy functions merge to the same value. This is because for an electron at the nucleus, the Fermi hole x .rr 0 / is spherically

246

12 Application of the Irrotational Component Approximation

Fig. 12.9 Variation of the path independent potential energy WxI .r / as a function of the radial distance r for different angles 

symmetric about it. As such the Pauli field E x .r/ and its solenoidal and irrotational components vanish at the nucleus. Thus, the different functions WxI .r/ all have the same value at the nucleus. Further, the slope of these functions there is zero. The fact that the functions WxI .r/ for different angles  all have the same value at the nucleus is also a reflection of the path-independence of these potential energy functions.

12.5 Endnotes on the Approximation In the earlier sections, the Irrotational Component Approximation at the Pauli level of electron correlation has been applied to the nonspherical density carbon atom in its degenerate 1s 2 2s 2 2pz2 state. This application leads to the following understandings. For the nonspherical density atom, the Fermi hole charge x .rr 0 / is spherically symmetric about the electron when it is at the nucleus. For other positions of the electron, the Fermi hole is neither spherically symmetric about the electron or the nucleus. (This is, of course, also the case for spherical density atoms.) For the nonspherical atom, both the scalar effective charge xeff .r/ and the vector vortex charge J x .r/ possess the symmetry of the atom, and are therefore nonspherically symmetric about the nucleus. For finite systems such as atoms and molecules, the Fermi hole

12.5 Endnotes on the Approximation

247

x .rr 0 / for all electron positions is localized to within the system. Thus, the scalar effective and vector vortex sources are so confined. The vector vortex source also vanishes in the classically forbidden region and along certain axes of symmetry. Thus, so does the solenoidal component E Sx .r/ of the Pauli field E x .r/. Therefore, in these regions and directions the field E x .r/ is conservative. As such the path-independent potential energy WxI .r/ in these regions and directions is equal to Wx .r/, the work done in the force field of the Fermi hole charge x .rr 0 /. As a result, over the substantial fraction of configuration space corresponding to these electron positions, the path-independent Pauli potential energy may be obtained directly from the Fermi hole x .rr 0 / as the work Wx .r/ instead of from the effective Fermi hole xeff .r/. One of the significant observations of the earlier calculation is that the solenoidal component E Sx .r/ of the Pauli field E x .r/ is negligible in comparison to its irrotational component E Ix .r/. In a fully self-consistent calculation, it is expected that the solenoidal component would be even smaller. This implies that essentially all the electron correlations due to the Pauli principle are accounted for by the irrotational component field E Ix .r/. In turn this means that the Irrotational Component Approximation is accurate. The earlier remarks are equally valid for the Coulomb hole charge c .rr 0 / and the corresponding scalar effective Coulomb hole ceff .r/, each of which is localized about the nucleus with a total charge of zero. One could reasonably expect that the corresponding solenoidal component E Sc .r/ of the Coulomb field E c .r/ is also negligible in comparison to its irrotational component E Ic .r/, and that it decays faster asymptotically. Thus, the asymptotic structure of the Coulomb field E c .r/ is that of its irrotational component E Ic .r/. Now because the charge of the Coulomb hole is zero, the Coulomb field E c .r/ will decay in the classically forbidden region very much faster than the Pauli field E x .r/. Therefore, the Coulomb potential energy Wc .r/ decays asymptotically faster than the Pauli potential energy Wx .r/. This behavior parallels that for spherically symmetric density atoms. At present, no rigorous comparative statement with regard to the magnitudes of the irrotational Z Itc .r/ and solenoidal Z Stc .r/ components of the CorrelationKinetic field Z tc .r/ for nonspherical density systems can be made. For model S systems in their ground state, Correlation-Kinetic effects are small. As such the use of the irrotational component Z Itc .r/ to obtain the corresponding path-independent potential energy WtIc .r/ would suffice. Again, as is the case for spherical density atoms, it is expected that the Correlation-Kinetic potential energy WtIc .r/ would decay asymptotically faster than both the Pauli Wx .r/ and Coulomb Wc .r/ potential energies. Finally, based on the earlier remarks, the asymptotic structure of the effective potential energy vee .r/ (see (6.64), (6.65)) for nonspherical density systems is that of WxI .r/ 1=r. As the potential energy WxI .r/ in this region equals that of Wx .r/ the work done in the field of the Fermi hole x .rr 0 /, the asymptotic structure of vee .r/ is once again due solely to Pauli correlations and known exactly. Since, the highest occupied eigenvalue is governed by this asymptotic structure, it is expected that in any fully self-consistent calculation, whether at the Pauli, Pauli-Coulomb or

248

12 Application of the Irrotational Component Approximation

fully correlated levels, accurate ionization potentials, electron affinities, transition energies and polarizabilities will also be obtained. The calculations described in this chapter indicate that essentially all the electron correlations are accounted for by the irrotational component of the effective field F eff .r/ from which the effective potential energy vee .r/ is obtained (see (6.65)). This in turn implies the accuracy of the Irrotational Component Approximation.

Chapter 13

Application of Q-DFT to Atoms in Excited States

This chapter is concerned with the application of Q-DFT to atoms in their excited states. As noted in the introduction (Chap. 1), the framework of the Q-DFT mapping from an interacting system of electrons to a model S system of noninteracting fermions with equivalent density, is general in that it is the same for both the ground and excited states of the real system. Hence, all the equations governing Q-DFT and consequently all the approximation schemes, are equally applicable to systems in their excited states. Additionally, within Q-DFT, the state of the S system is arbitrary: the model system may be constructed to be in a ground or excited state configuration [1–6] (see also Chap. 5). A benefit of the mapping from an excited state of an atom to a model S system in its ground state is that fewer single particle orbitals of the latter may have to be obtained. For example, in the mapping of any singly or doubly excited state of the He atom to an S system in its ground state, only one orbital of the latter need be determined. For an example of such a mapping from an excited state of the exactly solvable Hooke’s atom [7] to an S system in its ground state with equivalent density, see [1] and QDFT. The arbitrariness of the S system configuration within Q-DFT contrasts with that of traditional excited state density functional theory [8] where the mapping is restricted to an S system having the same excited state configuration as that of the atom. It is also observed (see [1] and Chap. 5) that if the mapping is to an S system of the same configuration as that of the interacting system, then the Correlation-Kinetic contributions are small in comparison to when the mapping is to a model system in a different configuration. This is the case for the mapping from both the ground and excited states of the interacting system. Thus, in approximation methods within Q-DFT in which one wishes to initially ignore Correlation-Kinetic effects, it is best to map an interacting system to an S system of the same configuration. In this chapter, the Q-DFT Pauli Approximation is applied to atoms in excited states. The equations of this approximation are given in Sect. 6.2.1. These fully selfconsistent results for excited states of atoms constitute a benchmark, in which only correlations due to the Pauli exclusion principle are considered; the correlations due to Coulomb repulsion and Correlation-Kinetic effects are absent. Recall that in contrast, Hartree–Fock theory incorporates Correlation-Kinetic effects due to the correlations between electrons of parallel spin. Thus, the difference between the

249

250

13 Application to Atomic Excited States

Q-DFT Pauli Approximation and Hartree–Fock theory results are an accurate estimate of these Correlation-Kinetic effects. (In “exchange-only” density functional theory [9,10], both Pauli correlations as well Correlation-Kinetic effects as obtained via lowest-order perturbation theory are accounted for.) See Sect. 6.2.2. and QDFT for further discussion of these points. In the calculations presented, the mapping from the excited state of the atom is to an S system in the same excited state configuration. It is observed, that as was the case for the ground state calculations of Chap. 10, the Q-DFT Pauli Approximation results for the excited states are essentially equivalent to those of Hartree–Fock theory. However, they do constitute an upper bound, the differences being a few parts per million. Thus, Correlation-Kinetic effects due to Pauli correlations are negligible for these excited states. As such the differences between the results obtained and those of correlated wave function calculations or of experiment may be attributed to Coulomb correlations and Correlation-Kinetic effects that arise due to Coulomb repulsion. In the later sections, the following applications of the Q-DFT Pauli Approximation to atoms in excited states are discussed: (a) the triplet 2 3 S state isoelectronic sequence of the He atom [11, 12]; (b) one-electron excited states of the Li atom: Li: [He] 3d , 4d , 5d , 4f , 5f [2]; (c) one-electron excited states of the Na atom: Na: 4s 1 , 5s 1 , 3p 1 , 4p 1 [13]; (d) multiplet structure associated with the ground state and two-electron excited state configurations of C and Si: the multiplets 3P , 1D, 1S [14]; (e) doubly excited autoionizing states of the He atom [15]. In the literature, there are innumerable calculations [16–20] of atoms in excited states that have been performed employing a hybrid of the Q-DFT Pauli Correlated Approximation and traditional Kohn–Sham DFT in an attempt to incorporate Coulomb correlations and Correlation-Kinetic effects. In these hybrid calculations, the idea is to employ the Q-DFT Pauli potential energy Wx .r/ in conjunction with the functional derivative vc .r/ of an approximate Kohn–Sham “correlation” energy functional Ecapprox Œ to generate the spin-orbitals. The Pauli–Coulomb energy Exc is then approximated as Exc  Ex C Ecapprox Œ, where Ex is the Pauli energy obtained from these orbitals. Although the results are quite accurate in comparison to both experiment and correlated wave function calculations, they do not rigorously fall under the rubric of Q-DFT in which the Coulomb correlation and Correlation-Kinetic contributions are obtained from their respective quantal sources and fields thereby leading to rigorous upper bounds to the total energy. Additionally, the approximate Kohn–Sham “correlation” energy functionals employed in these hybrid calculations - a parametrized Wigner-type local functional [21], and the nonlocal Lee–Yang–Parr [22] functional – are ground state energy functionals and not bidensity energy functionals [8] as must be the case for excited states. Furthermore, although the Lee–Yang–Parr “correlation” energy functional is one that is employed extensively in ground state calculations, there has recently been a critique [23–27] of the physical underpinnings of this functional. Hence, for the reasons given earlier, the results of these hybrid calculations are not discussed here. The applications described in this chapter clearly demonstrate the ease of performing atomic excited state calculations within the local effective potential energy

13.1 The Triplet 2 3 S State Isoelectronic Sequence of He

251

theory framework of Q-DFT. For a survey of other schemes for the calculation of excited state properties within traditional DFT as well as wave function based methods, the reader is referred to the review article of [28].

13.1 The Triplet 2 3 S State Isoelectronic Sequence of He The results [11] for the total energies as obtained by application of the Q-DFT Pauli Approximation to the triplet 2 3S isoelectronic sequence of the He atom are given in Table 13.1. The corresponding single-particle orbitals and single determinant wave function are obtained by solution of the differential equation (6.56). (The work on this state of the atom was originally performed by Singh and Deb [12]. However, in their numerical code, the asymptotic 1=r structure of the Pauli potential Wx .r/ was not explicitly incorporated [29]. Hence, the energy values in [12] differ slightly from those noted here.) For purposes of comparison, the total energies determined via a 55-term correlated wave function calculation [30, 31] are also considered. A study of the results of Table 13.1 show that the Q-DFT Pauli Approximation results are essentially exact. The differences from the “Exact” values range from 0:08% for He to 0:005% for Ne 8C . This negligible difference can be attributed to Coulomb correlations and Correlation-Kinetic effects. For the 2 3 S state of He, the sum of the Coulomb correlation Ec and Correlation-Kinetic energy Tc is: Ec C Tc D 0:0070 a.u. (see Table 13.1). The corresponding value ([32], Chap. 15) for the ground 1 1S state of the He atom is: Ec C Tc D 0:0421 a.u., an order of magnitude greater. The reason why the sum Ec C Tc is less for the triplet state in comparison with the ground state value is because the two electrons are generally further apart. Thus, one would expect that for singly excited states of atoms, this sum should always be less than its value for the ground state of the atom.

Table 13.1 Total energies of the 23 S state of the He atom isoelectronic sequence as determined by the Q-DFT Pauli Approximation, and “Exact” values obtained from a correlated wave function calculation. The quantities in parenthesis are the percent differences of the Q-DFT results. The negative values of the energies in Rydbergs are given Atom/Ion

Q-DFT Pauli Approximationa

He 4.3470 (0.080) LiC 10.2170 (0.044) Be2C 18.5894 (0.026) B3C 29.4626 (0.018) C4C 42.8361 (0.013) N5C 58.7099 (0.009) O6C 77.0837 (0.007) F7C 97.9576 (0.006) Ne8C 121.3316 (0.005) a See [11]; b see [30, 31]

“Exact”b 4.3505 10.2215 18.5943 29.4678 42.8415 58.7154 77.0893 97.9632 121.3373

252

13 Application to Atomic Excited States 8.50 Ne8+

F7+ 6.80

Radial density

O6+

N5+

5.10

C4+ B3+

3.40

Be2+ Li+ 1.70

He

0 0

1.20

2.40

3.60

4.80

6.00

r

Fig. 13.1 The radial probability density 4 r 2 .r/ of the He 2 3 S isoelectronic sequence in a.u. as determined by the Q-DFT Pauli Approximation [12]

The increase in accuracy down the isoelectronic sequence, is of course, a consequence of the diminution of the Coulomb correlation and Correlation-Kinetic effects which results because the core orbital shrinks more with increasing nuclear charge. This is explained [12] in Fig. 13.1 in which the radial probability density of the elements of the isoelectronic sequence are plotted. Observe, how sharply peaked the probability density for the core shell of Ne8C is in comparison to that of He, and how much wider the outer shell of the latter is.

13.2 One-electron Excited States of Li In Tables 13.2 and 13.3, the results of application [2] of the Q-DFT Pauli Approximation to the singly excited states of the Li atom ([He] 3d , 4d , 5d , 4f , 5f ) are given. These are fully self-consistent single-determinant calculations performed in the central field approximation with the S system in an excited state with the same configuration as that of the excited state of the atom. For purposes of comparison,

13.2 One-electron Excited States of Li

253

Table 13.2 Total energies of the excited states of the Li atom as determined by the Q-DFT Pauli Approximation, Hartree–Fock theory, and the “Exact” values as obtained by a configurationinteraction wave function. The negative values of the energies in a.u. are given Li ŒHe 3d ŒHe 4d ŒHe 5d ŒHe 4f ŒHe 5f a See [33]

Q-DFT Pauli Approximation

Hartree–Fock Theory

Configuration Interactiona

7.29194 7.26764 7.25640 7.26766 7.25641

7.29198 7.26767 7.25642 7.26767 7.25642

7.33552 7.31119 7.29993 7.31117 7.29992

Table 13.3 Ionization potentials of the excited states of the Li atom as determined by the Q-DFT Pauli Approximation. The second column corresponds to the negative of the highest occupied eigenvalue m . The remaining columns are ionization potentials as obtained by total energy differences Ek  Ekion as determined via Q-DFT, a CI calculation, and experiment. The values in a.u. are given

Li

Ionization Potential m Ek  Ekion Q-DFT Pauli CIa Approximation

ŒHe 3d 0.05572 ŒHe 4d 0.03137 ŒHe 5d 0.02008 ŒHe 4f 0.03125 ŒHe 5f 0.02000 a See [33]; b see [34]

0.05554 0.03124 0.02000 0.03126 0.02001

0.05561 0.03128 0.02002 0.03126 0.02000

Experimentb 0.05561 0.03127 0.02001 0.03124 0.01997

the results [2] of Hartree–Fock theory, as well as those of a configuration-interaction (CI) wave function calculation [33] are also considered. The Q-DFT results for the total energy constitute an upper bound to those of Hartree–Fock theory. The differences between the two sets of results (see Table 13.2) are 5 ppm for the Li: [He] 3d state and diminish further for the other higher excited states. This implies that the Correlation-Kinetic contributions due to the correlations arising as a result of the Pauli exclusion principle are negligible. The difference of the Q-DFT results taken with respect to the CI results (see Table 13.2) is 0:6% for the excited states considered. This difference can therefore be attributed solely to Coulomb correlations and Correlation-Kinetic effects arising from such correlations. The results for the ionization potential as determined via the highest occupied eigenvalue m and by total energy differences Ek Ekion , together with experimental values [34] are given in Table 13.3. The results of a CI calculation [33] obtained by total energy differences are also given. The accuracy of the Q-DFT Pauli Approximation results speaks for itself. Those of the highest occupied eigenvalue m are

254

13 Application to Atomic Excited States

Fig. 13.2 The Pauli potential energy Wx .r/ for the excited states of Li: [He]4d, 4f [2]

because the asymptotic structure of the electron-interaction potential energy vee .r/ for the fully interacting system is given by the Pauli potential energy Wx .r/ 1=r as r ! 1. (We expect that as for the ground state (see Chap. 7), the Coulomb Wc .r/ and Correlation-Kinetic Wtc .r/ potential energies will decay faster.) The plots of the Pauli potential energy Wx .r/ for the excited states Li: [He] 4d; 4f are given in Fig. 13.2. (Observe the similarity in structure of these Wx .r/ with that for the ground state of the Ar atom, Fig. 10.10). In a comparison with the CI results (see Table 13.3), it is evident that accurate ionization potentials can be obtained for excited states of atoms via the Q-DFT Pauli Approximation since the Coulomb correlation and Correlation-Kinetic contributions are negligible. Hence, as explained in Sect. 13.3 on the excited states of Na, accurate transition energies are also obtained via this approximation.

13.3 One-electron Excited States of Na The total ground state energy E and the highest occupied orbital eigenvalue m of the Na atom in its ground and various excited states: Na: 4s 1 , 5s 1 , 3p 1 , 4p 1 , as determined [13] by the Q-DFT Pauli Approximation and Hartree–Fock theory are

13.3 One-electron Excited States of Na

255

Table 13.4 Total energy E and highest occupied orbital eigenvalue m of the Na atom in its ground and various single electron excited states as determined [13] via the Q-DFT Pauli Approximation and Hartree–Fock theory. The negative values of these properties in a.u. are given

Na

Q-DFT Pauli Approximation E m

3s 1 4s 1 5s 1 3p 1 4p 1

161.851 161.738 161.712 161.777 161.725

0.195 0.080 0.042 0.116 0.054

Hartree–Fock Theory E m 161.859 161.747 161.714 161.786 161.727

0.182 0.070 0.037 0.110 0.053

Table 13.5 Transition energies in a.u. for the Na atom as determined by the Q-DFT Pauli Approximation from highest occupied eigenvalues differences, and Hartree–Fock theory from total energy differences. Experimental values are also given Transitions 3 2S ! 4 2 S 3 2S ! 5 2 S 3 2S ! 3 2 P 3 2S ! 4 2 P a See [35]

Hartree–Fock Theory

Q-DFT Pauli Approximation

Experimenta

0.112 0.145 0.073 0.132

0.115 0.153 0.079 0.141

0.117 0.151 0.077 0.138

given in Table 13.4. The Q-DFT total energies are an upper bound to and within 55 ppm of those due to Hartree–Fock theory. These results once again demonstrate the accuracy of Q-DFT in this approximation. And again, the results show that for these excited states Correlation-Kinetic effects due to Pauli correlations are negligible. Various transition energies as obtained from the Q-DFT highest occupied eigenvalues of Table 13.4 are given in Table 13.5 as are the experimental values [35] for these transitions. The Hartree–Fock theory values given, however, are obtained from total energy differences since this is the most accurate way of determining transition energies by this theory. (Note that the highest occupied eigenvalue of Hartree–Fock theory has the meaning of a removal energy only in the context of Koopmans’s theorem [36] which does not allow for the relaxation of the remaining orbitals.) The Q-DFT results are consistently superior to those of Hartree–Fock theory when compared to experiment. They also indicate that Coulomb correlation and Correlation-Kinetic effects are small. Thus, accurate transition energies can be obtained via the Q-DFT Pauli Approximation.

256

13 Application to Atomic Excited States

13.4 Multiplet Structure of C and Si In this section the energies of the multiplets of the open shell atoms C and Si in their ground and excited state configurations are given [14]. The calculations are performed within the Q-DFT Pauli Approximation in the central field model. The method for the determination of the multiplet structure as described in [14] is as follows. For a specific electronic configuration of the atom, the orbitals of the single Slater determinant are first determined via Q-DFT. Using these orbitals, Slater’s diagonal sum rule [37] is then employed to obtain the energies of the multiplets associated with that configuration. The diagonal sum rule of Slater follows from perturbation theory. It states that the sum of the diagonal matrix elements of the Hamiltonian with respect to the unperturbed wave functions equals the sum of the energy eigenvalues of the perturbed problem. The unperturbed wave function is obtained via the Q-DFT Pauli Approximation for a particular configuration of the atom. It is the interaction between the electrons in the open-shell of the atomic system that gives rise to the multiplet splitting in a particular open-shell configuration. Spin-orbit coupling is neglected. To explain how the multiplet structure is obtained [14], consider the configuration np 2 in which two optically active electrons are in the same p subshell. In Table 13.6, we list [38,39] all the possible sets of values of the quantum numbers ml and ms for the two electrons which do not violate the exclusion principle. For each set, we also

Table 13.6 Possible quantum numbers for an np 2 configuration P ml Number ml1 m s1 ml2 m s2

P

ms

C1

C 12

2

C1

3

C1

4

C1

1

5

C1

6

C1

C 12 C 12 C 12 C 12  12

0

C 12  12 C 12  12  12

7

C1

 12

1

C 12

0

0

8

C1

1

1

0

9

0

0

C1

10

0

0

0

11

0

C1

1

0

1

0

 12  12  12 C 12  12  12

0

1

1

 12

0

2

0

 12

1

1

1

12

0

13

1

 12 C 12 C 12 C 12 C 12 C 12

1

C 12

1

 12

14 15

C1

 12

0

C2

0

C1

C1

0

C1

C1

0

0

0

1

C1

0 1

C1 0 1 1

13.4 Multiplet Structure of C and Si

257

P P list the corresponding values of ml and ms . There are therefore 15 different determinants. The possible quantum states for two electrons with the configuration np 2 are only those associated with the terms 1S , 3P , and 1D [38, 39]. 1 On applying the diagonal sum rule, the energy of the P D term would P be that evaluated from the single determinant corresponding to ml D 2 and P ms D 0 (see P Number 1 of Table 13.6). Similarly, for the 3P state corresponding to ml D 1 and msPD 1, there is only P one wave function (see Number 2 of Table 13.6). For the cases ml D 0 and ms D 0, there are three wave functions (see Numbers 5, 7, 10 of Table 13.6). The eigenvalues of these three wave functions when added give the sum of the energies of the 3P , 1D, and 1S terms. Hence, the energy of the 1 S term can be obtained by subtracting the energies of the 3P and 1D terms from this sum. This method in general will give the energies of multiplets for any given open-shell configuration. However, in cases where a particular term occurs more than once in the configuration, the method gives only the sum of the energies of the like terms [14]. The total energy values for the three multiplets 1S , 3P , and 1D from the two equivalent p electrons in the ground state configurations of C: [He] 2s 2 2p 2 and Si: [Ne] 3s 2 3p 2 as obtained via the Q-DFT Pauli Approximation are given in Table 13.7. In these calculations, the same radial function is used for each multiplet. The Hartree–Fock theory results employing the Clementi–Roetti wave functions [40], which are different for each multiplet, are also given in Table 13.7. Observe that the Q-DFT total energy values constitute an upper bound to the Hartree–Fock theory results. As the radial probability density of the multiplets determined by the two theories are essentially equivalent [14], the differences between the energies constitutes an accurate estimate of Correlation-Kinetic effects due to the Pauli principle. This estimate in parts per million is noted in parentheses. Once again we note that these Correlation-Kinetic contributions are negligible.

Table 13.7 Term energies of the multiplets associated with the ground state configuration of C and Si as determined by the Q-DFT Pauli Approximation and Hartree–Fock theory [14]. The numbers in parentheses (ppm) are the differences between the two theories, and constitute an accurate estimate of the Correlation-Kinetic contribution due to the Pauli principle. The negative values of the total energy in Rydbergs are given Total energy Term Q-DFT Pauli Hartree–Fock Symbol Approximation Theory C Si C Si 3 P 75.3675 577.6763 75.3772 577.7086 (129) (56) 1 D 75.2522 577.5976 75.2627 577.6301 (140) (56) 1 S 75.0792 577.4471 75.0992 577.5171 (266) (121)

258

13 Application to Atomic Excited States

Table 13.8 Term energies of the multiplets associated with two-electron excited states of C and Si as determined by the Q-DFT Pauli Approximation [14]. The negative values of the total energy in Rydbergs are given C

Si

State

Energy

State

2s 2 3p 2 3 P 2s 2 4p 2 3 P 2s 2 5p 2 3 P

73.6362 73.2481 73.0828

Energy

3s 2 4p 2 3 P 3s 2 5p 2 3 P

576.6099 576.3274

2s 2 3p 2 1 D 2s 2 4p 2 1 D 2s 2 5p 2 1 D

73.5940 73.2264 73.0691

3s 2 4p 2 1 D 3s 2 5p 2 1 D

576.5768 576.3094

2s2 3p2 1 S 2s2 4p2 1 S 2s2 5p2 1 S

73.5307 73.1935 73.0487

3s 2 4p 2 1 S 3s 2 5p 2 1 S

576.5271 576.2823

First-order theory predicts [38] that the interval ratio .1S  1D/=.1D 3P / is 1:5, with the 1S as the highest term. For C the Q-DFT ratio is 1:5 thereby ensuring accurate excitation energies. The corresponding Hartree–Fock theory value is 1:43. For Si, the Q-DFT and Hartree–Fock theory ratios are 1:91 and 1:44, respectively. In the previous section, the results of single-electron excitations of the Na atom were given. The Q-DFT Pauli Approximation total energy values for each of the multiplets calculated after exciting the two p electrons in C and Si to higher p shells are given in Table 13.8. The configurations are C: [He] 2s 2 np 2 and Si: [Ne] 3s 2 np 2 , the highest value of n being 5. For the Q-DFT Pauli Approximation results for low-lying excited states of the open-shell atoms B, C, O, F, Na, Mg, Al, Si, P, and Cl, see [20]. The corresponding Hartree–Fock theory and experimental results are also given in this work.

13.5 Doubly Excited Autoionizing States of He Autoionizing states [38, 41, 42] are states that lie above the ionization threshold. They are discrete states embedded in the continuum. If there is no interaction between the discrete states and the configurations of the continuous spectrum, these states do not exhibit any special properties as a consequence of their location in the continuous spectrum. However, if there is an interaction, then there is a mixing of the eigenfunctions of the discrete state and the neighboring levels in the continuum. As a result, the discrete state acquires some of the character of the continuum. The most important feature of the states of the continuum is that they are unstable in the sense that one of the electrons moves in an orbit which extends to infinity. Hence,

13.5 Doubly Excited Autoionizing States of He

259

as a result of the interaction with the continuum states, the discrete level acquires to some extent the property of spontaneous ionization through one of the electrons moving to infinity. It is for this reason that these discrete states are referred to as autoionizing states. Consider the excitation of the two electrons of the He atom from their ground state to such a discrete excited state by the absorption of ultra-violet radiation of energy h. The decay of such an excited state could be via radiative transitions to bound states of the He atom. However, the state could also decay, as explained earlier, via a radiationless transition into a free electron and a HeC ion in the ground state. Such a transition is known as the Auger effect or autoionization. Various calculations have shown that the autoionizing transition is much more probable than a radiative transition to a bound state of the atom. The autoionizing states appear in the absorption spectrum as sharp peaks or resonances superimposed on the smooth continuum absorption in the neighborhood of the energy of autoionizing states. With He?? indicating a doubly excited state of the He atom, the above reaction in the vicinity of an autoionizing state may be expressed through the steps h C He.11S / ! He?? ! e  C HeC ;

(13.1)

where the autoionizing state He?? corresponds to a temporarily bound or resonant state of the compound system .e  C HeC /. Autoionizing states also manifest themselves in photoabsorption, electron impact, and ion impact experiments and have been observed in the solar flare and solar corona. For the appropriate references of these observations see [15]. The total energy of 22 doubly excited autoionizing states of the He atom as determined via the Q-DFT Pauli Approximation in the central field model are given in Table 13.9. The results for the resonance states of He up to N D 7 HeC threshold are considered. These states have all been identified in the literature as being autoionizing states. As is the case for the other calculations described in the chapter, these are also fully self-consistent single determinant calculations. For those multiplets which cannot be represented by a single determinant, Slater’s diagonal sum rule [37] is employed as explained earlier. There is no explicit mixing of the states with either the continuum states or other bound/metastable states. Figure 13.3 shows the radial probability density of the 7s7s.1S / state of He. As expected, there are seven maxima of the 7s7s configuration, the first shell not being visible on the scale of the figure. As expected, the radial probability density maximum is the greatest for the n D 7 shell, with decreasing maxima for the shells for n D 6 to n D 1. The results in the literature employing other methods using correlated Hylleraas and multiconfiguration Hartree–Fock (MCHF) type wave functions are also noted. A study of Table 13.9 shows that the Q-DFT Pauli Approximation results for the total energy are close to and consistently lie above those of the correlated Hylleraas and MCHF calculations. As such they constitute an upper bound to the true value. The difference between the Q-DFT results and those obtained by correlated wave

260

13 Application to Atomic Excited States

Table 13.9 Total energies for doubly excited autoionizing states of He as determined via the Q-DFT Pauli Approximation [15]. The results of correlated Hylleraas and multiconfiguration Hartree–Fock wave function calculations in the literature are also given. The negative values in Rydbergs are given Total Energy State

Q-DFT Pauli Approx

Literature Values

2s2p 3 P 2s3p 3 P 3s3p 3 P

1.5074 1.1426 0.6834

1.5209a , 1.5078b 1.1694a 0.7008a

2s2p 1 P

1.2912

1.3863c

2p2p 1 S 3p3p 1 S 4p4p 1 S 5p5p 1 S 6p6p 1 S 7p7p 1 S

1.2351 0.5733 0.3264 0.2100 0.1462 0.1076

1.2439c , 1.2106d 0.6349e , 0.6144d 0.3665d 0.2421d 0.1714d 0.1282f

2p2p 1 D 3p3p 1 D 4p4p 1 D 5p5p 1 D 6p6p 1 D 7p7p 1 D

1.3357 0.6182 0.3520 0.2264 0.1576 0.1160

1.4394 1.5557c , 1.5442d 2s2s 1 S 1 3s3s S 0.6400 0.7071g , 0.7058d 1 4s4s S 0.3600 0.4021e , 0.4024d 1 5s5s S 0.2304 0.2606d 1 6s6s S 0.1599 0.1817d 7s7s 1 S 0.1175 0.1350d a See [43]; b see [44]; c see [45]; d see [46]; e see [47]; f see [48]; g see [49]

functions are an estimate of the contributions due to Coulomb correlations and Correlation-Kinetic effects to the energy of these excited states. The two-electron excitation energies for the 22 doubly excited autoionizing states of Table 13.9 as obtained by the Q-DFT Pauli Approximation are given in Table 13.10. The set of results (I) are obtained with the reference ground state energy being that determined by the approximation. The set of results (II) are with reference to an accurate correlated wave function value [54] of the ground state energy. The various correlated Hylleraas and MCHF values in the literature are also included.

13.6 Endnote

261 0.10

0.08

4pr2r (r)

0.06

0.04

0.02

17.00

34.00

51.00

68.00

85.00

r

Fig. 13.3 The radial probability density 4 r 2 .r/ of the 7s7s(1 S) state of the He atom in a.u. as determined by the Q-DFT Pauli Approximation [15]

An examination of the results of set (I) shows them to be extremely close to the correlated wave function results of the literature. Thus, accurate two-electron excitation energies of such resonance states of the He atom are obtained if they are determined solely within the context of the Q-DFT Pauli Approximation. For the set of results (II), the ground state energy reference value incorporates all the electron correlations: Pauli, Coulomb, and Correlation-Kinetic. Thus, the difference between the Q-DFT Pauli Approximation set of results (II) and those of the correlated Hylleraas and MCHF values of the literature, are an accurate estimate of Coulomb correlation and Correlation-Kinetic contributions to these excitation energies.

13.6 Endnote It is evident from all the results described earlier of calculations on the excited states of atoms performed within the Q-DFT Pauli Approximation, that quite accurate results are obtained when only correlations due to the Pauli exclusion principle are considered. Correlation-Kinetic contributions due to the Pauli principle are also observed to be negligible. What remains then is to include the Coulomb correlation and Correlation-Kinetic contributions. The reader is referred to Sect. 6.3.2 and Chap. 18 for the description of possible schemes to incorporate these correlations.

262

13 Application to Atomic Excited States

Table 13.10 Two-electron excitation energies in Rydbergs for various doubly excited ionizing states of the He atom as determined by the Q-DFT Pauli Approximation. Set I is with reference to the ground state energy as determined by the approximation. Set II is with reference to the fully correlated ground state value [54]. The results of correlated Hylleraas and multiconfiguration Hartree–Fock theory wave function calculations in the literature are also given State

Excitation Energy Q-DFT Pauli I II

2s2p 3 P 2s3p 3 P 3p3p 3 P

Literature Values

4.2158 4.5806 5.0398

4.3000 4.6648 5.1240

4.2867a , 4.2893b , 4.2996c 4.6407b , 4.6381a 5.1067a

4.4320

4.5162

4.4296b , 4.4901c

2s2p

1

2p2p 3p3p 4p4p 5p5p 6p6p 7p7p

S S 1 S 1 S 1 S 1 S

4.4882 5.1500 5.3968 5.5132 5.5770 5.6156

4.5724 5.2342 5.4810 5.5974 5.6612 5.6998

4.6171b , 4.5968d 5.1721e , 5.1931d 5.4366e , 5.4409d 5.5654f 5.6360d 5.6792f

2p2p 1 D 3p3p 1 D

4.3875 5.1050

4.4718 5.1892

4.3922d , 4.4164g , 4.4338h 5.0954d , 5.1190g

4p4p 1 D 5p5p 1 D 6p6p 1 D 7p7p 1 D

5.3712 5.4968 5.5655 5.6072

5.4555 5.5810 5.6498 5.6914

5.3876d , 5.4084h 5.5344h

P

1

1

4.2837 4.3681 4.2518a , 4.2570g , 4.3140h 2s2s 1 S 1 3s3s S 5.0833 5.1675 5.0850f , 5.0992g , 5.1030h 1 4s4s S 5.3632 5.4475 5.3884f , 5.3964h , 5.4034g 1 5s5s S 5.4928 5.5771 5.5469f 1 6s6s S 5.5633 5.6475 5.6258f 1 7s7s S 5.6057 5.6899 5.6724d a b c d See [45]; see [50]; see [44]; see [48]; e see [51]; f see [46]; g see [52]; h see [53]

Finally, these calculations clearly demonstrate the ease, relative to other schemes [28], of performing atomic excited state calculations within the local effective potential energy framework of Q-DFT. It is reiterated that these are all fully self-consistent single determinant calculations.

Chapter 14

Application of the Multi-Component Q-DFT Pauli Approximation to the Anion–Positron Complex: Energies, Positron and Positronium Affinities

In this chapter we first consider the binding of a positron .e C / to various negative atomic ions (X  ). A positron is a positively charged electron. It is a particle which is identical to an electron in all of its properties except that the sign of its charge and its magnetic moment are opposite to that of the electron. As a consequence of the Coulomb attraction, a positron is bound to an anion, and in a manner similar to that of an electron in the Hydrogen atom, the positron can be in discrete energy states designated as 1s, 2s, 2p, 3p, 3d , etc. The anion–positron ŒX  I e C  complex is a multi-component system. The corresponding equations for this complex may be derived via Multi-Component QDFT [1]. (The derivation of these equations is not given here.) In this chapter the Q-DFT Pauli Approximation within this multi-component framework is applied [2] to the anion–positron system to determine the total energy, and to thereby obtain the positron affinity AP of the negative atomic ion. The positron affinity of the anion is defined as AP D E.X  /  EŒX  I e C ; (14.1) where E.X  / is the energy of the negative ion and EŒX  I e C  that of the anion– positron complex. Positive values of AP indicate binding of the positron i.e. for binding, the energy of the positron–anion system must be more negative than that of the ion itself. The results of these calculations show that there exist bound states of the positron when only correlations due to the Pauli principle are considered in the description of the electronic component of the system. Coulomb correlation between the electrons and those between the electrons and the positron as well as the corresponding Correlation-Kinetic effects are ignored in the Multi-Component Q-DFT Pauli Approximation. The energies of the anion–positron system obtained are upper bounds to those [3–5] of Hartree–Fock theory. The differences between these energies, which are in parts per million, are an estimate of the contribution to the Correlation-Kinetic energy due to Pauli correlations. The contribution of these effects to the total energy is therefore negligible. (Recall that this is also the case for neutral atoms and negative ions.) The differences between the Q-DFT Pauli Approximation results and those of Monte Carlo calculations [6] are an estimate of the Coulomb correlation and all the Correlation-Kinetic contributions. It turns

263

264

14 Application of Q-DFT to the Anion–Positron Complex

out that for the anion–positron systems considered, no definitive conclusions can be arrived at for the sum of these effects since the variational Monte Carlo and diffusion Monte Carlo results [6] for the positron affinity AP differ significantly. Another problem which may be addressed by that of anion–positron binding is the transient bound state of a positron with a neutral atom. A positron and an electron form a stable bound state called positronium (Ps). The system of an electron and a positron has a binding energy of 6.80 eV, exists in a singlet and triplet state, and has a very short lifetime. The issue of interest is whether and how the electronic density of say an anion is modified by the addition of a positron such that there exists a positronium in the anion–positron complex: a positronium “atom” in the anion–positron “molecule” [3]. There has been considerable work done in the field of positronium chemistry [7] (see also references in [2]). Here we focus on the property of positronium affinity APs a positive value of which means that the complex ŒX  I e C  is stable with respect to the breakup into a positronium (Ps) and a neutral atom .X /. The positronium affinity may be defined as APs D E.X / C E.Ps/  EŒX  I e C ;

(14.2)

APs D A C AP C E.Ps/;

(14.3)

or where E.X / is the energy of the neutral atom, E(Ps) the energy of positronium which is 6:8 eV, and A the electron affinity. The equations of the Multi-Component Q-DFT Pauli Approximation for the binding of positrons to anions are given in Sect. 14.1. Brief remarks on the Hartree– Fock theory derivation of these equations are made in Sect. 14.2. The Q-DFT results for the total energy E of the anion–positron complex for the anions Li , B , C , O , and F ; and the positron affinity AP of the anion for ground and excited eigen states of the positron are given in Sect. 14.3. The results for positronium affinities APs are given in Sect. 14.4. Comparisons with the results of Hartree–Fock theory and of Monte Carlo calculations are made where possible.

14.1 Equations of the Multi-Component Q-DFT Pauli Approximation The Schr¨odinger equation for the N -electron and one-positron anion–positron complex is H ‰.X x P / D E‰.Xx P /; (14.4) where the Hamiltonian H is the sum of the electronic He and positronic HP components (in a.u.): H D He C HP ;

(14.5)

14.1 Equations of the Multi-Component Q-DFT Pauli Approximation

He D 

265

N N N 1X 2 X 1 X 1 ; ri C v.r i / C 2 2 jr i  r j j i D1

i D1

(14.6)

i;j D1

i ¤j

X 1 1 ; HP D  rP2 C vP .r P /  2 jr i  r P j N

(14.7)

i D1

with v.r/ D Z=rI vP D Z=r; Z the atomic number; X D x 1 ; : : : ; x N I x D r the spatial r and spin  coordinates of the electrons; x P D r P P the spatial r P and spin P coordinates of the positron. The wave function ‰.Xx P / of the anion– positron complex depends upon the coordinates of the electrons and those of the positron. In the Multi-Component Q-DFT Pauli Approximation, Coulomb correlations between the electrons and those between the electrons and those between the electrons and the positron, and the corresponding Correlation-Kinetic contributions are ignored. The model system wave function ‰.X x p / is then a product of the single determinant ˆfi g of the model fermion orbitals i .x/ and the positron wave function P .x P /: ‰.X x P / D ˆfi gP .x P /I

i D 1; : : : ; N:

(14.8)

The differential equation for the model fermion orbitals i .x/ is 

 1  r 2 C v.r/ C vee .r/ C WH;P .r/ i .x/ D i i .x/I 2

i D 1; : : : ; N; (14.9)

where the electron interaction potential energy vee .r/ is the sum of the Hartree WH .r/ and Pauli Wx .r/ potential energy components: vee D WH .r/ C Wx .r/:

(14.10)

The Hartree potential energy WH .r/ is the work done in the Hartree field E H .r/ due to the density .r/: Z WH .r/ D 

r 1

E H .r 0 /  d` 0  Z

with E H .r/ D

Z

.r 0 / dr 0 ; jr  r 0 j

.r 0 /.r  r 0 / 0 dr : jr  r 0 j3

(14.11)

(14.12)

The alternate expression for WH .r/ in (14.11) is a consequence of the static nature of the charge .r/. The Pauli potential energy Wx .r/ is the work done in the Pauli field E x .r/: Z Wx .r/ D 

r

1

E x .r 0 /  d` 0 ;

(14.13)

266

14 Application of Q-DFT to the Anion–Positron Complex

where E x .r/ is obtained from its dynamical quantal source – the Fermi hole charge x .rr 0 / – via Coulomb’s law as Z E x .r/ D

.rr 0 /.r  r 0 / 0 dr : jr  r 0 j3

(14.14)

0 The Fermi hole is defined / D js .rr 0 /j2 =2.r/, where the Dirac denPasPx .rr ? 0 0 sity matrixPs .rr / D  .r/ i .r /, and the electron density .r/ D i  i P s .rr/ D i  ji .r/j2 . The last term WH;P .r/ on the left hand side of (14.9) is the Hartree potential energy of the positron. It is obtained from the positron orbital P .x/ which is the solution of the differential equation



 1  r 2 C vP .r/ C WH .r/ P .x/ D P P .x/; 2

as

Z WH;P .r/ D 

P .r 0 / 0 dr ; jr  r 0 j

(14.15)

(14.16)

P with P .r/ D  jP .r/j2 being the positron density. As a result of the terms WH;P .r/ (which depends upon the positron density P .r/) in (14.9), and of WH .r/ (which depends upon the electron density .r/) in (14.15), the two differential equations are coupled, and consequently must be solved self-consistently. These self-consistent calculations of this chapter are performed in the central field approximation so that the Pauli field E x .r/ is conservative. Note that the positron Hartree potential energy WH;P .r/ is the work done in the field E P .r/ determined via Coulomb’s law from the quantal source P .r/. It may be written in the form of (14.16) because the positron density P .r/ is also a static charge distribution. The total energy of the anion–positron system, absent the Correlation-Kinetic contribution, is E D Ts C Eext C EH C Ex C TP C Eext;P C EeP ;

(14.17)

where Ts is the kinetic energy of the noninteracting fermions: Ts D 

1X 2

Z i? .x/ri2 i .x/dx;

(14.18)

i

Eext , the external energy of the fermions: Z Eext D

.r/v.r/dr;

(14.19)

14.2 Remarks on Hartree–Fock Theory of Positron Binding to Anions

267

EH , the Hartree or Coulomb self-energy of the fermions: 1 EH D 2

Z

.r/.r 0 / drdr 0 D jr  r 0 j

Z .r/r  E H .r/dr;

(14.20)

Ex , the Pauli energy of the fermions: Z .r/r  E x .r/dr;

Ex

(14.21)

TP , the kinetic energy of the positron: 1 TP D  2

Z P .x P /rP2 P .x P /dx P ;

(14.22)

Eext;P , the external potential energy of the positron: Z Eext;P D

P .r/vP .r/dr;

(14.23)

and EeP , the energy of interaction between the model fermions and the positron: Z EeP D 

.r/P .r 0 / drdr 0 : jr  r 0 j

(14.24)

As noted above, the coupled differential equations for the fermion orbitals i .x/ and the positron orbital P .x P / are solved self-consistently. With the wave function ‰.X x P / known, the energy is determined via (14.17).

14.2 Brief Remarks on Hartree–Fock Theory of Positron Binding to Anions Since it is most meaningful to compare the results for the anion–positron complex as obtained via the Multi-Component Q-DFT Pauli Approximation to those of Hartree–Fock theory, we remark here on the latter. For the anion–positron system defined by the Hamiltonian H of (14.5), a wave function HF .X x P / of the form of (14.8) is assumed with the corresponding electron and positron spin-orbitals being iHF .x/ and P .x P /, respectively. The expression for the energy E HF is then obtained as the expectation of H taken with respect to HF .X x P /. This expression is the same as that of (14.17) with the exception that the kinetic energy Ts of the noninteracting fermions is replaced by the Hartree–Fock theory kinetic energy T HF . The T HF includes Correlation-Kinetic contributions that arise due to the Pauli principle according to which electrons of parallel spin are correlated and therefore kept apart.

268

14 Application of Q-DFT to the Anion–Positron Complex

Minimization of the energy expression E HF with respect to arbitrary variations ıiHF .x/ of the orbitals iHF .x/, together with the Lagrange multipliers ij to ensure the N.N C 1/=2 orthonormality conditions hiHF .x/jjHF .x/i D ıij is then performed. This leads to the differential equation (14.9) for the electrons with the exception that the local potential energy operator Wx .r/ is replaced by the nonlocal integral operator of Hartree–Fock theory (see QDFT). Minimization of E HF with respect to arbitrary variations ıP .x P / of the orbitals P .x P / together with a Lagrange multiplier to ensure normalization leads in turn to the differential equation (14.15) for the positron. The coupled differential equations for the electron and the positron orbitals are then solved self-consistently. The energy is determined via the expression for E HF . As this energy is obtained by application of the variational principle, it constitutes a rigorous upper bound to the true fully correlated value. The difference between the values of any property obtained by the Q-DFT Pauli Approximation and Hartree– Fock theory may then be attributed to the Correlation-Kinetic contributions arising from the Pauli principle.

14.3 Total Energy of the Anion–Positron Complex and Positron Affinities The total energies of the anion–positron system EŒX  I e C  with the positron in its ground and excited states as determined [2] by the Multi-Component Q-DFT Pauli Approximation are given in Table 14.1. In addition, the total ground state energy of the negative ion X  and the neutral atom X as obtained [2] by the Q-DFT Pauli Approximation are noted. The corresponding Hartree–Fock theory values [3, 4] for these properties are also given. The differences in parts per million between the Q-DFT and Hartree–Fock theory values for the anion–positron complex with the positron in its 1s ground state are given in Table 14.2. Since the Q-DFT orbitals for the anion–positron complex do not include the Correlation-Kinetic contributions that arise due to the Pauli principle, and therefore differ from the Hartree–Fock theory orbitals, the corresponding Q-DFT total energy values are an upper bound to those of the latter. This is the case whether the positron is in a ground or excited state. Once again we learn that these Correlation-Kinetic contributions lower the total energy. The differences in the total energies, however, are in parts per million, diminishing with the increase in the size of the anion (see Table 14.2). What this indicates is that the Correlation-Kinetic contributions to the total energy due to the Pauli principle are negligible for this multi-component system. This result is not surprising in light of the fact that the Correlation-Kinetic contribution for the negative ions (see Table 10.6) is also negligible. The positron affinities AP of the negative ions considered as obtained by the energy difference expression of (14.1) are given in Table 14.3. The Q-DFT [2], Hartree–Fock theory [3, 4], and Variational (VMC) and Diffusion (DMC) Monte Carlo [6] results are given. Observe that irrespective of its state, the positron is

14.3 Total Energy of the Anion–Positron Complex and Positron Affinities

269

Table 14.1 Total energy of the anion–positron EŒX  ; e C  system with the positron in its ground and excited states as obtained by the Multi-Component Q-DFT Pauli Approximation [2] and Hartree–Fock theory [3, 4]. The ground state energy of the negative ion X  and that of the corresponding neutral atom X as obtained by these theories are also noted [2]. The negative values in a.u. are given System Li 1s 2 2s 2 Li W 1s Li W 2s Li W 3s Li W 2p Li W 3p Li W 3d Li 1s 2 2s 1 B 1s 2 2s 2 2p 2 B W 1s B W 2s B W 2p B W 3s B W 3p B W 3d B 1s 2 2s 2 2p 1 C 1s 2 2s 2 2p 3 C W 1s C W 2s C W 2p C W 3s C W 3p C W 3d C 1s 2 2s 2 2p 2 O 1s 2 2s 2 2p 5 O W 1s O W 2s O W 2p O W 3s O W 3p O W 3d O 1s 2 2s 2 2p 4

Q-DFT-Pauli Approximation

Hartree–Fock Theory

7.4270 7.5286 7.4748 7.4528 7.5017 7.4653 7.4752 7.4316

7.4282 7.5299 7.4760

24.5156 24.6495 24.5733 24.6108 24.5477 24.5610 24.5694 24.5261

24.5192 24.6531 24.5769 24.6202

37.7041 37.8563 37.7671 37.8040 37.7384 37.7513 37.7584 37.6847

37.7088 37.8610 37.7718 37.8087

74.7849 74.9583 74.8534 74.8940 74.8214 74.8350 74.8403 74.8050

74.7897 74.9630 74.8582 74.9026

7.5030 7.4765 7.4328

24.5757 24.5292

37.7632 37.6887

74.8461 74.8095 (continued)

270

14 Application of Q-DFT to the Anion–Positron Complex

Table 14.1 (continued) System F 1s 2 2s 2 2p 6 F W 1s F W 2s F W 2p F W 3s F W 3p F W 3d F 1s 2 2s 2 2p 5 Cl ŒNe3s 2 3p 6 Cl W 1s Cl W 2s Cl W 2p Cl W 3s Cl W 3p ClŒNe3s 2 3p 5 

2

10

Br ŒAr4s 3d 4p

6

Br W 1s Br W 2s Br W 2p BrŒAr4s 2 3d 10 4p 5

Q-DFT-Pauli Approximation

Hartree–Fock Theory

99.4543 99.6383 99.5253 99.5641 99.4917 99.5048 99.5095 99.4046

99.4594 99.6434 99.5305 99.5692

459.5640 459.7071 459.6243 459.6625 459.5972 459.6107 459.4697

459.5769 459.7189 459.6373 459.6754

2572.523 2572.656 2572.5803 2572.6177 2572.229

2572.5363 2572.6695

99.5147 99.4095

459.4830

2572.6311

Table 14.2 Total energy differences in parts per million between the Multi-Component Q-DFT Pauli Approximation and Hartree–Fock theory results with the positron in its ground state. These differences are an accurate measure of the Correlation-Kinetic energy due to the Pauli exclusion principle System C ŒLi ; e1s 

C  ŒB ; e1s  C ŒC ; e1s  C  ŒO ; e1s C ŒF ; e1s  C  ŒCl ; e1s  C ŒBr ; e1s  

Differences (ppm) 173 146 124 59 51 26 5

bound when only Pauli correlations are considered and Correlation-Kinetic effects due to the Pauli principle ignored, as in the Q-DFT Pauli Approximation. Since these Correlation-Kinetic contributions to the total energy of the anion–positron complex are negligible (see Tables 14.1 and 14.2), the Q-DFT and Hartree–Fock theory results for the positron affinity AP are essentially equivalent. The VMC and

14.3 Total Energy of the Anion–Positron Complex and Positron Affinities

271

Table 14.3 Positron affinities AP from total energy differences (see (14.1)) as determined by the Multi-Component Q-DFT Pauli Approximation, Hartree–Fock theory, and Variational (VMC) and Diffusion (DMC) Monte Carlo calculations. The values in electronvolts are given System

Positron State n P lP

Positron Affinity AP Q-DFT Pauli Hartree–Fock Quantum Monte Approximationa Theoryb Carloc

Li

1s 2s 3s 3p 3d

2:765 1:301 0:7020 1:042 1:312

2.766 1.301

B

1s 2s 2p 3s 3p 3d

3:643 1:570 2:590 0:873 1:235 1:464

3.643 1.570 2.748

C

1s 2s 2p 3s 3p 3d

4:141 1:714 2:718 0:933 1:284 1:478

4.141 1.714 2.718

1s 2s 2p 3s 3p 3d

4:718 1:864 2:969 0:993 1:363 1:507

4.715 1.864 3.072

1s 2s 2p 3s 3p 3d 1s 2s 2p

5:007 1:932 2:988 1:018 1:374 1:502 3:894 1:641 2:680

5.007 1.935 2.988

1s 3:619 2s 1:559 2p 2:577 a From [2]; b from [3, 4]; c from [6]

3.624

O





F

Cl

Br

0.4598 (VMC) 6.5059 (DMC)

1.314 3.8366 (VMC) 6.013 (DMC)

1.537 4.3536 (VMC) 5.9399 (DMC)

1.480 2.2856 (VMC) 5.8610 (DMC)

1.535

1.505 3.864 1.643 2.680

2.580

5.306 (VMC) 6.1685 (DMC)

272

14 Application of Q-DFT to the Anion–Positron Complex

DMC results for each complex differ significantly from each other (see Table 14.3). Furthermore, the DMC values of AP are considerably larger than either those of Q-DFT or of Hartree–Fock theory. This implies that the sum of the Coulomb correlations between the electrons, those between the electrons and the positron, and the corresponding Correlation-Kinetic effects, as obtained from the difference between the DMC and Q-DFT values are very large. These contributions range from 57% for [Li ; e C ] to 19% for [F ; e C ] of the Q-DFT Pauli Approximation value. On the other hand, for the systems ŒB ; e C , [C ; e C ], and ŒF ; e C , the VMC values of AP indicate that the Coulomb correlation and Correlation-Kinetic contributions are only about 5% of the Q-DFT values. The VMC results of AP for ŒLi ; e C  and ŒO ; e C  are inexplicable.

14.4 Positronium Affinities The positronium affinity APs as obtained by the total energy expression of (14.2) (see Table 14.1 for E.X / and EŒX  I e C  values) as determined via the Q-DFT Pauli Approximation and Hartree–Fock theory are once again essentially equivalent [2]. This is because Correlation-Kinetic effects due to the Pauli principle are negligible so that the difference in the corresponding energies of the two theories is in parts per million. However, the positronium affinity calculated in this manner turns out to be negative [2] in all cases. Thus, employing the definition (14.2) with only Pauli correlations considered, leads to the conclusion that no system is stable with respect to dissociation into a neutral atom and positronium. For the determination of the positronium affinity APs via the alternative definition of (14.3) we employ m , the negative of the highest occupied eigenvalue of the anionic system as determined by the Q-DFT Pauli Approximation (see Table 10.7), to approximate the electron affinity A of the atom. The reason why this is meaningful is because the asymptotic structure of the effective potential energy of the anion in the classically forbidden region as determined in the Q-DFT Pauli Approximation is that of the fully correlated system. The corresponding m obtained, which depends upon this asymptotic structure, is then a reasonable approximation to the true electron affinity A (see discussion in Sect. 10.2.2). For the positron affinity AP , we employ P the negative of the positron eigenvalue of (14.15). The positron eigenvalues P for the ground and excited states as determined via the Multi-Component Q-DFT Pauli Approximation and Hartree–Fock theory are given in Table 14.4. With the values of m taken from Table 10.7, and those of P from Table 14.4, the positronium affinity as determined via (14.3) for the majority of the systems is again negative though less [2] than those obtained by the energy expression (14.2). However, for the positronium oxide, fluoride, chloride and bromide systems C C C C C ŒO ; e1s , ŒF ; e1s , ŒF ; e2p , ŒCl ; e1s , and ŒBr ; e1s , respectively, the Q-DFT positronium affinity APs is positive as indicated in Table 14.5. This means that even at the level of Pauli correlations, these systems are stable against the dissociation into a positronium and a neutral atom. The results of a Model-Potential quantum

14.4 Positronium Affinities

273

Table 14.4 Positron ground and excited state eigenvalues P as determined by the MultiComponent Q-DFT Pauli Approximation and Hartree–Fock theory. The negative values in electronvolts are given System

Positron Eigenvalues P

Positron State n P lP

Q-DFT Pauli Approximationa

Hartree–Fock Theoryb

W 1s W 2s W 3s W 3p W 3d

3.049 1.347 0.770 1.064 1.339

2.996 1.329

W 1s W 2s W 2p W 3s W 3p W 3d

3.785 1.584 2.557 0.876 1.212 1.442

3.778 1.582 2.799



W 1s W 2s W 2p W 3s W 3p W 3d

4.204 1.712 2.712 0.931 1.282 1.478

4.218 1.718 2.135

O

W 1s W 2s W 2p W 3s W 3p W 3d

4.784 1.869 2.917 0.993 1.350 1.498

4.769 1.865 3.079

W 1s W 2s W 2p W 3s W 3p W 3d W 1s W 2s W 2p

5.061 1.932 2.993 1.020 1.377 1.503 3.928 1.645 2.687

5.048 1.936 2.992

3.655 1.568 2.585

3.653

Li

B



C



F

Cl

Br

W 1s W 2s W 2p a From [2]; b from [3, 4]

1.331

1.540

1.480

1.532

1.502 3.922 1.644 2.687

2.588

274

14 Application of Q-DFT to the Anion–Positron Complex

Table 14.5 Positronium affinities APs as determined via the Multi-Component Q-DFT Pauli Approximation via expression (14.3), and a Model Potential quantum Monte Carlo calculation via expression (14.2). The values in electronvolts are given System

O F F Cl a

Positronium Affinity APs

Positron state n P lP

Q-DFT Pauli Approximationa

Quantum Monte Carlob

1s 1s 2p 1s

1.443 3.126 1.058 1.052

1.98 ˙ 0.17 1.91 ˙ 0.16 1.14 ˙ 0.11

From [2]; b from [8, 9]

Monte Carlo calculation [8, 9] for the positronium fluoride, chloride and bromide systems that employs a model potential for the core electrons are also given in Table 14.5. A comparison with the Q-DFT Pauli Approximation results indicates that for the positronium affinity APs the sum of the contributions of Coulomb correlation between the electrons, those between the electrons and the positron, and the corresponding Correlation-Kinetic effects, are quite significant. In a fully correlated calculation within Q-DFT, the contribution of correlations between the electrons and positron should be a major fraction of this sum [10], with the correlations between the electrons being less significant.

Chapter 15

Application of the Q-DFT Fully Correlated Approximation to the Helium Atom

In Chap. 9, the properties of atoms were determined by the Q-DFT Hartree Uncorrelated Approximation, the rigorously independent particle approximation with no electron correlations present (see Sect. 6.1.2). In Chap. 10, electron correlations due to the Pauli exclusion principle were introduced, and the Q-DFT Pauli Approximation applied to atoms. Recall (see Sect. 6.2.2), that in this latter approximation the Correlation-Kinetic contributions due to the Pauli principle are ignored. The calculations of the application of both these approximations to atoms are fully self-consistent. As a consequence of the inclusion of Pauli correlations, there is an improvement in results for atoms with atomic number Z > 2, e.g. the total energy is lower, and the negative of the highest occupied eigenvalue is closer to the experimental ionization potential. The obvious next steps in this study of atoms is to first additionally introduce Coulomb correlations as within the QDFT Pauli–Coulomb Approximation of Sect. 6.3.1, and then to incorporate the Correlation-Kinetic effects as in the Q-DFT Fully Correlated Approximation of Sect. 6.3.2. Such a study to separately examine the effects of Coulomb correlations and Correlation-Kinetic contributions employing approximate wave function functionals which are determined fully self-consistently (see Fig. 6.1), is in progress. Another approach to study the contributions of Coulomb correlations and Correlation-Kinetic effects to the properties of atoms is to employ highly accurate, albeit approximate, wave functions that exist in the literature within the Q-DFT Fully Correlated Approximation. The advantage of such calculations for atoms is that properties of the corresponding S systems, viz. the quantal sources, the resulting fields and potential energies, the components of the total energy, the total energy, and the highest occupied eigenvalues, are all determined “exactly” with an accuracy of either the same or second order as that of the wave function employed. The results of other approximate calculational schemes can then be compared with these “exact” values. The procedure for the mapping from the interacting to the S system employing accurate approximate wave functions approx .X / as the starting point, is as follows. With the S system described by the central field model, the density .r/ and the pair-correlation density g.rr 0 /, which are derived from the wave function (see (2.15) and (2.31)), are initially determined. From the quantal source g.rr 0 /, the electron-interaction field E ee .r/ is then obtained via Coulomb’s law

275

276

15 Application of Q-DFT to the Helium Atom

(see (2.44)). Next, the corresponding potential energy component Wee .r/ of the electron-interaction potential energy vee .r/ of the S system is obtained as the work done in this field. With the sum of Wee .r/ and the external potential energy v.r/, the S system differential equation is solved for an initial set of spin-orbitals i .x/. With these orbitals i .x/ and the wave function approx .X /, the corresponding initial Correlation-Kinetic Wtc .r/ potential energy component of vee .r/ is then determined. The sum of the functions v.r/, Wee .r/ and this Wtc .r/ are substituted into the S system differential equation, which is then solved to obtain a new set of orbitals i .x/. The procedure is continued till self-consistency is achieved. Thus, in these calculations, it is only the Correlation-Kinetic component Wtc .r/ that is determined in a fully self-consistent manner. Such calculations for the first twenty atoms of the Periodic Table are also in progress. In this chapter, we describe the results of application of the Q-DFT Fully Correlated Approximation in a calculation [1, 2] employing an accurate wave function approx .X / to the Helium atom in its ground state. The mapping is to a two modelfermion S system also in its ground state. For this mapping, the self-consistency procedure described earlier is not required because the self-consistent orbitals i .x/ are known explicitly in terms of the density .r/ (see QDFT or Sect. 5.2). In this chapter we therefore go beyond the Pauli Correlated Approximation of Chap. 10 to study properties of the atom associated additionally with Coulomb correlations and Correlation-Kinetic effects.

15.1 The Interacting System: Helium Atom in Its Ground State For the Helium atom in its ground state, the two electrons have opposite spin, so that on ignoring the spin coordinates, the Schr¨odinger equation (2.5) for the spatial part of the wave function .r 1 r 2 / may be written as HO .r 1 r 2 / D E .r 1 r 2 /;

(15.1)

where E is the energy, and the Hamiltonian of (2.1) is 1 Z Z 1 1 ;  C HO D  rr21  rr22  2 2 r1 r2 jr 1  r 2 j

(15.2)

r 1 and r 2 are the coordinates of the electrons relative to the nucleus, and the atomic number Z D 2. The wave function employed in the mapping to the S system is the analytical 39-parameter correlated wave function of Kinoshita [3] which is of the form approx

.r 1 r 2 / D

.stu/;

(15.3)

where s D r1 C r2 ; t D r2  r1 ; u D jr 1  r 2 j are the Hylleraas [4, 5] coordinates. The Kinoshita wave function is given in Appendix F together with expressions for

15.2 Mapping to an S System in Its 11 S Ground State

277

relevant properties of interest. This wave function leads to the same energy as that due to the 1,078-parameter correlated wave function of Pekeris [6] to seven significant figures, and to the expectations of various single particle operators from four-to-five significant figures. The wave function also satisfies the electron–nucleus coalescence condition (see Sect. 2.7) to three significant figures. (As noted in Sect. 8.1, the satisfaction of this constraint is a critical requirement for the electroninteraction potential energy function vee .r/ of the S system to be finite at the nucleus. If this constraint is not satisfied, then vee .r/ is singular at the nucleus as shown in Sect. 8.2.) The wave function of Kinoshita also satisfies the electron– electron coalescence condition (see Sect. 2.7) to two significant figures over most of the atomic region.

15.2 Mapping to an S System in Its 11 S Ground State The equations governing the mapping from the Helium atom in its ground state to a two model-fermion S system also in its ground state is similar to that for the twoelectron Hooke’s atom as described in QDFT (see also Sect. 5.2). Hence, we focus here principally on the results.

15.2.1 Coulomb Hole Charge Distribution c .rr 0 / We begin with a study of the structure of the Coulomb hole charge distribution c .rr 0 / as a function of the electron position at r. Such a study explains how the two electrons are correlated as a function of the non-uniform density .r/ of the atom. The Coulomb hole c .rr 0 / is defined as the difference between the Fermi-Coulomb hole xc .rr 0 / and the Fermi hole x .rr 0 / charge distributions. (See Sect. 3.2.3 for the various definitions.) As explained in that section, the Fermi hole charge x .rr 0 / for the model two-fermion S system in its ground state is x .rr 0 / D .r 0 /=2, independent of the electron position at r. In Fig. 15.1 we plot a cross-section through the Coulomb c .rr 0 /, Fermi x .rr 0 /, and Fermi-Coulomb xc .rr 0 / hole charge distributions for an electron at the nucleus r D 0. Also plotted for purposes of comparison is the electronic density .r 0 /. Observe that for this electron position all the hole charge distributions are spherically symmetric about the electron. At the position of the electron, the holes all also exhibit a cusp. (The cusp in the structure of the Fermi hole is a consequence of the cusp in the density.) At and about the electron position the Fermi–Coulomb hole is more negative than the Fermi hole as must be the case. This is a consequence of the fact that when Coulomb correlations are introduced, an electron creates a hole about it that is deeper than when only Pauli correlations are present. Thus, in the region about the electron, the Coulomb hole is negative. (This is the case for all electron positions.) However, near the surface of the atom .hri D 0:929 a.u.) and in the

278

15 Application of Q-DFT to the Helium Atom

Fig. 15.1 Cross sections of the Coulomb c .rr 0 /, Fermi x .rr 0 /, and Fermi-Coulomb xc .rr 0 / holes for the He atom as determined via the Kinoshita wave function for an electron at the nucleus r D 0, as indicated by the arrow. The nucleus is at the origin. The positive part of the Coulomb hole is not evident on the scale of the figure. The electronic density .r 0 / is also plotted

classically forbidden region, the Fermi-Coulomb hole must lie above the Fermi hole since both these distributions satisfy the same charge conservation constraint (see (2.41) and (3.19)) of total charge of negative one. In these regions the Coulomb hole is then positive. (The positive part of the Coulomb hole is not evident on the scale of Fig. 15.1, but is clearly exhibited in the figures to follow.) Thus the Coulomb hole is both positive and negative, and integrates to a total charge of zero (see (3.29)). The positive part of the hole indicates that for an electron at the nucleus the other electron is outside the surface in the classically forbidden region of the atom. Since the Fermi hole for this atom is independent of the electron position, we now focus our attention solely on the structure of the Coulomb hole for the other electron positions considered. In Fig. 15.2a we plot the Coulomb hole for an electron with a radial coordinate of r D 0:566 a.u. which is at the maximum of the radial probability density, and in Fig. 15.2b for an electron at r D 0:8 a.u. The polar angle  of the

15.2 Mapping to an S System in Its 11 S Ground State

279

Fig. 15.2 Coulomb hole c .rr 0 / for the He atom in three different directions corresponding to  0 D 0, 45ı , and 90ı with respect to the nucleus–electron direction. The electron position corresponds to a polar angle of  D 0ı . In (a) the radial coordinate of the electron, as indicated by the arrow, is at the maximum of the radial probability density for the He atom at r D 0:566 a.u., whereas in (b) the electron is at r D 0:8 a.u.

electron position is taken to be zero. The polar angles  0 of the Coulomb hole charge considered correspond to  0 D 0ı , 45ı , and 90ı with respect to the nucleus–electron direction. Since the electron position is along the  0 D 0ı direction, the Coulomb hole (for this spherically symmetric atom) is independent of its azimuthal angle  0 . For these electron positions the Coulomb hole is no longer spherically symmetric about the electron. Observe also the cusp in the structure of the Coulomb hole at the position of the electron. At and about these positions, the hole is principally negative. Its magnitude at the electron position diminishes with distance from the nucleus as a result of the decrease in the density. However, as the distance of the electron from the nucleus increases, a part of the hole that is positive emerges on the side of the atom opposite to that of the electron. The magnitude of this positive part is seen to increase relative to that of the negative part about the electron, with the

280

15 Application of Q-DFT to the Helium Atom

Fig. 15.3 Same as Fig. 15.2 except that in (a) the electron is at r D 1 a.u., in (b) at r D 1:5 a.u., and in (c) at r D 5 a.u.

positive part moving closer to the nucleus. This indicates that it is more probable for the other electron to be in this region. In Figs. 15.3(a)–15.3(c) we plot the Coulomb hole for an electron in the surface region of the atom at r D 1 a.u., and in the classically forbidden region at r D 1:5 and r D 5 a.u., respectively. The trends discussed earlier continue as the distance of the electron from the nucleus increases further. The positive part of the Coulomb hole increases in magnitude about the nucleus, and the negative part around the electron position continues to decrease. For the electron at r D 5 a.u. (Fig. 15.3c), most of the Coulomb hole is positive, localized about the nucleus, and essentially spherically symmetric about it. Thus, if one of the two electrons is in the asymptotic region, the other is localized about the nucleus. In Fig. 15.4 we plot the center of mass hr 0 i of the Coulomb hole c .rr 0 / as a function of electron position. The center of mass lies along the nucleus–electron direction. However, it is always on the side of the nucleus opposite to that of the electron, but also always close to the nucleus. For asymptotic positions of the electron, it

15.2 Mapping to an S System in Its 11 S Ground State

281

Fig. 15.4 Center of mass hr 0 i of the Coulomb hole c .rr 0 / for the He atom as a function of electron position

once again approaches the nucleus. The center of mass of the Fermi hole in this case is at the origin. However, for other atoms (see Fig. 2 of [7]), the center of mass of the Fermi hole for the most part lies on the same side of the nucleus as the electron. Furthermore, it follows the electron for positions within the atom, and lies considerably farther from the nucleus prior to approaching the origin for asymptotic electron positions.

15.2.2 Pauli–Coulomb E xc .r/, Pauli E x .r/, and Coulomb E c .r/ Fields, and the Pauli–Coulomb Exc , Pauli Ex , and Coulomb Ec Energies The Pauli–Coulomb E xc .r/, Pauli E x .r/, and Coulomb E c .r/ fields as determined from their respective quantal sources: the Fermi-Coulomb xc .rr 0 /, Fermi x .rr 0 /, and Coulomb c .rr 0 / holes, are plotted in Fig. 15.5. Observe that all the force fields vanish at the nucleus. This is a consequence of the fact that, for an electron at the nucleus, the charge distributions xc .rr 0 /, x .rr 0 /, and c .rr 0 / are all spherically symmetric about it. As such there is no force field at the position of the electron. The structure of the Coulomb field E c .r/ in the interior of the atom is similar to the Pauli field E x .r/, although it is about an order of magnitude smaller. However, both the fields E x .r/ and E c .r/ are negative throughout space. This is an interesting

282

15 Application of Q-DFT to the Helium Atom

Fig. 15.5 The Pauli–Coulomb Exc .r/, Pauli Ex .r/, and Coulomb Ec .r/ fields of the He atom. The function .1=r 2 / is also plotted

result since the corresponding quantal source charge distributions for these fields are strikingly different. The Fermi hole x .rr 0 / is negative for all electron positions and therefore the field E x .r/ is negative throughout. On the other hand, the Coulomb hole c .rr 0 / is both positive and negative and can be substantially one or the other depending on the position of the electron. The fact that the Coulomb field E c .r/ is negative is a consequence of the fact that the force field depends not only on the structure of its quantal source but also on the inverse of the square of the distance between the source charge and the electron. Thus the part of the charge that lies farther from the electron contributes less to the field than the charge that is closer. For example, for an electron at r D 1:5 a:u:, the positive part of the Coulomb hole (see Fig. 15.3b) is much larger in magnitude than its negative part. However, the positive part is localized about the nucleus far from the electron, and therefore its contribution to the force field is less than that of the negative charge closer to the electron, with the result that the net force field at the electron position is negative. This explains why the Coulomb field E c .r/ is negative. The fact that the Coulomb hole c .rr 0 / goes substantially positive for asymptotic positions of the electron does, however, cause the Coulomb field E c .r/ to decay asymptotically far more rapidly than the Pauli field E x .r/. As proved in Chap. 7 the decay of E c .r/ is of O.1=r 5 /, whereas that of E x .r/ is .1=r 2 /. This asymptotic structure may also be understood qualitatively to be a consequence of the fact that the total charge of the Coulomb hole c .rr 0 / is zero, whereas that of the Fermi hole x .rr 0 /

15.2 Mapping to an S System in Its 11 S Ground State

283

Table 15.1 Ground state properties of the Helium atom as obtained by the Q-DFT Fully Correlated Approximation mapping to an S system in its ground state Property

Value in a.u.

E Ts Eext EH Exc Ex Ec Tc I D m

2.90372 2.86708 6.75326 2.04914 1.10332 1.02457 0.07875 0.03664 1.808 .ryd/

is unity, and the fact that all these charge distributions are localized and essentially static about the nucleus for asymptotic positions of the electron. The Pauli–Coulomb field E xc .r/, which is the sum of the Pauli E x .r/ and Coulomb E c .r/ fields, is also negative throughout space and decays asymptotically as .1=r 2 / for the reasons described earlier. Observe that all the fields exhibit the single shell structure of the Helium atom in its ground state. The Pauli–Coulomb Exc , Pauli Ex , and Coulomb Ec energy components of the total energy E as obtained from the corresponding fields (see Sect. 3.4.1) are quoted in Table 15.1. The Hartree energy EH component obtained from the density .r/ or the Hartree Field E H .r/ (not plotted) is also given. Observe that Ec is an order of magnitude smaller than Ex . The reason for this is because the Coulomb field E c .r/ is an order of magnitude smaller than the Pauli field E x .r/ (see Fig. 15.5).

15.2.3 Pauli–Coulomb Wxc .r/, Pauli Wx .r/, and Coulomb Wc .r/ Potential Energies The Pauli–Coulomb Wxc .r/, Pauli Wx .r/, and Coulomb Wc .r/ potential energies determined as the work done in the corresponding fields E xc .r/, E x .r/, and E c .r/ are plotted in Fig. 15.6. As explained and proved in Chap. 8, these potential energy functions all approach the nucleus quadratically, and have zero slope at the origin as a consequence of the force fields vanishing there. In the interior of the atom, the structure of Wc .r/ is similar to that of Wx .r/, but is an order of magnitude smaller. Note that both the potential energies Wx .r/ and Wc .r/ are negative throughout space, as must be the case, because the fields from which they are obtained are each negative. Since the Pauli–Coulomb E xc .r/, Pauli E x .r/ and Coulomb E c .r/ fields decay asymptotically in the classically forbidden region as .1=r 2 /, .1=r 2 / and O.1=r 5 /, respectively, the decay of the Pauli–Coulomb Wxc .r/, Pauli Wx .r/, and Coulomb Wc .r/ potential energies is .1=r/, .1=r/, and O.1=r 4/ (see Chap. 7). All the potential energies are monotonic, which means that positive work

284

15 Application of Q-DFT to the Helium Atom

Fig. 15.6 The Pauli–Coulomb Wxc .r/, Pauli Wx .r/, and Coulomb Wc .r/ potential energies in the mapping to the S system in its ground state. The function .1=r/ is also plotted

must be done to move the electron in the force fields of the Fermi-Coulomb, Fermi and Coulomb hole charges. The potential energies all also exhibit the single shell structure of the model S system in its ground state.

15.2.4 Correlation-Kinetic Field Z tc .r/, Potential Energy Wtc .r/, and Energy Tc With the S system in its ground state, the spatial component of each spin orbital p is known explicitly in terms of the density: i .r/ D .r/=2. Thus, the kinetic energy Ts of the model fermions may be determined (see Table 15.1). The Dirac density matrix s .rr 0 / quantal source of the model fermions is also known. In turn, the interacting system density matrix .rr 0 / source is obtained from approx .rr 0 /. The Correlation-Kinetic field Z tc .r/ determined from the difference of these quantal sources is plotted in Fig. 15.7. Observe, as proved in Chap. 7, that the field Z tc .r/

15.2 Mapping to an S System in Its 11 S Ground State

285

Fig. 15.7 The Correlation-Kinetic field Z tc .r/ of the He atom

is finite at the nucleus. This field is not monotonic having both a negative and positive component with the latter being more significant over most of the model atom. The field decays asymptotically as a positive function, decaying as O.1=r 6 / as proved in Chap. 8. The field Z tc .r/ is also smaller than the Coulomb field E c .r/. As a consequence of this structure of Z tc .r/, the Correlation-Kinetic energy Tc obtained from it (see Sect. 3.4.2) is positive and about half the value of the Coulomb energy Ec (see Table 15.1). The Correlation-Kinetic potential energy Wtc .r/ of the S system obtained as the work done in the field Z tc .r/ is plotted in Fig. 15.8. Again, as a result of the structure of Z tc .r/, the potential energy Wtc .r/ is a positive function. It approaches the nucleus linearly (see Chap. 8), and decays asymptotically in the classically forbidden region as O.1=r 5 / (see Chap. 7). It is not monotonic since the field Z tc .r/ changes sign. For purposes of comparison, the Coulomb potential energy Wc .r/ is also plotted in Fig. 15.8. Observe that the functions Wtc .r/ and Wc .r/ are about the same magnitude but opposite in sign. This indicates that within the S system, there is a substantial cancelation of the Coulomb correlation and Correlation-Kinetic effect contributions. Coulomb correlations lower the energy whereas Correlation-Kinetic effects raise it. The net effect of these correlations is a lowering of the total energy (see Table 15.1). For the two-fermion model S system in its ground state, the sum of the potential energies Wc .r/ and Wtc .r/ is the Kohn–Sham theory “correlation” potential energy vc .r/ (see Chap. 5 of QDFT). This potential energy may also be determined directly by inversion of the Kohn–Sham differential equation as

286

15 Application of Q-DFT to the Helium Atom 0.10 He Atom Wc (r) vc (r) Wt (r)

0.05

Potentials (a.u.)

c

0.00

–0.05

0.1

0.0

–0.10 –.01

–0.15

0.01 2 3

0

1

0.1

2

23

3

1

23

10

4

r (a.u.)

Fig. 15.8 The Correlation-Kinetic Wtc .r/, Coulomb Wc .r/, and the Kohn–Sham theory “correlation” vc .r/ potential energies. The inset is on a logarithmic scale to indicate the structure near the nucleus of the model system atom.

vc .r/ D  C

r2 2

C

2  vH .r/  vx .r/: r

(15.4)

Here vH .r/ D WH .r/ the Hartree potential energy, and for this model two-fermion system the Kohn–Sham “exchange” potential energy vx .r/ D Wx .r/ the Pauli potential energy. The vc .r/ as determined via (15.4) employing the Kinoshita wave function, and assuming the eigenvalue  to be the negative of the first ionization potential [8], is also plotted in Fig. 15.8. The potential energy vc .r/ is negative and finite at the nucleus, approaching it linearly with very small slope. The function vc .r/ is not monotonic, going positive at r 0:3 a.u., and then becoming negative for r > 4 a.u., vanishing as a negative function of O.1=r 4 /. This is consistent with the derivation of this asymptotic structure in Chap. 7, and the fact that the polarizability ˛ is positive). (A calculation [9] of vc .r/ via (15.4) employing a more accurate 491-parameter correlated wave function confirms the above structure.) It is also evident that since the Coulomb potential energy Wc .r/ is negative and monotonic, the nonmonotonicity and the positiveness of the function vc .r/ is due to Correlation-Kinetic effects (see Fig. 15.8).

15.2.5 Total Energy and Ionization Potential The total energy E of the Helium atom in its ground state as obtained from the sum of the components Ts , Eext , EH , Ex , Ec , and Tc determined from the S system in

15.3 Endnotes

287

its ground state is quoted in Table 15.1. This result is an improvement over the QDFT Pauli Approximation value (see Table 10.1), and as must be the case, the same as that obtained via the Kinoshita wave function. The ionization potential I is the negative of the single S system eigenvalue . The latter may be determined from the S system differential equation as D

r2 2

C v.r/ C vee .r/;

(15.5)

which is an expression valid for arbitrary r, and where the Q-DFT components WH .r/, Wx .r/, Wc .r/, and Wtc .r/ of vee .r/ are as determined in the previous subsections. Equivalently, the eigenvalue may be determined by substituting the various components of vee .r/ into the S system differential equation and by numerically solving for the single orbital and eigenvalue. (Note that (15.4) and (15.5) are equivalent for this two-fermion model S system in its ground state.) The ionization potential thus obtained is quoted in Table 15.1. Observe that this value of I which is 1:808 ryd is an improvement over the Q-DFT Pauli Approximation result which is 1.836 ryd when compared with the experimental value of 1.807 ryd (see Table 10.4).

15.3 Endnotes As described earlier, it is possible within the Q-DFT Fully Correlated Approximation scheme to obtain all the individual Hartree WH .r/, Pauli Wx .r/, Coulomb Wc .r/, and Correlation-Kinetic Wtc .r/ components of the S system electroninteraction potential energy vee .r/, and thereby to determine all the individual components of the total energy E: the noninteracting kinetic Ts , external Eext , Hartree EH , Pauli Ex , Coulomb Ec , and Correlation-Kinetic Tc energies, and the ionization potential I of the Helium atom in its ground state. The results presented in this chapter are essentially “exact” and may be employed for purposes of comparison with other approximation methods. Observe that by additionally incorporating Coulomb correlations and Correlation-Kinetic effects, and thereby going beyond the Q-DFT Pauli Approximation, there is an improvement in the results for the density .r/, energy E, and ionization potential I . A further point to note is that all the properties determined by the Q-DFT Fully Correlated Approximation correspond to the same density .r/. This contrasts with the standard quantum chemistry definitions of various properties. For example, in quantum chemistry the correlation energy EcHF is defined (see (4) of Sect. 4.2.1) as the difference between the correlated wave function and Hartree–Fock theory values of the total energy. However, the correlated wave function and Hartree–Fock theory densities differ, and correspond in a sense to two distinct systems with different densities. In Q-DFT, all the properties are determined for the same density. For the Helium atom in its ground state, EcHF D 0:04204 a:u:. as obtained from the total energy values taken from Tables 10.2 and 15.1.

288

15 Application of Q-DFT to the Helium Atom

Since the mapping shown earlier is to a model two-fermion S system in its ground state, it is the same S system as that of Kohn–Sham theory. Hence, in this case the Kohn–Sham “exchange” energy ExKS is equivalent to the Pauli energy Ex . As such the Kohn–Sham “correlation” energy EcKS D Ec C Tc . From the values of Ec and Tc of Table 15.1 then, EcKS D 0:04211 a:u: Observe that the inequality EcKS < EcHF of (4.47) is satisfied (see (4) of Sect. 4.2.1). Finally, as noted previously (see (7) of Sect. 4.2.1), the S system electroninteraction potential energy function vee .r/ can be determined directly from a density derived from some accurate wave function approx .X /. Such calculations [10] for atoms show that the function vee .r/ is not monotonic but has a hump in the inter shell regions. Since both the Pauli Wx .r/ and Coulomb Wc .r/ components of the potential energy vee .r/ are monotonic, it becomes evident that these humps are a consequence solely of Correlation-Kinetic effects.

Chapter 16

Application of the Q-DFT Fully Correlated Approximation to the Hydrogen Molecule

We now turn our attention from atoms to the simplest molecule containing an electron-pair bond viz. the Hydrogen molecule (H2 ). In this chapter, we study the properties of this molecule from the perspective of Q-DFT, i.e. from the view point of the representation of all different electron correlations present in terms of the corresponding quantal sources, and of the resulting fields, potential energies and total energy components. We apply [1] the Q-DFT Fully Correlated Approximation to map the Hydrogen molecule in its ground state to an equivalent-density noninteracting fermion model S system “molecule” that is also in its ground state with configuration .g 1s/2 . We thereby consider correlations due to the Pauli principle, Coulomb repulsion, and Correlation-Kinetic effects. The application of this Q-DFT approximation to H2 can be performed in a manner similar to that described in Chap. 15 for the mapping of the Helium atom. As such, for our approximate ground state wave function approx .X /, we employ the accurate correlated wave function of Kolos and Roothaan [2]. Also, as the spins of the two model fermions are opposite, the single-particle molecular orbitals of the S system are known exactly in terms of the density, and therefore do not have to be determined in a numerically self-consistent manner. Hence, all the properties of the Q-DFT description of the H2 molecule determined are essentially exact. As this description of the H2 molecule differs in many ways from the conventional description [3], a considerable degree of new physics is gleaned. However, beyond the new understandings achieved, another attribute of these calculations is the knowledge that the structure of the corresponding Q-DFT properties for other diatomic molecules in their ground states will be qualitatively the same. Furthermore, these “exact” properties can be used as the basis for comparison and testing of other Q-DFT approximation methods prior to their application to more complex molecules.

16.1 The Interacting System: Hydrogen Molecule in Its Ground State The Hamiltonian of the Hydrogen molecule is HO D HO e C VOnn ;

(16.1)

289

290

16 Application of Q-DFT to the Hydrogen Molecule

where the electronic part HO e in atomic units is 1 1 1 1 1 1 1    C ; HO e D  r12  r22  2 2 r1a r2a r1b r2b r12

(16.2)

with 1 and 2 the electrons, a and b the nuclei, and r12 D jr 1  r 2 j with r 1 and r 2 the coordinates of the electrons. The nucleus–nucleus interaction potential energy Vnn D 1=R, where R D 2a is the internucleus separation. Each nucleus has a charge of Z D 1. As the molecule is in its ground state, the spin part of the wave function is antisymmetric. The symmetric spatial part of the wave function .r 1 r 2 R/ then satisfies the Schr¨odinger equation HO .r 1 r 2 R/ D E .r 1 r 2 R/:

(16.3)

The approximate wave function approx .r 1 r 2 R/ we employ in the mapping to the model S system is the 51-parameter correlated wave function of Kolos–Roothaan [2]. Thus, the symmetric spatial part of the approximate wave function is approx

X

.r 1 r 2 R/ D expŒı. 1 C 2 /

p Cmnjkp Π1m 2n j1 k2 C 1n 2m k1 j2 r12

mnjkp

(16.4) with 1 D .r1a C r1b /=R;

2 D .r2a C r2b /=R;

(16.5)

1 D .r1a  r1b /=R;

2 D .r2a  r2b /=R;

(16.6)

where the variational parameters are ı and the coefficients Cmnjkp . The values of these parameters are given in Appendix G. The plot of the electron density .r/ along the nuclear bond z-axis as obtained from this wave function is indistinguishable from that of Fig. 8.1 and is not repeated. The total energy E.H2 / D 1:174448 a.u. at the equilibrium internuclear separation of a D R=2 D 0:7005 a.u. The total energy of the Hydrogen molecular ion HC 2 at the equilibrium internuclear separation of the Hydrogen molecule is [4, 5] E.HC 2 /jaD0:7005 D 0:56998 a.u. Thus, the theoretical ionization potential of the H2 molecule corresponding to the wave function of (16.4) is I D E.HC 2 /jaD0:7005  E.H2 / D 0:60447 a.u.

16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration The Q-DFT equations governing the mapping from the Hydrogen molecule in its ground state to a model S system diatomic “molecule” of two noninteracting fermions also in their ground state are the same as those of the mapping from any ground state two-electron interacting system (See QDFT and Sect. 5.2). Hence, we focus here primarily on the description of the results.

16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration

291

16.2.1 Fermi–Coulomb xc .rr 0 /, Fermi x .rr 0 /, and Coulomb c .rr 0 / Hole Charge Distributions We begin with a study of the Fermi–Coulomb xc .rr 0 /, Fermi x .rr 0 /, and Coulomb c .rr 0 / hole charge distributions as a function of the electron position r.0; z/ on the nuclear bond axis z. Because of the cylindrical symmetry of the molecule, cylindrical coordinates are employed throughout. The Fermi–Coulomb hole charge xc .rr 0 / is obtained from the pair-correlation density g.rr 0 / (see Sect. 2.3.3). For the two model fermion S system in its ground state, the Fermi hole is defined as x .rr 0 / D .r 0 /=2, and as such is independent of the electron position at r (see Sect. 3.2.3). The Coulomb hole c .rr 0 / is the difference between the Fermi–Coulomb and Fermi hole charge distributions. The structure of both the Fermi–Coulomb and Coulomb holes is a description of how the two electrons are correlated as a function of the nonuniform electron density. Both structures describe, as a function of the position of one electron, the probability of where the other electron is most likely to be. In Fig. 16.1 we plot cross sections through the Fermi–Coulomb xc .rr 0 /, Fermi x .rr 0 /, and Coulomb c .rr 0 / hole sources as a function of r 0 D .0; z0 / for an electron at the origin r D .0; 0/ at the center of the nuclear bond. The electron position

Fermi - Coulomb, Fermi, Coulomb, Holes (a.u.)

0.1

0.0

–0.1 ρxc(r, r') ρx(r, r') ρc(r, r') –0.2 Electron at r = 0 (a.u.)

–0.3 –4

–3

–2

–1

0

1

2

3

4

z' (a.u.)

Fig. 16.1 Cross sections of the Fermi–Coulomb xc .rr 0 /, Fermi x .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for an electron at the center r D .0; 0/ of the bond. The electron position is indicated by the arrow

292

16 Application of Q-DFT to the Hydrogen Molecule

is indicated by the arrow. The three charge distributions, of course, have cylindrical symmetry about the bond axis. More significantly, they are symmetrical about the electron along the z0 axis. Observe that at the electron position, both the Fermi– Coulomb and Coulomb holes exhibit a cusp corresponding to the electron–electron cusp condition. (Based on the work of [6] it is known that the wave function does not satisfy this cusp condition exactly. It obviously satisfies it to a good degree as evidenced by the figure.) As expected, at the electron position, the Fermi–Coulomb hole is more negative than the Fermi hole. Thus, in the region about the electron, the Coulomb hole is negative. (This is also the case for all the other electron positions considered.) As both the Fermi–Coulomb and Fermi holes satisfy the same charge conservation sum rule, there must be regions where the former lies above the latter. This is clearly evident in the figure. Hence, in the outer and classically forbidden regions of the molecule, the Coulomb hole is positive. (The positive part of the Coulomb hole is more clearly evident in the figures that follow.) The Coulomb hole is both positive and negative as its total charge is zero. The positive part of the Coulomb hole is an indication that the other electron is equally likely to be in the classically forbidden region on either side of each nucleus. As the Fermi hole is independent of electron position, we now focus on the Fermi–Coulomb and Coulomb holes. In Figs. 16.2–16.4, we plot the cross sections of these holes for electron positions at r D .0; a=3/, r D .0; 2a=3/, r D .0; a/. Again, observe the cusp at the electron position for both the Fermi–Coulomb and

Fermi - Coulomb and Coulomb Holes (a.u.)

0.10 0.05 0.00 –0.05

ρxc(r, r')

–0.10

ρc(r, r')

–0.15 –0.20 –0.25 –0.30 Electron at r = 0 (0,a/3) (a.u.) –0.35 –4

–3

–2

–1

0 z' (a.u.)

1

2

3

4

Fig. 16.2 Cross sections of the Fermi–Coulomb xc .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for an electron at r D .0; a=3/ of the bond with the electron position indicated by the arrow

16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration

293

Fermi - Coulomb and Coulomb Holes (a.u.)

0.10 0.05 0.00 –0.05 –0.10 –0.15 –0.20

ρxc(r, r') ρc(r, r')

–0.25 –0.30 Electron at r = (0, 2a/3) –0.35 –3

–2

–1

0 z' (a.u.)

3

2

1

Fig. 16.3 Cross sections of the Fermi–Coulomb xc .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for the electron at r D .0; 2a=3/

Fermi - Coulomb and Coulomb Holes (a.u.)

0.10 0.05 0.00 –0.05 –0.10 ρxc(r, r')

–0.15

ρc(r, r')

–0.20 –0.25 –0.30 –0.35 Electron at r = (0, a) (a.u.) –0.40 –4

–3

–2

–1

0

1

2

3

4

z' (a.u.)

Fig. 16.4 Cross sections of the Fermi–Coulomb xc .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for the electron at r D .0; a/

294

16 Application of Q-DFT to the Hydrogen Molecule

Fermi - Coulomb and Coulomb, Holes (a.u.)

0.10 0.05 0.00 –0.05 –0.10 ρxc(r, r')

–0.15

ρc(r, r')

–0.20 –0.25 –0.30 –0.35

Electron at r = (0, 2a)

–0.40 –4

–3

–2

–1

0 z' (a.u.)

1

2

3

4

Fig. 16.5 Cross sections of the Fermi–Coulomb xc .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for the electron at r D .0; 2a/

Coulomb holes of each figure. Note also how the positive part of the Coulomb hole becomes more pronounced relative to the negative part as the electron is moved away from the center of the nuclear bond toward one nucleus. This is also evident from the plots of the Fermi–Coulomb hole where the reduction in probability of the second electron is less about the nucleus to the left. In Figs. 16.5–16.7, we plot the Fermi–Coulomb and Coulomb hole cross sections for an electron in the classically forbidden region at r D .0; 2a/, r D .0; 4a/, and r D .0; 6a/. The positive part of the Coulomb hole continues to increase about the left nucleus at the expense of the negative part as the electron is moved further from the molecule. Thus, even for the asymptotic position of an electron at r D .0; 6a/, the other electron is still mainly about the left nucleus. This fact is also indicated in the graph of the Fermi–Coulomb hole. Of course, for all electron positions, the Fermi–Coulomb hole xc .rr 0 / is negative as must be the case. (We note that the same cross section of the Fermi–Coulomb, Fermi, and Coulomb holes for an electron position 0:3 a.u. to the left of the right nucleus, which corresponds approximately to our Fig. 16.3 have been plotted by Baerends et al. [7] in their study of the dissociation of the molecule. However, in their figure, the electron– electron cusp in the Fermi–Coulomb and Coulomb holes is not present because the wave function employed by these authors is a configuration-interaction type wave function.)

Fermi - Coulomb and Coulomb Holes (a.u.)

16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration

295

0.15 0.10 0.05 0.00 –0.05 –0.10 ρxc(r, r')

–0.15

ρc(r, r')

–0.20 –0.25

Electron at r = (0, 4a)

–0.30 –4

–3

–2

–1

0

1

2

3

4

z' (a.u.)

Fig. 16.6 Cross sections of the Fermi–Coulomb xc .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for the electron at r D .0; 4a/

Fermi - Coulomb and Coulomb Holes (a.u.)

0.10 0.05 0.00 –0.05 –0.10

ρxc(r, r') ρc(r, r')

–0.15 –0.20 Electron at r = (0, 6a) (a.u.) –0.25 –4

–3

–2

–1

0

1 2 z' (a.u.)

3

4

5

6

7

Fig. 16.7 Cross sections of the Fermi–Coulomb xc .rr 0 /, and Coulomb c .rr 0 / holes along the nuclear bond axis for the electron at r D .0; 6a/

296

16 Application of Q-DFT to the Hydrogen Molecule

16.2.2 Electron Interaction E ee .r/ and Correlation-Kinetic Z tc .r/ Fields

Electron-interaction, Hatree, Pauli-Coulomb Fields (a.u.)

The electron-interaction field E ee .r/, and its Hartree E H .r/ and Pauli-Coulomb E xc .r/ components along the nuclear bond axis are plotted in Fig. 16.8. Observe that these fields all vanish at the center of the bond axis or origin. This is because their corresponding sources – the pair-correlation density g.rr 0 /, the density .r/, and the Fermi–Coulomb hole charge xc .rr 0 / – are symmetrical about the center of the nuclear bond for this electron position (see Figs. 8.1 and 16.1). The existence (nonzero value) of these fields for all other electron positions is a consequence of the fact that their sources are not symmetrical about the electron (see Figs. 8.1 and 16.2–16.7). The fields are also all antisymmetric about the center of the nuclear bond. (This is a reflection of the symmetry about the x–y plane at the center of the nuclear bond. As such, the potential energies obtained from these fields will be symmetric about this point.) In the positive half-space, there is a maximum in the electron-interaction and Hartree fields, and a minimum in the Pauli-Coulomb field. The Hartree and Pauli-Coulomb fields are of the same order of magnitude and opposite in sign, with the Hartree field being larger. This is because their sources .r/ and xc .rr 0 /, respectively, are of the same order of magnitude and opposite in sign. Asymptotically, in the z direction these fields decay as E ee .r/ 1=z2 ,

0.8

Eee

0.6

EH Exc

0.4 1/z2

0.2

2/z2

0.0 –0.2

–1/z2

–0.4 –0.6 –0.8 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

z (a.u.)

Fig. 16.8 The electron-interaction E ee .0; z/ field, and its Hartree EH .0; z/ and Pauli-Coulomb E xc .0; z/ components along the nuclear bond axis

16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration

297

0.4

Pauli and Coulomb Fields (a.u.)

0.3 0.2 0.1 0.0 –0.1 Ec

–0.2

Ex –0.3 –1/z2

–0.4 –0.5 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

z (a.u.)

Fig. 16.9 The Pauli Ex .0; z/ and Coulomb Ec .0; z/ fields along the nuclear bond axis. The function 1=z2 is also plotted

E H .r/ 2=z2 , and E xc .r/ 1=z2 as they must. Observe that with a slight translation to the right, the field E xc .r/ is strikingly similar to the corresponding field for the Helium atom (see Fig. 15.5). The Pauli E x .r/ and Coulomb E c .r/ field components of the Pauli–Coulomb field E xc .r/ along the nuclear bond axis are plotted in Fig. 16.9. Again, these fields vanish at the origin and are antisymmetric about it. Hence, the corresponding potential energies obtained from these fields will be symmetric. In the positive half-space, the Pauli field E x .r/ is negative as its source is a negative charge. The Coulomb field E c .r/, on the other hand, is positive in the internuclear region and negative throughout the region beyond the right nucleus. This structure is attributable to the fact that the Coulomb hole has both a positive and negative component. Asymptotically, the Pauli field decays as E x .r/ 1=z2 , whereas the Coulomb field E c .r/ has essentially vanished by about z D 5 a:u: (Once again in the positive half-space, the structure of these fields when translated slightly to the right, is similar to those of the Helium atom (see Fig. 15.5). In particular, observe that the Coulomb holes of the Hydrogen molecule for electron positions z > a (see Figs. 16.5–16.7) centered about the left nucleus are similar to those of the Helium atom for electron positions away from its nucleus (see Fig. 15.3).) As the model S system is constructed to be in its .g1s/2 ground state, the spatial component of each molecular orbital is known explicitly in terms of the density: 1=2 . Thus, the Dirac density matrix s .rr 0 / of the model fermions i .r/ D Œ.r/=2 is known. The interacting system density matrix .rr 0 / is in turn obtained from

298

16 Application of Q-DFT to the Hydrogen Molecule

Correlation-Kinetic Field (a.u.)

0.3

0.2

0.1

0.0

–0.1

–0.2 tc

–0.3 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

z (a.u.)

Fig. 16.10 The Correlation-Kinetic field Z tc .0; z/ along the nuclear bond axis

.r 1 r 2 R/ of (16.4). The corresponding Correlation-Kinetic field Z tc .0; z/ along the nuclear bond axis is plotted in Fig. 16.10. Note that this field too is antisymmetric about the nucleus. (Note: The Kolos–Roothaan wave function approx .r 1 r 2 R/ of (16.4) does not satisfy the electron–nucleus coalescence condition very accurately. As shown in Chap. 8, the satisfaction of this coalescence constraint is critical to the requirement that the electron-interaction potential energy vee .r/ be finite at the nucleus. As a consequence of the lack of rigorous satisfaction of the coalescence constraint, the kinetic fields of the interacting and model systems do not cancel exactly at each nucleus. Hence, in the determination of the field Z tc .r/, the calculations are performed up to close to each nucleus and the curves smoothed through each nucleus. A comparison of the results for the potential energy vee .r/ with those of Gritsenko et al. [8] who in their self-consistent calculations assumed vee .r/ to be finite at the nucleus, shows the two curves to be indistinguishable throughout space. The reason for this is that the densities of the two calculations are equivalent, and the vee .r/ of [8] is determined by working backwards from the density.) approx

16.2.3 Electron-Interaction Potential Energy vee .r/ The electron-interaction potential energy vee .r/ is the work done in the sum of the electron-interaction E ee .r/ and Correlation-Kinetic Z tc .r/ fields. For the two model-fermion S system in its ground state, the Pauli field E x .r/ component of

16.2 Mapping to an S System in Its .g 1s/2 Ground State Configuration

299

0.0

Potentials (a.u.)

–0.2

–0.4 Wc(0,z)

–0.6

Wx(0,z) –1/z

–0.8

–1/z

–1.0

0

1

2

3

4

5

6

7

z (a.u.)

Fig. 16.11 The Pauli potential energy Wx .0; z/ along the nuclear bond axis. The work done Wc .0; z/ in this direction in the force of the Coulomb field E c .0; z/, and the function 1=z, are also plotted

E ee .r/ is E x .r/ D E H .r/=2. Since the Hartree field E H .r/ is conservative, so is the Pauli field. Hence, the Pauli potential energy Wx .r/ component of vee .r/ is pathindependent. The potential energy Wx .r/ along the nuclear bond axis is plotted in Fig. 16.11. Observe that it decays asymptotically in this direction as Wx .r/ 1=z. The Coulomb E c .r/ and Correlation-Kinetic Z tc .r/ fields are separately not conservative. However, the sum of these fields is, i.e., r  ŒE c .r/ C Z tc .r/ D 0. The work done in the sum of the fields: Z r

vc .r/ D  E c .r 0 / C Z tc .r 0 /  d` 0 ; (16.7) 1

is path-independent. A plot of the potential energy vc .r/ along the nuclear bond axis is given in Fig. 16.12. The sum of the Hartree WH .r/ D 2Wx .r/, Pauli Wx .r/, and the Coulomb-Correlation-Kinetic vc .r/ potential energies is then the electroninteraction potential energy vee .r/. To obtain a quantitative sum of the separate Coulomb and Correlation-Kinetic contributions to vc .0; z/, we plot the work done Wc .0; z/ along the path of the nuclear bond in the force of the Coulomb field E c .r/ in both Figs. 16.11 and 16.12. The work done Wtc .0; z/ along the path of the nuclear bond in the force of the Correlation-Kinetic field Z tc .r/ is given in Fig. 16.12.

300

16 Application of Q-DFT to the Hydrogen Molecule 0.24 vc(0,z) Wc(0,z)

0.20 0.16

Wtc(0,z)

Potentials (a.u.)

0.12 0.08 0.04 0.00 –0.04 –0.08 –0.12 0

1

2

3

4

5

6

7

z (a.u.)

Fig. 16.12 The potential energy vc .0; z/, the sum of the Coulomb and Correlation-Kinetic potential energies, along the nuclear bond axis. The work done Wc .0; z/ of Fig. 16.11, and the work done Wtc .0; z/ in the force of the Correlation-Kinetic field Z tc .0; z/ of Fig. 16.10, are also plotted

Observe that the work Wc .0; z/ and Wtc .0; z/, and the potential energy vc .r/ when translated to the right nucleus are very similar to those of the corresponding properties of the Helium atom (see Fig. 15.8). The Coulomb correlation part Wc .0; z/ is negative throughout space and vanishes by about z D 5 a:u: The Correlation-Kinetic piece Wtc .0; z/ is much larger, positive throughout space, and decays asymptotically much more slowly. This structure is directly a consequence of the corresponding fields. (Note that whereas the Coulomb–Correlation-Kinetic work vc .r/ of (16.7) is a potential energy, the work Wc .0; z/ and Wtc .0; z/ do not represent such an energy.)

16.2.4 Total Energy and Ionization Potential The total energy E of the Hydrogen molecule in its ground state as obtained by the Q-DFT Fully Correlated Approximation, and its components Ts , Eext , Eee , Tc , and Vnn are given in Table 16.1. The value of the energy is, of course, the same as that obtained by the Kolos–Roothaan wave function of (16.4). The single eigenvalue m of (15.5) is valid for arbitrary r, and since v.r/ and vee .r/ vanish at infinity, the eigenvalue p m may be p obtained from the asymptotic value of the first term, i.e., m D r 2 .r/=2 .r/ for r ! 1.

16.3 Endnotes

301

Table 16.1 Ground state properties of the hydrogen molecule as obtained by the Q-DFT Fully Correlated Approximation mapping to an S system in its .g 1s/2 ground state Property E Ts Eext Eee Tc Vnn I D m

Value in a.u. 1.1744 1.1414 3.6501 0.5874 0.0330 0.7138 0.600

16.3 Endnotes The results of this chapter provide an understanding of Q-DFT as applied to the ground state of the Hydrogen molecule, and thereby of the various electron correlations in this molecule. First, there is the interesting structure of the Fermi–Coulomb xc .rr 0 / and Coulomb c .rr 0 / hole charge distributions as a function of electron position. For example, for asymptotic positions of the electron in the classically forbidden region along the nuclear bond axis, we learn that the second electron is localized about the nucleus that is further away. The symmetry of the molecule dictates that all the individual fields E ee .r/, E H .r/, E xc .r/, E x .r/, E c .r/, and Z tc .r/, must each be antisymmetric about the center of the nuclear bond. The corresponding electron-interaction potential energy vee .r/ – the work done in the sum of the fields E ee .r/ and Z tc .r/ – representative of all the correlations, must then be symmetric about this point. The potential energy vee .r/ is also finite at each nucleus, as must be the case. The Hartree E H .r/ and Pauli E x .r/ fields are the largest in magnitude, and opposite in sign, the former being twice as large as the latter. As such the principal contributions to the electron-interaction energy Eee and potential energy vee .r/ are due to the Hartree and Pauli correlation terms. The Coulomb E c .r/, and Correlation-Kinetic Z tc .r/ fields also tend to cancel each other, so that the contribution of their sum to the potential energy vee .r/ is an order of magnitude smaller. However, as the potential energy component vc .r/ representing the sum of these correlations is principally positive (see Fig. 16.12), it is evident that the Correlation-Kinetic effects are more significant. They are also more significant asymptotically, where the Coulomb correlation contributions to the potential energy vanish. Thus, Correlation-Kinetic effects play an important role in the structure of the local effective potential energy of the S system. The Correlation-Kinetic energy Tc , which is positive as must be the case, is however small and about the same as that for the Helium atom in its ground state (see Table 15.1). The significance of the Correlation-Kinetic contributions to the potential energy, and hence to the S system orbitals, has bearing on the construction of approximate “exchange-correlation” and “correlation” energy density functionals of traditional theory. It is evident that the qualitative features of the Q-DFT results obtained for the Hydrogen molecule – quantal sources, fields, and potential energies – will be similar

302

16 Application of Q-DFT to the Hydrogen Molecule

for other diatomic molecules. However, the fields, and hence the potential energies of these diatomics will have more structure as a consequence of the additional molecular subshells. The additional structure should be similar to that observed in atoms as the number of shells is increased. Finally, it is worth reiterating the striking similarity between the Q-DFT properties of the Hydrogen molecule and Helium atom for electron positions in the positive half space. It is interesting that in spite of the presence of a second nucleus, and therefore of a different symmetry, the quantal sources and fields representative of the various electron correlations in the Hydrogen molecule are so similar to those of the Helium atom. This speaks to the commonality of properties of these finite and distinct quantum systems as exhibited within Q-DFT.

Chapter 17

Application of Q-DFT to the Metal–Vacuum Interface

In this chapter we study select properties of the inhomogeneous electron gas at a metal–vacuum interface in the context of Local Effective Potential Energy Theory. The metal surface problem differs in fundamental ways from the atomic, ionic, and molecular systems studied thus far. The latter are few-electron, finite, discrete energy spectrum systems, whereas the physical system at hand is an extended, many-electron system with a continuous energy spectrum. Metal surface physics is a field unto itself. Here we are going to be concerned with certain intrinsic aspects of the subject. The model employed in the calculations is one in which the lattice of positive ions in the metal is smeared out and replaced by a uniform positive charge background or “jellium.” The jellium model is appropriate for the “simple” metals whose conduction band arises only from s and p shells. The semi-infinite jellium metal is confined to the negative half-space, the metal surface being defined as the position where the uniform positive background charge ends abruptly. In the plane parallel to the surface, the jellium extends to infinity. The positive half-space is the vacuum region. (The semi-infinite jellium metal surface model is distinct from the jellium slab metal model in which the positive charge background is finite. We will be concerned only with the semi-infinite jellium metal model.) Thus, in the case of the metal–vacuum interface, the external potential energy of the electrons is due to the half-space positive jellium charge distribution. As a consequence of the translational symmetry in the plane parallel to the surface, the electron density is nonuniform only in the direction perpendicular to the surface. It approaches the bulk-metal density value deep in the crystal, exhibits the Bardeen-Friedel [1, 2] oscillations at and about the surface due to the “impurity” which in this case is the surface, and vanishes asymptotically in the classically forbidden vacuum region. The semi-infinite jellium metal is charge neutral. In the jellium model of a metal surface, lattice properties recede in significance, and it is the electronic properties that are to the fore. The seminal work in the field is due to Bardeen [1] who attempted to perform a Hartree-Fock theory calculation of this electron inhomogeneity. (See QDFT for the Slater-Bardeen description of Hartree-Fock theory in which it is interpreted as an orbital-dependent theory.) In his paper, Bardeen attempts to determine the nonlocal exchange operator, but since computing facilities comparable to those existing today were unavailable to him, he made various approximations for the exchange

303

304

17 Application of Q-DFT to the Metal–Vacuum Interface

operator and hence for the orbital-dependent exchange potentials. One approximation was that he ignored the electron momentum parallel to the surface. In the final expressions, the electron momentum perpendicular to the surface was then replaced by the magnitude of the total momentum. Furthermore, Bardeen’s calculations were not self-consistent, and the exchange potential energy functions near the surface were fixed by employing values determined from model potential wave functions. It was only the electrostatic potential energy that was determined self-consistently. Coulomb correlation effects were introduced by assuming that the Coulomb correlation potential energies varied in the same way with momentum and position as did the exchange potential energies in the interior and at small distances from the surface. Furthermore, the combined exchange and Coulomb correlation potential energies each asymptotically approached the image potential (1=4x, where x is the distance from the surface) at large distances from the surface. In spite of the various approximations, Bardeen arrived at a reasonable value for the work function of Na, a medium density metal of Wigner-Seitz radius rs D 3:99. Many years later [3], via a variational calculation in which the nonlocal exchange operator was treated exactly, rigorous upper bounds to the Hartree-Fock theory value of the surface energy, and accurate corresponding work functions of metals were obtained. Values for the Hartree-Fock theory nonlocal surface exchange energy as a function of the density profile at the surface are given in [4], and numerically refined values of these energies in [5]. In these calculations, the density profile at the surface can be varied from rapidly to slowly varying. (Recently [6], fully selfconsistent calculations treating the nonlocal exchange operator exactly have been performed over metallic densities for the jellium slab-metal.) The present study is performed within local effective potential energy theory, in the context of Q-DFT and Kohn–Sham DFT, and of their interrelationship. For part of the study, we represent the effective potential energy of the model fermions vs .r/ at the surface by realistic analytical model potential energy functions. As such, the single particle orbital solutions of the corresponding Schr¨odinger equation are analytical. This then lends itself to the calculations of other properties being semianalytical or entirely analytical. Following a brief description of the equations of the jellium metal surface within local effective potential energy theory, and the general structure of the orbitals, two model effective surface potential energy functions are described. These are the Finite-Linear Potential and the Linear Potential Models [7, 8]. The accuracy of the analytical orbitals generated by these model potentials is then established [9–11] for properties such as the surface energy and work function by comparison with numerical fully self-consistent results [11–13] performed within the Local Density Approximation (LDA) for Kohn–Sham “exchange” and “correlation.” (For an understanding of the representation of electron correlations within the LDA, see QDFT.) We begin by a description of the structure of the density quantal source .r/ at and about the metal surface. We then study the dynamic structure of the nonlocal Fermi hole quantal source charge distribution x .rr 0 / as the electron is moved from within the metal, through the surface, and into the classically forbidden region [14–16]. We show that as the electron is moved from within the metal into the far

17 Application of Q-DFT to the Metal–Vacuum Interface

305

vacuum, the Fermi hole is not localized to the surface region, as originally thought to be the case [17–19, 36] but spreads out throughout the crystal. In fact it can be shown that irrespective of the electron position, there is always charge at minus infinity. The structure of the corresponding Pauli field E x .r/ due to this delocalized dynamic charge, and the resulting Pauli potential energy Wx .r/ of the model fermions obtained as the work done in this field are then determined [20,21]. (Recall that for atoms, the Fermi x .rr 0 / and Coulomb c .rr 0 / holes are localized about the nucleus. In the Hydrogen molecule, for asymptotic positions of the electron along the nuclear bond axis, the Coulomb hole c .rr 0 / is localized about the farthest nucleus.) The second component of this chapter focuses on the derivation of the analytical asymptotic structure of the effective potential energy function vs .r/ in the classically forbidden vacuum region [22–28]. We derive the asymptotic structure of the Q-DFT Pauli Wx .r/, Coulomb Wc .r/, and Correlation-Kinetic Wtc .r/ potential energies, as well as that of the Slater function VxS .r/ and the Kohn–Sham “exchange” vx .r/ and “correlation” vc .r/ potential energies. Thus, as in the case of finite atomic systems (see Chap. 7), we obtain the asymptotic structure of vs .r/ as a function of the different electron correlations contributing to it. For atomic systems, this structure is of significance because it is related to the ionization potential. For atoms we proved that the asymptotic structure was solely due to Pauli correlations, with the Coulomb correlation and Correlation-Kinetic effect contribution to this structure vanishing more rapidly. For the metal surface problem, the asymptotic structure is important for the understanding of image-potential-bound surface states [29, 30]. It has been commonly assumed that the asymptotic structure of vs .r/ is the classical image potential .1=4x/, that this structure is due entirely to Coulomb correlations, and that the structure is the same for all metals and therefore independent of the Wigner–Seitz radius rs . We show that the asymptotic structure of vs .r/ over the metallic range of densities .2  rs  6/ is image-potential-like with a coefficient that is approximately twice as large as that of the image potential, that Pauli and Coulomb correlations as well as Correlation-Kinetic effects contribute to this structure, and that the structure is dependent on the metal density. These conclusions are valid for fully self-consistent orbitals. We conclude with an endnote on image-potential-bound surface states, and note that the experimental results for these states need to be reinterpreted in light of the new asymptotic structure derived. In the literature, the ideas that for asymptotic positions of the electron in the vacuum region: (i) the Fermi hole ought to be localized about the surface becoming thinner and spreading out over the surface the further out the electron, and (ii) the asymptotic structure of vs .r/ is the image potential, stem from thinking of the asymptotic electron as an independent test charge external to the system. If the asymptotic electron were an independent external test charge, then the quantummechanical equivalent of the classical image charge ought to be localized about the surface. Furthermore, the potential energy of this external test charge due to this quantum-mechanical image charge would then have to be the image potential relative to the centroid of this charge, and therefore the same for all metals asymptotically. However, the asymptotic electron is not an independent external test charge,

306

17 Application of Q-DFT to the Metal–Vacuum Interface

but rather one of the electrons of the N -electron charge neutral metal, and must therefore be treated as such. Additionally, the asymptotic electron is a fermion. Hence, it has a Fermi hole, and as shown below, this Fermi hole is connected to it irrespective of how far the electron is in the classically forbidden region. Since the asymptotic electron is part of the N-electron system, it also has a Coulomb hole. A random-phase approximation calculation for the Fermi-Coulomb hole structure has been performed [31] for the unrealistic infinite-barrier model [1, 32, 33] of a metal surface. As the structure of the Fermi hole for these orbitals is known [15], the structure of the Coulomb hole in this approximation is also known. However, in this model, asymptotic positions of the electron in the classically forbidden region cannot be considered because the orbitals vanish at the barrier. There is no classically forbidden region in this model. Calculations of the structure of the Coulomb hole as a function of electron position in the vacuum for accurate realistic models of the surface have not yet been performed. Hence, the structure of the Coulomb hole charge as an electron is removed from within the metal to infinity in the vacuum region is unknown. Since the Fermi hole structure is known [14, 15] for realistic surface models, an RPA calculation of the Fermi-Coulomb hole for such orbitals would then lead to the structure of the Coulomb hole in this approximation valid for high density metals. (For RPA type calculations of the surface energy see for example [34]. As noted previously, this chapter is concerned with specific properties of the inhomogeneity at the metal–vacuum interface. For the broader perspective of the subject, the reader is referred to the books and review articles of references [29, 35–41]. However, for a good introduction with derivations, the text of Ref. [35] is recommended.

17.1 Jellium Model of a Metal Surface In the jellium model of a metal surface [1, 35], the charge of the ions of the metal is assumed smeared out into a uniform positive background of charge C .r/ D .x N C a/ ending abruptly at the surface at x D a, and where N D kF3 =3 3 is the bulk metal density, kF D 1=˛rs is the Fermi momentum, ˛ 1 D .9 =4/1=3 , and rs is the Wigner–Seitz radius. The jellium edge position at x D a is determined by the constraint of charge neutrality [1] or equivalently by the Sugiyama [42, 43] phase shift sum rule. The model S system fermions are confined to within the metal and about its surface by the local effective potential energy function vs .r/. This function approaches the bulk-metal value in the interior of the crystal, rises through the surface, and asymptotically approaches the vacuum reference level in the classically forbidden region. (See for example Fig. 17.3 of [37]. For the Finite Linear Potential model representation of this effective potential function, see Fig. 17.1). The external potential energy component of vs .r/ due to the jellium charge is

17.1 Jellium Model of a Metal Surface

307 s

Φ

FERMI LEVEL Ves +

Δφ

2 ½kF –μxctc

0

a

W

xF b

x

Fig. 17.1 Schematic representation of the Finite Linear Potential Model indicating all relevant energies and parameters. The hatched region represents the jellium background beginning at the surface at x D a. The barrier at x D L employed for normalization is not shown since in the calculations the limit as L ! 1 is always taken

Z vjell .r/ D

C .r 0 / 0 dr : jr  r 0 j

(17.1)

As a consequence of the translational symmetry in the plane parallel to the surface, the electron interaction field E ee .r/, its Hartree E H .r/, Pauli E x .r/ and Coulomb E c .r/ components, and the Correlation-Kinetic Z tc .r/ field are each separately conservative. The Hartree field E H .r/ is, of course, conservative because it is due to the static electronic density .r/. Thus the effective potential energy vs .r/ is vs .r/ D vjell.r/ C vee .r/;

(17.2)

vee .r/ D WH .r/ C Wx .r/ C Wc .r/ C Wtc .r/;

(17.3)

where and WH .r/, Wx .r/, Wc .r/, Wtc .r/ are, respectively, the work done in the fields E H .r/, E x .r/, E c .r/, Z tc .r/. In the surface physics literature, one also defines the electrostatic potential energy Ves .r/ as Ves .r/ D vjell .r/ C WH .r/ Z .r 0 /  C .r 0 / 0 dr : D jr  r 0 j

(17.4) (17.5)

This potential energy is obtained by solution of Poisson’s equation r 2 Ves .r/ D 4 T .r/;

(17.6)

where the total charge T .r/ D .r/  C .r/. Together with the facts that in the jellium model there is translational symmetry in the plane parallel to the surface, and that the effective potential energy of the model fermions is represented by a local (multiplicative) operator, the form of the S system orbitals is

308

17 Application of Q-DFT to the Metal–Vacuum Interface

r D

k .r/

2 i kk xk e k .x/; V

(17.7)

where .kk x k / are the momentum and position vectors parallel to the surface, .kx/ the components perpendicular to it, and V the volume of the crystal. Asymptotically in the metal interior k .x/ must be a phase shifted oscillatory function k .x/ D .2=L/1=2 sinŒkx C ı.k/, where for the normalization constant a fixed boundary at x D L is assumed. Surface properties are obtained in the limit as L ! 1. The expressions for the Dirac density matrix s .rr 0 / of (3.12) and the density .r/ are then Z 0 dk k2 k? .x/k 0 .x 0 / ei kk .xk x k / ‚.F  /; (17.8) s .rr 0 / D 2 2 .2 /

2 and .x/ D

1 2 2

Z

kf 0

.kF2  k 2 /jk .x/j2 dk;

(17.9)

where F D kF2 =2 is the Fermi energy. (Atomic units are used: jej D „ D m D 1. The unit of energy is 27.21 eV.) Note that irrespective of how the effective potential energy vs .r/ at the surface is modeled, the general expressions for all properties in terms of .kk x k ), .kx/, and k .x/ determined from the wave function of the form (17.7) will be the same. It is only the component of the wave function perpendicular to the surface k .x/ in these expressions that will differ. The jellium metal energy Ejell within the context of the S system, on ignoring the Coulomb self-energy of the positive charges, is then Z Ejell D Ts C

.r/vjel l .r/dr C EH C Ex C Ec C Tc

D Ts C Ees C Ex C Ec C Tc ;

(17.10) (17.11)

where the electrostatic energy Ees is Z Ees D 1 D 2

.r/vjell .r/dr C EH

(17.12)

Z T .r/Ves .r/dr;

(17.13)

and all the other energy terms – Pauli Ex , Coulomb Ec , and Correlation-Kinetic Tc – are defined as in Chap. 3. In the section to follow, we describe two model effective potential energy functions at a metal surface. We demonstrate the accuracy of the wave functions k .r/ generated by these functions for two properties: the surface energy  and the work function ˆ. The surface energy  is defined [35, 37] as the work done, per unit area of new surface formed, to split a crystal in two along a plane. Corresponding to the

17.1 Jellium Model of a Metal Surface

309

energy expression Ejell of (17.11), the surface energy may be written in terms of its components as  D s C es C x C c C tc ; (17.14) where s D .1=2A/f2Ts  Ts0 g; es D .1=2A/f2Ees 

0 Ees g;

(17.15) (17.16)

x D .1=2A/f2Ex  Ex0 g;

(17.17)

Ec0 g;

(17.18)

c D .1=2A/f2Ec 

tc D .1=2A/f2Tc  Tc0 g;

(17.19)

the unprimed energy components corresponding to the fragment, the primed components to the unsplit crystal, and A the surface area of each fragment. The work function ˆ is the minimum work done at 0ı K to remove an electron from the metal to infinity in the vacuum region. (The work function in metal surface physics is analogous to the first ionization potential of atomic and molecular physics.) The work function within local effective potential energy theory is thus (see Fig. 17.1) ˆ D W  F D Œvs .1/  vs .1/  F ;

(17.20) (17.21)

where W is the barrier height at the surface. Thus, the work function is comprised of a surface component and a bulk-metal component [44]. The barrier W confining the model fermions arises as follows. First there is the surface dipole barrier 4 formed due to the negative charge density which decays into the classically forbidden region and the positive jellium charge distribution ending abruptly at the surface. Then there are the contributions xctc ./ N as a result of the lowering of the energy due to correlations arising from the Pauli principle, Coulomb repulsion, and CorrelationKinetic effects. Thus, an alternate expression for the work function is [35, 37] (see Fig. 17.1) 1 ˆ D ŒVes .1/  Ves .1/  kF2  xctc ./ N 2 d D 4  Œ N T ./; N dN

(17.22) (17.23)

where the surface dipole barrier 4 is 4 D Ves .1/  Ves .1/; Z 1 xT .x/dx; D 4

1

(17.24) (17.25)

310

17 Application of Q-DFT to the Metal–Vacuum Interface

and T ./ N the total energy per fermion for the uniform bulk-metal density . N The T ./ N in turn is the sum of the metal-bulk kinetic k , Pauli x , Coulomb c , and Correlation-Kinetic tc components: N D k ./ N C x ./ N C c ./ N C tc ./; N T ./

(17.26)

N conwhere k D 3kF2 =10 and x D 3kF =4 . The many-body term xctc ./ tributing to the barrier is xctc ./ N D dfŒ N x ./ N C c ./ N C tc ./g=d N . N The sum Œc ./ N C tc ./ N is obtained as the difference between the total energy Ejell of the uniform system for a particular bulk density N (see (17.11), and the Hartree-Fock HF theory energy Ejell D Ts C Ees C Ex for the same bulk density . N The difference HF Ejell  Ejell D Ec C Tc because for the uniform gas, the S system and HartreeFock theory orbitals are plane waves. Since the occupation of states in k-space and hence the densities of the fully interacting S and Hartree-Fock theory systems are the same, the Ts and Ex are also the same. (Ees in the bulk is zero.) One expression for Œc ./ N C tc ./ N commonly employed in the literature is a parametrization [45] based on a stochastic Monte-Carlo calculation [46, 47] for the fully interacting uniform electron gas. This expression is Œc ./ N C tc ./ N D

 ryd; p 1 C ˇ1 rs C ˇ2 rs

(17.27)

where  D 0:284656, ˇ1 D 1:052944, ˇ2 D 0:333372. Another equivalent expression for the work function ˆ is [10, 35, 48, 49] ˆ D ŒVes .1/  Ves .a/  T ;

(17.28)

where once again there are the surface and bulk components to the definition. That the definitions for ˆ of (17.22) and (17.28) are equivalent follows by application to (17.23) of the Theophilou-Budd-Vannimenus theorem [50, 51] according to which N T ./=d N : N Ves .a/  Ves .1/ D d

(17.29)

As noted, the two definitions of the work function ˆ of (17.22) and (17.27) are equivalent in principle. They are also equivalent [48] in any fully self-consistent calculation in which the many-body effects are approximated such as in the LDA [12,13]. In energy variational calculations, the expression for ˆ of (17.22) is correct only to the same order in accuracy as that of the nonuniform electron density .r/ employed. Recall that it is the energy that is obtained correct to second order in the accuracy of the wave function. The density .r/, and hence the surface dipole barrier 4, are correct only to the same order as that of the wave function. However, it can be shown [10, 49] that for variational calculations, the work function ˆ expression of (17.27) is more accurate. The expression is neither stationary nor a bound on the work function. The accuracy lies between first and second order in the

17.2 Surface Model Effective Potential Energies and Orbitals

311

accuracy of the wave function. Hence, in calculations in which the surface energy is obtained variationally, it is the more accurate expression for ˆ of (17.27) that must be employed.

17.2 Surface Model Effective Potential Energies and Orbitals In this section we describe two model effective potential energy functions at a metal surface – the Finite Linear Potential (FLP) model [7] and the Linear Potential (LP) model [8]. As noted above, the solution of the Schr¨odinger equation for these model functions is analytical. A key advantage of this is that many of the spatial integrals can be performed analytically. What remains then are momentum space integrals from 0 to 1 in units normalized to the Fermi wave vector kF . Furthermore, essentially all metal surface properties may be expressed as universal functions of the parameters of the model potential energy functions. Thus, for example, if the parameters are determined by application of the variational principle for the energy, then all the other properties can be determined for these parameters directly from the corresponding universal functions of those properties. Another advantage of the analytical solutions is that it is then easier to understand the steps of each derivation.

17.2.1 The Finite Linear Potential Model In the Finite Linear Potential (FLP) model, the effective potential energy, vs .x/ of the S system is (see Fig. 17.1) vs .x/ D F xŒ.x/  .x  b/ C W .x  b/;

(17.30)

where F , the field strength, is defined in terms of the barrier height and slope parameters b and xF , respectively, as F D W=b D . 12 kF2 /=xF , and where W is the barrier height, 12 kF2 is the Fermi energy, and .x/ is the step function. We also specify the variation of the barrier height in terms of the parameter ˇ where ˇ 2 D W=. 12 kF2 / D b=xF . For the effective potential of (17.30), the solution of the Schr¨odinger equation for the electronic wave function is 8 < A sinŒkx C ı.k/ k .x/ D Bk Ai./ C Ck Bi./ : Dk exp.x/

for x  0 ; for 0  x  b; for x  b;

(17.31)

where k D .2E/1=2 ,  D Œ2.W  E/1=2 ,  D .x  E=F /.2F /1=3 , E is the energy, and where Ai./ and Bi./ are the linearly independent solutions of the Airy differential equation: d2 k =d 2  k D 0 [52, 53].

312

17 Application of Q-DFT to the Metal–Vacuum Interface

The constant A in the above equation is obtained by the normalization condition, whereas the phase factor ı.k/ and the coefficients Bk , Ck , and Dk are determined by the requirement of the continuity of the wave function and its logarithmic derivative at both x D 0 and x D b. Thus AD Bk D Ck D Dk D cot ı.k/ D

 1=2 2  ; L 1=2  0 A ; ƒ.0 / 1=2  X.b / 0 ; A ƒ.0 / Y .b / 1=2  p 0 A M.b / expŒ.b C 0 / b ; ƒ.0 / p .1= 0 /N.0 /=M.0 /I

(17.32) (17.33) (17.34) (17.35) (17.36)

where p b Ai.b /; p Y .b / D Bi 0 .b / C b Bi.b /;

(17.37)

M./ D Ai./  Bi./X.b /=Y .b /;

(17.39)

X.b / D Ai 0 .b / C

0

0

(17.38)

N./ D Ai ./  Bi ./X.b /=Y .b /I

(17.40)

ƒ.0 / D 0 M 2 .0 / C N 2 .0 /;

(17.41)

0 D k .b=2W / 2

2=3

D .k

2

=kF2 /.kF xF /2=3 ;

(17.42)

b D b.2W=b/1=3  0 D kF b.1=kF xF /1=3  0 ;

(17.43)

and where Ai 0 ./ and Bi 0 are the derivatives [52, 53] of the Airy functions.

17.2.2 The Linear Potential Model The Linear Potential (LP) model (see Fig. 17.2) is a special case of the FLP model. In this model the effective potential energy vs .x/ of the S system is vs .x/ D F x.x/;

(17.44)

where F is the field strength defined in terms of the slope parameter xF as F D .kF2 =2/=xF , 12 kF2 is the Fermi energy, and .x/ is the step function. The electronic wave function k .x/ is

17.2 Surface Model Effective Potential Energies and Orbitals

313

s=Fxθ

Ves

FERMI LEVEL

Φ

+

(x) Δφ

2 ½kF –μxctc

0 a

W x

xF

Fig. 17.2 Schematic representation of the Linear Potential model indicating all the relevant energies, jellium metal surface position, and parameters. The barrier at x D L employed for normalization is not shown since in the calculations the limit as L ! 1 is always taken

 k .x/ D

A sinŒkx C ı.k/ Ck Ai./

for x  0; for x  0;

(17.45)

where A D .2=L/1=2 is the normalization constant, Ck is a normalization factor, Ai./ is the Airy function,  D .x  E=F /.2F /1=3 , and E is the energy. The factor Ck and the phase shift ı.k/ are determined by the requirement of the continuity of the wave function and its logarithmic derivative at x D 0. Thus,

and

Ck D A sin ı.k; xF /ŒAi.0 /1

(17.46)

1 Ai 0 .0 / ; cot ı.k; xF / D p 0 Ai.0 /

(17.47)

where 0 D .k 2 =kF2 /.kF xF /2=3 , and where Ai 0 ./ is the derivative of the Airy function Ai./. Note: There are other model effective potential energy functions that are also used to study the inhomogeneous electron gas at a surface. The Infinite Barrier Potential [1, 32, 33], the Step Potential [1, 32, 33], and the Airy Gas [54] models are also all special cases of the FLP model. (As in the figures Figs. 17.1 and 17.2 for the FLP and LP models, respectively, the normalization barrier at x D L is usually not shown in the figures for the Infinite Barrier and Step Potential models because in the calculations, the limit as L ! 1 is finally taken.) The effective potential energy function vs .r/ for the Infinite Barrier Potential model is  vs .r/ D

0 1

for x  0 for x  0;

(17.48)

for the Step Potential model is vs .x/ D W .x/

(17.49)

314

17 Application of Q-DFT to the Metal–Vacuum Interface

and for the Airy Gas is  vs .r/ D

1 Fx

for x  0 for x  0:

(17.50)

(The Airy Gas model potential is also the same as the quantum-mechanical treatment of a particle in the homogeneous gravitational field near the earth’s surface, the latter reflecting the particle elastically [55].)

17.3 Accuracy of the Model Potentials We begin with the structure of the density .x/ at a metal surface. The density .x/ (see (17.9)) as obtained for the orbitals k .x/ of the LP model is plotted in Fig. 17.3 for different values of the field strength parameter F or equivalently xF . In the figure, three different profiles are shown corresponding to different values of yF D kF xF . The profile for yF D 0 is the same as that of the Infinite Barrier Potential model, and corresponds to a very rapidly varying density with jr j=2kF ./  1 at the jellium edge. The profile for yF D 7:4 corresponds to a very slowly varying density for which jr j=2kF ./ ' 0:1. The profile for yF D 3:4 is that of a typical high density metal [9] such as Al.rs D 2:07/. The density exhibits the Bardeen-Friedel [1, 2] oscillations inside the metal. These oscillations decay in the far interior of the metal to the bulk value. In the classically forbidden region the density vanishes asymptotically into the vacuum. (Note: Since in the LP model, the density profile can be changed from rapidly to slowly varying, the model has proved useful in the study of Kohn–Sham

ρ(x)/ρ

0.8 0.6

0 y F= .4 = 3 7.4 yF = yF

1.0

0.4 0.2 –0.8

–0.4

0

0.4

0.8

kF(x – α) / 2π

Fig. 17.3 Electron density .x/ profiles of the Linear Potential model normalized to the bulk value N for different values of the field strength parameter yF D kF xF (see Fig. 17.2 and the text for the definition of the parameter xF )

17.3 Accuracy of the Model Potentials

315

theory density gradient expansions for the noninteracting kinetic, “exchange” and “exchange-correlation” energy functionals [56–63].) We next demonstrate the accuracy of the FLP model by comparison with a fully self-consistent calculation [11–13] for the surface energy  and work function ˆ. The calculations are performed within the local density approximation (LDA) for the Pauli, Coulomb, and Correlation-Kinetic contributions. The FLP calculations are variational with the surface energies minimized with respect to the field strength or slope xF and barrier height b parameters. (Note that if the interacting uniform electron gas is represented by the Hamiltonian of the LDA, then the surface energies  thus obtained are rigorous upper bounds to the fully self-consistent values.) The work functions ˆ are then obtained by the variationally accurate expression of (17.28). For the sum of the Coulomb correlation and Correlation-Kinetic energy per particle, the expression of (17.27) is employed. The properties of interest to be determined for the above calculation are the density .x/, the jellium edge position at x D a, the total charge density T .x/, the surface dipole barrier 4, the electrostatic potential energy Ves .x/, the work function ˆ employing the variationally accurate expression of (17.28), the kinetic LDA k , electrostatic es , and the LDA Pauli, Coulomb, and Correlation-Kinetic xct c components of the surface energy s . The expression for these properties are [7, 35] .x/ D

L 2 2

Z

kF 0

3

3 aD  3 8kF kF

.kF2  k 2 /j Z

kj

2

dk;

(17.51)

kF

kı.k/dk;

(17.52)

0

T .x/ D .x/  .kF3 =3 2 /.x C a/; Z 1 4 D 4

xT .x/dx;

(17.53) (17.54)

1

Z Ves .x/ D 4  4

x

dx 1

0

Z

x0 1

dx 00 T .x 00 /;

(17.55)

( " )# Z Z kF kF4 80 3 2 kF 3 1C k k D kı.k/dk  k ı.k/dk 160

kF4 5 F 0 0 Z 

1 1

es

.vs .x/  vs .1//.x/dx;

1 D 2

Z

1 1

Z LDA xct D c

(17.56)

1 1

T .x/Ves .x/dx;

xctc ..x//  xctc ./ N .x/dx;

(17.57) (17.58)

316

17 Application of Q-DFT to the Metal–Vacuum Interface

Table 17.1 Metal surface energies and work functions as determined variationally employing the orbitals of the Finite Linear Potential Model. The Pauli and Coulomb correlations, and CorrelationKinetic effects are treated in the local density approximation. The results of fully self-consistent calculations within the same approximation for the electron correlations are also quoted Wigner-Seitz Parameters Surface Energies (erg=cm2 ) Work Functions (eV) radius rs (a.u.)

yF

yb

ya

1.5 2.0 2.5 3.0 4.0 5.0 6.0

4.20 3.33 2.66 2.13 1.35 0.91 0.62

4.66 4.25 3.79 3.11 2.58 2.09 1.67

1.68 1.33 1.06 0.84 0.48 0.25 0.06

Variational Formalism

Self-consistent Calculation

7007 835 101 229 168 102 64

7127 856 95 225 164 98 60

Variational Self-consistent Formalism Calculation 3.55 3.67 3.56 3.38 2.92 2.58 2.30

3.55 3.66 3.60 3.42 3.01 2.65 2.36

where xctc D x C c C tc . The expression for Ves .x/ of (17.55) is obtained by solution of Poisson’s equation d 2 Ves .x/=dx 2 D 4 T .x/ with the boundary conditions Ves .1/ D Ves0 .1/ D 0 and application of the charge neutrality condition. For the effective potential energy vs .x/ of the FLP model, all the spatial integrals of the above defined properties, with the sole exception of (17.58) can be performed analytically [7]. With a change of variables to y D kF x and k=kF D q, such that the jellium edge, slope, and barrier height parameters are now defined to be ya D kF a, yF D kF xF , and yb D kF b, the determination of all these properties reduces to simple numerical computations of momentum space integrals from 0 to 1. For the resulting semi-analytical expressions see [7]. The results [11] for the surface energy s and work function ˆ for 1:5  rs  6 as obtained variationally employing the FLP model orbitals and as determined fully self-consistently are quoted in Table 17.1. For rs  2:5 the surface energies are within 6 ergs=cm2 of the self-consistent results. For rs D 1:5 and 2:0, the difference is less than 2:5%. These energies of course lie above the self-consistent values as they must. The work functions differ by only hundredths of an electronvolt. Essentially the same degree of accuracy [9, 10] is obtained with the LP model wave functions. The results thus clearly demonstrate the accuracy of the orbitals generated by the model effective potential energy functions.

17.4 Structure of the Fermi Hole at a Metal Surface In this section we begin by deriving the expressions for the Fermi hole charge distribution x .rr 0 / for the inhomogeneous electron gas at metal surfaces employing orbitals of the general form of (17.7). (Note that the orbitals of the form of (17.7) are model fermion orbitals of the fully correlated S system. They are also the orbitals of electrons in the approximation when only correlations due to the Pauli principle are

17.4 Structure of the Fermi Hole at a Metal Surface

317

considered. Thus, the use of the nomenclature of electrons and model-fermions is equivalent and inter-changeable in this section.) Having demonstrated the accuracy of the LP and FLP models of a metal surface, we then study the structure of the Fermi hole as a function of electron position for the LP model wave functions. For the homogeneous (H ) electron gas, the orbitals are plane waves with the states occupied up to the Fermi level: 1 D  p e i kr ; V

H k .r/

(17.59)

where V is the volume of the crystal. The corresponding Dirac density matrix sH .rr 0 / and Fermi hole xH .rr 0 / are given by the expressions

and

where j.x/ D

sH .rr 0 / D j.kF R/; N

(17.60)

xH .rr 0 / D j 2 .kF R/; .=2/ N

(17.61)

3 3.sin x  x cos x/ D j1 .x/; x3 x

(17.62)

j1 .x/ is the first-order spherical Bessel function [52, 53], N D kF3 =3 2 is the bulk density defined in terms of the Fermi momentum kF , and R D jr 0  rj. Thus, as one might expect for the uniform electron gas, the Fermi hole charge about each electron as given (17.61) is spherically symmetric. For electron positions inside the metal far from the surface, the structure of the Fermi hole obtained by the surface orbitals of (17.7) must reduce to that of this expression.

17.4.1 General Expression for the Planar Averaged Fermi Hole x .xx 0 / To study the structure of the Fermi hole, relative to the jellium edge we next derive an expression for the hole averaged over the plane parallel to the surface. The planar average x .xx 0 / of the Fermi hole is defined as x .xx 0 / D

Z

Z dx k

where 0

I.xx / D

dx 0k x .rr 0 / D Z

Z dx k

I.xx 0 / ; 2.x/

dx 0k Œs .rr 0 /2 :

(17.63)

(17.64)

318

17 Application of Q-DFT to the Metal–Vacuum Interface

To be consistent with the literature, we rewrite (17.7) as 1 ‰k .r/ D p A where k .x/

D Bk .x/;

k .x/e

ikk x k

;

(17.65)

p B D  2=L:

(17.66)

Substituting for P‰k .r/ from (17.65) into the expression for the Dirac density matrix (s .rr 0 / D 2 k ‰k? .r 0 /‰k .r/ in k-notation of condensed matter physics) we have Œs .rr 0 /2 D

4 XX ‚.F  k;kk /‚.F  k 0 ;k0k / A2 k 0 k

 where kk 0 .xx

0

/D

kk 0 .xx

0

0

0

/e i.kk kk /.xk xk / ;

? ? 0 0 k .x/ k 0 .x / k .x / k 0 .x/;

(17.67)

(17.68)

.x/ is the step function and F is the Fermi energy. The two twofold integrals obtained on substitution of  2 from (17.67) into (17.64) may be solved by a change of variables to X D x k  x 0k , and by use of the definition of a two-dimensional delta function which is Z 0 1 dX e i.kk kk /X D ı .2/ .k0k  kk /; (17.69) 2 .2 / to obtain Z

Z dx k

0

0

dx 0k ei.kk kk /.xk xk / D A.2 /2 ı .2/ .k0k  kk /:

(17.70)

Thus, I.xx 0 / D

16 2 X X 0 kk 0 .xx / A k k0 XX  .F  k;kk /.F  k 0 ;k0 /ı .2/ .k0k  kk / k

k0k

kk

16 2 X X 0 kk 0 .xx / A 0 k k Z Z A2  dk dk0k .F  k;kk /.F  k 0 ;k0k /ı .2/ .k0k  kk / k .2 /4 A XX 0 0 D 2 (17.71) kk 0 .xx /H.kk /;

0 D

k

k

17.4 Structure of the Fermi Hole at a Metal Surface

319

where H.kk 0 / D

Z

Z dkk

dk0k .F  k;kk /.F  k 0 ;k0k /ı .2/ .k0k  kk /:

Now, since

X k

and

X

L D

B2 D

k

Z

(17.72)

kF

dk;

(17.73)

0

2

Z

kF

dk;

(17.74)

0

we have that I.xx 0 / D A

4

4

Z

Z

kF

kF

dk 0

dk 0 kk 0 .xx 0 /H.kk 0 /;

(17.75)

0

where the obvious definition of kk 0 .xx 0 / follows from (17.68) and (17.66). We next determine H.kk 0 /. Writing H.kk 0 / in component form and doing the integral over k0k we have Z

Z

0

H.kk / D

dkk Z

D

dk0k .kF2  k  kk2 /.kF2  k 02  kk02 /ı .2/ .k0k  kk /

dkk .kF2  k 2  kk2 /.kF2  k 02  kk2 / Z

kF

D 2

0

.kF2  k 2  kk2 /.kF2  k 02  kk2 /kk dkk :

(17.76)

Now since ( .kF2

k  2

kk2 /

D

1

for kk < .kF2  k 2 /1=2

0

for kk > .kF2  k 2 /1=2 ;

(17.77)

with a similar equation for .kF2  k 02  kk2 / and k replaced by k 0 , we must perform the integral of (17.76) up to the lesser of the values .kF2  k 2 /1=2 and .kF2  k 02 /1=2 . Thus, using the fact that k; k 0  kF , we may write H.kk 0 / as H.kk 0 / D .kF  k/.kF  k 0 /f.kF2  k 02 /Œ.kF2  k 2 /1=2  .kF2  k 02 /1=2  C.kF2  k 2 /Œ.kF2  k 02 /1=2  .kF2  k 2 /1=2 g D .kF  k/.kF  k 0 /Œ.kF2  k 02 /.k 0  k/ C .kF2  k 2 /.k  k 0 /: (17.78)

320

17 Application of Q-DFT to the Metal–Vacuum Interface

Substituting (17.78) into (17.75) we have I.xx 0 / D A

4

3

Z

kF 0 kF

Z

C 4

3

Z

dk.kF2

kF 0 kF

Z

C

0

Z

k / 2

dk.k 0  k/kk 0 .xx 0 / 0

0

0



dk .k  k /kk 0 .xx /

0

dk 0 .kF2  k 02 /

Z

Z dk.kF2

kF

0 kF

Z

0

DA

dk 0 .kF2  k 02 /

k / 2

k0

0 k

dkkk 0 .xx 0 / 0

dk 

kk 0

 .xx / ; 0

(17.79)

0

which, on interchanging k and k 0 in the first term, becomes 4 I.xx / D A 3

0

Z

kF 0

Z dk.kF2

Using the fact that

k

k / 2

dk 0 Œk 0 k .xx 0 / C kk 0 .xx 0 /:

(17.80)

0

k 0 k .xx 0 / D kk 0 .x 0 x/;

(17.81)

we may write the expression for the planar averaged Fermi hole charge density per unit surface area as 2 x .xx / D 3

.x/ 0

Z

kF 0

Z dk.kF2

k / 2

k

dk 0 Œkk 0 .x 0 x/ C kk 0 .xx 0 /: (17.82)

0

An alternate expression for x .xx 0 / may be obtained by first performing the k 0 .k/ integral in the first (second) term of (17.79) and then using (17.81). This equivalent expression is x .xx 0 / D

2 3

.x/

Z

Z

kF

kF

dk 0 .kF2  k 02 /Œkk 0 .xx 0 / C kk 0 .x 0 x/: (17.83)

dk 0

k

With a change of variables to y D kF x, y 0 D kF x 0 , q D k=kF , q 0 D k 0 =kF in (17.82) and (17.83) for the planar average, it is evident that the function x .yy 0 /= .3kF = / is a universal function of the slope parameter yF D kF xF which involves a Fermi-sphere integral from 0 to 1. Thus, for example, (17.82) is F .yy 0 I yF / x .yy 0 / D ; .3kF = / f .yI yF /

(17.84)

where F .yy 0 I yF / D

2 3

Z

Z

1

dq.1  q 2 / 0

0

q

dq 0 Œqq 0 .y 0 yI yF / C qq 0 .yy 0 I yF / (17.85)

17.4 Structure of the Fermi Hole at a Metal Surface

and

Z

321

1

f .yI yF / D

dq.1  q 2 /jq .yI yF /j2 :

(17.86)

0

For a different but equivalent expression for the planar averaged hole see [16]. (The Fermi hole at a surface may also be studied by considering a spherical average about the position of the electron. For such a study, the reader is referred to the original literature [15].)

17.4.2 Structure of the Planar Averaged Fermi Hole x .xx 0 / We begin by plotting the planar averaged Fermi hole x .yy 0 / for an electron deep in the metal bulk, i.e for the homogeneous (H ) electron gas. In Fig. 17.4 we plot F H .yy 0 / xH .yy 0 / D ; .3kF = / f H .y/

(17.87)

where F H .yy 0 / D

1 1 C Œ1  cos.2Y /  2Y sin.2Y /; 2 4Y 8Y4 f H .y/ D 1;

(17.88) (17.89)

ρxH(y-y')/(3kF/ π) (a.u.)

0.3

0.2

0.1

0.0 –6

–4

–2

0 2 (y-y') (a.u.)

4

6

Fig. 17.4 Structure of the planar averaged Fermi hole xH .yy 0 / plotted as the universal function xH .y  y 0 /=.3kF = / for an electron deep in the interior of the metal. This is the structure for the homogeneous (H) electron gas

322

17 Application of Q-DFT to the Metal–Vacuum Interface

and Y D kF .x  x 0 / D y  y 0 . As must be the case, the Fermi hole is symmetrical about the electron position. (In Fig. 17.4 and the figures for other electron positions to follow, the Fermi hole charge is considered positive.) In Figs. 17.5 and 17.6, we plot the planar averaged Fermi hole x .yy 0 / in terms of the universal function x .yy 0 /=.3kF = / relative to the jellium edge for different electron positions at y employing the wave functions of the LP model. The slope parameter used is yF D 3 corresponding to a typical metal surface density. The jellium edge position is at y D 1:187. (Note: y D 2 corresponds to 1 Fermi wave ˚ at rs D 4, F D 6:93 A; ˚ at length F D 2 =kF . At rs D 2, F D 3:46 A; ˚ rs D 6; F D 10:39 A.) In Fig. 17.5a, the electron position at y D 2 in the interior of the metal is similar to that of Fig. 17.4 because there the hole must be symmetric about the electron. In Fig. 17.5b, the electron is at the jellium edge, and in Fig. 17.5c it is at nearly a Fermi wave length outside in the vacuum region at y D 5. Observe that as the electron is removed past the jellium edge, the Fermi hole is left behind in the metal. Further, the hole begins to develop some distinct subsidiary structure which has obviously grown at the expense of the principal part of the hole. (Recall that the total charge of the Fermi hole is 1.) Thus, the principal amplitude of the hole diminishes in size, and the hole becomes wider spreading into the metal interior. In Figs. 17.6a– c, the Fermi hole is plotted for the electron at y D 7, 9, 11. As the electron is moved further away from the surface, the amplitude of the first peak continues to decrease and that of the subsidiary peaks to increase. Furthermore, although the hole remains principally within the jellium edge, its spatial extent in the metal continues to increase. Thus we observe that the Fermi hole charge is highly delocalized, with a width that is dependent upon the position of the electron. The further out the electron in the classically forbidden region, the deeper the hole spreads into the metal. A final fact not evident from Figs. 17.5 and 17.6 is that, irrespective of how far the electron is from the surface in the classically forbidden region, there is always a finite string of charge connecting the electron to the metal. This fact is significant because it is important to understand that the asymptotic electron is part of the N -electron Fermi gas of the metal. The structure of the Fermi hole is a consequence of the electron correlations due to the Pauli exclusion principle. As such the asymptotic electron cannot be thought of as an independent test charge. The limiting behavior and structure of the Fermi hole as the electron is removed to infinity may be understood by the following argument. It is only those electrons with momenta in the range k near the Fermi level that have energy to interact with an electron far outside the surface. The further away the electron, the smaller this range. The width of the Fermi hole, which is of the order .k/1 , thus becomes larger and larger. In the case when the electron is at infinity, it is only those electrons with Fermi momentum perpendicular to the surface that can possible interact with it. Therefore, from the general expression for the Fermi hole, we see that in this limit, x .xx 0 / j kF .x 0 /j2 , which is an oscillatory function inside the metal and spread throughout it. These oscillations of the Fermi hole charge as the electron is removed from within the solid are the Bardeen-Friedel oscillations.

17.4 Structure of the Fermi Hole at a Metal Surface

323

0.3 yF = 3

(a)

ELECTRON AT y = –2 JELLIUM EDGE

0.2

0.1

0.0 0.3

ρx(y-y')/(3kF/π) (a.u.)

ELECTRON AT JELLIUM EDGE

(b)

y =1.187 0.2

0.1

0.0 0.3 (c)

ELECTRON AT y= 5

0.2

0.1

0.0 –8

–6

–4

–2

0 y' (a.u.)

2

4

6

Fig. 17.5 Structure of the planar averaged Fermi hole x .yy 0 / plotted as the universal function x .yy 0 /=.3kF = / for different electron positions at y D 2; 1:187, and 5. The slope parameter of yF D 3 corresponds to typical metal surface densities

324

17 Application of Q-DFT to the Metal–Vacuum Interface

0.3

yF=3

(a)

ELECTRON AT y=7 JELLIUM EDGE 0.2

0.1

0.0

ρx (y-y')3kF/ π) (a.u.)

0.3

7 (b)

ELECTRON AT y=9

0.2

0.1

0.0 0.3

9 (c)

ELECTRON AT y =11

0.2

0.1

0.0 –8

–6

–4

–2

0

2

4

y'(a.u.)

Fig. 17.6 The same as in Fig. 17.5 but for electron positions at y D 7; 9, and 11

11

17.4 Structure of Fermi Hole in Planes Parallel to the Surface

325

To substantiate the above remarks, it is analytically proved [16] that for arbitrary position y of the electron, the planar averaged Fermi hole x .yy 0 / y 02 for y 0  0. This proof is independent of the model employed to represent the effective potential energy at the surface. This then is the asymptotic dependence even in the case when the surface potential energy is obtained fully self-consistently. That there is charge at y 0 D 1 can be seen by considering the center of mass of the Fermi hole. It is evident that the integral hy 0 i D

Z

C1

1

y 0 x .yy 0 /dy 0 lnjy 0 j;

(17.90)

is weakly divergent in the limit y 0 ! 1. If there were no charge at minus infinity, or equivalently if the extent of the Fermi hole were finite, the integral hy 0 i would R C1 converge. Recall that the integral 1 x .yy 0 /dy 0 does converge, and its value is unity. (Also note that for the homogeneous electron gas, hy 0 i D y.) Thus we see that there is a tail of the Fermi hole charge density extending all the way to minus infinity.

17.4.3 Structure of Fermi Hole in Planes Parallel to the Surface We next study the structure of the Fermi hole in the plane parallel to the surface. Substituting for k .r/ from (17.7) into the expression (3.21) for the definition of the Fermi hole x .rr 0 / we obtain x .rr 0 / D

  1 1 2 2X X .kF2  k 2  k2k /.kF2  k 02  k02 k / .r/=2 A2 L 0 0 k;kk k ;kk

e D

i.kk k0k /.x0k x k /

Z

1 2 6 .r/ Z 

0

kF

k? .x/k?0 .x 0 /k .x 0 /k 0 .x/ Z kF dk dk 0 k? .x/k?0 .x 0 /k .x 0 /k 0 .x/ 0

2 .kF k 2 /1=2

0

Z 

2 .kF k 02 /1=2

0

Z

kk dkk kk0 dkk0

2

de

ikk xk0 cos 

2

0 ikk0 xk0 cos  0

d e 0

Now the integral [64] Z

2 k 2 /1=2 .kF

0

Z kk dkk

2 0



0

Z

0

deikk xk cos 

(17.91)

 :

(17.92)

326

17 Application of Q-DFT to the Metal–Vacuum Interface

Z D 2

0

D 2

2 .kF k 2 /1=2

kk dkk J0 .kk xk0 /

.kF2  k 2 /1=2 J1 Œ.kF2  k 2 /1=2 xk0 ; xk0

(17.93)

(17.94)

so that (17.92) becomes ˇZ 2 2 1=2 2 ˇˇ kF ? 0 .kF  k / dk .x/ .x / x .rr / D 4 k k

.r/ ˇ 0 xk0 ˇ2 ˇ  J1 Œ.kF2  k 2 /1=2 xk0 ˇˇ : 0

(17.95)

Here J0 .x/ and J1 .x/ are the zeroth- and first-order Bessel functions, respectively. Changing to the dimensionless variables q D k=kF , q 0 D k 0 =kF and y D kF x, yk0 D kF xk0 , one gets x .yy 0 I y 0k / .=2/ N

ˇZ 2 1=2 36 ˇˇ 1 ? 0 .1  q / D dq .y/ .y / q q n .y/ ˇ 0 jy 0k j ˇ2

ˇ  J1 .1  q 2 /1=2 jy 0k j ˇˇ :

(17.96)

This is the average Fermi hole charge density at .y 0 y0k / for an electron at y. Here n .y/ is the electronic density normalized with respect to the bulk value N D kF3 =3 2 : Z 1 ˇ ˇ2 .y/ n .y/ D D3 dq.1  q 2 /ˇq .y/ˇ : (17.97) N 0 N in planes at In Fig. 17.7 we plot the structure of the Fermi hole x .yy 0 I y 0k /=.=2/ y 0 parallel to the surface as a function of the distance from the axis along which the electron is being removed. The structure is for the orbitals of the LP model for slope parameter yF D 3 for electron positions y D 8, 20, 50 outside the surface. The solid lines correspond to the quantum-mechanical charge distribution. These charge distributions are drawn for various peaks y 0 of the planar averaged Fermi hole x .yy 0 /=.3kF = /. Observe that for electron positions closer to the surface (Fig. 17.7a for y D 8), the radial charge distribution falls off rapidly. As the electron is removed further outside the metal (Fig. 17.7b for y D 20; Fig. 17.7c for y D 35), the planar charge spreads out more radially, and its fall-off rate diminishes. Thus, in the asymptotic limit, the Fermi hole also spreads out radially over the entire crystal. In the sections to follow, we determine the potential energy Wx .x/ of an electron due to the dynamic Fermi hole charge distribution x .xx 0 / described by Figs. 17.5– 17.7. It is of interest to compare this potential energy function with that of the classical image potential energy .1=4x/ of a test charge external to the metal.

17.4 Structure of Fermi Hole in Planes Parallel to the Surface

327

ELECTRON AT y=8

RADIAL EXCHANGE CHARGE DISTRIBUTION rx(y,y,½yII½)/r/2

0.3

(a)

QUANTUM-MECHANICAL DISTRIBUTION CLASSICAL DISTRIBUTION

0.2 AT y’=0.4 0.1

AT y’=3.4

0.0 0.14

ELECTRON AT y=20

0.12

(b)

0.10 0.08 AT y’= 0.8

0.06 0.14 0.02

AT y’= –6.4

0.00 0.08

ELECTRON AT y=35

0.06

(c)

AT y’=1.0

0.04 0.02 0.00

AT y’=–6.2 0

4

8

12

RADIAL DISTANCE ½yII½ (λF/2π)

Fig. 17.7 Structure of the radial distribution of the Fermi hole charge density x .yy 0 I jy 0k j/=.=2/ in planes at y 0 parallel to the surface as a function of the distance from the axis along which the electron is being removed. The solid lines correspond to the quantum-mechanical charge distribution, whereas the dashed lines represent the classical induced charge distribution

For a more fundamental understanding of this difference, we compare the quantummechanical and induced classical image charge distributions. The classical image charge, which is spread over the entire surface, has zero width in the direction perpendicular to it. For an electron at y outside the surface, the classical image charge density at jy0k j in the plane of the surface is given by the expression .jy 0k j/ D

.y 2

0 y 3 ; C jy 0k j2 /3=2

(17.98)

where 0 is the surface charge density at jy 0k j D 0, the axis of electron removal. The quantum-mechanical distribution of charge, on the other hand, is three dimensional,

328

17 Application of Q-DFT to the Metal–Vacuum Interface

and extends into the metal. Consequently, for purposes of comparison we assume the classical charge distribution to have the quantum-mechanical dependence in the direction perpendicular to the surface. In the planes parallel to the surface we assume the same analytical form as derived from classical electrostatics. Thus, in Fig. 17.7 we also plot the quantity xclassical .yy 0 I jy 0k j/

D

x .yy 0 ; jy 0k j D 0/.y  y 0 /3 Œ.y  y 0 /2 C jy 0k j2 3=2

:

(17.99)

It is clear from the figure that the classical charge distribution bears little resemblance to the quantum-mechanical distribution: it is always an overestimate. Thus in order to ensure the satisfaction of the charge conservation sum rule of the Fermi hole, the classical distribution would have to be cut off at some point in the plane. As a consequence of the differences between the shapes of the two charge distributions (and the fact that the classical distribution has a cutoff), one would expect that the corresponding potentials would also be different.

17.5 General Expression for the Pauli Field E x .x/ and Potential Energy Wx .x/ In the original study [20], the expression derived for the Pauli field E x .r/ due to the Fermi hole charge x .rr 0 / D js .rr 0 /j2 =2.r/ involved spatial integrations over all space. Thus, to obtain the asymptotic structure of the field in the vacuum region, the structure of the Fermi hole deep in the metal bulk had to be determined accurately. However, it turns out that as for the other properties, it is possible to obtain [21] an expression for the field in terms solely of momentum space integrals. Furthermore, this expression is general and valid for arbitrary metal. The potential energy Wx .r/, is then obtained as the work done in this field. Substituting for k .r/, s .rr 0 /, and .r/ from (17.7)–(17.9), the Pauli field E x .r/ is Z x  x 0 js .rr 0 /j2 (17.100) E x .r/ D dr 0 jr  r 0 j3 2.x/ Z ZZ 1 dkdk0 0 D dr 4 kk 0 .xx 0 / 2.x/ .2 /4 2     .x  x 0 / i qX k2 k 02  F  ; (17.101)  e  F  jr  r 0 j3 2 2 where kk 0 .xx 0 / D k? .x/k .x 0 /k?0 .x 0 /k 0 .x/;

(17.102)

17.5 General Expression for the Pauli Field Ex .x/ and Potential Energy Wx .x/

329

and q D kk  k0k , X D x k  x 0k . Now the integral Z

dx 0k

@ .x  x 0 / iqX e D jr  r 0 j3 @x

Z

dx 0k

eiqX jr  r 0 j 0

D 2 sgn.x  x 0 /eqjxx j ;

(17.103)

so that 1 4 2.x/

ZZ

dkdk0 .2 /4 2

Z

dx 0 kk 0 .xx 0 /     k2 k 02 0  F  : (17.104)  2 sgn.x  x 0 /eqjxx j  F  2 2

E x .r/ D 

The product of the  function is     k2 k 02  F   F  D .  kk /.0  kk0 /; 2 2

(17.105)

where 2 D kF2  k 2 , 02 D kF2  k 02 . The momentum space integrals of (17.104) may then be written as Z

Z

kF

kF

dk 0

dk

0

Z

0

Z

0

dk0k .kk /.0 kk0 /kk 0 .xx 0 /eqjxx j : (17.106)

dkk

Here there are two possibilities:  > 0 .k < k 0 / or  < 0 .k > k 0 /. Equation (17.106) is however symmetric with respect to an interchange of k and k 0 , so that we assume  > 0 .k < k 0 / and multiply the resulting equation by 2. Thus, we may rewrite (17.106) as Z 2

Z

kF

k0

dk 0

dk 0

0

Z

Z dkk

dk0k .  kk /.0  kk0 / 0

 kk 0 .xx 0 /e qjxx j ;

(17.107)

where it is understood that  > 0 . Substituting (17.107) into (17.104), the field Ex .x/ may be written in dimensionless coordinates normalized to the Fermi momentum as Z 1 Z k0 1 dk 0 dk ? Ex .z/ D 3  .z/k 0 .z/

.z/ 0 .2 / 0 .2 / k Z 1  dz0 k .z0 /k?0 .z0 / sgn.z  z0 / Ikk 0 .zz0 /; 1

(17.108)

330

17 Application of Q-DFT to the Metal–Vacuum Interface

where

Z

Z

0

Ikk 0 .zz / D

0

dk0k eqjzz j .  kk /.0  kk0 /:

dkk

(17.109)

In (17.108) and (17.109), we have used the same notation as before for the dimensionless variables so that now  D .1k 2 /1=2 , 0 D .1k 02 /1=2 , etc., and z D kF x. To determine the integral we change the variables to q D kk  k0k so that

K D

and

1 kk D K C q 2

and

1 .kk C k0k /; 2

1 k0k D K  q: 2

(17.110)

(17.111)

Since the Jacobian of the transformation J.q; K / D @.kk ; k0k /[email protected]; K / D 1, we can rewrite (17.109) as Z Z 1 1 0 Ikk 0 .zz0 / D dq eqjzz j dK .  jK C qj/ .0  jK  qj/; (17.112) 2 2 or equivalently as 0

Z

Ikk 0 .zz / D 2

where

Z F .q/ D

0

dq eqjzz j F .q/;

1 1 dK .  jK C qj/ .  jK  qj/: 2 2

(17.113)

(17.114)

The integral over K has a simple geometrical interpretation. It is the area of the hatched region of Fig. 17.8a corresponding to all points in the K -plane which belong simultaneously to circles of radii  and 0 whose centers are a distance q apart. Since in (17.109) we assumed  > 0 , there are three distinct cases for the K -integral as shown in Figs. 17.8(a, b, c) corresponding to the values of q being   0 < q <  C 0 , q <   0 , and q >  C 0 . For these cases we have from geometrical considerations 8 ˆ ˆ 0 02 <

 F .q/ D ˆ

02 Œ.2  02 /1=2  q ˆ : C S .q/ C S0 .q/

for q   C 0 for q    0

(17.115)

for   0 < q <  C 0 ;

where .2  X2 /1=2  X .2  X2 /1=2 ; X S0 .q/ D S .q/j!0 ;

S .q/ D 2 tan1

(17.116) (17.117)

17.5 General Expression for the Pauli Field Ex .x/ and Potential Energy Wx .x/

331

Fig. 17.8 The physical interpretation of the area of integration in (17.114). The hatched region corresponds to this area

X;0 .q/ D

  2  02 1 q˙ : 2 q

(17.118)

Thus, we may write Ikk 0 .zz0 / of (17.113) as  Z Ikk 0 .zz0 / D 2 02 C0 0

dq qe

0

dq e qjzz j

0

Z C

.2 02 /1=2

 ˚  0 S .q/ C S .q/ :

(17.119)

dk k? .z/k 0 .z/H.kk 0 I z/;

(17.120)

qjzz0 j

Substituting (17.119) into (17.108) we obtain 4 Ex .z/ D 2

 .3kF =2 / n .z/

Z

1

dk 0

0

Z

k0 0

332

17 Application of Q-DFT to the Metal–Vacuum Interface

where H.kk 0 I z/ D 02 Z C Z M.q; z/ D 2

1

1

Z

.2 02 /1=2

dq qM.q; z/ 0 C0

0

˚  dq qM.q; z/ S .q/ C S0 .q/ ; 0

dz0 sgn.z  z0 / eqjzz j k .z0 /k?0 .z0 /;

(17.121)

(17.122)

and where n .z/ is the density normalized to the bulk value N D kF3 =3 2 . The work done Wx .z/ is then Wx .z/ D .3kF =2 /

Z

z 1

Ex .z0 / dz0 : .3kF2 =2 /

(17.123)

For orbitals generated by model effective potentials, the spatial integral M.q; z/ of (17.122) can be determined analytically. Thus, for such orbitals, the field Ex .z/ is then entirely in terms of finite-region momentum–space integrals, and therefore accurately determined for all electron positions throughout space. Since the general structure of the orbitals in the asymptotic metal bulk (to within a phase factor) and vacuum regions is known, the integral M.q; z/ can also be determined analytically in these regions within a fully self-consistent calculation. For such a calculation, the integral M.q; z/ then has to be performed numerically only over a finite region of space about the surface. It is for this reason that a self-consistent calculation with Wx .z/ as the Pauli potential energy can be performed. Finally, the expression for the field Ex .z/ written in terms of the momentum space integrals also allows for the determination of the exact analytical structure of the work Wx .z/ in the vacuum region as derived below.

17.6 Structure of the Pauli Field E x .x/ and Potential Energy Wx .x/ The expression for the Pauli field Ex .z/ derived in the previous section in terms of momentum space integrals (17.120)-(17.122) allows for its easy and accurate determination. This is particularly the case for model potential orbitals that are analytical or semi-analytical. For the orbitals of the FLP model, the solution of the integral M.q; z/ of (17.122) is the following. For z  0,   k sin.k z C ı / kC sin.kC z C ıC / M.q; z/ D 2  2 2 q 2 C k q 2 C kC

17.6 Structure of the Pauli Field Ex .x/ and Potential Energy Wx .x/



kC sin ıC  q cos ıC k sin ı  q cos ı C 2 2 q 2 C k q 2 C kC 2Dk Dk 0 e.qC k C k0 /zb C q C k C k 0 Z zb 0 C2 dz0 eqz fBk Ai.k0 / C Ck Bi.k0 /g

333



e qz

0

fBk 0 Ai.k0 0 / C Ck 0 Bi.k0 0 /g:

(17.124)

For 0  z  zb , M.q; z/ D e

qz



Z C2 0

q cos ı C k sin ı q cos ıC C kC sin ıC  2 2 q 2 C k q 2 C kC z

0

dz0 e qz fBk Ai.k0 / C Ck Bi.k0 /g

 fBk 0 Ai.k0 0 / C Ck 0 Bi.k0 0 /g  2Dk Dk 0 e.qC k C k0 /zb eqz q C k C k 0 Z zb 0 C2 dz0 eqz fBk Ai.k0 / C Ck Bi.k0 /g z  fBk 0 Ai.k0 0 / C Ck 0 Bi.k0 0 /g :

(17.125)

For z  zb , M.q; z/ D e

qz



Z

q cos ı C k sin ı q cos ıC C kC sin ıC  2 2 q 2 C k q 2 C kC zb

C2 0

0

dz0 eqz fBk Ai.k0 / C Ck Bi.k0 /g

(17.126) fBk 0 Ai.k0 0 / C Ck 0 Bi.k0 0 /g 4Dk Dk 0 2Dk Dk 0 e.q k  k0 /zb  2 e. k C k0 /z :  q  k   k 0 q  .k C k 0 /2 In the above expression, k D k 0 k and ı D ı.k 0 / ı.k/. In Fig. 17.9, we plot the Pauli field Ex .z/ for rs D 2:0 and 6:0 for the orbitals of the FLP model from 2 Fermi wavelengths inside to 4 Fermi wavelengths outside the surface. The barrier height parameter ˇ (or equivalently yb ) is chosen from

334

17 Application of Q-DFT to the Metal–Vacuum Interface 0.2

Fig. 17.9 Structure of the Pauli field Ex .z/ at a metal surface for metals of Wigner– Seitz radius rs D 2:0 and 6:0

JELLIUM EDGE

rs = 2.0

2

ELECTRIC FIELD Ex (z)/(3kF /2p)

0.1

0.0 0.2 rs = 6.0 0.1

0.0

–2

0

2

4

z (IN FERMI WAVELENGTHS)

Table 17.1. Observe how the field is localized to the surface region, decaying into the classically forbidden region, and exhibiting the Bardeen–Friedel [1, 2] oscillations within the metal. The field at the surface is greater for rs D 6:0 than for rs D 2:0. The reason for this is that for electron positions near the jellium edge, the Fermi hole for low density metals is more localized in that region than it is for high density metals. The amplitude of the Bardeen–Friedel oscillations of the field Ex .z/ within the metal are also greater as expected for the lower density metal. In Fig. 17.10 we plot the work done Wx .z/ of (17.123) in the Pauli field Ex .z/ for rs D 2:0 and 6:0. In the classically forbidden region, the potential energy Wx .z/ approaches the exact asymptotic structure derived in the following section. In the interior of the metal, the Bardeen–Friedel oscillations, once again, are more pronounced for the low density metal. For rs D 2:0 the limiting value of Wx .1/=.3kF =2 / D 0:716, and for rs D 6:0, it is 0:823 (see Table 17.2.) The fact that the limiting value of Wx .1/=.3kF =2 / is more negative for lower density metals is a reflection of the greater Pauli field at the surface. The limiting value in the metal bulk of the Kohn–Sham theory “exchange” potential energy vx .1/=.3kF =2 / D 2=3. As explained in QDFT, the “exchange” potential energy vx .r/ is comprised of a Pauli component Wx .r/ and the .1/ lowest order Correlation-Kinetic component Wtc .r/. As such the limiting value .1/ of Wtc .1/=.3kF =2 / may be determined. These values are also given in Table 17.2. Observe from Table 17.2 that as the density becomes more slowly

17.7 Analytical Structure of the Pauli Potential Energy Wx .x/ Fig. 17.10 Structure of the Pauli potential energy Wx .z/ at a metal surface for metals of Wigner–Seitz radius rs D 2:0 and 6:0

335

0.0 JELLIUM EDGE POTENTIAL Wx (z)/(3kF/2p)

–0.3

rs=2.0

–0.6 –0.9 0.0

(At–¥)=–0.716

rs=6.0

–0.3 –0.6 –0.9

(At–¥)=–0.823

–2

0 2 z(IN FERMI WAVELENGTHS)

4

Table 17.2 The metal-bulk limiting values of the Pauli Wx .1/ and first-order Correlation.1/ Kinetic Wtc .1/ potential energies as a function of the Wigner–Seitz radius rs Wigner–Seitz Radius rs (a.u.)

Wx .1/=.3kF =2 /

Wtc1 .1/=.3kF =2 /

2.0 4.0 6.0

0.716 0.788 0.823

0.049 0.121 0.156

varying, the Correlation-Kinetic contribution becomes smaller. For very slowly varying densities .rs 1I ˇ ! 1/ the Correlation-Kinetic contribution becomes negligible. This is consistent with the fact that for slowly varying densities and within the local density approximation, the Kohn–Sham theory “exchange” potential energy vx .r/ and the work done Wx .r/ are equivalent (see QDFT). The exact .1/ asymptotic structure of Wtc .z/ in the classically forbidden region is derived below. For the structure of this potential energy at and about the surface, the reader is .1/ referred to [26]. (In [26], the structure of Wtc .r/ for atoms is also given.)

17.7 Analytical Structure of the Pauli Potential Energy Wx .x/ in the Classically Forbidden Region In this section we derive the analytical structure of the work Wx .x/ in the asymptotic classically forbidden vacuum region. For ease of understanding, the derivation is given in terms of the semi-analytical orbitals of the FLP-jellium model of a metal

336

17 Application of Q-DFT to the Metal–Vacuum Interface

surface described in Sect. 17.2.1. This analytical result is equally valid for fully self-consistently determined orbitals in the jellium model as shown below. To determine the asymptotic structure of Wx .x/, we employ the general expressions (17.120)–(17.123) derived for this potential energy. For the FLP orbitals, the solution of the integral M.q; z/ of (17.122) is given in (17.124)–(17.126). In the asymptotic large z region, the effective value of q 1=z due to the e qz factor in M.q; z/. Furthermore, since k k 0 for large z, the effective value of k D .k 0  k/ 1=z due to the e  k z factor. Expanding the expression for M.q; z/ for z  zb (see 17.126) in q we obtain for the asymptotic region   k q M.q; z/ eqz 2 cos ı C sin ı   2 2 z!1 q C k q 2 C k  sin ıC q cos ıC 2Dk Dk 0 . k C k0 /zb C eqz   C e 2 kC k C k 0 kC  Z zb 0 0 0 0 0 0 0 C2 dz fBk Ai.k / C Ck Bi.k /gfBk Ai.k 0 / C Ck Bi.k 0 /g 0

4Dk Dk 0 . k C k0 /z  e : k C k 0

(17.127)

Here k D k 0 k and ı D ı.k 0 / ı.k/. Next consider the contribution of M.q; z/ to the first integral of H.kk 0 I z/ of (17.121). The last term of (17.127) is exponentially small in the asymptotic vacuum region and does not contribute. The contribution of the second set of terms is 2

02 1=2

1  Œ1 C .2  02 /1=2 z e.  / z z2     Z zb ˚ ˚ sin ıC 2Dk Dk 0 . k C k0 /zb 1   C e C2 dz0 g g C 0 3 : kC k C k 0 z 0 (17.128) 1=2 z D .k z/1=2 z1=2 1 for large z since k 1=z. Now .2  02 /1=2 z k Thus, the contribution of (17.128) to the first integral of H.kk 0 I z/ is of 0.1=z2 /. The contribution of the first term of (17.127) to the first integral of H.kk 0 I z/ with qz D u and a D k z is

    Z 1 Z 1 u2 eu ue u 1 1 cos ı C0 2 : du 2 C a sin ı du 2 2 2 z u Ca u Ca z 0 0

(17.129)

We next note that the second term of H.kk 0 I z/ of (17.121) does not contribute to the leading order. This is because the integral is concentrated about its lower limit .0 / k .1=z/, because as noted previously q 1=z. In the limit q ! 0 , X ! , X0 ! 0 so that S;0 ! 0. Thus, H.kk 0 I z/ is given by (17.129). We R1 next consider the integral over k in (17.120) and rewrite it as .1=z/ 0 da. Again, since for large z, k k 0 , we have k z D k 0 z C ca where c D k 0 =k 0 , so that

17.7 Analytical Structure of the Pauli Potential Energy Wx .x/

337

k .z/ k 0 .z/ exp.ca/. Substituting this k .z/ into (17.120) and using the fact that cos ı 1 and sin ı 0 for k; k 0 1, the potential Wx .z/ of (17.123) is then  Z 1  Z 4 z dz0 Wx .z/ 0 02 2 0  3 dk .1  k / .z / 0 k .3kF =2 / 3 1 z02 n .z0 / 0    Z 1 Z 1 u2 e.caCu/ 1 C 0 : da du 2  u C a2 z2 0 0

(17.130)

The term in the first square parenthesis is the density normalized to the bulk value and cancels the corresponding term in the denominator. Finally, on solving the R1 double integral in the second square parenthesis, and noting that the integral 0 d k 0 p is concentrated near k 0 ! 1 for z ! 1 so that c ! 1= ˇ 2  1, we obtain the asymptotic structure of Wx .x/ to be ˛W .ˇ/ Wx .x/ ; D x!1 x

(17.131)

where the coefficient     .ˇ 2  1/ln.ˇ 2  1/ ˇ2  2 ˇ2  1 2 1 ˛W .ˇ/ D C (17.132) p ˇ2 ˇ2 ˇ2

ˇ2  1 with ˇ 2 D W=F . To prove that the asymptotic structure of Wx .x/ of (17.131) is valid for the selfconsistently determined effective potential, we divide the z axis into three parts: a metal bulk region, a surface region with a finite effective width, and the vacuum region. The critical point to note is that the asymptotic structure of the potential energy Wx .x/ in the vacuum region arises from orbitals deep in the metal bulk. In a self-consistent calculation the orbitals in this region are again of the form k .z/ D sinŒkx C ı.k/, so that the asymptotic structure of these potentials is 2 governed by the term e qz q cos ı =.q 2 C k / see (17.127), and leads to the corresponding coefficient and a .1=x/ decay. The contributions from the surface and vacuum regions is of 0.1=x 2 / as in the model calculation discussed above. (Note that the asymptotic structure of the Slater function VxS .x/ (see QDFT) and the Kohn–Sham “exchange” potential vx .x/ to be discussed in Sect. 17.8, also arises from the orbitals in the deep metal bulk.) In order to demonstrate the correctness of the derivation for the asymptotic structure of Wx .r/ as given by (17.131), we plot in Fig. 17.11 for Li metal (rs D 3:24) the electric field Ex .z/ and potential Wx .z/ outside the metal as determined by the general ‘EXACT’ expression (17.120-17.123) for these properties derived, as well as the quantities ˛W .ˇ/=z2 and ˛W .ˇ/=z. The calculation for the ‘EXACT’ results are performed for the orbitals of the finite-linear-potential model for which the relationship between rs and ˇ is determined via energy minimization in the local density approximation (see Sect. 17.3). The same value of ˇ is then employed in the

338

17 Application of Q-DFT to the Metal–Vacuum Interface 0.020 Li(rs = 3.24;β = 1.246)

2

E x (z)/(3k F/2π)

0.015

0.010

αW(β) z2

0.005 ‘EXACT’ 0.00

Wx (z)/(3kF/2π)

–0.03

‘EXACT’

–0.06 –

αW(β) z

–0.09

–0.12 1

2

3

4

z (IN FERMI WAVELENGTHS)

Fig. 17.11 Comparison of the ‘EXACT’ field Ex .z/ and potential energy Wx .z/ (as determined by the general expressions (17.120)–(17.123)) with the analytical expressions ˛W .ˇ/=z2 and ˛W .ˇ/=z, respectively, for asymptotic positions of the electron in the vacuum region for Li metal

analytical expressions. It is evident for both the field and potential energy that the analytical and ‘EXACT’ results merge by about four Fermi wavelengths from the surface.

17.8 Analytical Structure of the Lowest Order .1/ Correlation-Kinetic Potential Energy Wtc .x/ in the Classically Forbidden Region For atoms, the asymptotic structure of the effective potential energy vs .r/ in the classically forbidden region is solely due to Pauli correlations as obtained by the structure of Wx .r/. Neither Coulomb correlations nor Correlation-Kinetic effects

17.8 Analytical Structure of the Lowest Order Correlation-Kinetic Potential

339

contribute to this structure. (See Chap. 7). However, for the metal surface case, all three types of electron correlations contribute to the asymptotic structure of vs .r/ in the vacuum region. In this section we derive the analytical structure of the lowest.1/ order Correlation-Kinetic potential energy Wtc .r/ in the vacuum region far from the metal surface. We obtain the asymptotic structure of Wt.1/ .r/ via its relationship to the c Kohn–Sham “exchange” potential vx .r/ (see Chap. 5 of QDFT). The potential energy vx .r/ is the sum of the Pauli Wx .r/ and lowest-order Correlation-Kinetic Wt.1/ .r/ components: c .r/: (17.133) vx .r/ D Wx .r/  Wt.1/ c .r/ Determination of the asymptotic structure of vx .r/ then leads to that of Wt.1/ c since the asymptotic structure of Wx .r/ is now known. The Kohn–Sham (KS) exchange energy functional ExKS Œ of (4.36) may be expressed in terms of the Slater function VxS .r/ as ExKS Œ D

1 2

where

Z

Z VxS .r/ D

dr.r/VxS .r/;

(17.134)

x .rr 0 / 0 dr ; jr  r 0 j

(17.135)

and x .rr 0 / is the S system Fermi hole charge. Thus, the KS “exchange” potential energy of (4.34) is vx .r/ D

1 ıExKS Œ 1 D VxS .r/ C ı.r/ 2 2

Z

dr 0 .r 0 /

ıVxS .r 0 / : ı.r/

(17.136)

Since the leading term of the potential energy vx .r/ is the Slater function VxS .r/, we first determine the asymptotic structure of the latter. (It turns out that there is no need to calculate the second term of (17.136)).

17.8.1 Analytic Asymptotic Structure of the Slater Function VxS .x/ Following a calculational procedure and arguments entirely similar to that of the determination of the asymptotic structure of Wx .r/ (Sect. 17.7), the structure of the Slater function VxS .r/ is derived as [23–25] ˛S .ˇ/ VxS .x/ ; D x!1 x

(17.137)

340

17 Application of Q-DFT to the Metal–Vacuum Interface

where ˛S .ˇ/ D

  ln.ˇ 2  1/ ˇ2  1 ; 1  p ˇ2

ˇ2  1

(17.138)

where ˇ 2 D W=F . Once again, it can be rigorously proved that the expression (17.137) is equally valid for fully self-consistently determined orbitals. The reason for this, once more, is because the structure in the vacuum region is governed and arises from the orbitals deep in the metal interior. (As a point of information, the asymptotic structure of the Slater function VxS .r/ was originally determined [65] numerically from (17.135) by employing the Fermi hole obtained for the orbitals of the step-potential model.)

17.8.2 Analytical Asymptotic Structure of the Kohn–Sham “Exchange” Potential Energy vx .r/ We next determine the asymptotic structure of the Kohn–Sham “exchange” potential energy vx .r/. According to Sham [66], the integral equation relating vx .r/ to the “exchange” part of the self-energy †x .rr 0 / in the asymptotic classically forbidden region is 1 vx .r/ D 2‰k .r/ C

Z

1 ? 2‰k .r/

dr 0 †x .rr 0 I F /‰k .r 0 / Z

dr 0 ‰k? .r 0 /†x .r 0 rI F /;

(17.139)

P where the electron is at the Fermi level F . On substituting x .rr 0 / D s .r 0 r/= 2jr  r 0 j, the resulting expression turns out to be the orbital-dependent “exchange” function vx;k .r/ of Hartree–Fock theory (see QDFT): Z vx;k .r/ D where x;k .rr 0 / D 2

X

x;k .rr 0 / 0 dr ; jr  r 0 j

‰k?0 .r/‰k0 .r 0 /‰k .r 0 /=‰k .r/;

(17.140)

(17.141)

k0

is the orbital-dependent Fermi hole. (That asymptotically vs .r/ is vx;k .r/ for electrons at the Fermi level k D kF may also be derived via the optimized Potential Method (see QDFT)). Multiplying and dividing (17.140) by ‰k? .r/, we obtain

17.8 Analytical Structure of the Lowest Order Correlation-Kinetic Potential

Z vx;k .r/ D

dr 0 X ‰k? .r/‰k?0 .r 0 /‰k0 .r/‰k .r 0 / : jr  r 0 j 0 ‰k? .r/‰k .r/

341

(17.142)

k

Now the orbitals are of the form (see 17.65-17.66) ‰k .r/ D ‰kk .x k /ˆk .x/, p p ‰kk .x k / D .1= A/ expŒi kk  x k , ˆk .x/ D . 2=L/ k .x/. Since it is only the component of the electron momentum perpendicular to the surface that contributes to the asymptotic structure, we average the function vx;k .r/, with respect to the component of the wave function in the plane of the surface. This leads to the function vx;k .x/ which is dependent only on the component k of the electron momentum. Thus P ? kk ‰kk .x k /‰kk .x k /vx;k .r/ P vx;k .r/ D  (17.143) ? kk ‰kk .x k /‰kk .x k / Z dr 0 1 ? ˆk .x/ˆk .x/ jr  r 0 j   P P  ˆ?k .x/ˆk .x 0 / kk ‰k?k .x k /‰kk .x 0k / k0 ‰k?0 .r 0 /‰k0 .r/ P : (17.144) D ? kk ‰kk .x k /‰kk .x k / On solving the momentum space integrals analytically to the degree possible, and rewriting in terms of the variables z D kF x, z0 D kF x 0 , z0k D kF x 0k , R D jz0k j, q D k=kF , q 0 D k 0 =kF ,  D .1  q 2 /1=2 , 0 D .1  q 02 /1=2 , the above expression reduces to Z 1 Z 1 8 J1 .R/ vxq .z/ D 2 dz0 q .z0 / dR p .3kF =2 / 3 q .z/ 1 R .z  z0 /2 C R2 0 Z 1  dq 0 q?0 .z0 /q 0 .z/0 J1 .0 R/; (17.145) 0

where J1 .z/ is the first-order Bessel Function. At the Fermi level q D 1, so that (17.145) becomes Z 1 Z 1 4 vx1 .z/ 0 0 D dz 1 .z / dq 0 q?0 .z0 /q 0 .z/0 .3kF =2 / 31 .z/ 1 0 Z 1 J1 .0 R/  dR p ; (17.146) .z  z0 /2 C R2 0 where we have employed li m!0 J1 .R/ D R=2. Finally, since the R integral can be done analytically, we have the orbit-dependent function corresponding to the Fermi level electron with momentum perpendicular to the surface to be

342

17 Application of Q-DFT to the Metal–Vacuum Interface

Z 1 4 vx1 .z/ dz0 D 1 .z0 / .3kF =2 / 31 .z/ 1 jz  z0 j Z 1 0 dqq? .z0 /q .z/.1  ejzz j /:

(17.147)

0

(Originally [20], the asymptotic structure of vx1 .z/ was determined numerically employing the orbitals of the FLP model.) However, noting that 0

1  ejzz j D jz  z0 j

Z



0

dpepjzz j ;

(17.148)

0

the expression for vx1 .z/ reduces to Z

2 vx1 .z/ D .3kF =2 / 31 .z/ Z

where J.pz/ D 2

Z

1



dq q .z/ 0 1

1

dp J.pz/;

(17.149)

0

0

dz0 e pjzz j q? .z0 /1 .z0 /:

(17.150)

The integral J.pz/ is similar to that of M.qz/ of (17.122), and consequently, on employing the orbitals of the FLP model, has the same leading order term pe pz J.pz/ 2 cos ı ; 2 z!1 p C q

(17.151)

where q D 1  q and ı D ı.1/  ı.q/. Since z .q z/1=2 z1=2 1, and cos ı ! 1 as q ! 1, the integral over p in (17.149) for large z is Z

1

du 0

ueu ; u2 C a 2

(17.152)

where a D q z. In this case for large z, q .z/ q 0 D1 .z/eba with b D 1=q 0 . R q 0 D1 R1 Once again 0 dq D .1=z/ 0 da, so that vx1 .z/ 

1 3kF 2  2 31 .z/ z

Z

Z

1

1

da 0

du 0

ue bau u2 C a 2

(17.153)

or equivalently vx1 .x/ D 

1 VxS .x/ ˛s .ˇ/=2 D : x 2 x!1

(17.154)

Thus, the asymptotic structure of the KS “exchange” potential vx .r/ in the classically forbidden vacuum region is

17.8 Analytical Structure of the Lowest Order Correlation-Kinetic Potential

˛KS;x .ˇ/ vx .x/ D x!1 x where ˛KS;x .ˇ/ D

˛S .ˇ/ 2

343

(17.155)

(17.156)

with ˛S .ˇ/ given by (17.138). The above derivation also shows that for the semi-infinite metal surface system for which the energy spectrum is continuous, to leading order, the orbital-dependentfunctions vx;k .r/ for electrons within a shell of thickness .1=z/ about the Fermi level are the same. Thus, their average taken over this shell, which is the KS “exchange” potential energy, is equivalent in leading order to the orbital-dependent-function for electrons at the Fermi level. In the case of discrete spectrum systems such as atoms, it is of course more readily apparent that the asymptotic structure is due to the highest occupied orbital electrons. As was the case for the Pauli potential energy Wx .r/ and the Slater function VxS .r/, the analytical result for vx .r/ of (17.155) is equally valid for fully selfconsistently determined orbitals. The result may also be derived [27, 28] in an entirely independent manner. Observe that whereas the asymptotic structure of Wx .r/ and vx .r/ in the classically forbidden region of atoms is the same, this is not the case for the metal surface problem.

17.8.3 Analytical Asymptotic Structure of the Lowest-Order Correlation-Kinetic Potential Energy Wt1c .r/ With the asymptotic structure of Wx .r/ and vx .r/ known, it follows from (17.131), (17.133), and (17.155) that the asymptotic structure of the lowest-order CorrelationKinetic Potential energy Wt1c .r/ in the vacuum region is ˛t1 .ˇ/ Wt1c .x/ D c x!1 x

(17.157)

where ˛t1c .ˇ/ D 

 ˇ2  2 ˇ2  1 4 12  p 2 2ˇ 2ˇ 2

ˇ2  1   2 4  3ˇ 2  1C ln.ˇ  1/ : 4ˇ 2

(17.158)

Thus, Correlation-Kinetic effects contribute to the asymptotic structure of the effective potential energy vs .r/ in the classically forbidden region.

344

17 Application of Q-DFT to the Metal–Vacuum Interface

The existence of the lowest-order Correlation-Kinetic Wt1c .r/ contribution may also be understood by writing the Pauli potential energy Wx .r/ as 1 Wx .r/ D VxS .r/ C 2

Z

r

0

d`  1

Z

dr 00

.r 00  r 0 /x .r 0 r 00 / : jr 0  r 00 j

(17.159)

Note that the leading order term of vx .r/ (see (17.136)) and of Wx .r/ of (17.159) is the same. Furthermore, the asymptotic structure of vx .r/ is that of this leading term (see (17.154)). The second term of vx .r/ of (17.136) therefore does not contribute to its asymptotic structure. On the other hand, it is evident from the asymptotic structure of Wx .r/ of (17.131) that the second term of (17.159) does contribute to this structure. It is the difference between the potential energies Wx .r/ and vx .r/ that corresponds to the lowest-order Correlation-Kinetic Wt1c .r/ potential energy.

17.9 Analytical Structure of the Coulomb Wc .x/ and Second-and Higher-Order Correlation Kinetic Wtc2 .x/, Wtc3 .x/ : : : Potential Energies in the Classically Forbidden Region Having derived the asymptotic structure of the Pauli Wx .r/ and lowest-order Correlation-Kinetic Wt1c .r/ potential energies, we next determine the asymptotic structure of the sum of the Coulomb Wc .x/ and second-and higher-order Correlation-Kinetic Wt2c .x/, Wt3c .x/ : : : potential energies in the classically forbidden region. As derived in Chap. 5 of QDFT, the sum corresponds to the KS “correlation” potential energy vc .r/:  2 3 (17.160) vc .r/ D Wc .r/ C Wtc .r/ C Wtc .r/ C    : Recall from Sect. 4.2 that the S system effective potential energy vs .r/ is written within KS theory as vs .r/ D v.r/ C vH .r/ C vxc .r/; vxc .r/ D vx .r/ C vc .r/, with vxc .r/, vx .r/, and vc .r/ the “exchange-correlation,” “exchange,” and “correlation” potential energies. To obtain the asymptotic structure of vc .r/ we first derive [27, 28] new expressions for vxc .r/, vx .r/, and vc .r/ in the classically forbidden region. (These expressions differ from (7.74) and (17.139) derived by Sham [66].) To employ these expressions, we also derive the asymptotic structure of the model fermion orbitals, density, and Dirac density matrix. Employing the new expression for vx .r/, the previously derived result of (17.155) for vx .r/ as x ! 1 is reaffirmed [28].

17.9 Analytical Structure of the Coulomb Wc .x/ and Second-and Higher-Order Correlation

345

17.9.1 New Expression for Kohn–Sham “Exchange-Correlation” vxc .r/ Potential Energy in Classically Forbidden Region The S system Green’s function is Gs .r 1 r 2 !/ D

X

 i .r 1 /i? .r 2 /

i

 .F  i / .i  F / C ; !  i C i !  i  i

(17.161)

where the i .r/ and i are the model system fermion orbitals and eigenvalues, and F the highest occupied eigenvalue. Here we define  D 0C. Gs .r 1 r 2 !/ satisfies the following equation of motion   1 ! C r12  vs .r 1 / Gs .r 1 r 2 !/ D ı.r 1  r 2 /: 2

(17.162)

The Dyson equation relating the Green’s function for the real system to that for the S system is [67] G.r 1 r 2 !/ D Gs .r 1 r 2 !/ Z Q 0 r 00 !/G.r 00 r 2 !/dr 0 dr 00 ; C Gs .r 1 r 0 !/†.r

(17.163)

0 Q !/ D †xc .rr 0 !/  ı.r  r 0 /vxc .r/; †.rr

(17.164)

where and the “exchange correlation” part of the self-energy †xc .r 1 r 2 !/ is defined as †xc .r 1 r 2 !/ D †.r 1 r 2 !/  vH .r 1 /ı.r 1  r 2 /:

(17.165)

Here †.r 1 r 2 !/ is the standard proper self-energy of an electron. Equations (17.162) and (17.163) lead to

D.r 1 r 2 / G.r 1 r 2 !/  Gs .r 1 r 2 !/ Z Z Q 0 r 2 !/dr 0 ; (17.166) Q 1 r 0 !/G.r 0 r 2 !/dr 0 C G.r 1 r 0 !/†.r D  †.r where D.r 1 r 2 / D .1=2/r12 C vs .r 1 / C .1=2/r22  vs .r 2 /. We investigate the asymptotic structure of the KS “exchange correlation” potential energy vxc .r/ at a metal surface by studying the corresponding structure of the one-particle density matrix .r 1 r 2 / for the real system and the Dirac density matrix s .r 1 r 2 / for the S system. The density matrix is related to the Green’s function via 1 .r 1 r 2 / D

i

Z

1 1

G.r 1 r 2 !/ei! d!:

(17.167)

346

17 Application of Q-DFT to the Metal–Vacuum Interface

A similar relation holds between s .r 1 r 2 / and Gs .r 1 r 2 !/, which yields s .r 1 r 2 / D 2

X

l? .r 1 /l .r 2 /:

(17.168)

l<F

By performing an integration on (17.166) over the frequency one obtains

D.r 1 r 2 / .r 1 r 2 /  s .r 1 r 2 / Z Z 1 d!e i !

G.r 1 r 0 !/†xc .r 0 r 2 !/ D

i



†xc .r 1 r 0 !/G.r 0 r 2 !/ dr 0 C vxc .r 1 /  vxc .r 2 / .r 1 r 2 /: (17.169) Because the leading term of .r 1 r 2 / and s .r 1 r 2 / are the same in the classically forbidden region of a metal surface, .r 1 r 2 /  s .r 1 r 2 / is much smaller than s .r 1 r 2 /. Therefore



D.r 1 r 2 / .r 1 r 2 /  s .r 1 r 2 /  vxc .r 1 /  vxc .r 2 / s .r 1 r 2 /:

(17.170)

Thus the term on the left side of (17.169) can be dropped. For the same reason, G.r 1 r 2 !/  Gs .r 1 r 2 !/  G.r 1 r 2 !/ for r 1 in the classically forbidden region. Thus, one obtains xc .r 1 r 2 / vxc .r 1 /  vxc .r 2 / D (17.171) s .r 1 r 2 / in the asymptotic region, where xc .r 1 r 2 / Dx .r 1 r 2 /Cc .r 1 r2 / and x;c .r 1 r2 / D .2/ .1/ x;c .r 1 r 2 /  x;c .r 1 r 2 / with .1/ .r 1 r 2 / x;c

1 D

i

Z

Z e

i!

d!

Gs .r 1 r 0 !/†x;c .r 0 r 2 !/dr 0 ;

(17.172)

†x;c .r 1 r 0 !/Gs .r 0 r 2 !/dr 0 :

(17.173)

x;c .r 1 r 2 / : s .r 1 r 2 /

(17.174)

and .2/ .r 1 r 2 / D x;c

1

i

Z

Z ei! d!

Evidently, from 17.171, one also has vx;c .r 1 /  vx;c .r 2 / D

Equations (17.171) and (17.174) are the new asymptotic expressions for vxc .r/, vx .r/, and vc .r/. These expressions are, to the leading order, exact in the classically forbidden region.

17.9 Analytical Structure of the Coulomb Wc .x/ and Second-and Higher-Order Correlation

347

17.9.2 Analytical Asymptotic Structure of the Orbital k .x/, Dirac Density Matrix s .xx 0 /, and Density .x/ It is commonly accepted that the asymptotic structure of vxc .r/ in the classically forbidden region is of 0.1=x/ and that it is ˛KS;xc =x. With this fact, the differential equation for the component k .x/ of the S system orbital written as k .r/ of (17.7) in the direction of the inhomogeneity far from the metal surface is  

  1 @2 ˛KS;xc 1 k .x/ D k 2 k .x/; C W  2 @x 2 x 2

(17.175)

where W is the barrier height. p In this asymptotic region we write k .x/ as k .x/ D Pk .x/e  x , where  D 2W  k 2 , and Pk .x/ is a polynomial function that satisfies @2 @ 2˛KS;xc Pk .x/ D 0: Pk .x/  2 Pk .x/ C 2 @x @x x

(17.176)

From (17.176) it can be shown that the leading term of Pk .x/ is x ˛KS ;xc= . Therefore, asymptotically the orbital k .x/ decays as k .x/ x ˛KS;xc = e x :

(17.177)

This is the exact asymptotic structure of the one-particle orbital to the leading order. Note that we have dropped the constant coefficient for simplicity, and we will do the same to the Dirac density matrix below. Since we are concerned with the case of large x and x 0 , we put x k D x 0k . The Dirac density matrix is then s .xx 0 / D

1 2 2

Z

kF 0

dk.kF2  k 2 /k? .x/k .x 0 /:

(17.178)

At large x and x 0 , by use of (17.177), one has s .xx 0 /

Z

kF 0

0

.kF2  k 2 /.xx 0 /˛KS;xc = e .xCx / dk:

(17.179)

Equation (17.179) can be rewritten as s .xx 0 / .kF2  2W /

Z

kF

0

.xx 0 /˛KS;xc = e  .xCx / dk

0

  Z kF @ @ 0 ˛KS;xc =  .xC Q xQ 0 / .xx / e dk : C @xQ 0 @xQ 0 xDx; Q xQ 0 Dx 0

(17.180)

The leading contribution to s .xx 0 / arises from the region of k ! kF . The first term on the right side of the above expression can be evaluated as

348

17 Application of Q-DFT to the Metal–Vacuum Interface

 .ˇ 2  1/3=2 kF2 .xx 0 /˛KS;xc =.kF

p

ˇ 2 1/



p 1 0 2 ekF ˇ 1.xCx / (17.181) 0 xCx

at large x and x 0 . The second term yields p p 0 2 2 .ˇ 2  1/kF .xx 0 /˛KS;xc =.kF ˇ 1/ ekF ˇ 1.xCx / p   2 kF ˇ 2  1 :  C x C x0 .x C x 0 /2

(17.182)

Combination of these two terms yields the asymptotic structure of the Dirac density matrix: s .xx 0 / .xx 0 /˛KS ;xc=.kF

p

ˇ 2 1/

p 1 kF ˇ 2 1.xCx 0 / e : .x C x 0 /2



(17.183)

The density .x/ is the diagonal component of s .xx 0 /, which therefore has the asymptotic structure as .x/ x 2˛KS;xc =.kF

p

ˇ 2 1/2 2kF

e

p

ˇ 2 1x

:

(17.184)

(We note that the results of (17.183) and (17.184) for the asymptotic structure of the density matrix and density differ from those obtained by Almbladh and von Barth [68], even if one assumes ˛KS;xc D 1=4.)

17.9.3 Analytical Asymptotic Structure of the Kohn–Sham “Correlation” Potential Energy vc .x/ We next determine the asymptotic structure of the KS “correlation” potential energy vc .x/ in the classically forbidden region via the new expression of (17.174). We make use of the plasmon-pole approximation for the correlation part of the selfenergy. It is based on the assumption that the response of the electron system at a metal surface can be described by a frequency-dependent dielectric constant which has a zero point at !s , where !s is the surface plasmon frequency. The self-energy under this assumption is †c .rr 0 I !/ D

!s G.rr 0 I !  !s /; 2jr  r 0i j

(17.185)

where x 0i k D x 0k and xi0 D x 0 . Substituting †c .rr 0 I !/ into (17.172), one obtains c.1/ .r 1 r 2 / D

!s 2 i 

jr 0

Z

Z ei ! d!

Gs .r 1 r 0 I !/

1 Gs .r 0 r 2 I !  !s /dr 0 d!:  r 2i j

(17.186)

17.9 Analytical Structure of the Coulomb Wc .x/ and Second-and Higher-Order Correlation

349

In the above equation, we have replaced G.r 0 r 2 I ! !s / by Gs .r 0 r 2 I ! !s / which is exact to the leading order at large x1 and x2 . By using (17.161) for Gs .r 1 r 2 I !/, one carries through the frequency integration and obtains Z c.1/ .r 1 r 2 /

D !s

dr

XX 1 1 jr 0  r 2i j !s C l  j j

l

 Œ.F  l /.j  F /  .F  j /.l  F /  l .r 1 /l? .r 0 /j? .r 2 /j .r 0 /:

(17.187)

By using the relation, .F  l /.j  F /  .F  j /.l  F / D .F  l /  .F  j /;

(17.188)

one has  X X

c.1/ .r 1 r 2 / D !s Z 

l


j

X X  l

j
dr 0 l .r 1 /l? .r 0 /j? .r 2 /j .r 0 /

1 1 : !s C l  j jr 0  r 2i j

(17.189)

Operating by D.r 1 r 2 / on (17.189) leads to D.r 1 r 2 /c.1/ .r 1 ; r 2 / D !s Z

 X X l


j

X X  l

j
1 C !s c.1/ .r 1 r 2 / jr 0  r 2i j   1 C higher-order terms with factor r 1 ; (17.190) jr 1  r 0i j



dr 0 l .r 1 /l? .r 0 /j? .r 2 /j .r 0 /

where we have used the S system equation 

 1  r 2 C vs .r/ k .r/ D k k .r/: 2

(17.191)

Again the term on the left side of the above equation can be dropped since D.r 1 r 2 / c.1/ .r 1 r 2 /  !s c.1/ .r 1 r 2 /. Therefore c.1/ .r 1 r 2 / D

 X X j
Z



l



X X  j

l
dr 0 l .r 1 /l? .r 0 /j? .r 2 /j .r 0 /

1 : jr 0  r 2i j

(17.192)

350

17 Application of Q-DFT to the Metal–Vacuum Interface

By using the closure relation X

j? .r 2 /j .r 0 / D ı.r 2  r 0 /;

(17.193)

j

we have  c.1/ .r 1 r 2 / D

1 1  jr 1  r 2i j jr 2  r 2i j

 X

j .r 1 /j? .r 2 /;

(17.194)

j
which, by the use of (17.178), yields at large x1 , and x2 , c.1/ .x1 x2 / D

  1 1 1 s .x1 x2 /:  2 x1 C x2 2x2

(17.195)

  1 1 1 s .x1 x2 /:  2 x1 C x2 2x1

(17.196)

Similarly, one obtains c.2/ .x1 x2 / D

We note that no approximation for the one-particle orbital has been made in obtaining these results. Substituting c.1/ .x1 x2 / and c.2/ .x1 x2 / into (17.174) then leads to 1 1 vc .x1 /  vc .x2 / D  C ; (17.197) 4x1 4x2 which also means vc .x/ D .1=4x/. Thus, the final result for the asymptotic structure is vxc .x/ D ˛KS;xc =x, with ˛KS;xc D ˛KS;x C 1=4. Thus

.2/

.3/

Wc .x/ C fWtc .x/ C Wtc .x/ C    g

where ˛KS;c D

x!1

D

˛KS;c x

1 : 4

(17.198)

(17.199)

17.10 Analytical Asymptotic Structure of the Effective Potential Energy vs .x/ in the Classically Forbidden Region The S system effective potential energy vs .r/ is given in terms of its external v.r/, Hartree WH .r/, Pauli Wx .r/, Coulomb Wc .r/ and Correlation-Kinetic Wtc .r/ components as vs .r/ D v.r/ C WH .r/ C Wx .r/ C Wc .r/ C Wtc .r/;

(17.200)

17.10 Analytical Asymptotic Structure of the Effective Potential Energy

351

or equivalently in Kohn–Sham terms as vs .r/ D v.r/ C vH .r/ C vx .r/ C vc .r/;

(17.201)

with vx .r/ and vc .r/ defined in terms of Wx .r/, Wc .r/, and the perturbation expansion terms of Wtc .r/ by (17.133) and (17.160). In the asymptotic classically forbidden region, the sum of the potential energies Œv.x/ C WH .x/ decays exponentially. Thus, the asymptotic structure of vs .x/ is .x/x!1 vs .x/x!1 D ŒWx .x/  Wt.1/ c CŒWc .x/ C fWt.1/ .x/ C Wt.2/ .x/ C    gx!1 c c D

˛W .ˇ/  x

˛t.1/ .ˇ/ c x



˛KS;c x

(17.202) (17.203)

or equivalently vs .x/x!1 D Œvx .x/ C vc .x/x!1 ˛KS;xc ˛KS;x ˛KS;c  D ; D x x x

(17.204) (17.205)

where the coefficients ˛W .ˇ/ ˛t.1/ .ˇ/, ˛KS;x .ˇ/, ˛KS;c are given by (17.132), c (17.158), (17.156) and (17.199), respectively. In Fig. 17.12 we plot the decay coefficients ˛KS;xc .ˇ/, ˛KS;x .ˇ/, ˛KS;c .ˇ/, ˛W .ˇ/, ˛t.1/ .ˇ/, and ˛S .ˇ/ as functions of the barrier height parameter ˇ. In the c figure, the corresponding values of the Wigner–Seitz radius rs for the jellium model over the metallic range of densities is also given. The relationship between rs and ˇ is taken from the results of Sect. 17.3 and Table 17.1. The values of the coefficients for jellium metal corresponding to rs D 2, 4, 6 are given in Table 17.3. Observe [see row of ˛W .ˇ/] that there is a Pauli correlation contribution over the entire metallic density range, the contribution increasing with decreasing bulk density. The lowest-order Correlation-Kinetic con.1/ tribution ˛tc .ˇ/ also increases with decreasing bulk density, but is about an order of magnitude smaller. It is at present unclear what fraction the Coulomb and higher-order Correlation-Kinetic contributions to the “correlation” coefficient ˛KS;c .ˇ/ are. In light of the small values of ˛t.1/ .ˇ/, we conjecture that the principal c contribution to ˛KS;c .ˇ/ is due to Coulomb correlation. Note that the asymptotic structure of vs .x/ has contributions from all three types of electron correlations: Pauli, Coulomb, and Correlation-Kinetic. Each type of correlation contributes a term of 0.1=x/. The structure is image-potential-like but not the image potential. For metallic densities 2  rs  6, the decay coefficient (see (17.203) or (17.205) and Table 17.3) ranges from 0:445 to 0:524. This is approxip mately twice as large as the classical image potential value of 0:250. For ˇ D 2, which corresponds to rs 4:1 for which value of rs jellium metal is stable, we obtain vs .x/x!1 D 1=2x. It is entirely fortuitous that the Slater function

352

17 Application of Q-DFT to the Metal–Vacuum Interface WIGNER–SEITZ RADIUS rs(au) Structureless Pseudopotential model 0.7 2

4 3 Jellium model

0.6 2

0.4 0.3

4

6 5

6

aKS,xc

0.5 COEFFICIENTS

3

5

aW

aS

aKS,x

aKS,c

0.2 (1) atc

0.1 0 –0.1 1.1

1.2

1.3

1.4

1.5

1.6

BARRIER HEIGHT PARAMETER b=(W/E F)½ .1/

Fig. 17.12 The asymptotic structure coefficients ˛W .ˇ/, ˛tc .ˇ/, ˛KS;x .ˇ/, ˛KS;c .ˇ/, ˛KS;xc .ˇ/, p and ˛S .ˇ/ as a function of the barrier height parameter ˇ D W =F . The corresponding values of the Wigner–Seitz radius rs for both the jellium and structureless pseudopotential models are also given .1/

Table 17.3 The asymptotic structure coefficients ˛W .ˇ/, ˛tc .ˇ/, ˛KS;x .ˇ/, ˛KS;c .ˇ/, ˛KS;xc .ˇ/, and ˛S .ˇ/ as a function of the Wigner–Seitz radius rs Coefficients rs D 2 rs D 4 rs D 6 ˛W .ˇ/ 0.217 0.315 0.368 .1/ 0.022 0.066 0.094 ˛tc .ˇ/ 0.195 0.248 0.274 ˛KS;x .ˇ/ 0.250 0.250 0.250 ˛KS;c .ˇ/ ˛KS;xc .ˇ/ 0.445 0.498 0.524 ˛S .ˇ/ 0.390 0.496 0.548

coefficient ˛S .ˇ/ is a relatively close approximation to the decay coefficient ˛KS;xc .ˇ/ of vs .x/. Another important point to note is that the decay coefficient depends on the density of the metal. Each metal has a different asymptotic decay of the effective potential energy function. (This is in contrast to the image potential of classical physics of 1=4x which is the potential energy of an external test charge outside the metal surface, and which is the same for all metals.) The asymptotic electron in the vacuum region is not an external test charge but part of the N -electron system

17.11 Endnote on Image-Potential-Bound Surface States

353

of the metal. This is why in the quantum-mechanical calculation, the asymptotic structure of the effective potential energy function depends explicitly on the bulk metal density. Finally, we note that the results for the asymptotic structure of all various potential energies are equally valid for the stabilized jellium model [69, 70].

17.11 Endnote on Image-Potential-Bound Surface States The asymptotic structure of the effective potential energy vs .x/ and the many-body effects contributing to it are important to the understanding of the image-potentialbound surface states [29]. These states can be probed experimentally by scanning tunneling microscopy [71], inverse [72, 73], and two-photon photoemission [74]. In the interpretation of this data, it is assumed that the asymptotic structure of vs .x/ is the classical image potential, that this structure is due solely to Coulomb correlations, and that the structure is the same for all metals and therefore independent of the Wigner-Seitz radius rs . As we have derived, the asymptotic structure of vs .x/ is approximately twice as large as the image potential for the metallic range of densities, that Pauli and Coulomb correlations as well as Correlation-Kinetic effects contribute to this structure, and that the structure is dependent on the metal density. The expression for the energy spectrum of these states employed in the literature is of the form [29, 75]   1 Z2 2 E D V0 C jkk j  2 ; 2 

(17.206)

with V0 the vacuum level,  D n C ın , where n D 1, 2, 3, : : : and ın is the quantum defect, and where Z is the parameter in the long-range Z=x structure of the effective potential energy. In the above expression it is assumed that Z D 1=4, the image-potential value. However, our results show Z D ˛KS;xc .ˇ/. Thus, not only is the coefficient different from that of the image-potential value, but it is also depends on the bulk density of the metal. The experimental data therefore needs to be reinterpreted in light of these facts. As another consequence of our results, the localization property given by the relative probability function P .x/ D jk .x/j2 =jk .0/j2 , where x D 0 corresponds to the crystal surface, will also be different for each metal and image state.

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Chapter 18

Many-Body and Pseudo Møller-Plesset Perturbation Theory within Quantal Density Functional Theory

This chapter is comprised of two perturbation theory schemes in the context of Quantal Density Functional Theory. In the first we develop [1] a many-body perturbation theory within the framework of Q-DFT. In Q-DFT, the contributions of electron correlations due to the Pauli exclusion principle and Coulomb repulsion – the electron-interaction component, and those due to Correlation-Kinetic effects, are separately delineated. As such, a separate many-body perturbation series is developed for each of these two components. At lowest order of this Q-DFT many-body perturbation theory, only electron correlations arising solely from the Pauli principle are accounted for. (Correlation-Kinetic contributions due to the Pauli principle are also absent.) As with standard many-body perturbation theory, at lowest order, the energy thus obtained is a rigorous upper bound. The higher order Coulomb correlation and Correlation-Kinetic contributions can then be systematically and separately determined to any order. It turns out that the Q-DFT perturbation theory limited to lowest order is equivalent to the ad hoc Q-DFT Pauli Approximation (Sect. 6.2.1), thereby providing a formal justification for the latter. Including the perturbation series for Coulomb correlations but neglecting that for the Correlation-Kinetic effects is the Q-DFT perturbation theory counterpart to the ad hoc Q-DFT Pauli–Coulomb Approximation (Sect. 6.3.1). Additionally including the series for the Correlation-Kinetic effects is the corresponding counterpart to the ad hoc Q-DFT Fully Correlated Approximation (Sect. 6.3.2). The second perturbation theory to be described [2] involves employing key attributes of Møller-Plesset [3] perturbation theory – the applicability of Brillouin’s theorem [4] and the two-particle nature of the perturbation – in conjunction with Q-DFT to construct accurate wave functions in a self-consistent manner. In this way the wave functions are superior to those of Møller-Plesset perturbation theory because the various electron correlations – Pauli, Coulomb, and Correlation-Kinetic – are all intrinsically incorporated in the orbitals of the wave function. The selfconsistency indicated ensures that the density determined from the wave function is the same as that obtained from the Q-DFT model S system of noninteracting fermions. Once again, the separate Pauli, Coulomb, and Correlation-Kinetic contributions to the various properties can be determined. In contrast to Møller-Plesset theory, the total energy as obtained via the individual component terms of Q-DFT

355

356

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

or equivalently by the wave function, is a rigorous upper bound. We refer to this procedure as Pseudo Møller–Plesset perturbation theory within Q-DFT. The material in this chapter requires knowledge of the methods of both manybody theory and Møller–Plesset perturbation theory. We refer the reader to the texts [5, 6] for the former, and to [3, 7] for the latter.

18.1 Many-Body Perturbation Theory within Q-DFT In this section we develop a many-body perturbation theory within the framework of Q-DFT. We begin by writing the quantal sources – the density .r/, the singleparticle .rr 0 / and Dirac s .rr 0 / density matrices – in terms of the one-particle Green’s functions for the fully interacting and model S systems respectively, and the pair-correlation density g.rr 0 / in terms of the two-particle Green’s function. The perturbation series for the electron-interaction field E ee .r/, whose quantal source is the pair-correlation density g.rr 0 /, is then obtained by employing the Bethe–Salpeter equation [8, 9]. This equation expresses the two-particle Green’s function in terms of the one-particle Green’s function and an effective two-particle interaction. The perturbation series for the Correlation-Kinetic field Z tc .r/, whose quantal source is the difference Œ.rr 0 /  s .rr 0 / of the single-particle and Dirac density matrices, is obtained by relating the interacting and noninteracting system one-particle Green’s functions and the self-energy via Dyson’s equation. Various lower order approximations of this perturbation theory are then discussed. Finally, for completeness and comparison, the Sham-Schl¨uter [10, 11] integral equation for the S system electron-interaction potential energy vee .r/ in terms of the interacting and noninteracting system one-particle Green’s functions is derived from the Dyson equation. From this integral equation the corresponding integral equation for the Kohn–Sham “exchange” potential energy vx .r/ in the “exchange-only” approximation [12, 13] is then obtained. The latter integral equation for vx .r/ is also known as the Optimized Potential Method [14, 15] (see Chap. 5 of QDFT).

18.1.1 Quantal Sources in Terms of Green’s Functions The single-particle density matrix .rr 0 / quantal source of (2.19), on emphasizing the spin dependency, is  0 .rr 0 / D N

Z

‰ ? .r; X N 1 /‰.r 0  0 ; X N 1 /dX N 1 ;

(18.1)

where ;  0 are the spin indices. In second quantized notation 

 0 .rr 0 / D h‰j O  .r/ O  0 .r 0 /j‰i;

(18.2)

18.1 Many-Body Perturbation Theory within Q-DFT

357

 where O  .r/, O  .r/ are the creation and annihilation field operators. Now the single-particle Green’s function G 0 .rt; r 0 t 0 / is defined as 

G 0 .rt; r 0 t 0 / D i h‰0 jT f O  .rt/ O  0 .r 0 t 0 /gj‰0 i=h‰0j‰0 i;

(18.3)

where ‰0 is the ground state wave function of the interacting system in the Heisenberg picture or equivalently the Schr¨odinger wave function at time t D 0, i.e. the  solution of H j‰0 i D E0 j‰0 i; T is the time-ordering operator; O  .rt/, O  .rt/ are the creation and annihilation field operators in the Heisenberg picture. The single-particle density matrix is then  0 .rr 0 / D i lim0 G 0  .r 0 t 0 ; rt/; t !t C

(18.4)

and the density .r/ is .r/ D i

X 

lim G .rt 0 ; rt/:

t !t 0 C

(18.5)

The Dirac density matrix s; 0 .rr 0 / quantal source of (3.12) is similarly expressed in terms of the noninteracting fermion S system Green’s function Gs; 0 .rt; r 0 t 0 / as s; 0 .rr 0 / D i lim0 Gs; 0  .r 0 t 0 ; rt/; t !t C

(18.6)

with 

Gs; 0 .rt; r 0 t 0 / D ihˆ0 jT f O  .rt/ O  0 .r 0 t 0 /gjˆ0 i=hˆ0 jˆ0 i;

(18.7)

where jˆ0 i is the S system ground state Slater determinant. The density .r/ which is the same as that of the interacting system may correspondingly be written as .r/ D i

X 

lim Gs; .rt 0 ; rt/:

t !t 0 C

(18.8)

In terms of the Fourier transform G 0 .rr 0 !/.Gs; 0 .rr 0 !// of G 0 .rt; r 0 t 0 / .Gs; 0 .rt; r 0 t 0 //: G 0 .rr 0 !/ D

Z

0

d.t  t 0 /ei !.t t / G 0 .rt; rt 0 /;

(18.9)

the density .r/ may be expressed as X 1 Z d!ei ! G .rr!/; .r/ D i 2



(18.10)

358

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

or equivalently

X 1 Z .r/ D i d!e i ! Gs; .rr!/: 2



(18.11)

(Recall that for time-independent systems for which HO does not depend explicitly on time, the one-particle Green’s functions depend only on the time difference t  t 0 and not upon t and t 0 separately.) From conventional many-body theory [8], the equation satisfied by the Green’s function G 0 .rr 0 !/ is 1 Œ! C r 2  v.r/G 0 .rr 0 !/ 2 XZ  † .rr 00 !/G  0 .r 00 r 0 !/dr 00 D ı.r  r 0 /ı 0 ;

(18.12)



where v.r/ is the external potential energy operator, and † 0 .rr 0 !/ D

Z

0

d.t  t 0 /ei !.t t / † 0 .rt; r 0 t 0 /

(18.13)

is the Fourier transform of the self-energy † 0 .rt; r 0 t 0 /. With the definition Q .rr 0 !/ D † .rr 0 !/  ı.r  r 0 /vee .r/ı ; †

(18.14)

(18.12) may be rewritten as 1 Œ! C r 2  vs .r/G 0 .rr 0 !/ 2 XZ Q  .rr 00 !/G  0 .r 00 r 0 !/dr 00 D ı.r  r 0 /ı 0 ;  †

(18.15)



where vs .r/ is the S system effective potential energy (see (3.4)): vs .r/ D v.r/ C vee .r/;

(18.16)

with vee .r/ its effective “electron-interaction” energy component. The equation satisfied by the S system Green’s function Gs; 0 .rr 0 !/ is 1 Œ! C r 2  vs .r/Gs; 0 .rr 0 !/ D ı.r  r 0 /ı 0 : 2

(18.17)

The Green’s function G and Gs are therefore related by Dyson’s equation: G 0 .rr 0 !/ D Gs; 0 .rr 0 !/ XZ Q .yy 0 !/G  0 .y 0 r 0 !/dydy 0 : (18.18) Gs; .ry!/† C

18.1 Many-Body Perturbation Theory within Q-DFT

359

Dyson’s equation may be written in time-space as 0 0

0 0

G 0 .rt; r t / D Gs; 0 .rt; r t / C

XZ

Gs; .rt; y/



Q .y; y 0  0 /G  0 .y 0  0 ; r 0 t 0 /dyddy 0 d 0 : †

(18.19)

The two-particle density matrix .r 1 r 2 I r 01 r 02 / is defined as .r 1 r 2 I r 01 r 02 /

D N.N  1/=2

XZ

‰ ? .r 1 1 ; r 2 2 ; X N 2 /

1 2

‰.r 01 1 ; r 02 2 ; X N 2 /dX N 2 :

(18.20)

In terms of the diagonal matrix element .rr 0 I rr 0 /  .rr 0 / of the two-particle density matrix, the pair-correlation density quantal source g.rr 0 / is g.rr 0 / D

2.rr 0 / ; .r/

(18.21)

where 0

.rr / D N.N  1/=2

XZ

‰ ? .r; r 0  0 ; X N 2 /‰.r; r 0  0 ; X N 2 /dX N 2 :

 0

(18.22) The density matrix .rr 0 / can also be expressed as the expectation value of the field operators in second quantization form as 2.r 1 r 2 / D

X   h‰0 j O ˛ .r 1 / O ˇ .r 2 / O ˇ .r 2 / O ˛ .r 1 /j‰0 i;

(18.23)

˛ˇ

or in terms of the field operators in the Heisenberg picture as 2.r 1 r 2 / D lim 

X

lim

lim

t3 !t4 C t4 !t2 C t2 !t1 C

ˇ ˇ   h‰0 j O ˛ .r 3 t3 / O ˇ .r 4 t4 / O ˇ .r 2 t2 / O ˛ .r 1 t1 /j‰0 iˇˇ

:(18.24)

r 3 Dr 1 Ir 4 Dr 2

˛ˇ

.2/ Now the two-particle Green’s function G˛ˇ; .1; 2I 3; 4/ is defined as ı 



G˛ˇ; ı .1; 2I 3; 4/ D .i /2 h‰0 jT f O ˛ .1/ O ˇ .2/ O ı .4/ O  .3/gji=h‰0j‰0 i; (18.25) .2/

where 1  r 1 t1 , etc. Thus, the two-particle density matrix .r 1 r 2 /, and hence the pair-correlation density g.r 1 r 2 / can be expressed in terms of the two-particle Green’s function as

360

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

g.r 1 r 2 /.r 1 /  2.r 1 r 2 / D  lim

lim

lim ˇ X .2/ ˇ  G˛ˇ;˛ˇ .1234/ˇˇ

t3 !t4 C t4 !t2 C t2 !t1 C

: (18.26)

r 3 Dr 1 Ir 4 Dr 2

˛ˇ

18.1.2 Perturbation Series for the Electron-Interaction Field E ee .r/ We next determine the perturbation series for the electron-interaction field E ee .r/. To do so we employ the Bethe–Salpeter equation [8, 9]. This equation which .2/ expresses the two-particle Green’s function G˛ˇ; ı .12I 34/ in terms of the oneparticle Green’s function G˛ˇ .12/ and an effective two-particle interaction ˛ˇ ı .12I 34/ is .2/ .1; 2I 3; 4/ D G˛ .1; 3/Gˇ ı .2; 4/  G˛ı .1; 4/Gˇ .2; 3/ G˛ˇ; ı X Z C G 0  .30 ; 3/Gı 0 ı .40 ; 4/˛0 ˇ 0  0 ı 0 .10 ; 20 I 30 ; 40 / ˛0 ˇ 0  0 ı 0

G˛˛0 .1; 10 /Gˇˇ 0 .2; 20 /d10 d20 d30 d40 ;

(18.27)

where d1 D dr 1 dt1 , etc. Substituting (18.27) into (18.26) one obtains lim lim g.r 1 r 2 /.r 1 / D  lim t3 !t4 C t4 !t2 C t2 !t1 C X  fŒG˛˛ .r 1 t1 ; r 3 t3 /Gˇˇ .r 2 t2 ; r 4 t4 /  G˛ˇ .r 1 t1 ; r 4 t4 /Gˇ ˛ .r 2 t2 ; r 3 t3 / ˛ˇ

C

X Z

G 0  .30 ; 3/Gı 0 ı .40 ; 4/˛0 ˇ 0  0 ı 0 .10 ; 20 I 30 ; 40 /

˛0 ˇ 0  0 ı 0

ˇ ˇ : G˛˛0 .1; 1 /Gˇˇ 0 .2; 2 /d1 d2 d3 d4 gˇˇ r 3 Dr 1 ;r 4 Dr 2 0

0

0

0

0

0

(18.28)

 D˛;ıDˇ

Employing (18.5) it follows that the first term on the right-hand side of (18.28) is lim

lim

lim

t3 !t4 C t4 !t2 C t2 !t1 C

D .r 1 /.r 2 /:

X ˛ˇ

ˇ ˇ G˛˛ .r 1 t1 ; r 3 t3 /Gˇˇ .r 2 t2 ; r 4 t4 /ˇˇ

r 3 Dr 1 ;r 4 Dr 2

(18.29)

On dividing by .r 1 /, and on writing G˛ˇ .12/ D G.12/ı˛ˇ etc., the second term of (18.28) becomes on employing (18.4)

18.1 Many-Body Perturbation Theory within Q-DFT

lim

lim

lim

t3 !t4 C t4 !t2 C t2 !t1 C

D

X ˛ˇ

D since

361

ˇ X 1 ˇ G˛ˇ .r 1 t1 ; r 4 t4 /Gˇ ˛ .r 2 t2 ; r 3 t3 /ˇˇ .r 1 / r 3 Dr 1 Ir 4 Dr 2 ˛ˇ

1 G˛ˇ .r 1 t1 ; r 2 t1 C/Gˇ ˛ .r 2 t2 ; r 1 t2 C/ı˛ˇ ıˇ ˛ .r 1 /

.r 1 r 2 /.r 2 r 1 / ; 2.r 1 /

X

ı˛ˇ ıˇ ˛ D

(18.31)

XX ˛

˛ˇ

(18.30)

ı˛ˇ ıˇ ˛ D

X

ı˛˛ D

X

˛

ˇ

1 D 2:

(18.32)

˛

Again writing G 0  .30 3/ D G.30 3/ı 0  etc., and since X ˛0 ˇ 0  0 ı 0

ˇ ˇ ˛0 ˇ 0  0 ı 0 .10 20 I 30 40 /ı  0 ııı 0 ı˛˛0 ıˇˇ 0 ˇˇ

ˇ ˇ D ˛ˇ ı ˇˇ

 D˛IıDˇ

D ˛ˇ ˛ˇ .10 20 I 30 40 /;

(18.33)

 D˛IıDˇ

we have from (18.29), (18.31) and (18.33) the expression for the pair-correlation density g.r 1 r 2 / to be g.r 1 r 2 / D .r 2 /  

.r 1 r 2 /.r 2 r 1 / 2.r 1 /

1 lim .r 1 / t3 !t4 C 0

0

lim

lim

t4 !t2 C t2 !t1 C 0

0

0

XZ ˛ˇ 0

G.30 ; r 1 t3 /G.40 ; r 2 t4 /

˛ˇ ˛ˇ .1 ; 2 I 3 ; 4 /G.1; 1 /G.2; 2 /d10 d20 d30 d40 ;

(18.34)

where the effective two-particle interaction may be written diagrammatically as .see Fig: 18:1/

(18.35)

.see Fig: 18:2/

(18.36)

with

Fig. 18.1 Equation (18.35)

362

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

Fig. 18.2 Equation (18.36)

and where the double lines are associated with G and the single with Gs . Note that an extra factor i n is required for the nth-order interaction in the above diagrams. With the pair-correlation density quantal source g.rr 0 / given by (18.34)–(18.36), the electron-interaction field E ee .r/ may be determined via Coulomb’s law (see 2.44) to any order. The expression for the pair-correlation density g.rr 0 / of (18.34)–(18.36) and the field E ee .r/ may also be written in terms of the Fourier transformed self-energy P Q .rr 0 !/ and the S and interacting system Green’s functions Gs .rr 0 !/ and G.rr 0 !/ by substitution of Dyson’s equation (18.18). This is algebraically tedious and will not be done here. However, a similar procedure for the Correlation-Kinetic quantal source is shorter and will be demonstrated in Sect. 18.1.2.

18.1.3 Perturbation Series for the Correlation-Kinetic Field Z tc .r/ The quantal source of the Correlation-Kinetic field Z tc .r/ of (3.40) is the difference between the S system Dirac s .rr 0 / and the interacting system .rr 0 / singleparticle density matrices. The Correlation-Kinetic “force” ztc .r/ thus involves the difference between the noninteracting and interacting system kinetic-energydensity tensors ts;˛ˇ .r/ and t˛ˇ .r/, respectively. Thus, the Correlation-Kinetic field Z tc .r/ is   3 2 X @ ts;˛ˇ .r/  t˛ˇ .r/ Ztc ;˛ .r/ D .r/ @rˇ

(18.37)

ˇ D1

D

  3 @2 1 XX @ @2 C 2.r/ @rˇ @r˛0 @rˇ00 @rˇ0 @r˛00  ˇ D1

ˇ  ˇ  s .r 0 ; r 00 /  .r 0 ; r 00 / ˇˇ

: r 0 Dr 00 Dr

(18.38)

18.1 Many-Body Perturbation Theory within Q-DFT

363

Expressing the S system and interacting system density matrices in terms of the corresponding Green’s functions (18.6) and (18.4), the field is   3 1 XX @ @2 @2 Ztc ;˛ .r/ D C 0 00 2.r/ @rˇ @r˛0 @rˇ00 @rˇ @r˛ ˇ D1     0 lim00 i Gs; .r 00 t 00 ; r 0 t 0 /  G .r 00 t 00 ; r 0 t 0 / t !t C

:

r 0 Dr 00 Dr

(18.39) By employing Dyson’s equation (18.19) in (18.39) we have  X 3 i X @ @2 @2 C 0 00 lim Ztc ;˛ .r/ D 2.r/ @rˇ @r˛0 @rˇ00 @rˇ @r˛  t 0 !t 00 C ˇ D1 XZ dyddy 0 d 0 Gs; .r 00 t 00 ; y/ 

ˇ ˇ 0 0 0 0 0 0 ˇ Q † .y; y  /G  .y  ; r t /ˇ

:

(18.40)

r 0 Dr 00 Dr

This is the space–time expression for the Correlation-Kinetic field. In terms of the Fourier transforms of the Green’s functions and the self-energy (see (18.9)), we have   3 i X @ @2 @2 Ztc ;˛ .r/ D C 0 00 2.r/ @rˇ @r˛0 @rˇ00 @rˇ @r˛ ˇ D1 Z Z XX 1  dydy 0 d!1 d!2 d!3 .2 /3  Z 00 0 0  dd 0 0 lim00 ei!1 t ei!3 t ei.!2 !1 / ei.!3 !2 / t !t C ˇ ˇ Q .yy 0 !2 /G  .y 0 r 0 !3 /ˇ : (18.41) Gs; .r 00 y!1 /† ˇ r 0 Dr 00 Dr

Now since Z

dei.!2 !1 / D 2 ı.!2  !1 /;

(18.42)

d 0 ei.!3 !2 / D 2 ı.!3  !2 /;

(18.43)

and Z

364

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

we have Ztc ;˛ .r/ D

  3 i X @ @2 @2 C 2.r/ @rˇ @r˛0 @rˇ00 @rˇ0 @r˛00 ˇ D1 Z XXZ 0 1  d! lim ei!

dydy

!0C .2 /  ˇ ˇ 00 0 0 0 Q Gs; .r y!/† .yy !/G  .y r !/ˇˇ

:

(18.44)

r 0 Dr 00 Dr

For spin-independent interaction, Gs; .r 1 r 2 !/ D Gs .r 1 r 2 !/ı ; G  .r 1 r 2 !/ D G.r 1 r 2 !/ı  ; Q .r 1 r 2 !/ D †.r Q 1 r 2 !/ı : †

(18.45) (18.46) (18.47)

On substitution of (18.45)–(18.47) into (18.44), the sum X

ı ı ı  D



and then

P 

X

ı ı  D ı D 1;

(18.48)



1 D 2. Therefore,

Ztc ;˛ .r/ D

 Z 3 X i @2 @ @2 dydy 0 C 2 .r/ @rˇ @r˛0 @rˇ00 @rˇ0 @r˛00 ˇ D1 ˇ Z ˇ i!

00 0 0 0 Q  d! lim e  Gs .r y!/†.yy !/G.y r !/ˇˇ

!0C

; r 0 Dr 00 Dr

(18.49) or since the derivatives in the square parentheses are with respect to the vector r 0 and r 00 , we have Z Z Z 3 X i @ i!

0 Q Ztc ;˛ .r/ D d! lim e dy dy 0 †.yy !/

!0C 2 .r/ @rˇ ˇ D1   @ @ @ @ Gs .ry!/ G.y 0 r!/ C G.ry!/ G.y 0 r!/ :  @rˇ @r˛ @r˛ @rˇ (18.50) This then is the final expression for the Correlation-Kinetic field in terms of the Fourier transformed Green’s functions for the S and interacting systems and the self-energy.

18.1 Many-Body Perturbation Theory within Q-DFT

365

With the fields E ee .r/ and Z tc .r/ thus determined, the effective electroninteraction potential energy vee .r/ and the energy components Eee and Tc are then obtained from (3.67), (2.59), and (3.56), respectively, the orbitals i .x/ being generated by solution of the S system differential equation. We have thus derived a many-body perturbation theory within Q-DFT. In this theory there is a separate perturbation series for the electron-interaction (Pauli and Coulomb correlations) and Correlation-Kinetic components of the local effective electron-interaction potential energy vee .r/ and for the corresponding components Eee and Tc of the total energy. A simplification occurs for systems of symmetry such that the fields E ee .r/ and Z tc .r/ are separately conservative. These fields are then separately conservative at each order, and the corresponding components of the work done path-independent. For systems of arbitrary symmetry for which the individual fields are not conservative, the energy at each order is obtained directly in terms of the fields, but the potential energy is determined from their irrotational component.

18.1.4 Approximations within the Perturbation Theory The zeroth-order approximation of the above perturbation theory comprises of neglecting Coulomb correlations and Correlation-Kinetic effects, and considering only those correlations which arise due to the Pauli exclusion principle. Thus, at this lowest order G D Gs where   X .F  i / .i  F / C : (18.51) i .r/i? .r 0 / Gs .r; r 0 ; !/ D !  i C i !  i  i i

Additionally,  D 0 in (18.34), and Z tc .r/ D 0. Thus, in this case, we have from 0 0 (18.34) that g.rr 0 / D gs .rr / C x .rr 0 / where the Fermi hole x .rr 0 / D R / D .r 0 2 0 js .rr /j =2.r/ and gs .rr /dr 0 D N  1. Since the wave function in the zeroth-order approximation is a Slater determinant, the total energy is a rigorous upper bound. It is evident that this zeroth-order approximation is equivalent to the ad hoc Q-DFT Pauli Approximation of Sect. 6.2.1. The perturbation theory developed thus provides a formal justification for the latter approximation. The first-order approximation, which in addition to Pauli correlations, includes Coulomb correlations but no Correlation-Kinetic effects .Z tc .r/ D 0/ corresponds to .see Fig: 18:3/

(18.52)

The first-order Correlation-Kinetic field corresponds to assuming G D Gs in (18.50) and employing the lowest-order approximation for †.rr 0 !/, i.e. Q x .rr 0 !/ D †x .rr 0 !/  ı.r  r 0 /Œvee .r/  WH .r/; † where †x .rr 0 !/ D  s .r 0 r/=2jr  r 0 j.

(18.53)

366

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

Fig. 18.3 Equation (18.52)

For first-and higher-order calculations, the second term of (18.34) for g.rr 0 / may be represented by the Fermi hole x .rr 0 / as determined by the orbitals of the S system determined to that order.

18.1.5 Endnote For completeness we contrast the Q-DFT and Kohn-Sham many-body perturbation theories. On employing (18.10) and (18.11), one has on integrating the Dyson equation (18.18) over all frequencies that, Z

Z d!

Z dy

0 Q dy 0 Gs .ry!/†.yy !/G.y 0 r!/ D 0:

(18.54)

Q from (18.14) into the above equation , one obtains Substituting for † Z Z D

Z

dyvee .y/ d!Gs .ry!/G.yr! Z Z dy dy 0 d!Gs .ry!/†.yy 0 !/G.y 0 r!/:

(18.55)

This equation was first derived by Sham and Schl¨uter [10, 11]. It is an integral equation for the local electron-interaction potential energy vee .r/. The lowest order “exchange-only” approximation [12, 13] of the Sham-Schl¨uter equation is obtained by employing G.rr 0 !/ D Gs .rr 0 !/ and †x .rr 0 !/ D  s .r 0 r/=2jr  r 0 j, which leads to Z Z dyvx .y/ d!Gs .ry!/Gs .yr!/ Z Z Z 1 d!Gs .ry!/s .y 0 y/Gs .y 0 r!/: (18.56) D  dy dy 0 jy  y 0 j By using the S system Green’s function Gs of (18.51) in (18.56) one obtains XX i

j

i .r/j? .r/

1 .F  i / j  i

ZZ

dydy 0

s .y 0 y/ ?  .y/j .y 0 / 2jy  y 0 j i

18.2 Pseudo Møller–Plesset Perturbation TheoryWithin Q-DFT

D

XX i

i .r/j? .r/

j

1 .F  i /  j  i

367

Z dyvx .y/i? .y/j .y/; (18.57)

which is just the integral equation of the optimized potential method (OPM) for the KS exchange potential vx .r/. The OPM integral equation may be derived independently by minimizing the expectation value of the Hamiltonian as obtained via the S system Slater determinant with respect to arbitrary variations of the local effective potential vs .r/ [14, 15] (see QDFT). As proved in QDFT, the Kohn-Sham “exchange” potential energy vx .r/ and energy Ex Œ are representative of electron correlations due to the Pauli principle as well as that of the lowest order Correlation-Kinetic effect. Thus, each higherorder term of Sham–Schl¨uter theory implicitly corresponds to a contribution of Coulomb correlation and Correlation-Kinetic contributions. In contrast, in lowestorder Q-DFT perturbation theory, the correlations are strictly due to the Pauli principle. Furthermore, within Q-DFT many-body perturbation theory there is a separate perturbation series for Coulomb correlation and Correlation-Kinetic effects. Thus, within Q-DFT the contributions of the three types of electron correlations to properties can be separately and systematically studied to various orders.

18.2 Pseudo Møller–Plesset Perturbation Theory Within Q-DFT In this chapter we describe a method for the accurate self-consistent determination of a wave function by employing the principal attributes of Møller–Plesset perturbation theory in conjunction with Q-DFT. Since this wave function is determined by simultaneously constructing the model S system as defined within Q-DFT, the separate Pauli, Coulomb, and Correlation-Kinetic contributions to properties as well as the ionization potential are then obtained. We present here the broad outlines of this Pseudo Møller-Plesset Q-DFT perturbation theory to first order in the wave function. The generalization to the determination of the wave functions to higher-order follows readily.

18.2.1 Pseudo Møller–Plesset Q-DFT Perturbation Theory With v.r/ the external potential energy of both the interacting system electrons and model system noninteracting fermions, the differential equation for the latter as expressed within Q-DFT is 

 1  r 2 C v.r/ C vee .r/ i .r/ D i i .r/; 2

(18.58)

368

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

where the electron-interaction potential energy vee .r/ is the work done in the conservative effective field F eff .r/: Z

r

F eff .r 0 /  d` 0 ;

(18.59)

F eff .r/ D E ee .r/ C Z tc .r/;

(18.60)

vee .r/ D 

1

where

with E ee .r/ and Z tc .r/ being the electron-interaction and Correlation-Kinetic fields, respectively. For the definitions of the quantal sources and the corresponding fields, the reader is referred to Chap. 3. (Here the spin index is suppressed.) The Slater determinant of the orbitals i .r/ is ˆfi g, and leads to the density O .r/ D hˆ fi g j.r/jˆ fi gi D

X

ji .r/j2 ;

(18.61)

i

with .r/ O the density operator of (2.16). The Hartree–Fock theory equations as written in the Slater-Bardeen form (see Sects. 3.5.1 and 3.5.2 of QDFT) is HO HF;i .r/

i .r/

D i

i .r/;

(18.62)

where

1 HO HF;i .r/ D  r 2 C v.r/ C WH .r/ C vx;i .r/; 2 the orbital-dependent “exchange” function vx;i .r/ is Z vx;i .r/ D

x;i .rr 0 / 0 dr ; jr  r 0 j

(18.63)

(18.64)

the orbital-dependent Fermi hole charge x;i .rr 0 / is x;i .rr 0 / D 

N X j D1 spi nj Dspi ni

? 0 0 j .r / i .r / j .r/ i .r/

;

(18.65)

R and WH .r/ D dr 0 .r 0 /=jr  r 0 j is the Hartree potential energy. .0/ We define ‰0 f i g as the Slater determinant of the lowest occupied orbitals of the Hartree–Fock equation (18.62); ‰0.m/ f i g as the Slater determinant formed with only two excited orbitals of the Hartree–Fock equation; E0.0/ and E0.m/ as the total .0/ .m/ energies determined by the Slater determinants ‰0 f i g and ‰0 f i g, respectively. (Note that in ground state Hartree–Fock theory, the density .r/ is obtained

18.2 Pseudo Møller–Plesset Perturbation TheoryWithin Q-DFT

369

from the Slater determinant ‰0.0/ f i g with the orbitals i .r/ being determined self-consistently.) The perturbation VO .R/ in Møller-Plesset perturbation theory is VO .R/ D HO .R/  HO HF .R/;

(18.66)

where HO .R/ P is the fully interacting system Hamiltonian of (2.1)–(2.4), and HO HF .R/ D i HO HF;i .r/ with R D r 1 ; : : : ; r N . Thus, according to Møller-Plesset perturbation theory, the wave function of the fully interacting system to first order is ‰.R/ D ‰0.0/ f where the first-order term is D .m/ X ‰0 f ‰1 f 1 g D m¤0

ig

C ‰1 f

.0/ O i gjV .R/j‰0 f

E0.0/



E0.m/

ig

i g;

(18.67)

E .m/

‰0 f

i g:

(18.68)

Observe that in the matrix elements of (18.68) it is the Slater determinants ‰0.m/ f i g with only two excited orbitals that appear. The matrix element with Slater determinants having one excited state orbital i .r/ vanishes due to Brillouin’s theorem [4]; those with Slater determinants having three or more excited state orbitals vanish due to the two-particle nature of the perturbation. Note that Brillouin’s theorem is valid solely for eigenfunctions of the Hartree-Fock theory Hamiltonian. It is the above two attributes of Møller-Plesset perturbation theory, viz. the vanishing of the matrix elements of the singly and triply etc. excited Slater determinants that we employ in Pseudo Møller-Plesset Q-DFT. The self-consistent procedure for Pseudo Møller-Plesset Q-DFT is as follows. Step 1: We begin with say the ground state orbitals i .r/ of the Q-DFT Pauli Approximation differential equation (see Sect. 6.2.1). That is we initially consider only those correlations that arise due to the Pauli exclusion principle. With these orbitals we construct the Hartree–Fock theory Hamiltonian HO HF;i .r/ of (18.63). Thus, the corresponding density .r/ to be employed is due to the Slater determinant ˆfi g, and the corresponding orbital-dependent Fermi hole charge x;i .rr 0 / x;i .rr 0 / D 

N X i D1 spi nj Dspi ni

j? .r 0 /i .r 0 /j .r/ i .r/

:

(18.69)

With the Hartree-Fock theory single-particle Hamiltonian HO HF;i fi g thus determined and kept fixed, the Hartree–Fock equations are solved for both the occupied and unoccupied orbital states i .r/. These orbitals i .r/ are eigenfunctions of the Hartree–Fock Hamiltonian HO HF;i fi g as constructed by the Q-DFT Pauli Approximation orbitals i .r/. Equivalently, the Slater determinants ‰0.0/ f i g and ‰0.m/ f i g

370

18 Many-Body and Pseudo Møller–Plesset Perturbation Theory

P are eigenfunctions of the Hartree–Fock Hamiltonian HO HF fi g D i HO HF;i fi g. (Note that since HO HF;i fi g is constructed from the Q-DFT orbitals i .r/ and then kept fixed, the corresponding Slater determinants ‰0.0/ f i g and ‰0.m/ f i g differ from the self-consistently determined Hartree-Fock theory orbitals. Thus, the energy E0.0/ obtained from this ‰0.0/ f i g will lie above the Hartree-Fock theory ground state energy.) (The question arises as to why not employ the occupied and unoccupied orbitals i .r/ of the Q-DFT Pauli Approximation differential equation to construct the Slater determinants in the expression (18.68) for the wave function to first-order? The reason is that the matrix elements of (18.68) with singly and triply etc. excited state determinants will then not vanish. Brillouin’s theorem is not applicable to matrix elements obtained from Slater determinants generated by a local effective potential energy. However, since in ground state theory, Correlation-Kinetic effects are negligible, the Q-DFT Pauli Approximation and Hartree–Fock theory orbitals are essentially equivalent (see Chap. 10). Thus, the matrix elements of (18.68) taken with the singly and triply excited state Q-DFT Pauli Approximation Slater determinants will also be negligible.) Step 2: With the Slater determinants ‰0.0/ f i g and ‰0.m/ f i g as obtained in Step 1, the wave function ‰.R/ to first-order is then constructed. With this wave function the quantal sources – the pair-correlation density g.rr 0 /, the single-particle density matrix .rr 0 /, and the density .r/ – are then determined. The Dirac density matrix s .rr 0 / quantal source is that obtained from the Slater determinant ˆfi g of the Q-DFT differential equation which at this stage is that of the Q-DFT Pauli Approximation of Step 1. From these quantal sources, the fields E ee .r/, Z tc .r/, and F eff .r/ are determined. From F eff .r/ the local electron-interaction potential energy vee .r/ is then obtained. Keeping the local effective potential energy vs .r/ D v.r/ C vee .r/ fixed, the S system differential equation is then solved for the new lowest occupied orbitals i .r/. In this manner we have now additionally incorporated Coulomb correlations and Correlation-Kinetic effects. Step 3: With these new lowest occupied orbitals i .r/, the procedure for Step 1 is repeated. That is, first the Hartree–Fock theory Hamiltonian HO HF;i fi g is constructed. For this fixed Hamiltonian, the corresponding occupied and unoccupied .0/ orbitals i .r/ are determined, from which the Slater determinants ‰0 f i g and .m/ ‰0 f i g are obtained. Step 2 for the determination of the wave function ‰.R/ to first order, the quantal sources, fields, the potential energy vee .r/, and the new lowest occupied orbitals i .r/ of the S system differential equation, is next repeated. Step 4: The earlier procedure is continued until self-consistency is achieved, i.e. when the density as determined from the wave function ‰.R/ which is .r/ D h‰j.r/j‰i O Z D N ‰ ? .r; R N 1 /‰.r; R N 1 /dR N 1 ;

(18.70) (18.71)

18.2 Pseudo Møller–Plesset Perturbation TheoryWithin Q-DFT

371

with R N 1 D r 2 ; : : : ; r N ; dR N 1 D dr 2 ; : : : ; dr N , is equivalent to the density obtained from the S system Slater determinant ˆfi g as given by (18.61). In this manner the relationship between the wave function ‰.R/ and the model S system is a self-consistent one. With the properties of the S system fully determined, the total energy E and its components – kinetic, external, Hartree, Pauli, Coulomb and Correlation-Kinetic – may be obtained from the requisite quantal sources or fields. The ionization potential I is the negative of the highest occupied eigenvalue.

18.2.2 Endnote In Pseudo Møller–Plesset Q-DFT perturbation theory to any order in the wave function, the total energy is a rigorous upper bound to the true value. This is because the energy and its components are obtained from the various fields which in turn are determined self-consistently from the interacting and noninteracting system wave functions. Equivalently, the energy is the expectation of the interacting system Hamiltonian (2.1)–(2.4) taken with respect to the wave function to first or higher order. The wave function to any order, the S system orbitals, the quantal sources, fields, potential energies, and components of the total energy are all related in a selfconsistent manner up to that order of the wave function. If, on the other hand, the wave function to first-order were employed to obtain the energy to second-order in the perturbation expansion, the energy would then not lie within the realm of selfconsistency to first order. In other words, the energy would then not be due to the self-consistently generated fields as obtained from the wave function to first order. Additionally, as is the case in Møller–Plesset perturbation theory, the energy would no longer be an upper bound. In Møller–Plesset perturbation theory, the zeroth-order wave function is the Hartree–Fock theory Slater determinant. The zeroth-order wave function (see (18.67)) of Pseudo Møller–Plesset Q-DFT perturbation theory is not the Hartree–Fock theory determinant, and as such will give an energy that lies above the Hartree-Fock theory value. However, the wave function to first or higher-order will lead to an energy superior to that of Hartree-Fock theory. The reason for this is because the orbitals employed in the construction of the wave function are generated self-consistently by a local effective potential energy function that inherently incorporates all the electron correlations – Hartree, Pauli, Coulomb, and Correlation-Kinetic. For the same reason, this wave function will be superior to that of the same order as determined by Møller-Plesset perturbation theory.

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Chapter 19

Epilogue

This book is an extension of my volume on Quantal Density Functional Theory (QDFT) of electronic structure to the development of approximation methods and the application of the time-independent theory to realistic physical systems. Timeindependent Quantal Density Functional Theory (Q-DFT) of electronic structure is the mapping from an interacting system of electrons in their ground or excited state, and in the presence of an external field F ext .r/ such that F ext .r/ D r v.r/, to one of model noninteracting fermions with equivalent density .r/. (There is also a Q-DFT where the mapping is to a model system of noninteracting bosons.) The corresponding energy and ionization potential (or electron affinity) of the electrons are also thereby obtained. The description of both the interacting and model quantum-mechanical systems is in terms of “classical” fields and their quantal sources. In a manner similar to that of classical physics, this perspective makes the subject tangible. In this book I have attempted to demonstrate many different facets of time-independent Q-DFT as follows. (1) The applicability of Q-DFT to determine the electronic structure of matter. To do so I have chosen as examples disparate nonuniform electron gas systems with different symmetries: ground and excited states of atoms and ions of the Periodic Table, the anion–positron complex, the Hydrogen molecule, the metal-vacuum interface. (2) The applicability of the Q-DFT of the Density Amplitude for which the mapping is to a model system of noninteracting bosons is demonstrated for atomic examples. (3) As a consequence of the fact that within Q-DFT, the contributions of the electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and Correlation-Kinetic effects are explicitly defined, a key facet of Q-DFT demonstrated is the ability to study the electronic structure of a physical system systematically as a function of these different electron correlations. (4) As such approximation schemes within Q-DFT, both ad hoc variational–self-consistent and perturbative, are devised to account for these correlations in a systematic manner. (5) In Q-DFT, there is a rigorous physical interpretation of the equations governing the different electron correlations in terms of their quantal sources and fields. These quantal sources and fields are in turn defined in terms of the true interacting and model noninteracting fermion (or boson) wave functions. The physics then allows for the determination of the exact analytical structure of many properties, and the contribution of each of the different electron correlations to that property. Hence, as demonstrated, it is possible

373

374

19 Epilogue

to analytically determine properties at and near the nucleus of atoms, and in the classically forbidden region of atoms and molecules, and the far vacuum region of the metal–vacuum interface. (6) As noted in the Introduction, Q-DFT generalizes the concept of local effective potential energy theory beyond that originally understood on the basis of Slater theory, the Optimized Potential method, and Kohn–Sham density functional theory. This generalization of local effective potential energy theory too is demonstrated. (7) Finally, I have included recent new understandings of the fundamental theorem of time-independent Hohenberg–Kohn density functional theory, and of its extension to the time-dependent case due to Runge-Gross. Having demonstrated the earlier mentioned attributes of Q-DFT, the book then sets the stage for future work. There is the application of the various approximation methods of time-independent Q-DFT described in the book to atoms of the Periodic Table with all the electron correlations accounted for, and their application to study larger molecular and periodic condensed matter systems as a function of the different electron correlations. Time-dependent (TD) Q-DFT, described in QDFT, may be applied to determine the excitation energies of atoms, and to study atoms in the presence of strong electric fields. There is then the application to atoms and molecules of the recently developed Multi-Component TD Q-DFT to include the motion of the nuclei and thereby go beyond the Born-Oppenheimer Approximation employed in the book. From the fundamental perspective, there is the extension of TD Q-DFT to incorporate time-dependent electromagnetic fields, and the development of a relativistic Q-DFT. There are numerous other problems that may be addressed by Q-DFT, such as those of systems with a different dimensionality, and so on. The seeds of Q-DFT were sown by the original attempt to understand the physics of Kohn–Sham density functional theory. That understanding has evolved into the picture that is Q-DFT as described in the two volumes. Having explained the fundamental ideas and demonstrated the applicability of Q-DFT in these books, it is my hope that others will further both the theory as well as the approximation methods and applications. I hope you have enjoyed the book.

Appendix A

Quantal Density Functional Theory of Degenerate States

In this appendix, the Q-DFT of both ground and excited degenerate states, and for the cases of both the pure state and the ensemble v-representable densities, is described [1]. The interest in the ensemble cases stems from a ground state theorem due to Levy [2] and Lieb [3]. According to the theorem, most ensemble densities constructed from pure degenerate ground states are not interacting v-representable. In other words, no single ground state wave function of the Schr¨odinger Hamiltonian (2.1) will yield this ensemble density. Such ensemble densities are said to be ensemble v-representable. The translation of the theorem to the S system means that there is no single Slater determinant that leads to this ensemble density. At this time, the question of interacting v-representability of the ensemble density of degenerate excited states is still unanswered. Within Q-DFT, the densities concerned are always those obtained from the solution of the Schr¨odinger equation (2.5). Thus, the pure state density, and each component of the ensemble density, are interacting v-representable. The assumption of existence of an S system in Q-DFT therefore means that the pure state density, and each component of the ensemble density, are also noninteracting v-representable. We begin by (1) describing the Q-DFT of the individual degenerate pure state. For the mapping from a degenerate ground or excited state, the state of the S system is arbitrary in that it may be in a ground or excited state. In either case, the highest occupied eigenvalue is the negative of the ionization potential. For the ground and excited state ensemble cases, two different schemes within Q-DFT are described. Thus, (2) in the first, the corresponding noninteracting system ensemble density is obtained by constructing g S -systems, where g is the degeneracy of the state. Once again, the g S -systems may either be in a ground or excited state or a combination of the two. Next, (3) we describe the Q-DFT whereby the ensemble density is obtained from a single noninteracting fermion system whose orbitals could be degenerate. The construction of this model system is a consequence of the linearity of the “Quantal Newtonian” first law. Here the highest occupied eigenvalue is degenerate, and the ensemble density is obtained from the resulting Slater determinants as described by Ullrich and Kohn [4] whose work in turn is based on that of Chayes et al. [5]. Again for the mapping from an excited state, the noninteracting system may be in a ground or excited state. Examples demonstrating the above mappings within Q-DFT are then given. Finally, in QDFT a rigorous

375

376

A Quantal Density Functional Theory of Degenerate States

physical interpretation was provided of the energy functionals and functional derivatives of ground state Kohn–Sham density functional theory [6] (KS-DFT). The above mappings in turn explain the physics underlying all the various [4, 7, 9] Kohn–Sham theory degenerate state energy density and bidensity functionals and of their functional derivatives. (1) The Q-DFT of the bound individual degenerate pure state is as follows. The Schr¨odinger equation for a degenerate state whether ground or excited is h i HO ‰n; .X / D TO C VO C UO ‰n; .X / D En ‰n; .X /;

(A.1)

P P P TO D 12 i ri2 ; VO D i v.r i /; UO D 12 0i;j .1=jr i  r j j/, where ‰n; .X / and En are a bound degenerate state wave function and energy, n corresponds to the state, and  D 1; : : : ; gn the degeneracy, X D x 1 ; : : : ; x N , x D r, with  the spin coordinate. As the following equations are valid for arbitrary states, we drop the subscript n. The degenerate pure state density  .r/ D h‰ jj‰ O i, where O D P O j‰ i. ı.r  r /, and the energy E D h‰ j H i



i The corresponding differential equation for the S system of noninteracting fermions with the same density is

with

  1  r 2 C v.r/ C vee; .r/ i .x/ D i i .x/I 2 i D 1; : : : ; N;

(A.2)

˝ ˛ X  .r/ D ˆ fi g jjˆ ji .x/j2 ; O fi g D

(A.3)

i;

and ˆ fi g is the single Slater determinant of the orbitals i .x/. This is the S system wave function. The electron-interaction potential energy vee; .r/ is representative of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and Correlation-Kinetic effects. Correlation-Kinetic contributions to the potential energy are a consequence of the difference in kinetic energy between the interacting and noninteracting systems. The potential energy vee; .r/ is the work done to move a model fermion in the force of a conservative field F .r/: Z vee; .r/ D 

r 1

F .r 0 /  d` 0 ;

(A.4)

where F .r/ D E ee; .r/ C Z tc ; .r/. The fields E ee; .r/ and Z tc ; .r/ are not necessarily conservative. Their sum always is. The electron-interaction field E ee; .r/ is representative of the Pauli and Coulomb correlation: E ee; .r/ D e ee; .r/= .r/, where the electron-interaction “force” e ee; .r/ is obtained via Coulomb’s law as R e ee; .r/ D dr 0 P .rr 0 /.r  r 0 /=jr  r 0 j3 , where P .rr 0 / D< ‰ jPO .rr 0 /j‰ >, P with PO .rr 0 / D 0i;j ı.r  r i /ı.r 0  r j /. Equivalently, the field E ee; .r/ may be

A Quantal Density Functional Theory of Degenerate States

377

thought as being due to its quantal source, the pair-correlation density g .rr 0 / D P .rr 0 /= .r/. The Correlation-Kinetic field Z tc ; .r/ D Z s; .r/Z .r/, Z .r/ D z .rI Œ /= .r/, Z s; .r/ D zs; .rI Œs; /= .r/, and where Z .r/ and Z s; .r/ are the interacting and S system kinetic Pfields, respectively. The kinetic force z .r/ is defined by its component z ;˛ D 2 ˇ @t˛ˇ =@rˇ , with t˛ˇ .r; Œ / D 1 2 Œ@ =@r˛0 @rˇ00 C @2 =@rˇ0 @r˛00  .r 0 r 00 /jr 0 Dr 00 Dr the kinetic energy density tensor. The 4 source of the kinetic field Z .r/ is the spinless single particle density matrix P O i, XO D AO C iB, O AO D 1  .rr 0 / D h‰ jXj‰ j Œı.r j  r/Tj .a/ C ı.r j  2 P i 0 r /Tj .a/, BO D  2 j Œı.r j  r/Tj .a/  ı.r j  r 0 /Tj .a/, Tj .a/ is a translation operator, and a D r 0  r. The field Z s .r/ is defined in a similar manner O fi gi D in Dirac density matrix s; .rr 0 / D hˆ fi gjXjˆ Pterms? of the S system 0  .r/ .r /. i i; i The proof of (A.4) follows by equating the “Quantal Newtonian” first laws (2.11) and (3.5) for the interacting and S systems which are, respectively, r v.r/ D F .r/

and r v.r/ D F s; .r/;

(A.5)

where F .r/ D E ee; .r/ C D .r/ C Z .r/, F s; .r/ D r vee; .r/ C D .r/ C Z s; .r/, the differential density field D .r/ D d .r/= .r/, and d .r/ D  14 r r 2  .r/. Thus, one obtains r vee; .r/ D F .r/;

(A.6)

from which the interpretation of (A.4) follows. The total energy of the degenerate state  is then Z E D Ts; C

 .r/v.r/dr C Eee; C Tc; ;

(A.7)

where Ts; D hˆ fi gjTO jˆ fi gi is the S system kinetic energy, and the electroninteraction Eee; and Correlation-Kinetic Tc; energies in terms of the respective fields are Z Eee; D dr .r/r  E ee; .r/; (A.8) and Tc;

1 D 2

Z dr .r/r  Z tc ; .r/:

(A.9)

These expressions are independent of whether the fields E ee; .r/ and Z tc ; .r/ are conservative or not. The S system whereby the density and total energy equivalent to that of the interacting system degenerate state  is defined by (A.2)–(A.4) and (A.7). Irrespective of whether the interacting system is in a ground or excited state, the S system may be constructed to be either in a ground or excited state. Since the electroninteraction field remains unchanged, the difference between the corresponding

378

A Quantal Density Functional Theory of Degenerate States

potential energies is independent of the Pauli principle and Coulomb repulsion and due entirely to the corresponding correlation-kinetic fields Z tc ; .r/. Hence, the potential energy vee; .r/ is different depending on whether the S system is in a ground or excited state. In either case, the highest occupied eigenvalue of the S system differential equation is the negative of the ionization potential. This follows by equating the asymptotic structure of the density for the interacting and S systems (see Sect. 3.7). In the transformation from an excited pure degenerate state to an S system in its ground state, the fact that the interacting system wave function has nodes is of no relevance. By construction, the S system and interacting system density .r/ are equivalent, and the density .r/  0. Such a mapping for an excited pure nondegenerate state is demonstrated in [10, 11] and in QDFT. (2) We next describe the first of two ways of obtaining the ensemble density and energy of the degenerate states via Q-DFT. The interacting system ensemble density 0 O matrix operator D.XX / is defined as 0 O D.XX /D

g X

g X

! ‰ ? .X /‰ .X 0 /I

D1

! D 1I

0  !  1;

(A.10)

D1

where ! is a weighting factor, and g the degeneracy. Then the ensemble density ens .r/ and energy Eens are, respectively, ens .r/ D tr.DO / O D

g X

!  .r/;

(A.11)

! E .r/;

(A.12)

D1

and Eens D tr.DO HO / D

g X

D1

with  .r/ and E as defined previously. (There are ensemble densities that cannot be represented by a single Slater determinant. However, its pure state component density can always be reproduced by an S system.) For each degenerate state , the density  .r/ and energy E can be constructed from an S system as described in part (1). Thus, the ensemble density and energy of (A.11) and (A.12) may be obtained from g S -systems. Each S system contributing to the ensemble density may be in a ground or excited state. Note that the electron-interaction potential energy vee; .r/ for each of the g S -systems will be different. Further, vee; .r/ will be different depending on whether the particular S system is in a ground or excited state as explained previously. Thus, the ensemble density and energy within Q-DFT are obtained by replacing the  .r/ and E on the right-hand sides of (A.11) and (A.12) by the corresponding S system equivalents of (A.3) and (A.7), respectively.

A Quantal Density Functional Theory of Degenerate States

379

(3) The ensemble density and energy may also be determined from a noninteracting fermion system whose orbitals could be degenerate as constructed within Q-DFT. According to Chayes et al. [5], the ground state ensemble density may be determined as a unique weighted sum of squares of a finite number g of degenerate wave functions of this system. The potential energy vee .r/ of these noninteracting fermions is then determined via Q-DFT as follows. Rewrite the interacting and noninteracting system “Quantal Newtonian” first laws of (A.5) as g X

ens .r/r v.r/ D

! f .r/;

(A.13)

D1

where f .r/ D e ee; .r/  d .r/  z .r/, and ens .r/r v.r/ D

g X

! f s; .r/;

(A.14)

D1

where f s; .r/ D  .r/r vee; .r/  d .r/  zs; .r/. Equating (A.13) and (A.14) leads to g g X X !  .r/r vee; .r/ D ! q .r/; (A.15)

D1

D1

where q .r/ D e ee; .r/Cztc ; .r/, and ztc ; .r/ D zs; .r/z .r/. Equation (A.15) is a consequence of the linearity of the “Quantal Newtonian” first law. As we require a single effective potential energy vs .r/ D v.r/ C vee .r/, we replace vee; .r/, in (A.15) by vee .r/ to obtain r vee .r/ D Q.r/;

(A.16)

P where Q.r/ D Πg D1 ! q .r/=e ns .r/. Thus, the electron-interaction potential energy vee .r/ is the work done in the conservative field Q.r/: Z vee .r/ D 

r

Q.r 0 /  d` 0 :

(A.17)

1

Note that the components q .r/ are conservative so that Q.r/ is conservative, and hence vee .r/ is path-independent. For the occupation of orbitals we follow Ullrich-Kohn [4]. Accordingly, all levels are occupied except the highest (h) which is q-fold degenerate and partially occupied. The numbers of the model fermions in these levels are N h  2q. The ensemble density which is a weighted sum of the degenerate Slater determinants is e ns .r/ D

NX N h i;

ji .x/j2 C

g X i;

fi jih .xI R/j2 ;

(A.18)

380

A Quantal Density Functional Theory of Degenerate States

P with 0  fi  1, and fi D g D1 ! i; where i; D 1 if the orbital ih .xI R/ occurs in the determinant ˆ fi g, and 0 otherwise. Here the ih .xI R/ are appropriately rotated .R/ orbitals determined self-consistently together with the fi and the lower lying orbitals leading to the ensemble density. The ensemble energy is obtained from the g Slater determinants as in part (2). Once again for an excited state ensemble density, the corresponding noninteracting system may be in a ground or excited state. Note that the methodology of construction of the g S -systems of part (2) also follows from (A.15). We believe that it is easier to construct the g S -systems of part (2) than it is to construct the single noninteracting system of part (3). This is because each of the g S -systems may be constructed independently. The Q-DFT mapping from a pure degenerate excited state to an S system can be demonstrated via the first excited triplet state of the exactly solvable Hooke’s atom [12]. This atom is comprised of two electrons with a harmonic external potential energy v.r/ D 12 !r 2 . The triplet state wave functions are of the form ‰.r 1 r 2 / D C0 e

r 

!  ! r C C2 r2 1 C C1 2 2  3=2  ! C C3 r 3 Yl;m .; /; 2

!R2 !r 2 =4

e

(A.19)

where Yl;m .; / is the spherical harmonic with l D 1; m D 1; 0; 1; r D r 2  r 1 ; R D .r 2 C r 1 /=2, and C0 ; C1 ; C2 ; C3 are constants. For each degenerate wave function, the corresponding S system, in either a ground or excited state, can be determined in a manner similar to the transformation of the first excited singlet state [10, 11] (see QDFT). The ensemble density of these degenerate states then follows from the three S systems. (Note that the ensemble density for this two electron model is v-representable. However, the methodology for constructing the g S -systems is the same whether or not the density is v-representable.) Another example is that of the noninteracting Be atom [4]. Here the ensemble density, which is not v-representable, is the weighted sum of the density of four S systems in the states 1s 2 2s 2 , 1s 2 2pi2 .i D x; y; z/. This latter model atom is also an example [4] of the noninteracting fermion system that leads to the ensemble density with appropriately rotated highest occupied orbitals. From the above degenerate state Q-DFT description it is then possible to provide a rigorous physical interpretation for each energy functional and functional derivative of the corresponding KS-DFT. Within KS-DFT, the following cases have been considered: the mapping from (a) a pure degenerate ground state [7]; (b) a pure degenerate excited state [8]. In addition, maps to obtain the density and energy constructed from (c) an ensemble of pure degenerate ground states [4] and (d) an ensemble of pure degenerate excited states [9] have been developed. For case (a), the energy is a functional of the degenerate ground state density; for (b), the energy is a bidensity functional of the ground and excited state densities, with the functional derivative taken at the excited state density; for (c), the energy, which is a functional

A Quantal Density Functional Theory of Degenerate States

381

of the ground state ensemble density, is constructed by the ensemble generalization of the coupling constant scheme; and for (d) the energy is also a bidensity functional, in this instance of the ground state and excited state ensemble densities, with the functional derivative taken at the ensemble density. In each of the above cases, the local potential energy of the model fermion is the work done in a conservative field. The energy in turn may be expressed in terms of the components of this field. Thus, for example, the KS-DFT degenerate ground KS state electron-interaction energy functional Eee Œe ns  [4] of the ensemble density is the ensemble sum of the electron-interaction Eee; and Correlation-Kinetic Tc;

energies. The functional derivative is the work done to move the model fermion in the conservative field Q.r/. The same interpretation applies to the bidensity KS energy functional Eee Œgr ; e ns  [9] of degenerate excited state KS-DFT, and of its functional derivative.

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Appendix B

Generalization of the Runge–Gross Theorem of Time-Dependent Density Functional Theory

In this Appendix we generalize [1] the fundamental theorem of time-dependent (TD) density functional theory due to Runge and Gross [2] (RG) by a unitary or gauge transformation. For a system of N electrons in a TD external field F ext .rt/ such that F ext .rt/ D r v.rt/, with v.rt/ a scalar external potential energy operator, the TD Schr¨odinger equation is @‰.X t/ HO .Rt/‰.X t/ D i ; (B.1) @t where ‰.Xt/ is the wave function, X D x 1 ; : : : ; x N , x D r, r and  are the spatial and spin coordinates, and R D r 1 ; : : : ; r N . The Hamiltonian HO .Rt/ is the sum of the kinetic TO , electron-interaction potential energy WO , and external potential energy VO operators: HO .Rt/ D TO C WO C VO ; (B.2) P P P0 1 1 2 with TO D i . 2 ri /; WO D 2 i;j 1=jr i  r j j; VO D i v.r i t/. The TD density .rt/ is the expectation .rt/ D h‰.X t/j.r/j‰.Xt/i O ;

(B.3)

P where .r/ O D i ı.r  r i / is the density operator. The RG theorem is proved for the Hamiltonian HO .Rt/ of (B.2). It is proved on the assumption that the scalar operator v.rt/ is Taylor expandable about some initial time t0 . Furthermore, in the proof, the operators TO and WO , and the initial many-particle state ‰.t0 /, are assumed known and kept fixed. For fixed TO , WO , and ‰.t0 /, RG prove that the relationship between the density .rt/ and the external potential energy operator v.rt/ to within a function C.t/ is bijective: .rt/ $ v.rt/ C C.t/. Thus, knowledge of the density .rt/ of a system uniquely determines a Hamiltonian HO .Rt/ to within C.t/, and therefore via solution of the TD Schr¨odinger equation, the wave function ‰.t/. As a consequence of the bijectivity, the wave function is a functional of the density unique to within an arbitrary time-dependent phase factor ˛.t/. Hamiltonians that differ by C.t/ are considered equivalent in the sense that they lead to the same density .rt/.

383

384

B Generalization of the Runge–Gross Theorem

The TD unitary operator U we employ is UO D ei˛.Rt / ;

(B.4)

so that the transformed wave function ‰ 0 .X t/ is ‰ 0 .X t/ D UO  ‰.Xt/;

(B.5)

0 O .X t/i D .rt/. The unitary and the transformed density 0 .rt/ D h‰ 0 .X t/j.r/j‰ transformation thus preserves the density. The transformed Schr¨odinger equation is 0

@‰ .X t/ HO 0 .Rt/‰ 0 .X t/ D i ; @t

(B.6)

where the Hamiltonian HO 0 .Rt/ of the transformed system is d˛.Rt/ HO 0 .Rt/ D UO  HO .Rt/U C dt io d˛.Rt/ Xn h 1 D HO .Rt/  : UO  ri2 ; UO C 2 dt

(B.7) (B.8)

i

(Note that for the transformed system, the initial state and other boundary conditions too are transformed.) The solution of the commutator of (B.8) is the same as given in Sect. 4.3.1. Thus, the transformed Hamiltonian HO 0 .Rt/ is 1X d˛.Rt/ Oi CA O i  pO i C A O 2i /; C .pO i  A HO 0 .Rt/ D HO .Rt/ C dt 2

(B.9)

i

where pO i D i r i is the momentum operator, and where the vector potential energy O i D 0. O i D r i ˛.Rt/ so that r  A operator is defined as A The transformed Hamiltonian may also be written as 1X O i /2 C WO C VO 0 ; HO 0 .Rt/ D .pO i C A 2

(B.10)

i

where

d˛.Rt/ : (B.11) VO 0 D VO C dt Note that as is the case for the Hamiltonian HO .Rt/ of (B.2), there is no magnetic field in the transformed Hamiltonian HO 0 .Rt/. The vector potential energy operator O i as defined above is curl-free. A That HO .Rt/ and HO 0 .Rt/ represent the same physical system may also be seen by performing the following gauge transformation of HO .Rt/ to obtain HO 0 .Rt/ W / Oi ! A O0 D A O i C r i ˛.Rt/ with A O i D 0 so that and A VO ! VO 0 D V C d˛.Rt i dt

B Generalization of the Runge–Gross Theorem

Ψ(Xt)

385

Ĥ(Rt)

Ψ1′(Xt) = e -iα¹(Rt) Ψ(Xt)

Ĥ1′(Rt) = Ĥ(Rt) + dα1(Rt)/dt

+ ½∑i ( pˆ i · Âi1 + Âi1 · pˆ i + Â2i1)

ρ(rt) Ψ2′(Xt) = e -iα²(Rt) Ψ(Xt)

Same physical system

Ĥ2′(Rt) = Ĥ(Rt) + dα2(Rt)/dt + ½∑i ( pˆ i · Âi2 + Âi2 · pˆ i + Â2i2)

· · ·

· · ·

Fig. B.1 The generalization of the fundamental theorem of density functional theory demonstrating the bijectivity between the density of a physical system and the infinite set of Hamiltonians representing that physical system. The figure is drawn for (a) the most general time-dependent form of the gauge function ˛.Rt /. The figure reduces to the RG theorem for (b) when ˛.Rt / D ˛.t /. The figure further reduces to the most general form of the time-independent theorem when (c) ˛.Rt / D ˛.R/. Finally, the Hohenberg–Kohn theorem is recovered for (d) when ˛.Rt / D ˛, a constant

O 0 D 0. In quantum mechanO 0 D r i ˛.Rt/ and the magnetic field B 0 D r  A A i i ics it is well known [3] that the more general gauge transformation above with nonzero magnetic field B leaves the Schr¨odinger equation invariant provided the wave functions of the original and transformed Hamiltonians are related by the gauge transformation ˛.Rt/ of (B.5). The Hamiltonian HO 0 .Rt/ of (B.9), (B.10) is the most general form of the Hamiltonian for which the RG theorem is valid. It includes the scalar potential energy operator v.r i t/, the TD function C.Rt/ D d˛.Rt/=dt, the momentum operator pO i , O i D r i ˛.Rt/. Pictorially the and the TD curl-free vector potential energy operator A bijectivity of the RG Theorem in its general form is depicted in Fig. B.1. The bijectivity is .rt/ $ HO .Rt/ with HO .Rt/ of (B.2), or equivalently .rt/ $ HO j0 .Rt/ with HO j0 .Rt/ of (B.9), (B.10), depending on the gauge function ˛j .Rt/. The Hamiltonian HO .Rt/ and the Hamiltonian HO 0 .Rt/ all correspond to the same physical system. It is evident that the RG theorem in its original form is recovered from the above generalization for the special case when the gauge function ˛.Rt/ D ˛.t/ (see Fig. B.1). The functions C.t/ of RG are linked to the gauge function: C.t/ D d˛.t/=dt. Furthermore, the Hamiltonians HO 0 .Rt/ D HO .Rt/ C C.t/ all correspond to the same physical system because HO 0 .Rt/ is obtained from HO .Rt/ by a unitary or gauge transformation.

386

B Generalization of the Runge–Gross Theorem

It is also clear from the unitary or gauge transformation that in the general case the wave function ‰.X t/ must be a functional of both the density .rt/ and the gauge function ˛.Rt/ i.e., ‰.X t/ D ‰Œ.rt/I ˛.Rt/. This functional dependence of the wave function functional on the gauge function ˛.Rt/ ensures that it is gauge variant. Since the bijectivity is between the density .rt/ of a system and the Hamiltonians representing the same physical system (see Fig. B.1), the choice of gauge function is arbitrary. Thus, the choice ˛.Rt/ D 0 is equally valid. Thus, in the RG case, the choice of ˛.t/ D 0 leads to a wave function functional that can be a functional only of the density .rt/. In the RG case, Fig. B.1 shows that the bijectivity is between the density .rt/ and the infinite number of Hamiltonians HO .Rt/ C C.t/ representative of a physical system. Thus, the density uniquely determines the system Hamiltonian to within a function C.t/. It is, however, possible to construct [[4], QDFT] an infinite set of degenerate Hamiltonians fHO g that differ by a function C.t/, represent different physical systems, but yet possess the same density .rt/. In such a case, the density .rt/ cannot distinguish between the different physical systems. For such systems, the RG theorem is not valid. (See Corollary to the RG theorem in QDFT.) Finally, as a consequence of the unitary or gauge transformation, the following hierarchy exists in the fundamental theorem of density functional theory (see Fig. B.1). When the gauge function is ˛.Rt/, one obtains the most general form of the time-dependent theorem. For the gauge function ˛.t/, one recovers the original RG theorem. When the gauge function is ˛.R/, one obtains the most general form of the time-independent theorem. Finally, when the gauge function is the constant ˛, one recovers the original Hohenberg-Kohn theorem. (Note that the function C.t/ of the RG theorem does not reduce to the constant C of the Hohenberg-Kohn theorem.) This hierarchy makes the role of the phase factor as significant in density functional theory as it is in quantum mechanics.

Appendix C

Analytical Asymptotic Structure of the Correlation-Kinetic Potential Energy in the Classically Forbidden Region of Atoms

In this Appendix, we derive [1] the analytical asymptotic structure of the Correlation-Kinetic potential energy Wtc .r/ in the classically forbidden region of atoms. The derivation is in the framework of quasi-particle amplitudes as described by Almbladh and von Barth [2]. In second quantized notation, the single-particle density matrix .xx0 / is D .xx0 / D N j



E .x 0 / .x/jN ;

(C.1)

where jN i is the N -electron ground state wave function, .x/,  .x/ are respectively, the annihilation and creation field operators, and x D r. By inserting the complete set of the .N  1/-electron system eigen states fjN  1; sig into (C.1) we have X .xx0 / D fs .x/fs? .x 0 /; (C.2) s

where the matrix elements fs .x/ D hN  1; sj .x/jN i

(C.3)

are the quasi-particle amplitudes. These amplitudes are the interacting system counterparts of the one-electron orbitals of the noninteracting S system. Following Almbladh and von Barth, the differential equation for these amplitudes is   1 2  r C v.x/ fs .x/ C hN  1; sjOvH .r/ .x/jN i 2 D s fs .x/;

(C.4)

where s RD E0  Es.N 1/ (see (7.23)) is an exact one-particlePexcitation energy, vO H .r/ D dr 0 .r O 0 /=jr  r 0 j the Hartree operator, and .r/ O D   .x/ .x/ D P i ı.r  r i / is the density operator. By performing a multipole expansion of the 387

388

C Asymptotic Structure of Correlation-Kinetic Potential Energy

Coulomb interaction 1=jr  r 0 j, one obtains   1 2  r C v.x/ C hN  1jOvH .r/jN  1i  s fs .x/ 2 # " 1 X 1 X Dss 0 .r/ O C 3 Qss 0 .r/ O fs 0 .x/ D 0; C 2 r r 0 0 s

(C.5)

s

O and Qss 0 .r/ O are the dipole and quadrupole moment matrix elements where Dss 0 .r/ defined as ˛ ˝ Dss 0 .r/ O D rO  N  1; sjdjN  1; s 0 E D X Qss 0 .r/ O D rO˛ rˇ  N  1; sjQO ˛ˇ jN  1; s 0

(C.6) (C.7)

˛ˇ

with Z dD

rı .r/dr; O

ı .r/ O D .r/ O  hN  1j.r/jN O  1i ; Z

1 3r˛ rˇ  ı˛ˇ r 2 ı .r/dr; QO ˛ˇ D O 2

(C.8) (C.9) (C.10)

and rO D r=r. (In the asymptotic region, (C.5) and (7.22) are equivalent.) Equation (C.5) is the same as that derived by Almbladh and von Barth, but carried further to include the quadrupole moment term. As we have seen in Chap. 7, to study the asymptotic structure of vee .r/ to 0.1=r 5/, it is necessary to include the quadrupole term in the expansion of the Coulomb interaction. However, as also shown there, there is no quadrupole contribution of 0.1=r 5 / to the asymptotic structure of vee .r/. Thus, in the derivations to follow, we drop the quadrupole terms, while simultaneously bearing in mind that the results are correct to 0.1=r 5/. Following the same arguments as those below (7.24), we then have asymptotically  fs .x/ r!1 

.N 1/

1 Ds0 .r/ O f0 .x/; 2 r !s

(C.11)

.N 1/

 E0 is an excitation energy of the .N  1/-electron where !s D Es system. This, of course, is the same as (7.26) but without the quadrupole moment term. Substituting for fs .x/ from (C.11) into (C.2), we obtain " 0

.xx /

 f0 .x/f0? .x 0 / r;r 0 !1

# 0 ? X 1 Ds0 .r/D O s0 .rO 0 / : 1C !s2 r 2 r 02 s

(C.12)

C Asymptotic Structure of Correlation-Kinetic Potential Energy

389

Now ˛ 0 ˝ 0 ? X X X N  1jdˇ jN  1; s hN  1; sjd˛ jN  1i Ds0 .r/D O s0 .rO 0 / 0 D rO˛ rOˇ w2s !s2 s s ˛ˇ

(C.13) D

X

rO˛ rOˇ0 ˛ˇ ;

(C.14)

˛ˇ

where ˛ 0 ˝ X N  1jdˇ jN  1; s hN  1; sjd˛ jN  1i !s2 s + * ˇ ˇ ˇ ˇ 1P ˇ ˇ D N  1ˇdˇ h i2 d˛ ˇN  1 ; HO  E0.N 1/

˛ˇ D

(C.15) (C.16)

˛; ˇ are cartesian coordinates, rO˛ is a component of the unit vector r, O and P D jN  1ihN  1j is the projector onto the .N  1/-electron ground state. Note that the coefficient ˛ˇ is an expectation value with respect to the .N  1/-electron ground state. Other than that, ˛ˇ does not appear to have any physical interpretation. Thus, 2

.xx0 /



r;r 0 !1

3 X 1 f0 .x/f0? .x 0 / 41 C 2 02 rO˛ rOˇ0 ˛ˇ 5 : r r

(C.17)

˛ˇ

As the .N  1/-electron system is spherically symmetric, the coefficient ˛ˇ is diagonal, ˛ˇ D ı˛ˇ . Hence, the interacting system density matrix may be written as h i .xx0 / 0 f0 .x/f0? .x 0 / 1 C 2 02 : (C.18) r;r !1 r r Next we determine the asymptotic structure of the kinetic-energy-density tensor t˛ˇ .rI Œ/ of (2.54): " # ˇ ˇ 1X @2 @2 0 00 ˇ .r C ; r / t˛ˇ .r/ D ˇ 0 00 : 00 0 0 00 4  @r˛ @rˇ @rˇ @r˛ r Dr Dr

(C.19)

Substituting (C.18) into (C.19) we have "

@2 @2 C 00 0 @r˛0 @rˇ @rˇ @r˛00 h n  f0 .x 0 /f0? .x 00 / 1 C

1 t˛ˇ .r/ D 2

# io r 02 r 002 r 0 Dr 00 Dr

(C.20)

390

C Asymptotic Structure of Correlation-Kinetic Potential Energy

!   @ ? 00

1 @ 0 f0 .x / D 1 C 02 002 00 f0 .x / 0 2 @r˛ @rˇ r r  @ 1 Cf0 .x 0 /f0? .x 00 / 02 00 002 r @rˇ r     1 @ @ ? 00

0 f .x / f .x / 1 C C 0 2 @rˇ0 @r˛00 0 r 02 r 002  @ 1 Cf0 .x 0 /f0? .x 00 / 02 00 002 r @r˛ r r 0 Dr 00 Dr 



 

 @ @ ? f0 .x/ f0 .x/ 1 C 4 @r˛ @rˇ r   1 @ ? @ 1 f0 .x/ 2 C f0 .x/ 2 @rˇ r @r˛ r 2   1 @ 1 @ C f0 .x/ f0? .x/ 2 2 @r˛ r @rˇ r 2    1 @ 1 @ 1 ? C f0 .x/f0 .x/ 2 @r˛ r 2 @rˇ r 2   

1  @ @ ? C f0 .x/ f0 .x/ 1 C 4 2 @rˇ @r˛ r   1 @ ? @ 1 f .x/ 2 C f0 .x/ 2 @r˛ 0 r @rˇ r 2   1 @ 1 @ C f0 .x/ f0? .x/ 2 2 @rˇ r @r˛ r 2    1 @ 1 @ 1 ? : C f0 .x/f0 .x/ 2 @r˛ r 2 @rˇ r 2

1 D 2

(C.21)

(C.22)

As the quasi-particle amplitude and wave function expansion methods are equivalent, we know from the derivation in QDFT that the asymptotic structure of f0 .x/ depends only on r (see also [1–5]]). This structure is  f0 .x/ r!1 r  e 0 r .1 C A1 r 1 C A2 r 2 C A3 r 3 C : : :/;

(C.23)

where 1 C  D ZN  N C 1, ZN is the total charge of the nuclei, 02 =2 D E0.N 1/  E0 , and the Ai are coefficients. Usingthis fact and the relation @f .r/=@r˛ D

C Asymptotic Structure of Correlation-Kinetic Potential Energy

391

.r˛ =r/.@f .r/=@r/ in (C.22) the tensor of the interacting system is  ?   @f0 .x/

 1 r˛ rˇ @f0 .x/ 1 C t˛ˇ .r/ D 2 r2 @r @r r4

3 2 @ ? ? f0 .x/f0 .x/ C 6 f0 .x/f0 .x/  5 r @r r C 6 ı˛ˇ f0 .x/f0? .x/: 2r

(C.24)

We next need to determine the kinetic-energy-density tensor ts;˛;ˇ .rI Œs / of the S system. The asymptotic structure of the density .r/ and Dirac density matrix s .rr 0 / are governed by the highest occupied eigenfunction N .x/ of the S system differential equation. The diagonal element of the density matrix .rr 0 / is then  .r/ r!1

X 

h i f0 .x/f0? .x/ 1 C 4 ; r

(C.25)

D 2jN .x/j2 ;

(C.26)

h i  f0 .x/ 1 C 4 : N .x/ r!1 2r

(C.27)

so that asymptotically

As such, asymptotically, X

s .rr 0 / D



X 

D

X 

? N .x/N .x 0 /

(C.28)

h i ih f0 .x/f0 .x 0 / 1 C 4 1 C 04 : 2r 2r

(C.29)

The S system kinetic-energy-density tensor ts;˛;ˇ .r/ which is defined as " # ˇ ˇ 1X @2 @2 0 00 ˇ  C .r ; r / ts;˛ˇ .r/ D s ˇ 0 00 ; 00 0 0 00 4  @r˛ @rˇ @rˇ @r˛ r Dr Dr

(C.30)

then reduces on performing the various partial derivatives in the above manner to  ?   @f0 .x/

1 r˛ rˇ  @f0 .x/ 1 C ts;˛ˇ .r/ D C 2 r2 @r @r r4 4r 8

 2

4 @ 2 f0 .x/f0? .x/ C 10 f0 .x/f0? .x/ : (C.31)  5 1C 4 r 2r @r r

392

C Asymptotic Structure of Correlation-Kinetic Potential Energy

Thus, on neglecting terms of 0.1=r 8/ and higher-order, we have ts;˛ˇ .rI Œs /  t˛ˇ .rI Œ/   3r˛ rˇ ı˛ˇ 1 D C 6 f0 .x/f0? .x/: 2 r8 r

(C.32)

Observe the similarity of the interacting and noninteracting system tensors. An important difference, however, is the term with the delta function in the expression for t˛ˇ .rI Œ/ which is absent P in ts;˛ˇ .rI Œs /. This term contributes to the kinetic-energy-density t.r/ D ˛ t˛˛ .r/, and thus to the difference in kinetic energy of the two systems. Asymptotically, the difference Œts .r/  t.r/ is precisely P .3 =r 6 /  f0 .x/f0? .x/. Now the ˛th component of the Correlation-Kinetic field Z tc .r/ is .r/Ztc ;˛ .r/ D 2

3 X

@ ts;˛ˇ .rI Œs /  t˛ˇ .rI Œ/ ; @rˇ

(C.33)

ˇ D1

which on substituting for f0 .x/ from (C.23) becomes 80

3 X

jf0 .x/j2

ˇ D1

Since, asymptotically, .r/ the field Z tc .r/ as

P 

r˛ rˇ rˇ  : r8 r

(C.34)

jf0 .x/j2 , we obtain the asymptotic structure of

 80 Z tc .r/ r!1

r ; r7

(C.35)

and the work done in this field as Z

r

Ztc .r 0 /dr 0 Z r 80 1 0 D 80 dr D : 06 r 5r 5 1

Wtc .r/ D 

(C.36)

1

Thus, the Correlation-Kinetic potential energy decays as 0.1=r 5/.

(C.37)

Appendix D

The Pauli Field E x .r/ and Potential Energy Wx .r/ in the Central Field Approximation

In this appendix, we derive [1] the expression for the Pauli field E x .r/ and potential energy Wx .r/ for open-shell atoms in the Central Field Approximation. The Pauli field E x .r/ due to the Fermi hole charge x .rr 0 / is defined as (see 3.32) Z 1 E x .r/ D  x .rr 0 /r r dr 0 ; (D.1) jr  r 0 j where (3.21) x .rr 0 / D 

js .rr 0 /j2 ; 2.r/

(D.2)

with s .rr 0 / the Dirac density matrix (3.12), and .r/ D s .rr/ the density. Employing the identity i r  r r D @=@r, the radial component of the Pauli field is Ex;r .r/ D i r  E x .r/ Z @ 1 dr 0 : D  x .rr 0 / @r jr  r 0 j

(D.3) (D.4)

The spheral average of this radial component is then 1 Ex;r .r/ D  4

Z

x .rr 0 /

@ 1 dr 0 d r ; @r jr  r 0 j

(D.5)

which is spherically symmetric. For closed shell atoms, this is automatically the case. In the central field model of atoms, the single particle orbitals may be written as nlm .r/ D Rnl .r/Ylm ./;

(D.6)

where Rnl .r/ is the radial part of the orbital and Ylm ./, the angular part, is the spherical harmonic of order (lm).

393

394

D Pauli Field Ex .r/ and Potential Wx .r/ in Central Field Approximation

The Fermi hole x .rr 0 / of (D.2) is then x .rr 0 / D 

2 X Rnl .r/Rnl .r 0 /Rn0 l 0 .r/Rn0 l 0 .r 0 / .r/ nlm n0 l 0 m0

? Ylm . /Ylm . 0 /Yl 0 m0 . /Yl?0 m0 . 0 /;

(D.7)

and the density .r/ is .r/ D

1 X 2 .2l C 1/Rnl .r/: 2

(D.8)

nl

Employing the expansion 00 X r
Y 00 00 . /Yl m . / l 00 C1 ; jr  r 0 j 2l 00 C 1 l m r> 00 00

(D.9)

l m

where r< .r> / is the smaller (larger) of jrj and jr 0 j, the expression for the electric field becomes Z X 2 1 Ex;r .r/ D Rnl .r/Rnl .r 0 /Rn0 l 0 .r/Rn0 l 0 .r 0 / 00 .r/ 2l C 1 nlm n0 l 0 m0 l 00 m00

# 00 @ rl 00 C1 Z ?  Ylm . /Yl 0 m0 . /Yl?00 m00 . /d Z  Ylm . 0 /Yl?0 m0 . 0 /Yl 00 m00 . 0 /d 0 : "

(D.10)

Using the orthonormality condition of the spherical harmonics, which is [2] Z Yl?0 m0 . /Ylm . /d D ıl 0 l ım0 m ;

(D.11)

and the coupling rule for the spherical harmonics, which is [2] Yl1 m1 . /Yl2 m2 . / D

X  .2l1 C 1/.2l2 C 1/ 1=2 l

4 .2l C 1/

C.l1 l2 lI m1 m2 /C.l1 l2 lI 000/Yl;m1Cm2 . /; (D.12) where C are the Clebsch–Gordan [2] coefficients, the integrals over the solid angles in (D.10) which are complex conjugates of each other are

D Pauli Field Ex .r/ and Potential Wx .r/ in Central Field Approximation

395

Z

Yl?3 m3 . /Yl2 m2 . /Yl1 m1 . /d



.2l1 C 1/.2l2 C 1/ D 4 .2l3 C 1/

1=2

C.l1 l2 l3 I m1 m2 m3 /C.l1 l2 l3 I 000/ım3;m1 Cm2 : (D.13)

Substituting for these integrals, the expression for the spherically averaged field becomes Z X 1 Ex;r .r/ D Rnl .r/Rnl .r 0 /Rn0 l 0 .r/Rn0 l 0 .r 0 / 2 .r/ nlm n0 l 0 m0 l 00

# 00 @ rl 00 C1 "



.2l C 1/ 2 00 0 C .l l l I m; m0  m; m0 /C 2 .l l 00 l 0 I 000/: .2l 0 C 1/

(D.14)

For closed-subshell atoms, an occupancy of .2l C 1/ is assumed in the above expression for the electric field. For open-subshell atoms, a partial occupancy of Nl =.2l C 1/, where Nl is the number of electrons in the subshell, should be used. Since the field Ex;r .r/ is spherically symmetric, its curl vanishes. Hence, the work done Wx .r/ in this field is path-independent, spherically symmetric, and given by the integral Z r

Wx .r/ D 

1

Ex;r .r 0 /dr 0 :

(D.15)

With the expressions for the density .r/ of (D.8) and the spherically averaged field Ex;r .r/ of (D.14), the Pauli energy Ex may then be obtained from (3.47).

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Appendix E

Equations of the Irrotational Component Approximation as Applied to the Carbon Atom

In this appendix the analytical and integral expressions of the Irrotational Component Approximation as applied to the Carbon atom in its 1s 2 2s 2 2pz2 degenerate state are given [1]. The orbitals employed are Hydrogenic as given by (12.6). To simplify the expressions we define q D .r/; q 0 D .r 0  0 /; z D r cos ; z0 D r 0 cos  0 ; p D .1  3r/; p 0 D .1  3r 0 /. The functions f1 .rr 0 /, and f2 .qq 0 / up to f5 .qq 0 / are defined as 0

f1 .rr 0 / D e3r p  e3r p 0 f2 .qq 0 / D e3r z0  e

3r 0

z

f3 .qq 0 / D z0 p  zp 0

f4 .qq 0 / D z  z0 C 3z i r  .r  r 0 /

f5 .qq 0 / D z  z0 C 3r sin  i   .r  r 0 / :

(E.1) (E.2) (E.3) (E.4) (E.5)

The various vector dot products employed in the expressions are

i r  .r  r 0 / D r  r 0 cos  cos  0 C sin  sin  0 cos.   0 /

i   .r  r 0 / D r 0  sin  cos  0 C cos  sin  0 cos.   0 / i   .r  r 0 / D r 0 sin  0 sin. 0  /

(E.7) (E.8)

i   i  0 D cos  sin.   0 / i   i  0 D cos.   0 /

(E.9) (E.10)

i r  i  0 D sin  sin.   0 /:

(E.11)

(E.6)

E.1 Electron Density .r/ The electronic density .r/ D 2

3 X

ji .r/j2

(E.12)

i D1

397

398

E The Irrotational Component Approximation

has azimuthal symmetry so that 432 6r e A.q/

(E.13)

1 A.q/ D e6r C .p 2 C 9z2 /: 8

(E.14)

.r/ D .q/ D where

E.2 Fermi Hole x .rr 0 / The Fermi hole (without the negative sign) x .rr 0 / D js .rr 0 /j2 =2.r/ with 0

s .rr / D 2

3 X

i? .r/i .r 0 /

(E.15)

(E.16)

i D1

is independent of the azimuthal angles  and  0 so that x .qq 0 / D where

216 6r 0 B 2 .qq 0 / e

A.q/

1 0 B.qq 0 / D e3.rCr / C .pp 0 C 9zz0 /: 8

(E.17)

(E.18)

E.3 Gradient of Fermi Hole r x .rr 0 / r x .qq 0 / D i r

@x 1 @x C i @r r @

(E.19)

where   486B 6r 0 1 @x 3r 3r D f re e f C e .1 C 3r/f cos  C cos  1 2 3 @r

A2 8   486B 6r 0 3r 1 1 @x D e f2 sin  C pf3 sin  : e r @

A2 8

(E.20)

(E.21)

E.6 Vector Vortex Fermi Hole J x .r/

399

E.4 Pauli Field E x .r/ The Pauli field E x .r/ due to the Fermi hole x .qq 0 / of (E.17): Z E x .r/ D

x .rr 0 /

.r  r 0 / 0 dr jr  r 0 j3

(E.22)

has azimuthal symmetry and does not have an azimuthal component. Thus E x .q/ D i r Ex;r .q/ C i  Ex; .q/ Z

where Ex;r .q/ D

Z Ex; .q/ D

i r  .r  r 0 / 0 dr jr  r 0 j3

(E.24)

i   .r  r 0 / 0 dr : jr  r 0 j3

(E.25)

x .qq 0 / x .qq 0 /

(E.23)

E.5 Scalar Effective Fermi Hole eff x .r/ xeff .r/ D

xeff .q/ D

1 .r/ C 2 4

Z

r x .rr 0 / 

.r  r 0 / 0 dr jr  r 0 j3

Z 0  243 Be6r ˚ .q/ C Œ i r  .r  r 0 / re3r f1 2 2 0 3 2 2 A jr  r j 1 Ce3r f2 f4 C f3 f5 dr 0 : 8

(E.26)

(E.27)

E.6 Vector Vortex Fermi Hole J x .r/ The curl of the Pauli field E x .r/ of (E.23) also has azimuthal symmetry but only an azimuthal component: r  E x .r/ D i  Œr  E x .q/ Z

where Œr  E x .q/ D i  

r x .qq 0 /  .r  r 0 / 0 dr jr  r 0 j3

(E.28)

(E.29)

400

E The Irrotational Component Approximation

Z D

@x

i   .r  r 0 /

x  Œi r  .r  r 0 / 1r @ @ dr 0 : jr  r 0 j3

@r

(E.30)

The vector vortex Fermi hole J x .r/ D

1 r  E x .r/ 4

(E.31)

is then J x .q/ D i  Jx; .q/:

(E.32)

E.7 Irrotational Component E Ix .r/ of the Pauli Field E x .r/ As with the Pauli field E x .r/, its irrotational component Z E Ix .r/

D

xeff .r 0 /

.r  r 0 / 0 dr jr  r 0 j3

(E.33)

has azimuthal symmetry, and only radial and theta components: I I .q/ C i  Ex; .q/ E Ix .q/ D i r Ex;r

Z

where I .q/ Ex;r

D Z

I .q/ D Ex;

i r  .r  r 0 / 0 dr jr  r 0 j3

(E.35)

i   .r  r 0 / 0 dr : jr  r 0 j3

(E.36)

xeff .q 0 / xeff .q 0 /

(E.34)

E.8 Solenoidal Component E Sx .r/ of the Pauli Field E x .r/ The solenoidal component E Sx .r/ of the Pauli Field E x .r/ which is Z E Sx .r/ D

J x .r 0 / 

.r  r 0 / 0 dr jr  r 0 j3

(E.37)

also has azimuthal symmetry and only radial and theta components: S S .q/ C i  Ex; .q/ E Sx .q/ D i r Ex;r

(E.38)

S Ex;r .q/ D i r  E Sx .q/

(E.39)

where

E.9 The Potential Energy WxI .r/

Z D

n

401 0 00 /

.q q Œi  0  .r 0  r 00 / @x@r 0

0 00 /

.q q  Œi r 0  .r 0  r 00 / r10 @x@ 0

o

4 jr  r 0 j3 jr 0  r 00 j3 



 .i   i  0 / i   .r  r 0 /  .i   i  0 / i   .r  r 0 / dr 0 dr 00

(E.40)

and S Ex; .q/ D i   E Sx .q/

Z D

n

0 00 /

.q q Œi  0  .r 0  r 00 / @x@r 0

 Œi r 0  .r 0  r 00 / r10

(E.41) @x .q 0 q 00 / @ 0

o

4 jr  r 0 j3 jr 0  r 00 j3





 .i   i  0 / i r  .r  r 0 /  .i r  i  0 / i   .r  r 0 / dr 0 dr 00 :

(E.42)

E.9 The Potential Energy WxI .r/ The (path-independent) potential energy Z WxI .r/ D

xeff .r 0 / 0 dr jr  r 0 j

(E.43)

is WxI .q/

1 D 2

Z

Z 00 .q 0 / B.q 0 q 00 /e6r 243 0 dr C jr  r 0 j 2 2 A2 .q 0 /jr  r 0 j3 jr 0  r 00 j3

0  f i r 0  .r 0  r 00 / r 0 e3r f1 .r 0 r 00 / 0

C e3r f2 .q 0 q 00 /f4 .q 0 q 00 / 1 C f3 .q 0 q 00 /f5 .q 0 q 00 /gdr 0 dr 00 : 8

(E.44)

The six-dimensional integrals for the irrotational E Ix .r/ and solenoidal E Sx .r/ components of the field E x .r/, and that for the potential energy WxI .r/ are performed by the Monte Carlo method [2].

“This page left intentionally blank.”

Appendix F

Ground State Properties of the Helium Atom as Determined by the Kinoshita Wave Function

The Kinoshita [1] wave function for the ground state of the Helium atom, and various properties determined [2] from it are given in this appendix.

F.1 Wave Function

.r 1 r 2 /

The Kinoshita wave function for the ground state of the Helium atom is .r 1 r 2 / D

.stu/ D cN e1:860556s

6 X

u.i /Ti ;

(F.1)

i D1

where the Hylleraas [3, 4] coordinates are s D r1 C r2 , t D r2  r1 , u D jr 2  r 1 j, and the normalization constant cN D 1:364931021. The superscript .i / represents the power of u in each term of the summation, and the terms Ti are T1 D 0:001292t 2 C 0:002071t 2s;

(F.2) 2

2

2

T0 D 1  0:087026s C 0:030099t =s C 0:057891s C 0:226660t  0:011656s 3  0:022154st 2 C 0:001025s 4 C 0:005852s 2t 2 C 0:000555t 4  0:000200s 3t 2 ; T1 D 0:467736 C 0:008079s  0:155282t 2=s C 0:002128s 2

(F.3)

C 0:005221t 2 C 0:000252s 3; T2 D 0:049206=s  0:160571 C 0:039716t 2=s 2 C 0:028827s

(F.4)

C 0:003594t 2=s  0:002328s 2 C 0:003172t 2; T3 D 0:076577=s C 0:014535  0:000112s  0:001060t 2;

(F.5) (F.6)

T4 D 0:013293=s 2  0:012758=s  0:007997  0:000137s C 0:000105t 2;

(F.7)

T5 D 0:005655=s C 0:001326;

(F.8)

T6 D 0:000939=s  0:000002:

(F.9)

403

404

F Ground State Properties of the Helium Atom as Determined

F.2 Electron Density .r/ R 2 The electronic density is .r/ D 2 .rr 2 /dr 2 . By performing the angular integration over d rO 2 analytically we obtain "

Z .r/ D

2 4 cN

e

Œ2.1:860556/s

12 X

# yk .rr2 /wk .rr2 / r22 dr2 ;

(F.10)

kD2

where y2 D .lnb  lna/=.2rr2 /; p p y1 D . b  a/=.rr2 /; y0 D 2;

(F.12) (F.13)

i h yk D b .kC2/=2  a.kC2/=2 = Œrr2 .k C 2/

(F.14)

(F.11)

and for k D 1; 2; : : : 12:

a D r 2 C r22  2rr2 ;

(F.15)

bDr C

(F.16)

2

r22

C 2rr2 ;

and the terms wk are 2 w2 D T1 ; w1 D 2T0 T1 ;

(F.17) (F.18)

w0 D T02 C 2T1 T1 ; w1 D 2T0 T1 C 2T1 T2 ;

(F.19) (F.20)

w2 D T12  2T0 T2 C 2T1 T3 ; w3 D 2T0 T3 C 2T1 T4 C 2T1 T2 ;

(F.21) (F.22)

w4 D T22 C 2T0 T4 C 2T1 T5 C 2T1 T3 ;

(F.23)

w5 D 2T0 T5 C 2T1 T6 C 2T1 T4 C 2T2 T3 ;

(F.24)

w6 D

T32

C 2T1 T5 C 2T2 T4 C 2T0 T6 ;

w7 D 2T1 T6 C 2T2 T5 C 2T3 T4 ; w8 D 2T2 T6 C 2T3 T5 C

T42 ;

w9 D 2T3 T6 C 2T4 T5 ; T52

(F.25) (F.26) (F.27) (F.28)

w10 D C 2T4 T6 ; w11 D 2T5 T6 ;

(F.29) (F.30)

w12 D T62 :

(F.31)

F.4 Coulomb Field E c .r/

405

The expression for s, t, and u in Ti above are the same as those given in Sect. F.1, but with r1 replaced by r.

F.3 Coulomb Hole c .rr 0 / The Coulomb hole c .rr 0 / is determined by (3.28), where the Fermi-Coulomb hole xc .rr 0 / D Œ2j .rr 0 /j2 =.r/  .r 0 /, and the Fermi hole is x .rr 0 / D .r 0 /=2 as discussed in Sect. 3.2.3. The expressions for .rr 0 / and .r/ are given in Sects. F.1 and F.2, respectively.

F.4 Coulomb Field E c .r/ The Coulomb force field E c .r/ due to the Coulomb hole is defined by (3.33). By performing the angular integration over d rO 0 analytically, we obtain E c .r/ D iO r Ec .r/  2 Z 4 cN O eŒ2.1:860556/s D ir .r/  Z 2 r 0 0 2 0  2 dr .r / .r / ; r 0

12 X

! 0

0

zk .rr /wk .rr / .r 0 /2 dr 0

kD2

(F.32)

where 

i. h

.rr 0 /; z2 D h a3=2  b 3=2 =3 C d a1=2  b 1=2     1 1  C d .lnb  lna/ .2rr 0 /; z1 D h a b    

p p  1 1 Cd z0 D h p  p b a .rr 0 /; a b z1 D Œh .lna  lnb/ C d .a  b/ =.2rr 0 /;

(F.33) (F.34) (F.35) (F.36)

and for k D 2; : : : ; 12:    d .kC1/=2 h .k1/=2 .k1/=2 .kC1/=2 a a 3C .rr 0 /; b b zk D  h k1 kC1 (F.37) .r 0 /2 r 1 where h D 2  2r , and d D 2r . The expressions for s; a; b and wk are the same as those given previously, but with r2 replaced by r 0 .

406

F Ground State Properties of the Helium Atom as Determined

F.5 Coulomb Potential Energy Wc .r/ The Coulomb potential energy Wc .r/ is then obtained from (3.75) as Z Wc .r/ D 

r 1

Ec .r 0 /dr 0 :

(F.38)

Appendix G

Approximate Wave Function for the Hydrogen Molecule

The value of the parameter ı and the coefficients Cmnjkp of the Kolos–Roothaan [1] wave function of (16.4) for the ground state of the hydrogen molecule are listed in Table G.1.

Table G.1 Variational parameters in the normalized 51-parameter correlated wave function for the ground state of H2 No. of terms 50 ı D 0:995 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1

1 0 0 0 1 0 1 0 0 0 0 2 0 0 1 0 0 1 1 0 0 1 0 0

2 0 0 1 0 0 0 0 2 0 1 0 0 1 1 1 2 0 0 2 2 1 2 0

2 0 2 0 1 0 1 2 0 0 0 2 2 0 1 0 0 1 1 0 0 1 0 2

r12 0 0 0 0 1 0 0 0 2 0 0 1 1 0 2 0 1 2 1 1 2 2 1

Coefficients 2:065908 1:282032 0:144619 0:430253 0:787198 0:235454 0:148273 0:109859 0:212159 0:081387 0:182892 0:198555 0:324658 0:010794 0:077830 0:055114 0:130714 0:050854 0:014963 0:132980 0:000362 0:006992 0:050940 (continued)

407

408

G Approximate Wave Function for the Hydrogen Molecule

Table G.1 continued 1

1

1 1 0 1 1 1 2 0 0 1 0 1 1 1 0 3 2 0 0 0 0 2 1 2 1 0 0

1 0 0 0 0 2 0 0 0 0 1 0 2 1 2 0 1 0 0 0 0 0 0 0 0 2 2

No. of terms 50 2 2 1 1 0 1 0 3 3 1 3 1 2 3 1 2 3 3 2 1 3 3 1 3 1 3 1 3 3

1 0 2 0 2 0 0 2 0 2 1 0 2 1 0 0 1 2 0 0 2 0 2 0 2 0 0

ı D 0:995 r12 Coefficients 1 1 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 1 1 2 2 1 2

0:018027 0:017554 0:014601 0:015172 0:012656 0:000202 0:000856 0:009469 0:036963 0:022325 0:053233 0:004690 0:004707 0:017531 0:017270 0:000082 0:000031 0:094436 0:001789 0:000394 0:004475 0:000121 0:014893 0:000011 0:001016 0:003443 0:000225

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Chapter 3 1. 2. 3. 4. 5. 6.

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Chapter 5 1. Density Functional Theory of Atoms and Molecules, R. G. Parr and W. Yang, Oxford University Press, New York (1989) 2. Density Functional Theory, R. M. Dreizler and E. K. U. Gross, Springer-Verlag, Berlin (1990) 3. Quantal Density Functional Theory, V. Sahni, Springer-Verlag, Berlin, Heidelberg (2004) 4. V. Sahni, M. Slamet, Int. J. Quantum Chem. 100, 858 (2004) 5. V. Sahni, L. Massa, R. Singh, M. Slamet, Phys. Rev. Lett. 87, 113002 (2001) 6. M. Slamet, V. Sahni, Int. J. Quantum Chem. 85, 436 (2001) 7. M. Slamet, R. Singh, L. Massa, V. Sahni, Phys. Rev. A 68, 042504 (2003) 8. S. T. Epstein, C. M. Rosenthal, J. Chem. Phys. 64, 247 (1976) 9. O. Gunnarsson, B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976) 10. M. Levy, J. P. Perdew, in Density Functional Methods in Physics, Edited by R. M. Dreizler and J. da Provendencia, NATO ASI Series, Series B: Physics Vol. 123, (1985) 11. V. Sahni, X.-Y. Pan, Phys. Rev. Lett. 90, 123001 (2003) 12. N. R. Kestner, O. Sinanoglu, Phys. Rev. 128, 2687 (1962)

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Chapter 6 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.

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

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

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

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Chapter 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. 26. 27.

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Chapter 12 1. 2. 3. 4. 5.

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Chapter 13 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.

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Chapter 15 1. M. Slamet, V. Sahni, Phys. Rev. A 51, 2815 (1995) 2. V. Sahni, Top. Curr. Chem. 182, 1 (1996) 3. T. Kinoshita, Phys. Rev. 105, 1490 (1957)

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Chapter 16 1. 2. 3. 4. 5.

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Chapter 17 1. J. Bardeen, Phys. Rev. 49, 653 (1936) 2. J. Friedel, Phil. Mag. 7, 43 (1952); Nuovo Cimento 7, 287 (1958) (supplement) 3. V. Sahni, C.Q. Ma, Phys. Rev. B 22, 5987 (1980). (The product k?0 .x 0 / k0 .x/ missing from (13)) 4. C.Q. Ma, V. Sahni, Phys. Rev. B 20, 2291 (1979) 5. V. Sahni, J. Gruenebaum, J.P. Perdew, Phys. Rev. B 26, 4371 (1982) 6. H. Luo, W. Hackbusch, H-J. Flad, D. Kolb, Phys. Rev. B 78, 035136 (2008) 7. V. Sahni, C.Q. Ma, J.S. Flamholz, Phys. Rev. B 18, 3931 (1978) 8. V. Sahni, J.B. Krieger, J. Gruenebaum, Phys. Rev. B 15, 1941 (1977) 9. V. Sahni, J. Gruenebaum, Solid State Commun. 21, 463 (1977) 10. J.P. Perdew, V. Sahni, Solid State Commun. 30, 87 (1979) 11. V. Sahni, Surf. Sci., 213, 226 (1989) 12. N.D. Lang, W. Kohn, Phys. Rev. B 1, 4555(1970) 13. N.D. Lang, W. Kohn, Phys. Rev. B 3, 1215 (1971) 14. V. Sahni, K.-P. Bohnen, Phys. Rev. B 29, 1045 (1984) 15. V. Sahni, K.-P. Bohnen, Phys. Rev. B 31, 7651 (1985) 16. M.K. Harbola, V. Sahni, Phys. Rev. B 37, 745 (1988) 17. H.J. Juretschke, Phys. Rev. 92, 1140 (1953) 18. J. Bardeen, Surf. Sci. 2, 381 (1964) 19. H.J. Juretschke, Phys. Rev. B 36, 6168 (1987) 20. M.K. Harbola, V. Sahni, Phys. Rev. B 39, 10437 (1989)

k .x/

is

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30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66.

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73. 74.

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Chapter 18 1. Z. Qian, V. Sahni (unpublished) 2. L. Massa, V. Sahni, X.-Y. Pan (unpublished) 3. C. Møller, M. S. Plesset, Phys. Rev. 46, 618 (1934) (As an aside we note that in their paper, Møller and Plesset also proved that the density and consequently the expectation of all other single-particle operators as determined within Hartree-Fock theory are correct to secondorder.) 4. L. Brillouin, Actalit´es sci. et ind vol. 71 (1933); vol. 159 (1934); vol. 160 (1934) 5. S. Raimes, Many-Electron Theory (North-Holland, Amsterdam, 1972) 6. E.K.U. Gross, E. Runge, O. Heinonen, Many-Particle Theory (Adam Hilger, Bristol, 1991) 7. I. N. Levine, Quantum Chemistry, 5th edn. (Prentice-Hall, Upper Saddle River, NJ, 2000) 8. A. Fetter, J.D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, New York, 1975) 9. W. Jones, N.H. March, Theoretical Solid State Physics, Vol. 1 (Dover, New York, 1973) 10. L.J. Sham, M. Schl¨uter, Phys. Rev. Lett. 51, 1888 (1983) 11. L.J. Sham, Phys. Rev. B 32, 3876 (1985) 12. V. Sahni, J. Gruenebaum, J.P. Perdew, Phys. Rev. B 26, 4371 (1982) 13. V. Sahni, M. Levy, Phys. Rev. B 33, 3869 (1986) 14. R.T. Sharp, G.K. Horton, Phys. Rev. 30, 317 (1953) 15. J.D. Talman, W.F. Shadwick, Phys. Rev. A 14, 36 (1976)

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

V. Sahni, X.-Y. Pan, Phys. Rev. Lett. 90, 123001 (2003) M. Levy, Phys. Rev. A 26, 1200 (1982) E. Lieb, Int. J. Quantum Chem. 24, 243 (1983) C.A. Ullrich, W. Kohn, Phys. Rev. Lett. 87, 093001 (2001); 89, 156401 (2002). J.T. Chayes, L. Chayes, M.B. Ruskai, J. Stat. Phys. 38, 497 (1985) W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965) W. Kohn, in Highlights of Condensed Matter Theory, ed. by F. Bassani et al. (North-Holland, Amsterdam, 1985) M. Levy, A. Nagy, Phys. Rev. Lett. 83, 4361 (1999) A. Nagy, M. Levy, Phys. Rev. A 63, 052502 (2001) V. Sahni, L. Massa, R. Singh, M. Slamet, Phys. Rev. Lett. 87, 113002 (2001) M. Slamet, V. Sahni, Int. J. Quantum Chem. 85, 436 (2001) M. Taut, Phys. Rev. A 48, 3561 (1993).

422

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Appendix B 1. 2. 3. 4.

X.-Y. Pan, V. Sahni, Int. J. Quantum Chem. 108 2756 (2008) E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984) H.A. Kramers, Quantum Mechanics, North-Holland, Amsterdam (1957) X.-Y. Pan, V. Sahni, Int. J. Quantum Chem. 95, 387 (2003)

Appendix C 1. 2. 3. 4. 5.

Z. Qian, V. Sahni, Phys. Rev. A 57, 4041 (1998) C.-O. Almbladh, U. von Barth, Phys. Rev. A 31, 3231(1985) M. Levy, J.P. Perdew, V. Sahni, Phys. Rev. A 30, 2745(1984) J. Katriel, E.R. Davidson, Proc. Nat’l. Acad. Sci. USA 77, 4403 (1980) M.M. Morrell, R.G. Parr, M. Levy, J.Chem. Phys. 62, 549 (1975)

Appendix D 1. V. Sahni, Y. Li, M.K. Harbola, Phys. Rev. A 45, 1434 (1992) 2. M.E. Rose, Elementary Theory of Angular Momentum, (Wiley, New York, 1957)

Appendix E 1. M. Slamet, V. Sahni, and M.K. Harbola, Phys. Rev. A 49, 809 (1994) 2. G.P. Lepage, J. Comput. Phys. 27, 192 (1978)

Appendix F 1. 2. 3. 4.

T. Kinoshita, Phys. Rev. 105, 1490 (1957) M. Slamet, V. Sahni (unpublished) E.A. Hylleraas, Z. Phys. 48, 469 (1928) X.-Y. Pan, V. Sahni, L. Massa, e-print physics/0310128

Appendix G 1. W. Kolos, C.C.J. Roothaan, Rev. Mod. Phys. 32, 219 (1960).

Index

Airy functions, 311 Airy Gas Model, 313 anion, 263 anion–positron complex, 263 Asymptotic structure at and near nucleus kinetic “force” of S system, 164 Correlation-Kinetic field, 164 kinetic energy density of S system, 163 Kohn-Sham “exchange-correlation” potential, 165 of electron-interaction field, 159 of electron-interaction potential, 142 of kinetic “force”, 162 of kinetic energy density, 162 of the density, 157, 158 of the density matrix, 161 of wave function, 154, 156, 160 Asymptotic structure in the classically forbidden region at metal–vacuum interface Coulomb and higher-order CorrelationKinetic potential, 344, 350 effective potential, 351 Kohn–Sham “correlation” potential, 344, 350 Kohn–Sham “exchange” potential, 340 Kohn–Sham “exchange-correlation” potential, 345 lowest-order Correlation-Kinetic potential, 339, 343 of density, 348 of Dirac density matrix, 348 of orbitals, 347 Pauli potential, 335 Slater function, 339 Asymptotic structure in the classically forbidden region of atoms of Correlation-Kinetic field, 138

of Correlation-Kinetic potential, 126, 387 of Coulomb potential, 126, 136 of density, 50 of density matrix, 131 of electron-interaction field, 134 of electron-interaction potential, 126 of Hartree field, 135 of Hartree potential, 126 of Pauli field, 135 of Pauli potential, 126, 136 of S system density, 49 of wave function, 130 pair-correlation density, 132 Atomic shell structure, 168, 196 aufbau principle, 11, 168, 184, 188, 203, 207 Auger effect, 259 Autoionizing, 258 B system, 221 Correlation-Kinetic energy, 224 density matrix, 224 differential equation, 145, 222 effective field, 223 effective potential, 223 electron-interaction potential, 145, 224 kinetic “force”, 223 kinetic energy, 224 Bardeen-Friedel oscillations, 303, 314, 334 Bethe-Salpeter equation, 356, 360 bidensity energy functional, 250, 380 Born–Oppenheimer, 16 Brillouin’s theorem, 209, 355, 369 Central Field Approximation, 120, 393 chemical potential, 59 Clebsch–Gordan coefficients, 394 coalescence constraints, 29 differential form, 31 integral form, 31

423

424

Index

core charge, 177, 197 core radius, 177, 197 core–valence separations, 196 Correlated wave function Drake (He), 277 Kinoshita, 276 Kinoshita (He), 403 Kolos–Roothaan (H2 ), 290, 407 Komasa and Thakkar (H2 ), 146 Pekeris (He), 277 Coulomb hole, 40, 277, 294 Coulomb species, 55 Coulson, 178 cusp coalescence condition, 31

internal, 17, 36 irrotational component, 121 kinetic, 23, 42 nonconservative, 119 Pauli, 41, 192 Pauli–Coulomb, 22 self-interaction correction, 104, 172 solenoidal component, 121 Finite Linear Potential model, 311 “Force” differential density, 23 electron-interaction, 22 kinetic, 23 Fourier transform, 357

density amplitude, 221 Dirac density matrix, 38 for uniform electron gas, 317 dynamic charge distribution, 21 Dyson equation, 345, 358

gauge invariance, 71 gauge transformation, 69, 385 gauge variance, 71 Green’s function, 345, 356 for interacting system, 345 for S system, 345 single-particle, 357 two-particle, 359

effective Fermi hole charge, 236 effective Fermi–Coulomb hole charge, 122 electron density, 18, 37 Energy Correlation-Kinetic, 45 Coulomb, 44 electron-interaction, 16, 24 electrostatic, 308 external, 16, 45 external potential, 25 Hartree, 24 kinetic, 16, 25 Pauli, 44 Pauli-Coulomb, 24 ensemble density energy functional, 381 Euler’s theorem, 26 Euler–Lagrange equation, 59 Fermi hole, 39, 193 for uniform electron gas, 317 parallel to surface, 325 planar averaged, 317, 321 Fermi–Coulomb hole, 294 Fermi–Coulomb hole charge, 21 Field Correlation-Kinetic, 43 Coulomb, 42 differential density, 23, 42 effective, 37, 41, 47 electron-interaction, 22, 192 external, 15 Hartree, 22, 192

Hartree–Fock theory, 64, 108 Heisenberg picture, 357 Helmholtz’s theorem, 122 hierarchy within the fundamental theorem of density functional theory, 386 Hohenberg–Kohn theorem corollary, 57 generalization, 67 Hohenberg–Kohn theorems, 53 Hooke’s atom, 56, 76, 380 Hooke’s molecule, 56 Hooke’s species, 56 Hylleraas coordinates, 276, 403 idempotency, 38 image-potential-bound surface states, 353 integral virial theorem, 26 ionization potential, 50 Irrotational Component Approximation, 121, 235, 397 irrotational component of Pauli field, 242 jellium model, 303 kinetic-energy-density, 25, 45 kinetic-energy-density tensor, 23, 43

Index Kohn–Sham density functional theory, 59 “exchange” potential, 339 “correlation” energy functional, 63 “correlation” potential, 63 “exchange” energy functional, 62 “exchange” potential, 63 “exchange-correlation” energy functional, 62 “exchange-correlation” potential, 62 integral virial theorems, 63 electron interaction energy functional, 61 electron interaction potential, 61 Hartree energy functional, 62 Hartree potential, 62

Linear Potential model, 312 local charge distribution, 18 local density approximation, 315 local effective potential energy theory, 35

Madelung’s law, 168, 207 Metal surface models Finite-Linear Potential, 304, 311 Infinite Barrier Potential, 313 Linear Potential, 304, 312 Step Potential, 313 metal-vacuum interface, 303 mononegative ions, 214 multiplet structure, 256

N-representability, 59 node coalescence condition, 31 non idempotency, 19 noninteracting v-representability, 60 nonlocal charge distribution, 21

Operator annihilation, 357 creation, 357 density, 18 density matrix, 19 electron-interaction potential energy, 15 external potential energy, 15 kinetic, 15 momentum, 69, 384 pair-correlation, 20 translation, 19 unitary, 384 vector potential, 69, 384 optimized potential method, 367

425 orbital-dependent “exchange” function, 340, 368 orbital-dependent Fermi hole, 340, 368, 369 pair-correlation density, 20 pair-correlation function, 21, 40 Pauli in Nobel lecture, 178 Perturbation theory Møller-Plesset, 355 many-body, 355 pair-correlation density, 361 plasmon-pole approximation, 348 Poisson’s equation, 307 Polarizability dipole, 217 quadrupole, 217 positron, 263 positron affinity, 263, 268 positronium, 264 positronium affinity, 264, 272 Potential effective, 35 electron-interaction, 37, 47, 194 electrostatic, 307 Hartree, 48, 194 Pauli, 194 self-interaction correction, 105 Quantal density functional theory, 35, 113 Fully Correlated Approximation, 117, 275, 289 Hartree Uncorrelated Approximation, 106, 167 many-body perturbation theory, 355 Multi-Component Pauli Approximation, 263 Multi-Component theory, 263 of Hartree theory, 103 of Hartree–Fock theory, 108 of the density amplitude, 221 Pauli Approximation, 110, 187 Pauli–Coulomb Approximation, 114 perturbation series for Correlation-Kinetic field, 364 perturbation series for the electron-interaction field, 360 Pseudo Møller-Plesset perturbation theory, 356, 367 Quantal density functional theory of degenerate states of ensemble densities, 375, 378 of pure Slater densities, 375

426 “Quantal Newtonian” first law, 17 “Quantal Newtonian” second law, 17 quasi-particle amplitudes, 387

radial probability density, 175, 196, 252 radiationless transition, 259 Runge–Gross time-dependent density functional theory generalization of fundamental theorem, 383

Index static charge distribution, 18 Sum Rules for Coulomb hole, 40 for Fermi hole, 39 force, 48 torque, 48 sum rules for Fermi–Coulomb hole, 21 surface dipole barrier, 309 surface energy, 304, 309

Theophilou-Budd-Vannimenus theorem, 310 two-particle density matrix, 359 S system, 35 meaning of highest occupied eigenvalue, 49 nonuniqueness of effective potential, 74 nonuniqueness of wave function, 75, 85 Schr¨odinger theory, 15 self-energy, 345 self-interaction correction charge, 104, 168 Sham and Schl¨uter equation, 366 single-particle density matrix, 18 Slater, 178 Slater’s diagonal sum rule, 256 solenoidal component of Pauli field, 242 spherical harmonics, 394

unitary transformation, 67

v-representability, 59 vector potential, 123 vector vortex Fermi hole, 239 vector vortex source, 124 virial theorem, 26

Wigner–Seitz radius, 306 work function, 304, 309, 310