Quantal Density Functional Theory II: Approximation Methods and Applications

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Author: Viraht Sahni

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Quantal Density Functional Theory II

“This page left intentionally blank.”

Viraht Sahni

Quantal Density Functional Theory II Approximation Methods and Applications

With 115 Figures

123

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.


To my wife, Catherine, the love of my life


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

Schrödinger Theory from a “Newtonian” Perspective . . . . . . . .. . . . . . . . . . . 2.1 Time-Independent Schrödinger Theory . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2 Schrödinger 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

15 15

3

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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41 42 42 43

43 44 45 46 47 48 49 51

53 54 59 64 67 67 70

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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 Schrödinger 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 Schrödinger 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 Schrödinger theory from the new perspective of “classical” fields and quantal sources. This new perspective on time-independent Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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.


Chapter 2

Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger theory [2] from the same “Newtonian” perspective of “classical” fields and quantal sources [3, 4]. This perspective of Schrödinger 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/Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger Theory from a “Newtonian” Perspective

17

2.2 Schrödinger 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 Schrödinger 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 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


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


The asymptotic structure of the density may also be obtained for the interacting system by solution of the Schrödinger 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.


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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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


(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


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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger equation (2.5), to one of noninteracting fermions also in their ground state but whose density .r/ is the same

60


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 Schrödinger 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 Schrödinger 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


61

of the word. There are many local potential energies vs .r/; vee .r/ that generate the ground state density .r/ via the Schrödinger 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

EŒ D

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 EŒ of (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


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,


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 Œ D

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 Œ 0;

(4.37)

.r/r r vxc .r/dr D Tc Œ 0;

(4.38)

Z KS Exc Œ C

Z EH Œ C

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

(4.39)

64


Z ExKS Œ C

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

(4.40)

.r/r r vc .r/dr D Tc Œ 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 Œ satisfies the sum rule Z 2Ts Œ D

.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


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


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


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

(4.51)

so that the transformed time-independent Schrödinger 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)


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


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 Schrödinger 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


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


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].)


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


5.1 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 Schrödinger 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 Schrödinger 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/jî 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


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


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


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.


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


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


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


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)


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


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


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


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


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.)


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.


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 Schrödinger 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


101

Hartree, Hartree–Fock, and Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger theories is based on the

102


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 Schrödinger 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


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 Œ H / ; .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


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


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)


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 Œ HF / ; .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


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


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


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


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)


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


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.


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


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


‰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-


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


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)


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


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 Schrödinger 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


The Schrödinger 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


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


7.3 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
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 Schrödinger 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


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


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


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


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 Schrödinger 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 Schrödinger 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


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


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


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


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


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


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)


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


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.


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


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


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


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


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

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