· 2 © peter blöchl, 2000-february 18, 2016 source: permission to make digital or hard copies...

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Advanced Topicsof Theoretical Physics I

The electronic structure of matterPart 1: Introduction

Peter E. Blöchl

Caution! This is an unfinished draft version

Mistakes are unavoidable!

Institute of Theoretical Physics; Clausthal University of Technology;

D-38678 Clausthal Zellerfeld; Germany;

http://www.pt.tu-clausthal.de/atp/

http://www.pt.tu-clausthal.de/atp/

2

© Peter Blöchl, 2000-February 18, 2016Source: http://www2.pt.tu-clausthal.de/atp/phisx.html

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1To the title page: What is the meaning of ΦSX? Firstly, it sounds like “Physics”. Secondly, the symbols stand forthe three main pillars of theoretical physics: “X” is the symbol for the coordinate of a particle and represents ClassicalMechanics. “Φ” is the symbol for the wave function and represents Quantum Mechanics and “S” is the symbol for theEntropy and represents Statistical Physics.

http://www2.pt.tu-clausthal.de/atp/phisx.html

Foreword and Outlook

This book is work in progress. This book is not completed and not proofread. You can use this bookcurrently as a basis for following through the material presented in the course. However, you shouldnever blindly copy formulas an apply them. The chances that there are still errors is too large.

This book provides an introduction into the quantum mechanics of the interacting electron gas.It aims at students at the graduate level, that already have a good understanding on one-particlequantum mechanics. My aim is to include all proofs in a comprehensive and hopefully water-tightmanner. I am also avoiding special systems of units. I consider this important in view of a large varietyof notations, differing definitions etc. While the proofs have the advantage to train the student toperform the typical operations, it clutters the course with material far from applications. Therefore,I have placed many of the more detailed derivations into appendices. On the graduate level, thestudent should be able to follow the proofs without problem. It is strongly recommended to followthrough the derivations.

A recommended reading are the Lecture notes[1] by Wolf-Dieter Schöne, which has been usedin the preparation of this text. Another book that is very recommendable is the one by Fetter andWalecka[2]. Very useful has been also the book by Nolting[3]. While I did not follow the book ofMattuck[4]. A recommendable book for the reader, who wants to dig a little deeper into the density-functional theory is the Book by Eschrig[5]. Another excellent book on many particle physics, whichhowever takes a quite different route from this text is the book by Negele and Orland[6].

The goal of the lecture is that the student

• understands the Born-Oppenheimer approximation and its main limitations,

• is able to construct the band structure of a non-interacting problem using the tight-bindingmethod and is able to interpret a band structure,

• has a sound understanding of the connection between wave functions and the operator formal-ism of second quantization, i.e. creation and annihilation operators,

• understand the foundation of density functional theory, the dominant tool for first-principleselectronic structure calculations,

• understands the role of Green’s functions in many-particle physics and its relation to one-particlephysics, and that he

• understands the main theorems underlying the perturbation expansion fo Green’s functions,including Feynman diagrams.

I have tried to provide detailed and complete derivations of everything presented in this course.Some of the more extensive derivations are placed into an appendix. I strongly encourage the studentto understand and rationalize these derivations. Only this will make the student familiar with thelimitations of the theory and also with its flexibility. For a theoretical physicist this is of centralimportance, because he needs to be able to extend the existing theory, if necessary.

In the online version, there are many hyperreferences that allow to jump to the relevant informationin the book. To avoid the additional cost for color printouts just for the references they are not visible.They still work, when you click on them. Here is a list of links set:

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4

• List of contents

• references to equation numbers, figures, tables, sections, appendices, etc.

• citations will take you to the corresponding position in the list of references

Todo for the author

Topics:

1. Separate nuclear and electron dynamics

2. Spin-orbitals ans many-particle wave functions

3. non-interacting electrons: band structure

4. Hartree-Fock approximation

5. Density functional theory

6. Phonons

7. Transport theory: Boltzmann equation

• Open systems

• Decoherence

8. Magnetism

Contents

1 The standard model of solid-state physics 11

2 Born-Oppenheimer approximation and beyond 15

2.1 Separation of electronic and nuclear degrees of freedom . . . . . . . . . . . . . . . 15

2.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Classical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Non-adiabatic corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Non-diagonal derivative couplings . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Topology of crossings of Born-Oppenheimer surfaces . . . . . . . . . . . . . 23

2.3.3 Avoided crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4 Conical intersections: Jahn-Teller model . . . . . . . . . . . . . . . . . . . . 25

2.3.5 From a conical intersection to an avoided crossing . . . . . . . . . . . . . . 35

2.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Many-particle wave functions 39

3.1 Spin orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Symmetry and quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Levi-Civita Symbol or the fully antisymmetric tensor . . . . . . . . . . . . . 46

3.3.2 Permutation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.3 Antisymmetrize wave functions . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.4 Slater determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.5 Bose wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.6 General many-fermion wave function . . . . . . . . . . . . . . . . . . . . . . 50

3.3.7 Number representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Non-interacting electrons 51

4.1 Dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Crystal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.1 Reciprocal space revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.3 Non-interacting electrons in an external potential . . . . . . . . . . . . . . . 59

4.3 Bands and orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Calculating band structures in the tight-binding model . . . . . . . . . . . . . . . . 64

4.5 Thermodynamics of non-interacting electrons . . . . . . . . . . . . . . . . . . . . . 67

5

6 CONTENTS

4.5.1 Thermodynamics revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.2 Maximum entropy, minimum free energy principle and the meaning of the grand potential 68

4.5.3 Many-particle states of non-interacting electrons . . . . . . . . . . . . . . . 69

4.5.4 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5.5 Fermi distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.6 Thermodynamics of non-interacting electrons . . . . . . . . . . . . . . . . . 75

4.5.7 Brillouin-zone integrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6 Jellium model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6.1 Wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6.2 Band structure of the free electron gas . . . . . . . . . . . . . . . . . . . . 83

4.6.3 Density of states of the free electron gas . . . . . . . . . . . . . . . . . . . 84

4.7 Tour trough band structures and densities of states of real materials . . . . . . . . . 87

4.7.1 Free-electron like metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.7.2 Noble metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.7.3 Transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.7.4 Diamond, Zinc-blende and wurtzite-type semiconductors . . . . . . . . . . . 90

4.7.5 Simple oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8 What band structures are good for . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8.1 Chemical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Phonons 93

5.1 Phonons as quasi particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Connection to Born-Oppenheimer treatment . . . . . . . . . . . . . . . . . . . . . 93

5.3 Harmonic oscillator revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4 Lattice translation symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5.1 Left and right-moving waves . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5.2 From normal coordinates to phonon amplitudes . . . . . . . . . . . . . . . . 103

5.6 Phonon band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.6.1 Example for a phonon band structure . . . . . . . . . . . . . . . . . . . . . 107

5.7 Acoustic branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.8 Optic branches of the phonon band structure . . . . . . . . . . . . . . . . . . . . . 109

5.9 Thermodynamics of phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.9.1 Thermodynamics: Models for the density of states . . . . . . . . . . . . . . 109

5.9.2 Grand potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.9.3 Internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6 Boltzmann equation 113

6.1 Demonstration of the principle in real space . . . . . . . . . . . . . . . . . . . . . . 113

6.2 From coordinate space to phase space . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3 Collision term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4.1 Prefactor for the distributions . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4.2 General equilibrium distribution . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 Linearized Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

CONTENTS 7

6.5.2 Linear Response to external perturbations . . . . . . . . . . . . . . . . . . . 1206.6 Relaxation-time approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.7 Quasi Fermi level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7 Phonon scattering 125

7.1 Equation of motion for the phonon amplitudes . . . . . . . . . . . . . . . . . . . . 1257.2 Momentum and energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.2.1 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.2.2 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3.1 Compact notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.3.3 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.3.4 Initial random-phase approximation . . . . . . . . . . . . . . . . . . . . . . 1307.3.5 Repeated random-phase approximation . . . . . . . . . . . . . . . . . . . . 1347.3.6 Phonon scattering without lattice expansion . . . . . . . . . . . . . . . . . . 136

7.4 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.4.1 Units of quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.5 Quantum mechanical derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.5.1 From the Lagrangian to ladder operators . . . . . . . . . . . . . . . . . . . 1407.5.2 The phonon Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.5.3 Random-phase approximation and rate equation . . . . . . . . . . . . . . . 1437.5.4 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.5.5 Linearized Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 147

8 The Hartree-Fock approximation and exchange 151

8.1 One-electron and two-particle operators . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 Expectation values of Slater determinants . . . . . . . . . . . . . . . . . . . . . . . 153

8.2.1 Expectation value of a one-particle operator . . . . . . . . . . . . . . . . . . 1538.2.2 Expectation value of a two-particle operator . . . . . . . . . . . . . . . . . . 156

8.3 Hartree energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.4 Exchange energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.5 Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.6 Hartree-Fock of the free-electron gas . . . . . . . . . . . . . . . . . . . . . . . . . 1648.7 Beyond the Hartree-Fock Theory: Correlations . . . . . . . . . . . . . . . . . . . . 165

8.7.1 Left-right and in-out correlation . . . . . . . . . . . . . . . . . . . . . . . . 1658.7.2 In-out correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.7.3 Spin contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1678.7.4 Dynamic and static correlation . . . . . . . . . . . . . . . . . . . . . . . . . 167

9 Density-functional theory 169

9.1 One-particle and two-particle densities . . . . . . . . . . . . . . . . . . . . . . . . . 1709.1.1 N-particle density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1709.1.2 One-particle reduced density matrix . . . . . . . . . . . . . . . . . . . . . . 1709.1.3 Two-particle density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739.1.4 Two-particle density of the free electron gas . . . . . . . . . . . . . . . . . 176

8 CONTENTS

9.1.5 Pair-correlation function and hole function . . . . . . . . . . . . . . . . . . 178

9.2 Self-made density functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.3 Constrained search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.4 Self-consistent equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.4.1 Universal Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.4.2 Exchange-correlation functional . . . . . . . . . . . . . . . . . . . . . . . . 187

9.4.3 The self-consistency condition . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.4.4 Flow chart for a conventional self-consistency loop . . . . . . . . . . . . . . 190

9.4.5 Another Minimum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.5 Adiabatic connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.5.1 Screened interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.5.2 Hybrid functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.6 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.7 Xα method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.8 Local density functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.9 Local spin-density approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.10 Non-collinear local spin-density approximation . . . . . . . . . . . . . . . . . . . . . 198

9.11 Generalized gradient functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.12 Additional material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.12.1 Relevance of the highest occupied Kohn-Sham orbital . . . . . . . . . . . . 203

9.12.2 Correlation inequality and lower bound of the exact density functional . . . . 203

9.13 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.14 Reliability of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.15 Deficiencies of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

10 Magnetism 205

10.1 Charged particles in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . 205

10.2 Dirac equation and magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . 205

10.3 Magnetism of the free electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

10.4 Magnetic order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

A Adiabatic decoupling in the Born-Oppenheimer approximation 207

A.1 Justification for the Born-Oppenheimer approximation . . . . . . . . . . . . . . . . 207

B Non-adiabatic effects 211

B.1 The off-diagonal terms of the first-derivative couplings ~An,m,j . . . . . . . . . . . . . 211

B.2 Diagonal terms of the derivative couplings . . . . . . . . . . . . . . . . . . . . . . . 212

B.3 The diabatic picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

C Non-crossing rule 217

C.1 Proof of the non-crossing rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

D Landau-Zener Formula 221

E Supplementary material for the Jahn-Teller model 225

E.1 Relation of stationary wave functions for m and −m − 1 . . . . . . . . . . . . . . . 225

E.2 Behavior of the radial nuclear wave function at the origin . . . . . . . . . . . . . . . 227

CONTENTS 9

E.3 Optimization of the radial wave functions . . . . . . . . . . . . . . . . . . . . . . . 229E.4 Integration of the Schrödinger equation for the radial nuclear wave functions . . . . 231

F Notation for spin indices 235

G Some group theory for symmetries 237

G.0.1 Definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237G.0.2 Symmetry class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

H Sommerfeld expansion 239

I Why is the effective mass of semiconductors much smaller than that of metals? 241

J Anharmonic oscillator 245

K Time integrals for the phonon-phonon scattering elements 247

K.1 Expression 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247K.1.1 Product of two F (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

K.2 Expression 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

L Additional material for the Phonon Boltzmann equation 251

L.1 Sketch of the derivation of the Phonon Boltzmann equation . . . . . . . . . . . . . 252

M Downfolding 253

M.1 Downfolding with Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 255M.2 Taylor expansion in the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

N Time-inversion symmetry 257

N.1 Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257N.2 Pauli equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258N.3 Time inversion for Bloch states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

O Slater determinants for parallel and antiparallel spins 263

O.1 Spatial symmetry for parallel and antiparallel spins . . . . . . . . . . . . . . . . . . . 264O.2 An intuitive analogy for particle with spin . . . . . . . . . . . . . . . . . . . . . . . 266

P Hartree-Fock of the free-electron gas 267

P.1 Exchange potential as non-local potential . . . . . . . . . . . . . . . . . . . . . . . 267P.2 Energy-level shifts by the exchange potential . . . . . . . . . . . . . . . . . . . . . . 268

Q Dictionary 273

Q.1 Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Q.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

R Greek Alphabet 275

S Philosophy of the ΦSX Series 277

T About the Author 279

10 CONTENTS

Chapter 1

The standard model of solid-statephysics

Solid state physics has, in contrast to elementary particle physics, the big advantage that the funda-mental particles and their interactions are completely known. The particles are electrons and nuclei1

and their interaction is described by electromagnetism. Gravitation is such a weak force at small dis-tances that it is completely dominated by the electromagnetic interaction. The strong force, whichis active inside the nucleus, has such a short range that it does not affect the relative motion ofelectrons and nuclei.

Some argue that solid-state physics is not at the very frontier of research, because its particlesand interactions are fully known. However, a second look shows the contrary: While the constituentsare simple, the complexity develops via the interaction of many of these simple particles. The simpleconstituents act together to form an entire zoo of phenomena of fascinating complexity.

The dynamics is described by the Schrödinger equation:

i~∂t |Φ〉 = H|Φ〉 (1.1)

where the wave function depends on the coordinates of all electrons and nuclei in the system. Inaddition it depends on the spin degrees of freedom of the electrons. Thus, a wave function for Nelectrons and M nuclei has the form

Φσ1,...,σN (~r1, . . . , ~rN , ~R1, . . . , ~RM)

where we denote the electronic coordinates by a lower-case ~r and the nuclear positions by an uppercase~R. The symbols σj are the spin indices, which may assume values σj ∈ − 12 ,+ 12. We will often usethe notation σj ∈ ↓, ↑. The spin quantum number in z-direction is sz = ~σ.

Often we will also combine position and spin index of the electrons in the form ~x = (σ,~r), so that

Φ(~x1, . . . , ~xN , ~R1, . . . , ~RM)def=Φσ1,...,σN (~r1, . . . , ~rN , ~R1, . . . , ~RM)

For the integrations and summations we use the short-hand notation∫

d4xdef=∑

σ∈↑,↓

∫

d3r (1.2)

1The cautious reader may object that the nuclei are objects that are far from being fully understood. However, theonly properties that are relevant for us are their charge, their mass, and their size.

11

12 1 THE STANDARD MODEL OF SOLID-STATE PHYSICS

Most important is the antisymmetry of the wave function with respect to interchange of twoelectrons, which is the cause for the Pauli principle

PAULI PRINCIPLE

Φ(. . . , ~xi , . . . , ~xj , . . . , ~R1, . . . , ~RM) = −Φ(. . . , ~xj , . . . , ~xi , . . . , ~R1, . . . , ~RM) (1.3)

An electronic many-particle wave function for Fermions is antisymmetric under particle exchange. Asa consequence, no two electrons with the same spin can occupy the same point in space or the sameone-particle orbital.

The Standard Model

Now we arrive at the most important equation, which will be the basis of all that will be said in thisbook. This equation specifies the many-particle Hamiltonian of our standard model of solid-statephysics.

HAMILTONIAN OF THE STANDARD MODEL

The Hamiltonian for a many particle system, such as a molecule or a solid, is

H =

M∑

j=1

−~22Mj

~∇2~Rj︸︷︷︸

Ekin,nuc

+

N∑

i=1

−~22me

~∇2~ri︸︷︷︸

Ekin,e

+1

2

M∑

i 6=j

e2ZiZj

4πǫ0|~Ri − ~Rj |︸︷︷︸

EC,nuc−nuc

−N∑

i=1

M∑

j=1

e2Zj

4πǫ0|~ri − ~Rj |︸︷︷︸

EC,e−nuc

+1

2

N∑

i 6=j

e2

4πǫ0|~ri − ~rj |︸︷︷︸

EC,e−e

(1.4)

The electrons are characterized by their charge q = −e, their mass me and their position ~rj . Thenuclei are characterized by their masses Mj , their atomic numbers Zj and their positions ~Rj . The firstterm, denoted by Ekin,nuc , describes the kinetic energy of the nuclei. The second term, denoted byEkin,e , describes the kinetic energy of the electrons. The third term, denoted by EC,nuc−nuc , describesthe electrostatic repulsion between the nuclei. The fourth term, denoted by EC,e−nuc , describes theelectrostatic attraction between electrons and nuclei. The last term, denoted by EC,e−e , describesthe electrostatic repulsion between the electrons.

Deficiencies of the standard model

No model is perfect. Therefore, let me discuss what I consider its main deficiencies. The followingeffects are missing:

• Relativistic effects are important for the heavier elements, because the deep Coulomb potentialof the nuclei results in relativistic velocities for the electrons near the nucleus. Relativistic effectsare important for magnetic effects such as the magnetic anisotropy.

• Magnetic fields have only a small effect on solids. The dominant magnetic properties of

1 THE STANDARD MODEL OF SOLID-STATE PHYSICS 13

materials do not originate from magnetic interactions, but result from exchange effects of thespin-distribution. Important are, however, stray fields, which tend to force the magnetizationat a surface to be in-plane. Magnetic effects are also important to simulate the results ofNMR or Mössbauer experiments, which are sensitive to the magnetic interaction of nuclei andelectrons.

• Size, shape and spin of the nuclei play a role in certain nuclear experiments which measureisomer shifts, electric-field gradients and magnetic hyperfine parameters.

• Photons. The use of electrostatics is justified, because the electrons and nuclei are movingsufficiently slow that the electrostatic field responds instantaneously. Excitations by photonscan be treated explicitly.

These effects are not necessarily ignored, but can be incorporated into the standard model, whenrequired.

Dimensional Bottleneck

It is immediately clear that the standard model, as simple as it can be stated, is impossible to solve assuch. To make this argument clear let us estimate the amount of storage needed to store the wavefunction of a simple molecule such as N2. Each coordinate may be discretized into 100 grid points.For 2 nuclei and 14 electrons we need 100(3∗16) grid points. In addition, we need to consider the214 sets of spin indices. Thus, we need to store about 214 · 1003∗16 ≈ 10100 numbers. One typicalhard disk can hold about 100 GByte=1014 Byte, which corresponds to 1013 complex numbers. Acomplex number occupies 16 byte. This implies that we would need 1087 hard discs to store a singlewave function. A hard-disc occupies a volume of 5 cm×5cm×0.5 cm=1.25 ×10−5 m3. The volumeoccupied by the hard discs with our wave function would be 1082 m3 corresponding to a sphere witha radius of 1027 m or 1011 light years! I hope that this convinced you that there is a problem...

The problem described here is called the dimensional bottle-neck.In order to make progress we need to simplify the problem. Thus, we have to make approximations

and we will develop simpler models that can be understood in detail. This is what solid state physicsis about.

In the following we will perform a set of common approximations, and will describe their limitations.

14 1 THE STANDARD MODEL OF SOLID-STATE PHYSICS

Chapter 2

Born-Oppenheimer approximationand beyond

The Born-Oppenheimer approximation[7, 8] simplifies the many-particle problem considerably byseparating the electronic and nuclear coordinates. This method is similar, but not identical to themethod of separation of variables. We refer here to the formalism as described in appendix VIII ofthe Book of Born and Huang[8].

2.1 Separation of electronic and nuclear degrees of freedom

Our ultimate goal is to determine the wave function Φ(~x1, . . . , ~xN , ~R1, . . . , ~RM , t), which describesthe electronic degrees of freedom ~x1, . . . , ~xN and the atomic positions ~R1, . . . , ~RM on the basis ofthe time-dependent Schrödinger equation Eq. 1.1

i~∂t |Φ(t)〉 Eq. 1.1= H|Φ(t)〉 (2.1)

The Hamiltonian H is the one given in Eq. 1.4.In the Born-Oppenheimer approximation, one first determines the electronic eigenstates for a fixed

set of atomic positions. This solution forms the starting point for the description of the dynamics ofthe nuclei.

We will formulate an ansatz for the time-dependent many-particle wave functions. For this ansatzwe need the Born-Oppenheimer wave functions, which are the electronic wave function for a frozenset of nuclear positions.

Born-Oppenheimer wave functions and Born-Oppenheimer surfaces

Firstly, we construct the Born-Oppenheimer Hamiltonian HBO by removing all terms that depend onthe momenta of the nuclei from the many-particle Hamiltonian given in Eq. 1.4.

HBO(~R1, . . . , ~RM) =

N∑

i=1

−~22me

~∇2~ri︸︷︷︸

Ekin,e

+1

2

M∑

i 6=j

e2ZiZj

4πǫ0|~Ri − ~Rj |︸︷︷︸

EC,nuc−nuc

−N∑

i=1

M∑

j=1

e2Zj

4πǫ0|~ri − ~Rj |︸︷︷︸

EC ,e−nuc

+1

2

N∑

i 6=j

e2

4πǫ0|~ri − ~rj |︸︷︷︸

EC ,e−e

(2.2)

15

16 2 BORN-OPPENHEIMER APPROXIMATION AND BEYOND

The Born-Oppenheimer Hamiltonian acts in the Hilbert space of electronic wave functions, anddepends parametrically on the nuclear positions. In other words, the Born-Oppenheimer Hamiltoniandoes not contain any gradients acting on the nuclear positions.

In order to simplify the equations, I combine all the electronic coordinates and spin indices into avector ~x = (~x1, . . . , ~xN) and the nuclear positions into a 3M dimensional vector ~R = (~R1, . . . , ~RM).Thus, the many-particle wave function acting on electrons and nuclei will be written as Φ(~x, ~R, t) =〈~x, ~R|Φ(t)〉. I combine the nuclear masses into a diagonal matrix M with dimension 3M.

In addition, I introduce another notation, where the wave function is a regular function of theatomic positions and a quantum state |Φ(~R, t)〉 in the Hilbert space for the electrons, so thatΦ(~x, ~R, t) = 〈~x |Φ(~R, t)〉.

The full Hamiltonian of Eq. 1.4

H =1

2

(~

i~∇R)

M−1(~

i~∇R)

1 + HBO(~R) (2.3)

is obtained by adding the nuclear kinetic energy to the Born-Oppenheimer Hamiltonian HBO. Theunit operator 1 acts on the electronic Hilbert space.

The Born-Oppenheimer wave functions |ΨBOn (~R)〉 and the Born-Oppenheimer surfaces EBOn (~R)are obtained from the Born-Oppenheimer equation

[

HBO(~R)− EBOn (~R)]∣∣∣ΨBOn (~R)

⟩

= 0 . (2.4)

Eq. 2.4 corresponds to a Schrödinger equation for the electrons feeling the static potential fromthe nuclei, which are fixed in space. The Born-Oppenheimer surfaces EBOn (~R) are simply theposition-dependent energy eigenvalues of the Born-Oppenheimer Hamiltonian. For each atomic con-figuration we obtain a ground-state Born-Oppenheimer surface and infinitely many excited-stateBorn-Oppenheimer surfaces, which are labeled by the quantum number n. Similarly, the Born-Oppenheimer wave functions ΨBOn (~x1, . . . , ~xN , ~R1, . . . , ~RM) = 〈~x1, . . . , ~xN |ΨBOn (~R1, . . . , ~RM)〉 de-pend parametrically on the nuclear positions ~R.

The Born-Oppenheimer wave functions are chosen1 to be orthogonal for each set of atomicpositions, i.e.

⟨

ΨBOm (~R)∣∣∣ΨBOn (~R)

⟩

=

∫

d4Nx ΨBOm∗(~x, ~R)ΨBOn (~x, ~R) = δm,n (2.5)

Ansatz for the electronic-nuclear wave function

Now we are ready to write down the Born-Huang ansatz for the many-particle wave function of asystem of electrons and nuclei:

BORN-HUANG EXPANSION

Φ(~x, ~R, t) =∑

n

⟨

~x∣∣∣ΨBOn (~R)

⟩

φn(~R, t) (2.6)

The multi-valued nuclear wave function obeys the normalization condition

∑

n

∫

d3MR φ∗n(~R, t)φn(~R, t) = 1 (2.7)

The many-particle wave function Φ(~x, ~R, t) is expressed as a sum of products of the Born-Oppen-heimer wave functions |ΨBOn (~R)〉 and time-dependent nuclear wave functions φn(~R, t). It is better to

1They are orthogonal whenever the energies differ, and each set of degenerate states can be orthonormalized.

2 BORN-OPPENHEIMER APPROXIMATION AND BEYOND 17

think of Born-Oppenheimer wave functions and nuclear wave functions as multi-valued wave functionswith a component for each ground or excited state labelled by n. While the Born-Oppenheimer wavefunctions depend on the positions of the nuclei, the nuclear wave functions φn(~R1, . . . , ~RM , t) do nomore refer to the electronic positions, except for the state index n.

Some caution is required to keep the symbols for the different types of wave functions apart

Φ(~x, ~R, t) many-particle wave function of electrons and nucleiΨBOn (~x, ~R) Born-Oppenheimer wave function of electronsφn(~R, t) component of the nuclear wave function

Physical meaning of the Born-Huang expansion: In order to convey some of the physical meaningof the quantities defined in the Born-Huang ansatz, let me anticipate a few observations that will bederived and stated more rigorously in the following sections.

• One Born-Oppenheimer surface can be considered as the potential energy surface governingthe motion of the nuclei. The forces acting on the atoms are the negative gradients of theBorn-Oppenheimer surface. The Born-Oppenheimer surface is the object that is parameterizedin classical molecular dynamics simulations.

• The squared nuclear wave function describes the probability density pn(~R, t)def= φ∗n(~R, t)φn(~R, t)

for the nuclei to be at positions ~R and for the electrons to be in the n-th excited state.

• If the electrons remain in the ground state, only the nuclear wave function φ0(~R, t) with n = 0differs from zero, and the atomic motion is entirely determined by the single valued potentialenergy EBO0 (~R).

• An optical absorption event is usually sufficiently rapid, that the nuclei do not move by anyappreciable distance during that process. This is the Franck-Condon principle. In that casewe can estimate the absorption energy as the energy difference between two Born-Oppenheimersurfaces for the same nuclear coordinates.

The Born-Huang Expansion is not an approximation: It is important to realize that the Born-Huang expansion for the wave function is not an approximation. Every wave function can exactlybe represented in this form. For any given vector R and time t, the wave function Φ(~x, ~R, t) is afunction of the electronic coordinates ~x . This function can be expanded into any complete basisset|Ψn〉 in the form

Φ(~x, ~R, t) =∑

n

⟨

~x∣∣∣Ψn

⟩

cn(~R, t) (2.8)

with complex coefficients c(~R). Because the wave function changes with ~R and t, the coefficientsdepend on nuclear positions and time.

With the same right, one can choose a different complete basisset |Ψn(~R)〉 for every set ofnuclear coordinates, so that

Φ(~x, ~R, t) =∑

n

⟨

~x∣∣∣Ψn(~R)

⟩

cn(~R, t) (2.9)

The eigenstates of the Born-Oppenheimer approximation form such a complete basisset. Thuswe can replace |Ψn(~R)〉 by the Born-Oppenheimer wave functions. The corresponding coefficientsare the nuclear wave functions

Φ(~x, ~R, t) =∑

n

⟨

~x∣∣∣ΨBOn (~R)

⟩

φn(~R, t) (2.10)

which is nothing but the Born-Huang expansion Eq. 2.6.


Nuclear Schrödinger equation

When the Born-Huang expansion, Eq. 2.6, is inserted into the many-particle Schrödinger equationEq. 1.1 with the full Hamiltonian Eq. 1.4, we obtain

i~∂t

[∑

n

∣∣∣ΨBOn (~R)

⟩

φn(~R, t)

]

Eq. 1.1= H

[∑

n

∣∣∣ΨBOn (~R)

⟩

φn(~R, t)

]

Eq. 2.3=

[1

2

(~

i~∇R)

M−1(~

i~∇R)

+ HBO] [∑

n

∣∣∣ΨBOn (~R)

⟩

φn(~R, t)

]

Eq. 2.4=

∑

n

∣∣∣ΨBOn (~R)

⟩[1

2

(~

i~∇R)

M−1(~

i~∇R)

+ EBOn (~R)

]

φn(~R, t)

+∑

n

[(~

i~∇R) ∣∣∣ΨBOn (~R)

⟩]

M−1(~

i~∇R)

+1

2

(~

i~∇R)

M−1(~

i~∇R) ∣∣∣ΨBOn (~R)

⟩

φn(~R, t)

(2.11)

Now we multiply Eq. 2.11 from the left with the bra of the Born-Oppenheimer wave function⟨

ΨBOm (~R)∣∣∣ and integrate over the electronic coordinates. Hereby, we exploit the orthonormality

Eq. 2.5 of the Born-Oppenheimer wave functions.As the result, we obtain a Schrödinger equation for the nuclear motion alone.

i~∂tφm(~R, t)Eqs. 2.11,2.5

=

[~

i~∇RM−1

~

i~∇R + EBOm (~R)

]

φm(~R, t)

+∑

n

[⟨ΨBOm

∣∣~

i~∇R∣∣ΨBOn

⟩M−1

~

i~∇R +

1

2

⟨ΨBOm

∣∣~

i~∇RM−1

~

i~∇R∣∣ΨBOn

⟩]

︸︷︷︸

derivative couplings

φn(~R, t)

(2.12)

Note, that the brackets 〈. . .〉 are evaluated by integrating over the electronic degrees of freedomonly. Thus, the matrix elements still depend explicitly on the nuclear coordinates. Note also, thatthe gradients ~∇R in the parentheses act on the nuclear and not the electronic wave functions, whichis easily overlooked. The gradients ~∇R inside the matrix elements only act on the ket |ΨBOn 〉, butnot on the functions outside the matrix element.

Up to now, we did not introduce any approximations to arrive at the nuclear Schrödinger equationEq. 2.12. We are still on solid ground.

Derivative couplings

In the following, we will discuss the individual terms of Eq. 2.12. Before we continue, however, letus simplify the notation again: The general structure of Eq. 2.12 is

i~∂tφm(~R, t) =

[1

2~PM−1 ~P + EBOm (~R)

]

φm(~R, t) +∑

n

[

~Am,n(~R)M−1 ~P + Bm,n(~R)

]

φn(~R, t)

(2.13)

where ~P = ~

i~∇R is the momentum vector of the nuclear coordinates and the first- and second-

derivative couplings are defined by

~Am,n(~R) :=⟨

ΨBOm (~R)∣∣∣~

i~∇R∣∣∣ΨBOn (~R)

⟩

(2.14)

Bm,n(~R) :=1

2

⟨

ΨBOm (~R)∣∣∣

(~

i~∇R)

M−1(~

i~∇R) ∣∣∣ΨBOn (~R)

⟩

(2.15)


Second-order couplings from first-order derivative couplings: The first and the second-derivativecouplings are not independent of each other: The second-derivative coupling Bm,n(~R) can be ex-pressed by the first-derivative couplings ~Am,n(~R). This is shown as follows (see [9]):

~

i~∇RM−1 ~Am,n(~R) Eq. 2.14

=~

i~∇RM−1

⟨

ΨBOm (~R)

∣∣∣∣

~

i~∇R∣∣∣∣ΨBOn (~R)

⟩

= −⟨~

i~∇RΨBOm (~R)

∣∣∣∣M−1

~


⟩

+

⟨

ΨBOm (~R)

∣∣∣∣

(~

i~∇R)

M−1(~

i~∇R)∣∣∣∣ΨBOn (~R)

⟩

︸︷︷︸

2Bm,n(~R

Eq. 2.15= −

⟨~

i~∇RΨBOm (~R)

∣∣∣∣

(∑

k

∣∣∣∣ΨBOk (~R)

⟩⟨

ΨBOk (~R)

∣∣∣∣

)

︸︷︷︸

=1

M−1~


⟩

+ 2Bm,n(~R)

= −∑

k

⟨~

i~∇RΨBOm (~R)

∣∣∣∣ΨBOk (~R)

⟩

M−1⟨

ΨBOk (~R)

∣∣∣∣

~


⟩

+ 2Bm,n(~R)

= −∑

k

⟨

ΨBOk (~R)

∣∣∣∣

~

i~∇R∣∣∣∣ΨBOm (~R)

⟩∗M−1

⟨

ΨBOk (~R)

∣∣∣∣

~


⟩

+ 2Bm,n(~R)

= −∑

k

~A∗k,m(~R)M−1 ~Ak,n(~R) + 2Bm,n(~R)

Next, we exploit that the first-derivative couplings ~Am,n are hermitian in the band indices, i.e.~A∗k,m(~R) = ~Am,k(~R), which can be shown as follows: We start from the orthonormality conditionEq. 2.5 of the Born-Oppenheimer wave functions, which holds for all ~R. Thus the nuclear gradientof the scalar products vanishes

0Eq. 2.5= ~∇R

δm,n︷︸︸︷

〈ΨBOm |ΨBOn 〉= 〈~∇RΨBOm |ΨBOn 〉+ 〈ΨBOm |~∇R|ΨBOn 〉= 〈ΨBOn |~∇RΨBOm 〉∗ + 〈ΨBOm |~∇R|ΨBOn 〉

⇒ 0 = −〈ΨBOn |~

i~∇RΨBOm 〉∗ + 〈ΨBOm |

~

i~∇R|ΨBOn 〉

⇒ ~A∗k,m(~R) =⟨

ΨBOk

∣∣∣~

i~∇R∣∣∣ΨBOm

⟩∗=⟨

ΨBOm

∣∣∣~

i~∇R∣∣∣ΨBOk

⟩

= ~Am,k(~R) (2.16)

This allows us to rewrite Eq. 2.16 in the form

2Bm,n(~R) =~

i~∇RM−1 ~Am,n(~R) +

∑

k

~Am,k(~R)M−1 ~Ak,n(~R) (2.17)

Thus, we only need to evaluate the first-derivative couplings, while the second-derivative couplingscan be obtained from the former via Eq. 2.17.

Final form for the nuclear Schrödinger equation

Insertion of the second derivative couplings from Eq. 2.17 into the nuclear Schrödinger equationEq. 2.13 into the final form of Eq. 2.20.

Let me sketch here the line of thought schematically. I am dropping the matrix structure of theequation for the sake of clarity. The terms in Eq. 2.13 related to momenta and derivative couplings


have the form

(1

2~PM−1 ~P + ~AM−1 ~P + B

)

|φ〉 Eq. 2.17=1

2

(

~PM−1 ~P + 2~AM−1 ~P + (~PM−1 ~A) + ~AM−1 ~A)

|φ〉

=1

2

(

~PM−1 ~P + ~AM−1 ~P + ~PM−1 ~A+ ~AM−1 ~A)

|φ〉

=1

2

(

~P + ~A)

M−1(

~P + ~A)

|φ〉 (2.18)

We used the the chain rule for ~P , which is a gradient operator.

~PM−1 ~A|φ〉 = (~PM−1 ~A)|φ〉+ ~AM−1 ~P |φ〉 (2.19)

In this manner one arrives at our final form Eq. 2.20 for the nuclear Schrödinger equation.

SCHRÖDINGER EQUATION FOR THE NUCLEAR WAVE FUNCTIONS IN TERMS OFBORN-OPPENHEIMER WAVE FUNCTIONS

i~∂tφm(~R, t) =∑

n

[1

2

∑

k

(

δm,k~

i~∇R + ~Am,k(~R)

)

M−1(

δk,n~

i~∇R + ~Ak,n(~R)

)

+δm,nEBOm (~R)

]

φn(~R, t) (2.20)

or, combining the components of the nuclear wave functions in a vector-matrix notation – using asingle underline to denote a vector and a double underline for a matrix –

i~∂tφ(~R, t) =

(

1 ~P + ~A(~R))2

2M+ EBO(~R)

φ(~R, t) (2.21)

EBO denotes the diagonal matrix with the Born-Oppenheimer energies as diagonal elements and

~P = ~

i~∇R is the 3M-dimensional momentum operator of the nuclei. The kinetic energy expression in

Eq. 2.21 is mathematically not allowed, because I wrote a vector-matrix-vector expression as vector-squared-divided-by-matrix. I used it here to allude to the corresponding equation from electrodynamicsso that it can be memorized more easily, and because the matrix M is a diagonal matrix so that theexpression can be evaluated component by component.The first-derivative couplings ~Am,k(~R) are defined in Eq. 2.14.

~Am,n(~R)Eq. 2.14:=

⟨

ΨBOm (~R)∣∣∣~

i~∇R∣∣∣ΨBOn (~R)

⟩

=⟨

ΨBOm (~R)∣∣∣ ~P∣∣∣ΨBOn (~R)

⟩

(2.22)

As shown in Eq. 2.16, each vector component of the first-derivative couplings is a hermitian matrixin the indices m, n.

This form of the nuclear Schrödinger equation, i.e. Eq. 2.21, has been mentioned2 already in1969 by F. Smith[10].

We have not introduced any approximations to bring Eq. 2.12 into the new form given by Eq. 2.21.That is, we are still on solid grounds.

2see his Eq. 10. His ~P (R) are the first-derivative couplings.


Relation to electrodynamics

Interesting is the formal similarity of the nuclear Hamilton function in Eq. 2.14 with that of a chargedparticle in a electromagnetic field, which has the form

H(~p,~r) =1

2m

(

~p − q ~A(~r))2

+ qΦ(~r)

This analogy shows that the diagonal terms (in the band indices) of the derivative couplings act likea magnetic field ~B expressed by the vector potential ~A as ~B = ~∇ × ~A and an electric potential Φ.We will discuss this analogy later in somewhat more detail in the appendix B.2.

2.2 Approximations

2.2.1 Born-Oppenheimer approximation

The Born-Oppenheimer approximation amounts to ignoring the derivative couplings in Eq. 2.20

BORN-OPPENHEIMER APPROXIMATION FOR THE NUCLEAR WAVE FUNCTION

i~∂tφn(~R, t) =

M∑

j=1

−~22Mj

~∇2Rj + EBOn (~R)

φn(~R, t) (2.23)

This equation describes nuclei that move on a given total energy surface EBOn (~R), which is calledthe Born-Oppenheimer surface. The Born-Oppenheimer surface may be an excited-state surface orthe ground-state surface. The wave function may also have contributions simultaneously on differenttotal energy surfaces. However, within the Born-Oppenheimer approximation, the contributions ondifferent surfaces do not influence each other.

In other words, the system is with probability Pn =∫d3MR φ∗n(~R)φn(~R) on the excited state

surface EBOn (~R). In the Born-Oppenheimer approximation, this probability does not change withtime.

The neglect of the non-adiabatic effects is the essence of the Born-Oppenheimer approximation.In the absence of the non-adiabatic effects, we could start the system in a particular eigenstate ofthe Born-Oppenheimer Hamiltonian, and the system would always evolve on the same total energysurface EBOn (~R). Thus, if we start the system in the electronic ground state, it will remain exactlyin the instantaneous electronic ground state, while the nuclei are moving. Band crossings are theexceptions: Here, the Born-Oppenheimer approximation does not give a unique answer.

The separation of nuclear and electronic degrees of freedom have already been in use before theoriginal Born-Oppenheimer approximation[11]. Born and Oppenheimer [7] gave a justification for ne-glecting the non-adiabatic effects in many cases. As we will see the Born-Oppenheimer approximationbreaks down when Born-Oppenheimer surfaces come very close or even cross.

2.2.2 Classical approximation

Classical approximation in the Born-Oppenheimer approximation

The most simple approach to the classical approximation is to take the Hamilton operator in theBorn-Oppenheimer approximation from Eq. 2.23, constrain it to a particular total-energy surfacespecified by n, and to form the corresponding Hamilton function.

Hn(~P , ~R) =∑

j

~P 2j2Mj

+ EBOn (~R)


The Hamilton function defines the classical Hamilton equations of motion

∂t ~Rj = ~∇PjHn(~P , ~R, t) =1

Mj

~Pj

∂t ~Pj = −~∇RjHn(~P , ~R, t) = −~∇RjEBOn (~R)

This in turn leads to the Newton’s equations of motion for the nuclei

Mj∂2t~Rj = −~∇RjEBOn (~R)

This is the approximation that is most widely used to study the dynamics of the atoms ina molecule or a solid. A simulation of classical atoms using some kind of parameterized Born-Oppenheimer surface is called molecular dynamics simulation. When the Born-Oppenheimer sur-face is obtained on the fly from a quantum mechanical electronic structure method, the method iscalled ab-initio molecular dynamics.

The Born-Oppenheimer surface EBOn (~R) acts just like a total-energy surface for the motion ofthe nuclei. Once EBOn (~R) is known, the electrons are taken completely out of the picture. Withinthe Born-Oppenheimer approximation, no transitions between the ground-state and the excited-statesurface take place. This implies not only that a system in the electronic ground state remains inthe ground state. It also implies that a system in the excited state will remain there for ever.Transitions between different sheets EBOn (~R) of the Born-Oppenheimer energy are only possiblewhen non-adiabatic effects are included.

2.3 Non-adiabatic corrections

While the Born-Oppenheimer approximation is a widely used and tremendously successful approach,its limitations are important for relaxation processes, photochemistry, charge transfer processes andmany more interesting processes. They shall be briefly discussed here.

We will see that the derivative couplings governing the non-adiabatic effects are singular wheretwo Born-Oppenheimer surface meet. Unfortunately, these are exactly the points of physical interest,namely those where a system makes a non-radiative transition from one Born-Oppenheimer surfaceto another. Even worse, these singular points have non-local consequences, so-called topologicaleffects.

Interesting is a close formal similarity between non-adiabatic effects and field theory[12]. Relatedtopics are the Aharonov-Bohm effect[13] and Yang-Mills fields[14].

Under normal circumstances, the diagonal terms of the derivative couplings are of secondary im-portance. The diagonal term of the derivative couplings are discussed in the appendix B.2. Theybecome, however, important in the presence of conical intersections and in the presence of a geo-metrical phase.

2.3.1 Non-diagonal derivative couplings

The off-diagonal elements of the first-derivative couplings defined in Eq. 2.22 can be brought intothe form

~Am,n =

⟨

ΨBOm

∣∣∣

(~

i~∇RHBO(~R)

)∣∣∣ΨBOn

⟩

EBOn (~R)− EBOm (~R)for EBOm 6= EBOn (2.24)

The gradient acts only on the Born-Oppenheimer Hamiltonian but not further to the state on theright. This expression makes it evident that non-adiabatic effects are important when two Born-Oppenheimer surfaces become degenerate, i.e. when EBOm = EBOn .


Derivation of Eq. B.2 The off-diagonal elements of ~An,m(~R) of the first-derivative couplings areobtained as follows: We begin with the Schrödinger equation for the Born-Oppenheimer wave function

(

HBO(~R)− EBOn (~R))

|ΨBOn (~R)〉Eq. 2.4= 0

Because this equation is valid for all atomic positions, the gradient of the above expression vanishes.

0 = ~∇R(

HBO(~R)− EBOn (~R)) ∣∣∣ΨBOn (~R)

⟩

=(

~∇R, HBO(~R)) ∣∣∣ΨBOn (~R)

⟩

−∣∣∣ΨBOn (~R)

⟩

~∇REBOn (~R) +(


~∇R∣∣∣ΨBOn (~R)

⟩

Now, we multiply from the left with the bra 〈ΨBOm | and form the scalar products in the electronicHilbert space.3

0 =⟨ΨBOm

∣∣

(

~∇RHBO(~R)) ∣∣ΨBOn

⟩−⟨ΨBOm

∣∣ΨBOn

⟩~∇REBOn (~R) +

⟨ΨBOm

∣∣

(


~∇R∣∣ΨBOn

⟩

=⟨ΨBOm

∣∣

(

~∇R, HBO(~R)) ∣∣ΨBOn

⟩− δm,n~∇REBOn (~R)︸︷︷︸

=0for m 6= n

+

(

EBOm (~R)− EBOn (~R))⟨ΨBOm

∣∣ ~∇R

∣∣ΨBOn

⟩

If the indices m, n differ, the middle term containing gradient of the Born-Oppenheimer surfacesvanishes. If the energies EBOm , EBOn furthermore differ, we can divide by their difference which bringsus to the desired result.

~Am,nEq. 2.14=

⟨ΨBOm

∣∣~

i~∇R∣∣ΨBOn

⟩=〈ΨBOm |

(~

i~∇RHBO(~R)

)

|ΨBOn 〉EBOn (~R)− EBOm (~R)

for En 6= Em (2.25)

This is the proof for Eq. 2.24 given above. Unfortunately, no information can be extracted for thediagonal elements of the first-derivative couplings.

2.3.2 Topology of crossings of Born-Oppenheimer surfaces

The off-diagonal non-adiabatic effects Eq. 2.24 become important, when the different Born-Oppenheimersurfaces come close or even cross. Therefore we need to understand the topology of crossings ofBorn-Oppenheimer surfaces.

Non-crossing rule and conical intersections

The most important information on the topology of surface crossings is provided by the non-crossingrule of von Neumann and Wigner[15]. It is derived in appendix C on p. 217.

NON-CROSSING RULE

Consider a Hamiltonian H(Q1, . . . , Qm) which depends on m parameters Q1, . . . , Qm. For eachaccidental crossing of two eigenvalues, there is a three-dimensional subspace of the parameter spacein which the degeneracy is lifted except for a single point.If the hamiltonian is real for all parameters, the subspace where the degeneracy is lifted has twodimensions.

• In our case, the Hamiltonian is the Born-Oppenheimer Hamiltonian and the parameters are thenuclear positions.

3The Born-Oppenheimer wave functions depend on electronic and nuclear coordinates. The nuclear coordinates,however, play a different role, because they are treated like external parameters.


• Two energy surfaces that meet in a point within a two-dimensional space form two cones.Therefore the crossing is called a conical intersection.

• The hamiltonian is usually4 real in the absence of magnetic fields and relativistic effects. Spin-orbit coupling, a relativistic effect, and magnetic fields that act on the orbital motion of chargedparticles introduce complex contributions. The question whether the Hamiltonian is real isrelated to time inversion symmetry discussed in Appendix N on p. N.

• The limitation of the non-crossing rule to accidental crossings implies that there are no limita-tions on crossings that occur because two states have different symmetry.

• The non-crossing rule implies that there are no accidental crossings for dimeric molecules

Let me consider a conical intersection related to the motion of a single atom in a molecule or solid.Let me furthermore consider real Hamiltonians. For this example, the conical intersection would forma one-dimensional line. The two displacements perpendicular to this line lift the degeneracy. If weplot the energy surfaces in the plane of these two distortions, the Born-Oppenheimer surfaces wouldhave the shape of a double cone, a conical intersection. A change of a coordinate along the linewill change the shape of the cone, but it will not affect the qualitative cone-like topology.

More generally, there is a two-dimensional plane of coordinates relevant to the conical intersection.Distortions of distant nuclei will not affect the general shape of the conical intersection, while theydo have an effect on the total energy.

2.3.3 Avoided crossings

Often, the symmetries that are responsible for a degeneracy, which is a surface crossing, are onlyapproximately present. A small matrix element lifts the degeneracy “a little”. This is called an avoidedavoided crossing. Due to an avoided crossing two Born-Oppenheimer sheets may be come so closethat the non-adiabatic effects can no more be ignored. Thus, regions with avoided crossings areimportant for the relaxation of the system onto a Born-Oppenheimer surface with lower energy.

What is an avoided crossing? Let me start with a position-dependent Hamiltonian for a one-dimensional nuclear coordinate R, for which the energy levels cross for a certain position R0 = 0. Inaddition we add non-diagonal terms ∆.

The Hamilton has the form

H =P 2

2M+

(

|a〉|b〉

)(

E0 + FQ −∆−∆ E0 − FQ

)(

〈a|〈b|

)

︸︷︷︸

HBO

(2.26)

The Born-Oppenheimer Hamiltonian is obtained by stripping away the nuclear kinetic energy andby converting the coordinate operator Q into a parameter.

HBO(Q) =

(

|a〉|b〉

)(

E0 + FQ −∆−∆ E0 − FQ

)(

〈a|〈b|

)

(2.27)

Diagonalization of the Born-Oppenheimer Hamiltonian yields the two Born-Oppenheimer surfaces

EBO± (Q) = E0 ± ∆√

1 +

(F

∆Q

)2

(2.28)

where the minus sign applies for the lower and the plus sign for the upper Born-Oppenheimer surface.

4There are other interactions that are similar to a magnetic field. They are, however, not very important for solidstate physics.


R

E

2∆

The eigenstates of the Born-Oppenheimer Hamiltonian, the Born-Oppenheimer states are

|ΦBO+ (Q)〉 =(

|a〉|b〉

)(

− sin(γ(Q))cos(γ(Q))

)

and |ΦBO− (Q)〉 =(

|a〉|b〉

)(

cos(γ(Q))

sin(γ(Q))

)

(2.29)

where

γ(Q) = arctan

F

∆Q+

√

1 +

(F

∆Q

)2

=π

4+1

2arctan

(F

∆Q

)

(2.30)

is shown in Fig. 2.1.

-10 -5 0 5 100

π/2

π/4

Fig. 2.1: γ(Q) determining the eigenstates and derivative couplings for the avoided crossing. Thecoordinate is drawn in units of ∆F The dashed line is the derivative of φ(Q) in units of F

∆ .

The non-diagonal derivative couplings are

A−,+ = −A+,− =~

i

dγ(Q)

dQ(2.31)

and the diagonal derivative couplings vanish. From the red dashed line in Fig. 2.1, it is evident thatthe derivative couplings are large near the avoided crossing. The reason for this is that the wavefunction changes its character from one side of the derivative coupling to the other.

2.3.4 Conical intersections: Jahn-Teller model

Because I find the conical intersections a little mind-boggling, let me switch from the general andabstract to the more concrete case. That is will explain the conical intersection for a simple model,the Jahn-Teller model. The Jahn Teller model is the main model system for a conical intersection.

For a review on conical intersection see for example the paper of Yarkony [16]The Jahn-Teller model can be used to describe the Jahn-Teller effect in doped manganites.

Manganites such as CaxLa1−xMnO3 form a very interesting class of materials with a complex phasediagram and enormous variety of different physical properties that can be controlled by doping,


Fig. 2.2: Left: the two Jahn-Teller active modes of the octahedron in a perovskite ABO3. Thered spheres represent oxygen ions, the yellow atom in the center is the transition metal ion (B-typeion) on which the two eg orbitals reside. The green spheres are the A-type ions. The undistortedoctahedron is drawn in the center. Along the horizontal axis the positive and negative distortions ofthe Q2 mode are drawn. Along the vertical axis the positive and negative distortions of the Q3 modeare drawn. Right top: Jahn-Teller splitting of the two eg orbitals under the influence of a positiveand negative distortion of the Q3 mode. Left bottom: The two Born-Oppenheimer surfaces of theJahn-Teller model in the two-dimensional coordinate space of the two Jahn-Teller active modes. Thepoint of contact of the two Born-Oppenheimer surfaces is a conical intersection.

temperature, magnetic fields, etc. Their main structural element is a network of corner-sharingMnO6 octahedra. If Mn is in the 4+ charge state (MnIV ) as in CaMnO3, the Mn ions in the centerof the octahedra exhibit two degenerate d-orbitals of eg symmetry, which are unoccupied.

In the doped materials, electrons can be added to these pair of degenerate states. This electronin a given orbital induces a structural distortion of the octahedra, which lowers the filled orbital andraises the empty orbital. The distortions are the elongation and compression of the octahedra alongone axis into a prolate (cigar-shaped) and an oblate (pancake-shaped) form.

One can say that the electron and the octahedral distortion (a phonon) form a bound state. Thisbound state is called a polaron.

The Jahn-Teller distortions can be expressed in terms of the outward displacements ax , ayand az of the oxygen ligands in x,y, and z directions, respectively. These can be combined intoa vector (ax , ay , az). The two Jahn-Teller active displacements are[17] Q2 = 1√

2(1,−1, 0) and

Q3 = 1√6(−1,−1, 2). The positive and negative distortions Q3 can be described by a prolate

(cigar-shaped) and an oblate (pancake-shaped) distortion. The Jahn-Teller distortions span a twodimensional parameter space. They can be described by an amplitude and an angle parameter αspecifying the type of the distortion pattern.

In the following we discuss the conical intersection at a single octahedron. The relevant struc-tural distortions of the octahedra are described by the generalized nuclear coordinates x, y , and theelectronic wave function is described as a vector in a two-dimensional Hilbert space of the two egorbitals. I label the two eg orbitals as |a〉 and |b〉.

The atomic coordinates, that is the generalized coordinates for the two octahedral distortions,


α (ax , ay , az) (Q2,Q3)0 1√

2(1,−1, 0) (1, 0)

30 1√6(1,−2, 1) ( 12

√3, 12) oblate along y

60 1√2(0, 1,−1) ( 12 ,

12

√3)

90 1√6(−1,−1, 2) (0, 1) prolate along z

120 1√2(−1, 0, 1) (− 12 , 12

√3)

150 1√6(−2, 1, 1) (− 12

√3, 12) oblate along x

180 1√2(−1, 1, 0) (−1, 0)

210 1√6(−1, 2,−1) (− 12

√3,− 12) prolate along y

240 1√2(0, 1,−1) (− 12 ,− 12

√3)

270 1√6(1, 1,−2) (0, 1) oblate along z

300 1√2(1, 0,−1) ( 12 ,− 12

√3)

330 1√6(2,−1,−1) ( 12

√3,− 12) prolate along x

Table 2.1: . Jahn-Teller active distortions of the octahedron as function of the angular coordinate α.The second row (ax , ay , az) describes the bond expansions in x,y,z directions and the third describesthe mode amplitudes.

which couple to the two eg states are denoted by the two-dimensional vector5 ~R = (X,Z).The time-dependent Schrödinger equation

i~∂t |Φ(t)〉 =[~P 2

2M+ w ~R2 + g ~R ~σ

]

|Φ(t)〉 (2.32)

where g is the electron-phonon coupling parameter and w is the force constant for the restoringforce. ~σ act on the electronic degrees of freedom, where they are represented as the Pauli matrices

σx and σz . ~R = (X, Z) and ~P are the operators describing the nuclear octahedral distortions andtheir momenta.

As a little warning: In this problem only electrons of one spin direction take part. Because ofthat reason there is no spin coordinate, and we use the spatial coordinate ~r of the electrons. ThePauli matrices used here are no spin operators but matrices in the two-dimensional Hilbert space ofthe two eg orbitals (both having the same spin).

In components, this equation has the form

i~∂t

(

Φa(X,Z, t)

Φb(X,Z, t)

)

=

[−~22M

(∂2X + ∂

2Z

)+ w

(X2 + Z2

)](

Φa(X,Z, t)

Φb(X,Z, t)

)

+g

(

Z, X

X −Z

)(

Φa(X,Z, t)

Φb(X,Z, t)

)

(2.33)

Here we used the matrix elements Φa(X,Z, t) =∫d3r φa(~r)Φ(~r , X,Z, t) with the one-particle

orbitals φa and φb for the two eg orbitals. The full wave functions including nuclear and electronicwave functions is Φ(~r , X,Z, t) = φa(~r)Φa(X,Z, t) + φb(~r)Φb(X,Z, t).

Before we continue, let me mention for the interested reader the relation of the Jahn-Tellermodel with the Higgs mechanism, discussed in elementary particle physics as origin of the particlemass. The Higgs mechanism postulated a mexican-hat potential which results from a spontaneoussymmetry breaking. In the symmetry-broken state, the particle has two modes, of which the radial

5I am using uppercase symbols as these are nuclear coordinates. X and Z are used rather than X and Y , becauseof the two non-imaginary Pauli matrices σx and σz .


mode has a finite curvature and is identified with the Higgs mode, while the motion into the otherdirection has no curvature and is identified with the “massless” Nambu-Goldstone boson.

Born-Oppenheimer Hamiltonian and Born-Oppenheimer surfaces The Born-Oppenheimer Hamil-tonian is obtained by stripping the terms containing the nuclear momenta from the many-particleHamiltonian.6

HBO(~R) = g

(

Z X

X −Z

)

+ w

(

X2 + Z2, 0

0, X2 + Z2

)

= g ~R · ~σ + w ~R2111 (2.34)

The first term describes the electronic Hamiltonian, that is the splitting of the degenerate electronicstates upon distortion, while the second term is a simple restoring potential for the nuclear distortions.

The characteristic equation for the eigenvalues EBO± (~R) is(

+gZ + w ~R2 − EBO± (~R) gX

gX −gZ + w ~R2 − EBO± (~R)

)(

φa,±φb,±

)

= 0 (2.35)

This yields the two Born-Oppenheimer surfaces of the Jahn-Teller model

BORN-OPPENHEIMER SURFACES OF THE JAHN-TELLER MODEL

EBO± (~R) = ±g|~R|+ w ~R 2 (2.36)

The Born-Oppenheimer surfaces EBO± (~R) are shown in Fig. 2.2. The two surfaces meet at a singlepoint, which is the conical intersection.

Born-Oppenheimer wave functions To construct the eigenstates we take the first line of Eq. 2.35and construct the orthogonal vector according to the rule (u, v) ⊥ (−v , u), and then we normalizeit. This yields the Born-Oppenheimer states |ΨBO± 〉 with components ΨBOa,± = 〈a|ΨBO± 〉 and ΨBOb,± =〈b|ΨBO± 〉.

(

ΨBOa,±ΨBOb,±

)

=

(

−gXgZ + w ~R2 ∓ g|~R| − w ~R2

)

1√

g2X2 +(

gZ + w ~R2 ∓ g|~R| − w ~R2)2

=

(

−XZ ∓ |~R|

)

1√

X2 +(

Z ∓ |~R|)2

(2.37)

The bar ontop of the symbol for the Born-Oppenheimer state is there to distinguish it from the finalresult, which is multiplied with an additional phase factor.

Now we switch to polar coordinates

X(R,α) = R sin(α)

Z(R,α) = R cos(α) (2.38)

which yields(

ΨBOa,±ΨBOb,±

)

=

(

− sin(α)∓(1∓ cos(α))

)

1√

2(1∓ cos(α))(2.39)

6 ~R~σ = Xσx + Y σy .


We use the trigonometric identities

∣∣∣sin

(x

2

)∣∣∣ =

√

1

2

(

1− cos(x))

∣∣∣cos

(x

2

)∣∣∣ =

√

1

2

(

1 + cos(x))

(2.40)

to obtain(

ΨBOa,+ΨBOb,+

)

=

(

−2 sin(α2

)cos(α2

)

−2 sin2(α2

)

)

1

2| sin(α2

)| =

(

− cos(α2

)

− sin(α2

)

)

sgn(

sin(α/2))

(

ΨBOa,−ΨBOb,−

)

=

(

−2 sin(α2

)cos(α2

)

2 cos2(α2

)

)

1

2| cos(α2

)| =

(

− sin(α2

)

cos(α2

)

)

sgn(

cos(α/2))

(2.41)

where the sign function “sgn” is positive for positive arguments and negative for negative arguments.7

The resulting functions are sketched in Fig. 2.3.

Fig. 2.3: Angular dependence of the discontinuous, real Born-Oppenheimer wave functions for thelower (left) and upper(right) Born-Oppenheimer surfaces. The top graph is the coefficient for theorbital |a〉 and the lower graph is that for orbital |b〉. The radial dependence is only schematic and isnot calculated. Left and right wave functions have different orientations. The discontinuities are inopposite directions for the two Born-Oppenheimer wave functions.

The Born-Oppenheimer wave functions are discontinuous because of the sign function. The termwithout the sign-term is periodic with 4π, that is, it changes sign with every full turn. The sign term,restores the original 2π periodicity of the problem. It does so, however, at the price of introducing adiscontinuity.

The Born-Oppenheimer wave function are determined only up to a complex phase factor. Thismay lead to several alternative descriptions

• The Born-Oppenheimer wave functions are real, but discontinuous in the nuclear coordinatesas in the example above.

• The Born-Oppenheimer wave functions are real and continuous, but double valued. Doublevalued means that they change sign with a full turn around the conical intersection. That is,

7For an argument zero, the sgn function vanishes. Here we will not discuss this isolated point which will not causeproblems.


they obtain the original value only after two full turns. This result would be obtained if onedrops the sign term in the above equation.

• The Born-Oppenheimer wave function is continuous and single valued, but complex. This isobtained from the above result by replacing the sign term with a smooth phase factor with thesame periodicity as the sign function. This representation is created below.

In the literature the first choice, namely real Born-Oppenheimer wave functions are usually as-sumed, explicitly or implicitly. In order to avoid discontinuities, half-integer spins are introduced.While this may have some intellectual appeal, I find the resulting double-valuedness of the wave func-tion highly ambiguous. Therefore, I follow a different path and admit that the Born-Oppenheimerwave functions are complex-valued.

By replacing the sign function by8 ∓eiα/2, i.e.

|ΨBO+ 〉 = −|ΨBO+ 〉sgn(

sin(α/2))

eiα/2

|ΨBO− 〉 = +|ΨBO− 〉sgn(

cos(α/2))

eiα/2 (2.42)

we obtain9 Born-Oppenheimer states

(

ΨBOa,+ΨBOb,+

)

=

(

cos(α/2)

sin(α/2)

)

eiα/2 =1

2

(

eiα + 1

−i(eiα − 1)

)

(

ΨBOa,−ΨBOb,−

)

=

(

− sin(α/2)cos(α/2)

)

eiα/2 =1

2

(

+i(eiα − 1)eiα + 1

)

(2.43)

BORN-OPPENHEIMER STATES OF THE JAHN-TELLER MODEL

(

ΨBOa,+ΨBOb,+

)

=1

2

(

eiα + 1

−i(eiα − 1)

)

and

(

ΨBOa,−ΨBOb,−

)

=1

2

(

+i(eiα − 1)eiα + 1

)

(2.44)

The abstract states are obtained from these coefficients as

|ΨBO± (~R)〉 = |~a〉ΨBOa,±(~R) + |~b〉ΨBOb,±(~R) (2.45)

where α is determined by the nuclear coordinates ~R via Eq. 2.38

Eq. 2.42 is a gauge transformation, which removes the discontinuities except for the one at theconical intersection, i.e. for at the origin R = 0. At the conical intersection, the Born-Oppenheimerwave function is undetermined, which reflects that any unitary transformation of a degenerate setof eigenstates produces another, equally valid, set of eigenstates. We may fix this ambiguity by thechoice that for R = 0 the value α = 0 is taken.

Nevertheless the discontinuity at the origin remains. Discontinuities in the Born-Oppenheimerwave functions lead to δ-function-like contributions in the derivative couplings.

The important message is that the Born-Oppenheimer wave function near a conical intersec-tion cannot be chosen real without introducing steps in the wave function. This invalidates manyarguments which indicate that the derivative couplings may be negligible.

8The additional sign change has been introduced to make the equations appear simpler. It is permitted because itis a global sign change.

9Using cos(x) = 12(eix + e−ix ) and sin(x) = 1

2i(eix − e−ix )


Derivative couplings The next step towards the equation of motion for the nuclear wave functionsis to evaluate the derivative couplings.

~Am,nEq. 2.22=

⟨

ΨBOm (~R)∣∣∣~

i~∇~R

∣∣∣ΨBOn (~R)

⟩

(2.46)

with m, n ∈ −,+.We transform the gradient into polar coordinates defined in Eq. 2.38

(

∂R

∂α

)

=

(∂X∂R; ∂Z∂R

∂X∂α; ∂Z∂α

)(

∂X

∂Z

)

=

(

sin(α); cos(α)

R cos(α); −R sin(α)

)(

∂X

∂Z

)

⇒ ~∇~R =

(

∂X

∂Z

)

=

(

sin(α); + 1R cos(α)

cos(α); − 1R sin(α)

)(

∂R

∂α

)

(2.47)

~Am,nEq. 2.22=

⟨

ΨBOm (~R)∣∣∣~

i~∇∣∣∣ΨBOn (~R)

⟩

Eq. 2.44=

~

iR

(

cos(α)

− sin(α)

)⟨

ΨBOm (~R)∣∣∣ ∂α

∣∣∣ΨBOn (~R)

⟩

(2.48)

With the Born-Oppenheimer wave functions from Eq. 2.44 we obtain

⟨

ΨBO+ (~R)∣∣∣ ∂α

∣∣∣ΨBO+ (~R)

⟩

=(1

2(eiα + 1)

)∗( i2eiα)

+(

− i2(eiα − 1)

)∗(12eiα)

=i

2⟨

ΨBO+ (~R)∣∣∣ ∂α

∣∣∣ΨBO− (~R)

⟩

=(1

2(eiα + 1)

)∗(−12eiα)

+(

− i2(eiα − 1)

)∗( i2eiα)

= −12

⟨

ΨBO− (~R)∣∣∣ ∂α

∣∣∣ΨBO+ (~R)

⟩

=( i

2(eiα − 1)

)∗( i2eiα)

+(1

2(eiα + 1)

)∗(12eiα)

=1

2⟨

ΨBO− (~R)∣∣∣ ∂α

∣∣∣ΨBO− (~R)

⟩

=( i

2(eiα − 1)

)∗(−12eiα)

+(1

2(eiα + 1)

)∗( i2eiα)

=i

2(2.49)

which yields the first-derivative couplings

~A+,+ =~

2R

(

cos(α)

− sin(α)

)

=~

2R2

(

Z

−X

)

~A+,− =i~

2R

(

cos(α)

− sin(α)

)

=i~

2R2

(

Z

−X

)

~A−,+ =−i~2R

(

cos(α)

− sin(α)

)

=−i~2R2

(

Z

−X

)

~A−,− =~

2R

(

cos(α)

− sin(α)

)

=~

2R2

(

Z

−X

)

(2.50)


FIRST-DERIVATIVE COUPLINGS OF THE JAHN-TELLER MODEL

(~A++ ~A+−~A−+ ~A−−

)

=~

2R2

(

Z

−X

)

⊗(

1 +i

−i 1

)

(2.51)

The outer product between a vector and a matrix is defined as (~a⊗B)i ,j,k = aiBj,k . It can be writtena matrix

~a ⊗B =(

~a B1,1 ~a B1,2

~a B2,1 ~a B2,2

)

(2.52)

with vectors as matrix elements.

Nuclear wave function At this point, we know the Born-Oppenheimer surfaces from Eq. 2.36,the electronic Born-Oppenheimer wave functions from Eq. 2.44 and the derivative couplings fromEq. 2.50. Thus we can set up the Schrödinger equation Eq. 2.20 for the nuclei.

i~∂tφ±Eq. 2.20=

1

2M

[~

i~∇+ ~A±,±

]2

+1

2M~A±,∓ ~A∓,± + E

BO±

φ±

+1

2M

[~

i~∇+ ~A±,±

]

~A±,∓ + ~A±,∓

[~

i~∇+ ~A∓,∓

]

φ∓

(2.53)

Let me translate the Schrödinger equation into polar coordinates defined in Eq. 2.38 with Eq. 2.47.This yields

~

i~∇+ ~An,n =

~

i~∇

︷︸︸︷(

sin(α)

cos(α)

)

~

i∂R

︸︷︷︸

(a)

+

(

cos(α)

− sin(α)

)

~

iR∂α

︸︷︷︸

(b)

+

~An,n︷︸︸︷

~

2R

(

cos(α)

− sin(α)

)

︸︷︷︸

(c)

(2.54)

so that

(~

i~∇+ ~An,n

)2

=

−~2 ~∇2︷︸︸︷

−~2∂2R︸︷︷︸

(aa)

−~2

R∂R

︸︷︷︸

ba

− ~2

R2∂2α

︸︷︷︸

bb

+

nonBO︷︸︸︷

~2

iR2∂α

︸︷︷︸

bc+cb

+~2

4R2︸︷︷︸cc

=−~2

(R2∂2R + R∂R + ∂

2α

)

R2+~2

R2

(

−i∂α +1

4

)

~A±,∓(~

i~∇+ ~A∓,∓

)

= ±(~2

2R2∂α +

i~2

4R2

)

= ±i ~2

2R2

(

−i∂α +1

2

)

(~

i~∇+ ~A±,±

)

~A±,∓ = ±(~2

2R2∂α +

i~2

4R2

)

= ±i ~2

2R2

(

−i∂α +1

2

)

~A±,∓ ~A∓,± =~2

4R2(2.55)


In polar coordinates, Eq. 2.53 has thus the form

i~∂tφ± =

[−~2(R2∂2R + R∂R + ∂

2α

)

2MR2︸︷︷︸

−~2 ~∇2/2M

+~2(−i∂α + 14

)

2MR2︸︷︷︸

nonBO

]

+~2

8MR2︸︷︷︸

nonBO

±gR + wR2︸︷︷︸

EBO±

φ±(R,α, t)

±i ~2(−i∂α + 12

)

2MR2︸︷︷︸

nonBO

φ∓(R,α, t) (2.56)

This equation can be rewritten in two-component form

⇒ i~∂t

(

φ+

φ−

)

=−~2

(R2∂2R + R∂R + ∂

2α

)

2MR2︸︷︷︸

−~2 ~∇2/(2M)

(

φ+

φ−

)

+

((+gR + wR2

)φ+

(−gR + wR2

)φ−

)

+~2(−i∂α + 12

)

2MR2

(

1 +i

−i 1

)

︸︷︷︸

nonBO

(

φ+

φ−

)

(2.57)

Now, I shift a factor i into φ+, because this will lead to a radial Schrödinger equation with realcoefficients. iφ+ being a real function implies that φ+ is purely imaginary.

i~∂t

(

iφ+

φ−

)

=−~2

(R2∂2R + R∂R + ∂

2α

)

2MR2︸︷︷︸

−~2 ~∇2/(2M)

(

iφ+

φ−

)

+

((+gR + wR2

)iφ+

(−gR + wR2

)φ−

)

+~2(−i∂α + 12

)

2MR2

(

1 −1−1 1

)(

iφ+

φ−

)

=

−~2(R2∂2R + R∂R + ∂

2α

)

2MR2111

︸︷︷︸

−~2 ~∇2/(2M)

+gRσz + wR2111 +

~2(−i∂α + 12

)

2MR2(111− σx)

︸︷︷︸

nonBO

(

iφ+

φ−

)

(2.58)

This nuclear Schrödinger equation has a rotational symmetry, because the angular coordinate doesnot enter in the differential operator. Only the angular derivatives enter. Therefore, I decompose thenuclear wave function into angular-momentum eigenstates. The angular-momentum eigenstates arefurther divided into the eigenstates of the nuclear Hamiltonian. 10

φ±(α,R, t) =∑

m

eimα∑

j

e−i~Ej,mtR±,j,m(R)cj,m (2.59)

The energies Em,j are the those of the complete system including electronic and nuclear degrees offreedom. The angular-momentum quantum numbers m extend from −∞ to +∞. They are integer,so that the wave function is single valued.

Once the radial functions R±,m,j(R) and it energies Em,j are known, we can reconstruct the fullwave function my multiplying this nuclear wave function with the Born-Oppenheimer wave functions.

|Φ(t)〉 =∑

±,m,j

∫

dX

∫

dZ(

|a,X,Z〉ΨBOa,±(R,α) + |b,X,Z〉ΨBOb,±(R,α))

R±,j,m(R)eimαe−i~Ej,mtcj,m

(2.60)

10Another way of looking at this is: After multiplication with 2MR2, the Hamiltonian is the sum of an angular anda radial Hamiltonian. This allows to separate the angular and radial variables (see ΦSX:Quantum physics) and use aproduct ansatz.


Here, R =√X2 + Z2 and α = atan

(XZ

)are direct functions of the nuclear coordinates (X,Z).

We insert the Born-Oppenheimer wave functions from Eq. 2.44 to obtain the final form of theelectronic-nuclear wave function.

ELECTRONIC-NUCLEAR WAVE FUNCTION FOR THE JAHN-TELLER MODEL

The complete electronic nuclear wave functions expressed in the two electronic orbitals |~a〉 and |~a〉describing the two eg orbitals and the generalized nuclear coordinates (X,Z), which define the angularcoordinate α.

|Φ(t)〉 =∑

m,j

|Φj,m〉e−i~Ej,mtcj,m

(2.61)

The complex coefficients cj,m = 〈Φj,m|Φ(0)〉 are defined by the initial conditions. The eigenstates ofthe Hamiltonian are

|Φj,m〉 =∫

dX

∫

dZ1

2

(

|a,X,Z〉[

(eiα + 1)R+,j,m(R) + i(eiα − 1)R−,j,m(R)]

+ |b,X,Z〉[

−i(eiα − 1)R+,j,m(R) + (eiα + 1)R−,j,m(R)])

eimα (2.62)

The radial components are obtained from Eq. 2.58 with the ansatz Eq. 2.59 for the nuclear wavefunctions. Thus the defining equation is

−~2(R2∂2R + R∂R −m2

)

2MR2111

︸︷︷︸

−~2 ~∇2/(2M)

+ gRσz + wR2111

︸︷︷︸

EBO

−Ej,m111

+~2(m + 12

)

2MR2(111− σx)

︸︷︷︸

nonBO

(

iR+,j,m(R)R−,j,m(R)

)

Eq. 2.58= 0 (2.63)

The radial equation is a two-component system of coupled ordinary differential equations. It can besolved either by a power series ansatz, or by numerical integration of the differential equation.

The radial nuclear wave functions are shown in Fig. 2.4. For low energies the wave packet liesin the circular valley of the ground-state Born-Oppenheimer surface. With increasing energy thenumber of radial nodes increases. At the same time also the contribution of the excited-state Born-Oppenheimer surface grows. At higher energies, the wave function looks like a combination of aground-state and an excited-state Born-Oppenheimer wave function at about the same energy. Thusthe ground state contribution is spread over a wide region and has short oscillations which resultfrom the high kinetic energy. The excited state surface has no or less nodes and is localized aroundthe center, which is the minimum of the excited state Born-Oppenheimer surface.

What can be learned from this equation?

• At first we see that the eigenstates have contribution on both the upper and the lower sheetof the Born-Oppenheimer surface. This coupling, indicated in Eq. 2.63 as “nonBO”, is scaledwith the inverse mass. This seems to indicate that non-adiabatic effects become less importantwhen the nuclear masses are large (in comparison to the electron mass.) However, the non-adiabatic coupling also diverges at the position of the conical intersection. That is, it cannotbe ignored however large the nuclear masses are.

• An important observation is that the eigenstates are no angular-momentum eigenstates. Inaddition to the term eimα related to the nuclear wave function, there are also terms eiα fromthe Born-Oppenheimer wave function. Taken together, this implies that the eigenstates contain


0 1 2 30 1 2 3

Fig. 2.4: Radial nuclear wave functions R±,j,m(R) for the Jahn-Teller model for j = 0, 1, 2 and j = 9from to to bottom and for m = 0 left and m = 1 right. The ground-state component is shown asblack full line, and the component on the excited Born-Oppenheimer surface is shown as red dashedline. The energies for m = 0 are Ej,m = −0.89705,−0.69574,−0.49290,+0.80876 for j = 0, 1, 2, 9and m = 0 and Ej,m = −0.87415,−0.66431,−0.44920,+0.98900 for j = 0, 1, 2, 9 and m = 1. Themodel parameters are M = 1

2 , gJT = 2 and wJT = 1. The conical intersection is at E = 0 and theminimum of the ground-state Born-Oppenheimer surface lies at E = −1 and R = 1.

components from two angular momenta, namely Lz = ~m and Lz = ~(m + 1). It is as if theconical intersection would have its own pseudo-spin of± 12~, which may interact with the nuclear“orbital motion”.

• Even though the Schrödinger equation is purely real, the Born-Oppenheimer wave functions ata conical intersection cannot be chosen real and at the same time continuous.

• The stationary wave functions for m and −m − 1 are complex conjugates of each other, i.e.

Φj,−m−1(~r , X,Z) = Φ∗j,m(~r , X,Z) (2.64)

and the same holds for the radial functions(

R±,j,−m−1(R))∗= R±,j,m(R) ⇔

R−,j,−m−1(R) = −R−,j,m(R)R+,j,−m−1(R) = +R+,j,m(R)

(2.65)

and the energies are related by Ej,−m−1 = Ej,m. This implies that the corresponding solutionsof the stationary Schrödinger equation are complex conjugates of each other. The derivationof Eq. 2.64 and Eq. 2.65 is given in Appendix E.1 on p. 225.

2.3.5 From a conical intersection to an avoided crossing

We have seen that avoided crossings and conical intersections are the points where non-adiabaticeffects become important. In order to learn about the relation between the two, it is possible within the


vibrational levelsE

Born−Oppenheimer surfaces

R

E

Born−Oppenheimer surfaces

R

E

Fig. 2.5: Schematic for the optical absorption and desorption of a photon. Let R be one spatialcoordinate for the nuclei, such as a bond distance. A photon excites an electron from the ground-state sheet of the total energy surface to an excited state sheet. The atoms vibrate around theequilibrium structure of the excited energy surface. During that process the system dissipates energy.In other words, it thermalizes by emitting phonons. At some point the electron drops again on theground-state energy surface, while emitting a photon. The emitted photon has a lower energy thanthe absorbed photon, because some of the energy has been lost by creating for example phonos, i.e.heat. The system vibrates again about the equilibrium structure until it has dissipated its energy.Note that the transition proceeds between different energy levels of the total energy, which alsoinclude the vibrational levels. Thus vibrations or phonons are excited as well. Thus, to determinethe absorption probability not only the optical transition matrix elements of the electronic subsystemmust be considered, but also the overlap of the nuclear part of the wave function. Note that thefigure describes the case at zero temperature. At finite temperature, the system can be initiallyalready in a excited vibrational level. The left figure is appropriate for a classical description of thenuclear motion, while the right figure is an abstract energy level diagram appropriate for a quantumdescription.

Jah-Teller model to change the conical intersection into an avoided crossing with a single parameter.Here, I only point out how one arrives at it. Its investigation is left to the interested reader.

The Jahn-Teller model can easily be extended to a three-dimensional parameter space by addingthe third Pauli matrix in the Schrödinger equation Eq. 2.32.

i~∂t |Φ(t)〉 =[

P 2x + P2y + P

2z

2M+ w ~R2 + gX σx + gY σy + gZ σz

]

|Φ(t)〉 (2.66)

The conical intersection in the Born-Oppenheimer surface is at (X, Y, Z) = ~0.In the next step we can treat Y as a parameter, so that we obtain a two-dimensional model in

the coordinate space (X,Z) but an additional parameter Y = λ

i~∂t |Φ(t)〉 =[

P 2x + P2y

2M+ w ~R2 + gX σx + gZ σz + λ σy

]

|Φ(t)〉 (2.67)

For λ = 0 we obtain the Jahn-Teller model with the conical intersection. For non-zero values of λthe conical intersection is converted into an avoided crossing with a gap of 2|λ|.

2.4 Further reading

Here, a few reviews and other articles related to non-adiabatic effects. I have not read all of themcarefully yet myself.


• Article by Butler[18]. Interesting is also the PhD thesis of Florian Dufey[11].

• A good introduction given in the special volume of the Journal “Advances in Chemical Physics”from 2002[19]. In particular, I liked the article of Child[20] and that by Worth and Robb[9]

• the book “Beyond Born-Oppenheimer: Conical Intersections and Electronic Nonadiabatic Cou-pling Terms” by Michael Baer[21]. Baer uses a very compact notation, but he describes thetopological effects in a very elegant manner.

• Review article by Worth and Cederbaum[22]

• Herzberg[23]

• For the role of the so-called geometrical phase or Berry phase see Berry’s original paper[24].For the role of the Berry phase in chemical reactions see the short article by Clary[25].

Chapter 3

Many-particle wave functions

In the previous section we have shown how we can split the description into two parts, one for theelectrons and another one for the nuclei. The dynamics of the nuclei requires the knowledge ofthe electronic problem, namely the Born-Oppenheimer surfaces EBOn (~R) and the derivative couplings~Am,n(~R) obtained from the Born-Oppenheimer wave functions |ΨBO(~R)〉. The Born-Oppenheimerwave functions are states in the electronic Hilbert space and the atomic positions ~R are parametersand not coordinates in the quantum mechanical sense.

The next difficulty is to obtain the Born-Oppenheimer wave functions. For non-interacting elec-trons this is of the same complexity as a one-particle problem, while the interaction makes thisproblem usually intractable. Therefore we will start with non-interacting electrons before we turn tothe more complicated interacting systems.

In the present section, I will describe the basics of many-particle wave functions. I will start outhow to describe one-particle wave functions with spin. Then I will describe the underlying symmetryof identical particles, which results in the concept of Fermions and Bosons. Finally, I will make theturn back to one-particle wave functions, by showing how the wave function of a noninteractingsystem of fermions and bosons can be expressed by one-particle wave functions.

This will form the basis of the following sections that address non-interacting electrons.

3.1 Spin orbitals

Before I continue with many-particle wave functions, let me discuss one-particle orbitals. I willintroduce a notation for treating the electron spin which, on the one hand, simplifies the expressions,and on the other hand it is more rigorous.

When I first learned about spins and magnetic moments, I was puzzled by the special role, whichwas attributed to the z-coordinate. I learned that an electron can have a spin pointing parallel orantiparallel to the z-axis. The direction of the z-axis, however, is arbitrary. It seemed as if the physicschanged, when I turned my head.

Indeed, there is nothing special about the z-coordinate. Here, I present a more general formu-lation, which describes electrons by two-component spinor wave functions. With this formulation,one can form wave functions with a spin pointing in an arbitrary direction in each point in space.This description restores the rotational symmetry that appears to be broken in the simple-mindedformulation used commonly.

We use here two-component spinor wave functions, which are also called spin-orbitals

φ(~x) = φ(~r , σ) = 〈~r , σ|φ〉

Spin-orbitals actually consists of two complex wave functions, one for the spin-down φ(~r , ↓) and one

39

40 3 MANY-PARTICLE WAVE FUNCTIONS

for the spin up contribution φ(~r , ↑). One can write the spin orbitals also as two-component spinor

(

φ(~r , ↑)φ(~r , ↓)

)

def=

(

〈~r , ↑ |φ〉〈~r , ↓ |φ〉

)

PHYSICAL MEANING OF A SPIN WAVEFUNCTION

The square of a component of a spin wavefunction is the probability density Pσ(~r) for finding aparticle with the specified spin orientation at a specific position ~r .

This rule is completely analogous to the rule that the absolute square of a scalar wave functionis the probability density P (~r) = ψ∗(~r)ψ(~r) of finding a particle at a given position ~r .

Note, however, that the wave function contains much more information than that just men-tioned. Below, we will see that the spin wave function does not only provide the information on theprobabilities for the spin projection on the z-axis but also for any other axis.1

Relation to spin eigenstates

This notation is a little puzzling at first, because one usually works with spin-eigenfunctions, for whichone of the components vanishes. The spin is defined as

~S =~

2~σ (3.1)

where the three components of the vector ~σ are

σi =

∫

d3r

2∑

σ,σ′=1

|~r , σ〉σi ,σ,σ′〈~r , σ′|

Somewhat confusing are the different indices for the different vector spaces: The index i can havethe values x, y , z and refers to the components of the vector ~σ. Each component of the vector ~σ isa scalar operator. The indices σ, σ′ of σi ,σ,σ′ can assume the values ↑ or ↓, which are the coordinatesof the two-dimensional spinor space. That is, σi is a (2×2) matrix for each value of i . These (2×2)matrices are the Pauli matrices2

σx =

(

0 1

1 0

)

, σy =

(

0 −ii 0

)

, σz =

(

1 0

0 −1

)

(3.2)

1This is analogous to the fact that a wave function in configuration space contains information also on the mo-mentum.

2The matrices are made of the matrix elements σi ,σ,σ′ = 〈σ|σi |σ′〉 of the operator σi with σ ∈ ↑, ↓, σ′ ∈ ↑, ↓and i ∈ x, y , z.

3 MANY-PARTICLE WAVE FUNCTIONS 41

As a worked example, let us determine the expectation value of the operator Sx for a one-particlestate |φ〉. The states |~r , ↑〉 and |~r , ↑〉 are still eigenstates of Sz .

〈φ|Sx |φ〉 = 〈φ|[

~

2

∫

d3r

2∑

σ,σ′=1

|~r , σ〉σx,σ,σ′〈~r , σ′|]

|φ〉

=~

2

∫

d3r

[2∑

σ,σ′=1

〈φ|~r , σ〉σx,σ,σ′〈~r , σ′|φ〉]

=~

2

∫

d3r

[(

〈φ|~r , ↑〉〈φ|~r , ↓〉

)(

0 1

1 0

)(

〈~r , ↑ |φ〉〈~r , ↓ |φ〉

)]

=~

2

∫

d3r [〈φ|~r , ↑〉〈~r , ↓ |φ〉+ 〈φ|~r , ↓〉〈~r , ↑ |φ〉]

=~

2

∫

d3r [φ∗(~r , ↑)φ(~r , ↓) + φ∗(~r , ↓)φ(~r , ↑)]

We obtain the spin density, that is the probability that we find a particle at position ~r multiplied withthe average spin expectation value in x-direction of that particle.

For a spin orbital that is an eigenstate of Sz , one of the spinor components vanishes.

Sz |φ↑〉 = |φ↑〉(

+~

2

)

⇒(

〈~r , ↑ |φ↑〉〈~r , ↓ |φ↑〉

)

=

(

φ↑(~r , ↑)0

)

Sz |φ↓〉 = |φ↓〉(

−~2

)

⇒(

〈~r , ↑ |φ↓〉〈~r , ↓ |φ↓〉

)

=

(

0

φ↓(~r , ↓)

)

Note that the indices ↑, ↓ indicate the quantum number, while the pointers to the spinor componentsare treated as argument, so that they can be distinguished from the quantum numbers. The quantumnumber indicates that the orbital is an eigenstate to some symmetry operator.

The spin-orbitals have the advantage that they allow to describe orbitals, for which the spin doesnot point along the z-axis, but it can also point, for example, to the right or in any other direction.For a spin orbital the spin direction can actually vary in space. Such an orbital is called non-collinear,because the spins are not aligned.

Problem with conventional notation

In the literature, one usually uses another notation, namely

φσ(~r)def=φ(~r , σ)

Here, it is difficult to distinguish the role of σ as a coordinate and as quantum number. One usuallyuses one-particle orbitals with only one spin component, for which σ is a quantum number. In ournotation this would be – for a spin-up particle –

φ↑(~r , σ) = 〈~r , σ|φ↑〉=(

〈~r , ↑ |φ↑〉0

)

The orbital still has two components, but one of them, namely φ↑(~r , ↓), vanishes. With the commonnotation, it is difficult to write down expressions that do not break the rotational symmetry for thespin direction.


Magnetization

So-far we used the spin operator to determine the expectation value of the spin. Now we would liketo obtain the spin density or the magnetization respectively. In order to describe the principles, letus work out the charge density as a trivial example:

We know the projector P (~r) onto a certain point in space ~r , which is

P (~r)def=∑

σ

|~r , σ〉〈~r , σ| = |~r , ↓〉〈~r , ↓ |+ |~r , ↑〉〈~r , ↑ |

We obtain the charge density as expectation value of P (~r), multiplied with the electron chargeqe = −e.

ρ(~r) = 〈φ|[qP (~r)

]|φ〉 = q〈φ|

[∑

σ

|~r , σ〉〈~r , σ|]

|φ〉

= q∑

σ

〈φ|~r , σ〉〈~r , σ|φ〉

= q(

〈φ|~r , ↓〉〈~r , ↓ |φ〉+ 〈φ|~r , ↑〉〈~r , ↑ |φ〉)

Hence, the charge density is, up to the factor q, the sum of the spin-up and spin-down densities.After this introduction, we can analogously determine the magnetization.

The magnetization operator is obtained as product of the factor3 qm , the spin operator and the

projection operator onto a point in space.

mi(~r)def=

q

mP (~r)Si =

q~

2mP (~r)σi

=q~

2m

[∑

σ

|~r , σ〉〈~r , σ|]

︸︷︷︸

P (~r)

[∫

d3r ′∑

σ′,σ′′

|~r ′, σ′〉σi ,σ′,σ′′〈~r ′, σ′′|]

︸︷︷︸

σi

=q~

2m

∑

σ

∫

d3r ′∑

σ′,σ′′

|~r , σ〉〈~r , σ|~r ′, σ′〉︸︷︷︸

δ(~r−~r ′)δσ,σ′

σi ,σ′,σ′′〈~r ′, σ′′|

=q~

2m

∑

σ,σ′′

|~r , σ〉σi ,σ,σ′′〈~r , σ′′|

The factor µBdef= e~2me

is the Bohr magneton, which is approximately equal to the magnetic momentof an electron. Spin and magnetic moment of the electron point in opposite directions due to thenegative charge of the electron.

MAGNETIZATION OPERATOR

~m(~r) =q~

2m

∑

σ,σ′

|~r , σ〉~σσ,σ′〈~r , σ′| (3.3)

with the Pauli matrices ~σ defined in Eq. 3.2.

3The classical ratio of magnetic moment and angular momentum is q2m

, which however assumes a constant ratiothe mass- and charge distribution in space. While the “distribution” of electrons leads to such a constant ratio, theproperties of a single electron may differ in a classical picture. In a quantum mechanical description the electron ispoint-like with a certain gyromagnetic ratio, that follows directly from the relativistic Dirac equation.


More explicitly we obtain

ρ(r) = q

(

〈φ|~r , ↑〉〈φ|~r , ↓〉

)(

1 0

0 1

)(

〈~r , ↑ |φ〉〈~r , ↓ |φ〉

)

= q [φ∗(~r , ↑)φ(~r , ↑) + φ∗(~r , ↓)φ(~r , ↓)]

mx(r) =q~

2m

(

〈φ|~r , ↑〉〈φ|~r , ↓〉

)(

0 1

1 0

)(

〈~r , ↑ |φ〉〈~r , ↓ |φ〉

)

=q~

2me[φ∗(~r , ↑)φ(~r , ↓) + φ∗(~r , ↓)φ(~r , ↑)]

my (r) =q~

2m

(

〈φ|~r , ↑〉〈φ|~r , ↓〉

)(

0 −ii 0

)(

〈~r , ↑ |φ〉〈~r , ↓ |φ〉

)

= −i q~2m[φ∗(~r , ↑)φ(~r , ↓)− φ∗(~r , ↓)φ(~r , ↑)]

mz(r) =q~

2m

(

〈φ|~r , ↑〉〈φ|~r , ↓〉

)(

1 0

0 −1

)(

〈~r , ↑ |φ〉〈~r , ↓ |φ〉

)

=q~

2m[φ∗(~r , ↑)φ(~r , ↑)− φ∗(~r , ↓)φ(~r , ↓)]

The two-component spinor descriptions follows from the Pauli equation. The Pauli equation isthe non-relativistic limit of the Dirac equation, the relativistic one-particle equation for electrons.In the Dirac equation each particle has four components. Two components describe spin-up andspin-down electrons, and the two other components describe spin-up and spin-down positrons. Thepositron[26] is the anti-particle of the electron. In the non-relativistic limit, the electronic andpositronic components become independent of each other, so that the electrons can be described bya two-component spinor wave function.

A Slater determinant of N spin-orbitals corresponds to 2N Slater determinants of one-componentone-particle orbitals, according to the N spin indices.

3.2 Symmetry and quantum mechanics

In the following, we will be concerned with the symmetry of the Hamilton operator under permuta-tion of particles. Therefore, I will revisit the main symmetry arguments discussed in section 9 ofΦSX:Quantum Theory [27]. This is a series of arguments may be worthwhile to keep in mind.

A symmetry of an object is a transformation, that leaves the appearance of the object unchanged.For example a square is symmetric under four-fold rotation.

A physical system is characterized4 by its Hamiltonian, which determines the dynamics of thesystem by the Schrödinger equation. The system is symmetric under a given transformation, if everysolution of the Schrödinger equation, is transformed into another solution of the same Schrödingerequation.

Let me formalize this:

1. Definition of a transformation operator: An operator S can be called a transformation, if itconserves the norm for any state, that is if

∀|ψ〉〈ψ|ψ〉 |φ〉=S|ψ〉= 〈φ|φ〉 (3.4)

2. A transformation operator S is unitary, that is

S†S = 1 (3.5)4The argument is not limited to quantum systems. The Schrödinger equation can be replaced by any other equation

of motion governing the dynamics or other properties.


Proof:

∀|ψ〉〈ψ|S†S|ψ〉Eq. 3.4= 〈ψ|ψ〉 ⇒ S†S = 1

3. Definition of a symmetry: A system is called symmetric under the transformation S, if, forany solution of the Schrödinger equation describing that system, also S|Ψ〉 is a solution of thesame Schrödinger equation. That is, if

(i~∂t |Ψ〉 = H|Ψ〉

) |Φ〉def=S|Ψ〉⇒

(i~∂t |Φ〉 = H|Φ〉

)(3.6)

4. The commutator of the Hamilton operator with its symmetry operator vanishes, that is [H, S]− =0:

Proof:

i~∂t |Φ〉 = H|Φ〉|Φ〉def=S|Ψ〉⇒ i~∂t S|Ψ〉 = HS|Ψ〉

∂t S=0⇒ S (i~∂t |Ψ〉) = HS|Ψ〉i~∂t |Ψ〉=H|Ψ〉⇒ SH|Ψ〉 = HS|Ψ〉(HS − SH

)

︸︷︷︸

[H,S]−

|Ψ〉 = 0

Because this equation holds for any solution of the Schrödinger equation, it holds for any wavefunction, because any function can be written as superposition of solutions of the Schrödingerequation. (The latter form a complete set of functions.) Therefore

SYMMETRY AND COMMUTATOR

The commutator between the Hamilton operator H with its (time-independenta) symmetryoperators S vanishes.

[H, S]− = 0

Thus, one usually identifies a symmetry by working out the commutator with the Hamiltonian.

aFor time-dependent symmetry operators, the more general rule is [H, S]− = i~∂t S. An example for atime-dependent symmetry is that between two relatively moving frames of inertia.

5. The matrix elements of the Hamilton operator between two eigenstates of the symmetry op-erator with different eigenvalues vanish. That is

(S|Ψs〉 = |Ψs〉s ∧ S|Ψs ′〉 = |Ψs ′〉s ′ ∧ s 6= s ′

)⇒ 〈Ψs |H|Ψs ′〉 = 0

Thus, the Hamilton operator is block diagonal in a representation of eigenstates of its symmetryoperators. The eigenstates of the Hamilton operator can be obtained for each block individually.Because the effort to diagonalize a matrix increases rapidly with the matrix size, this simplifiesthe calculation of Hamilton eigenstates substantially, both for analytical and for numericalcalculations.

If the Hamiltonian is block diagonal, the time dependent Schrödinger equation does not mixstates from different subspaces. A wave function that starts out as an eigenstate of a symmetryoperator with a given eigenvalue, will always be an eigenstate with the same eigenvalue. In other


words, the eigenvalue of the symmetry operator is a conserved quantity. (Note, however, thatthe eigenvalue of a symmetry operator is usually complex.)

The eigenvalues of the symmetry operators are related to the quantum numbers.

Proof: In the following we will need an expression for 〈ψs |S, which we will work out first:

• We start by showing that the absolute value of an eigenvalue of a unitary operator is equalto one, that is s = eiφ where φ is real. With an eigenstate |ψs〉 of S we obtain

s∗〈ψs |ψs〉sS|ψs 〉=|ψs 〉s= 〈Sψs |Sψs〉 = 〈ψs | S†S︸︷︷︸

1

|ψs〉 = 〈ψs |ψs〉 ⇒ |s | = 1(3.7)

• Next, we show that the eigenvalues of the Hermitian conjugate operator S† of a unitaryoperator S are the complex conjugates of the eigenvalues of S.

⇒ |ψs〉 Eq. 3.5= S†S

︸︷︷︸

=1

|ψs〉S|ψs 〉=|ψs 〉s= S†|ψs〉s

⇒ S†|ψs〉s Eq. 3.7= |ψs〉 s∗s︸︷︷︸

=1

⇒ S†|ψs〉 = |ψs〉s∗⇒ 〈ψs |S = s〈ψs |

• With this, we are ready to show that the matrix elements of the Hamilton operator betweentwo eigenstates of the symmetry operator with different eigenvalues vanish.

0[H,S]−=0= 〈Ψs |[H, S]−|Ψs ′〉 = 〈Ψs |HS|Ψs ′〉 − 〈Ψs |SH|Ψs ′〉

S|Ψs 〉=|Ψs 〉s,etc.= 〈Ψs |H|Ψs ′〉s ′ − s〈Ψs |H|Ψs ′〉 = 〈Ψs |H|Ψs ′〉(s ′ − s)s 6=s ′⇒ 〈Ψs |H|Ψs ′〉 = 0

q.e.d

6. For a operator S with the property SN = 1 for a finite N, one can decompose any wave function|χ〉 into eigenstates |ψα〉 of the symmetry operation S with eigenvalue sα = ei

2πNα

|ψα〉 =N∑

n=0

Sn|χ〉s−nα =

N∑

n=0

(S†)n |χ〉s+nα (3.8)

3.3 Identical particles

Electrons with the same spin are indistinguishable. This says that there is no conceivable experimentthat discriminates between two electrons, except for their spin. Thus, there is a symmetry withrespect to exchange of two particles, and the Hamiltonian for indistinguishable particles commuteswith the permutation operator of two particles.

Editor: introduce symbol P for the permutation operator to avoid confusion with

a projection operator P (~r) used later.

The two-particle permutation operator is defined as

P(2)i ,j Ψ(. . . , ~xi , . . . , ~xj . . .) = Ψ(. . . , ~xj , . . . , ~xi . . .)

The eigenstates of the Hamiltonian are eigenstates of the permutation operator. Because(

P(2)i ,j)2

=

1, the permutation operator has the two eigenvalues, namely +1 and −1. Thus, the wave functions


are fully symmetric or fully antisymmetric with respect to the permutation of two particle coordinates.Particles with a symmetric wave function are called Bosons and particles with an antisymmetricwave function are called Fermions. Bosons are particles with integer spin such as photons, mesons,gluons, gravitons, etc, which are usually related to an interaction, while Fermions are particles withhalf-integer spin such as electrons, protons, neutrons, quarks, etc.

Unlike other symmetries, this symmetry is not a property of a specific Hamiltonian, but it isa property of the particles themselves. If the particles are indistinguishable, there is no conceivableHamiltonian, that is not symmetric under permutation of two of these particles. If we could constructonly one Hamiltonian that is not symmetric, we could design an experiment, just by realizing thisHamiltonian, that distinguishes two of such particles

The electron exchange is illustrated in Fig. 3.1. Except for the color, which distinguishes the twoelectrons, left and right situations are identical. Note, that for the particle exchange both spatialand spin-indices have to be exchanged simultaneously. Since the electrons are indistinguishable, nodistinguishing property like the color indicated can exist.

=

r1 r2 r1 r2

r1 r2 r1 r2=

r1 r2 r1 r2

=

Fig. 3.1: Demonstration of the particle exchange. For the two figures at the top, left and rightsituations are identical except for a color of the particles. For identical particles there is no prop-erty such as the color that allows one to distinguish them. Thus Ψ(~x1, ~x2) = −Ψ(~x2, ~x1) orΨ(~r1, σ1, ~r2, σ2) = −Ψ(~r2, σ2, ~r1, σ1). In the bottom figure, the particle positions are exchangedbut not their spin. Thus, the two configurations are not identical. In this case we may also viewelectrons with up and down spin as distinguishable particles, because they have different spin.

Fermions, such as electrons, have an antisymmetric wave function, that is

P(2)i ,j |Ψ〉 = −|Ψ〉

3.3.1 Levi-Civita Symbol or the fully antisymmetric tensor

In the following, I will represent wave functions by determinants. In order to work with determinants,the Levi-Civita symbol, also called the fully antisymmetric tensor, will be introduced. Here, I definethe fully antisymmetric tensor and derive some formulas, which will be needed later.

Definition

The Levi-Civita symbol5, also called the fully antisymmetric tensor

5Tullio Levi-Civita (1873-1941): Italian mathematician. Invented the covariant derivative. Made tensor-algebrapopular, which was used in Einstein’s theory of general relativity.


is a rank6 N Tensor defined by the following properties

• For ascending indices the Levi-Civita symbol has the value one:

ǫ1,2,3,...,N = 1

• The Levi-Civita symbol changes sign with any pairwise permutation of its indices

• The Levi-Civita symbol vanishes whenever at least two indices are pairwise identical

Relation to determinants

The determinant of a N × N matrix A is defined by the Levi-Civita symbol εi ,j,k,l ,... as

det[A] =∑

i ,j,k,...

εi ,j,k,...A1,iA2,jA3,k · · ·

Vector product (not needed)

A common and useful application of the Levi-Civita Symbol is to express the vector product in threedimensions by its components.

~a = ~b × ~c ⇔ ai =∑

j,k

εi ,j,kbjck

3.3.2 Permutation operator

The most simple way to construct a many-particle wave function, is to take the product of one-particle wave functions φj(~x), such as

Ψ(~x1, ~x2, . . .) = φa(~x1)φb(~x2) · · ·This is a product wave function.

Such a product wave function is neither symmetric nor antisymmetric with respect to permutation.Therefore, we need to antisymmetrize it. We use the N-particle permutation operator Pi1,...,iN thatis defined as follows

N-PARTICLE PERMUTATION OPERATOR

Pi1,...,iN :=∫

d4x1 · · ·∫

d4xN |~xi1 , . . . , ~xiN 〉〈~x1, . . . , ~xN | (3.9)

The permutation operator reorders the coordinates so that the first particle is placed onto the positionof the particle i1 and so on. This implies that the coordinates of the i1-th particle are placed on thefirst position, which is the one of the first particle.

In order to make the function of the operator more transparent, let us rewrite it as an operator ofreal-space functions. Consider a state |~y1, . . . , ~yN〉, which describes a particle distribution where thefirst particle is at position ~y2, the second at ~y2 and so on. Application of the permutation operatorPi1,...,iN yields

Pi1,...,iNΨ(~x1, . . . , ~xN) = 〈~x1, . . . , ~xN |Pi1,...,iN |Ψ〉 (3.10)

Eq. 3.9=

∫

d4y1 · · ·∫

d4yN 〈~x1, . . . , ~xN |~yi1 , . . . , ~yiN 〉︸︷︷︸

δ(~x1−~yi1 )···δ(~xN−~yiN )

〈~y1, . . . , ~yN |Ψ〉︸︷︷︸

Ψ(~y1,...,~yN)

(3.11)

6The rank is the number of indices that define a tensor element. Note, that there is another more limited definitionof the word rank.


Let me rename the coordinates so that the δ-functions can be resolved more conveniently

Pi1,...,iNΨ(~xi1 , . . . , ~xiN ) =∫

d4y1 · · ·∫

d4yN 〈~xi1 , . . . , ~xiN |~yi1 , . . . , ~yiN 〉︸︷︷︸

δ(~xi1−~yi1 )···δ(~xiN−~yiN )

〈~y1, . . . , ~yN |Ψ〉︸︷︷︸

Ψ(~y1,...,~yN)

(3.12)

= Ψ(~x1, . . . , ~xN) (3.13)

In the new state the first particle (first vector in the state) is at ~yi1 , that is at the position wherethe particle i1 has been before the permutation.

In the following it will be more convenient to work with the adjoint permutation operator.

P†i1,...,iNEq. 3.9=

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉〈~xi1 , . . . , ~xiN | (3.14)

Because the permutation operator is unitary, its adjoint is at the same time its inverse. The applicationof the adjoint permutation operator on an arbitrary wave function yields

P†i1,...,iNΨ(~y1, . . . , ~yN) = 〈~y1, . . . , ~yN |P†i1,...,iN |Ψ〉Eq. 3.14=

∫

d4x1 · · ·∫

d4xN 〈~y1, . . . , ~yN |~x1, . . . , ~xN〉︸︷︷︸

δ(~y1−~xi1 )···δ(~yN−~xiN )

〈~xi1 , . . . , ~xiN |Ψ〉︸︷︷︸

Ψ(~x1,...,~xN)

= Ψ(~yi1 , . . . , ~xiN )

3.3.3 Antisymmetrize wave functions

The exchange or permutation operator Pi1,...,iN applied to a fermionic wave function has the eigenvalue−1 if the sequence i1, . . . , iN is obtained by an odd number of permutations from regular order1, 2, 3, . . . N, and it has the eigenvalue +1 for an even number of permutations. These values arethose of the fully antisymmetric tensor or the Levi-Civita symbol.

EIGENVALUES OF THE PERMUTATION OPERATOR

A many-fermion state is an eigenstate of the permutation operators

Pi1,...,iN |ΨF 〉 = |ΨF 〉ǫi1,...,in (3.15)

with eigenvalues given by the fully antisymmetric tensor defined in section 3.3.1.A many-boson state can be defined similarly, but the eigenvalues are +1 for any permutation of theparticle coordinates.

Pi1,...,iN |ΨB〉 = |ΨB〉ǫ2i1,...,in (3.16)

Using Eq. 3.8 one can construct symmetry eigenstates. In a similar way we will construct Fermionicand Bosonic wave functions from a product state. In this case, however, we do not only have a singlesymmetry operation with a given multiplicity, but a set of non-commutating operations that froma symmetry group. Thus, Eq. 3.8 can not be applied directly: after having symmetrized the wavefunction with respect to one permutation, the symmetrization with respect to another permutationmay destroy the first symmetrization.

Editor: Here we need a proper derivation for symmetrized states using non-abelian

symmetry groups. I have started an appendix on symmetry groups for that purpose.

|ΦF 〉 =N∑

i1,...,iN=1

P†i1,...,iN |χ〉ǫi1,...,iN (3.17)


3.3.4 Slater determinants

The most simple many-particle wave function is a product state |φ1, . . . , φN〉, which can be writtenas

〈~x1, . . . , ~xN |φ1, . . . , φN〉 = 〈~x1|φ1〉 · · · 〈~xN |φN〉 = φ1(~x1) · · ·φN(~xN) (3.18)

A product state is not yet antisymmetric with respect to particle exchange, but with Eq. 3.8 it canbe antisymmetrized. Using the N-particle permutation operator Pi1,...,iN and the fully antisymmetrictensor ǫi1,...,iN we can antisymmetrize a product wave function.

|ΦF 〉 Eq. 3.19=

1√N!

N∑

i1,...,iN=1

P†i1,...,iN |φ1, φ2, . . . , φN〉ǫi1,...,iN (3.19)

The pre-factor 1√N!

is the required normalization for an orthonormal set of one-particle wavefunctions: there are N! distinct permutations of the N orbitals. Product states that differ by apermutation are orthonormal, if the one-particle states are orthonormal.

The corresponding wave function is

ΦF (~x1, ~x2, . . . , ~xN) =1√N!

N∑

i1,...,iN

ǫi1,...,iN P†i1,...,iNφ1(~x1) · · ·φN(~xN)

=1√N!

N∑

i1,...,iN

ǫi1,...,iNφ1(~xi1) · · ·φN(~xiN ) (3.20)

When one exploits the connection between Levi-Civita Symbol and the determinant discussed insection 3.3.1, the wave function Eq. 3.20 can also be written as a Slater determinant[28, 29]7

ΨF (~x1, . . . , ~xN) =1√N!det

∣∣∣∣∣∣∣∣∣∣

φ1(~x1) φ1(~x2) . . . φ1(~xN)

φ2(~x1) φ2(~x2) . . . φ2(~xN)...

......

...φN(~x1) φN(~x2) . . . φN(~xN)

∣∣∣∣∣∣∣∣∣∣

(3.21)

where ǫi ,j,k,... is the fully antisymmetric tensor defined in Sec. 3.3.1. The Slater determinant exploitsthe property of the determinant, that it changes its sign under permutation of two columns of amatrix. In the Slater determinant the exchange of two columns corresponds to the exchange of twoparticle coordinates.

An antisymmetric wave function can be constructed even from a non-orthogonal set of one-particle orbitals by forming the corresponding determinant. However, unless the one-particle orbitalsare orthonormal, the evaluation of matrix elements between Slater determinants is a nearly hopelessundertaking. Therefore, Slater determinants are usually built from an orthonormal set of one-particleorbitals.

As an example, the two-particle Slater determinant |Sa,b〉 of two one-particle states φa(~x) andφb(~x) is

Sa,b(~x1, ~x2) = 〈~x1, ~x2|Sa,b〉 =1√2

(

φa(~x1)φb(~x2)− φb(~x1)φa(~x2))

7According to a remark on Wikipedia, Heisenberg and Dirac proposed already in 1926 to write the antisymmetricwave function in a form of a determinant.


3.3.5 Bose wave functions

Analogous to the Slater determinant we can represent symmetrized product wave functions as apermanent. The permanent of a matrix [[[A] can be written as

perm[A] =N∑

i1,...,iN=1

ǫ2i1,...,iNA1,i1A2,i2 · · ·AN,iN

We see from Eq. 3.16 that the permanent plays for bosons the same role as the determinant forFermions.

A Bose wave function obtained by symmetrization of a product wave function has the form

ΨB(~x1, . . . , ~xN) =1√N!

N∑

i1,...,iN=1

ǫ2i1,...,iNφ1(~xi1) · · ·φiN (~xN)

If the one-particle states are orthonormal, also the resulting “Permanent-wave functions” built fromthe same one-particle basisset are orthonormal.

3.3.6 General many-fermion wave function

It is important to realize that not every antisymmetric wave function can be represented as a Slaterdeterminant. Slater determinants are derived from product wave functions and thus have a ratherrestricted form.

However, the Slater determinants that can be constructed from a complete, orthonormal set ofone-particle wave functions |φ1〉, |φ1〉, . . . form a complete orthonormal basis for antisymmetricmany-particle states. (without proof)8

A general N-particle state can be constructed as follows: Let us consider a complete and orthonor-mal one-particle basis set |φ1〉, |φ2〉, . . .. A subset of N such basis functions, namely |φi1〉, |φi2〉, . . . , |φiN 〉defines a particular Slater determinant |Si1,i2,...,iN 〉. A general antisymmetric N-particle state |Ψ〉 canbe written as a sum9. over all Slater determinants

|Ψ〉 =∞∑

i1,i2,...,iN=1;i1<i2<i3...

|Si1,i2,...〉ci1,i2,...

where the ci1,i2,... are the (complex) expansion coefficients.

3.3.7 Number representation

We have learned that a Slater determinant is defined by a subset of N states from a given one-particlebasis. Thus, we can form a vector with one-component for each one-particle orbital. If we set thecomponents equal to one for the orbitals in the set and equal to zero for all orbitals not in the set,we arrive at the number representation for Slater determinants

|Si1,...,in〉 = |0, 0, 1︸︷︷︸posi1

, 0, 0, 0, 1︸︷︷︸

posi2

, 1︸︷︷︸

posi3

, 0, . . .〉

This allows to write a general fermionic state in the form

|Ψ〉 =1∑

σ1=0

1∑

σ2=0

. . .

1∑

σ∞=0

|σ1, σ2, . . . , σ∞〉cσ1,σ2,··· ,σ∞ =∑

~σ

|~σ〉c~σ

8The one-particle states need not be orthonormal to fulfill the completeness condition. The requirement of orthonor-mal wave functions is a convenience to make the evaluation of matrix elements for many-particle states constructedfrom these one-particles states efficient. These will become clear later when the Slater-Condon rules are derived.

9The sum goes over all N-tuples (i1, . . . , iN) with 1 ≤ i1 < i2 < . . . < iN . The indices can go up to infinity.

Chapter 4

Non-interacting electrons

Because the electrons interact strongly by their Coulomb repulsion, their description is a most difficultquantum mechanical problem.

Before we enter into the difficulties of the many-particle physics, let us for the moment ignorethe interaction and study one-particle problems. At first, this sounds like intolerable approximation.However, if we consider, instead of bare electrons, quasi-particles that carry with them a distortionof the surrounding electrons, the effective interaction becomes screened. As a result, the interactionbecomes sufficiently small that the picture of non-interacting quasi electrons becomes appropriate formost physical systems. This picture breaks down in a fundamental way for materials with so-calledstrong correlations. Examples of materials with strong correlations are the High-Tc superconductors,manganates which exhibit colossal magnetoresistance, heavy Fermion systems, and many more.

4.1 Dispersion relations

In solid state physics we often deal with crystals, which exhibit discrete translation symmetry. As aconsequence of Bloch’s theorem, this is sufficient to make the momentum a conserved quantity fornon-interacting electrons. Thus we can characterize the states by their crystal momentum or wavevector.

The one-particle energy as function of the crystal momentum or the wave vector is the dispersionrelation or the band structure .

quasiparticles: The dispersion relation is a very general concept in solid state physics, which appliesto

• electrons, as free particles or in a material

• phonons, i.e. lattice vibrations,

• photons, i.e. light,

• holes, i.e. "missing electrons",

• excitons, i.e. electron-hole pairs,

• polariton, i.e. a photon interacting with electrons etc.,

• plasmons, i.e. charge oscillations,

• magnons, i.e. oscillations of the magnetization,

• polarons, i.e. electron-phonon pairs

51

52 4 NON-INTERACTING ELECTRONS

• Cooper pairs, i.e. electron pairs which are responsible for superconductivity.

All these entities are called quasiparticles, which behave as particles with a defined energy andmomentum. While we deal in solid-state physics with a small number of basic particles, electrons,nuclei with their electrodynamic interactions, the interplay between them produces a large zoo ofquasiparticles, which exceeds the complexity of real particles by far.

Free particles: Let us consider a free relativistic particle, which has the dispersion relation given byE2 − p2c2 = m20c4, that is

E±(~p) = ±√

(m0c2)2 + (pc)2 (4.1)

which is shown in Fig. 4.1 alongside with the bandstructure of electrons in silicon.

p

anti particles

particles

E

light

: E=c

p light: E=−cpm0c2

rest energy

nonrelativisticdispersion relationE=m0c2+p2/(2m0)

Γ X W L Γ X

-10

-5

0

5

10

Fig. 4.1: Left: Dispersion relation of a relativistic particle. Right: bandstructure of electrons insilicon as calculated with density-functional theory. (CP-PAW and PBE functional)

The dispersion relation of a relativistic particle has two branches, one for particles and one forantiparticles. For small momenta, the non-relativistic dispersion relation is obtained by a Taylorexpansion to quadratic order. For large momenta the dispersion relation approaches a straight linewith slope equal to the speed of light.

The energy as function of momentum can be considered as the Hamilton function H(~p,~r) of thecorresponding particle1. Because of translation symmetry of the free particle the Hamilton functionhas no spatial dependence. We can use Hamilton’s equations

.~r = ~v = ~∇pH(~p,~r) (4.2a).~p = ~F = −~∇rH(~p,~r) (4.2b)

Thus, we can obtain the velocity v of the particle with a given momentum graphically as the slopeof the dispersion relation. We observe that

• the particle is at rest for zero momentum, and that

• the velocity approaches the speed of light c for large momenta, but never exceeds it.

1Position and momenta are three dimensional.

4 NON-INTERACTING ELECTRONS 53

Observe also that the slope in the anti-particle branch, that is the velocity, is opposite to thedirection of the momentum. This is something to be taken seriously.2

The second Hamilton’s equation is Newton’s law. If the system is translationally symmetric, thereare no forces and the momentum is a constant of motion, i.e. it does not change with time. However,we may also consider external forces. A constant force implies that the momentum changes linearlywith time. By observing the change of the slope, that is the velocity, over time while the momentumis shifted to the right, we can extract the dynamics of the system under a constant force.

As an example, we may follow a skyrocket, which accelerates under the constant thrust createdby its rocket engines. Initially, that is for small momenta, the dispersion relation is quadratic, so thatthe velocity changes linear with momentum. Under a constant force, the velocity increases linearlywith time with an acceleration a obtained from ~F = m0~a. The curvature of the dispersion relationis the inverse mass. As the rocket continues to accelerate, its speed grows less and less, while themomentum still grows continuously. This captures that the speed of light is the absolute maximumvelocity for particles.

The dispersion relation of the relativistic particle has two branches. The upper one is identifiedwith particles and the other is identified with antiparticles. In order to avoid the problem, thatan antiparticle could accelerate indefinitely while emitting light, the concept of the Dirac sea hasbeen introduced. The problem finds a more satisfactory solution in the context of quantum fieldtheory. However, here we employ it, because it is analogous to the conventional view in solid statephysics. The idea is that all states of the antiparticle band are filled with electrons under normalcircumstances. Due to Pauli principle a fermionic particle cannot lower its energy by occupying analready filled state of the antiparticle band and thus emit light. Thus an antiparticle is a "missingelectron in the antiparticle band" or a hole.

A particle anti-particle pair can annihilate by emitting to quanta of light, which carry away energyand momentum of the particle antiparticle pair. This is an example of the interplay of two bandstructures, one for electrons and one for photons, i.e. light. Their only difference is the mass whichvanishes for photons. Let us consider the process in detail. Initially there is a particle, say withmomentum zero and a "hole" in the antiparticle branch. The particle can drop in the hole. In thatprocess the rest energy 2m0c2 of both particles and the momentum p′ = 0 needs to be carried away.Therefore we turn to the bandstructure for photons, that is for mass-less particles. We observe thatthe energy cannot be placed into a single photon, because that would also absorb a finite momentum.However it can be placed into two photons with opposite momentum and each having an energy equalto the rest mass of the annihilated particle. This is an example for a scattering process of particlesas they are present in solids.

Real band structure: The interpretation of the band structure of a real material is analogous tothat of the dispersion relation of the free particle. Therefore, the band structure of electrons insilicon are shown in Fig. 4.1 alongside the dispersion relation of the free particle.

There are two apparent difference to the free particle:

• there are even more more branches

• the momenta are drawn into several finite sections

The reason is the reduced symmetry of the solid compared to a particle in vacuum. The dispersionrelation of a free particle is isotropic, so that it is sufficient to draw it in one direction. In a crystal,the bands depend on the direction of the momentum.

Because the translational symmetry of a lattice is discrete, the coordinate axis for the momentumis folded up into a finite region. The resulting section of momentum space is thus finite. Furthermorethe folded bands add to the bands already present in this finite region. Each segment describes theband structure along a so-called high-symmetry line in momentum space.

Now, we can translate the observations made for the free particle to the band structure of silicon:2However, the interpretation of antiparticles can be changed so that the energies are positive.


• The bands at negative energies shown in red in Fig. 4.1 describe the valence electrons. Thesebands are filled with electrons as the antiparticle bands in the Dirac sea.

• The bands at positive energies shown in blue in Fig. 4.1 describe the conduction electrons.

• Valence and conduction states are separated by a band gap, a region where no electrons canexist in a perfect lattice.

• By heating up the crystal, a small number of electrons are created in the conduction band anda small number of holes, i.e. missing electrons, are formed in the valence band. In the followingwe often simply refer to electrons and holes. The electrons will relax towards the minima ofthe conduction band, near the point X.

• The curvature of the energy band at the minimum is related to the inverse effective mass ofthe conduction electron.

ǫ(~p) ≈ 12~pm−1eff ~p

mef f = ~2(

~∇k ⊗ ~∇k∣∣~k0ǫ)−1

(4.3)

The derivative is taken at the band extremum and for the band that forms it. This effective massis anisotropic, which can be seen from the curvature in Γ− X direction and the perpendiculardirection X −W . The holes are located at the maximum of the valence band, which lies atΓ. Flat bands indicate that the particle is heavy, while bands with large curvature indicate lightparticles. The effective masses of holes are obtained analytically, with the only difference thatthe sign in the definition changes.

• For metals, the slope of the bands at the Fermi level is a measure for the velocity of n electron,the Fermi velocity.

~vFermi(~k) =1

~~∇kǫ(~k) (4.4)

The Fermi velocity depends strongly on the particular band and the position at the Fermisurface. Typical Fermi velocity are 106 m/sec , which is only a factor 300 slower than thespeed of light. 3

• The number of thermally created holes and electrons and their effective masses determine theelectrical conductivity.

• A photon can be absorbed by the formation of electron-hole pairs, if its energy exceeds theband gap. Silicon is transparent to infrared radiation, but it reflects visible light. The bandgap of silicon is about Eg = 1.17 eV, while the photon energy of visible light ranges from 1.65to 3.26 eV. Because of the large speed of light, a photon has nearly infinite slope. Thus aexcitation or recombination is a vertical process.

• A lattice vibration can also be represented by a band structure. Opposite to the photon, thephonon has small energies and can have large momenta. This is consistent with the small speedof sound. Thus, the absorption of a phonon shifts the particle nearly horizontally in the bandstructure.

3Consider a free electron gas with dispersion relation ǫ(~p) = ~p2

2meAs shown below the particle number N in a

volume V is given by N = V∑

σ

∫

|~k |<kFd3k(2π)2

= 13π2

k3F V , so that kF =3

√

3π3 NV

. The velocity is vF = (~|kF |)/me =~

me3

√

3π3 NeV

. Let us estimate the quantities: Aluminium has a lattice constant of alat = 4.05 Å. Aluminium has theface centered cubic structure, with 4 atoms in the cubic unit cell. Each Al atom contributes three valence electrons.Thus the electron density is N/V = 12a−3lat = 0.18× 1030 m−3 With the electron mass of 0.911× 10−30 kg. Thus weobtain

vF =~

me

3

√

3π3Ne

V≈ 2× 106 m

s(4.5)


Correspondence principle: So far we have argued on the basis of classical particles. However,wave packets as in quantum mechanics behave completely analogous4. For waves, the dispersionrelation is usually drawn in terms of frequency ω(~k) as function of the wave vector ~k .

An example, which is often useful is the hanging linear chain. The hanging linear chain is a set ofpendula coupled by springs. In the continuous limit, the hanging linear chain has the same dispersionrelation as the relativistic particle.

The dispersion relation ω(~k) has the same interpretation as E(~p):

• the group velocity ~vg of a wave packet is given as the slope of the dispersion relation

~vg = ~∇kω(~k) (4.6)

• for a system that does not change with time, its frequency remains constant over time. If wenow consider a dispersion relation ω(~k,~r) with a weak spatial dependence, we can investigatethe motion of a wave packet centered at ~r(t) having a wave vector ~k(t)

0 =d

dtω(

~k(t), ~r(t))

=.~k ~∇kω(~k,~r)︸︷︷︸

.~r

+.~r ~∇rω(~k,~r) =

.~r(.~k + ~∇rω(~k,~r)

)

⇒.~k = −~∇rω(~k,~r) (4.7)

Thus, we arrive at equations of motion for a wave packet that is completely analogous to Hamil-ton’s equations for particles.

.~r = ~∇kω(~k) (4.8a).~k = −~∇rω(~k,~r) (4.8b)

This is the basis for the correspondence principle, which says that localized wave packets behavelike classical particles with energy and momentum given by

E = ~ω and ~p = ~~k (4.9a)

The constant ~ is at first an arbitrary constant. Experiments determine it for quantum particles tobe equal to Planck’s quantum. The correspondence principle, i.e. the analogy of particles and wavepackets can be considered as one of the main pillars for the formulation of quantum physics.

4.2 Crystal lattices

In solid state physics, one deals mostly with materials that are periodic in space, so-called crystals.Crystals exhibit a discrete translational symmetry

V (~r + ~t) = V (~r) (4.10)

4See Chapter 3 of “ΦSX: Quantum Physics.”[27]


where the translation vectors ~t form a regular lattice. More specifically, we should require for theone-particle Hamilton operator acting on the electrons

〈~r + ~t|H|~r + ~t〉 = 〈~r |H|~r〉 (4.11)

In addition to the discrete translation symmetry, a crystal has also a point-group symmetry, whichallows to divide lattices into 230 crystallographic space groups . A set of useful tools for crystal-lography can be found on the Bilbao crystallographic server http://www.cryst.ehu.es.

Besides crystals, there are also solids without translational symmetry. Among them are quasicrys-tals, amorphous material and glasses. Even crystals are rarely perfect.

They usually have

• point defects such as substitutional defects (foreign atoms), vacancies (missing atoms), in-terstitials (additional atoms)

• line defects that are so-called dislocations

• planar defects such as grain boundaries, twin boundaries and interfaces

• other defects such as precipitates, i.e. small crystallites within another material.

4.2.1 Reciprocal space revisited

From ΦSX: Quantum mechanics of the chemical bond [30]. Shortened version of section “Real and

reciprocal lattice in three dimensions”

Let me explain my notation and remind you of the some of the concepts related to real and reciprocallattices.

Fig. 4.2.1 demonstrates the concepts developed in the following. Corresponding to the threespatial directions there are now three primitive lattice vectors, which we denote by ~T1, ~T2, ~T3. Notethat in two or three dimensions there is no unique choice of primitive lattice vectors. The threeprimitive lattice vectors span the primitive unit cell.

A general lattice vector ~ti ,j,k can be expressed by the primitive lattice vectors ~T1, ~T2, ~T3 as

~ti ,j,k = i ~T1 + j ~T2 + k ~T3=

Tx,1 Tx,2 Tx,3

Ty,1 Ty,2 Ty,3

Tz,1 Tz,2 Tz,3

︸︷︷︸

T

i

j

k

where i , j, k are arbitrary integers.It is often convenient to combine the lattice vectors into a 3×3 matrix T as shown above. Often

the atomic positions ~Rn are provided in relative positions ~X which are defined as

~Rn = T ~Xn ⇔ ~Xn = T−1 ~Rn

A potential is called periodic with respect to these lattice translations, if V (~r + ~ti ,j,k) = V (~r).

http://www.cryst.ehu.es


RECIPROCAL LATTICE

The reciprocal lattice is given by those values of the wave vector ~G, for which the correspondingplane waves ei ~G~r have the same periodicity as the real-space lattice.The primitive reciprocal-lattice vectors ~gn for n = 1, 2, 3 are defined in three dimensions as

~gn~Tm = 2πδn,m (4.12)

The reciprocal lattice vectors can be obtained as

~g1 = 2π~T2 × ~T3

~T1

(

~T2 × ~T3) , ~g2 = 2π

~T3 × ~T1~T1

(

~T2 × ~T3) , ~g3 = 2π

~T1 × ~T2~T1

(

~T2 × ~T3)

The general reciprocal lattice vectors ~Gi ,j,k are multiples of the primitive lattice vectors ~g1, ~g2, ~g3.

~Gi ,j,k = i~g1 + j~g2 + k~g3

• The reciprocal lattice vectors are perpendicular to the lattice planes of the real-space lattice.

~g1||~t2 × ~t3 and ~g2||~t3 × ~t1 and ~g3||~t1 × ~t2

• The length of the primitive reciprocal lattice vectors is 2π divided by the distance ∆n of thelattice planes.

|~gn| =2π

∆n


∆1g

g2

2T

T1

C3 C6

t−2,3,0 ∆2

1

Fig. 4.2: Translational and rotational symmetry of a single graphite sheet and demonstration ofreciprocal lattice vectors. Graphite is a layered material, which consists of sheets as the one shown.The yellow balls indicate the carbon atoms and the blue sticks represent the bonds. In graphite thesetwo-dimensional sheets are stacked on top of each other, where every second sheet is shifted suchthat an atom falls vertically below the point indicated by C6. Here, we only consider the symmetryof a single sheet. The elementary unit cell is indicated by the red parallelogram, which is spanned bythe elementary lattice vectors ~T1 and ~T2. The third lattice vector points perpendicular to the sheettowards you. An example for a general lattice vector is ~t3,−2,0 = 3~T1− 2~T2+0~T3. The lattice planesindicated by the dashed lines. The lattice planes are perpendicular to the sheet. The distance of thelattice planes are indicated by ∆1 and ∆2. The elementary reciprocal lattice vectors are ~g1 and ~g2.The third reciprocal lattice vector points perpendicular out of the plane towards you. Note that thereciprocal lattice vectors have the unit “inverse length”. Thus, their length is considered irrelevant inthis real space figure. The axis through C3 standing perpendicular to the plane is a three-fold rotationaxis of the graphite crystal. The axis through C6 perpendicular to the plane is a 6-fold rotation axis ofthe graphite sheet, but only a three-fold rotation axis of the graphite crystal. (Note that a rotationaxis for the graphite crystal must be one for both sheets of the crystal). In addition there is a mirrorplane lying in the plane. Furthermore there are several mirror planes perpendicular to the plane: Onepassing through every atom with one of the three bonds lying in the plane and one perpendiculareach bond passing through the bond center.

4.2.2 Bloch Theorem

From section 10.5.3 of ΦSX: Quantum Theory [27].

Symmetry: For every lattice-translation vector ~t, we define a lattice-translation operator

S(~t)def=

∫

d3r |~r 〉⟨~r − ~t

∣∣ (4.13)

which satisfies

〈~r |S(~t)|ψ〉 = 〈~r − ~t|ψ〉 . (4.14)


Let us investigate the eigenstates of the translation operator. Because the lattice-translationoperator is unitary, the absolute value of the eigenvalue is one, that is

S(~t)|ψϕ〉 = |ψϕ〉eiϕ (4.15)

where the phase shift depends on the translation vector.Because S(n~t) = Sn(~t), a similar relation must hold for the eigenvalues, i.e. ϕ(n~t) = nϕ(~t).

This requirement can be satisfied by choosing the form ϕ(~t) = ~k~t. In addition, this form works forsuperpositions of linearly independent lattice vectors. The vector-parameter ~k is the so-called Blochvector or simply wave vector.

Thus, the eigenvalue equation for the lattice-translation operator is (see also Eq. 10.1 of [27])

S(~t)|φ~k〉 = |φ~k〉e−i~k~t (4.16)

Let us now construct the eigenstates of the lattice translation operator:

〈~r |S(~t)|φ~k〉Eq. 4.16= 〈~r |φ~k〉e−i

~k~t (4.17)Eq. 4.14⇒ 〈~r − ~t|φ~k〉 = 〈~r |φ~k〉e−i

~k~t

⇒ 〈~r − ~t|φ~k〉e−i~k(~r−~t)

︸︷︷︸

u~k (~r−~t)

= 〈~r |φ~k〉e−i~k~r

︸︷︷︸

u~k (~r)

(4.18)

Thus, the function

u~k(~r)def= |φ~k〉e−i

~k~r (4.19)

which is defined for an eigenstate of the lattice translation operator S(~t) with eigenvalue ei~k~t isinvariant with the lattice translations. That is

u~k(~r − ~t) = u~k(~r) . (4.20)

Hence the eigenstates of the lattice-translation operators can be written as product of a periodicfunction and a plane wave. Without limitation of generality, this Bloch vector can be chosen fromwithin the first reciprocal unit cell.

Lattice symmetry implies that the eigenstates of the lattice Hamiltonian can be chosen to beeigenstates of the lattice-translation operator, which immediately implies Bloch theorem.

BLOCH THEOREM

The eigenstates of a system with lattice symmetry can be written as product of a periodic functionu~k(~r), i.e.

uk(~r + ~t) = u~k(~r) , ‘ (4.21)

and a modulating plane wave ei~k~r , that is

ψk(~r) = u~k(~r)ei~k~r (4.22)

where the wave vector ~k , the Bloch vector, is a good quantum number of the system.

4.2.3 Non-interacting electrons in an external potential

Another way to obtain the Bloch theorem is to set up the Hamiltonian in a plane wave basis. Onthe one hand, this approach to the problem connects to some of the principles used in a plane


wave pseudopotential method. On the other hand it shows that lattice potential leads to a couplingbetween points in the reciprocal space, which are separated by reciprocal lattice vectors.

Let us now consider the consequences of a weak external potential.

V (~r) =∑

~G

V~Gei ~G~r (4.23)

Since the potential shall be periodic, only reciprocal lattice vectors ~G occur in the sum. The Hamil-tonian can be written as

H =∑

σ

∫

d3r |~r , σ〉(

−~22me

~∇2 +∑

G

VGei ~G~r

)

〈~r , σ|

Let us now work out the matrix elements of the Hamiltonian between two plane waves,

φ~k,σ(~r , σ′)

Eq. ??= 〈~r , σ′|φ~k,σ〉 =

δσ,σ′√Ωei~k~r , (4.24)

which are the eigenstates of the free electron gas.

Hσ,σ′(~k, ~k ′) := 〈φ~k,σ|H|φ~k ′,σ′〉

=∑

σ′′

∫

Ω

d3r 〈φ~k,σ|~r , σ′′〉︸︷︷︸

φ∗~k,σ(~r ,σ′′)

(

−~22me

~∇2 +∑

G

VGei ~G~r

)

〈~r , σ′′|φ~k ′,σ′〉︸︷︷︸

φ~k,σ(~r ,σ′′)

Eq. ??=

δσ,σ′

Ω

∫

Ω

d3r e−i~k~r

(

−~22me

~∇2 +∑

G

VGei ~G~r

)

ei~k ′~r

=(~~k)2

2meδσ,σ′

1

Ω

∫

Ω

d3r ei(~k ′−~k)~r

︸︷︷︸

δ~k ′ ,~k

+∑

~G

V~Gδσ,σ′1

Ω

∫

Ω

d3r e−i~k~rei

~G~rei~k ′~r

︸︷︷︸

δ~k ′+~G,~k

= δσ,σ′∑

~G

[

(~~k)2

2meδ~G,~0 + V~G

]

δ~k ′+~G,~k (4.25)

We have used here the normalization condition5 corresponding to a finite, even though very large,volume Ω with side-lengths Lx , Ly , Lz . Correspondingly, the spectrum is discrete with wave vectors~ki ,j,k =

2πLxi + 2πLy j +

2πLzk .

Thus, there is only a coupling between states that are separated by a reciprocal lattice vector.This is one way to prove Bloch’s theorem.

BLOCH’S THEOREM

Bloch’s theorem says that the eigenstates in a periodic potential can be written as a product of afunction u~k,σ(~r , σ

′) that is periodic with the periodicity of the potential and a phase factor ei~k~r .

This can be shown as follows: Once we diagonalize the Hamiltonian, the eigenstates will bea superposition of the basis states, the plane waves, that are connected by a Hamiltonian matrixelement. Thus, the final wave function has the general form

ψ~k(~r) =∑

~G

ei(~k+~G)~rck,G =

∑

~G

ei~G~rck,G

︸︷︷︸

u~k (~r)

ei~k~r = u~k(~r)e

i~k~r (4.26)

5Other choices would normalize the wave function within the unit cell, or they would set the average density equalto one.


Because the function uk(~r) has only Fourier components at the reciprocal lattice vectors, it is periodicwith the real space lattice symmetry.

2πa

E

k

Fig. 4.3: Schematic demonstration how a periodic potential results in a coupling between states withthe same wave vector in the reduced zone scheme

ggg g

ε εε

k k k

Fig. 4.4: Extended, periodic and reduced zone scheme for a one-dimensional free particle.

This is the reason why band structures are not represented in an extended zone scheme but ina reduced zone scheme.

• in the extended zone scheme the band structure of free electrons would be a single parabola,and would extend to infinity in reciprocal space.

• in the periodic zone scheme, the free electron would consists of many parabola centered atthe reciprocal lattice points. Thus, the reciprocal zone scheme is periodic with the periodicityof the reciprocal lattice.

• in the reduced zone scheme, only the irreducible part of the periodic zone scheme is shown,that is one single repeat unit. The band structures shown in Fig. 4.12 are in a reduced zonescheme. There are several choices for the repeat units of the reciprocal lattice. One choice issimply the primitive unit cell of the reciprocal lattice. However, the shape of the unit cell does


not have the point-group symmetry of the reciprocal lattice. Therefore one instead choosesthe Wigner Seitz cell6 of the reciprocal lattice, which is called the Brillouin zone.

In the reduced zone scheme, all wave vectors connected by a reciprocal lattice vector, fall on topof the same point in the Brillouin zone. Thus, a Hamiltonian that has lattice periodicity couples allpoints that lie at the same point in the Brillouin zone. This is demonstrated in Fig. 4.3.

From perturbation theory we know that the splitting of two interacting states is large when thetwo states are close or even degenerate. Thus the largest effect in the band structure occurs rightat the boundary of the irreducible zone, where the degeneracy of the free electron gas is lifted by theperiodic potential. Thus, local band gaps appear at the surface of the Brillouin zone. This band gap,however, is usually warped, so that there is no energy window, that completely lies in all local bandgaps. In that case the material remains a metal.

4.3 Bands and orbitals

From ΦSX:Quantum mechanics of the chemical bond[30]

So far we discussed the band structures as a variation of the dispersion relation of the free-electronbands. However, this point of view obscures the relation to chemical binding and the local picture ofthe wave functions.

The band structure can also be constructed starting from local orbitals. This is demonstratedhere for the two-dimensional square lattice with atoms having s- and p-orbitals in the plane.

In order to simplify the work, we restrict ourselves to the high symmetry points Γ, X and M ofthe two-dimensional reciprocal unit cell.

The real space lattice vectors are

~T1 =

(

1

0

)

a and ~T2 =

(

0

1

)

a

The corresponding reciprocal space lattice vectors are

~g1 =

(

1

0

)

2π

aand ~g2 =

(

0

1

)

2π

a

The high-symmetry points are

kΓ = 0 =

(

0

0

)

2π

a; kX =

1

2~g1 =

(

1

0

)

π

aand kM =

1

2(~g1 + ~g2) =

(

1

1

)

π

a

The first step is to construct Bloch waves out of the atomic orbitals, by multiplying them withei~k~t , where ~t is the lattice translation of the atom relative to the original unit cell. This is illustrated

in Fig. 4.5 on page 63.

• At the Γ, the phase factor for a lattice translation of the orbital is one. Thus, we repeat theorbital from the central unit cell in all other unit cells.

• At the X point, the phase factor for a lattice translation along the first lattice vector is ei~kX ~T1 =eiπ = −1 and the phase factor for the lattice translation along the second lattice vector ~T2 isei~kX ~T2 = +1. Thus, if we go to the right the sign of the orbital alternates, while in the vertical

direction the orbital remains the same.6The Wigner Seitz cell of a lattice consists of all points that are closer to the origin than to any other lattice point.

Thus, it is enclosed by planes perpendicular to the lattice vectors cutting the lattice vector in half.


MXΓ

Γ X

M

Γ X M Γ

g1

g2

εs

hppσ

hppπ

hppσ

hppπ

εp

ssσ2h

Fig. 4.5: Illustration of a band structure in real and reciprocal space for a planar square lattice.Bottom left: The reciprocal lattice is spanned by the reciprocal lattice vectors ~g1 and ~g2.The yellowsquare is the corresponding Brillouin zone. The green inset is the Brillouin zone, whose corners arethe high symmetry points Γ, X and M. On the top left, the corresponding basis functions made fromone s-orbital and two p-orbitals in a Bloch-basis are shown. These states are also eigenstates ofthe Hamiltonian, because the coupling vanishes due to point group symmetry at the high-symmetrypoints. On the top right, a schematic band structure is shown and compared to a free-electron band(red).

• At the M point, the phase factor alternates in each lattice direction, resulting in a checkerboardpattern.

Next we need to set up the Hamilton matrix elements for each k-point, using the Slater-Kosterparameters. For a back-on-the envelope estimate the nearest neighbor matrix elements will be suffi-cient. The matrix elements are firstly ǫs and ǫp, the energies of the orbitals without any hybridization.Secondly we need the hopping matrix elements hssσ, hppσ, hppπ. In principle we would also need hspσ,but that matrix element will not be needed in our example.

In our simple example, the Bloch states are already eigenstates of the Hamiltonian so that weonly need to calculate their energy. We obtain7

Γ X M

ǫp + |hppσ| − |hppπ| ǫp + |hppσ|+ |hppπ| ǫp − |hppσ|+ |hppπ|ǫp + |hppσ| − |hppπ| ǫp − |hppσ| − |hppπ| ǫp − |hppσ|+ |hppπ|

ǫs − 2|hssσ| ǫs ǫs + 2|hssσ|

7Note, that the hopping matrix elements have the opposite sign than the overlap matrix elements. Thus, a positiveoverlap leads to a lowering of the energy level of the bond. In order to avoid confusion, I am introducing explicitly theabsolute value of the hopping matrix elements, which have the same sign as the corresponding overlap matrix elements.


If there is a band structure available, we can estimate the Hamilton matrix elements from that.Otherwise we just make an educated guess.

4.4 Calculating band structures in the tight-binding model


Now we wish to calculate the band structure. Let us begin with a tight-binding basisset with theusual assumption that the tight-binding orbitals shall be orthogonal. The tight-binding orbitals shallbe represented by kets |~t, α〉, where ~t denotes a discrete lattice-translation vector and α denotes thetype of the orbital such as s, px , py , pz , . . . and a spin quantum number, as well as a site index R ofan atom in the first unit cell of the lattice.

A Bloch wave can be represented as

|Ψ~k,n〉 =∑

~t,α

|~t, α〉〈~t, α|Ψ~k,n〉

Eq. 4.22=

∑

~t,α

|~t, α〉〈~t, α|∫

d4x |~x〉 ei~k~r 〈~x |u~k,n〉︸︷︷︸

ψ~k (~x)=u~k (~x)ei~k~r

=∑

~t,α

|~t, α〉ei~k~t〈~t, α|∫

d4x |~x〉ei~k(~r−~t)〈~x |u~k,n〉

u(~x)=u(~r−~t,σ)=

∑

~t,α

|~t, α〉ei~k~t〈~t, α|∑

σ

∫

d3r |~r , σ〉ei~k(~r−~t)〈~r − ~t, σ|u~k,n〉

~r→~r+~t=

∑

~t,α

|~t, α〉ei~k~t∑

σ

∫

d3r 〈~t, α|~r + ~t, σ〉︸︷︷︸

〈~0,α|~r ,σ〉

ei~k~r 〈~r , σ|u~k,n〉︸︷︷︸

〈~r ,σ|Ψ~k,n〉

=∑

~t,α

|~t, α〉ei~k~t〈~0, α|ψ~k,n〉 (4.27)

This shows us how we can represent the entire wave function with a vector ~cn with elements cα,n =〈~0, α|Ψk,n〉, which has the dimension of the number of orbitals in the elementary unit cell.

Our next goal is to find an equation for this vector. We will use the matrix elements of theHamilton operator in the basis of tight-binding orbitals

〈~t, α|H|~t ′, β〉 = h~t,α,~t ′,β (4.28)

Let us insert the ansatz Eq. 4.27 into the Schrödinger equation

H|Ψk,n〉 = |Ψk,n〉ǫk,nEq. 4.27⇒ H

∑

~t,α

|~t, α〉ei~k~t〈~0, α|Ψk,n〉 =∑

~t,α

|~t, α〉ei~k~t〈~0, α|Ψk,n〉ǫk,n

〈~0,β|⇒∑

~t,α

〈~0, β|H|~t, α〉︸︷︷︸

h~0,β,~t,α

ei~k~t〈~0, α|Ψk,n〉 =

∑

~t,α

〈~0, β|~t, α〉︸︷︷︸

δ~0,~tδα,β

ei~k~t〈~0, α|Ψk,n〉ǫk,n

⇒

∑

~t,α

h~0,β,~t,αei~k~t

︸︷︷︸

hα,β(~k)

〈~0, α|Ψk,n〉︸︷︷︸

cα,n(~k)

= 〈~0, β|Ψk,n〉︸︷︷︸

cβ,n(~k)

ǫk,n

⇒ h(~k)~cn(~k) = ~cn(~k)ǫn(~k) (4.29)


Thus, we obtain

SCHRÖDINGER EQUATION IN K-SPACE

h(~k)~cn(~k) = ~cn(~k)ǫn(~k) (4.30)

with cα,n(~k)def= 〈~0, β|Ψ~k,n〉, respectively |Ψ~k,n〉

Eq. 4.27=

∑

~tα,n

|~t, α〉ei~k~tcα,n(~k) (4.31)

and

hα,β(~k)def=∑

~t,α

h~0,α,~t,βei~k~t (4.32)

Thus, we obtained a k-dependent eigenvalue equation in matrix form with a finite and usuallysmall dimension. This problem can be broken down into smaller subproblems, if we exploit thesymmetry arguments that we used earlier for the molecules. Symmetry can be used especially at thehigh-symmetry points and lines in reciprocal space, where the band structure is usually represented.

Worked example

Let us consider the example of a linear chain atoms with one s-type orbital. There shall be two atomsper unit cell. The lattice constant shall be a. The orbitals are denoted by (t/a, α), where α ∈ 1, 2denote the two orbitals and t/a is the number of lattice displacements.

The two atoms have orbital energies ǫ1 and ǫ1. We denote the hopping matrix element betweenorbitals (i , 1) and (i , 2) by hssσ and the one between (i , 2) and (i + 1, 1) by h′ssσ.

For the sake of simplicity we only consider wave functions with one spin direction. The bands ofthe two spin direction are identical, because there is no magnetic field. When we consider one spindirection, the wave function component of the other spin direction is zero.

The infinite Hamiltonian has the form

h =

......

...

. . . ǫ1 hssσ 0 0 . . .

. . . hssσ ǫ2 h′ssσ 0 . . .

. . . 0 h′ssσ ǫ1 hssσ 0 0 . . .

. . . 0 0 hssσ ǫ2 h′ssσ 0 . . .

. . . 0 h′ssσ ǫ2 hssσ . . .

. . . 0 0 hssσ ǫ1 . . ....

......

...

The Hamilton matrix elements can also be written as

hi ,j =

(

ǫ1 hssσ

hssσ ǫ2

)

δi ,j +

(

0 0

h′ssσ 0

)

δi ,j+1 +

(

0 h′ssσ0 0

)

δi ,j−1 (4.33)

where i and j are the indices of the lattice translations ~tj = aj . The components of the 2 × 2 andthe orbital in the unit cell.

When we transform the Hamiltonian into Bloch representation using Eq. 4.32, we obtain the


k-dependent Hamiltonian with ~tj = aj

h(~k)Eq. 4.32=

∑

j

h(0, j)eikaj =

(

ǫ1 hssσ + h′ssσe

ika

hssσ + h′ssσe

−ika ǫ2

)

The eigenvalues ǫn(k) and eigenvectors cα(k) are obtained from the characteristic equationdet |h(k)− ǫS| = 0.

(ǫ1 − ǫ)(ǫ2 − ǫ)−(

hssσ + h′ssσe

+ika)(

hssσ + h′ssσe

−ika)

= 0

ǫ2 − 2 ǫ1 + ǫ22

ǫ+( ǫ1 + ǫ22

)2

−( ǫ1 + ǫ22

)2

+ ǫ1ǫ2 = h2ssσ + h

′2ssσ + 2hssσh

′ssσ cos(ka)

(

ǫ− ǫ1 + ǫ22

)2

=( ǫ1 − ǫ22

)2

= h2ssσ + h′2ssσ + 2hssσh

′ssσ cos(ka)

ǫn(k) =ǫ1 + ǫ22

±√( ǫ1 − ǫ22

)2

+ h2ssσ + h′2ssσ + 2hssσh

′ssσ cos(ka) (4.34)

The first band n = 1 is obtained with ± = − and the second band is obtained with ± = +.

Fig. 4.6: Bandstructure of the alternating linear chain. (Energy vs. k)

Let us investigate some special cases

• chain of decoupled molecules: We set one hopping parameter to zero, i.e. h′ssσ = 0 . Theresult are two k-independent bands for the molecular orbitals

ǫn(k) =ǫ1 + ǫ22

±√( ǫ1 − ǫ22

)2

+ h2ssσ (4.35)

• both atoms and hopping parameters are identical, i,e, ǫ1 = ǫ2 and hssσ = h′ssσ.

ǫn(k) = ǫ1 ±√

2h2ssσ + 2h2ssσ cos(ka)

= ǫ1 ± |hssσ|√

2 + 2 cos(ka)︸︷︷︸√

4cos2(ka/2)

= ǫ1 ±∣∣∣2hssσ cos

(1

2ka

)∣∣∣ (4.36)

This is the bandstructure of a mono-atomic chain

ǫ(k) = ǫ1 + 2hssσ cos(ka/2)

folded back into the first Brillouin zone.


• for alternating orbital energies or alternating hopping matrix elements, The band gap at thezone boundary opens, so that an insulator is obtained if the orbitals are half filled.

Note that hssσ is negative, because it is approximately equal to the overlap matrix elementmultiplied with the potential in the bond-center. The potential is negative.

4.5 Thermodynamics of non-interacting electrons

What makes the many-electron problem so complicated is the interaction between the electrons. Asimilar problem is known from classical mechanics. While the Kepler problem, a two-body problem,can be solved analytically, no such solution exists for the three-body problem, where the gravitationalattraction between the planets is considered as well.

4.5.1 Thermodynamics revisited

General setup

For a system described by a Hamiltonian H, which is coupled to a heat bath and reservoirs for theobservables Xi, the free energy8 F (T, fj) is (Eq. 1.61 of [31])

F (T, fj) = −kBT ln[

Tr(

e−β(H+∑

j fj Xj ))]

(4.39)

The heat bath is characterized by the temperature T and the other reservoirs are characterized bythe other intrinsic variables fj .

From the free energy, we can obtain (Eqs. 1.65, 1.66, 1.67 of [31]) the expectation values forthe observables X and the internal energy U, the expectation value of the energy of the system by

S(T, fk) = −∂F (T, fk)

∂T(4.40)

Xi(T, fk) = +∂F (T, fk)

∂fi

U(T, fk) = F (T, fk) + TS(T, fk)−∑

i

fiXi(T, fk) (4.41)

Specific relations

For our purposes we will start from a grand canonical ensemble, which describes a system in thermalequilibrium with a heat bath and a particle reservoir.

extensive variable X f intensive variable

volume V p pressureparticle number N −µ chemical potential

8 The partition function can be expressed in a basisset of many-particle states |Ψn〉, which are simultaneouseigenstates of the many-particle Hamiltonian H and the observables Xj . (Which implies that the Xj are conservedquantities of the Hamiltonian.)

Z(T, fj) def= Tr

(

e−β(H+fj Xj ))

=∑

n

〈Ψn|e−β(H−µN)|Ψn〉

=∑

n

〈Ψn|Ψn〉e−β(En−µNn) =∑

n

e−β(En−fjXj,n) (4.37)

where H|Ψn〉 = |Ψn〉En and Xj |Ψn〉 = |Ψn〉Xj,n. The partition function in turn defines the Free energy as

F (T, fj) = −kBT ln(Z(T, fj)) (4.38)


The thermodynamic potential for the grand canonical ensemble is the grand potential Ω(T, V, µ).

Ω(T, V, µ) = −kBT ln[

Tr(

e−β(H+µN))]

(4.42)

which yields

S(T, V, µ) = −∂Ω(T, V, µ)∂T

(4.43)

N(T, V, µ) = −∂Ω(T, V, µ)∂µ

(4.44)

U(T, V, µ) = Ω(T, V, µ) + TS(T, V, µ) + µN(T, V, µ) (4.45)

These equations can be put together to a Legendre transform to a quantity depending on theextensive variables

U(S, V, N) = statT,µ

(

Ω(T, V, µ) + TS + µN)

(4.46)

which yields the fundamental relation of thermodynamics

dU = TdS︸︷︷︸

heat

+

Work︷︸︸︷

dΩ

dV︸︷︷︸−p

dV + µdN (4.47)

which in turn defines Heat and Work.The pressure is defined as the derivative of the internal energy with respect to volume at fixed

entropy and particle number

pdef= −∂U(S, V, N)

∂V= −∂Ω(T, V, µ)

∂V(4.48)

4.5.2 Maximum entropy, minimum free energy principle and the meaning ofthe grand potential

The grand potential is related to the total energy of a composite system which consists of thematerial under study with an internal energy U(S, V, N), the the heat-bath with energy Uhb and aparticle reservoir with energy Upr = µNpr .

Consider a starting configuration where the entropy and the particle number on the system understudy vanish initially, that is (S = 0, N = 0). Also the particle number of the particle reservoir andthe energy of the heat bath vanish, i.e. (Uhb = 0, Npr = 0).

As the system is brought into contact with the heat bath and its reservoirs, particles are transferredfrom the reservoir to the system and heat ThbdShb is transferred from the heat bath to the system.Thus the energy of the heat bath is

Uhb(Shb) =

∫ Shb

0

dS′ Thb = ThbShb (4.49)

The energy conservation between heat bath and our system has the form

∂U

∂SdS + ThbdShb = 0

⇒ ∂U

∂SdS − ThbdS + Thb(dShb + dS) = 0

dStot :=S+Shb⇒ dUhb = −∂U

∂SdS = −ThbdS + ThbdStot (4.50)


where dStot is the total amount of entropy produced in the process. This entropy production iswhat drives the system into thermal equilibrium with its reservoirs. The entropy is produced duringthe heat transfer between heat bath and our system. Entropy is generally generated when heat istransferred along a temperature gradient or between two systems at different temperatures.

The total energy Etot and total number of particles is conserved, i.e. dEtot = 0, where the totalenergy is

Etot(Thb, V, µpr , S, N, Uhb, Npr ) = U(S, V, N)︸︷︷︸

system

+ Uhb︸︷︷︸

heat bath

+ µprNpr︸︷︷︸

particle reservoir

(4.51)

Using the particle conservation Npr = −N and Eq. 4.50, the total energy obtains the form

Etot(Thb, V, µpr ;S, Stot , N) = U(S, V, N)− ThbS + ThbStot − µprN (4.52)

As the system evolves, the total entropy dStot increases until it reaches its maximum. While itevolves, the process obeys energy conservation, that is dEtot = 0. Hence the state evolves on ahypersurface defined by Etot = const in the S,N, Stot space. As a result, the total entropy can beexpressed as function of S and N.

−ThbStot(S, V, N) = U(S, V, N)− ThbS − µprN − U(0, V, 0) (4.53)

This is only valid though, if the state (S, V, N) is connected by an energy conserving process to theinitial state (S = 0, V, N = 0) defined above!

The thermal process comes to halt at a maximum of the total entropy. This is when an infinites-imal heat transfer from the heat bath does no more produce entropy in the system, which in turnimplies that our system has reached the same temperature as the heat bath. The maximum entropyprinciple translates into a minimum energy principle for the grand potential

Ω(Thb, V, µpr ) = U(0, V, 0)− TStot(S, V, N)= min

S,N

(

U(S, V, N)− ThbS − µprN)

− Etot (4.54)

The grand potential is therefore the total energy of the system under study and the reservoirs(heat bath and particle reservoir) which are in contact with our system, minus the heat generatedrespectively transported out of the entire system.

Ω(THB, V, µpr ) = minS,N

(

U(S, V, N)− ThbS − µprN)

(4.55)

4.5.3 Many-particle states of non-interacting electrons

Nevertheless, we should start with the most simple case, namely the non-interacting electron gas.The Hamiltonian for a non-interacting electron gas in a potential v(~r) has the form

H =

N∑

i=1

hi =

N∑

i=1

p2i2me

+ v(~ri)

=

N∑

i=1

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉(−~22me

~∇2i + v(~ri))

〈~x1, . . . , ~xN | (4.56)

Here, ~pi is the momentum operator acting on the i-th particle, and ~ri is the position operator of thei-th particle.

Even though the electrons do not interact, they do not move completely independently. Thereason is, again, the Pauli-principle or the antisymmetry requirement for the wave function. If one


electron is at a certain position, no other electron with the same spin can approach it. There is norepulsive potential. The two electrons simply do not “like” each other.

In practice, this means that we have to use an antisymmetric wave function. Luckily, the eigen-states of a system of non-interacting electrons (with a non-degenerate ground state) is a single Slaterdeterminant. In contrast, the wave function for interacting electrons is, in general, a superpositionof many Slater determinants.

Here, we mention a few facts about the states of non-interacting electrons without proving them.Some are fundamental facts of statistical mechanics and others will be shown later in this book, inchapter 8.

Hamilton eigenstates: The eigenstates are Slater determinants made of eigenstates φn(~x) of the

one-particle Hamiltonian h = ~p2

2m + v(~r)

[−~22m

~∇2 + v(~r)]

φn(~x)

︸︷︷︸

〈~x |h|φn〉

= φn(~x)ǫn (4.57)

The Slater determinant, which is an eigenstate of the many-particle system, can be representedby its number representation. The occupations numbers, which may be zero or one, are combinedinto a vector ~σ. The i-th element of ~σ vector is the occupation number of the i-th eigenstate of theone-particle Hamiltonian.

Thus the eigenstates of the many-particle Hamiltonian can be labeled by the vector ~σ

|Φ~σ〉 = |σ1, σ2, . . .〉 (4.58)

Energy eigenvalues: A Slater determinant |Φ~σ〉 has the energy

E~σ =

∞∑

n=1

σnǫn

where the ǫni are the one-particle energies defined above in Eq. 4.57.

Particle number: A Slater determinant |Φ~σ〉 has the particle number

N~σ =

∞∑

n=1

σn

Editor: We came up to here on Wednesday, Dec. 2, 2015

Grand potential: The grand canonical ensemble describes a system in thermal equilibrium whichis in contact with a heat bath and a particle reservoir. A heat bath is characterized by a temperatureT and the particle reservoir is characterized by a chemical potential µ. The temperature is oftenrepresented by the thermodynamic beta β = 1

kBT. kB is the Boltzmann constant.

The thermodynamic potential for the grand-canonical ensemble is the grand potential. The


grand potential for a system of non-interacting fermions is

Ω(T, µ)Eq. 4.42= −kBT ln Tr[e−β(H−µN)]

= −kBT ln[∑

~σ

exp

(

−β(

E~σ − µN~σ))

︸︷︷︸

partition sum

]

= −kBT ln[∑

~σ

e−β∑∞i=1 σi (ǫi−µ)

]

= −kBT ln[ ∞∏

i=1

1∑

σi=0

e−βσi (ǫi−µ)]

= −kBT ln[ ∞∏

i=1

(

1 + e−β(ǫi−µ))]

= −kBT∞∑

i=1

ln(

1 + e−β(ǫi−µ))

(4.59)

The interesting observation is that the thermodynamic potential of non-interacting particles, hereelectrons, is a sum over one-particle states.

4.5.4 Density of States

The grand potential turns out to be a simple sum over one-electron states with an integrand thatdepends on the one-particle energy ǫi , the chemical potential µ and the temperature T . This impliesthat the thermodynamics quantities, that are derivatives of the grand potential can be described asa sum over one-particle states as well. Therefore it is convenient to split off the spectrum of the oneparticle states. We can represent the one-particle spectrum by the density of states

DENSITY OF STATES

The one-particle density of states is defined as

D(ǫ)def=∑

n

δ(ǫ− ǫn) (4.60)

where δ(x) is Dirac’s δ-function and the one-particle energies are denoted by ǫn.

Note that the density of state as defined here is a macroscopic quantity. In practice one often usesthe density of state per volume or per unit cell.

With the density of states, we can separate the thermodynamic information from the system-specific information. The density of states is specific for the system, while the Fermi-distribution isindependent of the system and contains the thermodynamic information.

Projected Density of States: The total density of states discussed above is a special case of thedensity of states operator

D(ǫ) =∑

n

|ψn〉δ(ǫ− ǫn)〈ψn| (4.61)

which allows one to represent any thermal expectation value of one-particle operators by

〈A〉 =∑

n

f (ǫn)〈ψn|A|ψn〉 =∫

dǫ f (ǫ) Tr[D(ǫ)A

](4.62)


Grand potential expressed by the Density of States:

Ω(T, µ) = −kBT∞∑

n=1

ln(

1 + e−β(ǫn−µ))

=

∫ ∞

−∞dǫ D(ǫ)

[

−kBT ln(

1 + e−β(ǫ−µ))]

(4.63)

4.5.5 Fermi distribution

The mean occupations fi are given by the Fermi-distribution function evaluated for the one-particleenergy ǫi of the state.(see “ΦSX: Statistical Physics”)

fT,µ(ǫ)def=

1

1 + eβ(ǫ−µ)(4.64)

where β = 1/(kBT ). The Fermi function is shown in Fig. 4.7. In the absence of temperature effectswe set the temperature to zero, so that the Fermi distribution function is turned into a step function.

( () )

1

0

41

21

kBT− ε µ

kBT− ε µ

µ

f

ε

~exp

−

BT− ε µ

1−exp~ − k

Fig. 4.7: Fermi distribution fT,µ(ǫ) =(1 + eβ(ǫ−µ)

)−1.

The chemical potential of the electrons is often called[32] the Fermi energy or Fermi level.While I will use the term in this way, it should be noted that the Fermi energy is sometimes[33]defined as the chemical potential at zero temperature.

Derivatives of the Fermi function: In the following, we will also need the derivatives of the Fermidistribution with respect to temperature and chemical potential. They can be expressed by the energyderivative of the Fermi function. 9

∂T fT,µ(ǫ) = −ǫ− µT

∂ǫfT,µ(ǫ) (4.66)

∂µfT,µ(ǫ) = −∂ǫfT,µ(ǫ) (4.67)

9

∂ǫ

[

1 + eβ(ǫ−µ)]−1= β

[d

dx

∣∣∣∣x=β(ǫ−µ)

[

1 + eβ(ǫ−µ)]−1]

∂µ

[

1 + eβ(ǫ−µ)]−1= −β

[d

dx

∣∣∣∣x=β(ǫ−µ)

[

1 + eβ(ǫ−µ)]−1]

= −∂ǫfT,µ(ǫ)

∂T

[

1 + eβ(ǫ−µ)]−1=(

− 1Tβ(ǫ− µ)

)[ d

dx

∣∣∣∣x=β(ǫ−µ)

[

1 + eβ(ǫ−µ)]−1]

= − ǫ− µT

∂ǫfT,µ(ǫ) (4.65)


-5 -4 -3 -2 -1 0 1 2 3 4 5

0.1

0.2

Fig. 4.8: Derivative of the Fermi distribution with respect to the chemical potential. Vertical axisβ∂µfT,µ, horizontal axis β(ǫ−µ). With lowering temperatures the function becomes larger but morenarrow. The integral equals always 1.

In the low-temperature limit, the energy derivative of the Fermi function approaches the negativedelta function at the Fermi level. 10

limT→0

∂ǫfT,µ(ǫ) = −δ(ǫ− µ) (4.68)

Sommerfeld expansion: In order to evaluate the integrals involving the Fermi function one can usethe Sommerfeld expansion derived in appendix H on p. 239.

SOMMERFELD EXPANSION

∫ ∞

−∞dǫ g(ǫ)f (ǫ) =

(∫ µ

−∞dǫ g(ǫ)

)

+(πkBT )

2

6

(

∂ǫ|µ g)

+(7πkBT )

4

360

(

∂3ǫ∣∣µg)

+O(

(kBT )6)

(4.69)

Note however, that the Sommerfeld expansion is a low-temperature expansion, which breaks down ifthe function to be integrated is not analytical within a few kBT away from the chemical potential.

Expansion for |ǫ − µ| > kBT : If the Fermi level lies outside the energy spectrum, we expand theFermi distribution function into powers of Boltzmann factors11 e−β(ǫ−µ).

fT,µ(ǫ) =

∑∞n=1(−1)n

[

e−β(ǫ−µ)]n

for ǫ− µ≫ kBT

1−∑∞n=1(−1)n[

eβ(ǫ−µ)]n

for ǫ− µ≪ −kBT(4.71)

This implies that the Fermi distribution approaches the Boltzmann distribution far from the chemicalpotential. If the Fermi level lies outside the energy spectrum, the electrons form a very dilute gas.Because the probability that two particles come close together is then vanishingly small, it does notmatter whether the particles are identical or distinct. Thus the Fermi-distribution describing identicalparticles and the Boltzmann distribution describing distinguishable particles become similar in a dilutegas.

10The delta function is the negative derivative of the Heaviside step function.11 We use:

1

1 + ex= e−x

1

1 + e−x= e−x

∞∑

n=0

(

−e−x)n= e−x

∞∑

n=1

(−1)n+1(

e−x)n

(4.70)


In the following we will often have to do with integrals of the form∫ ∞

ǫ0

dǫ fT,µ(ǫ)√ǫ− ǫ0 (4.72)

which can be addressed as follows 12:

∫ ∞

ǫ0

dǫ fT,µ(ǫ)√ǫ− ǫ0 =

∞∑

n=1

(−1)n+1∫ ∞

ǫ0

dǫ√ǫ− ǫ0

[

e−β(ǫ−µ)]n

=

∞∑

n=1

(−1)n+1∫ ∞

ǫ0

dǫ√ǫ− ǫ0

[

e−β(ǫ−ǫ0)−β(ǫ0−µ)]n

=

∞∑

n=1

(−1)n+1[

e−β(ǫ0−µ)]n(1

βn

) 32∫ ∞

0

dx√xe−x

︸︷︷︸12

√π

=

√π

2(kBT )

32

∞∑

n=1

(−1)n+1n32

[

e−β(ǫ0−µ)]n

(4.73)

NUMBER OF ELECTRONS IN A DILUTE, ELECTRON GAS WITH A GIVEN EFFECTIVEMASS

∫ ∞

ǫ0

dǫ fT,µ(ǫ)4πΩ

(2π~)3(2m∗)

32

√ǫ− ǫ0 = 2Ω

(2πm∗kBT(2π~)2

) 32

e−β(ǫ0−µ) +O

([

e−β(ǫ0−µ)]2)

(4.74)

Here Ω is the volume considered. m∗ is the effective mass, and ǫ0 is the band-minimum.

Thermal smoothening of the density of states: An expression of interest for numerical calculationis

∫

dǫ g(ǫ)fT,µ(ǫ) =

∫ µ

−∞dµ′∂µ

∫

dǫ g(ǫ)fT,µ(ǫ)

=

∫ µ

−∞dµ′

∫

dǫ g(ǫ)(

−∂ǫfT,µ(ǫ))

=

∫ µ

−∞dǫ gT (ǫ)

=

∫ ∞

−∞dǫ gT (ǫ)f0,µ(ǫ) (4.75)

where

gT (ǫ)def=

∫ ∞

−∞dǫ′ g(ǫ′)

(

−∂ǫfT,ǫ(ǫ′))

(4.76)

Thus, we can smoothen the function g(ǫ) using Eq. 4.76 and then treat the resulting functiongT (ǫ) like at zero temperature. The disadvantage is that g depends on temperature. The advantageis two-fold: Firstly, the zero-temperature expressions are easier to comprehend. Secondly, noise fromexperiment or numerical calculations is averaged over by the smoothening.

12∫∞0 dx xne−x = Γ(n+ 1), where Γ(x) is the Gamma function. Γ(1) = 1, Γ( 1

2) =√π. From Γ(x + 1) = xΓ(x) we

obtain Γ( 32) = 1

2Γ( 12) = 1

2

√

(π). (see Bronstein[34]).


4.5.6 Thermodynamics of non-interacting electrons

As shown in section 8.5 of “ΦSX: Statistical Physics”[31], we can express the thermodynamic quan-tities of non-interacting Fermions by the density of states

Grand potential

For a grand canonical ensemble, the free energy is the grand potential Ω(T, µ), which is obtained as

Ω(T, µ)Eq. 4.42= −kBT

∞∑

n=1

ln(

1 + e−β(ǫn−µ))

=

∫ ∞

−∞dǫ D(ǫ)

[

−kBT ln(

1 + e−β(ǫ−µ))]

(4.77)

The function in the integral has the form.

−kBT ln(

1 + e−β(ǫ−µ))

≈

ǫ− µ for ǫ− µ≪ −kBT−kBT ln(2) + 12β(ǫ− µ) for |ǫ− µ| ≪ kBT

0 for ǫ− µ≫ kBT

(4.78)

Using two expressions derived below, namely Eq. 4.82 and Eq. 4.87, we obtain with Eq. 4.78 anintuitive form for the grand potential In the low-temperature limit.

Ω(T = 0, µ) =

∫ µ

−∞dǫ D(ǫ)

(

ǫ)− µ)

=

∫ µ

−∞dǫ D(ǫ)ǫ)

︸︷︷︸

E

−µ∫ µ

−∞dǫ D(ǫ)

︸︷︷︸

N

= E − µN (4.79)

where E is the total energy, i.e. the low-temperature limit of the internal energy, and N is the particlenumber.

Particle number and chemical potential

The particle number N(T, µ) is obtained as

N(T, µ)Eq. 4.44= −∂Ω(T, µ)

∂µ(4.80)

=

∫ ∞

−∞dǫ D(ǫ)

[

−∂µ(

−kBT ln(

1 + e−β(ǫ−µ)))]

(4.81)

=

∫ ∞

−∞dǫ D(ǫ)fT,µ(ǫ) (4.82)

Usually, the particle number is provided instead of the chemical potential. The particle numberfollows typically from the charge neutrality condition. In order to extract the chemical potential onneeds to evaluate the particle number as function of chemical potential and find that value for whichN(T, µ) = N.

In the following we will need the first order shift of the chemical potential with increasing tem-perature. At low-temperatures we can estimate the particle number as

N(T, µ)Eq. 4.69=

(∫ µ

−∞dǫ D(ǫ)

)

︸︷︷︸

N(T=0,µ)

+(πkBT )

2

6

(

∂ǫ|µD)

+O(

(kBT )4)

(4.83)


The requirement that the particle number remains unchanged gives us an expression for the temper-ature shift of the chemical potential.

0 =dN

dT

∣∣∣∣T=0,µ(T=0)

=∂N

∂T

∣∣∣∣T=0,µ(T=0)

+dµ

dT

∂N

∂µ

∣∣∣∣T=0,µ(T=0)

= πkB(πkBT )

3

(

∂ǫ|µD)

+dµ

dTD(µ) +O

(

(kBT )2)

dµ

dT= −(πkB)

2

3

(

∂ǫ|µD)

D(ǫF )T +O(T 2) (4.84)

Thus the chemical potential moves quadratically away from its low-temperature value. This equationis only valid for metals, which have a finite density of states at the Fermi level.

Entropy

S(T, µ)Eq. 4.43= −∂Ω(T, µ)

∂T

=

∫ ∞

−∞dǫ D(ǫ)

[

−∂T(

−kBT ln(

1 + e−β(ǫ−µ)))]

=

∫ ∞

−∞dǫ D(ǫ)

[(

kB ln(

1 + e−β(ǫ−µ)))

︸︷︷︸

1/(1−f )

+

(

kBT1

1 + e+β(ǫ−µ)︸︷︷︸

f

( 1

kBT 2(ǫ− µ)

)

︸︷︷︸1Tln(f −1−1)

)]

=

∫ ∞

−∞dǫ D(ǫ)

[

kB

(

− ln(1− f ) + f ln(1− ff))]

=

∫ ∞

−∞dǫ D(ǫ)

[

−kB(

f ln(f ) + (1− f ) ln(1− f ))]

(4.85)

The entropy can be obtained from the occupations f = fT,µ(ǫ).

0.25 0.5 0.75

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-10 -5 0 5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.9: Entropy −kB(f ln(f ) + (1 − f ) ln(1 − f )) as function of occupation f and (right) entropy−kB(f (ǫ) ln(f (ǫ)) + (1− f (ǫ)) ln(1− f (ǫ))) as function of one-particle energy.

Eq. 4.85 tells us completely filled or completely empty orbitals do not contribute to the entropyof a system. A half occupied orbital contributes kB ln(2) to the entropy. This is exactly the entropyof a yes-no decision, because the orbital can be either filled or empty with equal probability.

Internal energy

The internal energy U is the expectation value of the energy of the system.


Before we continue, we express the grand potential by the occupations, because that will simplifythe equations.

Ω(T, µ) =

∫ ∞

−∞dǫ D(ǫ)

[

−kBT ln(

1 + e−β(ǫ−µ))]

=

∫ ∞

−∞dǫ D(ǫ)

[

kBT ln (1− fT,µ(ǫ))]

(4.86)

U(T, µ)Eq. 4.45= Ω(T, µ) + TS(T, µ) + µN(T, µ)

=

∫ ∞

−∞dǫ D(ǫ)

[

kBT ln (1− f )︸︷︷︸

←Ω

−kBT(

f ln(f ) + (1− f ) ln(1− f ))

︸︷︷︸

←TS

+µf

]

=

∫ ∞

−∞dǫ D(ǫ)

[

f ·[

kBT ln

(1− ff

)

︸︷︷︸

β(ǫ−µ)

+µ]

︸︷︷︸ǫ

]

=

∫ ∞

−∞dǫ D(ǫ) fT,µ(ǫ) ǫ (4.87)

where, again, f = fT,µ(ǫ). The internal energy has a simply interpretation: it is the sum of theenergies of the occupied states.

In order to obtain the internal energy as function of particle number, we need to know µ(T,N),which needs to be obtained from the expression for N(T, µ) obtained above.

U(T,N) = Ω(T, µ(T,N)) + TS(T, µ(T,N)) + µN

=

∫ ∞

−∞dǫ D(ǫ) fT,µ(T,N)(ǫ) ǫ (4.88)

The temperature dependent term can be separated out as follows

U(T,N) =

∫ ∞

−∞dǫ D(ǫ) fT,µ(T,N)(ǫ) ǫ

=

∫ ∞

−∞dǫ D(ǫ) fT=0,µ(T=0)(ǫ) ǫ

︸︷︷︸

U(T=0,N)

+

∫ ∞

−∞dǫ D(ǫ)

(

fT,µ(T )(ǫ)− fT=0,µ(T=0)(ǫ))

ǫ (4.89)

The weight function for the temperature dependent term is shown in Fig. 4.10.The low-temperature limit of the above equation is

U(T,N) = U(0, N) +(πkBT )

2

6D(ǫF ) (4.90)

which can be verified by comparison with the Sommerfeld expansion Eq. 4.69.

Pressure

The pressure p is defined as the derivative of the total energy upon contraction.

pEq. 4.48= −∂U(S, V, N)

∂V(4.91)

Note, that it is not allowed to calculate the pressure as derivative of U(T, V, µ)! The pressure is thederivative of the total energy with fixed occupations.


-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

0.1

0.2

0.3

Fig. 4.10: Weight function of the thermal part to the internal energy. ǫ(fT,µ(T,N)(ǫ)−fT=0,µ(T=0,N)(ǫ))as function of the deviation from the zero-temperature chemical potential in units of kBT . The one-sided integral of this function (shaded area) is π2/12. The orange line shows a function for atemperature-dependent chemical potential.

However, we can calculate the pressure from the grand potential

p = −∂Ω(T, V, µ)∂V

=

∫ ∞

−∞dǫ

dD(ǫ)

dV

[

+kBT ln(

1 + e−β(ǫ−µ))]

(4.92)

Why does this give the same result as the derivative of U(S, V, N): The grand potential describesthe energy of the system and the reservoirs. If the volume expansion changes the entropy or theparticle number of the system, the corresponding energy change is exactly balanced to first order byan opposite energy change of the reservoirs. Thus including the energies of the reservoirs producesthe same energy change as keeping entropy and particle number fixed.

Bulk modulus

The bulk modulus is the measure of how a material can resist compression.

B0 = −V∂P

∂V= V

∂2U(S, V, N)

∂V 2(4.93)

Thus it is a second derivative of the internal energy.

Heat capacity

The heat capacity is the amount of energy required to raise the temperature by one unit.

CV =∂U(T, V, N)

∂T(4.94)

When calculating the heat capacity from U(T, V, µ) we need to account for the shift of the


chemical displacement with temperature

cV =∂U(T, V, µ)

∂T+∂U(T, V, µ)

∂µ

dµ(T, V, N)

dT

=

∫ ∞

−∞dǫ D(ǫ)

∂fT,µ(ǫ)

∂Tǫ+

∫ ∞

−∞dǫ D(ǫ)

∂fT,µ(ǫ)

∂µǫdµ(T, V, N)

dT

=

∫ ∞

−∞dǫ D(ǫ)

ǫ− µT

∂fT,µ(ǫ)

∂µǫ+

∫ ∞

−∞dǫ D(ǫ)

∂fT,µ(ǫ)

∂µǫdµ(T, V, N)

dT

=1

T

∫ ∞

−∞dǫ D(ǫ)ǫ

∂fT,µ(ǫ)

∂µ

[

ǫ− µ+ T dµ(T, V, N)dT

]

=1

T

∫ ∞

−∞dǫ D(ǫ)

∂fT,µ(ǫ)

∂µ

[


]2

+1

T

[

µ− T dµ(T, V, N)dT

] ∫ ∞

−∞dǫ D(ǫ)

∂fT,µ(ǫ)

∂µ

[


]

(4.95)

The particle number conservation provides us with dµ/dT .

0 =dN

dT=

∫

dǫD(ǫ)[∂fT,µ(ǫ)

∂T+∂fT,µ(ǫ)

∂µ

dµ

dT

]

=

∫

dǫD(ǫ)[(ǫ− µ)

T

∂fT,µ(ǫ)

∂µ+∂fT,µ(ǫ)

∂µ

dµ

dT

]

=1

T

∫

dǫD(ǫ)∂fT,µ(ǫ)

∂µ

[

ǫ− µ+ T dµdT

]

(4.96)

Thus the second term in the specific heat vanishes completely so that

cV =1

T

∫ ∞

−∞dǫ D(ǫ)

∂fT,µ(ǫ)

∂µ

[


]2

(4.97)

where

Tdµ

dT= −

∫dǫD(ǫ)

∂fT,µ(ǫ)∂µ

(

ǫ− µ)

∫dǫD(ǫ)

∂fT,µ(ǫ)∂µ

(4.98)

The function used to obtain the specific heat from an integration over the density of states is shownin Fig. 4.11.

-15 -10 -5 0 5 10 15

0.1

0.2

0.3

0.4

Fig. 4.11: Weight function g(ǫ) = β ∂f∂µ(ǫ−µ)2 which yields the specific heat cV = kB∫dǫ D(ǫ)g(ǫ)

without a thermal shift of the chemical potential.


If the density of states is approximately constant with kBT from the chemical potential, we canapproximate the expression by

cV ≈1

TD(µ)

∫ ∞

−∞dǫ (ǫ− µ)2

(

−∂ǫf (ǫ))

=1

TD(µ)(kBT )

2(

−∫ ∞

−∞dx x2∂x

1

1 + ex

)

=((πkB)

2

3D(µ)

)

︸︷︷︸γ

T (4.99)

In the last step we used an integral out of the Sommerfeld expansion, which is described in appendix H.The parameter

γdef= lim

T→01

TcV (4.100)

is called the Sommerfeld coefficient of the specific heat.At low temperatures the heat capacity is proportional to the temperature, with a coefficient

proportional to the density of states at the electron chemical potential.It turns out that vibrations enter only with a higher power with temperature. Hence a low-

temperature measurement of the heat capacity provides us with a direct measure of the density ofstates at the chemical potential.

Note that Eq. 4.99 is restricted to low temperatures, or to density of states that are approximatelyconstant within thermal energies from the chemical potential. Eq. 4.97 on the other hand is onlyrestricted to non-interacting Fermions electrons. It is neither restricted to low temperatures nor doesit make any assumptions on the density of states.

Magnetic susceptibility

In a magnetic field, the energy of an electron is shifted by an energy

∆E = −~m~B , (4.101)

where ~m is the magnetic moment of the electron and ~B is the magnetic field. The magneticmoment of an electron is related to its spin by its gyromagnetic ratio γe , that is

~m = γe ~S = γe1

2~~eS =

1

2geµB~eS (4.102)

The factor ge = 2.0023 . . . is the g-factor of the free electron. The Bohr magneton µB =e~2me

isa unit for the magnetic moment. The Bohr magneton is nearly (up to a factor 12ge) identical to themagnetic moment of the electron.

Depending on the spin alignment with respect to the magnetic field the electron is raised ordecreased. This is the so-called Zeeman splitting.

The Zeeman splitting between the energies of spin-up and spin-down electrons in the field of 1Tesla corresponding to a strong permanent magnet is approximately 0.1 eV.

When the Zeeman energy is added to the dispersion relation, the band structure of the spin up-electrons is shifted by a small amount with respect to that of the down spin electrons. If the electronnumber is initially the same for both spin directions, the system can reduce its energy by changingthe spin of some of its electrons. Thus the total energy depends, via the Zeeman splitting, on themagnetic field.


The magnetization is

~M =

(1

2geµB

)

︸︷︷︸

magnetic moment

1

2V

(

N(T, µ− 12geµB|~B|)− N(T, µ+

1

2geµB|~B|)

)

︸︷︷︸

spin density

=

( |geµB|2

)2D(µ)

VB +O(|~B|2) (4.103)

where N(T, µ) is the total electron number, including both spin directions.From the field dependence of the magnetic susceptibility, we obtain the magnetic susceptibility

of a free electron gas

χ =d | ~M|d |~B|

=

( |geµB|2

)2D(µ)

V(4.104)

, which is the Pauli spin susceptibility.

4.5.7 Brillouin-zone integrations

In order to evaluate expectation values or the density of states, it is necessary to sum over Blochstates. Because the wave vector k is continuous, it is not a-priori clear how the states need to beweighted, that is how the transition from discrete states to a continuum is performed.

Solid in a box

One way to do the transition is to enclose the solid into a box and then let the box go to infinity.The potential for a solid in a box consists of the periodic potential superimposed with the potentialof a box. That is, the potential outside the box is infinite.

The wave function inside the box can be represented as Bloch waves.

ψk,n(~r) = u~k,n(~r)ei~k~r (4.105)

Let us consider the box with side-lengths Lx , Ly , Lz along the three cartesian coordinates.13 Thebox shall occupy the volume with

0 < x < Lx0 < y < Ly0 < z < Lz

For a free particle in the box we enforce the boundary conditions by

ψ(~r) =

√

8

L3sin(kxx) sin(kyy) sin(kzz) (4.106)

with

~ki ,j,k =

πLxi

πLyj

πLzk

(4.107)

with positive integers i , j, k .

13I allow for different side lengths to allow for the study of quasi-one-dimensional or quasi-twodimensional systems.


Periodic boundary conditions

For a solid in its pure sense the description of particles in a box is inconvenient, because the surfacesdistract from the essence of the bulk material. One of the inconveniences is that the eigenstatesare standing waves and not propagating waves. The ability to act as medium for currents is howeveressential for the description.14

Therefore, one has introduced periodic boundary conditions15. This implies that we connectthe right side of the box with the left side, the back with the front and the bottom with the top.In other words, we consider solutions that extend over all space, but they repeat each other after acertain distance in each direction.

This implies that we require a true periodicity for the wave function. While the wave function isperiodic with the unit cell of the crystal, the wave function is periodic with a much larger unit cell.We may use N1 unit cells along the first lattice vector N2 unit cells along the second and N3 alongthe third lattice vector. Thus the wave function are periodic in a supercell containing N1N2N3 unitcells of the crystal.

ψ~k,n(~r + N1~T1) = ψ~k,n(~r)

ψ~k,n(~r + N2~T2) = ψ~k,n(~r)

ψ~k,n(~r + N3~T3) = ψ~k,n(~r) (4.108)

For Bloch states this simply restricts the wave vector to

~ki ,j,k =~g1N1i +

~g2N2j +

~g3N3k (4.109)

with integer i , j, k . In order not to duplicate wave functions we further more need to restrict thek-points into one reciprocal unit cell, that is

0 ≤ i < N1 and 0 ≤ j < N2 and 0 ≤ k < N3 and (4.110)

As a consequence of the periodic boundary conditions, only certain wave vectors are allowed. Becausethe allowed wave vectors form a discrete set, also the wave functions form a discrete set, so that thedensity of states can be expressed as a sum of δ-functions. according to Eq. 4.60 and Eq. 4.61. Ina second step we make the transition to an infinite system by letting N1, N2, N3 go to infinity.

We still need to define the normalization of the wave functions. Let us define the wave functionsso that they are normalized in the supercell.

In the following we use the matrix T of real space lattice vectors, and the matrix g of reciprocalspace lattice vectors. They obey the relation g†T = 2π111 so that det(g) det(T ) = (2π)3.

The density of states for a certain energy is

D(ǫ0)Eq. 4.60=

∑

n

N1∑

i=1

N2∑

j=1

N3∑

k=1

|ψ~k,n〉δ(

ǫ(~ki ,j,k , σ)− ǫ0)

〈ψ~k,n|

=∑

n

N1∑

i=1

N2∑

j=1

N3∑

k=1

d3k︷︸︸︷

det |g|N1N2N3

N1N2N3 det |T |(2π)3

|ψ~k,n〉δ(

ǫ(~ki ,j,k , σ)− ǫ0)

〈ψ~k,n|

=Ω

(2π)3

∑

n

∫

d3k |ψ~k,n〉δ(

ǫ(~k, σ)− ǫ0)

〈ψ~k,n| (4.111)

where the wave functions are normalized within the volume Ω = N1N2N3 det |T | of the super cell.

14Such a description is also possible using particles in the box, but requires extra measures such as alternating andspatially varying currents.

15For periodic boundary conditions see also ΦSX: Statistical Physics, Chapter: Non-interacting Particles.


Thus, we replaced the sum over states by an integral over k-space.The total density of states can be obtained by forming the trace of D(ǫ), i.e.

Dtot(ǫ0) = Tr[D(ǫ0)

] Eq. 4.111=

Ω

(2π)3

∑

n

∫

d3k δ(

ǫ(~k, σ)− ǫ0)

(4.112)

Note, that the density of states is defined as an extensive quantity16, that is, it is proportionalto the volume Ω. This is understandable, because the energy levels come closer together as thesystem size is increased until the discrete spectrum is converted at infinite volume into a continuousspectrum. A little inconvenient is that the density of states is multiplied with an, in principle, infinitevolume. This can be avoided by defining the density of states per volume g(ǫ)

g(ǫ)def=1

ΩD(ǫ)

Eq. 4.112=

∑

n

∫d3k

(2π)3δ(

ǫ(~k, σ)− ǫ0)

(4.113)

Outlook: Numerics

A short excursion: It is worth remembering the form of the Brillouin-zone integral

〈A〉 = Ω∑

σ

∫

ǫ(~k)<ǫF

d3k

(2π)3Aσ(~k) (4.114)

where Aσ(~k) := 〈ψσ(~k)|A|ψσ(~k)〉 is the expectation value of a one-particle operator with a spin statefor a given Bloch vector.

4.6 Jellium model

We use model systems to understand the qualitative behavior of a system before we attempt todescribe a real system in all its complexity. For the electron gas, the model system is the freeelectron gas or the jellium model. The jellium model consists of electrons and a spatially constant,positive charge background, which ensures overall charge neutrality.

4.6.1 Wave functions

The wave functions of free electrons have, in the extended zone scheme

ψ~k,σ(~r , σ′) = δσ,σ′

1√Ωei~k~r (4.115)

where Ω is the normalization volume.

4.6.2 Band structure of the free electron gas

The dispersion relation of free electrons forms a parabola

ǫ~k,σ = ǫ0 +(~~k)2

2me(4.116)

where ǫ0 is the value of the constant potential. The dispersion relation of electrons in a solid is oftencalled the band structure.

16The notion of extensive and intensive quantities originates from thermodynamics. An extensive quantity scaleswith the size of the system, while an intensive does not.


21

21

21( , , )

21(1, ,0)

Γ Γ K XX W L

(1,0,0)

(0,0,0) (1,1,0)

(0,0,0)

1 1 1

2

2

21 2 11

2 210eV

5eV

15eV

Γ X W L Γ K X

-10

-5

0

5

10

15

E[e

V]

Fig. 4.12: Band structure of free, non-interacting electrons. The lattice is an fcc-cell with a latticeconstant of 4.05 Å corresponding to aluminum. The high symmetry points are given in units of2πalat

. The numbers indicate the degeneracy beyond spin-degeneracy. Below the band structure ofaluminum calculated by density functional theory is shown in comparison.

In Fig. 4.12, the band structure of free and non-interacting electron gas is compared to that ofa real material, namely aluminium. We observe that the free electron gas already provides a fairlygood description of some realistic systems.

The jellium model approximates real metals surprisingly well, even though the potential of thenuclei is far from being a constant. In a hand-waving manner we can say that the valence electronsare expelled from the nuclear region by the Pauli repulsion of the core electrons, so that they movearound in a region with fairly constant potential.

4.6.3 Density of states of the free electron gas

Now we can calculate the density of states for the free, non-interacting electron gas.

D(ǫ)Eq. 4.112=

Ω

(2π)3

∑

σ︸︷︷︸

2

∫

d3k δ

ǫ0 +

(~~k)2

2me︸︷︷︸

ǫ~k,σ

−ǫ

(4.117)

It is convenient to exploit that the delta function is the derivative of the Heaviside step functionand to determine the density of states as derivative of the number of states at zero temperature.

N(T = 0, µ) =

∫ ∞

−∞dǫ D(ǫ)fT=0,µǫ =

∫ µ

−∞dǫ D(ǫ)

⇒ D(ǫ) =∂N(T = 0, µ)

∂µ

∣∣∣∣µ=ǫ

(4.118)


The number of states function at zero temperature for the free electron gas is

N(T = 0, µ) = Ω

∑

σ∈↑,↓

︸︷︷︸

=2

∫d3k

(2π)3θ

(

µ− ǫ0 −(~~k)2

2me

)

= Ω

∑

σ∈↑,↓

︸︷︷︸

=2

1

(2π)34π

3k3F = Ω

∑

σ∈↑,↓

︸︷︷︸

=2

4π

3

(2me(µ− ǫ0)(2π~)2

) 32

(4.119)

where

kF (µ)def=1

~

√

2me(µ− ǫ0) (4.120)

is the Fermi wave vector. ~kF is the Fermi momentum.17

Thus we obtain the density of states of the free electron gas

DENSITY OF STATES OF THE FREE ELECTRON GAS

D(ǫ)Eqs. 4.118,4.119

= 2πΩ

∑

σ∈↑,↓

︸︷︷︸

=2

(2me(2π~)2

) 32 √

ǫ− ǫ0 (4.121)

The Fermi momentum is very useful as upper cutoff for the energy integration for the zero-temperature results. It can directly be related to the electron density n = N

Ω . From Eq. 4.119 weobtain

n =N

Ω

T=0=

N(T = 0, µ)

Ω

Eq. 4.119=

∑

σ∈↑,↓

︸︷︷︸

=2

1

(2π)34π

3k3F

⇒ kF (n) = 3

√

6π2n∑

σ∈↑,↓=

3√3π2n (4.122)

With the Fermi momentum from Eq. 4.122, we can determine the Fermi energy

ǫF = ǫ0 +(~kF )

2

2me= ǫ0 +

(3π2~3)23

2men23 (4.123)

While the Fermi energy of the free electron gas depends on the mass, the potential and the electron-electron interaction, the Fermi-momentum of a free electron gas only depends on the density. Thereason is that the dispersion relation of a free electron gas is isotropic, and the volume of the volumeinside the Fermi surface is determined by the number of states.

The density of the electron gas is often represented by a rs , which is the radius of a sphere whichoccupies the mean volume per electron. If the electron density is n = N

Ω , we obtain

n =14π3 r3s

⇒ rs =3

√

3

4πn(4.124)

17In Hartree atomic units ~ = 1 so that the difference between Fermi momentum and Fermi wave vector is notapparent. Therefore the distinction between Fermi wave vector and Fermi momentum is rarely made explicit.


The electron-gas parameter or Seitz radius18 The Seitz radius rs is in common use because itprovides a characteristic length scale of the system.

18I found the name Seitz radius in [35].


4.7 Tour trough band structures and densities of states of real

materials

4.7.1 Free-electron like metals

The free electron like metals are the main group elements with a valence of one, two and three. Thisincludes the alkali metals, the alkaline earth metals and tri-valent metals such as Aluminium.

Ba

Ra

Ge

Sn

Tl

Ga

Pb

In

Se

PoBi

Sb

As

Te

At

I

Br

Hg

Cd

Zn

Au

Ag

CuCr Mn Fe Co Ni

Y ZrSr Mo

TaHf

Ac

La

H

Tc

ReW Os Ir Pt

Ru PdRhNb

He

Li Be

Sc

Mg

Ti

Na

V

CB ON F Ne

Ar

Rn

Kr

Xe

ClSPSiAl

Ca

Cs

Fr

K

Rb

Fig. 4.13: Location of free-electron like metals in the periodic table marked in green.

The band structure of these materials is very similar to that of the free electron gas. The maindifference is the lifted degeneracies at the zone boundaries, where the free-electron bands cross eachother.

Γ X W L Γ K X

-10

-5

0

5

10

15

E[e

V]

21

21

21( , , )

21(1, ,0)

Γ Γ K XX W L

(1,0,0)

(0,0,0) (1,1,0)

(0,0,0)

1 1 1

2

2

21 2 11

2 210eV

5eV

15eV

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10E(eV)

Fig. 4.14: Band structure (left) and density of states (right) of the free-electron-like metal aluminiumand. For comparison, the band structure (middle) of the free-electron gas in an fcc-cell with a latticeconstant of 4.05 Å corresponding to aluminum is shown. The bands are colored blue, green magentafor the bands derived from s, p and d-bands of a corresponding tight-binding model. The highsymmetry points marked in the free-electron band structure are given in units of 2πalat . The numbersindicate the degeneracy beyond spin-degeneracy. The function marked in black in on the right-handside is the total density of states of aluminium. The blue, green yellow areas ontop of the totaldensity of states are the projections on s, p and d-type orbitals inside an atom-centered sphere. Theband structure and density of states of Al has been calculated with density-functional theory usingthe CP-PAW code. Parts of this figure are identical to Fig. 4.12.

The valence of the element determines the number of electrons per unit cell, and thus the positionof the Fermilevel.

The density of states shows the typical square-root-like behavior of the free electron gas.


4.7.2 Noble metals

Rb

Ba

Ra

Ge

Sn

Tl

Ga

Pb

In

Se

PoBi

Sb

As

Te

At

I

Br

Hg

Cd

Zn

Au

Ag

CuCr Mn Fe Co Ni

Y ZrSr Mo

TaHf

H

La

Nb Tc

ReW Os Ir Pt

Ru PdRh

Ac

He

Li Be

Sc

Mg

Ti

Na

V

CB ON F Ne

Ar

Rn

Kr

Xe

ClSPSiAl

Ca

Cs

Fr

K

Fig. 4.15: Location of noble metals in the periodic table marked in green.

The noble metals has a wide free-electron-like band, which is interrupted by a five rather flatbands of the d-electrons. As an example the band structure of silver and its density-of-states isshown in Fig. 4.16. The free electron band actually starts at 8 eV below the Fermi level. From -7 eVto -3 eV the five d-band cross the free-electron band. This is actually not a true crossing, but thefree-electron like band convert at the bottom into d band, while on the top a d-band converts intoa free -electron band. This behavior is a consequence of the non-crossing rule.

The large band width of the free-electron band results in a large Fermi-velocity of the electronsat the Fermi level. This is one explanation for the good conductivity of the noble metals.

Γ X W L Γ K X

-8

-6

-4

-2

0

2

4

6

E[e

v]

-8 -6 -4 -2 0 2 4 6 8E(eV)

Fig. 4.16: Band structure and density of states of silver (Ag). The energy zero is the Fermi level.


4.7.3 Transition metals

Rb

Ba

Ra

Ge

Sn

Tl

Ga

Pb

In

Se

PoBi

Sb

As

Te

At

I

Br

Hg

Cd

Zn

Au

Ag

CuCr Mn Fe Co Ni

Y ZrSr Mo

TaHf

H

La

Nb Tc

ReW Os Ir Pt

Ru PdRh

Ac

He

Li Be

Sc

Mg

Ti

Na

V

CB ON F Ne

Ar

Rn

Kr

Xe

ClSPSiAl

Ca

Cs

Fr

K


The band structure of the transition metals is determined by the narrow d-bands. Below and abovethe the d-states, one finds the free-electron like s-p bands. At the Fermi level, these free-electronbands are not important.

The transition metals form either closed-packed lattices such as fcc and hcp19. However in themiddle of the transition metal series the bcc-lattice is dominant. The origin of this behavior canbe understood from the density of states. The d-density of states in the closed packed lattices isblock-like. In contrast, the density of states of d-electrons in the body-centered lattice consists of anupper and a lower parts with high density of states, which are separated by a region of low density ofstates. This quasi-band gap results in a lowering of the lower half of the states. As a consequencethis structure is favorable when the Fermi-level is in the middle, the quasi band gap.

N Γ H N P Γ-10

-8

-6

-4

-2

0

2

4

6

8

10

E[e

v]

-10 -8 -6 -4 -2 0 2 4E(eV)

Fig. 4.18: Band structure and density of states of iron (Fe). The energy zero is the Fermi level.

2) iron, bcc vs fcc, ferrite vs austenite, magnetism

19fcc denotes the face-centered cubic lattice, hcp denotes the hexagonal closed-packed lattice, and bcc stands forbody-centered lattice.


4.7.4 Diamond, Zinc-blende and wurtzite-type semiconductors

The typical semiconductor materials have the diamond structure with four-fold coordinated atoms ina tetrahedral arrangement. This is the structure of the elemental semiconductors diamond, siliconand germanium.

Ba

Ra

Ge

Sn

Tl

Ga

Pb

In

Se

PoBi

Sb

As

Te

At

I

Br

Hg

Cd

Zn

Au

Ag

CuCr Mn Fe Co Ni

Y ZrSr Mo

TaHf

Ac

La

H

Tc

ReW Os Ir Pt

Ru PdRhNb

He

Li Be

Sc

Mg

Ti

Na

V

CB ON F Ne

Ar

Rn

Kr

Xe

ClSPSiAl

Ca

Cs

Fr

K

Rb


Γ X W L Γ K X-5

0

5

10

-4 -2 0 2 4 6 8 10 12 14E(eV)

Fig. 4.20: Left: band structure of silicon. The valence bands (occupied states) are drawn in blackand the conduction bands (unoccupied states) are drawn in blue. Right: density of states of silicon.the colored areas correspond to the weight of s (blue), p (green) and d (red) electrons. The emptystates (above +8-eV) are drawn in a lighter color. Based on density functional calculations using theCP-PAW code.

Besides the elemental semiconductors which are four-valent, there exists a variety of compoundsemiconductors such as GaAs, which consist, for example out of tri-valent and penta-valent ions.These compounds are named III-V compounds. There are also II-VI compounds such as ZnS andZnSe.

Compound semiconductors such as GaAs distribute the ions onto the lattice sites of he diamondstructure, which results in the zinc-blende structure.

However there is also a hexagonal variation of the diamond structure, the wurtzite lattice. Also inthe Wurtzite lattice, the atoms are tetrahedrally coordinated. The difference to the diamond latticebecomes evident only in the second nearest neighbor shell. Therefore the properties in the cubic andthe hexagonal variants are very similar.


4.7.5 Simple oxides

Oxides are a tremendously important class of materials. They are the major corrosion end productsunder atmospheric conditions, and thus they are the major component of rocks. They also include awide range of functional materials such as the high-Tc superconductors. You look at them when youstare out of the window, you lie on them on the beach, because glass and sand are both the oxideof silicon. The most simple oxides are, as the name says, simple oxides. They are the oxides of themost electropositive elements, for which the cations do not contain transition metals, and for whichthe valence electrons are used up completely to for O−2 anions.

Simple oxides form typically a closed backed lattice of oxygen ions, with cations in the voids,between the anion spheres.

These materials are a good starting point for understanding the more complex oxides. One cansimplify them even further, by making the same trick as for the jellium model: when we smear out thecation cores to a homogenous positive charge background and leave only the electrons to saturatethe oxygen valence shell, we obtain, what I call jellium oxide or jelly-O. The oxygen ions form aclosed packed lattice. The electronic structure of Jelly-O is almost indistinguishable to those of thesimple oxides.

Γ X W L Γ K X-20

-15

-10

-5

0

5

10

E[e

V]

-20 -15 -10 -5 0 5 10E(eV)

Fig. 4.21: Electronic structure of jellium oxide an fcc array of O2− ions with a homogeneous positivecharge background. Left: band structure. The filled bands are black, the empty ones are blue. Right:Density of states. Based on density functional calculations using the CP-PAW code.

The band structure is characterized by a low lying band of oxygen s-character. Above, and wellseparated from the s-band one finds the oxygen p-band which forms the valence shell. In contrastto the semiconductors, the s and p-bands are now well separated. This is, on the one hand, due tothe large s-p splitting of the atom, and on the other hand it is due to the smaller band width of theoxygen p-band caused by the larger distance between oxygen ions.

Very characteristic for the oxygen valence band is the double-peaked structure. It is causedentirely by the hopping between the oxygen ions and is not to be mistaken by a bond to the cations.

The empty conduction band is very similar to a free-electron like band.

4.8 What band structures are good for



4.8.1 Chemical stability

Jahn-Teller distortion, Peierls distortion (polyacetylene, silicon)

Abrupt interfaces

While a dispersion relation is strictly defined only for translationally invariant systems, where theenergy eigenstates are also momentum eigenstates, we can also understand what happens at theinterface between materials. As a particle passes from one material to the other, it may gain or loosemomentum perpendicular to the interface, but we can use energy conservation and conservation ofthe momentum parallel to the interface to obtain some information on the interface crossing.

Lifetime broadening

Sometimes the material has some randomness or other interactions that break translational symmetry.If these perturbations of translational symmetry are small, the eigenstates of the Hamiltonian are nomore momentum eigenstates. If we still project the energy eigenstates onto momentum eigenstates,each momentum will obtain contribution from several energies. Instead of sharp energies for eachmomentum we have a broadened distribution. The width of the energy distribution is a measure forthe life time of a momentum: Scattering processes will redirect the particle.

Chapter 5

Phonons

A recommended reading is chapter 3 of the 1st Volume Theoretical Solid state physics by Jones andMarch.[36]

5.1 Phonons as quasi particles

Just as electrons in a lattice can be described as entities with a given energy and momentum, alsothe vibrations of the atoms about their equilibrium positions can be combined into vibrational modeswith a given momentum ~p = ~~k and energy E(~k) = ~~ω(~k). Thus the vibrations can be characterizedby a dispersion relation, the phonon band structure.

Each vibrational mode has the form of a harmonic oscillator. If the vibrations are quantized, theintensity of the vibrations increases in discrete steps En = ~ω(~k)(n+ 12). Thus the vibrations come inquanta with specific energy and momentum. Therefore one describes these entities as quasiparticles.These quasi particles are named them phonons

The study of phonons provides insight into the ladder of quantization, namely

1. We can consider phonons as classical particles with a given energy and momentum. This ishow one looks at phonons in the Boltzmann equation.

2. We can consider phonons as vibrational modes or distortion fields. This description followswhen one treats the atoms as classical particles. The “distortion fields” are analogous to thewave function of the Schrödinger equation of quantum mechanics. For this reason one oftentalks of the first quantization as a classical field theory.

3. When the quantum mechanical nature of the atoms is considered, we obtain a quantum fieldtheory. This is what is described as second quantization.

The second quantization of electrons is at the center of the course on advanced Solid State Theory.The analogies from moving up and down the ladder of quantization is very useful to understandcomplex many particle physics.

In contrast to electrons, phonons are bosons. They are thus a model for the description of otherquasi particles with integer spin such as photons.

5.2 Connection to Born-Oppenheimer treatment

In the Born-Oppenheimer treatment we arrived at a Schrödinger equation 2.21 for the wave functiondescribing the nuclear motion given on p. 20. It contains the Born-Oppenheimer surfaces En and thederivative couplings ~Am,n.

93

94 5 PHONONS

At sufficiently low temperature, the probability amplitude of the nuclear positions will have a dom-inant contribution at the global minimum of the ground-state Born-Oppenheimer surface. Therefore,we limit the description to the electronic ground state and the global minimum ~R0 of the correspond-ing Born-Oppenheimer surface.

We can do a Taylor expansion of the Born-Oppenheimer surfaces and the vector coupling coeffi-

cients in the displacements δRdef= ~R − ~R0 from this minimum ~R0.

The energy surface obtains the form

EBO0 (~R) = EBO0 (~R0) +

1

2!

(

δ ~R)

C(

δ ~R)

+1

3!

∑

α,β,γ

Wα,β,γ

(

δRα

)(

δRβ

)(

δRγ

)

+O(

(~R − ~R0)4)

(5.1)

where the force constants Cα,β and the coefficients Wα,β,γ are

Cα,βdef=

∂2EBO0∂Rα∂Rβ

∣∣∣∣~R0

Wα,β,γdef=

∂3EBO0∂Rα∂Rβ∂Rγ

∣∣∣∣~R0

(5.2)

The expansion can be carried further to higher orders1.

1. The constant EBO0 (~R0) = 〈ΨBO0 (~R0)|HBO|ΨBO0 (~R0)〉 is the “electronic energy” of the system.

2. The quadratic term of Eq. 5.1 in the displacements corresponds to a harmonic oscillator. Thevibrational modes of this harmonic oscillator are independent of each other and will be attributedto phonons.

3. The higher order terms, proportional to W , will be interpreted in terms of phonon-phononscattering. They describe the interaction between phonons.

Where are the electrons? The electronic system is described by the Born-Oppenheimer Hamilto-nian HBO(~R0). We can include the electronic excitations by adding the energy

∆Ee = 〈Ψ|HBO(~R0)|Ψ〉 − EBO0 (~R0) (5.3)

which depends on the many-electron wave functions |Ψ〉. Note that these wave functions are Born-Oppenheimer wave functions.

Electron-phonon coupling enters in two different ways: Once we can include deviations from theequilibrium positions in the Hamiltonian acting on the electrons2

∆Ee−ph =∑

α

δRα

(

〈Ψ| ∂HBO

∂Rα

∣∣∣∣~R0

|Ψ〉 − ∂EBO0∂Rα

∣∣∣∣~R0

)

(5.4)

In a similar fashion we can also include the derivative couplings. I do not want to proceed further inthat direction, but I felt it important to show how the description ties into our known framework.

5.3 Harmonic oscillator revisited

The harmonic oscillator has the Hamilton function

H(~P , ~R, t) =1

2~PM−1 ~P +

1

2~RC ~R (5.5)

1The expansion of the up to third order is actually problematic, because it produces an energy that is not boundedfrom below. For the sake of simplicity, I will nevertheless not go beyond the third order.

2The derivative of he groundstate energy vanishes of course. I included in order to make the concept evident andto demonstrate how to proceed with higher orders.

5 PHONONS 95

where ~R is a 3M dimensional vector of the atomic positions ~P is the 3M-dimensional vector ofcanonical momenta. M is the mass tensor, which is a diagonal matrix with the atom masses on thediagonal. C is the force-constant matrix formed from the second derivatives of the potential energy.

Newtons equation of motion of the harmonic oscillator

M..~R = −C ~R (5.6)

are obtained from the Hamilton function via the Hamilton’s equation of motion.Below, I will use the action principle, which is why I want to turn to Lagrange functions. The

Lagrange function for the Harmonic oscillator is3

L(~V , ~R, t) = stat~P

[

~P ~V −H(~P , ~R, t)]

=1

2~VM~V − 1

2~RC ~R (5.7)

The action principle yields the Euler-Lagrange equation

d

dt

∂L∂Vα︸︷︷︸

Pα

=∂L∂Rα︸︷︷︸

−∂H/∂Rα

(5.8)

d

dtRα = Vα

︸︷︷︸

∂H/∂Pα

(5.9)

which are equivalent to Hamilton’s equations.4

The Hamilton function is obtained from the Lagrange function as

H(~P , ~R, t) = stat~V

[

~P ~V − L(~V , ~R, t)]

(5.10)

The stationary condition defines the canonical momentum as

Pαdef=

∂L∂Vα

(5.11)

The harmonic oscillator is solved by two variable transformations The first transformation intro-duces mass weighted coordinates

L = 12

(√M

.~R)2

−12

(√M ~R

)( 1√MC1√M

)

︸︷︷︸

D

(√M ~R

)

(5.12)

where

Ddef=1√MC1√M

(5.13)

is the dynamical matrix.The second transformation requires the eigenvectors of the dynamical matrix. The dynamical

matrix is diagonalized yielding eigenvalues dn and eigenvectors ~un

D~Un = ~Undn (5.14)

3I denote with stat~P the stationary point of the following expression against variations of ~P . This condition

Vα = ∂H/∂Pα defines ~P (~V , ~R, t), which is inserted to eliminate the explicit dependence on ~P .

4The last equation us a definition of ~V as the velocity. It is usually suppressed by treating.~R itself as a variable. I

prefer to make this explicit.

96 5 PHONONS

Because the dynamical matrix is hermitian, the eigenvalues are real and the eigenvectors can bechosen orthonormal.

The eigenvectors can be combined in a unitary matrix with matrix elements Ui ,n, which are thecomponents of the eigenvectors ~Un. The orthonormality of the eigenvectors ~U∗m ~Un = δm,n translatesinto the requirement that U is unitary, i.e. U†U = 111.

Thus the dynamical matrix has the form

D = UdU† (5.15)

This yields

L = 12

(

U†√M

.~R)†(U†√M

.~R)

−12

(

U†√M ~R

)∗d(

U†√M ~R

)

=1

2

.~Y† .~Y − 1

2~Y d~Y

=

3M∑

n=1

[1

2

.Y2

n −1

2YndnYn

]

(5.16)

Thus the Lagrangian decomposes into a set of decoupled one-dimensional harmonic oscillators. Thecorresponding coordinate transform is

~Ydef= U†

√M ~R ⇔ ~R =

1√MU ~Y (5.17)

The trajectories are

Yn(t) = Ane−iωnt + A∗ne

iωnt (5.18)

with the circular frequency

ωn = +√

dn (5.19)

and the complex amplitudes An. Thus, the general trajectories of the atomic positions are

~R(t) =∑

n

1√M~UnYn(t) =

∑

n

1√M~Un

(

Ane−iωnt + A∗ne

iωnt)

(5.20)

5.4 Lattice translation symmetry

In the following I will show how to deal with a Lagrangian of the form

L(~V , δ ~R, t) = 12~VM~V − 1

2δ ~RCδ ~R − 1

3!

∑

α,β,γ

Wα,β,γδRαδRβδRγ (5.21)

In a lattice, the goal is to exploit translation symmetry. We will proceed analogously to the descriptionof electrons in a lattice by using Bloch’s theorem.

Lattice vectors, periodic boundary conditions and k-point grid: Let me remind us of the nota-tion: The lattice is characterized by the primitive lattice vectors ~T1, ~T2, ~T3, which can be combinedinto a the matrix T . A general lattice vector is denoted by a lowercase ~t, and is an integer multipleof the primitive lattice vectors.

The primitive reciprocal lattice vectors are denoted as ~g1, ~g2, ~g3, which can be combined into thematrix g. The general reciprocal lattice vectors are denoted by the uppercase symbol ~G and is aninteger multiple of he primitive reciprocal lattice vectors.

5 PHONONS 97

Real-space and reciprocal lattice vectors are related by

~gi ~Tj = 2πδi ,j (5.22)

As for the description of electrons we will in the following introduce periodic boundary conditionson a supercell. Furthermore, instead of summing the energy terms over the infinite lattice, we willperform the sums only over one supercell. As a consequence of the periodic boundary conditions weonly need to consider a discrete lattice of k-points.

We denote the primitive lattice translation vectors of the unit cell by ~T (S)j . The lattice vectors ofthe supercell are multiples of those of the real lattice, i.e.

~T(S)j = ~TjNj for j = 1, 2, 3 (5.23)

The restriction imposed by the periodic boundary conditions can be removed by performing the limitNj →∞.

The reciprocal lattice of the supercell define a grid of k-points. We denote the reciprocal latticevectors of the supercell by ~g(S)j . They are obtained as

~g(S)j = ~gj

1

Njfor j = 1, 2, 3 (5.24)

which is easily shown by

~g(S)i~T(S)j =

1

Ni~gi ~Tj︸︷︷︸

2πδi ,j

Nj = 2πδi ,j (5.25)

A sum over k-points includes all reciprocal lattice vectors of the super cell in the first Brillouinzone ΩG of the reciprocal lattice of the crystal.

~ki ,j,k = ~g(S)1 i + ~g

(S)2 j + ~g

(S)3 k (5.26)

we denote the sum as∑

~k∈ΩG

(5.27)

The sum has N = N1N2N3 terms, that is the number of real space unit cells of the crystal within asupercell.

In the limit of a very large supercell, the sum over k-points becomes an integral.

∑

k∈ΩG=∑

~k

1

N det |G|︸︷︷︸

(∆k)3

N det |T |(2π)3

︸︷︷︸

Ω/(2π)2

︸︷︷︸

=1

= Ω∑

~k

(∆k)3

(2π)3→ Ω

∫

ΩG

d3k

(2π)3(5.28)

where Ω = N det |T | is the size of the supercell for which all extensive quantities are calculated.Instead of the Brillouin zone, which is centered at the Γ-point, we could also have chosen the

first reciprocal crystal unit cell. In that case, for example, the reverted k-point −~k lies outside thefirst unit cell. We exploit the periodicity in the reciprocal lattice and bring it back into the first unitcell by adding an appropriate reciprocal lattice vector which leads to −~k + ~g1 + ~g2 + ~g3.

Notation for displacement vectors The displacements are grouped according to specific unit cellsof the lattice. The unit cell can be characterized by a translation vector ~t which describes its positionrelative to the origin. A specific displacement is specified by an index (i , ~t), of which i characterizes

98 5 PHONONS

a particular atom in the unit cell and a cartesian direction for the corresponding displacement, and~t, which is the lattice translation vector.

The Lagrangian of an anharmonic crystal with N unit cells has the form

L(~V , δ ~R) = 12

∑

α,~t

MαV2α,~t− 12

∑

α,β,~t,~t ′

δRα,~t Cα,~t,β,~t ′ δRβ,~t ′

− 13!

∑

α,β,γ,~t,~t ′, ~t ′′

Wα,~t,β,~t ′,γ,~t ′′ δRα,~t δRβ,~t ′ δRγ,~t ′′ (5.29)

If the lattice has a translational symmetry, we can exploit

Cα,~t,β,~t ′ = Cα,~0,β,~t ′−~tWα,~t,β,~t ′,γ,~t ′′ = Wα,~0,β,~t ′−~t,γ,~t ′′−~t (5.30)

Eigenstates of the lattice translation: The lattice translation operator Sj is defined to shift thedisplacement vector by one primitive lattice vector ~Tj . It transforms the displacement vector δ ~R hav-ing components δRα,~t into the shifted displacement vector δ ~R′ with components δR′

α,~t ′= δRα,~t ′−~Tj .

Let us introduce a notation that is analogous to the bra-ket notation of Dirac for quantum states.Here, we apply the notation to displacement vectors rather than to vectors in Hilbert space. In thisnotation the displacement vector has the form

δRα,~t = 〈α, ~t|δR〉 (5.31)

The lattice translation operator S(~t) then produces

δR′α,~t= 〈α, ~t|Sj |δR〉 = 〈α, ~t − ~Tj |δR〉 = δRα,~t−~Tj (5.32)

A closed expression for the lattice translation operator is

S =∑

α,~t

|α, ~t〉〈α, ~t − ~Tj | (5.33)

A simultaneous eigenvector of the translation along the three basic lattice translation vectors~T1, ~T2, ~T3 obeys 5 (see also Eq. 10.1 of [27])

Sj |δR〉 = |δR〉e−i~k ~Tj (5.34)

⇒ δRα,~t−~TjEq. 5.32= 〈α, ~t|Sj |δR〉 Eq. 5.34

= 〈α, ~t|δR〉e−i~k ~Tj = δRα,~te−i~k ~Tj for j = 1, 2, 3

⇒ δRα,~t = δRα,~t−~Tj e+i~k ~Tj

⇒ δRα,~te−ik~t = δRα,~t−~Tj e

−i~k(~t−~Tj ) = δRα,0e−i~k~0

⇒ δRα,~t = δRα,~0ei~k~t (5.35)

Hence, an eigenstate of the lattice translation vector has the form

δRα,~t = δRα,~0ei~k~t (5.36)

Thus the eigenstates of the lattice translation operator have the form of a Bloch state, namely a aproduct of a displacement vector with lattice periodicity and a phase factor.

5In order to bring the eigenvalues into this form we exploited that the eigenvalue of a unitary matrix is complex andhas the absolute value 1. Thus the eigenvalues can be written eiφj wit real phases φj . From ~k ~Tj = φj we can obtaindirectly the wave vector ~k = ~φT−1.

5 PHONONS 99

BLOCH THEOREM FOR LATTICE VIBRATIONS

The eigenmodes of a lattice are eigenstates of the lattice translation. Thus, they have the form givenin Eq. 5.36, namelya

δRα,~t =1√Mα

Uαei~k~t (5.37)

which is the product of a displacement vector Uα, which is periodic with the lattice, and a phasefactorei~k~t .

This theorem is analogous to the Bloch theorem for electrons Eq. 4.22 given on p. 59 and to Eq. 4.31on p. 65.

aThe mass weighting at this point is not essential but has been included for the sake of consistency of symbols.

Block diagonalization of the dynamical matrix: Let us now consider the Lagrangian in terms ofEigenstates of the translation operator

δRα,t(t) =∑

k∈ΩG ,n

1√Mα

Uα,n(~k)ei~k~tQn(~k, t) (5.38)

The sum is performed over all k-points on the grid define above as the reciprocal lattice vectors ofthe supercell. We already did the mass weighting. The vectors ~Un(~k) remain undetermined at themoment except that they shall be orthonormal

~U∗m(~k)~Un(~k) = δm,n (5.39)

and complete within the space of distortions in one unit cell

EharmEq. 5.29=1

2

∑

~t,~t ′

δ ~R~tC~t,~t ′δ~R~t ′

Eq. 5.38=1

2

∑

~t,~t ′

∑

~k,~k ′

∑

m,n

Q∗m(~k)e−i~k~t ~U∗m(~k)

1√MC~t,~t ′︸︷︷︸

C~0,~t′−~t

1√M~Un(~k)e

i ~k ′~t ′Qn(~k)

=1

2

∑

~k,~k ′

[∑

~t

ei(~k ′−~k)~t

]

︸︷︷︸

N δ~k, ~k ′

∑

m,n

Q∗m(~k)~U∗m(~k)

[∑

~t ′

1√MC~0,~t ′−~t

1√Mei~k ′(~t ′−~t)

]

︸︷︷︸

D(~k)

~Un(~k)Qn(~k)

=1

2N∑

~k∈ΩG

∑

m,n

Q∗m(~k)~U∗m(~k)D(~k)~Un(~k)Qn(~k) (5.40)

where N = N1N2N3.We observe exactly what we expected, namely that the contributions from different wave vec-

tors are completely independent, and that the harmonic energy can be expressed as a sum overcontributions from k-points.

We introduced here the k-dependent dynamical matrix6

Dα,β(~k)def=∑

~t

Dα,~0,β,~tei~k~t (5.41)

6Compare the equation for the k-dependent dynamical matrix with the one, Eq. 4.32 for the k-dependent Hamiltonmatrix.

100 5 PHONONS

We diagonalize the k-dependent dynamical matrix7, which then determines the eigenvalues dnand the eigenvectors ~Un, i.e.

D(~k)~Un(~k) = ~Un(~k)dn(~k) (5.43)

The eigenvalues dn(~k) are identified with the squared frequencies.8

Thus, we obtain a dispersion relation for phonons

ωn(~k) =

√

dn(~k) (5.44)

With these definitions, the harmonic potential energy obtains the simple form

EharmEq. 5.40=1

2N

∑

~k∈ΩG ,nω2n(~k)Q

∗n(~k)Qn(~k) (5.45)

As shown below, also the kinetic energy will be a sum over the states introduced above, whichshows that the eigenmodes of the harmonic problem have the form of Bloch waves.

LATTICE VIBRATIONS IN TERMS OF NORMAL COORDINATES

The displacements can be expressed in terms of lattice eigenmodes and the normal coordinatesa

Qn(~k, t) as

δRα,~t(t) =∑

~k

∑

n

1√Mα

Uα,n(~k)ei~k~tQn(~k, t)

⇔ Qn(~k, t) =1

N∑

α,~t

e−i~k~tU∗α,n(~k)δRα,~t (5.46)

This is a complete Ansatz for the displacements and therefore no approximation. The eigenmodesand frequencies are, nevertheless, defined solely through the harmonic terms.

aCompare Eq. 3.2.10 of Jones and March[36]. Our definitions for the normal cordinates differ by a factor√N .

7The eigenvectors are orthonormal and the eigenvalues are real because the k-dependent dynamical matrix ishermitian. The latter property is shown as follows:

D(~k)Eq. 5.41=

∑

~t

D~0,~tei~k~t =

(∑

~t

D~t,~0e−i~k~t

)†=(∑

~t

D~0,−~te−i~k~t

)†=(∑

~t

D~0,~tei~k~t)† Eq. 5.41

= D†(~k) (5.42)

where we used translational invariance and that the dynamical matrix is symmetric and real, i.e. Dα,~t,β,~t ′ = Dβ,~t ′,α,~t =

D∗β,~t ′,α,~t

.8This identification rests on the equations of motion derived below.

5 PHONONS 101

Lagrangian in Bloch waves: For the sake of completeness let us calculate the kinetic energy andthe anharmonic terms in terms of normal coordinates Qn(~k, t).

EkinEq. 5.29=1

2

∑

~t,~t ′

δ.~R~tM~t,~t ′δ

.~R~t ′

Eq. 5.38=1

2

∑

~t,~t ′

∑

~k,~k ′

∑

m,n

.Q∗m(~k)e

−i~k~t ~U∗m(~k)1√MMδ~t,~t ′

1√M

︸︷︷︸

111δ~t,~t′

~Un(~k)ei ~k ′~t ′

.Qn(~k)

=1

2

∑

~k,~k ′

[∑

~t

ei(~k ′−~k)~t

]

︸︷︷︸

N δ~k, ~k ′

∑

m,n

.Q∗m(~k) ~U

∗m(~k)~Un(~k)︸︷︷︸

δm,n

.Qn(~k)

=1

2N

∑

~k∈ΩG ,n

.Q∗n(~k)

.Qn(~k) (5.47)

EanharmEq. 5.29=

1

3!

∑

~t,~t ′, ~t ′′

∑

α,β,γ

Wα,~t,β,~t ′,γ,~t ′δRα,~tδRγ,~t ′δRbeta,~t ′′

Eq. 5.38=

1

3!

∑

~t,~t ′, ~t ′′

∑

α,β,γ

Wα,~t,β,~t,γ,~t︸︷︷︸

=Wα,~0,β,~t′−~t,γ, ~t′′−~t

∑

~k,~k ′, ~k ′′∈ΩG

∑

l ,m,n

1√Mα

Uα,l(~k)ei~k~tQl(~k)

1√Mβ

Uβ,m(~k′)ei

~k ′~t ′Qm(~k ′)1

√Mγ

Uγ,n(~k′′)ei

~k ′′ ~t ′′Qn( ~k ′′)

=1

3!

∑

~k,~k ′, ~k ′′∈ΩG

∑

l ,m,n

∑

~t

ei(~k+~k ′+ ~k ′′)~t

︸︷︷︸

N ∑~G δ~k+~k ′+ ~k ′′−~G

[∑

α,β,γ

[∑

~t ′, ~t ′′

Wα,~0,β,~t ′−~t,γ,~t ′′−~t√MαMβMγ

ei~k ′(~t ′−~t)ei

~k ′′(~t ′′−~t)]

×Uα,l(~k)Uβ,m(~k ′)Uγ,n(~k ′′)]

Ql(~k)Qm(~k ′)Qn( ~k ′′)

=1

3!N

∑

~k,~k ′, ~k ′′∈ΩG

∑

l ,m,n

Xl ,m,n

(

~k ′, ~k ′′)

Ql(~k)Qm(~k ′)Qn( ~k ′′) (5.48)

with

ANHARMONIC SCATTERING TERMS

a

Xl ,m,n(~k, ~k ′, ~k ′′)def=

∑

~G

δ~k+~k ′+ ~k ′′−~G

×∑

α,β,γ

[∑

~t ′, ~t ′′

Wα,~0,β,~t ′,γ,~t ′′√MαMβMγ

ei~k ′~t ′ei

~k ′′ ~t ′′]

Uα,l(~k)Uβ,m(~k ′)Uγ,n(~k′′) (5.49)

acompare with Eq. (3.13.17) of Jones and March, Vol.1[36].

The first term in Eq. 5.49 expresses momentum conservation, which is described in detail in sec-tion 7.2.1 on p. 126.

102 5 PHONONS

From Eqs. 5.47, 5.45, and 5.48, we obtain the Lagrangian

L = N∑

~k∈ΩG ,n

(1

2

.Q∗n(~k)

.Qn(~k)−

1

2ω2n(~k)Q

∗n(~k)Qn~k)

)

−∑

~k,~k ′, ~k ′′∈ΩG

∑

l ,m,n

1

3!Xl ,m,n

(

~k, ~k ′, ~k ′′)

Ql(~k)Qm(~k ′)Qn( ~k ′′)

(5.50)

Real displacements: The Lagrangian Eq. 5.55 does not seem to be purely real. The real property,however, is hidden in the requirement that he distortions are purely real.

The ansatz Eq. 5.38

δRα,t(t)Eq. 5.38=

∑

~k∈ΩG ,n

1√Mα

Uα,n(~k)ei~k~tQn(~k, t) (5.51)

can be made real by requiring

Qn(~k, t) = Q∗n(−~k, t) (5.52)

Uα,n(~k) = U∗α,n(−~k) (5.53)

The second relation follows from the fact that the dynamical matrix Dα,~t,β,~t ′ is real. A conse-quence of this property is that inverting the wave vector in the k-dependent dynamical matrix is thesame as taking the complex conjugate of it.

D(~k)Eq. 5.41=

∑

~t

D~0,~tei~k~t =

(∑

~t

D~0,~te−i~k~t

)∗= D∗(−~k) (5.54)

This property directly produces complex conjugate eigenvectors for the inverted wave vector.We use the relation between the amplitudes with opposite wave vector to express the Lagrangian

without referring to the complex conjugates.

L Eq. 5.50= N

∑

~k,n

(1

2

.Qn(−~k)

.Qn(~k)−

1

2ω2n(~k)Qn(−~k)Qn~k)

)

−∑

~k,~k ′, ~k ′′∈Ωg

∑

l ,m,n

1

3!Xl ,m,n

(

~k, ~k ′, ~k ′′)

Ql(~k)Qm(~k ′)Qn( ~k ′′)

(5.55)

Euler-Lagrange equations: The Euler-Lagrange equations are obtained as

0 =d

dt

∂L∂.Qn(−~k)

− ∂L∂Qn(−~k)

Eq. 5.55= N

..Qn(~k) + ω

2n(~k)Qn(~k) +

∑

~k ′, ~k ′′∈ΩG

∑

l ,m

1

2Xn,l,m

(

−~k, ~k ′, ~k ′′)

Ql(~k ′)Qm( ~k ′′)

(5.56)

Editor: Previously, there has been a sign error: the first wave vector in X carries

now a minus sign.

This is now the starting point of the discussion of lattice vibrations.

5 PHONONS 103

5.5 Phonons

Let us ignore the scattering term in Eq. 5.55 for a moment and just consider the harmonic contributionin the Lagrangian

Lharm = N∑

~k,n

1

2

( .Qn(−~k)

.Qn(~k)− ω2n(~k)Qn(−~k)Qn~k)

)

(5.57)

The Euler-Lagrange equations are

..Qn(~k) = −ω2n(~k)Qn(~k) (5.58)

We obtain the solution

Qn(~k, t) = An(~k)e−iωn(~k)t + Bn(~k)e

iωn(~k)t (5.59)

5.5.1 Left and right-moving waves

The distortions resulting from Eq. 5.59 are composed of two terms, one of which has a phase velocityparallel to the wave vector, and a second term with a phase velocity in the opposite direction.

We divide the displacements Eq. 5.38 into left and right-moving partial waves

δRα,~tEqs. 5.59=

∑

~k∈ΩG ,n

1√Mα

Uα,n(~k)ei [~k~t−ωn(~k)t]An(~k) +

∑

~k

∑

n

1√Mα

Uα,n(~k)︸︷︷︸

U∗α,n(−~k)

ei [~k~t+ωn(~k)t]

︸︷︷︸(

ei [−~k~t−ωn(~k)t])∗

Bn(~k)

︸︷︷︸(

Uα,n(−~k)ei [−~k~t−ωn(~k)t]B∗n(~k))∗

=∑

~k

∑

n

1√Mα

Uα,n(~k)ei [~k~t−ωn(~k)t]An(~k) +

∑

~k

∑

n

( 1√Mα

Uα,n(~k)ei [~k~t−ωn(~k)t]B∗n(−~k)

)∗

(5.60)

We have rearranged the terms so that each wave vector contains the contributions propagating witha phase velocity in the direction of the wave vector.

Now it is straightforward to impose the requirement that the distortions are real, namely by

An(~k) = B∗n(−~k) (5.61)

Thus the trajectories are

δRα,~t = 2Re

[∑

~k

∑

n

1√Mα

Uα,n(~k)ei [~k~t−ωn(~k)t]An(~k)

]

(5.62)

5.5.2 From normal coordinates to phonon amplitudes

Eq. 5.62 gives us the trajectories for a known set of coefficients An(~k). Now we approach the inverseproblem, namely to calculate the coefficients from a given trajectory.

This will be important to decompose the trajectories into its components after a scattering event.We start from any trajectory expressed in normal coordinates Qn(~k, t) and decompose it into

traveling plane waves.

Qn(~k, t) = An(~k, t)e−iωn(~k)t + Bn(~k)e

iωn(~k)t

.Qn(~k, t) = −iωn(~k)An(~k, t)e−iωn(~k)t + iωn(~k)Bn(~k, t)eiωn(~k)t (5.63)

104 5 PHONONS

where Bn(~k, t) = A∗n(−~k, t). The time argument for the coefficients refers to the time at which themodes are matched to the trajectory.

The problem can be inverted to obtain the coeffciients

(

Qn(~k, t).Qn(~k, t)

)

=

(

e−iωn(~k)t eiωn(~k)t

−iωn(~k)e−iωn(~k)t iωn(~k)eiωn(~k)t

)(

An(~k, t)

Bn(~k, t)

)

⇒(

An(~k, t)

Bn(~k, t)

)

=1

2iωn(~k)

(

iωn(~k)eiωn(~k)t −eiωn(~k)t

iωn(~k)e−iωn(~k)t e−iωn(~k)t

)(

Qn(~k, t).Qn(~k, t)

)

=1

2

(

eiωn(~k)t −eiωn(~k)t

e−iωn(~k)t e−iωn(~k)t

)(

Qn(~k, t)1

iωn(~k)

.Qn(~k, t)

)

(5.64)

Thus we can extract the coefficients via

An(~k, t) = eiωn(~k)t

1

2

(

1− 1

iωn(~k)∂t

)

Qn(~k) (5.65)

Bn(~k, t) = e−iωn(~k)t 1

2

(

1 +1

iωn(~k)∂t

)

Qn(~k) (5.66)

Harmonic oscillator: Let us consider first the algebraic treatment of the quantum harmonic oscilla-tor. The standard approach is to introduce creation and annihilation operators out of momentum andposition operators. Here we do the analogous transformation, but on the level of classical mechanics.

We start from the Hamilton function of the harmonic oscillator

H(p, x) =p2

2m+1

2mω20x

2 =1

2m

(

p + imωx)(

p − imωx)

= ~ω0b+b− (5.67)

where

b−def= eiω0t

1√2m~ω0

(

ip +mω0x)p=m

.x= eiω0t

1√2m~ω0

(

im.x +mω0x

)

= eiω0t√

m

2~ω0

(

ω0 + i∂t

)

x(t)

b+def= e−iω0t

1√2m~ω0

(

−ip +mω0x)

= e−iω0t√

m

2~ω0

(

ω0 − i∂t)

x(t) (5.68)

The requirement that the energy is real is imposed by

b+ = b∗− (5.69)

b±(t) are constant of motion of the harmonic oscillator. This can be verified by evaluating it for

5 PHONONS 105

a general trajectory.

b−(t) = eiω0t

√m

2~ω0

(

ω0 + i∂t

)(

Aeiω0t + Be−iω0t)

︸︷︷︸

x(t)

= eiω0t√

m

2~ω0

(

A(ω0 − ω0)eiω0t + (ω0 + ω0)Be−iω0t)

=1

~

√

2m~ω0B

b+(t) = e−iω0t

√m

2~ω0

(

ω0 − i∂t)(

Aeiω0t + Be−iω0t)

︸︷︷︸

x(t)

= e−iω0t√

m

2~ω0

(

A(ω0 + ω0)eiω0t + (ω0 − ω0)Be−iω0t

)

=1

~

√

2m~ω0A (5.70)

The variables b±(t) pick out the coefficients. Thus, the trajectory has the form

x(t) =~√2m~ω0

(

b+(t)eiω0t + b−(t)e

−iω0t)

(5.71)

The introduction of the coordinates b±(t) is first and foremost a variable transform from x(t) tob±(t). While the transform has been motivated by a specific physical system, the harmonic oscillator,it can be used irrespective of whether the Hamilton function describes a harmonic oscillator of includeshigher order terms in x . Beyond this, the new variables have the special property that the are constantsof motion for the harmonic system.

The occurrence of ~ does not indicate any quantum effects. We have introduced it merely toshow the analogy to quantum mechanics. Our treatment of phonons is, so far, completely classical.

Back to phonons: We can use the same procedure for the phonons by introducing

bn,−(~k, t)def=1

~

√

2N~ωn(~k)An(~k, t) (5.72)

Eq. 5.65= eiωn(k)t

√

2N~ωn(~k)

1

2

(

ωn(~k) + i∂t

)

Qn(~k, t)

bn,+(~k, t)def=1

~

√

2N~ωn(~k)Bn(~k, t)

Eq. 5.66= e−iωn(k)t

√

2N~ωn(~k)

1

2

(

ωn(~k)− i∂t)

Qn(~k, t) (5.73)

For later reference, let me mention that the amplitude bn,−(~k) will turn into the annihilation operatorbn(~k), while the reaction operator b†n(k) corresponds to b∗n,−(~k) = bn,+(−~k).

106 5 PHONONS

PHONON AMPLITUDES

The phonon amplitudes are a

bn,σ(~k) = e−iσωn(~k)t

√

N2~ωn(~k)

(

ωn(~k)Qn(~k)− iσ.Qn(~k)

)

(5.74)

Qn(~k) =~

√

2N~ωn(~k)

∑

σ∈±bn,σ(~k)e

iσωn(~k)t (5.75)

acompare with Eq. 11 of Peierls[37].

For the derivation of the back transformation Eq. 5.75, see below.9

Trajectories: The displacement vectors for a given set of phonon amplitudes are then obtainedusing Eq. 5.62 and Eq. 5.72.

δRα,~t(t)Eq. 5.46=

∑

~k

∑

n

1√Mα

Uα,n(~k)ei~k~tQn(~k, t)

Eq. 5.75=

∑

~k

∑

n

1√Mα

Uα,n(~k)ei~k~t

[~

√

2N~ωn(~k)

∑

n

∑

σ∈±bn,σ(~k)e

iσωn(~k)t]

]

=∑

~k

∑

n

1√Mα

Uα,n(~k)

[~

√

2N~ωn(~k)

∑

σ∈±bn,σ(~k)e

i [~k~t+σωn(~k)t]

]

(5.77)

Each phonon amplitude is the prefactor of a well-defined displacement wave moving (with phasevelocity) into the direction of the wave vector if σ = − and opposite to the wave vector for σ = +.

The condition that the trajectories are real imposes bn,+(~k) = b∗n,−(−~k), so that the trajectorycan be expressed with the help of Eq. 5.53 as

δRα,~t(t) = 2Re

∑

~k

∑

n

1√Mα

Uα,n(~k)

[~

√

2N~ωn(~k)bn,−(~k)e

i [~k~t−ωn(~k)t]]

(5.78)

In this way we recognize again that the general solution consists of wave with a phase velocity in thedirection of the wave vector.

It is noteworthy that the ratio between displacements δRα,t and phonon amplitudes bn,σ(~k)depends on the frequency. This is a result of the convention used in quantum mechanics.

9The back transformation Eq. 5.75 is obtained as follows

b∗n(−~k)Eq. 5.74= e−iωn(~k)t

√

N2~ωn(~k)

(

ωn(~k)Q∗n(−~k)− i

.Q∗n(−~k)

)

Eq. 5.52= e−iωn(~k)t

√

N2~ωn(~k)

(

ωn(~k)Qn(~k)− i.Qn(~k)

)

⇒ bn(~k)e−iωn(~k)t + b∗n(−~k)eiωn(

~k)t =

√

N2~ωn(~k)

2ωn(~k)Qn(~k)

⇒ Qn(~k) =~

√

2N~ωn(~k)

(

bn(~k)e−iωn(~k)t + b∗n(−~k)eiωn(

~k)t)

(5.76)

5 PHONONS 107

Phonon energy:

~ωn(~k)b∗n,−(~k)bn,−(~k)

Eq. 5.74= ~ωn(~k)

[

e−iωn(k)t√

N2~ωn(~k)

(

ωn(~k)Q∗n(~k)− i

.Q∗n(~k)

)]

×[

e+iωn(k)t

√

N2~ωn(~k)

(

ωn(~k)Qn(~k) + i.Qn(~k)

)]

=N2

(

ωn(~k)Qn(−~k)− i.Qn(−~k)

)(

ωn(~k)Qn(~k) + i.Qn(~k)

)

=N2

[ .Qn(−~k)

.Qn(~k) + ω

2n(~k)Qn(−~k)Qn(~k)

−iωn(~k)( .Qn(−~k)Qn(~k)−Qn(−~k)

.Qn(~k)

)]

(5.79)

When we perform the k-sum, the last term drops out because it changes sign upon inverting thewave vector.

Thus we obtain the total energy of the harmonic system as a sum of phonon contributions.

Eharmtot =∑

~k,n

∂Lharm

∂.Qn(~k)

.Qn(~k)− Lharm

Eq. 5.57= N

∑

~k,n

1

2

[ .Qn(−~k)

.Qn(~k) + ω

2n(~k)Qn(−~k)Qn(~k)

]

Eq. 5.79=

∑

~k,n

~ωn(~k)b∗n,−(~k)bn,−(~k) (5.80)

This is the total energy for non-interacting phonons in a rigorously classical treatment. Note thatthis term includes kinetic and potential energy for the harmonic system.

Eq. 5.80 looks like a quantum expression

E =∑

~k,~n

~ωn(~k)(

b†n(~k)bn(~k) +1

2

)

(5.81)

whch is that of a harmonic oscillator for each phonon. In the classical limit the spacing of the energylevels and the zero-point energy vanish. This is exactly what we see in the above result which ispurely classical. Note that the factor ~ is inconsequential, as it is absorbed by a factor ~−

12 in the

definition of the variables bn(~k).

In analogy with the quantum expression we can identify nn(~k)def= b∗n,−(~k)bn,−(~k) with the number

of bosons. In the classical theory we consider here, the number of bosons is a continous number.The concept of individual particles is a result of quantization and emerges only in the quantum theory.

5.6 Phonon band structures

5.6.1 Example for a phonon band structure

In Fig. 5.1 the phonon band structures of gold and silicon are shown.

5.7 Acoustic branches

Acoustic and optical branches All phononic band structures have three branches with a frequencythat vanishes at the Γ-point, i.e. for ~k = 0, and that, for small enough wave vectors, grows

108 5 PHONONS

Fig. 5.1: Top: Phonon band structure and density of states of gold.[38]. Bottom: Phonon bandstructure and density of states of silicon[39] (Copyright Physical Review B)

approximately linearly with increasing wave vectors. They are called acoustic branches of the phononband structure. The other branches have a finite frequency at the Gamma point and are called opticbranches of the phonon band structure.

The number of acoustic branches is identical to the dimension of the of the system.The acoustic branch result from the fact that a global translations of the entire lattice does not

cost any energy. Such a translation could be described an equal displacement of all atoms into anyspatial direction. Thus there are three eigenvector of the dynamical matrix at the Γ-point, i.e. at~k = ~0, with zero frequency.

It is important to exclude these modes from the sum, because they are not vibrations and thusmust not be represented by anharmonic oscillators. Thermodynamically these modes would be treatedas the translational motion of the macroscopic crystal.

Longitudinal and transversal acoustic branches: The three acoustic modes can be classified10 aslongitudinal and transversal acoustic waves. For longitudinal waves, the displacement vector pointsparallel to the wave vector, while for transversal waves, the displacement is orthogonal to the wavevector. The transversal wave is a shear wave.

Sound: In the long-wavelength limit, the distortions of the material can be described by the theoryof elasticity. The elastic distortions are described by a displacement field ~u(~r), which gives thedisplacement of an atom with equilibrium position at ~r .

The frequency spectrum of the acoustic modes near the Γ point can be approximated

ωn(~k) =

√

~kAn~k (5.82)

where the index n refers to a particular acoustic branch, and the matrix A is a second derivative ofthe corresponding eigenvalue dn(~k) of the dynamical matrix.

10given sufficiently high symmetry

5 PHONONS 109

Propagating elastic waves are called sound waves. Their velocity is

~vn = An~k

ωn(~k)(5.83)

Each acoustic branch has its own speed of sound. The speed of sound of the transversal modes isoften a factor two smaller than that of the longitudinal modes.

Editor: Mention S (shear/secondary) and P-waves (pressure/primary). include drawing

of the distortion patterns.

Elasticity: Editor: This needs to be done

5.8 Optic branches of the phonon band structure

Reststrahl absorption

(see Optical Properties of Ions in Solids, by Baldassare Di Bartolo)An example for the interaction of quasi-particles is the conversion of light into vibrations. The

absorption of light into phonons in a one-phonon process is called reststrahl absorption.This process can be investigated by superimposing the one-particle band structures of phonons

with those of light. Where the band structures of photons and phonons intersect, a conversionsatisfies energy and momentum conservation.

The speed of light is 3×108 m/s, while the speed of sound is of the order of 103−104m/s, whichis about 105 times smaller than the speed of light. Because of the large speed of light, the photonbands are very steep, so that they intersect the phonon band structure very close to the Γ-point.This tells us that the reststrahl absorption takes place at the energy of the optical phonons.

In addition to energy and momentum conservation the transformation requires a non-vanishingmatrix element. In this case it is a electrostatic dipole, which can couple to the electric field of thephoton. The dipole is due to the displacement of positive and negative ions. (This implies that thereststrahl absorption for non-ionic materials will be small.) Because the electric field of the light waveis perpendicular to the wave vector, is will couple to the transversal optical phonons with frequencyωTO.

In the crystal the photon and phonon bands will create an avoided crossing. That is, near thecrossing the light will travel as a polariton, which is a mixed photon-phonon quasi particle.

5.9 Thermodynamics of phonons

5.9.1 Thermodynamics: Models for the density of states

As shown for the thermodynamic properties of non-interacting Fermions one can extract analogouslythe thermodynamic properties of non-interacting phonons from a given density of states.

Einstein Model for the density of states:

The most simple model for the density of states is the Einstein Model, which replaces the phononicdensity of states by delta function at the so-called Einstein frequency ωE .

The basic approximation of the Einstein model, is that the dynamical matrix is replaced by themean diagonal element of the dynamical matrix. Thus the Einstein frequency is the

ωE =1∑

α

∑

α

√

Dα,~0,α,~0 (5.84)

110 5 PHONONS

The density of states of the Einstein model is

DE(ǫ) = Ω

∑

α

det |T |δ(ǫ− ~ωE) (5.85)

where∑

α is the number of degrees of freedom in one unit cell and Ω is the volume of the supercell.We used here that that the number of modes is N∑α and the N = Ω/ det |T |.

Debye model for the density of states

The Debye model uses a model of a linear dispersion relation

ωn(~k) = c |~k | for n = 1, 2, 3 (5.86)

which is simply cut off[36] beyond a so-called Debye frequency ωD. The Debye frequency can beroughly identified with the highest frequency in the phonon dispersion. Thus, it is large for a materialsuch as diamond, which has strong bonds and light atoms, while it is very low for a material like gold,with heavy atoms and relatively weak metallic bonds.

The Debye model considers only one speed of sound c . The different speed of sound betweenthe longitudinal and the transversal modes is not accounted for. The linear dispersion relation canbe considered as a consequence of assuming one group velocity throughout the Brillouin zone.

A second implicit assumption of the Debye model is that it treats all atoms as identical. Therefore,there are only three branches of the phonon band structure Eq. 5.86 in the extended zone scheme.

The density of states is obtained as derivative of the number of states function. We consider threebranches of the phonon band structure, one for each polarization, each having a slope determined bythe speed of sound c .

• For energies ǫ = ~ω below the Debye energy ~ωD the particle number is∫ ǫ

0

dǫ′D(ǫ′) = 3Ω

∫d3k

(2π)3θ(~c |~k | − ǫ) = 3Ω 1

(2π)34π

3

( ǫ

~c

)3

(5.87)

where Ω is the volume of the supercell. The volumetric phonon density is N/Ω.

• Above the Debye energy, there are no further states, so that the number of states is constant.On the one hand we can calculate the number of states by integrating the density of statesEq. 5.87 up to the Debye energy.

∫ ǫ

0

dǫ′D(ǫ′) = 3Ω1

(2π)34π

3

(~ωD~c

)3

(5.88)

On the other hand, the total number of states is the total number of degrees of freedom, thatis three times the number of atoms. For M atoms in the unit cell of the crystal, the volume

density of atoms is natdef= Mdet |T | .

∫ ǫ

0

dǫ′D(ǫ′) = 3ΩM

det |T | = 3natΩ (5.89)

The density of states D(ǫ) of the Debye model is

DD(ǫ) =12πΩ

(2π~c)3ǫ2θ(~ωD − ǫ) = Ω

9nat~ωD

(ǫ

~ωD

)2

θ(~ωD − ǫ) (5.90)

where nat = M/ det |T | is the number of atoms per volume.We provide here the density of states defined as number of states per unit energy interval for the

sake of consistency with the description of Fermions. In the present case it may seem more naturalto use the number of states per unit frequency interval.

5 PHONONS 111

Debye temperature: The Debye frequency is often expressed[36] by the Debye temperature viathe relation

kBTD = ~ωD (5.91)

Thus the density of states of the Debye model is, in terms of the Debye temperature,

DD(ǫ) = Ω9natkBTD

(ǫ

kBTD

)2

θ(kBTD − ǫ) (5.92)

5.9.2 Grand potential

In order to extract thermodynamic properties we start with the partition function and the grandpotential. In constract to the previous case of electrons we need to consider here bosons.

In constract to the previous discussion, we switch now to the quantumm mechanical description.This implies that we add the zero point energy and discretize the energy levels.

The energy of the phonons is

Eph[nλ] =∑

λ

~ωλ(nλ +1

2) (5.93)

where nλ is the number of excitations of the phonon λ λ combines the wave vector and the bandindex, i.e. ωλ = ωn(~k). Each eigenstate of the phonon Hamiltonian can be characterized by a vectorof occupation numbers ~n with components nλ.

The partition sum is

Z(T, µ) =∑

~n

e−β[E(~n)−µN(~n)] (5.94)

The total number of excitations is N(~n) =∑

λ nλ

Because phonons do not have particle conservation there is no such thing as a phonon reservoir.Therefore the chemical potential is set to zero.

Z(T, µ) =∏

λ

∞∑

nλ=0

e−β(ǫλ−µ)nλ =∏

λ

1

1− e−β(ǫλ−µ) (5.95)

Thus we obtain the grand potential as

Ω(T, µ) = −kBT ln[Z(T, µ)] =∑

λ

kBT ln[1− e−β(ǫλ−µ)] (5.96)

5.9.3 Internal energy

U(T, µ) =

∫ ∞

0

dǫ D(ǫ)ǫb(ǫ) (5.97)

where

bT,µ =1

eβ(ǫ−µ) − 1 (5.98)

is the Bose distribution.

112 5 PHONONS

Chapter 6

Boltzmann equation

6.1 Demonstration of the principle in real space

Consider a particle-conservation law1 of the form

∂tρ(~r , t) + ~∇~j(~r , t) = 0 (6.1)

where ρ(~r , t) is the particle density and ~j(~r , t) is the particle-current density.The current of particles, which follow a given velocity field ~v(~r , t), is given by

~j(~r , t) = ρ(~r , t)~v(~r , t) (6.2)

which brings the particle conservation law into the form

∂tρ(~r , t) + ~v ~∇ρ+ ρ(

~∇~v)

= 0 (6.3)

If we add another assumption, namely that the flow ~v(~r , t) is incompressible, that is ~∇~v = 0we obtain

∂tρ(~r , t) + ~v ~∇ρ = 0 (6.4)

A violation of the particle number can be described by sources and sinks. A source producesparticles, while a sink removes them.

∂tρ(~r , t) + ~∇~j(~r , t) = S(~r , t) (6.5)

where S(~r , t) is the source density, to which sources and sinks contribute with opposite sign.

6.2 From coordinate space to phase space

The state of particles is characterized by position and momentum. The dispersion relation ǫ(~p) canbe identified with the Hamilton function of the system. The Hamilton function provides a “local bandstructure”, which is allowed to vary in space. The dynamics of particles is determined by Hamilton’sequation.

We have seen that the position and wave vector of wave packets obey similar equations of motion,which are related to Hamilton’s equation through the correspondence principle 2 ǫ = ~ω and ~p = ~~k .

1This is the general form for a conservation law.2The correspondence principle is a general property of wave packets an is not limited to quantum systems. While

we have chosen Planck’s constant ~ as proportionality constant, any other value would be equally justified.

113

114 6 BOLTZMANN EQUATION

Because the dynamics of particles and waves is expressed by Hamilton’s equations can can bedescribed in phase space, let me translate Eq. 6.5 into phase space. The dispersion relation ǫ(~p),respectively ω(~k) classifies the states by its momentum or wave vector, and it provides the energyas function of momentum.

3

The particles or wave packets are now characterized by position and momentum. The velocityfield in phase space is given by the Hamilton’s equations of motion and can be evaliated from aHamilton function.

( .~r.~p

)

=

(~∇pH(~p,~r)−~∇rH(~p,~r)

)

=:

(

~v(~r , ~p)~F (~r , ~p)

)

(6.8)

The first equation determines the velocity field ~v(~r , ~p) and the second the force field ~F (~r , ~p).Liouville’s’s theorem says that the flow in phase space is incompressible. It is proven by expressing

the divergence of the flow in phase space by the Hamilton function.

(~∇r~∇p

)(~∇pH(~p,~r)−~∇rH(~p,~r)

)

︸︷︷︸

divergence of phase-space flow

= ~∇r ~∇pH − ~∇p ~∇rH = 0 (6.9)

Because Hamiltons equation provide us with a velocity field in phase space, which furthermore isincompressible, we can to translate Eq. 6.5 into phase space. The density ρ(~r , t) translates into thephase-space density f (~r , ~p, t). Its equation of motion is

∂t f (~r , ~p, t) + ~v(~r , ~p)︸︷︷︸

~∇pH(~r ,~p)

~∇r f + ~F (~r , ~p)︸︷︷︸

−~∇rH(~r ,~p)

~∇pf = S(~r , ~p, t) (6.10)

Thus, we have a description for the dynamics of a distribution of non-interacting particles.

Many bands: A complicated band structure usually develops several bands ǫn(~r , ~p). Each bandacts like a separate particle type, which is characterized by the band index n and which has its owndispersion relation. Thus, each band is described by its own Hamilton function Hn(~r , ~p).

∂t fn(~r , ~p, t) + ~vn(~r , ~p, t)︸︷︷︸

~∇pHn(~r ,~p)

~∇r fn(~r , ~p, t) + ~F (~r , ~p, t)︸︷︷︸

−~∇rHn(~r ,~p)

~∇pfn(~r , ~p, t) = Sn(~r , ~p, t) (6.11)

Boltzmann equation: When the source and sink terms describe collisions with external obstaclesor with other particles, we arrive at the Boltzmann equation.

3From the dispersion relation we obtain the equation of motion. the effective mass m∗,

(

m∗(~p))−1

i ,j

def=d2ǫ(~p)

dpidpj=dvg,i (~p)

dpj(6.6)

which relates the force to a change in momentum or wave vector.

..r i =

∑

j

dvg,i

dpj

.pj =

∑

j

dvg,i

dpj

(

−∂H∂rj

)

(6.7)

6 BOLTZMANN EQUATION 115

BOLTZMANN EQUATION

[

∂t + ~vn(~r , ~p)︸︷︷︸

~∇pHn(~r ,~p)

~∇r + ~Fn(~r , ~p)︸︷︷︸

−~∇rHn(~r ,~p)

~∇p]

fn(~r , ~p, t) = Sn(~r , ~p, t)︸︷︷︸

( ∂fn(~r ,~p)∂t )scatt

(6.12)

The left-hand side of the equation is called the drift term , while the right-hand side is the collisionterm.

Usually, the collision term is denoted as(∂fn(~r , ~p)

∂t

)

scatt

def= Sn(~r , ~p, t) (6.13)

I will not use this notation. It is, however, important to know this notation for comparison with theliterature.

A compact way of writing the Boltzmann equation is

d

dtfn

(

~r(t), ~p(t), t)

= Sn

(

~r(t), ~p(t))

(6.14)

which folds up to the full Boltzmann equation if we insert Hamiltons equations of motion for thetrajectories.

6.3 Collision term

The sources and sinks are due to scattering of particles. A particle may collide with an obstacleand thus change its momentum. This can be described as a deletion of a particle with the initialmomentum and the simultaneous creation of another particle with the final momentum. The deletionis described by the sink term and the creation by a source term.

The form of the collision term depends on the physical interaction. Usually, the collisions areconsidered to be local: In a scattering event, two particles may collide at a given position in space.4

The scattering term can have different forms:

• The collision of particles with obstacles such as defects has the form

Sn(~r , ~p, t) =∑

m

∫

d3p′(

Lnm(~p, ~p′)fm(~r , ~p, t)− Lmn (~p′, ~p)fm(~r , ~p, t)

)

(6.15)

The intrinsic scattering probability Lnm(~p′, ~p) is the probability that a particle of band m withmomentum ~p is scattered into band n with momentum ~p′.

• three-particle scattering plays a role for phonons that scatter due to anharmonic terms. Three-particle scattering violates particle number conservation and is forbidden for particles that obeya particle conservation law such as electrons.

Sn(~r , ~p, t) =∑

a,b

∫

d3p′∫

d3p′′(

Lna,b(~p, ~p′, ~p′′)fa(~r , ~p′, t)fb(~r , ~p′, t)

−2Lab,n(~p′, ~p′′, ~p)fb(~r , ~p′, t)fn(~r , ~p′, t))

(6.16)

• four-particle scattering is the dominant collision term for electrons, because the particle-numberconservation law excludes a scattering term involving three particles.

4The notion of local collisions developed from the classical picture of point particles. For wave packets the collisionprocess has a more complex interpretation.


The collision term as a functional of the distribution itself

Sn(~r , ~p, t, [f ]) (6.17)

Note that these terms are valid for the classical formulation and will be modified for quantumparticles.Editor: This needs to be done.

6.4 Equilibrium

Under the assumption that the scattering term does not affect the equilibrium distribution, i.e.Seqn (~r , ~p, [f

eq]) = 0, any distribnution of the form

f eqn (~r , ~p) = G(

Hn(~r , ~p))

(6.18)

is stationary. G(ǫ) can, in principle, be an arbitrary function of energy.The change of the distribution with time is

∂t feqn (~r , ~p, t) = −~vn~∇r f eqn − ~Fn~∇pf eqn + Seqn

=(

~∇pHn)

︸︷︷︸

~vn

dG

dǫ

∣∣∣∣Hn

(

~∇rHn)

+(

−∇~rHn)

︸︷︷︸

~Fn

dG

dǫ

∣∣∣∣Hn

(

~∇pHn)

︸︷︷︸

=0

+Seqn

= Seqn (~r , ~p, t) (6.19)

The condition for the distribution Eq. 6.18 to be stationary is that the net scattering vanishes forthe equilibrium distribution.

Seqn (~r , ~p, [feq]) = 0 (6.20)

Note that the requirements above lead to stationary distributions in the Boltzmann equation.The criterion for a distribution to be in thermal equilibrium is that it maximizes the entropy. Whilewe can expect a distribution fulfilling the maximum entropy is also stationary, there may be stationarydistributions that are not in thermal equilibrium and thus do not maximize the entropy.

The equilibrium distributions for non-interacting particles are the Boltzmann, the Bose and theFermi distribution. (see ΦSX:Statistical Physics[31]) The function G(ǫ) can be for example5

• the Boltzmann distribution G(ǫ) ∝ e−β(ǫ−µ) when the particles are distinguishable,

• the Bose distribution G(ǫ) ∝(

eβ(ǫ−µ) − 1)−1

when the particles are Bosons, or

• the Fermi distribution G(ǫ) ∝(

eβ(ǫ−µ) + 1)−1

when the particles are Fermions.

Thus, the equilibrium distributions are, with the prefactors from section 6.4.1,

f eqn (~r , ~p) =1

(2π~)3

e−β[Hn(~r ,~p)−µ] for distinguishable particles(

eβ[Hn(~r ,~p)−µ] − 1)−1

for Bosons(

eβ[Hn(~r ,~p)−µ] + 1)−1

for Fermions

(6.21)

5The symbol ∝ stands for “proportional to”.


6.4.1 Prefactor for the distributions

A material with volume Ω = N det(T ), where T contains the primitive lattice vectors, has N statesper band. The momentum-volume per state is thus

∆p3 = ~3∆k3 = ~3det(g)

N= ~3

(2π)3

det(T )

1

N=(2π~)3

Ω(6.22)

Thus a distribution function is

fn(~r , ~p) =1

(2π~)3fT,µ

(

ǫn(~r , ~p))

(6.23)

where fT,µ(ǫ) may be the Boltzmann, Bose, or Fermi distribution.

6.4.2 General equilibrium distribution

Let us investigate here the assumption that the net source density vanishes in thermal equilibrium.One instructive example is when the collision term has the form

Seqn (~r , ~p) = −[(

~∇pΣn(~r , ~p))

~∇r +(

−~∇rΣn(~r , ~p))

~∇p]

f eqn (~r , ~p, t) + ∆Seqn (~r , ~p)

with ∆Seqn (~r , ~p) = 0 (6.24)

The functions Σn(~r , ~p), which themselfes functionals of the distribution function, produce a simpleshift of the band structure.

We can combine Σ with H and thus obtain the equilibrium distributions are

f eqn (~r , ~p) =1

(2π~)3

e−β[Hn(~r ,~p)+Σn(~r ,~p)−µ] for distinguishable particles(

eβ[Hn(~r ,~p)+Σn(~r ,~p)−µ] − 1)−1

for Bosons(

eβ[Hn(~r ,~p)+Σn(~r ,~p)−µ] + 1)−1

for Fermions

(6.25)

This ansatz can be inverted, so that we start from any equilibrium distribution f eq and producethe corresponding term Σn(~r , ~p).

Σn(~r , ~p) = −Hn(~r , ~p) + µ+ kBT

ln

([

(2π~)3f eqn (~r , ~p)]−1)

for distinguishable particles

ln

([

(2π~)3f eqn (~r , ~p)]−1+ 1

)

for bosons

ln

([

(2π~)3f eqn (~r , ~p)]−1− 1)

for fermions

(6.26)

The relevance of this observation is that the collision term can produce distributions that cannotbe represented by distribution functions of non-interacting particles such as the Boltzmann, Bose, orFermi distribution expressed in terms of the band structure given by H.

However, by introducing a shift Σ of the band structure, also the stationary distributions of inter-acting particles and of stationary non-equilibrium states can be expressed in the form of equilibriumdistributions of non-interacting particles. The effective band structure Hn(~p) + Σn(~p), however,simply encodes that distribution and does not have any other physical meaning.

Whatever the equilibrium distribution of the system is, there is a variety of collision terms obeying∆Seq[f eq] = 0, and all these collision terms produce the same stationary.


6.5 Linearized Boltzmann equation

Let us assume that the solution of the Boltzmann equation relaxes into a final stationary distribution,which we call the equilibrium distribuion f eqn (~r , ~p). It obeys

[

~vn(~r , ~p)︸︷︷︸

~∇pHn(~r ,~p)

~∇r + ~Fn(~r , ~p)︸︷︷︸

−~∇rHn(~r ,~p)

~∇p]

f eqn (~r , ~p) = Seq(~r , ~p) (6.27)

where the equilibrium source density is obtained from the equilibrium distribution f eqn (~r , ~p).If the deviation from the equilibrium distribution is small, the collision term can be linearized in

the deviation f − f eq from the equilibrium distribution.

Sn(~r , ~p, t) ≈ Seqn (~r , ~p)−∑

m

∫

d3p′ Bn,m(~r , ~p, ~p′)

(

fm(~r , ~p′, t)− f eqm (~r , ~p′))

(6.28)

where

Bn,m(~r , ~p, ~p′) =∂Sn(~r , ~p)

∂fm(~r , ~p′)

∣∣∣∣f eqn (~r ,~p)

(6.29)

LINEARIZED BOLTZMANN EQUATION

[

∂t + ~vn(~r , ~p)︸︷︷︸

~∇pHn(~r ,~p)

~∇r + ~Fn(~r , ~p)︸︷︷︸

−~∇rHn(~r ,~p)

~∇p]

fn(~r , ~p, t)

= Seq(~r , ~p)−∑

m

∫

d3p′ Bn,m(~r , ~p, ~p′)

(

fm(~r , ~p′, t)− f eqm (~r , ~p′))

(6.30)

Denoting the deviation from the equilibirium distribution with δf , so that

fn(~r , ~p, ~t) = feqn (~r , ~p) + δfn(~r , ~p, ~t) (6.31)

we obtain a linear integro-differential equation for δf .[

∂t + ~vn(~r , ~p)︸︷︷︸

~∇pHn(~r ,~p)

~∇r + ~Fn(~r , ~p)︸︷︷︸

−~∇rHn(~r ,~p)

~∇p]

δfn(~r , ~p, t) = −∑

m

∫

d3p′ Bn,m(~r , ~p, ~p′)δfm(~r , ~p′, t) (6.32)

6.5.1 Dynamics

Liouville operator: The linearized Boltzmann equation can be expressed in terms of linear operatorsquite analogous to the ones used in quantum mechanics. The difference is that they act on a Hilbertspace of functions defined in phase space rather than in real space. The scalar prodoct is thereforedefined as an integral over all phase space.

For a given Hamilton function H(~r , ~p, t) the Liouville operator is

i Ldef=

3∑

j=1

∂H

∂pj

∂

∂rj− ∂H∂rj

∂

∂pj= ~v(~r , ~p)~∇r + ~F (~r , ~p)~∇p (6.33)

The factor i has been introduced because the Liouville operator defined this way is hermitean. Theproof is analogous to that in quantum mechanics. Important are the boundary conditions: The


boundary conditions for the distributions are (1) that the distributions vanish for infinite momentumand (2) that they are either periodic in real space or that they vanish for infinite positions.

6

In the case of many bands, the distribution is a vector function in phase space. Analogous tothe Dirac’s Bracket notation, we introduce here a similar notation for square integrable function overphase space. A ket is denoted as |f ) and a bra by (f | The scalar product has the form

(f |g) =∑

n

∫

d3r

∫

d3p f ∗n (~r , ~p)gn(~r , ~p) (6.35)

While the distributions need to be real, we define the algebra over complex functions in order to beanalogous to quantum mechanics.

The components are obtaines as

fn(~r , ~p)def= (n,~r , ~p|f ) (6.36)

A general operator A transforms a vector distribution in phase space into another vector distri-bution in phase space.

|f ′) = A|f )!⇒ f ′n(~r , ~p, t) =

∑

m

∫

d3r ′∫

d3p′ An,m(~r , ~p, ~r ′, ~p′)fm(~r , ~p) (6.37)

Thus we generalize the Liouville operator to

|f ′) = L|f )!⇒ f ′n(~r , ~p, t) =

3∑

j=1

[∂Hn∂pj

∂

∂rj− ∂Hn∂rj

∂

∂pj

]

fn(~r , ~p) (6.38)

Analogously we define the collision term

|f ′) = B|f )!⇒ f ′n(~r , ~p, t) =

∑

m

∫

d3p′ Bn,m(~r , ~p, ~p′)fm(~r , ~p) (6.39)

Using this notation the linearized Boltzmann equation obtaines the form

∂t

∣∣∣δf (t)

)

=(

i L+ B)

|δf ) (6.40)

Because the linearized Boltzmann equation is linear, its time dependent solution can be representedin the form

fn(~r , ~p, t) = feqn (~r , ~p) +

∑

γ

Aγqm,γ(~r , ~p) e−γt (6.41)

where the qm,γ(~r , ~p) obey the stationary eigenvalue equation(

i L+ B − γ1)

|qγ) = 0 (6.42)

6Editor: not for students. This is a sketch! For a problem with three (d=3) spatial dimensions, the phasespace has six (2d) dimensions. If we consider Nb different particle types, we can generalize the phase space to a 2d ·Nbdimensions. In our case each band in the band structure acts like such a particle type and Nb is the number of bands.

A distribution given by the components fn(~r, ~p, t) translates into

Γ(~r1, ~p1, . . . , ~rNb , ~pNb ) =

Nb∏

m=1

fm(~rm, ~pm) (6.34)


This equation will have a set of solutions with eigenvalues γ and eigenvectors qm,γ(~r , ~p). Thiseigenvalue problem can in principle be addressed with standard linear algebra techniques. Thus oneobtains a systematic approach to solving the general time-dependent problem of the Boltzmannequation.

In the absence of the collision term, the eigenvalues γ are purely imaginary. This implies that norelaxation into an equilibrium state takes place. The equilibration is entirely due to the collision term.

For the long-time behavior, we can restrict the solution to the modes with smallest real part ofthe eigenvalue. A small real part of the eigenvalue indicates that the system does not thermalizerapidly. This in turn indicates that this mode stores energy for long periods of time, which againmakes them accessable for technological exploitation.

Particular interesting is to explore whether there are vanishing eigenvalues γ. They indicate thatthe system cannot relax into a unique equilibrium state. In this case the final state depends on theinitial conditions. This does happen in model systems, where symmetries result in conservation laws,which prohibit thermalization. Hence its analysis is able to extract conservation laws. It also shows,where the system is non-ergodic. Ergodicity is the fundamental requirement for the approach to awell defined thermodynamic equilibrium.

6.5.2 Linear Response to external perturbations

Consider the system that is perturbed by external forces. We divide the Hamiltonian into an unper-turbed term H(0) and a perturbation H(1) which we scale by a parameter λ.

Hn(~r , ~p, t) = H(0)n (~r , ~p) + λH

(1)n (~r , ~p, t) (6.43)

This division affects velocities and forces as

~vn(~r , ~p, t) = ~v(0)n (~r , ~p) + λ~v

(1)n (~r , ~p, t)

~Fn(~r , ~p, t) = ~F (0)n (~r , ~p) + λ~F(1)n (~r , ~p, t) (6.44)

Because we derive the perturbing terms from a Hamiltonian, we ensure that Liouville’s theorem isstill obeyed.

The equilibrium distribution of the unperturbed system is f (0)n (~r , ~p), which satisifies the Boltzmannequation to zeroth order

~v (0)n ~∇r f (0)n + ~F (0)n~∇pf (0)n = Seq(~r , ~p) (6.45)

The first order term in λ of the Boltzmann equation is[

∂t + ~v(0)n~∇r + ~F (0)n

~∇p]

f (1)n (~r , ~p, t) +[

~v (1)n ~∇r + ~F (1)n~∇p]

f (0)n (~r , ~p) = −∑

m

∫

d3p′ Bn,m(~p, ~p′)f(1)m

⇒[

∂t + ~v(0)n~∇r + ~F (0)n

~∇p]

f (1)n +∑

m

∫

d3p′ Bn,m(~p, ~p′)f(1)m = −~v (1)n ~∇r f (0)n − ~F (1)n

~∇pf (0)n

(6.46)

If we make the assumption that the problem is translationally invariant in space and time, we canset ∂t f

(1)n = 0 and ~∇r f (1)n = 0 and ~Fn = ~∇rHn = 0.

With these assumptions the problem simplifies to

∑

m

∫

d3p′ Bn,m(~p, ~p′)f(1)m (~p) = −

[

~v (1)n ~∇r + ~F (1)n~∇p]

f (0)n (~r , ~p) (6.47)

With the the inverse K of the collision term defined by

∑

o

∫

d3p′′ Km,o(~p, ~p′′)Bo,n( ~p′′, ~p′) = δm,nδ(~p − ~p′) (6.48)


we obtain the result for the induced change of the distribution

LINEAR RESPONSE OF THE DISTRIBUTION FUNCTION

f (1)n (~p) = −∑

m

∫

d3p′ Kn,m(~p, ~p′)[

~v (1)n ~∇r + ~F (1)n~∇p′]

f (0)n (~r , ~p′) (6.49)

where the inverse of the collision term K is defined by Eq. 6.48.This equation requires as assumption

[

∂t + ~v(0)m~∇r + ~F (0)m

~∇p]

f (1)n = 0 (6.50)

which is satisfied if the perturbed distribution is stationary and translationally invariant and if theforces in the unperturbed system vanish.

With the response of the distribution function at our disposal, we can evaluate the current den-sities. The equilibrium distribution does not contribute to the net currents.7

• the charge current density of particles with charge q is

~jq(~r) = q

∫

d3p∑

n

fn(~r , ~p, t)~vn(~r , ~p, t) (6.51)

• the heat current density is the energy density current

~jQ(~r) =

∫

d3p∑

n

fn(~r , ~p, t)~vn(~r , ~p, t)ǫn(~r , ~p, t) (6.52)

• the momentum current

jq(~r) = q

∫

d3p fn(~r , ~p, t)(

~p ⊗ ~v(~r , ~p, t))

(6.53)

6.6 Relaxation-time approximation

In the previous section we have seen that the most difficult part to obtain the linear response viaEq. 6.49 is the inverse of the collision term.

A particular simple approximation is the relaxation-time approximation, which approximates theinverse of the collision term by a diagonal matrix.

Km,n(~p, ~p′) ≈ τn(~p)δm,nδ(~p − ~p′) (6.54)

where the matrix elements τn(~p) are the relaxation times.With Eq. 6.54 the linearized Boltzmann equation Eq. 6.30 obtains the following form:

RELAXATION TIME APPROXIMATION OF THE BOLTZMANN EQUATION

[

∂t + ~vn(~r , ~p)︸︷︷︸

~∇pHn(~r ,~p)

~∇r + ~Fn(~r , ~p, t)︸︷︷︸

−~∇rHn(~r ,~p)

~∇p]

fn(~r , ~p, t) = Seq(~r , ~p)− 1

τn(~p)

(

fn − f eqn (~r , ~p))

(6.55)

where the distribution has the full arguments fn(~r , ~p, t).

7If it would have currents, the system may be stationary but not in thermal equilibrium. This is a subtle argumentfrom irreversible thermodynamics, which says that spontaneous currents are related to an increase in entropy and thatthey are actually caused by an increase of entropy. In thermal equilibrium the entropy is maximized, so that there areno more currents allowed.


With the relaxation-time approximation the solution can be immediately written down as

fn(~r , ~p, t) = feqn (~r , ~p) +

(

fn(~r , ~p, 0)− f eqn (~r , ~p))

exp

(

− t

τn(~p)

)

(6.56)

With the assumptions of Eq. 6.49, we obtain the current densities in response to external forces.

LINEAR RESPONSE OF THE DISTRIBUTION FUNCTION IN THE RELAXATION TIMEAPPROXIMATION

f (1)n (~p) = −τn(~p)[

~v (1)n ~∇r + ~F (1)n~∇p′]

f (0)n (~r , ~p′) (6.57)

This equation requires as assumption[

∂t + ~v(0)n~∇r + ~F (0)n

~∇p]

f (1)n = 0 (6.58)

which is satisfied if the perturbed distribution is stationary and translationally invariant and if theforces in the unperturbed system vanish.

6.6.1 Electrical conductivity

Let us evaluate the electric conductivity in the relaxation-time approximation. The perturbation isthe Lorentz force due to a homogeneous electruc field ~E acting on the particles with charge q.

With the perturbation

~F (1)n (~r , ~p) = q ~E (6.59)

we obtain the induced current as

~jqEq. 6.51= q

∑

n

∫

d3p fn(~r , ~p, t)~vn(~r , ~p, t)

Eq. 6.57= −q

∑

n

∫

d3p τn(~p)~v(0)n (~r , ~p)

[

~v (1)n ~∇r + ~F (1)n~∇p]

f (0)n (~r , ~p)

=

[

q2∑

n

∫

d3p τn(~p)~v(0)n (~r , ~p)⊗

(

−~∇pf (0)n (~r , ~p))]

︸︷︷︸σ

~E (6.60)

where ⊗ identified the dyadic product (~a ⊗ ~b)i ,j = aibj .With the additional assumption that the equilbrium distribution is the Fermi distribution, we

obtain8

σ = q2∑

n

∫d3p

(2π~)3τn(~p)~v

(0)n (~r , ~p)⊗

(

−~∇pf (0)n (~r , ~p))

= q2∑

n

∫d3p

(2π~)3

(

− ∂fT,µ(ǫ)

∂ǫ

∣∣∣∣ǫn(~p)

)

τn(~p)~v(0)n (~r , ~p)⊗ ~v (0)n (~r , ~p)

︸︷︷︸

~∇pǫn(~r ,~p)

)

(6.61)

8Eq. 6.61 can be compared with Eq. 6.4.27 of Jones and March[40].


Thus we obtain9 the low-temperature limit of the electric conductivity as

σT=0= q2

∑

n

∫d3p

(2π~)3δ(ǫ− µ)τn(~p)~v (0)n (~r , ~p)⊗ ~v (0)n (~r , ~p) (6.63)

Remember here, that we use a special notation for the spin multiplicity, which in our case is partof the sum over bands. If the sum is limited to pairs of spin-degenate bands, a factor of two needsto be added to account for spin multiplicity.

Only charge carriers near the Fermi level contribute to the current. The product ℓn = |τn~vn| canbe identified with the mean free path of the particle.

Using the density of states defined in Eq. 4.112 and doing a number of averages, we obtain

σEq. 4.112=

1

3

D(µ)

Ωτ~v2 (6.64)

The factor 13 enters as follows: first we evaluate the trace of the conductivity tensor. Then we divideby three to obtain the mean matrix diagonal element.

6.7 Quasi Fermi level

A useful way to represent the distribution is to express it by a quasi Fermi level and a local temperature.

fn(~r , ~p, t) = G(

Hn(~r , ~p, t)−Φn(~r , ~p, t))

≈ G(

Hn(~r , ~p, t))

− ∂G(ǫ)∂ǫΦn(~r , ~p, t)) (6.65)

where G(ǫ) is a temperature-dependent Boltzmann, Fermi or Bose distribution.Instead of using the occupations as fundamental variable now the quasi-Fermi level is used, re-

spectively its deviation Φn(~r , ~p, t) from the equilibrium value.

9We use

∂ǫfT,µ(ǫ) = ∂ǫ

[

1 + eβ(ǫ−µ)]−1= −

[

1 + eβ(ǫ−µ)]−2

βeβ(ǫ−µ) = −β[

e12 β(ǫ−µ) + e

12 β(ǫ−µ)

]−2

= − β

4 cosh2( 12β(ǫ− µ))

≈ −δ(ǫ− µ) (6.62)

Chapter 7

Phonon scattering

Editor: This chapter is yet a collection of notes. It is not in a stage ready reading

7.1 Equation of motion for the phonon amplitudes

Let us now derive the equations of motion for the anharmonic case. We start from the definition ofthe phonon amplitudes Eq. 5.74 and form the derivative.

bn,σ(~k)Eq. 5.74= e−iσωn(

~k)t

√

N2~ωn(~k)

(


)

⇒ i~.bn,σ(~k) = i~e−iσωn(

~k)t

√

N2~ωn(~k)

[

−iσωn(~k)(


)

+(

ωn(~k).Qn(~k)− iσ

..Qn(~k)

)]

= i~e−iσωn(~k)t

√

N2~ωn(~k)

[

−iσ(

ω2n(~k)Qn(~k) +..Qn(~k)

)]

Eq. 5.56= −σ~e−iσωn(~k)t

√

N2~ωn(~k)

[∑

~k ′, ~k ′′∈ΩG

∑

l ,m

1

2Xn,l,m

(

−~k, ~k ′, ~k ′′)

Ql(~k ′)Qm( ~k ′′)

]

Eq. 5.75= i~e−iσωn(

~k)t

√

N2~ωn(~k)

[

+iσ∑

~k ′, ~k ′′∈ΩG

∑

l ,m

1

2Xn,l,m

(

−~k, ~k ′, ~k ′′)

×

~

√

2N~ωl(~k ′)

∑

σ′∈±bl ,σ′e

iσ′ωn(~k ′)t

~

√

2N~ωm( ~k ′′)

∑

σ′′∈±bm,σ′′e

iσ′′ωm( ~k ′′)t

]

= −σ[∑

~k ′, ~k ′′∈ΩG

∑

l ,m

N2

(~

2N) 32Xn,l,m

(

−~k, ~k ′, ~k ′′)

√

ωn(~k)ωl(~k ′)ωm( ~k ′′)

×∑

σ′,σ′′∈±bl ,σ′bm,σ′′e

i [σ′ωn(~k ′)+σ′′ωm( ~k ′′)−σωn(~k)]t]

(7.1)

125

126 7 PHONON SCATTERING

EQUATION OF MOTION FOR THE PHONON AMPLITUDES

The phonon amplitudes follow the equatiuon of motiona

i~.bn,σ(~k) = −σ

∑

~k ′, ~k ′′∈ΩG

∑

l ,m

Yn,l,m(−~k, ~k ′, ~k ′′)∑

σ,σ′∈+,−ei [σ

′ωl (~k ′)+σ′′ωm( ~k ′′)−σωn(~k)]tbl ,σ′(~k ′)bm,σ′′( ~k ′′)

(7.2)

with

Yn,l,m(~k, ~k ′, ~k ′′) =N2

(~

2N

) 32 Xn,l,m

(

~k, ~k ′, ~k ′′)

√

ωn(~k)ωl(~k)ωm(~k)(7.3)

where X is given by Eq. 5.49 on p. 101.

Xl ,m,n(~k, ~k ′, ~k ′′)def=(∑

~G

δ~k+~k ′+ ~k ′′−~G

)

×∑

α,β,γ

[∑

~t ′, ~t ′′

Wα,~0,β,~t ′,γ,~t ′′√MαMβMγ

ei~k ′~t ′ei

~k ′′ ~t ′′]

Uα,l(~k)Uβ,m(~k′)Uγ,n(~k

′′) (7.4)

acompare with Eq.1.2 of Brout and Prigogine[41]. Note that the frequencies occur with opposite sign. The differentsign can be traced to the notation adopted by Ziman.

An interesting side aspect is that start from a second order differential equation for the displace-ments δRα,t and we arrive at a first-order differential equation of a complex variable b(t). This hassome analogy with the Schrödinger equation. The difference is that the Schrödinger equation is a lin-ear equation, while the equaltion of motion for the phonon amplitudes is nonlinear. The non-linearityis a consequence of the interaction, that is scattering between particles.

7.2 Momentum and energy conservation

7.2.1 Momentum conservation

Important is the selection rule via the factor

∑

~G

δ~k+~k ′+ ~k ′′−~G

(7.5)

which vanishes unless the wave vectors obey a selection rule. This selection rule implies that phononscan only scatter, if they obey momentum conservation. Momentum conservation is a result of thelattice translation symmetry. The momentum ~~G is a momentum that is transferred to the crystalas a whole.

For each pair of wave vectors ~k ′, ~k ′′ in the reciprocal unit cell, there is exactly one reciprocallattice vector ~G of the crystal, which selects a third wave vector ~k in the first reciprocal unit cell.

It is interesting how the momentum conservation emerged out of the crystal symmetry. Note,that this momentum is not related to the momenta of the atoms. The momentum of a mode hasthe form1

~Pn(~k, t) =∑

α,~t

Mα,~tδ.Rα,~t =

∑

α,~t

√

MαUα,ne−i~k~t .Qn(~t) = N δ~k

(∑

α

√

MαUα,n

) .Qn(~t) (7.6)

1Without restriction of generality, we consider here a complex partial wave.

7 PHONON SCATTERING 127

Thus only the modes with ~k = 0 can have a nonzero true momentum, which can be anything, andthe with a finite wave vector ~k have vanishing true momentum ~Pn(~k).

This is a nice illustration that the momentum conservation follows from a symmetry, and notfrom some god-given fundamental momentum conservation.

Normal and Umklapp processes: A scattering event where the crystal momentum ~G in Eq. 7.5vanishes is called a normal process, while one with a finite crystal momentum in in Eq. 7.5 is calledan umklapp process. The term “umklapp process” has been coined by Peierls implying that the totamomentum of the phonons in the scattering event may “flip over”.

The umklapp processes are important to bring a system into thermal equilibrium.Editor: This needs to be worked on.

Another way of looking at it is the following:The sum over ~k and the one over ~q are not limited to the first reciprocal unit cell, but such that

~k + ~q and ~k − ~q lie in the first reciprocal unit cell Ωg.

The wave vectors ~k, ~k ′, ~k ′′ vary all three within the first unit cell. This allows combinations where~k ′+ ~k ′′−~k add up to a reciprocal lattice vector other than ~G = ~0. For this purpose we have introducedthe reciprocal lattice vector ~G. As a result ~k + ~G can lie anywhere in reciprocal space (albeit on thereciprocal lattice of the supercell). Therefore, we rename ~k + ~G by the symbol ~k , but we let this ~kto lie also outside the first unit cell of the reciprocal lattice. The only conditions that remain is that~k + ~q and ~k − ~q lie in the first unit cell of the reciprocal lattice.

k

G

k’Γ

k’’

7.2.2 Energy conservation

The factor

ei [σ′ωl (~k ′)+σ′′ωm( ~k ′′)−σωn(~k)]t (7.7)

is responsible for energy conservation. If the frequencies do not add up to zero, this factor oscillatesrapidly in time, so that the net effect on the phonon amplitude nearly cancels. If the frequencies addup to zero, i.e. σωn(~k) = σ′ωl(~k ′) + σ′′ωm( ~k ′′) the factor is a constant, so that the oscillations arein resonance and the response has time to build up.

At this point it is a hand waving argument. In the following we will obtain a more precise meaningof it.

7.3 Scattering

7.3.1 Compact notation

For the following discussion it will be convenient to use a more compressed notation, namely tointegrate the mode index n, the wave vector ~k and the index σ into a single variable λ = (~kλ, nλ, σλ).


bλ = bnλ,σλ(~kλ)

ωλ = σλωnλ(~kλ)

Yλ,λ′,λ′′ = σλYn−λ,nλ′ ,nλ′′ (−~kλ, ~kλ′ , ~kλ′′)Uα,λ = Uα,nλ(~kλ) (7.8)

For each mode we also define the corresponding “negative element”, namely the complex conju-gate, by

b−λdef= b∗λ = bnλ,−σλ(−~kλ)

n−λ = nλ

σ−λ = −σλ~k−λ = −~kλω−λ = −ωλUα,λ = U∗α,−λ

Y−λ,−λ′,−λ′′ = −Y ∗λ,λ′,λ′′ (7.9)

The displacements in the new notation are

δRα,~t(t)Eq. 5.77=

∑

λ

1√Mα

Uα,λ

√

~

2N~|ωλ|bλe

i [~kλ~t+ωλt] (7.10)

With these definitions, the equations of motion Eq. 7.2 obtain the form

EQUATION OF MOTION FOR PHONON AMPLITUDES

i~.bλ

Eq. 7.2= −

∑

λ′,λ′′

Y−λ,λ′,λ′′ei [ωλ′+ωλ′′+ω−λ]tbλ′bλ′′ (7.11)

7.3.2 Perturbation theory

Now we can integrate the equation of motion. Let us tackle this problem by perturbation theory,that is, we scale the scattering matrix by a scale factor γ and expand the solutions in orders of γ.

bλ(t) =

∞∑

n=0

γnb(n)(t) (7.12)

The equation of motion has the form

bλ(t)Eq. 7.11=

i

~

∫ t

0

dt ′∑

λ′,λ′′

γYλ,λ′,λ′′ei [ωλ′+ωλ′′+ω−λ]t

′bλ′(t

′)bλ′′(t′) (7.13)

Finally, the scale factor γ is set to unity.

Phonon amplitudes in first order The phonon amplitudes in first order are

b(1)λ (t) =

i

~

∑

µ,µ′

Y−λ,µ,µ′[ ∫ t

0

dt ′ ei [ωµ+ωµ′+ω−λ]t]

︸︷︷︸

tF (1)([ωµ+ωµ′+ω−λ]t)

b(0)µ b(0)µ′ (7.14)

t→∞→ iπ

~

∑

µ,µ′

Y−λ,µ,µ′δ(ωµ + ωµ′ + ω−λ)b(0)µ b

(0)µ′ (7.15)


In the last equation we used Eq. 7.19 derived below.Some caution is required, because we treat the sum over phonons as discrete sum, and at the same

time we use the delta function, which requires a continuous integral over frequencies. This problemdisapears, when we make the limit of an infinitely large system, where the spectrum is continuous.This problem hints at an important difference between finite and infinite systems that is importantfor the approach to thermal equilibrium.

Phonon amplitudes in second order The phonon amplitudes in second order are2

b(2)λ (t) =

2i

~

∫ t

0

dt ′∑

µ,µ′

Y−λ,µ,µ′ei [ωµ+ωµ′+ω−λ]t

′b(0)µ b

(1)µ′ (t)

=2i

~

∫ t

0

dt ′∑

µ,µ′

Y−λ,µ,µ′ei [ωµ+ωµ′+ω−λ]t

′b(0)µ

i

~

∑

ν,ν ′

Y−µ′,ν,ν ′

[∫ t ′

0

dt ′′ ei [ων+ων′+ω−µ′ ]t′′]

b(0)ν b(0)ν ′

= − 2~2

∑

µ,µ′,ν,ν ′

Y−λ,µ,µ′Y−µ′,ν,ν ′

[∫ t

0

dt ′ ei [ωµ+ωµ′+ω−λ]t′∫ t ′

0

dt ′′ ei [ων+ων+ω−µ′ ]t′′]

︸︷︷︸

t2F (2)([ωµ+ωµ′+ω−λ]t,[ων+ων′+ω−µ′ ]t)

b(0)µ b(0)ν b(0)ν ′

(7.16)

The result is a product of two or three initial phonon amplitudes, a frequency dependent term, fromthe time integration and a product of the anharmonic terms. Let us deal with those contributionsone-by-one.

Time dependence

As shown in appendix K, the integration can be expressed for large times in terms of delta functions.The first time integral has the form

F (1)(ωt)def=1

t

∫ t

0

dt ′ eiωt′

(7.17)

It has the form

F (1)(x)Eq. K.2=sin(x)

x− i cos(x)− 1

x(7.18)

For large times we use the limiting expression

limt→∞

tF (1)(ωt)Eq. K.4= πδ(ω) (7.19)

For the second-order terms we will use

limt→∞

t2F (1)(ωt)F (1)(−ωt) Eq. K.8= 2πtδ(ω) (7.20)

The second time integral is3

F (2)(ωt, ω′t)def=1

t2

∫ t

0

dt ′ eiωt′∫ t ′

0

dt ′′ eiω′t ′′ (7.21)

In the large-time limit is has the form

limt→∞

t2F (2)(ωt,−ωt) Eq. K.15= πtδ(ω) (7.22)

2compare Eq. 3.1 of Brout and Prigogine[41].3 12F (2) is the average of a plane wave ei(ωt

′+ω′t ′′) with wave vector (ω,ω′) over a triangle in the t ′, t ′′-plane withcorners (0, 0), (t, 0), (t, t).


7.3.3 Density matrix

Our goal is to predict the time evolution of the phonon numbers. This will allow us to understandhow heat, carried by phonons, is redistributed inside the material. For this purpose the relevantquantity is the number of phonons. The phonon number in turn is related to products of two phononamplitudes.

nλ(t) = b∗λ(t)bλ(t) = b−λ(t)bλ(t) (7.23)

Therefore we evaluate the one-phonon-reduced density matrix

ρλ,λ′(t) = bλ(t)bλ′(t) (7.24)

for a given initial state after a certain time interval.Up to second order in the anharmonicity, the density matrix is

ρλ,λ′(t) =

∞∑

n=0

ρ(n)λ,λ′(t)

= b(0)λ (t)b

(0)λ′ (t)

+ b(1)λ (t)b

(0)λ′ (t) + b

(0)λ (t)b

(1)λ′ (t)

+ b(2)λ (t)b

(0)λ′ (t) + b

(1)λ (t)b

(1)λ′ (t) + b

(0)λ (t)b

(2)λ′ (t) + . . . (7.25)

In zeroth order, we obtain the initial value,

ρ(0)λ,λ′(t) = b

(0)λ b

(0)λ′ (7.26)

which does not depend on time.In first order, we obtain

ρ(1)λ,λ′(t) = b

(0)λ b

(1)λ′ (t) + b

(1)λ (t)b

(0)λ′

=i

~

∑

µ,µ′

Y−λ,µ,µ′tF(1)([ωµ + ωµ′ + ω−λ]t)b

(0)µ b

(0)µ′ b

(0)λ′

t→∞→ iπ

~

∑

µ,µ′

Y−λ,µ,µ′δ(ωµ + ωµ′ + ω−λ)b(0)µ b

(0)µ′ b

(0)λ′ +

(

λ↔ λ′)

(7.27)

In the last step we used Eq. 7.15.To second order we obtain

ρ(2)λ,λ′(t) = −

2

~2

∑

µ,µ′,ν,ν ′

Y−λ,µ,µ′Y−µ′,ν,ν ′t2F (2)([ωµ + ωµ′ + ω−λ]t, [ων + ων ′ + ω−µ′ ]t)b

(0)µ b(0)ν b

(0)ν ′ b

(0)λ′

+1

2

(i

~

)2∑

µ,µ′

∑

ν,ν ′

Y−λ,µ,µ′Y−λ′,ν,ν ′t2F (1)([ωµ + ωµ′ + ω−λ]t)F

(1)([ων + ων ′ + ω−λ′ ]t)b(0)µ b

(0)µ′ b

(0)ν b

(0)ν ′

+ (λ↔ λ′) (7.28)

The term denoted with (λ↔ λ′) indicates that the same expression as above needs to be addedwith the two indices interchanged.

7.3.4 Initial random-phase approximation

The initial conditions for a trajectory, that is positions and velocities of all atoms are uniquely deter-mined by the set of phonon amplitudes. The phonon amplitude bλ =

√nλe

iφλ consist of a absolutevalue, which reflects the phonon number nλ = b∗λbλ, and a phase φλ.


If one is only interested in the phonon number, it seems appropriate to average over all phasesφλ of the initial state. This is the so-called initial random-phase approximation [37].

Not all phases are independent. The phases of bλ and b−λ are opposite, because they are complexconjugates of each other.4 States with opposite indices λ have opposite phases.

φλ = −φ−λ (7.29)

When we average over all initial phases, the products with independent phases in two termsaverage to zero, and only products that can be decomposed into contractions, that is pairs withcancelling phases, i.e. opposite indices, contribute to a final result.

We obtain

⟨

b(0)λ b

(0)λ′

⟩

~φ= δλ+λ′n

(0)λ

⟨

b(0)λ b

(0)λ′ b

(0)λ′′

⟩

~φ= 0

⟨

b(0)λ b

(0)λ′ b

(0)λ′′ b

(0)λ′′′

⟩

~φ= δλ+λ′δλ′′+λ′′′

(

1− δλ+λ′′′)

︸︷︷︸

(a,−a,b,−b)−(a,−a,a,−a)

n(0)λ n

(0)λ′′

+ δλ+λ′′δλ′+λ′′′(

1− δλ+λ′)

︸︷︷︸

(a,b,−a,−b)−(a,−a,−a,a)

n(0)λ n

(0)λ′

+ δλ+λ′′′δλ′+λ′′(

1− δλ+λ′′)

︸︷︷︸

(a,b,−b,−a)−(a,a,−a,−a)

n(0)λ n

(0)λ′ (7.30)

Explanation of the double-counting correction: The term with four amplitudes can be contractedin three possible ways: connect the first amplitude to any of the three remaining ones and pairthe remaining two. This gives three different possibilities, namely (a,−a, b,−b), (a, b,−a,−b), and(a, b,−b,−a). If there are four phonons have the same “absolute index” |a|, that is, two equalphonons and two opposite ones such as (a, a,−a,−a), (a,−a, a,−a), and (a,−a,−a, a), they oc-cur each in two of the three terms considered so far (a, a,−a,−a) ∈ (a, b,−a,−b), (a, b,−b,−a),(a,−a, a,−a) ∈ (a,−a, b,−b), (a, b,−b,−a), and (a,−a,−a, a) ∈ (a,−a, b,−b), (a, b,−a,−b).Thus, they are accounted twice, so that one needs to be removed again. We choose to integrate(a,−a, a,−a) into the pattern (a,−a, b,−b), (a,−a,−a, a) into (a, b,−a,−b) and (a, a,−a,−a)into (a, b,−b,−a).

We can convince ourselfes of the need to take the double counting into account. For that purposewe consider a specific example, namely

⟨

b(0)a b(0)−ab

(0)a b

(0)−a

⟩

~φ

Eq. 7.30= δa−aδa−a

︸︷︷︸

=1

(

1− δa−a)

︸︷︷︸

=0

n(0)a n(0)a + δa+aδ−a−a︸︷︷︸

=0

(

1− δa−a)

︸︷︷︸

=0

n(0)a n(0)−a

+ δa−aδ−a+a︸︷︷︸

=1

(

1− δa+a)

︸︷︷︸

=1

n(0)a n(0)−a = n

(0)a n(0)a (7.31)

Without the double-counting correction, an additional, erroneous factor two would have occured.Eq. 7.30 shows that the random-phase approximation eliminates the first order terms for the

density matrix Eq. 7.25.

4This is a consequence of the fact that displacements are real.


In the random-phase approximation the second-order term of the density matrix reduces to

⟨

ρ(2)λ,λ′(t)

⟩

~φ= − 2~2

∑

µ,µ′,ν,ν ′

Y−λ,µ,µ′Y−µ′,ν,ν ′t2F (2)([ωµ + ωµ′ + ω−λ]t, [ων + ων ′ + ω−µ′ ]t)

⟨

b(0)µ b(0)ν b(0)ν ′ b

(0)λ′

⟩

~φ

+1

2

(i

~

)2∑

µ,µ′

∑

ν,ν ′

Y−λ,µ,µ′Y−λ,ν,ν ′t2F (1)([ωµ + ωµ′ + ω−λ]t)F

(1)([ων + ων ′ + ω−λ]t)⟨

b(0)µ b(0)µ′ b

(0)ν b

(0)ν ′

⟩

~φ

+ (λ↔ λ′)

= − 2~2

∑

ζ,µ′

Y−λ,ζ,µ′Y−µ′,−ζ,−λ′t2F (2)([ωζ + ωµ′ + ω−λ]t, [ω−ζ + ω−λ′ + ω−µ′ ]t)

×(

1− δζ+λ′)

n(0)ζ n

(0)λ′ ← (A): (µ, ν, ν ′, λ′)→ (ζ,−ζ,−λ′, λ′)

− 2~2

∑

ζ,µ′

Y−λ,ζ,µ′Y−µ′,−λ′,−ζt2F (2)([ωζ + ωµ′ + ω−λ]t, [ω−λ′ + ω−ζ + ω−µ′ ]t)

×(

1− δζ−λ′)

n(0)ζ n

(0)λ′ ← (B): (µ, ν, ν ′, λ′)→ (ζ,−λ′,−ζ, λ′)

− 2~2

∑

ζ,µ′

Y−λ,−λ′,µ′Y−µ′,ζ,−ζt2F (2)([ω−λ′ + ωµ′ − ωλ]t, [ωζ + ω−ζ

︸︷︷︸

=0

−ωµ′ ]t)

×(

1− δζ+λ′)

n(0)ζ n

(0)λ′ ← (C): (µ, ν, ν ′, λ′)→ (−λ′, ζ,−ζ, λ′)

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−12

1

~2

∑

ζ,ζ′

Y−λ,ζ,−ζY−λ′,ζ′,−ζ′t2F (1)([ωζ + ω−ζ

︸︷︷︸

=0

−ωλ]t)F (1)([ωζ′ + ω−ζ′︸︷︷︸

=0

−ωλ′ ]t)

×(

1− δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (D) (µ, µ′, ν, ν ′)→ (ζ,−ζ, ζ′,−ζ′)

−12

1

~2

∑

ζ,ζ′

Y−λ,ζ,ζ′Y−λ′,−ζ,−ζ′t2F (1)([ωζ + ωζ′ − ωλ]t)F (1)([ω−ζ + ω−ζ′ − ωλ′ ]t)

×(

1− δζ+ζ′)

n(0)ζ n

(0)ζ′ ← (E) (µ, µ′, ν, ν ′)→ (ζ, ζ′,−ζ,−ζ′)

−12

1

~2

∑

ζ,ζ′

Y−λ,ζ,ζ′Y−λ′,−ζ′,−ζt2F (1)([ωζ + ωζ′ − ωλ]t)F (1)([ω−ζ′ + ω−ζ − ωλ′ ]t)

×(

1− δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (F) (µ, µ′, ν, ν ′)→ (ζ, ζ′,−ζ′,−ζ)

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+ (λ↔ λ′) (7.32)

• terms (A) and (B) are identical except for the double counting correction. (We use Ya,b,c =Ya,c,b.)

• terms (E) and (F) are identical except for the double counting correction. (We use Ya,b,c =Ya,c,b.)


⟨

ρ(2)λ,λ′(t)

⟩

~φ= − 2~2

∑

ζ,µ′

Y−λ,ζ,µ′Y−µ′,−ζ,−λ′t2F (2)([ωζ + ωµ′ − ωλ]t, [ω−ζ + ω−λ′ − ωµ′ ]t)

×(

2− δζ+λ′ − δζ−λ′)

n(0)ζ n

(0)λ′ ← (A)+(B)

− 2~2

∑

ζ,µ′

Y−λ,−λ′,µ′Y−µ′,ζ,−ζt2F (2)([ω−λ′ + ωµ′ − ωλ]t,−ωµ′t)

(

1− δζ+λ′)

n(0)ζ n

(0)λ′ ← (C)

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−12

1

~2

∑

ζ,ζ′

Y−λ,ζ,−ζY−λ′,ζ′,−ζ′t2F (1)(−ωλt)F (1)(−ωλ′t)

(

1− δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (D)

−12

1

~2

∑

ζ,ζ′

Y−λ,ζ,ζ′Y−λ′,−ζ,−ζ′t2F (1)([ωζ + ωζ′ − ωλ]t)F (1)([ω−ζ + ω−ζ′ − ωλ′ ]t)

×(

2− δζ+ζ′ − δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (E)+(F)

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+ (λ↔ λ′) (7.33)

The expression obtained here is valid in the second-order perturbation theory in the anharmonicity.The second-order perturbation theory implies that the anharmonic terms must be weak. Higher orderterms such as quartic interactions, describing scattering events in four phonons, are not included.

The initial random-phase approximation implies that we start from a specific set of phonon num-bers n(0)λ , but there is already an statistical average over the phases of the initial states. As aconsequence, the initial density matrix has nonzero elements ρλ,λ′ only if λ′ = −λ

While the phonons are propagated, they scatter among each other. After a certain while, a newdensity matrix in terms if phonon numbers is obtained. This density matrix is no more diagonal. Thisimplies that the relative phase factors between the phonons are important to describe the final state.

The relevance of phase factors is usually attributed to quantum effects. Note, however, that ourtheory of phonons is purely classical. The role of phase factors emerges naturally from the wavenature of the underlying theory, be it classical or quantum in nature.

What are the phase factors? Consider a wave packet f (x, t) with a narrow spectral width, thatis a~k contributes only near ~k0.

f (x, t) =∑

k

a~kei [~k~r−ω(~k)t]

= ei [~k0~r−ω(~k0)t]

∑

k

a~kei(~k−~k0)[~r−∇kω(~k)t]+O(|~k−~k0|2)

≈ ei [~k0~r−ω(~k0)t]∑

k

a~kei(~k−~k0)[~r−~vgt]

︸︷︷︸

χ(~r−~vgt)

= ei [~k0~r−ω(~k0)t]χ(~r − ~vgt) (7.34)

where χ(~r) is the envelope function, which modulates the leading wave ei [~k0~r−ω(~k0)t]. The wavepacked moves with the group velocity ~vg.

1. Multiplication of the wave packet with a constant phase factor leaves the position and theshape of the envelope function unchanged but it displaces the leading wave.

2. Multiplication of the wave packet with a phase factor that changes linearly with ~k , i.e. φ =~d(~k − ~k0) leaves the shape of the envelope function unchanged but it displaces the envelopefunction ins space by ~d .


3. Other phase factors will change the shape of the envelope function. If the phases are continu-ous5 functions of ~k , however, these terms will only affect envelope functions that are “not verysmooth”. That is their effect vanishes in the limit of extremely broad envelope functions.

7.3.5 Repeated random-phase approximation

Our goal is to construct a theory that provides the temporal evolution of the phonon numbers nλ(t).In principle this can be done by working out Eq. 7.33 as function of time and pick out the diagnoalelements as phonon numbers, i.e.

nλ(t) = nλ(0) +⟨

ρ(1)λ,−λ(t)

⟩

~φ︸︷︷︸

=0

+⟨

ρ(2)λ,−λ(t)

⟩

~φ+ . . . (7.35)

If the time span gets longer, it is probably important to go to higher order in the anharmonic terms.Another approach is taken in the Boltzmann equation: Here Eq. 7.35 is used to evaluate the

phonon number after only a short time interval. The resulting phonon numbers are then used asinitial conditions for the next time interval, and so on. If the time intervals are short, one can extractan equation of motion, namely

∂tnλ(t) =nλ(t + ∆)− nλ(t)

∆=

⟨

ρ(2)λ,−λ(∆)

⟩

~φ

∆(7.36)

In this method, the phase information is thrown away in every time step. This loss of informationis reflected in an increase of the entropy. The entropy growth is one of the fundamental hallmarksof the Boltzmann equation. It is important for transport theory because the entropy growth is thedriving force of all thermodynamic processes near equilibrium.

The question regarding the origin of the entropy growth is very fundamental and of great interest:Namely it can be shown that the entropy for a closed quantum system is a constant of motion, thatis, it does not grow. Because this question is far from being settled I will leave the subject and referto the scientific literature. Van Hove[42] has been one of the pioners of this topic.

Here, we will simply do the repeated random-phase approximation,[37] namely we will extractthe phonon numbers after a short period of time and use those for the next iteration.

The phonon numbers are obtained via Eq. 7.35. We furthermore use the large-time limitsfor the time dependence, namely Eq. K.12 , i.e. t2F (2)(ωt,−ωt) →= πtδ(ω) and Eq. 7.20,i.e.t2F (1)(ωt)F (1)(−ωt) = 2πtδ(ω).

5This assumption is not necessarily valid.


n(2)λ (t) =

⟨

ρ(2)λ,−λ(t)

⟩

~φ

= −2πt~2

∑

ζ,ζ′

Y−λ,ζ,ζ′Y−ζ′,−ζ,λ︸︷︷︸

σλσζ′ |Y−λ,ζ,ζ′ |2

δ(ωζ + ωζ′ − ωλ)(

2− δζ−λ − δζ+λ)

n(0)ζ n

(0)λ ← (A)+(B)

−2πt~2

∑

ζ,ζ′

Yλ,−ζ,−ζ′Yζ′,ζ,−λ︸︷︷︸

σλσζ′ |Y−λ,ζ,ζ′ |2

δ(ω−ζ + ω−ζ′ − ω−λ)︸︷︷︸

δ(ωζ+ωζ′−ωλ)

(

2− δ−ζ+λ − δ−ζ−λ)

︸︷︷︸

2−δζ−λ−δζ+λ

n(0)ζ n

(0)λ ← (A’)+(B’)

−2πt~2

∑

ζ,ζ′

Y−λ,ζ,ζ′Y+λ,−ζ,−ζ′︸︷︷︸

−|Y−λ,ζ,ζ′ |2

δ(ωζ + ωζ′ − ωλ)(

2− δζ+ζ′ − δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (E)+(F)+(E’)+(F’)

−2πt~2

∑

ζ,ζ′

Y−λ,λ,ζ′Y−ζ′,ζ,−ζδ(ωζ′)(

1− δζ−λ)

n(0)ζ n

(0)λ ← (C)

−2πt~2

∑

ζ,ζ′

Yλ,−λ,−ζ′Yζ′,−ζ,ζ︸︷︷︸

Y ∗−λ,λ,ζ′Y∗−ζ′ ,ζ,−ζ

δ(ω−ζ′)︸︷︷︸

δ(ωζ′ )

(

1− δ−ζ+λ)

︸︷︷︸

1−δζ−λ

n(0)ζ n

(0)λ ← (C’)

−2πt~2

∑

ζ,ζ′

Y−λ,ζ,−ζY+λ,ζ′,−ζ′δ(ωλ)(

1− δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (D)+(D’) (7.37)

We used Yλ,λ′,λ′′ = −Y ∗(−λ,−λ′,−λ′′) and that σλYλ,λ′,λ′′ is invariant under interchange of theindices.

The primed terms (A’)+(B’),(C’),(D’),(E’)+(F’) are the terms obtained by interchanging λ andλ′ in the expression for the density matrix. The terms (E)+(F) and (D) are already symmetricunder interchange of λ and λ′ so that (E’),(F’) and (D’) are accounted for simply by multiplicationof (E)+(F) and (D) by a factor of two. In the occupation number the interchange of λ and λ′

translates into a inversion of the index λ. For the terms (A’)+(B’) and (C’), the sum indices ζ, ζ′

have been inverterted alongside with the index λ, which does not change the result, because positiveand negative terms are included in the sum.

n(2)λ (t) = −

4πt

~2

∑

ζ,ζ′

σλσζ′ |Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)(


n(0)ζ n

(0)λ ← (A)+(B)+(A’)+(B’)

+2πt

~2

∑

ζ,ζ′

|Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)(

2− δζ+ζ′ − δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (E)+(F)+(E’)+(F’)

−4πt~2

∑

ζ,ζ′

Re[Y−λ,λ,ζ′Y−ζ′,ζ,−ζ]δ(ωζ′)(

1− δζ−λ)

n(0)ζ n

(0)λ ← (C)+(C’)

−2πt~2

∑

ζ,ζ′

Re[Y−λ,ζ,−ζY+λ,ζ′,−ζ′ ]δ(ωλ)(

1− δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (D)+(D’) (7.38)

The term (A)+(B) can be compared with Eq. 3.7 of Brout and Prigogine[? ].Editor: There is

a discrepancy of a factor of two. There may be a mistake regarding the definition

of the phonon number including both λ and −λ.


Now we can limit λ to values with σλ = +.

n(2)λ (t)

σλ=+= −4πt

~2

∑

ζ

[∑

ζ′

σζ′ |Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)](


n(0)ζ n

(0)λ ← (A)+(B)+(A’)+(B’)

+2πt

~2

∑

ζ,ζ′

|Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)(

2− δζ+ζ′ − δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (E)+(F)+(E’)+(F’)

−4πt~2

∑

ζ,ζ′

Re[Y−λ,λ,ζ′Y−ζ′,ζ,−ζ]δ(ωζ′)(

1− δζ−λ)

n(0)ζ n

(0)λ ← (C)+(C’)

−2πt~2

∑

ζ,ζ′

Re[Y−λ,ζ,−ζY+λ,ζ′,−ζ′ ]δ(ωλ)(

1− δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (D)+(D’) (7.39)

7.3.6 Phonon scattering without lattice expansion

The terms (C),(C’),(D),(D’) describe the interaction of the phonon gas with an an homogeneousexpansion or contraction of the lattice. These effects differ from the phonon-phonon scattering. Letme therefore divide the change of occupation into two terms n(2a), which describes phonon-phononscattering, while n(2b) describes the interaction of the phonon gas with an expansion of the lattice.

Let me consider the first two terms only. In order to make the equations more transparent, allterms will be expressed by those with σ > 0.

n(2a)λ (t) = −4πt

~2

∑

ζ,ζ′

σζ′ |Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)(


n(0)ζ n

(0)λ ← (A)+(B)+(A’)+(B’)

+2πt

~2

∑

ζ,ζ′

|Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)(

2− δζ+ζ′ − δζ−ζ′)

n(0)ζ n

(0)ζ′ ← (E)+(F)+(E’)+(F’)

= −4πt~2

∑

ζ,ζ′:σζ=σζ′=+

×

|Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)[(

2− δζ−λ)

n(0)λ

︸︷︷︸

(a)

− 12

(

2− δζ−ζ′)

n(0)ζ′

︸︷︷︸

(b)

]

n(0)ζ

+|Y−λ,−ζ,ζ′ |2δ(ω−ζ + ωζ′ − ωλ)[(

2− δζ−λ)

n(0)λ

︸︷︷︸

(c)

− 12

(

2− δζ−ζ′)

n(0)ζ′

︸︷︷︸

(d)

]

n(0)ζ

+|Y−λ,ζ,−ζ′ |2δ(ωζ + ω−ζ′ − ωλ)[

−(

2− δζ−λ)

n(0)λ

︸︷︷︸

(e)

− 12

(

2− δζ−ζ′)

n(0)ζ′

︸︷︷︸

(f )

]

n(0)ζ

+|Y−λ,−ζ,−ζ′ |2 δ(ω−ζ + ω−ζ′ − ωλ)︸︷︷︸

=0

[

−(

2− δζ−λ)

n(0)λ −

1

2

(

2− δζ−ζ′)

n(0)ζ′

]

n(0)ζ

=4πt

~2

∑


|Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)[

−(

2− δζ−λ)

n(0)λ n

(0)ζ

︸︷︷︸

(a)

+1

2

(

2− δζ−ζ′)

n(0)ζ′ n

(0)ζ

︸︷︷︸

(b)

]

+|Y−λ,ζ,−ζ′ |2δ(ωζ − ωζ′ − ωλ)[(

2− δζ′−λ)

n(0)λ n

(0)ζ′

︸︷︷︸

(c)

−(

2− δζ−λ)

n(0)λ n

(0)ζ

︸︷︷︸

(e)

−(

2− δζ−ζ′)

n(0)ζ n

(0)ζ′

︸︷︷︸

(d)+(f )

]

(7.40)

• terms (d) and (f ) are identical because they only differ by an exchange of ζ with ζ′


• The term (a) describes the loss of phonons λ as they collide with a phonon σ and form a newphoton σ′

•

λ

’ζ’

ζ’ ζ’

ζ’ ζ’

λ

ζ

λζζ

λ λζ

ζζ

λ

ζ

Interpretation: Editor: This is in preparation

• The terms (A) + (B) + (A′) + (B′) describe for ...

The terms (C),(C’),(D),(D’) couple phonons with frequency close to zero. Those are the acousticphonons, which also describe the elastic constants. I believe that these terms describe thermalexpansion of the lattice, respectively the cooling of the phonon-gas upon expansion of the lattice.The cooling effect can occur in two ways, namely by (C)+(C’) changing the phonon numbers or by(D)+(D’) reducing the energy of existing phonons.Editor: check this!

7.4 Kinetic equation

Now we are ready to write down a kinetic equation that determines the dynamics of the phononnumbers. We evaluate Eq. 7.36 and obtain

∂tnλ = Lλζ′,ζ′nζnζ′ + L

λ,ζζ′

(

nζ′ + nζnζ′ + nλnζ′)

− 12Lζ,ζ

′

λ

(

nλ + nλnζ + nλn′ζ

)

− Lζ′λ,ζ(

nλnζ

)

(7.41)

where

Lλλ′,λ′′ = −Lλ′,λ′′λ

def=4π

~2Y−λ,λ′,λ′′δ(ωλ′ + ωλ′′ + ω−λ) (7.42)

∂tnλ =4π

~2

∑


|Y−λ,ζ,ζ′ |2δ(ωζ + ωζ′ − ωλ)[

−(

2− δζ−λ)

n(0)λ n

(0)ζ

︸︷︷︸

(a)

+1

2

(

2− δζ−ζ′)

n(0)ζ′ n

(0)ζ

︸︷︷︸

(b)

]

+|Y−λ,ζ,−ζ′ |2δ(ωζ − ωζ′ − ωλ)[(

2− δζ′−λ)

n(0)λ n

(0)ζ′

︸︷︷︸

(c)

−(

2− δζ−λ)

n(0)λ n

(0)ζ

︸︷︷︸

(e)

−(

2− δζ−ζ′)

n(0)ζ n

(0)ζ′

︸︷︷︸

(d)+(f )

]

(7.43)

Now we consider the phonon numbers after a certain period of time. The time shall be largeenough that the

nλ(t)− nλ(0)t

→ ∂tnλ (7.44)


When we perform a final random-phase approximation

∂tnλ =8π

~2

∑

λ′,λ′′;σλ>0,σλ>0

δ(ωλ′ + ωλ′′ − ωλ)|Yλ,λ′,λ′′ |2[1

2nλ′nλ′′ − nλnλ′′

]

+8π

~2

∑

λ′,λ′′;σλ>0,σλ>0

δ(ωλ′ − ωλ′′ − ωλ)|Yλ,λ′,−λ′′ |2[

nλ′nλ′′ − nλnλ′′ − nλnλ′]

(7.45)

CLASSICAL KINETIC EQUATION FOR PHONON NUMBERS

∂tnσ(k) = +~π

4

∑

~k1,~k2,σ1>0,σ2>0

δ

∑

j∈1,2ωσj (~kj)− ωσ(~k)

∣∣∣Aσ,σ1,σ2(−~k,~k1, ~k2)

∣∣∣

2

/Nωσ(~k)ωσ1(~k1)ωσ2(~k2)

×[1

2n(0)σ1 (

~k1)n(0)σ2 (

~k2)− n(0)σ (~k)n(0)σ2 (~k2)]

+~π

4

∑

~k1,~k2,σ1>0,σ2>0

δ(

ωσ1(~k1)− ωσ2(~k2)− ωσ(~k))

∣∣∣Aσ,σ1,σ2(−~k,~k1,−~k2)

∣∣∣

2

/Nωσ(~k)ωσ1(~k1)ωσ2(~k2)

×[

n(0)σ1 (~k1)n

(0)σ2 (

~k2)− n(0)σ (~k)n(0)σ2 (~k2) + n(0)σ (~k)n(0)σ1 (~k1)]

The processes can all be described as the collision of two phonons or antiphonons that result in athird phonon or antiphonon.

1. 1 + 2→ 0 create

2. 0 + 2→ 1 annihilate

3. 1 + 2→ 0 create

4. 0 + 2→ 1 annihilate

5. 0 + 1→ 2 create

There is a sign error!

For an easier comparison with the quantum expression Eq. 7.50 derived later, when we expressthe result with the intrinsic probabilities of transition defined in Eq. 7.51.

∂tnσ(k) =∑

~k1,~k2,σ1>0,σ2>0

L~k,σ~k1,σ1;~k2,σ2

[1

2n(0)σ1 (

~k1)n(0)σ2 (

~k2)− n(0)σ (~k)n(0)σ2 (~k2)]

+∑

~k1,~k2,σ1>0,σ2>0

L~k,σ;~k2,σ~k1,σ1

[

n(0)σ1 (~k1)n

(0)σ2 (

~k2)− n(0)σ (~k)n(0)σ2 (~k2) + n(0)σ (~k)n(0)σ1 (~k1)]

7.4.1 Units of quantities

For testing purposes I list here the units of the quantities involved:


UNITS

uj,~t ∼ m displacement

c ∼ kgs2 force constant

d ∼ kgms2 anharmonicity

D ∼ 1s2 dynamical matrix

u ∼ 1√kg

eigenmode

Q ∼ 1 amplitude

A ∼ 1ms2√kg

anharmonic coupling

~ ∼ kgm2

s Planck quantum

Y ∼ 1s2 anharmonic coupling

b± ∼ 1 phonon amplitude

Γ ∼ 1s rate constant


7.5 Quantum mechanical derivation

7.5.1 From the Lagrangian to ladder operators

We begin with the Lagrangian Eq. ??, which has the simplified form

L = 12

~

2ω

.Q2

− 12

~ω

2Q2

In order to simplify the derivation I drop anharmonicity and the sum over phonons.Let us derive the Hamiltonian

H = P.Q− L with P =

∂L∂.Q

The momentum is

P =∂L∂.Q=~

2ω

.Q so that

.Q =

2ω

~P

and the Hamiltonian

H = P.Q− L with

=2ω

~P 2 − 1

2

~

2ω

(2ω

~P

)2

+1

2

~ω

2Q2

=1

2

2ω

~P 2 +

1

2

~ω

2Q2

= ~ω

[1

~2P 2 +

1

4Q2]

Let us introduce the variables Eq. ??

b∗ =1

2

(

1 +1

iω∂t

)

Q =1

2

(

Q+2

i~P

)

b =1

2

(

1− 1iω∂t

)

Q =1

2

(

Q− 2i~P

)

which correspond to our b∗ = bf+σeiωσt and b = bf−σe

−iωσt .Let us now replace momentum and coordinates into operators.

b† =1

2

(

Q+2

i~P

)

b =1

2

(

Q− 2i~P

)

The products of the phonon amplitudes are

b†b =1

2

(

Q+2

i~P

)1

2

(

Q− 2i~P

)

=1

4

(

Q2 +4

~2P 2 +

2

i~[P , Q]−

)

=1

4Q2 +

1

~2P 2 − 1

2

)

bb† =1

2

(

Q− 2i~P

)1

2

(

Q+2

i~P

)

=1

4

(

Q2 +4

~2P 2 − 2

i~[P , Q]−

)

=1

4Q2 +

1

~2P 2 +

1

2


which yields the known commutator relations

[b†, b

]

+= 2

(1

4Q2 +

1

~2P 2)

=2

~ωH

[b, b†

]

− = 1

This the Hamiltonian has the form

H =~ω

2

[b†, b

]

+=~ω

2

(

b†b + bb†︸︷︷︸

1−b†b

)

= ~ω

(

b†b +1

2

)

The basis of energy eigenstates can be constructed from the ground state |0〉 via

|n〉 = 1√n!

(b†)n |0〉

with

H|n〉 = |n〉~ω(

n +1

2

)

7.5.2 The phonon Hamiltonian

Let us return again to the Lagrangian Eq. ??, but let us now consider the anharmonicity and thephonons. This will allow us to write down the many-phonon Hamiltonian/.

L = 12

∑

k,σ

~

2ωσ(~k)

.Qσ(~k, t)

.Q∗σ(~k, t)−

1

2

∑

~k,σ

~ωσ(~k)

2Qσ(~k, t)Q

∗σ(~k, t)

− 13!

(~

2

) 32 ∑

~k,~k ′, ~k ′′

∑

σ,σ′,σ′′

Aσ,σ′,σ′′(~k, ~k ′, ~k ′′)/√N

√

ωσ(~k)ωσ′(~k ′)ωσ′′( ~k ′′)Qσ(~k, t)Qσ′(~k ′, t)Qσ′′( ~k ′′, t)

Now we construct the Hamiltonian by introducing the canonical momentum

Pσ(~k, t) =∂L

∂.Qσ(~k)

=~

4ωσ(~k)

.Q∗σ(~k, t)

so that

H =∑

k,σ

Pσ(~k).Qσ(~k) + P

∗σ (~k)

.Q∗σ(~k)− L

=1

2

∑

k,σ

8ωσ(~k)

~Pσ(~k)Pσ(~k) +

1

2

∑

~k,σ

~ωσ(~k)

2Qσ(~k)Q

∗σ(~k)

+1

6

(~

2

) 32 ∑

~k,~k ′, ~k ′′

∑

σ,σ′,σ′′

Aσ,σ′,σ′′(~k, ~k ′, ~k ′′)/√N

√

ωσ(~k)ωσ′(~k ′)ωσ′′( ~k ′′)Qσ(~k, t)Qσ′(~k ′)Qσ′′( ~k ′′)

Qσ(~k, t) = b†σ(~k) + bσ(−~k)

Pσ(~k, t) = 2i~

(

b†σ(~k)− bσ(−~k))

⇒ [P , Q]− =~

iwith [b, b†] = 1


H =∑

k,σ

Pσ(~k).Q∗σ(~k) + P

∗σ (~k)

.Qσ(~k)− L

=1

2

∑

k,σ

8ωσ(~k)

~Pσ(~k)P

∗σ (~k) +

1

2

∑

~k,σ

~ωσ(~k)

2Qσ(~k)Q

∗σ(~k)

+1

6

(~

2

) 32 ∑

~k,~k ′, ~k ′′

∑

σ,σ′,σ′′

Aσ,σ′,σ′′(~k, ~k ′, ~k ′′)/√N

√

ωσ(~k)ωσ′(~k ′)ωσ′′( ~k ′′)Qσ(~k, t)Qσ′(~k ′)Qσ′′( ~k ′′)

With the transformation

Qσ(~k, t) = b+σ (~k, t)e

+iωσ(~k)t + b−σ (~k, t)e−iωσ(~k)t

defined and the condition for real amplitudes Eq. ?? b−σ (−~k)def=(

b+σ (~k))∗

, we may translate

Qσ(~k, t) = b†I,σ(~k, t)e+iωσ(

~k)t + bI,σ(−~k, t)e−iωσ(~k)t

= ei~H0t b†I,σ(~k, t)e

− i~H0t + e

i~H0t bI,σ(−~k, t)e−

i~H0t

which expresses the operators bI in the interaction picture with the harmonic terms defining theunperturbed Hamiltonian H0.

The corresponding Hamiltonian is6

H =∑

k,σ

~ωσ(~k)

(

b†σ(~k)bσ(~k) +1

2

)

+1

6

(~

2

) 32 ∑

~k,~k ′, ~k ′′

∑

σ,σ′,σ′′

Aσ,σ′,σ′′(~k, ~k ′, ~k ′′)/√N

√

ωσ(~k)ωσ′(~k ′)ωσ′′( ~k ′′)

(

b†σ(~k) + bσ(~k)

)(

b†σ′(~k′) + bσ′(~k ′)

)(

b†σ′′( ~k′′) + bσ′′( ~k ′′)

)

=

±∑

k,σ

1

2~ωσ(~k)b−σ(~k)bσ(~k)

+1

6

(~

2

) 32 ∑

~k,~k ′, ~k ′′

±∑

σ,σ′,σ′′

Aσ,σ′,σ′′(~k, ~k ′, ~k ′′)/√N

√

ωσ(~k)ωσ′(~k ′)ωσ′′( ~k ′′)bσ(~k)bσ′(~k ′)bσ′′( ~k ′′) (7.46)

We have extended the sum over positive and negative values of σ, where we introduced

bσ =

b†σ(~k) for σ > 0

bσ(−~k) for σ < 0

The many-phonon wave function has the form

|Φ(t)〉 =∑

~n

|~n〉c~n(t) =∑

~m

∏

~k,σ>0

(

b†σ(~k)

)m~k,σ

|O〉c~m(t)

where |~n〉 is the number representation forming the many-particle basisset and |O〉 is the vacuumstate.

6Compared to Eq. Chernatynskiy-B4 the creation and annihilation operators translate as

b†σ(~k)→ ia∗−~k,σ and bσ(~k)→ −ia~k,σ

In addition there is an unexplained factor√

VN . Note also that the k-vectors in our notation differ in sign from those

of Chernatynskiy and Ziman.


7.5.3 Random-phase approximation and rate equation

The random-phase approximation for quantum systems is derived in the appendix ’Random-Phase

approximation” of ΦSX:Quantum physics. In our case the perturbation is the interaction betweenthe phonons. While the interaction is time-independent we make a Gedankenexperiment, where theinteraction is switched on and switched off at some later time. We choose ω = 0 and W thephonon-phonon interaction operator.

In the limit of the Fermi’s golden rule we obtain

RATE EQUATION USING FERMI’S GOLDEN RULE

∂tP~n(t) =∑

~m

(

Γ~n←~mP~m(t)− Γ~m←~nP~n(t))

with

Γ~n←~m =2π

~

∣∣〈φn|W |φm〉

∣∣2δ(En − Em − ~ω) = Γ~m←~n

Note that this approximation rests on a number of seemingly contradicting assumptions.

Here P~n(t) are the probabilities of many-phonon states |~n〉. In our case the interaction is time inde-pendent and therefore ~ω = 0.

In the random-phase approximation the phonon density matrix collapses into a diagonal matrixin the phonon-number representation. As in the classical case the random-phase approximationadditionally ensures that the off-diagonal elements do not develop. This approximation implies theabsence of entangled phonons.

The change of occupation numbers can be obtained from the rate equation

∂nλ∂t=∑

~n

nλ∂tP~n(t) =∑

~n, ~m

nλ

(

Γ~n←~mP~m(t)− Γ~m←~nP~n(t))

=∑

~n, ~m

(nλ −mλ)Γ~n←~mP~m

Let us now consider all the potential transitions from a given configuration |~n〉 that change theoccupation number of the phonon λ:

• There is one matrix element having a product of three creation operators and one with threeannihilation operators. Because the argument of δ(~ωλ + ~ωλ′′ + ~ωλ′) never vanishes, thesematrix elements do not contribute. In other words such a process always violates energyconservation.

• There are matrix elements where two phonons are created and one is annihilated. There aretwo possibilities, namely that phonon λ is created or that it is annihilated.

• There are matrix elements where two phonons are annihilated and one is created There are twopossibilities, namely that phonon λ is created or that it is annihilated.


dnλdt=1

2

∑

λ′,λ′′

∑

~n

P~n

×2π

~

∣∣∣∣〈nλ + 1, nλ′ − 1, nλ′′ − 1|W |nλ, nλ′ , nλ′′〉

∣∣∣∣

2

δ(~ωλ − ~ωλ′ − ~ωλ′′)

+2π

~

∣∣∣∣〈nλ + 1, nλ′ + 1, nλ′′ − 1|W |nλ, nλ′ , nλ′′〉

∣∣∣∣

2

δ(~ωλ + ~ωλ′ − ~ωλ′′)

+2π

~

∣∣∣∣〈nλ + 1, nλ′ − 1, nλ′′ + 1|W |nλ, nλ′ , nλ′′〉

∣∣∣∣

2

δ(~ωλ − ~ωλ′ + ~ωλ′′)

−2π~

∣∣∣∣〈nλ − 1, nλ′ + 1, nλ′′ + 1|W |nλ, nλ′ , nλ′′〉

∣∣∣∣

2

δ(−~ωλ + ~ωλ′ + ~ωλ′′)

−2π~

∣∣∣∣〈nλ − 1, nλ′ − 1, nλ′′ + 1|W |nλ, nλ′ , nλ′′〉

∣∣∣∣

2

δ(−~ωλ − ~ωλ′ + ~ωλ′′)

−2π~

∣∣∣∣〈nλ − 1, nλ′ + 1, nλ′′ − 1|W |nλ, nλ′ , nλ′′〉

∣∣∣∣

2

δ(−~ωλ + ~ωλ′ − ~ωλ′′)

The factor 12 compensates the double counting of pairs (λ′, λ′′) and (λ′′, λ′). The transitions 3 and4 as well as the transitions 5 and 6 can be combined into the same term after exchanging λ′ and λ′′

in one of the terms.

NEGLECT OF DOUBLE EXCITATIONS AND ELASTIC DISTORTIONS

In the above equation we implicitly assumed that all three participating phonons are distinct, that isλ 6= λ′ 6= λ′′. This implies that we ignore cases where two phonons with the same quantum numberare simultaneously deleted or created.Respectively we ignore that a three-phonon process creates and annihilates the same phonon, resultingin an effective single phonon scattering event. because of the energy and momentum conservationthe latter effect does only occur for phonons at the Gamma point with zero energy. This it is probablya process such as thermal expansion.

The Hamiltonian Eq. 7.46 is of the form

W =∑

λ,λ′,λ′′

Xλ,λ′,λ′′(b†λ + b−λ)(b

†λ′ + b−λ′)(b

†λ′′ + b−λ′′) (7.47)

where −λ stands for −~k, σ.We need to evaluate matrix elements of the form

⟨

nλ + 1, nλ′ + 1, nλ′ − 1〉∣∣∣∣b†λb

†λ′ bλ′′

∣∣∣∣nλ, nλ′ , nλ′′

⟩

=

√(

nλ + 1

)(

nλ′ + 1

)

nλ′′

⟨

nλ + 1, nλ′ + 1, nλ′ − 1〉∣∣∣∣b†λbλ′ bλ′′

∣∣∣∣nλ, nλ′ , nλ′′

⟩

=

√(

nλ + 1

)

nλ′nλ′′

where we used

|n〉 = 1√n!

(b†)n |0〉

b†|n〉 = 1√

(n + 1)!

(b†)n+1 |0〉

√n + 1 = |n + 1〉

√n + 1

b|n〉 = 1√

(n − 1)!(b†)n−1 |0〉

√n = |n − 1〉

√n

b2|n〉 = b|n − 1〉√n = |n − 2〉

√

n(n − 1)


but we ignored matrix elements where two or more operators act on the same phonon.

dnλdt=1

2

∑

λ′,λ′′

∑

~n

P~n

2π

~

∣∣∣∣6Xλ,−λ′,−λ′′

∣∣∣∣

2

(nλ + 1)nλ′nλ′′δ(~ωλ − ~ωλ′ − ~ωλ′′)

+2π

~2

∣∣∣∣6Xλ,λ′,−λ′′

∣∣∣∣

2

(nλ + 1)(nλ′ + 1)nλ′′δ(~ωλ + ~ωλ′ − ~ωλ′′)

−2π~

∣∣∣∣6X−λ,λ′,λ′′

∣∣∣∣

2

nλ(nλ′ + 1)(nλ′′ + 1)δ(~ωλ − ~ωλ′ − ~ωλ′′)

−2π~2

∣∣∣∣6X−λ,−λ′,λ′′

∣∣∣∣

2

nλnλ′(nλ′′ + 1)δ(~ωλ + ~ωλ′ − ~ωλ′′)

We have changed the sign of the arguments of the delta functions in the last two terms. The factorsix enters because there are several terms of the Hamiltonian connecting initial and final states. Theydiffer in the permutation of the three indices.

ASSUMPTION OF UNCORRELATED PHONON OCCUPATIONS

Next we assume that the mean occupation numbers of the phonons are uncorrelated, that is

〈nλnλ′〉 =∑

~n

P~nnλnλ′ =

(∑

~n

P~nnλ

)(∑

~n

P~nnλ′

)

= 〈nλ〉〈nλ′〉 for λ 6= λ′

respectively,

〈nλnλ′nλ′′〉 =∑

~n

P~nnλnλ′nλ′′ =

(∑

~n

P~nnλ

)(∑

~n

P~nnλ′

)(∑

~n

P~nnλ′′

)

= 〈nλ〉〈nλ′〉〈nλ′′〉 for λ 6= λ′ 6= λ′′

The mean occupation number will be described by the same symbol as the occupation number becausethe correct meaning becomes obvious from the context.


This yields

dnλdt=1

2

∑

λ′,λ′′

2π

~

∣∣∣∣6Xλ,−λ′,−λ′′

∣∣∣∣

2

δ(~ωλ − ~ωλ′ − ~ωλ′′)︸︷︷︸

Lλλ′ ,λ′′

(nλ + 1)nλ′nλ′′

+22π

~

∣∣∣∣6Xλ,λ′,−λ′′

∣∣∣∣

2

δ(~ωλ + ~ωλ′ − ~ωλ′′)︸︷︷︸

Lλ,λ′

λ′′

(nλ + 1)(nλ′ + 1)nλ′′

− 2π~

∣∣∣∣6X−λ,λ′,λ′′

∣∣∣∣

2

δ(~ωλ − ~ωλ′ − ~ωλ′′)︸︷︷︸

Lλ′ ,λ′′λ

nλ(nλ′ + 1)(nλ′′ + 1)

−2 2π~

∣∣∣∣6X−λ,−λ′,λ′′

∣∣∣∣

2

δ(~ωλ + ~ωλ′ − ~ωλ′′)︸︷︷︸

Lλ′′λ,λ′

nλnλ′(nλ′′ + 1)

This result can be expressed in a more convenient form

dnλdt=∑

λ′,λ′′

1

2Lλλ′,λ′′(nλ + 1)nλ′nλ′′ + L

λ,λ′λ′′ (nλ + 1)(nλ′ + 1)nλ′′

−12Lλ

′,λ′′λ nλ(nλ′ + 1)(nλ′′ + 1)− Lλ

′′λ,λ′nλnλ′(nλ′′ + 1)

where the L, defined below in Eq. 7.48, are called intrinsic probabilities of transition. (Comparewith Chernatynski Gl. A5.)

Lλλ′,λ′′ =2π

~

∣∣∣∣6Xλ,−λ′,−λ′′

∣∣∣∣

2

δ(~ωλ − ~ωλ′ − ~ωλ′′)

Lλ,λ′

λ′′ =2π

~

∣∣∣∣6Xλ,λ′,−λ′′

∣∣∣∣

2

δ(~ωλ + ~ωλ′ − ~ωλ′′)

Lλ′,λ′′λ =

2π

~

∣∣∣∣6X−λ,λ′,λ′′

∣∣∣∣

2

δ(~ωλ − ~ωλ′ − ~ωλ′′)

Lλ′′λ,λ′ =

2π

~

∣∣∣∣6X−λ,−λ′,λ′′

∣∣∣∣

2

δ(~ωλ + ~ωλ′ − ~ωλ′′)

It may be helpful to notice that the lower indices of L obtain a negative index of X and a negativesign for ~ω in the delta function.

With

Xν,ν ′,ν ′′Eq. 7.47=1

6

(~

2

) 32 Aν,ν ′,ν ′′/

√N√

ωνων ′ων ′′

we obtain the intrinsic probabilities of transition. (Compare with Chernatynski Gl. A5.)

Lλλ′,λ′′ =2π

~

(~

2

)3 |Aν,−ν ′,−ν ′′ |2 /Nωνω−ν ′ω−ν ′′

δ(~ωλ − ~ωλ′ − ~ωλ′′)

Lλ′,λ′′λ =

2π

~

(~

2

)3 |A−ν,ν ′,ν ′′ |2 /Nω−νων ′ων ′′

δ(~ωλ − ~ωλ′ − ~ωλ′′) (7.48)


Using the symmetry property that the three phonon scattering matrix A becomes complex con-jugate, when the sign of all k-vectors is reversed, we find that the two remaining intrinsic scatteringprobabilities are identical, that is

Lλλ′,λ′′ = Lλ′,λ′′λ (7.49)

QUANTUM PHONON-BOLTZMANN EQUATION

∂tnλ + ~vλ~∇nλ =∑

λ′,λ′′

1

2Lλλ′,λ′′(nλ + 1)nλ′nλ′′ + L

λ,λ′λ′′ (nλ + 1)(nλ′ + 1)nλ′′

−12Lλ

′,λ′′λ nλ(nλ′ + 1)(nλ′′ + 1)− Lλ

′′λ,λ′nλnλ′(nλ′′ + 1)

(7.50)

with λ = (~k, σ), λ′ = (~k ′, σ′), and λ = ( ~k ′′, σ′′), ~vλ := ~∇~kωλ, and

L~k,σ~k ′,σ′, ~k ′′,σ′′

Eq. 7.48=2π

~

(~

2

)3

∣∣∣Aσ,σ′,σ′′(~k,−~k ′,− ~k ′′)

∣∣∣

2

/Nωσ(~k)ωσ′(−~k ′)ωσ′′(− ~k ′′)

δ

(

~ωσ(~k)− ~ωσ′(~k ′)− ~ωσ′′( ~k ′′))

L~k ′,σ′, ~k ′′,σ′′

~k,σ

Eq. 7.48=2π

~

(~

2

)3

∣∣∣Aσ,σ′,σ′′(−~k, ~k ′, ~k ′′)

∣∣∣

2

/Nωσ(−~k)ωσ′(~k ′)ωσ′′( ~k ′′)

δ

(

−~ωσ(~k) + ~ωσ′(~k ′) + ~ωσ′′( ~k ′′))

(7.51)

Eq. 7.50 is identical to Chernatynskiy-A5. The different signs are explained by his definitionresulting in ∂t fλ + ~vλ~∇fλ = −

∑

λ′,λ′′ Pλ,λ′,λ′′ .

7.5.4 Classical limit

The classical limit of the phonon Boltzmann equation is obtained for dilute phonon gases, that is ifnλ << 1. In this case all triple products of phonon occupations are ignored.

∂tnλ + ~vλ~∇nλ =∑

λ′,λ′′

1

2Lλλ′,λ′′nλ′nλ′′ + L

λ,λ′λ′′

[

n′′λ + nλ′nλ′′ + nλnλ′′

]

−12Lλ

′,λ′′λ

[

nλ + nλnλ′ + nλnλ′′

]

− Lλ′′λ,λ′nλnλ′

(7.52)

Pay attention to the terms linear in the occupations.!!!

Normally the high-temperature limit is identical to the classical limit. Here, the opposite seemsto be true. Only at low temperatures the phonon gas is dilute.

A good description of the classical limit seems appropriate for the optical phonons below the Debyetemperature. The current-carrying acoustic modes have occupations that are larger than classicalresult.

7.5.5 Linearized Boltzmann equation

We introduce a new variable Φλ (Chernatynsky A6), which plays the role of a quasi Fermi level.

nλ = neq(ǫλ −Φλ) ≈ neqλ −Φλ

∂neqλ∂ǫλ


where

neq =

(eβ(ǫ−µ) − 1

)−1for Bosons

(eβ(ǫ−µ) + 1

)−1for Fermions

(eβ(ǫ−µ)

)−1for a Boltzmann gas

is the Bose, Fermi or Boltzmann distribution.For phonons we use the Bose distribution with µ = 0 and ǫλ = ~ωλ.

∂ǫneqλ = ∂ǫ

1

eβ(ǫ−µ) − 1 = −βeβ(ǫ−µ) 1

(eβ(ǫ−µ) − 1)2 = −β(neq)2

(

(neq)−1 + 1

)

= −βneq(

neq + 1

)

so that

nλ = neq(ǫλ −Φλ) ≈ neqλ + βΦλneqλ (1 + neqλ )

︸︷︷︸

δnλ

This yields

δnλ =

[

βΦλ(1 + neqλ )

]

neqλ

δ(1 + nλ) =

[

βΦλneqλ

]

(1 + neqλ )

δ

[

(1 + neqλ )neqλ′ n

eqλ′′

]

= β

[

Φλn(eq)λ +Φλ′(1 + n

(eq)λ′ ) + Φλ′′(1 + n

(eq)λ′′ )

]

(1 + neqλ )neqλ′ n

eqλ′′

δ

[

(1 + nλ)(1 + nλ′)nλ′′

]

= β

[

±Φλn(eq)λ ±Φλ′n(eq)λ′ +Φλ′′(1 + n(eq)λ′′ )

]

(1 + neqλ )(1 + neqλ′ )n

eqλ′′

Following Chernatynskiy (Eq. Chenatynskiy-B10), we define the equilibrium transition rate Λ

Λλλ′,λ′′ := Lλλ′,λ′′(n

eqλ + 1)n

eqλ′ n

eqλ′′

Λλ′,λ′′λ := Lλ

′,λ′′λ neqλ (n

eqλ′ + 1)(n

eqλ′ + 1) (7.53)

The intrinsic probabilities of transition L have been defined in Eq. 7.48.We obtain Eq. 7.50 to first order in the shift of the quasi-chemical potential in the form

∂tnλEq. 7.50=

∑

λ′,λ′′

[

+1

2Lλλ′,λ′′(nλ + 1)nλ′nλ′′ − Lλ

′′λ,λ′nλnλ′(nλ′′ + 1)

+ Lλ,λ′

λ′′ (nλ + 1)(nλ′ + 1)nλ′′ −1

2Lλ

′,λ′′λ nλ(nλ′ + 1)(nλ′′ + 1)

]

=∑

λ′,λ′′

[

+1

2Λλλ′,λ′′ − Λλ

′′λ,λ′ + Λ

λ,λ′λ′′ −

1

2Λλ

′,λ′′λ

]

+β∑

λ′,λ′′

[

+1

2Λλλ′,λ′′

(

Φλn(eq)λ +Φλ′(1 + n

(eq)λ′ ) + Φλ′′(1 + n

(eq)λ′′ )

)

− Λλ′′λ,λ′(

Φλ′′n(eq)λ′′ +Φλ(1 + n

(eq)λ ) + Φλ′(1 + n

(eq)λ′ )

)

+ Λλ,λ′

λ′′

(

Φλn(eq)λ +Φλ′n

(eq)λ′ +Φλ′′(1 + n

(eq)λ′′ )

)

− 12Λλ

′,λ′′λ

(

Φλ′n(eq)λ′ +Φλ′′n

(eq)λ′′ +Φλ(1 + n

(eq)λ )

)]

+O(Φ2)

= β∑

λ′,λ′′

[

+1

2Λλλ′,λ′′

(

Φλ′ +Φλ′′

)

− Λλ′′λ,λ′(

Φλ +Φλ′

)

+ Λλ,λ′

λ′′ Φλ′′ −1

2Λλ

′,λ′′λ Φλ

]

+O(Φ2)


In the last step we exploited the equilibrium condition that says that the equilibrium contributionis stationary so that the zeroth order term vanishes. Using exactly the same condition all termsproportional to neq cancel as well.

As a consequence of detailed balance (to be proven!!!)

Λλλ′,λ′′ = Λλ′,λ′′λ (7.54)

Proof: We start with the definition of the equilibrium transition rate Eq. 7.53.

Lλλ′,λ′′(neqλ + 1)n

eqλ′ n

eqλ′′

Eq. 7.53= Λλλ′,λ′′

?= Λλ

′,λ′′λ

Eq. 7.53= Lλ

′,λ′′λ neqλ (n

eqλ′ + 1)(n

eqλ′ + 1)

The intrinsic probabilities of transition are invariant against replacing upper and lower indices accordingto Eq. 7.49. Thus we need to show

(neqλ + 1)neqλ′ n

eqλ′′?= neqλ (n

eqλ′ + 1)(n

eqλ′ + 1) ⇔ neqλ + 1

neqλ

neqλ′

neqλ′ + 1

neqλ′′

neqλ′′ + 1

?= 1

For the Bose distribution neqλ +1

neqλ= eβ~ωλ . Thus

eβ(~ωλ−~ωλ′−~ωλ′′ = 1 ⇔ ~ωλ − ~ωλ′ − ~ωλ′′ = 0

This is nothing but the energy conservation condition inherent in the definition of the intrinsic prob-abilities of transition L. This completes the proof of Eq. 7.54.

Eq. 7.54 simplifies the expression further to the scattering term in the linearized Phonon Boltz-mann equation

LINEARIZED QUANTUM PHONON-BOLTZMANN EQUATION

β~ωλneqλ (1 + n

eqλ )βkB

︸︷︷︸

∂T neqλ

(

∂tT + ~vλ~∇T)

+ βneqλ (1 + neqλ )

︸︷︷︸

∂ǫneqλ

(

∂tΦλ + ~vλ~∇Φλ)

= β∑

λ′,λ′′

[

Λλ′′λ,λ′

(

Φλ′′ −Φλ −Φλ′)

+1

2Λλ

′,λ′′λ

(

Φλ′ +Φλ′′ −Φλ)]

(7.55)

with λ = (~k, σ), λ′ = (~k ′, σ′), and λ = ( ~k ′′, σ′′), and

Λ~k,σ~k ′,σ′, ~k ′′,σ′′

Eq. 7.53=2π

~

(~

2

)3

∣∣∣Aσ,σ′,σ′′(~k,−~k ′,− ~k ′′)

∣∣∣

2

/Nωσ(~k)ωσ′(−~k ′)ωσ′′(− ~k ′′)

×δ(

~ωσ(~k)− ~ωσ′(~k ′)− ~ωσ′′( ~k ′′))(neqσ (~k) + 1

)neqσ′ (

~k ′)neqσ′′ ( ~k′′)

Λ~k ′,σ′, ~k ′′,σ′′

~k,σ

Eq. 7.53=2π

~

(~

2

)3

∣∣∣Aσ,σ′,σ′′(−~k, ~k ′, ~k ′′)

∣∣∣

2

/Nωσ(−~k)ωσ′(~k ′)ωσ′′( ~k ′′)

×δ(

−~ωσ(~k) + ~ωσ′(~k ′) + ~ωσ′′( ~k ′′))

neqσ (~k)(neqσ′ (

~k ′) + 1)(neqσ′′ (

~k ′′) + 1)

(7.56)

This equation Eq. 7.55 compares to Eq. (Chernatynskiy-A9) and Eq. (Chernatynskiy-B10). Notethat his P is defined with the opposite sign. There is a deviation between our Eq. 7.56 andEq. (Chernatinskyi-B10): In the matrix element with one lower index the argument of the Delta


function has the opposite sign for ~ωλ′ and ~ωλ′′ , while we obtain the same sign. Our sign stemsfrom the fact that one phonon with λ is annihilated, while two phonons with λ′, λ′′ are created. Theexpression of Chernatinskiy is derived from his Eq. (Chernatinskiy-B9), where the sign conventionis similar to our expression. This error may be caused by copying the delta function term from B9without changing the indices.

The equation Eq. 7.55 is also equivalent to Eq. (Broido-3)[43], if we translate

W+λ,λ′,λ′′ = Λλ,λ′λ′′ and W−λ,λ′,λ′′ = Λλ

λ′,λ′′

This translation is reversed in the definition of Broido’s W± Eq. (Broido-4) when compared toour Λ. This can be done on the basis of detailed balance, which has been applied in the derivationbefore. Note that there is an error in Eq. (Broido-4), which is corrected in Eq.2 of a later of hispapers[44]. Therefore we compare to the latter paper.

W+λ,λ′,λ′′ =2π

~N

(~

2

)3 |A−λ,−λ′,λ′′ |2ωλωλ′ωλ′′

δ(~ωλ + ~ωλ′ − ~ωλ′′)(neqλ + 1)(neqλ′ + 1)neqλ′′ (= Λλ,λ′

λ′′ )

W−λ,λ′,λ′′ =2π

~N

(~

2

)3 |A−λ,λ′,λ′′ |2ωλωλ′ωλ′′

δ(~ωλ − ~ωλ′ − ~ωλ′′)(neqλ + 1)neqλ′ neqλ′′ (= Λλλ′,λ′′)

The agreement is there when the sign of the indices of the three-phonon scattering matrix elementsare given the opposite sign. Such a change does not affect the result, because the three phonon scat-tering matrix elements are only made complex conjugate by such a sign change. Broido furthermoreassumes inversion symmetry of the frequencies in k-space.

Chapter 8

The Hartree-Fock approximation andexchange

The Hartree-Fock method[28, 45, 46] is an electronic structure method that is the work-horseof quantum chemistry. Today it plays an important role as a starting point of more accurate andinvolved methods. The Hartree-Fock method had a predecessor, the Hartree method[47, 48], which,however, does not play an important role in practice.

The basic idea of the Hartree-Fock method is to restrict the wave functions to single Slaterdeterminants[28]. For this Slater determinant, the energy is determined as expectation value ofthe true many-particle Hamiltonian. That is, the Hartree-Fock approximation takes the completeelectron-electron interaction into account. Furthermore the one-particle orbitals are optimized toyield the lowest energy.

One can easily show that the Hartree-Fock energy always provides an upper bound for the energy.The Hamilton operator is, up to a constant, positive definite and the ground state is the state withthe lowest energy. As we restricted our wave function to single Slater determinants, the wave functionis, normally, not the ground state and therefore higher in energy than the ground-state energy.

After having defined Slater determinants in the previous section, in this section we will gain somefamiliarity with them. We will also see why systems of non-interacting many-particle problems canbe described by a set of one-particle wave functions alone, something that we implicitly assumedin the section about non-interacting electrons. Furthermore, we will become familiar with the mostimportant contributions of the electron correlation, namely Hartree energy and exchange.

8.1 One-electron and two-particle operators

The Born-Oppenheimer Hamiltonian Eq. 2.4 for an N-electron system has the form

E = ENN + 〈Ψ|N∑

i=1

hi +1

2

∑

i 6=jWi ,j |Ψ〉

Where

ENN(~R1, . . . , ~RM)def=1

2

M∑

i 6=j

e2ZiZj

4πǫ0|~Ri − ~Rj |(8.1)

is the electrostatic repulsion between the nuclei, and the operators acting on the electrons can bedivided into one-particle terms hi and two-particle terms Wi ,j .

151

152 8 THE HARTREE-FOCK APPROXIMATION AND EXCHANGE

The one-particle terms describe the kinetic and potential energy of the electrons in an externalpotential vext(~r)

hi =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉[−~22me

~∇2~ri + vext(~ri)]

〈~x1, . . . , ~xN | (8.2)

The external potential describes the electrostatic attraction between electrons and nuclei.

vext(~r ; ~R1, . . . , ~RM) = −M∑

j=1

e2Zj

4πǫ0|~r − ~Rj |(8.3)

The two-particle operator describes the interaction between the electrons

Wi ,j =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉e2

4πǫ0|~ri − ~rj |〈~x1, . . . , ~xN | (8.4)

The indices on hi and Wi ,j determine, onto which electron coordinates the operator acts.The one-particle operators act on the coordinates of one particle at a time. The kinetic energy

and the external potential are examples for one-particle operators, because they can be determinedfor each particle individually and then be summed up. The interaction energy Eq. 8.4 on the otherhand depends on the coordinates of two particles simultaneously and is therefore a true two-particleoperator. It is the two-particle term that is the cause for the dazzling complexity of many-particlephysics.

Let us consider a general operator in real space

A =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉〈~x1, . . . , ~xN |︸︷︷︸

1

·A∫

d4x ′1 · · ·∫

d4x ′N |~x ′1, . . . , ~x ′N〉〈~x ′1, . . . , ~x ′N |︸︷︷︸

1

=

∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N |~x1, . . . , ~xN〉

·〈~x1, . . . , ~xN |A|~x ′1, . . . , ~x ′N〉〈~x ′1, . . . , ~x ′N |A general matrix element has therefore 2N arguments.

• If the matrix element of an operator has the special form

〈~x1, . . . , ~xN |A|~x ′1, . . . , ~x ′N〉 =N∑

i=1

A(~xi , ~x′i )∏

j 6=iδ(~xj − ~x ′j)

we call the operator a one-particle operator.

• If furthermore the function A(~x, ~x ′) has the special form that it is diagonal in the primed andunprimed arguments, i.e.

A(~x, ~x ′) = a(~x)δ(~x − ~x ′)we call the one-particle operator local.

• If the matrix element of an operator has the special form

〈~x1, . . . , ~xN |A|~x ′1, . . . , ~x ′N〉 =1

2

N∑

i ,j=1

A(~xi , ~xj , ~x′i , ~x′j)∏

k 6=i ,jδ(~xk − ~x ′k)

we call the operator a two-particle operator.

8 THE HARTREE-FOCK APPROXIMATION AND EXCHANGE 153

• If furthermore the function A(~x1, ~x2, ~x ′1, ~x ′2) has the special form that it is diagonal in theprimed and unprimed arguments, i.e.

A(~x1, ~x2, ~x ′1, ~x ′2) = a(~x1, ~x2)δ(~x1 − ~x ′1)δ(~x2 − ~x ′2)

we call the two-particle operator local.

8.2 Expectation values of Slater determinants

Let us now work out the expectation values for the one-particle and two-particle operators for aSlater determinant.

8.2.1 Expectation value of a one-particle operator

Here, we will work out the expectation value of a one-particle operator for a Slater determinant. Anexample for such a one-particle operator is the non-interacting part of the Hamiltonian.

Explicit example for the two-particle wave function

The two-particle Slater determinant of two one-particle orbitals |φa〉 and |φb〉 has the form

〈~r1~r2|Ψ〉︸︷︷︸

Ψ(~r1,~r2)

=1√2

[

〈~r1|φa〉〈~r2|φb〉 − 〈~r1|φb〉〈~r2|φa〉]

︸︷︷︸

φa(~r1)φb(~r2)−φb(~r1)φa(~r2)

(8.5)

To avoid unnecessary complication we drop the spin-coordinates, that is we consider spin-less fermions.As an specific example for a one-particle operator, we consider the particle-density operator n(~r)

defined as

n(~r) =

∫

d3r1

∫

d3r2 |~r1, ~r2〉[

δ(~r − ~r1) + δ(~r − ~r2)]

〈~r1, ~r2|


The electron density is

n(~r) = 〈Ψ|n(~r)|Ψ〉

=

∫

d3r1

∫

d3r2 Ψ∗(~r1, ~r2)

[

δ(~r − ~r1) + δ(~r − ~r2)]

Ψ(~r1, ~r2)

Ψ(~r1,~r2)=−Ψ(~r2,~r1)=

∫

d3r1

∫

d3r2 Ψ∗(~r1, ~r2)δ(~r − ~r1)Ψ(~r1, ~r2)

+

∫

d3r1

∫

d3r2 Ψ∗(~r2, ~r1)δ(~r − ~r2)Ψ(~r2, ~r1)

~r1↔~r2= 2

∫

d3r1

∫

d3r2 Ψ∗(~r1, ~r2)δ(~r1 − ~r)Ψ(~r1, ~r2)

Eq. 8.5= 2

∫

d3r1

∫

d3r21√2

[

φa(~r1)φb(~r2)− φa(~r2)φb(~r1)]∗

· δ(~r − ~r1)1√2

[

φa(~r1)φb(~r2)− φa(~r2)φb(~r1)]

=

∫

d3r1

∫

d3r2 φ∗a(~r1)φ

∗b(~r2)δ(~r − ~r1)φa(~r1)φb(~r2)

−∫

d3r1

∫

d3r2 φ∗a(~r1)φ

∗b(~r2)δ(~r − ~r1)φa(~r2)φb(~r1)

−∫

d3r1

∫

d3r2 φ∗a(~r2)φ

∗b(~r1)δ(~r − ~r1)φa(~r1)φb(~r2)

+

∫

d3r1

∫

d3r2 φ∗a(~r2)φ

∗b(~r1)δ(~r − ~r1)φa(~r2)φb(~r1)

= 〈φa|~r〉〈~r |φa〉〈φb|φb〉︸︷︷︸

=1

−〈φa|~r〉〈~r |φb〉〈φb|φa〉︸︷︷︸

=0

−〈φb|~r〉〈~r |φa〉〈φa|φb〉︸︷︷︸

=0

+〈φb|~r〉〈~r |φb〉〈φa|φa〉︸︷︷︸

=1

= φ∗a(~r)φa(~r) + φ∗b(~r)φb(~r)

Let us mention a few observations:

• The number of terms is drastically reduced just because we have used an orthonormal set ofone-particle wave functions. This is the sole reason for using orthonormal basissets in many-particle physics.

• The sum over particles is turned into a sum over orbitals. This is true for a Slater determinant,but not for a general many particle states.

• The antisymmetry of the wave function ensures that the same expectation value is obtained ifwe work out a property of the first or the second electron.1

General derivation

Now, we will determine the same result in its full generality: First, we evaluate only the matrix elementof the one-particle operator h1, that acts only on the coordinates of only one particle, namely thefirst. This result is then generalized for the other particles and then summed up. Thus, we obtainthe expectation value of the non-interacting part of the Hamiltonian.

1The wave function is, up to a sign, the same when the coordinates of the coordinates of the first and the secondelectron are interchanged. The sign drops out for expectation values, because the wave function enters twice. Example:Using the antisymmetry of the wave function ψ(x, x ′) = −ψ(x ′, x) one can convert the density ρ1 of the first particleρ1(x) =

∫dx ′ ψ∗(x, x ′)ψ(x, x ′) = (−1)2

∫dx ′ ψ∗(x ′, x)ψ(x ′, x) = ρ2(x) into the density ρ2(x) of the second.


To be concise, we write here the detailed expressions for the matrix elements of the one-particleHamilton operator, we have in mind

〈φ|h|φ〉 =∫

d4x

∫

d4x ′ 〈φ|~x〉

δ(~r − ~r ′)δσ,σ′(

− ~2

2me~∇2 + v(~r)

)

︸︷︷︸︷︸︸︷

〈~x |h|~x ′〉〈~x ′|φ〉

=∑

σ

∫

d3r φ∗(~r , σ)

[−~22me

~∇2 + vext(~r)]

φ(~r , σ) (8.6)

The one-particle Hamiltonian hj acting on the j-th particle in a many particle wave function Ψ〉yields the expectation value

〈Ψ|hj |Ψ〉 =∫

d4x1 · · ·∫

d4xN Ψ∗(~x1, . . . , ~xN)

(−~22me

~∇2j + vext(~rj))

Ψ(~x1, . . . , ~xN)

so that the matrix element for two product wave functions is

〈φ1, . . . , φN |hj |ψ1, . . . , ψN〉 = 〈φ1|ψ1〉 · · · 〈φj |h|ψj 〉 · · · 〈φN |ψN〉 (8.7)

The Slater determinant, given in Eq. 3.21, has the form

〈~x1, . . . , ~xN |Ψ〉 =1√N!

N∑

i1,...,iN=1

ǫi1,...,iN 〈~x1|φi1〉 · · · 〈~xN |φiN 〉 (8.8)

Thus, the matrix element of a one-particle operator Eq. 8.2 is

〈Ψ|N∑

i=1

hi |Ψ〉 = N〈Ψ|h1|Ψ〉

Eq. 8.8= N · 1

N!

N∑

i1,i2,...,iN

N∑

j1,j2,...,jN

ǫi1,i2,...,iN 〈φi1 , . . . , φiN |h1|φj1 , . . . , φjN 〉ǫj1,j2,...,jN

Eq. 8.7= N

1

N!

N∑

i1,i2,...,iN

N∑

j1,j2,...,jN

ǫi1,i2,...,iN ǫj1,j2,...,jN 〈φi1 |h1|φj1〉〈φi2 |φj2〉︸︷︷︸

δi2 ,j2

· · · 〈φiN |φjN 〉︸︷︷︸

δiN ,jN

= N1

N!

N∑

i1,j1=1

〈φi1 |h1|φj1〉N∑

i2,...,iN

ǫi1,i2,...,iN ǫj1,i2,...,iN

︸︷︷︸

δi1 ,j1 (N−1)!

=

N∑

j=1

〈φj |h1|φj 〉

In the last step we exploited, that the sum over j1 only contributes when j1 = i1. There is only oneorbital left if all the orbitals but the first are determined. Thus, the first orbital must be the samefor both Levi-Civita Symbols.

Thus, we obtain the expectation value of the one-particle Hamiltonian for a Slater determinant|Ψ〉 as


EXPECTATION VALUE OF A ONE-PARTICLE OPERATOR WITH A SLATER DETERMINANT

〈Ψ|N∑

i=1

hi |Ψ〉 =N∑

i=1

〈φi |h|φi 〉 (8.9)

Here, |Ψ〉 is a N-particle Slater determinant built from the one-particle orbitals |φi 〉, and∑N

i=1 hi isa one-particle operator.

Expectation values of common one-particle operators with Slater determinants

Analogously, the expectation value of any one-particle operator with a Slater determinant can berepresented by a sum over one-particle orbitals. Thus, we obtain the expression for the density

n(~r) = 〈Ψ|N∑

i=1

∑

σ

(

|~r , σ〉〈~r , σ|)

i|Ψ〉 =

N∑

i=1

∑

σ

φ∗i (~r , σ)φi(~r , σ) (8.10)

where

(

|~r〉〈~r |)

i

def=

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉δ(~r − ~ri)〈~x1, . . . , ~xN |

The expectation value of the kinetic energy for a Slater determinant has the form

Ekin = 〈Ψ|N∑

i=1

~p2i2me|Ψ〉 =

N∑

i=1

∑

σ

∫

d3r φ∗i (~r , σ)−~22me

~∇2φi(~r , σ) (8.11)

8.2.2 Expectation value of a two-particle operator

Let us now turn to the two-particle term. We introduce the symbol

W (~x, ~x ′) =e2

4πǫ0|~r − ~r ′|

The interaction is independent of the spin index. Furthermore it is local in the two coordinates.2

We proceed as for the expectation value of a one-particle operator and relate the interaction energybetween all particles to the interaction between the first two electrons. This is allowed because ofthe indistinguishability of the electrons: The interaction between any two electrons is identical to any

2A general two-particle operator has the form

W1,2 =

∫

dx1 · · ·∫

dxN

∫

dx ′1 · · ·∫

dx ′N |~x1, . . . , ~xN〉〈~x1, . . . , ~xN |W1,2|~x ′1, . . . , ~x ′N〉〈~x ′1, . . . , ~x ′N |

=

∫

dx1 · · ·∫

dxN

∫

dx ′1

∫

dx ′2|~x1, . . . , ~xN〉〈~x1, . . . , ~xN |W1,2|~x ′1, ~x ′2, ~x3, . . . , ~xN〉〈~x ′1, ~x ′2, ~x3, . . . , ~xN |

=

∫

dx1 · · ·∫

dxN

∫

dx ′1

∫

dx ′2|~x1, . . . , ~xN〉〈~x1, ~x2|W1,2|~x ′1, ~x ′2〉〈~x ′1, ~x ′2, ~x3, . . . , ~xN |

Thus, a general two-particle operator is nonlocal in the two particle coordinates and therefore its matrix elementsdepends on four arguments.


other pair. There are N(N−1)2 ordered pairs and therefore the interaction energy is

EW = 〈Ψ|1

2

N∑

i 6=jWi ,j |Ψ〉 =

N(N − 1)2

〈Ψ|W1,2|Ψ〉

=N(N − 1)2

1

N!

N∑

i1,i2,...,iN

N∑

j1,j2,...,jN

ǫi1,i2,...,iN ǫj1,j2,...,jN 〈φi1 , . . . , φiN |W12|φj1 , . . . , φjN 〉

=1

2 · (N − 2)!N∑

i1,i2,...,iN

N∑

j1,j2,...,jN

ǫi1,i2,...,iN ǫj1,j2,...,jN 〈φi1 , φi2 |W12|φj1 , φj2〉〈φi3 |φj3〉︸︷︷︸

δi3 ,j3

· · · 〈φiN |φjN 〉︸︷︷︸

δiN ,jN

=1

2 · (N − 2)!N∑

i1,i2,j1,j2

〈φi1 , φi2 |W12|φj1 , φj2〉N∑

i3,...,iN

ǫi1,i2,i3...,iN ǫj1,j2,i3...,iN

︸︷︷︸

(N−2)!ǫi1 ,i2 ǫj1 ,j2 for i1, i2 ∈ j1, j2

=1

2

N∑

i1,i2

∑

j1,j2∈i1,i2〈φi1 , φi2 |W12|φj1 , φj2〉 ǫi1,i2ǫj1,j2

︸︷︷︸

δi1 ,j1 δi2 ,j2−δi1 ,j2 δi1 ,j2

=1

2

N∑

i1,i2

(

〈φi1 , φi2 |W12|φi1 , φi2〉 − 〈φi1 , φi2 |W12|φi2 , φi1〉)

Note, that we exploited in the last step that the two interaction terms cancel when the indices i1and i2 are identical. Therefore they need not be excluded from the sum. The latter formula is moreconvenient to discuss the physical implications, while the former is commonly used in the quantumchemical literature.

Thus, we obtain

〈Ψ|12

N∑

i 6=jWi ,j |Ψ〉 =

1

2

N∑

i ,j=1

[

〈φi , φj |W |φi , φj 〉 − 〈φi , φj |W |φj , φi 〉]

(8.12)

Hartree Exchange

Wijij Wijji

i

i

j

j

j

j

i

i

Fig. 8.1: The left diagram describes that two particles are scattered by the Coulomb interaction.The right diagram describes the same process, but the two electrons are exchanged. The secondprocess is possible, because the two electrons are indistinguishable so that we cannot detect if thetwo electrons are still the same or not.

To be concise, we write here the detailed expressions for the matrix elements used in the above


equation Eq. 8.12

〈φa, φb|W |φc , φd〉 =∑

σ,σ′

∫

d3r

∫

d3r ′ φ∗a(~r , σ)φ∗b(~r′, σ′)

e2

4πǫ0|~r − ~r ′|φc(~r , σ)φd(~r ′, σ

′)

(8.13)

Homework: Work out the expectation value of the Coulomb interaction between two electronsfor a two-electron Slater determinant. Consider once a Slater determinant from non-orthonormalone-particle orbitals and once for orthonormal one-particle orbitals. Investigate the role of the spinindices.

8.3 Hartree energy

The surprising fact of Eq. 8.12 is the appearance of two terms. Therefore let us try to give somephysical meaning to the two contributions.

The first interaction term in Eq. 8.12 is the so-called Hartree energy. The Hartree energy turnsout to be the classical electrostatic interaction of the electron density.

EH =1

2

∑

i ,j

∑

σ,σ′

∫

d3r

∫

d3r ′ φ∗i (~r , σ)φ∗j (~r′, σ′)

e2

4πǫ0|~r − ~r ′|φi(~r , σ)φj(~r ′, σ

′)

=1

2

∫

d3r

∫

d3r ′[∑

i ,σ

φ∗i (~r , σ)φi(~r , σ)

]

︸︷︷︸

n(~r)

e2

4πǫ0|~r − ~r ′|

∑

j,σ′

φ∗j (~r ′, σ′)φj(~r ′, σ

′)

︸︷︷︸

n(~r ′)

(8.14)

The equation can be simplified using the definition of the charge density ρ(~r), or the electron densityn(~r), as

ρ(~r) = −en(~r) = −eN∑

i=1

∑

σ

φ∗i (~r , σ)φi(~r , σ)

︸︷︷︸

n(~r)

Thus, we obtain the final expression for the Hartree energy

HARTREE ENERGY

EH =1

2

∫

d3r

∫

d3r ′ρ(~r)ρ(~r ′)

4πǫ0|~r − ~r ′|(8.15)

8.4 Exchange energy

Exchange energy

The Hartree energy is clearly not the correct electrostatic energy, as it describes the interaction ofN electrons with N electrons. However, each electron can only interact with N − 1 electrons. Thus,the Hartree term also includes, incorrectly, the interaction of each electron with itself.


This so-called self interaction is subtracted by the so-called exchange energy, the second in-teraction term in Eq. 8.12

EX = −1

2

∑

i ,j

∑

σ,σ′

∫

d3r

∫

d3r ′ φ∗i (~r , σ)φ∗j (~r′, σ′)

e2

4πǫ0|~r − ~r ′|φj(~r , σ)φi(~r ′, σ

′)

(8.16)

The exchange term consists of the electrostatic interaction of densities3

ni ,j(~r)def=∑

σ

φi(~r , σ)φ∗j (~r , σ) (8.17)

The exchange energy can be written as

EX = −1

2

∑

i ,j

∫

d3r

∫

d3r ′e2nj,i(~r)ni ,j(~r ′)

4πǫ0|~r − ~r ′|(8.18)

while the Hartree term has the form

EH =1

2

∑

i ,j

∫

d3r

∫

d3r ′e2ni ,i(~r)nj,j(~r ′)

4πǫ0|~r − ~r ′|. (8.19)

Self interaction

All pair densities with different indices integrate to zero and the one with identical indices integrateto one, that is

∫

d3r ni ,j(~r)Eq. 8.17= 〈φj |φi 〉 = δi ,j

Thus, we can expect that the dominant contribution to the exchange energy results from the diagonalterms. This so-called self-interaction energy

ESI = −1

2

∑

i

∑

σ,σ′

∫

d3r

∫

d3r ′ φ∗i (~r , σ)φ∗i (~r′, σ′)

e2

4πǫ0|~r − ~r ′|φi(~r , σ)φi(~r ′, σ

′)

= −12

∑

i

∫

d3r

∫

d3r ′e2ni i(~r)ni i(~r ′)

4πǫ0|~r − ~r ′|

describes the electrostatic interaction of each electron with itself, that has been included in the Hartreecorrection. There is a well-known electronic structure method, based on density-functional theory,which uses the same trick to get rid of artificial self-interaction terms.[49] This theory subtractsthe self energy from the Hartree term and thus includes the so-called self-interaction correctionESIC = −ESI .

Exchange and spin alignment

Only electron pairs with the same spin contribute to the exchange energy. This can be shown whenone considers one-particle spin-orbitals with defined spin along the z-axis. That is, each one-particlespin-orbital has only one spin component, the other being zero.

3In the expression for ni ,j the complex conjugate of the right orbital is taken on purpose. This choice is analogousto that of the one-particle density matrix.


• If the two orbitals have the same spin, pointing for example into the ~ez direction, they can bewritten as

φi(~x) =

(

φi(~r , ↑)0

)

and φj(~x) =

(

φj(~r , ↑)0

)

The resulting exchange-energy would be formed from the pair densities

ni ,j(~r)Eq. 8.17=

∑

σ

φi(~r , σ)φ∗j (~r , σ) = φi(~r , ↑)φ∗j (~r , ↑) (8.20)

so that its contribution to the exchange energy has the form

∆EXEq. 8.18= −1

2

∑

i ,j

∫

d3r

∫

d3r ′e2nj,i(~r)ni ,j(~r ′)

4πǫ0|~r − ~r ′|

Eq. 8.20= −1

2

∫

d3r

∫

d3r ′e2φ∗i (~r , ↑)φj(~r , ↑)φ∗j (~r ′, ↑)φi(~r ′, ↑)

4πǫ0|~r − ~r ′|

• If the two orbitals have the opposite spin, they can be written as

φi(~x) =

(

φi ,↑(~r)

0

)

and φj(~x) =

(

0

φj,↓(~r)

)

The contribution to the exchange energy vanishes, because the pair densities vanish

ni ,j(~r)Eq. 8.17=

∑

σ

φi(~r , σ)φj(~r , σ) = φi(~r , ↑) · 0 + 0 · φj(~r , ↓) = 0

Note that this behavior of the exchange term is different from the one of the Hartree term: Thereis a Coulomb interaction between electrons of opposite spin, but no exchange term, because spin-upand spin-down electrons are distinguishable.

HUND’S RULE AND FERROMAGNETISM

Exchange acts only between electrons pairs with the same spin. Because exchange counteracts theCoulomb repulsion due to its changed sign, it favors if many electrons have the same spin.A result is Hund’s 1st rule: For a given electron configuration, the term with maximum multiplicity

has the lowest energy. The multiplicity is equal to , where is the total spin angular momentum for

all electrons. The term with lowest energy is also the term with maximum .

The same effect is responsible for the tendency of some solids to become ferromagnetic. Due tothe finite band width of metals, spin-alignment leads to some loss of kinetic energy. The balance ofkinetic energy and exchange determines whether ferromagnetism wins.

8.5 Hartree-Fock equations

The Hartree-Fock energy is an upper bound

The Hartree-Fock method yields an upper estimate EHF for the ground-state energy EGS.The ground state of an N-electron system is that state that minimizes the total energy for a

given Hamiltonian, i.e.

EGS = min|Φ〉,E

[

〈Φ|H|Φ〉 − E(

〈Φ|Φ〉 − 1)]

(8.21)


for all normalized many-particle wave functions |Φ〉. The normalization constraint has been addedwith the method of Lagrange multipliers and E is the corresponding Lagrange multiplier.4

The Hartree-Fock method restricts this search for the minimum to a subset S of wave functions,namely those that can be expressed in the form of a single Slater determinant.

EHF = min|Φ〉∈S,E′

[

〈Φ|H|Φ〉 − E′(

〈Φ|Φ〉 − 1)]

(8.22)

Because the search for the minimum is limited in the Hartree-Fock, the Hartree-Fock energy is alwaysabove the ground-state energy.

EGS ≤ EHF (8.23)

Hartree-Fock equations

Because every Slater determinant |Φ〉 ∈ S can be expressed by a set of one-particle orbitals |φj 〉, theHartree-Fock energy can be obtained by a minimum condition for the latter.

EHF = min|φ1〉,...,|φn〉,ΛΛΛ

[

ENN +

N∑

i=1

〈φi |h|φi 〉+1

2

N∑

i ,j=1

[

〈φi , φj |W |φi , φj 〉 − 〈φi , φj |W |φj , φi 〉]

−∑

i ,j

Λi ,j [〈φi |φj 〉 − δi ,j ]]

The last term ensures the orthonormality of the one-particle orbitals using the method of Lagrangemultipliers. The translation of the matrix elements between Slater determinants into those of theone-particle orbitals has been given in section 8.2. The matrix elements of one-particle Hamiltonianand interaction are defined in Eq. 8.6 and Eq. 8.13. The energy ENN is the Coulomb repulsion ofthe nuclei as defined in Eq. 8.1.

The minimum condition yields the Hartree-Fock equations. The derivation of the Hartree-Fockequations is analogous to the derivation of the Euler-Lagrange equations. Special is that here wework with fields, the one-particle wave functions, and that the variational parameters are complex.It is advantageous to use the Wirtinger derivatives, which are described briefly in ΦSX: KlassischeMechanik.

Wirtinger derivatives The essence of Wirtinger derivatives is that we can write the identity

df (Re[c ], Im[c ]) =df

dRe[c ]dRe[c ] +

df

d Im[c ]d Im[c ] =

df

dcdc +

df

dc∗dc∗ = df (c, c∗)

for the first variation of f . Thus, we can form the variations of a function of complex arguments, asif the variable and its complex conjugate were completely independent variables.For the functional derivatives we use

dΨ∗n(~r , σ)dΨ∗k(~r0, σ0)

= δ(~r − ~r0)δσ,σ0δn,k

dΨn(~r , σ)

dΨ∗k(~r0, σ0)= 0

4The expression is written a little sloppy, because the expression need not be be a minimum with respect to theLagrange multiplier. The requirement is only that the gradient vanishes.


−~22me

~∇2 +M∑

j=1

−e2Zj4πǫ0|~r − ~Rj |

+

∫

d3r ′e2n(~r ′)

4πǫ0|~r − ~r ′|

φi(~r , σ)

−N∑

j=1

∑

σ′

∫

d3r ′e2φ∗j (~r

′, σ′)φi(~r ′, σ′)

4πǫ0|~r − ~r ′|φj(~r , σ)−

N∑

j=1

φj(~r , σ)Λj,i = 0

Note that this is not a simple Schrödinger equation, because it mixes all orbitals.However, we can rewrite the equations to obtain a true Hamiltonian, which acts on all wave

functions in the same way:

−~22me

~∇2 +M∑

j=1

−e2Zj4πǫ0|~r − ~Rj |

+

∫

d3r ′e2n(~r ′)

4πǫ0|~r − ~r ′|

φi(~r , σ)

−∑

σ′

∫

d3r ′

N∑

j=1

e2φj(~r , σ)φ∗j (~r ′, σ′)

4πǫ0|~r − ~r ′|

︸︷︷︸

−VX,σ,σ′ (~r ,~r ′)

φi(~r ′, σ′)−

∑

j

φj(~r , σ)Λj,i = 0 (8.24)

This form actually shows that the one-particle orbitals can be superimposed so that the matrixΛΛΛ becomes diagonal. It can be shown that the system of equations is invariant under a unitarytransformation of the one-particle orbitals. Thus, one can use the matrix that diagonalizes ΛΛΛ, tobring the equations into a diagonal form. Hence, we can write the Hartree-Fock equations as trueeigenvalue equations.

The following observation provides some physical insight: The nonlocal exchange potential

VX,σ,σ′(~r , ~r ′) =

−N∑

j=1

e2φj(~r , σ)φ∗j (~r ′, σ′)

4πǫ0|~r − ~r ′|

(8.25)

contains a spin and angular momentum (ℓ,m) projection. Thus, every occupied atomic orbital isshifted towards lower energies, while empty atomic orbitals do not experience this exchange potential.The more compact the orbital, the larger the denominator and the stronger is the stabilization.

Non-local potential

However, the price to pay for the eigenvalue equation Eq. 8.24 is that we have to deal with a trulynon-local potential in the exchange term. What is a non-local potential? Consider an arbitraryoperator A

〈ψ|A|φ〉 = 〈ψ|[∫

d3r |~r〉〈~r |]

︸︷︷︸

1

A

[∫

d3r ′ |~r ′〉〈~r ′|]

︸︷︷︸

1

|φ〉

=

∫

d3r

∫

d3r ′ 〈ψ|~r〉︸︷︷︸

ψ∗(~r)

〈~r |A|~r ′〉︸︷︷︸

A(~r ,~r ′)

〈~r ′|φ〉︸︷︷︸

φ(~r ′)

=

∫

d3r

∫

d3r ′ ψ∗(~r)A(~r , ~r ′)φ(~r ′)

Thus, a general one-particle operator has always two sets of coordinates in its real space represen-tation. We call them non-local because they act on two positions at the same time. It is a special


property of most potentials, that they are local. For a local operator or local potential V the realspace matrix elements are diagonal in real space, that is

V (~r , ~r ′)def=〈~r |V |~r ′〉 = V (~r)δ(~r − ~r ′)

so that its matrix elements can be represented by a single real space integral.

Self-consistency cycle

The Hartree-Fock equations are self-consistent equations. This means that the Hamiltonian de-pends itself on the wave functions. Thus, the problem has to be solved iteratively.

• We start out with trial wave functions |φi 〉 to determine the Hamiltonian.

• Then, we obtain new wave functions from the eigenvalue problem Eq. 8.24.

• New and old wave functions are mixed, which is often necessary to obtain convergence of thecycle.

• The mixed wave functions are used as new trial functions to repeat the cycle.

If the new and old wave functions or the new and old Hamiltonian are identical, the wave functionsare consistent with the Hamiltonian and the Hartree-Fock equations are solved.

In the quantum chemical literature the term self-consistent equations is used synonymous withHartree-Fock equations. This is misleading because there are many other schemes that also requirea self-consistent cycle.

Fock operator

We can now write the Hamilton operator of the Hartree-Fock equation, namely the so-called Fockoperator

FOCK OPERATOR

HFock =∑

σ

∫

d3r |~r , σ〉

−~22me

~∇2 −M∑

j=1

Zje2

4πǫ0|~r − ~Rj |+

∫

d3r ′n(~r ′)e2

4πǫ0|~r − ~r ′|

〈~r , σ|

−n∑

i ,j=1

∑

σ,σ′

∫

d3r

∫

d3r ′ |~r , σ〉e2φ∗j,σ(~r)φj,σ′(~r

′)

4πǫ0|~r − ~r ′|〈~r ′, σ′|

+[

1−n∑

j=1

|φj 〉〈φj |]

C[

1−n∑

j=1

|φj 〉〈φj |]

The operator C in the last term is completely arbitrary. Because of the projection operator on bothsides, it only acts on the “unoccupied” one-particle wave functions. Because this Hamilton operatoris never applied to the unoccupied states, it is also irrelevant. We included this term here to showthat the Hartree-Fock operator contains a lot of arbitrariness and that furthermore the action of theoperator on the unoccupied states is arbitrary.

Note, that the Hartree-Fock total energy is not identical to the sum of one-particle energies asit is the case for non-interacting particles.


8.6 Hartree-Fock of the free-electron gas

Let us now study the free-electron gas in the Hartree-Fock approximation. Our goal is to determinethe changes of the dispersion relation due to the interaction.

First we determine non-local potential as defined in Eq. 8.24.

Vx(~x, ~x ′)Eq. 8.24= −

∑

j

e2φ∗j (~x)φj(~x′)

4πǫ0|~r − ~r ′|

(8.26)

Eq. P.3=

e2δσ,σ′

4πǫ0s

1

(2π)34π

3k3F

[

3(kF s) cos(kF s)− sin(kF s)

(kF s)3

]

(8.27)

where s = |~r − ~r ′| and kF is the Fermi momentum.

-15 -10 -5 0 5 10 15

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

-15 -10 -5 0 5 10 15-1

0

Fig. 8.2: The shape of the non-local exchange potential (right) for a free electron gas as calculatedin the Hartree-Fock method. The dashed line corresponds to a Coulomb interaction. The function3 x cos(x)−sin(x)x3 is shown on the left-hand side.

Now we need to evaluate the expectation values of this potential in order to obtain the energyshifts:

dǫ~k,σ =1

Ω

∫

d3r

∫

d3r ′ Vx(|~r − ~r ′|, σ, σ′)ei~k(~r−~r ′)

= − e2

4πǫ0

2kFπ

[1

2+1− a24a

ln

∣∣∣∣

1 + a

1− a

∣∣∣∣

]

Here, a = |~k |/kF . The function in parenthesis is the so-called Lindhard function. It is shown inFig. 8.3.

We find, as shown in Fig. 8.4, that the occupied band width is larger in Hartree-Fock than forthe non-interacting free electrons. However, the experimental band width of real free-electron-likematerials is smaller. For potassium the band width of the free electron gas is 2 eV. The Hartree-Fockresult is 5.3 eV, while the band width of real potassium is in the range 1.5-1.6 eV.[1]

A second problem[1] of the Hartree-Fock theory is that the density of states at the Fermi levelvanishes. This is again in contradiction with experiment, which is described pretty well with

√ǫ-

behavior of the density of states. The experimental result is closer to the free electron gas. Thediscrepancy is related to the fact that the electron can not respond to the potential describing theinteraction. Note that the wave functions are still plane waves.5

5This argument as such is not conclusive. The band structure is obtained for a fixed potential. The potentialdepends explicitly on the number of electrons. In an experiment the Fermi level is measured for example by extractingan electron, which changes the electron number and therefore the interaction potential of Hartree-Fock.


0 1 20.0

0.5

1.0

Fig. 8.3: The Lindhard function f (x) = −(12 +

1−x24x ln

∣∣ 1+x1−x∣∣

)

. Note that the slope for x = 1

(corresponding to k = kF ) is infinite. For x >> 1 the function approaches zero.

EFnon−interacting

E

kHartree−Fock

kF

Fig. 8.4: Dispersion relation of the free electron gas as calculated with Hartree-Fock and withoutinteractions. The two dispersion relations are shifted so that their Fermi levels agree.

8.7 Beyond the Hartree-Fock Theory: Correlations

The result obtained by the Hartree-Fock approximation always lies above the correct result. Theenergy difference is called electron correlation. The term was coined by Per-Olov Löwdin. [P.-O.Löwdin, “Correlation Problem in Many-Electron quantum mechanics”, Adv. Chem. Phys. 2, 207(1959)]

In this article I will point out a number of failures of Hartree-Fock theory, in order to introduceinto the terminology of correlations.

8.7.1 Left-right and in-out correlation


Fig. 8.5: Per-Olov Löwdin,Swedish physicist andtheoretical chemist 1916-2000. Courtesy of theAmerican PhilosophicalSociety.

The notation of left-right correlation and in-out correlation has beentermed by Kolos and Roothaan[50], who performed accurate calcula-tions for the hydrogen molecule.

Left-right correlation describes that the two electrons in one bondare most likely on the opposite sides of the bond. This principle isviolated by a Slater determinant build from bonding orbitals.

Let us consider a hydrogen molecule. We start from a one-particlebasis |L, ↑〉, |L, ↓〉, |R, ↑〉, |R, ↓〉, where L refers to the left atom and Rrefers to the right atom. First we transform the basis onto bonding andantibonding orbitals.

|B, ↑〉 def=1√2

(

|L, ↑〉+ |R, ↑〉)

(8.28)

|B, ↓〉 def=1√2

(

|L, ↓〉+ |R, ↓〉)

(8.29)

|A, ↑〉 def=1√2

(

|L, ↑〉 − |R, ↑〉)

(8.30)

|A, ↓〉 def=1√2

(

|L, ↓〉 − |R, ↓〉)

(8.31)

Now we form a Slater determinant from the two bonding orbitals

Ψg(~r1, σ1, ~r2, σ2) =1√2

(

〈~r1, σ1|B, ↑〉〈~r2, σ2|B, ↓〉 − 〈~r1, σ1|B, ↓〉〈~r2, σ2|B, ↑〉)

=1

2

〈~r1, σ1|(

|L, ↑〉+ |R, ↑〉)

〈~r2, σ2|(

|L, ↓〉+ |R, ↓〉)

−〈~r1, σ1|(

|L, ↓〉+ |R, ↓〉)

〈~r2, σ2|(

|L, ↑〉+ |R, ↑〉)

=1

2

(

〈~r1, σ1||L, ↑〉〈~r1, σ1||R, ↑〉)(

〈~r2, σ2|L, ↓〉+ 〈~r2, σ2|R, ↓〉)

−(

〈~r1, σ1|L, ↓〉〈~r1, σ1|R, ↓〉)(

〈~r2, σ2|L, ↑〉+ 〈~r2, σ2|R, ↑〉)

=1

2

(

〈~r1, σ1|L, ↑〉〈~r2, σ2|L, ↓〉 − 〈~r1, σ1|L, ↓〉〈~r2, σ2|L, ↑〉)

+(

〈~r1, σ1|L, ↑〉〈~r2, σ2|R, ↓〉 − 〈~r1, σ1|R, ↓〉〈~r2, σ2|L, ↑〉)

+(

〈~r1, σ1|R, ↑〉〈~r2, σ2|L, ↓〉 − 〈~r1, σ1|L, ↓〉〈~r2, σ2|R, ↑〉)

+(

〈~r1, σ1|R, ↑〉〈~r2, σ2|R, ↓〉 − 〈~r1, σ1|R, ↓〉〈~r2, σ2|R, ↑〉)

=1√2[(↑↓, 0) + (↓, ↑) + (↑, ↓) + (0, ↑↓)]

Thus, the Slater determinant of the bonding orbitals contains determinants where both electrons areeither both localized to the right or with both electrons localized on the left side. These ionizedconfiguration lie high in energy of the bond is long. This is the explanation that the Hartree-Focktheory in this form results in a too large dissociation energy.

In order to obtain the correct limit, namely 1√2(↑, ↓) + (↓, ↑) we need to mix in the Slater deter-

minant with the antibonding orbitals.

Ψu(~r1, σ1, ~r2, σ2) =1√2[(↑↓, 0)− (↓, ↑)− (↑, ↓) + (0, ↑↓)]

8.7.2 In-out correlation

• “in-out” correlation


• left-right correlation

• angular correlation

• radial correlation

• dynamic

• nondynamic

• near-degeneracy

• long-range

• shortrange

8.7.3 Spin contamination

8.7.4 Dynamic and static correlation

In quantum chemistry the definition of dynamic and static correlation is opposite to that used byphysicists. In Quantum chemistry, static correlation is the effect of very few Slater determinants thatcontribute to the ground state with approximately equal weight. That static correlation is present ifthe dominant Slater determinant has a weight less than about 95% of the complete wave function.

Dynamic correlation on the other hand are the small contribution of many Slater determinants,lowering the energy. Dynamic correlation is important to describe the cusp condition: The Coulombhole has a cusp in the density at the site of the observer electron. Because all one-particle orbitalsare smooth, many Slater determinants are required to build up this cusp as superposition of smoothwave functions.

Chapter 9

Density-functional theory

Density-functional theory[51, 52] is an extremely powerful technique. It is based on an exact theoremthat the ground-state energy is determined solely by the density alone.

The theorem says

• that all ground-state properties are unique functionals of the electron density and

• that the electron density can be obtained from a Schrödinger equation in an effective potential.

density-functional theory maps the interacting electron system onto a system of non-interactingelectrons in an effective potential. The effective potential depends in turn on the electron distributionand describes the interacting between the electrons in an effective way.

density-functional theory provides a total energy functional of the form

E[n(~r)] =∑

n

fn〈φn|p2

2me|φn〉

︸︷︷︸

Ts [n(~r)]

+

∫

d3r n(~r)vext(~r)

+1

2

∫

d3r

∫

d3r ′e2n(~r)n(~r ′)

4πǫ0|~r − ~r ′|︸︷︷︸

EHartree

+Exc [n(~r)]

Ts is not the kinetic energy of the many-electron system, but the one of the non-interacting referencesystem with density n(~r). We will see later that it can be written as functional of the density. Excis another functional of the density, called the exchange and correlation energy functional. Thisfunctional is not known in practice. Therefore, one uses approximate functionals, that are derivedfrom known analytical properties of the functional and/or even experimental data.

As a result of the minimum principle we obtain the following Kohn-Sham equation for the one-particle orbitals

[p2

2me+ vef f (r)− ǫn

]

|φn〉 = 0

where the effective potential is defined as

vef f (~r) = vext(~r) +e2

4πǫ0

∫

d3r ′n(~r ′)

|~r − ~r ′|+δExcδn(~r)︸︷︷︸

µxc (~r)

The proofs will be given later. Let us start out with the language used in the field of density-functional theory and the physical pictures that emerge.

169

170 9 DENSITY-FUNCTIONAL THEORY

9.1 One-particle and two-particle densities

In order to get used with the terminology, let me introduce a number of definitions related to densitymatrices[53].

9.1.1 N-particle density matrix

Let us start with the N-particle density matrix of an N-particle system

ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)def=Ψ(~x1, . . . , ~xN)Ψ

∗(~x ′1, . . . , ~x ′N)

The N-particle density matrix is formed from the matrix elements

ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N) = 〈~x1, . . . , ~xN |ρ(N)|~x ′1, . . . , ~x ′N〉

of the density operator

ρ(N)def= |Ψ〉〈Ψ| (9.1)

This is the same density operator, that has been used in statistical mechanics a lot. In Eq. 9.1,however, we restricted ourselves to the density matrix of a pure state. A pure state can be describedby a single many-particle wave function, whereas a statistical mixture is an ensemble of many-particlestates, of which each contributes with a certain probability.

The N-particle density matrix contains all the information of the wave function itself, becauseone can determine the expectation value of any operator from it:

We use the representation of the unity operator in a N-particle Hilbert space of spin-orbitals

1 =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉〈~x1, . . . , ~xN |

and obtain

〈Ψ|A|Ψ〉 = 〈Ψ|A|Ψ〉 = 〈Ψ|1A1|Ψ〉

=

∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N

· 〈~x1 . . . , ~xN |Ψ〉〈Ψ|~x ′1 . . . , ~x ′N〉︸︷︷︸

ρ(N)

〈~x ′1 . . . , ~x ′N |A|~x1 . . . , ~xN〉

=

∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N

·ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)〈~x ′1 . . . , ~x ′N |A|~x1 . . . , ~xN〉 (9.2)

For us the expectation value of the Hamilton operator plays a special role. The energy expectationvalue can be constructed from the N-particle density matrix. However, the special form of theHamilton operator relevant to us allows to use simpler, so-called reduced density matrices. We willdefine those in the following.

Fertig: WS06/07 8 Doppelstunde 27.Nov.06

9.1.2 One-particle reduced density matrix

From the N-particle density matrix we can form several contractions that are physically important,because they allow to represent expectation values of one-particle operators and the Coulomb inter-action in an elegant fashion.

9 DENSITY-FUNCTIONAL THEORY 171

For example, we do not need the full N-particle density matrix to determine the expectation valueof a one-particle operator, if we know the so-called one-particle density matrix.

A general one-particle operator has the matrix elements

〈~x1, . . . , ~xN |A|~x ′1, . . . , ~x ′N〉 =N∑

i=1

A(~xi , ~x′i )

N∏

k(k 6=i)δ(~xk − ~x ′k) (9.3)

This allows us to simplify the expression for the expectation value as follows

〈Ψ|A|Ψ〉 Eq. 9.2,9.3=

∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N

ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)N∑

i=1

A(~xi , ~x′i )

N∏

k(k 6=i)δ(~xk − ~x ′k)

(1)=

∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N

ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)NA(~x1, ~x ′1)N∏

k=2

δ(~xk − ~x ′k)

=

∫

d4x1

∫

d4x ′1 A(~x1, ~x′1)

·N∫

d4x2 · · ·∫

d4xN

∫

d4x ′2 · · ·∫

d4x ′N

·ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)N∏

k=2

δ(~xk − ~x ′k)

=

∫

d4x1

∫

d4x ′1 A(~x1, ~x′1)

·N∫

d4x2 · · ·∫

d4xN ρ(N)(~x1, ~x2, . . . , ~xN ; ~x ′1, ~x2 . . . , ~xN)

︸︷︷︸

ρ(1)(~x1,~x ′1)

(9.4)

In (1), we exploited that the density matrix describes identical particles. Thus, it is invariant underpermutation of two particle indices. (Simultaneously on both sides.)

Thus, if we define the one-particle reduced density matrix1 ρ(1)(~x, ~x ′), as2

ρ(1)(~x, ~x ′)def= N

∫

d4x2 · · ·∫

d4xN ρ(N)(~x, ~x2, . . . , ~xN ; ~x ′, ~x2, . . . , ~xN)

= N

∫

d4x2 · · ·∫

d4xN Ψ(~x, ~x2, . . . , ~xN)Ψ∗(~x ′, ~x2, . . . , ~xN) (9.5)

we can determine the expectation values of any one-particle operator as

〈Ψ|A|Ψ〉 Eq. 9.4=

∫

d4x

∫

d4x ′ ρ(1)(~x, ~x ′)A(~x ′, ~x) (9.6)

The integral in the first equation of the definition Eq. 9.5 provides the density of a single electron.This density is the same for all the electrons. Therefore, we obtain the total density by multiplicationby the number N of electrons, which explains the factor in front of the integral.

1 The word “reduced” is usually dropped. However the context must avoid confusion with the density matrix of aone-particle system.

2Note that this definition only holds for density matrices of indistinguishable particles.


Electron density

The operator for the electron density at a position ~r is

n(~r) =∑

σ

|~r , σ〉〈~r , σ| (9.7)

It has the matrix elements

〈~r , σ|n(~r0)|~r ′, σ′〉 = δ(~r − ~r0)δ(~r ′ − ~r0)δσ,σ′ (9.8)

in a one-particle, real-space and spin representation.The matrix elements of the operator in a N-particle, real-space and spin representation are

〈~x1, . . . , ~xN |n(~r0)|~x ′1, . . . , ~x ′N〉 Eq. 9.3=

N∑

i=1

δ(~ri − ~r0)δ(~r ′i − ~r0)δσi ,σ′i∏

k(k 6=i)δ(~xk − ~x ′k)

Thus, we obtain the electron density as

n(~r0) = 〈Ψ|n(~r0)|Ψ〉 Eq. 9.6=

∫

d4x

∫

d4x ′ ρ(1)(~x, ~x ′)δ(~r − ~r0)δ(~r ′ − ~r0)δσ,σ′

=∑

σ

ρ(1)(~r0, σ,~r0, σ) (9.9)

Magnetization

The magnetization operator has the form

~m =q

me~S =

~q

2me~σ (9.10)

Its matrix elements in a one-particle real-space representation are

〈~r , σ|mj |~r ′, σ′〉 =~q

2meδ(~r − ~r ′)σj,σ,σ′ (9.11)

where ~S is the spin operator, q is the electron charge and me is its mass. With σj we denote thePauli matrices defined in Eq. 3.2 for j ∈ x, y , z. Unfortunately, it is a little difficult not to getconfused by the many σ’s.

The matrix elements in the N-particle Hilbert space are

〈~x1, . . . , ~xN |mn(~r)|~x ′1, . . . , ~x ′N〉 Eq. 9.3=

N∑

i=1

δ(~ri − ~r ′ i)~q

2meσj,σ,σ′

︸︷︷︸

〈~ri ,σi |mj |~r ′ i ,σ′i 〉

∏

k(k 6=i)δ(~xk − ~x ′k)

Thus, we obtain the magnetization as

~m(~r0) = 〈Ψ| ~m(~r0)|Ψ〉 =∫

d4x

∫

d4x ′ ρ(1)(~x, ~x ′)δ(~r0 − ~r)~σσ,σ′

=∑

σ,σ′

ρ(1)(~r0, σ,~r0, σ′)~σσ′,σ


Kinetic energy

The matrix elements for the operator that provides the kinetic energy in a given direction at a givenpoint are

〈~x1, . . . , ~xN |T |~x ′1, . . . , ~x ′N〉 =N∑

i=1

δ(~xi − ~x ′ i)−~22me

~∇′2i∏

k(k 6=i)δ(~xk − ~x ′k)

This expression is weird: The matrix element is a differential operator that points no-where! Notreally. Let us investigate the meaning of this expression:

Consider a one-particle state |f 〉 and let us define a second one as |g〉def= ~p2

2me|f 〉. Then we find

g(~r) = 〈~r |g〉 = 〈~r | ~p2

2me|f 〉 = 〈~r | ~p

2

2me

∫

d3r ′ |~r ′〉〈~r ′|︸︷︷︸

1

|f 〉 =∫

d3r ′ 〈~r | ~p2

2me|~r ′〉〈~r ′|f 〉︸︷︷︸

f (~r ′)

On the other hand, we know that g(~r) = −~22me

~∇2f (~r), so that we can formally identify the kineticenergy operator in a one-particle Hilbert space as

〈~r | ~p2me|~r ′〉 = δ(~r − ~r ′)−~

2

2me~∇′2

It is, however, important that the gradient stands in front of the function depending on ~r ′. I do notrecommend to use tricks like this, because one has to be awfully careful.

The gradient ~∇′i acts exclusively on the i-th primed position ~r ′ i . Note that the gradient cannotbe interchanged with the δ-function preceding it, because a gradient does not commutate with afunction of the corresponding coordinate.

Thus, we obtain the kinetic energy T as

Tdef= 〈Ψ|T |Ψ〉 =

∫

d4x

∫

d4x ′ δ(~x − ~x ′)−~2

2me

(

~∇′)2

ρ(1)(~x ′, ~x)

=∑

σ

∫

d3r lim~r→~r ′−~22me

~∇′2ρ(1)(~r ′, σ,~r , σ)

9.1.3 Two-particle density

In order to evaluate the Coulomb interaction energy between electrons, we need to know the two-particle density.

The matrix elements of the interaction in a real space representation are

〈~x1, . . . , ~xN |W |~x ′1, . . . , ~x ′N〉 =1

2

∑

i 6=jw(~xi , ~xj)

N∏

i=1

δ(~xi − ~x ′ i)

where w(x, x ′) = e2

4πǫ0|~r−~r ′| .


〈Ψ|W |Ψ〉 =∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N

ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)〈~x ′1, . . . , ~x ′N |W |~x1, . . . , ~xN〉

=

∫

d4x1 · · ·∫

d4xN

∫

d4x ′1 · · ·∫

d4x ′N

ρ(N)(~x1, . . . , ~xN ; ~x ′1, . . . , ~x ′N)1

2

∑

i 6=jw(~xi , ~xj)

N∏

i=1

δ(~xi − ~x ′ i)

=

∫

d4x1 · · ·∫

d4xN ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)

1

2

∑

i 6=jw(~xi , ~xj)

=

∫

d4x1 · · ·∫

d4xN ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)

N(N − 1)2

w(~x1, ~x2)

=1

2

∫

d4x

∫

d4x ′ w(~x, ~x ′)

N(N − 1)∫

d4x1 · · ·∫

d4xN ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)δ(~x − ~x1)δ(~x ′ − ~x2)

︸︷︷︸

n(2)(~x ′,~x)

=1

2

∫

d4x

∫

d4x ′ n(2)(~x, ~x ′)w(~x, ~x ′)

Thus, we define the two-particle density3. n(2)(~x, ~x ′) as

n(2)(~x, ~x ′)def= N(N − 1)

∫

d4x1 · · ·∫

d4xN

ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)δ(~x − ~x1)δ(~x ′ − ~x2) (9.12)

so that we can determine the interaction energy as

〈Ψ|W |Ψ〉 = 12

∫

d4x

∫

d4x ′ n(2)(~x, ~x ′)w(~x, ~x ′)

The two-particle density can also be expressed as expectation value as follows

n(2)(~x, ~x ′)Eq. 9.12=

∫

d4x1 · · ·∫

d4xN Ψ(~x1, . . . , ~xN)∑

i 6=jδ(~x − ~xi)δ(~x ′ − ~xj)Ψ∗(~x1, . . . , ~xN)

= 〈Ψ|∑

i 6=jδ(~x − ~xi)δ(~x ′ − ~xj)|Ψ〉 (9.13)

The two-particle density describes the probability density that there is one electron at ~r and another

one at ~r ′. It integrates up to the number of ordered pairs, namely N(N − 1).Note that the two-particle density is often confused with the one-particle density matrix. Both

have the same number of arguments, but they are fundamentally different quantities.The q-particle densities contain the information only about the probability to find a q-tuple of

electrons with a given set of coordinates. It is, however, not possible to extract for example themomentum or the kinetic energy from a given q-particle density. In contrast, the q-electron density

3There are different conventions in use, which differ by a factor 1/2. We follow here the choice of McWeeny[53],who does not have the factor 1

2. Here, the factor 1

2, which compensates for the double counting of two pairs at ~x

and ~x ′ (one obtained by exchanging the two particles from the other) must be done explicitly. On the other hand thetwo-particle density as defined by McWeeny has the simple physical interpretation of a density for one particle at ~x andanother one at ~x ′, irrespective of their order.


matrix contains the complete information on q-tuple of electrons. For a q-particle system, the q-particle density matrix contains the complete information on the system, that is the same informationcontained in the wave function. Already from the one-particle density matrix we can determine themomentum or the kinetic energy.

In the following we list various densities and density-matrices in comparison, to show the difference

• one-particle density matrix

ρ(1)(~x1, ~x ′1) = N

∫

d4x2 · · ·∫

d4xN ρ(N)(~x1, ~x2, . . . , ~xN ; ~x ′1, ~x2, . . . , ~xN)

• two-particle density matrix

ρ(2)(~x1, ~x2, ~x ′1~x ′2) = N(N − 1)∫

d4x3 · · ·∫

d4xN ρ(N)(~x1, ~x2, ~x3, . . . , ~xN ; ~x ′1, ~x ′2, ~x3, . . . , ~xN)

• one-particle density

n(1)(~x1) = N

∫

d4x2 · · ·∫

d4xN ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)

• two-particle density

n(2)(~x1, ~x2) = N(N − 1)∫

d4x3 · · ·∫

d4xN ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)

• N-particle density

n(N)(~x1, . . . , ~xN) = ρ(N)(~x1, . . . , ~xN ;~x1, . . . , ~xN)

The q-particle density can directly be obtained from the q-particle density matrix

n(q)(~x1, . . . , ~xq) = ρ(q)(~x1, . . . , ~xq, ~x1, . . . , ~xq)

The density matrix has always two arguments per particle while the density has only one per particle.The two-particle density is so important for many-particle physics, because we can directly express

the interaction energy by the two-particle density. The Coulomb interaction between electrons canbe written as

EH + EX = 〈Ψ|1

2

∑

i 6=j

e2

4πǫ0|~r − ~r ′||Ψ〉 = 1

2

∑

σ,σ′

∫

d3r

∫

d3re2n(2)(~x, ~x ′)

4πǫ0|~r − ~r ′|(9.14)

In chapter 8 we derived the expression for the expectation value of a two-particle operator with aSlater determinant. According to Eq. 9.13, the two-particle density can be expressed as an expectationvalue. For a single Slater determinant, the two-particle density would have the form

n(2)(~r , ~r ′)Eqs. 8.12,9.13

=

N∑

i ,j

∫

d4x1

∫

d4x2 φ∗i (~x1)φ

∗j (~x2)δ(~r − ~r1)δ(~r − ~r2)φi(~x1)φj(~x2)

−N∑

i ,j

∫

d4x1

∫

d4x2 φ∗i (~x1)φ

∗j (~x2)δ(~r − ~r1)δ(~r − ~r2)φj(~x1)φi(~x2)

= n(~r)n(~r ′)−N∑

i ,j

∑

σ,σ′

φ∗i (~r , σ)φj(~r , σ)φ∗j (~r′, σ′)φi(~r ′, σ

′) (9.15)


9.1.4 Two-particle density of the free electron gas

Let us work out the two-particle density of the free electron gas in the Hartree-Fock approximation:For the free electron gas the one-particle orbitals are, according to Eq. ??,

φ~k,σ(~r , σ′) =

1√Ωei~k~rδσ,σ′

Now we need to determine the two-particle density according to Eq. 9.15. The first term is trivialbecause the density is constant. The exchange term is more complicated. let us therefore work outthe expression C(~x, ~x ′) which occurs in the exchange term of Eq. 9.15.

C(~x, ~x ′)def=

N∑

i ,j

φ∗i (~x)φj(~x)φ∗j (~x′)φi(~x ′)

Eq. 4.114=

∑

σ1,σ2

Ω2∫

|~k1|<kF

d3k1(2π)3

∫

|~k2|<kF

d3k2(2π)3

φ∗~k1,σ1(~r , σ)φ~k2,σ2(~r , σ)φ∗~k2,σ2(~r ′, σ′)φ~k1,σ1(

~r ′, σ′)

Eq. ??=

∫

|~k1|<kF

d3k1(2π)3

∫

|~k2|<kF

d3k2(2π)3

ei(~k2−~k1)(~r−~r ′)

∑

σ1,σ2

δσ,σ1δσ,σ2δσ′,σ2δσ′,σ1

︸︷︷︸

=δσ,σ′δσ′ ,σ=δσ,σ′

= δσ,σ′

(∫

|~k1|<kF

d3k1(2π)3

e−i~k1(~r−~r ′)

)(∫

|~k2|<kF

d3k2(2π)3

ei~k2(~r−~r ′)

)

~k1→~k= δσ,σ′

∣∣∣∣

∫

|~k|<kF

d3k

(2π)3ei~k(~r−~r ′)

∣∣∣∣

2

Now we use a tremendously useful theorem, that allows to decompose plane waves into radial func-tions and spherical harmonics Yℓ,m

ei~k~r = 4π

∞∑

ℓ=0

ℓ∑

m=−ℓi ℓjℓ(|~k ||~r |)Yℓ,m(~r)Y ∗ℓ,m(~k) (9.16)

where the jℓ(z) are the spherical Bessel functions , also named Bessel functions of half-integerorder. The Bessel function j0(x) is shown in the following figure

1π 2π 3π 4π 5π

x1

x1−

1

0

0

j0(x)=sin(x)x

In the last result we see that it depends only on the relative positions for the two electrons.Therefore we introduce

~udef=~r − ~r ′


Since we integrate over a spherical volume in reciprocal space, only the terms with ℓ = 0 contributea finite value, because all other spherical harmonics integrate to zero.

C(~x, ~x ′)~r−~r ′→~u= δσ,σ′

∣∣∣∣

∫

|~k<kF

d3k

(2π)3ei~k~u

∣∣∣∣

2

Eq. 9.16= δσ,σ′

∣∣∣∣∣

∫

|~k<kF

d3k

(2π)34π

∞∑

ℓ=0

ℓ∑

m=−ℓi ℓjℓ(|~k ||~u|)Yℓ,m(~u)Y ∗ℓ,m(~k)

∣∣∣∣∣

2

= δσ,σ′

∣∣∣∣∣∣∣∣∣

1

(2π)3

∫

k<kF

dk k24πj0(|~k ||~u|)1

4π︸︷︷︸

Y0,0(~u)Y ∗0,0(~k)

∣∣∣∣∣∣∣∣∣

2

|~k|→k= δσ,σ′

1

(2π)6

(

4π

∫

k<kF

dk k2sin(k |~u|)k |~u|

)2

k|~u|→x= δσ,σ′

1

(2π)6

(4π

|~u|3∫

x<kF |u|dx x sin(x)

)2

The integral can be solved using

d

dx(x cos(x)− sin(x)) = cos(x)− x sin(x)− cos(x) = x sin(x)

Now we continue

C(~x, ~x ′)k|~u|→x= δσ,σ′

1

(2π)6

∣∣∣∣

1

|~u|3 (kF |~u| cos(kF |~u|)− sin(kF |~u|))∣∣∣∣

2

=δσ,σ′k

6F

(2π)6

(kF |~u| cos(kF |~u|)− sin(kF |~u|)

(kF |~u|)3)2

For the sake of esthetics4, let scale the expression in parenthesis such that it starts with value 1 atu = 0. (See footnote below5)

limx→0

x cos(x)− sin(x)x3

= limx→0

∞∑

n=0

((−1)n2n!

x2n−2 − (−1)n(2n + 1)!

x2n−2)

= limx→0

∞∑

n=0

(1

(2n)!− 1

(2n + 1)!

)

(−1)nx2n−2

= limx→0

∞∑

n=0

(1− 1) x−2 −(1

2− 16

)

x0 +O(x2) =1

3

4esthetics=Ästetik”5

ex =

∞∑

n=0

1

n!xn

⇒ eix =∞∑

n=0

1

n!inxn =

∞∑

n=0

(1

(2n)!(−1)nx2n + i 1

(2n + 1)!(−1)nx2n+1

)

= cos(x) + i sin(x)

⇒ cos(x) =∞∑

n=0

(−1)n2n!

x2n ; sin(x) =

∞∑

n=0

(−1)n+1(2n + 1)!

x2n+1


We rescale the term in parenthesis and obtain

C(~x, ~x ′) = δσ,σ′

(1

(2π)34π

3k3F

)

︸︷︷︸12n

2(

3kF |~u| cos(kF |~u|)− sin(kF |~u|)

(kF |~u|)3)2

With n = 2 · 1(2π)3

4π3 k3F we obtain the final result

n(2)(~r , ~r ′) = n2 −∑

σ,σ′

C(~x, ~x ′)

=∑

σ,σ′

(n↑ + n↓2

)2

1− δσ,σ′(

3kF |~r − ~r ′| cos(kF |~r − ~r ′|)− sin(kF |~r − ~r ′|)

(kF |~r − ~r ′|)3

)2

= (n↑ + n↓)2

1− 12

(

3kF |~r − ~r ′| cos(kF |~r − ~r ′|)− sin(kF |~r − ~r ′|)

(kF |~r − ~r ′|)3

)2

The result is shown in Fig. 9.1. We observe that the two-particle density for identical spin directionvanishes at small distances in accordance with the Pauli principle, which excludes that two identicalparticles can be at the same position. The two-particle density for opposite spin is that of twouncorrelated electrons, namely the product of the two spin densities. For large distances the two-particle densities for both spin directions approach the uncorrelated value.

n(r1)h(r1,r2)

|r2−r1|0

n2n(2)(r2,r1)

n(2)(r2,r1)

0

Fig. 9.1: Two-particle density of the non-interacting free electron gas.

9.1.5 Pair-correlation function and hole function

Another quantity that is useful is the pair-correlation function


PAIR-CORRELATION FUNCTION

g(~x, ~x0) =n(2)(~x, ~x0)

n(~x0)

The pair-correlation function describes the probability to find an electron with coordinates ~x , giventhat there is one at ~x0. The pair-correlation function is the electron density seen by one electron thatis located at ~x0. Consider an N-electron system. Pick out one electron and determine the density ofthe remaining N − 1 electrons.

Given that an electron will see at large distances nothing but the charge density itself, it makessense to define a more short-ranged quantity, namely the exchange-correlation hole functionh(~x, ~x0)

HOLE FUNCTION H( ~X, ~X0)

g(~x, ~x0) = n(~x) + h(~x, ~x0)

The physical picture of the hole function is that each electron “sees” the total density n(~x) and thedensity h(~x, ~x ′) of “one missing electron”. The density h(~x, ~x ′) is like a hole in the total charge densitydue to the fact that the remaining electrons do not come near to the electron in question. Thus, theCoulomb repulsion between the electrons is reduced relative to the Hartree energy by the attractionof the electron to its exchange-correlation hole.

Accurate calculations of the exchange-correlation hole have been obtained by quantum MonteCarlo calculations[54].

The two-particle density can now be expressed by the density and the exchange-correlation holefunction as

n(2)(~x, ~x0) = n(~x0) [n(~x) + h(~x, ~x0)]

Now we can also rewrite the interaction energy Eq. 9.14

Eint =1

2

∑

σ,σ′

∫

d3r

∫

d3r ′e2n(~x)n(~x ′)

4πǫ0|~r − ~r ′|︸︷︷︸

EH

+∑

σ

∫

d3r n(~x)1

2

∑

σ′

∫

d3r ′e2h(~x ′, ~x)

4πǫ0|~r − ~r ′|︸︷︷︸

Uxc

9.2 Self-made density functional

In order to understand, how density functionals are made, let us demonstrate the principle on a simpleexample. We will make our own, first-principles, density functional. It will not be terribly good, butit will not be a complete disaster either. However, it will be our own!

The typical process of creating a density functional is that we take all available information on thehole function, that may be relevant, and find an expression that reproduces this information as bestas possible. As more and more information about the hole function becomes available, the functionalsare continuously improved.

Apparently all the information about the interaction is contained in the hole function h(~r , ~r ′). Letus collect all we know so far about the hole function:


Fig. 9.2: Top: Exchange-correlation hole in the 110 plane of silicon[55]. The cross indicates theposition of the reference electron. In the bond, the exchange-correlation hole is centered on thereference electron, while in the tail region it is located off-center. Bottom: Valence charge densityof silicon in the (110) plane.

KNOWN PROPERTIES OF THE HOLE FUNCTION

• The hole function vanishes for large distances between the electrons

lim|~r−~r0|→∞

h(~x, ~x0) = 0

• The integral over the hole function with equal spins is equal to one, that is∫

d3r h(~x, ~x0) = −δσ,σ0

This statement follows from the fact that an electron in an N-electron system, always seesthe N − 1 other electrons. Thus, the correlation function ~g integrates to N − 1 and the holefunction integrates to −1. The argument is easily extended to different spin directions.

• The hole function for two electrons at the same position and spin must cancel the spin density


Thus, we know the height, the width and the volume of the hole function. What is unknown isthe shape. Let us assume the most simple shape, namely a spherical box centered at the referenceelectron, that is

h(~x, ~x0) = −n(~x0)θ(

3

√

3

4πn(~x0)− |~r − ~r0|

)

δσ,σ0 (9.17)

where θ(x) is the Heaviside function.6

(r0)=φ−e

φ(r)−e

h(r,r0)r0

4 0πεe2

n(r0)2π

g (r,r0)n(r)

r

r

r

3334

Fig. 9.3: Schematic representation of the xc-hole and the xc-potential energy. The electron densityn(~r) is assumed to be constant. The electron shall be located at ~r0. The two-particle density n(~r ,~r0)describes the density of the N − 1 electrons as experienced from the reference electron at ~r0. Thetwo-particle density differs from the total density by the exchange hole h(~r − ~r ′). If we use the modelthat the exchange hole is a homogeneously charged sphere, the interaction energy of the referenceelectron and the exchange hole can be calculated as v(~r0).

6The Heaviside function is defined by

θ(x) =

0 for x < 012

for x = 0

1 for x > 0


Now we can determine the exchange energy as

Exc =

∫

d3r∑

σ

n(~x)1

2

∑

σ′

∫

d3r ′e2h(~x ′, ~x)

4πǫ0|~r − ~r ′|

=

∫

d3r∑

σ

n(~x)1

2

∑

σ′

∫

d3r ′e2

4πǫ0

−n(~x)δσ,σ′θ(3

√3

4πn(~x) − |~r ′ − ~r |)

|~r ′ − ~r |

=

∫

d3r1

2

e2

4πǫ0

∑

σ

n(~x)

∫

d3r ′−n(~x)θ

(3

√3

4πn(~x) − |~r ′ − ~r |)

|~r ′ − ~r |

=

∫

d3r1

2

e2

4πǫ0

∑

σ

n(~x)

− 3

2 3√

34πn(~x)

= −∫

d3r3

43

√

4π

3

e2

4πǫ0

∑

σ

n(~x)43

The integral over ~r ′ is easily solved directly. Here, we take a little detour, that may add somephysical insight. The integral corresponds to the potential in the center of a homogeneously charged

sphere with radius s0 = 3

√34πn , with a total charge of one electron. The potential of a homogeneously

charged sphere can be determined by fitting the potential outside and inside.

Φ(~s) =

∫

d3s ′enθ(s0 − |~s ′|)4πǫ0|~s ′|

=e

4πǫ0

1|~s| for s > s0

1s0

[

32 − 12

(|~s|s0

)2]

for s < s0

Thus, we obtain the exchange-correlation energy per electron within our model as

ǫxc(n(~r , ↑), n(~r , ↑)) = −3

43

√

4π

3

e2

4πǫ0

∑

σ n(~r , σ)43

∑

σ n(~r , σ)

If the spin-densities of both spin directions are equal, we obtain, introducing the symbol nt(~r) forthe total electron density,

ǫxc(1

2nt(~r),

1

2nt(~r)) = −

3

43

√

4π

3

e2

4πǫ0·(nt(~r)

2

) 13

and if the all electrons have the same spin, we obtain

ǫxc(nt(~r), 0) = −3

43

√

4π

3

e2

4πǫ0· nt(~r)

13

The exchange-correlation energy is more favorable, if the electrons are all parallel. This is a reflectionof Hund’s rule.

In Fig. 9.4 the exchange-correlation energy per electron as function of the electron density iscompared to the Hartree-Fock and the numerically correct result. Note that our model approachesthe exact result for low densities. The deviation of our model is somewhat better than the Hartree-Fock result. Still, the errors are of order few eV, which is in the range of to binding energies. In anactual calculation of bond formation or bond-breaking the density changes only by a small amountand only in a local region, so that the errors will be acceptable.

Since we can now determine the interaction energy, at least in our model, we can write down a


0.0 0.1 0.2Na

0 eV

-10 eV

-20 eV

n

ǫxcunpolarized electron gas

LSD.

............................................................ HF.

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

model.

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

.....

..

..

.

..

..

.

..

..

..

..

..

..

...........

.

.........

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

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

...........

.........

...........

...........

...........

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

..........

...........

..........

..........

. .........

. .........

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

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

..

..

..

..

..

.

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

.....

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

....

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

.....

..

....

..

..

..

.

..

..

.

....

..

..

..

.

.

..

..

..

..

.

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

.

.........

.

........

.........

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

...........

...........

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

...........

...........

...........

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

..........

..........

..........

..........

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

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

Fig. 9.4: Exchange-correlation energy per electron for a free electron gas as function of the density.The density is given in Hartree atomic units. The dashed line denoted “model” denotes the resultobtained from the model of a homogeneously charged sphere as exchange-correlation hole. Thedashed line denoted “HF” is the result obtained from Hartree-Fock. The correct result, which is theresult from density-functional theory is denoted by “LSD”. Note that the hard-sphere model describesthe exchange-correlation energy per electron well in the low-density region, where the electrostaticrepulsion dominates over the kinetic energy cost required to deform the exchange-correlation hole.The correct result lies in between the hard-sphere model and the Hartree-Fock result. This canbe understood as follows: The exchange hole of the free electron gas (Hartree-Fock) does not yetadjust to the interaction. It corresponds to the limit where the kinetic energy determines the shapeof the hole. The interaction makes the hole more compact than the true exchange-correlation holeand thus lowers the total energy. In the hard-sphere model, the hole is chosen to optimize theinteraction energy. It corresponds to the limit where the kinetic energy cost to deform the hole isnegligible. However, it does not consider the kinetic energy cost at all. Thus, it underestimates thetrue exchange-correlation energy.

total energy expression

E[n(~x)] =∑

n

fn〈ψn|p2

2me|ψn〉

︸︷︷︸

Ekin

+

∫

d3r nt(~r)vext(~r)

+1

2

∫

d3r

∫

d3r ′e2nt(~r)nt(~r ′)

4πǫ0|~r − ~r ′|︸︷︷︸

Hartree

−∫

d3r3

43

√

4π

3

e2

4πǫ0

∑

σ

n(~r , σ)43

︸︷︷︸

Exc

The effective potential is obtained as the derivative of the potential energy

vef f (~x) =δEpotδn(~x)

= vext(~r) +

∫

d3r ′e2nt(~r ′)

4πǫ0|~r − ~r ′|− e2

4πǫ0

3

√

4π

3n(~x)

13

︸︷︷︸

µxc (~x)

Note, that the exchange-correlation potential differs from the exchange-correlation energy per elec-tron. Both scale in the same way, but they differ by a factor.

There is a second observation. The effective potential is spin dependent. This spin dependence isresponsible that we can calculate whether materials are magnetic or not. The potential has nothing todo with a magnetic interaction, even though it looks exactly like a magnetic field in the Pauli equation.


In fact the origin for the magnetization of materials is unrelated to the magnetic interactions. Itstrue origin is the so-called exchange interaction, which again roots in the Pauli principle. Note,however, that the exchange interaction is not a new physical form of interaction between particles.

Fertig: WS06/07 9 Doppelstunde 4.Dec.06

9.3 Constrained search

Density-functional theory says that all ground-state properties can be expressed, at least in principle,

as functionals of the charge density alone. This is a nontrivial statement, because the wave functioncontains far more information than the density. It also provides a dramatic simplification as thedensity, in contrast to the wave function, can be handled on a computer: The density is a functionin three dimensions, while the wave function is a function in 3N dimensions.

The original proof of existence for density functionals goes back to Hohenberg and Kohn[56].Later, Levy[57, 58] and Lieb[59] have given a recipe how the density functional can actually beconstructed . This proof contains the proof of existence. For this reason, I will not present theoriginal proof of Hohenberg and Kohn, but show the one by Levy.

n1(r) n2(r) n3(r) n4(r)

|ΨΑ>

|ΨΕ> |ΨΗ>

|ΨΚ>

|ΨΜ>

|ΨΒ>|ΨΙ >|ΨΧ>

F1[n1(r)]

F0[n1(r)]F0[n2(r)]

F1[n2(r)]

F0[n3(r)]

F1[n3(r)]

F0[n4(r)]

F1[n4(r)]

Fig. 9.5: Sketch of Levy’s derivation of a density functional.

Let us start with a Hamilton operator

Hλ = T + Vext + λW

where

T =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉(

N∑

i=1

−~22me

~∇2i

)

〈~x1, . . . , ~xN |

is the kinetic energy operator,

V =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉(

N∑

i=1

vext(~ri)

)

〈~x1, . . . , ~xN |

represents the external potential, and

W =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉

1

2

∑

i 6=j

e2

4πǫ0|~ri − ~rj |

〈~x1, . . . , ~xN |


is the interaction energy operator. We have scaled the electron-electron interaction by a couplingparameter λ, which allows us later to switch the electron-electron interaction on an off. This is notrequired for the proof of Levy, but later on, we will often refer to Hamiltonians without interaction.

The idea of the proof is very simple:

1. Group all normalized, antisymmetric many-particle wave functions |Ψ〉 according to theirdensity. Each such subset A[n(~r)] contains all wave functions in the Fock space F with thesame density.

A[n(~r)] = |Ψ〉 ∈ F : 〈Ψ|n(~r)|Ψ〉 = n(~r)

The Fock space is the union7 of all N-particle Hilbert spaces HN , that is

Fdef=

∞⋃

N=0

HN

The Fock space will be introduced in more detail later. If the reader feels uncomfortable withthis concept, he may replace the Fock space with the N-particle Hilbert space for a specific N.

2. For each subset A[n(~r)], that is for each density, we determine that wave function |Ψ(0)λ [n(~r)]〉that provides the lowest energy 〈Ψ|Hλ|Ψ〉. The minimum of the total energy defines the totalenergy functional Eλ[n(~r)] of the electron density.

Eλ[n(~r)] = min|Ψ〉∈A[n(~r)]

〈Ψ|Hλ|Ψ〉 = 〈Ψ(0)λ [n(~r)]|Hλ|Ψ(0)λ [n(~r)]〉

3. The energy can be divided into universal density functional Fλ[n(~r)], which is a generalproperty of the electron gas, and a system-specific term, that depends on the external potential.

Eλ[n(~r)] = Fλ[n(~r)]︸︷︷︸

〈Ψ(0)λ [n]|T+λW |Ψ(0)λ [n]〉

+

∫

d3r n(~r)vext(~r)

︸︷︷︸

〈Ψ(0)λ [n]|V |Ψ(0)λ [n]〉

Because the energy related to the external potential only depends on the density, it is a constantwithin each group of a given density. Thus, we can directly determine the universal functionalFλ[n(~r)] as minimum of 〈Ψ|T + λW |Ψ〉 for all wave functions with the specified density.

UNIVERSAL DENSITY FUNCTIONAL

Fλ[n(~r)]def= min|Ψ〉∈A[n(~r)]

〈Ψ|T + λW |Ψ〉

4. To find the ground-state energy E(0)λ of an electron gas with N electrons in a given externalpotential, we minimize the total energy functional under the constraint of a specified totalnumber of electrons.

E(0)λ = min

n(~r):∫d3r n(~r)=N

Fλ[n(~r)] +

∫

d3r n(~r)vext(~r)

The Lagrange parameter µ is the chemical potential of the electrons. This minimization withrespect to the density provides us with the ground-state density and the ground-state energy.If we have remembered the wave function |Ψ0[n(~r)]〉 that minimized the total energy within agroup with given density n(~r), we can, in principle work our way back to the ground-state wavefunction, which contains all information about the electronic ground state.

7germ.:union=Vereinigung


This concludes the proof by Levy on the existence of a universal functional of the density, that allowsto determine the ground-state energy and, in principle, all other properties of the electronic groundstate.

9.4 Self-consistent equations

In the following I will discuss, how this proof can be used for a practical scheme to find the electronicground-state energy and density under the assumption that the universal functional is known.

We will be lead to a system of equations, that must be fulfilled simultaneously for the electronicground state. Then we will show, how this system of equations can be solved iteratively.

9.4.1 Universal Functional

Let us write down the universal functional Fλ[n(~r)] discussed in the proof of Levy more explicitly.

Fλ[n(~r)] = extr

η

extr

γ(~r)

min|Ψ〉

〈Ψ|T + λW |Ψ〉+∫d3r γ(~r)

[

〈Ψ|n(~r)|Ψ〉 − n(~r)]

−η[

〈ψ|ψ〉 − 1]

Instead of explicitly grouping the wave functions we enforce the constraints by the method of Lagrangemultipliers γ(~r) and η. Note, that we need not specify the chemical potential here, because theparticle number is fixed by the specification of the density n(~r).

The operator n(~r) yields the one-particle density of a many-particle wave function.8 It is definedby

n(~r) =

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉(

N∑

i=1

δ(~r − ~ri))

〈~x1, . . . , ~xN |

This means that we minimize the total energy 〈ψ|Hλ|ψ〉 under the constraint that the density ofthe wave function is identical to n(~r) specified beforehand, namely.

〈Ψ|n(~r)|Ψ〉 = n(~r)

The corresponding Lagrange multiplier is γ(~r). It is position dependent, because there is a constraintfor every position. In addition, we need to impose the constraint that the wave function is normalized,that is

〈Ψ|Ψ〉 = 1

This normalization condition is automatically fulfilled only if we restrict the search to N particle wavefunction and the density n(~r) integrates to N. However, if we allow for wave functions with variableparticle number, the norm-condition must be imposed as an extra condition.

8Thus

〈Ψ|n(~r)|Ψ〉 =∫

d4x1 · · ·∫

d4xN 〈Ψ|~x1, . . . , ~xN〉(

N∑

i=1

δ(~r − ~ri ))

〈~x1, . . . , ~xN |Ψ〉

=

N∑

i=1

∫

d4x1 · · ·∫

d4xN Ψ∗(~x1, . . . , ~xN)δ(~r − ~ri )Ψ(~x1, . . . , ~xN)


The equilibrium condition results in a Schrödinger equation[

T + λW +

∫

d3r γ(~r)n(~r)− η]

|Ψ〉 = 0

The wave function obeys a Schrödinger equation in an external potential γ(~r). The energy in thatpotential is η.

9.4.2 Exchange-correlation functional

Now we split off from the functional F1[n], that is the universal density functional of the interactingelectron gas, two terms that are fairly easy to evaluate. One is the Hartree energy EH and the otheris the kinetic energy of a non-interacting electron gas with the same density, which is simply F0[n].As a result, we obtain the exchange and correlation energy

EXCHANGE-CORRELATION FUNCTIONAL

Exc[n(~r)]def=F1[n(~r)]− F0[n(~r)]−

1

2

∫

d3r

∫

d3r ′e2n(~r)n(~r ′)

4πǫ0|~r − ~r ′|︸︷︷︸

EH

Note that F1 contains the kinetic energy of the interacting electron system and the electrostatic inter-action energy, while F0 is the kinetic energy of a non-interacting electron gas with the same density.Even though the wave functions of the interacting and the non-interacting electron gases have thesame density, they are different, because they minimize different Hamiltonians.

The total energy functional of the interacting electron system can be written with the help of theexchange and correlation energy as

E1[n(~r)] = F0[n(~r)] +1

2

∫

d3r

∫

d3r ′e2n(~r)n(~r ′)

4πǫ0|~r − ~r ′|

+

∫

d3r n(~r)vext(~r) + Exc[n(~r)] (9.18)

It seems as if we have not gained much because now we have to deal with two functionals F0 andExc . The advantage is that F0 is the functional for non-interacting electrons, which we can evaluate,and that Exc is a small term that can be approximated. Important is also the Exc can be expressedin physically intuitive terms, so that we can make reasonable assumptions about it.

9.4.3 The self-consistency condition

In order to determine the minimum of the total energy, we need to be able to determine the densitywhich fulfills

δE1δn(~r)

Eq. 9.18=

δF0δn(~r)

+

∫

d3r ′e2n(~r ′)

4πǫ0|~r − ~r ′|+ vext(~r) + µxc(~r)− µ = 0 (9.19)

where the exchange-correlation potential is defined as

µxc(~r)def=δExcδn(~r)

The exchange-correlation potential µxc should not be confused with the chemical potential µ.We can form all the functional derivatives of Eq. 9.19 with the exception of the functional

derivative of F0. In the following we will determine it.


We will make use of a Legendre transform of the universal density functional of the non-interacting electron gas.

As a reminder of the Legendre transform, let us take a little detour and repeat as examplefor a Legendre transform. We look into how the enthalpy H(p), which is a function of thepressure, is constructed from the total energy E(V ), given as function of the volume. Westart to define the enthalpy as the minimum total energy of a system connected to a volumereservoir. The latter is characterized by a pressure.

H(p) = minVE(V ) + pV

as minimum condition we obtain

dE

dV+ p = 0 ⇒ p(V (0)) = − dE

dV

∣∣∣∣V (0)

This equation is resolved to obtain the optimum volume V (0)(p) for a given pressure. Thus,we can express the enthalpy as a function of the pressure alone, namely

H(p) = E(V (0)(p)) + pV (0)(p)

Similarly we can obtain the energy from the enthalpy by a Legendre backtransform.

E(V ) = maxpH(p)− pV

which yields the minimum condition

dH

dp− V = 0

from which we obtain the pressure for a given volume p(0)(V ), which we insert to obtain theenergy as function of volume

E(V ) = H(p(0)(V ))− p(0)(V )V

Thus, we showed how the Legendre transform works back and forth. The Legendre transformis unique if the function is either fully concave or fully convex.

After this little detour, let us Legendre transform the universal density functional.

Q[u(~r ] = minn(~r)

F0[n] +

∫

d3r n(~r)u(~r)

The minimum condition with respect to the density yields

δF0δn(~r)

∣∣∣∣n(0)([vef f ],~r)

+ u(~r) = 0

which specifies a density n(0)([u], ~r) for each u(~r), so that

Q[u] = F0[n(0)] +

∫

d3r n(0)(~r) u(~r)

The functional Q[u] has an intuitive interpretation and can be calculated in a straightforwardmanner. It is simply the ground-state total energy of a non-interacting electron gas in an external

potential vef f (~r)def=u(~r)+µ, where the one-particle orbitals are occupied up to the chemical potential


µ. Because the ground state of a non-interacting system is represented by a Slater determinant wecan express Q by one-particle orbitals.

Q[u] =∑

n

〈φn|~p2

2me+ vef f |φn〉 =

∑

n

ǫn

where the one-particle orbitals obey the Schrödinger equation[−~22me

~∇2 + vef f (~r)− ǫn]

φn(~r) = 0

and have energies smaller than the chemical potential µ. Note that the chemical potential drops outof the equation. It has been introduced here to draw the connection to the known physical quantities.

After we know how to obtain the Legendre transform Q[u], the universal density functional ofthe non-interacting electron gas F0 can be obtained by a Legendre backtransform

F0[n] = maxu

Q[u]−∫

d3r n(~r)u(~r)

(9.20)

We see again that F0 corresponds to the kinetic energy of a non-interacting electron gas.The minimum condition defines an effective potential u(0)([n], ~r) as function of the density

δQ

δu(~r)

∣∣∣∣u(0)(~r)

− n(~r) = 0 (9.21)

so that

F0[n(~r)] = Q[u(0)([n], ~r)]−

∫

d3r n(~r)u(0)([n], ~r)

Now we can form the derivative with respect to the density

δF0δn(~r)

=

∫

d3r ′(

δQ

δu(~r ′)− n(~r ′)

)

︸︷︷︸

=0

δu(0)(~r ′)δn(~r)

− u(0)(~r)

Eq. 9.21= −u(0)(~r) (9.22)

Thus, we found an expression for the derivative of F0[n] and a recipe for constructing it by solving aSchrödinger equation for a non-interacting electron gas.

Now, we can insert this result, namely Eq. 9.22, into the minimum condition Eq. 9.19 of the totalenergy

δE

δn(~r)= −u(0)([n], ~r)︸︷︷︸

δF0δn(~r)

+

∫

d3r ′e2n(~r ′)

4πǫ0|~r − ~r ′|+ vext(~r) + µxc(~r)− µ = 0 (9.23)

which is an equation that specifies the effective potential vef f (~r) = u(~r)+µ, that yields the ground-state density, as

vef f (~r)︸︷︷︸

u(~r)+µ

= vext(~r) +

∫

d3r ′e2n(~r ′)

4πǫ0|~r − ~r ′|+ µxc(~r) (9.24)

where n(~r) is the ground-state density of the non-interacting electron gas in the potential vef f (~r)and a chemical potential µ.


9.4.4 Flow chart for a conventional self-consistency loop

Thus, we can now list the equation system that defines the ground-state energy[

~p2

2me+ vef f (~r)− ǫn

]

|φn〉 = 0

n(~r) =

N∑

n=1

φ∗n(~r)φn(~r)

vef f (~r) = vext(~r) +

∫

d3r ′e2n(~r ′)

4πǫ0|~r − ~r ′|+ µxc(~r)

In Fig. 9.6 the steps required to solve this coupled system of equations is shown.

veff(r )

n(r )

veff(r )

v (r )extveff(r ) d3r’4 0πε ’|r −r |e2n(r ’)

xc(r )µ+= +

|φn

nn(r )=Σ *φn φn(r ) (r )

p2

2meveff εn =0|φn−

veff=veffoutin ?α )veff+αin veff

outveff=(1−in

in

out

out

in

Fig. 9.6: Self-consistency cycle

9.4.5 Another Minimum principle

The total energy functional has the form

E1[n] = F0[n] +1

2

∫

d3r

∫

d3r ′e2n(~r)n(~r ′)

4πǫ0|~r − ~r ′|+

∫

d3rn(~r)vext(~r) + Exc[n(~r)]


Because the ground state of a non-interacting electron gas is a Slater determinant, we can definethe functional F0[n] by minimizing over Slater determinants only. This allows us to describe F0 asminimum over one-particle wave functions.

The functional F0 can therefore be written as

F0[~n] = extr

Λn,m

extr

vef f (~r)

min|φn〉

∑

n

〈φn|~p2

2me|φn〉

+

∫

d3r vef f (~r)

[∑

n

φ∗n(~r)φn(~r)− n(~r)]

−∑

n,m

(〈φn|φm〉 − δn,m) Λm,n

where vef f plays the role of a Lagrange multiplier. This allows us to express the derivative of theuniversal density functional of the non-interacting electrons as

δF0δn(~r)

= vef f (~r)

In practical calculation we (1) chose an arbitrary potential vef f . Then (2) we solve the Schrödingerequation, to obtain the one-particle wave functions, and the kinetic energy of the non-interactingelectrons. Now (3) we evaluate the density. The kinetic energy and the density allows us (4) toevaluate the total energy. The density provides us (5) with a new expression for vef f . Out of thelast and the new effective potential one creates (6) a better choice for the effective potential, whichis used as new start potential for (1). If the effective potentials, the one used for the Schrödingerequation and the one obtained from the density, are identical, we have found the desired ground-stateenergy and density.


9.5 Adiabatic connection

We have seen so far that one can reasonably estimate the exchange-correlation hole, and evaluatethe energy of the electron in the potential of the exchange-correlation hole. However, it is nearlyhopeless to estimate the kinetic energy contribution to the exchange-correlation energy.

If there would be no cost to deform the exchange-correlation hole, the latter would simply adjustsuch that the electrostatic energy is optimized. One can easily estimate how the hole would looklike: It would have the shape of a hard sphere centered at the reference electron. Its radius wouldbe determined by the sum rule that says that there is exactly one positive charge in the exchange-correlation hole. The reason for this shape is that the electron would repel its neighbors as much aspossible, in order to reduce its electrostatic interaction.

However as the exchange-correlation hole is deformed, there is a price to pay, namely that thekinetic energy goes up. In the extreme case of a hard-sphere like hole, the wave functions would havesteps, that is infinite curvature. The kinetic energy is just a measure of the curvature of the wavefunction. Thus, the deformation of the hole costs kinetic energy. The final shape of the XC-hole istherefore a balance between electrostatic gain and kinetic energy cost.

The adiabatic connection[60, 61, 62] is a theorem that allows to relate the net energy gain,including the kinetic energy cost, to the electrostatic energy gain only, but for variable strengths of theinteraction energy. While this theorem looks like mystery, it cannot be applied directly. Neverthelessit provides an important route that allows important approximations to be made. The theorem isdemonstrated in the following:

We consider a given electron density for which the non-interacting functional F0[n(~r)] is exactlyknown.

Fλ[n(~r)] = 〈Ψλ|T + λW |Ψλ〉

+

∫

d3r µλ(~r)(

〈Ψλ|n(~r)|Ψλ〉 − n(~r))

− Eλ(

〈Ψλ|Ψλ〉 − 1)


where |Ψλ〉 is the many electron wave function that minimizes Fλ under the constraint that the densityof the wave function is equal to n(~r). The function µλ(~r) represents the Lagrange parameters. Theoperator

n(~r)def=

∫

d4x1 · · ·∫

d4xN |~x1, . . . , ~xN〉(

N∑

i=1

δ(~r − ~ri))

〈~x1, . . . , ~xN |

is the operator that extracts the electron density from the wave function.The wave functions that minimize the functional obey the Schrödinger equation

[

T + λW +

∫

d3r µλ(~r)n(~r)− Eλ]

|Ψλ〉 = 0 (9.25)

This means that as the interaction strength is increased, a local potential, µλ(~r), is switched on thatensures that the electron density remains unchanged.

If we know the derivatives of the functional with respect to the interaction strength λ, and if weknow the functional for the non-interacting electron gas, we can determine the functional for theinteracting electron gas as

F1[n(~r)] = F0[n(~r)] +

∫ 1

0

dλdFλ[n(~r)]

dλ

= 〈Ψ0|T |Ψ0〉+∫ 1

0

dλdFλ[n(~r)]

dλ

The first term is simply the kinetic energy of the non-interacting electron gas with the same densityas the interacting system.

Now we can show that the derivative of the functional with respect to the interaction strengthcan be determined from the interaction energy alone. The proof rests on the Hellmann-Feynmantheorem.

dFλ[n(~r)]

dλ=

d

dλ

[

〈Ψλ|T + λW |Ψλ〉

+

∫

d3r µλ(~r)(

〈Ψλ|n(~r)|Ψλ〉 − n(~r))

− Eλ(

〈Ψλ|Ψλ〉 − 1)]

= 〈dΨλdλ|T + λW |Ψλ〉+ 〈Ψλ|W |Ψλ〉+ 〈Ψλ|T + λW |

dΨλdλ〉

+

∫

d3rdµλ(~r)

dλ(〈Ψλ|n(~r)|Ψλ〉 − n(~r))︸︷︷︸

=0

+

∫

d3r µλ(~r)

(

〈dΨλdλ|n(~r)|Ψλ〉+ 〈Ψλ|n(~r)|

dΨλdλ〉)

−dEλdλ

(

〈Ψλ|Ψλ〉 − 1)

︸︷︷︸

=0

−Eλ(

〈dΨλdλ|Ψλ〉+ 〈Ψλ|

dΨλdλ〉)

= 〈Ψλ|W |Ψλ〉

+〈dΨλdλ|[

T + λW +

∫


|Ψλ〉︸︷︷︸

=0

+ 〈Ψλ|[

T + λW +

∫


︸︷︷︸

=0

|dΨλdλ〉

Eq. 9.25= 〈Ψλ|W |Ψλ〉


Thus, we can express the functional for the finite interaction strength as

F1[n(~r)] = 〈Ψ0|T |Ψ0〉+∫ 1

0

dλ 〈Ψλ|W |Ψλ〉︸︷︷︸

EH [n(~r)]+Exc [n(~r)]

Thus, we have obtained an explicit expression for the exchange-correlation energy, that does notdirectly refer to the kinetic energy.

Exc [n(~r)] =

∫ 1

0

dλ 〈Ψλ|W |Ψλ〉 −1

2

∫

d3r

∫

d3r ′e2n(~r)n(~r ′)

4πǫ0|~r − ~r ′|︸︷︷︸

EH

This expression has the problem that we still need the wave functions |Ψλ〉 for all possible values forthe interaction strength.

The interaction energy can further be expressed by the exchange-correlation hole for a given valueof the interaction, namely

〈Ψλ|W |Ψλ〉 =1

2

∫

d3r

∫

d3r ′e2n(~r)n(~r ′)

4πǫ0|~r − ~r ′|+

∫

d3r n(~r)1

2

∫

d3r ′e2hλ(~r , ~r ′)

4πǫ0|~r − ~r ′|

so that

Exc =

∫

d3r n(~r)

∫ 1

0

dλ1

2

∫

d3r ′e2hλ(~r , ~r ′)

4πǫ0|~r − ~r ′|︸︷︷︸

=:ǫxc

(9.26)

The variable ǫxc , which itself is a functional of the density is the exchange-correlation energy perelectron.

9.5.1 Screened interaction

It is possible9 to reformulate the exchange-correlation energy by the exchange-only energy with ascreened interaction. This is a form that reminds of the so-called GW approximation, where theself-energy is expressed by Green’s-function times screened interaction.

ExcEq. 9.26=1

2

∫

d3r

∫

d3r ′ n(r)h0(r, r′)

∫

dλhλ(~r , ~r ′)

h0(~r , ~r ′)

e2

4πǫ0|~r − ~r ′|︸︷︷︸

vscr (~r−~r ′)

=1

2

∑

i ,j

∫

d3r

∫

d3r ′ φ∗i (~r)φj(~r)vscr (~r − ~r ′)φ∗j (~r ′)φi(~r ′)

where the screened interaction is defined as

vscr (~r − ~r ′) def=

∫

dλhλ(~r , ~r ′)

h0(~r , ~r ′)

e2

4πǫ0|~r − ~r ′|

Note that, in contrast to the GW approximation, our expression is exact, but, on the other hand, itis limited to the ground state.

Note also that the screened interaction in this formulation only enters the exchange term, butnot the Hartree term. It also includes the kinetic energy correction.

The qualitative behavior can be understood easily:

9This is an idea from the author, that has not been crosschecked properly. Therefore some caution is required.


h0

ha

h0

rs

1

|r−r’|

hx

Fig. 9.7: Schematic diagram of the effect of the screening of the interaction. The golden line h0 isthe hole function of the non-interacting reference system, while the red line hλ is the hole functionin presence of a scaled interaction. The ratio scaled the Coulomb interaction in the exchange term.It suppresses the long-range tail of the interaction, and enhances the interaction for intermediatedistances. rs is the average electron radius 4π3 r

2s ∗ n = 1. (ha and hx should be hλ; Problem with the

drawing program.)

• The hole function hλ(~r , ~r ′) for a finite interaction is generally more compact than the onobtained without interaction. Thus, the screened interaction vscr is generally more short rangedthan the unscreened interaction.

• The screened interaction for ~r = ~r ′ is identical to the unscreened interaction, because the holefunction cancels exactly the spin density, independent of the strength of the interaction.

• Due to the compression of the hole upon increasing interaction and because of the sum-rulethat the hole integrates to exactly one positive charge, the interaction must be enhanced forintermediate distances.

• The screening will be more effective, if the price for the distortion of the wave function islow, that is for a low density. This implies that the unscreened interaction is a reasonableapproximation in the high-density region around the atomic center, while the tail region of amolecule is better described by a more short-ranged interaction.

9.5.2 Hybrid functionals

The adiabatic connection scheme is exploited in the construction of so-called hybrid functionals.These density functionals describe the exchange-correlation energy as a mixture of the correct ex-change, obtained as in the Hartree-Fock method and another local or gradient corrected functional.

The performance of the hybrid functionals is probably among the best. They have the disadvan-tage that they require that the Hartree Fock exchange must be evaluated, which is computationallymore challenging than the evaluation of the regular density functionals.

The idea behind the hybrid functionals is that the local functionals are appropriate if the interactionenergy is very large. However if the interaction energy is small, the exchange hole is given by HartreeFock exchange.

If for example the kinetic energy dominates, the shape of the exchange-correlation hole cannotadjust, and we can approximate hλ(~r , ~r ′) ≈ h0(~r , ~r ′) by the hole of the non-interacting electron gas.


Thus, we obtain

ǫxc ≈1

2

∫

d3r ′e2h0(~r , ~r ′)

4πǫ0|~r − ~r ′|

If the electrostatic interaction dominates over the kinetic energy then h1(~r , ~r ′) would be a hardsphere, and therefore fairly local. Therefore, the hybrid functionals try to approximate the integralby a superposition of HF-exchange and a local density functional.

d3r4 0|r−r’|πεe2h (r,r’)λ

21

Hartree−Fock(hole of non−interacting electrons)

xcε

0 1 λ

9.6 Functionals

According to the Bible, Jacob had a dream in which he saw a ladder descending from Heaven toEarth and angels climbing and descending the ladder10. In John Perdew’s dream, the angels are usersof DFT who climb the ladder to gain greater precision (at greater cost), but who also need to beable to descend the ladder depending upon their needs[? ].

10Jacob’s ladder has been described in the Book of Genesis (28:11-19). It is a ladder from earth to heaven on whichthe angels descend and ascend.


JACOB’S LADDER OF DENSITY FUNCTIONALS

The different stairs of Jacob’s Ladder of density functionals are

1. LDA (local density approximation): the exchange-correlation hole is taken from the free elec-tron gas. These functionals exhibit strong overbinding. While the van-der Waals bond is notconsidered in these functionals, the overbinding mimicked van-der-Waals bonding, albeit for thewrong reason.

2. GGA (generalized gradient approximation): These functionals not only use the electron densitybut also its gradient to estimate the asymmetry of the exchange-correlation hole with respectto the reference electrons. As a result, surfaces are energetically favored compared to LDAand the overbinding is strongly reduced.

3. meta-GGA: in addition to the gradient also the kinetic energy density is used as a parameter.The kinetic energy is a measure for the flexibility of the electron gas.

4. Hybrid functionals: Hybrid functional include a fraction of exact exchange. This functionalsimprove the description of left-right correlations.

5. Exact: An exact density functional can be obtained using the constrained search formalismusing many-particle techniques.

9.7 Xα method

Even before the invention of density-functional theory per se, the so-called Xα method has beenintroduced. Today, the Xα method has mostly historical value. The Xα method uses the expressionfor the exchange of a homogeneous electron gas instead of the exchange-correlation energy. However,the exchange energy has been scaled with a parameter, namely Xα, that has been adjusted to Hartree-Fock calculations. The results are shown in Fig. 9.8.

The rationale behind the Xα-method is a dimensional argument. Choose a given shape for theexchange-correlation hole, but scale it according to the density and the electron sum rule. Thenthe exchange-correlation energy per electron always scales like n

13 . Each shape corresponds to a

pre-factor.Consider a given shape described by a function f (~r) with

f (~0) = 1∫

d3r f (~r) = 1

Now, we express the hole function by the function f by scaling its magnitude at the origin such thatthe amplitude of the hole cancels the electron density. Secondly, we stretch the function in space sothat the sum rule, which says that the hole must integrate to −1, is fulfilled. These conditions yieldthe model for the exchange correlation hole.

h(~r0, ~r) = −n(~r0)f (~r − ~r0n(~r0)

13

)

The corresponding exchange-correlation energy per electron is

ǫxc(~r) = −1

2

∫

d3r ′e2

4πǫ0|~r − ~r ′|f (~r − ~r ′n(~r)

13

)


10 20 30 40

Z

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Xα

Kastenform

homogenes Elektronengas

rr

r rrrrrr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r

Fig. 9.8: Xα value obtained by comparison of the atomic energy with exact Hartree-Fock calculationsas function of atomic number of the atom.[63, 64]

If we introduce a variable transform

~y =~r − ~r ′n(~r)

13

we obtain

ǫxc(~r) = −n(~r)

13

2

∫

d3y ′e2

4πǫ0|~y |f (~y) = −Cn 13

where C is a constant that is entirely defined by the shape function f (~r).In general, the Xα method yields larger band gaps than density functional theory. The latter

severely underestimates band gaps. This is in accord with the tendency of Hartree-Fock to overesti-mate band gaps. In contrast to Hartree-Fock, however, the Xα method is superior for the descriptionof metals because it does not lead to a vanishing density of states at the Fermi level.

9.8 Local density functionals

The first true density functionals were constructed for the homogeneous electron gas, by extractingthe exchange-correlation energy per electron ǫxc from a calculation of the interacting homogeneouselectron gas.


A breakthrough came about when Ceperley and Alder[65] performed quantum Monte-Carlo calcu-lations of the homogeneous electron gas as function of the density. Quantum Monte-Carlo calcula-tions are computationally expensive, but provide the energy of an interacting electron gas in principleexactly, that is with the exceptions of numerical errors. These results have been parameterized byPerdew and Zunger[49] and combined with so-called RPA results for the high-density limit. RPAstands for Random-Phase Approximation[66], which is accurate in the high density limit.

These density functionals exhibit very good results for solids. Electron densities are nearly perfect.Bond distances are typically underestimated by 1-3 % and bond angles agree with experiment withinfew degrees. Binding energies, on the other hand, are strongly overestimated. The errors are inthe range of electron volts and thus comparable to bond energies. Thus, these functionals havebeen useless for studying chemical reactions. However, the results for solid state processes such asdiffusion have been very good. This has lead to a long-standing misunderstanding between solid statephysicists and chemists about the usefulness of density-functional theory. The density functionalsworked find for solids, which was what the physicists are interested in, while they were a disaster forthe binding energies of molecules, the major interest of chemists. This difficulty has been overcomeby the gradient corrected density functionals discussed later.

The overestimate of the binding energies covers up the lack of Van der Waals interactions. Forexample, the binding energy and the bond distances of nobel gases are in very good agreementwith experiment. However, the agreement is good for the wrong reason. The weak van der Waalsinteraction is not described properly in local density functionals, but the overbinding compensates forthis fact.

For a long time it was surprising that the approximate density functionals work at all. They havebeen derived from a completely homogeneous electron gas and are applied to an electron density ofreal materials, which are far from homogeneous. Jones and Gunnarsson gave one explanation, namelythat density functionals observe an important sum rule, namely that the exchange-correlation holeintegrates to minus one electron charge. Fig. 9.9 compares the exchange hole of a free-electron gaswith that of a nitrogen atom, demonstrating that despite their very different shape, their contributionsto the exchange energy are similar.

9.9 Local spin-density approximation

In order to describe magnetic systems, or so-called open-shell molecules, one uses the local spin-density approximation, where the spin-dependent density n(~x) = n(~r , σ) is used instead of the totaldensity n(~r) =

∑

σ n(~x). As a result, we obtain one-particle wave functions with spin up and spindown character and one obtains two effective potentials, one for the spin-up electrons and anotherfor the spin-down electrons.

The difference between the effective potentials acts like a magnetic field, even though its originis purely electrostatic, namely exchange and correlation also called the exchange interaction.

9.10 Non-collinear local spin-density approximation

While the local spin-density formalism only allows one-particle wave function to have either purelyspin-up or purely spin down character, the non-collinear formulation allows the wave functions tohave a mixed spin-up and spin-down character. As a result, the magnetization can not only vary inmagnitude but also in direction.

9.11 Generalized gradient functionals

For a long time one has hoped to obtain better functionals by including also a dependence on thegradient of the density. The first attempts started from a model with a weakly oscillating electron


0 0

r−r0

0−0.3a 0−0.2a 00.1a0−0.1a r0

r−r000.3a00.2a00.1a

r−r0

−1.5a0 −1.0a0 −0.5a0 r0 00.5a

00.5a 01.0a 01.5a 02.0a

Exact

LSD

R

R

Exact

LSD

LSD

ExactLSD

Exact

R

Fig. 9.9: exchange-correlation hole of an electron in the nitrogen atom (full line) and in the localdensity-functional approximation (dashed) [67]. In the left row, the reference electron is located0.13 a0 from the nucleus and on the right it is located further, at 0.63 a0. Even though the shapeof the DFT-exchange hole deviates grossly from the correct exchange hole, the spherical averagesshown in the bottom figures are similar, which is attributed to the particle sum rule, obeyed in thelocal density approximation (LDA). The arrows pointing down indicate the exact and the DFT result

of[∫d3r |h(~r ,~r0)||~r−~r0|

]−1. Similar results are available for neon[49].

density. The resulting functionals however were worse than the local functionals.The reason for this failure was that when the expansion for slowly varying densities are extrapolated

to strongly oscillating densities, the density of the exchange-correlation hole was overcompensatingthe total density.

Later one realized that one should introduce a dimension-less scaled gradient defined as

s =rs |~∇n|n

The scaled gradient allows to switch off the gradient corrections in a physical sense, that is if thehole runs into the danger of producing a negative correlation function g(~x ′, ~x).

One way to construct the density is to use the results for slowly varying densities, which providesthe parameter β.

ǫxc(n, s) = ǫhomxc

[

1 + β(n)s2]

for s << 1 (9.27)

Surprisingly, the scaled gradient is largest in the tails of the wave functions, where the densityfalls off exponentially, that is

n(~r) ≈ Ae−λ~e~r

where ~e is the unit vector pointing into the direction in which the density falls off. Let us determine


rs| n|

∆

rsrs r0

n

−h(r0,r)

r

n

the scaled gradient for this density

s =3

√

3

4π

Aλ exp(−λ~e~r)A43λ exp(− 43λ~e~r)

=3

√

3

4πλn−

13 →∞

If we use only the gradient correction with the small-s expansion, the exchange-correlation energyper particle clearly becomes infinite.

However, in the tail region we can make use of other information. Consider an electron that isfar outside of a molecule. It is clear that it “sees” a molecule with N − 1 electrons. Thus, the hole isentirely localized on the molecule and the center of the hole is far from the reference electron. If thehole is at the molecule and the reference electron is far from it, we can estimate its interaction by

ǫxc =1

2r(9.28)

where r is the distance of the electron from the molecule.Let me demonstrate the construction of such a gradient corrected functional using the example

of Becke’s gradient correction for exchange[68]. We make an ansatz for the exchange energy perelectron as

ǫxc = Cn13F (s) (9.29)

where C is the pre-factor for the exchange energy. The function F (s) is determined such that smallgradient expansion, Eq. 9.27, is reproduced, and that the exchange energy per electron in the tailregion is correct.

To consider the tail region we require

ǫxcEq. 9.29= Cn

13F (s)

Eq. 9.28=

1

2r⇒ F (s) =

1

2Crn13

for s →∞

First we express the radius by the density and insert the result in Eq. 9.30 to obtain an expressionfor F (n(s)).

n(r) = Ae−λr ⇒ r(n) = − 1λln[ n

A

]

⇒ F (s) =1

2Crn13

=−λ

2Cn13 (ln[n]− ln[A]))


Next we express the density by the scaled gradient and insert the result in the above equation toobtain an expression for F (s).

s(n) =3

√

3

4πλn−

13 ⇒ n(s) =

3λ3

4πs−3

⇒ F (s) =−λ

2Cn13 (ln[n]− ln[A]))

=−λ

2C 3

√3λ3

4π1s

(

ln[ 3λ3

4π s−3]− ln[A]

)

=−s

2C 3

√34π

(

ln[ 3λ3

4πA3 s−3]− 3 ln[s]

)

s→∞≈ 1

6C3

√

4π

3

s2

s ln[s]

This equation gives us the large gradient limit of F (s).In order to connect the low gradient limit with the large gradient limit we choose

F (s) = 1 +βs2

1 + 6Cβs sinh−1(s)

The result is shown in Fig. 9.10.

s

F(s)

0

1

−3 3

Fig. 9.10: The function F (s) from Becke’s gradient corrected functional [68] for the exchange energyand from the small gradient expansion (dashed). Note that the gradient correction of Becke is smallerthan the expansion for large gradients.

The gradient correction plays the role of a surface energy, as it contributes mostly in the tail region.While the exchange-correlation energy per electron falls off exponentially in the local functionals, itfalls off as 1

2r in the gradient corrected functionals. This effect lowers the energy in the tail region


of a molecule compared to the local functionals. If we break a bond, the surface area of a moleculeincreases, because the bond is transformed into a tail region. Thus, the gradient correction favorsthe dissociation of the bonds. As a consequence, gradient corrected functionals avoid the artificialoverbinding of the local functionals. This argument also explains that local functionals perform fairlywell in solids: If an atom diffuses, there is no additional surface created so that the gradient correctionis minor.

Up to know there is an entire suite of different gradient corrected functionals. They are calledGeneralized Gradient Approximations (GGA) to differentiate them from the gradient expansions.The most common functionals are the Perdew-Wang-91-GGA, which has been superseded by thesimpler Perdew-Burke-Ernzerhof functional. Both yield nearly identical results.

As shown in Fig. 9.11, the performance of gradient corrected functionals is extremely good alsofor binding energies, which were unsatisfactory in the local density functionals. This has drawn alsothe chemists into the field of density-functional calculations.

0eV 10eV 20eV 30eV0eV

10eV

20eV

30eV

Fig. 9.11: Accuracy of the atomization energies calculated with the Becke88-Perdew86 densityfunctional compared to experiment.


9.12 Additional material

9.12.1 Relevance of the highest occupied Kohn-Sham orbital

According to Perdew[69? ], the energy of the highest occupied Kohn Sham orbital is the ionizationpotential of the material.

This statement has been disputed by Kleinman[? ].

9.12.2 Correlation inequality and lower bound of the exact density functional

9.13 Functionals

• [X-α] Uses a local expression for exchange as derived from the free electron gas, and scaledwith a factor α.

• [Barth-Hedin]

• [PZ (Perdew-Zunger)] parameterization of Ceperley and Alder’s quantum Monte Carlo dataof the free electron gas.

• [BP86 (Becke-Perdew)]

• [PW91 (Perdew-Wang)] First completely parameter free version of a gradient corrected den-sity functional

• [PBE (Perdew-Burke-Ernzerhof)]

• [B3LYP (Becke-Lee-Yang-Parr)] Hybrid functional.

• [GGA] (Generalized gradient approximation) General expression for gradient corrected func-tionals.

• [Hybrid functionals] General expression functionals employing a mixture of Hartree-Fock anda density functional


9.14 Reliability of DFT

If we knew the exact density functional, we would be able to determine exactly

• the total energy

• the electron density and

• the energy of the highest occupied orbital

The argument that DFT predicts the highest occupied orbital, while the orbital energies of allother one-particle orbitals are, in principle, without physical meaning goes as follows: Far from thesurface of a crystal the density of a one-particle orbital falls off as e−λz where λ is related to theorbital energy by λ = 1

~

√−2meǫ, where the orbital energy is measured relative to the vacuum level,that is limz→∞ vef f (~r). For large distances the highest occupied orbital, which has the slowest decay,will dominate the electron density. Thus, if we would know the electron density very accurately farfrom a surface, we would be able to determine the dominant exponential decay constant and thus


the energy of the highest occupied orbital.11 Thus, the energy of the highest occupied orbital canbe determined from the density, which is an exact prediction of DFT.

Of course, even though exact DFT predicts these quantities accurately, it remains to be seen ineach case if that also holds for the approximate functionals used in practice.

• Bond energies: Local functionals overestimate binding by often more than 1 eV. Energies arefairly good as long as the effective surface area of the system is not changed. This poor resultis dramatically improved by the GGA’s.

• Structures: bond lengths are underestimated in LDA by 0-2 %. They are overestimated byGGA’s by 0-2 %.

• Dipole moments standard deviation 0.1 Debye[70]

9.15 Deficiencies of DFT

• Missing Van der Waals interaction: The van-der-Waals interaction is not included. Surpris-ingly the description of van der Waals bonds is nearly perfect with truly local density functionals.This however is an artifact of the tendency to overbinding in truly local density functionals onthe one hand, and the lack of van der Waals interactions on the other. Since the overbindingof covalent bonds introduces errors in the range of 1 eV, it is generally not a good idea to usetruly local density functionals instead of their gradient corrected counterparts.

A density functional for the van der Waals interaction has been developed by Dion et al.[71].It is named vdWDF (van-der-Waals density functional).

• Band-gap Problem: If we compare the spectrum of Kohn Sham states with optical absorptionspectra we observe that the band gaps are too small. This is not a deficiency of density-functional theory, but an over-interpretation of the Kohn Sham spectrum. The excited statesshould not be calculated from the same potential as the ground state.

• Reduced band width: Band width of alkaline metals.

• Mott insulators

• unstable negative ions

• Broken symmetry spin states: singlet O2 Molecule

• Transition states:

• High-energy spectrum: The high energy part if the Kohn-Sham spectrum is too low comparedto the measured optical spectrum. The reason is that the exchange-correlation hole cannotfollow the very fast electrons.????

11There is a caveat in the argument. Consider an insulator which may have surface states in the band gap. In thatcase the density would probe the highest occupied surface state, which may lie above the highest occupied bulk state.

Chapter 10

Magnetism

Noncollinear magnetism See (1) C.Herring, Magnetism, exchange interactions among itinerant

electrons, Academic press, New York and London 1966. (2) L. Sandratskii, Phys Stat Sol. (b) 135,p.167 (1986) (3) L. Sandratskii J. Phys. F: Metal Phys. 15. L43 (1986) (4) Lizarraga Jurado, R.,Non-collinear Magnetism, in d- and f-electron systems, PhD Thesis, Upsala University (2006), ISBN91-554-6540-4, https://www.diva-portal.org/smash/get/diva2:168187/FULLTEXT01.pdf [72]

10.1 Charged particles in a magnetic field

10.2 Dirac equation and magnetic moment

Spin-orbit coupling responsible for magnetic anisotropy.

• Non-relativistic magnetic moment ~m of a charge rotating about a center

~mcl =q

2m~L (10.1)

• Dirac equation defines the gyromagnetic ratio of g = 2.

• Quantum field theory leads to g = 2.00232 . . .

10.3 Magnetism of the free electron gas

Landau levels

10.4 Magnetic order

Stoner criterion, Pauli paramagnetism,ferromagnetism, ferrimagnetism, antiferromagnetism, Ising model, Heisenberg model, Weiss re-

gions.

205

https://www.diva-portal.org/smash/get/diva2:168187/FULLTEXT01.pdf

206 10 MAGNETISM

Appendix A

Adiabatic decoupling in theBorn-Oppenheimer approximation

A.1 Justification for the Born-Oppenheimer approximation

In the following, I will present an argument, that I believe is close to the original proof of Born andOppenheimer. We will investigate an artificial model that we can solve exactly and compare theresults of the Born-Oppenheimer approximation with the exact result.

Imagine a Hamiltonian in which all interactions are harmonic instead of Coulombic. Secondly,consider only a one-dimensional electron and a one-dimensional nucleus. The electron coordinate isdenoted by x and the nuclear coordinate by R. The corresponding momenta are pel for the electroncoordinate and Pnuc for the nuclear coordinate. Thus, the full Hamiltonian is

H =P 2nuc2M

+p2el2m+1

2ax2 + bxR +

1

2cR2

=m

M

P 2nuc2m+p2el2m+1

2ax2 + bxR +

1

2cR2 (A.1)

Exact solution of the model system

From the description of the multidimensional harmonic oscillator (see ΦSX:Quantum Physics andΦSX:Klassische Mechanik), we know that we need to know only the two classical eigenfrequenciesω+ and ω− to specify the spectrum of the two-dimensional quantum mechanical harmonic oscillator.

Ej,n = ~ω+

(

n +1

2

)

+ ~ω−

(

j +1

2

)

(A.2)

In order to obtain the classical frequencies ω±, let us consider the classical harmonic oscillator,which has the Hamilton function

H(Pnuc , pel , R, x)Eq. A.1=

P 2nuc2M

+p2el2m+1

2ax2 + bxR +

1

2cR2

Hamilton’s second equation yield( .pel.P nuc

)

=

(

m..x

M..R

)

= −(

a b

b c

)(

x

R

)

=

(

Fel

Fnuc

)

By diagonalizing the dynamical matrix D

D =1

m

(

a√

mM b√

mM b

mM c

)

207

208 A ADIABATIC DECOUPLING IN THE BORN-OPPENHEIMER APPROXIMATION

we obtain the eigenvalues ω2 of the dynamical matrix

mω2 =a + m

M c

2±√(a − m

M c

2

)2

+m

Mb2

Let us investigate first two terms in an expansion in orders of mM .

mω2+ = a +m

M

b2

a+O

(m

M

)2

mω2− =m

M

(

c − b2

a

)

−(m

M

)2(

c2

2a+a

2

(2b2 − ac2a2

)2)

+O(m

M

)4

The higher frequency ω+ can be attributed to electronic excitation energies ∆E = ~ω+. Thus, ~ω+is similar size as excitation energies, that is typically in the range of few eV. The lower frequencyω− can be attributed to the atomic motion. The energy ~ω− is comparable to vibrational excitationenergies, that is in the range of about 10 meV.

Thus, the eigenvalue spectrum is

Ej,nEq. A.2= ~

√√√√√c − b2

a − mM

[

2c2

a +a2

(2b2−ac2a2

)2]

M

(

j +1

2

)

+ ~

√

a + mMb2

a

m

(

n +1

2

)

+O

((m

M

)2)

(A.3)

Born-Oppenheimer solution of the model system

We can now apply the Born-Oppenheimer approximation in order to see how it deviates from theexact result:

HBO =p2el2m+1

2ax2 + bRx +

1

2cR2

=p2el2m+1

2a(x +

b

aR)2 +

1

2

(

c − b2

a

)

R2

The force constant for the electrons is a, so that we obtain a frequency ωBO+ =√

am . Knowing that

the energy eigenvalues of the harmonic oscillator are ~ωBO+ (n+12) we can immediately determine the

Born-Oppenheimer energy surfaces.

EBOn (R) =1

2

(

c − b2

a

)

R2 + ~

√a

m

(

n +1

2

)

Now we can determine the eigenvalues of the nuclear Hamiltonian

Hnuc =P 2nuc2M

+ EBOn (R)

which has eigenvalues

Ej,n = ~

√

c − b2

a

M

(

j +1

2

)

+ ~

√a

m

(

n +1

2

)

(A.4)

We see that the spectrum obtained from the Born-Oppenheimer approximation Eq. A.4 provides thecorrect result, Eq. A.3, if m

M vanishes.

A ADIABATIC DECOUPLING IN THE BORN-OPPENHEIMER APPROXIMATION 209

The corrections to the Born-Oppenheimer approximations, that is the difference between Eq. A.3and Eq. A.4, are of order m

M .This ratio, namely m

M , is typically in of order 10−4 and definitely smaller than 10−3, because themass of a single nucleon (proton or neutron) is about 1823 times the mass of an electron. Thereforewe can assume that the Born-Oppenheimer corrections can be ignored in most cases.

The underlying reason for the validity of the Born-Oppenheimer approximation is that oscillatorsdo not affect each other strongly, if they are out of resonance. In a solid, the oscillators are theelectronic excitations and the phonons, that is lattice vibrations. Electronic excitation energies arelarger than the optical band gap of the material, which is typically in the range of 1 eV. Latticevibrations have wavenumbers below than 800 cm−1 and thus their energies lie below 10 meV. Exceptfor metals or molecules with a nearly degenerate ground state this separation of frequencies is real.

Illustration of the out-of-resonance condition

To illustrate this effect let us consider the classical model of two coupled harmonic oscillators shown inFig. A.1. The large pendulum represents the nuclear vibrations, while the small pendulum representsthe electronic degrees of freedom. For the electrons, the nuclei seem static. The dynamics of thenuclei is perturbed little, because the forces from the electrons are almost averaged out. Two effectsmust be considered though: Only the center of gravity of electrons and nuclei performs a smoothoscillation, that is unperturbed by the electrons. The nucleus has in addition a small high-frequencyoscillation with small amplitude. The long oscillation period is not given by the mass of the nucleusalone, but by the total mass of electrons and nucleus.

ω+

ω−

x

Y

t

t

Fig. A.1: Schematic demonstration of the classical analog for the Born-Oppenheimer approximation

210 A ADIABATIC DECOUPLING IN THE BORN-OPPENHEIMER APPROXIMATION

Appendix B

Non-adiabatic effects

This section contains some derivations to which we refer in the section on non-adiabatic effects.

B.1 The off-diagonal terms of the first-derivative couplings ~An,m,j

The non-Born-Oppenheimer terms in the Schrödinger equation for the nuclei Eq. 2.20 are related tothe derivative couplings defined in Eq. 2.22 on p. 20

~Am,nEq. 2.22:= 〈ΨBOm |

~

i~∇R|ΨBOn 〉 = 〈ΨBOm |P |ΨBOn 〉 (B.1)

Below, we will relate the first-derivative couplings to the derivatives of the Born-Oppenheimer Hamil-tonian and the Born-Oppenheimer surfaces.

FIRST-DERIVATIVE COUPLINGS

The first-derivative couplings can be expressed as

~Am,n,j(~R)Eq. B.3=

⟨

ΨBOm (~R)∣∣∣

[Pj , H

BO(~R)]

−

∣∣∣ΨBOn (~R)

⟩

EBOn (~R)− EBOm (~R)for En 6= Em (B.2)

where Pj = ~

i~∇Rj is the momentum operator for the j-th nucleus.

Eq. B.2 makes it evident that non-adiabatic effects are dominant, when two Born-Oppenheimersurfaces EBOn (~R) become degenerate. Furthermore, these expressions, Eq. B.2, can be calculatedwithout the need to evaluate the derivative of Born-Oppenheimer wave functions with respect toatomic positions. We only need the expectation values of the derivatives of the Born-OppenheimerHamiltonian, and that of the Born-Oppenheimer surfaces.

Derivation of Eq. B.2 The off-diagonal elements of ~An,m,j(~r) of the first-derivative couplings areobtained as follows: We begin with the Schrödinger equation for the Born-Oppenheimer wave function

(


|ΨBOn (~R)〉Eq. 2.4= 0

Because this equation is valid for all atomic positions, the gradient of the above expression vanishes.

0 = ~∇R(

HBO(~R)− EBOn (~R)) ∣∣∣ΨBOn (~R)

⟩

=

[

~∇R, HBO(~R)]

−

∣∣∣ΨBOn (~R)

⟩

−∣∣∣ΨBOn (~R)

⟩

~∇REBOn (~R) +(


~∇R∣∣∣ΨBOn (~R)

⟩

211

212 B NON-ADIABATIC EFFECTS

Now, we multiply from the left with the bra 〈ΨBOm | and form the scalar products in the electronicHilbert space.1

0 =⟨ΨBOm

∣∣

[

~∇R, HBO(~R)]

−

∣∣ΨBOn

⟩−⟨ΨBOm

∣∣ΨBOn

⟩~∇REBOn (~R) +

⟨ΨBOm

∣∣

(


~∇R∣∣ΨBOn

⟩

=⟨ΨBOm

∣∣

[

~∇R, HBO(~R)]

−

∣∣ΨBOn

⟩− δm,n~∇REBOn (~R) +

(

EBOm (~R)− EBOn (~R))⟨ΨBOm

∣∣ ~∇R

∣∣ΨBOn

⟩

Thus, we obtain

~Am,nEq. B.1=

⟨ΨBOm

∣∣~

i~∇R∣∣ΨBOn

⟩=〈ΨBOm |

[~

i~∇R, HBO(~R)

]

−|ΨBOn 〉

EBOn (~R)− EBOm (~R)

=〈ΨBOm |

[

P , HBO(~R)]

−|ΨBOn 〉

EBOn (~R)− EBOm (~R)for En 6= Em (B.3)

This is the proof for Eq. B.2 given above. Unfortunately, no information can be extracted for thediagonal elements of the first-derivative couplings.

B.2 Diagonal terms of the derivative couplings

As we have seen, the non-adiabatic effects are determined entirely by the derivative couplings~Am,n,j(~R) defined in Eq. 2.22 on p. 20, where m and n refer to the particular sheet of the Born-Oppenheimer surfaces, j refers to a particular atom and ~R is the 3N-dimensional vector describingthe nuclear coordinates.

The most simple approximation including some of the non-adiabatic effects is to include only thediagonal terms of the first derivative couplings. In this approximation, the nuclear motion on thedifferent Born-Oppenheimer surfaces are decoupled as in the Born-Oppenheimer approximation: Thiscan be seen by inspecting Eq. 2.22: The derivative couplings are the only non-diagonal terms.

Within this diagonal approximation, we can do again the classical approximation and see that thederivative couplings act like a magnetic field.

Hn(~P , ~R) =

3M∑

j=1

1

2Mj

(

~Pj + ~An,n,j(~R))2

+ EBOn (~R) (B.4)

For convenience, we use 3M dimensional vectors for the momenta and coordinates of the nuclei.Similarly, we attribute a matrix Mj to each of the 3M nuclear coordinates. The index n refers to aparticular sheet of the Born-Oppenheimer surfaces.

Hamilton’s equations for this system have the form

.~P j = −~∇~Rj

H(~R, ~P ) = −∑

k

(

~∇j ⊗ ~An,n,k

) 1

Mk

(

~Pk − ~An,n,k

)

︸︷︷︸.Rk

−~∇EBOn (B.5)

.~Rj = ~∇~Pj

H(~R, ~P ) =1

Mj

(

~Pj + ~An,n,j

)

(B.6)

With ⊗ I denote the dyadic or outer product defined by (~a⊗~b)i ,j = aibj . Note here that the gradientEq. B.5 does not act beyond the closed parenthesis, that is, it does not act on the vector potentialstanding on the right.

1The Born-Oppenheimer wave functions depend on electronic and nuclear coordinates. The nuclear coordinates,however, play a different role, because they are treated like external parameters.

B NON-ADIABATIC EFFECTS 213

The two Hamilton’s equations Eq. B.5 and Eq. B.6 can be combined into the form of Newton’sequations

Mj

..~Rj

Eq. B.6=

.~P j +

∑

k

( .~Rk ~∇~Rk

)

~An,n,j(~R)

Eq. B.5=

.~P j

︷︸︸︷

−∑

k

(

~∇j ⊗ ~An,n,k

) 1

Mk

(

~Pk + ~An,n,k

)

︸︷︷︸.~Rk

−~∇jEBOn +∑

k

( .~Rk ~∇k

)

~An,n,j(~R)

Eq. B.6=

∑

k

( .~Rk ~∇k

)

~An,n,j(~R)−∑

k

(

~∇j ⊗ ~An,n,k

) .~Rk − ~∇jEBOn

Eq. B.6=

∑

k

.~RkFk,j − ~∇jEBOn

(B.7)

where Fk,j is something like a generalized field-strength tensor, which captures the non-adiabaticeffects

Fk,j =(

~∇k ⊗ ~An,n,j

)

−(

~∇j ⊗ ~An,n,k

)†(B.8)

Like the Hamilton function, also the equation of motion has great similarities with equation ofmotion of a charged particle (with charge q = −1) in an electromagnetic field. There, is howevera major difference, namely the velocity of one particle contributes to the force acting on another.There would be generalized magnetic field for each pair of atoms. The field strength tensor is not athree-by-three matrix, but a 3M × 3M-matrix, where M is the number of nuclei.


Reminder of electrodynamics The equation of motion of a charged particle in a magnetic field is

Mj

..~Rj = q

(

~E(~R) +.~R × ~B(~R)

)

= q(

~E +.~RF)

(B.9)

The magnetic field ~B and the electric field ~E are connected to the vector potential ~A and the electricpotential φ by

~B = ~∇× ~A

~E = −~∇Φ−.~A (B.10)

The electric field-strength tensor F is

F =(

~∇⊗ ~A)

−(

~∇⊗ ~A)†=

0 Bz −ByBy 0 Bx

By −By 0

(B.11)

The components the field-strength tensor are

Fi ,j = ∂iAj − ∂jAi (B.12)

and those of the magnetic field are

Bi =∑

j,k

ǫi ,j,k∂jAk (B.13)

where ǫi ,j,k is the Levi-Civita symbol or fully antisymmetric tensor. The Lorentz force can be writtenin terms of the Field strength tensor using

.~RF =

.~R × ~B (B.14)

We can work out the analogue of the field-strength tensor using the notation with 3M dimensionalvectors that combine cartesian coordinate with the particle index.

∂iAn,n,j − ∂jAn,n,i = ∂Ri 〈ΨBOn |~

i∂Rj |ΨBOn 〉 − ∂Rj 〈ΨBOn |

~

i∂Ri |ΨBOn 〉

=~

i

(

〈∂RiΨBOn |∂RjΨBOn 〉 − 〈∂RjΨBOn |∂RiΨBOn 〉)

= 2~ Im

(

〈∂RiΨBOn |∂RjΨBOn 〉)

Using again the vector-matrix notation for cartesian coordinates and indices for the atoms, we obtain

Fk,j = 2~ Im⟨~∇kΨBOn

∣∣⊗∣∣~∇jΨBOn

⟩(B.15)

This result seems to indicate that the non-adiabatic effects can be made to vanish, if the Born-Oppenheimer wave functions are chosen to be real. This is, however, not so: While real wave functionscan be chosen in the absence of magnetic fields as a consequence of time-inversion symmetry2, thereare cases where such a choice unavoidable produces a step in the wave function. This is related tothe existence of a geometrical phase. Key words related to the geometrical phase is the Jahn-Tellereffect and the Berry phase.

B.3 The diabatic picture

Editor: This section is not yet completed. It may be used alongside Baer’s paper.

2See appendix N

B NON-ADIABATIC EFFECTS 215

When the Born-Oppenheimer surfaces approach each other, the derivative couplings diverge.Therefore not only the Born-Oppenheimer approximation fails in those regions of configuration space,the formulation becomes including the non-adiabatic corrections becomes intractable. In order toavoid this problem, the diabatic representation of the nuclear wave functions has been developed,which avoids these problems. Unfortunately, the construction of the diabatic states is non-trivial andit is a non-local operation. The diabatic representation has been described well in the paper by Baer[73], which we follow here:

We start from the nuclear Hamiltonian in the adiabatic picture, which we have derived earlier

i~∂tφadiam (~R, t)

Eq. 2.20=

∑

n

[ M∑

j=1

1

2Mj

∑

k

(

δm,k~

i~∇Rj + ~Am,k,j(~R)

)(

δk,n~

i~∇Rj + ~Ak,n,j(~R)

)

+δm,nEBOm (~R)

]

φadian (~R, t) (B.16)

Now we introduce a position-dependent, unitary transformation of the nuclear wave function S(~R)that depends on the atomic positions.

φadian (~R, t) =∑

m

φdiam (~R, t)Sm,n(~R)

We want to insert it into Eq. B.16. Therefore, we first investigate the action of the kinetic energyterm on the transformation. We obtain

∑

m

(

δn,m ~Pj + ~An,m,j

)

Sm,o(~R) =~

i~∇jSn,o(~R) +

∑

m

Sn,m(~R)δm,o ~Pj +∑

m

~An,m,jSm,o(~R)

=∑

m

Sn,m(~R)δm,o ~Pj , (B.17)

if we can find a transformation Sn,m so that

~

i~∇jSn,o(~R) +

∑

m

~An,m,jSm,o(~R) = 0 (B.18)

Now we insert this transformation into Eq. B.16

i~∂t∑

n

Sm,nφdian (~R, t) =

∑

n,o

[ M∑

j=1

1

2Mj

∑

k

(

δm,k~

i~∇Rj + ~Am,k,j(~R)

)(

δk,n~

i~∇Rj + ~Ak,n,j(~R)

)

+δm,nEBOm (~R)

]

Sn,o(~R)φdiao (~R, t)

=∑

n,o

Sm,n(~R)

[ M∑

j=1

1

2Mj

∑

k

(

δn,k~

i~∇Rj)(

δk,o~

i~∇Rj)

+∑

k

S†n,k(~R)EBOk (~R)Sk,o(~R)

]

φdiao (~R, t)

i~∂tφdian (~R, t) =

∑

o

[ M∑

j=1

1

2Mjδn,o ~P

2j +

∑

k

S†n,k(~R)EBOk (~R)Sk,o(~R)

]

φdiao (~R, t) (B.19)

That is, in the diabatic picture the nuclear Schrödinger equation Eq. 2.20 has the simple form

i~∂tφdian (~R, t) =

[ M∑

j=1

−~22Mj

~∇2Rj + EBOn (~R)]

φn(~R, t) +∑

m

Wn,m(~R)φm(~R, t) (B.20)

where Wn,m(~R) =∑

k

S†n,k(~R)EBOk (~R)Sk,o(~R)− EBOn (~R)δn,o (B.21)


Instead of a vector potential, the non-adiabatic effects are capture in a shifts of the total energysurfaces and coupling terms between them.

The adiabatic-to-diabatic transformation can be obtained

Sm,n(~R) =∑

o

exp

− i~

M∑

j=1

∫ ~R

~R0

d ~R′j ~Aj(~R)

m,o

So,n(~R0) (B.22)

if the integral is independent of the path taken. The integral is performed along a path connecting~R0 and ~R.

The requirement that the integral in Eq. B.22 is independent of the path is violated at coni-cal intersections. Thus, in the presence of a conical intersection the transformation to a diabaticrepresentation can only be approximated.

From here on, I recommend to follow the paper by Baer [73], and for references to newer devel-opments the review by Worth and Robb[9].

Appendix C

Non-crossing rule

Here we provide the proof of the non-crossing rule discussed in section 2.3.2 on p. 2.3.2.

NON-CROSSING RULE

Consider a Hamiltonian H(Q1, . . . , Qm) which depends on m parameters Q1, . . . , Qm. For eachaccidental crossing of two eigenvalues, there is a three-dimensional subspace of the parameter spacein which the degeneracy is lifted except for a single point.If the hamiltonian is real for all parameters, the subspace where the degeneracy is lifted has twodimensions.

The non-crossing rule has been derived by von Neumann and Wigner[15].

C.1 Proof of the non-crossing rule

Consider a Hamiltonian H(~Q), which depends on a set of coordinates ~Q. Let us explore the changesof the Hamiltonian in the neighborhood of ~Q0.

H(~Q) = H(~Q0) +(

~Q− ~Q0

)

~∇~Q

∣∣~Q0H +O

(

|~Q− ~Q0|2)

(C.1)

1. A gradient of a scalar function of ~Q defines a single direction in the coordinate space. Anydisplacement perpendicular to this direction will leave the scalar function unchanged to firstorder.

2. For a double valued function, the gradients of the first and the second component span atwo-dimensional space.

3. The number of free parameters of an object determines the maximum dimension of the spacein which it can be varied without affecting its defining properties.

This argument is generalized to a Hamiltonian, once with a doubly degenerate state and oncewith the degeneracy split into two non-degenerate states. Below, we will see that the space spannedwithout the degeneracy has a higher dimension than that with the degeneracy. The number ofadditional dimensions is 2 for real hamiltonians and 3 for complex hamiltonians. Thus, there are two,respectively three directions, along which the degeneracy is lifted. Displacements orthogonal to thesetwo or three distortions leave the degeneracy intact.

Thus, one can identify a two-dimensional, respectively three-dimensional space of displacementsthat are relevant for the intersection. Within this subspace, the intersection is zero-dimensional, thatis, a point. This is why one calls this intersection a conical intersection. Orthogonal to this two,respectively three-dimensional subspace, are the “irrelevant” directions, which may affect the positionor value of the conical intersection, but they do not lift the degeneracy.

217

218 C NON-CROSSING RULE

Free parameters of a hamiltonian with a specified multiplet structure: What is left, is todetermine the number of independent parameters that change a hamiltonian with a given multipletstructure without, however, lifting any degeneracies.

Every Hamiltonian can be constructed from its eigenvalues Ek and eigenvectors. The eigenvectorsform the unitary matrix U

Hi ,j =∑

k

U†i ,kEkUk,j (C.2)

Below, we will determine the number κ(n) of free parameters for a unitary matrix of dimension n.Note that κ(n) differs for real and complex unitary matrices.

Let us assume that the eigenvalues form ℓ multiplets with degeneracies gj for j = 1, . . . , ℓ.If we sum the number ℓ of free parameters for the energies Ek and the number κ(n) of parameters

of a unitary transformation U, we overcount the number of free parameters, because a number ofunitary matrices U do not affect the Hamiltonian at all.

The overcounting can be quantified as the number of independent unitary transformations thatleave the Hamiltonian invariant. These are the unitary transformations V with

∑

k

V †m,iHi ,jVj,n = Hi ,j . (C.3)

Thus, thus the desired unitary matrices V commutate with the Hamiltonian, i.e.

[H, V ]− = 0 . (C.4)

This requirement can be transformed into

0 = [H, V ]− =[UHU†,UV U†

]

− =[E, V ′′′

]

− (C.5)

where (E)i ,jdef= Eiδi ,j and V ′′′

def= UV U†.

Any unitary transformation that transforms only states within the same multiplet leaves thematrix E invariant.1 Any unitary transformation between different multiplets mixes energies and thuschanges the matrix E. Hence, the overcounting corresponds to the free parameters of all unitarytransformations that act only within the multiplets. Thus we need to subtract

∑ℓj=1 κ(gj).

Thus, the number of free parameters of a Hamiltonian with ℓ multiplets with degeneracies gj is

D(g1, . . . , gℓ) = κ

ℓ∑

j=1

gj

︸︷︷︸n

+ ℓ−ℓ∑

j=1

κ(gj) (C.6)

Degrees of freedom for a real unitary matrix Let me now work out the number κ(n) of a realunitary matrix with dimension n.

1. a general real matrix of dimension n has n2 matrix elements and thus n2 degrees of freedom.

2. the requirement of orthonormal vectors accounts for n(n−1)2 real-valued conditions∑

k U∗k,iUk,j =

0 with j > k . This corresponds to 12n(n− 1) degrees of freedom. Thus the number of degreesof freedom is reduced to n2 − 12n(n − 1).

3. the normalization of the vectors, i.e.∑

k U∗k,iUk,i = 1 with i = 1, . . . , n, contributes n real-

valued conditions. This reduces the number of degrees of freedom further to n2− 12n(n−1)−n =12n(n − 1).

1Consider one multiplet. The energies for all states in the multiplet are the same, that is E = E111, where E is theenergy of the multiplet. U†EU = EU†U = E111 = E.

C NON-CROSSING RULE 219

Thus, the number of degrees of freedom to fix a general unitary matrix is

κreal(n) =n(n − 1)2

(C.7)

Degrees of freedom for a complex unitary matrix Let me now work out the number κcomplex(n)of a complex unitary matrix with dimension n.

1. a general complex matrix of dimension n has n2 complex matrix elements and thus 2n2 realdegrees of freedom.

2. the requirement of orthonormal vectors accounts for n(n−1)2 complex-valued conditions∑

k U∗k,iUk,j =

0 with j > k . This corresponds to n(n − 1) degrees of freedom. Thus the number of degreesof freedom is reduced to 2n2 − n(n − 1).

3. the normalization of the vectors, i.e.∑

k U∗k,iUk,i = 1 with i = 1, . . . , n, contributes n real-

valued conditions. This reduces the number of degrees of freedom further to 2n2−n(n−1)−n =n2.

Thus, the number of degrees of freedom to fix a general unitary matrix is

κcomplex(n) = n2 (C.8)

Compare parameters with and without degeneracy: Now we need to understand how the numberof free parameters changes when a double degenerate states splits into two nondegenerate levels.In the presence of the double degeneracy, there shall be p multiplets plus one doublet. When thedegeneracy is lifted, the doublet splits into two singlets. The dimension of both hamiltonians isn = 2 +

∑pj=1 gj .

D(g1, . . . , gp, 1, 1)−D(g1, . . . , gp, 2) Eq. C.6=

[

κ(n) + (p + 2)−p∑

j=1

κ(gj)− 2κ(1)]

︸︷︷︸

split

−[

κ(n) + (p + 1)−p∑

j=1

κ(gj)− κ(2)]

︸︷︷︸

degenerate

= 1− 2κ(1) + κ(2)Eqs. C.7, C.8=

2 for a real hamiltonian

3 for a complex hamiltonian(C.9)

220 C NON-CROSSING RULE

Appendix D

Landau-Zener Formula

not finished!!!

The transition between two adiabatic total-energy surfaces has been analyzed by Landau andZener1 [75? , 74]. for a one-dimensional model. Here, we follow the analysis of Wittig[74].

The model of Landau and Zener considers two crossing Born-Oppenheimer surfaces in only onedimension X.

HBO(X) =

(

−F1X 0

0 −F2X

)

The Born-Oppenheimer surfaces are assumed to depend linearly on the atomic coordinate X.The non-adiabatic effects are approximated by non-diagonal elements of the Hamiltonian, that

are considered independent of the atomic position. Thus, the full Hamiltonian has the form

H(X) =

(

−F1X H12

H∗12 −F2X

)

The nuclear dynamics is considered classical, and with a constant velocity ∂tX(t) = V , thatis X(t) = V t. This turns the position-dependent Hamiltonian into an effectively time-dependentHamiltonian.

H(t) =

(

−F1V t H12

H∗12 −F2V t

)

The time dependent electronic Hamiltonian is described by the following Ansatz(

Φ1

Φ2

)

=

(

A(t)e−i~

∫dt EBO1 (X(t))

B(t)e−i~

∫dt EBO2 (X(t))

)

=

(

A(t)e+i~

∫dt F1V t

B(t)e+i~

∫dt F2V t

)

=

(

A(t)e+i~

F1V

2t2

B(t)e+i~

F2V

2t2

)

The lower bound of the time integral has been ignored because they are a multiplicative factor, thatcan be absorbed in the initial conditions.

Insertion into the time dependent Schrödinger equation yield

i~∂t

(

A(t)eiF1V t2/(2~)

B(t)eiF2V t2/(2~)

)

=

(

−F1V t H12

H∗12 −F2V t

)(

A(t)eiF1V t2/(2~)

B(t)eiF2V t2/(2~)

)

(

∂tA(t)

∂tB(t)

)

=

(

− i~H12B(t)e

i(F2−F1)V t2/(2~)

− i~H∗12A(t)e

−i(F2−F1)V t2/(2~)

)

1According to Wittig[74] the result of Landau[? ] has an error of 2π

221

222 D LANDAU-ZENER FORMULA

thus

∂tA(t) = −i

~H12B(t)e

i(F2−F1)V t2/(2~)

∂tB(t) = −i

~H∗12A(t)e

−i(F2−F1)V t2/(2~) ⇒ A(t) = −~i

H12|H12|2︸︷︷︸

1/H∗12

ei(F2−F1)V t2/(2~)∂tB(t)

Insertion of the second equation into the first yields a differential equation for B(t).

∂t

(

−~i

H12|H12|2

ei(F2−F1)V t2/(2~)∂tB(t)

)

︸︷︷︸

A(t)

= − i~H12B(t)e

i(F2−F1)V t2/(2~)

− H12|H12|2

(F2 − F1)V tei(F2−F1)V t2/(2~)∂tB(t) +

(

−~i

H12|H12|2

ei(F2−F1)V t2/(2~)∂2tB(t)

)

= − i~H12B(t)e

i(F2−F1)V2~

t2

− 1

|H12|2(F2 − F1)V t∂tB(t) +

(

−~i

1

|H12|2∂2tB(t)

)

= − i~B(t)

i

~(F2 − F1)V t ∂tB(t) + ∂2tB(t) = −

1

~2|H12|2B(t)

∂2tB(t) +i

~(F2 − F1)V t ∂tB(t) +

1

~2|H12|2B(t) = 0

Thus, we have a differential equation for the probability amplitude on the second sheet. We now lookfor a solution of this differential equation, respectively its value B(t = +∞) with the initial conditionB(−∞) = ∂tB(−∞) = 0.

This is a differential equation of the form..x + iαt

.x + βx = 0

So far, the derivation is standard. Wittig showed a solution to the problem using contour integrals.

1

t

..x

x+ iα

.x

x+β

t= 0

iα

∫ tf

ti

dt ∂t ln[x(t)] = −β∫ tf

ti

dt1

t−∫ tf

ti

dt1

t

..x

x

iα ln[xfxi] = −β

∫ ∞

−∞dt1

t−∫ ∞

−∞dt1

t

..x

x

The integral of the first term on the right hand side yields∫ ∞

−∞dt1

t= ±iπ

depending on whether the integration is performed in the upper or lower half plane.Thus, we obtain

ln[xf ] = ln[xi ]∓ πβ

α+i

α

∫ ∞

−∞dt1

t

..x

x

xf = xie∓πβ/α exp

(i

α

∫ ∞

−∞dt1

t

..x

x

)

If the transition probability is small, that is xf − xi << 1, we can ignore the last term and obtainthe estimate

xf ≈ xie∓πβ/α

D LANDAU-ZENER FORMULA 223

When we insert the parameters α = (F2−F1)V/~ and β = |H12|2/~2 we obtain the Landau-Zenerestimate.

LANDAU ZENER ESTIMATE FOR THE TRANSITION PROBABILITY AT A BAND CROSSING

The transition probability is (Eq. 16 of Wittig)

P = exp

2π

|H12|~

︸︷︷︸

Rabi frequency

|H12|V |F1 − F2|︸︷︷︸

τ

τ is of order of the time it takes to cross the region where the Born-Oppenheimer surfaces are closerthan the avoided crossing surface.

================================The result of their analysis is the Landau-Zener formula, which gives the probability for a

transition for a one-dimensional model

P = e−2πω12τ with ω12 =H12~

and τ =|H12|

v |F1 − F2|(D.1)

where v is the velocity and H12 is the coupling between the two crossing total-energy surfaces. ω12is the Rabi frequency at the crossing point and τ is a measure for the duration of the interactionbetween the surfaces. The model hamiltonian underlying the Landau-Zener formula has the form

224 D LANDAU-ZENER FORMULA

Appendix E

Supplementary material for theJahn-Teller model

This appendix extends the description of the Jahn-Teller model described in section 2.3.4 on p. 25.The stationary wave functions of the Jahn-Teller problem have been given in Eq. 2.62 on p. 34

|Φj,m〉 Eq. 2.62=

∫

dX

∫

dZ1

2

(

|a,X,Z〉[


+ |b,X,Z〉[

−i(eiα − 1)R+,j,m(R) + (eiα + 1)R−,j,m(R)])

eimα

(E.1)

where the radial nuclear wave functions obey the one-dimensional, two-component radial Schrödingerequation Eq. 2.63 for the generalized nuclear coordinates (X,Z) of the Jahn-Teller model,

0Eq. 2.63=

−~2(R2∂2R + R∂R −m2

)

2MR2111

︸︷︷︸

−~2 ~∇2/(2M)

+ gRσz + wR2111

︸︷︷︸

EBO

+~2(m + 12

)

2MR2(111− σx)

︸︷︷︸

nonBO

−Ej,m111(

iR+,j,m(R)R−,j,m(R)

)

(E.2)

E.1 Relation of stationary wave functions for m and −m − 1In Eq. 2.64 and Eq. 2.65 on p. 35 we mentioned that the electron-nuclear wave functionsΦj,m(~x,X,Z)and Φj,−m−1(~x,X,Z) are complex conjugates of each other.1

Φj,−m−1(~x,X,Z) = Φ∗j,m(~x,X,Z) (E.3)

and that the same holds for the radial functions

(

R±,j,−m−1(R))∗= R±,j,m(R) ⇔

R−,j,−m−1(R) = −R−,j,m(R)R+,j,−m−1(R) = +R+,j,m(R)

(E.4)

Here, we provide the detailed derivation.

1More precisely, they can be chosen to be complex conjugates of each other.

225

226 E SUPPLEMENTARY MATERIAL FOR THE JAHN-TELLER MODEL

Complex conjugation of the radial nuclear wave function: We start from the equation Eq. 2.63for R±,j,−m−1

−~2(R2∂2R + R∂R − (−m − 1)2

)

2MR2111

︸︷︷︸

−~2 ~∇2/(2M)

+ gRσz + wR2111

︸︷︷︸

EBO

+~2((−m − 1) + 12

)

2MR2(111− σx)

︸︷︷︸

nonBO

−Ej,m111(

iR+,j,−m−1,(R)R−,j,−m−1(R)

)

Eq. 2.63= 0

⇒−~2

(R2∂2R + R∂R −m2

)+

2A︷︸︸︷

2~2(m +1

2)

2MR2111 + gRσz + wR

2111

+~2(m + 12

)

2MR2(−111︸︷︷︸

−A+ σx︸︷︷︸

B

)− Ej,m111(

iR+,j,−m−1,(R)R−,j,−m−1(R)

)

= 0

(E.5)

The term denoted 2A combines with the term −A. The term B is the only coupling term betweenthe two components of the wave function. A sign change of this term can be absorbed by a signchange of one component relative to the other. Thus we arrive at

−~2(R2∂2R + R∂R −m2

)


2111

+~2(m + 12

)

2MR2(111− σx)− Ej,m111

(−iR+,j,−m−1,(R)R−,j,−m−1(R)

)

= 0

(E.6)

This is the equation for the index m. Thus, if the solution for m is known, we immediately know thesolution for −m − 1 by the transformation Eq. 2.65, which completes the proof of Eq. E.4.

Complex conjugation of the electron-nuclear wave function: Next, we need to show that alsothe electron nuclear wave function Eq. E.1 obey

〈~x,X,Z|Φj,m〉 = 〈~x,X,Z|Φj,−m−1〉∗ (E.7)

We start with Eq. E.1 with m replaced by −m − 1. We use that R+,j,m is purely imaginary andthat R−,j,m is purely real.

〈~x,X,Z|Φj,−m−1〉 Eq. E.1=1

2

(

〈~x |a〉[

(eiα + 1)R+,j,−m−1(R) + i(eiα − 1)R−,j,−m−1(R)]

+ 〈~x |b〉[

−i(eiα − 1)R+,j,−m−1(R)︸︷︷︸

R∗+,j,m(R)

+(eiα + 1)R+,j,−m−1(R)︸︷︷︸

R∗+,j,m(R)

])

ei(−m−1)α

Eq. E.4=

[1

2

(

〈~x |a〉∗[

(e−iα + 1)R+,j,m(R)− i(e−iα − 1)R−,j,m(R)]

+ 〈~x |b〉∗[

i(e−iα − 1)R+,j,−m−1(R) + (e−iα + 1)R−,j,−m−1(R)])

ei(m+1)α]∗

=

[1

2

(

〈~x |a〉∗[


− 〈~x |b〉∗[

+i(eiα − 1)R+,j,m(R) + (eiα + 1)R−,j,m(R)])

eimα]∗

Eq. E.1= 〈~x,X,Z|Φj,m〉∗ (E.8)

E SUPPLEMENTARY MATERIAL FOR THE JAHN-TELLER MODEL 227

which completes the proof that the electron-nuclear wave functions for m and −m − 1 are complexconjugates of each other.

E.2 Behavior of the radial nuclear wave function at the origin

In order to obtain a solution of a radial Schrödinger equation such as Eq. 2.63, it is always helpful toknow the behavior at the origin, which will be shown here.

At the origin the derivative couplings and the kinetic energy dominate the behavior of the wavefunction. Let me analyze this behavior. We start with a Taylor expansion:

R± =∑

n

a±,nRn (E.9)

Insertion of the power-series ansatz into Eq. 2.63 yields

0 =

−~2(n(n − 1) + n −m2

)

2M111 +~2(m + 12

)

2M(111− σx)

︸︷︷︸

nonBO

(

ia+,n

a−,n

)

+gσz

(

ia+,n−3a−,n−3

)

+ w

(

ia+,n−4a−,n−4

)

− E(

ia+,n−2a−,n−2

)

=~2

2M

(−n2 +m2

)111 +

(

m +1

2

)

(111− σx)︸︷︷︸

nonBO

(

ia+,n

a−,n

)

+gσz

(

ia+,n−3a−,n−3

)

+ w

(

ia+,n−4a−,n−4

)

− E(

ia+,n−2a−,n−2

)

(E.10)

This equation can be written as a recursive equation for the coefficients with increasing order(

ia+,n

a−,n

)

=2M

~2

(−n2 +m2

)111 +

(

m +1

2

)

(111− σx)︸︷︷︸

nonBO

−1

×

w

(

ia+,n−4a−,n−4

)

+ gσz

(

ia+,n−3a−,n−3

)

− E(

ia+,n−2a−,n−2

)

(E.11)

We start the expansion with zero coefficients, i.e. a±,n = 0 for all n below a minimum. Theiteration produces non-zero elements only if the inverted matrix is singular. Thus we need to lookfor the indices where an eigenvalue of

(−n2 +m2

)111 +

(m + 12

)(111− σx) vanishes.

Thus the leading order of the nuclear wave function is determined by the zero of the determinantof the above matrix.

det

∣∣∣∣∣

−n2 +m2 +m + 12 −(m + 12

)

−(m + 12

)−n2 +m2 +m + 12

∣∣∣∣∣= 0

⇒(

−n2 +m2 +m + 12

)2

−(

m +1

2

)2

= 0

⇒ n2 = m2 +m +1

2±(

m +1

2

)

⇒ n = ±(m + 1) or n = ±m (E.12)

Thus we obtain four solutions for the leading order n, of which two correspond to regular solutionsof the differential equation and two further ones correspond to the irregular solutions.


m nreg,1 nreg,2 ni r r,1 ni r r,2

m ≥ 0 m m + 1 −m −m + 12 2 3 -2 -31 1 2 -1 -20 0 1 0 -1-1 0 1 0 -1-2 1 2 -1 -2-3 2 3 -2 -3

-|m| |m|-1 |m| -|m|+1 -|m|

What is the meaning of having two orders where the power series may start? They correspondto two independent solutions. Two independent solutions rather than one occur because the nuclearSchrödinger equation of the Jahn-Teller problem is a two-component equation. Giving the leadingorder n0 the next higher power determined by the recursion Eq. E.11 is n0+2. An independent powerseries expansion can be started with n0 + 1.

Next we need to determine the null vectors. We distinguish the cases with m ≥ 0 and with m < 0.(Note that the eigenvectors of the inverse are the eigenvectors of the original matrix.)

• first we consider m ≥ 0:

0 =(−n2 +m2

)111 +

(

m +1

2

)

(111− σx)︸︷︷︸

nonBO

(

ia+,n

a−,n

)

n=m=

(

m +1

2

)(

1 −1−1 1

)(

ia+,m

a−,m

)

⇒(

ia+,m

a−,m

)

=

(

1

1

)

Am (E.13)

where Am is an arbitrary coefficient. Now we consider the second regular solution

0 =(−n2 +m2

)111 +

(

m +1

2

)

(111− σx)︸︷︷︸

nonBO

(

ia+,n

a−,n

)

n=m+1= −

(

m +1

2

)(

1 1

1 1

)(

ia+,m

a−,m

)

⇒(

ia+,m

a−,m

)

=

(

−11

)

Bm · Am(E.14)

where B is another arbitrary coefficient.

Thus the wave function for m ≥ 0 has the form(

iR+(R)R−(R)

)

= ARm

(

1− BR +O(R2)1 + BR +O(R2)

)

(E.15)

• now we consider m < 0:

0 =(−n2 +m2

)111 +

(

m +1

2

)

(111− σx)︸︷︷︸

nonBO

(

ia+,n

a−,n

)

n=−m−1= −

(

m +1

2

)(

1 1

1 1

)(

ia+,m

a−,m

)

⇒(

ia+,m

a−,m

)

=

(

−11

)

Am (E.16)

where Am is an arbitrary coefficient.


0 =(−n2 +m2

)111 +

(

m +1

2

)

(111− σx)︸︷︷︸

nonBO

(

ia+,n

a−,n

)

n=−m=

(

m +1

2

)(

1 −11 −1

)(

ia+,m

a−,m

)

⇒(

ia+,m

a−,m

)

=

(

1

1

)

Bm · Am (E.17)

where Bm is another arbitrary coefficient.

Thus the wave function for m ≥ 0 has the form(

iR+(R)R−(R)

)

= AmRm

(

−1 + BmR +O(R2)1 + BmR +O(R

2)

)

(E.18)

We observe that we can choose the coefficients Am, Bm such that the first terms of the powerseries expansion for m and −m − 1 are complex conjugates of each other. We will see below insection E.1 that this is a general property of the solutions.

Another observation is that the nuclear wave function is not differentiable in all orders at theorigin. Rather the m = solution exhibits a kink at the origin.

E.3 Optimization of the radial wave functions

There are different ways to solve the nuclear radial Schrödinger equation numerically, which arediscussed in the following two sections.

The one described here exploits Ritz’ variational principle. It starts from an expression for thetotal energy as functional of a nuclear wave function. By minimizing this energy we obtain the groundstate. Minimizing the same functional again again, but with the restriction that it is orthogonal tothe ground state, will yield the first excited state. Similarly, we can successively build up all radialwave functions.

We write down the total energy of an ensemble of electron and nuclear states. The states canbe expressed entirely by the nuclear radial functions. The occupations, which are denoted as Pm,j .The are Boltzmann weights.

In order to deal with real functions we introduce(

f+

f−

)

def=

(

iR+R−

)

(E.19)

The total energy functional to be minimized is obtained from the differential equation. Here weinclude all excited states, but give them probabilities Pm,j . The probabilities can be chosen fairlyarbitrarily. While they are inspired by the expressions for a thermal ensemble, they are used here onlyas a numerical tool. Without proof let me state the following: (1) If all probabilities differ from eachother, the resulting wave functions are eigenstates of the stationary Schrödinger equation. (2) Thestates are ordered such that the energies for states with higher probability are lower.

E[~fj,m(R)] =∑

m,j

Pm,j

∫ ∞

0

dR (2πR)︸︷︷︸

ang. integr.

×

~f ∗j,m(R)

[−~2(R2∂2R + R∂R −m2

)


2111 +~2(m + 12)

2MR2(111 + σx)

]

~fj,m(R)

+∑

m

∑

j,j ′

Λm,j ′,j

[∫ ∞

0

dR (2πR)︸︷︷︸

ang. integr.

~f ∗j,m~fj ′,m

]

− δj,j ′

(E.20)


This expression acts as potential energy in a fictitious Lagrangian

L[~fj,m(R),.~f j,m(R)] =

∑

m,j

Pm,j

∫

drmf

.~f j,m(R)

.~f j,m(R)− E[~fjm(R)] (E.21)

and derive an equation of motion for the radial wave function in the spirit of the Car-Parrinellomethod[76]. The time defining the “velocity of ~fm,j ”, is not a real time but simply a coordinatedescribing dynamics of the wave functions during the optimization process.

The first variation of the total energy with respect to the radial functions is

dE =∑

m,j

Pm,j

∫ ∞

0

dR (2πR) d~f ∗j,m~yj,m + c.c. (E.22)

which yields an effective gradient of the functional

~yj,m =

[−~2(R2∂2R + R∂R −m2

)


2111 +~2(m + 12)

2MR2(111 + σx)

]

~fj,m(R)

+∑

j ′

~fj ′,mΛm,j ′,j

(E.23)

The Euler Lagrange equations for the Lagrangian Eq. E.21 produce an equation of motion

mf

..~f m,j(R) = −~yj,m(R) (E.24)

We did not consider here the constraint term, because we will enforce this constraint separately. Thisequation of motion is augmented by a friction term and discretized using the Verlet algorithm[77].

In the Verlet algorithm[77] a differential equation is discretized by replacing the first and secondderivatives by the differential quotients

.x → x(t + ∆)− x(t − ∆)

2∆..x → x(t + ∆)− 2x(t) + x(t − ∆)

∆2(E.25)

This allows to iteratively obtain the solution of a differential equation m..x = F (x)−γ .x for x(j∆).

x(t + ∆) = x(t)2

1 + a− x(t − ∆)1− a

1 + a+ F

∆2

m(1 + a)(E.26)

with a = γ∆2m .

We start from an arbitrary wave function ~fj,m(R) for each m and j . The wave function is firstorthogonalized to all functions with lower j , and then it is normalized. Starting from this initial wavefunction we perform a Car-Parrinello like optimization in the space orthogonal to the lower-lyingstates.

1. calculate the gradient ~yj,m(R)

~yj,m =

[−~2(R2∂2R + R∂R −m2

)


2111 +~2(m + 12)

2MR2(111 + σx)

]

~fj,m(R)

(E.27)

2. orthogonalize ~yj,m(R) to all lower-lying functions

~yj,m ← ~yj,m −j−1∑

j ′=1

~fj ′,m

(∫ ∞

0

dR R ~f ∗j ′,m(R)~yj,m(R) (E.28)


3. propagate with the Verlet algorithm

~f = ~f (t)2

1 + a− ~f (t − ∆)1− a

1 + a− ~y ∆2

2mf (1 + a)(E.29)

4. normalize

~f (t + ∆) = ~f + ~f (t)λ

1!= 2π

∫ ∞

0

dR R ~f ∗(+)~f (+)

=(

2π

∫ ∞

0

dR R ~f ∗ ~f (+))

− 1︸︷︷︸

C

+2λRe(

2π

∫ ∞

0

dR R ~f ∗~f (0))

︸︷︷︸

B

+λ2 2π

∫ ∞

0

dR R ~f ∗(0)~f (0)︸︷︷︸

A

λ = −BA

[

1−√

1− A2C

B2

]

(E.30)

Since the function is known for the next time step, we can begin again with the first point of theiteration sequence.

E.4 Integration of the Schrödinger equation for the radial nu-

clear wave functions

Editor: This is not finished!

0 =

− ~2

2M∂2R −

~2

2MR∂R +

~2[m(m + 1) + 12

]

2MR2± g|~R|+ w ~R2 − Em,j

R±,m,j ±i~2(m + 12)

2MR2R∓,m,j

0 =

−R2∂2R − R∂R +[

m(m + 1) +1

2

]

+2M

~2

[

−Em,jR2 ± gR3 + wR4]

R±,m,j ± i(

m +1

2

)

R∓,m,j

Now we introduce an exponential Ansatz R(x) = ex

R±,m,j(ex) = f±,m,j(x) (E.31)

R(x) = ex ⇒ dR

dx= ex = R ⇒ dx

dR= 1/R

R∂R = ∂x

R2∂2R = R2∂R1

RR∂R = R

2∂R1

R∂x = R

2(

− 1R2+1

R2R∂R

)

∂x = −∂x + ∂2xR2∂2R + R∂R = ∂

2x (E.32)

This yields

0 =

−∂2x +[

m(m − 1) + 12

]

+2M

~2

[

−Em,jR2 ± gR3 + wR4]

f±,m,j(x)∓ i(

m +1

2

)

f∓,m,j(x)

We solve this equation numerically on a grid using the Verlet algorithm[77], which replaces thesecond derivative with the differential quotient

∂2x f =f (+)− 2f (0) + f (−)

α2(E.33)


where α is the grid spacing.An important aspect is that one needs to integrate from the outside inward. The reason is that in

the outer region, there is one solution that falls of towards the outside and one solution that increasesexponentially. We are interested in the solution falling off.

When we integrate outward, any little numerical error will admix a fraction of the exponentiallyincreasing solution. This small fraction will now be exponentially amplified, leading to a poor solution.On the other hand, when we integrate inward, every error will admix a partial solution that falls offin the inward direction, so that any error heals immediately.

Therefore we choose the strategy to integrate from the outside inward and match the solutionsin the center.

For a given energy, we calculate the Schrödinger equation outward from the inside with two

distinct boundary conditions and obtain(

f(i)1,1(x), f

(i)2,1(x)

)

and(

f(i)1,2(x), f

(i)2,2(x)

)

. At the same time

I evaluate the energy derivatives yielding(.f(i)

1,1(x),.f(i)

2,1(x))

and(.f(i)

1,2(x),.f(i)

2,2(x))

. The boundary

conditions are(

f(i)1,1(x), f

(i)2,1(x)

)

= (1, 0) and(

f(i)1,2(x), f

(i)2,2(x)

)

= (0, 1) at the first grid point. The

derivatives are set to zero. For the energy derivative functions value and derivative at the first gridpoints are set to zero.

The condition that the two functions match with value and derivative can be approximated bythe condition that the two functions match on two subsequent points xk on the grid.

2∑

α=1

(

f(i)i ,α (xk) + (ǫ− ǫ0)

.f(i)

i ,α(xk)

)

cα =

2∑

α=1

(

f(o)i ,α (xk) + (ǫ− ǫ0)

.f(o)

i ,α (xk)

)

cα+2 (E.34)

This value must be fulfilled for k = 1, 2 and i = 1, 2. This is an eigenvalue equation

f(i)1,1(xa) f

(i)1,2(xa) −f (o)1,1 (xa) −f (o)1,2 (xa)

f(i)2,1(xa) f

(i)2,2(xa) −f (o)2,1 (xa) −f (o)2,2 (xa)

f(i)1,1(xb) f

(i)1,2(xb) −f (o)1,1 (xb) −f (o)1,2 (xb)

f(i)2,1(xb) f

(i)2,2(xb) −f

(o)2,1 (xb) −f

(o)2,2 (xb)

+ (ǫ− ǫ0)

.f(i)

1,1(xa).f(i)

1,2(xa) −.f(o)

1,1(xa) −.f(o)

1,2(xa).f(i)

2,1(xa).f(i)

2,2(xa) −.f(o)

2,1(xa) −.f(o)

2,2(xa).f(i)

1,1(xb).f(i)

1,2(xb) −.f(o)

1,1(xb) −.f(o)

1,2(xb).f(i)

2,1(xb).f(i)

2,2(xb) −.f(o)

2,1(xb) −.f(o)

2,2(xb)

= 0

(E.35)

We define the matrix

Adef=

.f(i)

1,1(xa).f(i)

1,2(xa) −.f(o)

1,1(xa) −.f(o)

1,2(xa).f(i)

2,1(xa).f(i)

2,2(xa) −.f(o)

2,1(xa) −.f(o)

2,2(xa).f(i)

1,1(xb).f(i)

1,2(xb) −.f(o)

1,1(xb) −.f(o)

1,2(xb).f(i)

2,1(xb).f(i)

2,2(xb) −.f(o)

2,1(xb) −.f(o)

2,2(xb)

−1

f(i)1,1(xa) f

(i)1,2(xa) −f

(o)1,1 (xa) −f

(o)1,2 (xa)

f(i)2,1(xa) f

(i)2,2(xa) −f

(o)2,1 (xa) −f

(o)2,2 (xa)

f(i)1,1(xb) f

(i)1,2(xb) −f

(o)1,1 (xb) −f

(o)1,2 (xb)

f(i)2,1(xb) f

(i)2,2(xb) −f

(o)2,1 (xb) −f

(o)2,2 (xb)

(E.36)

and diagonalize it using the LAPACK routine ZGEEV. The eigenvalues are an estimate for the nextenergy eigenvalue. The eigenvalues are ǫ − ǫ0. Using the eigenvalues only one can zoom into thecorrect energy. That is, ǫ0 is adjusted to the new estimate for the energy eigenvalue until ǫ − ǫ0nearly vanishes for the chosen eigenvector.

Finally, one can evaluate the right-hand eigenvector, which provides the coefficients of the innerand outer partial solutions.

The final solution is then constructed

~f (x) =(

~f(i)1 + (ǫ− ǫ0)

.~f(i)

1

)

c1 +(

~f(i)2 + (ǫ− ǫ0)

.~f(i)

2

)

c2

~f (x) =(

~f(o)1 + (ǫ− ǫ0)

.~f(o)

1

)

c3 +(

~f(o)2 + (ǫ− ǫ0)

.~f(o)

2

)

c4 (E.37)


Depending on the value of x , either the equation with the inner or the outer partial solutions is used.The solution is then normalized so that

2π

∫

dR R · R2m(

f ∗+,m,j(x(R))f+,m,j(x(R)) + f∗−,m,j(x(R))f−,m,j(x(R))

)

= 1

2π

∫ ∞

−∞dx

dR

dxR · R2m

(

f ∗+,m,j(x(R))f+,m,j(x(R)) + f∗−,m,j(x(R))f−,m,j(x(R))

)

= 1

2π

∫ ∞

−∞dx e(2m+2)x

(

f ∗+,m,j(x)f+,m,j(x) + f∗−,m,j(x)f−,m,j(x)

)

= 1

2πα∑

xi

e(2m+2)xi(

f ∗+,m,j(xi)f+,m,j(xi) + f∗−,m,j(xi)f−,m,j(xi)

)

= 1 (E.38)

The factor 2πr results from the angular integration. Please check!

Appendix F

Notation for spin indices

We use a notation that combines the continuous spatial indices ~r with the discrete spin indices into afour dimensional vector ~x . This shortcut notation has a more rigorous basis[78], which we will showhere.

We introduce an artificial spin coordinate q. This is not a spatial coordinate, but it is a coordinatein some other abstract one-dimensional space. On this space, we assume that there is a complete1

and orthonormal basis with only two functions, namely α(q) and β(q). They obey∫

dq α∗(q)α(q) = 1∫

dq β∗(q)β(q) = 1∫

dq α∗(q)β(q) = 0

A wave function may now depend on a spatial coordinate and this fictitious spin variable q. Theyform together a four dimensional vector

~xdef=(~r , q)

An electronic wave function depends on all these variables:

〈~x |ψ〉 = 〈~r , q|ψ〉 = ψ(~r , q)

We can now decompose this vector into its components

ψ(~r , ↑) def=

∫

dq α∗(q)ψ(~r , q)

ψ(~r , ↓) def=

∫

dq β∗(q)ψ(~r , q)

1This is where the argument is weak: a complete set of functions on a finite interval is always infinite. Probablyone needs Grassman variables

235

236 F NOTATION FOR SPIN INDICES

Appendix G

Some group theory for symmetries

Symmetry operators are important to break down a Hamiltonian into a block diagonal matrix, whichgreatly simplifies the construction of the Hamilton eigenvalues. For this purpose we need to deter-mine distinct sets of states which are eigenstates with respect to the symmetry operators of theHamiltonian.

The following follows closely the Chemwikihttp://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Group_Theory%3A_Theory

G.0.1 Definition of a group

The symmetry operators form a group.A group consists of a set of elements and an operation. In our case the elements are the symmetry

operations and the operations is the operator multiplication. To be a group the following propertiesneed to be fulfilled.

• Closure: For any two members A and B of the group, their product C = AB is a member ofthe group.

• Associativity: The order in which the multiplications are performed does not affect the result.

∀A,B,C∈G (AB)C = A(BC) (G.1)

• Identity: There is one element 1 in the group, called identity, which leaves all elements in thegroup unchanged when operated on it from the left or from the right.

∃1∈G∀A∈G 1A = A1 (G.2)

• Inverse: For each element A in the group, there is one element in the group, called the inverseA−1 of A, which produces the identity when operated on A.

∀A∈G∃A−1∈GA−1A = AA−1 = 1 (G.3)

G.0.2 Symmetry class

A similarity transform1 with an invertible operator X transforms an operator A into an operator Bby

X−1AX = B (G.4)

1Source: https://en.wikipedia.org/wiki/Matrix_similarity,http://mathworld.wolfram.com/SimilarityTransformation.html.

237

http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Group_Theory%3A_Theory

https://en.wikipedia.org/wiki/Matrix_similarity

http://mathworld.wolfram.com/SimilarityTransformation.html

238 G SOME GROUP THEORY FOR SYMMETRIES

To construct the symmetry class CA containing operator A, calculate all operators obtained by asimilarity transform with any of the operators in the group.

CA = X−1AX∀X∈G (G.5)

If B is in the symmetry class CA containing A, then A is member of the symmetry class CB.Proof: If B is member of CA, then there is one group operation X so that B = X−1AX. Thus

also A = XBX−1. Because also the inverse X−1 is member of the group, A is member of the classCB containing B.

The classes produced by two operators in the same class are identical.

B ∈ CA ⇒ CB = CA (G.6)

Appendix H

Sommerfeld expansion

Here we sketch the derivation for the Sommerfeld expansion we refer to in section 4.5.5 on p. 72.Many integrations involving the Fermi functions have the form

I =

∫ ∞

−∞dǫ g(ǫ)f (ǫ) (H.1)

where f (ǫ) = (1 + eβ(ǫ−µ))−1 is the Fermi function and g(ǫ) is some other function which is differ-entiable to infinite order.1

The Sommerfeld expansion is a method to evaluate these integrals given that integral and thederivatives of g(ǫ) are known at the Fermi level. The Sommerfeld expansion is a low-temperatureexpansion.

Let us define the integrand of g(ǫ)

G(ǫ)def=

∫ ǫ

−∞dǫ g(ǫ) (H.2)

Then

I =

∫ ∞

−∞dǫ g(ǫ)f (ǫ)

=

∫ ∞

−∞dǫ[

∂ǫ

(

G(ǫ)f (ǫ))

− G(ǫ)∂ǫf (ǫ)]

= G(∞) f (∞)︸︷︷︸

=0

−G(−∞) f (−∞)︸︷︷︸

=1

−∫ ∞

−∞dǫ G(ǫ)∂ǫf (ǫ)

= −G(−∞)−∞∑

n=0

1

n!

(

∂nǫ |µ G)∫ ∞

−∞dǫ (ǫ− µ)n∂ǫ

1

1 + eβ(ǫ−µ)︸︷︷︸

f (ǫ)

= −G(−∞)−∞∑

n=0

1

n!

(

∂nǫ |µ G)

(kBT )n

∫ ∞

−∞dx xn∂x

1

1 + ex

= G(µ)− G(−∞)−∞∑

n=1

(kBT )2n

(2n)!

(

∂2nǫ∣∣µG)∫ ∞

−∞dx x2n∂x

1

1 + ex

=

(∫ µ

−∞dǫ g(ǫ)

)

+

∞∑

n=1

(kBT )2n

(

− 1

(2n)!

∫ ∞

−∞dx x2n∂x

1

1 + ex

)

︸︷︷︸an

(

∂2n−1ǫ

∣∣µg)

(H.3)

1This may apply to some models. For real systems this assumption is not obeyed.

239

240 H SOMMERFELD EXPANSION

Following Ashcroft and Mermin[79] (Ashcroft-Eq. 2.70 and Appendix C)

an = −1

(2n)!

∫ ∞

−∞dx x2n∂x

1

1 + ex=(

2− 1

22n−2

)

22n−1π2n

(2n)!Bn

(H.4)

with the Bernoulli numbers B1 = 16 and B2 = 1

30 .

a1 =π2

6

a2 =7π4

360(H.5)

Thus we obtain

SOMMERFELD EXPANSION

∫ ∞

−∞dǫ g(ǫ)f (ǫ) =

(∫ µ

−∞dǫ g(ǫ)

)

+(πkBT )

2

6

(

∂ǫ|µ g)

+7(πkBT )

4

360

(

∂3ǫ∣∣µg)

+O(

(kBT )6)

(H.6)

which is equivalent to Eq. 4.69 on p. 73.

Appendix I

Why is the effective mass ofsemiconductors much smaller thanthat of metals?

Let me respond to the question why the effective masses of metals are comparable to the free-electronmass, while the effective masses of the semiconductors are usually much larger.

The first part of the answer is that the two masses are different concepts.

• The effective mass of a semiconductor is the related to the curvature at the band edges andtherefore describes the transport properties of charge carriers in the valence and conductionband.

• the effective mass of metals is related to the heat capacity, which in turn is a measure of thedensity of states at the Fermi level.

For the elemental metals such as Na(1e), K(1e), Al(3e), Cu(1e), Ag(1e), Au(1e) etc. theelectronic structure is a variation of the free electron band structure. Thus one expects their effectivemasses to be similar (order of magnitude...) to that of a free electron.

The following discussion shall rationalize the underlying process leading from free electrons tothe band gap of semiconductors. While I believe that the underlying concepts are more or less valid,please take the discussion with a grain of salt. I have not spent too much time on it.

The concepts can be followed from the band structures of the free electron metal Aluminiumand the semiconductor silicon shown in Fig. I.1. If one follows the lowest bands from Γ to X andback, one will see some similarities between the two band structures. The uppermost green band ofaluminium at the Γ point (at about +12 eV) translates into the valence band top of silicon (at 0 eV).1 The band gap of silicon can loosely be attributed to an avoided crossing of the free-electron bands.

Let us try to model this: For the semiconductors the band gap develops out of an avoided crossingof “free-electron-like” bands, and this determines the effective mass. A true free electron would havea true band crossing. The resulting curvature of the bands is infinite. Hence the effective mass wouldvanish completely.

Let us make a crude model to elucidate the principles. Consider a free electron band

E(k) =~2k2

2me(I.1)

The Fermi level shall be at the Fermi momentum pF . The Fermi velocity is thus vF = ∂E∂p =

pFme

.

1The Fermi level in silicon is higher up because it has 8 electrons in the unit cell instead of the three electrons ofaluminium.

241

242 I WHY IS THE EFFECTIVE MASS OF SEMICONDUCTORS MUCH SMALLER THANTHAT OF METALS?

Γ X W L Γ K X

-10

-5

0

5

10

15

E[e

V]

Γ X W L Γ X

-10

-5

0

5

10

Fig. I.1: Comparison of the band structure of aluminium (left) and that of silicon (right). The Fermilevel is at energy zero.

Now, we introduce a lattice so that pF is equal to the reciprocal lattice vector. Because theband structure is periodically folded in, the Fermi momentum will be folded onto the Gamma point(~G = 0). Furthermore there will be several bands crossing at Γ. If the lattice potential lifts thedegeneracy, the band crossing will turn into an avoided crossing with some band gap Eg.

Fig. I.2: Intersection of two free-electron dispersions ǫ(~k − ~G) for two different reciprocal latticevectors ~G. At their intersection the upper and lower bands are formed. The sharp edge of the upperband will turn into an avoided crossing with a large curvature, i.e. light effective mass.

Imagine an avoided crossing of two bands with the slope given by some Fermi velocity vF and aband gap Eg.

E±(~k) = ±√(

~vF )2k2 +(1

2Eg

)2

(I.2)

The curvature at the band edges is ∂2E∂k2

∣∣∣k=0= 2(~vF )

2/Eg =~2

m∗ , so that the effective mass is

m∗ =Eg

2v2F=Egm

2e

2p2F(I.3)

In this model, the effective mass is about proportional to the band gap, even though the “free-electronband” has a curvature determined by the true free electron mass. It is also clear that the effectivemass is much larger than the free-electron mass.

For similar semiconductors with a direct band gap at Gamma, I expect that the effective masswill be approximately proportional to the band gap.

I WHY IS THE EFFECTIVE MASS OF SEMICONDUCTORS MUCH SMALLER THAN THATOF METALS? 243

For band gaps such as silicon which has the band minimum near the X-point, an analogousconstruction tells that the longitudinal effective electron mass should be similar to the free electronmass, while the transversal mass should again be very small.

244 I WHY IS THE EFFECTIVE MASS OF SEMICONDUCTORS MUCH SMALLER THANTHAT OF METALS?

Appendix J

Anharmonic oscillator

L(x, v) = 12m

.x2 − 12mω20x

2 − 13!gx3 (J.1)

The Euler Lagrange equations are

m..x = −mω20x −

g

2x2 ⇒ ..

x + ω20x = −g

2mx2 (J.2)

We introduce the amplitudes

bσ(t) = e−iσω0t 1√

2m~ω0

(

ω0 − iσ∂t)

x(t) (J.3)

for which

x(t) =

√2m~ω02ω0

∑

σ

bσ(t)eiσω0t (J.4)

i~∂tbσ(t) = e−iσω0t1√2m~ω0

[

−iσω0(

ω0x(t)− iσ∂tx(t))

+(

ω0∂tx(t)− iσ∂2t x(t))]

= e−iσω0t1√2m~ω0

[

−iσω20x(t)−ω0.x(t) + ω0

.x(t)

︸︷︷︸

=0

−iσ∂2t x(t)]

Eq. J.2= −iσe−iσω0t 1√

2m~ω0

[

− g

2mx2(t)

]

Eq. J.4= iσe−iσω0t

1√2m~ω0

g

2m

[(√2m~ω02ω0

)2∑

σ,σ′

bσb′σei [σω0+σ′ω0]t

]

= iσg√2m~ω0

8mω20

∑

σ′,σ′′

bσ′bσ′′ei [σ′ω0+σ′′ω0−σω0]t (J.5)

245

246 J ANHARMONIC OSCILLATOR

Appendix K

Time integrals for thephonon-phonon scattering elements

K.1 Expression 1

The first time integral has been defined in Eq. 7.17 via

F (1)(ωt)def=1

t

∫ t

0

dt ′ eiωt′

(K.1)

It has the form

F (1)(x) =1

x

∫ x

0

dx ′ eix =eix − 1ix

=sin(x)

x− i cos(x)− 1

x(K.2)

For large arguments, F (1) turns into a δ-function. This will lead us to the concept of energyconservation.

If the sums over states are transformed into integrations over frequencies, we obtain integrals ofthe type

∫

dω g(ω)tF (1)(ωt) =

∫

d(ωt) g

(ωt

t

)

F (1)(ωt) =

∫

d(x) g(x

t

)

F (1)(x)

limt→∞

∫

dω g(ω)tF (1)(ωt) = g(0)

∫

dx F (1)(x)

=(∫

dx F (1)(x))

︸︷︷︸π

∫

dω g(ω)δ(ω) (K.3)

We used the identity∫∞−∞ dx

sin(x)x = π from [34]. The imaginary part vanishes because the integrand

is odd under inversion.Thus we can write

limt→∞

tF (1)(ωt) = πδ(ω) (K.4)

K.1.1 Product of two F (1)

Furthermore we will need

F (1)(x)F (1)(−x) = eix − 1x

e−ix − 1−x = −21− cos(x)

x2= −22 sin

2(x/2)

x2= −sin

2(x/2)

(x/2)2(K.5)

247

248 K TIME INTEGRALS FOR THE PHONON-PHONON SCATTERING ELEMENTS

Fig. K.1: The function Re[tF (1)(ωt)] as function of ω for two values of t. The time for the bluecurve is twice that of the red one.

This function is narrowly peaked at the origin. Its integral is

∫

dx F (1)(x)F (1)(−x) = −∫

dxsin2(x/2)

(x/2)2= −2

∫

dxsin2(x)

x2= −2π (K.6)

Thus we obtain

limt→∞

∫

dω g(ω)tF (1)(ωt)F (1)(−ωt) = limt→∞

∫

dx g(x

t

)

F (1)(x)F (1)(−x)

= g(0)

∫

dx F (1)(x)F (1)(−x)

= 2πg(0) (K.7)

so that

limt→∞

t2F (1)((ωt)F (1)(−ωt) = 2πtδ(ω) (K.8)

K.2 Expression 2

as defind in Eq. 7.21

F (2)(ωt, ω′t)def=1

t2

∫ t

0


0

dt ′′ eiω′t ′′ (K.9)

F (2)(ωt, ω′t) =1

t2

∫ t

0


0

dt ′′ eiω′t ′′

=1

t2

∫ t

0

dt ′ eiωt′ 1

iω′

(

eiω′t ′ − 1

)

=1

t

1

iω′t

∫ t

0

dt ′(

ei(ω+ω′)t ′ − eiωt ′

)

=−1

(ω′t)(ωt + ω′t)

(

ei(ωt+ω′t) − 1

)

− −1(ω′t)(ωt)

(

eiωt − 1)

(K.10)

K TIME INTEGRALS FOR THE PHONON-PHONON SCATTERING ELEMENTS 249

F (2)(x, x ′) = −ei(x+x ′) − 1x ′(x + x ′)

+eix − 1xx ′

=[

−cos(x + x′)− 1

x ′(x + x ′)+cos(x)− 1

xx ′

]

+ i[

−sin(x + x′)

x ′(x + x ′)+sin(x)

xx ′

]

(K.11)

Important will be the function for opposite arguments

F (2)(x,−x) = limx ′→−x

−ei(x+x ′) − 1x ′(x + x ′)

+eix − 1xx ′

= limx ′→−x

1

x ′

[

−ei(x+x ′) − 1(x + x ′)

+eix − 1x

=1

−x

[

−i + cos(x) + i sin(x)− 1x

]

=1− cos(x)

x2+ i

x − sin(x)x2

(K.12)

∫

dx F (2)(x,−x) =∫

dx1− cos(x)

x2=

∫

dx2 sin2( 12x)

x2=

∫

dxsin2(x)

x2= π (K.13)

Thus we obtain

limt→∞

∫

dω g(ω)tF (2)(ωt,−ωt) = limt→∞

∫

dx g(x

t

)

F (2)(x,−x)

= g(0)

∫

dx F (2)(x,−x)

= πg(0) (K.14)

so that

limt→∞

t2F (2)((ωt,−ωt) = πtδ(ω) (K.15)

-20 -15 -10 -5 0 5 10 15 20

Fig. K.2: F (2)(x,−x) as function of x . blue: real part; red: imaginary part.

250 K TIME INTEGRALS FOR THE PHONON-PHONON SCATTERING ELEMENTS

Appendix L

Additional material for the PhononBoltzmann equation

251

252 L ADDITIONAL MATERIAL FOR THE PHONON BOLTZMANN EQUATION

L.1 Sketch of the derivation of the Phonon Boltzmann equation

1. Set up Newton’s equations of motion for the distortions Qλ(t).

2. Introduce phonon amplitudes b±λ (t)

Qλ(t) = b+λ (t)e

iωλt + b−λ (t)e−iωλt

The state can be described alternatively by the phonon amplitudes, or by the density matrix

ρλ,λ′(t) = b+λ (t)b

−λ′(t)

Both contain the full information of trajectories The off-diagonal elements of the density matrixdescribe the entanglement of the phonons.

3. Perturbation expansion in the anharmonic terms (We go to second order, which is the minimumto obtain forward and back-reaction.) This is not required for a derivation of the Boltzmannequation, but it ensures a simple scattering operator.

4. Initial random-phase approximation: The initial random phase apprixmation removes theoff-diagonal elements from the initial density matrix.

The state is now an enseble described by a density matrix average⟨

ρλ,λ′(t)

⟩

=

⟨

b+λ (t)b−λ′(t)

⟩

initial phases

5. Long-time limit, which turns the oscillatory functions into delta functions of the frequencydifference. The result is energy conservation. The initial random-phase approximation isrequired to obtain the desired form of the long-time limit.

6. Final random-phase approximation: The final randomization of the phases deletes the off-diagonal elements of the density matrix. This is the point where irreversibility is introduced.

A thorough deviation should analyze ρλ,λ′(t).

• After a decoherence time the off-diagonal elements should vanish

• the diagonal elements obey the mapping

ρλ,λ(t) =∑

λ′

Mλ,λ′(t)ρλ′,λ′(0)

The mapping shall be of exponential form M(t) = e−At , that is

A(t) = −1tln[M(t)] for t > Tdecoh

It is possible that a unitary transformation of the phonons fulfills the requirements, if this isnot the case in the original basis.

7. Formulation of a rate equation: Let us combine the occupations nλ(t) = ρλ,λ(t) into a vector~n(t). A rate equation gives

∂t~n(t) = −A~n(t) ⇔ ~n(t) = e−At~n(0)

Appendix M

Downfolding

Often, a Schrödinger equation can be divided into an important and an unimportant portion of Hilbertspace. This could be a division into an impurity, which is important, and which is embedded into an“unimportant” bath, which is spatially separated.

It could also be that we need to focus onto certain bands in a band structure of non-interactingelectrons in a periodic potential, which the “unimportant” states shall not be completely ignored.

Another application is that of a many-particle problem, where few Slater-determinants are impor-tant. They constitute what chemists call “static correlation”, while other Slater determinants are ofsecondary importance. Their contribution is termed “dynamic correlation” in the context of quantumchemistry.1

Let us consider a Hilbert (or Fock-) space, which is divided into an important subspace A =|χAj 〉

with states |χAj 〉 and an unimportant subspace B =|χBj 〉

with states |χAj 〉. The matrix elements

of the hamiltonian H are

HAA,i,j = 〈χAi |H|〉χAj 〉HAB,i,j = 〈χAi |H|〉χBj 〉HBA,i,j = 〈χBi |H|〉χAj 〉HBB,i,j = 〈χBi |H|〉χBj 〉 (M.1)

The stationary Schrödinger equation has the form

(

HAA − ǫ111AA HAB

HBA HBB − ǫ111BB

)(

~cA

~cB

)

= 0 (M.2)

with the norm

~c2A + ~c2B = 1 (M.3)

We use the symbols 111AA and 111BB for the unit matrices in the subspaces A and B, respectively.From the second line of the Schrödinger equation Eq. M.2, we obtain

HBA~cA + (HBB − ǫ111BB)~cB = 0⇒ ~cB = −(HBB − ǫ111BB)−1HBA~cA (M.4)

1In the context of solid state physics and Green’s functions the notion of static and dynamic correlation is entirelydifferent: Here a static correlation is attributed to a self energy that is frequency independent, while dynamic correlationsare related with a frequency dependency of the self energy.

253

254 M DOWNFOLDING

The result Eq. M.4 is inserted into the first line of the Schrödinger equation Eq. M.2

(HAA − ǫ111AA)~cA +HAB~cB = 0Eq. M.4⇒

[

HAA −HAB(HBB − ǫ111BB)−1HBA︸︷︷︸

M(ǫ)

−ǫ111AA]

~cA = 0 (M.5)

Let us introduce a new symbol, namely

M(ǫ)def= HAA −HAB(HBB − ǫ111BB)−1HBA (M.6)

Eq. M.5 is an equation that acts only in A, the subspace of the important degrees of freedom. Thisis good. The price to pay is that the equation is now non-linear in the energy. We will come to thatlater.

We still need to translate the norm condition:

~c2A + ~c2B = 1

Eq. M.4⇒ ~cA

[

111AA +HAB︸︷︷︸

H†BA

(HBB − ǫ111BB)−2HBA]

~cA = 1 (M.7)

Let us introduce another new symbol, namely

O(ǫ)def= 111AA +HAB(HBB − ǫ111BB)−2HBA (M.8)

The matrix O(ǫ) is a matrix in the subspace A can be considered an “overlap operator”. It accountsfor the fact that the complete wave function has also a contribution in subspace B, and thereforethe probability weight inside A is less than one.

It turns out that the overlap operator O can be expressed as a derivative of the matrix M.

O(ǫ) = 111AA − ∂ǫM(ǫ) (M.9)

An eigenstate |ψ〉 of the full Hamiltonian can be expressed now in the form

|ψ〉 =∑

j∈A|χA,j 〉cA,j +

∑

k∈B|χB,k〉cB,k

Eq. M.4=

∑

j∈A|χA,j 〉cA,j −

∑

k,l ,m∈B

∑

j∈A|χB,k〉(HBB − ǫ111BB)−1l ,mHBA,m,jcA,j

=∑

j∈A

(

|χA,j 〉 −∑

k,l ,m∈B|χB,k〉(HBB − ǫ111BB)−1l ,mHBA,m,j

)

cA,j (M.10)

Thus, we arrived at a definition of energy-dependent basis states

|χA,j(ǫ)〉 = |χA,j 〉 −∑

k,l ,m∈B|χB,k〉(HBB − ǫ111BB)−1l ,mHBA,m,j (M.11)

The “unimportant” contribution from subspace B have been downfolded into the states connectedto subsystem A.

The nonlinear matrix equation is recovered as

M(ǫ)− 111 = 〈χA,j |H − ǫ111|χA,k(ǫ)〉 = 〈χA,j(ǫ)|H − ǫ111|χA,k(ǫ)〉 = 0 (M.12)

M DOWNFOLDING 255

M.1 Downfolding with Green’s functions

The Green’s function is defined as(

ǫ111−H)

G(ǫ) = 111 (M.13)

Note, that there are different conventions for the Green’s function, that differ in the sign.

(

ǫ111AA −HAA −HAB−HBA ǫ111BB −HBB

)(

GAA(ǫ) GAB(ǫ)

GBA(ǫ) GBB(ǫ)

)

=

(

111AA 000AB

000BA 111BB

)

(M.14)

From the lower left submatrix of the above equation we obtain

−HBAGAA(ǫ) + (ǫ111BB −HBB)GBA(ǫ) = 000⇒ GBA(ǫ) = (ǫ111BB −HBB)−1HBAGAA(ǫ) (M.15)

This result we insert into the upper left submatrix(

ǫ111AA −HAA)

GAA(ǫ)−HABGBA(ǫ) = 111AA

⇒[

ǫ111AA −HAA −HAB(

ǫ111BB −HBB)−1HBA

]

GAA(ǫ) = 111AA (M.16)

This allows one to calculate the sub-block for the Green’s function for subsystem A directly

⇒ GAA(ǫ) =

[

ǫ111AA −HAA −HAB(

ǫ111BB −HBB)−1HBA

]−1(M.17)

This method is uses for example to describe non-interacting electrons near interfaces: It allowsto remove part of the system and replace it by another one.

M.2 Taylor expansion in the energy

In order to solve the equation

[M(ǫ)− 111ǫ] ~c = 0 (M.18)

we need to follow the eigenvalues of M(ǫ) − ǫ111 and determine those energies where one or moreeigenvalues become zero. This can be done by computing the determinant of M(ǫ)− 111ǫ as functionof energy and determine the zeros. The corresponding eigenvectors with zero eigenvalue are thesolutions that occur at the specific energies.

This process is much more complicated that the diagonalization of an energy-independent Hamil-tonian. Therefore one usually searches for approximate solutions.

The most simple approach is the linearization about some energy ǫν[M(ǫν) + (ǫ− ǫν) ∂ǫ|ǫν M − 111ǫ

]~c = 0

[(

M(ǫν)− ǫν ∂ǫ|ǫν M)

−(

111− ∂ǫ|ǫν M)

ǫ]

~c = 0[(

M(ǫν) + ǫν [O(ǫν)− 111])

−O(ǫν)ǫ]

~c = 0 (M.19)

Thus, for each energy ǫν we have a linear Schrödinger equation. Those eigenstates that lie closeto the expansion energy will be accurate.

The Taylor expansion however has one important caveat: Every Taylor expansion has a con-vergence radius beyond which the Taylor expansion does no more converge to the correct solution.Usually the convergence radius is given by the distance of the next pole or discontinuity from theexpansion point. In the present case the eigenstates of HBB, eigenstates of the isolated subsystemB introduce a singularity. Thus, at these energies the Taylor expansion will become unreliable.

256 M DOWNFOLDING

Appendix N

Time-inversion symmetry

Time inversion symmetry says that it is not possible from a conservative classical trajectory to findout if it time is running forward or backward in time. If we take a physical trajectory ~x(t) and let thetime run backwards, i.e. ~x ′(t) = ~x(−t), the new trajectory ~x ′(t) still fulfills the equations of motion.

Electrodynamics and gravitation obey time-inversion symmetry exactly. However, while time time-inversion symmetry is one of the most fundamental properties of natural laws, it is, taken alone, not afundamental symmetry of nature: The weak interaction, which is, for example, responsible for the βdecay of nuclei, violates it. Time inversion must be replaced by the weaker CPT-inversion symmetry.This is the so-called CPT-theorem posed by Gerhart Lüders and Wolfgang Pauli. The CPT theoremsays that the fundamental laws of nature must obey a symmetry under simultaneous application ofthree operations:

• charge inversion (C)

• space inversion (P for Parity)

• time inversion (T)

The CPT theorem is based on the assumptions of Lorentz invariance, causality, locality and theexistence of a Hamilton operator that is bounded by below. Electrodynamics and gravitation aresymmetric under the three symmetry operations individually.

As we are not concerned with weak interactions we can assume exact time inversion symmetry.

N.1 Schrödinger equation

Let us now investigate what time inversion symmetry implies in quantum mechanics:Let us consider the Schrödinger equation in a magnetic field

i~∂tΨ(~r , t) =

[

( ~i~∇− q ~A)22m

+ qΦ

]

Ψ(~r , t) (N.1)

Let us take the complex conjugate of the Eq. N.1

−i~∂tΨ∗(~r , t) =[

(− ~i ~∇− q ~A)22m

+ qΦ

]

Ψ∗(~r , t) =

[

( ~i~∇+ q ~A)22m

+ qΦ

]

Ψ∗(~r , t) (N.2)

Next we look for the equation obeyed by ψ(~r ,−t), if Eq. N.2 holds

i~∂tΨ∗(~r ,−t) =

[

( ~i~∇+ q ~A(~r ,−t))2

2m+ qΦ(~r ,−t)

]

Ψ∗(~r ,−t) (N.3)

257

258 N TIME-INVERSION SYMMETRY

One can see immediately that Eq. N.3 is identical to Eq. N.1, when the time and simultaneouslythe vector potential ~A are reverted. Thus, the Schrödinger equation is symmetric under the timeinversion symmetry as stated below:

TIME-INVERSION

~A(~r , t)→ −~A(~r ,−t) (N.4)

Φ(~r , t)→ Φ(~r ,−t) (N.5)

Ψ(~r , t)→ Ψ∗(~r ,−t) (N.6)

For the time-independent Schrödinger equation we obtain

Ψ(~r , t) = Ψǫ(~r)e− i~ǫt (N.7)

so that

Ψ∗(~r ,−t) Eq. N.7= Ψ∗ǫ(~r)e

− i~ǫt (N.8)

Thus, the time-inversion symmetry applied to energy eigenstates has the effect that the wave functionis turned into its complex conjugate.

Let us look at the problem from stationary Schrödinger equation.

[

( ~i~∇− q ~A)22m

+ qΦ− ǫ]

Ψǫ(~r) = 0 (N.9)

We take the complex conjugate of this equation

[

( ~i~∇+ q ~A)22m

+ qΦ− ǫ]

Ψ∗ǫ(~r) = 0 (N.10)

We observe that the complex conjugate of the wave function solves the same Schrödinger equationwith the magnetic field reversed.

This implies that, in the absence of a vector potential ~A, the complex conjugate is also a solutionof the original wave function. When these two degenerate solutions are superimposed, one obtainsanother solution of the same Schrödinger equation. Hence, also the real part and the imaginarypart are solutions of the Schrödinger equation. Thus, in the absence of a magnetic field, the wavefunctions can be assumed to be purely real.

REAL WAVE FUNCTIONS

In the absence of magnetic fields, the wave function of the non-relativistic Schrödinger equation canbe chosen as real functions.

N.2 Pauli equation

The proper theory of electrons is the Dirac equation, which describes electrons by a four componentspinor, that describes spin up and spin-down electrons as well as their antiparticles, the positrons.In the non-relativistic limit, electrons and positrons become independent. In this limit electrons andpositrons obey the so-called Pauli equation.

N TIME-INVERSION SYMMETRY 259

The Pauli equation has the form

i~∂t |ψ(t)〉 =[

(~p − q ~A)22me

+ qΦ− q

me~S ~B

]

|ψ〉 (N.11)

The wave function is now a two-component spinor with a spin-up and a spin-down component. Thespin-operator is represented by the Pauli matrices

~SEq. 3.1=~

2(σx , σy , σz)

The Pauli matrices are given in Eq. 3.2 on 40. The magnetic field is related to the vector potentialvia ~B = ~∇× ~A.

Expressed with explicit spinor components, the Pauli equation has the form

i~∂tψ(~r , σ, t) =∑

σ′

(

( ~i~∇− q ~A)22me

+ qΦ

)

δσ,σ′

︸︷︷︸

H0

− q

me~B~Sσ,σ′

ψ(~r , σ′, t) (N.12)

We proceed as we did for the Schrödinger equation by taking the complex conjugate of the Pauliequation Eq. N.12

−i~∂tψ∗(~r , σ, t) =∑

σ

[(

(− ~i ~∇− q ~A)22me

+ qΦ

)

δσ,σ′ −q

me~B~S∗σ,σ′

]

ψ∗(~r , σ′, t) (N.13)

Now we revert the time argument1 in Eq. N.13 and perform the transformation of the potentials~A′(~r , t) = −~A(~r ,−t) and Φ′(~r , t) = Φ(~r ,−t). The transformed magnetic field is ~B′(~r , t) = ~∇ ×~A′(~r , t) = −~∇× ~A(~r ,−t) = −~B(~r ,−t).

i~∂tψ∗(~r , σ,−t) =

∑

σ′

[(

( ~i~∇− q ~A′(~r , t))22me

+ qΦ′(~r , t)

)

δσ,σ′

︸︷︷︸

H′0

+q

me~B′(~r , t)~S∗σ,σ′

]

ψ∗(~r , σ′,−t)

(N.14)

We observe, that it is no more sufficient to replace the wave function by its complex conjugate of thetime-reverted function as in the Schrödinger equation. This is because the complex conjugate of thespin operator ~Sσ,σ′ is not identical to its complex conjugate. The y -component is purely imaginary,because the corresponding Pauli matrix is purely imaginary. Thus, the y -component changes its sign,when the complex conjugate is taken.

In order to find the transformation of the wave functions, let us rewrite the equation in compo-nents. The original equation Eq. N.12 written in components looks like

(i~∂t − H0)(

ψ(~r , ↑, t)ψ(~r , ↓, t)

)

Eq. N.12= − ~q

2me

(

Bz Bx − iByBx + iBy −Bz

)(

ψ(~r , ↑, t)ψ(~r , ↓, t)

)

(N.15)

The last equation, Eq. N.14, written in component notation has the form

(i~∂t − H′0)(

ψ∗(~r , ↑,−t)ψ∗(~r , ↓,−t)

)

Eq. N.14= − ~q

2me

(

−B′z −B′x − iB′y−B′x + iB′y B′z

)

︸︷︷︸

+ qme~B′ ~S∗

(

ψ∗(~r , ↑,−t)ψ∗(~r , ↓,−t)

)

(N.16)

1Consider a replacement t = −t ′. Thus, ∂t ′ = −∂t and ψ(t) = ψ(−t ′). After the transformation we drop theprime.


In order to get an idea on how to proceed, let us consider the special case B′x = B′y = 0. Inthis case the diagonal elements of the equation could be brought into the form of Eq. N.15 byinterchanging the spin indices of the wave functions.

Guided by this idea, we interchange the spin indices of Eq. N.16. This interchanges the columnsand the rows of the 2× 2 matrix. We obtain

(

i~∂t − H′)(

ψ∗(~r , ↓,−t)ψ∗(~r , ↑,−t)

)

= − ~q2me

(

B′z −B′x + iB′y−B′x − iB′y −B′z

)(

ψ∗(~r , ↓,−t)ψ∗(~r , ↑,−t)

)

The off-diagonal elements of the equation still differ by the sign from those in Eq. N.15. Thisproblem can be remedied by replacing the upper spinor component by its negative. While rewritingthe equation for the new convention, the sign of the off-diagonal elements of the matrix is changed.

(

i~∂t − H′)(

−ψ∗(~r , ↓,−t)ψ∗(~r , ↑,−t)

)

= − ~q2me

(

B′z B′x − iB′yB′x + iB

′y −B′z

)(

−ψ∗(~r , ↓,−t)ψ∗(~r , ↑,−t)

)

(N.17)

Thus, we arrived at the desired form Eq. N.15.By comparing Eq. N.17 with the original Pauli equation Eq. N.15, we see that the for any solution

of the Pauli equation also the result of the time inversion operation Eq. N.18 for two-componentspinors and electromagnetic fields is a solution of the same Pauli equation.

TIME-INVERSION FOR TWO-COMPONENT SPINORS AND ELECTROMAGNETIC FIELDS

ψ′(~r , ↑, t) = −ψ∗(~r , ↓,−t)ψ′(~r , ↓, t) = +ψ∗(~r , ↑,−t)~A′(~r , t) = −~A(~r ,−t)Φ′(~r , t) = Φ(~r ,−t) (N.18)

If we investigate the resulting transformation of the spin-expectation values, we see that the spinis inverted. This is expected under time-inversion symmetry, if we interpret the spin as a angularmomentum. If we invert the time, the particle is spinning in the opposite direction.

N.3 Time inversion for Bloch states

Often we require the implications of this symmetry for wave in a representation of Bloch waves. ABloch state is given by as product of a periodic function u~k,n and a phase factor ei~k~r , so that

ψ~k,n(~r , σ, t) = u~k,n(~r , σ)ei(~k~r−~ωnt) (N.19)

If time inversion symmetry is obeyed, for example in the absence of magnetic fields with staticpotentials, we can use ψ(~r , ↑, t) = −ψ∗(~r , ↓,−t) and ψ(~r , ↓, t) = −ψ∗(~r , ↑,−t), so that

u~k,n(~r , ↑)ei(~k~r−~ωt) = ψ(~r , ↑, t) = −ψ∗(~r , ↓,−t) = −u∗−~k,n(~r , ↓)e

i(~k~r−~ωt)

u~k,n(~r , ↑) = −u∗−~k,n(~r , ↓) (N.20)

and

u~k,n(~r , ↓)ei(~k~r−~ωt) = ψ(~r , ↓, t) = ψ∗(~r , ↑,−t) = u∗−~k,n(~r , ↑)e

i(~k~r−~ωt)

u~k,n(~r , ↓) = u∗−~k,n(~r , ↑) (N.21)

N TIME-INVERSION SYMMETRY 261

Thus, we obtain for the periodic parts of the wave functions(

u~k,n(~r , ↑)u~k,n(~r , ↓)

)

=

(

0 −11 0

)(

u−~k,n(~r , ↑)u−~k,n(~r , ↓)

)∗

(N.22)

Appendix O

Slater determinants for parallel andantiparallel spins

Why can spin-up and spin-down electrons be treated as non-identical particle, even through they areonly spinor-components of identical particles.

Here, we try to give an answer by showing that a one-particle orbitals can be occupied indepen-dently of each other with one spin-up and one spin-down electron. That is for each placement ofthe two electrons into one-particle orbitals there is a Slater determinant. For two spin-up electronsor for two spin-down electrons, the same one-particle orbital can only be occupied once.

This implies that the statistics of spin up and spin-down electrons is such that spin up electronsare treated as a class of identical particles and spin-down electrons are treated as a different class ofindependent particles.

Let us consider two spatial one-particle orbitals χa(~r) and χb(~r). The two orbitals shall beorthonormal. Out of these two orbitals, we construct four spin orbitals, namely

φa↑(~r , σ) = χa(~r)δσ,↑φa↓(~r , σ) = χa(~r)δσ,↓φb↑(~r , σ) = χb(~r)δσ,↑φb↓(~r , σ) = χb(~r)δσ,↓

or using α ∈ a, b and s ∈ ↑, ↓.

φα,s(~r , σ) = χα(~r)δs,σ

Out of these one-particle spin orbitals, we construct the Slater determinants

Φα,β,s,s ′(~r , σ, ~r ′, σ′) =

1√2

(

φα,s(~r , σ)φβ,s ′(~r ′, σ′)− φβ,s ′(~r , σ)φα,s(~r ′, σ′)

)

=1√2

(

χα(~r)χβ(~r ′)δs,σδs ′,σ′ − χβ(~r)χα(~r ′)δs,σ′δs ′,σ)

Some of these Slater determinants are zero states, but that does not matter at the moment.These Slater determinants are products of a spatial two-particle wave function and a two-particle

wave function in the spin space. It will be convenient to introduce such two-particle states that are

263

264 O SLATER DETERMINANTS FOR PARALLEL AND ANTIPARALLEL SPINS

symmetric and antisymmetric under particle exchange. (Exchange of the arguments.) These are

Ψ+α,β(~r ,~r ′)

def=1√2

(

χα(~r)χβ(~r ′) + χβ(~r)χα(~r ′))

Ψ−α,β(~r , ~r′)

def=1√2

(

χα(~r)χβ(~r ′)− χβ(~r)χα(~r ′))

G+s,s ′(σ, σ′)

def=1√2

(

δs,σδs ′,σ′ + δs ′,σ′δs,σ

)

G−s,s ′(σ, σ′)

def=1√2

(

δs,σδs ′,σ′ − δs ′,σ′δs,σ)

The Slater determinants have the form

Φα,β,s,s ′(~r , σ, ~r ′, σ′) =

1√2

[1

2

(

Ψ+α,β +Ψ−α,β

)(

G+s,s ′ + G−s,s ′

)

− 12

(

Ψ+α,β −Ψ−α,β)(

G+s,s ′ − G−s,s ′)]

=1√2

[

Ψ+α,β(~r ,~r ′)G−s,s ′(σ, σ

′) + Ψ−α,β(~r , ~r′)G+s,s ′(σ, σ

′)]

(O.1)

Now we can analyze the result using

G−s,s ′(σ, σ′) ≡ 0 for s = s ′

Ψ−α,β(~r , ~r′) ≡ 0 for α = β

Let us now consider two spin up electrons. Because of Eq. O.1 and G−↑,↑ = 0, there is only asingle non-zero Slater determinant, namely

|Φa,b,↑,↑〉 = Ψ−a,b(1√2G+s,s ′

)

The Slater determinant |Φb,a,↑,↑〉 differs from |Φa,b,↑,↑〉 only by a sign change. Thus, we see thatthe spatial wave function is antisymmetric. Hence, the Pauli principle exists in the spatial coordinates.

To show the difference let us investigate the states with different spin.

|Φa,a,↑,↓〉 =( 1√2Ψ+a,a

)

G−↑,↓

|Φa,b,↑,↓〉 =1√2

(

Ψ+a,bG−↑,↓ +Ψ

−a,bG

+↑,↓

)

|Φb,a,↑,↓〉 =1√2

(

Ψ+a,bG−↑,↓ −Ψ−a,bG+↑,↓

)

|Φb,b,↑,↓〉 =( 1√2Ψ+b,b

)

G−↑,↓

We see that the two electrons with different spin can be placed without restriction into the twospatial one-particle orbitals.

This indicates that electrons that the statistics of electrons with different spin is identical to thatof two non-identical particles. The statistics of electrons with like spin is like identical particles.

O.1 Spatial symmetry for parallel and antiparallel spins

Here, we show that

• the spatial wave functions for two electrons with parallel spin is antisymmetric

• the spatial wave functions for two electrons with anti-parallel spin is symmetric

O SLATER DETERMINANTS FOR PARALLEL AND ANTIPARALLEL SPINS 265

under exchange of the coordinates.A common misconception is that the Slater determinant of two one-particle spin-orbitals with

opposite spin describes two electrons with anti-parallel spin. For such a state the z-component ofthe spin vanishes, but it is a superposition of a two states with Sz = 0, namely one with antiparallelspin and the other with parallel spin with a total spin lying in the xy-plane.

In order to determine states with parallel and antiparallel spin, we need to determine the spin

eigenstates and eigenvalues. If the eigenvalue for S2tot = ( ~S1 + ~S2)2 vanishes, the wave function has

antiparallel spin. If the eigenvalue is 2~2 the spins are parallel.Using the ladder operators S± we write the total spin as

S2tot = ( ~S1 + ~S2)2

= ~S21 + ~S22 + 2 ~S1 ~S2

= ~S21 + ~S22 + 2

[1

2S1,+S2,− +

1

2S1,−S2,+ + S1,z S2,z

]

= ~S21 + ~S22 + S1,+S2,− + S1,−S2,+ + 2S1,z S2,z

]

The ladder operators are defined as

S+ = Sx + i Sy

S− = Sx − i Syand obey

S−| ↑〉 = | ↓〉~ and S−| ↓〉 = |∅〉S+| ↑〉 = |∅〉 and S+| ↓〉 = | ↑〉~

Thus(

S1+S2− + S1−S2+)

|G±↑,↓〉 =(

S1+S2− + S1−S2+) 1√2

(

| ↑, ↓〉 ± | ↓, ↑〉)

=1√2

(

| ↓, ↑〉 ± | ↑, ↓〉)

~2

= |G±↑,↓〉(

±~2)

(

S1+S2− + S1−S2+)

|G±↑,↑〉 = |∅〉(

S1+S2− + S1−S2+)

|G±↓,↓〉 = |∅〉

S1,z S2,z |G±↑,↓〉 = |G±↑,↓〉(

−~2

4)

S1,z S2,z |G±↑,↑〉 = |G±↑,↑〉(

+~2

4)

S1,z S2,z |G±↓,↓〉 = |G±↓,↓〉(

+~2

4)

~S2i |G±s,s ′〉 = |G±s,s ′〉3~2

4

S2tot |G±↑,↓〉 = |G±↑,↓〉[3~2

4+3~2

4± ~2 − ~

2

2

]

= |G±↑,↓〉(1± 1)~2

S2tot |G±↑,↑〉 = |G±↑,↑〉[3~2

4+3~2

4+~2

2

]

= |G±↑,↑〉2~2

S2tot |G±↓,↓〉 = |G±↓,↓〉[3~2

4+3~2

4+~2

2

]

= |G±↓,↓〉2~2

266 O SLATER DETERMINANTS FOR PARALLEL AND ANTIPARALLEL SPINS

Thus, we see that |G+↑,↓〉, |G+↑,↑〉 and |G+↓,↓〉 describe two electrons with parallel spin. Note that|G−s,s ′〉 vanishes for parallel spin.

The only solution with antiparallel spin is |G−↑,↓〉.From Eq. O.1 we know that the antisymmetric spin wave function is always combined with the

symmetric spatial orbital and vice versa. Hence, the wave functions describing two electrons withanti-parallel spin are symmetric in their spatial coordinates, while the ones with parallel spin have anantisymmetric spatial wave function.

O.2 An intuitive analogy for particle with spin

Consider balls that are painted on the one side green and on the other side red.If we place two such balls on a table with the green side up, turn around and look at them again,

we cannot tell if the two balls have been interchanged.No we place them with opposite colors up. If we only allow that the positions of the two balls are

interchanged, but exclude that they are turned around, we can tell the two spheres apart from theirorientation. Hence, we can treat them as non-identical spheres.

However, if we consider exchanges of position and orientation, we are again unable to tell, whetherthe spheres have been interchanged or not.

Thus, if we can exclude that the spins of the particles –or the orientation of our spheres– change,we can divide electrons into two classes of particles, namely spin-up and spin-down electrons. Particleswithin each class are indistinguishable, but spin-up and spin-down electrons can be distinguished.When the spin is preserved, the Hamiltonian is invariant with respect to spin-rotation, and spin is agood quantum number.

However, if there is a magnetic field, which can cause a rotation of the spin direction, the divisioninto spin-up and spin-down particles is no more a useful concept. In our model, if the orientation ofthe spheres can change with time, we cannot use it as distinguishing feature.

Appendix P

Hartree-Fock of the free-electron gas

Here, we derive the changes in the dispersion relation ǫ(~k) of the free-electron gas due to the exchangepotential discussed in section 8.6 on p. 164.

Because of the translational symmetry, we can assume that the charge density is spatially con-stant.1 Since the problem is translationally invariant, we can furthermore deduce that the eigenstatesare plane waves.

P.1 Exchange potential as non-local potential

Let us evaluate the non-local potential as defined in Eq. 8.24 using plane waves φ~k,σ(~r , σ′) =

〈~r , σ′|φ~k,σ〉 = 1Ωe

i~k~rδσ,σ′ as defined in Eq. ?? as basis functions.

Vx(~x, ~x ′)Eq. 8.24= −

∑

j

e2φ∗j (~x)φj(~x′)

4πǫ0|~r − ~r ′|

Eq. ??=

−e24πǫ0Ω

∑

~k

θ(kF − |~k |)

1︷︸︸︷

(2π)3

Ω︸︷︷︸

→d3k

Ω

(2π)3ei~k(~r ′−~r)δσ,σ′

|~r − ~r ′|

=−e2

4πǫ0|~r − ~r ′|δσ,σ′

(2π)3

∫

d3k θ(kF − |~k |)ei~k(~r ′−~r) (P.1)

We see that the integral is isotropic in ~r ′ − ~r . Thus, we can assume, without loss of generality,that the distance vector points in z-direction, that is ~r ′ − ~r = ~ezs.

kF

kz

kF2 −kz

2

1We assume here that there is no symmetry breaking, which is not guaranteed.

267

268 P HARTREE-FOCK OF THE FREE-ELECTRON GAS

∫

~k≤kFd3k eikz s =

∫ kF

−kFdkze

ikz s

∫

k2x+k2y<k

2F−k2z

dkxdky

︸︷︷︸

π(k2F−k2z )

=

∫ kF

−kFdkz

(π(k2F − k2z )

)eikz s

= πk2F

∫ kF

−kFdkz e

ikz s − π∫ kF

−kFdkz k

2z eikz s

= πk2F

∫ kF

−kFdkz e

ikz s + πd2

ds2

∫ kF

−kFdkz e

ikz s

= π

[

k2F +d2

ds2

] [1

i seikz s

]kF

−kF

= π

[

k2F +d2

ds2

]eikF s − e−ikF s

i s

= 2πk3F

[

1 +d2

d(kF s)2

]sin(kF s)

kF s

= 2πk3F

[(

1 +d2

dx2

)sin(x)

x

]

x=kF s

= 2πk3F

[sin(x)

x− sin(x)

x− 2 cos(x)

x2+2 sin(x)

x3

]

x=kF s

=4π

3k3F

[

−3cos(kF s)(kF s)2

+ 3sin(kF s)

(kF s)3

]

(P.2)

Now we insert the result of the integral into the expression for the nonlocal potential.

Vx(s, σ, σ′)Eqs. P.1,P.2

=−e2δσ,σ′4πǫ0s

1

(2π)34πk3F

[

−cos(kF s)(kF s)2

+sin(kF s)

(kF s)3

]

︸︷︷︸∫d3k θ(kF−|~k|)ei~k(~r ′−~r)

=e2δσ,σ′

4πǫ0s

1

(2π)34π

3k3F

[

3(kF s) cos(kF s)− sin(kF s)

(kF s)3

]

(P.3)

The non-local exchange potential acts only between electrons of the same spin. It is important torealize that this non-local potential is not an interaction potential between two electrons, but it is aone-particle potential acting on each electron individually. Rather is is a potential that produces alarge energy correction for a particle if its wave function is concentrated in one region, as comparedto a very delocalized electron density. Nevertheless this potential expresses an interaction: The shapeof this non-local potential depends on the state of all the other electrons.

P.2 Energy-level shifts by the exchange potential

Now we need to evaluate the expectation values of this potential in order to obtain the energy shifts:

dǫ~k,σ =1

Ω

∫

d3r

∫

d3r ′ Vx(|~r − ~r ′|, σ, σ′)ei~k(~r−~r ′)

~sdef=~r−~r ′=

1

Ω

∫

d3r

︸︷︷︸

1

∫

d3s Vx(|~s |, σ, σ′)ei~k~s

P HARTREE-FOCK OF THE FREE-ELECTRON GAS 269

-15 -10 -5 0 5 10 15

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

-15 -10 -5 0 5 10 15-1

0

Fig. P.1: The shape of the non-local exchange potential (right) for a free electron gas as calculatedin the Hartree-Fock method. The dashed line corresponds to a Coulomb interaction. The function3 x cos(x)−sin(x)x3 is shown on the left-hand side.

Now we decompose the plane wave into spherical harmonics(see appendix “Distributionen, d-Funktionenund Fourier transformationen in the text book of Messiah[80])

ei~k~r = 4π

∞∑

ℓ=1

ℓ∑

m=−ℓi ℓjℓ(|~k ||~r |)Y ∗ℓ,m(~k)Y ∗ℓ,m(~r)

Thus, we obtain

dǫ~k,σ =

∫

d3s Vx(|~s |, σ, σ′)4π∞∑

ℓ=1

ℓ∑

m=−ℓi ℓjℓ(|~k ||~s |)Y ∗ℓ,m(~k)Y ∗ℓ,m(~s)

= 4π

∞∑

ℓ=1

ℓ∑

m=−ℓi ℓY ∗ℓ,m(~k)

∫

d3s Vx(|~s |, σ, σ′)jℓ(|~k ||~s |)Y ∗ℓ,m(~s)

Because the non-local potential is isotropic, only the term with ℓ = 0 contributes.

dǫ~k,σ = 4π Y ∗0,0(~k)︸︷︷︸

1√4π

∫

d3s Vx(|~s |, σ, σ) j0(|~k ||~s |)︸︷︷︸

j0(x)=sin(x)x

Y ∗0,0(~s)︸︷︷︸

1√4π

=

∫

d3s Vx(|~s |, σ, σ)sin(|~k ||~s |)|~k ||~s |

sdef=|~s|= 4π

∫

ds s2Vx(s, σ, σ)sin(|~k |s)|~k |s

=4π

|~k |

∫

ds Vx(s, σ, σ)s sin(|~k |s)

Eq. 8.27=

4π

|~k |

∫

dse2

4πǫ0s

1

(2π)24π

3k3F

[

3kF s cos(kF s)− sin(kF s)

(kF s)3

]

s sin(|~k |s)

=4π

|~k |e2

4πǫ0

1

(2π)24π

3k3F

∫

ds

[

3kF s cos(kF s)− sin(kF s)

(kF s)3

]

sin(|~k |s)

rdef=kF s=

4π

kF |~k |e2

4πǫ0

1

(2π)24π

3k3F

∫ ∞

0

dr

[

3r cos(r)− sin(r)

r3

]

sin(|~k |kFr)

Now we need to solve the integral

I =

∫ ∞

0

dx

[

3x cos(x)− sin(x)

x3

]

sin(ax)


where a = |~k |/kF . The difficulty with this integral is that we cannot take the two terms apart,because the individual parts of the integrand diverge at the origin. Thus, during the derivation, wehave to deal with divergent expressions.

∂xsin(x)

xn−1= −(n − 1)sin(x)

xn+cos(x)

xn−1

⇒ sin(x)xn

=1

(n − 1)

[cos(x)

xn−1− ∂x

sin(x)

xn−1

]

(P.4)

Eq. P.4⇒∫ ∞

0

dxsin(x)

x3sin(ax) =

1

2

∫ ∞

0

dxcos(x)

x2sin(ax)− 1

2

∫ ∞

0

dx sin(ax)∂xsin(x)

x2

=1

2

∫ ∞

0

dxcos(x)

x2sin(ax)

−12

[

sin(ax)sin(x)

x2

]∞

0

+a

2

∫ ∞

0

dxsin(x)

x2cos(ax)

=a

2+1

2

∫ ∞

0

dxcos(x) sin(ax) + a sin(x) cos(ax)

x2(P.5)

Eq. P.4⇒∫ ∞

0

dxsin(x)

x2cos(ax) =

∫ ∞

0

dxcos(x)

xcos(ax)−

∫ ∞

0

dx cos(ax)∂xsin(x)

x

=

∫ ∞

0

dxcos(x)

xcos(ax)

−[

cos(ax)sin(x)

x

]∞

0

− a∫ ∞

0

dxsin(x)

xsin(ax)

= 1 +

∫ ∞

0

dxcos(x) cos(ax)− a sin(x) sin(ax)

x(P.6)

P HARTREE-FOCK OF THE FREE-ELECTRON GAS 271

Thus, we obtain

I = 3

∫ ∞

0

dxx cos(x)− sin(x)

x3sin(ax)

= 3

∫ ∞

0

dxcos(x) sin(ax)

x2− 3

∫ ∞

0

dxsin(x) sin(ax)

x3

Eq. P.5= 3

∫ ∞

0

dxcos(x) sin(ax)

x2− 3a2− 32

∫ ∞

0

dxcos(x) sin(ax) + a sin(x) cos(ax)

x2

= −3a2+3

2

∫ ∞

0

dxcos(x) sin(ax)

x2− 3a2

∫ ∞

0

dxsin(x) cos(ax)

x2

= −3a2+3a

2

∫ ∞

0

dycos( 1a y) sin(y)

y2− 3a2

∫ ∞

0

dxsin(x) cos(ax)

x2

Eq. P.6= −3a

2+3a

2

[

1 +

∫ ∞

0

dycos( 1a y) cos(y)− 1a sin( 1a y) sin(y)

y

]

−3a2

[

1 +

∫ ∞

0

dxcos(ax) cos(x)− a sin(ax) sin(x)

x

]

= −3a2+3a

2

[∫ ∞

0

dxcos(x) cos(ax)− 1a sin(x) sin(ax)

x

]

−3a2

[∫ ∞

0

dxcos(ax) cos(x)− a sin(ax) sin(x)

x

]

= −3a2+3a

2(−1a+ a)

[∫ ∞

0

dxsin(x) sin(ax)

x

]

= −3a2+3

2(a2 − 1)

[∫ ∞

0

dxsin(x) sin(ax)

x

]

We recognize that the integral is the fourier transform of sin(x)x . From the Fourier transform tablesof Bronstein[34] we take

√

2

π

∫ ∞

0

dx sin(xy)sin(ax)

x

Bronstein=

1√2πln

∣∣∣∣

y + a

y − a

∣∣∣∣

a→1;y→a⇒∫ ∞

0

dx sin(ax)sin(x)

x=

1

2ln

∣∣∣∣

a + 1

a − 1

∣∣∣∣=1

2ln

∣∣∣∣

1 + a

1− a

∣∣∣∣

Thus, we obtain

I = −3a2+3

4(a2 − 1) ln

∣∣∣∣

1 + a

1− a

∣∣∣∣

Now we are done with the Integral. We need to insert the result in the correction for the energyeigenvalues

dǫ~k,σ =4π

kF |~k |e2

4πǫ0

1

(2π)24π

3k3F

[

−3a2+3(a2 − 1)4

ln

∣∣∣∣

a + 1

a − 1

∣∣∣∣

]

=4π

k2F

e2

4πǫ0

1

(2π)24π

3k3F

[

−32+3(a2 − 1)4a

ln

∣∣∣∣

a + 1

a − 1

∣∣∣∣

]

= − e2

4πǫ0

2kFπ

[1

2+1− a24a

ln

∣∣∣∣

1 + a

1− a

∣∣∣∣

]

The function in parenthesis is shown in Fig. P.2 and the resulting dispersion relation is discussed insection 8.6 on p. 164.


0 1 20.0

0.5

1.0

Fig. P.2: The function f (a) = −(12 +

1−a24a ln

∣∣ 1+a1−a∣∣

)

as function of a = k/kF . Note that the slope

for k = kF is infinite. For k >> kF the function approaches zero.

Appendix Q

Dictionary

caveat Vorbehaltcoincidence Übereinstimmungconvolution Faltungcross section Wirkungsquerschnittesthetics Ästetikfactorial Fakultät (mathematische Operation)faculty Fakultät (akademische Verwaltungseinheit)for the sake of ... um ... willen (gen)formidable gewaltig, eindrucksvollpropagator Zeitentwicklungsoperatorcommutate kommutierenrank Rangscattering amplitude Streuamplitudewithout loss of generality ohne Beschränkung der Allgemeinheitwlog oBdA (ohne Beschränkung der Allgemeinheit)

Q.1 Explanations

• caveat: Latin, a modifying or cautionary detail to be considered

Q.2 Symbols

A list of mathematical symbols can be found on the internet https://en.wikipedia.org/wiki/List_of_mathematical_symbols_by_subjeWhile it is the goal to adhere to a standard set of symbols, this may not have been achieved.

• A vector is represented as ~a, a matrix is shown as bold-face symbol a and an operator isindicated by a hat a.

• Natural numbers: N, integer numbers Z, real numbers R, complex numbers C.

• c∗ complex conjugate of the complex number c .

• O(xn) stands for all terms of order n and higher of x .

• A† is the hermitian conjugate or adjunct of the matrix A

273

https://en.wikipedia.org/wiki/List_of_mathematical_symbols_by_subject

274 Q DICTIONARY

• AT is the transpose of matrix A

• θ(x) Heaviside function.

• ∀x . . . “for all x . . . ”. ∃x . . . “there exists one x with . . .”

• [A,B]− = [A,B] = AB−BA commutator of A and B. A and B may be matrices or operators.

• [A,B]+ = AB + BA anti-commutator of A and B. A and B may be matrices or operators.

• ~a · ~b scalar product between two vectors.

• ~a × ~b vector product between two three dimensional vectors ~a and ~b.

~a × ~b =

aybz − azbyazbx − axbzaxby − aybx

(Q.1)

• ~a ⊗ ~b outer product or dyadic product of two vectors ~a and ~b.(

~a ⊗ ~b)

i ,j= aibj (Q.2)

• det |A| determinant of matrix A

• perm|A| permanent of matrix A. The permanent is computed like the determinant with theexceptions that all terms are summed up without the sign changes for permutations.

• δi ,j Kronecker delta

δi ,j =

1 for i = j

0 else(Q.3)

• δ(~x) Dirac’s delta function defined by

∀f (x)∫

dx f (x)δ(x) = f (0) (Q.4)

which holds for all differentiable functions f (x). The delta function of a vector is the productof the delta functions of its components.

• ǫi1,...,iN Levy-Civita Symbol or fully antisymmetric tensor

Appendix R

Greek Alphabet

A α alpha N ν nuB β beta Ξ ξ ksiΓ γ gamma O o, omicron∆ δ delta Π π, piE ǫ, ε epsilon P ρ, rhoZ ζ zeta Σ σ, ς sigmaH η eta T τ tauΘ θ, ϑ theta Υ υ upsilonI ι iota Φ φ,ϕ phiK κ kappa X χ chiΛ λ lambda Ψ φ phiM µ mu Ω ω omega

275

276 R GREEK ALPHABET

Appendix S

Philosophy of the ΦSX Series

In the ΦSX series, I tried to implement what I learned from the feedback given by the students whichattended the courses and that relied on these books as background material.

The course should be self-contained. There should not be any statements “as shown easily...” if,this is not true. The reader should not need to rely on the author, but he should be able to convincehimself, if what is said is true. I am trying to be as complete as possible in covering all materialthat is required. The basis is the mathematical knowledge. With few exceptions, the material is alsodeveloped in a sequence so that the material covered can be understood entirely from the knowledgecovered earlier.

The derivations shall be explicit. The novice should be able to step through every single step ofthe derivation with reasonable effort. An advanced reader should be able to follow every step of thederivations even without paper and pencil.

All units are explicit. That is, formulas contain all fundamental variables, which can be inserted inany desirable unit system. Expressions are consistent with the SI system, even though I am quotingsome final results in units, that are common in the field.

The equations that enter a specific step of a derivation are noted ontop of the equation sign.The experience is that the novice does not immediately memorize all the material covered and thathe is struggling with the math, so that he spends a lot of time finding the rationale behind a certainstep. This time is saved by being explicit about it. The danger that the student gets dependent onthese indications, is probably minor, as it requires some effort for the advanced reader to look up theassumptions, an effort he can save by memorizing the relevant material.

Important results and equations are highlighted by including them in boxes. This should facilitatethe preparations for examinations.

Portraits of the key researchers and short biographical notes provide independent associationsto the material. A student may not memorize a certain formula directly, but a portrait. Fromthe portrait, he may associate the correct formula. The historical context provides furthermore anindependent structure to organize the material.

The two first books are in german (That is the intended native language) in order to not addcomplications to the novice. After these first books, all material is in English. It is mandatory that thestudent masters this language. Most of the scientific literature is available only in English. Englishis currently the language of science, and science is absolutely dependent on international contacts.

I tried to include many graphs and figures. The student shall become used to use all his sensesin particular the visual sense.

I have slightly modified the selection of the material commonly tought in most courses. Sometopics, which I consider of mostly historical relevance I have removed. Others such as the Noethertheorem, I have added. Some, like Chaos, Stochastic processes, etc. I have not added yet.

277

278 S PHILOSOPHY OF THE ΦSX SERIES

Appendix T

About the Author

Prof. Dr. rer. nat Peter E. Blöchl studied physics at Karlsruhe University of Technology in Germany.Subsequently he joined the Max Planck Institutes for Materials Research and for Solid state Researchin Stuttgart, where he worked on development of electronic-structure methods related to the LMTOmethod and on first-principles investigations of interfaces. He received his doctoral degree in 1989from the University of Stuttgart.

Following his graduation, he joined the renowned T.J. Watson Research Center in YorktownHeights, NY in the US on a World Trade Fellowship. In 1990 he accepted an offer from the IBMZurich Research Laboratory in Ruschlikon, Switzerland, which had just received two Nobel prices inPhysics (For the Scanning Tunneling Microscope in 1986 and for the High-Temperature Supercon-ductivity in 1987). He spent the summer term 1995 as visiting professor at the Vienna Universityof Technology in Austria, from where was awarded the habilitation in 1997. In 2000 he left theIBM Research Laboratory after a 10-year period and accepted an offer to be professor for theoreticalphysics at Clausthal University of Technology in Germany. Since 2003, Prof. Blöchl is member ofthe Braunschweigische Wissenschaftliche Gesellschaft (Academy of Sciences).

The main thrust of Prof. Blöchl’s research is related to ab-initio simulations, that is, parameter-free simulation of materials processes and molecular reactions based on quantum mechanics. Hedeveloped the Projector Augmented Wave (PAW) method, one of the most widely used electronicstructure methods to date. This work has been cited over 15,000 times. It is among the 100 mostcited scientific papers of all times and disciplines, and is among the 10 most-cited papers out of morethan 500,000 published in the 120-year history of Physical Review. Next to the research related tosimulation methodology, his research covers a wide area from biochemistry, solid state chemistry tosolid state physics and materials science. Prof. Bloechl contributed to 8 Patents and published about100 research publications, among others in well-known Journals such as “Nature”. The work of Prof.Blöchl has been cited over 23,000 times, and he has an H-index of 41.

279

280 T ABOUT THE AUTHOR

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Index

acion principle, 95acoustic branch, 108adiabatic connection, 191adjunct, 273Aharonov-Bohm effect, 22antisymmetric tensor, 46avoided crossing, 24

band structure, 51, 83Berry phase, 37Berry phase, 214Besselfunction

half-integer order, 176spherical, 176

Bloch vector, 59Bloch’s theorem, 60Bohr magneton, 42, 80Boltzmann constant, 70Boltzmann distribution, 116Boltzmann equation, 114, 115Born-Huang expansion, 16Born-Oppenheimer Approximation, 21Born-Oppenheimer approximation, 15Born-Oppenheimer surface, 21Born-Oppenheimer surfaces, 16Born-Oppenheimer wave functions, 15, 16Bose distribution, 111, 116Boson, 46Brillouin zone, 62, 97Brillouin-zone integral, 83bulk modulus, 78

chemical potential, 70collision term, 115commutator, 274complex number

set of, 273conical intersection, 24, 28, 217conical intersection, 24, 25contraction, 131, 170correspondence principle, 113correspondence principle, 55crystallographic space groups, 56

Debye frequency, 110

Debye temperature, 111delta function, 274density matrix

N-particle, 170density of states, 71density of states

free electron gas, 85density operator, 170derivative coupling, 18determinant, 274dimensional bottle-neck, 13Dirac equation, 43Dirac equation, 258Dirac sea, 53Dirac’s delta function, 274dispersion relation, 51dispersion relation, 83dispersion relation

phonons, 100drift term, 115dyadic product, 274dynamical matrix

k-dependent, 99

Einstein frequency, 109electron-gas parameter, 86equilibrium transition rate, 148exchange and correlation energy, 187exchange energy, 159exchange interaction, 184, 198exchange-correlation hole function, 179extended zone scheme, 61

Fermi distribution, 116Fermi energy, 72, 85Fermi level, 72Fermi momentum, 85Fermi wave vector, 85Fermi-distribution function, 72Fermion, 46ferromagnetism, 160Fock Operator, 163Fock space, 185force, 53Franck-Condon principle, 17

INDEX 287

free electron gas, 83free electron gas

density of states, 85fully antisymmetric tensor, 46, 274fundamental relation, 68

g-factor, 80Generalized gradient approximations, 202geometrical phase, 214GGA, 202grand canonical ensemble, 70grand potential, 70group, 237GW approximation, 193gyromagnetic ratio, 80

Hamilton function, 52Hamilton’s equations, 52harmonic oscillator, 94Hartree method, 151Hartree Energy, 158Hartree energy, 158Hartree-Fock method, 151Hartree-Fock equations, 161Heat, 68Heaviside function, 274Hellmann-Feyman theorem, 192hermitean conjugate, 273Higgs mechanism, 27hole, 54Hund’s rule, 160, 182hybrid functionals, 194

incompressible flow, 113integer number

set of, 273intrinsic probabilities of transition, 146intrinsic probabilities of transition, 138, 146

Jacob’s ladder, 196Jahn-Teller effect, 214Jahn-Teller model, 25jellium model, 83jellium oxide, 91jelly-O, 91

Kronecker delta, 274

Legendre transform, 188Levi-Civita symbol, 46Levy-Civita Symbol, 274Lindhard function, 164Liouville operator, 118Liouville’s’s theorem, 114local operator, 163

local potential, 163longitudinal

waves, 108

magnetic anisotropy, 12mass, 53mass

effective, 54maximum entropy principle, 69mean free path, 123minimum energy principle, 69molecular dynamics, 22molecular dynamics

ab-initio, 22momentum conservation

phonons, 126

natural numbersset of, 273

non-crossing rule, 23, 88, 217non-local potential, 162normal coordinates, 100number of bosons, 107number representation, 70

oblate, 26occupation numbers, 70one-particle operator, 152one-particle reduced density matrix, 171one-particle wave function, 47optic branch, 108outer product, 274

pair-correlation function, 178partition function, 67Pauli matrix, 40Pauli equation, 43, 259Pauli principle, 12periodic boundary conditions, 82periodic zone scheme, 61permanent, 274permutation operator, 45, 47phase

geometrical, 37phonon amplitudes, 106phonon band structure, 93phonons, 93point-group symmetry, 56polariton, 109polaron, 26positions

relative, 56positron, 43product wave function, 47

288 INDEX

prolate, 26

quantum field theory, 93quantum numbers, 45quasiparticles, 52

Rabi frequency, 223random-phase approximation, 198random-phase approximation

initial, 131random-phase approximation

repeated, 134rank, 47real number

set of, 273Reciprocal lattice vectors, 57reduced zone scheme, 61relaxation times, 121relaxation-time approximation, 121reststrahl absorption, 109RPA, 198

scalar product, 274second quantization, 93Seitz radius, 86self interaction, 159self-interaction energy, 159simple oxides, 91sink, 113Slater determinant, 49Slater determinant, 70Sommerfeld coefficient, 80Sommerfeld expansion, 73, 239sound waves, 109source, 113source density, 113spectrum

one-particle, 71spin multiplicity, 123spin-orbitals, 39spinor, 39standard model, 11stray fields, 13strong correlations, 51symmetry, 43, 44

tensorfully antisymmetric, 46

tensorfully antisymmetric, 274

thermodynamic beta, 70time-inversion symmetry, 258

for spinors, 260transformation operator, 43

transpose, 274transversal

waves, 108two-particle density, 174two-particle operator, 152

umklapp process, 127unit cell

primitive, 56universal density functional, 185

vector product, 274velocity, 52velocity

group velocity, 55Verlet algorithm, 230vertical process, 54

Wigner Seitz cell, 62Wirtinger derivatives, 161Work, 68

Yang-Mills fields, 22

Zeemann splitting, 80

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