LINEAR RESPONSE DENSITY FUNCTIONAL THEORY FOR METAL SURFACES
WITH APPLICATION TO SECOND HARMONIC GENERATION
By
Justin Droba

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Applied Mathematics – Doctor of Philosophy
2014

ABSTRACT
By
Justin Droba
This thesis is a study of electronic excitations at metal surfaces, as described within the context of density
functional theory (DFT). Before presenting any physics, the document develops an adaptive spline collocation method that is the workhorse for most of the numerical computations presented afterwards. The
background and mathematical foundations of DFT are briefly explored. From there, two different techniques for computing ground state densities—orbital-free density functional theory and Kohn-Sham density
functional theory—are fully realized. Development of algorithms for numerical computation is the primary
component of the presentation, but both techniques are also given rigorous theoretical treatment in full
proofs of asymptotic results. What follows next is a rigorous derivation of linear response theory from first
principles. Great effort is then spent to develop a scheme for numerical computation of excited responses
via linear response theory. Finally, the thesis concludes by demonstrating the application of the techniques
developed in the previous chapters to a nonlinear optical phenomenon called second harmonic generation.

To my parents,
who have been with me since the beginning.
To my wife Yuqi,
who will be with me until the end.

iii

ACKNOWLEDGMENTS
It is said that it takes a village to raise a child. I suppose the academic analog is that it takes a department
to produce a PhD. Perhaps a whole department is a bit of an exaggeration, but I certainly would not have
been able to reach this end without the selfless contributions of so many different people.
First and foremost, I must thank my parents, who have supported me financially and emotionally for the
entirety of my long academic career. As far back as I can remember, they constantly inspired me to push
myself to levels I thought attainable only by others and willingly sacrificed luxury to give me the finest
education possible. It is certainly no exaggeration to say that without their love, support, and frienship, I
never would have even come to Michigan State University to pursue my PhD, let alone complete it.
Had I not come to Michigan State, I never would have met my wife Yuqi. It is hard to fathom the odds
defied in two people coming from two places so far apart, meeting in the insignificant town in the middle
of Michigan, and discovering perfection in each other. When I reached my lowest points, when it seemed
all hope was lost and I was on the verge of breaking, it was she who gave me the strength to perservere.
Without her providing something greater for which to strive, I fear I would given up before reaching the end.
My greatest academic thanks goes to Prof. Gang Bao, who served as my advisor for four years. His generous
support of research assistantships semester after semester allowed me to pursue research without distraction
and freed up time to chase other academic ventures that greatly enriched my graduate experience. On top
of that, he arranged for three visits to China, including a multiweek stay at Zhejiang University.
The man with the second greatest impact on my research is Prof. Hiroshi Ajiki. My project had stalled
out before I arrived under his care at the Photon Pioneers Center at Osaka Unversity in June 2012. His
guidence and tutelage revitalized my research and set me on a path to achievement. The trip would not
have been possible without the East Asia and Pacific Summer Institutes (EAPSI) fellowship program and
the sponsoring agencies, the National Science Foundation (NSF) and Japan Society for the Promotion of
Science (JSPS), who provided the tremendous funds that paid for my twelve-week stay in Japan. A special
thanks goes to Prof. Di Liu, who provided valuable suggestions that ultimately led to a successful proposal.
I would like to express my gratitude to my other professors at Michigan State, in particular Jeffrey Schenker
and Keith Promislow, for keeping their doors open and allowing me to pick their brains when I was stuck
on my research. I thank Andrew Christlieb, Jianliang Qian, and Casim Abbas for their service on my
committee. I am greatful to Prof. Aklilu Zeleke for his continued friendship and for retaining me as mentor

iv

for the REU for four summers, even when my travels limited my ability to participate fully. I would also like
to acknowledge Professors Neepa Maitra and Ansgar Liebsch, who had no reason to take the time to answer
the silly questions of a nagging graduate student far removed from their circles—but chose to do so anyway.
My journey through graduate school would not have been nearly as enjoyable without the good company
that continuously surrounded me, sharing in the good moments and providing advice and emotional support
in the bad. I would especially like to acknowledge my friends, who each contributed in a unique way:
❼ Eric Wolf, for reading my proposals and fellowship applications and providing suggestions that helped

them to success, for always being willing to take time away from his own work to provide some thinking
points when I was stuck on a problem, and for in so many ways being the best friend I’ve ever had.
❼ Kazuko Fuchi, without whose Japanese help my initial emails to Prof. Ajiki may have gone ignored.
❼ Jackie Dresch and Richard Shadrach, for being such amicable officemates for my first four years.
❼ Adam Giambrone and Faramarz Vafaee, my next-door neighbors after our move to the fifth floor, for

always being ready and able to serve as a most welcome diversion from work.
❼ Luke Williams, for so many engaging philosophical conversations over lunches at Charlie Kang’s.
❼ The alumni students and postdocs of Prof. Bao’s group not already mentioned: Guanghui Hu, Xian-

liang Hu, Jun Lai, Peijun Li, Junshan Lin, Songting Luo, Russell Richins, Yuliang Wang, Xiang Xu,
Zhengfu Xu, Hai Zhang, and Xinghui Zhong. I hope our paths will cross again one day.
❼ Jaylan Jones, who provided me with the light of hope in one of my darkest moments.

July 1, 2014

v

TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

KEY TO ABBREVIATIONS AND SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter 1 Introduction . . . . . . . . . . . . . . . . . .
1.1 Atomic Units . . . . . . . . . . . . . . . . . . . . . .
1.2 Conventions . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Asymptotic Behavior . . . . . . . . . . . . .
1.2.2 Complex Numbers and Conjugation . . . .
1.2.3 Defined Quantities . . . . . . . . . . . . . .
1.2.4 Function Notation . . . . . . . . . . . . . . .
1.2.5 Norm, Absolute Value, and Inner Product
1.2.6 Superscripts . . . . . . . . . . . . . . . . . .
1.2.7 Theorems and Lemmata . . . . . . . . . . .
1.2.8 Vector Quantities . . . . . . . . . . . . . . .
1.3 Special Functions . . . . . . . . . . . . . . . . . . . .
1.3.1 Dirac Delta . . . . . . . . . . . . . . . . . . .
1.3.2 Fourier Transforms . . . . . . . . . . . . . .
1.3.3 Step and Sign Functions . . . . . . . . . . .
1.4 Dirac Notation . . . . . . . . . . . . . . . . . . . . .

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1
3
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Equations
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9
9
13
16
17

Chapter 3 Density Functional Theory . . . . . . . . . . . . . . . . . .
3.1 The Big Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . . . . .
3.3 The Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Kohn-Sham Potential in Self-Contained Systems . . . . . .
3.4 The Exchange-Correlation Functional . . . . . . . . . . . . . . . . .
3.4.1 The Local Density Approximation (LDA) . . . . . . . . . . .
3.4.2 The Wigner-Seitz Radius rs . . . . . . . . . . . . . . . . . . .
3.4.3 Exchange Kernel . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Correlation Kernel of Wigner . . . . . . . . . . . . . . . . . .
3.4.5 Correlation Kernel of Vosko, Wilk, and Nusair . . . . . . . .
3.4.6 Correlation Kernel of Perdew and Wang . . . . . . . . . . .
3.4.7 Analysis of the Perdew-Wang Exchange-Correlation Kernel

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20
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32

Chapter 4 The Ground State: Orbital-Free DFT .
4.1 The Jellium Model . . . . . . . . . . . . . . . . . . .
4.1.1 Properties of Exact Electron Density . . .
4.2 The Kinetic Energy Functional . . . . . . . . . . .
4.3 Orbital-Free Density Functional Theory . . . . . .
4.4 Asymptotic Behavior of the Orbital-Free Density
4.5 Numerical Implementation . . . . . . . . . . . . . .

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37
37
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53

Chapter 2 Spline Collocation Method for Differential
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Application to Nonlinear Equations . . . . . . . . . . .
2.3 Application to Linear Equations . . . . . . . . . . . . .
2.4 Local Adaptivity . . . . . . . . . . . . . . . . . . . . . .

vi

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4.6
4.7

Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
56

Chapter 5 The Ground State: Kohn-Sham DFT . . . . .
5.1 Introduction to Solid-State Physics . . . . . . . . . . . . .
5.1.1 The Fermi Surface and Fermi Sphere . . . . . . .
5.1.2 Three Important Physical Quantities . . . . . . .
5.1.3 Fermi-Dirac Statistics . . . . . . . . . . . . . . . . .
5.2 Effect of Jellium . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Boundary Conditions for Wavefunctions . . . . . . . . . .
5.3.1 Asymptotic Conditions at +∞ . . . . . . . . . . .
5.3.2 Asymptotic Conditions at −∞ . . . . . . . . . . .
5.3.3 Summary and Consequences . . . . . . . . . . . . .
5.4 Asymptotic Behavior of the Kohn-Sham Density . . . . .
5.4.1 Friedel Oscillations . . . . . . . . . . . . . . . . . .
5.5 Electrostatic Potential: A Deceptively Hard Problem . .
5.6 Self-Consistent Field Iteration (SCF) . . . . . . . . . . . .
5.7 Numerical Implementation . . . . . . . . . . . . . . . . . .
5.7.1 Determining a Wavefunction ψk (Step 3) . . . . .
5.7.1.1 Finding the Phase . . . . . . . . . . . . .
5.7.1.2 Renormalizing . . . . . . . . . . . . . . .
5.7.2 Forming the Density (Step 4) . . . . . . . . . . . .
5.7.3 Setting the Cut-off Filter (Step 5) . . . . . . . . .
5.7.4 Solving the Nonlinear Poisson Equation (Step 7)
5.8 Computational Results . . . . . . . . . . . . . . . . . . . .
5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6 Green’s Functions and Spectra . . . .
6.1 Derivation of Green’s Function . . . . . . . . . .
6.2 Spectral Theory . . . . . . . . . . . . . . . . . .
6.2.1 The Operator H . . . . . . . . . . . . . .
6.2.2 Review of Functional Analysis . . . . .
6.2.3 Characterization of the Spectrum of H
6.2.4 Spectral Representation for G . . . . . .
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . .

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. 91
. 91
. 94
. 94
. 95
. 96
. 101
. 103

Chapter 7 The Excited State: Linear Response . . . . . . . . . . . . . . . .
7.1 Time-Dependent Density Functional Theory . . . . . . . . . . . . . . . . .
7.1.1 The Runge-Gross Theorem . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.3 Adiabatic Local Density Approximation (ALDA) . . . . . . . . . .
7.2 Linear Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Frequency Domain Representation . . . . . . . . . . . . . . . . . . .
7.2.2 Spectral (“Lehmann”) Representation using Kohn-Sham Orbitals
7.2.2.1 Bra-kets Involving n(x): Two-electron Systems . . . . . .
7.2.2.2 Bra-kets Involving n(x): N -Electron Systems . . . . . . .
7.2.3 Representation within Jellium Model . . . . . . . . . . . . . . . . .
7.3 Linear Response Density for Jellium Surface . . . . . . . . . . . . . . . . .
7.4 The Driving Function ξ1 (x; ω) . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Contributions from the
√ Bulk Metal . . . . . . . . . . . . . . . . . .
7.4.1.1 Case I: k ≥ √2ω . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1.2 Case II: k < 2ω . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Contributions from the Surface . . . . . . . . . . . . . . . . . . . . .
7.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Asymptotic Behavior of φ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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vii

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58
58
58
59
60
62
63
64
65
67
67
70
71
74
78
78
78
79
79
80
82
83
89

104
104
104
106
109
111
111
113
114
116
117
119
124
125
127
128
129
130
131

7.6
7.7

7.8
7.9

7.5.1 Computing ξ2 (x; ω) . . . . . . . . . . . . . . . .
Putting It Together . . . . . . . . . . . . . . . . . . . .
Numerical Implementation . . . . . . . . . . . . . . . .
7.7.1 The Nystr¨om Method for Integral Equations .
7.7.2 Formulation with Simpson’s Rule . . . . . . . .
7.7.3 Computing G(x, y; εk + ω) and G(x, y; εk − ω)
7.7.4 Computing I1 (x; εk ± ω) and I2 (x; εk ± ω) . .
Computational Results . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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133
134
135
135
136
139
140
141
146

Chapter 8 Second Harmonic Generation . . . . . . .
8.1 The Mathematics of Second Harmonic Generation
8.1.1 The Classical Model . . . . . . . . . . . . .
8.2 The Case for Density Functional Theory . . . . . .
8.2.1 The Complete TD-DFT Approach . . . . .
8.3 Intensity Formula . . . . . . . . . . . . . . . . . . .
8.4 Dynamical Force Sum Rules . . . . . . . . . . . . .
8.5 The Hybrid TD-DFT Approach . . . . . . . . . . .
8.5.1 a(ω) in the Hybrid TD-DFT Approach . .
8.5.2 Comparison with Full Solution . . . . . . .
8.6 Computational Results . . . . . . . . . . . . . . . .
8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

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147
148
149
151
152
153
155
155
157
158
159
160

APPENDICES . . . . . . . . . . . . .
Appendix A Variational Calculus .
Appendix B Solid State Constants .
Appendix C The Spline Class . . .

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161
162
166
167

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

viii

LIST OF TABLES
Here
Table 1.1

Unitary Quantities in Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Table 1.2

Important SI Quantities in Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Table 3.1

Parameters for Perdew-Wang Exchange Kernel . . . . . . . . . . . . . . . . . . . . . . .

32

Table 4.1

TFDW Decay Rate and Frequency of Oscillation . . . . . . . . . . . . . . . . . . . . .

49

Table 5.1

Ground State Energies for Jellium Surfaces . . . . . . . . . . . . . . . . . . . . . . . . .

60

Table 5.2

Gaussian Quadrature Nodes and Weights . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Table 5.3

Error Per Step of SCF Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Table 5.4

Adaptive Splines in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Table 7.1

Values of kc and ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Table 7.2

SI Wavelengths and Frequencies in Atomic Units . . . . . . . . . . . . . . . . . . . . . 141

Table 7.3

Adaptive Splines in Action, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Table 7.4

Computation Time for Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Table 8.1

Computed Values for a(ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Table B.1

Values of rs , kF , and n
¯ for Common Materials . . . . . . . . . . . . . . . . . . . . . . . 166

ix

LIST OF FIGURES
Here
Figure 2.1

Placement of Spline Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Figure 2.2

Adaptive Evaluation Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.3

Adaptively Addable Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 3.1

DFT Publications Per Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Figure 3.2

Periodic Table with rs Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Figure 4.1

Reduction of Dimension in Semi-infinite Jellium . . . . . . . . . . . . . . . . . . . . . .

38

Figure 4.2

∆(¯
n) for Relevant rs Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Figure 4.4

Orbital-free Density for rs = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Highlight of Density Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Figure 4.5

Orbital-free Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Figure 4.6

Spline Distribution for OF-DFT Computation . . . . . . . . . . . . . . . . . . . . . . .

56

Figure 5.1

Placement of N electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 5.2

Fermi-Dirac Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Figure 5.3

Failure of Standard Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Figure 5.4

Self-consistent Field Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Figure 5.5

Complete SCF Iteration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Figure 4.3

Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12

Ground State Density n0 for rs = 3 and ς = 0 . . . . . . . . . . . . . . . . . . . . . . . .

Density After Each Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electrostatic Potential φ for rs = 3 and ς = 0 . . . . . . . . . . . . . . . . . . . . . . . .

Wavefunctions ψk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Screened Density nς for rs = 3 and ς = 5 × 10−4 . . . . . . . . . . . . . . . . . . . . . . .

Electrostatic Potential φ for rs = 3 and ς = 5 × 10−4 . . . . . . . . . . . . . . . . . . . .
Difference between n0 and nς for rs = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

84
85
86
86
87
87
88

Figure 6.1

Ground State Density n0 for rs = 2 and ς = 0 . . . . . . . . . . . . . . . . . . . . . . . .

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

95

Figure 6.2

Detail of Kohn-Sham Potential V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Figure 5.13

Kohn-Sham Potential V

88

Figure 7.2

ξ1 (x; ω) for Several ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 7.3

Spectacular Failure of Direct Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 7.1

Figure 7.4

Incorrect ξ1 (x; ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Linear Response Function χ1 (x, y; ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
x

Figure 7.5
Figure 7.6

Linear Response for 1064 nm Incident Photons . . . . . . . . . . . . . . . . . . . . . . . 142
Linear Response at Various Frequencies, rs = 3 . . . . . . . . . . . . . . . . . . . . . . . 143

Figure 7.7

Comparison of Low-frequency Responses . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Figure 7.8

Comparison of Low-frequency Responses, Part II . . . . . . . . . . . . . . . . . . . . . 144

Figure 8.1

“Demonstration” of SHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 8.2

Examples of Metal Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 8.3

SHG in Dielectrics vs. Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 8.4

Complete TD-DFT Approach to SHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure 8.5

Geometry of Incidence and Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 8.6

Hybrid TD-DFT Approach to SHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 8.7

Nonlinear Response from Full TD-DFT Approach . . . . . . . . . . . . . . . . . . . . . 158

Figure 8.8

Nonlinear Response from Hybrid TD-DFT Approach . . . . . . . . . . . . . . . . . . . 159

Figure C.1

Combining Spline Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xi

LIST OF ALGORITHMS
Algorithm 2.1 Spline Collocation Method for Nonlinear BVPs . . . . . . . . . . . . . . . . . . . . .

15

Algorithm 2.2 Spline Collocation Method for Linear ODEs . . . . . . . . . . . . . . . . . . . . . . .

17

Algorithm 2.3 Adaptive Spline Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Algorithm 5.1 Complete SCF Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Algorithm 5.2 Adjustment of γk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Algorithm 5.3 Determining the Cut-off Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

xii

KEY TO ABBREVIATIONS AND SYMBOLS
Algorithm
Symbol

Meaning

Detailed in

BVP

Boundary value problem

Chapter 2

DFT

Density functional theory

Chapter 3

i.s.d

In the sense of distributions

Section 1.3

Kohn-Sham density functional theory

Chapter 5

LDA

Local density approximation

Section 3.4.1

LRT

Linear response theory

Section 7.1.2

ODE

Ordinary differential equation

Chapter 2

Orbital-free density functional theory

Chapter 4

SCF

Self-consistent field iteration

Section 5.6

SHG

Second harmonic generation

Chapter 8

Time-dependent density functional theory

Section 7.1

Thomas-Fermi-Dirac-von Weizs¨
acker

Section 4.2

Rd

Set of d-tuples of real numbers

Standard

N

Set of natural numbers

Standard

C

Set of complex numbers

Standard

iR

Set of purely imaginary numbers

Standard

Set of k-times continuously differentiable functions on A

Standard

Set of functions that vanish at infinity

Standard

Set of infinitely-differentiable and compactly-supported

Standard

KS-DFT

OF-DFT

TD-DFT
TFDW

C k (A)
C0 (A)

Cc∞ (A)
Lp (A)

Lploc (A)
L∞ (A)
H k (A)

functions
For 1 ≤ p < ∞, the set of p-integrable functions on A

Standard

The set of essentially bounded functions on A

Standard

The set of L2 functions with weak derivatives up to and

Standard

The set of functions in Lp (K) for each compact subset K ⊆ A

Standard

αk

including order k in L (A)

Coefficient of right asymptotic behavior of ψk

Section 5.3.1

γk

Phase shift of left asymptotic behavior of ψk

Section 5.3.2

∆V

Surface barrier potential

Equation (5.3)

δ(x)

Dirac delta distribution

Section 1.3

ε(ω)

Dielectric constant

Equation (7.47)

2

xiii

εk , ε k

Energy of Kohn-Sham orbital corr. to momentum k or k

Equation (5.11)

Correlation kernel for local density approximation

Section 3.4.1

Exchange kernel for homogeneous electron gas

Equation (3.23)

Exchange-correlation kernel under LDA

Section 3.4.1

Positive infinitesimal

End of Section 6.2.4

Unit step function

Section 1.3

θ

Polar angle (angle with respect to surface normal)

Figure 8.5

λ

Chapter 4: strength of von-Weizs¨
acker gradient correction

Equation (4.3)

Chapter 5: strength of cut-off filter

Equation (5.35)

Chapter 7: free parameter for Poisson integral equation

Theorem 7.3

λk

Exponent of right asymptotic behavior of ψk

Section 5.3.1

µ

Exponent of left asymptotic behavior of ϕ1 , ϕ2

Equation (6.3)

µ±k

Exponent of left asymptotic behavior of ϕi (⋅; εk ± ω)

c (n)

x (n)

xc (n)

η

Θ(x)

ν

ξ1 (x; ω)
ξ2 (x; ω)
σ(ω)

σ(A)
σess (A)

Section 7.4.1

Exponent of right asymptotic behavior of ϕ1 , ϕ2

Equation (6.4)

Driving function for n1 integral equation

Equation (7.56)

Additional driving function for n1 integral equation

Equation (7.71)

Zeroth moment of first-order density n1

Equation (7.46)

Spectrum of the operator A

Section 6.2.2

Essential spectrum of the operator A

Section 6.2.2

ς

Boundary condition for electrostatic potential

Equation (5.27)

τa

Tolerance for adaptive spline method

Algorithm 2.3

τsc

Tolerance for SCF convergence

Algorithm 5.1

Chemical exchange potential and Lagrange multiplier

Equation (4.7)

ΦW

Work function

Equation (5.4)

φ(x)

Electrostatic potential for ground state density

Equation (3.14)

Quasi-electrostatic potential of linear response density

Equation (7.52)

Electrostatic potential for linear response density

Equation (7.19)

Linear response external potential

Equation (7.20)

Linear response exchange-correlation under ALDA

Equation (7.21)

Spherical angle (angle with respect to plane of incidence)

Figure 8.5

Basis functions for Green’s function G(x, y; ε)

Equation (6.5)

Linear response function

Section 7.2

σpt (A)

υ

φ1 (x; ω)

φest (x; ω)

φext (x; ω)
φxc (x; ω)
φ

ϕ1 (x), ϕ2 (x)
χ1 (x, y, ω)

Point spectrum of the operator A

xiv

Section 6.2.2

Ψ(x)

Many-body wavefunction

Equation 3.1

Kohn-Sham wavefunction corr. to energy εk , εk , resp.

Equation (5.10)

ω

Angular frequency (incident laser or Fourier domain)

Section 7.3

ωp

Plasmon frequency

Equation (7.48)

a(ω)

Parameter that determines second harmonic intensity

Equation (8.9)

Ap

Amplitude of contribution to SHG from p-polarization

Equation (8.7)

As

Amplitude of contribution to SHG from s-polarization

Equation (8.8)

Time-harmonic magnetic induction

Section 8.1

ψk (x), ψk (x)

B(x, ω)
B ± (x, k; ω)
c

D(x, ω)
D
d⊥ (ω)

E(x, ω)
E0

EF
EH [n]

Eext (⋅)
Exc [n]
fˆ(ξ)

F{f }(ξ)
F{f }(ω)
F (ε)

Fc (x)
fk , fk

fxc (x)

G(x, y; ε)
GL (x − y)
gxc (x)

H(x, ω)
kF
H

Bulk contribution to ξ1 corr. to G(⋅, ⋅; εk ± ω)
Chapter 5: location of cut-off filter

Equation (5.34)

Outside Chapter 5: speed of light

Section 1.1

Time-harmonic electric displacement

Section 8.1

Surface dipole barrier

Equation (5.2)

Effective surface location of normal comp. of E

Equation (7.50)

Time-harmonic electric field

Section 8.1

Magnitude of Eext

Equation (7.44)

Fermi energy

Equation (5.1)

Energy functional corr. to Hartree interaction

Equation (3.9)

Electric field due to external potential

Equation (7.44)

Energy functional corr. to exchange-correlation

Sections 3.3 & 3.4

Ordinary Fourier transform of f

Section 1.3

Same as fˆ above

Section 1.3

Special non-unitary angular Fourier transform of f

Section 1.3

Fermi-Dirac distribution

Equations (5.5) & (5.6)

Cut-off filter for density splitting

Equation (5.35)

Fermi-Dirac occupation factor corr. to k, k, resp.

End of Section 7.2.2

Derivative in n of Vxc

Equation (3.31)

Green’s function for Schr¨
odinger operator H − ε

Green’s function for Laplace operator L

Section 7.4

Equation (6.6)
Sections 3.3 & 7.1

Second derivative in n of Vxc

Equation (3.33)

Time-harmonic magnetic field

Section 8.1

Fermi wavevector

Section 5.1.1

Hamiltonian of Kohn-Sham Schr¨
odinger operator

Equation (6.1)

xv

I1 (x; ε)

Helper integral used in computing ξ1 and ξ2

Equation (7.64)

Helper integral used in computing ξ1 and ξ2

Equation (7.65)

Iω

Intensity of fundamental-frequency radiation

Section 8.3

I2ω

Intensity of second-harmonic radiation

Section 8.3

Time-harmonic current

Section 8.1

Laplace operator scaled by 1/4π (except in Chapter 6)

Section 7.1

Density operator (time-independent)

Equation (7.26)

Density operator in Heisenberg picture

Equation (7.25)

Ground state density

Chapter 3

Linear response density

Section 7.3

Nonlinear response density

Section 8.2.1

Positive background charge in jellium system

Equation (4.1)

Screened density

Section 5.8

Induced portion of ground state density

Equation (5.31)

Quantum contribution to ground state density

Equation (5.37)

Average electron density

Section 3.4.2

I2 (x; ε)

J(x, ω)
L

n(x)
n(x, t)
n0 (x)

n1 (x; ω)
n2 (x; ω)
n+ (x)

nσ (x)

nind (x)
nqu (x)
n
¯

R2

Left endpoint (as −R1 ) of computational domain
Right endpoint of computational domain

Section 7.4

rs

Wigner-Seitz radius

Section 3.4.2

R1

S ± (x, k; ω)

Section 7.4

Surface contribution to ξ1 corr. to G(x, y; εk ± ω)

Sign (signum) function

Section 1.3

True kinetic energy functional (interacting system)

Section 3.3

Kinetic energy functional for noninteracting system

Section 3.3

Kohn-Sham potential for self-contained system

Equation (3.15)

Kohn-Sham potential (general)

Equation (3.11)

LDA exchange-correlation potential (Perdew-Wang)

Equation (3.30)

W

Wronskian of Green’s basis functions ϕ1 and ϕ2

Equation (6.9)

˜
x

Component of x parallel to the surface

Section 5.2

sgn(x)
T [n]

Ts [n]
V (x)

Veff (x)
Vxc (x)

xvi

Section 7.4

Chapter

1

Introduction
The project detailed in this dissertation originated when the author was charged with a very simple-sounding
task by his advisor: investigate modeling and computation for second harmonic generation at metal/dielectric
interfaces. In the course of his preliminary literature review, the author encountered a six-page paper on the
application of density functional theory (DFT) to the phenomenon by Weber and Liebsch [113]. Dissatisfied
with the conventional techniques, the author was intrigued by seemingly simpler and more flexible framework
DFT offered. However, as the length of this document testifies, the DFT approach is anything but simple.
The original motivation for this work, second harmonic generation, does not make an appearance until nearly
one hundred forty pages in. As time progressed, the purpose of the research behind this dissertation gradually
shifted from attempting to describe a single phenomenon to becoming a comprehensive study in electronic
excitations at metal surfaces, as described within the context of DFT. The subject of electronic excitations
is rather broad and can be used to describe a host of physical phenomena, such as surface plasmons, van der
Waals attraction, and nonlocal optics. Such applications are not the focus of this dissertation and second
harmonic generation is offered as merely a novel demonstration of the techniques developed herein.
Trained as a mathematician and an engineer but never a physicist, the author undertook this project with
no previous familiarity with the background material. Therefore, a foremost goal of this dissertation is
self-containment. The author’s intent is that a mathematically advanced reader lacking knowledge of DFT
and solid state physics can acquire all the knowledge necessary to understand the document from within it.
There are two separate yet intertwined primary goals of this dissertation:
1. Theoretical: To present complete and rigorous derivations of all the fundamental equations and
expressions used in the study. Too often, works of physics assert the validity of “mystery equations” as

1

though they were gospel. A major effort of this thesis has been to rigorize important concepts. While
this endeavor is likely of more interest to mathematicians—and this is a thesis of mathematics—it
serves the physics community by preserving the mathematical justification of critical quantities.
2. Computational: To develop fundamentally sound methods of numerical computation for the processes described in the theory sections and to implement them in redistributable, easy-to-use codes.
This thesis is divided into the following chapters:
❼ Chapter 2 details the adaptive spline collocation method for solving ordinary differential equations and

boundary value problems. The method is the workhorse for most of the computations in this thesis,
the only exception being the integral equations of Chapter 7.
❼ Chapter 3 is a background chapter that lays the theoretical foundations of DFT. In many ways, it is

the true “introductory chapter,” at least as the term is commonly applied to research publications. It
introduces and analyzes the behavior of the Perdew-Wang correlation kernel that is used throughout
the thesis. No computations are performed in this chapter. A true expert both on DFT and solid state
physics will find little new information here, but the discussion on the Wigner-Seitz radius in Section
3.4.2 is critical to the understanding of this thesis for those familiar with only DFT.
❼ Chapter 4 first introduces the jellium model in which all computations are framed. A technique

known as orbital-free density functional theory is detailed. The end of the chapter contains sample
computations and the first of many uses the adaptive spline method of Chapter 2 in the thesis. The
biggest contributions of the chapter are the asymptotic results contained in Section 4.4.
❼ Chapter 5 opens with additional solid state physics background needed to formulate Kohn-Sham density

functional theory for jellium surfaces. Derivation of the Kohn-Sham system and boundary conditions is
done from first principles. The proof of Theorem 5.2 may be considered the most substantial theoretical
result of the thesis. The electrostatic potential for semi-infinite jellium systems is notoriously difficult
to compute numerically. Section 5.5 details an algorithm that overcomes these issues due to long-range
Coulomb interaction. A full self-consistent field iteration is then formulated with this fix. The last
section before the Conclusion contains a myriad of numerical results generated by the aforementioned
iteration process in conjunction with the adaptive spline method.
❼ Chapter 6 is the most mathematical chapter. The beginning section concerns the Green’s function

for the Kohn-Sham Schr¨odinger opertor. The remainder is comprised almost entirely of theorems and
proofs for the spectrum of the operator, which culminates in a spectral representation for the Green’s
function. The representation is essential to a derivation in the following chapter.

2

❼ Chapter 7 is the longest, most technical, and most difficult chapter of the entire thesis. It begins with

a complete rigorous derivation of linear response theory and the linear response function for jellium
surfaces. The lengthy demonstration is not found in a single published work and is therefore a valuable
compilation and rigorization of widely used results. A numerical scheme based on the adaptive spline
method for computing the linear response is then introduced. The analytic computations of Section 7.4
have never been demonstrated before and represent a landmark advancement in efficiency and simplicity
for such calculations. Finally, the chapter concludes with numerical results for linear response densities.
❼ Chapter 8 wraps up the thesis with a presentation of a tangible application of the results of the

preceeding chapters. The application is second harmonic generation, a simple process in nonlinear
optics. The chapter briefly discusses the history of the discovery of the process, as well as an overview
of classical modeling techniques before demonstrating how DFT can be used to characterize it.
The remainder of this introductory chapter is dedicated to explicating the conventions and notation used
throughout the document. The author has spent considerable time and effort ensuring that notation is clear,
free of contradictions, and consistent with standard symbology whenever possible. The comprehensive list on
the preceeding pages serves as not only a legend of symbols but also an index of concepts for the document.

1.1

Atomic Units

All computations are done in Hartree atomic units, denoted a.u. In this system, the following four fundamental quantities are taken to have value one:
Symbol
me
e

̵
h

ε0

Physical meaning

Value in SI Units
9.109383 × 10−31 kg

Mass of electron
Charge of electron
Reduced Planck’s constant
Permittivity of free space

1.602177 × 10−19 C

1.054572 × 10−34 J⋅s

8.987552 × 108

kg⋅m3/s2 ⋅C2

Table 1.1: Unitary Quantities in Atomic Units. These values are redefined to be 1 in atomic units.
Another important constant is the speed of light c. In atomic units, c = 137.036.

While virtually every measurable quantity—including time and force—has a corresponding value in atomic
units, of concern to us are only the following:

3

Dimension
Electric field
Energy
Frequency
Intensity
Length

Value in SI Units

Comments

5.1422065 × 1011 V/m

One a.u. is strength of field between e− and nucleus

4.359744 × 10−18 J

4.134137 × 10

16

Hz

Denoted with “a.u.” (no unique name)

3.509455 × 1020 W/m2
5.291772 × 10

−11

One Hartree, denoted Eh ; equal to 27.211396 eV

m

Measured at peak; also called electric flux density
One Bohr, denoted a0

Table 1.2: Important SI Quantities in Atomic Units. This table lists correspondence of 1 a.u. to SI
units for quantities that will appear in this thesis.
Aside from energy and length, none of the above has a name in atomic units; the above table gives the value
of 1 a.u. in SI units. Otherwise, employment of these units in this thesis will be transparent to the reader.

1.2
1.2.1

Conventions
Asymptotic Behavior

A function ψ is said to behave asymptotically like g as x → ∞ and is denoted ψ ∼ g if
lim (ψ(x) − g(x)) = 0

x→∞

1.2.2

Complex Numbers and Conjugation

i always denotes the imaginary unit i =

√

−1. It is never used as an index except for iterations, when it will

always appear as superscript (i). Complex conjugation is denoted by a superscipt ∗, so that
z = a + bi ⇒ z ∗ = a − bi

Hermitian adjoints do not appear in this thesis save for very briefly in Section 6.2.3, so no confusion can occur
there. To avoid confusion with the critical parameter n
¯ , we will not use an overbar to indicate conjugation.

1.2.3

Defined Quantities

When quantities are equated by definition, ≜ is used to indicate the assignment.

4

1.2.4

Function Notation

Within function notation, semicolons separate independent variables from parameters. In f (x, y; ω), x and

y are the independent variables and ω is an input parameter that determines some properties of f . ω is
allowed to be changed but not vary freely: if f (x, y; ω) = x + y + ω 2 , then f (x, y; 3ω) = x + y + 9ω 2 .

1.2.5

Norm, Absolute Value, and Inner Product

When used on a vector quantity, ∣ ⋅ ∣ denotes the standard Euclidean norm: ∣x∣ = ∑ x2k . Per physics con2

vention, we will frequently write x2 for ∣x∣ . For scalar quantities, it denotes absolute value. The norm of a
2

functional space H is denoted ∣∣ ⋅ ∣∣H ; if H is a Hilbert space, its inner product is ⟨ ⋅ , ⋅ ⟩H .

1.2.6

Superscripts

In an iterative process, superscripts in parentheses indicate the iteration number from which a quantity
comes. For example, if the quantity x is computed via an iterative process, then x(5) refers to the result

after five such steps. The same notation is used for derivatives or order higher than three, below which
primes are used, but the distinction should be clear from context.

1.2.7

Theorems and Lemmata

A “lemma” refers to a result used to prove another result; they are generally not referenced outside the
section in which they appear. A “theorem” signifies a major result; frequently they are recalled multiple
times, sometimes many pages after their appearance.

1.2.8

Vector Quantities

As per standard, vector quantities are denoted with boldface type.

1.3
1.3.1

Special Functions
Dirac Delta

δ(x) denotes the Dirac delta distribution. For f ∈ C(R),
∫

+∞

−∞

f (x)δ(x − a) dx = f (a)

5

Likely to accompany usage of δ is the abbreviation “i.s.d,” which stands for “in the sense of distributions.”
Because of its ubiquity in this document, δ never refers to anything else, including the Kronecker delta.

1.3.2

Fourier Transforms

The preferred Fourier transform is the “ordinary Fourier transform”:
F{f }(ξ) = fˆ(ξ) = ∫

+∞

−∞

f (x)e−2πixξ dx

This definition of the Fourier transform is technically valid only for f ∈ L2 (R) (via limiting argument), but

we will often take Fourier transforms as (tempered) distributions for f ∈ L2loc (R). In such a case, the choice of

normalization is taken to be the above and the transform is understood to mean the (tempered) distribution
fˆ that satisfies the Plancherel identity

∫

+∞

−∞

⟨fˆ, ϕ⟩S(R) = ⟨f, ϕ⟩
ˆ S(R)

fˆ(x)ϕ(x) dx = ∫

+∞

−∞

f (x)ϕ(x)
ˆ
dx

for all ϕ ∈ S(R), the set of Schwarz functions. The integral on the right is understood i.s.d and is more

properly integration against a general measure. Fourier transforms of tempered distributions may be differentiated and all convolution formulas and identities hold for tempered distributions.
We will occasionally employ a non-unitary, angular frequency Fourier transform:
F{f }(ω) = ∫

+∞

−∞

f (x)eiωx dx

Take careful note of the positive sign in the exponential. This sign is used to be consistent with sign
convention of time-harmonic fields. The change in sign has the effect of conjugating transforms of familiar
functions, as the quantities we will transform will be real valued. The notation fˆ is not used to denote this
transform so that there is no confusion with the ordinary transform.

1.3.3

Step and Sign Functions

Θ(x) denotes the unit step function:

⎧
⎪
⎪
⎪
⎪ 0
Θ(x) = ⎨
⎪
⎪
⎪
1
⎪
⎩
6

x<0
x≥0

sgn(x) denotes the sign (or signum) function:
⎧
⎪
⎪
⎪
⎪ -1
sgn(x) = ⎨
⎪
⎪
⎪
1
⎪
⎩

x<0
x≥0

The values for both of these functions at x = 0 often differ in other sources, but the specific choice of this
value is not important in this thesis.

1.4

Dirac Notation

On occasion, mostly notably in Chapters 6 and 7, we employ the Dirac notation commonplace in quantum
mechanics. Let H be a Hilbert space. An element ψ ∈ H is called a ket vector or just a “ket” and is denoted

∣ ψ⟩. An element of the continuous dual space ϕ ∈ H ∗ is called a bra vector, or a “bra,” and is denoted ⟨ϕ ∣.

By the Riesz Representation Theorem, H ∗ is identified with H up to isometric isomorphism. If ϕ∗ ∈ H ∗ ,
then there is an element ϕ ∈ H such that

ϕ∗ (ψ) = ⟨ψ, ϕ⟩H

Therein lies the motivation for Dirac notation: rather than work with elements from the dual space, which
are functionals, we may take the inner product with the corresponding element from the main space. The
action of the functional described by the bra ⟨ϕ ∣ is written ⟨ϕ ∣ ψ⟩ and is expressed in standard math notation
⟨ϕ ∣ ψ⟩ = ⟨ψ, ϕ⟩H

If A is an operator on a subspace of H , then the matrix element of A corresponding to ϕ and ψ is given by
⟨ϕ ∣ A ∣ ψ⟩ = ⟨Aψ, ϕ⟩H

Dirac notation is intuitive for physicists because it allows them to bypass the heavy mathematical machinery
of dual spaces and easily manipulate elements of abstract Hilbert spaces as though they were vectors in Cd .
In particular, outer products become incredibly easy to represent and manipulate. Mathematically, the outer
product of ψ ∈ H and ϕ ∈ H ∗ is the operator Pϕ,ψ (φ) = ⟨φ, ϕ⟩ψ. In Dirac notation, this operator is easily
represented as Pϕ,ψ = ∣ ψ⟩⟨ϕ ∣ so that its action is obtained merely by writing symbols next to each other:
Pϕ,ψ ∣ φ⟩ = ∣ ψ⟩⟨ϕ ∣∣ φ⟩ = ∣ ψ⟩⟨ϕ ∣ φ⟩
7

⇐⇒

Pϕ,ψ (φ) = ⟨φ, ϕ⟩ψ

If ϕn is a complete orthonormal sequence, then the Fourier series representation of ψ is
ψ = ∑ ⟨ψ, ϕn ⟩ϕn
n

or in Dirac notation

⟨ψ ∣ = ∑ ∣ ϕn ⟩⟨ϕn ∣ ψ⟩
n

which allows us to write the identity operator compactly as as
ˆ
1 = ∑ ∣ ϕn ⟩⟨ϕn ∣
n

The action of the right side is immediately clear just by juxtaposing symbols. This representation is called
a completeness relation. Mathematics notation does not allow for such an elegant representation if H ≠ Cd .

It is perhaps the ease of writing completeness relations that is the most attractive feature of Dirac notation.
To facilitate seemless transition from Dirac notation back to mathematics notation, if H = L2 (Rd ), then
⟨ϕ ∣ ψ⟩ = ∫

⟨ϕ ∣ A ∣ ψ⟩ = ∫

Rd
Rd

ϕ∗ (x)ψ(x) dx

ϕ∗ (x)Aψ(x) dx

8

Chapter

2

Spline Collocation Method for
Differential Equations
Before diving into the main content of this thesis, we open by presenting a method for solving ordinary
differential equations that forms the basis for virtually all of the numerical computations of this work.

2.1

Overview

Definition 2.1 (Spline). Let a = x1 < x2 < ⋯ < xN +1 = b and S ∶ [a, b] → R be a piecewise-defined function,
⎧
⎪
⎪
s1 (x)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ s2 (x)
S(x) = ⎨
⎪
⎪
⎪
⋮
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ sN (x)

x1 ≤ x < x2

x2 ≤ x < x3
⋮

xN ≤ x ≤ xN +1

S is called a spline of degree m if each sk is a polynomial of degree m and S ∈ C m−1 [a, b].

While the definition calls S the spline, in a form of terminology overloading, we also refer to the pieces sk
as “splines.” Because polynomials are C ∞ , the only potential impediments to the global m − 1 continuous
differentiability of S occur at the transition points xk , called “nodes” or “knots.” In particular, the continuity
requirement means that we must have

lim

x→xk+1

∂p
∂p
[s
(x)]
=
lim
[sk+1 (x)]
k
x→xk+1 ∂xp
∂xp

9

for all 0 ≤ p ≤ m − 1 and 1 ≤ k ≤ N − 1.

Splines are most often used in the context of data interpolation and curve fitting, particularly in image
processing, but following the work of Albasiny and Hoskins [1], they can also be used to solve differential
equations and boundary value problems. Because an m degree spline can be written in the form
sk (x) = am xm + am−1 xm−1 + ⋯ + a1 x + a0
the spline collocation method for differential equations is nothing more than a search for coefficients. When
used for curve fitting, the spline interpolates a set of given values. When used for solving differential
equations, the spline interpolates the differential equation itself.
The specifics of the method are best explained by illustrative example. Consider the boundary value problem
u′′ (x) = f (x, u(x), u′ (x))
u(a) = α, u(b) = β

We first discretize the interval [a, b] into the mesh a = x1 < x2 < ⋯ < xN < xN +1 = b, which need not be
uniformly spaced. For this second-order equation, it is natural to use cubic splines to seek a C 2 solution.
Each spline sk must satisfy the following conditions:

DE conditions ∶
C 0 conditions ∶

C 1 conditions ∶

Boundary conditions ∶

s′′k (xk ) = f (xk , sk (xk ), s′k (xk ))

s′′k (xk+1 ) = f (xk+1 , sk (xk+1 ), s′k (xk+1 ))
sk (xk+1 ) = sk+1 (xk+1 )

s′k (xk+1 ) = s′k+1 (xk+1 )
s1 (x1 ) = α

1≤k≤N

1≤k≤N

1≤k ≤N −1

1≤k ≤N −1

sN (xN +1 ) = β

There are N splines in total, giving 4N total coefficients. The above gives 2N DE conditions, N − 1 C 0

conditions, and N − 1 C 1 conditions. The two boundary conditions complete the set of 4N equations. Figure

2.1 below provides a visual illustration of the placement of the conditions at each node.

10

Spline

x1

Conditions DE

s1

x2
DE x2
C 0, C 1

s2

x3
DE x2
C 0, C 1

s3

x4

s4

DE x2
C 0, C 1

x5

s5

DE x2
C 0, C 1

x6

s6

DE x2
C 0, C 1

x7

s7

DE x2
C 0, C 1

x8

Node

DE

Figure 2.1: Placement of Spline Conditions. Example mesh with N = 7 splines. Each interval is
labeled with the spline defined (only) there; beneath each node are the conditions that must be satisfied
there. Boundary conditions are omitted because these can be placed anywhere.

We note that there is no need to enforce continuity conditions corresponding to the order of the differential
equation. Continuing the example, at xk+1 the DE conditions
s′′k (xk+1 ) = f (xk+1 , s(xk+1 ), s′ (xk+1 ))

s′′k+1 (xk+1 ) = f (xk+1 , sk+1 (xk+1 ), s′k+1 (xk+1 ))

imply that s′′k (xk+1 ) = s′′k+1 (xk+1 ). Thus, the DE conditions in conjunction with the C 0 and C 1 conditions
automatically result in a C 2 solution. For an mth order equation, the DE conditions always result in

continuity of the mth derivative, assuming f is continuous. For a quartic solution to this equation, we would
need to write C 3 conditions beyond the previously prescribed C 0 and C 1 and the C 2 we get automatically.
The previous paragraph highlights the advantage of choosing polynomials of one degree higher than the
order of the differential equation, but it is also possible to seek solutions of higher degree. Doing so results
in additional degrees of freedom not covered by just the increased C m conditions, and requires the invention of additional constraints—for example, additional regularity at endpoints, as is common in ordinary
spline interpolation. However, we are always limited by the smoothness of the governing ODE; if f is only
continuous, for instance, it would be unwise to seek quartic solutions to u′′ = f (x, u, u′ ).

There are three particularly attractive benefits to the spline collation method. The first, and perhaps
greatest, boasting point is that the method produces analytic-like solutions that can be evaluated at any
point in the computational domain. This thesis will employ the spline method in a heirarchal process, a
progression that requires the result from A to compute B, B to compute C, and so on. The spline method
allows computation of each step without need to look upward—A can be computed on the best mesh for A
without consideration of what will be needed for B yet compatibilty of meshes will still be guaranteed.
Secondly, the representation of the solution as polynomials allows elementary calculus operations to be
performed easily and accurately. For instance, the definite integral of a spline S over [ξ1 , ξ2 ] ⊂ [xk , xk+1 ] for
11

some k is able to be computed exactly:
∫

ξ2
ξ1

S(x) dx = ∫

ξ2
ξ1

m

p
p
sk (x) dx = ∑ a(k)
p [ξ1 − ξ2 ]
p=1

Integration over intervals not wholly contained in one of the [xk , xk+1 ] are handled by decomposing the

integral into a combination of subintervals that are. Of particular interest is the access to accurate derivative
information, something most methods for differential equations do not provide. We shall use this ability many
times over, most notably in formulating an adaptive version of the method in Section 2.4.
The third benefit is their excellent accuracy, which mirrors the accuracy when splines are used for interpolation, given in a result in Stoer and Bulirsch [103, p.105]:
Theorem 2.1. Suppose that g ∈ C 4 [a, b] and ∣g (4) (x)∣ ≤ L for x ∈ [a, b]. Let a = x1 < ⋯ < xN +1 = b be a
partition of the interval [a, b] and K be a constant such that for k = 1, 2, . . . , N
h
≤K
xk+1 − xk

h = max (xk+1 − xk )
1≤k≤N

If S is the cubic spline function that interpolates the values of the function g at the knots x1 , ⋯, xN +1 and
satisfies S ′ (x) = g ′ (x) at x = a, b, then there exist constants Ck ≤ 2 independent of the partition such that
max ∣g (m) − S (m) ∣ ≤ Ck LK ⋅ h4−m

x∈[a,b]

The corresponding result for the spline method for differential equations is given by Loscalzo and Talbot [70]

Theorem 2.2. Let u be the true solution to the differential equation
u′′ = f (x, u, u′ )

with f ∈ C 2 of all its arguments and S be the cubic spline approximation computed by the spline collocation
method. Then

max ∣u(m) (x) − S (m) (x)∣ = O(h4−m )

x∈[a,b]

for m = 0, 1, 2, 3. At the nodes xk , the derivatives S ′′′ (xk ) are to be interpreted as average values of the left

12

and right limits
S ′′′ (xk ) ≜

1
[ lim S ′′′ (x) + lim+ S ′′′ (x)]
x→xk
2 x→x−k

We have given the accuracy result only for cubic splines; as all differential equations to which we will apply
this method are second-order, we will always use cubic splines. Several of the equations we will solve have a
discontinuous source term. Navigating the discontinuity will require specialized results and, unfortunately,
the bag of tricks is not so deep that enough boundary conditions may be summoned to accomodate high-order
splines. Furthermore, as detailed in [70], splines of higher degree can become unstable as h → 0.

2.2

Application to Nonlinear Equations

To improve numerical conditioning, it is better to write the splines in the shifted form
sk (x) = ak (x − xk )3 + bk (x − xk )2 + ck (x − xk ) + dk

(2.1)

for x ∈ [xk , xk+1 ]. Another advantage of this form is that evaluation at the node xk is trivial.

The approach to applying the spline method to nonlinear equations is first to collect all the necessary
equations as outlined in Figure 2.1. We assume f to be nonlinear in u and u′ , so we will need to employ
a nonlinear solver. The venerable Newton’s Method for systems [103, p.269] is the method of choice. To
implement it, we will also need to find the Jacobian matrix of the collection of equations in terms of all the
coefficients of all the splines; this will require computing derivatives of each of the equations in ak , bk , etc.
We begin with the DE conditions for sk , which we denote Dk1 for xk and Dk2 at xk+1 . (The k on the D
corresponds to the k of sk , not of the spatial coordinate x.) Let hk ≜ hk+1 − hk . By (2.1), we have
sk (xk ) = dk

sk (xk+1 ) = ak h3k + bk h2k + ck hk + dk

s′′k (xk ) = 2bk

s′′k (xk+1 ) = 6ak hk + 2bk

s′k (xk ) = ck

s′k (xk+1 ) = 3ak h2k + 2bk hk + ck

To help simplify the notation, let sk and sk+1 denote the ordered triples
sk ≜ (xk , sk (xk ), s′k (xk ))

= (xk , dk , ck )

sk+1 ≜ (xk+1 , sk (xk+1 ), s′k (xk+1 )) = (xk+1 , ak h3k + bk h2k + ck hk + dk , 3ak h2k + 2bk hk + ck )
13

With this notation, the DE conditions for sk become
Dk0 = 2bk − f (sk )

(2.2)

Dk1 = 6ak hk + 2bk − f (sk+1 )

(2.3)

To compute derivatives in the coefficients, it is easiest to use the chain rule
∂f
∂f ∂u
∂f ∂u′
=
⋅
+ ′⋅
∂ak ∂u ∂ak ∂u ∂ak
∂f
∂f
+ 3h2k ⋅ ′
= h3k ⋅
∂u
∂u

and similarly for bk , ck , and dk . In the above evaluation, we have sk = u but written partials of f in u to

keep the notation cleaner. We can then write
∂Dk0
∂ak
∂Dk0
∂bk
∂Dk0
∂ck
∂Dk0
∂dk

∂Dk1
∂ak
∂Dk1
∂bk
∂Dk1
∂ck
∂Dk1
∂dk

=0
=2
=

∂f
(sk )
∂u′
∂f
=
(sk )
∂u

= 6hk − h3k ⋅

∂f
∂f
(sk+1 ) − 3h2k ⋅ ′ (sk+1 )
∂u
∂u
∂f
∂f
= 2hk − h2k ⋅
(sk+1 ) − 2hk ⋅ ′ (sk+1 )
∂u
∂u
∂f
∂f
= −hk ⋅
(sk+1 ) − ′ (sk+1 )
∂u
∂u
∂f
= − (sk+1 )
∂u

It is convenient to use gradient notation, defining ∇ck = [
derivatives more compactly as

∂
∂
∂
∂
,
,
,
]. We can write the above
∂ak ∂bk ∂ck ∂dk

∂f
(sk ) ⋅ [0, 0, 0, 1]
−
∂u
∂f
(sk+1 ) ⋅ [h3k , h2k , hk , 1] −
∇ck Dk1 = [6hk , 2, 0, 0] −
∂u
∇ck Dk0 = [0, 2, 0, 0]

∂f
(sk ) ⋅ [0, 0, 1, 0]
∂u′
∂f
(sk+1 ) ⋅ [3h2k , 2hk , 1, 0]
∂u′

−

(2.4)
(2.5)

Proceeding just as for the DE conditions, the C 0 and C 1 continuity conditions are
Ck0 = ak h3k + bk h2k + ck hk + dk − dk+1
Ck1 = 3ak h2k + 2bk hk + ck − ck+1

with derivatives
∇ck Ck0 = [h3k , h2k , hk , 1]

∇ck+1 Ck0 = −[0, 0, 0, 1]
14

(2.6)

∇ck Ck1 = [3h2k , 2hk , 1, 0]

∇ck+1 Ck1 = −[0, 0, 1, 0]

(2.7)

We have omitted equations for the boundary conditions. While the model problem has been written with
Dirichlet conditions at both ends, the spline collation method is capable of handling Neumann conditions at
either or both ends. The equations corresponding to the boundary conditions are incredibly simple, so it is
left to the reader to derive them himself if he so desires to see them.
Let c be the vector of coefficients, stacked in descending powers and grouped by k:
c = (a1 , b1 , c1 , d1 , ⋯, aN , ⋯, dN )

T

With the pieces finally in place, we present the complete algorithm below.
Algorithm 2.1. Spline Collocation Method for Nonlinear BVPs.
1. Generate an initial guess for the coefficient vector c(0) . Enter the Newton iteration:
2. For each k, add the contribution to the Jacobian matrix
⎛
⎜
⎜
⎜
⎜
J[index ∶ 4k, index ∶ 4(k + 1)] = ⎜
⎜
⎜
⎜
⎜
⎝

∇ck Dk0
∇ck Dk1
∇ck Ck0

∇ck Ck1

0
0
∇ck+1 Ck0

∇ck+1 Ck1

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

where index = 1 + 4(k − 1) and 0 = [0, 0, 0, 0]. The components are given in (2.4), (2.5), (2.6), and (2.7).

This 4 × 8 block comprises all the nonzero entries in rows index to 4k.

3. For each k, add the contribution to the equation vector

⎛
2bk − f (sk )
⎜
⎜
⎜
6ak hk + 2bk − f (sk+1 )
⎜
F[index ∶ 4(k − 1)] = ⎜
⎜
⎜ ak h3k + bk h2k + ck hk + dk − dk+1
⎜
⎜
⎝
3ak h2k + 2bk hk + ck − ck+1

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

4. Add boundary conditions to the first and/or last rows of J and F, depending on whether the conditions
are imposed on the left or the right.

15

5. Update c by solving
Jc(i+1) = Jc(i) − F
6. Repeat Steps 2 through 5 until
∣∣ c(i+1) − c(i) ∣∣ < τn

where τn is the Newton solve tolerance, typically 10−12 or smaller. Any vector norm on R4N may be
used, although the max norm (

∞

) is the most natural choice.

The matter of generating a suitable starting vector in Step 1 is highly context dependent. When used as
part of an adaptive algorithm, as we shall develop in Section 2.4, the initial guess can come from a solution
computed on the previous coarse grid. Before that, a trial solution may be generated based on intuition
from the underlying physical problem or from something like a linearization of the problem.
Steps 2 and 3 are worded to suggest use of for or while loops, but such an implementation is likely to be
slow. In fact, by exploiting patterns k of the components, the Jacobian matrix and equation vector can be
assembled very quickly. We leave details of that for documentation in source code.

2.3

Application to Linear Equations

We will also apply this method to ordinary differential equations. In this thesis, all such ODEs are secondorder linear equations of the form
u′′ (x) + g(x)u(x) = 0

u ∼ b(x) as x → −∞ or x → −∞
Fortunately, for problems of this form, the spline method is much simpler to implement than for nonlinear
BVPs. For linear equations, obviously there is no need for Newton’s method. By characterizing in terms of
the moments Mk = u′′ (xk ) and applying continuity of the first derivative, we can derive a simple recurrence

relation involving u(xk ), u(xk−1 ), and u(xk−2 ). The two starting values necessary to transform the method

into a linear multi-step method are generated by evaluating the boundary function at the first or last two
grid points. Consequently, there are no matrices to be formed and no equations to be solved.

Denote uk ≜ u(xk ) and gk ≜ g(xk ). The algorithm below has been modified from Albasiny [1] for use on a
nonuniform grid. It is presented with the boundary condition on the left; a boundary condition on the right
proceeds similarly and is explained afterwards.

16

Algorithm 2.2 assumes that a mesh a = x1 < x2 < ⋯ < xN +1 = b has already been chosen and that the domain
is large enough that the asymptotic condition is approximately achieved at the finite endpoint.
Algorithm 2.2. Spline Collocation Method for Linear ODEs.
1. Set u1 = b(x1 ) and u2 = b(x2 ).

2. For each k = 3, 4, . . . , N + 1, set
−1

h2
hk−1
(1 + k−1 gk )
uk =
hk−2
6

3. For each k, set Mk = −gk uk .

[uk−1 (1 +

h2
h2
hk−2
hk−2 hk−1
− ( k−2 +
) gk−1 ) + uk−2 (1 + k−2 gk−2 )]
hk−1
3
3
6

4. The coefficients of the spline (2.1) are given in terms of the Mk by [103, p.98]
ak =

ck =

Mk+1 − Mk
6hk

uk+1 − uk hk (2Mk + Mk+1 )
−
hk
6

bk =

Mk
2

dk = uk

If the boundary condition is instead posed as x → +∞, then set uN = b(xN ) and uN −1 = b(xN −1 ) in Step 1;
in Step 2, proceed over k = N − 2, N − 3, . . . , 1 and replace k by k + 2 in the formula for uk .

2.4

Local Adaptivity

That the spline collocation method can be formulated on unstructured grids can be exploited to transform
the method into a locally adaptive one. In this section, we describe how to do this.
Suppose we have computed a spline solution S on the mesh {x1 , . . . , xN +1 }. By virtue of the DE conditions
imposed at the nodes, S ′′ (xk ) = f (xk , S(xk ), S ′ (xk )) to machine precision. However, inside each interval

[xk , xk+1 ], S is not likely to satisfy the differential equation to such precision. The error estimate of Theorem

2.2 is helpful, but remember that big O notation provides an estimate in the limit h → 0; the constant

multiplying h4 could be quite large, thus overwhelming a finite h.

To ensure that computed solutions are just as accurate inside the mesh as at the nodes, we take advantage
of the spline method’s ability to work on nonuniform grids and add in local adaptivity. At the midpoint
nodes xk+1/2 ≜ 21 (xk + xk+1 ), depicted below in Figure 2.2, consider the residual error:
Ek+1/2 ≜ ∣S ′′ (xk+1/2 ) − f (xk+1/2 , S(xk+1/2 ), S ′ (xk+1/2 ))∣
17

Spline

s1

x1

✩

s2

x2

✩

s3

x3

s4

x4

✩

✩

s5

x5

s6

x6

✩

✩

s7

x7

✩

x8

Node

Figure 2.2: Adaptive Evaluation Points. The red × marks, which correspond to the midpoints
between the existing nodes, are the test points for Algorithm 2.3.
Because the spline solution is an analytic expression known globally, the derivatives appearing in Ek+1/2 can

be computed easily without need of estimation. If Ek+1/2 > τa , refine the mesh by adding the nodes xk+1/3 and
xk+2/3 . This augmentation results in three splines where there had been one, on the intervals [xk , xk+1/3 ],
[xk+1/3 , xk+2/3 ], and [xk+2/3 , xk+1 ]. The process is illustrated in Figure 2.3 below.
Spline

x1

s1
✬

✬

x2

s2
✬

✬

x3

s3
✬

x4

✬

s4
✬

✬

x5

s5
✬

x6

✬

s6
✬

✬

x7

s7
✬

✬

x8

Node

Figure 2.3: Adaptively Addable Points. The green plus marks, which correspond to the one-third
and two-thirds points between the nodes, are the points that may be added via adaptivity, Algorithm 2.3.

Algorithm 2.3. Adaptive Spline Method Choose an adaptive tolerance τa .
1. Generate an uniform grid of N + 1 points xk = a + (k − 1)h where h =

b−a
.
N

2. Execute Algorithm 2.1 or 2.2 to compute the spline solution on this uniform grid.
3. Generate a test grid composed of the midpoints of the current grid
1
xk+1/2 = (xk + xk+1 )
2
4. Compute the residual error at each test point

and set E = max Ek+1/2 .

Ek+1/2 ≜ ∣S ′′ (xk+1/2 ) − f (xk+1/2 , S(xk+1/2 ), S ′ (xk+1/2 ))∣

5. If E < τa , terminate the procedure. Otherwise, for each k such that Ek+1/2 > τk , add the points
1
xk+1/3 = (xk + xk+1 )
3

2
xk+2/3 = (xk + xk+1 )
3

to the existing mesh {x1 , x2 , ⋯, xN +1 }.

18

6. Compute the spline solution on the new grid via Algorithm 2.1 or 2.2 and return to Step 3.

19

Chapter

3

Density Functional Theory
Density functional theory (DFT) is a computational method in quantum mechanics that attempts to characterize the electronic structure of a system of atoms or molecules by investigating its electron density. As the
name implies, the objects of study are functionals—maps from a function space into a scalar field—of the
electron density. DFT has been popular in computational physics since nearly its inception, but it is only in
past twenty years or so that the technique has been considered accurate enough for deployment in quantum
chemistry applications [27, p.252]. Since acceptance in that community, the number of publications about

19,619

16,338

17,909

201
2
201
3

14,431

201
1

14,012

201
0

12,306

10,678

9,842

200
7
200
8
200
9

6,664

8,887

200
5
200
6

5,532

3,240

3,821

200
2
200
3
200
4

2,247

2,735

200
0
200
1

1,840

1,596

1,139

783

199
7
199
8
199
9

594

199

5
199
6

454

368

199
4

359

199

104

199
0
199
1

2
199
3

DFT has exploded from a few hundred per year to nearly 20,000, as shown below in Figure 3.1.

Year
Figure 3.1: DFT Publications Per Year. This chart depicts the number of DFT publications per year
as counted by searching for articles with topic containing “density functional theory” (as string literal) or
“DFT” on the ISI Web of Science (http://apps.webofknowledge.com). At the time of completion of this
thesis in early June, over 7000 articles had already been published on the subject in 2014.

20

This chapter provides an overview of ground state density functional theory. DFT has been adapted for
time-dependent situations, one example of which will be seen in Chapter 7, but the term “density functional
theory” specifically refers only to ground state computations. The intent of the next sections is not to provide
a complete exposition on the subject; for a comprehensive examination from first principles, the reader is
advised to consult one of the many available texts, such as Engel and Dreizler [24], Eschrig [25], or Parr and
Yang [83]. The works of Elliot Lieb, in particular [62] and [64], may be of special interest to mathematicians.

3.1

The Big Idea

A system of N electrons in the ground state is described by the time-independent Schr¨
odinger equation
ˆ
HΨ(x
1 , ⋯, xN ) = EΨ(x1 , ⋯, xN )

ˆ = − 1 ∑ ∇2x + ∑ V (xk ) +
H
2 k=1 k k=1
N

N

∑ W(xk , xk′ )
N

(3.1)

k=1
k<k′

where W quantifies the interaction of electrons with each other. In a theoretical environment, there is no issue
with this equation. Because each xk ∈ R3 , the many-body wavefunction Ψ has domain R3N . Consequently,
if N is larger than three or so, attempting numerical computation of Ψ is a foray into the impossible.

The goal of density functional theory is to resolve this difficulty by working directly with the electron density,
which can be defined in terms of the many-body wavefunction
n(x) = N ∫

R3 ×⋯×R3

∣Ψ(x, x2 , . . . , xN )∣ dx2 ⋯dxN
2

(3.2)

As electrons are fermions, the wavefunction is anti-symmetric in each of its arguments, so it does not matter
over which N − 1 coordinates the integral is taken. Of course, the goal is to avoid the wavefunction entirely,
so the above characterization is not terribly helpful.

3.2

The Hohenberg-Kohn Theorems

Density functional theory can trace its conceptual roots all the way back to 1927 in the works of Llewellyn
Thomas [107] and Enrico Fermi [32, 33, 34], who independently argued that the distribution of electrons of
an atom can be described in a statistical setting. Thomas wrote
Electrons are distributed uniformly in the six-dimensional phase space for the motion of an
electron at the rate of two for each h3 of volume.

21

Without realizing what would follow some forty years later, Thomas established the groundwork for description of systems as electron densities. Assuming uniform distribution of electrons in space, his description is
crude in comparison to the sophistication of the tag team of (3.1) and (3.2). Nevertheless, by considering
regions of electrons in lieu of discrete point charges, this idea represents a breakthrough in concept.
The true genesis of DFT would not come until 1964 in the innoculously titled paper “Inhomogeneous Electron
Gas” by Pierre Hohenberg and Walter Kohn [52], a work that has amassed nearly 29000 citations❸ since
its publication. In a mere seven pages understandable to anyone with only basic knowledge of quantum
mechanics, they spawned an entirely new field that would eventually lead Kohn to the award of the 1998
Nobel Prize in Chemistry. Below are the two theorems that begot density functional theory. Because of the
surprising ease and elegance with which these landmark results are proven, the proofs are included.
Theorem 3.1 (Hohenberg-Kohn I). If two systems of electrons with potentials V1 and V2 generate the
same ground state density, then V1 − V2 ≡ constant.

Proof. Suppose that V1 and V2 generate the same ground state density but with respective wavefunctions
Ψ1 and Ψ2 . If V1 − V2 ≠ constant, then Ψ1 ≠ Ψ2 because they satisfy different Schr¨
odinger equations. Let
ˆ k = − 1 ∇2 + Vk (x) + W(x)
H
2

ˆ k Ψk (x) = Ek Ψk (x), where x ∈ R3N , for k = 1, 2. Because the ground state energy is the minimum
so that H

among all possible wavefunctions that can describe a system, we have
ˆ 2 Ψ2 ⟩ 2 3N < ⟨Ψ1 , H
ˆ 2 Ψ1 ⟩ 2 3N
E2 = ⟨Ψ2 , H
L (R )
L (R )

⇒ E2 < E1 + ∫

R3N

n(x)[V1 (x) − V2 (x)] dx

We can switch the indices and repeat the above to obtain the second inequality
E1 < E2 + ∫

R3N

n(x)[V2 (x) − V1 (x)] dx

Adding the two yields the contradiction
E1 + E2 < E1 + E2
Corollary 3.2. The ground state density alone is sufficient to determine all properties of the system.
❸ All

citation numbers as counted by Google Scholar.

22

∎

Theorem 3.3 (Hohenberg-Kohn II). There exists a universal functional F [n] such that
E[n] = F [n] + ∫

R3

V (x)n(x) dx

which obtains a minimum on the set
N = {n ∶ R3 → R ∣ n(x) ≥ 0 and ∫

R3

n(x) dx = N }

(3.3)

at the ground state density n0 . The value E[n0 ] is the ground state energy. The functional F is universal
in the sense that it is valid for any number of particles and any external potential.

Proof. By (3.2), the density is a functional of the many-body wavefunction. Therefore, define
1
F [n] ≜ ⟨∇Ψ, ∇Ψ⟩L2 (R3N ) + ⟨Ψ, WΨ⟩L2 (R3N )
2

(3.4)

It is clear that F [n] is universal in the claimed sense. For a fixed potential V , we can define
E[n] = F [n] + ∫

R3

V (x)n(x) dx

which obviously corresponds to the ground state energy E if the correct n is input. We now show that E[n]
assumes its minimum at the ground state density n0 . Let Ψ0 be the wavefunction corresponding to n0 and
let n ∈ N correspond to wavefunction Ψ. Then because

1
E[n] = E[Ψ] = ⟨Ψ, V Ψ⟩L2 (R3N ) + ⟨∇Ψ, ∇Ψ⟩L2 (R3N ) + ⟨Ψ, WΨ⟩L2 (R3N )
2

we have E[n0 ] < E[n]. Positivity is automatic from (3.2) and that n integrates to N follows from the unit
normalization of Ψ:
∫

R3

n(x) dx = N ∫

R3

∫

R3 ×⋯×R3

∣Ψ(x, x2 , . . . , xN )∣ dx2 ⋯dxN dx = N ∫
2

∣Ψ(y)∣ dy = N
2

R3N

∎

As Hohenberg and Kohn readily admitted, the universal functional is not easily constructed. It has been
defined in terms of the wavefunction Ψ, which we cannot easily find from a given density—and the whole
point of DFT is to sidestep the wavefunction entirely. The Hohenberg-Kohn Theorems alone are not enough
to escape Schr¨odinger wave theory, even if they do establish the theoretical foundation of DFT. If we seek a
computable theory, it seems there is more work that must be done.

23

3.3

The Kohn-Sham Equations

While the impact of the first DFT paper by Hohenberg and Kohn is nearly immeasurable, it can be argued
that Kohn’s subsequent publication the next year with Lu Jeu Sham [55] is even more monumental. In that
work, Kohn and Sham brought the last piece to the DFT puzzle: a practical method for computation of
ground state densities, without which the Hohenberg-Kohn theorems might have passed into obscurity.
Consider a system of noninteracting electrons so that in (3.1) the Hamiltonian has W ≡ 0. Then if we define

the single-particle wavefunctions ψk to be the solutions to

1
(− ∇2 + V (x)) ψk (x) = εk ψk (x)
2
and put Ψ to be the Slater determinant
ψ1 (x1 )

ψ1 (x2 )

⋯

ψ1 (xN )

ψN (x1 )

ψN (x2 )

⋯

ψN (xN )

ψ2 (x1 )

1
Ψ(x1 , x2 , . . . , xN ) = √ det
N!

⋮

ψ2 (x2 )
⋮

⋯
⋱

ψ2 (xN )
⋮

(3.5)

then this Ψ will be the solution to (3.1) as well with E = ∑ εk . If we suppose that the ψk are orthogonal,

then by (3.2), the density corresponding to this wave function will be
n0 (x) = ∑ ∣ψk (x)∣
N

2

k=1

In the universal functional (3.4), the first quantity represents the kinetic energy
1
T [n] ≜ ⟨∇Ψ, ∇Ψ⟩L2 (R3N )
2

(3.6)

For the noninteracting system with wavefunction (3.5) and orthogonalized ψk , this kinetic energy will be
given exactly by
Ts [n] ≜

1
∑ ⟨∇ψk , ∇ψk ⟩L2 (R3 )
2 k=1
N

(3.7)

The matter of the kinetic energy is precisely the issue that plagues Hohenberg-Kohn theory: T is known only
in terms of the many-body wavefunction Ψ, which we cannot compute. The single-particle wavefunctions
ψk , however, exist on R3 , which is certainly in the realm of feasible computation.

24

Kohn and Sham proposed a method that exploits the computational ease of Ts : instead of the full interacting system, we consider a noninteracting system with the same density. The kinetic energy of the
noninteracting system will be given exactly by (3.7). Of course, the universal functionals for the interacting
and noninteracting systems cannot be same. Kohn and Sham proposed a separation of the form
E[n] = ∫

R3

Vext (x)n(x) dx + Ts [n] + EH [n] + Exc [n]

(3.8)

so that the first two pieces correspond to the noninteracting system. The third term is the classical Hartree
contribution to the energy from the Coulombic interaction of electrons among themselves. It is given by
EH [n] =

1
∫ ∫ GL (x − y)n(x)n(y) dx dy
2 R3 R3

(3.9)

1
where GL denotes the Green’s function for the Laplace operator L ≜ − 4π
∇2 ; in three-dimensions, GL (x−y) =
−1

∣x − y∣ . The remainder of the differences between the interacting and noninteracting system are contained
in Exc ; this somewhat mysterious term, called the exchange-correlation functional, will be explored in great

detail in the next section. To find the minimum of (3.8), we seek the solution to the Euler-Lagrange equation
given by taking the first variation (see Appendix A):
δTs [n]
+ Veff (x) = υ
δn(x)

(3.10)

where υ is the Lagrange multiplier associated with the constraint ∫ n(x) dx = N and Veff is
Veff (x) = Vext (x) + ∫

R3

GL (x − y)n(y) dy +

δExc [n]
δn(x)

(3.11)

While it seems we’ve finally made it, it’s the same old story: we don’t know Ts in terms of n. Fortunately,
the brilliance of Kohn and Sham comes to rescue us from this quagmire in which we have been repeatedly
ensnared: they showed that (3.10) is precisely the equation obtained from the usual density functional theory
when a noninteracting system is subject to the external potential Vext = Veff .

The Kohn-Sham approach to DFT is therefore to solve the celebrated Kohn-Sham equations
1
(− ∇2 + V (x)) ψk (x) = εk ψk (x)
2

25

(3.12)

with V = Veff as in (3.11) and form the density via

n0 (x) = ∑ ∣ψk (x)∣
N

2

(3.13)

k=1

Because the Kohn-Sham energies εk are unknown, (3.12) are eigenvalue problems.
The derivation of the Kohn-Sham equations presented here lacks rigor and is valid only for non-degenerate
ground states—a restriction shared by the Hohenberg-Kohn Theorems—and then only for such densities
that are the ground-state of some noninteracting system. Extension of the result to remove this limitation
has been performed by Levy [59, 60], Levy and Perdew [61], and Lieb [63], among others.

3.3.1

Kohn-Sham Potential in Self-Contained Systems

In (3.11), the “external potential” comes from any force not directly attributable to the electrons themselves.
In a self-contained system, the only such source is the Coulomb repulsion from the positively charged nuclei:
Eext [n] = − ∫

Rd

∫

Rd

GL (x − y)n+ (y)n(x) dy dx

where n+ (x) is the positive charge distribution due to the nuclei. Taking the first variation yields the

potential

Vext (x) = − ∫

Rd

GL (x − y)n+ (y) dy

within Veff , we may fold this contribution into the one from the Hartree electron-electron interaction. Rather
than integrate against the Green’s function, we may solve the corresponding Poisson equation:
∇2 φ(x) = −4π(n(x) − n+ (x))

(3.14)

This φ represents the total electrostatic potential. In conjunction with (3.14), we may write
V (x) = φ(x) +

3.4

δExc [n]
δn(x)

(3.15)

The Exchange-Correlation Functional

The exchange-correlation functional Exc appears nowhere in the pioneering work of Hohenberg and Kohn.
It is not a natural part of DFT but rather an artificial introduction due to Kohn and Sham, in part due
to the “universal functional” Hohenberg and Kohn postulated but could not produce. The purpose of the

26

exchange-correlation functional is to compensate for two defects in the Kohn-Sham formulation:
1. The kinetic energy Ts [n] of the noninteracting system is not the same as that of the true system.

2. The Hartree energy EH is a strictly classical expression. Proper description of the interaction of
electrons among themselves should include quantum-level effects.
As the name implies, there are two pieces to the correction. The exchange part is the energy change that
occurs when spatial coordinates of two electrons are interchanged; this phenomenon is easily quantified.
The more elusive of the two, the correlation portion, attempts to characterize behavior that is lost by
representing the electrons as a density rather than point charges. When one point charge moves, the others
adjust accordingly. Electrons are free to move in small regions without changing the overall density, so the
individuality of the particles is lost. It is this direct influence that “correlation” attempts to recapture.

3.4.1

The Local Density Approximation (LDA)

Consider a homogeneous electron gas (HEG) of uniform density n
¯ . Then if

HEG
n)
xc (¯

quantifies the ground

state energy per electron from exchange and correlation for an HEG, the total energy due to the effect is
HEG
Exc
=∫

Rd

n
¯

HEG
n) dx
xc (¯

(3.16)

An idea also due to Kohn and Sham, the issue of exchange-correlation energy for an inhomogeneous electron
gas can be resolved by replacing the constant density n
¯ in (3.16) by the spatially varying density n(x):
LDA
Exc
[n] = ∫

Rd

n(x)

HEG
xc (n(x)) dx

(3.17)

This approximation is known as the local density approximation (LDA) because Exc depends only on local
coordinates of the density. A more sophisticated description known as the generalized gradient approach
(GGA) includes contribution from the gradient of the density. However, the LDA is sufficient for the
majority of applications and, for this reason, it is the most commonly used exchange-correlation functional.
LDA
The LDA exchange-correlation potential Vxc
is the first variation of the exchange-correlation functional.

Following standard variational calculus, it is given by the expression
LDA
Vxc
(x) ≜

=

HEG
δExc
[n]
∂
[n
=
δn
∂n

HEG
xc (n)]

∂ HEG
HEG
xc
(n(x))
xc (n(x)) + n(x)
∂n

27

(3.18)

Because we work solely in the LDA in this thesis, we will drop the superscript LDA from all terms from here
forward. Furthermore, because all exchange-correlation expressions technically describe only the homogeneous electron gas, we drop the superscript HEG as well. Due to its appearance in a functional defined by
integration, we dub

xc (n)

the exchange-correlation kernel.

Per the introduction to this section,

xc

can be separated into exchange and correlation portions so that
xc (n)

=

x (n) + c (n)

(3.19)

In the coming subsections, we will explore expressions for the exchange kernel

3.4.2

x

and correlation kernel

c.

The Wigner-Seitz Radius rs

We now introduce two parameters: the average electron density n
¯ and the dimensionless Wigner-Seitz radius
rs . The two are related by

1

3 3
rs = (
)
4π¯
n

⇐⇒

n
¯=

3
4πrs3

(3.20)

Physically, rs is the inverse ratio of the Bohr radius and the radius of the sphere occupied by a single “average
electron.” Values for elemental metals are known and contained in the periodic table on the next page.
It is more convenient to write the correlation kernel in terms of rs instead of the density n. In doing so,
rs becomes a spatially varying function of x, as opposed to its usual usage as a fixed constant set by the
material under study. In this usage, for each x, rs (x) and n(x) are related just as in (3.20).

In what might be considered an abuse of notation, in later chapters we will use rs as a constant. We will

commonly write Vxc (rs ), often right next to Vxc (x) = Vxc (n0 (x)). When evaluating, the meaning of the
input should be placed in context and adjusted appropriately before insertion into the formula.

3.4.3

Exchange Kernel

The exact exchange energy for a homogeneous electron gas can be obtained using a perturbation approach
within Hartree-Fock theory [25, Ch.7.1]:
1

3 3 3
4
Ex [n] = − ( ) ∫ n /3 (x) dx
d
4 π
R

28

(3.21)

1

2

H

He
Transition Metal

3

3.25

4

Li

Be

Lithium

Beryllium

11

3.93

12

Number

Lanthanide/Actinide

1.88

3

rs

3.93

Na

Ordinary Metal

Name

5

C

13

2.07

Mg

Al

Sodium

Magnesium

Aluminium

4.86

20

3.27

21

2.37

22

1.92

23

1.78

24

25

1.86

2.15

26

1.85

27

2.08

28

1.80

29

2.67

30

2.31

31

Ca

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

Ga

Potassium

Calcium

Scandium

Titanium

Vanadium

Chromium

Manganese

Iron

Cobalt

Nickel

Copper

Zinc

Gallium

41

42

43

5.20

38

3.56

39

2.61

40

2.11

1.80

1.61

1.76

44

1.76

45

1.95

46

1.99

47

3.02

48

2.59

14

49

32

33

Ge

2.41

50

51

2.40

2.53

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Rubidium

Strontium

Yttrium

Zirconium

Niobium

Molybden.

Technetium

Ruthenium

Rhodium

Palladium

Silver

Cadmium

Indium

Tin

Antimony

55

56

74

75

76

2.64

72

2.08

73

1.80

1.62

1.58

1.56

77

1.77

78

2.01

79

3.01

80

2.71

81

2.48

82

2.30

83

2.67

Ba

La

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Caesium

Barium

Lanthanum

Halfnium

Tantalum

Tungsten

Rhenium

Osmium

Iridium

Platinum

Gold

Mercury

Thallium

Lead

Bismuth

87

88

Fr

Ra

89

36

Br

52

Kr

53

Te

Cs

Ar

35

Se

Zr

18

Cl

34

As

Y
57

Ne

17

S

Sr
3.69

10

F

16

P

Rb
5.63

9

O

15

Si

2.19

K
37

8

N

Sodium

Na
19

7

B

Nonmetal/Unknown rs

2.65

6

Symbol

54

I

84

Xe

85

Po

86

At

Rn

2.72

Ac

58

Actinium

2.64

59

2.64

60

2.64

Ce

Pr

Nd

Cerium

Praseodym.

Neodymium

91

92

90

2.37

2.16

2.03

61

62

Pm
93

1.98

2.61

63

3.38

64

2.61

Eu

Gd

Samarium

Europium

Gadolinium

94

95

96

Sm
2.01

Th

Pa

U

Np

Pu

Thorium

Protactin.

Uranium

Neptunium

Plutonium

Am

Cm

65

2.58

66

2.57

Tb

Dy

Terbium

Dysprosium

97

Bk

98

67

2.56

Ho
Holmium
99

Cf

68

2.54

Er
Erbium
100

Es

Fm

69

2.53

Tm
Thulium
101

Md

70

3.22

Yb
Ytterbium
102

No

71

2.51

Lu
Lutetium
103

Lr

Figure 3.2: Periodic Table with rs Values. All elements considered, we have 1.61 ≤ rs ≤ 5.63. For transition metals, we have 1.61 ≤ rs ≤ 3.01,
with most elements under rs = 2, with the notable exceptions of gold, silver, and copper. Ordinary metals average about rs ≈ 2.4 except for alkali
metals, which all have rs > 3. Lanthanides hover around rs ≈ 2.6, except for europium and ytterbium; the few actinides for which rs is available
typically have rs ≈ 2. This diagram constitutes a derivative work of LATEX code written by Ivan Griffin and posted freely on texample.net and is
fully compliant with the terms and conditions of the LATEX Project Public License version 1.3. All rs values taken from [7, p.89].

29

The exchange kernel is given by the variational derivative of Ex :
1

3 3 3 1/3
x (n) = − ( ) n
4 π
or in terms of rs ,

(3.22)

1

3 9 3 1
)
x (rs ) = − (
4 4π 2
rs

(3.23)

The exchange kernel of this section is the one used in every published work the author of this thesis has ever
encountered; the matter is considered completely resolved. The correlation kernel, however, is trickier and
no perfect solution exists. The next subsections detail three popular expressions for

c.

By no means do just

these comprise an exhaustive complication. There are results due to Perdew and Zunger [86] and Cole and
Perdew [17], certainly as well as others of which this author is unaware.

3.4.4

Correlation Kernel of Wigner

Before the development of modern correlation kernels, the standard workhorse was a kernel that goes all the
way back to 1934 to the work of Wigner [115], a paper that has been cited over 2300 times. The Wigner
exchange kernel is a compact and elegant expression:
Wig
(n)
c

=−

n /3
1.410 + 17.73n1/3
1

(3.24)

Despite predating DFT by thirty years and incorrect behavior in both high- and low-density limits, this
kernel is surprisingly accurate across the range of relevant physical values 1 ≤ rs ≤ 6. For this reason, as well
as the kernel’s simplicity of form, it is still used today, especially in computational demonstration [87]. In

particular, this functional has been employed by authors [15, 87, 113] wishing to compare their work to that
of the past, especially that of Lang and Kohn, who performed many of the initial studies on DFT for solid
state applications [56, 57, 58].
Various authors have made improvements on the Wigner kernel by correcting the poor behavior as n → 0

and n → ∞. Gell-Mann and Brueckner [40] addressed the high-density limit and Nozieres and Pines [80]

crafted an improvement based on interpolation. Because the modern kernels of the coming sections are of
more interest to this thesis, these results are mentioned only in passing.

30

3.4.5

Correlation Kernel of Vosko, Wilk, and Nusair

In formulating the first truly modern exchange kernel, Vosko, Wilk, and Nusair [112] took a fundamentally
different approach. Instead of relying on pure theory, they used Pad´e interpolation of high quality data
from Monte Carlo experiments from both para- and ferro-magnetic electron states to obtain a kernel that
is accurate for not only a range of rs values but also for a range of spin polarizations. For the unpolarized
spin (paramagnetic) case, they wrote
VWN
(rs )
c

⎡
⎢
2b
Q
rs
)+
tan−1 ( √
)
= A⎢⎢ log (
X(rs )
Q
2 rs + b
⎢
⎣
√
⎤
⎧
⎫
⎪
⎪
( rs − x0 )2
bx0 ⎪
2(2x0 + b)
Q
⎪⎥⎥
−1
⎨
log
(
−
)
+
tan
(
)
⎬
√
⎥
⎪
X(x20 ) ⎪
X(rs )
Q
2 rs + b ⎪
⎪
⎩
⎭⎥⎦

√
√
where X(rs ) = rs + b rs + c and Q = 4c − b2 . The reader interested in the precise values of the constants
b,c, and x0 is directed to consult the original work for their values.

The exchange kernel of Vosko, Wilk, and Nusair is widely used and is generally the kernel of choice when the
Wigner form is not used. The original paper [112] has been cited almost 14000 times; any attempt to compile
a coherent list of example usage would be an exercise in futility. While

VWN
c

works well for ground-state

only computations, the unwieldiness of the formula in regards to differentiation makes it cumbersome to use
in excited-state computations. Consequently,

3.4.6

VWN
c

is not the kernel of choice of this thesis.

Correlation Kernel of Perdew and Wang

Following the lead of Vosko, Wilk, and Nusair, Perdew and Wang [85] produced the following kernel from
Monte Carlo data from Ceperley and Alder [14]:

PW
c

⎛
⎞
1
⎟
= −2A(1 + α1 rs ) log ⎜1 +
3/2
1/2
2A (β1 rs + β2 rs + β3 rs + β4 rsp+1 ) ⎠
⎝

(3.25)

Like the trio before them, Perdew and Wang’s expression can be adapted for different spin polarizations,
although such generality is unnecessary here. They provided two sets of values for the constants p, A, α1 ,
and β1 ∼ β4 , one in the RPA❸ and one “beyond RPA.” These values are presented in Table 3.1 below.

By standing on the shoulders of giants, Perdew and Wang were able to improve upon the already excellent
accuracy of Vosko, Wilk, and Nusair. In addition, their formula is much easier to handle. For these two
❸ In the random phase approximation (RPA), electrons are assumed to respond only to an external potential and a screening
potential. Other effects average out and only the total potential at wavevector k contributes.

31

reasons, this is the kernel with the “beyond RPA” parameters that is used in all computations in this thesis.

RPA
(rs )
c
c (rs )

p➜

A➜

α1

β1

β2

β3

β4

0.75

0.031091

0.082477

5.1486

1.6483

0.23647

0.20614

1.00

0.031091

0.21370

7.5957

3.5876

1.6382

0.49294

Table 3.1: Parameters for Perdew-Wang Exchange Kernel. These are the parameters in c of
(3.25). Parameters marked with ➜ are constrained to exact values from Vosko, Wilk, and Nusair [112].

3.4.7

Analysis of the Perdew-Wang Exchange-Correlation Kernel

Combining (3.25) with (3.23), the complete exchange-correlation kernel of Perdew and Wang is
1
⎛
⎞
3 9 3 1
1
⎜
⎟
)
−
2A(1
+
α
r
)
log
1
+
xc (rs (n(x))) = − (
1
s
1/2
3/2
4 4π 2
rs
2A (β1 rs + β2 rs + β3 rs + β4 rsp+1 ) ⎠
⎝

(3.26)

The kernel itself is not what is needed to perform DFT calculations. What we really need is the exchangecorrelation potential Vxc , given by (3.18). Of use in computing it are the chain rules
1

∂rs
1 3 3 1
4πrs4
=− ( )
=
−
∂n
3 4π
9
n4/3

(3.27)

1

∂ 2 rs 4 3 3 1
64π 2 rs7
=
(
)
=
7
∂n2 9 4π
27
n /3

(3.28)

Employing these rules in (3.18) for the Perdew-Wang kernel, we have
Vxc (rs ) =

xc (rs ) +

= −(

where

3 ∂ xc (rs ) ∂rs
=
4πrs3 ∂rs
∂n

xc (rs ) −

rs ∂ xc (rs )
3 ∂rs

9 3 1
1
2Ars
P ′ (rs )
(1 + α1 rs ) 2
)
− 2A (1 + 23 α1 rs ) log (1 +
)−
2
4π
rs
P (rs )
3
P (rs ) + P (rs )
1

P (rs ) ≜ 2A (β1 rs/2 + β2 rs + β3 rs/2 + β4 rsp+1 )
1

3

3
1
1
1
P ′ (rs ) = 2A ( β1 rs− /2 + β2 + β3 rs/2 + (p + 1)β4 rsp )
2
2

32

(3.29)
(3.30)

′
Necessary for excited state computations is fxc = Vxc
(n), found by the chain rule:
1
⎡
⎢1
1
4A
P ′ (rs )
9 3 4Aα1
⎢
fxc (rs ) = ⎢ 2 ( 2 ) −
log (1 +
)+
3
P (rs )
3 P 2 (rs ) + P (rs )
⎢ rs 4π
⎣

2 ⎤
(2P (rs ) + 1)(P ′ (rs )) ⎞ ⎥ −4πrs4
⎛
2Ars
P ′′ (rs )
⎥⋅(
−
(1 + α1 rs )
−
)
⎥
2
3
9
⎝ P 2 (rs ) + P (rs )
(P 2 (rs ) + P (rs )) ⎠ ⎥⎦

(3.31)

where P and P ′ are as before and

1
3
3
1
P ′′ (rs ) = 2A (− β1 rs− /2 + β3 rs− /2 + p(p + 1)β4 rsp−1 )
4
4

′′
The second derivative gxc = Vxc
(n) will be needed later as well. By differentiating fxc with help from the

chain rule,

′′
gxc (rs ) = Vxc
(n) =

∂ 2 Vxc ∂rs 2 ∂Vxc ∂ 2 rs
(
) +
∂rs2
∂n
∂rs ∂n2

We can assemble the first term by differentiating the quantity in the brackets [

(3.32)
] contained in (3.31) and

using (3.27). The computational burden may be reduced for the second term by recognizing that
∂Vxc
9
=−
fxc (rs )
∂rs
4πrs4
Substituting the second chain rule expression (3.28) into the above, we have
gxc (x) =

∂ 2 Vxc ∂rs 2 16πrs3
(
) −
fxc
∂rs2
∂n
3

We can then compute the missing piece directly, finally obtaining:
1
2
⎡
(2P + 1)(P ′ ) ⎞
⎛ P ′′
16π 2 rs8 ⎢⎢ 2
9 3 4Aα1 P ′
2A
gxc (rs ) =
−
(
) +
+
(1 − 2α1 rs )
−
2
81 ⎢⎢ rs3 4π 2
3 P2 + P
3
⎝P2 + P
(P 2 + P ) ⎠
⎣

P ′ (2P + 1)(2P + P ′′ ) + 2(P ′ )
⎛ P ′′′
2Ars
−
(1 + α1 rs )
+
2
3
⎝P2 + P
(P 2 + P )
2
3 ⎤
2(2P + 1) (P ′ ) ⎞ ⎥ 16πrs3
⎥−
+
fxc
3
3
⎠ ⎥⎥
(P 2 + P )
⎦
3

(3.33)

Because exchange-correlation measures effects due to electrons, as the number of electrons in a region decreases, the magnitude of exchange-correlation should also decrease. Consequently, a “correct” Vxc should
decay to zero as n → 0, or equivalently, as rs → +∞.
33

Theorem 3.4. For the Perdew-Wang Vxc given in (3.30), we have
lim Vxc (rs ) = 0

rs →+∞

Proof. We first return to formal expression (3.29) for Vxc and separate into the exchange and correlation
pieces, as in (3.19):

A quick examination of

Vxc (rs ) = [1 −

x (rs ),

rs ∂
]
3 ∂rs

x (rs ) + c (rs ) −

given in (3.23), reveals that

∂k

x (rs )
∂rsk

rs ∂ c (rs )
3 ∂rs

(3.34)
−(k+1)

is of the form Crs

; accordingly, we

need not perform further asymptotic analysis of the exchange contribution. Instead, we focus on

c (rs )

= −2A(1 + α1 rs ) log (1 +

1
)
P (rs )

(3.35)

∂ c (rs )
1
P ′ (rs )
= −2Aα1 log (1 +
) − 2A(1 + α1 rs ) 2
∂rs
P (rs )
P (rs ) + P (rs )

(3.36)

Let qk (rs ) be a polynomial in rs of degree k ≥ 1. Then we can directly compute
log (1 + P (rs ) )
1
)] = lim
lim [qk (rs ) log (1 +
1
rs →+∞
rs →+∞
P (rs )
q (r )
1

k

s

q 2 (rs )
P ′ (rs )
= lim k′
rs →+∞ q (rs ) P 2 (rs ) + P (rs )
k

Because P is a polynomial-type expression of “degree” p + 1 and qk has degree 1, from the above expression
we have that

lim [qk (rs ) log (1 +

rs →+∞

1
rsp+2k
=0
)] ∼ lim 2p+k+1
rs →+∞ r
P (rs )
s

when p > k − 1. Further, we note that when this condition holds, log (1 +
we turn attention to

lim [qk (rs )

rs →+∞

P ′ (rs )
]
P 2 (rs ) + P (rs )

1
)
P (rs )

vanishes like rsk−1−p . Now

Unlike in the previous case, in which the function whose limit we were computing contained terms with only
positive powers of rs , P ′ contains terms with negative exponents. We need not pay special attention to

these, as they may be split off into terms of the form 1/(rsa P(rs )) with a > 0 and P of all positive exponents;

34

such terms clearly vanish at infinity. Using the ∼ symbol to characterize the slowest rate of decay, we have
lim [qk (rs )

rs →+∞

P ′ (rs )
rsp+k
=0
]
∼
lim
rs →+∞ r 2p+2
P 2 (rs ) + P (rs )
s

(3.37)

whenever p > k − 2. For choices of p given in Table 3.1, per (3.35) and (3.36) and the above results, we have
lim

rs →+∞

c (rs )

=0

lim

rs →+∞

rs ∂ c (rs )
=0
3 ∂rs

Consequently,
lim Vxc (rs ) = 0

rs →+∞

Before we begin to analyze the behavior of fxc , it is convenient to write it in a form similar to (3.34):
fxc (rs ) =

∂
rs ∂
[(1 −
)
∂rs
3 ∂rs

=[

x (rs ) + c (rs ) −

rs ∂ c (rs ) ∂rs
]⋅
3 ∂rs
∂n

2 ∂ x (rs ) rs ∂ 2 x (rs ) 2 ∂ c (rs ) rs ∂ 2 c (rs ) ∂rs
−
+
−
]⋅
3 ∂rs
3
∂rs2
3 ∂rs
3 ∂rs2
∂n

=−

8πrs4 ∂ x (rs ) 4πrs4 ∂ 2 x (rs ) 8πrs4 ∂ c (rs ) 4πrs5 ∂ 2 c (rs )
−
+
+
27
∂rs
27
∂rs2
27
∂rs
27
∂rs2

In the brief discussion that follows (3.34), we stated that

∂k

∎

x (rs )
∂rsk

−(k+1)

∝ rs

(3.38)

; therefore, via the first two terms

of (3.38), it is clear that ∣fxc ∣ → +∞ as rs → +∞. However, this is not the behavior of concern. Whenever
fxc will appear, it will always do so as nfxc . Consequently, it is Theorem 3.5 that really matters.
Theorem 3.5. Using the Perdew-Wang correlation kernel, fxc as defined in (3.31) has
lim n(rs ) fxc (rs ) = 0

rs →+∞

Proof. Using (3.20) and (3.38), we have
n(rs ) fxc (rs ) = −

2rs ∂ x (rs ) rs2 ∂ 2 x (rs ) 2rs ∂ c (rs ) rs2 ∂ 2 c (rs )
+
−
+
9
∂rs
3
∂rs2
9
∂rs
9
∂rs2

The first two terms vanish as rs → +∞ per the discussion before the theorem statement. The proof of
Theorem 3.4 shows that
(3.36) and compute

∂

c (rs )
∂rs

→ 0. Consequently, we need only focus on the last term; we proceed from

2⎤
⎡
(2P (rs ) + 1)(P ′ (rs )) ⎥
⎢
∂ 2 c (rs )
P ′ (rs )
P ′′ (rs )
⎢
⎥
= −4Aα1 2
− 2A(1 + α1 rs ) ⎢ 2
−
⎥
2
∂rs2
P (rs ) + P (rs )
2 (r ) + P (r ))
⎢ P (rs ) + P (rs )
⎥
(P
s
s
⎣
⎦

35

(3.39)

When we compute the limit

rs2 ∂ 2 c (rs )
rs →+∞ 9
∂rs2
lim

the result of (3.37) handles the first term from (3.39), as k = 2 in this case. Next, because P ′′ is polynomial-

type with degree p − 1, we have

lim [qk (rs )

rs →+∞

rsp+k−1
P ′′ (rs )
=0
]
∼
lim
rs →+∞ r 2(p+1)
P 2 (rs ) + P (rs )
s

whenever p > k − 3. Finally, we address the last term:

2⎤
⎡
3p+k+1
(2P (rs ) + 1)(P ′ (rs )) ⎥
⎢
⎥ ∼ lim rs
lim ⎢⎢qk (rs )
=0
⎥
2
rs →+∞ ⎢
2 (r ) + P (r ))
⎥ rs →+∞ rs4(p+1)
(P
s
s
⎣
⎦

again whenever p > k − 3. Combining all these results, we have

rs2 ∂ 2 c (rs )
=0
rs →+∞ 9
∂rs2
lim

That completes the proof of the theorem.

36

∎

Chapter

4

The Ground State: Orbital-Free DFT
4.1

The Jellium Model

In the Drude-Sommerfeld model, also known as the free electron model, the valence of electrons of a crystalline
metallic solid are assumed to be detached from their nuclei and thus free to wander about as gas particles
in a box. An extension of Drude-Sommerfeld, which neglects to include even Coulombic electron-electron
repulsion, the jellium model is the simplest model for delocalized, interacting electrons of a metal solid.
The ansatz that powers the jellium model is that the positive charge, due to the nuclei sans the wandering
valence electrons, is uniformly distributed in space. In the macroscopic view, the symmetric arrangement of
the nuclei in the metal crystal structure creates an environment where positive charges smear out in space.
Ignoring the location of the nuclei and neglecting the structure of the atomic lattice makes jellium especially
suited for studies concerning effects due to the quantum behavior of electrons.
The name “jellium” is attributed to Conyers Herring, co-receipient with Phillip Nozieres of the 1984/5 Wolf
Prize in physics for their “fundamental theory of solids, especially of the behavior of electrons in metals”
[36], who likened the uniform charge distribution to a spreading of “positive electron jelly” on bread [28].
Jellium is well-suited for studying electronic excitations at metal surfaces. Combined with the LDA in a
DFT setting, the use of jellium is attractive for several reasons [67, p.10]:
1. It provides an accurate, self-consistent description of the density distribution in the surface region.
2. Electronic excitations in the bulk are absent for input frequencies below the bulk plasma frequency.
Electronic surface excitations can be examinined in the jellium model without concern that bulk excitations will interfere with and invariably obscure or corrupt the surface excitation processes.

37

3. Since the jellium model is translationally invariant parallel to the surface, the reduction of dimension
it provides makes the computational effort significantly simpler than for three-dimensional systems.
We model a solid as a block that extends infinitely in all but one direction. In atomic units, the unit length is
the Bohr radius and has value 5.291772×10−11 m. As 1 mm of material is 1.89×107 a.u, a macroscopic sample

is, practically speaking, infinite. A true free-space formulation greatly simplifies the problem formulation
and the underlying analysis at the cost of complicating the numerics, as we shall see.
Mathematically, if we assume the metal to lie in the negative half-space {x ∈ R3 ∶ x3 < 0}, the positive
background charge n+ (x) = n+ (x) of the positive nuclei is given by
⎧
⎪
⎪
⎪
¯
⎪ n
n+ (x) = n
¯ Θ(−x) = ⎨
⎪
⎪
⎪
0
⎪
⎩

x≤0

(4.1)

x>0

where n
¯ is the average electron radius of (3.20). This is the reduction of dimension of the third point above.

z

x

x

y
Figure 4.1: Reduction of Dimension in Semi-infinite Jellium. The surface is assumed to extend
infinitely in the y-, z-, and negative x-directions. By assuming uniformity of charge in space, jellium
transforms the three-dimensional block on the right to the one-dimensional line on the left in computations.

The term “bulk” refers to the portion of the metal that is too distant to feel surface effects, typically no
more than a handful of atoms in. Mathematically, this region begins around x ≈ −10 and ends at −∞.

4.1.1

Properties of Exact Electron Density

The exact electron density of a jellium system and its corresponding electrostatic potential (3.14) satisfy
several properties. The first is a theorem due to Budd and Vannimenus [12] that links the electrostatic
potential at the surface to the electron energy in the bulk material.

38

Theorem 4.1 (Budd-Vannimenus). The electrostatic potential φ satisfies
φ(0) − φ(−∞) = n
¯

∂εT
(¯
n)
∂n

where εT is the total energy per electron in the system:
3
2
5
εT (n) = (3π 2 ) /3 n /3 + εxc (n)
5
In the presence of an external electric field, the Budd-Vannimenus Theorem takes a slightly different form.
The generalization below is due to Theophilou [106].
Theorem 4.2 (Budd-Vannimenus in Electric Field). For a semi-infinite jellium surface in the presence
of a strong electric field, the Budd-Vannimenus Theorem takes the form
φ(0) − φ(−∞) = n
¯

∂εT
2πς 2
(¯
n) +
∂n
n
¯

where the strength of the field is E0 = 2πς. This corresponds to the boundary condition φ′ (+∞) = −4πς.
Corollary 4.3. In the local density approximation, the result of Theorem 4.2 becomes
2/3
1
2πς 2
φ(0) − φ(−∞) = (3π 2 n
¯ ) + Vxc (¯
n) − εxc (¯
n) +
5
n
¯

Proof. Under the local density approximation, Vxc (n) =
Vxc (n) = n

⇒n

∂
[nεxc (n)].
∂n

(4.2)

As a result,

∂εxc
+ εxc (n)
∂n

∂εxc
= Vxc (n) − εxc (n)
∂n

After differentiating the first term of εT in n directly and evaluating at n
¯ , we arrive at (4.2).
There is also a result about the value of the density at the surface from Perdew [84].

∎

Theorem 4.4. In the high-density limit, the value of the density at the surface edge results from the ThomasFermi model:

3 /2
n(0) = n
¯( )
5
3

The high-density limit [76] means that rs should be “small.” The limit rs → 0 refers to an extreme high-

density limit, so the high-density limit does not imply rs must be infinitesimal. A close examination of

39

Kohn-Sham densities of Chapter 5 would reveal that Theorem 4.4 is satisfied reasonably well even for rs = 3.
However, the range of validity of this relation is not an investigation point of this study.

It is important to understand that the density obtained from DFT is merely an approximation of the exact
density, and these theorems only hold for the latter. Consequently, we cannot expect computed densities to
satisfy them automatically. Rather, they will be called upon to overcome computational difficulties.

4.2

The Kinetic Energy Functional

The Kohn-Sham approach detailed in Section 3.3 is the most popular method of resolving the conundrum of
kinetic energy. As we shall see in the next chapter, specifically in Section 5.6, it comes at a high computational
price—the equations that construct the density are implicit and must be solved by iterative methods.
An alternative approach that bypasses Schr¨
odinger wavefunctions entirely is to write an explicit functional
for the kinetic energy in terms of the density. This is, of course, no easy task—if it were, there would be
no reason for the Kohn-Sham equations to exist. For a crude solution, we hearken back all the way to the
origins of DFT and write the Thomas-Fermi kinetic energy functional
TTF [n] =

3
2
5
(3π 2 ) /3 ∫ n /3 (x) dx
d
10
R

As one might expect, use of TTF alone does not lead to terribly good results. In the absence of the nonlinear
exchange-correlation term, it is possible to solve the resultant Euler-Lagrange equation for a single-atom
system semi-analytically. It can then be shown that n ∼ ∣x∣

−3/2

as ∣x∣ → 0 so that the density blows up near

the nucleus. This is clearly incorrect—the positive nucleus should repel electrons, not collect them!

The second purpose of Exc was to quantify the non-classical electron-electron interaction. Even with a
perfectly exact kinetic energy, we still have a need for this correction. It seems reasonable that a plausible
revision to the Thomas-Fermi functional is to include the exchange energy (3.21), giving
1

3
3 3 3
2
5
4
TTFD [n] = (3π 2 ) /3 ∫ n /3 (x) dx − ( ) ∫ n /3 (x) dx
d
d
10
4
π
R
R

This functional is called the Thomas-Fermi-Dirac kinetic energy functional, so named because the exchange
energy is actually due to Dirac [20]. While this seems like an improvement, the density under TTFD has the
same problems at the origin as before and further underestimates the kinetic energy [83, p.113].

40

The primary shortcoming of both TTF and TTFD is that they are based on homogeneity of the density; we
recall the quote from Thomas at the beginning of Section 3.2 and remember that the Dirac exchange energy
is for the homogeneous electron gas. An attempt to incorporate inhomogeneous effects first came from von
Weizsacker [111], who suggested a gradient correction to Thomas-Fermi kinetic energy. Application of his
idea within DFT results in the Thomas-Fermi-Dirac-von Weizsacker (TFDW) functional:
√
2
3 3 3
λ
3
2
5
4
TTFDW [n] = (3π 2 ) /3 ∫ n /3 (x) dx + ∫ ∣∇ n(x)∣ dx − ( ) ∫ n /3 (x) dx
d
d
d
10
8 R
4 π
R
R
1

(4.3)

von Weizs¨
acker used the value λ = 1 in his work, but the canonical value has been established to be λ = 91 .

Still, this value is not set in stone; following the lead of Chizmeshya and Zaremba [15], who employed the
TFDW functional for use in second harmonic generation, we will take λ = 14 . Regardless of the value of λ,

the TFDW functional constitutes a monumental improvement over the others of this section. In addition to
providing more accurate kinetic energy estimates, it results in densities that are finite near the nucleus.
The TFDW functional (4.3) incorporates the exchange energy yet ignores the correlation energy. Because
the correlation contribution to Exc attempts to capture important behavior that is still lost in representing
as densities, it is more natural to remove the exchange from the kinetic energy, replacing Ts [n] by
T [n] =

√
2
3
λ
2
5
(3π 2 ) /3 ∫ n /3 (x) dx + ∫ ∣∇ n(x)∣ dx
d
d
10
8 R
R

(4.4)

in (3.8) and including the entirety of Exc .

The functionals here each produce very different mathematical problems of uniqueness, existence, and solution space of minimizers to (3.8). For an extensive compilation of such knowledge, see Lieb [62]. Despite the
document’s age, the author believes its information is still current and its open questions remain so.

4.3

Orbital-Free Density Functional Theory

The kinetic energy functionals of Section 4.2 form the basis for a technique known as orbital-free density
functional theory (OF-DFT). It is so named because it lacks the wavefunctions that correspond to the nonphysical orbitals of the Kohn-Sham equations. The benefit of OF-DFT over Kohn-Sham DFT (KS-DFT) is
the explicit equations it produces; the cost is inaccuracy in kinetic energy.
With (4.4) in hand, formulation of orbital-free density functional theory proceeds in exactly the same manner

41

as the Kohn-Sham approach does. We take the first variation of the energy functional
E[n] = ∫

R3

n+ (x)n(x) dx + T [n] + EH [n] + Exc [n]

to yield the Euler-Lagrange equation for the ground state density n0
δT [n]
+ φ(x) + Vxc (n0 (x)) = υ
δn0 (x)
Veff (x)

This semi-complete form highlights the similarity with (3.10) that appeared in the derivation of the KohnSham equations. The same Veff reappears in the above, in the form (3.15) because the jellium system is a
self-contained one. φ is the total electrostatic potential and satisfies the Poisson equation
∇2 φ(x) = −4π(n0 (x) − n+ (x))
However, in stark contrast to the Kohn-Sham approach, the kinetic energy is known explicitly in terms of
the density, and we don’t need any hocus pocus to find the variation

δT [n]
.
δn0 (x)

Instead, we can directly apply

the formula for the first variation of a functional defined by integration (see Appendix A):
δT [n]
1
λ ∇n0 (x)
λ ∇2 n0 (x)
2/3
2
= (3π 2 ) /3 n0 (x) + (
) −
δn0 (x) 2
8 n0 (x)
4 n0 (x)
2

Consequently, the Euler-Lagrange equation becomes the coupled system

1
λ n′′0 (x)
λ n′ (x)
2/3
2
(3π 2 ) /3 n0 (x) + ( 0
) −
+ φ(x) + Vxc (n0 (x)) − υ = 0
2
8 n0 (x)
4 n0 (x)
2

φ′′ (x) = −4π(n0 (x) − n+ (x))

(4.5)
(4.6)

Because the background charge n+ depends only on the depth into the metal x, the equations reduce to
one-dimension and partial derivatives transform into ordinary ones. Finally, the Lagrange multiplier υ is the
chemical exchange potential [79] and is equal to
1
1
2
υ = (3π 2 n
¯ ) /3 + Vxc (¯
n) = kF2 + Vxc (¯
n)
2
2

(4.7)

Traditionally, this Lagrange multiplier is denoted µ, which is also the symbol ordinarily used for the Fermi
level, which coincides with the Fermi energy

1 2
k
2 F

at zero temperature. To make the distinction between

42

Fermi level and (4.7), which includes an exchange-correlation contribution, υ is used for the multiplier.
The Lagrange multiplier accounts for the integral constraint in the set of minimization (3.3) of the energy
functional, but the positivity one has so far been unenforced. In KS-DFT, this requirement is handled
automatically by (3.13), but in OF-DFT, it is not. To that end, let
n0 (x) = Y 2 (x)

⇒

n′0 = 2Y Y ′

⇒

n′′0 = 2Y Y ′′ + 2(Y ′ )2

which yields the set of equations
1
2
7
Y ′′ = (3π 2 ) /3 λ−1 Y /3 + 2λ−1 Y (Vxc (Y 2 ) + φ − υ)
2
φ′′ = −4π(Y 2 − n+ )

(4.8)
(4.9)

Deep inside the metal, we require that n0 → n
¯ as x → −∞; electrons cannot get infinitely far away from the

surface, so n0 must vanish at +∞. We also normalize so that the electrostatic potential vanishes inside the
metal, or φ(−∞) = 0. These assertions give the set of boundary conditions
Y (−∞) =

√

φ(−∞) = 0

n
¯

Y (+∞) = 0

The last of these corresponds to the charge condition
φ′ (+∞) = −4π ∫

+∞

−∞

φ′ (+∞) = −4πς

(4.10)

(n0 (x) − n+ (x)) dx = −4πς

As will be discussed in the next chapter, ς represents screening charge; for a charge-neutral system, ς = 0.

4.4

Asymptotic Behavior of the Orbital-Free Density

To prove a result about the asymptotic behavior of the density determined by (4.8)–(4.9), we will call upon
the following theorem, a mainstay in theory of control of time-varying systems. The result is necessary
because integrability of f is not sufficient to imply existence of a limit at infinity.
Theorem 4.5 (Barb˘
alat’s Lemma). If f ∶ [a, +∞) → R is uniformly continuous and
lim ∫
t→+∞

t

a

f (s) ds

43

exists and is finite, then f → 0 as t → 0.

∎

Proof. See, for example, Khalil [53, p.323].

Corollary 4.6. If f(t) has a finite limit as t → +∞ and either f ′ is uniformly continuous or f ′′ is bounded,
then f ′ → 0 as t → +∞.

Proof. By the Fundamental Theorem of Calculus,
f (t) = f ′ (a) + ∫

t
a

f ′ (s) ds

If f ′ is uniformly continuous and f has a finite limit as t → +∞, then all the hypothesis of Barb˘
alat’s Lemma
are satisfied and f ′ → 0. If f ′′ is bounded, then f ′ is Lipschitz, and hence, uniformly continuous.

∎

Theorem 4.7. The TFDW OF-DFT density n0 satisfies
lim n′0 (x) = 0

x→−∞

lim n′′0 (x) = 0

x→−∞

Proof. By evaluating (4.8) at the boundary conditions Y (−∞) =

√

n
¯ and φ(−∞) = 0, we see that Y ′′ (−∞) = 0.

Continuity of the right-hand side of (4.8) implies Y ′′ is too, so we have Y ′′ ∈ C0 (−∞, a] for at least a < 0.
Since Y has a finite limit at −∞ by imposition, Y ′ → 0 as x → −∞ by Corollary 4.6. Then because n′0 = 2Y Y ′

and n′′0 = 2Y Y ′′ + 2(Y ′ )2 , we have both n′0 → 0 and n′′0 → 0 as x → −∞.

∎

Theorem 4.7 is nice and ultimately necessary for the next result, but it is not terribly enlightening. What

we really seek are results that characterize the behavior of the density itself. Below are two such results
that give the asymptotic behavior at each infinite endpoint. The first of these was shown by Utreras-Diaz
[109]; the proof here mirrors his except that this one contains considerably more detail and makes a farther
reaching conclusion. In particular, analytic expressions for the decay factor and oscillation period are part
of the theorem statement here, while Utreras-Diaz left them as unknowns.
Theorem 4.8. As x → −∞, the TFDW OF-DFT density n0 exhibits oscillatory exponentially decay:
n0 (x) ∼ n
¯ + Aeθx cos(ζx + γ)

where the amplitude A and phase shift γ are unknown, but
√√
α2 + D 2 + α
θ=
>0
2

ζ=
44

√√

α2 + D 2 − α
2

where α and D are given by
2/3
4 1
α = ( (3π 2 n
¯) + n
¯ fxc (¯
n))
λ 3

D = Im

√

α2 −

64π¯
n
λ

Furthermore, n′0 , φ, φ′ , and φ′′ all decay to zero in the same manner with rate θ and frequency ζ.
Proof. Rather than work with Y, it is more convenient this time to write (4.8) and (4.9) directly in n0 :
n′′0 =

λ (n′0 )2
4n0 1
2/3
2
[ (3π 2 ) /3 n0 +
+ Vxc (n0 ) + φ − υ]
λ 2
8 n0

φ′′ = −4π(n0 − n+ )

(4.11)
(4.12)

Our analysis is based on linearization. Define ρ1 ≜ n0 and ρ2 ≜ n′0 so that (4.11) becomes
ρ′1 = ρ1
ρ′2 =

λ ρ22
4ρ1
2/3
2
[(3π 2 ) /3 ρ1 +
+ Vxc (ρ1 ) + φ − υ] ≜ f (ρ1 , ρ2 , φ, φ′ )
λ
8 ρ1

Because n+ = n
¯ for x < 0, Corollary 4.6 gives that φ′ → 0 as x → −∞. The boundary conditions n0 (−∞) =

n
¯ and φ(−∞) = 0 plus Theorem 4.7 imply that the equilibrium point about which we are linearizing is
(ρ1 , ρ2 , φ, φ′ ) = (¯
n, 0, 0, 0). We first compute the partial of f with respect to ρ1 :

∂f
4 1
4ρ1 1
λ ρ2 2
λ ρ22
2/3
2
2
−1/3
= [ (3π 2 ) /3 ρ1 +
+ Vxc (ρ1 ) + φ − υ] +
[ (3π 2 ) /3 ρ1 − ( ) + fxc (ρ1 )]
∂ρ1 λ 2
8 ρ1
λ 3
8 ρ1

′
where fxc = Vxc
as in (3.31). Now we evaluate:

2/3
4 1
∂f
(¯
n, 0, 0, 0) = [ (3π 2 n
¯) + n
¯ fxc (¯
n)]
∂ρ1
λ 3

wherein we used the value of υ in (4.7). Partials of f with respect to ρ2 and φ are easily seen:
∂f
(¯
n, 0, 0, 0) = 0
∂ρ2

∂f
(¯
n, 0, 0, 0) = 4¯
nλ−1
∂φ

Noting that (4.12) is already linear, after defining φ1 ≜ φ, and φ2 ≜ φ′ , we can cast the linearization of (4.8)

45

and (4.9) into the matrix system
⎛ ρ1 ⎞ ⎛ 0
⎜ ⎟ ⎜
⎜ ⎟ ⎜
⎟ ⎜
∂ ⎜
⎜ ρ2 ⎟ ⎜ α
⎜ ⎟=⎜
⎟ ⎜
∂x ⎜
⎜φ1 ⎟ ⎜ 0
⎜ ⎟ ⎜
⎜ ⎟ ⎜
⎝φ2 ⎠ ⎝−4π

where

α≜

1

0

0 4¯
nλ−1

0

0

0

0

⎛ ρ1 ⎞
0 ⎞ ⎛ ρ1 ⎞
⎟⎜ ⎟
⎜ ⎟
⎟⎜ ⎟
⎜ ⎟
⎜ ρ2 ⎟
⎜ ρ2 ⎟
0⎟
⎟⎜ ⎟
⎜ ⎟
⎟⎜ ⎟ ≜ A⎜ ⎟
⎟⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜φ 1 ⎟
1⎟
⎟ ⎜φ1 ⎟
⎜ ⎟
⎟⎜ ⎟
⎜ ⎟
⎝φ 2 ⎠
0⎠ ⎝φ2 ⎠

(4.13)

4 1
2
( (3π 2 n
¯ ) /3 + n
¯ fxc (¯
n))
λ 3

The solution to (4.13) may be characterized by the eigenvalues of A, which can be computed analytically:
√
1√
2α + 2 α2 − 64π¯
nλ−1
2
√
1√
λ3 =
2α − 2 α2 − 64π¯
nλ−1
2

λ1 =

λ2 = −λ1
λ4 = −λ3

We now need to examine the quantities λ1 and λ3 closer. First, define the function of n
¯
∆(¯
n) ≜ α2 − 64π¯
nλ−1 =

2
16 1
64π¯
n
2
[ (3π 2 n
¯ ) /3 + n
¯ fxc (¯
n)] −
2
λ 3
λ

Plotted below in terms of rs , which from the periodic table in Figure 3.2 we know ranges from 1.61 ≤ rs ≤ 5.63
for elemental metals, we see that ∆(¯
n) < 0 for relevant values of λ in Figure 4.2 below.
∆(rs (¯
n)) for Various Parameters

∆(rs )

0

−10

−20
1

1.5

2

2.5

3

3.5
rs

4

4.5

λ = 1/9
λ = 1/4
λ = 1/2
5

λ = 3/4
λ=1
5.5

6

Figure 4.2: ∆(¯
n) for Relevant rs Values. For physical values of rs and appropriate λ, we have
∆(¯
n) < 0. For λ = 91 , we have ∆ < 0 for rs > 1.9427 or n
¯ = 3.2560 × 10−2 . For λ = 41 , ∆ crosses zero at
rs = 1.2380, well below the lowest possible physical value of the parameter.

Because

√

√
∆(¯
n) ∈ iR, define D ≜ Im ∆ with D > 0. Then we have
√

√
2√
λ3 =
α − iD
2

2√
λ1 =
α + iD
2

46

Using the formula for the square root of a complex number (principle branch),
√

1 √√ 2 2
sgn(b) √√ 2 2
a + bi = √
a +b +a + i √
a +b −a
2
2

we may express λ1 and λ3 as
√√
1 √√ 2
[
α + D2 + α + i
α2 + D 2 − α ]
2
√√
1 √√ 2
λ3 = [
α + D2 + α − i
α2 + D 2 − α ]
2
λ1 =

¯ 1 . Because Re(λ1 ), Re(λ3 ) > 0 and λ2 = −λ1 , λ4 = −λ3 and ρ1 → n
We observe that λ3 = λ
¯ , we know that the
modes corresponding to λ2 and λ4 must be not active in the solution for ρ1 . Therefore, we have
ρ1 (x) ∼ n
¯ + K1 eλ1 x + K2 eλ3 x = K1 eλ1 x + K2 eλ1 x
¯

Defining
θ≜

√√

√√
α2 + D 2 − α
ζ≜
2

α2 + D 2 + α
2

so that λ1 = θ + iζ, we have

ρ1 (x) ∼ n
¯ + K1 eθx [cos ζx + i sin ζx] + K2 eθx [cos ζx − i sin ζx]

(4.14)

∼n
¯ + eθx [(K1 + K2 ) cos ζx + i(K1 − K2 ) sin ζx]
˜ 1 cos ζx + K
˜ 2 sin ζx]
∼n
¯ + eθx [K

˜ 1, K
˜ 2 ∈ R. Thus, we may write
We note that K1 , K2 ∈ C but in the above expression, we must have K
ρ1 (x) ∼ n
¯+e

θx

√

⎡
⎤
⎢
⎥
˜2
˜1
K
K
⎢
⎥
2
2
˜
˜
K1 + K2 ⎢ √
cos ζx + √
sin ζx⎥
⎢ ˜2 ˜2
⎥
2
2
˜
˜
⎢ K1 + K2
⎥
K1 + K2
⎣
⎦

Accordingly, there exists γ ∈ R such that

˜2
K
sin γ = − √
˜2 +K
˜2
K
1
2

˜1
K
cos γ = √
˜2 +K
˜2
K
1
2

47

(4.15)

so that (4.15) becomes
ρ1 (x) = n0 (x) ∼ n
¯ + eθx

√
˜ 2 [cos γ cos ζx − sin γ sin ζx]
˜2 +K
K
2
1

∼n
¯ + Aeθx cos(ζx + γ)

after applying the standard difference formula from elementary trigonometry.
Because ρ2 = n′0 , φ1 = φ, and φ2 = φ′ all decay to zero, the linearized solution for each takes the form

(4.14), except without the n
¯ . We may repeat the analysis that led to the asymptotic form of ρ1 . The
constants K1 and K2 come from the first component of the eigenvector corresponding to λ1 ; θ and ζ
come from the eigenvalue and are independent of the entries of the eigenvector. Consequently, each of
n′0 , φ, φ′ ∼ eθx cos(ζx + γ) as x → −∞, with each having its own proportionality constant. That φ′′ also
exhibits this behavior is clear from the dynamics (4.12). This completes the proof.

∎

Remark 4.1. Utreras-Diaz asserted that the decay is still exponentially oscillatory even the absence of
exchange-correlation. However, this is not the case for all metals. If Vxc ≡ 0, then ∆(¯
n) becomes
∆(¯
n) =

Using the relation kF = (3π 2 n
¯ ) /3 , we can write

16
64π¯
n
4
(3π 2 n
¯ ) /3 −
9λ2
λ

1

∆(kF ) =

4
16 3 1
k ( kF − )
3λ F 3λ
π

which is positive until kF = 12λπ −1 . For λ = 91 , this happens at kF = 0.4244 or rs = 4.5219, well-within the

range of physical values. Utreras-Diaz expressed surprise at his result and had pegged exchange-correlation
as the likely source of oscillation. His intuition was correct but his analysis clearly erred somewhere. For
λ = 14 , the transition occurs at kF = 0.9549 or rs = 2.0097, which is at the bottom of the physical range.

∎

Remark 4.2. It is possible to compute both the amplitude A and γ analytically, but the formulas are so
cumbersome and unwieldy that they are not illustrative. They have accordingly been omitted here.
The table below gives values of the decay rate θ and frequency of oscillation ζ for common materials.

∎

The following result appears in several sources [15],[67, p.14], yet no hint of its derivation or reference thereto
is ever provided. We have therefore produced a rigorous proof of this very important property.

48

Material

rs

θ

ζ

Platinum

2.01

1.4132

0.6589

Zinc

2.31

1.2354

0.6686

Copper

2.67

1.0726

0.6614

Silver/Gold

3.01

0.9530

0.6469

Sodium

3.93

0.7283

0.5990

Potassium

4.86

0.5832

0.5537

Table 4.1: TFDW Decay Rate and Frequency of Oscillation. The values for the decay rate θ and
frequency of oscillation ζ for common materials. This values differ from those appearing in Utreras-Diaz
because he used the Wigner correlation kernel; these were computed with Perdew-Wang.
Theorem 4.9. As x → +∞, the TFDW OF-DFT density n0 for charge-neutral systems decays exponentially:
2(υ − φ(+∞)) ⎞
⎟
λ
⎠

⎛
n0 ∼ B exp ⎜−2x
⎝

where B is an undetermined constant.

(4.16)

Proof. The starting point is the original Euler-Lagrange equations (4.5)–(4.6). The method of attack for
proving this theorem is the method of dominant balance [5, p.83]: suppose that n0 ∼ eθx for some θ < 0.
Then we can characterize the behavior of the various terms appearing in the top equation as follows:
Terms that approach constants:
Terms that go to 0 exponentially:

(

n′0 (x)
n′′ (x)
) , 0
, φ(x), υ
n0 (x)
n0 (x)
2

2/3

n0 (x), Vxc (n0 (x))

The proof of Theorem 3.4 shows that the slowest decaying piece of Vxc behaves as rs−1 . While φ → φ(+∞) ≠ 0,

it does so at an exponential rate; this is apparent from (4.6), after integrating twice, if n0 ∼ eθx is assumed.
Therefore, for large x, we must have

λ n′0 (x)
λ n′′0 (x)
(
) −
+ φ(+∞) − υ = 0
8 n0 (x)
4 n0 (x)
2

Define K ≜ 8λ−1 (φ(+∞) − υ) > 0. (See Remark 4.4 afterwards for a demonstration of positivity.) We then

must solve

(

n′′ (x)
n′0 (x)
) =2 0
−K
n0 (x)
n0 (x)
2

49

(4.17)

Motivated by the observation that

∂ n′0
n′
n′′
[ ] = −( 0) + 0
∂x n0
n0
n0
2

n′
∂
n′
[log n0 ] = 0 to obtain
we subtract 2 ( 0 ) from both sides of (4.17) and use that
n0
∂x
n0
2

⎡ ′′
2
2⎤
⎢ n (x)
n′0 (x)
n′ (x) ⎥⎥
) = 2 ⎢⎢ 0
−( 0
) ⎥−K
n0 (x)
n0 (x) ⎥
⎢ n0 (x)
⎣
⎦
2
∂
∂
− [ ( log n0 )] = 2 2 [ log n0 ] − K
∂x
∂x

−(

Making the substitution g ≜

∂
[ log n0 ],
∂x

what we must solve is then
1
1
g′ + g2 = K
2
2

(4.18)

To solve (4.18), introduce another substitution:
√
g(x) ≜ − K + h(x)

⇒ g ′ (x) = h′ (x)

√
g 2 = K − 2h K + h2

⇒

Then (4.18) becomes
√
1
h′ − h K + h2 = 0
2

h′
√
=1
h K − 12 h2

⇒

We may solve by converting to partial fractions,

and then integrating to obtain

Solving for h yields

1
1
1
√ ( + √
) h′ = 0
K h 2 K −h

√
√
log h − log (2 K − h) = x K + A
√
h(x) = 2 K

Aex
Aex

√

√

K

K

+1

Using the definitions of h and g yields the following simple equation for n0 :
√

√
√
Aex K
∂
[log n0 ] = − K + 2 K
√
∂x
Aex K + 1
50

which may be integrated directly and then solved for n0 :
n0 = Be−x

√

K

(Aex

√

K

+ 1)

2

Finally, because n0 → 0, we know that A = 0 (B remains undetermined), so we have
⎛
n0 = B exp ⎜−2x
⎝

2(υ − φ(+∞)) ⎞
⎟
λ
⎠

∎

Remark 4.3. The method of dominant balance supposes behavior of the form eS(x) and then solves a
differential equation for S. We know that without the exchange-correlation term, the TFDW density decays
exponentially [83, p.133]. It is therefore a reasonable supposition that the result should still hold in its
presence, so we put S(x) = θx from the outset to reach the inevitable conclusion with greater celerity.

∎

Remark 4.4. In the above proof, we asserted the positivity of K without explanation. That K > 0 follows

readily from the work function ΦW , which will not be detailed until Section 5.1.2. The author did not wish
to clutter the proof with a lengthy aside on this concept. Based on its physical interpretation, ΦW is always
positive. If λ = 1, then the exponential decay rate is exactly 2ΦW . By (4.7), we have
1
φ(+∞) − υ = φ(+∞) − kF2 − Vxc (¯
n)
2

= D − Vxc (−∞) − EF = ΦW

The physical significance is that the gradient correction forces the density to decay twice the energy it takes
to remove them. As λ decreases to zero, the rate of decay increases, meaning that a weak gradient correction
forces the electrons to remain closer to the surface. As λ → 0, electrons improperly bunch up at the surface,
reflecting that the λ → 0 limit of the TFDW model is Thomas-Fermi-Dirac.

∎

Because n+ is discontinuous, n0 − n+ ∉ H 2 (R) because requisite absolute continuity is lacking at the origin.
However, as a direct consequence of Theorems 4.8 and 4.9 in tandem, we can assert the following.

Corollary 4.10. n0 − n
¯ ∈ W 2,p (−∞, 0) and n0 ∈ W 2,p (0, +∞) for 1 ≤ p ≤ ∞, where W 2,p is the Sobolev space
W 2,p (A) = {f ∈ Lp (A) ∶ f ′ , f ′′ ∈ Lp (A)}

Remark 4.5. The specifics of Vxc were not used in any of the proofs of Theorems 4.9 and 4.8, aside from
requiring that the decay of Vxc be dominated by some homogeneous term so that the overall decay remains

51

exponential and that Vxc vanish as n → 0. As seen in the proof of Theorem 3.4, the exchange energy decays

like rs−2 . Neglecting correlation but including exchange, which is the Thomas-Fermi-Dirac model, leads to a
density that exhibits polynomial decay ∣x∣

−6

as ∣x∣ → +∞ (for a single atom) [83, p.112].
−2 −C∣x∣

Without correlation, the TFDW density decays as ∣x∣

e

for single-atom systems [83, p.133]. Here,

with correlation, we have a purely exponential rate of decay. Again recalling the proof of Theorem 3.4, the
correlation exhibits slower decay than the exchange, which accounts for the difference.
What can be concluded from all this is that it is the gradient correction that give the exponential decay;
correlation is not insignificant, however, as its inclusion slows the decay rate by a factor of ∣x∣ .
2

∎

The section has one final asymptotic result, which concerns the value of the electrostatic potential at +∞.
Theorem 4.11. For a charge neutral system (i.e., φ′ (+∞) = 0) normalized so that φ(−∞) = 0,
φ(+∞) = 4π ∫

+∞

−∞

y[n0 (y) − n+ (y)] dy

(4.19)

Proof. The Green’s function for the Poisson equation in 1D is G(x − y) = −2π ∣x − y∣; we may obtain the

solution to (4.6) by integrating against G. The exponential decay of both n0 − n+ → 0 at both endpoints
makes the integral

φ(x) = −2π ∫

+∞

−∞

∣x − y∣ [n0 (y) − n+ (y)] dy

well-defined and absolutely convergent (not a principle value). Then for x > 0,
−

x
+∞
1
(φ(x) − φ(−x)) = ∫ (x − y)[n0 (y) − n+ (y)] dy + ∫
(y − x)[n0 (y) − n+ (y)] dy
2π
−∞
x

−∫

= 2x ∫
Now consider
lim x ∫

x→+∞

−∞

−∞
−x

−∞

−x

−x

(−x − y)[n0 (y) − n+ (y)] dy − ∫

[n0 (y) − n+ (y)] dy − 2 ∫

x

−x

+∞

−x

(y + x)[n0 (y) − n+ (y)] dy

y[n0 (y) − n+ (y)] dy

−x

∫−∞ [n0 (y) − n+ (y)] dy
x→+∞
1/x

[n0 (y) − n+ (y)] dy = lim

Because n0 − n+ is integrable due to its exponential decay, the numerator tends to zero as x → +∞. It is
therefore valid to apply L’Hopital’s Rule, followed by the change of variables x ↦ −x:
lim x ∫
x→+∞

−x

−∞

n0 (x) − n+ (x)
= lim x2 (n0 (x) − n
¯) = 0
x→−∞
x→−∞
1/x2

[n0 (y) − n+ (y)] dy = lim

52

because the exponential squashes the x2 . Consequently,
lim (φ(x) − φ(−x)) = 4π lim ∫
x→+∞
x→+∞

Then because φ is finite at both +∞ and −∞, we have

φ(+∞) = φ(−∞) + 4π lim ∫
x→+∞
= 4π ∫

∞

−∞

x
−x

x

−x

y[n0 (y) − n+ (y)] dy

y[n0 (y) − n+ (y)] dy

y[n0 (y) − n+ (y)] dy

with the last line following from that φ(−∞) = 0.

4.5

∎

Numerical Implementation

The numerical method we will use to solve (4.8)–(4.9) is the adaptive spline method of Chapter 2 detailed
in Algorithm 2.3. However, a challenge in applying the spline method is that (4.9) contains a discontinuous
term in n+ . It is well-known that attempts to interpolate across discontinuities fail dramatically. A simple

fix for jump discontinuities (as n+ has) is to place the discontinuity at a node and break into right-hand

and left-hand problems, using the appropriate limit values to make the function seem continuous to the
interpolation technique. However, doing so requires the value of the function to be known at the break.
Fortunately, Theorems 4.2 and 4.4 provide just the information needed to remedy the issue. First, let
f (Y, φ) ≜

1
2
2
7
(3π 2 ) /3 Y /3 + Y (Vxc (Y 2 ) + φ − υ)
2λ
λ

Instead of (4.8) and (4.9) with the boundary conditions (4.10), we consider the left and right problems
Left problem:

Right problem:

2/3

where φBV ≜ 15 (3π 2 n
¯)

Y ′′ = f (Y, φ)

Y (−∞) =

φ′′ = −4π(Y 2 − n
¯)

√

n
¯

φ(−∞) = 0
√ 3 3/4
Y (0) = n
¯ (5)

Y ′′ = f (Y, φ)
φ′′ = −4πY 2

φ(0) = φBV

Y (0) =

√

φ(0) = φBV

Y (+∞) = 0

3/4

n
¯ ( 35 )

φ′ (+∞) = 0

+ Vxc (¯
n) − εxc (¯
n). Computation will be restricted to charge-neutral systems for the

scope of this demonstration so that ς = 0 or φ′ (+∞) = 0.

Of the two spline method algorithms, Algorithm 2.1 is the one applicable to nonlinear BVPs. To complete

53

the recipe for the spline method detailed in Section 2.2, we need the derivatives
∂f
7
2
2Y 2
2
4
=
(3π 2 ) /3 Y /3 + (Vxc (Y 2 ) + φ − υ) +
fxc (Y 2 )
∂Y
6λ
λ
λ
∂f 2Y
=
∂φ
λ

Note that Theorem 3.5 ensures that Y 2 fxc (Y 2 ) will behave for large x when Y → 0. The derivatives in Y of
the equations for φ′′ are trivial to compute, so they are not presented here.

To extend the spline method from single equations to systems, let ak , bk , ck , dk be the coefficients of the k th
spline for Y and a
¯k , ¯bk , c¯k , d¯k be those for φ. Then we interweave these as sets to form the coefficient vector
c = (a1 , b1 , c1 , d1 , a
¯1 , ¯b1 , c¯1 , d¯1 , ⋯, aN , ⋯, dN , a
¯N , ⋯, d¯N )

T

The equations go in corresponding procession, with the two DE, the C 0 , and then C 1 conditions for the k th
spline for Y first, followed by the ones for φ. This arrangement retains the banded structure of the Jacobian
matrix J that allows for efficient solution, although the bandwidth will double from four to eight.
Per Step 1 of Algorithm 2.1, Newton’s method needs an initial guess. For this, we use the trial density
n(0) =

1
1 + e10x

and then perform standard spline interpolation on the initial mesh. We then take φ(0) ≡ 0.

To extend the mechanism of adaptive grid generation to the system, we form the errors

Ek+1/2 = max { ∣Yk′′ (xk+1/2 ) − f (Yk (xk+1/2 ), φk (xk+1/2 ))∣ , ∣φ′′k (xk+1/2 ) + 4π(Yk (xk+1/2 ) − n+ (xk+1/2 ))∣ }
where Yk and φk denote the k th spline in the numerical solution for Y and φ, respectively. Points are then
added just as before. In this manner, Y and φ are solved on the same grid.
After computation, we form n0 by squaring the spline solution for Y using spline arithmetic. See Appendix
C for details. This results in sixth degree polynomials, although they remain only C 2 and O(h4 ) accurate.

54

4.6

Computational Results

In this example, rs = 3 and the infinite domain was truncated to [−20, 10]. With the exponential decay of
the density on both sides, even this domain is larger than necessary. An adaptive tolerance of τa = 10−7 was
used. The results are compiled below in Figures 4.3, 4.4, and 4.5.

10
8

×10−3

Orbital-free Density n0

n0 (x)

6
4
2

0
−20

n0 (x)
n+ (x)

−15

−10

−5
x

0

5

10

Figure 4.3: Orbital-free Density for rs = 3. This figure depicts the TFDW OF-DFT ground state
density There is little overshoot of the background charge and convergence to limits is incredibly rapid.
The density itself is rather unremarkable and there is little to say about it. The profile decays rapidly to its
limits n
¯ in the bulk metal and zero in the vacuum. Figure 4.4 highlights the oscillations.
×10−3

Density Oscillations

¯)
e−θx (n0 (x) − n

8
4
0

−4

−8
−20

−16

−12

x

−8

−6

−2

Figure 4.4: Highlight of Density Oscillations. By depicting e−θx (n0 (x) − n
¯ ), this figure highlights the
oscillations in the density. Computed per Theorem 4.8, θ = 0.9562. By equating A cos(ζx + γ) with the
numerical solution evaluated at the points x1 = −18 and x2 = −15, we find A = −6.337 × 10−3 and
γ = −0.04312. Although not shown, substitution of these values into A cos(ζx + γ), with ζ = 0.6474 per
Theorem 4.8, yields excellent agreement with the depicted plot throughout the domain.
The corresponding electrostatic potential settles into its limit values just as quickly as the density. There
is large a dip near the surface. The low point does not correspond to the peak overshoot in n0 , which is
located approximately one unit farther into the bulk material.

55

8

φ(x)

6

×10−2

Orbital-free Electrostatic Potential φ

4
2
0
−2
−20

−15

−10

−5
x

0

5

10

Figure 4.5: Orbital-free Electrostatic Potential. This figure shows the electrostatic potential
corresponding to Figure 4.3. Convergence is very rapid, with limits nearly attained by x = −5 and x = 5.
The number line of Figure 4.6 portrays the distribution of splines across the domain. On the very left, there
is no refinement of the initial mesh, which had spacing h = 0.02.
−20

−18

−16

−14

−12

−10

−8

−6

x

−4

−2

0

2

4

6

8

10

Figure 4.6: Spline Distribution for OF-DFT Computation. Solving the right problem to tolerance
τa = 10−7 required 1262 splines; the left needed 960. As we see, the splines are concentrated near the origin,
where the transition is sharpest. Very few are needed in flat areas at both ends of the domain.

4.7

Conclusion

Because of the liberties that must be taken in the kinetic energy, OF-DFT is less often deployed than its
cousin KS-DFT. For very large systems, however, when the Kohn-Sham method becomes intractable, OFDFT offers a very reasonable approximation, especially if the alternative is nothing at all. The adaptive
spline method presented here is capable of finding highly accurate solutions in just a few seconds.
The primary contributions of this chapter are the proofs of Theorems 4.8 and 4.9. A less advanced version
of the first was presented in a published source, and due to an error in computation, reached an erroneous
conclusion about the source of the oscillations. The more illustrative proof produced here allowed the mistake
to be discovered and provides support for the original author’s intuition that exchange-correlation plays an
important role in the oscillatory nature of TFDW densities.
The result of the second theorem appears in [15], but no reference to its proof is provided. Only a heuristic

56

argument based on the nature of electrons near the Fermi level was given to support the validity of the claim.
Making a rigorous proof of this fundamental result available serves mathematical and physics audiences alike.
The computational methodology here marks an advancement in the OF-DFT techniques for jellium of
Chizmeshya and Zaremba [15]. Their work includes a third equation in total charge Q = φ′ to contain the

1 ′
φ (+∞) so that standard BVP methods can be applied. Placing the
boundary condition Q(+∞) = ς = − 4π

boundary condition directly on φ′ reduces the size of the computation by one-third. Furthermore, the novelty
of employing the Budd-Vanninemus Theorem and Theorem 4.4 to circumvent the discontinuity in n+ must

not be discounted. The decomposition into left and right problems expands the set of usable numerical
solvers from those that can handle discontinuities to virtually anything that works on BVPs.

57

Chapter

5

The Ground State: Kohn-Sham DFT
5.1

Introduction to Solid-State Physics

The previous chapter introduced a solid-state physics model for interacting electrons within a metal solid
called jellium. This section discusses a few needed concepts from solid-state physics in brevity.

5.1.1

The Fermi Surface and Fermi Sphere

The most rudimentary concepts of solid-state physics are inherently linked to the construction of an N electron atom. The Pauli Exclusion Principle states that no more than two electrons can occupy the same
energy level (orbital) at the same time. Following this principle, electrons are shuffled into the discrete
energy levels described by quantum mechanics as follows:
1. Place two electrons at energy level zero.
2. Proceed to the orbital corresponding to the next energy level and place two electrons.
3. Repeat Step 2 until the supply of electrons has been exhausted.
If k is the momentum of a particle, then its energy ε is given by
1
ε = k2
2
so that we may index particles by energy or (magnitude of) momentum interchangeably. This fact, along
with Figure 5.1 on the next page, motivates a slew of “Fermi” terminology:
❼ Fermi wavevector: kF , momentum at the highest energy level.

58

kF

k4
k3
k2

k1

k1

Step 1
Step 2
Step N /2

Figure 5.1: Placement of N electrons. In Step 1, two electrons are placed at momentum k = 0. In
Step 2, two electrons are placed at k1 . The process continues until step N /2 when the final electrons are
placed at momentum level kF , called the Fermi wavevector.
❼ Fermi sphere: sphere of radius kF (outermost layer). It encloses all occupied energy levels.
❼ Fermi surface: surface of the Fermi sphere. It separates occupied from unoccupied orbitals.
❼ Fermi energy: energy level of the highest occupied particles. It is given by

1
EF = kF2
2

(5.1)

Some care is needed with the term “Fermi energy.” It is more properly the energy difference between the
highest and lowest occupied energy levels and can only be defined at zero temperature. Zero-temperature
models are excellent approximations for ordinary finite temperatures and are accordingly used in this thesis.

5.1.2

Three Important Physical Quantities

In this section, we introduce some physical quantities of critical importance in solid-state and surface physics.
Because of their limited role in this thesis, characterization here will be restricted to a brief description and
mathematical definition. Comprehensive discussions can be found in Ashcroft [2], for example.
Definition 5.1 (Surface dipole barrier). The surface dipole barrier D is a dipole layer formed at the
surface due to the spill-out of electrons from the material into the vacuum. It is given by
D ≜ φ(+∞) − φ(−∞)

(5.2)

Definition 5.2 (Surface barrier potential). The surface barrier potential is the total height of the barrier
potential seen by an electron on the surface. It is given by

59

∆V ≜ V (+∞) − V (−∞)

(5.3)

= D − Vxc (−∞)

with the second line coming because Vxc vanishes as x → +∞.

Definition 5.3 (Work function). The work function is the energy required to remove an electron from
the Fermi level (or electron with energy equal to EF at zero temperature) from a solid to a point far enough
away from the surface that it is no longer influenced by electric fields in the vacuum emanating from the
surface. Mathematically, it is defined by the expression
ΦW ≜ ∆V − EF

(5.4)

= D + Vxc (−∞) − EF

Per the last line, D can be viewed as the surface contribution to the work function, while the bulk contribution
is represented in the Vxc term. The EF appears because it is the energy difference between the current state
and one far away; if electromagnetic effects didn’t exist, for instance, EF would be precisely the energy
required to remove an electron. Table 5.1 lists values of these three quantities for various jellium surfaces.

rs

EF

ΦW

D

∆V

Example Material (actual rs )

2.0

12.52

3.89

6.80

16.41

Platinum (1.99)

2.5

8.01

3.72

3.83

11.73

Cadmium (2.59)

3.0

5.57

3.50

2.32

9.07

Gold (3.01)

3.5

4.09

3.26

1.44

7.35

Strontium (3.56)

4.0

3.13

3.06

0.91

6.19

Sodium (3.93)

Table 5.1: Ground State Energies for Jellium Surfaces. Adapted from Lang and Kohn [57], this
table contains values of the Fermi energy EF , the work function ΦW , dipole barrier D, and surface barrier
potential ∆V for jellium surfaces of various average densities rs . All quantities are in eV. Reproduced with
express permission from the American Physical Society and Norton Lang.

5.1.3

Fermi-Dirac Statistics

As fermions (identical particles that obey Pauli Exclusion Principle), electrons in metals obey Fermi-Dirac
statistics, first noted by Sommerfeld [102]. Fermi-Dirac statistics describe the probability of finding a fermion
at energy level ε via the distribution
F (ε) =

1

e(ε−µ)/kB T
60

+1

(5.5)

where µ is chemical potential or Fermi level❸ , kB is Boltzman’s constant, and T is absolute (Kelvin) temperature. The Fermi level is the energy level that has a 50% probability of occupation—which happens when
the exponential is zero. At zero temperature, this value coincides with the Fermi energy so that
1
µ = EF = kF2
2

With this knowledge, as we take T → 0+ in (5.5), we can deduce the zero-temperature limit of the Fermi-Dirac

distribution, expressed in terms of k,

⎧
⎪
⎪
⎪
⎪ 1
F (ε) = ⎨
⎪
⎪
⎪
0
⎪
⎩

∣k∣ < kF

(5.6)

∣k∣ > kF

Fermi-Dirac Distribution at Various Temperatures
T → 0+ Limit
kB T = µ/100
kB T = µ/50
kB T = µ/20
kB T = µ/5

1
0.8
F( )

0.6
0.4
0.2

0
0

0.2

0.4

0.6

0.8

1

1.2
/µ

1.4

1.6

1.8

2

2.2

Figure 5.2: Fermi-Dirac Distribution. Considering kB T = µ/f , for fixed µ, low T corresponds to large
f . For gold (µ ≈ 0.2046), the values f = 100, 50, 20, 5 depicted correspond to temperatures of 373○ C,
1019○ C, 2978○ C, and 12650○ C, respectively. A temperature of 20○ C corresponds to f ≈ 220.5.
Figure 5.2 shows what an excellent approximation (5.6) makes to (5.5) for ordinary temperatures. In atomic
units, kB = 3.1668 × 10−6 Rydberg/K. As room temperature is around 300 K, it follows that kB T ≈ 10−3 ;

since µ = EF ∈ (0, 0.5) per the table in Appendix B, all cases of interest have f > 200, with f as in the figure.
Thus, the full Fermi-Dirac distribution is well-approximated by the T → 0+ limit for such cases.

Suppose that there are N (ε) dε states at energy level ε. Then the total number of electrons n is [117, p.136]

❸ The

n=∫

0

∞

F (ε)N (ε) dε

Lagrange multiplier of the previous chapter was denoted υ to avoid confusion with this Fermi level.

61

(5.7)

Inserting the Fermi-Dirac distribution (5.6) and taking the volume element in k space dε ↦
n=

1
N (ε) dk
∫
(2π)3 ∣k∣<kF

dk
, we have
(2π)3

The state counter N is given in terms of the Kohn-Sham wavefunctions, indexed in momentum
N (ε) = 2 ∣ψk (x)∣

2

The factor of 2 comes from that two electrons are allowed for each value of k (or ε). Because the wavefunctions
only give us a probabilistic sense of whether an electron is at position x, the left-hand side of (5.7) is more
appropriately interpreted as the ground state density rather than the actual number of electrons, so
n0 (x) =

5.2

1
2
∣ψk (x)∣ dk
∫
4π 3 ∣k∣<kF

(5.8)

Effect of Jellium

Let k = (k, k2 , k3 ) . Denote the components parallel to the metal surface using a tilde, mathematically
T

˜ = k − (k ⋅ ν) ν, where ν denotes the unit normal vector pointed outward of the surface. Because
defined as k

˜ = (0, k2 , k3 )T .
the surface is assumed to occupy the yz-plane, ν = (1, 0, 0)T , we have k

Because of the symmetry of the jellium surface, the wave functions ψk (indexed by vector subscripts) can be
decomposed into a plane wave parallel to the surface and a wave function ψk (indexed by scalar subscript)
depending only on the perpendicular direction:
ψk (x) = eik⋅˜x ψk (x)
˜

(5.9)

˜⋅x
˜ = k2 x2 + k3 x3 , substituting (5.9) into Kohn-Sham equations (3.12) yields
Because k
(−

1 ∂2
1
1
˜
+ k22 + k32 + V (x) − εk ) eik⋅˜x ψk (x) = 0
2
2 ∂x
2
2

˜ 2 + 1 k 2 + V (−∞), each ψk satisfies a one-dimensional Schr¨odinger
Because we have εk = 21 k2 + V (−∞) = 21 k
2

equation of the form

(−

1 ∂2
+ V (x) − εk ) ψk (x) = 0
2 ∂x2

62

(5.10)

where
1
εk = k 2 + V (−∞)
2

(5.11)

We are able to reduce the dimension from x to x in V by noting that because of (5.9) the Kohn-Sham
representation of the density (5.8) becomes
n0 (x) ≡ n0 (x) =

1
ψ 2 (x) dk
∫
4π 3 ∣k∣<kF k

We can remove the absolute value because the ψk are real-valued (because εk is). Now, because the integrand
depends only on the perpendicular component of k, we can integrate analytically over the parallel momenta:
√

√

kF
kF −k2
kF −k2 −k3
1
√
ψk2 (x) ∫ √ 2
dk2 dk3 dk
n0 (x) = 3 ∫
∫
2 −k 2 −k 2
4π −kF
− kF −k2
− kF
3
√
2 −k 2 √
kF
kF
1
2
= 3∫
kF2 − k 2 − k32 dk3 dk
ψk (x) ∫ √ 2
2π −kF
− kF −k2
2

2

2

Using the integral formula
∫

√

a2 − x2 dx =

the inner integral may be computed exactly:
n0 (x) =

x√ 2
a2
x
a − x2 +
tan−1 ( √
)
2
2
a2 − x2

kF
1
(k 2 − k 2 ) ψk2 (x) dk
∫
4π 2 −kF F

By virtue of (5.11), the integrand is even in k, so we can write
n0 (x) =

5.3

kF
1
(kF2 − k 2 )ψk2 (x) dk
∫
2
2π 0

(5.12)

Boundary Conditions for Wavefunctions

Next, we will use the representation (5.12) to formulate asymptotic boundary conditions for the ψk ’s. The
following result will of great usefulness in doing so:
Theorem 5.1. Let V ∈ C(R) be such that

lim V (x) = V (+∞)

lim V (x) = V (−∞)

x→+∞

x→−∞

63

and let ψ be a solution to the Schr¨
odinger equation
(−
Then ψ behaves like

1 ∂2
+ V (x) − ε) ψ = 0
2 ∂x2

⎧
iˆ
κx
−iˆ
κx
⎪
⎪
⎪
⎪ β1 e + β2 e
ψ∼⎨
⎪
ˆ
ˆ
iλx
−iλx
⎪
⎪
⎪
⎩ α1 e + α2 e

x → −∞

x → +∞

ˆ 2 = 2ε − 2V (−∞). Note that either or both of κ
ˆ may be imaginary, as no
where κ
ˆ 2 = 2ε − 2V (+∞) and λ
ˆ or λ

assumption is made on ε in relation to V (−∞) and V (+∞).

It is easy to understand the intuition of this result: by continuity of solutions to differential equations in
their coefficients, as x reaches +∞, it seems reasonable that ψ should approach the solution to
(−

1 ∂2
+ V (+∞) − ε) ψ = 0
2 ∂x2

and similarly for −∞. A rigorous proof of this theorem, however, requires far more machinery than the
simple ideas from which it was conceptualized and involves asymptotics of Jost solutions for Strum-Liouville
equations. For details, see Theorem 9.2.3 in Promislow and Kapitula [88, p.256].

5.3.1

Asymptotic Conditions at +∞

ˆ 2 via (5.11)
To apply Theorem 5.1 with ε = εk , we first compute λ
k

ˆ 2 = k 2 − 2V (−∞) + 2V (+∞) = 2 ( 1 k 2 − ∆V )
λ
k
2

where ∆V is the surface barrier potential of (5.3). Based on its physical interpretation, the work function
(5.4) is always positive, so ∆V > EF . Because k < kF
λk ≜

√

2V (+∞) − 2εk

ˆ 2 < 0. Define
⇒ 12 k 2 < EF , we have 12 k 2 − ∆V < 0, so λ
k
⇒

ˆ k = iλk
λ

with the square root taken to produce a positive imaginary part. Then the asymptotic form of ψk becomes
ψk ∼ α1 (k) e−λk x + α2 (k) eλk x

64

We note that all quantities in the above expression are real. Because we require that n0 → 0 as x → +∞, we

must have α2 (k) = 0 for all k, as λk > 0 and this exponentially growing mode would preclude the possibility
of satisfying the required decay condition. We therefore have as x → +∞
ψk ∼ αk e−λk x

5.3.2

(5.13)

Asymptotic Conditions at −∞

Again appealing to Theorem 5.1, we compute

κ
ˆ 2 = k 2 + 2V (−∞) − 2V (−∞) = k 2

so that as x → −∞ we have the asymptotic behavior

ψk ∼ β1 (k) eikx + β2 (k) e−ikx

Because ψk is real-valued, we must have β1 = β2∗ . Thus, we may write the above as
ψk ∼ βk eikx + βk∗ e−ikx

We may substitute (5.14) into (5.12) to obtain

kF
1
2
(kF2 − k 2 ) [2 ∣βk ∣ + β12 e2ikx + (β1∗ )2 e−2ikx ] dk
∫
2
2π 0
⎡
⎤
kF
kF
⎥
1 ⎢⎢ kF
2
2
2
2
2
2
2ikx
∗ 2
2
2
−2ikx
2 ∣βk ∣ (kF − k ) dk + ∫
βk (kF − k ) e
dk + ∫
(βk ) (kF − k ) e
dk ⎥⎥
∼ 2 ⎢∫
2π ⎢ 0
0
0
⎥
⎣
⎦
≜A
≜B

n0 ∼

To prove that A, B → 0, we will need the following famous lemma [43, p.105]:

Lemma 5.1 (Riemann-Lebesgue). If f ∈ L1 (Rd ), then its Fourier transform fˆ(ξ) has
Claim. A, B → 0 as x → −∞.

lim fˆ(ξ) = 0

∣ξ∣→∞

Proof. Because ψk = ψ−k so that βk = β−k , by sending k ↦ −k
A=∫

0

−kF

x
βk2 (kF2 − k 2 ) e−2ikx dk = ∫ βk2 (kF2 − k 2 ) [Θ(k + kF ) − Θ(k)] e−2ikx dk = fˆ ( )
π
R
≜ f (k)
65

(5.14)

where fˆ denotes the ordinary Fourier transform of f if this quantity is well-defined. By continuity of k ↦ βk ,
f is bounded and compactly supported, so f ∈ L1 (R). Hence, by the Riemann-Lebesgue Lemma, fˆ → 0 as

∣x∣ → ∞, so we have A → 0 as desired. Turning our attention to B, we have that

x
B = ∫ (βk∗ )2 (kF2 − k 2 ) [Θ(k) − Θ(k − kF )] e−2ikx dk = gˆ ( )
π
R
≜ g(k)

Hence, by the same argument as above, B → 0.

Because n0 → n
¯ as x → −∞, we have

n
¯=

kF
1
2
∣βk ∣ (kF2 − k 2 ) dk
∫
2
π 0

∎

(5.15)

We return to (5.14) and express it in a more familiar form:
ψk ∼ ∣βk ∣ (eiϑk eikx + e−iϑk e−ikx )
= 2 ∣βk ∣ cos(kx − ϑk )

= −2 ∣βk ∣ sin(kx − γk )

where ϑk ≜ angle(β1 ) and γk ≜ ϑk − π2 . If we choose ∣βk ∣ ≡
n will be correct:

lim n0 (x) =

x→−∞

√1
2

for each k, via (5.15), the asymptotic value for

kF
kF3
1
2
2
(k
)
−
k
dk
=
=n
¯
∫
F
2π 2 0
3π 2

The last equality follows from Appendix B. Therefore, taking each ψk to have asymptotic behavior
√
2
ψk ∼ − √ sin(kx − γk ) = − 2 sin(kx − γk )
2

(5.16)

as x → −∞ yields the correct result for the density n. Furthermore, because ψk appears only as ψk2 in (5.12)
(or as ψk (x)ψk (y) in spectral forms), we may remove the negative sign; because the ψk are generated by
linear equations, we can absorb the factor of

1
2

in (5.12) into the ψk , giving

ψk ∼ sin(kx − γk )

66

(5.17)

5.3.3

Summary and Consequences

As shown in the preceding subsections, in order for the wavefunctions ψk to produce the correct asymptotic
behavior of the ground state density n0 , they must behave like
⎧
⎪
⎪
⎪
⎪ sin(kx − γk )
ψk ∼ ⎨
⎪
−λk x
⎪
⎪
⎪
⎩ αk e

x → −∞

x → +∞

In deriving the boundary condition at −∞, we began with (5.12), which has a factor of

1
2

out front. How-

ever, in the above boundary conditions, we incorporated that factor into the ψk , so with this choice of
normalization, the density is instead given by
n0 (x) =

5.4

kF
1
(kF2 − k 2 ) ψk2 (x) dk
∫
π2 0

(5.18)

Asymptotic Behavior of the Kohn-Sham Density

The following definition and lemma appear in Lighthill [69, p.56]:
Definition 5.4. A distribution f is said to be well-behaved at infinity if there exists R > 0 such that

f − F ∈ L1 ((−∞, −R)) ∩ L1 ((R, +∞)), where F is a linear combination of the functions for β, k ∈ R
eikx ∣x∣

β

eikx ∣x∣ sgn(x)

eikx ∣x∣ log ∣x∣

F (x) sin(2πxy) ∼

F (0) F ′′ (0) F (4) (0)
−
+
−⋯
2πy
(2πy)3 (2πy)5

β

β

eikx ∣x∣ log ∣x∣ sgn(x)
β

Lemma 5.2. If F and all its derivatives exist as functions for x ≥ 0 and are well-behaved at infinity, then
0

∞

0

∞

∫

∫

F (x) cos(2πxy) ∼ −

F ′ (0)
F ′′′ (0) F (5) (0)
+
−
+⋯
(2πy)2 (2πy)4 (2πy)6

(5.19)
(5.20)

We will use the above lemma in proving the following asymptotic formula for the density n0 . The result
appears frequently throughout the literature (e.g., [54, p.89]), but always without reference to any proof.
Theorem 5.2. As x → −∞, the Kohn-Sham density given by (5.18) has asymptotic behavior
n0 (x) ∼ n
¯ (1 +

1
3 cos(2xkF − 2γkF )
)+O( 3)
(2xkF )2
x

Proof. We begin by using the asymptotic form of the wavefunction given in (5.17) in (5.18)
n0 ∼

kF
1
(kF2 − k 2 ) sin2 (kx − γk ) dk
∫
2
π 0

67

(5.21)

kF
1
(kF2 − k 2 )(1 − cos(2kx − 2γk )) dk
∫
2
2π 0
kF
kF
1
1
(kF2 − k 2 ) dk − 2 ∫
(kF2 − k 2 ) cos(2kx − 2γk ) dk
∼ 2∫
2π 0
2π 0
kF
1
(kF2 − k 2 ) cos(2kx − 2γk ) dk
∼n
¯− 2∫
2π 0

∼

(5.22)

We applied the standard power-reducing formula to sin2 to reach line two from line one; to obtain the final
line, we have once again used that kF3 = 3π 2 n
¯ . Formally, we may write
∫

kF

0

(kF2 − k 2 ) cos(2kx − 2γk ) dk = ∫

0

∞

(kF2 − k 2 ) cos(2kx − 2γk ) dk − ∫

∞

kF

(kF2 − k 2 ) cos(2kx − 2γk ) dk

where the integrals of the right-hand side are to be interpreted as distributions (i.s.d), as they do not converge
to actual functions. We will compute asymptotic behavior of these distributions. Write
∫

0

∞

(kF2 − k 2 ) cos(2kx − 2γk ) dk = ∫

0

∞

(kF2 − k 2 ) cos 2γk cos 2kx dk + ∫

0

∞

(kF2 − k 2 ) sin 2γk sin 2kx dk

The functions
F1 (k) ≜ (kF2 − k 2 ) cos 2γk

F2 (k) ≜ (kF2 − k 2 ) sin 2γk

satisfy all the hypotheses for Lemma 5.2, so we may apply the results (5.19) and (5.20):
∫

0

∞

(kF2 − k 2 ) cos(2kx − 2γk ) dk =

The relevant derivatives are given by

F2 (0) F1′ (0) F2′′ (0)
1
−
−
+O( 4)
2
3
2x
(2x)
(2x)
x

∂γk
⋅ (kF2 − k 2 ) sin 2γk
∂k
∂γk
∂ 2 γk
∂γk 2
2
2
)
F2′′ (k) = −2 (4k
− (kF2 − k 2 ) ⋅
)
cos
2γ
−
2
(1
+
2(k
−
k
⋅
(
) ) sin 2γk
k
F
∂k
∂k 2
∂k
F1′ (k) = −2k cos 2γk − 2

Because γ0 = 0, we have

F2 (0) = 0

F1′ (0) = 0

F2′′ (0) = 2kF2

68

∂ 2 γk
∣
∂k 2 k=0

And so
∫

0

∞

(kF2 − k 2 ) cos(2kx − 2γk ) dk = − (

kF2 ∂ 2 γk
1
1
∣ ) 3 +O( 4)
2
4 ∂k k=0 x
x

(5.23)

To resolve the integral over [kF , ∞), we first make the change of variables k ↦ k − kF so that we have
∫

∞

kF

(kF2 − k 2 ) cos(2kx − 2γk ) dk = ∫

=∫

0

0

∞

∞

(kF2 − (k + kF ) ) cos(2kx + 2xkF − 2γk+kF ) dk
2

(kF2 − (k + kF ) ) cos(2xkF − 2γk+kF ) cos 2kx dk

+∫

2

0

∞

(kF2 − (k + kF ) ) sin(2xkF − 2γk+kF ) sin 2kx dk
2

Defining
G1 (k) ≜ (kF2 − (k + kF ) ) cos(2xkF − 2γk+kF )
2

G2 (k) ≜ (kF2 − (k + kF ) ) sin(2xkF − 2γk+kF )
2

we can apply Lemma 5.2
∫

∞

kF

(kF2 − k 2 ) cos(2kx − 2γk ) dk =

The required derivatives are

G2 (0) G′1 (0) G′′2 (0)
1
−
−
+O( 4)
2x
(2x)2
(2x)3
x

G′1 (k) = −2(k + kF ) cos(2xkF − 2γk+kF ) + 2

G′′2 (k) = 2 (4(k + kF )

∂γk+kF
∂k

∂γk+kF
2
⋅ (kF2 − (k + kF ) ) sin(2xkF − 2γk+kF )
∂k
∂ 2 γk+kF
2
− (kF2 − (k + kF ) ) ⋅
) cos(2xkF − 2γk+kF )
∂k 2

− 2 (1 + 2(kF2 − (k + kF ) ) ⋅ (
2

∂γk+kF 2
) ) sin(2xkF − 2γk+kF )
∂k

so that we have
G2 (0) = 0

G′1 (0) = −2kF cos(2xkF − 2γkF )

G′′2 (0) = (8kF

∂ 2 γk+kF
∣ ) cos(2xkF − 2γkF ) − 2 sin(2xkF − 2γkF )
∂k 2 k=0

69

The derivative of γk term can be explained by rewriting γk using standard function notation:
γk ≡ γ(k)

⇒

γk+kF ≡ γ(k + kF )

⇒
⇒

so that

Plugging in these evaluations gives
∫

∞

kF

∂γk
= γ ′ (k)
∂k

∂γk+kF
= γ ′ (k + kF )
∂k
∂ 2 γk+kF
= γ ′′ (k + kF )
∂k 2

∂ 2 γk
∂ 2 γk+kF
′′
∣
=
γ
(k
)
=
∣
F
∂k 2 k=0
∂k 2 k=kF

(kF2 − k 2 ) cos(2kx − 2γk ) dk = −

2kF cos(2xkF − 2γkF ) G′′2 (0)
1
−
+O( 4)
(2x)2
(2x)3
x

(5.24)

Combining (5.23) and (5.24) and substituting into (5.22), we have the asymptotic behavior for n0 :
n0 (x → −∞) = n
¯+
with
ζ≜

kF cos(2xkF − 2γkF ) ζ
1
+ 3 +O( 4)
2
2
π
(2x)
x
x

(5.25)

kF2 ∂ 2 γk
kF
1
∂ 2 γk
∣
+ 2 cos(2xkF − 2γkF ) ⋅
∣
− 2 sin(2xkF − 2γkF )
2
2
2
8π ∂k k=0 2π
∂k k=kF 8π

To further simplify (5.25), we can multiply and divide the second term by 3kF2 and again use that kF3 = 3π 2 n
¯.

Furthermore, because γk is not known analytically as a function of k, the x−3 term is not particularly helpful.
∎

Discarding it gives the form claimed in (5.21).

5.4.1

Friedel Oscillations

Theorem 5.2 says that the Kohn-Sham density exhibits Friedel Oscillations [38], which are alternating regions
of positive and negative charge due to impurities in the electron jelly. These oscillations always look like
ρ(x) ∼ ρ0 (x) +

A cos(2kF ∣x∣ + γ)
∣x∣

(5.26)

The source of this phenomenon is the nonzero de Broglie wavelength of electrons at low temperatures, which
causes a scattering-like effect that smears out charge in space; at high temperatures, the range of electron
wavelength widens, and there is no coherent collective smearing of charge. Only electrons near the Fermi
energy can participate in this effect, as the oscillations are due to electrons jumping to unoccupied states

70

above their current energy level and no such states exist for electrons below the Fermi energy. This is
reflected in the period of oscillation

π
kF

, which is half the de Broglie wavelength of the outermost electrons.

The de Broglie wavelength of particles at the Fermi energy is also known as the Fermi wavelength λF =

2π
.
kF

Within the jellium model, it can be thought of as a Gibbs-type phenomenon. The positive background
density terminates abruptly at the interface, yet the electron density is continuous. Because of the charge
neutrality condition
∫

+∞

−∞

(n0 (x) − n+ (x)) dx = 0

one can think of the continuous n0 , which is a sum of of sine waves for x ≫ 0, as attempting to approximate
the discontinuous n+ . A Gibbs-like overshoot occurs near the interface and ripples through the domain.

5.5

Electrostatic Potential: A Deceptively Hard Problem

The total electrostatic potential associated with the density is the solution to the Poisson equation
φ′′ (x) = −4π(n0 (x) − n+ (x))

(5.27)

φ(−∞) = 0, φ (+∞) = −4πς
′

The right boundary condition is the total charge condition and enforces the requirement that
∫

+∞

−∞

(nς (x) − n+ (x)) dx = ς

(5.28)

For a charge-neutral system, ς = 0; ς ≠ 0 describes systems with surface screening charge. A na¨ıve approach
to solving (5.27) would be to proceed we did in the orbital-free computation: truncate the infinite domain

to a large finite one [R1 , R2 ] and enforce the boundary conditions φ(R1 ) = 0 and φ′ (R2 ) = −4πς. A more
sophisticated attempt would make use of the known analytic behavior of n0 from (5.13) and (5.17):
kF
⎧
⎪
⎪
(kF2 − k 2 ) sin2 (kx − γk ) dk
⎪
⎪
⎪ ∫0
n0 (x) ∼ ⎨
kF
⎪
⎪
⎪
(kF2 − k 2 ) αk2 e−2λk x dk
⎪
∫
⎪
⎩ 0

x≪0
x≫0

With the help of Fourier transforms, distributions, and complex contour integrals, the exact values of φ(R1 )

and φ′ (R2 ) can be computed exactly. Following either the na¨ıve or sophisticated path for truncating the
domain and then using any standard ODE solver (finite difference, linear multi-step method, finite element,
etc.) results in an incorrect solution, depicted below in Figure 5.3.

71

Na¨ıve Attempt to Solve Poisson System
0.5

φ(x)

0.4
0.3
0.2
−20

−15

−10

−5
x

0

5

10

Figure 5.3: Failure of Standard Methods. A second-order finite difference method was used to solve
(5.27) in the na¨ıve formulation described above with ς = 0. Note the linear behavior on the left.
Liebsch cites “long-range Coulomb potential due to charge imbalance” [67, p.39] as the reason for the
difficulty in solving (5.27) and advocates an approached pioneered by Manninen et al. [75]. In this technique,
a parameter λ > 0 is chosen and the Poisson equation is reformulated as an implicit integral equation
φ(x) = ∫

+∞

−∞

e−λ∣x−y∣ [

2π
λ
(n0 (y) − n+ (y)) + φ(y)] dy
λ
2

(5.29)

The exponential decay of the kernel suppresses the problematic interactions. When combined with integration
by parts, the integral equation allows for an exact (in principle) determination on a truncated interval.
Furthermore, following the work of Causley [13], the integral may be computed on a uniform grid of N
points in O(N ) time. Unfortunately, the integral operator is incapable of seeing boundary conditions other
than ς = 0, so despite its upsides, it is unsuitable for cases of nonzero screening charge.

One may be tempted to conclude that the integral equation formulation includes these “long-range Coulomb
effects” by concentrating wide-ranging Coulombic effects over (−∞, +∞) into a finite interval, thereby trans-

forming the infinite domain into a finite one. Accordingly, it would seem that a suitable fix would be to
employ Fourier transforms. Information over all of R could be incorporated into the solution by decomposing into three contributions over (−∞, R1 ), [R1 , R2 ], and (R2 , +∞). Over the “numerical” domain [R1 , R2 ],

an FFT would suffice and the analytic asymptotic forms would be used over the semi-infinite pieces. This
method also fails, indicating that the core problem is far deeper than complications from truncation.
The issue is a fundamental incompatibility between the boundary conditions for the Poisson equation and
the spatial distribution of the electron density. Illustrated in [99], consider a simple integration of (5.27):
φ′ (x) = φ′ (0) − 4π ∫

x

0

72

(n0 (y) − n+ (y)) dy

φ(x) = φ(0) + xφ′ (x) − 4π ∫

x
0

y(n0 (y) − n+ (y)) dy

If we are to have ∣φ(±∞)∣ < ∞, then xφ′ (x) → 0 as ∣x∣ → ∞ and thus
φ(+∞) = −4π ∫

+∞

−∞

φ′ (0) − 4πς = 4π ∫

0

y(n0 (y) − n+ (y)) dy

+∞

(n0 (y) − n+ (y)) dy

If these relations do not hold, then it is impossible to obtain the correct φ by solving (5.27) with a known righthand side; it is exceptionally unlikely everything will just fall into place if we use an intermediate density
from the Kohn-Sham process. Because the density depends on the electrostatic potential and potential
depends on the density, the problem is one of self-consistency—or rather, lack there of.
To remedy the incompatibility, we follow the procedure detailed in [87] to replace the long-range Coulomb
interaction with a screened interaction over finite range. To do this, we consider a splitting of the density:
n0 (x) = nind (x) + nqu (x)

(5.30)

The induced density nind is given by
nind (φ(x)) =

3/2
2 /2
(n
[E
−
φ(x)
−
V
(x))]
F
xc
0
3π 2
3

(5.31)

The remainder nqu represents quantum corrections to the quasiclassical electron distribution. With this
separation of the density, the Poisson equation (5.27) becomes
φ′′ (x) + 4πnind (x) = 4π(n+ (x) − nqu (x))

(5.32)

It is critical to note that the representation (5.31) is valid only if
∂nind
<0
∂φ

(5.33)

This condition generally fails to hold when the density is small (such as in the vacuum) and thus the
significance of the exchange-correlation potential falls to the same level as that of the electrostatic potential.
While an expression for the induced density may exist, it is not given by (5.31). Because the electron density
decays rapidly away from the interface, Coulombic effects from the vacuum are negligible. Consequently, we

73

may take nind ≡ 0 when (5.33) no longer holds. Let c be the first point such that
∂nind
(c) = 0
∂φ

and define for λ > 0

Fc (x) ≜

1
1 + eλ(x−c)

(5.34)

(5.35)

For λ large enough, Fc is a C ∞ approximation to the step function Θ(c − x) so that Fc ≈ 0 for x > c and
Fc ≈ 1 for x < c. While it is possible to construct a smooth function that is exactly 1 for −R < x < c and
exactly 0 outside this range, this Fc is precise enough. Employing this filter, nind is redefined to be
nind (φ(x)) =

3/2
2 /2
(n
⋅ Fc (x)
[E
−
φ(x)
−
V
(x))]
F
xc
0
3π 2
3

(5.36)

nqu is still defined as the difference
nqu = n0 − nind

(5.37)

and the modified Poisson equation (5.32) remains the same. How to find c is the subject of Section 5.7.3.

5.6

Self-Consistent Field Iteration (SCF)

The Schr¨
odinger equations (5.10) that the wavefunctions satisfy are linear differential equations. However,
the wavefunctions generate the density n0 , which is part of the potential V = φ + Vxc (n0 ) that generates
the wavefunctions. In physics, this circular dependence is known as self-consistency. With the Kohn-Sham

equations, we generate a density (via wavefunctions) with an a priori potential—the density does not generate
the potential used to compute that density, so the two are not self-consistent.
To square this otherwise circular process, we perform what is known as a self-consistent field iteration (SCF):
we assume that we know the potential and then use that to compute the density. That newly computed
density is then used to generate a new potential, which in turn is used to generate another density, depicted
below in Figure 5.4. The process can be thought of as a fixed-point iteration and is continued until the
potential and density generate each other in some approximate sense.

The above figure is a mere overview of the process. Not only does the density appear explicitly in V through
the Vxc term but it also does so implicitly through φ, which satisfies the Poisson equation (5.27), in which
n0 appears on the right-hand side. The difficulty in solving for φ detailed at length in Section 5.5 can be

74

Compute Wavefunctions

Form Potential

V = φ + Vxc (n0 )

∂
(− 12 ∂x
2 + V − εk ) ψk = 0
2

n0 =

Form Density
1
π2

∫0

kF

(kF2 − k 2 ) ψk2 dk

Figure 5.4: Self-consistent Field Iteration. The potential V is needed to compute the density n0 and
the density is needed to compute the potential. In contrast to orbital-free techniques, the Kohn-Sham
process is not capable of solving for these quantities simultaneously, so iteration becomes necessary.
directly attributed to the issue of self-consistency—the “form potential” step is easier said than done! In
order for the iteration of Figure 5.4 to work, we need to incorporate the fixes of the previous section.
While based on the work of Posvyanskii and Shul’man [87], the algorithm below differs slightly from theirs.
The excellent starting density and potential afforded by OF-DFT allows rearrangment of the steps.

Algorithm 5.1. Complete SCF Iteration.
(0)

1. Compute initial n0

and φ(0) using OF-DFT techniques of Chapter 4. Enter iteration:

2. Form the Kohn-Sham effective potential
(i)

V (i+1) (x) = φ(i) (x) + Vxc (n0 (x))
3. For an appropriate set of discrete k values, compute the wavefunctions
(−

1 ∂2
+ V (i+1) (x) − εk ) ψk (x) = 0
2 ∂x2

4. Form the density from the wavefunctions computed in Step 3 via
(i+1)

n0

(x) =

kF
1
(kF2 − k 2 )ψk2 (x) dk
∫
2
π 0

5. Determine the cut-off filter of (5.35). (See Algorithm 5.3 in Section 5.7.3.)

75

6. Update the quantum contribution nqu to the density via the formula
(i+1)

n(i+1)
qu (x) = n0

(x) −

3/2
2 /2
(i+1)
(i)
(n
⋅ Fc(i+1) (x)
(x))]
[E
−
φ
(x)
−
V
F
xc
0
3π 2
3

(i+1)

nind (x)

7. Solve the nonlinear Poisson equation
∂xx φ(i+1) +

3/2
2 /2
(i+1)
[EF − φ(i+1) − Vxc (n0 )] ⋅ Fc(i+1) (x) = 4π(n+ − n(i+1)
qu )
3π
7

φ(i+1) (−∞) = 0, ∂x φ(i+1) (+∞) = −4πς

8. Compute the L∞ error between successive densities (“previous-to-current-iterate error”)
(i+1)

E = ∣∣ n0

(i)

− n0 ∣∣

L∞

If E < τsc , terminate the process. Otherwise, return to Step 2.

A visual illustration of Algorithm 5.1 as a flowchart is depicted in Figure 5.5 on the next page. The description
above merely tells what is to be computed; the details of how are the subject of the next section.
The iteration indexing used in Algorithm 5.1 warrants clarification. In Step 2, the (i + 1)th potential is
(i)

computed using n0 and φ(i) . It may seem that this potential should be numbered V (i) , but doing so leads

to an index mismatch. In the utilized numbering scheme, the (i)th V is used to compute the (i)th density,
which is then used to compute the (i)th electrostatic potential. The progression is more intuitively
(0)

(1)

→ φ(1) → V (2) → ⋯

(0)

(1)

→ φ(1) → V (1) → ⋯

{n0 , φ(0) } → V (1) → n0

instead of
{n0 , φ(0) } → V (0) → n0
(i+1)

In Step 6, the underbrace is labeled nind

(i+1)
nqu

+

(i+1)
nind ,

(i+1)

even though it is generated by φ(i) . We do this so that n0

=

even though doing so is really an index mismatch. In Step 7, we solve an equation that

involves the unknown nind (φ(i+1) ). Because double use of nind would be overly confusing, Algorithm 5.1
eliminates the symbol nind . It is provided in Step 6 merely to connect to a previously introduced concept.

76

Initialize n0 , φ

Compute Wavefunctions

Form Potential

V

(i+1)

= φ

(i)

+

(i)
Vxc (n0 )

∂2
(− 21 ∂x
2

+V

(i)

− εk ) ψk = 0

(i+1)
n0

Form Density

=

1
π2

∫0

kF

(kF2 − k 2 ) ψk2 dk

else

Compute Error
(i+1)

E = ∣∣ n0

(i)

− n0 ∣∣

∞

E < τsc

STOP

Solve Nonlinear Poisson

Determine Cut-off Filter
(see Algorithm ??)

Update Quantum Density

∂xx φ(i+1) + 4πnind (φ(i+1) ) =
(i+1)
4π(n+ − nqu )

(i+1)

nqu

(i+1)

= n0

Figure 5.5: Complete SCF Algorithm. This is a visual representation of Algorithm 5.1.

77

− nind (φ(i) )

5.7
5.7.1

Numerical Implementation
Determining a Wavefunction ψk (Step 3)

Recall that ψk is subject to the asymptotic boundary conditions (5.13) and (5.17). Unfortunately, both of
those expressions contain unknown values: αk in the right condition and γk in the left one. By the linearity
of (5.10), if aψ is a solution, then so is ψ. Consequently, we can remedy the issue by first solving
(−

1 ∂2
+ V (x) − εk ) ψ˜k (x) = 0
2 ∂x2

(5.38)

ψ˜k ∼ e−λk x as x → +∞

and then renormalizing ψ˜k to form the correct ψk :

ψk =

1 ˜
ψk
A

(5.39)

where A is the amplitude of
ψ˜k ∼ A sin(kx − γk ) as x → −∞

(5.40)

To solve (5.38), we use the adaptive spline method prescribed in Algorithm 2.3. The system (5.38) fits the
mold of Section 2.3 exactly, so we employ Algorithm 2.2 in conjunction with the adaptive one.
5.7.1.1

Finding the Phase

The kink in the renormalization technique (5.39) is that γk is unknown, so (5.40) alone doesn’t determine
A. Fortunately, the spline method’s ability to compute accurate derivatives comes to the rescue.
If ψ˜ behaves as in (5.40), then ψ˜k′ behaves like
ψ˜k′ ∼ Ak cos(kx − γk ) as x → −∞
By assumption, the asymptotic conditions are achieved to satisfactory accuracy at x1 . Put
f1 ≜ ψ˜k (x1 )

f2 ≜ ψ˜k′ (x1 )

Then we have
f1 1
= tan (kx1 − γk )
f2 k

⇒ γk

= kx1 − tan−1 (
78

kf1
)
f2

In order to adjust γk so that it lies between −π and π, we instead take

We further wish to have 0 ≤ γk ≤

γk = kx1 mod π − tan−1 (

kf1
)
f2

(5.41)

π
, so we follow Algorithm 5.2 below to make that happen.
2

Algorithm 5.2. Adjustment of γk
1. If γk as computed by (5.41) has γk >

2. If γk < 0, then put γk ↤ −γk , A ↤ −A.

π
, put γk ↤ π − γk .
2

π π
π 3π
Because the inverse tangent has range (− , ), (5.41) produces a value in (− , ). After Step 1 of
2 2
2 2
π π
Algorithm 5.2, we will have γk ∈ (− , ) and Step 2 takes care of negative values.
2 2
5.7.1.2

Renormalizing

With γk in hand, we can find the amplitude A by taking
A=
and then taking ψk as in (5.39).

5.7.2

ψ˜k (x1 )
sin(kx1 − γk )

Forming the Density (Step 4)

To form the density via (5.18) the integral must be approximated by a sum of finitely many k values. The
question becomes how many values of k should be used and how should they be distributed in [0, kF ]? At

each k value, we must solve an ODE, so using more values than necessary creates a high computational
burden; using too few can lead to an inaccurate result, which is ultimately self-defeating.
The answer comes in the form of Gaussian quadrature, which allows for computation of highly accurate
approximations to integrals with just a handful of function evaluations [3, p.270]. The idea is to write
∫

1
−1

N

g(x) dx ≈ ∑ wq g(xq )
q=1

for a set of nodes xq and weights wq . Table 5.2 below presents these values for a few N -point rules.

79

Points

Nodes xq

±0.9061798459
±0.5384693101

5

0

±0.9324695142
±0.6612093865

6

±0.2386191861
±0.9491079123
±0.7415311856

7

±0.4058451514
0

Weights wq
0.2369268851
0.4786286705
0.5688888889
0.1713244924
0.3607615730
0.4679139346
0.1294849662
0.2797053915
0.3818300505
0.4179591837

Table 5.2: Gaussian Quadrature Nodes and Weights. This table lists the nodes xq and weights wq
for high-order Gauss-Legendre quadrature rules.
Gaussian quadrature rules are always written for integrals over [−1, 1]. Integrals over [a, b] are transformed

into ones over [−1, 1] via change of variables:
b

b−a

1

∫a g(x) dx ≈ 2 ∫−1 g (

a + b + x(b − a)
) dx
2

(5.42)

Then the weights and nodes can be extracted from Table 5.2.
This formula is especially useful in formulating a composite rule for even greater accuracy. To compute the
integral over [a, b], we divide up the interval into d equally spaced intervals of the form [ξp , ξp+1 ] with ξ1 = a
and ξd+1 = b. Then we write

∫

a

b

d

g(x) dx = ∑ ∫
p=1

ξp+1
ξp

g(x) dx

and apply the change of variables (5.42) to each of the integrals on the right-hand side.

5.7.3

Setting the Cut-off Filter (Step 5)

The next order of business is to describe how to determine the value of c for the cut-off filter (5.35) of Step
5 of Algorithm 5.1. The definition (5.34) is inconvenient because nind as defined in (5.31) is implicit: n0
depends on nind and n0 appears in the right-hand side within Vxc . The “inverse”
φ(nind ) = EF −

2/3
1
2/3
(3π 2 ) nind − Vxc (nind + nqu )
2

80

(5.43)

is a more convenient subject of analysis. The relevant condition (5.34) for c then becomes
∂φ
(c) = 0
∂nind
The needed derivative can be computed explicitly:
G(nind ) ≜

2/3
∂φ
1
−1/3
′
= − (3π 2 ) nind − Vxc
(n0 )
∂nind
3
2/3
1
−1/3
= − (3π 2 ) nind − fxc (n0 )
3

(5.44)

We therefore need to find the value of c such that G(nind (c)) = 0, solving the above nonlinear equation which
is parameterized in x. An important distinction is to be made here: we are looking for an x value that makes

(5.44) zero. We could solve this for a value of nind = nc and then find c such that nind (c) = nc ; this is a poor
method because it requires solving nonlinear equations twice. It is more efficient to solve directly for c.

The method of choice for solving nonlinear equations is Newton’s method. Newton’s method can be a bit
temperamental, especially when a reasonably good initial guess is not used. Fortunately, we can generate a
suitable starting value by evaluating (5.44) on the mesh {xm } that constitutes the nodes of the splines of

the density n0 and locating the first index m such that xm−1 < 0 and xm > 0. We can then make c(0) to be
the midpoint of these values. To implement Newton’s method, we need the derivative of (5.44) in x:
2/3
∂G 1
∂fxc ∂n0
−4/3 ∂nind
= (3π 2 ) nind ⋅
−
⋅
∂x 9
∂x
∂n ∂x

We have written all the derivatives in Leibnitz notation at the moment for the sake of clarity. Recall that
′
fxc is a function of n and that gxc = fxc
was previously defined in (3.32). Therefore, the above expression

can be simplified using usual prime notation:

2/3
1
−4/3
G′ (x) = (3π 2 ) nind (x)n′ind (x) − gxc (n0 (x)) n′0 (x)
9

(5.45)

The remaining necessary derivative n′ind can be computed by differentiating (5.31) directly:
n′ind (x)

=−

√

2

π2

[EF − φ − Vxc (n0 )]

1/2

⋅ (φ′ + fxc (n0 (x)) n′0 (x))

(5.46)

Thanks to the spline method, we have access to accurate values for n′0 and φ′ anywhere within the computational domain. Having assembled all the needed pieces, the complete algorithm is:

81

Algorithm 5.3. Determining the Cut-off Filter.
Let x1 < x2 < ⋯ < xN +1 be the nodes of the splines for the density n0 .

1. Find the first index m such that G(xm−1 ) < 0 and G(xm ) > 0. Put c(0) =

2. Newton Step:

c(i) = c(i−1) −

xm−1 + xm
.
2

G (c(i−1) )

G′ (c(i−1) )

G and G′ are given by formulas (5.44) and (5.45) with help from (5.46).
3. Repeat Step 2 until

for some desired tolerance τ ≈ 10−12 .

∣c(i) − c(i−1) ∣ < τ

The last matter is the value of λ in (5.35). While a larger λ will make Fc more like a step function, the
cutoff filter doesn’t need to be so precise. Local adaptivity can handle a steep transition with as little effort
as possible, but the overhead is ultimately unnecessary. Taking λ = 50 is a good compromise; such a value

approximates a step function reasonably well while not being too sharp (cf. the red curve in Figure 5.2).

5.7.4

Solving the Nonlinear Poisson Equation (Step 7)

The last detail to be resolved is the crafting of a solution to the nonlinear Poisson equation
φ′′ (x) +

3/2
2 /2
[EF − φ(x) − Vxc (n0 (x))] ⋅ Fc (x) = 4π(n+ (x) − n0 (x))
3π
7

φ(−∞) = 0, φ′ (+∞) = −4πς

Because of the nonlinearity in φ, we appeal to Algorithm 2.1 when the adaptive Algorithm 2.3 calls for
it. There is one major wrinkle we must iron out before proceeding: the background charge function n+ is
discontinuous at x = 0, so a na¨ıve application of the spline method will not work well.

To remedy the issue with the discontinuity, we appeal to the Budd-Vannimenus Theorem (or more explicitly,
Corollary 4.3) as in the orbital-free computation to decompose the problem into
Left problem:

φ′′ (x) = fL (x, φ)

Right problem:

φ(−∞) = 0, φ(0) = φBV

φ′′ (x) = fR (x, φ)

φ(0) = φBV , φ′ (+∞) = −4πς

82

2/3
1
2πς 2
where φBV ≜ (3π 2 n
¯ ) + Vxc (¯
n) − εxc (¯
n) +
and
5
n
¯

fL (x, φ) ≜ 4π(¯
n − nqu (x)) −

fR (x, φ) ≜ −4πnqu (x)

3/2
2 /2
[EF − φ(x) − Vxc (n0 (x))] ⋅ Fc (x)
3π
7

The location of the cutoff filter c occurs near the surface but always within the metal so that we always have
c < 0. Thus, Fc ≡ 0 wipes out the nonlinear term in the right problem. Even though the right problem is a

linear equation, we still use Algorithm 2.1. In applying the nonlinear algorithm to linear equations, all the
derivative terms that appear in (2.4) and (2.5) vanish and the Jacobian matrix is the system matrix for a
formulation of the method for linear BVPs. The “Newton iteration” will converge in one step.
For the left side, though, the derivative contributions contained in (2.4) and (2.5) do not vanish. Since fL
does not involve φ′ , we need only compute
1/2
2 /2
∂fL
=−
[EF − φ(x) − Vxc (n0 (x))] ⋅ Fc (x)
∂φ
π
5

The solutions to the left and right problems are glued together by interpolating the complete set of nodes
and computed values with a cubic spline. We cannot simply juxtapose the two half-solutions because the
composite result will not be differentiable at 0. Thanks to Theorem 2.1, the interpolation error is same order
as local truncation error, so resplining does not introduce additional error.

5.8

Computational Results

For the simulations depicted in this section, τa = 10−7 was used for the adaptive tolerance and τsc = 10−6 for

convergence of the SCF iteration. The infinite domain was truncated to [−35, 10]. A composite seven-point
rule of d = 4 divisions (28 k points total) was used in forming the density. Rules for other domains were also

investigated. Five divisions were needed for [−50, 10]; d = 4 was insufficient to damp the strong oscillations
that appear for x ≪ 0. Two divisions were enough for [−20, 10] while [−120, 10] required ten.

For rs = 3 and ς = 0, we were able to obtain a ground state density profile after 12 iterations of SCF

(Algorithm 5.1), terminating with a final previous-to-current-iterate error of 8.987029 × 10−7 . Table 5.3
below lists this error after each iteration. Figure 5.7 on the next page shows all the densities.

Figure 5.6 below shows the final density. When x ≪ 0, we see the dying Friedel oscillations described in
Section 5.4.1 about the limit value n
¯ , plotted as the broken black line depicting the background charge n+ .
83

Iteration
1
2
3
4

Error
5.203366 × 10

Iteration
−4

5

5.520106 × 10−5

6

1.204258 × 10−4

7

9.385186 × 10−5

8

Error
2.595343 × 10

Iteration
−5

3.137688 × 10−6

9

1.900946 × 10−5

6.433941 × 10−6

10

2.475704 × 10−5

4.434425 × 10−6

11

1.200144 × 10−5

Error

8.987029 × 10−7

12

Table 5.3: Error Per Step of SCF Iteration. The desired error 10−6 was achieved after 12 iterations.
We also see that some electrons spill out into the vacuum and that the density decays exponentially from
just outside the surface; this is the cause of the surface dipole barrier (see Definition 5.1).

10
8

×10−3

Ground State Density n0

n0 (x)

6
4
2

0
−35

n0 (x)
n+ (x)

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.6: Ground State Density n0 for rs = 3 and ς = 0. The density profile is shown in red; the
broken black line is positive background charge. Algorithm 5.1 converged to this solution after 12 iterations.

Figure 5.8 depicts the corresponding electrostatic potential φ, computed as described in Section 5.7.4. The
potential displays Friedel oscillations for x < 0, although they are not as pronounced as the ones in the

density. This is the behavior that standard numerical methods applied directly to the Poisson equation fail
to reproduce, although for x > 0 they do well qualitatively—but not quantitatively! (recall Figure 5.3).

We take a detour to highlight one of the benefits of adaptivity: each ψk can be computed on the best grid
for that function. While each ψk can be computed quickly, even on large grids, because we must compute
twenty to thirty wavefunctions per iteration and expect perform a dozen or more iterations, even a savings
of one second per ψk can cut the time required to complete the SCF process significantly.
Table 5.4 presents the number of splines required to drive the residual error of wavefunctions below τa = 10−7 .

Only twelve are shown in the table, but in all, twenty-eight were used for n0 . Because ψk ∼ sin(kx − γk ) for
84

10

×10−3

Ground State Densities Through the Iterations

9

8

7

6

n0

(i)

5

4

3

2

1

0
−35

OF-DFT
Iteration #1
Iteration #2
Iteration #3
Iteration #4
Iteration #5
Iteration #6
Iteration #7
Iteration #8
Iteration #9
Iteration #10
Iteration #11
Iteration #12
−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.7: Density After Each Iteration. The starting density computed by the orbital-free methods of Chapter 4 is indicated by the solid
black line. Note the much larger overshoot of Kohn-Sham densities compared to the orbital-free density.

85

10
8

×10−2

Electrostatic Potential φ

φ(x)

6
4
2
0

−2
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.8: Electrostatic Potential φ for rs = 3 and ς = 0. This potential corresponds to the density
depicted in Figure 5.6. The boundary condition ς = 0 forces φ to approach a limit on the right end.
x ≪ 0, fewer splines are needed for small k, as ψk oscillates slower in the asymptotic region as k decreases.
The qualitative differences of low and high k wavefunctions is shown in Figure 5.9 beneath the table.

k

Splines

k

Splines

k

Splines

0.0041

2991

0.1559

3171

0.3674

6009

0.0475

2889

0.2074

3333

0.4591

6853

0.0800

2831

0.2723

4131

0.5273

7189

0.1124

2785

0.3158

5449

0.6191

7831

Table 5.4: Adaptive Splines in Action. This table lists the number of splines needed to compute the
wavefunction ψk to residual error of τa = 10−7 . For rs = 3, kF = 0.6397. In all, 28 k-values were used.
1

Wavefunction ψk for k = 0.0475

Wavefunction ψk for k = 0.5598

ψk (x)

0.5
0

−0.5

−1
−35 −30 −25 −20 −15 −10 −5
x

0

5

10−35 −30 −25 −20 −15 −10 −5
x

0

5

10

Figure 5.9: Wavefunctions ψk . Two ψk , corresponding to k = 0.0475 (left) and k = 0.5598 (right), used
to form the density in Figure 5.6. The period of oscillation is 2π
, so small k means slower oscillation.
k

Below in Figure 5.10 is a density profile computed when ς = 5 × 10−4 was taken as the boundary condition

for φ′ (+∞). Because this density does not satisfy charge neutrality, we refer to it as a “screened density,”
86

as ς should more properly be interpreted as screening charge❸ than actual charge.

10
8

×10−3

Screened Density nς

nς (x)

6
4
2

0
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.10: Screened Density nς for rs = 3 and ς = 5 × 10−4 . This result was obtained after 15
iterations of SCF and concluded with the same final error of 5.8948 × 10−7 .
The screened profile looks nearly identical to the ground state one, but the corresponding electrostatic
potential, depicted below in Figure 5.11, is obviously different: instead of flattening out to a steady-state
value, φ decreases linearly on the right end of the domain. Since φ′ (+∞) = −4πς < 0, this should be expected.
Physically, this result shows that the effect of screening increases as we move farther from the surface.

8

φ(x)

6

×10−2

Electrostatic Potential φ

4
2
0
−2
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.11: Electrostatic Potential φ for rs = 3 and ς = 5 × 10−4 . This potential corresponds to the
density depicted in Figure 5.10. Note how φ decreases linearly at the right end of the domain.

The last figure for rs = 3 shows that there are quantitative differences between n0 and nς . As Figure 5.12

shows, the largest differences are concentrated near zero. Since screening is a surface effect, it is natural that
the greatest contributions to screening occur in the immediate region of the interface. Because the ultimate

❸ Electrons each possess their own electric fields. These fields cause electrons to repel each other, which creates little
“bubbles” in which no other electron is present. At far enough distances, this bubble can be perceived as a sheet of positive
charge that serves to negate the electrons’ electric fields. The term “screening charge” refers to the collective effect of this.

87

goal for nς is in describing second harmonic generation in Chapter 8, ς has been limited to small values
(weak-electric fields), which explains the small difference between n0 and nς . As ς increases, the screened
density deviates from the ground state one significantly; see, for example, Gies and Gerhardts [42].
×10−5

16

nς (x) − n0 (x)

12

Difference Between nς and n0

8
4
0

−4
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.12: Difference between n0 and nς for rs = 3. That largest difference occurs inside the
metal in the immediate vicinity of the surface reflects the nature of screening as a surface phenomenon.
The last result is the ground state density for rs = 2 in Figure 5.13 below. Lower tolerances of τa = 5 × 10−7

and τsc = 2 × 10−5 were used for this run. Significantly many more splines are required for high accuracy,
leading to instabilities and long computation times. The result is unremarkable when compared to the rs = 3,

except that Friedel oscillations are much less pronounced.

3

×10−2

Ground State Density n0

n0 (x)

2
1
0

−35

n0 (x)
n+ (x)

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 5.13: Ground State Density n0 for rs = 2 and ς = 0. The density profile is shown in blue; the
broken black line is positive background charge. Convergence took 8 iterations and error 1.5584 × 10−5 .
Finally, a word on computation time. The workstation used was a quad-core Intel Core i7 875K overclocked
to 3464 MHz with 16GB of DDR3-1333 RAM, running MATLAB 2010a and Windows Vista Business x64.
From the beginning of the first wavefunction computation to solving for φ, each iteration required 77 seconds
on average. To obtain convergence to τsc = 10−6 , the process took 15 minutes and 27 seconds in totality.
88

5.9

Conclusion

Kohn-Sham calculations for the ground state in jellium systems are not new. However, despite the saturation
of the topic, this chapter has put forth both noteworthy theoretical and significant numerical advancements.
Outside the detailed presentation of the setup of KS-DFT for jellium systems in the early sections, the
primary theoretical contribution of this chapter is the proof of the asymptotic formula in Theorem 5.2.
Friedel oscillations are a distinguishing feature of jellium densities, and while the result of the theorem is
common knowledge in the field, the proof has not been disseminated in any measure of publicity.
Numerically, the foremost contribution of this work is the adaptive spline collation method for ψk . The
spline method facilitates obtainment of highly accurate results without need for analysis from the user aside
from selection of a tolerance. The method maintains maximum efficiency by tailoring grids for each equation.
Previous works universally used the Numerov method [81] on fixed grids, the use of which in atomic structure
calculations can be traced back to Hartree [49, p.71] and has perpetuated for forty years [66, 87].
The direct method employed in this thesis for solving the Poisson equation for the electrostatic potential is
a very welcome advancement. Since its first development by Manninen [75], conversion of the differential
equation to the implicit integral equation (5.29) has been a staple in virtually every DFT for jellium computation. The method carries several significant disadvantages, all of which are resolved by the nonlinearization
of the Poisson equation presented here:
1. (5.29) must be solved iteratively and concurrent with the Kohn-Sham system.
2. (5.29) leads to instability unless some sort of mixing scheme is used on successive density iterates.
3. (5.29) cannot handle the nonzero boundary conditions of charged systems.

The density splitting scheme was first proposed by Shul’Man [99] and solidified into a coherent algorithm by
Posvyanskii and Shul’man [87]. This thesis greatly improves upon the latter’s method in several ways:
1. Invocation of the Budd-Vanninemus Theorem allows for satisfactory sidestepping of the discontinuity
in the background charge. Posvyanskii and Shul’man used a relaxation method that complicates the
equation by introducing a time variable and then seeks the steady-state solution.
2. The adaptive spline method provides accurate derivative information that facilitates computation of
the phase shifts γk so that ψk can be properly normalized. Posvyvanskii and Shul’man stuck with the
traditional Numerov method, using a Taylor series to find the needed values.

89

3. Algorithm 5.3 provides a means to set the cutoff filter for arbitrary Vxc . Posvyanskii and Shul’man
used the Wigner correlation kernel (3.24), exploiting its simple form to find c analytically.

90

Chapter

6

Green’s Functions and Spectra
The subject of Green’s functions, especially in the context of quantum mechanics [22] and even timedependent density functional theory [110, p.86], is a deep one of paramount importance. In physics, particularly in many-body theory [71], the term “Green’s function” is used liberally to refer to any one of several
different, unrelated concepts, reaching as far as correlation functions of creation and annhilation operators.
Throughout this thesis, “Green’s function” will used strictly in the mathematical sense: it is the fundamental
solution to a linear differential equation. If G is the Green’s function of a differential operator A, then a

solution to Au = f may be obtained by integrating G(x, y)f (y) in y. This chapter investigates the Green’s
function for the Schr¨odinger operator H that generates the Kohn-Sham wavefunctions via (5.10):
H≜−

6.1

1 ∂2
+ V (x)
2 ∂x2

(6.1)

Derivation of Green’s Function

Recalling Theorem 5.1 and the techniques used to generate boundary conditions for ψk in Sections 5.3.1 and
5.3.2, we know that solutions to (H − ε)ϕ = 0 behave like
⎧
iµx
−iµx
⎪
⎪
⎪
⎪ ae + be
ϕ∼⎨
⎪
iνx
−iνx
⎪
⎪
⎪
⎩ ce + de

x → −∞

x → +∞

where the two exponents are❸ , with help from Theorem 3.4 and the boundary condition φ(−∞) = 0,

(6.2)

❸ This definition of ν differs the one used in other sources, e.g., [67]. Because φ(+∞) ≪ 1, giving V (+∞) ≈ 0, this quantity
is traditionally neglected without explanation. We include it because there is little reason not to.

91

µ≜

ν≜

√

√

2ε − 2V (−∞) =
2ε − 2V (+∞) =

√
√

2ε − 2Vxc (rs )

(6.3)

2ε − 2φ(+∞)

(6.4)

We allow ε ∈ C, so we must specify the branch of the complex square root to be used. We take the branch

cut removing the negative real axis and taking the argument θ of z to lie in (0, 2π). Then we take
√

z=

√ iθ/2
re

This choice of argument ensures that Im(µ) ≥ 0 and Im(ν) ≥ 0. We can generate two linearly independent
solutions (“basis functions”) ϕ1 and ϕ2 by imposing the asymptotic conditions
⎧
iµx
−iµx
⎪
⎪
⎪
⎪ ae + be
ϕ1 ∼ ⎨
⎪
iνx
⎪
⎪
⎪
⎩ e

⎧
−iµx
⎪
⎪
⎪
⎪e
ϕ2 ∼ ⎨
⎪
iνx
−iνx
⎪
⎪
⎪
⎩ ce + de

x → −∞
x → +∞

x → −∞
x → +∞

(6.5)

Let G(x, y; ε) be the Green’s function to the operator H − ε. It satisfies the equation
(H − ε)G(x, y; ε) = −δ(x − y)

(6.6)

and must have the following properties [21, p.36]:
1. (Continuity) G(⋅, ⋅; ε) is continuous on R × R

2. (Continuity of derivative)
3. (Jump condition)

∂G
(⋅, y; ε) is continuous on R/{y}
∂x
+

x=y
∂G
∂G
∂G
1
(x, y; ε)∣
= lim+
(x, y; ε) − lim−
(x, y; ε) = 1 = −2.
x→y ∂x
x→y ∂x
∂x
−2
x=y −

4. (Symmetry) G(x, y; ε) = G(y, x; ε)

In 1D, the Green’s function “switches” between the two linearly independent solutions. To this end, put
⎧
⎪
⎪
⎪
⎪ A(y)ϕ1 (x), −∞ < x ≤ y
G(x, y; ε) = ⎨
⎪
⎪
⎪
y<x≤∞
⎪
⎩ B(y)ϕ2 (x),

(6.7)

By Properties (1) and (3), the basis functions ϕ1 and ϕ2 must satisfy
A(y)ϕ1 (y) − B(y)ϕ2 (y) = 0

A(y)ϕ′1 (y) − B(y)ϕ′2 (y) = 2
92

(6.8)

To solve the system of equations (6.8), we shall employ Cramer’s Rule. First, recall the Wronskian
⎡ ϕ (y)
⎢ 1
W (ϕ1 , ϕ2 )(y) ≜ det ⎢⎢
⎢ ϕ′ (y)
⎣ 1

ϕ2 (y) ⎤⎥
⎥
⎥
ϕ′2 (y) ⎥⎦

(6.9)

and that scaling a column of a matrix changes its determinant by the same factor. Therefore,
⎡
⎢ ϕ1 (y)
⎢
−W (ϕ1 , ϕ2 )(y) = det ⎢
⎢ ′
⎢ ϕ1 (y)
⎣

By Cramer’s Rule, the coefficients A and B are given by
⎡ 0
⎢
1
det ⎢⎢
A(y) = −
W (ϕ1 , ϕ2 )(y)
⎢ 2
⎣

⎤
−ϕ2 (y) ⎥⎥
⎥
⎥
−ϕ′2 (y) ⎥⎦

−ϕ2 (y) ⎤⎥
2ϕ2 (y)
⎥=−
⎥
W
(ϕ
1 , ϕ2 )(y)
−ϕ′2 (y) ⎥⎦

⎡ ϕ (y)
⎢ 1
1
B(y) = −
det ⎢⎢
W (ϕ1 , ϕ2 )(y)
⎢ ϕ′ (y)
⎣ 1

0 ⎤⎥
2ϕ1 (y)
⎥=−
⎥
W
(ϕ
1 , ϕ2 )(y)
2 ⎥⎦

By properties of determinants −W (ϕ1 , ϕ2 ) = W (ϕ2 , ϕ1 ). Using this fact on the above solutions A and B,
substitution of the result into (6.7) yields the revised formula for G:

⎧
ϕ1 (x)ϕ2 (y)
⎪
⎪
, −∞ < x ≤ y
2
⎪
⎪
⎪
⎪ W (ϕ2 , ϕ1 )(y)
G(x, y; ε) = ⎨
⎪
ϕ1 (y)ϕ2 (x)
⎪
⎪
2
,
y<x≤∞
⎪
⎪
⎪
⎩ W (ϕ2 , ϕ1 )(y)

The derivation will be complete after an appeal to the well-known Abel’s Identity [7, p.118]:
Lemma 6.1 (Abel’s Identity). Let ζ1 and ζ2 be two linearly independent solutions to
ζ ′′ (y) + p(y)ζ ′ (y) + q(y)ζ(y) = 0
Then the Wronksian W satisfies
W (ζ1 , ζ2 )(y) = W (ζ1 , ζ2 )(y0 ) exp (− ∫

In the case of H − ε, p ≡ 0, so for any y0 ∈ R

y
y0

p(t) dt)

W (ϕ2 , ϕ1 )(y) = W (ϕ2 , ϕ1 )(y0 ) ≡ constant
93

In light of this, the constant W may be computed in the limit y → −∞ or y → +∞ by appealing to the

asymptotic boundary conditions (6.5):

W (ϕ2 , ϕ1 ) = lim [ϕ2 (y)ϕ′1 (y) − ϕ′2 (y)ϕ1 (y)]
y→−∞

= lim [e−iµx (iaµeiµy − ibµe−iµy ) + iµe−iµy (aeiµy + be−iµy )]
y→−∞

= 2iµa

(6.10)

or
W (ϕ2 , ϕ1 ) = lim [ϕ2 (y)ϕ′1 (y) − ϕ′2 (y)ϕ1 (y)]
y→+∞

= lim [(aeiνy + be−iνy )iνeiνy − (iaνeiνy − ibνe−iνy )eiνy ]
y→+∞

= 2iνc

(6.11)

Because ϕ1 , ϕ2 , and W (which can now be written sans arguments) all depend on ε, it is preferable to write

6.2
6.2.1

⎧
2
⎪
⎪
ϕ1 (x; ε)ϕ2 (y; ε) −∞ < x ≤ y
⎪
⎪
⎪ Wε
G(x, y; ε) = ⎨
⎪
2
⎪
⎪
ϕ1 (y; ε)ϕ2 (x; ε),
y<x≤∞
⎪
⎪
⎩ Wε

(6.12)

Spectral Theory
The Operator H

The nature of the Kohn-Sham potential V has a large effect on the spectrum of H. Consequently, an

examination of the specifics of V is necessary before we can proceed farther.

Depicted in Figure 6.1 is the Kohn-Sham potential for rs = 3. Perhaps more important than any properties
it possesses are the properties it does not have:
❼ V is not monotonic
❼ As V (±∞) ≠ 0, V is not integrable, nor is V − cΘ(x) or V − cΘ(−x) for any constant c
❼ V attains both positive and negative values

By applying standard results from scattering theory and the theory of one-dimensional Schr¨
odinger operators
[100, p.63], if V possessed one of these properties, the spectrum of H would be easily obtained.
94

0.1

Kohn-Sham Potential V (x)

V (x)
V (−∞)

0
V (x)

−0.1
−0.2

−0.3
−20

−15

−10

−5
x

0

5

10

Figure 6.1: Kohn-Sham Potential V . Depicted for rs = 3 is the Kohn-Sham potential V = φ +Vxc (n0 )
and the limit value V (−∞). n0 and φ are the ones shown in Figures 5.6 and 5.8, respectively.

Figure 6.2 below shows the detail of the potential and showcases the oscillations in V due to the Friedel
oscillations of density. The diagram depicts just rs = 3. Other rs values of interest have similar potentials,
although there are quantitative differences, of course. For rs = 2, for example, the smaller and less pronounced
oscillations in the density lead to a corresponding effect in the potential.

−0.24

V (x)
V (−∞)

−0.24

V (x)

−0.25
−0.25

Kohn-Sham Potential V (x) (Detail)

−0.25
−10

−9

−8

−7

−6

x

−5

−4

−3

−2

Figure 6.2: Detail of Kohn-Sham Potential V . This highlight of V (for rs = 3) shows the oscillations
in the Kohn-Sham potential due to the Friedel oscillations of n0 .

6.2.2

Review of Functional Analysis

This section briefly reviews important background from functional analysis. For an in-depth discussion, the
reader is directed to consult one of the many texts on the subject, such as Rudin [92] or Conway [18]. The
theorem and definition statements appearing throughout Section 6.2 are adapted from Teschl [105].
Definition 6.1 (Spectrum). Let B be a Banach space over C and A ∶ D → B be a linear operator defined

on a dense subspace D ⊆ B. The spectrum of A is the set

σ(A) = {λ ∈ C ∶ A − λ does not have a bounded inverse}
95

The operator A − λ can fail to have a bounded inverse in several different ways:
−1

1. (A − λ)

exists but is not bounded

2. range(A − λ) ≠ B or range(A − λ) is not dense in B and A − λ is not bounded below.

3. A − λ is not one-to-one

If A is bounded, then its inverse, if it exists, must also be bounded, so item 1 is only possible for unbounded

operators (e.g., any differential operator). If item 3 fails, then there exists ψ ∈ B/{0} such that (A − λ)ψ = 0;

in this case, λ is called an eigenvalue and ψ is an eigenstate (physics) or eigenvector (mathematics).
The spectrum can be decomposed into several important subsets, defined below.
Definition 6.2 (Point Spectrum). The point spectrum σpt (A) is the set of eigenvalues of A.

Definition 6.3 (Essential Spectrum). The essential spectrum, denoted σess (A), is the spectrum minus
the set of eigenvalues that are isolated points with finite-dimensional eigenspace:

σess (A) = σ(A)/{λ ∶ 0 < dim range(A − λ) < ∞ and ∃r > 0 s.t. Br (λ)/{λ} ∩ σ(A) = ∅}

6.2.3

Characterization of the Spectrum of H

Definition 6.3 of the essential spectrum is not terribly useful, as it requires knowledge of all eigenvalues and
their eigenspaces. In practice, such information is incredibly difficult to obtain. An important result credited
to Weyl concisely characterizes the essential spectrum of a self-adjoint operator defined on a dense subspace
of a Hilbert space in a way that can be easily checked.
Definition 6.4 (Symmetric and Self-adjoint Operators). Let A ∶ D → H be a densely-defined operator.

A is called symmetric if for all ψ, ϕ ∈ D,
The adjoint operator A∗ is defined by

⟨ψ, Aϕ⟩H = ⟨Aψ, ϕ⟩H

˜ ϕ⟩ ∀ϕ ∈ D(A)}
D(A∗ ) = {ψ ∈ H ∣ ∃ψ˜ ∈ H ∶ ⟨ψ, Aϕ⟩H = ⟨ψ,
H
A∗ ψ = ψ˜

If A = A∗ , then the operator is said to be self-adjoint.
96

Remark 6.1. While all self-adjoint operators are symmetric, the converse is not true. The domain of the
adjoint operator may be different from that of the original operator. A self-adjoint operator is therefore a
symmetric operator such that D(A) = D(A∗ ).

Remark 6.2. Although not a direct consequence of the definition, self-adjoint operators have σ(A) ⊆ R.

∎

Lemma 6.2 (Weyl Criterion). λ ∈ σess (A) if and only if there is a sequence ψn such that ∣∣ ψn ∣∣ = 1,

⟨ψm , ψn ⟩ = 0 for n ≠ m, and ∣∣ (A − λ)ψn ∣∣ → 0. Such a sequence is called a singular Weyl sequence for λ.
Before proceeding to the critical result in Theorem 6.1, we need another definition and lemma.

Definition 6.5 (Relatively bounded). An operator B ∶ D(B) → H is called relatively bounded with
respect to another operator A ∶ D(A) → H if D(A) ⊆ D(B) and there are constants a, b ≥ 0 such that
∣∣ Bψ ∣∣H ≤ a ∣∣ Aψ ∣∣H + b ∣∣ ψ ∣∣H

for all ψ ∈ D(A). The A-bound of B is the value

a∗ = inf {a ∣ ∃b ≥ 0 such that ∣∣ Bψ ∣∣H ≤ a ∣∣ Aψ ∣∣H + b ∣∣ ψ ∣∣H ∀ψ ∈ D(A)}

Remark 6.3. If B is a bounded operator, then it is necessarily relatively bounded with respect to any other

operator A (then D(B) = H and we can take a = 0).

∎

Lemma 6.3 (Kato-Rellich). Suppose that A is self-adjoint and B is symmetric and relatively bounded
with A-bound less than one. Then A + B, D(A + B) = D(A) is also self-adjoint.

The Weyl Criterion and Kato-Rellich lemma are the easiest route to proving the result below, which appears
as an exercise in [105, p.148].
∂
2
Theorem 6.1. Let q ∈ L∞ (R) be real-valued and A = − 21 ∂x
2 + q(x) with domain D = H (R). Then A is
2

self-adjoint. If

−u′′ (x) + q(x)u(x) = λu(x)

has a solution for which u and u′ are bounded but u ∉ L2 (R), then λ ∈ σess (H).

∂
Proof. First, we prove that A is self-adjoint. Define the free Schr¨
odinger operator H0 ≜ − 21 ∂x
2 and the
2

multiplication operator Qψ(x) ≜ q(x)ψ(x). It is well known that H0 is self-adjoint when D(H0 ) = H 2 (R)
[105, p.168]. Because q ∈ L∞ (R), Q is a bounded operator on L2 (R):

∣∣ Qψ ∣∣L2 (R) = ∫ ∣q(x)ψ(x)∣ dx ≤ ∣∣ q ∣∣L∞ (R) ∣∣ ψ ∣∣L2 (R)
2

2

R

97

2

2

(6.13)

Furthermore, because q is real-valued, A is also symmetric: for any ψ, ϕ ∈ L2 (R)
∗

⟨ψ, Qϕ⟩L2 (R) = ∫ ψ(x)(q(x)ϕ(x)) dx = ∫ (q(x)ψ(x))ϕ∗ (x) dx = ⟨Qψ, ϕ⟩L2 (R)
R

R

Consequently, by Remark 6.3 and the Kato-Rellich lemma, A is self-adjoint with domain D(A) = H 2 (R).

Now, let (u, λ) be a solution and value satisfying the properties of the theorem statement. For n ∈ N, let

Kn ≜ [−n, n] and ϕn ∈ Cc∞ (R) be such that

⎧
⎪
⎪
⎪
⎪ 1
ϕn (x) = ⎨
⎪
⎪
⎪
0
⎪
⎩

x ∈ Kn
x ∉ Kn

A continuous function meeting the above specifications exists by Urysohn’s lemma [90, p.179]; a Cc∞ version
can be constructed by convolving with a standard mollifier [26, p.629]. Define
ψn ≜

1
u ⋅ (ϕn+1 − ϕn )
cn

where cn is chosen so that ∣∣ ψn ∣∣L2 (R) = 1, which is well defined because u ∈ L∞ (R). Note that
cn = ∣∣ u ⋅ (ϕn+1 − ϕn ) ∣∣L2 (R) ≤ ∣∣ u2 ∣∣L∞ (R) ∣∣ ϕn+1 − ϕn ∣∣L2 (R)

1/2

= ∣∣ u ∣∣L∞ (R) [∫ (ϕn+1 − ϕn ) dx]
2

2

R

= ∣∣ u ∣∣L∞ (R) [∫
2

−n

−n−1

dx + ∫

n+1

n

1/2

dx]

=

√

2 ∣∣ u ∣∣L∞ (R)
2

so the cn are uniformly bounded. Next, observe that ψn forms an orthonormal sequence, as ψn and ψm have
disjoint support whenever n ≠ m. Because

∂2
[uϕn ] = u′′ ϕn + 2u′ ϕ′n + uϕ′′n
∂x2

we have
cn (A − λ)ψn = (A − λ)u ⋅ (ϕn+1 − ϕn ) − 2u′ ϕ′n+1 − uϕ′′n+1 + 2u′ ϕ′n + uϕ′′n
= −2u′ ⋅ (ϕ′n+1 − ϕ′n ) − u ⋅ (ϕ′′n+1 − ϕ′′n )

98

because (A − λ)u = 0. As a consequence of (6.13), we have the estimate
∣∣ (A − λ)ψn ∣∣L2 (R) ≤

√

2 ∣∣ u′ ∣∣L∞ (R) ∣∣ ϕ′n+1 − ϕ′n ∣∣L2 (R) +
2

√

2 ∣∣ u ∣∣L∞ (R) ∣∣ ϕ′′n+1 − ϕ′′n ∣∣L2 (R)
2

Because ϕ′n and ϕ′′n vanish outside of very small regions near the points n and −n, we have ϕ′n → 0 and
ϕ′′n → 0 pointwise as n → 0. An application of Fatou’s Lemma [91, p.23] gives
∣∣ (A − λ)ψn ∣∣L2 (R) → 0

Therefore, ψn forms a singular Weyl sequence for λ, so λ ∈ σess (H).

∎

Corollary 6.2. [V (−∞), +∞) ⊂ σess (H). In particular, each εk defined by (5.11) is an element of σess (H).

Proof. Let ε ∈ [V (−∞), +∞), then µ2 ≥ 0. Consequently, following Theorem 5.1 the solution u corresponding
to ε is oscillatory as x → −∞; hence, it is is bounded near −∞ but not square integrable. Since ν 2 < 0 as
well, u can be chosen to decay exponentially as x → +∞, so it is bounded everywhere.

Theorem 6.1 motivates a defining a new concept highly applicable to the Kohn-Sham wavefunctions ψk :

∎

Definition 6.6 (Extended Eigenvector). For λ ∈ σess (A)/σpt (A), ψ is called an extended eigenvector
(eigenstate) if (A − λ)ψ = 0 but ψ ∉ D(A).

There are a number of issues in the above definition, chief among them is that if ψ ∉ D(A), then to which
space does it belong? Is it even possible to define A on another space? These matters can only be by

considering the specifics of the space H , the operator A, and its domain D.

In the proof of Theorem 6.1, we were able to construct a Weyl sequence by beginning with a classical solution
u ∈ C 2 (R) ∩ L∞ (R). The construction fundamentally depended on the ability to define the operator A on a

space different but “not too different” from H 2 (R). In this case, we were able work with Cc∞ (R) functions
because both L2 (R) and H 2 (R) are completions of Cc∞ (R) in their respective norms.

In essence, what we have done is replace the original Hilbert space with what is known as a rigged Hilbert
Space. Without digressing into the rigors, the idea to take a subspace Φ ⊆ H with a continuously embedded
topology. By using the inner product on H to form linear functionals and identifying H with its continuous
dual space H ∗ by Riesz Representation, we extend H by sandwiching between dual spaces: Φ ⊆ H ⊆ Φ∗ .
In the context of H 2 (R), we consider the dense subspace Φ = Cc∞ (R), whose dual space is the set of ordinary
distributions. It is i.s.d. under which extended eigenvectors are to be understood.

99

Remark 6.4. Per Corollary 6.2, each ψk is an extended eigenvector for H corresponding to εk .

We return to “regular” eigenvalues and eigenvectors and characterize the point spectrum of H:

∎

Theorem 6.3. pt spec(H) = ( − ∞, V (−∞))

Proof. The proof is by construction of an eigenvector. Let ε ∈ ( − ∞, V (−∞)). Invoking the highly useful
Theorem 5.1, the asymptotic behavior of solutions is

⎧
iµx
−iµx
⎪
⎪
⎪
⎪ ae + be
ψ∼⎨
⎪
iνx
−iνx
⎪
⎪
⎪
⎩ ce + de

x → −∞

By (6.4), it is clear that µ2 < 0. Put τ ≜ ε − V (−∞) < 0. Then

x → +∞

1 2
ν = ε − V (+∞) = ε − V (−∞) + V (−∞) − V (+∞)
2
= τ − ∆V < 0

because ∆V > 0 as well. Therefore, if we set a = 0 and d = 0, we will have a solution that behaves like
⎧
−iµx
⎪
⎪
⎪
⎪ be
ψ∼⎨
⎪
iνx
⎪
⎪
⎪
⎩ ce

x → −∞
x → +∞

so that ψ decays exponentially as ∣x∣ → +∞, meaning that ψ ∈ L2 (R). Now because ψ ′′ = 2(V (x) − ε)ψ and
as we saw in the proof of Theorem 6.1, V − ε ∈ L∞ (R)
Because the H 2 norm can be taken to be

⇒ 2(V (x) − ε)ψ ∈ L2 (R), we have ψ ′′ ∈ L2 (R).

∣∣ ψ ∣∣H 2 (R) = ∣∣ ψ ∣∣L2 (R) + ∣∣ ψ ′′ ∣∣L2 (R)
2

2

2

we can conclude ψ ∈ H 2 (R) for each ε < V (−∞), meaning each such ε is an eigenvalue.

By the proof of Corollary 6.2, if ε ≥ V (−∞), then we cannot find a square integrable solution to (H − ε)ψ = 0

because we can only choose two values of a, b, c, and d to set to zero. Because ν 2 < 0, to have any chance
of integrability, d must be zero. But then we will be left with at least non-decaying (but bounded) term on

the left, so we cannot find a square integrable solution for ε ≥ V (−∞).
Corollary 6.4. The spectrum of H is all of R.

100

∎

The discrete spectrum consists of all isolated eigenvalues whose eigenspaces are finite-dimensional; it is the
complement of the essential spectrum. As H has no isolated eigenvalues, its discrete spectrum is empty; the

essential spectrum, continuous spectrum, and spectrum are all the same sets for H.

6.2.4

Spectral Representation for G

Unlike in finite-dimensional spaces, it is not true that every self-adjoint operator A on a Hilbert space

possesses a complete set of eigenvectors. When it does, eigenvectors corresponding to distinct eigenvalues
are orthogonal and we have the completeness relation❸

∑ ∣ ψn ⟩⟨ψn ∣ = ˆ1
n

when ∣∣ ψn ∣∣H = 1 for each n and ˆ1 denotes the identity operator. Considering A on a rigged Hilbert space
and including extended eigenvectors, we have an equivalent completeness relation

∑ ∣ ψm ⟩⟨ψm ∣ + ∫
m

σess (A)

∣ ψλ ⟩⟨ψλ ∣ dλ = ˆ
1

(6.14)

where ∣ ψm ⟩ are in the discrete spectrum and ψλ is normalized to 1 in the rigged space. The identity is to be
taken in the rigged space; elements from the original space belong to the rigged space, but not vice versa.
If ψ ∶ R → C is continuous, then

∫ ψ(y)δ(x − y) dy = ψ(x)
R

By considering the duality pairing on distributions (which coincides with our rigged Hilbert space), we encode
the above relation in the symbol❹ ⟨x ∣, which acts on a state ∣ ψ⟩ as
⟨x ∣ ψ⟩ = ψ(x)

We can then write δ(x − y) = ⟨x ∣ y⟩. As in Economou [22, p.4], we may write
⟨x ∣ G(ε) ∣ y⟩ = G(x, y; ε)

(6.15)

⟨x ∣ H ∣ y⟩ = δ(x − y)H

where G(ε) and H are the operators such that

❸ This
❹ In

(H − ε)G(ε) = −ˆ
1

(6.16)

subsection employs Dirac notation. See Section 1.4 for an explanation.
quantum mechanics, ⟨x ∣ is the eigenvector of the position operator. Such abstract machinery is not needed here.

101

We now show that (6.16) is equivalent to (6.6) by computing the matrix elements
⟨x ∣ HG ∣ y⟩ − ε⟨x ∣ G(ε) ∣ y⟩ = ⟨x ∣ ˆ
1 ∣ y⟩ = −δ(x − y)
Inserting the completeness relation
1
∫ ∣ z⟩⟨z ∣ dz = ˆ
R

inbetween the H and G gives us

∫ ⟨x ∣ H ∣ z⟩⟨z ∣ G(ε) ∣ y⟩ dz − ε⟨x ∣ G(ε) ∣ y⟩ = −δ(x − y)
R

∫ δ(x − z)H⟨z ∣ G(ε) ∣ y⟩ dz − ε⟨x ∣ G(ε) ∣ y⟩ = −δ(x − y)
R

H⟨x ∣ G(ε) ∣ y⟩ dz − ε⟨x ∣ G(ε) ∣ y⟩ = −δ(x − y)

which is the same as (6.6) after restoring to coordinate representation via (6.15). The operator H is the

coordinate representation of H. Since we will work with (6.16) from here forward, we will instead write❸
G(ε) = (ε − H)−1 =

1
ε−H

For ε ∉ σ(H) = R, G(ε) is bounded, and therefore, continuous. In the completeness relation (6.14) for H, the

sum term is vacuous; inserting it into the above, we have
G(ε) =

1
1
∣ ψλ ⟩⟨ψλ ∣ dλ
∫ ∣ ψλ ⟩⟨ψλ ∣ dλ = ∫
ε−H R
R ε−H

For an analytic function f , Borel functional calculus [105, chap.3] defines f (A)∣ ψ⟩ = f (λ)∣ ψ⟩ whenever

(λ, ψ) are an extended eigenpair. Thus,

G(ε) = ∫

R

1
⋅ ∣ ψλ ⟩⟨ψλ ∣ dλ
ε−λ

which we may write in coordinate representation as

G(x, y; ε) = ∫

R

ψλ (x)ψλ∗ (y)
dλ
ε−λ

We are interested in evaluating the above integral for ε ∈ R, for which there is a nonintegrable singularity.
❸ The very right-hand side, being an arithmetic expression involving operators, cannot be taken literally, may be considered
an abuse of notation but is standard. This relation indicates that G(ε) is the resolvent of ε − H.

102

Therefore, the integral must be interpreted as a principal value. The Cauchy Principle Value
p.v. ∫

R

ψλ (x)ψλ∗ (y)
ψλ (x)ψλ∗ (y)
dλ = lim+ ∫
dλ
η→0
ε−λ
ε−λ
R/[ε−η,ε+η]

is one choice, but it is preferrable to use a branch cut by defining
p.v. ∫

R

ψλ (x)ψλ∗ (y)
ψλ (x)ψλ∗ (y)
dλ = lim+ ∫
dλ
η→0
ε−λ
ε − λ + iη
R

In this expression, η → 0+ is implicit and therefore we can drop the limit and write
G(x, y; ε) = ∫

R

ψλ (x)ψλ∗ (y)
dλ
ε − λ + iη

(6.17)

One can view the above as G(⋅, ⋅; ε + iη) with η taken as a positive infinitesimal in the original formula; this

resolves the issue with the singularity but requires us to evaluate at an “incorrect value.” Often, physicists
will often use the formula with finite η to incorporate the effects of phonon damping, which is the interaction
of electrons with the positively charged nucleus and full electron shell [72, p.157].

6.3

Conclusion

This chapter has two purposes: to introduce the Green’s function and to rigorize its spectral representation
(6.17). The Green’s function will be an important computational tool in the next chapter on linear response.
The spectral result (6.17) appears often in physics literature, but any justification is woefully omitted.
Establishment of this result required an examination into the spectrum of the underlying operator H and
involved many topics from advanced branches of mathematics. Perhaps such machinery is out of place in
physics texts but it is certainly most welcome in a thesis of mathematics.

103

Chapter

7

The Excited State: Linear Response
7.1

Time-Dependent Density Functional Theory

The subject of time-dependent density functional theory (TD-DFT) is a deep one despite its young age.
Ground state DFT can trace its genealogy to Thomas-Fermi theory of the 1920s and was formalized with
the Hohenberg-Kohn theorems in 1964. TD-DFT has no such illustrious pedigree and was not conceptualized
until 1984, sharing a birth year with the author of this thesis. Here, we explore just one technique from this
rapidly growing field; Marques et al. have compiled an excellent set of lecture notes [77] with contributions
from the pioneers of TD-DFT that surveys the wide variety of topics encompassed by TD-DFT.
1
First introduced way back in Section 3.3, let GL be the Green’s function to the Laplace operator L = − 4π
∇2

so that results generalize to Rd for d = 1, 2, 3. To remind the reader,
d=1∶

d=2∶

7.1.1

d=3∶

The Runge-Gross Theorem

GL (x − y) = −2π ∣x − y∣

GL (x − y) = −2 log ∣x − y∣
GL (x − y) = ∣x − y∣

−1

As discussed in Section 3.2, ground state DFT is powered by the two Honenberg-Kohn theorems, without
which the field would not exist. In the same vein, TD-DFT owes its existence to three theorems from the
work of Erich Runge and Eberhard K. U. Gross, who published the seminal paper “Density Functional
Theory for Time-Dependent Systems” [94] exactly twenty years after the Hohenberg-Kohn Theorems.
Theorem 7.1 (Runge-Gross I). For a given initial state ∣ Ψ0 ⟩, for t ∈ [t0 , t1 ] there is a one-to-one

104

correspondence between the set of external potentials analytic in time,
∞

V = {V (x, t) ∣ V (x, t) = ∑
k=0

and the set of densities

(t − t0 )k ∂ k V
∣
}/{c(t) ∣ c ∶ [t0 , t1 ] → R}
k!
∂tk t=t0

N = {n(x, t) ∣ n(x, t) = ⟨Ψ(t) ∣ n(x) ∣ Ψ(t)⟩ and i

∂Ψ(x, t)
= HΨ(x, t), Ψ(x, t0 ) = ∣ Ψ0 ⟩ where
∂t

1
H = − ∇2 + ∫ n(x)V (x, t) dx + W for some V ∈ V}
2
Rd

The operator W quantifies the interaction of electrons among themselves. n(x) is the density operator. (It
will be explored in the coming section.)

Remark 7.1. The quotient space in the definition of V means that potentials differing by a time-varying

function cannot be distinguished by this theorem. This is the analog of the first Hohenberg-Kohn Theorem,

which can only identify potentials up to an additive constant. The “constant” is always with respect to
spatial variance, which in the case of TD-DFT, means time-varying functions.

∎

Remark 7.2. It is insufficient to specify the initial density n0 (x) = n(x, t0 ) = ⟨Ψ0 ∣ n(x) ∣ Ψ0 ⟩. The initial

state ∣ Ψ0 ⟩ is what is necessary, unless the system is in its ground state until time t0 . At t0 , the Honenberg-

Kohn Theorem would apply, allowing creation of a unique state that corresponds to the given initial density.
This idea is, in essence, the backbone of the linear response theory of the next subsection.

∎

One might suppose that the theorem is proved in a manner similar to the Hohenberg-Kohn Theorems by
attempting to minimize the action integral
A(t0 , t1 ) ≜ ∫

t1
t0

̵ t − H ∣ Ψ(t)⟩ dt
⟨Ψ(t) ∣ ih∂

(7.1)

Unfortunately, the Rayleigh-Ritz variational principle used to prove the Hohenberg-Kohn Theorems fails
because (7.1) has no minimum, so the proof of the Runge-Gross Theorem is significantly more involved. We
omit the proof here and refer the interested reader to either the original paper or Englel and Dreizler [24,
p.311], whose proof is slightly more comprehensive and detailed than that of Runge and Gross.
While variational principles cannot be used to prove the one-to-one correspondence, a consequence of this
bjiection is that states can be constructed from the density up to some phase function:
∣ Ψ(t)⟩ = e−iα(t) ∣ Ψ[n, Ψ0 ](t)⟩
105

Consequently, we do have a variational principle:
Theorem 7.2 (Runge-Gross III). If the potential V is chosen such that V − W ≠ c(t) for any W ∈ V, the
action integral is a functional of the density:

A[n] = B[n] − ∫

t1
t0

∫Rd n(x, t)V (x, t) dx dt

The functional B is a universal functional of the density in the sense that the same dependence on n holds

for all such choices of V . The exact density of the system is given by the stationary point of the system, or
the solution to the Euler-Lagrange equation
δA
(x, t) = 0
δn
The reader has surely noticed that we proceeded from Runge-Gross I to Runge-Gross III. Despite omission of
an intermediate theorem appearing in the original work, the labeling of Runge and Gross has been retained.

7.1.2

Linear Response Theory

Applying Runge-Gross III, we can write the density n(x, t) as a functional of the external potential Vext :
n(x, t) = n[Vext ](x, t)

(7.2)

Linear response theory (LRT) considers potentials of the form

⎧
⎪
⎪
⎪
⎪ V0 (x)
Vext (x, t) = ⎨
⎪
⎪
⎪
V (x) + V1 (x, t)
⎪
⎩ 0

t ≤ t0

t > t0

= V0 (x) + V1 (x, t)Θ(t − t0 )

The system is in the ground state up to time t0 , at which time a time-varying potential turns on. This
describes precisely a host of physical situations, among them incident laser fields in particular. More importantly, though, is that this switching allows the ground state density to uniquely determine the initial state
of the time-varying one, as noted in Remark 7.2.
It is more notationally convenient and better suited to our end to perform the derivation of LRT in frequency
domain; for a time-domain derivation, see, for example, Gross, Dobson, and Petersilka [45, p.101]. After a
Fourier transform in time, LRT begins by writing a functional Taylor series for (7.2):

106

n(x, ω) = n0 (x) + n1 (x, ω) + n2 (x, ω) + ⋯

(7.3)

n0 is the ground state density under the potential V0 ; it may be found using the KS-DFT of Chapter 5. As
before, V0 is the positive background charge due to the nuclei sans valence electrons.
Following Taylor’s Theorem for functionals, the first frequency-dependent term of (7.3) is given by
n1 (x, ω) = ∫

χ(x, y, ω) ≜

Rd

χ(x, y, ω)V1 (y, ω) dy

δn[Vext ](x, ω)
∣
δVext (y, ω) V =V0

(7.4)
(7.5)

If we had a closed-form expression for n[Vext ], then we would be able to perform the above integration and

obtain the linear density immediately. Unfortunately, all the functionals encountered so far have been in
terms of n. In principle, these functionals can be inverted, but this is, practically speaking, impossible to
execute. Instead, we take a different course of action and consider the Kohn-Sham potential
V (x) = Vext (x) + ∫

Rd

GL (x − y)n(y) dy + Vxc (n(x))

(7.6)

Our goal is to use (7.6) and make repeated use of the variational chain rule to transform χ into something
we can actually compute explicitly. We begin with
δn[Vext ](x, ω)
δn[V ](x, ω) δV (x′ , ω)
=∫
dx′
′
δVext (y, ω)
Rn δV (x , ω) δVext (y, ω)
= ∫ χ1 (x, x′ , ω)

where we have defined
χ1 (x, y, ω) ≜

δV (x′ , ω)
dx′
δVext (y, ω)

δn[V ](x, ω)
∣
δV (y, ω) V =V0

(7.7)

(7.8)

While we do not have a closed form expression for the functional n[V ], χ1 can be obtained by appealing to
the underlying state and using standard perturbation theory. This is the subject of the next section.

The remaining variational derivative in the integrand of (7.7) will come from the chain rule. For a fixed x,
we may treat V of (7.6) as a functional of n. Consequently, by the variational chain rule for functionals,
δV (x′ , ω)
δV (x′ , ω) δn[Vext ](x′′ , ω) ′′
=∫
dx
′′
δVext (y, ω)
δVext (y, ω)
Rd δn(x , ω)
= δ(x′ − y) + ∫

Rd

(GL (x′ − x′′ ) +
107

δVxc [n](x′ ) δn(x′′ , ω)
)
dx′′
δn(x′′ )
δVext (y, ω)

(7.9)

The δ comes from the special case of the variational chain rule for functionals described in Appendix A
applied to the first term from (7.6):
δ(x′ − y) = ∫

We recall (7.5) and define

Rd

δVext (x′ , ω) δn(x′′ , ω)
dx′′
δn(x′′ , ω) δVext (y, ω)

K(x, y, ω) ≜ GL (x − y) + fxc (x, y, ω)

fxc (x, y, ω) ≜

δVxc [n](x, ω)
∣
δn(y, ω)
n=n0

(7.10)
(7.11)

It is tempting to jump to the conclusion that fxc is as in (3.31). For the moment, the reader should put that
connection aside and pretend that (7.11) is different. With these, (7.9) simplifies to
δV (x′ , ω)
= δ(x′ − y) + ∫ K(x′ , x′′ , ω)χ(x′′ , y, ω) dx′′
δVext (y, ω)
Rd

(7.12)

Next, we substitute (7.12) into (7.7) and equate the result with (7.5)
χ(x, y, ω) = χ1 (x, y, ω) + ∫

Rd

χ1 (x, x′ , ω) ∫

Rd

K(x′ , x′′ , ω) χ(x′′ , y, ω) dx′′ dx′

(7.13)

Finally, we multiply both sides of (7.13) by V1 (y, ω) and integrate over y:
∫

Rd

χ(x, y, ω)V1 (y, ω) dy = ∫

Rd

χ1 (x, y, ω)V1 (y, ω) dy

+∫

Rd

V1 (y, ω) ∫

Rd

χ1 (x, x′ , ω) ∫

Rd

K(x′ , x′′ , ω)χ(x′′ , y, ω) dx′′ dx′ dy

≜I

We freely interchange the order of integration within I, performing the y integral first:
I=∫

=∫

Rd
Rd

χ1 (x, x′ , ω) ∫

χ1 (x, x′ , ω) ∫

Rd
Rd

K(x′ , x′′ , ω) [∫

Rd

V1 (y, ω)χ(x′′ , y, ω)dy] dx′′ dx′

K(x′ , x′′ , ω)n1 (x′′ , ω) dx′′ dx′

after using (7.4). This finally gives the expression (after some switching of dummy variables)
n1 (x, ω) = ∫

=∫

Rd
Rd

χ1 (x, y, ω) [V1 (y, ω) + ∫

Rd

χ1 (x, y, ω) V1,scf (y, ω) dy

108

K(y, y′ , ω) n1 (y′ , ω) dy′ ] dy

(7.14)

where V1,scf (y, ω) denotes the self-consistent potential
V1,scf (y, ω) ≜ V1 (y, ω) + ∫

7.1.3

Rd

K(y, y′ , ω)n1 (y′ , ω) dy′

(7.15)

Adiabatic Local Density Approximation (ALDA)

The exchange-correlation functional appearing in fxc should, in principle, depend on the frequency. In the
derivation of the preceding section, it appears we pulled a Fourier transform inside a nonlinear functional
F{n[Vext ](x, ⋅)}(ω) = n[F{Vext (x, ⋅)}(ω)] = n[V0 (x) + V1 (x, ω)]

when we implicitly assumed that n0 (x) corresponds to V0 (x) and n1 (x, ω) corresponds to V1 (x, ω). The

hang-up is the exchange-correlation portion, as it is the only nonlinear piece of the functional. To be proper,
we should have written the functional Taylor series in time-domain and then transformed at the end.
Runge-Gross III speaks of a universal functional B[n]; much like the second Hohenberg-Kohn theorem, it
says nothing of what it looks like. Recalling KS-DFT, one might imagine that B looks something like
B[n] = Ts [n] +

1 t1
∫ ∫ ∫ W(x − y)n(x, t)n(y, t) dx dy dt + Axc [n]
2 t0 Rd Rd

where Axc [n] is a time-dependent version of the exchange-correlation functional. The extension of Ts [n] to

time-varying systems is easy to visualize, but then the question becomes: how can one incorporate explicit
time-dependence into exchange and correlation? Formulating reasonable approximations in the ground state

is a daunting enough task. An easy solution is to take existing exchange-correlation functionals and merely
replace the ground-state density with the time-varying one:
AALDA
[n] = ∫
xc

t1

t0

∫

Rd

n(x, t)

xc (n(x, t)) dx dt

Then the first variation is given by
δAALDA
[n](x, t)
∂[n xc (n)]
xc
= δ(t − τ )δ(x − y)
∣
δn(y, τ )
∂n
n=n(x,t)
= δ(t − τ )

δExc [n](x)
∣
δn(y)
n=n(x,t)

(7.16)

This replacement of time-varying functionals with static ones is known as the adiabatic local density approximation (ALDA). Although it is quite robust and sufficient in most applications of TD-DFT, it fails for some

109

computations, such as those involving charge transfer [73] and double excitations [74, 108].
Inspired by (7.16), if we now write the functional Taylor series for Exc [n] to first order,
Exc [n0 + n1 ] = Exc [n0 ] + ∫

Rd

δExc [n](x)
∣
n1 (y, t) dy
δn(y)
n=n0

after taking the first variation and using that
δExc [n](x)
= Vxc (n0 (x))δ(x − y)
δn(y)
we have

(7.17)

⎡
⎤
δAALDA
[n](x, t)
δVxc [n](x)
⎢
⎥
xc
⎥
= δ(t − τ )δ(x − y) ⎢Vxc (n0 (x)) + n1 (y, t)
∣
⎢
⎥
δn(y, τ )
δn(y)
⎣
n=n0 ⎦

or after Fourier transforming in τ ↦ ω ′ and then t ↦ ω and recognizing (7.11)

δAALDA
[n](x, ω)
xc
= δ(x − y)δ(ω − ω ′ )[Vxc (n0 (x)) + n1 (y, ω)fxc (x, y)]
δn(y, ω ′ )

which is precisely the result of the previous section. The ω argument has been removed from fxc because it
is a static kernel. The δ’s will disappear when x = y and ω = ω ′ are taken in the Euler-Lagrange equation.
This discussion has provided insight to simplify (7.15). In light of (7.17), we can compute fxc from (7.11)
fxc (x, y) =

∂Vxc
∣
δ(x − y)
∂n n=n0 (x)

′
(n0 (x))δ(x − y)
= Vxc

(7.18)

which upon substitution into (7.15) yields
′
(n0 (y))n1 (y, ω) + ∫
V1,scf (y, ω) = V1 (y, ω) + Vxc

Rd

GL (y − y′ )n1 (y′ , ω) dy′

Rather than integrate against the Green’s function for the Laplace operator, we can solve the corresponding
differential equation. Let φest be the solution to
∇2 φest (x, ω) = −4πn1 (x, ω)

(7.19)

We denote it “est” because it represents the electrostatic potential corresponding to the linear response
density. To symmetrize our notation in subscripted varieties of φ, define

110

φext (x, ω) ≜ V1 (x, ω)

′
(n0 (x))n1 (x, ω)
φxc (x, ω) ≜ Vxc

(7.20)
(7.21)

In terms of these newly defined potentials, we can write the equation (7.14) for n1 (x, ω) instead as
n1 (x, ω) = ∫

Rd

χ1 (x, y, ω)[φext (y, ω) + φest (y, ω) + φxc (y, ω)] dy

(7.22)

Finally, perhaps much to the relief of the anxious reader, we note that the fxc of this section is, in fact, the
same one we computed way back in (3.31) and have used many times since. And therefore we have
φxc (x, ω) = fxc (x)n1 (x, ω)

7.2

(7.23)

Linear Response Function

In this section, we derive a spectral (“Lehmann”) representation for the linear response function χ1 in terms
of ground state orbitals. We will begin in the time-domain and derive a frequency-domain representation in
Section 7.2.1. From there, we will find an expression using Kohn-Sham orbitals in Section 7.2.2. Finally, we
will arrive at the response function specific to the jellium model in the final subsection.

7.2.1

Frequency Domain Representation

While the primary sources include Wehrum and Hermeking [114], Fetter and Walecka [35], and Senatore and
Subbaswamy [97], the derivation here follows the outline of Gross and Maitra [46], filling in all the details left
out of their presentation. We begin with the time-domain representation of the first-order response function:
X1 (x, y, t − τ ) = −iΘ(t − τ )⟨Ψ0 ∣ [n(x, t), n(y, τ )] ∣ Ψ0 ⟩

(7.24)

where brackets denote the standard commutator of two (self-adjoint) operators
[A, B] = AB − BA,

Ψ0 denotes the t = 0 state, and n(x, t) denotes the density operator in the Heisenberg picture [11, p.88]
n(x, t) = eiHt n(x)e−iHt
ˆk)
n(x) = ∑ δ(x − x
k

111

(7.25)
(7.26)

We have overloaded the symbol n, using it for both time-dependent and independent operators, as the latter
is the former with t set to zero. The notation for n(x) is perhaps a bit perplexing. As we shall see when we
apply it to states, the notation implicitly encodes how it operates.
What we seek is the linear response function, which is the angular frequency Fourier transform❸ of X1 :
χ1 (x, y, ω) = F{X1 }(ω)

We begin by analyzing the commutator term of (7.24). Expanding by definition, we have
[n(x, t), n(y, τ )] = n(x, t)n(y, τ ) − n(y, τ )n(x, t)

Because the two terms are merely permutations of (x, t) and (y, τ ), we need only examine one of them and
assert the result for the second. By inserting the completeness relation over excited (exc) states

∑ ∣ Ψj ⟩⟨Ψj ∣ = ˆ1
exc

(7.27)

j

we have

⟨Ψ0 ∣ n(x, t)n(y, τ ) ∣ Ψ0 ⟩ = ⟨Ψ0 ∣ n(x, t) ˆ
1 n(y, τ ) ∣ Ψ0 ⟩

= ∑ ⟨Ψ0 ∣ n(x, t) ∣ Ψj ⟩⟨Ψj ∣ n(y, τ ) ∣ Ψ0 ⟩
exc
j

In deriving the spectral representation for the Green’s function at the beginning of Section 6.2.4, we discussed
extended states and rigged Hilbert spaces. The derivation of this section is also done in this context, although
no appeal to the structures of the underlying L2 or distribution space will be necessary.
Using the definition of the time-dependent density operator (7.25), the first matrix element becomes
⟨Ψ0 ∣ n(x, t) ∣ Ψj ⟩ = ⟨Ψ0 ∣ eiHt n(x)e−iHt ∣ Ψj ⟩

= ⟨e−iHt Ψ0 ∣ n(x) ∣ e−iHt Ψj ⟩

Because HΨj = Ej Ψj , by Borel functional calculus again, we have e−iHt Ψj = e−iEj t Ψj , so
⟨Ψ0 ∣ n(x, t) ∣ Ψj ⟩ = ⟨e−iE0 t Ψ0 ∣ n(x) ∣ e−iEj t Ψj ⟩

❸ See

= ei(E0 −Ej )t ⟨Ψ0 ∣ n(x) ∣ Ψj ⟩

Section 1.3 for the definition of this transform. It does not conform to usual mathematical conventions.

112

(7.28)

Therefore, using the transitivity of symbols, we have
⟨Ψ0 ∣ n(x, t)n(y, τ ) ∣ Ψ0 ⟩ = ∑ ei(E0 −Ej )(t−τ ) ⟨Ψ0 ∣ n(x) ∣ Ψj ⟩⟨Ψj ∣ n(y) ∣ Ψ0 ⟩
exc
j

Applying this result to (7.24), we have

X1 (x, y, t, τ ) = ∑ −iΘ(t − τ )[ei(E0 −Ej )(t−τ ) ⟨Ψ0 ∣ n(x) ∣ Ψj ⟩⟨Ψj ∣ n(y) ∣ Ψ0 ⟩
exc
j

− ei(Ej −E0 )(t−τ ) ⟨Ψ0 ∣ n(y) ∣ Ψj ⟩⟨Ψj ∣ n(x) ∣ Ψ0 ⟩]

Finally, because we assume the perturbation turns on adiabatically [48, p.14], we insert an η → 0+ infinitesimal
term into the exponentials and take the Fourier transform with respect to t′ = t − τ :
F{Θ(t′ )eηt ei(E0 −Ej )t }(ω) =
′

′

1
1
=
i(ω − (E0 − Ej )) − η i(ω + (Ej − E0 ) + iη)

(7.29)

Taking χ1 = F{X1 } and applying the above, we obtain the general spectral representation

⎡ ⟨Ψ0 ∣ n(x) ∣ Ψj ⟩⟨Ψj ∣ n(y) ∣ Ψ0 ⟩ ⟨Ψ0 ∣ n(y) ∣ Ψj ⟩⟨Ψj ∣ n(x) ∣ Ψ0 ⟩ ⎤
⎢
⎥
⎥
−
χ1 (x, y, ω) = ∑ ⎢
⎢
⎥
ω − (Ej − E0 ) + iη
ω + (Ej − E0 ) + iη
⎦
j ⎣
exc

7.2.2

(7.30)

Spectral (“Lehmann”) Representation using Kohn-Sham Orbitals

Within the TD-DFT LRT framework, the wavefunctions of (7.30) are expressed in terms of Kohn-Sham
orbitals ψk . Because the system is noninteracting, Ψ0 is a Slater determinant❸ [89, p.14]; in Dirac notation,
∣ Ψ0 ⟩ = ∣ ψ1 ψ2 ⋯ψN ⟩
or in coordinate notation

1
Ψ0 (x1 , x2 , . . . , xN ) = √ det
N!

ψ1 (x1 )

ψ1 (x2 )

⋯

ψ1 (xN )

ψN (x1 )

ψN (x2 )

⋯

ψN (xN )

ψ2 (x1 )
⋮

ψ2 (x2 )
⋮

⋯
⋱

= ∑sgn(P )ψ1 (xk1 )ψ2 (xk2 )⋯ψN (xkN )

ψ2 (xN )
⋮

P

❸ In Section 3.3, we associated the ground state wavefunction with a Slater determinant. There, we constructed a noninteracting system with such a wavefunction. Here, we assert the reverse: that Ψ for a noninteracting system has this form.

113

P = {k1 , k2 , . . . , kN } is a permutation of {1, 2, . . . , N } and sgn(P ) = (−1)s(P ) , where s(P ) counts the number
of switches from numerical order in P . The energy of the ground state is then
E0 = ∑εk
k

An excited state Ψj is generated by moving an electron from the occupied orbital ψm to an unoccupied one
ψj . The resultant wave function is
∣Ψj ⟩ = ∣ψ1 ⋯ψm−1 ψj ψm+1 ⋯ψN ⟩
and energy of the excited state is
Ej = εj +

which means that the change in energy is given by

∑ εk

k≠m

Ej − E0 = εj − εm
7.2.2.1

(7.31)

Bra-kets Involving n(x): Two-electron Systems

To resolve the bra-ket terms, such as ⟨Ψj ∣ n(x) ∣Ψ0 ⟩, it is illustrative to derive the result for a two-electron

system. Larger systems result in an unwieldy number of terms (a Slater determinant contains N ! terms),
which in turn requires cumbersome and difficult-to-read notation to represent compactly.
When N = 2, the wavefunction of the initial state is

1
Ψ0 (x1 , x2 ) = √ (ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 ))
2

Generate an excited state by moving an electron from 1 ↦ j. Then denoting Ψj ≡ Ψ1↦j
1
Ψ1↦j (x1 , x2 ) = √ (ψj (x1 )ψ2 (x2 ) − ψj (x2 )ψ2 (x1 ))
2

We have n(x) = δ(x − x1 ) + δ(x − x2 ), so let us concentrate on the first term. We have

(7.32)

1
⟨Ψ1↦j ∣ δ(x − x1 ) ∣Ψ0 ⟩ = ⟨ψj (x1 )ψ2 (x2 ) − ψj (x2 )ψ2 (x1 ) ∣ δ(x − x1 ) ∣ ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 )⟩
2
1
= [A1 + B1 + C1 + D1 ]
2

114

with
A1 ≜ ⟨ψj (x1 )ψ2 (x2 ) ∣ δ(x − x1 ) ∣ ψ1 (x1 )ψ2 (x2 )⟩

C1 ≜ ⟨ψj (x2 )ψ2 (x1 ) ∣ δ(x − x1 ) ∣ ψ1 (x1 )ψ2 (x2 )⟩

B1 ≜ ⟨ψj (x1 )ψ2 (x2 ) ∣ δ(x − x1 ) ∣ ψ1 (x2 )ψ2 (x1 )⟩

We first compute A1 :

A1 = ∫

R3

= (∫

∫

R3

R3

D1 ≜ ⟨ψj (x2 )ψ2 (x1 ) ∣ δ(x − x1 ) ∣ ψ1 (x2 )ψ2 (x1 )⟩

ψj∗ (x1 )ψ2∗ (x2 )ψ1 (x1 )ψ2 (x2 )δ(x − x1 ) dx1 dx2

ψj∗ (x1 )ψ1 (x1 )δ(x − x1 ) dx1 ) (∫

= ψj∗ (x)ψ1 (x)

R3

since ⟨ψ2 ∣ ψ2 ⟩ = 1. We proceed to B1 :
B1 = (∫

R3

ψj∗ (x1 )ψ2 (x1 )δ(x − x1 ) dx1 ) (∫

R3

ψ2∗ (x2 )ψ2 (x2 ) dx2 )

ψ2∗ (x2 )ψ1 (x2 ) dx2 ) = 0

because ⟨ψ2 ∣ ψ1 ⟩ = 0. C1 and D1 are similar to the previous case:
C1 = (∫

D1 = (∫

R3
R3

ψ2∗ (x1 )ψ1 (x1 )δ(x − x1 ) dx1 ) (∫

ψ2∗ (x1 )ψ2 (x1 )δ(x − x1 ) dx1 ) (∫

R3
R3

again because both ⟨ψj ∣ ψ2 ⟩ = 0 and ⟨ψj ∣ ψ1 ⟩ = 0. Therefore,

ψj∗ (x2 )ψ2 (x2 ) dx2 ) = 0

ψj∗ (x2 )ψ1 (x2 ) dx2 ) = 0

1
⟨Ψ1↦j ∣ δ(x − x1 ) ∣ Ψ0 ⟩ = ψj∗ (x)ψ1 (x)
2

Next, we follow the same procedure to calculate
⟨Ψ1↦j ∣ δ(x − x2 ) ∣Ψ0 ⟩

Define A2 –D2 as before, except with the x1 in the δ replaced by x2 . Then
A2 = (∫

R3

C2 = (∫

R3

B2 = (∫

R3

ψj∗ (x1 )ψ1 (x1 ) dx1 ) (∫

R3

ψ2∗ (x1 )ψ1 (x1 ) dx1 ) (∫

R3

ψj∗ (x1 )ψ2 (x1 ) dx1 ) (∫

R3

ψ2∗ (x2 )ψ2 (x2 )δ(x − x2 ) dx2 ) = 0

ψ2∗ (x2 )ψ1 (x2 )δ(x − x2 ) dx2 ) = 0

ψj∗ (x2 )ψ1 (x2 )δ(x − x2 ) dx2 ) = 0
115

(7.33)

D2 = (∫

R3

ψ2∗ (x1 )ψ2 (x1 ) dx1 ) (∫

R3

ψj∗ (x2 )ψ1 (x2 )δ(x − x2 ) dx2 ) = ψj∗ (x)ψ1 (x)

so that we have
1
⟨Ψ1↦j ∣ δ(x − x2 ) ∣ Ψ0 ⟩ = ψj∗ (x)ψ1 (x)
2

which, when combined with (7.33), gives

⟨Ψ1↦j ∣ n(x) ∣ Ψ0 ⟩ = ψj∗ (x)ψ1 (x)

(7.34)

To confirm the pattern, we instead consider promoting an electron from ψ2 to ψj . This has the effect of
switching j ⇆ 1 and 2 ⇆ j in (7.32), while Ψ0 remains as before. In A1 –D2 , we make the replacements
ψj∗ ↦ ψ1∗ and ψ2∗ ↦ ψj∗ because Ψj is the bra. This gives us that
A1 = 0

B1 = 0

A2 = ψj∗ (x)ψ2 (x)

C1 = 0

B2 = 0

D1 = ψj∗ (x)ψ2 (x)

C2 = 0

D2 = 0

so that
⟨Ψ2↦j ∣ n(x) ∣ Ψ0 ⟩ = ψj∗ (x)ψ2 (x)
7.2.2.2

(7.35)

Bra-kets Involving n(x): N -Electron Systems

The general pattern indicated by the results of (7.34) and (7.35) is apparent: when an excited state Ψj is
created by moving an electron from orbital m to j, we have
⟨Ψj ∣ n(x) ∣ Ψ0 ⟩ = ψj∗ (x)ψm (x)
∗
⟨Ψ0 ∣ n(x) ∣Ψj ⟩ = ψm
(x)ψj (x)

(7.36)
(7.37)

The sum in (7.30) is over all excited states. A unique excited state can be generated by moving any originally
occupied state m to an unoccupied one j [48, p.16]. Therefore, for each target j, there are m possible starting
points, meaning each j induces a summation over m. In other words, (7.30) becomes
χ1 (x, y, ω) =

⎡ ⟨Ψ0 ∣ n(x) ∣ Ψj ⟩⟨Ψj ∣ n(y) ∣ Ψ0 ⟩

∑ ∑ ⎢⎢⎢

unocc occ
j

m

⎣

ω − (Ej − E0 ) + iη

116

−

⟨Ψ0 ∣ n(y) ∣ Ψj ⟩⟨Ψj ∣ n(x) ∣ Ψ0 ⟩ ⎤⎥
⎥
⎥
ω + (Ej − E0 ) + iη
⎦

Applying the energy change (7.31) and simplifying the bra-kets with (7.36) and (7.37), the above becomes
χ1 (x, y, ω) =

∑ ∑[

unocc occ
j

m

∗
ψm
(x)ψj (x) ψj∗ (y)ψm (y)

ω − εj + εm + iη

−

∗
ψm
(y)ψj (y) ψj∗ (x)ψm (x)

ω + εj − εm + iη

]

By introducing occupation factors from Fermi-Dirac statistics, whereby fk = 1 indicates that state k is

occupied and fk = 0 indicates it is unoccupied, we can write
χ1 (x, y, ω) = ∑ ∑ fm (1 − fj ) [
all all
j

m

∗
ψm
(x)ψj (x) ψj∗ (y)ψm (y)

ω − εj + εm + iη

−

∗
ψm
(y)ψj (y) ψj∗ (x)ψm (x)

Finally, in the second term, we make the change of variables j ⇆ m:

∑∑ fm (1 − fj )
j

m

∗
ψm
(y)ψj (y) ψj∗ (x)ψm (x)

ω + εj − εm + iη

= ∑∑ fj (1 − fm )
m

j

ω + εj − εm + iη

]

(7.38)

∗
ψj∗ (y)ψm (y) ψm
(x)ψj (x)

ω + εm − εj + iη

so that after switching the order of summation in the first term, we can combine the terms of (7.38):
χ1 (x, y, ω) = ∑∑ (fm − fj )
m

7.2.3

j

∗
ψm
(x)ψm (y) ψj (x)ψj∗ (y)

Representation within Jellium Model

ω + εm − εj + iη

(7.39)

Within jellium, states correspond to the orbital’s momentum❸ Restoring this notation in (7.39) yields
χ1 (x, y, ω) = ∑∑ (fk − fk′ )
k

k′

ψk∗ (x)ψk (y) ψk′ (x)ψk∗′ (y)
εk − εk′ + ω + iη

(7.40)

Summation is more properly integration in momentum space, so it is more appropriate to write (7.40) as
ψk∗ (x)ψk (y) ψk′ (x)ψk∗′ (y) ′
ψk∗ (x)ψk (y) ψk′ (x)ψk∗′ (y)
dk dk − ∫ fk′ ∫
dk dk′
εk − εk′ + ω + iη
εk − εk′ + ω + iη
R3
R3
R3

χ1 (x, y, ω) = ∫

fk ∫

χ1 (x, y, ω) = ∫

fk ∫

R3

In the second integral, we can make the change of variables k ⇆ k′ and obtain that
R3

ψk∗ (x)ψk (y) ψk′ (x)ψk∗′ (y) ′
ψk (x)ψk∗ (y) ψk∗′ (x)ψk′ (y) ′
dk dk + ∫ fk ∫
dk dk
εk − εk′ + ω + iη
εk − εk′ − ω − iη
R3
R3
R3

We recognize that the second integral is the complex conjugate of the first one with ω replaced by −ω. It is
therefore sufficient to work out the details of
I(x, y, ω) ≜ ∫

R3

fk ∫

ψk∗ (x)ψk (y) ψk′ (x)ψk∗′ (y) ′
dk dk
εk − εk′ + ω + iη
R3

❸ The true wavefunction in jellium is ψ (x). However, symmetry allows computation of the one-dimensional, scalar-indexed
k
ψk (x). The distinction between the computational tool of ψk (x) and the actual wavefunction must not be forgotten.

117

˜ ) and perpendicular to the surface as in Section 5.2, and applying
Separating into components parallel (i.e., x
the wavefunction decomposition (5.9), we can write
I(x, y, ω) = ∫

R3

fk ∫

R3

˜′ ˜

ei(k −k)(˜x−˜y)

ψk∗ (x)ψk (y) ψk′ (x)ψk∗′ (y) ′
dk dk
εk − εk′ + ω + iη

Define (formally) the Fourier transform parallel to the surface:
˜ }(˜
F{f
q) ≜ ∫

Then the parallel Fourier transform of I is

R2

e−i˜q⋅(˜x) f (˜
x) d˜
x

(7.41)

∗
∗
˜+k
˜ ′ ) ψk (x)ψk (y) ψk′ (x)ψk′ (y) dk′ dk
δ(˜
q−k
εk − εk′ + ω + iη
R3
R3
⎤
⎤
⎡
⎡
˜+k
˜′)
⎥ ′⎥
⎢
⎢
δ(˜
q−k
′⎥
∗
∗
⎢
⎢
dk ⎥ dk ⎥⎥ dk
= ∫ fk ψk (x)ψk (y)⎢∫ ψk′ (x)ψk′ (y) ⎢∫ 1
⎥
⎥
⎢ R
⎢ R2 2 k2 − 12 k′ 2 + εk − εk′ + ω + iη
R3
⎦
⎦
⎣
⎣
⎡
⎤
∗
⎢
⎥
ψk′ (x)ψk′ (y)
= ∫ fk ψk∗ (x)ψk (y)⎢⎢∫
dk ′ ⎥⎥ dk
1 2
3
′
˜
˜
2k
⋅
q
−
q
+
ε
−
ε
+
ω
+
iη
⎢
⎥
R
k
k
2
⎣ R
⎦

˜
˜ ; ω) = ∫
F{I}(x,
y, q

fk ∫

We now employ a long-wavelength limit ∣˜
q∣ → 0 relevant for incident photons [67, p.42] to obtain that
˜
I(x, y, ω) = F{I}(x,
y, 0; ω) = ∫

R3

=∫

R3

⎡
⎢

fk ψk∗ (x)ψk (y)⎢⎢

fk ψk∗ (x)ψk (y)G(x, y; εk + ω) dk

∫R

⎢
⎣

⎤
⎥
ψk′ (x)ψk∗′ (y)
′⎥
dk ⎥ dk
εk − εk′ + ω + iη
⎥
⎦

where we have used the spectral representation (6.17) of G. The fk come from Fermi-Dirac statistics (5.6):
I(x, y, ω) = ∫

∣k∣<kF

ψk∗ (x)ψk (y)G(x, y; εk + ω) dk

To resolve the integral over k, we mirror the procedure that began with (5.8), went through (5.12), and
produced (5.18). Doing so yields
I(x, y, ω) =

kF
1
(kF2 − k 2 )ψk (x)ψk (y)G(x, y; εk + ω) dk
∫
2
π 0

We have removed the conjugation from the wavefunction because it is real valued. Because χ1 (x, y, ω) =
I(x, y, ω) + I ∗ (x, y, −ω), we have the expression for the linear response function
χ1 (x, y, ω) =

kF
1
(kF2 − k 2 )ψk (x)ψk (y)[G(x, y; εk + ω) + G∗ (x, y; εk − ω)] dk
∫
π2 0

118

(7.42)

Remark 7.3. The wavefunction decomposition (5.9) was again responsible for a reduction of dimension,
allowing x and y to be replaced by x and y in (7.42). For the ground state density, the absolute value
unilaterally eliminated the exponential factor. This time, the reduction is only allowed in the q → 0 limit,

∎

which is a macroscopic approximation that valid because of the symmetry of the jellium surface.
Remark 7.4. In Liebsch’s derivation [67] of (7.42), he uses the summation
G(x, y; ε) = ∑
k

ψk (x)ψk∗ (y)
ε − εk + iη

which also appears in other sources too [78, p.182]. Because all the states are members of the continuous
spectrum, this summation is an integral. His derivation is fundamentally correct as long as this interpretation
∎

is made; the above proceeds in proper rigor, even if the difference little more than a symbol.

7.3

Linear Response Density for Jellium Surface

We will now formulate how to compute the linear response density for the jellium system. We return to the
integral equation for the linear response density n1 (x; ω) given in (7.22)
n1 (x; ω) = ∫

+∞

−∞

χ1 (x, y; ω)[φext (y; ω) + φest (y; ω) + φxc (y; ω)] dy

(7.43)

χ1 is as in (7.42) and φest and φxc are as in (7.19) and (7.21), respectively, with dimension reduced. We
assume that the source of the external potential is a uniform, normally incident time-varying electric field

or in frequency domain

Eext (t) = (E0 eiωt , 0, 0)T

(7.44)

Eext (ω ′ ) = (E0 , 0, 0)T δ(ω − ω ′ )

If Eext is monochromatic, then the only frequency contribution to n1 from Eext occurs at ω ′ = ω. If it is
polychromatic, then we may write it as a sum of waves like (7.44) and consider each frequently separately.
This explains the change of notation from (x, ω) in (7.22) to (x; ω) in (7.43)—now, we consider ω fixed.

Following elementary electromagnetics, there is a potential φext (x; ω) such that −∇φext (x; ω) = Eext (ω). In

this simple one-dimensional problem, it is easy to see that φext (x; ω) = −E0 x does the trick. However, to
correspond to the long-wavelength limit of inelastic electron scattering [67], we choose the normalization
φext (x; ω) = −2πx
119

(7.45)

We now see the motivation for the symbol swap performed at the end of Section 7.1.3. It desirable for the
external potential to share the standard symbology of using φ for potentials in electromagnetics.
Much as long-range Coulomb interactions wreaked havoc in the ground state, necessitating the splitting of
density described at length in Section 5.5, we cannot solve the Poisson equation (7.19) directly. We therefore
construct another elaborate procedure that begins by defining the zeroth moment of the linear density:
σ(ω) ≜ ∫

+∞

−∞

n1 (x; ω) dx

(7.46)

Because the normal component of the induced electric displacement D(x; ω) = ε(ω)E(x; ω) must be continuous across the metal/vaccuum interface, we have

lim ε(ω)Ex (x; ω) = lim+ ε(ω)Ex (x; ω)

x→0−

x→0

where ε(ω) is the dielectric constant of the metal. In the Drude-Sommerfeld model,
ε(ω) = 1 − (

ωp 2
)
ω

(7.47)

where ωp denotes the plasma frequency (sometimes called the bulk-plasmon frequency)
ωp2 =

3
rs3

(7.48)

The limits denote approaching from inside the metal and vacuum, respectively. Then because
lim Ex (x) = −2π(1 − σ(ω))

x→0−

lim Ex (x) = −2π(1 + σ(ω))

x→0+

we have
σ(ω) =

ωp2
ε(ω) − 1
= 2
ε(ω) + 1 ωp − 2ω 2

(7.49)

so that σ(ω) is always known a priori from the linear density. Next, we define
d⊥ (ω) ≜

+∞
1
xn1 (x; ω) dx
∫
σ(ω) −∞

(7.50)

This quantity, first introduced by Feibelman, is a complex number (because n1 is). The real part measures

120

the effective surface location of the normal component of the electric field, while the imaginary part represents
the phase lag of this component in the surface region [31, p.325].
Following Liebsch [66], we now derive the splitting that filters out all those problematic bulk Coulomb effects
by defining the bulk potential φb and quasi-electrostatic potential φ1
φb (x; ω) ≜ −2π(1 − σ(ω))x + 2πσ(ω)d⊥ (ω)

φ1 (x; ω) ≜ −4π ∫

+∞

−∞

(7.51)

(x − y)Θ(x − y)n1 (y; ω) dy

(7.52)

Recall that in one-dimension, GL (x − y) = −2π ∣x − y∣ = −2π(x − y)sgn(x − y). Consequently, as the solution
to the Poisson equation (7.19), φest may be defined formally by integration against this Green’s function
φest (x; ω) = −2π ∫

+∞

−∞

(x − y)sgn(x − y)n1 (y; ω) dz

By writing sgn(x) = 2Θ(x) − 1, we can write the above formal solution as
φest (x; ω) = φ1 (x; ω) + 2π ∫

+∞

−∞

(x − y)n1 (y; ω) dy

(7.53)

Now we observe that
∫

+∞

−∞

(x − y) n1 (y; ω) dz = x ∫

+∞

−∞

n1 (y) dy − ∫

= σ(ω)(x − d⊥ (ω))

+∞

−∞

yn1 (y; ω) dy

Combining this result with (7.53) and using (7.45), we have
φest (x; ω) + φext (x; ω) = φ1 (x; ω) + 2πσ(ω)(x − d⊥ (ω)) − 2πx
= φ1 (x; ω) + φb (x; ω)

Before we can go further, we need to break from the action to state and prove two lemmata.
Lemma 7.1. If ψk is a Kohn-Sham wavefunction corresponding to εk , then
∫

+∞

−∞

ψk (y)G(x, y; ε) dy =

121

ψk (x)
ε − εk + iη

(7.54)

Proof. Using the spectral representation of G given in (6.17), we have for any ε
∫

+∞

−∞

ψk (y)G(x, y; ε) dy = ∫

+∞

−∞

ψk (y) ∫

R

ψλ (x)ψλ (y)
dλ dy
ε − ελ + iη

+∞
ψλ (x)
ψλ (y)ψk (y) dy] dλ
[∫
R ε − ελ + iη
−∞
ψλ (x)δ(λ − k)
ψk (x)
=∫
dλ =
ε − ελ + iη
ε − εk + iη
R

=∫

To obtain the final line, we have called upon the orthogonality of states (understood i.s.d.)
∫

+∞

ψλ (y)ψk (y) dy = δ(λ − k)

−∞

∎

Lemma 7.2. The linear response function χ1 (x, y; ω) satisfies
∫

+∞

−∞

χ1 (x, y; ω) dy = 0

Proof. Recalling (7.42) and reversing the order of integration, we define
I≜∫

0

kF

(kF2 − k 2 )ψk (x) ∫

+∞

−∞

ψk (y)[G(x, y; εk + ω) + G∗ (x, y; εk − ω)] dy dk

The conjugate from G∗ in the second term filters out because yψk (y) ∈ R, so we have
I=∫

kF

0

(kF2 − k 2 ) ψk (x) [∫

+∞

−∞

ψk (y)G(x, y; εk + ω) dy + [∫

+∞

−∞

∗
ψk (y)G(x, y; εk − ω) dy] ] dk

Applying the result of Lemma 7.1 to the above, we have
I=∫

kF
0

(kF2 − k 2 ) ψk (x) [

which completes the proof.

ψk (x)
ψk (x)
+
] dk = 0
−ω + iη ω − iη

∎

By definition,
φ1 (x; ω) = −4π ∫

x
−∞

(x − y)n1 (y; ω) dy → 0

as x → −∞, so all long-range Coulomb interaction from the bulk in (7.43) is contained in
ξ1 (x; ω) ≜ ∫

+∞

−∞

χ1 (x, y; ω)φb (y; ω) dy

= −2π(1 − σ(ω)) ∫

+∞

−∞

yχ1 (x, y; ω) dy + 2πσ(ω)d⊥ (ω) ∫

+∞

−∞

122

χ1 (x, y; ω) dy

(7.55)

Application of Lemma 7.2 to (7.55) gives a compact form for ξ1 :
ξ1 (x; ω) = −2π(1 − σ(ω)) ∫

+∞

−∞

yχ1 (x, y; ω) dy

(7.56)

Juxtaposing (7.56) and (7.54) and substituting them into (7.43), we have
n1 (x; ω) = ξ1 (x; ω) + ∫

∞

−∞

χ1 (x, y; ω)[φ1 (y; ω) + φxc (y; ω)] dy

We had dubbed φ1 the “quasi-electrostatic potential.” To see why this name is appropriate, we differentiate
(7.52) twice with help from the Fundamental Theorem of Calculus and chain rule:
φ′1 (x; ω) = −4π ∫

x
−∞

n1 (y; ω) dy

⇒

φ′′1 = −4πn1

Despite being related by the Poisson equation, φ1 is obviously not the true linear electrostatic potential—
the two differ by a term in the kernel of the operator

∂2
.
∂x2

Because the operator annihilates the difference

function, the two satisfy the same differential equation. However, because of this kernel term, φ1 behaves
inappropriately for an electrostatic potential as x → +∞, as will be shown in Section 7.5.

Even though we’ve made significant progress in isolating long-range Coulomb effects, we have not removed
all of them—and, of course, we can only relocate them, not remove them from the system. φb contains the

long-range effects from the bulk material, but does nothing to the Friedel oscillations in n1 . Those make
computing φ1 via direct integration unfeasible. A direct solve of the Poisson equation is possible [96], but
great care must be taken to ensure numerical stability. Instead, we call upon the theorem below.
Theorem 7.3. If u is a solution to the free Poisson equation over all of R
−u′′ (x) = 4πf (x)

then for any λ > 0, u also solves the integral equation
u(x) = ∫

+∞

−∞

λ
2π
e−λ∣x−y∣ [ u(y) +
f (y)] dy
2
λ

Proof. Let λ > 0. Then we may add λ2 u to both sides of the Poisson equation to give
(−

∂2
+ λ2 ) u = 4πf + λ2 u
∂x2
123

(7.57)

∂
2
We now compute the Green’s function of the Helmholtz (“with the good sign”) operator Z = − ∂x
2 +λ
2

(−

∂2
+ λ2 ) GZ (x, y) = δ(x − y)
∂x2

by Fourier transform in x:
−2πiξy
ˆ Z (ω, y) = e
G
4πξ 2 + λ2

⇒

GZ (x, y) =

1 −λ∣x−y∣
e
2λ

Consequently, the solution to the equivalent Poisson equation (7.57), treating the right-hand side as known,
is given by integration against GZ :
u(x) = ∫

+∞

−∞

λ
2π
f (y)] dy
e−λ∣x−y∣ [ u(y) +
2
λ

∎

Remark 7.5. Theorem 7.3 is precisely the technique used by Manninen [75] for computing φ in the ground
state (cf. (5.29)). The technique is especially convenient here because a pair of coupled integral equations is
∎

easier solved than a system of an integral equation and a differential equation.
As a consequence of this theorem, we can obtain a pair of coupled integral equations for φ1 and n1 :
n1 (x; ω) = ξ1 (x; ω) + ∫
φ1 (x; ω) = ∫

7.4

+∞

−∞

∞

−∞

χ1 (x, y; ω) [φ1 (y; ω) + fxc (y)n1 (y; ω)] dy

λ
2π
e−λ∣x−y∣ [ φ1 (y; ω) +
n1 (y; ω)] dy
2
λ

(7.58)
(7.59)

The Driving Function ξ1 (x; ω)

Substituting the definition of χ1 given in (7.42) into (7.56) and then interchanging the order of integration,
ξ1 (x) = −2π(1 − σ(ω)) ∫

0

kF

(kF2 − k 2 )ψk (x) ∫

+∞

−∞

yψk (y)[G(x, y; εk + ω) + G∗ (x, y; εk − ω)] dy dk

(7.60)

We cannot perform a numerical computation over an infinite domain, so we decompose into three pieces:
B + (x, k; ω) ≜ ∫
S + (x, k; ω) ≜ ∫

V + (x, k; ω) ≜ ∫

−R1

−∞
R2

−R1
∞

R2

yψk (y)G(x, y; εk + ω) dy

B − (x, k; ω) ≜ ∫

yψk (y)G(x, y; εk + ω) dy

V − (x, k; ω) ≜ ∫

yψk (y)G(x, y; εk + ω) dy

S − (x, k; ω) ≜ ∫

124

−R1

−∞
R2
−R1
∞
R2

yψk (y)G∗ (x, y; εk − ω) dy

yψk (y)G∗ (x, y; εk − ω) dy

yψk (y)G∗ (x, y; εk − ω) dy

The symbols for these integrals have been selected to correspond to the physical region over which the integral is defined: the bulk metal, the surface, and the vacuum, respectively. Because both ψk and G(x, ⋅; ε)
decay exponentially into the vacuum, the contributions from V + and V − can be neglected. Therefore,
ξ1 (x; ω) = −2π(1 − σ(ω)) ∫

kF

0

(kF2 − k 2 )ψk (x)[B + (x, k; ω) + S + (x, k; ω) + B − (x, k; ω) + S − (x, k; ω)] dk

B + and B − can be computed analytically—and it is imperative to a correct result for ξ1 that these contributions not be neglected—but S + and S − must be computed numerically from calculated solutions.

7.4.1

Contributions from the Bulk Metal

In this section, we will derive analytic expressions for the contributions to ξ1 from deep within the bulk
metal, encoded in B + and B − . We are concerned only with x ∈ [−R1 , R2 ], so we may assume that x ≥ y in

the integrals that define B + and B − . This means that G does not “switch” in the middle of the integral, so
G(x, y; εk ± ω) =

2
ϕ1 (y; εk ± ω)ϕ2 (x; εk ± ω)
Wk±

where the superscript plus indicates the value should be taken from G(⋅, ⋅; εk + ω); minus, from G(⋅, ⋅; εk − ω).

We open with the simpler “plus case,” which requires computing

−R1
2
yψk (y)ϕ1 (y; εk + ω)ϕ2 (x; εk + ω) dy
∫
Wk+ −∞
−R1
2
yψk (y)ϕ1 (y; εk + ω) dy
= + ϕ2 (x; εk + ω) ∫
Wk
−∞

B + (x, k; ω) =

The interval of integration is contained in the asymptotic region, so the asymptotic expressions (5.17) and
(6.5) may replace ψk and ϕ1 , respectively. Then
B + (x, k; ω) =

+∞
+
2
ϕ2 (x; εk + ω) ∫
yΘ(−y − R1 ) sin(ky − γk )e−iµk y dy
+
Wk
−∞

≜I

Because µ+k ∈ R, I does not converge to a function in k. However, because the result will be integrated
against a smooth❸ function in k, we may compute I as a distribution with some help from the ordinary

❸ The integrand (k 2 − k)2 ψ is smooth enough in k, but the compact region of integration is equivalent to multiplying
k
F
the integrand by the characteristic function χ[0,kF ] (k) = Θ(k − kF ) − Θ(k), which is not smooth. For full rigor, the integral
in momentum space should be taken in the limit against Cc∞ (R) functions that approximate the integrand. In the end, the
distribution I can be applied to piecewise continuous functions, so the same result is obtained as when proper rigor is obeyed.

125

Fourier transform and the convolution derivative formula
∂
[f (ξ) ∗ g(ξ)] = f ′ (ξ) ∗ g(ξ) = f (ξ) ∗ g ′ (ξ)
∂ξ
By defining
f (y) ≜ Θ(−y − R1 )

g(y) ≜ sin(ky − γk ) = sin (k (y − γk /k))

we have that
I=

i ∂ ˆ
i ˆ′
=
[f (ξ) ∗ gˆ(ξ)] +
[f (ξ) ∗ gˆ(ξ)] +
ξ=µk /2π
ξ=µk /2π
2π ∂ξ
2π

where the derivative enters because of the y term. By making a change of variables y ↦ −y in the Fourier
integral that defines fˆ and applying shift rules, we have that

fˆ(ξ) = e2πiR1 ξ F ∗ {Θ(y)}(ξ)

gˆ(ξ) = e−2πiγk /k F{ sin ky}(ξ)

The transforms of the step and sine functions are well-known. Accordingly,
1
1
fˆ(ξ) = e2πiR1 ξ (−
+ δ(ξ))
2
πiξ
k
k
1
) − eiγk δ (ξ +
)]
gˆ(ξ) = [e−iγk δ (ξ −
2i
2π
2π
Then
I=

1 −iγk ˆ′
k
k
[e
f (ξ) ∗ δ (ξ −
) − eiγk fˆ′ (ξ) ∗ δ (ξ +
)]
4π
2π
2π ξ=µ+ /2π

µ+ + k
1 −iγk ˆ′ µ+k − k
=
[e
f (
) − eiγk fˆ′ ( k
)]
4π
2π
2π

k

by the sifting property of the Dirac delta. Differentiating i.s.d directly, we have
1
2R
1
fˆ′ (ξ) = e2πiR1 ξ [ −
+
+ 2πiRδ(ξ) + δ ′ (ξ)]
2
ξ
iπξ 2

√
Neither k + µ+k nor k − µ+k vanish, as that would require k = ± k 2 + 2ω, which is impossible because ω > 0.

126

Accordingly, the contributions from δ and δ ′ disappear and
+
µ+ ± k
i
R1
fˆ′ ( k
) = −2πeiRµk ±iRk [ +
+ +
]
2π
µk ± k (µk ± k)2

Thus,

+
+
1
R1
i
1
R1
i
I = eiR1 µk ei(R1 k+γk ) [ +
+
] − eiR1 µk e−i(R1 k+γk ) [ +
+
]
2
µk + k (µ+k + k)2
2
µk − k (µ+k − k)2

so that the desired computation is given by
+

eiR1 µk i(R1 k+γk )
R1
i
R1
i
B+ = ϕ2 (x; εk + ω) ⋅
(e
[ +
+ +
] − e−i(R1 k+γk ) [ +
+ +
])
+
2
Wk
µk + k (µk + k)
µk − k (µk − k)2
The minus case is complicated by a kc =

(7.61)

√
2ω at which µ−k will change from purely imaginary to real. Table

7.1 below gives the value of kc for various input frequencies ω and choices of rs . For inputs ω above the
cutoff frequency ωc = 12 kF2 = EF , we have kc > kF so that µ−k remains real throughout the integral.
rs

kF

0.1ωp

0.2ωp

0.3ωp

0.4ωp

ωc

2.0

0.9596

0.3500

0.4949

0.6062

0.6999

0.7518ωp

2.5

0.7677

0.2960

0.4187

0.5127

0.5921

0.6725ωp

3.0

0.6397

0.2582

0.3651

0.4472

0.5164

0.6139ωp

3.5

0.5483

0.2300

0.3253

0.3984

0.4600

0.5683ωp

4.0

0.4798

0.2081

0.2943

0.3604

0.4162

0.5316ωp

Table 7.1: Values of kc and ωc . The middle block lists the value of kc for several frequencies per rs
value. For the listed frequencies, kc always lies in [0, kF ]. The last column lists the ω above which kc > kF .
7.4.1.1

Case I: k ≥

√
2ω

Because µ−k ∈ R in this case, the analysis mirrors the plus case exactly. Therefore,
B − = ϕ2 (x; εk − ω) ⋅

−

eiR1 µk i(R1 k+γk )
R1
i
R1
i
(e
[ −
+ −
] − e−i(R1 k+γk ) [ −
+ −
])
−
2
Wk
µk + k (µk + k)
µk − k (µk − k)2

127

(7.62)

7.4.1.2

Case II: k <

√

2ω

In this case, the asymptotic behavior of G is quantitatively different. Rather than become oscillatory, G
decays exponentially as per (6.5), so
B − (x, k; ω) =

−R1
−
2
y sin(ky − γk )eκ˜ ω y dy
ϕ
(x;
ε
−
ω)
2
k
∫
Wk−
−∞

where µ
˜−k ≜ Im µ−k . This integral can be computed using a tabular method to integrate by parts multiple

times, but the process is tedious. Instead, by converting the sine term to an exponential
B − (x, k; ω) =

+∞
−
2
yΘ(−y − R1 )eµ˜k y e−iky dy
ϕ2 (x; εk − ω) ⋅ Im ∫
−
Wk
−∞

≜I

we can again use Fourier transforms to evaluate the integral by defining

f (y) ≜ Θ(−y − R)eµ˜k y = e−˜µk R [Θ(−y − R1 )eµ˜k (y+R1 ) ]
−

−

−

and recognizing that
I=

i ˆ′ k
f ( )
2π
2π

By making a change of variables y ↦ −y and applying shift rules, fˆ is expressed
−
fˆ(ξ) = e−˜µk R1 ∫

+∞

−∞

Θ(y − R1 )e−˜µk (y−R) e2πiξy dy
−

= e−˜µk R1 e2πiR1 ξ F ∗ {Θ(y)e−˜µk y }(ξ)
−

−

This time, f ∈ L2 (R), so its Fourier transform is defined as a function:

Differentiating directly, we have

−
e2πiR1 ξ
fˆ(ξ) = e−˜µk R1 −
µ
˜k − 2πiξ

−
fˆ′ (ξ) = 2πie−˜µk R1 e2πiRξ [

Then
I = −eiR1 k−R1 µ˜k [
−

R1
1
+ −
]
µ
˜−k − 2πiξ (˜
µk − 2πiξ)2

R1
1
+ −
]
µ
˜−k − ik (˜
µk − ik)2
128

gives us that
B − (x, k; ω) = −

−
2
1
R1
ϕ2 (x; εk − ω) ⋅ Im (ei(R1 k+γk )−R1 µ˜k [ −
+ −
])
−
Wk
µ
˜k − ik (˜
µk − ik)2

z
In order to draw parallels with (7.62), we make use of the identity Im z = Re ( ):
i
Im (ei(R1 k+γk )−R1 µ˜k [
−

−
1
i
R1
R1
+ −
] ) = Re (ei(R1 k+γk )−R1 µ˜k [ −
+
])
µ
˜−k − ik (˜
µk − ik)2
i˜
µk + k (i˜
µ−k + k)2

= Re (ei(R1 k+γk )+iR1 µk [
−

i
R1
])
+ −
+ k (µk + k)2

µ−k

with the last line coming by reversing µ
˜−k = Im µ−k and that Re µ−k = 0. We then have
B − (x, k; ω) = −

−
2
R1
i
ϕ2 (x; εk − ω) ⋅ Re (ei(R1 k+γk )+iR1 µk [ −
+
])
−
Wk
µk + k (µ−k + k)2

(7.63)

Feibelman performed computations similar in nature to the ones in this section [29, 30], although his purpose
was to find the asymptotic behavior of n1 . His methodology differed significantly from the one here: in
evaluating integrals like the ones that define B + , he chose to do the integrals in k first. He converted the
sine term to complex exponentials and then wrote
e−i(µk ±k)y = ±
+

∂ −i(µ+k ±k)y
i µ+k
⋅
[e
]
y µ+k ± k ∂k

From there, he iterated integration by parts twice. While functional, this method leads to an incredibly
tedious and painstaking computation—the author of this thesis first attempted this method, with an incorrect
solution resulting after nine dense pages of typeset text. Using Fourier transforms is in every way superior:
it is easy to understand and produces an elegant solution with little effort.

7.4.2

Contributions from the Surface

Because no asymptotic behavior is needed to compute S + and S − there is no difference aside from a
plus/minus symbol. We therefore handle the general expression
G(x; ε) = ∫

R2
R1

yψk (y)G(x, y; ε) dy

so that
S + (x, k; ω) = G(x; εk + ω)

S − (x, k; ω) = G(x; εk − ω)
129

Because the real part of G has a cusp at x = y, accurate numerical integration requires breaking the integral
at x and designing two quadrature rules, one for [−R1 , x] and another for [x, R2 ]. Computing these two
integrals separately for each grid point x is incredibly inefficient, as the same computations are performed
twice. We therefore seek to craft a more efficient method of computing S + and S − . To this end, define
I1 (x; ε) ≜ ∫

I2 (x; ε) ≜ ∫

R2

x
R2
x

yψk (y)ϕ1 (y; ε) dy

(7.64)

yψk (y)ϕ2 (y; ε) dy

(7.65)

Then
G(x; ε) = ∫
=

=

x
−R1

yψk (y)G(x, y; ε) dy + ∫

R2
x

yψk (y)G(x, y; ε) dy

R2
R2
R2
2
yψk (y)ϕ2 (y) dy)
yψk (y)ϕ1 (y) dy] + ϕ1 (x) ∫
yψk (y)ϕ1 (y) dy − ∫
(ϕ2 (x)[ ∫
Wε
x
x
−R1

2
(ϕ2 (x; ε)[I1 (−R1 ; ε) − I1 (x; ε)] + ϕ1 (x; ε) I2 (x; ε))
Wε

(7.66)

Using the result of (7.66), the needed quantities S + and S − are
S + (x, k; ω) =

S − (x, k; ω) =

7.4.3

2
(ϕ2 (x; εk + ω)[I1 (−R1 ; εk + ω) − I1 (x; εk + ω)] + ϕ1 (x; εk + ω) I2 (x; εk + ω))
Wk+
2
(ϕ2 (x; εk − ω)[I1 (−R1 ; εk − ω) − I1 (x; εk − ω)] + ϕ1 (x; εk − ω) I2 (x; εk − ω))
Wk−

(7.67)
(7.68)

Summary

Figure 7.1 below displays several examples of ξ1 as assembled from (7.61)–(7.63) and (7.67)–(7.68).
Real and Imaginary Parts of ξ1 (x; ω) for Several ω

Re ξ1 (x; ω)

0.6
0.3
0

−0.3

ξ1 (x; 0.1 ωp )

Im ξ1 (x; ω)

−0.6
0.3

ξ1 (x; 0.2 ωp )

ξ1 (x; 0.3 ωp )

0

−0.3
−0.6

−0.9
-35

ξ1 (x; 0.1 ωp )
-30

-25

ξ1 (x; 0.2 ωp )
-20

-15

ξ1 (x; 0.3 ωp )
-10

-5

0

5

10

Figure 7.1: ξ1 (x; ω) for Several ω. This figure depicts the real (top) and imaginary (bottom) parts of
ξ1 (x; ω) for ω = 0.1, 0.2, 0.3ωp , all for rs = 3.
130

Note that ξ1 (x; ω) displays rather unpredictable behavior and does not vanish as x → −∞. Because ξ1
contributes only locally to n1 (x; ω), there are no concerns as there would be if ξ1 were in an integral.

The importance of the analytic contributions B + and B − cannot be understated. Depicted below in Figure
7.2 is the result (for ω = 0.1ωp ) if these bulk contributions are not included.

Failure of ξ1 (x; ω) without Bulk Contributions

Im ξ1 (x; ω)

Re ξ1 (x; ω)

3
2
1
0
−1
−2
3
2
1
0
−1
−2
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 7.2: Incorrect ξ1 (x; ω). When the bulk contributions are not included, the result is a very
different (and very wrong!) ξ1 (x; ω), shown for ω = 0.1ωp and rs = 3.

7.5

Asymptotic Behavior of φ1

We return to φ1 ’s integral definition, reproduced from (7.52) below:
φ1 (x; ω) = −4π ∫

x
−∞

(x − y)n1 (y; ω) dy

Because n1 → 0 exponentially [29], for x ≫ 0 the integral in the above definition of φ1 is well-approximated
by one over all of R. Consequently, we conclude that φ1 behaves asymptotically like
φ1 (x; ω) ∼ −4π(x ∫

+∞

−∞

n1 (y; ω) dy − ∫

∼ −4πσ(ω)(x − d⊥ (ω))

+∞

−∞

yn1 (y; ω) dy)

(7.69)

after applying the definitions (7.46) and (7.50). As a result, φ1 grows unbounded. The electrostatic potential
should approach a constant as we move farther and father away from the surface. Because φ1 does not display

131

this behavior, it cannot represent the electrostatic potential. The cost of negating long-range Coulomb effects
from the bulk by incorporating is that φ1 inherited the external potential’s linear growth.
One might expect the exponentially decaying kernels of (7.58) and (7.59) for x ≫ 0 to snuff out (7.69), but

failure to account for it results in a fantastically incorrect result, as depicted below in Figure 7.3.

15
10

Erroneous Computation of n1 (x; ω)

×10−3

Re n1 (x)

5
0

−5

−10

−15
−35

−30

−25

−20

−15

−10

x

−5

0

5

10

Figure 7.3: Spectacular Failure of Direct Solution. Attempt to solve (7.58) and (7.59) directly
without consideration of the asymptotic behavior (7.69) of φ1 results in this trainwreck of a solution.

Therefore, to avoid a disaster like the one depicted in the figure, we define
φ˜1 (x) ≜ φ1 (x) + 4πσ(ω)xΘ(x)

(7.70)

which has φ˜1 → 4πσ(ω)d⊥ (ω) for x ≫ 0. Substituting (7.70) into (7.58) and (7.59) gives
n1 (x; ω) = ξ1 (x; ω) − 4πσ(ω) ∫

0

+∞

yχ1 (x, y; ω) dy + ∫

φ˜1 (x; ω) = 4πσ(ω)xΘ(x) − 2πλσ(ω) ∫

0

+∞

+∞

−∞

ye−λ∣x−y∣ dy + ∫

χ1 (x, y; ω) [φ˜1 (y; ω) + fxc (y)n1 (y; ω)] dy
+∞

−∞

2π
λ
e−λ∣x−y∣ [ φ˜1 (y; ω) +
n1 (y; ω)] dy
2
λ

To clean up these cumbersome expressions, we define the function
ξ2 (x; ω) ≜ −4πσ(ω) ∫

0

+∞

yχ1 (x, y; ω) dy

and compute analytically the integral
−2πλσ ∫

0

+∞

ye−λ∣x−y∣ dy = −

2πσ(ω) −λ∣x∣
e
− 4πσ(ω)xΘ(x)
λ

132

(7.71)

so that we can write more concisely
n1 (x; ω) = ξ1 (x; ω) + ξ2 (x; ω) + ∫

7.5.1

2πσ(ω) −λ∣x∣
φ˜1 (x; ω) = −
e
+∫
λ

Computing ξ2 (x; ω)

+∞

χ1 (x, y; ω) [φ1 (y; ω) + fxc (y)n1 (y; ω)] dy
−∞
+∞
λ
2π
e−λ∣x−y∣ [ φ˜1 (y; ω) +
n1 (y; ω)] dy
2
λ
−∞
˜

(7.72)
(7.73)

Just as in Section 7.4, we employ the definition (7.42) of χ1 to expand ξ2 as we did ξ1 :
ξ2 (x; ω) = −2π(1 − σ(ω)) ∫

kF
0

(kF2 − k 2 ) ψk (x) ∫

0

+∞

yψk (y) [G(x, y; εk + ω) + G∗ (x, y; εk − ω)] dy dk

Similar to what was done in Section 7.4.2, we focus on integrals of the form
G(x; ε) = ∫

0

∞

yψk (y)G(x, y; ε) dy = ∫

R2
0

yψk (y)G(x, y; ε) dy

Because G’s cusp at x = y will occur within the domain of integration if x > 0, we distinguish two cases.

Case I: x ≤ 0. Because y ≥ 0 in the integral, we have x ≤ y throughout, so G does not switch. Therefore,
G(x; ε) =

R2
2
2
yψk (y)ϕ2 (y; ε) dy =
ϕ1 (x; ε) ∫
ϕ1 (x; ε)I2 (0; ε)
Wε
Wε
0

Case II: x ≥ 0. G will switch at x. Following the exact same procedure that led to (7.66), we have
G(x; ε) =

2
(ϕ2 (x; ε)[I1 (0; ε) − I1 (x; ε)] + ϕ1 (x; ε) I2 (x; ε))
Wε

Combining the results of the two cases, the complete recipe for ξ2 (x; ω) is
ξ2 (x; ω) = −2π(1 − σ(ω)) ∫

kF

0

(kF2 − k 2 ) ψk (x)P(x; k) dk

(7.74)

with
⎧
2
2
⎪
⎪
ϕ (x; εk + ω)I2 (0; εk + ω) + − ϕ1 (x; εk − ω)I2 (0; εk − ω)
⎪
⎪
+ 1
⎪
W
W
⎪
ω
ω
⎪
⎪
⎪
⎪
⎪ 2
⎪
P ≜ ⎨ + (ϕ2 (x; εk + ω)[I1 (0; εk + ω) − I1 (x; εk + ω)] + ϕ1 (x; εk + ω) I2 (x; εk + ω))+
Wω
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
(ϕ2 (x; εk − ω)[I1 (0; εk − ω) − I1 (x; εk − ω)] + ϕ1 (x; εk − ω) I2 (x; εk − ω))
⎪
⎪
⎪
Wω−
⎩
133

x≤0
x≥0

This formula for ξ2 reuses things we must compute to obtain ξ1 , so we obtain it essentially “for free.”

7.6

Putting It Together

We now summarize the results of the last ten pages or so of derivations and give a complete description of all
the components necessary to determine the linear response density. To obtain n1 , we must solve the coupled
system of inhomogeneous Fredholm integral equations of the second kind
n1 (x; ω) = ξ1 (x; ω) + ξ2 (x; ω) + ∫

R2

−R1

χ1 (x, y; ω) [φ˜1 (y; ω) + fxc (y)n1 (y; ω)] dy

(7.75)

R2
λ
2πσ(ω) −λ∣x∣
2π
e−λ∣x−y∣ [ φ˜1 (y; ω) +
φ˜1 (x; ω) = −
e
+∫
n1 (y; ω)] dy
λ
2
λ
−R1

where λ is freely chosen. (7.72) and (7.73) have been truncated to the finite domain [−R1 , R2 ]. The driving
function ξ1 is

ξ1 (x; ω) = −2π(1 − σ(ω)) ∫

kF
0

(kF2 − k 2 )ψk (x)[B + (x, k; ω) + S + (x, k; ω) + B − (x, k; ω) + S − (x, k; ω)] dk

where B + is given in (7.61); B − , in (7.62) for k ≥

√

2ω and (7.63) for k <

√
2ω. S + and S − are given by

(7.67) and (7.68), respectively. The helper functions I1 and I2 are written in (7.64) and (7.65), respectively.

Finally, ξ2 is listed above in (7.74), with its auxiliary function just beneath it.

The “correct” φ1 , of course, can be obtained from φ˜1 by reversing (7.70), but this is unnecessary. None of
the potentials are terribly interesting, being little more than facilitators for obtaining the density.
Remark 7.6. As the Fourier transform of the time-dependent linear density, n1 (x; ω) is complex-valued
except at zero-frequency. If fˆ(ξ) ∈ R for all ξ, then f (x) = f ∗ (−x), which means that a real-valued f must
be even. As n1 (x, t) ≡ 0 for t < t0 , it is certainly not even in t, so its Fourier transform must be complex.

Mathematically, the injection of complex numbers into an otherwise real system comes from two sources:
the left and right boundary conditions of G(x, y; εk + ω). Because µ+k ∈ R for each k, eixµk ∈ C and so
√
G(x, y; εk + ω) ∈ C. On the right side, (νk+ )2 = k 2 + 2ω − 2∆V , so for k > 2∆V − 2ω, we have νk− ∈ R.
∎
+

134

7.7
7.7.1

Numerical Implementation
The Nystr¨
om Method for Integral Equations

The goal of this section is to introduce a general method for solving Fredholm integral equations of the
second kind called the Nystr¨om method. The source of this presentation follows Atkinson [4, ch.4], although
in slightly less generality. Let f ∈ C[a, b] and suppose an N -point quadrature scheme of the form
∫

b

a

f (x) dx ≈ ∑wq f (xq )
N

(7.76)

q=1

is pre-defined. We assume that this scheme converges to the true value of the integral as N → ∞ for every
continuous f ; it may come from Gaussian quadrature, as in Section 5.7.2, or a Newton-Cotes rule. Suppose
K(x, y) ∈ C(Ω) where Ω = [a, b] × [a, b] and consider the integral equation for x ∈ [a, b]
λu(x) = g(x) + ∫

b

a

K(x, x′ )u(x′ ) dx′

(7.77)

By applying the quadrature rule (7.76) to the integral of (7.77), we have an approximate solution u
˜
λ˜
u(x) = g(x) + ∑wq K(x, xq )u(xq )
N

q=1

Allowing x to be one of the xq , we have

λ˜
u(xp ) = g(xp ) + ∑wq K(xp , xq )u(xq )
N

(7.78)

q=1

If p runs from 1 to N , the result is an N × N linear system of the form A˜
u = g with the system matrix
⎛λ − w1 K(x1 , x1 )
⎜
⎜
⎜ −w1 K(x2 , x1 )
⎜
A ≜ λ − KW = ⎜
⎜
⎜
⋮
⎜
⎜
⎝ −w1 K(xN , x1 )

−w2 K(x1 , x2 )

⋯

−w2 K(xN , x2 )

⋯

λ − w2 K(x2 , x2 )
⋮

⋯
⋱

−wN K(x1 , xN ) ⎞
⎟
⎟
−wN K(x2 , xN ) ⎟
⎟
⎟
⎟
⎟
⋮
⎟
⎟
λ − wN K(xN , xN )⎠

(7.79)

˜ and g have
where K is the matrix with elements Kpq = K(xp , xq ) and W = diag(w1 , . . . , wN ). The vectors u
˜p = u
entries u
˜(xp ) and gp = g(xp ).

Beyond its ease of implementation, the Nystr¨
om method has two benefits: first, its accuracy is the same
as that of the underlying quadrature method. Second, it comes with a built in interpolation formula for

135

˜ is the numerical solution, then for any x
evaluation away from the quadrature points x1 , . . . , xN : if u
u(x) =

⎡
⎤
N
⎥
1 ⎢⎢
⎥
g(x)
+
w
K(x,
x
)˜
u
∑ q
q
q⎥
⎢
λ⎢
⎥
q=1
⎣
⎦

(7.80)

In practice, the above Nystr¨
om interpolation formula produces results superior to values from interpolation
methods applied to the computed solution, such as splines.

7.7.2

Formulation with Simpson’s Rule

Considering nothing but efficiency of the quadrature rule, the optimal choice for (7.76) would be a Gaussian
rule. Unfortunately, the cusps in G, which in turn become cusps in χ1 (see Figure 7.4), render it either
impossible (if cusps are accommodated) or useless (if not) to formulate the Nystr¨
om method using Gaussian
quadrature. Without special treatment of the cusps, the quadrature rule will not converge as N → ∞.
Re χ1 (x, 0; 0.1ωp )

1

Real and Imaginary Parts of χ1 (x, 0; 0.1ωp )

×10−2

0

−1

×10−3

Im χ1 (x, 0; 0.1ωp )

−2
2
0

−2
−4

−6
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 7.4: Linear Response Function χ1 (x, y; ω). Plotted in this figure are the real (top) and
imaginary (bottom) parts of χ1 (x, 0; 0.1ωp ) for rs = 3 (ω = 0.033 a.u. or photons at 1367 nm). Note the
cusp in the real part at x = y = 0; the imaginary part lacks this because δ affects only the real part of G.
Construction of the Nystr¨om method begins by writing a N point composite Gauss rule for [−R1 , R2 ].
Suppose these nodes are {xq }N
q=1 . We generate the linear system determined by (7.79) by marching through

the quadrature points, beginning at x1 . As described at the beginning of Section 7.4.2, to deal with the
cusps at y = x1 , we must break the integrals of (7.75) into one over [−R1 , x1 ] and another over [x1 , R2 ].
N
This necessitates writing two new quadrature rules: let {ζq }M
q=1 be the nodes for [−R1 , x1 ] and {ζm }q=M +1

136

for [x1 , R2 ], with weights wq corresponding to ζq . Then the first equation for n1 like (7.78) is
′
n1 (x1 ) = ξ(x1 ) + ∑wq χ1 (x1 , ζq ) [φ˜1 (ζq ) + Vxc
(ζq )n1 (ζq )]
N

q=1

Because of the nonlinear spacing of Gaussian points, we will have ζp ∉ {xq }N
q=1 for every p. Consequently,

there are N + 1 unknowns in the above equation. While using N − 1 total ζm values momentarily fixes that
′ N
issue, when we move to x2 and write quadrature rules {ζq′ }M
q=1 for [−R1 , x2 ] and {ζq }m=M +1 for [x2 , R2 ], we

will have ζp′ ∉ {ζq , xq }N
q=1 , meaning we generate an entirely new set of unknowns. As a result, we cannot

formulate a linear system using this approach to accommodating the cusps.

An alternative approach is to write rules for each interval [xq , xq+1 ] so that the cusps in the integrand will

always occur at endpoints. Because there are N − 1 such intervals, to obtain N total unknowns for each
of the N nodes xq (for a square matrix A), we can only write one-point rules for each [xq , xq+1 ], except

for a single two-point rule to bring the total nodes to N . The one-point Gaussian rule coincides with the
midpoint (rectangle) method, which offers only middling O(h2 ) accuracy. As the primary benefit of Gaussian
quadrature is its superb accuracy with respect to the number of points used, the large initial mesh being
locked to one-point rules necessitates destroys any purpose for using Gaussian quadrature.
We instead consider a quadrature rule based on Simpson’s rule. For N odd and xq = a+(q −1)h with h =
the rule takes the form

∫

b
a

f (x) dx ≈

h
[f (x1 ) +
3

N −1

∑

q=2
m even

4f (xq ) +

b−a
,
N −1

N −2

∑ 2f (xq ) + f (xN +1 )]

q=3
q odd

and is O(h4 ) accurate, assuming f ∈ C 4 [a, b]. Because this differentiability is lost only at y = x, at which

point the integral is broken, we have the regularity necessary to guarantee this error estimate. Thus, in stark
contrast to Gaussian quadrature, Simpson’s rule cooperates with splitting at x.
Consider the abstract example (7.77). Let xp be one of the Simpson nodes. Then we can write
λu(x) = g(x) + ∫

xp
a

K(x, x′ )u(x′ ) dx′ + ∫

b

xp

K(x, x′ )u(x′ ) dx′

First, suppose that p is odd with p ≥ 3. Then the two subrules on [a, xp ] and [xp , b] take the form
∫

xp

a

f (x) dx ≈

h
[f (x1 ) + 4f (x2 ) + 2f (x3 ) + ⋯ + 4f (xp−1 ) + f (xp )]
3
137

∫

b
xp

f (x) dx ≈

h
[f (xp ) + 4f (xp+1 ) + 2f (xp+2 ) + ⋯ + f (xN )]
3

when these result are combined, we get
∫

b
a

f (x) dx = ∫

xp

a

≈

f (x) dx + ∫

b
xp

f (x) dx

h
[f (x1 ) + ⋯ + 4f (xp−1 ) + 2f (xp ) + 4f (xp+1 ) + ⋯ + f (xN )]
3

which is precisely the rule on the whole of [a, b]. This shows that the standard Simpson’s rule automatically
handles cusps that occur at odd numbered grid points inside the domain.

Next, suppose that p is even with p ≥ 4. Simpson’s rule requires an odd number of points (or even number
of subintervals), and when p is even, there are an odd number of nodes each to the left and right of xp .
Assuming f is defined a bit beyond [a, b], we instead write an approximation for the integral over [a−h, b+h].
Then {a − h, x1 , . . . , xp } and {xp , . . . , xN , b + h} will both contain an odd number of elements and
∫

∫

a−h
b+h

xk

f (x) dx ≈

h
[f (a − h) + 4f (x1 ) + 2f (x2 ) + ⋯ + 4f (xp−1 ) + f (xk )]
3
h
f (x′ ) dx ≈ [f (xp ) + 4f (xp+1 ) + 2f (xp+2 ) + ⋯ + 4f (xN ) + f (b + h)]
3

xk

so that upon combination, we generate the rule on whole of [a − h, b + h].

Na¨ıvely gluing the even and odd cases together haphazardly would be unwise, as we would generate two
different approximations to the same integral from splitting. If f (a) = 0 and f (b) = 0, then we define
⎧
⎪
⎪
⎪
⎪ f (x)
f˜(x) ≜ ⎨
⎪
⎪
⎪
0
⎪
⎩

so that
∫

b
a

f (x) dx = ∫

b
a−2h

x ∈ [a, b]
x ∉ [a, b]

f˜(x) dx = ∫

b+h
a−h

f˜(x) dx

For splitting at the odd numbered points, we compute the integral after the first equals sign; for even
numbered points, after the second equals sign. Assuming K(x, x′ )u(x′ ) vanishes at the endpoints,
p even ∶ λu(xp )= g(xp ) +

N −1

4h
2h
∑ K(xp , xq )u(xq ) +
∑ K(xp , xq )u(xq )
3 q even
3 q odd
N

N −1

2h
4h
p odd ∶ λu(xp )= g(xp ) +
∑ K(xp , xq )u(xq ) +
∑ K(xp , xq )u(xq )
3 q even
3 q odd
N

138

(7.81)

This modified scheme also handles the cases of p = 1 and p = 2, which were previously excluded. Note

that the grid points should be taken so that x1 = a and xN = b. The extended points outside [a, b] are be

numbered x−2 = a − 2h, x−1 = a − h, xN +1 = b + h. While N + 2 points were used to write each rule, only N of
these contribute, so we may consider it an N point rule, since the spacing is still given by h =

b−a
.
N −1

As we saw in Figure 7.4, χ1 (−R1 , y; ω) ≈ 0 and χ1 (x, y; ω) → 0 exponentially for x ≫ 0. φ˜1 → 0 as x → −∞

and approaches a constant for x ≫ 0. The kernel e−λ∣x−y∣ clearly decays, so the only potential trouble spot
is fxc (y)n1 (y; ω). As n1 and n0 display similar decay properties as x → +∞, an invocation of Theorem 3.5,

combined with the boundedness fxc of x → −∞ and the decay of n1 inside the metal let us conclude that
both integrands of (7.75) vanish in y at −R1 and R2 . Accordingly, (7.81) is applicable to (7.75).

Let {xq }N
q=1 uniformly spaced nodes on [−R1 , R2 ] such that xq = 0 for some q. We need 0 to be a node because

we require I1 (0; εk ± ω) to compute ξ2 and it is much more efficient to include it as a node. Application of

the scheme (7.81) to the system (7.75) yields the linear system
⎛1 − (χ ⋅ W)F
xc
⎜
⎜
2π
⎝ − λ Λ⋅W

˜ ξ, and λ are N × 1 vectors with entries
where n, φ,

−χ ⋅ W ⎞ ⎛ n ⎞ ⎛ ξ ⎞
⎟⎜ ⎟ = ⎜ ⎟
⎟⎜ ⎟ ⎜ ⎟
˜⎠ ⎝λ⎠
1 − λ2 Λ ⋅ W⎠ ⎝φ

np = n1 (xp ; ω)

ξ p = ξ1 (xp ; ω) + ξ2 (xp ; ω)

˜p = φ˜1 (xp ; ω)
φ

λp = −

and χ, Λ, and W are N × N matrices with entries
χpq = χ1 (xp , xq ; ω)

Wpq

Λpq = e−λ∣xp −xq ∣

⎧
⎪
⎪
⎪
⎪
=⎨
⎪
⎪
⎪
⎪
⎩

2πσ(ω) −λ∣xp ∣
e
λ

2h
3
4h
3

if p − q is even
if p − q is odd

Finally, Fxc = diag(fxc (x1 ), . . . , fxc (xN )). The dot between matrices indicates elementwise multiplication

(i.e., C = A ⋅ B has entries Cpq = Apq Bpq ).

7.7.3

Computing G(x, y; εk + ω) and G(x, y; εk − ω)

The procedure for computing G is the same for both εk + ω and εk − ω, just with different parameters. The
basis functions ϕ1 and ϕ2 are found by applying Algorithm 2.3 to the following problems:

139

(−

1 ∂2
+ V (x) − εk ∓ ω) ϕ1 (x; εk ± ω) = 0
2 ∂x2

(−

ϕ1 ∼ eiνk x as x → +∞
±

1 ∂2
+ V (x) − εk ∓ ω) ϕ2 (x; εk ± ω) = 0
2 ∂x2
ϕ2 ∼ e−iµk x as x → −∞
±

where, as has been lurking in the background of this chapter all along,
µ±k =
νk± =

√

√

k 2 ± 2ω

k 2 ± 2ω − 2∆V

The infinite asymptotic conditions are imposed as starting values per Algorithm 2.2:
ϕ1 (xN −1 ; εk ± ω) = eiνk xN −1

ϕ2 (x1 ; εk ± ω) = eiµk x1

±

±

ϕ1 (xN ; εk ± ω) = eiνk xN
±

7.7.4

ϕ2 (x2 ; εk ± ω) = eiµk x2
±

Computing I1 (x; εk ± ω) and I2 (x; εk ± ω)

Aside from the different basis functions in the integrand, I1 and I2 of (7.64) and (7.65) can be computed
in an identical manner. Further, it makes no procedural difference whether we are calculating for εk + ω or
εk − ω, as there’s no switching, changing realities, or other potential trouble spots that must be sidestepped.

Since all integrands are well-behaved and the spline method that computes ψk , ϕ1 , and ϕ2 gives solutions
that are known on all of [−R1 , R2 ], Gaussian quadrature may be used to perform the integrations. “Exact”

integration, or multiplying out the spline representations and then integrating each polynomial analytically,
is also an option. However, because ψk and ϕm are defined on different grids and the meshes for ϕm can be
quite large, the multiplication will be quite slow (see Appendix C for details on spline multiplication). The
gain in accuracy is marginal at best and cannot justify the protracted computational cost.
The Gauss rule of choice is the seven-point composite rule previously used in forming n0 in Section 5.7.2.
We chop up the domain of integration [x, R2 ] into d divisions, where d = ⌈ R22−x ⌉, giving seven points per
two units of space. While we have no alternative but to march through each x in the mesh for [−R1 , R2 ], at
least we ensure that each step will be computed swiftly and accurately.

140

7.8

Computational Results

Finally, we present the culmination of all the hard work of this chapter by showing example numerical
computations for the linear response density at a variety of input frequencies. For easier cross-comparison
across different materials, it is natural to represent frequencies as fractions of ωp , since atomic frequency
units are rather enigmatic. Table 7.2 below lists equivalent wavelengths for photons.

rs

0.05ωp

0.1ωp

0.2ωp

0.3ωp

0.4ωp

1064 nm

Example Material

2.0

1488

744

372

248

186

0.0699ωp

Platinum (1.99)

2.5

2080

1040

520

347

260

0.0977ωp

Cadmium (2.59)

3.0

2734

1367

683

456

342

0.1285ωp

Gold (3.01)

3.5

3445

1722

861

574

431

0.1619ωp

Strontium (3.56)

4.0

4209

2104

1052

701

526

0.1978ωp

Sodium (3.93)

Table 7.2: SI Wavelengths and Frequencies in Atomic Units. The energy of a photon in eV is
eV-nm
E(eV) = 1239.841876
. One a.u. of angular frequency is 27.211396 eV, so a photon with frequency ω
λ (nm)
a.u. has wavelength λ = 45.563332ω −1 nm. The fourth column gives the frequency that corresponds exactly
to 1064 nm. The color of text indicates the color of light: infrared, red, green, blue, violet, and ultraviolet.
A case of particular interest is ω = 0.0428 a.u., which is the frequency of photons at 1064 nm. This is

most common variant of the venerable Nd:YAG laser, which has been one of the most widely available and
commonly employed solid-state lasers since its invention at Bell Labs [41]. The fractions of the plasma
frequency corresponding to this important wavelength are contained in the fourth column of Table 7.2.
The sample computations are all for rs = 3. As seen in the table, the first four tenths of ωp for this rs value

correspond to four different colors of light. This is more novelty than demonstrative, however; a stronger
motivator for choosing this rs value is that it closely approximates gold, always a material of interest.
For the sample computations presented in this section, an adaptive tolerance of τa = 5 × 10−7 was selected

for the spline methods used to compute the basis functions ϕ1 and ϕ2 . This is slightly less than what was
used for the ψk of the ground state, but as seen in Table 7.3, the number of splines required to reach even

this tolerance is significantly higher. For all computations, a mesh of 901 points was used on the domain
[−35, 10], giving spacing h = 0.05 and including the origin as a mesh point as required.

Figure 7.5 below depicts the first example: ω = 0.1285ωp , which corresponds to 1064 nm for rs = 3.

141

Im n1 (x, 0.1285ωp ) Re n1 (x, 0.1285ωp )
4
3
2
1
0
−1

Real and Imaginary Parts of n1 (x, 0.1285ωp )

×10−1

×10−2

2
1
0
−1
−2
−3
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 7.5: Linear Response for 1064 nm Incident Photons. This figure depicts the linear response
density n1 (x; ω) for rs = 3 and ω = 0.0482 = 0.1285ωp , which corresponds to photons of wavelength 1064 nm.
The profile is quite different from what we see in the ground state. There is a large peak in the real part at
approximately x = 0.25; this indicates that the maximum linear response occurs just outside the metal. The

order of magnitude of the imaginary part is one less than the real part. Even this near the end of the IR
spectrum (which ends around 700-750 nm), the real part is the primary contributor to the response.
A small amount of corruption in the solution can be see at the very left endpoint; it is more pronounced in

the imaginary part than the real part. The lengthy discussion about accommodating the cusps in the space
variable of χ1 technically applies to the integral in k as well. Per Section 7.4.1, a fully proper treatment
would involve splitting the integral in k at 2ω instead of plowing right through this singularity. To ensure
consistency, the ψk constructing χ1 must also construct n0 , so we cannot use different sets of k values.
Consequently, we would have to customize the ground state computation for each ω. As the corruption is
minor and confined near an arbitrarily chosen endpoint, repair is not worth doubling the computation time.
The results of runs for ω = 0.05, 0.1, 0.2, 0.3, and 0.4ωp are shown on the composite figure on the next page.
The real part of the response densities varies little in shape or magnitude until 0.4ωp . The imaginary part

increases steadily in magnitude with ω, finally becoming an equal contributor in the ultraviolet regime. In
the infrared and visible regions of the spectrum, the response profile is dominated by the real part.
Physically, the induced dipole moment (first moment) of the imaginary part of n1 ,
α(ω) = ∫

Rd

xd n1 (x; ω) dx

142

Im n1 (x; 0.1ωp )

Re n1 (x; 0.1ωp )

0.1ωp

−2
2

×10−1

Im n1 (x; 0.2ωp )

Re n1 (x; 0.2ωp )

0.2ωp

−2
2

×10−1

Im n1 (x; 0.3ωp )

Re n1 (x; 0.3ωp )

×10−1

0.3ωp

0

×10−1

6
4
2
0
−2
−35

2

0

×10−1

Re n1 (x; 0.4ωp )

6
4
2
0
−2

−2
0

×10−1

6
4
2
0
−2

0.05ωp

0

×10−1

6
4
2
0
−2

×10−1

−2

×10−1

Im n1 (x; 0.4ωp )

6
4
2
0
−2

2

Im n1 (x; .05ωp )

Re n1 (x; .05ωp )

×10−1

4

0.4ωp

2
0

−30

−25

−20

−15

x

−10

−5

0

5

10

−2
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 7.6: Linear Response at Various Frequencies, rs = 3. The linear response densities are shown above for 0.05ωp , 0.1ωp , 0.2ωp , 0.3ωp ,
0.4ωp . The color of the plot indicates the corresponding photon wavelength, which are 1823 nm, 1367 nm, 683 nm, 456 nm, and 342 nm, respectively.
Real parts are on the left in solid lines; imaginary parts, the right in broken lines. Note the different scale in y on the imaginary plot of 0.4ωp .

143

gives the excitation spectrum [116]; in one-dimension, it is proportional to d⊥ (ω). The imaginary part of
α(ω) then gives the absorption, which is of course expected to be weaker at low frequencies.

An especially interesting comparison to make is among low-frequency responses and difference of screened
and ground state densities nς − n0 from way back in Figure 5.12. The comparison is made in Figure 7.7.
Comparison of Low-frequency Responses
0.3
n(x)

0.2
0.1

nς (x) − n0 (x)
Re n1 (x; 0.05 ωp )
Re n1 (x; 0.10 ωp )

0
−0.1
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 7.7: Comparison of Low-frequency Responses. To facilitate comparison, all quantities have
been normalized to integrate to one. As in Figure 7.7, ς = 5 × 10−4 .
There are two striking observations to make in the above figure. First, there is very little difference in the
responses at 0.05ωp and 0.10ωp ; further analysis puts their absolute difference on the order of 10−3 , which is

also what it is between 0.10ωp and 0.1285ωp . Second, the linear response at low-frequencies is remarkably
similar to the difference nς − n0 , except for the slightly smaller magnitude and slight offset in phase.

The more appropriate comparison for the low-frequency response is nς − n−ς , normalized to integrate to one,
(x;ω)
and Re n1σ(ω)
, which also integrates to one per (7.46). Figure 7.8 below displays the striking outcome.

Revised Comparison of Low-frequency Response
0.3
n(x)

0.2
0.1

nς (x) − n−ς (x)
Re n1 (x; 0.1 ωp )/σ(ω)

0
−0.1
−35

−30

−25

−20

−15

x

−10

−5

0

5

10

Figure 7.8: Comparison of Low-frequency Responses, Part II. All quantities have been normalized
to have integral one. This figure justifies future claims of adiabacity we will make later.

144

Because n1 (x; 0.05ωp ) is nearly identical, Figure 7.8 shows only n1 (x; 0.1ωp ) for the sake of clarity. We see
that nς − n−ς agrees with n1 (x; 0.1ωp ) magnificently, placing the peak in the correct location just beyond the

surface and nearly matching in phase. Figure 7.8 will justify an adiabatic assumption in the next chapter.

The presentation of results concludes by once more highlighting the hard work of the adaptive spline method.
Table 7.3 shows the number of splines required to compute ϕ1 and ϕ2 for both εk + ω and εk − ω for each

frequency shown in a figure this section. The table shows that ϕ2 ( ⋅ ; εk + ω) is always the hard one to

compute, while ϕ2 ( ⋅ ; εk − ω) dramatically increases in difficulty with ω, although only for low k. The large

number of splines required can be attributed to exponentially growing behavior at one of the endpoints; for

0.0475

6940

48139

3670

153143

239248

0.2723

52511

3604

62161

2602

38515

3636

121969

21176

0.5273

24949

5644

30069

4490

16037

5726

46637

3694

0.0475

57859

3214

124561

26866

39613

3664

225625

1101174

0.2723

47279

3688

76957

2562

27683

5104

151123

204336

0.5273

22065

5912

37691

3650

14219

5850

56651

3304

0.0475

55333

3436

133433

51242

28035

5028

294575

1602716

0.2723

44057

3732

93543

3008

22417

5900

219727

936386

0.5273

20217

5944

41621

3706

23267

6252

75275

2562

0.30ωp

106681

G(x, y; εk − ω)
ϕ1
ϕ2

0.40ωp

2594

G(x, y; εk + ω)
ϕ1
ϕ2

0.05ωp

64579

G(x, y; εk − ω)
ϕ1
ϕ2

0.10ωp

G(x, y; εk + ω)
ϕ1
ϕ2

0.13ωp

k

0.20ωp

high k, the growth factor is smaller, so accurate solutions are easier to compute.

Table 7.3: Adaptive Splines in Action, Part II. For rs = 3 (kF = 0.6397), this table shows the number
of splines required to compute ϕ1 and ϕ2 for both εk + ω and εk − ω for various f ω with adaptive tolerance
τa = 5 × 10−7 . The value of k extends across the thick center line.
Frequency

Computation Time

Frequency

Computation Time

0.05ωp

10 min, 36 sec

0.20ωp

16 min, 11 sec

0.10ωp

11 min, 14 sec

0.30ωp

36 min, 26 sec

0.13ωp

11 min, 47 sec

0.40ωp

52 min, 31 sec

Table 7.4: Computation Time for Linear Response. This table shows the time required to compute
n1 . An adaptive tolerance of τa = 5 × 10−7 was used. Time required to compute n0 is not included.
Finally, Table 7.4 depicts the total computation time for determining the linear response density from start
to finish. The workstation is the same as used for the ground state (a Core i7 at 3464 MHz with 16GB of

145

RAM). Table 7.3 indicated that an exceptional number of splines are required to reach the adaptive tolerance
for certain ω; consequently, most of the CPU time is devoted to G(x, y; εk + ω) and G(x, y; εk − ω).

7.9

Conclusion

As attested to by the bevy of citations throughout this chapter, this thesis is certainly not the first to tackle
linear response calculations for jellium surfaces. However, it contains a number of notable improvements in
compution that make the contributions of this work significant.
Sections 7.1 and 7.2 contain a rigorous and complete derivation of linear response theory and the linear
response function from first principles, a presentation with scope the author was unable to locate in any
single publication. While the resultant formulas and hints of their origin are readily available in virtually
every source discussing the concepts, the mathematically inclined will appreciate the effort made here.
In terms of analytic calculations, Section 7.4 presented a streamlined computation based on Fourier transforms that renders the cumbersome methods of Feibelman [29, 30] wholly obsolete. Authors such as Liebsch
[67, p.170] have indicated the necessity of analytic-based contributions from inside the bulk metal, but none
has provided a shred of the specifics of the execution aside from Feibelman.
Finally, the modifications of (7.58) and (7.59) of Section (7.5) that enable direct solution of the integral
equations via (7.75) are a very welcome upgrade to the established iterative methods of Liebsch [65, 66, 67].
Continued use of the adaptive spline method to compute G(x, y; εk ±ω) allows for seamless vertical integration
with the ground state computations, as the grid for each G can be chosen independently.

146

Chapter

8

Second Harmonic Generation
Second Harmonic Generation (SHG), in which a material converts electromagnetic radiation at frequency
ω to that at 2ω—for example, red light becomes green or green light becomes blue-violet—is the simplest
yet perhaps most studied nonlinear optical phenomenon. The humbling story of its discovery is every bit as
interesting as the numerous applications [47, 50, 82] SHG has found throughout the years.
In the early 1960s at the University of Michigan in Ann Arbor, Robert Terhune bet his colleague Peter
Franken a nickel—the standard wager in physics at the time—that he could not generate optical harmonics.
A favorite story among the co-discoverers [51], Franken never collected the monetary spoils from his victory,
which Terhune did pay in the form of check never cashed, and was content to settle for the satisfaction of
revolutionizing the world by birthing the field of nonlinear optics.
For the disarray he was willing to unleash upon Terhune’s checkbook, Franken would not go without impunity.
When the results of the celebrated experiment were first published in Physical Review Letters [37], the editor
mistakingly erased the photographic evidence that proved their discovery, reproduced below.

Figure 8.1: “Demonstration” of SHG. Where the arrow points should be a small dot indicating the
second harmonic. Its conspicuous absence is due to the editor’s misguided assumption that the mark was
blemish. Reproduced with express permission from the American Physical Society and Gabriel Weinreich.

147

8.1

The Mathematics of Second Harmonic Generation

Mathematically, SHG is modeled by the time-harmonic❸ Maxwell’s equations in Gaussian units
∇ × E(x, ω ′ ) = −

iω ′
B(x, ω ′ )
c

∇ × H(x, ω ′ ) =

iω ′
D(x, ω ′ )
c

where E is the electric field, H and B are the magnetic field and induction (respectively), and D is electric
displacement; the frequency ω ′ is taken to be either ω or 2ω depending on whether the fundamental or
second harmonic is of interest. For nonmagnetic materials, we have B = H. The nonlinearity necessary to
describe nonlinear optical phenomena is contained in the constitutive relation for D
D(x, ω ′ ) = ε(ω ′ )E(x, ω ′ ) + 4πPNL (x, ω ′ )

PNL is the nonlinear polarization and has form highly dependent on the material under consideration.
Franken’s first demonstration of SHG was in quartz, a dielectric material. The mathematical formulation
for such materials is well-established [8, 98] and takes the polarization as a power series in the electric field,
PNL (x, ω ′ ) = χ(2) (ω ′ ) ∶ EE + χ(3) (ω ′ ) ∶ EEE + ⋯

The optical susceptibilities χ(k) are k +1-rank tensors and are not to be confused with the response functions

of TD-DFT, which are denoted with subscripts. As a rank-3 tensor, χ(2) has twenty-seven components,

many of which zero out with the crystal’s symmetry structure. For centrosymmetric materials, χ(2) ≡ 0. As
a second-order process, SHG is governed by χ(2) and is theoretically impossible in centrosymmetric crystals.

(a)

Metal

Lattice

Chromium

bcc

Copper

fcc

Gold

fcc

Iron

bcc

Platinum

fcc

Silver

fcc

(b)

Figure 8.2: Examples of Metal Crystal Structure. Two common metal crystal lattice structures are
shown here. (a) is a body-centered cubic (bcc) lattice and (b) shows a face-centered cubic (fcc) as Bravais
lattice. In both cases, the middle of the cube (in yellow) is the point of inversion symmetry; in bcc, this
point is a node in the lattice. The table on the very right indicates metals that have these structures.
❸A

time-harmonic quantity is of the form A(x, t) =

1
Re (A(x)eiωt ).
2

148

Of the thirty-two crystal classes, eleven of them display inversion symmetry, so SHG is not possible in every
nonlinear optical material, although third-order effects are. All metals possess crystal classes among these,
of which two examples and list of metals possessing them are displayed in Figure 8.2 on the previous page.
Consequently, χ(2) vanishes in metals, so SHG should not be possible. Despite this, Bloembergen et al. [6]
experimentally demonstrated SHG in metals as a reflected wave. Figure 8.3 depicts the difference in process.

ω

ω

2ω

ω
ω

Nonlinear
Material

2ω

Metal

(a)

(b)

Figure 8.3: SHG in Dielectrics vs. Metals. (a) SHG in dielectrics: the dominant second-harmonic
wave is a transmitted wave. (b) SHG in metals: it is a reflected wave because of attenuation.

The deficiency in describing SHG in centrosymmetrc materials is not an inherent limitation of the power
series expansion. Rather, the polarization is taken in the electric dipole approximation, which ignores the
much weaker contributions from electric quadrupoles and magnetic effects, and these dipole moments vanish
in centrosymmetric materials. More properly, the nonlinear polarization should be [9, p.28]
(2)

ΠNL (x, ω ′ ) = χ(2) (ω ′ ) ∶ EE −∇ ⋅ [ χQ (ω ′ ) ∶ EE ] +
P(2)

Q(2)

(2)

µ0
(2)
∇ × [ χM (ω ′ ) ∶ EE ] + ⋯
iω ′
M(2)
(2)

where χQ denotes the (electric) quadrupole susceptibility and χM denotes the magnetic dipole susceptibility. The unwieldiness of the above form and the difficulty of experimentally determining the additional
susceptibilities means that different modeling techniques are needed for centrosymmetric materials.

8.1.1

The Classical Model

In seeking a revised theory that theoretically describe their experimental finding of SHG in metals, Bloembergen et al. considered the Drude-Sommerfeld free-electron model for metals and began with the classical
hydrodynamic equation
∂v
e
1
+ (v ⋅ ∇)v = − (E + v × H)
∂t
m
c

149

(8.1)

where e and m are the the electron charge and mass❸ , respectively, and v is the average electron velocity.
By expanding each quantity in powers of eiωt , they obtained at the fundamental frequency
PNL (ω) = 0
and a complicated nonlinear polarization at the second harmonic
PNL (2ω) =

e
n
¯ e3
ie3 n
¯
(E(ω) ⋅ ∇)E(ω) +
E(ω)(∇ ⋅ E(ω)) +
E(ω) × H(ω)
2
4
2
2
4m ω
8πmω
4m cω 3

(8.2)

Applying the vector identity
1
E × (∇ × E) = ∇(E ⋅ E) − (E ⋅ ∇)E
2

in conjunction with the Faraday’s law −
PNL (2ω) =

c
∇ × E(ω) = B(ω) = H(ω), (8.2) is more commonly written
iω

n
¯ e3
e
e3 n
¯
(E(ω)
⋅
∇)E(ω)
+
E(ω)(∇
⋅
E(ω))
−
∇(E(ω) ⋅ E(ω))
2
4
2
2
2m ω
8πmω
8m ω 4

While also appearing in Shen [98], Brevet [9, Ch.9] has a thorough presentation of the derivation of (8.2).
Bloembergen et al. argued that second harmonic waves originated from electric quadrupoles in the bulk
and drown out the surface contributions in highly reflective media. A year later, Brown and Matsuoka [10]
observed that “atomically clean silver surfaces generate approximately four times as much harmonic light as
those subjected to adsorption” [10], indicating that domination of quadrupoles was not entirely correct.
Rudnick and Stern [93] examined the nature of second harmonic generation at metal surfaces and found that
the response is composed of three currents: one from the bulk, one parallel to the surface, and one normal to
the surface. The third of these had been incorrect in the established theory. The bulk current is expressed
in the third term of (8.2) and contributes only because of the surface. The neglected quadrupoles had been
pegged as the suspect for nearly ten years [104], so it was a remarkable flash of insight to argue that SHG is
intrinsically due to the break in symmetry at the surface and not just predictive failure of previous models.
The hydrodynamic approach to SHG at metal surfaces has been improved a number of times since Rudnick
and Stern’s observations. To incorporate the surface effects lacking from Bloembergen’s model, Sipe et al.
[101] included a quantum pressure term −∇p to the right-hand side of (8.1); Corvi and Schaich [19] further

added an Ohmic damping term − τ1 v. Because of its simplicity and high level of customization, polarizations

developed from hydrodynamic models are still regularly employed in studies today [16, 95].
❸ In

order to present the original results verbatim, this subsection (and this subsection only) does not use atomic units

150

8.2

The Case for Density Functional Theory

Acknowledged in a multitude of sources [44, 93, 101], the hydrodynamic approach is a rather unrealistic
approximation to electron dynamics at the metal surface. Its biggest flaw is that it must assume that the
electron density is constant up to the surface (much like the background charge in jellium); of course, the
profile should vary continuously as we’ve seen in DFT computations. While the hydrodynamic picture could
be improved by incorporating the actual electron density, it would still remain a semi-classical model. If
electron density is to be used, it is better to formulate a fully quantum mechanical approach.
The polarization P is related to the total current J by
J = Jf + ∇ × M +

∂P
∂t

where Jf is the free current injected into the system by external sources and M is the magnetization current
density, a measure of the strength of magnetic dipoles. In non-magnetic materials, M = 0, a fact implicitly

assumed in writing the constitutive relation B = H, which is otherwise B = H + 4πM. In nonmagnetic

materials in the absence of free current, polarization is just the time-integral of the current.
Continuity of charge relates the current to the total charge ρ(x, t) via

Using that J =

∂P
∂t

∇ ⋅ J(x, t) +

∂ρ
(x, t) = 0
∂t

(8.3)

and that in Fourier space, differentiation in t is equivalent to multiplication by iω, we can

eliminate the current and have the relation
∇ ⋅ P(x, ω ′ ) + ρ(x, ω ′ ) = 0

The total charge is easily given in terms of the electron density as ρ(x, t) = en(x, t) = n(x, t) so that if we
can obtain the density in Fourier space, we can obtain the polarization via
∇ ⋅ P(x, ω ′ ) = −n(x, ω ′ )

Per previous discussion, in the case of second harmonic generation, we dub the part of the polarization at ω
the “linear polarization” and the portion at 2ω the “nonlinear polarization” so that
PNL (ω) = 0

∇ ⋅ PNL (2ω) = −n(x, 2ω)
151

Once we have PNL (2ω), we can solve a linear Maxwell equation with source term to obtain the field E(2ω).

8.2.1

The Complete TD-DFT Approach

In the derivation of the TD-DFT linear response of Section 7.3, we saw that n1 (x; ω) contains all of the ω

frequency content of n(x, ω ′ ), which is why we denoted it n1 (x; ω). Consequently, we cannot hope to extract
the frequency information of n(x, ω ′ ) at 2ω from linear response theory alone.

The linear response density n1 is the first term in the functional Taylor series (7.3). The second is given by
n2 (x, ω) = ∫

Rd

V1 (x′ , ω) ∫

Υ(x, x′ , x′′ , ω, ω) ≜

Rd

Υ(x, x′ , x′′ , ω, ω)V1 (x′′ , ω) dx′′ dx′

(8.4)

δ 2 n[Vext ](x, ω)
∣
δVext (x′ , ω)δVext (x′′ , ω) Vext =V0

In SHG, V1 is solely due to the normally incident laser field and is given explicitly by the expression V1 (x, t) =

E0 xeiωt so that V1 (x, ω ′ ) = E0 xδ(ω ′ − ω). Therefore n2 contains all the 2ω frequency content of n(x, ω ′ ).

Mimicking the procedures of Section 7.1.2 and 7.2, we can massage (8.4) into something computable for the
jellium system. After considerable effort, we obtain a form comparable to what we had for n1 :
n2 (x; ω) = ζ(x; ω) + ∫

+∞

−∞

χ1 (x, y, 2ω)[φ2 (y; ω) + fxc (x)n2 (y; ω)] dy

ζ(x; ω) ≜ ∫ φ1,scf (y; ω) ∫ χ2 (x, y, y ′ ; ω, ω)φ1,scf (y ′ ; ω) dy ′ dy +
R

R

(8.5)

1
2
∫ χ1 (x, y; 2ω)gxc (y)n1 (y; ω) dy
2 R

where φ1,scf (x; ω) = φest (x; ω) + φext (x; ω) + φxc (x; ω) with these components of (7.19), (7.20), and (7.21),

respectively; φ2 is electrostatic potential corresponding to n2 and therefore solves φ′′2 (x; ω) = −4πn2 (x; ω).
Finally, the nonlinear response function is given by the expression❸
χ2 (x, y, y ′ ; ω, ω) =

kF
1
(kF2 − k 2 )[ψk (x)ψk (y) G(x, y; εk + 2ω)G(y, y ′ ; εk + ω)+
∫
π2 0

ψk (x)ψk (y) G(x, y; εk − 2ω)G(y, y ′ ; εk − ω)+

ψk (y)ψk (y ′ )G(x, y; εk + ω)G(y, y ′ ; εk − ω)] dk

We see that just like the linear response density, the nonlinear response density n2 satisfies an inhomogeneous
Fredholm integral equation of the second kind, identical in form to (7.58). Remember that we arrived at
❸ A slightly different expression appears in [68]. By switching dummy variables to combine two of the terms, one may obtain
that the expression given here has identical action to the χ2 of Liebsch and Schaich, although the two are not equal as functions.
While the author did not complete his own derivation, he believes that anything with a minus ω ′ should be conjugated.

152

(7.43) from (7.58) by folding together φest and φext . Because the external potential contains no frequency
content at 2ω, the φ2 of (8.5) does represent the electrostatic potential, unlike φ1 of (7.58).
Not only are n1 and n2 determined by integral equations with the same kernel but also n1 is an input in
the source term ζ(x; ω). The ground state also makes an appearance through the wavefunctions needed to
compute χ2 . We thus have a hierarchal process: the nonlinear response requires the linear response, which
requires the ground state. The progression is depicted visually in Figure 8.4 below.
Formula

n0 , ψ k
Ground
State
Density

Formula

χ1

Integral
Equation

Linear
Response
Function

n1

χ2

Linear
Density

Nonlinear
Response
Function

Integral
Equation

n2

Maxwell
Equation

Nonlinear
Density

E(2ω)
Electric
Field

Figure 8.4: Complete TD-DFT Approach to SHG. This diagram illustrates the hierarchy clearly.
While the integral equation (8.5) can be solved in a manner as that for the linear density, the source term
ζ(x; ω) requires considerably more effort to obtain. Each of the three terms of χ2 has a contribution deep
within the bulk metal that must be determined analytically ´
a la Section 7.4. That each term contains a
product of Green’s functions makes the analysis much more difficult. Numerically, we must compute an
additional pair of Green’s functions. The computation of Green’s functions comprised nearly all the runtime
for the linear response calculation. While reuse of G(x, y; εk ± ω) from the previous step can prevent a true
doubling of CPU time, it should be expected that the process will take at least as long as the linear response.

8.3

Intensity Formula

Figure 8.4 shows the procedure concluding with obtainment of the electric field via Maxwell’s equations. In
practice, the electric field does not provide much insight and what we really seek is just the field’s intensity.
It is possible to obtain the conversion efficiency without solving any partial differential equations.
The intensity formula is limited to giving information about the p-polarized component of the generated
second harmonic field. The incident wave can be of mixed s-p polarization but must have a nonzero p
component. Figure 8.5 illustrates clearly the geometry of incident and reflected fields.

153

z

z
2ω
ν

θ

φ

ω

x

y

x

y
(a)

(b)

Figure 8.5: Geometry of Incidence and Reflection. In (a), θ denotes the polar angle, or the angle
with respect to the surface normal ν. φ denotes the angle with respect to the plane of incidence. (This φ is
not electrostatic potential!) Radiation with φ = 0○ is called p-polarized; with φ = 90○ , s-polarized. In (b),
the dashed green line is the p-polarized component of the second harmonic wave, in green.
Let I2ω and Iω be the intensities of the second- and first-harmonic electric fields just outside the surface:
I2ω ≜

2
c
∣ lim+ E(x, 2ω)∣
2π x→0

Iω ≜

2
c
∣ lim+ E(x, ω)∣
2π x→0

The conversion ratio is given by a handy formula due to Sipe et al. [101], transcribed below in atomic units
ε(ω)[ε(ω) − 1] tan θ
I2ω
8π
(Ap cos2 φ + As sin2 φ)
= 2 3
Iω2
ω c [ε(2ω) + s(2ω)][ε(ω) + s(ω)]2

2

(8.6)

The amplitudes As and Ap , due to the s- and p-polarized components of the incident wave, are
Ap ≜ a(ω)
As ≜

where

2s(ω)s(2ω)
1
ε(2ω) 2
sin θ +
cos2 θ +
ε(ω)
ε(ω)
2

[ε(ω) + s(ω)]

(8.7)

2

2ε(ω)[1 + s(ω)]

(8.8)

2

s(ω) = sec θ

√

ε(ω) − sin2 θ

Two parameters, b(ω) and d(ω), which would be factors in the second and third terms of (8.7), respectively,
and in (8.8), have been replaced by their values −1 and 1, per Corvi and Schaich [19], who also showed that
the parameterization (8.6) is very nearly what is obtained from a full solution of Maxwell’s Equations.

The parameter a(ω), which is linked to the normal component of the polarization at the second harmonic

154

frequency, is therefore all that is necessary to obtain the intensity via (8.6). It can be found via [67, p.230]
a(ω) = −

8.4

4¯
n

σ 2 (ω)

∫

+∞

−∞

xn2 (x; ω) dx

(8.9)

Dynamical Force Sum Rules

Like n0 , n1 and n2 exhibit slowly decaying Friedel oscillations which make direct numerical evaluation of
the integral in (8.9) impractical. Per (5.26), n1 and n2 behave something like Ax−2 cos(2kF + γ) so that
when multiplied by x and integrated in (7.50) or (8.9), the resultant integrals converge only conditionally.

An accurate quadrature scheme would have to be written on an impractically enormous interval. This
section contains two dynamical force sum rules that make it possible to find d⊥ (ω) and a(ω) from numerical
solutions. The first of these is due to Liebsch [66] and the second is due to Liebsch and Schaich [68].
Lemma 8.1 (Dynamical Force Sum Rule I). The parameter d⊥ (ω) is equivalently given by
d⊥ (ω) =

ε(ω) + 1 ∞
xn1 (x; ω) dx
∫
ε(ω)
0

(8.10)

assuming that the jellium surface occupies the negative half-space.
∎

Proof. See Liebsch [66] or [67, p.172].
The second dynamical force sum rule is directly applicable to a(ω).
Lemma 8.2 (Dynamical Force Sum Rule II). The first moment of n2 (x; ω) is equivalently given by
∫

+∞

−∞

xn2 (x; ω) dx =

ε(2ω) − 1 ∞
σ 2 (ω)
xn2 (x; ω) dx +
∫
ε(2ω)
2¯
n
0

(8.11)

assuming that the jellium surface occupies the negative half-space.
Proof. See Liebsch and Schaich [68] or Liebsch [67, p.232].

8.5

∎

The Hybrid TD-DFT Approach

The first application of DFT to SHG was performed by Weber and Liebsch [113], who used KS-DFT to
estimate a(ω). Chizmeshya and Zaremba [15] built upon that work by instead using Thomas-Fermi-Diracvon Weizs¨
acker theory within OF-DFT, just as shown in Chapter 4. Both studies began with the electrostatic

155

perturbation expansion
ρ(x) = ρ0 (x) + (

E0
E0 2
) ρ1 (x) + ( ) ρ2 (x) + ⋯
2π
2π

(8.12)

where ρ0 is the electron density in the absence of an external electric field and E0 = 2πς is the magnitude of
a static electric field normally incident upon the jellium surface. The induced densities can be found via
1
[ρ+ (x) − ρ− (x)]
2ς
1
ρ2 (x) = 2 [ρ+ (x) + ρ− (x) − 2ρ0 (x)]
2ς

ρ1 (x) =

(8.13)
(8.14)

where ρ+ and ρ− are computed with E0 = 2πς > 0 and E0 = −2πς < 0, respectively. If instead the applied

field is of the form E(t) = E0 eiωt with ω sufficiently low, say ω ≲ 0.1ωp , then the system may be assumed to

respond adiabatically and the time varying quantity E(t) may replace E0 in (8.12):
ρ(x, t) = ρ0 (x) + (

E(t)
E(t)
) ρ1 (x) + (
) ρ2 (x) + ⋯
2π
2π
2

= ρ0 (x) + ςρ1 (x)eiωt + ς 2 ρ2 (x)e2iωt + ⋯

which after a Fourier transform in time becomes
ρ(x, ω) = ρ0 (x)δ(ω ′ ) + ςρ1 (x)δ(ω ′ − ω) + ς 2 ρ2 (x)δ(ω − 2ω ′ ) + ⋯

(8.15)

In the macroscopic view, the bulk metal feels a screening charge of ς = E0 /2π. The applied field may be

viewed as originating from a uniform sheet of charge parallel to the surface infinitely far away. Near infinity,
ˆ . Since φ = −∇Etot , the
the total electric field between the surface and this uniform sheet is Etot = 4πςx x
screening charge generates the boundary condition φ′ (+∞) = −4πς. Consequently, in (8.12), ρ0 corresponds
to the ground state density n0 and ρ+ and ρ− correspond to the screened densities nς and n−ς , respectively.
Via the preceding paragraph, (8.15) allows us to associate ρ1 and ρ2 with the TD-DFT linear and nonlinear
response densities n1 (x; ω) and n2 (x; ω). To place the linear term ρ1 , first observe that from (8.13),
∫ ρ1 (x) dx =
R

1
[∫ (nς (x) − n+ (x)) dx − ∫ (n−ς (x) − n+ (x)) dx] = 1
2ς R
R

by the charge condition (5.28). Remember that n1 (x; ω) integrates to σ(ω) per (7.46). Consequently,
ρ1 (x) = −

1
n1 (x; ω)
σ(ω)

156

By making the appropriate replacements in ρ2 ,
∫ ρ2 (x) dx =
R

1
[∫ (nς (x) − n+ (x)) dx + ∫ (n−ς (x) − n+ (x)) dx − 2 ∫ (n0 (x) − n+ (x)) dx] = 0
2ς 2 R
R
R

It can be shown that n2 (x; ω) as defined in (8.5) integrates to zero, so there are no issues with normalization

as there were for ρ1 and n1 (x; ω). Accordingly, ρ2 = n2 (x; ω) and we have the approximation
nς (x) = n0 (x) +

ς
n1 (x; ω) + ς 2 n2 (x; ω) + ⋯
σ(ω)

Liebsch and Schaich computed solutions to (8.5) in [68]. An inspection of the limited results presented there
reveals that there are only small differences between low-frequency responses, just as we saw for n1 (x; ω) in

Figure 7.7. Furthermore, in Figure 7.8, we saw very near agreement of the ρ1 of (8.13) and n1 (x; ω) up to
0.1ωp . It is not a stretch to conclude that a full solution of the nonlinear response equations is unnecessary

and we can generate a reasonable approximation to n2 (x; ω) by rearranging (8.12):
n2 (x; ω) ≈

1
ς
[nς (x) − n0 (x) −
n1 (x; ω)]
ς2
σ(ω)

(8.16)

We dub this method the hybrid TD-DFT approach because it combines linear response theory and electrostatics. A comparison of this method with a full solution to (8.5) is the subject of Section 8.5.2.
While the hybrid TD-DFT approach requires nς to be available, the complete TD-DFT approach must
compute n1 (x; ω) and n2 (x; ω) for each frequency. The screened density nς is independent of the frequency

and need only be computed once for a given rs and ς, just like n0 . Therefore, the hybrid approach need only
compute n1 (x; ω) once nς is available, requiring only half the work for subsequent inputs ω.

8.5.1

a(ω) in the Hybrid TD-DFT Approach

The nonlinear density is a means to the end; it is necessary only to compute the a(ω) parameter that appears
in Ap . The dynamical force sum rules of Section 8.4 allow for direct expression of a(ω) without reference to
n2 (x; ω) in the hybrid TD-DFT approach. First, by Lemma 8.2,
a(ω) = −

4¯
n ε(2ω) − 1 ∞
xn2 (x; ω) dx − 2
∫
ε(2ω)
0

σ 2 (ω)

157

Then substituting (8.16) into the above and using Lemma 8.1, we have
∞
∞
ς
1 4¯
n ε(2ω) − 1
x(nς (x) − n0 (x)) dx −
xn1 (x; ω)] dx − 2
[∫
∫
2
2
ς σ (ω) ε(2ω)
σ(ω) 0
0
⎤
∞
1 4¯
n ε(2ω) − 1 ⎡⎢
ε(ω)d⊥ (ω)
⎥
⎢ς
= −2 + 2 2
x(nς (x) − n0 (x)) dx⎥
−∫
⎥
ς σ (ω) ε(2ω) ⎢⎣ σ(ω)(ε(ω) + 1)
0
⎦

a(ω) = −

(8.17)

With the formula (8.17) in hand, we can bypass forming n2 and proceed directly to the intensity formula.
The diagram below is an updated version of Figure 8.4 to reflect the new procedure.

Ground
State
Density

Screened
Density

n0 , ψ k

nς

Formula

χ1

Integral

n1

Equation

Linear
Response
Function

Formula

a(ω)

Intensity
Formula

I2ω

Linear
Response
Density

Figure 8.6: Hybrid TD-DFT Approach to SHG. The computational burden is significantly less than
the complete approach once nς has been computed and stored permanently.

8.5.2

Comparison with Full Solution

The formula for a(ω) in (8.17) is handy, but an important consideration is its accuracy in comparison to a
full solution to the nonlinear response equation (8.5). Below is the full solution for rs = 3 and ωp = 0.1ωp :

Figure 8.7: Nonlinear Response from Full TD-DFT Approach. This figure depicts the full
solution n2 (x; 0.1ωp ) to (8.5) computed by Liebsch and Schaich [68] for rs = 3. Reproduced with express
permission from the American Physical Society and Ansgar Liebsch.
Below in Figure 8.8 is the hybrid TD-DFT version of n2 (x; ω) from (8.16), assembled from the ground state
density of Figure 5.6 and screening density with ς = 5 × 10−4 of Figure 5.10.
158

Real and Imaginary Parts of n2 (x, 0.1ωp )

Re n2 (x, 0.1ωp )

30
20
10
0

Im n2 (x, 0.1ωp )

−10
20
10

0

−10

−20
−30

−25

−20

−15

−10
x

−5

0

5

10

Figure 8.8: Nonlinear Response from Hybrid TD-DFT Approach. Shown is n2 (x; 0.1ωp ) for
rs = 3 and ς = 5 × 10−4 . Unlike the linear response density, which has a single spike just outside the surface,
n2 has three such peaks alternating in sign.
Comparing the real parts, we see that the hybrid TD-DFT method reproduces the spike just outside the
surface nearly perfectly in both placement and magnitude but falters inside the metal. In particular, the
negative spike occurring near the surface inside the metal is significantly weaker than it is in the full TDDFT solution. The leftmost peaks match in location around x ≈ −5 but differ slightly in magnitude. Friedel

oscillations in the bulk material are equivalent phase and magnitude in both approaches.

The imaginary part of the hybrid TD-DFT’s nonlinear density is significantly different than the full-solution’s.
As both n0 and nς are real valued, the only contribution to Im n2 (x; ω) in (8.16) comes from a scaled version
of n1 (x; ω). As n1 contributes to n2 nonlinearly in (8.5), significant divergence is to be expected.

Because all of the quantities ε(ω ′ ), σ(ω), n
¯ , and ς are real numbers, the real and imaginary parts of a(ω)

are determined solely from the corresponding parts of xn2 (x; ω). Because of Lemma 8.2, the convergence of

the real parts of the hybrid and full TD-DFT approaches in the region [0, +∞) means that we should expect
good agreement between the real parts of a(ω) computed by both approaches.

8.6

Computational Results

The table below shows sample results for a(ω) computed via (8.17).
There is no “correct value” for a(ω). For rs = 3, the Kohn-Sham based electrostatic approach of Weber

and Liebsch [113] found a(0) = −12.9 and the TFDW techniques of Chizmeshya and Zaremba [15] placed
159

ς
5.0 × 10

−4

2.5 × 10−4
1.0 × 10

−4

Average

0.05ωp

−16.3473 + 0.7321i
−16.9031 + 1.4643i
−21.4864 + 3.6607i
−18.2456 + 1.9524i

0.10ωp

−14.0104 + 1.2906i
−11.7983 + 2.5811i
−8.1691 + 6.4528i

−11.3259 + 3.4415i

0.1285ωp

−15.3935 + 1.3427i
−14.1679 + 2.6853i

−13.5819 + 6.7133i

Average
-15.2504 + 1.1218i
-14.2898 + 2.2436i
-14.4125 + 5.6089i

-14.3811 + 3.5804i

Table 8.1: Computed Values for a(ω). This table shows the result of the hybrid TD-DFT method

a(0) = −14.9. The methodologies of both studies are unclear. Neither mentions how a single value of a(0)

was distilled from presumably many values of ς, just that “ς was taken on the order of 10−4 .” To maintain
a legitimate comparison with these studies, ς was chosen on the same order in Table 8.1.

Unfortunately, Liebsch and Schaich did not present a table of their values for a(ω) for the full TD-DFT
approach, instead presenting only a small graph from which it is difficult to extract precise values. A careful
inspection of their figure places Re a(0.05ωp ) ≈ −13, Re a(0.10ωp ) ≈ −14 and Re a(0.1285ωp ) ≈ −15. This is
more or less what we obtain from the hybrid TD-DFT method, although the result is a bit sensitive to ς.

8.7

Conclusion

The DFT approach for SHG is a truly multiphysics formulation, mixing together the quantum mechanics of
DFT, the solid-state physics of jellium and other surface properties, and the optics of Maxwell’s equations.
The hybrid approach tosses one more ingredient—electrostatics—into the boiling cauldron.
A full solution of the nonlinear response equation (8.5) may be considered the optimal approach, but the hybrid TD-DFT approach presented in this chapter is certainly not without merit. If the fully static approaches
of Weber and Liebsch and Chizmeshya and Zaremba produce values deemed acceptable, then incorporation
of any frequency dependence can be considered an improvement. At the very least, the hybrid approach
corrobrates the validity of the static approximation by confirming adiabacity up to the linear response.
The hybrid TD-DFT approach is well-suited to describe processes that depend on the response outside the
surface, where there is excellent agreement with the full nonlinear response. In particular, because second
harmonic generation at a metal surface is not a bulk phenomenon, the hybrid approach is up to the task.
Insight into the nature of SHG that can be obtained from all this: the process is dominated by the linear
response and ground state. Nonlinearity is largely a property of the bulk material and because of the
dynamical force sum rules is not needed to describe all surface phenomena.

160

APPENDICES
blank space

161

Appendix

A

Variational Calculus
This appendix provides a brief overview of the key results from variational calculus employed in the body of
this thesis. For comprehensive instruction in the subject, see, for example, Gelfand and Fomin [39].
Let F (A; B) denote the set of functions from A to B. We will limit discussion here to the set F (Rd ; R). It
is possible to generalize the results here to F (Rd ; Rd ), but doing so requires tensor calculus.

Definition A.1. A functional is a map from a vector space of functions to its underlying scalar field. On
F (Rd ; R), it is a rule that takes a function and returns an element in R.

On Hilbert spaces, the powerful Riesz Representation Theorem gives a complete characterization of all
possible linear functionals. Here, we allow functionals to be nonlinear and define an analog to standard
calculus on the space of functionals. Without delving into too much detail or rigor, we present some essential
results from this variational calculus that are employed on several occasions in the body of this thesis.
Definition A.2 (Variational (Functional) Derivative). For a functional F ∶ F (Rd ; R) → R, the varia-

tional (or functional) derivative is defined by

lim
∫Rd δρ ϕ(x) dx = η→0
δF

where ϕ ∈ F (Rd ; R).

F [ρ + ηϕ] − F [ρ]
∂
= [ F [ρ + ηϕ]]
η
∂η
η=0

δF
is called the “first-variation of F ”
δρ
and is a function. The process of taking a functional derivative is often called “taking the variation.” It is
The functional derivative of a functional is again a functional, and

also possible to compute variations of higher-order by nesting the above definition, which is equivalent to
taking a derivative of the corresponding order in η on the very right side of the above definition.

162

For simple functionals defined by integration, a handy formula is available for the first variation:
Theorem A.1 (Formula for First Variation). If f ∶ R3 × R × Rd → R is C 1 in its second and third

arguments, then the functional F ∶ C 1 (Rd ) → C(Rd ) defined by
F [ρ] = ∫

has first-variation

By letting dk ≜

Rd

f (x, ρ(x), ∇ρ(x)) dx

δF ∂f
∂f
=
−∇⋅
δρ ∂ρ
∂∇ρ

∂f
∂ρ
and writing f (⋅, ⋅, ∇ρ(x)) = f (⋅, ⋅, d1 , ⋯, dn ), the derivative
is interpreted
∂xk
∂∇ρ
∂f
∂f
∂f T
=[
, ⋯,
]
∂∇ρ
∂d1
∂dn

One may also interpret this as “taking the gradient with respect to the gradient vector.”
Proof. Let ϕ ∶ Rd → R be such that ϕ → 0 as ∣x∣ → +∞ (e.g., ϕ is compactly supported). We proceed by
direct computation, beginning with the definition:

∫Rd δρ ϕ(x) dx = [ ∂η ∫Rd f (x, ρ(x) + ηϕ, ∇ρ(x) + η∇ϕ) dx]
η=0
δF

∂

Differentiating under the integral by use of the chain rule, we have

δF
∂f
∂f
=∫ ( ϕ+
⋅ ∇ϕ) dx
d
δρ
∂ρ
∂∇ρ
R

∂f
∂f
and
of f (x, ρ(x) + ηϕ, ∇ρ(x) + η∇ϕ), we differentiate f in its second
∂ρ
∂∇ρ
and third arguments, respectively, and then evaluate at ρ + ηϕ and ∇ρ + η∇ϕ, respectively. By the product

To compute the derivatives

rule for divergence,

so that

∇⋅[

∂f
∂f
∂f
ϕ] =
⋅ ∇ϕ + ϕ∇ ⋅ [
]
∂∇ρ
∂∇ρ
∂∇ρ

∫Rd ∂∇ρ ⋅ ∇ϕ dx = ∫Rd (∇ ⋅ [ ∂∇ρ ϕ] − ϕ∇ ⋅ ∂∇ρ ) dx
∂f

∂f

= −∫

R

d

ϕ∇ ⋅

163

∂f
dx
∂∇ρ

∂f

with the second line coming from the Divergence Theorem and that ϕ vanishes at infinity. Then

∫Rd δρ ϕ dx = ∫Rd ( ∂ρ − ∇ ⋅ ∂∇ρ ) ϕ dx
∂f

δF

Because ϕ was chosen arbitrary, we conclude that

∂f

δF
is exactly as claimed in the theorem statement.
δρ

Of great use in the derivation of linear response theory are the following two chain rules.

∎

Theorem A.2 (Variational Chain Rule for Functions). If f ∶ R → R is a differentiable function, then
the functional chain rule of F [f (ρ)] is given by the expression

δF [f (ρ)] δF [f (ρ)] ∂f (ρ)
=
(x)
δρ(x)
δf (ρ(x)) ∂ρ

The symbol chasing of the above might be a bit confusing. The variational derivative of F should be taken
in the symbol f and the partial of f should be taken in the symbol ρ. For example, if ϕ, ρ ∶ Rd → R and
F [ϕ] = ∫

Rd

ϕ2 (x) dx

f (ρ(x)) = e−ρ(x)

Then the variational derivative of F [f (ρ)] is

δF [f (ρ)]
δ
∂f
= [ ∫ f 2 (ρ(x)) dx] ⋅
(x)
d
δρ(x)
δf R
∂ρ
= 2f (ρ(x)) ⋅ (−e−ρ(x) )
= −2e−2ρ(x)

We can confirm the result by direct computation:
F [ρ] = ∫

Rd

e−2ρ(x) dx

⇒

δF [ρ]
(x) = −2e−2ρ(x)
δρ

We can extend the chain rule for functionals. Suppose that F is a functional that depends on some function
G(ϕ), which itself is a functional G[ρ](ϕ), so that F is a functional of ρ: F [G[ρ](ϕ)]

Theorem A.3 (Variational Chain Rule for Functionals). The variational chain rule for functionals of
F [G[ρ](ϕ)] is given by the expression

δF [ρ]
δF [G] δG[ρ](ϕ)
=∫
dϕ
δρ(x)
δG(ϕ) δρ(x)
164

There is a special case of the above theorem worth investigating. Consider the functional F [G[ρ(x)](y)] =
ρ(x0 ). Then the variational chain rule for functionals yields
δ(x − x0 ) =

δρ(x0 ) δF [ρ]
=
δρ(x)
δρ(x)

=∫
=∫

R

d

R

d

δF [G] δG[ρ](y)
dy
δG[y] δρ(x)

δρ(x0 ) δG[ρ](y)
dy
δG[y] δρ(x)

We state one final result of great relevance to us—it forms the backbone of linear response theory in timedependent density functional theory:
Theorem A.4 (Functional Taylor Series). Given a functional F [y(x)] ∶ F (Rd ; R) → R, its functional
Taylor series is

F [y + δy] = F [y] + ∫ K1 (x1 )δy(x1 ) dx1 +
+

1
∬ K2 (x1 , x2 )δy(x1 )δy(x2 ) dx1 dx2
2!

1
∭ K2 (x1 , dx2 , dx3 )δy(x1 )δy(x2 )δy(x3 ) dx1 x2 x3 + ⋯
3!

where the n-th kernel K is the n-th variation of F :
Kn (x1 , . . . , xn ) ≜

δ n F [y]
δy(x1 )⋯δy(xn )

Again, note that the coefficients are given as integrals against variations, very different from Taylor’s Theorem
for functions on Rd .

165

Appendix

B

Solid State Constants
The Wigner-Seitz radius rs , Fermi wavevector kF , and average electron density n
¯ recur extraordinarily
frequently throughout this thesis. These critical values can all be expressed in terms of one another:

1/3

kF = (3π 2 n
¯)
n
¯=

=(

kF3
3
=
4πrs3 3π 2

rs = (

1/3

9π
)
4rs3

/3
3
9π
) =( 3 )
4π¯
n
4kF
1

1/3

The table below gives the numerical values, along with the Fermi energy EF , for common materials.

rs

kF

2.0

0.9596

2.5

0.7677

3.0

0.6397

3.5

0.5483

4.0

0.4798

4.5

0.4265

5.0

0.3838

n
¯
2.9842 × 10−2
1.5279 × 10

−2

8.8419 × 10−3
5.5681 × 10

−3

3.3730 × 10−3

2.2620 × 10

−3

1.9099 × 10−3

EF

Example Material (actual rs )

0.4604

Platinum (1.99)

0.2947

Cadmium (2.59)

0.2046

Gold (3.01)

0.1503

Strontium (3.56)

0.1151

Sodium (3.93)

0.0909

Potassium (4.86)

0.0737

Rubidium (5.20)

Table B.1: Values of rs , kF , and n
¯ for Common Materials.

166

Appendix

C

The Spline Class
The adaptive spline methods of this this thesis are powered by a custom Matlab class called Spline. While
Matlab provides some built-in spline functionality, this self-developed class is far more flexible, powerful,
and broad in scope than provided code. This appendix details the capabilities of Spline.
The Spline class operates on splines are written in the shifted form
(k)

(k)

(k)

(k)

sk (x) = ad (x − xk )d + ad−1 (x − xk )d−1 + ⋯ + a1 (x − xk ) + a0

(C.1)

for x ∈ [xk , xk+1 ]. It is critical for correct operation that coefficients passed in via the first class constructor
be consistent with this form. If the second class constructor is used, compatible coefficients will be generated.

Class Constructors
The Spline provides two configurations of the class constructor:
❼ Spline(X L,X R,C)

X L and X R are column vectors of length num that represent the left and right endpoints of the splines.
C is a (deg+1)×num column vector that contains the coefficients of the splines. The member variables
deg and num are set automatically from the lengths of these input vectors. C may be complex-valued.
❼ Spline(x,f,fp0,fpn)

x and f are vectors of equal length that represent a mesh and set of function values on that mesh. The
make method is used perform cubic spline interpolation on the values f. The other two inputs fp0 and
fpn represent the value of the function’s derivative at the first and last entries of x, respectively.

167

Creation of null objects is possible by passing no arguments into the constructor. Such functionality is
useful in preallocating memory for arrays of Spline objects.

Member Variables
❼ C: Column vector of length (deg+1)×num that contains the coefficients of the splines.
❼ deg: Scalar value giving the degree of this instance of the Spline object
❼ num: Total number of splines contained in this instance of the Spline object.
❼ X L: Column vector of length num that contains the left endpoints defining the splines.
❼ X R: Column vector of length num that contains the right endpoints defining the splines.

Member Functions
❼ antidiff: Takes an input value a and a location as a string, which must be ‘begin’ or ‘end’, and

returns the antiderivative S of S, given piecewise by
S(x) = {

1 (k)
1 (k)
a (x − xk )d+1 + ad−1 (x − xk )d + ⋯ + a0 (x − xk ) + αk
d+1 d
d

x ∈ [xk , xk+1 ]

The values αk are computed to make S continuous to give it regularity one higher than S has. These

values are found by making S(x1 ) = a if ‘begin’ was input or S(xnum+1 ) = a if the input was ‘end’.
Then the method then marches spline by spline from either 2:num or num-1:1 to set the remaining αk
by enforcing C 0 conditions at the nodes. Consequently, the method will be slow if the mesh is large.
❼ diff: Takes an input k and returns the k th derivative of S. If k > deg S, the user is warned that the

requested derivative vanishes identically but the operation is still allowed. Unless a high-order spline

is created by an external method or as a result of antidiff, S is only C 2 . The diff method will allow
inputs of k > 2 and will return the correct piecewise result, which may not be continuous. However, if

further Spline methods are called upon a discontinuous S, incorrect results may be produced.

❼ evaluate: For sorted vector input x = (x1 , ⋯, xM ), returns the vector (S(x1 ), ⋯, S(xM )) . The
T

method can accept either a column or a row vector for x but always returns a column.

❼ integrate: Takes two input values a and b and returns the value of the definite integral from a to b.

First, the method calls the shrink method to temporarily downsize the domain to [a, b] so that the

168

reduced spline has x1 = a and xM = b. The definite integral is computed exactly by the formula
∫

b

a

S(x) dx =

N −1

∑∫

k=1

xk+1

xk

sk (x) dx =

M −1

∑ ∑

k=1

d

1
d+2−m
a(k)
m hk
d
+
2
−
m
m=1

where hk = xk+1 − xk . If called with no arguments, integration over the whole domain is performed; if

called with just a, integration is performed from a to the end of the domain. The method is vectorized
so that it runs quickly even if num is large, in contrast to antidiff.
❼ get domain: Returns the two-vector [X L(1), X R(end)].
❼ get mesh: Returns the set of complete set of nodes [X L; X R(end)] as an num+1 column vector.
❼ make: Takes inputs of a vector x (of length N ) and corresponding function values f along with values
′
f0′ and fN
that represent derivative information at the first and last values of x. The make method

performs cubic spline interpolation on the input data and creates a 4N vector of coefficients. This
method’s primary function is as a helper method for one of the class constructors.

❼ plot: A highly flexible method for plotting that may be called in a variety of argument configurations:

– No arguments: The method generates its own test mesh with help from get mesh; if the mesh
contains more than 2000 points, then a uniformly spaced mesh of 2000 points is used instead.
– One argument: A column or row vector x specifies the points to use to generate the plot.
– Two arguments: Two configurations are possible. The first argument may be either an empty
vector [ ], in which case a test mesh will be automatically generated, or a vector x of points to
use. The second may be either a string compatible with the Matlab plot command, such as
‘--r’, or a 3 × 1 row-vector containing values in [0, 1] specifying the color in the RGB palette.

– Three arguments: The first argument is either x or [ ] as above. The second argument must be
a 3 × 1 vector specifying a standard RGB color. The third must be a string specifying a standard
Matlab line style: -, :, -., --, .’, o, x, +, *, s, d, v, ^, <, >, p, or h.

– Four arguments: The first three are as above. The fourth is a number that specifies the line width.
If S is complex-valued, then the plot will display as a 2 × 1 subplot with the real part on the top and
imaginary part on the bottom, both with the same user-specified mesh, color, and style options.

❼ respline: Takes an input value N and transfers S from its current mesh to an N -point uniformly

spaced mesh (on the same domain) by re-interpolating values from evaluate.
❼ shrink: Takes input values a and b and permanently reduces the domain to [a, b]. Information on

the original domain outside [a, b] is discarded. This method was designed as a helper for integrate.
169

❼ times power: Takes an input k (which must be an integer) and performs the multiplication xk S,

preserving continuity. This method iteratively calls times x k times.
❼ times x: Globally multiplies S by x without using spline arithmetic. The regularity is preserved.

Spline Arithmetic
Spline objects may be added, subtracted, and multiplied using the standard binary operators +, -, and *.
Addition and subtraction of scalar constants is also supported, as is multiplication by scalar values. Division
in any form is not supported; to divide a Spline by a constant value a, instead multiply by 1/a.
The primary challenge in performing spline arithmetic is that splines may be defined on different meshes.
Accordingly, both splines must be transferred to the same mesh before any arithmetic can be done. It is
highly desirable to preserve the original precision, so transfer must be done without loss of information (e.g.,
via respline). The process is best explained visually, shown below in Figure C.1.
x1

x2

x3 x5

x6 x7

x8 x9 x10

x11

x12

x13

x14

x15
x16

+
x′2

x′3

x′4 x′5

x′6

x′′1 x′′2 x′′4 x′′5 x′′6

x′′7 x′′8

x′7 x′8

x′9

x′10 x′11

x′12 x′13 x′14 x′15 x′16

=

x′1

x′′3

x′′9 x′′10

x′′11

x′′13 x′′14

x′′12

x17

x′′15

x′′16

x′′17

x′′18 x′′19 x′′20

x′′21

x19
x18

x′17
x′′22 x′′24

x′′23 x′′25

x′18

x′′26

Figure C.1: Combining Spline Meshes. Addition of meshes is performed by including all unique
points between the two meshes. The bottommost mesh is the “sum” of the two above it. Points in red are
unique to the top mesh, points in blue are unique to the bottom, and points in purple are shared by both.
Because the coefficients assume the shifted form (C.1), coefficients cannot merely be copied to new subintervals. For example, for S defined on the red mesh in Figure C.1, a new point x′2 was added between x1 and
x2 in transferring to the bottom mesh. For such an S and a point y ∈ [x′2 , x2 ], S(y) was originally given by
(1)

(1)

(1)

(1)

S(y) = ad (y − x1 )d + ad−1 (y − x1 )d−1 + ⋯ + a1 (y − x1 ) + a0

On the combined mesh, S should be evaluated by an expression of the form

170

(2)

(2)

(2)

(2)

S(y) = bd (y − x′2 )d + bd−1 (y − x′2 )d−1 + ⋯ + b1 (y − x′2 ) + b0
(2)

because y now falls within the interval that defines the second spline. The new bm coefficients are computed
(1)

from the am by first using the age-old trick of adding and subtracting the same thing
(1)

(1)

S(y) = ad (y − x′2 − (x1 − x′2 )) + ad−1 (y − x′2 − (x1 − x′2 ))
d

d−1

(1)

(1)

+ ⋯ + a1 (y − x′2 − (x1 − x′2 )) + a0

and then expanding each power with the binomial formula. In general, if xm is the largest point from the
original mesh such that xm ≤ x′′k , where x′′k is a point from the combined mesh, then
⎛ (k) ⎞
⎛ (m) ⎞
⎜bd ⎟
⎜ad ⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜ ⋮ ⎟ = [A ⋅ Z] ⎜ ⋮ ⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜ (k) ⎟
⎜ (m) ⎟
b
a
⎝ 0 ⎠
⎝ 0 ⎠

where A and Z are (d + 1) × (d + 1) matrices with entries
Apq

⎧
d−q+1
⎪
⎪
⎪
⎪ (d − p + 1)
=⎨
⎪
⎪
⎪
0
⎪
⎩

if p ≥ q

Zpq

else

⎧
⎪
′′
p−q
⎪
⎪
⎪ (xk − xm )
=⎨
⎪
⎪
⎪
0
⎪
⎩

if p ≥ q
else

and the dot indicates elementwise multiplication (i.e., [A ⋅ Z]pq = Apq Zpq ). The algorithm must march

element by element through the new mesh, although it remembers the smallest xm from the original mesh

from x′′k−1 so that the original mesh need not be searched each time. Consequently, spline arithmetic can
run slowly, especially when splines on large meshes containing many mutually disjoint points are combined.

Once the same mesh, arithmetic on splines of different degrees is slightly complicated by the different
dimensions of the corresponding C vectors. Fortunately, this issue is much easier resolved than the matter
of different meshes. To add or subtract, the spline of the lower degree is converted to “higher degree” by
padding zeros. For example, if S of degree three and T of degree five are to be combined, S.C is replaced by
(1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

(num)

[0, 0, a3 , a2 , a1 , a0 , 0, 0, a3 , a2 , a1 , a0 , ⋯, 0, 0, a3

(num)

, a2

(num)

, a1

(num)

, a0

]

T

Because S and T are on the same mesh, num = S.num = T.num. The coefficients of the sum or difference of
S and T is then given by the corresponding operation applied to T.C and the new S.C.
Multiplication is slightly less efficient. First, a new vector to represent the coefficients of the product of

171

length (S.deg+T.deg+1)×num is preallocated. Then the entries of P.C, where P is the product spline, on
each [xk , xk+1 ] for k = 1:num are found by circularly convolving the appropriate entries of S.C and T.C:
P.C(p:p+S.deg+T.deg) = conv(S.C(s:s+S.deg),T.C(t:t:T.deg))

where p = 1+(S.deg+T.deg+1)×(k-1), s = 1+(S.deg+1)×(k-1), and t = 1+(T.deg+1)×(k-1).
Addition and subtraction of a scalar value a from S is handled without creation of a spline to represent the
constant: all that need be done is to correspondingly add or subtract a from S.C(S.deg+1:S.deg+1:end).
Multiplication of a scalar requires nothing more than scaling S.C by the desired value.

Child Classes
The spline method is used to compute several different types of functions in this thesis, each with a different
set of parameters associated with it. Spline serves as a parent class for each of these data types, which add
additional member variables to store these distinguishing values. The child classes of Spline are:
❼ Density: used for representing electron densities n0 .

Additional member variables:
– rs: Stores the Wigner-Seitz radius rs .
– sigma: Stores the screening charge ς.
– type: A string, generally ‘OF’ or ‘KS’, that indicates the DFT formulation.
– kvals: For KS-DFT densities, contains the values of k that index the wavefunctions ψk generating
the density. For OF-DFT densities, this field is left empty.
– weights: For KS-DFT densities, contains the Gaussian quadrature weights of the corresponding
entries of kvals. For OF-DFT densities, this field is left empty.
– lambdas: For KS-DFT densities, contains the exponents λk of right boundary conditions of the
wavefunctions ψk generating the density. For OF-DFT densities, this field is left empty.
– alphas: For KS-DFT densities, contains the coefficients αk of right boundary conditions of the
wavefunctions ψk generating the density. For OF-DFT densities, this field is left empty.
– gammas: For KS-DFT densities, contains the phase shifts γk of wavefunctions ψk generating the
density. For OF-DFT densities, this field is left empty.
– gamma kF: For KS-DFT densities, contains the phase shift at k = kF (which is not used to generate
the density). For OF-DFT densities, this field is left empty.

172

Additional functionality:
– When arithmetic is performed two Density objects, the result is a Density object. The values
of rs, kvals, weights, and lambdas—which are assumed to be the same—are preserved in the
result. When a Density object is multiplied by a scalar, all member fields are preserved.
❼ ExcitedState: used for representing the linear response density n1 (x; ω)

Additional member variables:

– rs: Stores the Wigner-Seitz radius rs .
– type: A string that indicates the nature of the response (e.g., ‘LR’ for linear response)
– f omega p: Frequency as a fraction of the plasma frequency, ω = f ωp . The f is stored here.

❼ WaveFun: used for representing Kohn-Sham wavefunctions ψk .

Additional member variables:
– k: Value of k of this wavefunction.
– gamma: Value of phase shift γk of this wavefunction.
– alpha: Value of coefficient αk in right-hand asymptotic behavior.
– lambda: Exponential factor λk appearing in right-hand asymptotic behavior.

173

REFERENCES
blank space

174

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