Diagrammatic techniques for time-dependent density ... · Dynamical problems can be tackled using...

Diagrammatic techniques fortime-dependent

density-functional theory

Den Naturwissenschaftlichen Fakultätender Friedrich-Alexander-Universität Erlangen-Nürnberg

zurErlangung des Doktorgrades

vorgelegt von

Ralf Stubner

aus Schwabach

Als Dissertation genehmigt von den Naturwissenschaftlichen Fakultätender Friedrich-Alexander-Universität Erlangen-Nürnberg.

Tag der mündlichen Prüfung: 6. Dezember 2005

Vorsitzender derPromotionskomission: Prof. Dr. D.-P. Häder

Erstberichterstatter: Prof. Dr. O. Pankratov

Zweitberichterstatter: Prof. Dr. P.-G. Reinhard

ii

Contents

Important acronyms and symbols v

Zusammenfassung vii

1 Introduction 1

2 Fundamental concepts 72.1 Many-body theory . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Green functions . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Perturbation expansion and Feynman diagrams . . . 12

2.1.4 Linear-response theory . . . . . . . . . . . . . . . . . 17

2.2 Static density-functional theory . . . . . . . . . . . . . . . . . 20

2.3 Diagrammatic representation of the xc potential . . . . . . . 26

2.4 Time-dependent density-functional theory . . . . . . . . . . 37

3 The exchange-correlation kernel 493.1 Diagrammatic representation of the xc kernel . . . . . . . . 49

3.1.1 Derivation of the xc kernel via differentiation . . . . 49

3.1.2 Derivation of the xc kernel via expansion of the re-sponse function . . . . . . . . . . . . . . . . . . . . . . 55

3.1.3 Generalizations of the diagrammatic rules . . . . . . 62

3.2 The xc kernel as “mass operator” . . . . . . . . . . . . . . . 66

3.3 Different models for the xc kernel . . . . . . . . . . . . . . . 78

4 Particle–hole excitation energies 834.1 Eigenvalue equation for the excitation energies . . . . . . . . 83

4.2 Expansion in terms of the irreducible elements . . . . . . . . 86

iii

Contents

4.2.1 Cancellations in the perturbation expansion . . . . . 96

4.2.2 Need for consistent perturbation theory . . . . . . . 98

4.3 Expansion in terms of the interaction . . . . . . . . . . . . . 100

4.3.1 First-order correction . . . . . . . . . . . . . . . . . . . 102

4.3.2 Second-order correction . . . . . . . . . . . . . . . . . 104

5 Excitons in extended systems 1095.1 Excitonic effects in the exchange-correlation kernel . . . . . 111

5.2 Model system . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3 Short-range interaction . . . . . . . . . . . . . . . . . . . . . . 124

5.3.1 Solution of the BSE . . . . . . . . . . . . . . . . . . . . 125

5.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.3 Validity of the first-order approximation . . . . . . . 135

5.3.4 Static long-range exchange-correlation kernel . . . . 138

5.4 Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . 141

5.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Summary 149

Bibliography 151

Acknowledgments 161

Curriculum vitae 163

iv

Important acronyms and symbols

ALDA adiabatic local-density approximation (see section 2.4)

BSE Bethe-Salpeter equation (see chapter 5)

DFT density-functional theory (see section 2.2)

EXX exact exchange (see section 2.2)

GLPT Görling-Levy perturbation-theory

HK Hohenberg-Kohn (see section 2.2)

KS Kohn-Sham (see section 2.2)

LDA local-density approximation (see section 2.2)

MBT many-body theory

OEP optimized-effective potential (see section 2.2)

QP quasiparticle

RPA random phase approximation

TDDFT time-dependent density-functional theory (see section 2.4)

xc exchange-correlation

χ exact density–density reponse function

χ proper or irreducible density–density reponse function

χQP density–density reponse function of independent QPs

χS density–density reponse function of independent KS particles

v

Important acronyms and symbols

χxc xc part of χ, i.e., χ = χS + χxc

Πxc xc part of χ, i.e., χ = χS + Πxc

ΠQP quasiparticle part of χQP, i.e., χQP = χS + ΠQP

ΠEx excitonic part of χ, i.e., χ = χQP + ΠEx

fxc xc kernel

f QPxc quasiparticle part of the xc kernel

f Exxc excitonic part of the xc kernel

VC Coulomb interaction

V sum of Coulomb interaction and xc kernel

vi

Zusammenfassung

Die Dichtefunktionaltheorie (DFT) ist eine der wichtigsten ab initio Me-thoden zur Berechnung der Eigenschaften von Vielteilchensystemen. DerErfolg der DFT beruht auf dem Kohn-Sham-Ansatz (KS), der das Vielteil-chenproblem durch ein effektives Einteilchenproblem ersetzt. Die dabeinotwendigen Näherungen für die Effekte von Austausch und Korrelatio-nen (xc) sind nicht nur gut, sondern vor allem gut verstanden. Allerdingsist die DFT zunächst eine Theorie für den Grundzustand. Die Eigenschaf-ten angeregter Zustände sind nur schwierig zu bestimmen.

Die zeitabhängige Dichtefunktionaltheorie (TDDFT) kann hier Abhilfeschaffen. Für lineare Antworten des Systems erlaubt die TDDFT dendirekten Zugriff auf angeregte Zustände. Dabei sind wieder Näherungenfür die im xc-Kern fxc enthaltenen xc-Effekte erforderlich. Jedoch sinddiese Näherungen bisher nicht gut verstandenen.

Ein alternativer Zugang zu den Eigenschaften von Vielteilchensyste-men ist durch die auf Greensche Funktionen basierende Vielteilchentheo-rie (MBT) gegeben. Insbesondere in Verbindung mit diagrammatischenMethoden erlaubt die MBT physikalisch motivierte Näherungen.

In dieser Arbeit werden diagrammatische Techniken der MBT für dieWeiterentwicklung der TDDFT angewendet. Es ist das Ziel dieser Ar-beit, beim Entwickeln, Testen und vor allem Verstehen der notwendigenNäherungen für die xc-Effekte zu helfen. Es wird gezeigt, dass eine dia-grammatische Entwicklung des xc-Kerns möglich ist. Hierzu werden ex-plizite diagrammatische Regeln aufgestellt, um fxc in beliebiger OrdnungStörungstheorie anzugeben.

Die abgeleiteten diagrammatischen Regeln für fxc erlauben es, die ana-lytische Struktur des xc-Kerns zu untersuchen. Dabei wird gezeigt, dassfxc in jeder Ordnung Störungstheorie keine Teilchen–Loch-Divergenzen des

vii

Zusammenfassung

nichtwechselwirkenden KS-Systems enthält. Daher kann fxc als „Massen-operator“ für die exakte Dichte–Dichte-Antwortfunktion dienen.

Allerdings unterscheidet sich fxc wesentlich von der Selbstenergie, dieals Massenoperator für die exakte Greensche Funktion fungiert. Dieszeigt sich bei der Berechnung der Vielteilchenkorrekturen zu den KS-Anregungsenergien mit Hilfe der störungstheoretischen Entwicklung fürfxc. Diese Korrekturen werden bis zur zweiten Ordnung in der Wechsel-wirkung und bezüglich der ein- und zweiteilchen-irreduziblen Elementeberechnet. Dabei wird gezeigt, dass die Korrekturen zu den KS-Anre-gungsenergien durch eine konsistente Störungstheorie gewonnen werdenmüssen, wenn eine störungstheoretische Näherung für fxc benutzt wird.

Die Berechnung optischer Anregungsspektren von Festkörpern ist eineder vielversprechendsten Anwendungen der TDDFT im Bereich linea-rer Antwort. Hier könnte die TDDFT die numerisch aufwendige Bethe-Salpeter-Gleichung (BSE) ersetzen, die für die Beschreibung der exzito-nischen Korrelationen sonst verwendet wird. Es wird gezeigt, dass derxc-Kern sich exakt und eindeutig in einen Quasiteilchen- und einen Exzi-tonenanteil aufspalten lässt. Die für fxc gefundenen diagrammatischenRegeln gelten separat für beide Anteile. Damit lässt sich der Exzitonen-anteil durch die Dreipunktfunktion Λ ausdrücken. Die Integralgleichung,der Λ genügt, bietet eine exakte Übersetzung der BSE in die Sprache derTDDFT. Der Kern der Gleichung für Λ ist in manchen Fällen klein oderverschwindet ganz, weil sich Terme gegenseitig aufheben. In diesen Fäl-len ist eine Näherung erster Ordnung für Λ ausreichend.

Die Aufhebung in der Gleichung für Λ wird anhand eines Modellhalblei-ters untersucht. Für den Fall einer kurzreichweitigen Wechselwirkungzwischen den Quasiteilchen werden alle relevanten Gleichungen analy-tisch gelöst. Es wird gezeigt, dass die Aufhebung in der Gleichung für Λ

für Energien nahe der Bandlücke am effektivsten ist. Daher ist für dieseEnergien der Exzitonenanteil von fxc – unabhängig von der Wechselwir-kunsstärke – in erster Ordnung sehr ähnlich zum exakten Ergebnis, wasfür schwache Wechselwirkung zu einer sehr guten Beschreibung der ex-zitonischen Korrelationen in der Dichte–Dichte-Antwortfunktion führt.

viii

1 Introduction

Calculation of electronic and structural properties of atomic, molecular,and condensed-matter systems is one of the central problems of quantumphysics and chemistry. The ability to predict such properties has manyapplications ranging from fundamental research to technology. The maindifficulty lies in the intrinsic many-body nature of these problems. Al-ready the helium atom with two electrons surrounding the nucleus isonly accessible via approximations. The complexity increases tremen-dously if larger atoms, molecules, or solids are considered. A wide rangeof semi-empirical methods have been developed over the years, whichallow to cast these questions into manageable forms. During the lastdecades ab initio methods have gained in popularity for such calculations,too. Compared to empirical methods, ab initio methods do not requireexperimental data as input, which increases their predictive power. How-ever, this comes at the cost of a higher the complexity of the calculations.

One of the most important ab initio methods is the density-functionaltheory (DFT), which was pioneered by Hohenberg, Kohn, and Sham inthe 1960’s. Especially for solid state systems, DFT based calculationsare the “state of the art”. In DFT it is, in principle, sufficient to knowthe density of the interacting particles, since all physical quantities canbe expressed as functionals of the density. The density can be obtainedfrom the self-consistent solution of a set of effective one-particle equationsthat are known as Kohn-Sham (KS) equations. The set of KS equationsis similar in complexity to the Hartree approximation. However, whileexchange-correlation (xc) effects are neglected in the Hartree approach,DFT contains them in an in principle exact fashion.

In actual DFT calculations approximations to the xc energy that de-scribes the xc effects are, of course, necessary. The reason for the tremen-

1

1 Introduction

dous success of DFT is the possibility to use local approximations to thexc energy such as the local-density approximation (LDA). These approx-imations to the xc energy are well understood. Both when and why theseapproximations are reliable is known.

The main problem of DFT is that it is first and foremost a theory forthe ground state of a system. Only properties such as the total energyor the equilibrium geometry can be obtained directly. When solving theKS equations one obtains a set of eigenvalues and eigenfunctions thatone might want to identify with the possible excitations of the system.Such an identification is, however, not justified. At best, the “KS par-ticles” may serve as a zero-order approximation for the real excitationspectrum. The situation becomes even more complicated when one triesto describe dynamic problems, as these are beyond the reach of DFT. Forexample, while we are able to describe the binding of molecules in DFT,it is difficult to obtain their excitation spectrum. And it is impossible toinvestigate how the bonds between the atoms break when the moleculeis subjected to the strong electric field of an intense laser beam.

Dynamical problems can be tackled using the time-dependent density-functional theory (TDDFT). Here again, the complicated many-bodysystem is mapped onto the now time-dependent density, which in turncan be calculated with the help of an effective system of independentparticles. This way it is possible to describe the evolution of an interact-ing many-body system by solving a set of time-dependent Schrödinger-equations for the fictitious noninteracting KS particles. However, findinggood approximations for the dynamic xc effects is much more difficultthan in the static case.

The simplest problem in TDDFT is given by a system in its groundstate exposed to a weak perturbation that can be described by linear-response theory. While there is no real dynamics in this case, access toproperties of excited states can be obtained. The xc effects are containedin the so called “exchange-correlation kernel” within the framework ofTDDFT in the linear-response regime. Already the xc kernel is muchmore complicated than the xc energy of the static case, mainly because it

2

is not possible to consistently construct local approximations beyond theadiabatic limit.

A possible application of TDDFT in the linear response regime is thecalculation of the electronic excitation spectra of solids. Of special inter-est are particle–hole excitations which determine the material’s opticalproperties. Especially in semiconductors and insulators it is not suffi-cient to describe these excitations using independent particles, since theinteraction between particles and holes introduces excitonic correlationeffects. Probably the best known excitonic effect is the appearance ofa series of Rydberg states close to the fundamental absorption edge ofsemiconductors and insulators. Besides these bound excitons there arealso “unbound” excitonic effects. These appear above the single-particlethreshold and often substantially modify the absorption spectrum. Thiseffects is known as Sommerfeld factor. Conventionally, one solves theBethe-Salpeter equation (BSE) to describe excitonic correlations, which iscomputationally very expensive. TDDFT would be a viable replacement,if reliable approximations for the xc kernel were known, since the math-ematical structure of the relevant equations in TDDFT is much simpler.

Not only finding, but also testing approximations for the dynamicalxc effects is difficult. Most often these approximations are used in nu-merical calculations of the dielectric function ε(ω). While comparison ofε(ω) compared with experimental data does tell us where a particular ap-proximation is reliable, it does not aid our understanding. We still do notknow why the particular approximation gives good results, which makesit difficult to judge its range of validity.

In the past, a conventional many-body theory (MBT) has been suc-cessfully used to clarify the approximations involved in the static DFT.The Green-functions-based conventional MBT is another – in principleexact – approach to the many-body problem. Especially in connectionwith diagrammatic techniques, it is particularly suitable for approximatesolutions which capture the essential physics.

(TD)DFT and MBT can be used as complementary approaches. Theformer reduces realistic problems to a manageable form, but requires ap-

3

1 Introduction

proximations for the xc effects. It does not give us any systematic wayto obtain these approximations. This gap can be filled by MBT. WhileMBT is in general more difficult to use in calculations for realistic systemsthan (TD)DFT, MBT can help constructing, testing, and understandingapproximations for the xc effects. At least for model systems one cantry to calculate the unknown properties from (TD)DFT using techniquesfrom MBT. Often the physical insight provided by the diagrammatictechniques also suggest approximations for these properties. This allowsto not only compare the results of a (TD)DFT calculation with exact re-sults or experimental data, but to compare at the same time exact andapproximate input quantities.

Outline of the present work

This work discusses the connection between on the one hand conven-tional MBT and on the other hand TDDFT in the linear-response regime.A particular focus is laid on the development of diagrammatic techniques.Another important aspect is the use of model systems that often providemore insight into why certain approximations may be suitable for realisticsystems.

The work is organized as follows. In chapter 2 a brief review of thefundamental concepts that are necessary for this work is given. Green-functions-based MBT is introduced as well as the diagrammatic tech-niques used in perturbative treatments. The main principles of DFT aswell as the possibilities to apply diagrammatic techniques within DFTare discussed. In addition, TDDFT with a particular focus on the linear-response regime and possible approximations is reviewed. In this contextwe also discuss different excitonic correlations in solids.

Chapter 3 is devoted to the central quantity of TDDFT in the linear-response regime, the exchange-correlation kernel. Diagrammatic rulesfor constructing the xc kernel in any order of the perturbation theory arederived. For actual applications of such a perturbative expansion it isnecessary to know the analytic properties of the xc kernel. The diagram-matic rules for the xc kernel are used to analyze its analytic properties.

4

In chapter 4 we apply the findings of chapter 3 to the calculation ofparticle–hole excitation energies. We find that the analytic structure ofthe perturbative expansion of the xc kernel has important consequencesfor actual calculations of particle–hole excitation energies that have notbeen taken into account up to now.

Chapter 5 is concerned with the effects of the particle–hole interactionin extended systems, where excitons can be formed. We present a modelsystem which allows an exact analytic solution. This system is used to testvarious approximations and to investigate the reason for the successesand failures in different parameter ranges.

The final chapter 6 summarizes the findings of this work.Some results of this work have been published in Tokatly, Stubner, and

Pankratov (2002) as well as in Stubner, Tokatly, and Pankratov (2004).

5

2 Fundamental concepts

In this chapter the fundamental theories required in this work shall bereviewed. All these theories are concerned with the many-body problemin quantum mechanics. A large class of problems in atomic, molecular,and condensed matter physics can be formulated as the problem of manyinteracting electrons in an external potential, where one has to solve theSchrödinger equation with the Hamiltonian

H = T + W + Vext. (2.1)

Here, the kinetic energy T, the interaction W, and the external potentialVext are given by

T = −12

∫d3r ψ†(r)∇2ψ(r), (2.2a)

W =12

∫d3r∫

d3r′ ψ†(r)ψ†(r′)w(r, r′)ψ(r′)ψ(r), (2.2b)

Vext =∫

d3r vext(r)ψ†(r)ψ(r), (2.2c)

where the field operators ψ†(r) and ψ(r) create or destroy a particle atposition r, respectively. The interaction potential w(r, r′) is normally theCoulomb potential VC(r, r′) = 1/|r − r′|. The external potential vext istypically defined by the atomic nuclei in the system. Note that through-out this work Hartree atomic units h = m = e = 4πε0 = 1 are used andthat spin is neglected for notational simplicity.

In general (large N), solving the Schrödinger equation with the Hamil-tonian (2.1) is extremely difficult. For a system with N particles, the exactwave function depends on 3N spatial coordinates. This makes it practi-cally impossible to obtain the exact wave function for real systems. Over

7


time, a wide range of methods for dealing with this problem have beendeveloped. Some of which will be discussed in this chapter.

In section 2.1 some important concepts of standard MBT will be re-viewed. We will then cover DFT in its static (section 2.2) and its dynamic(section 2.4) incarnation. The question of how to use diagrammatic tech-niques of MBT for DFT is the topic of section 2.3.

2.1 Many-body theory

A thorough account of standard MBT can be found in may textbooks(Abrikosov et al., 1963; Fetter and Walecka, 1971; Mahan, 1990) and willnot be repeated here. In addition, Mattuck (1976) has given a particularaccessible introduction to diagrammatic techniques in MBT. Hedin andLundqvist (1969) reviewed Green function techniques with a particularfocus on calculations in solids. A short review of some key concepts isuseful for the following, though.

2.1.1 Quasiparticles

One of the central ideas of MBT is the concept of quasiparticles (QPs),which was first introduced by Landau (1957a,b). A very good accountof this concept and its uses in actual calculations was given by Nozières(1964) as well as Pines and Nozières (1966).

In a system of strongly interacting fermions, it is extremely difficultto describe the state of the system exactly. Elementary excitations of thesystem are often easier to handle, though.

Imagine a homogeneous electron gas in the ground state at zero tem-perature, i.e., all states up to the Fermi energy are occupied. If we add oneadditional electron with momentum p and an energy above the Fermilevel, it will repel the other electrons in its surrounding, forming the socalled screening cloud. When moving through the electron gas, the addi-tional electron will have to drag with it this cloud of repelled electrons,changing its mobility as compared with a free particle. One can interpret

8


this as a change in the mass of the particle, i.e., the particle acquires aneffective mass.

A filled Fermi sea with one additional particle above the Fermi level isnot an (energy) eigenstate of the many-particle system. The state of theextra electron will therefore be characterized by a finite lifetime due tocollisions with the other electrons in the system.

Another effect of the screening cloud is also immediately obvious. Ifthere are two or more such additional electrons present, screening weak-ens the interaction between them as compared to bare electrons. Theeffective interaction is often so weak that it can be treated perturbatively.

Instead of trying to account for all the complicated interactions be-tween the additional particle and the particles in the Fermi sea, we canconsider a gas of QPs that differ from the original particles by their mass,finite lifetime, and reduced interaction. Similar considerations apply toquasiholes, which are formed when a particle is removed from the Fermisea. The hole left behind effectively carries a positive charge attractingelectrons, which will again form a screening cloud.

This qualitative picture of (almost) noninteracting QPs with an effec-tive mass can be described by Landau’s Fermi liquid theory. The inter-esting thing in the present context is that this very intuitive picture ofQPs can be rigorously justified within the framework of a Green func-tions perturbation theory (Nozières and Luttinger, 1962; Luttinger andNozières, 1962). It is therefore useful to keep this picture in mind wheninterpreting the often involved Green function calculations.

In Landau’s Fermi liquid theory exists a one-to-one correspondencebetween particles in the noninteracting system and the QPs. However,the interaction between the particles, which brought about the changeof the properties of the QPs can also lead to new types of excitationsthat are not linked to a single particle. These collective excitations aresometimes described as QPs, too. Plasma oscillations (plasmons) area typical example for this. Another example that is interesting in thecontext of the present work are excitons. As noted above, there exist aresidual interaction between QPs. In the case of a quasiparticle and a

9


quasihole, this interaction is attractive. In semiconductors and insulators,where due to the band gap the screening of the interaction is less efficient,this interaction can lead to bound excitonic states. Bound excitons aswell as unbound excitons will be discussed in greater detail at the end ofsection 2.4 and in chapter 5.

2.1.2 Green functions

The one-particle Green function of an interacting system is defined as

G(rt, r′t′) = −i〈ΨN,0|T ψH(rt)ψ†H(r′t′)|ΨN,0〉, (2.3)

where |ΨN,0〉 is the exact normalized N-particle ground state in the Hei-senberg picture which fulfills

H|ΨN,0〉 = EN,0|ΨN,0〉. (2.4)

The field operators ψH(rt) and ψ†H(rt) create or destroy a particle at posi-

tion r and time t, respectively. They are the Heisenberg operators corre-sponding to the field operators of Eq. (2.2):

ψH(rt) = eiHtψ(r)e−iHt (2.5a)

ψ†H(rt) = eiHtψ†(r)e−iHt. (2.5b)

Wick’s time-ordering operator T arranges the subsequent terms in as-cending time order from right to left with a sign change for any inter-change of fermion field operators. We can therefore rewrite Eq. (2.3) as

G(rt, r′t′) = −i Θ(t− t′)〈ΨN,0|ψH(rt)ψ†H(r′t′)|ΨN,0〉

+ i Θ(t′ − t)〈ΨN,0|ψ†H(r′t′)ψH(rt)|ΨN,0〉.

(2.6)

For t > t′ the Green function G(rt, r′t′) describes the probability am-plitude that an additional particle is added to the system at (r′t′) andremoved at (rt). Note that since we are dealing with identical particles,the question whether or not the same particle is removed is meaningless.

10


For t < t′ the Green function describes the amplitude that at (rt) a parti-cle is removed, i.e., a hole is created, and the particle is then added againat (r′t′). This is very similar to the quasiparticles and -holes discussedin the previous section, and we can expect to find their properties in theGreen function.

The analytic properties of the Green function G are best studied byinserting a unit operator ∑N′ ,m |ΨN′ ,m〉〈ΨN′ ,m| formed from exact eigen-states of the interacting system with energy EN′ ,m between the field oper-ators in Eq. (2.6). After Fourier transformation into the frequency domainwe find

G(ω, r, r′) = ∑m

(〈ΨN,0|ψ(r)|ΨN+1,m〉〈ΨN+1,m|ψ†(r′)|ΨN,0〉

ω − (EN+1,m − EN,0) + iη

+〈ΨN,0|ψ†(r′)|ΨN−1,m〉〈ΨN−1,m|ψ(r)|ΨN,0〉

ω − (EN,0 − EN−1,m)− iη

), (2.7)

where the infinitesimally small factors iη are required for convergence.From Eq. (2.7) we see that G(ω) has simple poles at the energies of one-particle excitations of the interacting system, when ω is treated as a com-plex variable. For an added particle [first term in Eq. (2.7)], the poles liebelow the real axis. For a removed particle [second term in Eq. (2.7)], thepoles lie above the real axis.

The energies of one-particle excitations can be interpreted as the QPenergies considered in section 2.1.1. These energies can be obtained fromthe real part of the poles of G(ω), while the imaginary part determinesthe QP life time. The residual of the pole is a measure of the excitation’sspectral weight (e.g., Abrikosov et al., 1963, section 7.3.). As mentionedin the preceding section, the different properties of QPs can indeed befound from the Green function.

From the above considerations it is clear that G(ω) is not analytic inboth the upper and the lower ω half plane. For contour integration it isoften useful to use functions that are analytic in at least one of the halfplanes. One therefore defines the so called retarded and advanced Green

11


functions

GR(rt, r′t′) = −i Θ(t− t′)〈ΨN,0|ψH(rt), ψ†H(r′t′)|ΨN,0〉 (2.8a)

GA(rt, r′t′) = i Θ(t′ − t)〈ΨN,0|ψH(rt), ψ†H(r′t′)|ΨN,0〉 (2.8b)

in addition to the time-ordered or causal Green function of Eq. (2.3). Thecurly braces in Eq. (2.8) denote an anticommutator.

An analysis similar to above reveals that GR(ω) and GA(ω) have sim-ple poles at the energies of one-particle excitations, too. However, forGR(ω) all these poles lie in the lower half plane. In contrast, all poles ofGA(ω) lie in the upper half plane. For real ω the different Green func-tions differ only in their imaginary parts. One can therefore calculate theother two, when one knows one of them.

Of course, for calculating G according to Eq. (2.7) one already needsto know the excited states we want to obtain from the Green function.However, Eq. (2.7) is greatly simplified for the case of noninteractingparticles with one-particle states ϕm(r) and energies εm, where the causalGreen function is given by

G0(ω, r, r′) =unocc

∑m

ϕ∗m(r′)ϕm(r)ω − εm + iη

+occ

∑m

ϕ∗m(r′)ϕm(r)ω − εm − iη

. (2.9)

This simple result for the noninteracting Green function suggests a per-turbative approach to calculating G, where the interaction between theparticles is treated as perturbation. This shall be considered in the nextsection.

2.1.3 Perturbation expansion and Feynman diagrams

In order to derive a perturbation expansion for the Green function, wefirst split the Hamiltonian (2.1) as H = H0 + W and treat the interac-tion W as perturbation. In such a situation it is best to use the interactionpicture instead of the Heisenberg picture used above. We therefore define

12


the field operators in the interaction picture

ψ(rt) = eiH0tψ(r)e−iH0t (2.10a)

ψ†(rt) = eiH0tψ†(r)e−iH0t. (2.10b)

A similar transformation is used for the interaction

W(t) = eiH0tWe−iH0t. (2.11)

The theorem by Gell-Mann and Low (1951) can be used to relate thematrix element of a Heisenberg operator to the matrix element of thecorresponding interaction operator. Applying it to the Green function ofEq. (2.3) we obtain

G(rt, r′t′) = −i〈ΦN,0|T Sψ(rt)ψ†(r′t′)|ΦN,0〉

〈ΦN,0|S|ΦN,0〉, (2.12)

where |ΦN,0〉 is the ground state of the noninteracting system correspond-ing to H0. The S-matrix is defined as

S = T exp−i∫ ∞

−∞dt W(t)

=

∞

∑n=0

(−i)n

n!

∫ ∞

−∞dt1 . . .

∫ ∞

−∞dtn T W(t1) . . . W(tn).

(2.13)

This gives a perturbative expansion for the Green function

G(rt, r′t′) =−i

〈ΦN,0|S|ΦN,0〉

∞

∑n=0

(−i)n

n!

∫ ∞

−∞dt1 . . .

∫ ∞

−∞dtn

〈ΦN,0|T W(t1) . . . W(tn)ψ(rt)ψ†(r′t′)|ΦN,0〉. (2.14)

From Eq. (2.2b) we see that W contains a product of field operators.Hence, in Eq. (2.14) we have to evaluate matrix elements of a time-orderedproduct of a certain number of field operators. This can be quite tedious,but is simplified by the Wick theorem (1950). The idea behind this the-orem is to move all annihilation operators to the right where they give

13


zero when acting on the ground state. The result of these considerationsis that the matrix element of a time-ordered product of any number offield operators with respect to the noninteracting ground state can beexpressed as a sum of products of noninteracting Green functions.

This is quite obvious for the zero-order term, since the free Green func-tion is given by

G0(rt, r′t′) = −i〈ΦN,0|T ψ(rt)ψ†(r′t′)|ΦN,0〉. (2.15)

In the first-order, the matrix element in Eq. (2.14) contains six field op-erators, giving three free Green functions. These introduce two internalpoints that have to be integrated over with the interaction w(r1, r2) as rel-ative weight. There are six possible ways to distribute these four pointsover three Green functions. Keeping track of all these terms is difficult,especially for higher order corrections. This task can be simplified by us-ing Feynman diagrams, where we draw a directed solid line between anypoints connected by a Green function G0 in a particular term. The inter-nal points appearing in an interaction are connected by a dashed line. Ifwe restrict ourselves to topologically distinct diagrams, i.e., diagrams thatcannot be obtained from each other by interchanges of the operators W,there are four diagrams for the first-order correction:

. (2.16)

The first two diagrams in Eq. (2.16) are connected, whereas the other twoare disconnected. These disconnected diagrams without the single Greenfunction also appear in the expansion of 〈ΦN,0|S|ΦN,0〉 in the denomina-tor of Eq. (2.14), canceling the terms in the numerator we just found. Itis therefore enough to draw all topologically distinct, connected Feynmandiagrams. These diagrams can be translated into formulas by associat-ing a free Green function iG0 with every solid line and the interaction−iw(r1, r2) with every dashed line. One has to integrate over all internalvariables. In addition, the overall sign of a particular diagram is (−1)F

14


where F is the number of closed fermionic loops in the diagram.† Similarrules apply when one works in momentum instead of position space.

The fermion loops in the first and third diagram in Eq. (2.16) are pro-portional to the density of the unperturbed system. Therefore, the first di-agram in Eq. (2.16) represents the (lowest order) Hartree part of the inter-action, whereas the second diagram represents the “exchanged Hartree”contribution, i.e., the Fock exchange.

If we do the same for the second-order correction to the Green function,we obtain ten different diagrams. Some representative examples are

. (2.17)

Again, these diagrams fall into two different groups. The last one is (one-particle) reducible, since it is possible to separate it into two by cutting onefermionic line. The other two are (one-particle) irreducible. All diagramsfrom the first order correction to the Green function, i.e., the first twodiagrams from Eq. (2.16), are irreducible, too. The same distinction canbe applied to higher orders, too. The diagrams in the expansion forG are either irreducible or parts of irreducible diagrams connected byunperturbed Green functions.

We can define the self-energy Σ(rt, r′t′) as the sum of all irreduciblediagrams with the two Green functions at the ends cut off. The expansionfor the Green function can then be written as

G = G0 + G0 · Σ · G0 + G0 · Σ · G0 · Σ · G0 + . . . , (2.18)

where the dots indicate the necessary integrations. By formal summationof Eq. (2.18) we obtain the Dyson equation

G = G0 + G0 · Σ · G. (2.19)

† There exist other “dictionaries” for translating Feynman diagrams into formulas, mainlydiffering in the way how prefactors i and −i are used. The result is, of course, always thesame.

15


In diagrammatic form

= + Σ , (2.20)

where the thick lines represent the exact Green function G.Equation (2.19) becomes particular simple in a spatially homogeneous

system, where all quantities only depend on one momentum and onefrequency. The unperturbed Green function (2.9) reduces to

G0(ω, k) =1

ω − εk + iηk, (2.21)

where ηk = η for unoccupied states above the Fermi energy and ηk = −η

for occupied states. The exact Green function according to Eq. (2.19) isthen given by

G(ω, k) =1

ω − εk − Σ(ω, k) + iηk. (2.22)

Here the real part of Σ(ω, k) determines the difference between free andquasiparticle excitation energies, whereas the imaginary part is responsi-ble for their finite life-time. Physically, we can interpret the self-energyas the energy of the bare particle interacting with itself via the screeningcloud that the particle generates in the many-body system.

In this section we have treated the interaction perturbatively. Onemight ask, if this is legitimate at all. Is the interaction between the par-ticles “small” in some sense? Often, this is not the case. This does notinvalidate the perturbative expansion, though. It rather shows a strengthof the diagrammatic approach to MBT: The ability to single out the im-portant terms in the expansion and sum only these terms to all orders.A nice example for this is the first diagram in Eq. (2.17). This diagramdiverges for a homogeneous electron gas, and the same is true for thehigher order terms with several of these “pair bubbles” in the interaction.It is, however, possible to sum all these diverging contributions to obtaina finite result. This technique is known as “partial summation”. Here we

16


define a new effective interaction

= + . (2.23)

In terms of this effective interaction we can calculate the self-energy as

, (2.24)

which incorporates the Fock exchange and all diagrams with pair bubblesin the interaction. This approximation for the screening of the interactionis known as random phase approximation (RPA) and is useful for a high-density electron gas.

2.1.4 Linear-response theory

In the previous sections we have discussed single-particle excitations,which determine the poles of the one-particle Green function. Collec-tive excitations, as introduced in section 2.1.1, cannot be obtained fromthe one-particle Green function directly. They can be found from the sys-tem’s response to an appropriate weak perturbation, as we will see inthis section.

Consider an interacting many-particle system in its ground state. Gen-erally speaking, if some probe coupling to the physical quantity A intro-duces a weak perturbation, the system might respond by changes in thequantity B. Changes that are linear in the perturbation are then describedby the B–A response function.

For definiteness let us consider the change in density δn(rt) inducedby a change in the external potential δvext(rt), which couples to the den-sity. We are therefore interested in the density–density response func-tion χR(rt, r′t′) defined by

δn(rt) =∫

d3r′∫

dt′ χR(rt, r′t′)δvext(r′t′). (2.25a)

17


The “R” indicates that χR is a retarded quantity as required by causality.It is convenient to introduce the notation

δn = χR · vext (2.25b)

for integrals of this type, where the dot indicates the necessary integra-tions.

Using conventional time-dependent perturbation theory, one can showthat χR is given by

χR(rt, r′t′) = −i Θ(t− t′)〈ΨN,0|[nH(rt), nH(r′t′)]|ΨN,0〉, (2.26)

with the density operator nH(rt) = ψ†H(rt)ψH(rt) in Heisenberg represen-

tation. Note that in difference to Eq. (2.8a) the commutator – indicated bybrackets – is required here, since nH has bosonic commutation relation-ships. Similar to section 2.1.2, we can insert a complete set of eigenstatesto obtain

χR(ω, r, r′) = ∑m

(〈ΨN,0|n(r)|ΨN,m〉〈ΨN,m|n(r′)|ΨN,0〉

ω − (EN,m − EN,0) + iη

− 〈ΨN,0|n(r′)|ΨN,m〉〈ΨN,m|n(r)|ΨN,0〉ω + (EN,m − EN,0) + iη

), (2.27)

with n(r) = ψ†(r)ψ(r). We see that χR(ω) has simple poles at the exci-tation energies of those states of the N-particle system that are coupledto the ground state through the density operator. Note that χR is inde-pendent of the external perturbation δvext(rt) and only depends on theunperturbed system. This important property holds for other linear re-sponse functions, too.

For a noninteracting system Eq. (2.27) reduces to

χR0 (ω, r, r′) = ∑

m,n( fn − fm)

ϕ∗n(r)ϕm(r)ϕ∗m(r′)ϕn(r′)ω − (εm − εn) + iη

, (2.28)

where fm is the Fermi occupation number.

18


Similar to the discussion in section 2.1.3, we would like to use a pertur-bative expansion for χR. However, χR is not suitable for a diagrammaticanalysis since Wick’s theorem can be applied only to time ordered prod-ucts. One therefore defines the time-ordered density–density responsefunction χ(rt, r′t′) as

χ(rt, r′t′) = −i〈ΨN,0|T nH(rt), nH(r′t′)|ΨN,0〉. (2.29)

The frequency space representation χ(ω, r, r′) can be obtained by insert-ing a complete set of eigenstates in Eq. (2.29). The result is very similar toEq. (2.27), the only modification being the use of −iη in the second term.We can therefore conclude that χ(ω) and χR(ω) are identical for ω > 0and we will therefore drop the distinction.

The lowest order term in an expansion for χ corresponding to nonin-teracting particles is just the “pair bubble” mentioned at the end of theprevious section, which explains why χ is also referred to as polarizationoperator or susceptibility. In diagrammatic form the first-order term isgiven by

+ + + + + .

(2.30)The first four diagrams in Eq. (2.30) are the first-order self-energy cor-rections for both the upper and the lower line. The fifth diagram is theCoulomb energy of the induced charge density, while the last diagramrepresents the interaction between particles and holes.

The fifth diagram in Eq. (2.30) is special in that it is possible to separateit into two parts by cutting one interaction line. Similar to the self-energyabove, this suggests the definition of the proper or irreducible polariza-tion operator χ that is related to the full polarization operator χ as

χ = χ + χ ·VC · χ, (2.31)

19


where the dot indicating the necessary integrations as introduced in thecontext of Eq. (2.25). The first four diagrams and the last diagram inEq. (2.30) form the first-order approximation to χ. If we approximate χ

by its zero-order term, i.e., the pair bubble χ0, we obtain the RPA resultfor the response function

χRPA = χ0 + χ0 ·VC · χRPA, (2.32)

which is closely related to the RPA result for the screened interaction ofEq. (2.23). In a homogeneous system we can Fourier transform Eq. (2.32)with respect to both space and time, converting the integrations into mul-tiplications. We then obtain for the RPA density–density response func-tion

χRPA(ω, q) =χ0(ω, q)

1− χ0(ω, q)VC(q). (2.33)

Compared with χ0(ω) we find an additional pole in χRPA(ω) when thedenominator is zero. We can identify this pole with the plasmon.

One can see from Eq. (2.30) that Feynman diagrams can look quite com-plicated, when (almost) all the arrows are drawn. Retaining the arrows isnot necessary in many cases, though. It is therefore more convenient todraw most diagrams without arrows but follow the convention that theupper line in any polarization insertion goes from left to right, while thelower line goes from right to left.

2.2 Static density-functional theory

Compared with the techniques discussed up to now, density-functionaltheory (DFT) takes a very different approach to the quantum mechani-cal many-body problem. The fundamental idea behind DFT is that fordescribing the ground-state properties of a quantum mechanical many-body system, the exact many-body wave function is actually not required.The one-particle density n(r) contains already all required informationand can therefore be considered as the basic variable. This was proved

20


by Hohenberg and Kohn (1964) who found that the mapping from exter-nal potentials to densities is invertible. Over the last decades DFT hasbecome one of the main tools in many fields of atomic, molecular, andcondensed matter physics. The formalism and applications have beenreviewed in many places (e.g., Jones and Gunnarsson, 1989; Dreizler andGross, 1990), hence only a brief overview will be given here.

The foundation of DFT is given by the Hohenberg-Kohn (HK) theo-rem (Hohenberg and Kohn, 1964) which states that the full many-bodyground-state |ΨN,0〉 is a unique functional of the density n(r), |ΨN,0〉 =|ΨN,0[n]〉. The proof of this statement uses Ritz’s variational principleto show that two ground states |ΨN,0〉 and |Ψ′

N,0〉 corresponding to twoexternal potentials vext(r) and v′ext(r) cannot give rise to the same den-sity n(r), if vext(r) and v′ext(r) differ by more than a constant. Thus whilethe density n(r) trivially is a functional of the external potential vext(r),the converse is also true, i.e., vext(r) is a unique functional of n(r). Sincevext(r) fixes the Hamiltonian, the same holds for |ΨN,0〉.

A corollary to the HK theorem is that the ground state expectationvalue of any quantum mechanical operator O is a unique functional ofthe density

O[n] = 〈ΨN,0[n]|O|ΨN,0[n]〉. (2.34)

This is of particular interest when we consider the Hamiltonian H. UsingRitz’s variational principle again, one finds that for all N-particle densi-ties the energy functional

E[n] = 〈ΨN,0[n]|H|ΨN,0[n]〉 (2.35)

has a minimum at E[nGS] = EGS, where EGS and nGS are the exact ground-state energy and density, respectively.

If E[n] were known, EGS and nGS could be determined by minimization.Of course, the exact energy functional is not known. A particular fruit-ful approximation scheme was given by Kohn and Sham (1965). These

21


authors separated E[n] as

E[n] = T0[n] +∫

d3r n(r)(

vext(r) +12

vH[n](r))

+ Exc[n]. (2.36)

Here, T0[n] is the kinetic energy of a system of N noninteracting particleswith density n and vH[n] is the classical Hartree potential for electronswith density n. The difference from the exact energy functional is in-corporated into the exchange-correlation (xc) energy Exc. Equation (2.36)can be seen as a definition of Exc, which has to account for both xc effectsand the difference between the kinetic energy of interacting and nonin-teracting particles of a given density.

Minimizing E[n] given by Eq. (2.36) is equivalent to solving the Schrö-dinger equation for N noninteracting particles(

−12∇2 + vS[n](r)

)ϕk(r) = εk ϕk(r). (2.37)

Here, vS[n] is the KS potential

vS[n](r) = vext(r) + vH[n](r) +δExc[n]δn(r)

=: vext(r) + vH[n](r) + vxc[n](r),(2.38)

where the xc potential vxc[n] has been introduced. The eigenfunctions ϕn

are known as KS orbitals. The density n(r), which is the same for theinteracting and the noninteracting system, can be calculated from

n(r) =N

∑k=1

|ϕk(r)|2. (2.39)

Equations (2.37)–(2.39), which are known as KS equations, have to besolved in a self-consistent procedure, since the KS potential depends onthe density.

There exists an alternative way to derive the KS equations. The HK the-orem is true for any interaction W between the particles. The external

22


potential which maintains a given particle density is a unique functionalof the density. For noninteracting particles, this external potential is noth-ing but the KS potential of Eq. (2.38). Note that it is not clear per se thatsuch a potential exists, since the HK theorem only ensures its unique-ness. This problem is known as the v-representability assumption, whichunderlies the KS method (Levy, 1982).

We have seen that the main idea behind the KS scheme is to map thesystem of N interacting particles onto a fictitious system of N noninteract-ing particles in a self-consistent potential with identical density, while allthe complicated many-body effects are lumped into the local xc potential.The Schrödinger equation for noninteracting particles can be efficientlysolved. The KS scheme is therefore very attractive, if good approxima-tions for Exc[n] or equivalently vxc[n] are known. Kohn and Sham (1965)introduced the LDA

ELDAxc [n] =

∫d3r n(r)εxc(n(r)), (2.40)

where εxc(n) is the xc energy per particle of a homogeneous electrongas of density n, which can be obtained from MBT calculations. Thexc potential in LDA depends on the local density only, while in generalvxc[n] is a functional depending on the whole density distribution.

The LDA is strictly valid only for the case of slowly varying density,which is not realized in atoms, molecules, or solids. Kohn and Shameven remarked that “we do not expect an accurate description of chemicalbonding”. In actual calculations the LDA gives surprisingly good resultsin many cases, though. The reasons for this success have been reviewedby Jones and Gunnarsson (1989). The exact xc energy can be expressedas the Coulomb interaction between an electron and the xc hole densitynxc,

Exc[n] =12

∫d3r n(r)

∫d3r′

1|r − r′|nxc(r, r′ − r). (2.41)

The xc hole density nxc can be expressed in terms of an integral overthe coupling constant of the pair correlation function, but the precise

23


definition is irrelevant for the present discussion. Due to the isotropicnature of the Coulomb interaction the xc energy in Eq. (2.41) dependsonly on the spherical average of nxc. Therefore, Exc depends only weaklyon the details of nxc, especially if the important sum rule∫

d3r′ nxc(r, r′ − r) = −1 (2.42)

is fulfilled, which is the case for the LDA (2.40).There are, of course, several shortcomings of the LDA. For example,

in a finite system the KS potential should asymptotically behave as 1/r,where r is the distance from the system. In LDA, the KS potential de-creases exponentially often inhibiting the binding of additional electrons,which is possible in reality, though. This wrong decay is also the reasonwhy the ionization energy, which is given by the negative of the eigenen-ergy of the highest occupied KS state according to a theorem by Janak(1978), is often wrong.

Another shortcoming of the LDA and any approximation related toit can be seen in extended systems with a band gap. Perdew and Levy(1983) as well as Sham and Schlüter (1983, 1985) noted that the exactband gap Egap is related to the KS band gap Egap

S , i.e., the differencebetween the energies of the highest occupied and the lowest unoccupiedKS orbital, as

Egap = EgapS + ∆xc, (2.43)

where ∆xc is the discontinuity of the xc potential on addition of an elec-tron:

∆xc = vN+1xc (r)− vN

xc(r). (2.44)

Note that while both the xc potential for the N + 1 particle system vN+1xc

and for the N particle system vNxc are functions of r, their difference is

a position independent constant. The addition of an electron produces achange in density of the order of the inverse volume of the system, whichis negligible in extended systems. Due to the simple density dependence

24


in LDA, this implies that ∆xc is equal to zero when the LDA is used.The importance of ∆xc in different systems still to be studied, but it isgenerally believed to make a sizable contribution to Egap (e.g., Godbyet al., 1988). The KS band gap in the LDA is typically about 50 % of thetrue band gap.

The discontinuity of the xc potential (2.44) implies an extremely non-local behavior of vxc[n]. It does not matter where an electron is addedto the conduction band, the xc potential changes in the whole crystal bythe same amount. Such a behavior appears to be rather unphysical atfirst glance. One should not be too surprised, though, since we are tryingto capture all the complicated many-body physics by means of a localpotential.

One could try to design xc energy functionals with a more involvedexplicit density dependence, but the more fruitful approach is to usexc energy functionals that depend explicitly on the KS orbitals. The func-tional dependence on the density is then only implicit, which complicatesthe calculation of the xc potential. In the present context the optimized-effective potential (OEP) method (Sharp and Horton, 1953; Talman andShadwick, 1976) is of particular interest. The OEP was originally de-signed as a local simplification to the Hartree-Fock method with its non-local exchange potential. The nonlocality is a consequence of minimizingthe expectation value of the Hamiltonian with a single Slater determinantas a trial wave function. In the OEP method one imposes the additionalrequirement that the single particle wave functions in the Slater determi-nant are the eigenfunctions of a single-particle Hamiltonian with a localpotential, which is the optimized-effective potential.

It can be shown that the OEP is equivalent to the xc potential obtainedby approximating Exc[n] with the exchange energy of Hartree-Fock typeusing the KS orbitals (Sahni et al., 1982). The OEP method has the disad-vantage, though, that the OEP is determined by a complicated integralequation. This integral equation has to be solved numerically which hasup to date been possible only for systems with one spacial coordinatesuch as radial symmetric atoms. For more complex systems like solids

25


approximations like the one proposed by Krieger, Li, and Iafrate (1992)have to be used (Bylander and Kleinman, 1995). An alternative formula-tion is given by the exact exchange (EXX) method (Görling, 1996b), whichcan also be applied to solids (Städele et al., 1997, 1999) or molecules (Hi-rata et al., 2001; Della Sala and Görling, 2001; Yang and Wu, 2002). Thegeneral trend in these calculations is that structural properties are com-parable to the LDA results, while the KS energies are closer to the realQP energies. This improvement is especially pronounced for unoccupiedKS orbitals.

Overall, DFT is an extremely successful method in many fields ofphysics. The main reason for this success is that there exist practical ap-proximations with known limitations for the xc energy functional. Orbitaldependent functionals help to remove some of these limitations and canbe systematically improved, as will be discussed in the next section. Themain strength of DFT lies in the calculation of the static ground-stateproperties, though. The generalization to dynamic problems is the sub-ject of section 2.4.

2.3 Diagrammatic representation of the xc potential

In this section we are following the path laid out by Tokatly and Pankra-tov (2001) to derive a diagrammatic expansion of the xc potential vxc(r).The discussion is quite detailed, since these diagrammatic techniques willbe essential for the remainder of this work and no such discussion is avail-able in the literature.

In order to derive a diagrammatic expansion of the xc potential wetreat the KS system with the KS Hamiltonian

HS = T + VS (2.45)

as “unperturbed” system in the many-body perturbation theory. TheKS potential operator VS is defined as

VS = VH + Vxc + Vext, (2.46)

26


where VH describes the electrostatic Hartree interaction, Vxc is the xc po-tential operator Vxc =

∫d3r vxc(r)n(r), and Vext is defined by Eq. (2.2c).

The KS orbitals ϕk(r) and eigenvalues εk can be obtained by solving theKS equations (2.37)– (2.39) and are assumed to be known.

The real Hamiltonian is of the form of Eq. (2.1), where the interactionbetween the particles W is given by the Coulomb interaction VC. It isconvenient to split this Hamiltonian as H = HS + V, where V accountsfor the difference between the exact Coulomb interaction and the KS self-consistent potential:

V = VC − VH − Vxc. (2.47)

This interaction can be understood as a “reduced” two-particle Coulombinteraction where the Hartree part and the one-particle xc part vxc havebeen subtracted from the full two-particle interaction.

In order to derive a diagrammatic expansion for the xc potential vxc,we are going to use a perturbation theory which uses the KS systemas unperturbed starting point. Hence V (2.47) is used as perturbationinstead of VC as in section 2.1.3. Since the density of the KS systemexactly matches the true particle density it must remain unchanged bythe perturbation (2.47). in any order of the perturbation theory. Thisis equivalent to the coupling constant method used in the Görling-Levyperturbation-theory (GLPT) (Görling and Levy, 1993, 1994, 1995). Thediagrammatic expansion we are going to derive can therefore be seen asdiagrammatic formulation of GLPT. This is of particular interest sinceone can then use powerful diagrammatic techniques to derive systematicimprovements for EXX, which is the first-order term in the GLPT.

It is convenient to use the Matsubara finite-temperature (imaginary-time) technique (Abrikosov et al., 1963, chapter 3; Fetter and Walecka,1971, part 3) to formulate the many-body perturbation-theory with theinteraction (2.47). Note that the Matsubara technique is used only forits computational convenience. The limit temperature T → 0 is taken inall final formulas. The Matsubara formalism is formally similar to theT = 0 formalism with a real time discussed in section 2.1.3. Therefore,

27


only the most important formulas together with references to the corre-sponding formula in the T = 0 formalism will be given here. Note thata quantity depending on imaginary times can be Fourier transformed toimaginary frequencies. The corresponding retarded quantity dependingon real frequencies can be obtained by analytic continuation.

The KS field operators in the interaction representation [c.f. Eq. (2.10)]are defined as

ψ(r, τ) = eHSτψ(r)e−HSτ (2.48a)

ψ†(r, τ) = eHSτψ†(r)e−HSτ (2.48b)

with imaginary time variable τ. In order to simplify the notation, thechemical potential µ is set equal to zero. The time-independent fieldoperators are introduced as

ψ(r) = ∑k

ϕk(r)ck (2.49a)

ψ†(r) = ∑k

ϕ∗k (r)c†k , (2.49b)

where ck destroys a particle in the k-th KS state and c†k creates it. In terms

of the time-dependent field operators of Eq. (2.48), the KS temperatureGreen function GS [c.f. Eq. (2.15)] is given by

GS(r1, τ1, r2, τ2) = −〈T ψ(r1, τ1)ψ†(r2, τ2)〉0, (2.50)

where 〈. . . 〉0 represents the thermal averaging over the KS ground stateand T is the time-ordering operator. As a function of an imaginary fre-quency iωn with ωn = πT(2n + 1) and temperature T this Green functiontakes the form

GS(r1, r2, iωn) = ∑k

ϕk(r1)ϕ∗k (r2)iωn − εk

. (2.51)

The exact one-particle Green function G [c.f. Eq. (2.14)] is given by

G(r1, τ1, r2, τ2) = −〈T ψ(r1, τ1)ψ†(r2, τ2)σ〉0

〈σ〉0, (2.52)

28


with the Matsubara S-matrix [c.f. Eq. (2.13)]

σ = T exp−∫ τ

0dτ′ V(τ′)

, (2.53)

where V(τ) is the operator of the perturbation (2.47) in the interactionrepresentation [c.f. Eq. (2.11)]

V(τ) = eHSτVe−HSτ . (2.54)

Similar to the procedure outlined in section 2.1.3, the perturbation se-ries for G can be obtained by expanding the S-matrix in powers of theinteraction (2.47).

At this stage, if one simply treated vxc as a given external potential, theperturbative expansion would contain all common diagrams for the two-particle interaction (except the Hartree graphs), the diagrams describingthe scattering by the one-particle potential vxc, and all possible combina-tions of these two types of graphs. For example, the first order of this ex-pansion would consist of the exchange (Fock) diagram and the diagramdescribing a single scattering event by vxc. However, vxc is a “mean-field”-type description of the xc processes and contains all powers of thetwo-particle interaction. This must be sorted out in the formulation ofthe perturbation theory. In other words, vxc has also to be expanded interms of the Coulomb interaction as vxc = ∑∞

k=1 v(k)xc with v(k)

xc containingk Coulomb interactions.

Finally, collecting the terms of the same order in the expansion of theS-matrix we obtain the perturbative expansion for G

G = GS +∞

∑k=1

G(k) (2.55)

in terms of the Coulomb interaction. The connection between the fullGreen function G and the KS Green function GS can be, of course, ex-pressed in the form of a Dyson equation

G = GS + GS · ΣS · G. (2.56)

29


The KS self-energy ΣS incorporates all common xc diagrams minus theinteraction with the local xc potential vxc.

It is apparent that the first-order correction G(1) in the expansion (2.55)consists of the Fock diagram and the first-order scattering diagram, butthe latter should contain only the first order v(1)

xc rather than the full vxc.In diagrammatic form we obtain:

G(1) = −v(1)

xc. (2.57)

Here the solid lines stand for KS Green functions whereas the dashedlines represent the Coulomb interaction. Similarly one can obtain higherorder corrections to GS.

In order to have a closed diagrammatic expansion, we also need a di-agrammatic representation of v(k)

xc in every order of the interaction. Thiscan be achieved by the requirement that the density is unchanged in anyorder of the perturbation theory. To achieve this, the correction to theGreen function of any given order must not induce a density change.The density change due to a certain correction to the Green function canbe calculated in the same way as the density itself is calculated fromthe Green function by evaluation at a given point and summation overfrequencies:

δn(k)(r) = T+∞

∑n=−∞

G(k)(r, r, iωn) = 0. (2.58)

In the diagrammatic language, the density change resulting from adiagram for a correction to the Green function can be obtained by joiningits external points. For instance, applying Eq. (2.58) to G(1) (2.57), we seethat

δn(1) = δn(1)C − δn(1)

xc = −

v(1)xc

= 0 (2.59)

has to be fulfilled, i.e., the density change due to the exchange diagramδn(1)

C and the density change due to first-order scattering by the xc poten-

30


tial δn(1)xc must be equal. The latter can be expressed via the KS suscepti-

bility χS as δn(1)xc = χS · v(1)

xc , leading to

v(1)xc = χ−1

S · δn(1)C = , (2.60)

where the wiggly line stands for the inverse KS response function χ−1S .

Note that Eq. (2.59) is the diagrammatic form of the OEP equation, whilev(1)

xc as defined in Eq. (2.60) is the diagrammatic form of the EXX poten-tial.

If the condition (2.58) holds in every order of the perturbation theory,it is equivalent to the well known Sham-Schlüter equation (Sham andSchlüter, 1983). In diagrammatic form the Sham-Schlüter equation isgiven by (Sham, 1985)

−

vxc

= 0, (2.61)

where the thick lines represent the full Green function and the shadedcircle represents the self-energy excluding Hartree effects. Equation (2.59)can be recovered from Eq. (2.61) as the first-order expansion (Sham, 1985).

If we insert v(1)xc [Eq. (2.60)] into Eq. (2.57) we arrive at the final form of

the first-order correction to the Green function

G(1) = − . (2.62)

Equation (2.62) with the external KS Green functions removed gives thefirst-order KS self-energy Σ

(1)S .

In a similar fashion we can write down the second-order correction tothe Green function G(2), which depends on the just obtained v(1)

xc and the

31


unknown v(2)xc . In diagrammatic form, G(2) is given by

G(2) = + +

+ − −

− + −v(2)

xc.

(2.63)

The first four diagrams in Eq. (2.63) are the “standard” second-orderxc diagrams. The next four graphs account for the different possibilitieshow v(1)

xc can contribute to G(2). The last diagram describes the scatteringby v(2)

xc . The four “standard” diagrams follow the usual diagrammaticrules, as well as the last diagram, which is an obvious consequence of theperturbation (2.47). Compared with the first-order exchange diagram,the first two diagrams make corrections to the bare interaction and theinteraction vertex, while the other two “standard” diagrams apply thisfirst-order correction to the Green functions. According to Eq. (2.62) thefirst-order exchange is not the only first-order correction to the Greenfunction, which leads to the other four diagrams. For example, the thirddiagram in Eq. (2.63) (the “rainbow diagram”) contains the standard Fockexchange with an additional interaction line above it. The correspondingterm where this Fock diagram is replaced by −v(1)

xc is the first graph inthe third line of Eq. (2.63).

A diagrammatic expression for v(2)xc is obtained by using Eq. (2.58). As

above, the density change due to G(2) is calculated in the diagrammatic

32


language by closing all diagrams of G(2) in a loop. Similar to the caseof v(1)

xc , solving for v(2)xc is done by applying χ−1

S . Diagrammatically this

means attaching a wiggly line to all graphs which do not contain v(2)xc . In

this way we arrive at the graphical expression for v(2)xc :

v(2)xc = + + +

− − − + .

(2.64)

Note that, since the diagrams of v(2)xc are constructed by closing the

diagrams for G(2) in a loop, there is a strict one-to-one correspondencebetween the diagrams for v(2)

xc and those entering G(2) (excluding the onecontaining v(2)

xc ). This means that for every such diagram for G(2), thereis a diagram in v(2)

xc which cancels the variation of the density due to theoriginal diagram in G(2). This can be most easily seen by grouping thedifferent graphs contributing to G(2) appropriately, as it has been donein Fig. 2.1.

A consequence of this one-to-one correspondence between graphs inG(2) and v(2)

xc is that the relationship between different diagrams in G(2)

seen above is also present in v(2)xc . The whole second line of Eq. (2.64) is

related to the the third and forth diagrams via the interchange of the ex-change diagram with −v(1)

xc . For example, the third diagram in Eq. (2.64)is obtained from the “rainbow diagram” of Eq. (2.63) by closing it in aloop and attaching a wiggly line. Similarly to the “rainbow diagram” itcontains – at the very top of the graph – a part which is the standardFock exchange. Replacing this part with −v(1)

xc produces the third dia-gram in the second line, which was obtained by closing the first graph in

33


G(2) = − + −

+ − + −

− + − +

− + + −

Figure 2.1: Diagrammatic expansion of G(2).

34


the third line of Eq. (2.63) in a loop and attaching a wiggly line. As wesaw above, that graph can be obtained by replacing the Fock exchangein the “rainbow diagram” by −v(1)

xc . Hence, the two operations “closingdiagrams in loops” and “replacing Fock exchange by −v(1)

xc ” are inter-changeable. Note that, since the Fock exchange and v(1)

xc are related toeach other, there exists an alternative view of this replacement procedure:Given a graph such as the third diagram in Eq. (2.64), we can separateit into two by cutting the two fermionic lines just before and after theexchange interaction. Closing both diagrams and attaching a wiggly linebetween the new external points gives the third diagram in the secondline of Eq. (2.64):

→ → . (2.65)

A similar relation exists between the forth diagram and the remainder ofthe second line of Eq. (2.64).

In principle, this procedure can be continued the same way to higherorders. Given diagrammatic representations for all orders of vxc up tothe n-th order, one can construct G(n+1) with v(n+1)

xc as the only unknownquantity. A diagrammatic expression for v(n+1)

xc can then be derived usingEq. (2.58). However, this is a redundant procedure since the observationsoutlined above for G(2) and v(2)

xc can be easily generalized. Obviously,v(n+1)

xc is equal to the density change induced by all other diagrams inG(n+1) with a wiggly line attached to the external point. These other dia-grams in G(n+1) are of two possible types. On the one hand, there are –not necessarily irreducible – “standard” xc diagrams. On the other hand,there are diagrams which contain lower orders of vxc. The latter diagramsare not independent of the former, but can be constructed by searchingfor all configurations where parts of the “standard” xc diagrams of or-

35


der n + 1 are “standard” xc diagrams of lower order and replacing allthese parts with an appropriate diagram from lower order vxc. Most con-veniently, this can be done after the “standard” xc diagrams have beenclosed in a loop. One searches then for possibilities to separate any givengraph into two by cutting two fermionic lines. If these fermionic lines arenot both connected to the external point of the graph, this is equivalent tofinding fragments which are lower order “standard” xc diagrams. Join-ing these cut fermionic lines and connecting the two parts with a wigglyline leads to the correct replacement with a diagram from vxc in lowerorder, when the sign of the diagram is changed. Note that there are di-agrams where more than one cutting of this type is possible. In thesecases, one has to do all possible cuttings in the parent graph and in thederived graphs to produce the correct diagrams for vxc. When applyingthe cutting rules in this recursive fashion, one must not cut fermioniclines attached to the same wiggly line, as this would lead to the samediagram but with a negative sign.

Putting everything together, we arrive at the following rules for con-structing the diagrammatic representation for vxc in any order:

1. Draw all graphs for the xc contribution to the particle density ac-cording to the usual rules and attach a wiggly line to the externalpoint of each graph.

2. If possible, separate any given graph into two by cutting two fer-mionic lines. Join these cut fermionic lines for each part separately,creating two external points. Connect these external points by awiggly line, and change the sign of the graph. Do not cut linesattached to the same wiggly line.

3. Repeat the last rule for all possible cuttings in all graphs, includingthose obtained from this rule.

4. Keep only nonequivalent graphs.

Note that the change of sign is a direct consequence of the definition (2.47)that contains a minus sign in front of Vxc. Inspecting the diagrams for v(2)

xc

36

2.4 Time-dependent density-functional theory

in Eq. (2.64), one can see how these rules work. The diagrams in the firstline are the “standard” xc contributions as referred to in the first rule.There is only one possibility to separate the third diagram into two parts,which leads to the third diagram in the second row. This is also illustratedin Eq. (2.65). The other three diagrams in the second row follow from thelast diagram of the first row, which offers two possibilities for separation.Here the recursive nature of the third rule comes into play. The lastrule prevents the double counting of the last diagram in the second row.Using these rules, we can now construct the xc potential and the Greenfunction in any order of the perturbation theory.


What happens if we replace Eq. (2.2c) by

Vext(t) =∫

d3r vext(rt)ψ†(r)ψ(r) (2.66)

making the external potential time dependent? Is the now time depen-dent many-body state |ΨN(t)〉 a functional of the time-dependent den-sity n(rt)? Obviously, the HK theorem from section 2.2 does not applyhere. A generalization of the HK theorem was provided by Runge andGross (1984) laying the formal foundation for time-dependent density-functional theory (TDDFT). Since then, TDDFT has been reviewed bymany authors (e.g., Gross et al., 1995; Burke and Gross, 1998; Dobson,1998; van Leeuwen, 2001; Werschnik et al., 2005), hence only a brief over-view will be given here.

In order to prove the time-dependent analogue of the HK theorem,one has to show that the two time-dependent many-body states |ΨN(t)〉and |Ψ′

N(t)〉 that evolve from the same initial state |φN〉 under the influ-ence of two time-dependent potentials vext(rt) and v′ext(rt), which differby more than a time-dependent constant, lead to different densities n(rt)and n′(rt). The time-dependent external potential and hence the time-dependent many-body state are functionals of the time-dependent den-

37


sity and the initial state, if this condition is fulfilled. For notational sim-plicity the dependence on the initial state will not be indicated below.

Of course, there exists no minimum principle for the time-dependentdevelopment as it was used in the static case. Therefore, the proof pro-vided by Runge and Gross (1984) uses two steps. In the first step itis proved that different potentials vext(rt) and v′ext(rt) lead to differentcurrent densities j(rt) and j′(rt). This can be understood on physicalgrounds, since the current density is proportional to the momentumdensity. Changes in the momentum density are caused by force den-sity which is proportional to the gradient of the external potential. Thesecond step uses the difference of the continuity equations for the twosystems,

∂

∂t(n(rt)− n′(rt)

)= −∇

(j(rt)− j′(rt)

), (2.67)

to prove that the densities differ, if the current densities are distinct. Thiscompletes the proof given by Runge and Gross. We can conclude thatthe time-dependent many-body state is indeed a functional of the time-dependent density. Similar to Eq. (2.34), the same holds for the expecta-tion value of any operator.

For practical applications of this theorem, a time-dependent versionof the KS scheme outlined in section 2.2 would be useful. Similar to thestatic case, it can be derived by invoking the time-dependent HK theoremwith the interaction set to zero. We thereby obtain the time-dependentKS potential vS[n](rt) such that(

−12∇2 + vS[n](rt)

)ϕi(rt) = i

∂

∂tϕi(rt), (2.68)

with

n(rt) =N

∑i=1

|ϕi(rt)|2 (2.69)

being the exact time-dependent density of the interacting system. Asin the static case, the KS potential is split into external, Hartree and

38


xc potential, i.e.,

vS[n](rt) = vext(rt) + vH[n](rt) + vxc[n](rt). (2.70)

Note that the existence of vS[n] is guaranteed only together with someadditional though reasonable assumptions (van Leeuwen, 2000). The firsttwo terms in Eq. (2.70) are known explicitly, leaving the unknown vxc[n]that has to be approximated. One can expect that this is considerablymore difficult than in the static case, since in general the xc potential is anonlocal functional of the density in both space and time. As was recentlyshown by Lein and Kümmel (2005), the discontinuity of the static vxc

when a particle is added affects the dynamic xc potential, too.An additional complication is the initial state dependence of vS[n]. The

KS potential is a functional of both the density and the initial state. Thiscan be simplified, if we restrict our considerations to systems where wecan split vext(rt) as

vext(rt) = v0(r) + δv(rt), (2.71)

with δv(rt) = 0 for t < t0 and assume that the system is in the groundstate corresponding to v0(r) with density n0(r) for t < t0. The densityfor t > t0 can be written as

n(rt) = n0(r) + δn(rt). (2.72)

Equation (2.71) describes a very broad class of systems and is applica-ble to most experimental problems, where a system is perturbed by atime-dependent external probe. If the perturbation δv is small, we canuse linear-response theory as introduced in section 2.1.4. The relationbetween δv and δn is then given in terms of the – generally unknown –density–density response function χ:

δn(rt) =∫

d3r′∫

dt′ χ(rt, r′t′)δv(r′t′). (2.73)

Since the KS system has the same density as the interacting system,there exists a analogous relation for KS particles:

δn(rt) =∫

d3r′∫

dt′ χS(rt, r′t′)δvS(r′t′). (2.74)

39


As the KS particles are noninteracting, the density–density response func-tion χS of the KS system is given by the expression for independent parti-cles [Eq. (2.28)]. However, the change in the KS potential δvS is unknown.According to Eq. (2.70) vS depends self-consistently on the density. Up tolinear order we find

δvS = δv +δvH

δn[n0] · δn +

δvxc

δn[n0] · δn

= δv + VC · χ · δv + fxc · χ · δv,(2.75)

where the dots indicate the necessary integrations over space and timevariables. Note that in a time-dependent theory the Coulomb interactionis given by VC(rt, r′t′) = δ(t− t′)/|r − r′|. In Eq. (2.75) the xc kernel fxc

has been introduced as a functional derivative of the xc potential withrespect to the density

fxc(rt, r′t′) =δvxc(rt)δn(r′t′)

[n0]. (2.76)

For TDDFT in the linear response regime the xc kernel fxc is the centralunknown quantity. Since in general vxc is a nonlocal functional of thedensity, fxc is in general a nonlocal quantity in both space and time.

Petersilka et al. (1996) combined Eqs. (2.73)–(2.75) and found

χ = χS + χS · (VC + fxc) · χ. (2.77)

Thus, the exact density–density response function χ is related to its KScounterpart χS via an RPA-type equation. Therefore TDDFT providesan in principle exact way to calculate χ. As we have seen in section 2.1.4,χ(ω) has poles at energies corresponding to excitations which conservethe particle number. In order to use TDDFT for the calculation of excita-tion energies one needs first a reliable static DFT ground state, includingthe unoccupied KS orbitals that enter χS. Second, an approximationto fxc that captures the essential physics is required. If we neglect fxc inEq. (2.77), we recover the RPA result (2.32) with KS orbitals and energiesas starting point.

40


We have seen in the discussion of static DFT that vxc has some unusualproperties. This is not surprising, since vxc describes all the complicatedmany-body effects in a local potential. The situation with fxc is similar.All the many-body effects, which make the difference between χ and χS,are described by an effective interaction modifying the Coulomb interac-tion in an RPA-type equation. Some unusual behavior of fxc is thereforeexpected. Indeed, the nonlocality range of fxc can be of the order ofthe size of the system under consideration (Tokatly and Pankratov, 2001),which is related to the discontinuity of vxc on addition of a particle.

ApproximationsWhat sort of approximations can be used for vxc or fxc? In their pioneer-ing work Zangwill and Soven (1980a,b) calculated the photo-absorptionin rare gases in a self-consistent field manner. They used what later be-came known as adiabatic local-density approximation (ALDA), simplysubstituting the time-dependent density in the LDA xc potential, i.e.,vALDA

xc (rt) = vLDAxc

(n(r, t)

). The resulting xc kernel is local in space and

time

f ALDAxc (rt, r′t′) = δ(r − r′) δ(t− t′)

dvLDAxc

dn(r)(n(r, t)

). (2.78)

Yet, locality in both space and time may be a very strong approxima-tion. Can we expect ALDA to reasonably describe the dynamics of ahomogeneous electron gas, where in the static case LDA is exact? In ahomogeneous system we can Fourier transform Eq. (2.77) with respectto both space and time. The integrations in Eq. (2.77) become simplemultiplications and we can solve for χ as

χ(ω, q) =χS(ω, q)

1− χS(ω, q)(VC(q) + fxc(ω, q)

) . (2.79)

Comparing Eq. (2.79) with the RPA result (2.33) we see that fxc intro-duces a correction but does not change the overall expression. In the the-ory of the homogeneous electron gas the dynamic Hubbard local-fieldfactor G(ω, q) is introduced to describe the corrections to the RPA re-sult (e.g., Mahan, 1990, chapter 5). The functions fxc(ω, q) and G(ω, q)

41


are related as G(ω, q) = − fxc(ω, q)/VC(q). This relation implies that inthe homogeneous electron gas fxc(ω, q) depends only weakly on the fre-quency and its nonlocality is of short range (see e.g., Sturm and Gusarov,2000). It is therefore not surprising that ALDA works quite well for thehomogeneous electron gas, though approximate inclusion of the nonlo-cality further improves the results (Lein et al., 2000; Tatarczyk et al., 2001).

However, in the late nineties it has been realized that ALDA cannotserve as a starting point for the dynamic xc response of an inhomoge-neous electron gas. To fulfill the harmonic potential theorem (Dobson,1994), any dynamic fxc has to be of long range for an inhomogeneoussystem (Vignale, 1995a,b). For extended systems with an energy gap fxc

has to be long-ranged even in the static case (Gonze et al., 1995; Ghosezet al., 1997).

One can see this also in practical calculations. ALDA has been success-fully applied to various finite systems such as atoms or molecules (Peter-silka et al., 1996; Casida et al., 1998; van Gisbergen et al., 1998; Grabo et al.,2000). Typically in these systems already the RPA response functioncalculated with KS eigenvalues and eigenfunctions gives good results.The correction due to f ALDA

xc is quite small, which indicates that Hartreeeffects dominate. Unfortunately, f ALDA

xc remains insignificant also in ex-tended systems like semiconductors or insulators, where KS-RPA givesa very poor description of the absorption spectra (Gavrilenko and Bechst-edt, 1997; Kim et al., 2003). Thus while a correct accounting for xc effectsbecomes crucial in extended systems, the ALDA kernel f ALDA

xc fails toprovide even a reasonable starting approximation.

In order to regain the ability to use local approximations that includedynamic xc effects, a change of the basic variables has been suggested.Vignale and Kohn (1996, 1998) introduced the time-dependent current-density-functional theory (TDCDFT), which uses the current density j(rt)instead of the particle density n(rt) as a basic variable. For example, TD-CDFT has been successfully used to calculate the polarizability of conju-gated polymers (van Faassen et al., 2002). Vignale et al. (1997) further re-fined this theory by switching to the velocity variable v(rt) = j(rt)/n(rt).

42


This additional change of variables allowed to cast the xc effects in amore physical viscoelastic form and has been used to describe collectivemodes in quantum wells (Ullrich and Vignale, 1998). The currently mostpromising formulation of the time-dependent theory has been recentlyset forward by Tokatly (2005a,b), who formulated a general quantum hy-drodynamics in the reference frame of a co-moving observer. This lead toa “geometric” formulation of TDDFT based on the dynamic deformationof the electron liquid allowing to rigorously derive local approximationsfor the dynamic xc effects.

If one wishes to stay with the density as the basic variable, one has toaccount for the nonlocal nature of vxc or fxc. Similar to the static DFT, onecan resort to functionals that explicitly depend on the KS orbitals. An ex-ample is the time-dependent generalization of the OEP method that wasgiven by Ullrich et al. (1995). However, an implementation of this schemefor vxc would be very demanding. With some additional approximations,which are especially suitable for small atoms, it can be used in the linear-response regime (Petersilka et al., 1996). Görling (1998a,b) derived theexact xc kernel at the EXX level, i.e., in first-order GLPT. Its adiabaticlimit has been used to calculate optical properties of silicon (Kim andGörling, 2002a,b).

Absorption spectrum of silicon

The absorption spectrum of silicon can be regarded as a test case for the-oretical methods used to calculate optical properties of insulating solids.The experimental spectrum (dots in Fig. 2.2) exhibits two prominentpeaks at about 3.5 eV and 4.5 eV. The latter is called E2 and is caused bytransitions at and close to the X point in the fcc Brillouin zone (Phillips,1966). There has been considerable debate about the nature of the E1 peak.For example, Phillips (1966) attributed this peak to excitonic, i.e., two-particle effects, while Heine and Jones (1969) stayed within the single-particle picture. This debate was resolved when Hanke and Sham (1980)showed that the E1 peak could only be described when the particle–holeinteraction was taken into account. The E1 peak is indeed an effect of

43


•••••

•••••••••••••••••••••••••••••••••••

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

•••••

E1

E2

2 3 4 5 60

20

40

60

ω (eV)

=εM

Figure 2.2: Imaginary part of the macroscopic dielectric function of silicon asobtained from ALDA (dotted line), RPA using GW energies (dashed line),and BSE (solid line). The dots indicate experimental data. Adapted fromSottile et al. (2003).

unbound excitons.It is worth noting here that excitons, which are caused by the particle–

hole interaction, can modify the absorption spectrum of typical semi-conductors in several ways (e.g., Phillips, 1966, part III and referencestherein). The probably best known effect is the series of Rydberg statesbelow the single-particle threshold which leads to sharp peaks in the ab-sorption spectrum for energies within the band gap. These “parabolic ex-citons” are very similar to the hydrogen atom, but with a typical bindingenergy of only a few tens of millielectronvolts. Similar to the scatteringstates of the hydrogen atom, parabolic excitons introduce an absorptionenhancement just above the single-particle threshold. This unbound exci-ton effect is known as “Sommerfeld absorption enhancement” or “Som-merfeld factor”.

Besides these well known excitonic effects at a parabolic critical point

44


in the joint density of states, excitons can also occur at hyperbolic criti-cal points (“hyperbolic exciton” or “saddle-point exciton”) or at almostdegenerate critical points of different nature (“hybrid excitons”). In bothcases excitons can considerably change the structure of the absorptionspectrum close to the critical point(s). The E1 peak in silicon is mostlikely of the hybrid type (Phillips, 1966; Hanke and Sham, 1980) and isdifficult to describe using simple models. These effects are referred to as“unbound excitons”, since hyperbolic critical points can only occur abovethe single-particle threshold.

Hanke and Sham (1980) obtained their results using an approximatetight-binding representation for the QP states and taking the particle–hole interaction into account by solving the Bethe-Salpeter equation (BSE)which will be discussed in more detail in chapter 5. In the “state of theart” procedure solving the BSE is the final step in the calculation of theabsorption spectrum (see Onida et al., 2002, for a recent review). As afirst step, a DFT calculation in the LDA is performed. The absorptionspectrum for silicon calculated with KS orbitals and energies is shownby the dotted line in Fig. 2.2. Here, the ALDA was used, though RPAgives qualitatively similar results (Gavrilenko and Bechstedt, 1997; Kimet al., 2003). The whole spectrum is red-shifted by about 0.7 eV, whichis attributed to the error in the band gap mentioned in section 2.2. TheE2 peak is well described, while there is just a shoulder in place of theE1 peak.

The next step on the way to the true absorption spectrum involves thecalculation of QP corrections within the GW formalism (Hedin, 1965).In this formalism the electronic self-energy is approximated by a Fockexchange diagram with the screened interaction similar to Eq. (2.24) (dif-ferent levels of self-consistency may be employed, see Aulbur et al., 2000,for a recent review). The absorption spectrum using RPA and the QP en-ergies from the GW calculation is indicated by the dashed line in Fig. 2.2.The spectrum is slightly blue-shifted, but the overall agreement with theexperimental data is better than in LDA. However, the E1 peak is stillmissing.

45


The third and final step is the incorporation of the particle–hole inter-action by solving of the BSE. The result can be seen as the solid linein Fig. 2.2. In addition to the correct position of the main absorption re-gion, the E1 peak is correctly described. The outlined procedure leads tohighly accurate results, as has been shown for a number of simple sys-tems, mostly bulk semiconductors (see Onida et al., 2002, and referencestherein). However, this method is extremely laborious, and for more com-plex systems the calculations become prohibitively expensive.

TDDFT in the linear-response regime would be an attractive alterna-tive, since one only has to solve an RPA-type equation for the density–density response function χ(ω) (2.77). Indeed, Kim and Görling (2002a,b)obtained a very good description of the absorption spectrum for siliconincluding the E1 peak by using fxc in the EXX approximation. Theseauthors particularly stressed the importance of incorporating the nonlo-cality of fxc. De Boeij et al. (2001) arrived at a similar conclusion. In thecontext of time-dependent current-density-functional theory the nonlocaleffects were also shown to be crucial for unbound excitons (Kootstra et al.,2000). The importance of the nonlocality of fxc was highlighted in a par-ticularly clear way by the work of Reining and coworkers (Reining et al.,2002; Botti et al., 2004) who were able to describe the contributions of un-bound excitons in several diamond and zinc-blende type semiconductorswith a static xc kernel proportional to 1/|r − r′|.

It can be easily understood that a nonlocal fxc should contribute toexcitonic effects. The density–density response function χ can be calcu-lated by means of Eq. (2.77), where in a crystalline solid χ, χS, VC, andfxc are matrices in reciprocal space. The matrix structure of χ is respon-sible for local-field effects in the macroscopic dielectric function (Adler,1962; Wiser, 1963). However, these are relatively small in typical semi-conductors and can be neglected for a qualitative analysis. Keeping onlydiagonal elements with zero reciprocal lattice vectors is equivalent to ap-proximating the solid as a homogeneous system, for which χ is given byEq. (2.79).

In a homogeneous system the macroscopic dielectric function εM(ω) is

46


given by the limit q→ 0 of the dielectric function ε(ω, q) defined by

1ε(ω, q)

= 1 + VC(q)χ(ω, q). (2.80)

Using Eq. (2.79) we thus obtain for the macroscopic dielectric function

εM(ω) = 1− limq→0

VC(q)χS(ω, q)1− fxc(ω, q)χS(ω, q)

. (2.81)

An excitonic peak in εM(ω) appears when the denominator vanishes.However, it is well known that χS is proportional to q2 in the limit q→ 0for systems with an energy gap (Adler, 1962; Wiser, 1963). Hence just likethe Coulomb interaction VC(q) = 4π/q2 the xc kernel fxc has to behaveas 1/q2 in this limit to counterbalance χS. Otherwise the xc kernel wouldhave no effect on εM(ω) at all. For the static long-ranged xc kernel ofReining et al., fxc(ω, q) = 4πβ/q2 with some constant β, the macroscopicdielectric function reads

εM = 1− 4παS(ω)1− 4πβαS(ω)

, (2.82)

where αS(ω) is the macroscopic polarizability of the Kohn-Sham systemαS(ω) = limq→0 χS(ω, q)/q2. For a typical αS(ω) close to the band edgethis formula suggests the existence of only one excitonic peak. One could,e.g., fit β to the E1 peak in silicon, but then miss all bound excitons inthe band gap. Phenomenologically this problem can be circumvented byintroducing a frequency-dependent β. One would need to introduce veryrapid oscillations in the region of the Rydberg series of bound excitonicstates, though.

Overall, the last 20 years have seen a rapid progress in both under-standing and application of TDDFT. However, the situation is not assettled as for the static DFT. In many cases it is still unclear which ap-proximations for vxc or fxc should be used, or what errors are introducedby them.

47


As was shown in section 2.3, in the static case it is possible to dia-grammatically expand the full Green function in terms of the KS Greenfunctions. If one has such an expansion for G, one can easily give the cor-responding expansions for the energy and for any correlation functionlike the density–density response function χ, which is a ground-stateproperty. However, this is not necessarily correct for the xc kernel fxc.Although it also is a ground-state property, fxc is more like the inverse ofa correlation function. It is not clear per se that it is possible to diagram-matically expand fxc in a similar fashion. This question is considered inthe next chapter.

For the particular problem of excitonic effects in solids, a direct com-parison of the TDDFT formalism with the BSE is probably the mostpromising path in the quest for a good approximation to fxc (Reininget al., 2002; Sottile et al., 2003; Adragna et al., 2003; Marini et al., 2003).Simply comparing the calculated spectra, it was found that it is often suf-ficient to use an approximation to fxc which is of the first-order in thescreened particle–hole interaction. Although these results are very en-couraging, it is unclear why this approximation is so efficient and whatits range of validity is. This will be discussed in greater detail with aspecial focus on diagrammatic techniques in chapter 5.

48

3 The exchange-correlation kernel

In this chapter we investigate how diagrammatic techniques from MBTcan be used for TDDFT in the linear response regime. For this we firstderive a diagrammatic expansion for the xc kernel fxc using two differentapproaches. This diagrammatic expansion is then used to analyze theanalytic properties of fxc. Finally we discuss possibilities to constructconsistent approximations to fxc. A brief account of some of these resultshas been given in Tokatly, Stubner, and Pankratov (2002).

3.1 Diagrammatic representation of the xc kernel

In this section two equivalent ways for deriving a diagrammatic represen-tation of the xc kernel fxc are presented. The first one uses the definitionof fxc as functional derivative of the xc potential vxc with respect to thedensity. The second one uses the fact that fxc acts as an effective in-teraction in an RPA-type equation for the response function as seen insection 2.4.

3.1.1 Derivation of the xc kernel via differentiation

In section 2.4 fxc was defined as the functional derivative of vxc withrespect to the density n


. (3.1)

This can be used to derive a diagrammatic representation of fxc in everyorder of the perturbation theory with respect to the interaction, sincea diagrammatic representation of vxc in every order of the perturbation

49


theory was found in section 2.3. By using a chain rule, this differentiationcan be expressed as


=∫

d3r′′∫

dt′′δvxc(rt)

δvS(r′′t′′)δvS(r′′t′′)δn(r′t′)

, (3.2)

where vS is the KS potential. This simplifies the calculation, as the KS re-sponse function χS is defined as χS = δn/δvS, so that the last term isjust the inverse response function χ−1

S . For the differentiation δvxc/δvS

we have to keep in mind that functional differentiation of a Green func-tion with respect to its generating potential inserts an additional externalpoint, e.g., δGS(rt, r′t′)/δvS(r′′t′′) = GS(rt, r′′t′′)GS(r′′t′′, r′t′). Consider-ing v(1)

xc from Eq. (2.60), we see that there are three Green functions whichcould acquire an external point

2

1 3

, (3.3)

leading to three diagrams for f (1)xc :

. (3.4)

Here solid, dashed and wiggly lines represent KS Green functions, theCoulomb interaction and the inverse KS response function, respectively.Equation (3.4) does not give the whole f (1)

xc , though, as one also has toconsider the differentiation of the wiggly line with respect to vS. This isgiven by δχ−1

S /δvS = −(χ−1S )2δχS/δvS, or in diagrammatic form

δ

δvS= − − . (3.5)

50


Putting Eqs. (3.4) and (3.5) together a diagrammatic representation forf (1)xc is obtained:

f (1)xc = − + −

+ .

(3.6)

This is the diagrammatic form of the first-order fxc found by Görling(1998a,b). The first four diagrams describe self-energy corrections in theupper and the lower line of the internal loop. The last diagram describesthe particle–hole interaction. Note that for the self-energy insertionsone always needs diagrams that compensate the induced density change.These are the second and forth diagrams of Eq. (3.6).

The same technique can be used to derive f (2)xc , as a diagrammatic ex-

pression for v(2)xc is given by Eq. (2.64). However, the number of graphs

increases enormously. As we have seen from Eqs. (3.3) and (3.4), eachsolid line in a diagram for vxc leads to one diagram in fxc. Equation (3.5)tells us, that each wiggly line in a diagram for vxc gives another two di-agrams for fxc. The “standard” xc diagrams in the first line of Eq. (2.64)all have five solid lines and one wiggly line, giving in total 28 diagramsin f (2)

xc . The first three diagrams in the second line of Eq. (2.64) come withseven solid and two wiggly lines, contributing 33 diagrams in f (2)

xc . Thelast diagram of v(2)

xc as nine solid and three wiggly lines, which adds an-other 15 diagrams to f (2)

xc . Overall there are 76 diagrams in f (2)xc . Figure 3.1

on pages 52–54 shows all these diagrams.There are several patterns that can be found within these diagrams.

These patterns allow the proper grouping of the diagrams. Similar to thefirst order expression (3.6), there are particle–hole interaction diagrams(vertex corrections) and self-energy insertions. The latter are symmetricfor the upper and lower line insertions. Another similarity is that all self-energy corrections have one (or several) accompanying diagram that com-pensates the induced density change. Besides the “genuine” second or-

51


Pure vertex correction:

+ + + +

+ + + −

Second order self-energy insertions in the upper line:

+ − + −

+ − − +

Second order self-energy insertions in the lower line:

+ − + −

+ − − +

Figure 3.1: Diagrammatic representation for f (2)xc (continued on the next two

pages).

52


Two first order self-energy insertions in the upper line:

+ − − +

− + + −

− + + −

Two first order self-energy insertions in the lower line:

+ − − +

− + + −

− + + −

Figure 3.1: Diagrammatic representation for f (2)xc (continued).

53


First order self-energy insertions in upper and lower line:

+ − −

− + +

− + +

+ − −

First order vertex and first order self-energy insertion in the upper line:

+ − + −

− + − +

First order vertex and first order self-energy insertion in the lower line:

+ − + −

− + − +

Figure 3.1: Diagrammatic representation for f (2)xc (continued).

54


der diagrams, there are also graphs that contain, e.g., two Fock exchangediagrams or one Fock exchange and one first order interaction. It is inter-esting to note that all these diagrams are accompanied by diagrams withthe same corrections distributed over two internal loops, which are linkedby a wiggly line. In addition, the sign of a particular diagram with k wig-gly lines is given by (−1)k. This is easy to understand for the “one-loop”graphs, since all but the two external wiggly lines stem from scatteringby the xc potential vxc, which enters the perturbating interaction definedin Eq. (2.47) with a negative sign. However, the internal wiggly line in the“two-loop” graphs does change the sign, too. These “two-loop” graphswill be discussed in greater detail in the next section.

In principle one could use the general rules presented in section 2.3 toconstruct vxc in any order of the perturbation theory and obtain the corre-sponding fxc by differentiation. However, as we have seen on example ofthe second order, this procedure quickly gets cumbersome. Hence directrules for constructing fxc in any order of the perturbation theory shouldbe more efficient. These rules are derived in the next section.

3.1.2 Derivation of the xc kernel via expansion of the responsefunction

The diagrammatic form of f (1)xc in Eq. (3.6) has been found by Tokatly and

Pankratov (2001), who compared the expansion for the response functionχ as obtained from TDDFT with that obtained from MBT. In this sectionit is shown that a similar technique can be used to obtain an expressionfor fxc in every order of the perturbation theory. In addition, diagram-matic rules for constructing fxc in every order of the perturbation theoryare derived.

In order to obtain a diagrammatic representation for the xc kernel, wecompare the RPA-type equations for the response function χ in TDDFT[c.f. Eq. (2.77)]

χ(ω) = χS(ω) + χS(ω) · (VC + fxc(ω)) · χ(ω), (3.7)

55


and in MBT [c.f. Eq. (2.31)]

χ(ω) = χ(ω) + χ(ω) ·VC · χ(ω), (3.8)

where the dots indicate the convolutions in position or momentum space.Solving Eqs. (3.7) and (3.8) for fxc we obtain

fxc = χ−1S − χ−1, (3.9a)

which is equivalent to

χ(ω) = χS(ω) + χS(ω) · fxc(ω) · χ(ω). (3.9b)

The proper polarization operator χ(ω) can be split as χ(ω) = χS(ω) +Πxc(ω) in the KS part χS and the xc part Πxc, where the latter containsall interaction effects. Expanding in Eq. (3.9a) the inverse of χ in terms ofΠxc, we obtain the following series for fxc:

fxc = χ−1S − (χS + Πxc)−1 = (1 + χ−1

S ·Πxc)−1 · χ−1S ·Πxc · χ−1

S

= χ−1S ·Πxc · χ−1

S − χ−1S ·Πxc · χ−1

S ·Πxc · χ−1S

+ χ−1S ·Πxc · χ−1

S ·Πxc · χ−1S ·Πxc · χ−1

S . . . .

(3.10)

This expansion can also be expressed as an integral equation

fxc = χ−1S ·Πxc · χ−1

S − χ−1S ·Πxc · fxc. (3.11)

In order to derive an expansion of fxc in terms of the interaction, wehave to expand Πxc in terms of the coupling constant as Πxc = Π

(1)xc +

Π(2)xc + Π

(3)xc + . . . . Inserting this expansion in Eq. (3.10) and collecting

the terms of the same order in the coupling constant we arrive at the

56


following sequence of contributions to fxc:

f (1)xc = χ−1

S ·Π(1)xc · χ−1

S (3.12a)

f (2)xc = χ−1

S ·Π(2)xc · χ−1

S − χ−1S ·Π

(1)xc · χ−1

S ·Π(1)xc · χ−1

S (3.12b)

f (3)xc = χ−1

S ·Π(3)xc · χ−1

S − χ−1S ·Π

(2)xc · χ−1

S ·Π(1)xc · χ−1

S (3.12c)

− χ−1S ·Π

(1)xc · χ−1

S ·Π(2)xc · χ−1

S

+ χ−1S ·Π

(1)xc · χ−1

S ·Π(1)xc · χ−1

S ·Π(1)xc · χ−1

S

. . . .

We see that the n-th order contribution f (n)xc is a sum of different terms.

One term is the n-th order of the xc part of the proper polarizability Π(n)xc

with inverse KS response functions attached to both external points. Theother terms contain combinations of Πxc in lower orders connected byinverse KS response functions and with inverse KS response functionsattached to the remaining external points. One has to consider all possi-ble combinations of lower-order terms for Πxc such that their orders addup to the desired order n. Note that similar to the observations made atthe end of the previous section, the sign of each term is given by (−1)k

for k occurrences of χ−1S .

The expression for fxc in Eq. (3.12) is suitable for a diagrammatic inter-pretation, as it contains only inverse KS response functions and differentorders of Πxc. A graphical representation for the latter can easily be con-structed using the diagrammatic technique described in section 2.3, asΠxc is (a part of) a correlation function. The first order contribution Π

(1)xc

is given by

Π(1)xc = − + − + . (3.13)

57


Some representative examples of the second order contributions are

Π(2)xc = + + − . . . . (3.14)

When constructing these different orders of Πxc, one has to keep inmind that we are using KS orbitals as basis functions. Therefore, everystandard self-energy insertion should be accompanied by a complemen-tary diagram containing an interaction with vxc. As explained in sec-tion 2.3, this diagram counterbalances the density change induced by thestandard self-energy insertion.

With the diagrammatic form of Πxc and Eqs. (3.12) we achieve a graph-ical representation of fxc in all orders of the perturbation theory via in-sertion of wiggly lines (i.e. the inverse KS response function χ−1

S ). Forexample, for the first order

f (1)xc = − + −

+ ,

(3.15)

which is the same as the above result of Eq. (3.6). Some representativeexamples from f (2)

xc are

f (2)xc = +

+ −

− + + . . . .

(3.16)

58


All 76 diagrams of f (2)xc shown in Fig. 3.1 can be obtained this way.

In principle, one could go on constructing higher orders of Πxc andbuilding different orders of fxc according to Eq. (3.12). However, similarto the situation for vxc described in the previous chapter, it is possible tofind a general recipe for the direct construction of fxc in any order of theperturbation theory. This procedure also sheds some light on the generalstructure of fxc.

Let us first take a closer look at Π(2)xc , which consists of both two-

particle reducible and two-particle irreducible diagrams. Two-particleirreducible diagrams are those diagrams, which cannot be separated intotwo parts by cutting two fermionic lines with different frequencies, ex-cept for the trivial case of cutting two lines connected to the same ex-ternal point. This is the case when the diagram contains second ordercontributions to the one- and two-particle irreducible elements (self en-ergy and vertex). Note that cutting two fermionic lines with different fre-quencies is equivalent to cutting one upper and one lower fermionic lineof the considered diagram. For two-particle reducible diagrams such aseparation is possible, as they contain two first-order contributions to theirreducible elements. Examples for this can be seen in Eq. (3.14), wherethe first graph for Π

(2)xc is two-particle irreducible, as it contains a second

order self-energy insertion. Similar considerations apply to the secondgraph, which contains a second order vertex. The third and fourth graph,however, are two-particle reducible, as they contain both a first orderself-energy insertion and a first order vertex.

As it can be seen from Eqs. (3.12) and (3.16), f (2)xc contains not only

diagrams from Π(2)xc with two wiggly lines attached, but also “two-loop”

diagrams with two diagrams from Π(1)xc connected by a wiggly line. For

every “two-loop” diagram, there is a two-particle reducible diagram inΠ

(2)xc which contains the same vertex and self-energy elements. Examples

are the second and the third line of Eq. (3.16). This statement is obviousfor most diagrams, and there is a one-to-one correspondence betweenthose. Only the case where there is a self-energy insertion in both the

59


upper and the lower line in a graph for Π(2)xc requires a closer look,† as

there are two “two-loop” diagrams containing self-energy insertions in aupper and a lower line. The first “loop” has the self-energy either in theupper or the lower line, with the second “loop” reversed:

(3.17)

However, the diagram from Π(2)xc in question is two-particle reducible in

two different ways. One can either cut the upper line to the right ofthe self-energy insertion and the lower line to the left of the self-energyinsertion, or do it the other way round. Together with the mentionedone-to-one correspondence for the other graphs, this leads to the follow-ing conclusion: All “two-loop” diagrams from f (2)

xc can be derived fromthose diagrams of f (2)

xc , which consist of a two-particle reducible diagramof Π

(2)xc with wiggly lines attached. Specifically, one separates the par-

ticular diagram from f (2)xc into two parts by cutting two fermionic lines

(one upper and one lower line), joins the loose ends of each part andconnects these parts with a wiggly line. Due to the structure of the ex-pansion (3.10), one also has to change the sign. The graphs in the secondline and those in the third line of Eq. (3.16) exemplify this. In both casesthe second diagram can be obtained from the first one by application ofthese cutting rules.

Not only the “two-loop” and the two-particle reducible diagrams areconnected like this. Similar rules also apply to diagrams containing stan-dard self-energy insertions, which are related to those diagrams contain-ing the appropriate interaction with vxc needed for keeping the densityfixed. The latter diagrams can be obtained from the former ones by cut-ting the fermionic lines before and after the self-energy insertion and† As one can see from Fig. 3.1 there are many diagrams of this sort. They can all be treated

in the same fashion as the current example.

60


connecting the two parts with a wiggly line. The sign has to be changedtoo. Examples are the two graphs in the third line of Eq. (3.16), whichcan be constructed from those graphs directly above them. Hence, wecan start, e.g., with the first graph in the second line of Eq. (3.16) andderive all other diagrams in the second and third line using the cuttingrules as described above.

Up to now we have focused on f (2)xc . However, these rules are not lim-

ited to the second order in the perturbation theory. The second connec-tion described above, which relates diagrams with standard self-energyinsertions to those with scattering by the xc potential, is a direct conse-quence of our choice of the basic propagators. Whenever there is a self-energy insertion in a diagram, there must be another diagram present tocounter the density change. This is valid for fxc in every order of the per-turbation theory. For f (1)

xc , this can be directly seen from Eq. (3.15). Forf (2)xc a close inspection of the diagrams depicted in Fig. 3.1 also shows the

same feature.Similarly, the first connection described above, which relates “two-

loop” diagrams to two-particle reducible contributions to Πxc, is a directconsequence of the expansion of fxc in Eq. (3.10). A diagram from Π

(n)xc

is two-particle reducible only if it contains two or more self-energy inser-tions or vertices of a lower order than n. The same self-energy insertionsand vertices can be found in lower orders of Πxc, which contribute tof (n)xc when connected with wiggly lines according to Eq. (3.10). Exactly

these “two-loop” (or “multi-loop”) diagrams are obtained from the “par-ent” graph, which contains Π

(n)xc , when the described cutting rules are

applied.Summarizing it can be concluded that in order to construct fxc in any

order of the coupling constant, one does not need to use the expan-sion (3.10), but instead can apply the following diagrammatic rules:

1. Draw all diagrams for the xc-part of the proper polarization op-erator Πxc(ω) up to the desired order, using standard self-energyinsertions, and attach a wiggly line representing χ−1

S (ω) to both

61


external points.

2. If possible, separate any given graph into two by cutting two fermi-onic lines. Join the external fermionic lines of these parts, connectthem by a wiggly line, and change the sign. Do not cut lines at-tached to the same wiggly line.

3. Repeat the last rule for all possible cuttings in all graphs, includingthose obtained from this rule.

4. Keep only nonequivalent graphs.

It is interesting that these rules are almost identical to those for vxc

outlined on page 36 in section 2.3. The only difference being the firstrule, where the “parent” graphs are constructed. For vxc one shouldtake a loop with one external point whereas a loop with two externalpoints is used for fxc. This suggests that similar rules hold for the generalfunctional derivative δmExc/δnm of the xc energy Exc with respect to thedensity n. This has not been proved up to now, though.

3.1.3 Generalizations of the diagrammatic rules

In the previous section the diagrammatic expansion of fxc using a pertur-bation theory in terms of the bare interaction was presented. Two simplebut for the following investigations useful generalizations of these ideasare discussed in this section.

The second rule for constructing fxc describes how diagrams with inter-nal wiggly lines are derived from parent diagrams by cutting fermioniclines. As mentioned above, one may cut two fermionic lines carryingeither different or the same frequencies. If one cuts two fermionic lineswith different frequencies, the one- and two-particle irreducible elements,i.e., self-energy or vertex insertions, are not affected. This is exemplifiedin both the second and the third line of Eq. (3.16), where the “two-loop”diagrams are derived from the “one-loop” diagrams by cutting the upper

62


and the lower fermionic line between the self-energy and the vertex inser-tion. Cutting two fermionic lines carrying the same frequency is possiblebefore and after a group of self-energy insertions in a particular line. Ver-tex insertions are not affected by this sort of cutting at all. The resultingdiagram still has the form of a self-energy insertion, which is necessaryfor keeping the density fixed as we have seen in section 2.3. An examplefor this is the relation between the second and the third line of Eq. (3.16).

Since either the one- and two-particle irreducible insertions are unal-tered by the cutting rules or necessary additional terms of the same typeare produced, it is possible to group appropriate graphs from differentorders and carry out a partial summation. The sum of all self-energyinsertions is the KS self energy ΣS introduced in Eq. (2.56) on page 29.For example, the first two diagrams in Eq. (3.15) and the first diagram inEq. (3.16) are all part of that group, which after summation gives a “one-loop” diagram containing one interaction with ΣS in the upper fermionicline:

ΣS= − + + . . . . (3.18)

Similarly, the last diagram in Eq. (3.15) and the second diagram inthe first line of Eq. (3.16) are part of that group, which gives a “one-loop” diagram with one interaction with the irreducible vertex I aftersummation:

I = + + . . . . (3.19)

Here, the irreducible vertex I is defined as the sum of all diagrams de-scribing particle–hole scattering which are two-particle irreducible in theparticle–hole channel.

Therefore, instead of starting with parent graphs which are of n-th or-der in the interaction, one can also start with parent graphs which are

63


of n-th order in the irreducible elements. The cutting rules for derivingthe other diagrams of the same order are unaltered. This leads to a per-turbative expansion for fxc in terms of the irreducible elements insteadof the bare interaction. The xc kernel in the first order with respect toirreducible elements is given by:

f (1)xc =

ΣS+

ΣS

+ I . (3.20)

Obviously, one can always go back to a perturbation theory in terms ofthe interaction by expanding the irreducible elements in powers of theinteraction. Note that the irreducible self-energy ΣS already insures thatthe density is kept constant. Hence, one only has to cut fermionic linescarrying different frequencies when deriving further diagrams from theparent graphs. For example, in the first order xc kernel (3.20) no cuttingis necessary or possible. This is different in the second order:

f (2)xc =

ΣS ΣS +ΣS ΣS

+ΣS

ΣS

+ IΣS

+ IΣS

+ IΣS

+ IΣS

+ I I + cut graphs.

(3.21)

Here all the parent graphs displayed in Eq. (3.21) can be separated bycutting two fermionic lines with different frequencies. Only in the caseof the third diagram on the first line one has to keep in mind that there aretwo possibilities to separate this graph, similar to the situation depictedin Eq. (3.17).

64


A further generalization is possible, if we consider the diagrammaticexpansion of the effective interaction V = VC + fxc instead of only fxc. Infact, the diagrammatic rules for fxc derived in the previous section applyto V as well. Only the group of diagrams used as parent graphs has tobe changed. Instead of the xc contributions to the proper polarizationoperator, the diagrams for χxc = χ − χS should be used. This can beseen as follows. Consider some diagram for χ with at least one Coulombinteraction in the annihilation channel. Normally it is possible to separatethis graph into two parts by cutting two fermionic lines which are bothconnected to this interaction. Similar to the situation for internal wigglylines described above, connecting these two parts according to our rulesproduces the starting diagram with an opposite sign. However, whereasthis was explicitly forbidden in the case of two fermionic lines beingattached to the same wiggly line, it is allowed here. A simple examplefor this is

− , (3.22)

which is obviously equal to zero. The consequence is that every diagramfor χ with at least one Coulomb interaction in the annihilation channeldoes not contribute to V. The only exception is the first order contribu-tion to χ which produces VC, after wiggly lines are attached.

The perturbative expansion for χ does not have to be in terms of theinteraction, but can also be in terms of irreducible elements. This conse-quently leads to a perturbative expansion for V in terms of the irreducibleelements. The only change compared to the expansion for fxc in terms ofirreducible elements is the definition of the vertex I that now consists ofall two-particle irreducible four-point functions in the particle–hole chan-nel, i.e., the first order annihilation diagram together with the scatteringdiagrams considered above.

One might ask why one should use the diagrammatic rules for con-struction of V, when all additional diagrams cancel. However, the rulesdescribed above can be used to study analytic properties of both fxc andof V at the same time. This is done in the next section.

65


3.2 The xc kernel as “mass operator”

In this section we investigate the possibility of divergences at KS exci-tation energies in finite-order approximations to fxc for systems with adiscrete energy spectrum. The “danger” of these divergences may beseen on example of a simplified fxc with only vertex interactions takeninto account, as studied by Tokatly et al. (2002). The general graph ofΠ

(n)xc for this fxc simply contains a chain of n four-point vertex-functions

connected by unperturbed particle–hole propagators:

Π(n)xc = I I . . . I . (3.23)

The n-th order contribution to fxc is obtained from this diagram for Π(n)xc

by application of the diagrammatic rules for fxc derived in section 3.1.2.One thereby finds in this “vertex only” model

f (n)xc = I

(−

)I . . . I . (3.24)

The particle–hole propagators between the vertices in Π(n)xc (3.23) as well

as the contracted propagators at both ends show first order divergencesat all particle–hole excitation energies. However, as shown by Tokatlyet al. (2002), the terms with wiggly lines that appear in Eq. (3.24) exactlycancel these divergences. In this section, it is shown that a similar argu-mentation can be applied to the general fxc.

To understand the potential “danger” of divergences at KS excitationenergies, it is best to go back to the simpler and more familiar case ofthe one particle Green function. We have seen in section 2.1.3 that theexact Green function G can be expanded in a perturbation series usingthe unperturbed Green function G0 and the interaction. As it is in generalnot possible to sum all these terms, an approximate solution is required.Conventional perturbation theory would suggest to retain only terms upto some given order in the interaction to obtain an approximate Green

66


function. This approach is, however, problematic as any finite order ap-proximation for G contains divergences due to the poles of G0 at theunperturbed particle energies. Even worse, it contains poles of secondor higher order at the unperturbed system’s energies. Therefore suchan approximate Green function is useless since it has completely wronganalytic properties.

A meaningful approximate Green function can be obtained using theDyson equation, though. With the self energy Σ defined as the sum of allone-particle irreducible self-energy insertions, the exact Green functionis given by

G = G0 + G0 · Σ · G. (3.25)

Solving this integral equation gives the correct analytic properties for G,because Σ is free of unperturbed particle divergences. Since Σ is thesum of all one-particle irreducible elements, this is not only true for theexact Σ, which cannot be obtained in general, but also for any finite-orderapproximation to Σ. Hence the correct way to calculate an approximateGreen function is to solve Eq. (3.25) using an approximation to Σ.

The relation between G, G0, and Σ in Eq. (3.25) looks formally similarto the relation between χ, χS, and fxc in Eq. (3.9b). Since Σ is sometimescalled the “mass operator” for G, one could therefore call fxc the “massoperator” for χ. Similar to the situation with Σ, a perturbative expansionof fxc is only meaningful if fxc is free of divergences at KS excitation ener-gies in every order of the perturbation theory. Otherwise the exact fxc wouldbe required, which obviously is free from KS particle–hole divergences.However, as we saw in Eq. (3.12), fxc does not reduce to irreducible el-ements. It is therefore not clear per se that fxc is free from unperturbedparticle–hole divergences in every order of the perturbation theory. In theremainder of this section it will be proved that this is indeed the case.Similar arguments hold for V, which acts as “mass operator” for χ. Inthe following only properties of fxc will be discussed, but as the diagram-matic structure of V is identical to the one of fxc as seen in the previoussection, an analogous argumentation can be made for V.

67


The proof that fxc is finite at unperturbed particle–hole excitation en-ergies in every order of the perturbation theory is organized as follows.First, we identify the part of Π

(n)xc that is resonant at KS excitation en-

ergies. Taking this part as a “chain structure” similar to Eq. (3.23), it isshown that all divergences cancel each other. Finally, it is shown that theresonant part of Π

(n)xc can indeed be represented by such a chain struc-

ture. Note that in this section the standard zero temperature techniquewith real time is used. The chemical potential µ is still set to zero.

General graph of Π(n)xc

Our starting point is a general graph contributing to Πxc with n selfenergies ΣS or vertices I and causal Green functions GS, where we havein the beginning energy ε + ω in the upper branch and energy ε in thelower branch. Here, ω is the transferred or external frequency, whileε is an internal frequency that is integrated over. After the first vertex,we have energy ε′ + ω in the upper branch and energy ε′ in the lowerbranch, where we have to integrate over ε′ and so on. After performingall the integrations over energy in this n-th order order (with respectto irreducible elements) for Πxc, one sees that it diverges close to theKS particle–hole excitation energy ωij = εi − ε j:

. . . ∝1

(ω −ωij)n+1 .

(3.26)Here the circles represent the KS self energy ΣS and the squares repre-sent the irreducible interaction I. As Π

(n)xc with two wiggly line attached

contributes to f (n)xc , we see that there are contributions to fxc that diverge

at the KS particle–hole excitation energies.When doing the frequency integrations, one has to consider the poles

in ΣS(ω), I(ω), and GS(ω). In general, if ΣS and I have poles as a func-tion of ω, they are not at the same energies as the ones in GS. Therefore,they will not lead to divergences at particle–hole excitation energies. Aswe are only interested in the behavior close to particle–hole excitation

68


energies, we hence use frequency-independent ΣS and I.The next step is to replace the causal KS Green functions GS by re-

tarded Green functions GR in the upper branch and advanced Greenfunctions GA in the lower branch:

GS = GR − i2 Θ(−ω)=GR (3.27a)

GS = GA − i2 Θ(+ω)=GA. (3.27b)

After this replacement in the expression for the graph in Eq. (3.26) andthe appropriate multiplication, we arrive at a sum of different terms con-tributing to the general graph from Π

(n)xc . These terms can be organized

in three groups. First, there is one term which only contains GR and GA.This term results in a divergence at particle–hole excitation energies andis therefore the one we are interested in. Second, there are several termswhich contain GR, GA and either one =GR or one =GA. These termsproduce divergences at particle–particle, hole–hole, and negative particle–hole excitation energies and are therefore unimportant for the currentconsiderations. Finally, there are terms which contain several =GR or=GA. These terms look dangerous at first sight, as they contain powersof delta functions. However, a careful consideration of these terms showsthat these singularities cancel. Of course, it has to be like this, as theproduct of the causal Green functions does not have these singularities,and the replacement (3.27) cannot introduce them.

Hence, if we are interested in the behavior of Π(n)xc close to particle–hole

excitation energies, it is sufficient to consider a general graph with GR inthe upper branch, GA in the lower branch and the frequency indepen-dent ΣS and I. The resonant part of Π

(n)xc is given by the sum of all these

general graphs. We will see at the end of this section that this sum can bewritten as a chain structure with appropriately defined four-point func-tions. This is similar to the chain structure discussed in the beginning ofthis section [Eq. (3.23) and (3.24)], though the particle–hole propagatorswere constructed from causal Green functions there, whereas retardedand advanced Green functions are used here.

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We therefore write Π(n)xc as a sum of this chain structure and terms

which are finite at particle–hole excitation energies. Using the diagram-matic rules for fxc of section 3.1.2, an expression for f (n)

xc can be derived.The terms in this expression can be grouped into three classes. First, thechain structure mentioned above with χ−1

S attached to the ends plus allgraphs which can be derived from that chain structure using our cuttingrules, i.e., chains of lower order connected with χ−1

S . Second, contribu-tions from the nonresonant part of Π

(n)xc . Finally, cross terms between

lower order chains and nonresonant terms together with all graphs ob-tained from application of the cutting rules.

Not surprisingly, the divergences in f (n)xc come from the resonant parts

of Π(n)xc and from the resonant parts of lower-order terms of Πxc. As

all the cutting rules apply to the resonant parts too, it is sufficient toshow that there are no divergences at particle–hole excitation energieswhen one considers the sum of Π

(n)xc and all graphs derived from this

parent graph by application of our cutting rules. The only exceptionsare the divergences from the contracted particle–hole propagators at thebeginning and the end of Π

(n)xc , which shall be treated separately.

We therefore consider only the resonant part of Π(n)xc that can be repre-

sented by a chain structure, as will be shown in the final step of the proof.This chain structure takes the form

R

A

R

A

Ω Ω . . .A

R

Ω , (3.28)

where the solid lines with an “A” stand for iGA, those with an “R” foriGR. The structure of the four-point vertex Ω will be investigated later inthis section. Using the cutting rules for fxc, all the internal particle–holepropagators K(ω, ε) = −GR(ε + ω)GA(ε) are replaced by

J(ω, ε, ε′) = K(ω, ε)δ(ε− ε′)− K(ω, ε)iχ−1S (ω)K(ω, ε′), (3.29)

as the wiggly line represents iχ−1S in the zero temperature technique.

Here, ε and ε′ are both internal frequencies that one has to integrate

70


over. In diagrammatic form:

R, ε + ω

A, ε

→

R, ε + ω

A, ε

−

A, ε

R, ε + ω

A, ε′

R, ε′ + ω

. (3.30)

Next we will discuss how to handle singularities of J(ω, ε, ε′) as a func-tion of the external frequency ω. This will be done separately for nonde-generate and degenerate transitions. Note that it does not matter whethera transition is degenerate because one or both of the states are degenerate,or because two otherwise unrelated transitions have the same energy.

Nondegenerate transitionDivergences in J at a nondegenerate KS excitation energy ω = ωij couldoccur, if all retarded Green functions belong to the unoccupied state i andall advanced Green functions belong to the occupied state j. Integratingthis “dangerous” part Jij over the internal frequencies ε and ε′ we obtain

∞∫−∞

dε

2π

∞∫−∞

dε′

2πJij(ω, ε, ε′) = −i

|ij〉〈ij|ω −ωij + iη

+ i|ij〉ij|χ−1

S (ω)|ij〈ij|(ω −ωij + iη)2 .

(3.31)Here |ij〉 is the state of the resonant particle–hole pair, i.e., ϕi(r)ϕ∗j (r′)in position space representation, while |ij is the same state with equalcoordinates of the particle and the hole, i.e., ϕi(r)ϕ∗j (r) in position spacerepresentation. The matrix element in the second fraction of Eq. (3.31) istherefore given by

ij|χ−1S (ω)|ij =

∫d3r

∫d3r ϕ∗i (r)ϕj(r)χ−1

S (ω, r, r′)ϕi(r′)ϕ∗j (r′). (3.32)

Obviously, both terms in Eq. (3.31) diverge at ω = ωij (when the limitη → 0 is taken). However, χ−1

S goes to zero at ω = ωij such that thesetwo divergences cancel. To prove this, we first single out the divergingterm in the KS response function

χS(ω) = χr(ω) +|ijij|

ω −ωij + iη, (3.33)

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where χr is the regular part,

χr(ω) = ∑kl 6=ij

|klkl|ω −ωkl + iη

. (3.34)

As mentioned by Gonze and Scheffler (1999), the inverse of such a de-composition can be calculated using the Sherman-Morrison formula (seePress et al., 1986, p. 66), which gives

χ−1S (ω) = χ−1

r (ω)− χ−1r (ω)|ijij|χ−1

r (ω)ω −ωij + iη + ij|χ−1

r (ω)|ij. (3.35)

Inserting Eq. (3.35) into Eq. (3.31) and taking the limit η → 0 we obtainfor ω = ωij

∞∫−∞

dε

2π

∞∫−∞

dε′

2πJij(ωij, ε, ε′) = −i

|ij〉〈ij|ij|χ−1

r (ωij)|ij, (3.36)

which is indeed finite.

Degenerate transitionIf the KS transition at ω = ω0 is P-fold degenerate, few changes have tobe made to the above derivation. In particular, the singular contributionto fxc (3.31) becomes a P× P matrix with entries

∞∫−∞

dε

2π

∞∫−∞

dε′

2πJij,kl(ω, ε, ε′) = −i

|ij〉〈kl|ω −ω0 + iη

(δij,kl −

ij|χ−1S (ω)|kl

ω −ω0 + iη

).

(3.37)For this to be finite, every entry in the large parentheses in Eq. (3.37) hasto be of order O(ω − ω0). Similar to Eq. (3.33) we can single out thediverging contribution to χS as

χS(ω) = χr(ω) +P

∑ij=1

|ijij|ω −ω0 + iη

, (3.38)

72


where the summation takes into account all P degenerate particle–holestates. It is most convenient to study this problem in an explicit matrix no-tation using some N dimensional basis |n〉, e.g., the N lowest KS states.Equation (3.38) can then be written as

S = R + U ·VT, (3.39)

where S and R are N × N matrices representing χS and χr, respectively.The P degenerate particle–hole states are used for the columns of theN × P matrices U and V with elements 〈n|ij and 〈n|ij/(ω − ω0 + iη),respectively. A matrix of the form found in Eq. (3.39) can be inverted bysuccessive application of the Sherman-Morrison formula leading to theWoodbury formula (see Press et al., 1986, p. 66)

S−1 = R−1 − R−1 ·U ·(

1 + VT · R−1 ·U)−1

·VT · R−1, (3.40)

where only the P × P matrix in parentheses has to be inverted, if R−1

is assumed to be known. Defining Q := VT · R−1 · U, the P × P matrixgiven by the parentheses in Eq. (3.37) can be calculated as

1−VT · S−1 ·U = 1−Q−Q · (1 + Q)−1 ·Q

≈ 1−Q−(

1−Q−1 + (Q−1)2)·Q

= Q−1,

(3.41)

where the matrix inversion was done perturbatively. This is legitimate,as Q has elements ij|χ−1

r (ω)|kl/(ω − ω0 + iη). The numerator of theelements of Q is well behaved at ω = ω0, which makes every elementof Q−1 of order O(ω − ω0). This is also the criterion we found fromEq. (3.37) for the divergences to cancel each other. One can also seenicely how for a nondegenerate transition the inversion of Q simplifiesto a division leading to Eq. (3.36).

Thus, the two-particle propagator modified according to Eqs. (3.29)or (3.30) is free of divergences at degenerate as well as nondegenerateKS particle–hole excitation energies, if all K’s in Eq. (3.29) are resonant.

73


A divergence could also occur, when only one of the two K’s in the sec-ond term of Eq. (3.29) is resonant. This divergence is canceled, however,by the zero of χ−1

S at ω = ωij (or ω = ω0). The same happens to the di-vergences in the contracted particle–hole propagators at the ends of Π

(n)xc .

Those are canceled by the zeros of χ−1S that are attached to Π

(n)xc when

f (n)xc is constructed.

Putting all of the above together we arrive at the conclusion that f (n)xc is

finite at KS particle–hole excitation energies, provided we can prove thechain structure for Π

(n)xc . This last point is tackled next.

Chain structure of Π(n)xc

On the next few pages it is proved that the resonant part of Π(n)xc can in-

deed be written in a chain structure of the form displayed in Eq. (3.28). Ofcourse, this statement is trivial, if we consider the simplified Π

(n)xc which

contains only vertices (3.23), as the four-point function Ω in Eq. (3.28) canbe identified with the vertex function.

Consider now the resonant part of a graph for Π(n)xc which contains

only self-energy terms. Specifically, consider a graph with m1 self-energyinsertions in the upper and m2 self-energy insertions in the lower branch(m1 + m2 = n). We first go from the frequency domain to the time do-main by Fourier transformation. In the time domain, both branches ofthe graph are time ordered due to the properties of the retarded and ad-vanced Green functions. There would be no time ordering within thebranches, if we used causal Green functions. Note that the self-energyinsertions in the time domain depend only on one time, as they are set tobe constant in the frequency domain. However, there is no time orderingbetween the two branches. If we artificially introduce such a time order-ing between the two branches, we have to consider (m1 + m2)!/(m1!m2!)different graphs to capture all possible time orderings. To make this ar-tificial time ordering more explicit, we introduce an intersection into abranch at every time, at which there is a self-energy insertion in the otherbranch. This intersection does not change the diagram itself, as we haveGR(0, τ) = GR(0, t)GR(t, τ) for 0 ≤ t ≤ τ and equivalently for GA. For

74


the simple case of m1 = 2 and m2 = 1 this gives

ΣS ΣS

ΣS

→

ΣSΣS

ΣS+

ΣS ΣS

ΣS

+ΣS ΣS

ΣS, (3.42)

where dots represent the introduced intersections and dotted lines indi-cate the synchronization between self energy insertions and these inter-sections. Introducing this time ordering between the different branchesis similar to drawing Goldstone diagrams (e.g., Mattuck, 1976, section 9.5).

The structure of these artificially time ordered graphs with self-energyinsertions in one branch and intersections, which basically do nothing,in the other branch, motivate the following definition of the four-pointfunction

Ω(r1, r′1; r2, r′2) = ΣS(r1, r′1)δ(r2 − r′2)− ΣS(r2, r′2)δ(r1 − r′1). (3.43)

Technically, the minus sign in Eq. (3.43) is a consequence of capturingthe behavior of a two-point function within a four-point function. Aloop with a single self-energy insertion in the upper is very similar to aloop with the self-energy insertion in the lower line. However, the twodiagrams produce opposing signs when calculated. The minus sign inthe definition of Ω conserves this difference. Physically, we will see inthe next chapter that the matrix element of Ω contributes to the differ-ence between the exact particle–hole excitation energies and those in theKS system. For the particle state, a positive matrix element of the self-energy would increase the excitation energy, while for the hole state, theexcitation energy would decrease. The minus sign in Eq. (3.43) reflectsthis difference.

A chain structure with n four-point functions Ω will lead to 2n arti-ficially time-ordered graphs. Especially, there are (m1 + m2)!/(m1!m2!)graphs with m1 self-energy insertions in the upper and m2 self-energy in-sertions in the lower branch, which are exactly the graphs mentioned inthe previous paragraph. The other graphs are responsible for the other

75


possible distributions of n self-energy insertions over the two branches.For example, for the simple case n = 3 considered in Eq. (3.42), therecould also be only one self-energy insertion in the upper and two in thelower branch, again with three different time orderings. In addition, allthree self-energy insertions could be in the upper or in the lower branch,which already fixes the time ordering.

Hence, the sum of the resonant parts of graphs for Π(n)xc which contain

only self-energy terms can be expressed as a chain of n four-point func-tions Ω with unperturbed particle–hole propagators between them. Thischain, which is still in the time domain, can be Fourier transformed intothe frequency domain, where the cutting rules for the construction of f (n)

xc

lead to a cancellation of the divergences in the particle–hole propagator,as we have seen above.

Finally, we want to look at what has to be changed in the above ar-guments if vertex terms are added. Consider a diagram for Π

(n)xc with

a given number of vertices. The number of self-energy insertions be-tween consecutive vertices and between the outer vertices and the ex-ternal points is known, too. After Fourier transformation into the timedomain we see that there is a strict time ordering in each branch sepa-rately and that the vertices “synchronize” the two branches. But as abovethere is no relative time ordering for the self-energy insertions of the twobranches between two vertices (or an outer vertex and the appropriateexternal point). However, we can apply the arguments developed abovefor diagrams with only self-energy insertions to the region between twovertices. By doing so, we obtain a chain structure with two different four-point functions: the vertex I and the four-point function Ω defined inEq. (3.43). To get the resonant part of the full Π

(n)xc , one would have to

consider all possible numbers and distributions of I and Ω. However, itis more instructive to define a new four-point function Ω = Ω + I. Thisnew four-point function takes into account the different numbers and dis-tributions of I and Ω, so that the resonant part of the full Π

(n)xc can be

written as a chain of n four-point functions Ω connected by unperturbedparticle–hole propagators.

76


Thus we have proved that the resonant part of the full Π(n)xc can be

written as a chain structure, for which we know that it leads to a f (n)xc

which is finite at particle–hole excitation energies. Therefore, we arriveat the important conclusion that fxc is finite at particle–hole excitationenergies in every order of the perturbation theory and hence can be inter-preted as a “mass operator” for χ. This result is one of the main con-clusions of this work. Similar results hold for V, where the only changeis the addition of the bare interaction in the annihilation channel to I.Therefore, χxc = χ − χS can be written as a chain structure for energiesclose to KS particle–hole excitation energies. Hence, V is also finite atparticle–hole excitation energies in every order perturbation theory and canbe interpreted as a “mass operator” for χ.

It is worth noting that although we consider the KS system as an un-perturbed starting point (hence KS particle–hole excitation energies), thiswas not used in the proof of the chain structure of Π

(n)xc . We can therefore

conclude that this result is more general. Whenever we do a perturba-tive expansion of the (proper) polarization operator in terms of the two-particle irreducible elements close to the particle–hole excitation energiesof the unperturbed system, we can write it as a chain structure (3.28),regardless of the unperturbed system used. This can also be interpretedphysically. If we add a particle and a hole to the system and let thempropagate with an energy which is close to an excitation energy of theunperturbed system, it does not matter how the scattering processes be-tween the particle and the hole or with the medium occur precisely. Onlythe total number of such scattering processes is important. These effectivescattering processes are captured in the four-point function Ω.

There is, however, one important difference between fxc or V as “massoperator” for χ or χ and the common mass operator for the Green func-tion. While Σ is a one-particle irreducible quantity, which is obviouslyfree of unperturbed particle divergences even in finite-order approxima-tions, this is not the case for fxc. In this section we were able to provethat the divergences caused by the KS particle–hole propagators in thedifferent terms of f (n)

xc do cancel each other. However, this cancellation is

77


not complete. Rather there are still traces of the two-particle propagatorleft in f (n)

xc . As shown above, the denominator that causes the divergenceis replaced by ij|χ−1

r |ij (or an appropriate matrix inverse). This de-nominator is problematic, since in general χ−1

r (ω, r, r′) goes to zero for|r − r′| → ∞, and the wave functions of the particle–hole states are in-versely proportional to the system volume due to normalization. Hencethe matrix element ij|χ−1

r |ij should vanish with the increase of thesystem size. This implies that ij|χ−1

r |ij must not occur in any physicalquantity calculated with this fxc This remainder of the particle–hole prop-agator makes perturbative approximations for fxc more problematic then,e.g., for the self energy. The consequences of this will become apparentin chapter 4.

3.3 Different models for the xc kernel

We have seen in the previous section that fxc can be regarded as a sensible“mass operator” for χ. Hence it makes sense to introduce ad hoc approx-imations for it in the same way as the self energy Σ in a Green functionmay be replaced by the approximate expressions from Hartree-Fock orGW. However, not all approximations observe general constraints suchas conservation laws. Baym and Kadanoff (1961) formulated the recipefor the so called “conserving approximations”, which automatically re-spect the microscopic conservation laws for particle number, energy, mo-mentum and angular momentum. An important feature of these approx-imations is that the same type of approximation must be used in theGreen function and in the of correlation functions.

Regarding fxc the situation is similar. The approximation for the one-particle KS system of the static DFT should be the same as the approxi-mation used in a subsequent TDDFT of the response function. In otherwords, the approximations for vxc and fxc should be consistent. This canbe ensured by deriving fxc directly from vxc via the functional differentia-tion with respect to the density. The diagrammatic techniques developed

78


in this chapter are an efficient tool for this.A simple example is the first order vxc discussed in section 2.3, which

is equivalent to the OEP or EXX potential:

vxc = . (3.44)

As shown in section 3.1.1, functional differentiation of this diagramgives the first-order approximation for fxc (3.6):

fxc = − + −

+ .

(3.45)

Of course, it is not surprising that Eqs. (3.44) and (3.45) that are both offirst order in the bare interaction are consistent. The consistency questiongets more nontrivial, once one uses self-consistent approximations thatcontain an infinite set of particular diagrams. For example, a self energyof GW-type†

Σ = with = + , (3.46)

leads to the first order xc potential

vxc = . (3.47)

† Since none of the Green functions in Eq. (3.46) are “dressed”, this is normally called G0W0

approximation (e.g., Aulbur et al., 2000).

79


The fxc consistent with this choice of vxc is

fxc = − + −

+ + + .

(3.48)

This is a nontrivial generalization of the first-order result (3.45). Replac-ing the bare interaction with the screened one in Eq. (3.45) is not enough.There are two additional diagrams of second order in the screened inter-action. Without the direct functional differentiation it would be difficultto tell which – if any – of the 76 diagrams of Fig. 3.1 should be includedwith screened interaction lines.

Of course, in terms of a perturbation theory with respect to the irre-ducible elements, both Eq. (3.45) and Eq. (3.48) are of first order, i.e., theycan be written in the form of Eq. (3.20). For example, in the case ofEq. (3.48) this can be done by identifying the KS self-energy as

ΣS = − (3.49)

and the irreducible interaction as

I = + + . (3.50)

The xc kernel of Eq. (3.48) is related to the polarization operator stud-ied by Richardson and Ashcroft (1994) and has been successfully usedto calculate the correlation energy and the plasmon excitations in a ho-mogeneous electron gas (Lein et al., 2000; Tatarczyk et al., 2001). TheOEP kernel and approximations to it have been especially successful insmall atoms (Petersilka et al., 1996, 2000; Grabo et al., 2000). Both thesexc kernels are of first order with respect to irreducible elements, which

80


is reasonable, since fxc can be seen as the “mass operator” for χ. At firstglance one should not expect problems due to the incomplete cancella-tion of the particle–hole propagator [see Eq. (3.36)], as it appears only inhigher-order corrections to fxc. Unfortunately, it is not true. The remain-der from this incomplete cancellation does affect practical applications ofthe first-order xc kernels. This will become apparent in the next chapter,where the particle–hole excitation energies are calculated.

81

4 Particle–hole excitation energies

A particle–hole excitation occurs when one particle is lifted above theFermi level leaving a hole behind. For example, this happens in photo-absorption experiments. As a zero-order approximation for the excitationenergy one can use the difference of the KS energies of the initial and thefinal state. However, this KS particle–hole excitation energy has to becorrected, since QP and KS energies are not the same. In addition, theinteraction between the two QPs has to be considered.

In this chapter we calculate these corrections to the KS particle–hole ex-citation energies. The first- and the second-order perturbation theory interms of irreducible elements (section 4.2) and in terms of the interaction(section 4.3) will be used. The expansion of the particle–hole excitationenergy with respect to the irreducible elements has been touched uponin Tokatly, Stubner, and Pankratov (2002).

Particle–hole excitation energies can be calculated from TDDFT in dif-ferent ways. Petersilka et al. (1996) derived an integral operator whoseeigenvalues vanish at the exact excitation energies. Casida (1996) derivedan eigenvalue problem for the square of the excitation energy. FollowingTokatly and Pankratov (2001), we will use an eigenvalue problem for theexcitation energy itself.

4.1 Eigenvalue equation for the excitation energies

We start with the equation of motion for the quantum mechanical densitymatrix ρ with the Hamiltonian H:

iddt

ρ = [H, ρ], (4.1)

83


where the square brackets indicate the commutator. In the equilibriumstate of the KS system Eq. (4.1) is solved by the ground state KS Hamil-tonian HS and the ground-state KS density matrix

ρS(r, r′) = ∑j

f j ϕj(r)ϕ∗j (r′), (4.2)

which is time independent. The summation in Eq. (4.2) extends over allKS states. Nonzero contributions come from the occupied states only,due to the Fermi occupation number f j.

In the linear response formalism we decompose the density matrix as

ρ(t, r, r′) = ρS(r, r′) +(ρ1(r, r′) exp(−iωt) + c.c.

)(4.3)

for an excited state with excitation energy ω. The variation of the densitymatrix introduces a variation of the density. The KS Hamiltonian is alsochanged, since it is a functional of the density:

H(r, t) = HS(r) +(∫

dt′∫

d3r′δvS(rt)δn(r′t′)

n1(r′) exp(−iωt′) + c.c.)

= HS(r) +(

exp(−iωt)∫

d3r′ V(ω, r, r′)n1(r′) + c.c.)

,(4.4)

where n1(r′) = ρ1(r′, r′) and V(ω, r, r′) = VC(r, r′) + fxc(ω, r, r′). If weinsert Eqs. (4.3) and (4.4) into Eq. (4.1), we obtain for the terms linear inρ1 and proportional to exp(−iωt)

ωρ1(r, r′) =(

HS(r)− HS(r′))

ρ1(r, r′)

+∫

d3r′′(V(ω, r, r′′)− V(ω, r′, r′′)

)n1(r′′)ρS(r, r′). (4.5)

The change in the density matrix ρ1 can be expanded in terms of theKS orbitals as

ρ1(r, r′) = ∑ij

aij ϕi(r)ϕ∗j (r′). (4.6)

84

4.1 Eigenvalue equation for the excitation energies

Inserting Eqs. (4.6) and (4.2) into Eq. (4.5) gives a matrix equation for theexcitation energy ω and the expansion coefficients aij

(ω −ωij)aij − f ji ∑klij|V(ω)|klakl = 0, (4.7)

with the KS excitation energy ωij = εi − ε j and f ji = f j − fi. The notation|kl was introduced in the context of Eq. (3.31), such that

ij|V(ω)|kl =∫

d3r∫

d3r′ ϕ∗i (r)ϕj(r)V(ω, r, r′)ϕk(r′)ϕ∗l (r′). (4.8)

For an excitation, which can be unambiguously related to a particle–hole excitation of the KS system, it is possible to use a perturbation the-ory to obtain approximate excitation energy from Eq. (4.7).† Using thecommon Rayleigh-Schrödinger perturbation theory to the second orderwe obtain for the excitation energy of the transition that is linked to thenondegenerate transition from the occupied KS state j to the unoccupiedKS state i:

ω = ωij + ij|V(ωij)|ij+ ij|V(ωij)|ijij| ∂V∂ω

(ωij)|ij

+ ∑kl 6=ij

ij|V(ωij)|kl flkkl|V(ωij)|ijωij −ωkl

. (4.9)

The third term in Eq. (4.9) is a consequence of the energy dependentperturbation in Eq. (4.7).

If the KS transition is degenerate, one would normally first diagonalizethe perturbation V in the subspace of the degenerate states and then ap-ply the common second order perturbation theory. In some situations itis better to revert the order of these two steps by using “quasidegenerate”perturbation theory (Löwdin, 1951). Since the degenerate case does notgive any new physical insight here, the further discussion is restricted tothe nondegenerate case, though.

† This is not the case, e.g., for plasmons, which also can be obtained from Eq. (4.7).

85


The perturbation V = VC + fxc in Eqs. (4.7) and (4.9) is not generallyknown, but its perturbative expansion has been derived in the precedingchapter. If we insert the second-order expansion V = V(1) + V(2) intoEq. (4.9) and group the terms of the same order, we obtain for the firstand second order energy shift

∆ω(1)ij = ij|V(1)(ωij)|ij (4.10a)

∆ω(2)ij = ij|V(2)(ωij)|ij+ ij|V(1)(ωij)|ijij|∂V(1)

∂ω(ωij)|ij

+ ∑kl 6=ij

ij|V(1)(ωij)|kl flkkl|V(1)(ωij)|ijωij −ωkl

. (4.10b)

The expansion of V can have two different meanings. We can either ex-pand in terms of the interaction or in terms of the irreducible elements ΣS

and I (see section 3.1.3). Since it is always possible to obtain the expan-sion in terms of the interaction from the expansion in terms of irreducibleelements, the latter is studied first. The expansion of V in terms of theinteraction will be considered in section 4.3.

4.2 Expansion in terms of the irreducible elements

In this section we consider the corrections to the KS particle–hole exci-tation energies using a perturbative expansion of V in terms of the irre-ducible elements. We also find some general results for the structure ofthe perturbation expansion as well as an important consequence for thepractical use of perturbative approximations for V or fxc.

The first and the second order in terms of the irreducible elements ΣS

and I of the xc kernel were given in Eqs. (3.20) and (3.21) on page 64

in diagrammatic form. As noted in section 3.1.3, the same diagrams canbe used for V(1) and V(2), when the bare interaction in the annihilationchannel is added to the irreducible interaction I. From Eq. (4.10) we seethat V is required in the vicinity of the KS excitation energy ωij. We

86


can therefore make use of the chain structure derived in section 3.2. Indiagrammatic form V(1) and V(2) can be expressed as

V(1) = Ω (4.11a)

and

V(2) = Ω Ω − Ω Ω . (4.11b)

The four-point function Ω is defined as

Ω(r1, r′1; r2, r′2) = ΣS(r1, r′1)δ(r2 − r′2)− ΣS(r2, r′2)δ(r1 − r′1)

+ I(r1, r′1; r2, r′2),(4.12)

where, as mentioned above, I includes the bare interaction in the annihi-lation channel as VC(r1 − r2)δ(r1 − r′2)δ(r2 − r′1). Note that ΣS and I arefrequency dependent, but have to be evaluated on the KS particle–hole“mass shell” in the end. The concept of evaluating on the KS particle–hole “mass shell” shall be explained in the following with two examples.

Consider the first diagram from Eq. (3.20), where the fermionic loopcontains one interaction with ΣS in the upper line. Just like the bareloop χS is a sum of terms with first-order divergences, a diagram witha self-energy insertion in one of the lines is a sum of terms with eithersecond-order divergences or two first-order divergences. Second order-divergences occur if the states before and after the interaction with ΣS arethe same. For ω close to ωij the leading order in the Laurant expansionin ω of the first diagram from Eq. (3.20) stems from the term with theresonant particle–hole pair ij both in the beginning and in the end of theloop. In the notation of this work this can be expressed as

ΣS=

χ−1S |ij〈i|ΣS(ε j + ω)|i〉ij|χ−1

S(ω −ωij)2 +O

(1

ω −ωij

). (4.13)

The resonant state |ij occurs at both ends, while the wiggled lines are

87


directly translated into χ−1S . The matrix element of ΣS is defined as

〈k|ΣS(ω)|l〉 =∫

d3r∫

d3r′ ϕ∗k (r)ΣS(ω, r, r′)ϕl(r′). (4.14)

Evaluating on the KS particle–hole mass shell means that ΣS(εi) is usedin Eq. (4.13), which is equivalent to setting ω = ωij in ΣS.

Evaluation on the KS particle–hole mass shell has a similar meaningfor the irreducible interaction I, which appears in the third diagram inEq. (3.20). Note that the four lines attached to I in general carry fourdifferent frequencies. Since the transfered frequency is conserved, onlythree of the are independent. Here, the transferred energy and the ener-gies of the left and right lower line are chosen as independent variables.Similar to the case with one self-energy insertion discussed above, a di-agram with one vertex shows second-order divergences. For ω close toωij the leading order in the Laurant expansion in ω is again due to theterm with the resonant particle–hole pair ij at both ends of the loop. Inthe notation of this work this can be expressed as

I =χ−1

S |ij〈ij|I(ε j, ε j, ω)|ij〉ij|χ−1S

(ω −ωij)2 +O(

1ω −ωij

),

(4.15)This is very similar to Eq. (4.13). The main difference is that in the matrixelement of I, which is defined as

〈kl|I(ε, ε′, ω)|mn〉 =∫

d3r1

∫d3r′1

∫d3r2

∫d3r′2

× ϕ∗k (r1)ϕ∗n(r2)I(ε, ε′, ω; r1, r′1; r2, r′2)ϕm(r′1)ϕl(r′2), (4.16)

both states of the particle–hole pairs enter on both sides. Evaluating onthe KS particle–hole mass shell means that ω = ωij is set in the matrixelement of I. For the frequency derivatives of the irreducible elements,the derivative is evaluated first and then ω = ωij is set.

Using the matrix elements of ΣS (4.14) and I (4.16) we can express thematrix element of Ω as defined in Eq. (4.12) as

〈kl|Ω|mn〉 = 〈k|ΣS|m〉δln − 〈n|ΣS|l〉δkm + 〈kl|I|mn〉. (4.17)

88


Several terms in Eq. (4.24) contain matrix elements ij|χ−1r |ij that ap-

peared in section 3.2 in the context of the cancellation of divergences. Asnoted at the end of that section, such matrix elements must not occur inany physical quantity. Thus we have to find a way how to cancel theseterms with other contributions to the second-order energy shift (4.10b).For the terms of Eq. (4.24) this can be straightforwardly done by rear-ranging the second contribution to V(2) in Eq. (4.11b):

Ω Ω = Ω Ω

=i

jΩ Ω

+ Ω Ωr ,

(4.25)

where the labels i and j at the polarization loop indicate the resonantpart.

Comparing Eq. (4.25) with Eq. (4.23) we see that one part of V(2) cancelsthe last term in Eq. (4.10b) completely. Hence the second-order energyshift (4.10b) is determined by a modified first term and the second term.

Let us consider the second term in Eq. (4.10b) next. The differentiationwith respect to frequency can be obtained from Eq. (4.18) by taking intoaccount the frequency dependence via Eq. (4.19). Alternatively one coulduse Eq. (4.20), since differentiation and the integration needed for thematrix element commute in this case. We thereby obtain

ij|∂V(1)

∂ω(ωij)|ij = 〈ij|∂Ω

∂ω|ij〉 − 2

〈ij|Ω|ij〉ij|χ−1

r (ωij)|ij

+ ∑kl 6=ij


r (ωij)|ij+ ij|χ−1r (ωij)|kl〈kl|Ω|ij〉

ij|χ−1r (ωij)|ij(ω −ωkl)

, (4.26)

which has to be multiplied by ∆ω(1)ij [Eq. (4.21)] to obtain the second

term in Eq. (4.10b). In Eq. (4.26) the first term takes care of the frequencydependence of the irreducible elements. One expects such a term in the

91


second and the third term in Eq. (4.26) to ∆ω(2)ij . Finally, the second-order

energy shift is given by

∆ω(2)ij = 〈ij|Ω|ij〉〈ij|∂Ω

∂ω|ij〉+ ∑

kl 6=ijflk〈ij|Ω|kl〉〈kl|Ω|ij〉

ωij −ωkl. (4.30)

It is interesting to note that the first-order [Eq. (4.21)] and second-orderenergy shift (4.30) are exactly what one would expect from a frequency-dependent perturbation Ω. This can be interpreted in the light of theresults of section 3.2, where we found that for ω close to KS excitationenergies the polarization operator χ can be written as a chain structurewith the four-point function Ω.

The excitation energies can also be obtained from direct investigationof χ by looking at its poles. If the energy shift is small compared tothe level spacing in the system, i.e., the exact transition energy and theKS transition energy are close to each other compared with other excita-tion energies of the system, we can approximate χ in the region of theseenergies by a single pole

χ(ω) ≈ |ijij|ω −ωij −∆(ω)

. (4.31)

If ∆(ω) is small compared with ωij, we can approximate further

χ(ω) ≈ |ijij|ω −ωij

(1 +

∆(ω)ω −ωij

+(

∆(ω)ω −ωij

)2)

, (4.32)

i.e., χ(ω) is a sum of terms that diverge at ω = ωij with increasing orderof divergence. This is already similar to the results of section 3.2, wherewe were able to express Πxc(ω) close to ωij as a sum of chain structures.All these chain structures diverged at ω = ωij with increasing orderof divergence. To compare this with the above results obtained fromTDDFT, we have to expand the exact energy shift ∆(ω) to second orderin the irreducible elements as ∆(ω) = ∆(1)(ω) + ∆(2)(ω). In addition, we

93


have to expand ∆(ω) with respect to ω to second order around ω = ωij

∆(ω) ≈ ∆(1)(ωij) + ∆(1)(ωij)∂∆(1)

∂ω(ωij) + ∆(2)(ωij). (4.33)

Inserting Eq. (4.33) into Eq. (4.31) we obtain

χ(ω) ≈ |ijij|ω −ωij

(1 +

∆(1)(ωij)ω −ωij

+∆(1)(ωij)ω −ωij

∂∆(1)

∂ω(ωij)

+∆(2)(ωij)ω −ωij

+(

∆(1)(ωij)ω −ωij

)2)

. (4.34)

We see in Eq. (4.34) that close to ωij the response function should con-tain an unperturbed part which has a first order divergence at ω = ωij.This is, of course, just χS close to ωij. There is one term with a secondorder divergence that contains the correct first-order energy shift (4.21).This term can be identified with a chain with only one interaction Ω. Thenext two terms contain the correct second-order energy shift (4.30). Thefirst of these describes the energy dependence of the first order correction.The second is generated by those chains with two interactions Ω, wherethe internal lines are nonresonant, just as in Eq. (4.30). Finally, there isa term with a third order divergence. This term can be identified with achain of two interactions Ω where all lines are resonant. This term makesthe divergent contribution to V, which had to be canceled above. Herewe can dismiss it, as the square of the first-order energy shift is not avalid contribution to the second-order energy shift.

As expected, the TDDFT result for first- and second-order energy shift[Eqs. (4.21) and (4.30)] are consistent with what one obtains from directinvestigation of the response function. We can take this connection onestep further by expressing these energy shifts directly in terms of the ir-reducible elements ΣS and I, which can then be linked directly to thediagrams in the perturbative expansion of χ. For ∆ω

(1)ij this is given by

Eq. (4.21). The three terms in Eq. (4.21) reflect the three first-order dia-grams for χ in a perturbation theory with respect to irreducible elements.

94


These diagrams where used in Eq. (3.20) on page 64. Inserting Eq. (4.17)into Eq. (4.30) we obtain for the second-order shift

∆ω(2)ij = ∆ω

(1)ij

(〈i|∂ΣS

∂ω(εi)|i〉 − 〈j|∂ΣS

∂ω(ε j)|j〉+ 〈ij| ∂I

∂ω(ε j, ε j, ωij)|ij〉

)+ ∑

kl 6=ij

flk δl j

ωij −ωkl

(〈i|ΣS(εi)|k〉〈k|ΣS(εi)|i〉

+ 〈i|ΣS(εi)|k〉〈kl|I(ε l , ε j, ωij)|ij〉

+ 〈ij|I(ε j, ε l , ωij)|kl〉〈k|ΣS(εi)|i〉)

+ ∑kl 6=ij

flk δikωij −ωkl

(〈j|ΣS(ε j)|l〉〈l|ΣS(ε j)|j〉

− 〈j|ΣS(ε j)|l〉〈kl|I(ε l , ε j, ωij)|ij〉

− 〈ij|I(ε j, ε l , ωij)|kl〉〈l|ΣS(ε j)|j〉)

+ ∑kl 6=ij

flkωij −ωkl

〈ij|I(ε j, ε l , ωij)|kl〉〈kl|I(ε l , ε j, ωij)|ij〉. (4.35)

These terms can be related to the different diagrams in the second-orderapproximation to χ, as used in Eq. (3.21) on page 64. The diagram withtwo self-energy insertions in the upper line is responsible for the secondline of Eq. (4.35). Similarly, the diagram with two self-energy insertionsin the lower line is responsible for the fifth line of Eq. (4.35). The last lineof Eq. (4.35) is linked to the diagram with two irreducible interactions,whereas the four possible combinations of ΣS and I are the reason forthe other four lines in the summations of Eq. (4.35). This accounts forseven of the nine possible combinations of ΣS and I in a second orderchain. The two possibilities for a self-energy insertion in both lines – rep-resented by a single diagram in the conventional diagrammatic techniqueof Eq. (3.21) – do not contribute to Eq. (4.35). It is not possible to have anonresonant contribution from such a diagram, if the lines in the begin-ning and the end are resonant. It therefore contributes only to the lastterm in Eq. (4.34) when both external pairs are resonant.

95


4.2.1 Cancellations in the perturbation expansion

We have seen in this section that several terms in the second-order expan-sion cancel each other. These cancellations are not a peculiarity of thesecond-order perturbation theory. We will show now that some of thesecancellations can be proved for every order in the perturbation expansion,similar to the cancellation of divergences investigated in section 3.2.

The eigenvalue equation (4.7) can be also considered using Brillouin-Wigner perturbation theory. Within this theory, the energy shift betweenthe unperturbed state |ij and the exact state |i j derived from it is givenby

ω −ωij = f jiij|V(ω)|i j. (4.36)

We are interested here in the case where j stands for an occupied KS state,while i stands for an unoccupied one, i.e., f ji = 1. The exact state can beexpanded as

|i j = |ij+ ∑kl 6=ij

|kl flkkl|ω −ωij

V(ω)|i j

= |ij+ ∑kl 6=ij


V(ω)|ij

+ ∑kl 6=ij

mn 6=ij


V(ω)|mn fnmmn|

ω −ωijV(ω)|ij+ . . . .

(4.37)

The main difference to the more familiar Rayleigh-Schrödinger perturba-tion expansion is that in Eq. (4.37) the exact excitation energy ω enterseverywhere on the right hand side. Similar to Eq. (4.22), χr(ω) as definedin Eq. (3.34) can be identified in Eq. (4.37), which allows for a formal sum-mation of the complete perturbation expansion:

|i j = |ij+ χr(ω) · V(ω)|ij+ χr(ω) · V(ω) · χr(ω) · V(ω)|ij+ . . .

=(1− χr(ω) · V(ω)

)−1|ij.(4.38)

96


Inserting Eq. (4.38) into Eq. (4.36) we obtain

ω −ωij = ij|V(ω) ·(1− χr(ω) · V(ω)

)−1|ij. (4.39)

Instead of calculating the matrix element of the perturbation between theexact and the unperturbed state, we now have to calculate the matrixelement of a new perturbation with respect to the unperturbed state only.However, the new perturbation

R(ω) := V(ω) ·(1− χr(ω) · V(ω)

)−1 (4.40)

is unusual in that it depends on the investigated transition since it con-tains χr(ω). On the other hand, the advantage of this new perturbationis that it already incorporates some effects of the perturbation expansion.Calculating the energy shift from Eq. (4.39) to second order gives

ω = ωij + ij|R(ωij)|ij+ ij|R(ωij)|ijij| ∂R∂ω

(ωij)|ij, (4.41)

which is simpler than Eq. (4.9).What does the perturbation R(ω) look like? This is most easily an-

swered by transforming Eq. (4.40) into an equation for V(ω):

V(ω) = R(ω)− R(ω) · χr(ω) · V(ω). (4.42)

This looks remarkably similar to Eq. (3.11) on page 56 if it is formulatedfor V instead of fxc. With χxc = χ− χS we can express V as

V = χ−1S − χ−1 = χ−1

S − (χS + χxc)−1. (4.43)

An expansion in terms of χxc then gives

V = χ−1S · χxc · χ−1

S − χ−1S · χxc · V

= χ−1S · χxc · χ−1

S − χ−1S · χxc · χ−1

S · χS · V

= χ−1S · χxc · χ−1

S − χ−1S · χxc · χ−1

S ·(

χr +|ijij|ω −ωij

)· V.

(4.44)

97


The relation between Eqs. (4.42) and (4.44) is similar to the so called “ver-tex renormalization” (e.g., Abrikosov et al., 1963, section 18), and we cancombine them as

R = χ−1S · χxc · χ−1

S − χ−1S · χxc · χ−1

S|ijij|ω −ωij

R. (4.45)

Note that the modified V(2) depicted in Eq. (4.27) is just R(2), i.e.,

R(2) = χ−1S · χ

(2)xc · χ−1

S − χ−1S · χ

(1)xc · χ−1

S|ijij|ω −ωij

χ−1S · χ

(1)xc · χ−1

S , (4.46)

since χ(2)xc is a chain structure with two interactions with Ω.

It follows from the above discussion that the cancellation between thelast term in Eq. (4.10b) and parts of the first term in Eq. (4.10b) is notaccidental. The elimination of χr(ω) from the perturbation expansionoccurs in every order of the perturbation theory. It has not been possibleso far to prove the same for the other cancellations that we had foundexplicitly in the second-order calculation, though.

4.2.2 Need for consistent perturbation theory

The above calculation of pair-excitation energies shows that a delicatebalance of different terms in the perturbation expansion is important forgetting correct results. Otherwise, terms containing the matrix elementij|χ−1

r (ωij)|ij appear. As discussed at the end of section 3.2, this matrixelement must not appear in any physical quantity. This has importantimplications if approximate V or fxc are used.

In section 3.3, two such approximations [Eqs. (3.45) and (3.48)] werediscussed, which are both of the first order with respect to irreducible ele-ments. Therefore both of them do not generate the first term in Eq. (4.10b)which stems from the second-order part of the interaction V. Hence only

98


the second term and the third term are present. However, as was dis-cussed in the context of Eq. (4.24), the third term in Eq. (4.10b) is containsij|χ−1

r (ωij)|ij in the denominator. The same holds for parts of the sec-ond term. Thus using a first-order approximation for V to calculate thesecond-order energy correction according to Eq. (4.10b) gives a wrong re-sult. The reason for this is the incomplete cancellation of the two-particlepropagator found in section 3.2. If this cancellation were complete, fxc

or V would consist of two-particle irreducible elements, and the second-order energy shift would be given by Eq. (4.30) or (4.35).

The important consequence of these findings is that TDDFT calcula-tions of excitation energies must use a consistent perturbation theory. Forexample, a first-order approximation to fxc may only be used for cal-culating the energy shifts in the first-order perturbation theory (4.10a).Any higher order corrections or an exact solution of the eigenvalue prob-lem (4.7) generates wrong terms and hence uncontrollable errors. Thisis in sharp contrast to using an approximate self-energy Σ in the Dysonequation (2.19). In this case it is possible and customary to employ finite-order approximations for Σ to calculate the one-particle Green function.

Some researchers encountered this problem in their numerical calcula-tions. For example, Petersilka et al. (2000) calculated excitation energiesfor small atoms using a certain approximation to the OEP xc kernel.They found that the first-order correction gave better results than the ex-act solution of the eigenvalue equation, in accordance with the discussionabove. Yet these calculations also contain some other approximations,and it is difficult to determine which one is responsible for this. It wouldbe desirable to investigate the effects of a nonconsistent perturbation the-ory as discussed here under more controlled conditions. Especially thedependence on the system size is interesting, since the matrix elementij|χ−1

r |ij should vanish for larger systems, as discussed at the end ofsection 3.2. As we have seen above in Eqs. (4.24) and (4.26), this termappears in the denominator of the unphysical terms in the perturbationexpansion.

99


4.3 Expansion in terms of the interaction

In this section we consider the corrections to the KS excitation energiesin a perturbative expansion in terms of the interaction. For this we goback to Eq. (4.10) and interpret the expansion of V as an expansion interms of the interaction. In this case, Eq. (4.10) is equivalent to a GLPTcalculation up to the second order. This is of particular interest, sinceAppel et al. (2003) also gave an expression for the energy shift in second-order GLPT starting from the eigenvalue equation for the square of theexcitation energy derived by Casida (1996). Using the notation from thiswork, their Eq. (8) reads

∆ω(2)ij = ij|V(2)(ωij)|ij+ 2 ∑ph

kl 6=ij

ωkl |ij|V(1)(ωij)|kl|2

ω2ij −ω2

kl

+ ij|V(1)(ωij)|ijij|∂V(1)

∂ω(ωij)|ij −

|ij|V(1)(ωij)|ij|2

2ωij, (4.47)

where ∑ph stands for a summation over particle–hole pairs. In Eqs. (4.9)and (4.10b) the summation also includes hole–particle, particle–particle,and hole–hole pairs. However, the latter two do not contribute, as thedifference of the Fermi occupation numbers flk = fl − fk equals zero.The first and third term in Eq. (4.47) can be readily identified with thefirst and second term in Eq. (4.10b). However, identifying the second andforth term in Eq. (4.47) with the third term in Eq. (4.10b) needs a closerinspection. The two terms in Eq. (4.47) can be rewritten as

∑ph

kl 6=ij

|ij|V(1)(ωij)|kl|2

ωij −ωkl

(1−

ωij −ωkl

ωij + ωkl

)−|ij|V(1)(ωij)|ij|2

2ωij=

∑ph

kl 6=ij

|ij|V(1)(ωij)|kl|2

ωij −ωkl− ∑ph

kl

|ij|V(1)(ωij)|kl|2

ωij + ωkl. (4.48)

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The third term in Eq. (4.10b) can be split explicitly into summations in-cluding only particle–hole or hole–particle pairs:

∑ph

kl 6=ij

|ij|V(1)(ωij)|kl|2

ωij −ωkl− ∑hp

kl 6=ij

|ij|V(1)(ωij)|kl|2

ωij −ωkl=

∑ph

kl 6=ij

|ij|V(1)(ωij)|kl|2

ωij −ωkl− ∑ph

lk

|ij|V(1)(ωij)|kl|2

ωij + ωlk, (4.49)

where it is used that a hole–particle pair kl is always different from theparticle–hole pair ij. In addition, note that ωkl = −ωlk and that a summa-tion over hole–particle pairs kl is the same as a summation over particle–hole pairs lk. The first terms in Eqs. (4.48) and (4.49) are the same.The second terms are very similar after interchanging the summationindices k and l. The only difference is the order of k and l in the matrixelement: |kl compared to |lk. According to Eq. (4.8) this differenceimplies that ϕk(r′)ϕ∗l (r′) is used in the matrix element of Eq. (4.48) butϕl(r′)ϕ∗k (r′) in Eq. (4.49). These expressions are, of course, the same forreal wave functions. It is always possible to choose the eigenfunctions tobe real, if the system obeys the time-reversal symmetry. This symmetrymay be broken either by an external magnetic field or by the spin–orbitinteraction. Neither Casida (1996) nor the above derivation takes theseinto account. Hence, for the systems under consideration here Eq. (4.10b)– published in Tokatly et al. (2002) – and Eq. (4.47) given by Appel et al.(2003) are indeed identical expressions for the second-order energy shiftin GLPT.

In principle, the next step would be to express the matrix elements ofV(1) and V(2) in Eq. (4.10) in terms of the interaction. However, we haveseen in section 4.2 that there are several cancellations in Eq. (4.10b). Tonot repeat the work done in section 4.2, it is better to start from Eqs. (4.21)and (4.30). These equations express the first- and second-order energycorrections in terms of matrix elements of the four-point function Ω. Ifwe expand Ω up to second-order in the interaction as Ω = Ω(1) + Ω(2)

101


and collect all first- and second-order terms we find

∆ω(1)ij = 〈ij|Ω(1)|ij〉 (4.50a)

∆ω(2)ij = 〈ij|Ω(2)|ij〉+ 〈ij|Ω(1)|ij〉〈ij|∂Ω(1)

∂ω|ij〉

+ ∑kl 6=ij

flk〈ij|Ω(1)|kl〉〈kl|Ω(1)|ij〉

ωij −ωkl. (4.50b)

Inserting

〈kl|Ω(1)|mn〉 = 〈k|Σ(1)S |m〉δln − 〈n|Σ(1)

S |l〉δkm + 〈kl|I(1)|mn〉 (4.51a)

〈kl|Ω(2)|mn〉 = 〈k|Σ(2)S |m〉δln − 〈n|Σ(2)

S |l〉δkm + 〈kl|I(2)|mn〉 (4.51b)

we can also obtain the first- and second-order energy shift in terms ofthe first- and second-order irreducible elements, similar to Eqs. (4.21)and (4.35). The evaluation of the matrix elements in Eq. (4.51) is donein the following two subsections

4.3.1 First-order correction

The KS self-energy to first order in the interaction can be identified in thefirst-order expansion of the Green function given in Eq. (2.62)

Σ(1)S = − . (4.52)

Since vxc is considered as a known one-particle potential (see section 2.3),v(1)

xc and v(2)xc are regarded as known.

The other contribution in the first order is the Fock exchange term,which can be expressed via the nonlocal operator vF:

〈k|vF|m〉 = −∑α

fαVαkmα. (4.53)

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Note that Greek indices here and in the following equations are used forinternal KS states and therefore require a summation. The interactionmatrix element Vijkl is defined as

Vijkl =∫

d3r∫

d3r′ ϕ∗i (r)ϕ∗j (r′)VC(|r − r′|)ϕk(r)ϕl(r′). (4.54)

Note that Vijkl = Vjilk holds, as VC depends on the relative distance |r− r′|only. The matrix element of Σ

(1)S is then given by

〈k|Σ(1)S |m〉 = 〈k|vF − v(1)

xc |m〉. (4.55)

The first-order term in the expansion of the irreducible interaction I isgiven in diagrammatic form by

I(1) = + . (4.56)

The matrix element of I(1) between a particle–hole pair kl and a particle–hole pair mn, i.e., k and m in the upper lines and l and n in the lowerlines, is given by

〈kl|I(1)|mn〉 = −Vknml + Vnkml . (4.57)

One can see clearly that one of these terms is the “exchange term” ofthe other, which also explains the sign difference. This holds in bothdirections, though, and the nomenclature depends on the point of view(Hanke and Sham, 1980). If one wants to describe excitons, the first termdescribing the particle–hole interaction is “direct” whereas the secondterm is “exchanged”. For describing the screening in a material, it is theother way round. Another way to understand the sign difference comesfrom the diagrammatic formalism. If we attach two fermionic lines toboth sides of I(1) and close them, the diagram with the particle–holeinteraction will have one fermion loop, whereas the screening diagramwill have two fermion loops. Since each fermion loop gives a factor −1,this has to be incorporated into the matrix element (4.57).

103


Inserting the matrix elements of Σ(1)S [Eq. (4.55)] and I(1) [Eq. (4.57)]

into Eq. (4.51a) we find

〈kl|Ω(1)|mn〉 = 〈k|vF − v(1)xc |m〉δln − 〈n|vF − v(1)

xc |l〉δkm −Vknml + Vnkml(4.58)

for the matrix element of Ω, and

∆ω(1)ij = 〈i|vF − v(1)

xc |i〉 − 〈j|vF − v(1)xc |j〉 −Vijij + Vjiij (4.59)

for the first-order energy shift (4.50a). Equation (4.59) is, of course, notnew. It has been found earlier by several authors (Görling, 1996a; Filippiet al., 1997; Gonze and Scheffler, 1999; Tokatly and Pankratov, 2001).

4.3.2 Second-order correction

In contrast to the first-order result (4.59), the second-order correction ofEq. (4.50b) has not been obtained before. According to Eq. (4.50b) weneed for this the second-order irreducible elements, the frequency deriva-tive of the first order, and matrix elements of the first-order irreducibleelements between resonant and nonresonant states. The latter can beobtained directly from Eq. (4.58). Since the first-order irreducible ele-ments are frequency independent, the second term in Eq. (4.50b) vanishes.Hence, the only hurdle is the second-order of the irreducible elements.

The second-order of the KS self energy can be readily identified fromthe second-order expansion of the Green function (2.63) on page 32:

Σ(2)S

= + + Σ(1)S

− v(2)xc . (4.60)

The first two of these terms are related as exchange terms and can begrouped together. These two terms are also energy dependent. Using the

104


Matsubara formalism, we obtain at imaginary frequency iω:

〈k|Σ(2)S (iω)|m〉 = ∑

αβγ

fβα fβγ

εγ − iω + ωαβVγαmβ(−Vβkαγ + Vβkγα)

+ ∑αβ

fβα

ωαβ〈β|vF − v(1)

xc |α〉Vαkmβ − 〈k|v(2)xc |m〉,

(4.61)

where the terms are arranged in the same order as in Eq. (4.60). The resultfor real frequencies is obtained by analytic continuation. The diagramsin Eq. (4.60) show very nicely the process involved in the formation ofthe self energy: screening of the interaction, introduction of vertex correc-tions and self-consistency for the Green function.

The usual rules can be used to draw the second-order diagrams of theirreducible interaction in the particle–hole channel:

I(2) = + + + + + . (4.62)

Here, the first four as well as the last two diagrams are related as ex-change terms and can be grouped together. All these terms are energydependent. As usual with four-point functions, one has to make a choicewhich energies are treated as independent. Here, the transferred en-ergy iω and the energies of the left and right lower line iε and iε′ arechosen. Note that the Matsubara formalism is used again, so that theresults for real frequencies are obtained by analytic continuation. Evalu-ating the frequency summations in Eq. (4.62) gives

〈kl|I(2)(iε, iε′, iω)|mn〉 =

= ∑αβ

fβα

i(ε− ε′)−ωαβ

(−VkαmβVβnαl + VkαβmVβnαl

+ VkαmβVβnlα −VkαβmVβnlα)

+ ∑αβ

fα + fβ − 1i(ω + ε + ε′)− εα − εβ

(VnkβαVβαlm −VknβαVβαlm

),

(4.63)

105


where the order of the terms is the same as in Eq. (4.62). Actually, the firstdiagram in Eq. (4.62) describes the first-order screening of the particle–hole interaction. The next two terms are vertex corrections.

If we combine the matrix elements of the second-order self-energy Σ(2)S

[Eq. (4.61)] and the second-order vertex I(2) [Eq. (4.63)], we can get thegeneral matrix element of Ω(2). Evaluating this matrix element on theKS particle–hole mass shell for the resonant particle–hole pair ij we ob-tain

〈ij|Ω(2)|ij〉 = 〈i|Σ(2)S (εi)|i〉 − 〈j|Σ(2)

S (ε j)|j〉+ 〈ij|I(2)(ε j, ε j, ωij)|ij〉

= ∑αβγ

fβα fβγ

(Vγαiβ(−Vβiαγ + Vβiγα)

ωγi + ωαβ−

Vγαjβ(−Vβjαγ + Vβjγα)ωγj + ωαβ

)

+ ∑αβ

fβα

ωαβ〈β|vF − v(1)

xc |α〉(Vαiiβ −Vαjjβ

)− 〈i|v(2)

xc |i〉+ 〈j|v(2)xc |j〉

−∑αβ

fβα

ωαβ

(−ViαiβVβjαj + ViαβiVβjαj + ViαiβVβjjα −ViαβiVβjjα

)+ ∑

αβ

fα + fβ − 1ωiα + ωjβ

(VjiβαVβαji −VijβαVβαji

),

(4.64)

where similar contributions from the two self-energy terms have beengrouped together. Since fαβ/ωαβ is never negative, one sees that thefirst-order screening of the interaction (first term in the second-to-lastline) reduces the particle–hole interaction as expected. The vertex cor-rections increase the interaction. The last line, which represents the lasttwo diagrams in Eq. (4.62), constitutes another interesting term. Thisterm diverges when the sum of the energies of the two intermediatestates εα + εβ is the same as the sum of the energies of the resonantparticle–hole pair εi + ε j. The trivial case, where αβ is resonant, is notallowed due to the Fermi occupation numbers in the numerator. Such anaccidental degeneracy is unlikely in finite systems, but can be importantin extended systems.

106


Equation (4.64) represents the first term for the second-order energyshift (4.50b). The second term from Eq. (4.50b) is zero, as noted above.The last term can be directly derived by using the matrix elements of Ω(1)

from Eq. (4.58):

∑αβ 6=ij

fβα〈ij|Ω(1)|αβ〉〈αβ|Ω(1)|ij〉

ωij −ωαβ=

= ∑α 6=i

1− fα

ωiα

(〈i|vF − v(1)

xc |α〉〈α|vF − v(1)xc |i〉

− 〈i|vF − v(1)xc |α〉(Vαjij −Vjαij)− (Vijαj −Vjiαj)〈α|vF − v(1)

xc |i〉)

+ ∑β 6=j

fβ

ωβj

(〈j|vF − v(1)

xc |β〉〈β|vF − v(1)xc |j〉

+ 〈j|vF − v(1)xc |β〉(Vijiβ −Vjiiβ) + (Viβij −Vβiij)〈β|vF − v(1)

xc |j〉)

+ ∑αβ 6=ij

fβα

ωij −ωαβ(Viβαj −Vβiαj)(Vαjiβ −Vjαiβ).

(4.65)

This completes our derivation of the second-order correction to the KS ex-citation energy ωij, which is given by the sum of Eqs. (4.64) and (4.65).Note that in Eq. (4.65) only unoccupied states contribute for diagramswith a self-energy insertion in the upper line. For a self-energy insertionin the lower line, only occupied states are important.

Using the matrix elements of Σ(1)S [Eq. (4.55)], I(1) [Eq. (4.57)], Σ

(2)S

[Eq. (4.61)], and I(2) [Eq. (4.63)], one can explicitly give an analytic ex-pression for f (2)

xc .

107

5 Excitons in extended systems

In this chapter we want to investigate excitonic effects in TDDFT in thelinear response regime. As we have seen in section 2.4, this is one ofthe most interesting possibilities, where TDDFT can be applied to the abinitio calculation of properties of solids. After a brief recollection of theexcitonic effects within the framework of a standard MBT, we will focuson the excitonic part of the xc kernel fxc, for which an exact equation canbe derived. After that, a model system which allows analytic solutionswill be used to test approximations for the excitonic fxc. To some extent,the results of this chapter have been published in Stubner, Tokatly, andPankratov (2004).

As discussed in section 2.1.1, excitons are a consequence of the resid-ual attractive interaction between quasiparticles and quasiholes. In semi-conductors and insulators bound excitons are possible, though unboundscattering states of particle–hole pairs are also strongly altered by theinteraction. To calculate these states, one has to consider the correlatedpropagation of a particle and a hole. This propagation is described by theparticle–hole propagator. In the absence of an interaction between parti-cle and hole, the particle–hole propagator is simply the product of a freeparticle propagator and a free hole propagator. Obviously, the excitoniceffects cannot be found in this zero-order part. One has to take into ac-count the interaction between the particle and the hole. The particle–holeirreducible interaction is given by an infinite series of diagrams:

= + + + + + + . . . (5.1)

Here the solid lines are the QP Green functions and the dashed lines rep-resent the screened interaction. This is different from previous chapters,

109


where they represent the KS Green function and the bare interaction.This change of notation will help to highlight the similarity to the dia-grammatic technique used in previous chapters, which justifies the slightchance for confusion.

Bound states, such as excitons, are not contained in the irreducibleinteraction, as it only a single scattering event. Particles staying in closecontact for a long time experience many scattering processes. One cansum over the multiple scattering processes by calculating the scatteringmatrix T, which solves the integral equation

T = + T . (5.2)

If one solves Eq. (5.2) by iteration, one can see that T describes repeatedscattering of the particle–hole pair. Scattering states needed to describethe unbound exciton effects mentioned in section 2.4 are contained inthe T-matrix, too. The integral equation (5.2) is called Bethe-Salpeterequation (BSE) (Salpeter and Bethe, 1951). Note that instead of formulat-ing the BSE as an integral equation for T, one can equivalently use anintegral equation for the particle–hole propagator. Sometimes it is thisequation that is referred to as BSE.

In the context of a connection to TDDFT we are not interested inthe four-point particle–hole propagator but rather in response functions.Therefore we consider a modified BSE with two of the four external linesof T being contracted to form the three-point function Γ. Hence Eq. (5.2)becomes an integral equation for Γ

Γ = + Γ . (5.3)

The function Γ is related to the proper polarization operator χ as

χ = χQP + ΠEx = + Γ . (5.4)

110

5.1 Excitonic effects in the exchange-correlation kernel

Here we introduce the QP response function χQP and the excitonic con-tribution to the response function ΠEx. For the present purposes Eq. (5.3)is equivalent to Eq. (5.2) and will be referred to as BSE in the following.


In TDDFT the proper polarization operator χ is given by [c.f. Eq. (3.9b)]

χ(ω) = χS(ω) + χS(ω) · fxc(ω) · χ(ω), (5.5)

where χS represents the density–density response function of the non-interacting KS particles, i.e., a bare loop of two KS Green functions. Al-though Eq. (5.5) looks like a common RPA equation, χ is the exact properpolarization operator and thus formally includes all self-energy and ver-tex diagrams. Hence fxc contains both quasiparticle and correlation (e.g.,excitonic) effects. As in this chapter we are interested in excitonic corre-lations, it is tempting to separate the quasiparticle and excitonic contri-butions to fxc, as alluded by Reining et al. (2002) and Sottile et al. (2003).This separation is indeed possible without approximations, if we write

fxc = χ−1S − χ−1 = χ−1

S − χ−1QP︸︷︷︸

=: f QPxc

+ χ−1QP − χ−1︸︷︷︸=: f Ex

xc

, (5.6)

where χQP is the density–density response function for noninteractingquasiparticles. By definition, f QP

xc and f Exxc are the kernels in the following

RPA-type equations:

χQP(ω) = χS(ω) + χS(ω) · f QPxc (ω) · χQP(ω) (5.7a)

χ(ω) = χQP(ω) + χQP(ω) · f Exxc (ω) · χ(ω). (5.7b)

The newly introduced quantities f QPxc and f Ex

xc indeed describe quasiparti-cle and excitonic effects, respectively. This can be visualized by applyingthe diagrammatic rules for fxc as derived in section 3.1.

111


Writing χQP = χS + ΠQP we can express f QPxc as

f QPxc = χ−1

S ·ΠQP · χ−1S − χ−1

S ·ΠQP · χ−1S ·ΠQP · χ−1

S . . . . (5.8)

This looks the same as the fxc expansion in Eq. (3.10) on page 56, exceptthat ΠQP contains only self-energy insertions. Thus in the diagrammaticexpansion for f QP

xc no vertex insertions are allowed. This clearly describesthe quasiparticle corrections. All the former results for fxc, includingthe rules for constructing finite-order approximations and the possibil-ity for partial summation, stay the same. In addition, according to thearguments of section 3.2, f QP

xc acts as a mass operator for χQP and is fi-nite at KS excitation energies in every order of the perturbation theory.Commonly, the QP corrections to the KS levels have been successfullydescribed using the GW method (Aulbur et al., 2000). It remains unclearwhether expressing them in terms of f QP

xc would bring any additionalinsight.

In this chapter we focus on f Exxc , not on f QP

xc . The diagrammatic rep-resentation of f Ex

xc can be derived similar to Eq. (5.8). Separating χ intoquasiparticle and excitonic parts χ = χQP + ΠEx we find for f Ex

xc :

f Exxc = χ−1

QP ·ΠEx · χ−1QP − χ−1

QP ·ΠEx · χ−1QP ·ΠEx · χ−1

QP . . . . (5.9)

This looks again almost the same as the fxc expansion in Eq. (3.10) onpage 56. This time, however, we have χ−1

QP instead of χ−1S . In addition,

ΠEx contains only vertex insertions but with QP Green functions as basicpropagators instead of the KS Green functions in Eq. (3.10). This canbe easily accounted for in the diagrammatic technique by letting solidand wiggly lines represent the QP Green function and the inverse QP re-sponse function χ−1

QP, respectively. This convention is used throughoutthe remainder of this chapter. Obviously, f Ex

xc describes excitonic correla-tions. Similar to f QP

xc , all previous general results for fxc, including therules for constructing finite-order approximations and the possibility forpartial summation, hold for f Ex

xc , too. Following the arguments of sec-tion 3.2, f Ex

xc acts as a mass operator for χ and is finite at the excitation

112


energies of the noninteracting QP system in every order of the perturba-tion theory. Note that f Ex

xc is the same as the f FQPxc of Del Sole et al. (2003)

and Adragna et al. (2003).We can now use the diagrammatic rules from section 3.1 with the

above-mentioned modifications to construct the n-th order correction tof Exxc . We have to draw all loops with n particle–hole interactions to con-

struct the n-th order correction first. These diagrams serve as parentgraphs for the construction of the n-th order f Ex

xc . Yet it is much simplerto use a perturbation theory in terms of the irreducible interaction here,as there is only one diagram with n irreducible interactions. This diagramwith its chain structure is the parent graph for the n-th order f Ex

xc with re-spect to the irreducible interaction. To the ends of the diagram we haveto attach wiggly lines representing χ−1

QP. Next, we work out all possibili-ties to separate this parent graph into two by cutting two fermionic lines.Then we join the external fermionic lines of these parts, connect them bya wiggly line and change the sign of the resulting diagram. Obviously,the only way to separate the parent graph is to cut between adjacent irre-ducible interactions. The cutting does not alter the chain structure of thediagrams and we arrive at

f Exxc

(n)=

(−

). . . .

(5.10)The summation of all orders of diagrams of this type can be cast in anintegral equation. We thereby obtain for f Ex

xc

f Exxc = Λ (5.11)

with

Λ = + Λ − Λ . (5.12)

113


If one inserts f Exxc obtained from Eq. (5.11) after solving Eq. (5.12) into

the equation for the response function χ (5.7b), the result will be the sameas the χ obtained from the BSE (5.3). Equations (5.11) and (5.12) give inthis sense the exact “translation” of the BSE into the TDDFT language.

The ladder approximationNote that generally the irreducible interaction is not known. An approx-imation for it is needed. We are interested here in systems with a bandgap with a low density of particles and holes. In such systems, the scat-tering matrix T is dominated by the so called “ladder diagrams” (Fetterand Walecka, 1971; Sham and Rice, 1966). We can restrict T to these “lad-der diagrams” when we approximate the irreducible interaction by thescreened interaction, i.e., only the first term of the infinite series (5.1). Thediagrammatic form of the integral equation for T (5.2) then becomes

T = + T . (5.13)

Similarly, if the ladder approximation is used in the integral equation forΓ, the exact equation (5.3) reduces to

Γ = + Γ . (5.14)

Sometimes the ladder approximation is referred to as the BSE instead ofthe general equations for T (5.2) or Γ (5.3). The ladder approximationwill be used in the calculations in this chapter.

Our goal is to compare BSE and TDDFT. As the latter requires somesort of f Ex

xc [Eq. (5.7b)], the problem is to find the f Exxc that exactly cor-

responds to the ladder approximation in the BSE. If we replace in theintegral equation for Λ (5.12) the irreducible interaction by the screenedinteraction, i.e., use the ladder approximation, it reduces to

Λ = + Λ − Λ . (5.15)

114


Similarly to Eqs. (5.11) and (5.12) Λ obtained from Eq. (5.15) determinesf Exxc according to Eq. (5.11) such that χ (5.7b) will be identical to the χ

obtained from the BSE in the ladder approximation (5.14).

Unfortunately, solving Eq. (5.15) is at least as difficult as solving theBSE (5.14). However, one can hope that the two-point kernel f Ex

xc is moresuitable for approximations. An indication in this direction can be seendirectly from the diagrams of Eq. (5.15). Comparing this equation toEq. (5.14) we see that it can be obtained from the BSE by the followingsubstitution of the particle–hole propagator

→ − . (5.16)

The same replacement was used in section 3.2 to prove the cancellationof divergences in fxc at KS excitation energies. Similarly, it facilitates thecancellation of divergences in f Ex

xc at QP excitation energies. There is oneimportant difference, though. In section 3.2 we were concerned with pos-sible divergences at KS excitation energies. These divergences would becompletely unphysical, hence it is very important that they are removed.The situation is different here. The divergence at QP excitation energiesmust be present in any finite-order approximation to the response func-tion or to the vertex Γ. This makes it necessary to sum the diagrams up toan infinite order. In contrast, in the equation for Λ (5.15) the divergencesat QP excitation energies in the second and third term on the right handside should cancel each other. Then the different finite-order approxima-tions would not add new divergences at QP excitation energies to theone already present in the first term on the right hand side of Eq.(5.15),which is the first-order approximation to Λ. This gives the hope that atleast close to QP excitation energies much of the physics of the vertex Λ

is already contained in the first-order approximation.

Alternatively, we can interpret the difference between the BSE and theequation for Λ as a modification of the interaction process in the second

115


diagram of the right-hand side of Eq. (5.14):

→ − . (5.17)

As f Exxc is responsible for translating the ladder diagrams into the an-

nihilation channel, f Exxc is nontrivial only if the ladder channel and the

annihilation channel are distinguishable. This is easily seen from the re-placement depicted in Eq. (5.17). Indeed, the ladder and the annihilationchannel coincide for a static point interaction in a system with one energyband, e.g., a simple metal. The many-band case is still nontrivial (see sec-tion 5.3 below). For a point interaction the upper and lower quasiparticlepropagators in Eq. (5.17) should be contracted at the interaction point toform a polarization loop which cancels the wiggly line. As a result thetwo diagrams of Eq. (5.17) cancel exactly. In Eq. (5.15) this means that thelast two terms cancel and Λ reduces to the first term, which is a productof the bare interaction and χS. The excitonic part of the xc kernel thenreduces to the bare interaction. One can therefore expect that in systemswith a short-ranged and almost static effective interaction, the two termsof Eq. (5.17) will cancel to a large extent, and a low-order approximationto f Ex

xc will be sufficient. Conditions like this can be found, e.g., in simplemetals. On the contrary, in semiconductors the screening is suppressedand the effective interaction is long-ranged. Further research is needed toverify to what extent the cancellation may be efficient systems in general.

The discussed cancellation effects been, in fact, observed in numericcalculations by Marini et al. (2003). The success of the lowest-order fxc

found by Marini et al. implies that de facto the cancellation can be efficientin materials with a band gap as well.

At this point, it is worth highlighting the differences between the equa-tion for the excitonic part of fxc derived by Marini et al. (2003) (Eqs. (4)and (5) therein) and the integral equation (5.12). The iterative equationof Marini et al. is for the two-point xc kernel and is based on finite-orderapproximations to the excitonic part of the response function and is thusanalogous to the diagrammatic fxc expansion of section 3.1 and Tokatly

116

5.2 Model system

et al. (2002). Using this scheme, one has to calculate different finite-orderapproximations to ΠEx and subtract them appropriately in order to calcu-late finite-order approximations to f Ex

xc . As noted above, these finite-orderapproximations to ΠEx show divergences that must cancel in finite-orderapproximations to f Ex

xc . One therefore has to calculate these divergingterms with very high precision in order to correctly describe their differ-ences.

In contrast to this, the cancellation of diverging terms occurs in thekernel of the integral equation (5.12). Having accounted for these diver-gences once, one can solve Eq. (5.12) to any desired accuracy withouthaving to deal with subtracting singular terms from each other.

Meanwhile Bruneval et al. (2005) have found an interesting alternativeexpression for f Ex

xc and Λ. Using a functional derivative technique founda relation which can be cast into the following equation

Λ =δΣ

δn. (5.18)

Combining this functional derivative technique with the diagrammatictechnique presented in this work should be fruitful for future investiga-tions of the structure of f Ex

xc and Λ.

5.2 Model system

In this section we consider a model system which reveals both boundand unbound excitonic effects and where both the BSE and the equationfor Λ can be solved analytically. With the exact f Ex

xc at hand we can verifyunder what circumstances the first-order approximation to f Ex

xc is suffi-cient. The approximate kernel must describe bound excitons as well asunbound excitonic effects. A simple system with a bound exciton is givenby the two-band Dirac model with a static density–density interaction.We consider the two dimensional version of the model to avoid technicaldifficulties with diverging integrals. It is know that the response functionof the 3d Dirac model exhibits a so called “ultra-violet catastrophe”, i.e.,

117


the momentum integration diverges for short wave lengths. There existseveral techniques in quantum field theory for renormalizing such diver-gences in response functions (e.g., Ryder, 1996). One simple possibility isto subtract the static part of the response function. However, here we arealso interested in f Ex

xc , which is more like an inverse response function. Itis not straightforward to apply these renormalization procedures to f Ex

xc .By restricting the calculation to the 2d case we avoid dealing with theseproblems, since there is no ultra-violet divergence in this case.

The model Hamiltonian is given by

H =∫

d2r ψ†(r)Hψ(r) +12

∫d2r∫

d2r′ n(r)V(r − r′)n(r′), (5.19)

where n(r) = ψ†(r)ψ(r) is the density operator, ψ(r) is a two-componentfield operator, and V(r − r′) describes the interaction between the parti-cles. The Hamiltonian H of the free particles reads

H = kxσx + kyσy + ∆σz =

(∆ k−k+ −∆

), (5.20)

where σx,y,z are Pauli matrices and k± = kx ± iky with the 2d momen-tum operator k = (kx, ky). The band gap is equal to 2∆. The sameHamiltonian appears in a spherically symmetric two-band k·p theory(Kane, 1966). The energy spectrum of the noninteracting particles hastwo branches which we label c and v for the unoccupied conductionband and the occupied valence band:

Ec/v(k) = ±Ek with Ek =√

∆2 + k2. (5.21)

This spectrum is presented in Fig. 5.1 in a schematic fashion. The eigen-vectors of H are

Ψck =

(uk

k+k vk

), Ψvk =

(− k−

k vkuk

)(5.22)

with

uk =

√12

(1 +

∆

Ek

), vk =

√12

(1− ∆

Ek

). (5.23)

118

5.2 Model system

Ec/v(k)

k

2∆ω

c

vFigure 5.1: Schematic drawing of the energy spec-

trum. The valence band is filled, while theconduction-band states are unoccupied.

We will solve this model in the ladder approximation, i.e., ignoring self-energy terms and higher-order corrections to the irreducible scatteringmatrix. This implies that H describes the free QPs and V is the screenedinteraction between them. Therefore the one-particle Green function cor-responding to H is regarded as the QP Green function:

GQP(ω, k) =ΨckΨ†

ckω − Ek + iη

+ΨvkΨ†

vkω + Ek − iη

. (5.24)

Note that Ψvk and Ψck are two-component vectors and GQP is a 2× 2 ma-trix. To lowest order in the wave vector q, the quasiparticle responsefunction χQP(ω, q) is given by

χQP(ω, q) = −i∫ dε

2π

∫ d2k(2π)2 tr GQP(ε + ω, k + q)GQP(ε, k)

= − q2

16π∆

(ω2 + 1

2ω3 ln1 + ω

1− ω− 1

ω2

),

(5.25)

where ω = ω/(2∆). Here we see explicitly the q2-dependence of theresponse functions mentioned in section 2.4. The real and the imaginarypart of this function are displayed in Fig. 5.2. A nonvanishing imaginarypart occurs only above the quasiparticle gap ω > 1, when the argumentof the logarithm becomes negative. This is expected, since a nonzeroimaginary part of the polarization operator indicates a finite absorption.The real part shows a logarithmic divergence at ω = 1, which is typicalfor a parabolic critical point in the joint density of states of a 2d system.

119


0 0.5 1 1.5 2−8

−6

−4

−2

0

ω

<χQP(ω)q2/16π∆

0 0.5 1 1.5 2

−3

−2

−1

0

ω

=χQP(ω)q2/16π∆

Figure 5.2: The real and the imaginary part of the exact χQP.

We are now in the position to solve the BSE in this model analytically.However, it is helpful to introduce some technical issues first. Due to thematrix structure of GQP one has to compute traces when calculating χQP.Similarly, the three-point functions Γ and Λ are 2× 2 matrices and traceshave to be calculated over internal indices in Eqs. (5.14) and (5.15). Sinceall these matrices have nonvanishing off-diagonal elements, evaluationof these traces becomes tedious. Therefore it is convenient to choose theeigenstates of H (5.20) as the basis functions. The Green function GQP

120

5.2 Model system

becomes then a diagonal matrix with the two elements

Gc/v(ω, k) =1

ω ∓ Ek ± iη. (5.26)

Now we can abandon the matrix notation altogether and use the twoscalar Green functions Gc and Gv instead of the diagonal 2 × 2 matrixGQP. Every full line is now either a conduction- or a valence-state prop-agator (5.26). This, of course, increases the number of diagrams we haveto draw, because we must consider all combinations of conduction- andvalence-band states. However, in all diagrams the “upper” and “lower”Green functions that constitute a particle–hole propagator must alwaysbe of a different type (c or v). The diagrams with c–c or v–v two-particlepropagators vanish due to the integration over frequency since the polesof both “upper” and “lower” Green function are in the same half of thecomplex plane. These diagrams thus do not contribute to the polariza-tion. This is expected, since in our model [Eqs. (5.19) and (5.20)] onlytransitions between different bands are allowed. The same holds for thethree-point functions Γ and Λ. Instead of one equation for the 2× 2 ma-trix Γ as depicted in Eq. (5.14) we have two coupled equations for thescalar functions Γcv and Γvc. For Γcv the upper line is a conduction-bandstate and the lower line a valence-band state, whereas for Γvc it is viceversa.

The diagonal representation of GQP is achieved by a unitary transfor-mation of the field operators in Eq. (5.19). This transformation, however,affects the interaction term in Eq. (5.19) and the interaction with an exter-nal field, which is need in a linear response calculation. It generates the“bare” vertex, describing interaction with an external field at the externalpoints of polarization diagrams, as well as the momentum-dependent “in-teraction vertices”. In the original representation (5.20) all these verticeswere unit matrices. They become nontrivial matrices in the diagonal rep-resentation. However, only three matrix elements of these vertices areessential. One is the “bare” vertex, which to the lowest order in the trans-

121


ferred wave vector q computes to

γ(q, k) =

v k

c k + q

= Ψ†ck+qΨvk =

12Ek

(u2

kq− −(

vkk−k

)2q+

). (5.27)

Note that γ(q, k) is linear in q because of the opposite parity of c- and v-eigenfunctions at k = 0. Thus the linear q-dependence stems from the off-diagonal momentum operator in the Hamiltonian (5.20). This is actuallythe cause of the q2-dependence of the response function that was referredto in conjunction with Eq. (5.25) as well as at the end of section 2.4. Theother two matrix elements are interaction vertices. They can be called as“charges”, since in quantum electrodynamics one associates the chargecarried by a particle with the vertex. These vertices are

g1(k, k′) =c k c k′

= Ψ†ckΨck′ = ukuk′ +

k−k

vkvk′k′+k′

g2(k, k′) =c k v k′

= Ψ†ckΨvk′ =

k−k

vkuk′ − ukvk′k′−k′

.

(5.28)

Other possible combinations of valence- and conduction-band states dif-fer from Eqs. (5.27) and (5.28) only by sign changes or complex conjuga-tion. From now on we use the diagrammatic technique with two differenttypes of full lines representing conduction- and valence-band states. Withthese lines we associate the scalar Green functions of Eq. (5.26). Verticesare associated with the scalar functions of Eqs. (5.27) and (5.28).

There is an alternative interpretation of the basis transformation. Con-sider one of the traces that has to be calculated for χQP

tr(Ψck+qΨ†ck+q)(ΨvkΨ†

vk), (5.29)

where parentheses show the grouping of the matrix multiplication. Onethus has to calculate the outer products of two vectors, multiply the re-sulting matrices and in the end take the trace. However, this grouping

122

5.2 Model system

can be changed as follows

tr Ψck+q(Ψ†ck+qΨvk)Ψ†

vk = (Ψ†ck+qΨvk)(Ψ†

vkΨck+q). (5.30)

Now one computes inner products of two vectors and multiplies the re-sulting scalar functions. Taking the trace is accounted for automatically inEq. (5.30). The change from a matrix Green function to two scalar Greenfunctions outlined above is in effect a way to incorporate this regroupinginto the formalism.

When we work in the basis of the conduction- and valence-band states,the BSE (5.14) has to be split into two scalar equations for Γcv(ω, q, k) andΓvc(ω, q, k). Note that while Γcv and Γvc depend on two momenta, theydepend only on one frequency for a frequency-independent interaction.These two three-point functions are in fact not independent but relatedby the replacement q→ −q, ω → −ω, and complex conjugation. We cantherefore derive one equation for Γcv:

Γcv(ω, q, k) = Γ1(ω, q, k)

+ ∑k′

Vk,k′g1(k, k′)Γcv(ω, q, k′)g∗1(k′, k)

2Ek′ −ω

+ ∑k′

Vk,k′g2(k, k′)Γ∗cv(−ω,−q, k′)g2(k′, k)

2Ek′ + ω(5.31a)

Γ1(ω, q, k) = ∑k′

Vk,k′g1(k, k′)γ(q, k′)g∗1(k′, k)

2Ek′ −ω

+ ∑k′

Vk,k′g2(k, k′)γ∗(−q, k′)g2(k′, k)

2Ek′ + ω. (5.31b)

Note that we omit the q-dependence in the “charges” g1 and g2 as wellas in the energy denominators, as we are only interested in the lowest-order expansion in q, which stems from the external vertices. Here Vk,k′

is the matrix element of the interaction between the particles. Now weinvestigate this equation for two different types of interaction, a generalshort-range interaction and the long-ranged Coulomb interaction.

123


5.3 Short-range interaction

In this section we investigate the effects of a short-ranged interactionbetween the QPs, i.e., an interaction with a characteristic length scaleshorter than ∆−1. The most significant difference to the more familiarcase of a long-ranged Coulomb interaction is that there is only one boundexcitonic state instead of an infinite Rydberg series.

In the calculation we are going to split the integrands in Eq. (5.31)into a low- and a high-energy part. The latter will be used to definean effective interaction, which acts on the low-energy contributions. Thisinteraction can be characterized by the physical (renormalized) scatteringlength, in terms of which all final results will be expressed. In principle,one could do this renormalization of the interaction now. However, itis more transparent to postpone this renormalization and formally use amomentum independent bare interaction Vk,k′ = V in Eq. (5.31).

A momentum independent interaction is just a contact interaction inreal space. In the light of the discussion in the context of Eq. (5.17) onemight think that such a contact interaction makes f Ex

xc trivially equal tothe interaction itself. However, this is not that case in the two-component(“relativistic”) model. The interactions in the ladder channel and in theannihilation channel in such a system are always distinct, even for a con-tact interaction. In both the ladder and the annihilation channel the inter-action lines in the diagrams reduce to points, and the diagrams look thesame. However, in the ladder channel we have to associate appropriatecharges from Eq. (5.28) with these points, since the diagram still describesa single fermionic loop. We have to use the bare vertex from Eq. (5.27) in theannihilation channel, which describes a series of separate fermionic loops.If one uses matrix Green functions one has to calculate a single trace forthe single fermionic loop in the ladder channel. For the annihilation chan-nel separate traces for the separate fermionic loops have to be calculated.Hence, for a multi-band system like our model even a contact interactionproduces a nontrivial f Ex

xc .

124


5.3.1 Solution of the BSE

In order to solve the BSE, an ansatz for Γcv is used that separates the ω-,q-, and k-dependence. From Eq. (5.31) we see that q enters only throughthe bare vertex γ(q, k) [Eq. (5.27)]. Therefore, Γcv has to be the sum of twoterms that are proportional to q+ and q−, respectively. The prefactors forthese terms are functions of ω and k. In the case of the bare vertex γ(q, k)[Eq. (5.27)], these prefactors are a function of k only. Note that the firstprefactor depends only on the absolute value k (s-like), while the secondone has a d-like angular dependence.

Inspecting Eq. (5.31) again for the case of a momentum independent in-teraction we see that that the k-dependence of Γcv is given by the chargesfrom Eq. (5.28). The angular dependences in g1(k, k′) and g2(k, k′) are s-and p-like. The total angular k′-dependence in Eq. (5.31) has to be s-like,otherwise the integral over k′ vanishes. Carefully considering all possibletypes of coupling one finds that the k-dependence of the two terms of Γcv

have s- and d-like character in the same way as the bare vertex γ. Eventhe radial k-dependence of these two terms is almost the same as for γ.One therefore arrives at the following ansatz for Γcv:

Γcv(ω, q, k) = u2kq−Γ

(s)cv (ω) +

(vk

k−k

)2q+Γ

(d)cv (ω). (5.32)

Now the frequency dependence of the s- and d-like parts has to be deter-mined.

Inserting this ansatz into Eq. (5.31) we obtain two coupled equations:

Γ(s)cv (ω) = V ∑

k

12Ek

(u4

k(2Ek −ω)

−v4

k(2Ek + ω)

)

+ V ∑k

u4k

2Ek −ωΓ

(s)cv (ω) + V ∑

k

v4k

2Ek + ωΓ

(d)cv

∗(−ω) (5.33a)

125


Γ(d)cv (ω) = V ∑

k

12Ek

(u4

k(2Ek + ω)

−v4

k(2Ek −ω)

)

+ V ∑k

v4k

2Ek −ωΓ

(d)cv (ω) + V ∑

k

u4k

2Ek + ωΓ

(s)cv

∗(−ω). (5.33b)

From these equations it follows immediately that Γ(s)cv (ω) = Γ

(d)cv

∗(−ω)

and Eq. 5.33 reduces to

Γ(s)cv (ω) = V ∑

k

12Ek

(u4

k(2Ek −ω)

−v4

k(2Ek + ω)

)

+ V ∑k

(u4

k2Ek −ω

+v4

k2Ek + ω

)Γ

(s)cv (ω)

=: Vγ1(ω) + VK(ω)Γ(s)cv (ω), (5.34)

where γ1(ω) and K(ω) are defined by the second equation. Here K(ω)describes the frequency dependence of a particle–hole propagator, whileγ1(ω) describes a contracted one.

At this point it is convenient to perform the renormalization of theinteraction mentioned at the beginning of this section. This is done bysplitting K(ω) into a low- and a high-energy part. For the latter theintegrand in the expression for K(ω) is replaced by its limiting value fork → ∞, i.e., it is equal to ∑k 1/(4Ek). The low-energy part K(ω) is thengiven by

K(ω) = K(ω)−∑k

14Ek

= ∑k

(2∆ + ω)2

4Ek(4E2k −ω2)

. (5.35)

The high energy part logarithmically diverges at large k. This divergencecan be removed by the standard “vertex renormalization” (e.g., Abrikosovet al., 1963, section 18), leading to the renormalized interaction

V =V

1−V ∑k 1/(4Ek)=:

4π

∆a, (5.36)

126


where the dimensionless scattering length a is introduced. With thisrenormalized interaction, Γ

(s)cv (ω) fulfills the following equation

Γ(s)cv (ω) = Vγ1(ω) + VK(ω)Γ

(s)cv (ω). (5.37)

Equation (5.37) has the obvious solution

Γ(s)cv (ω) =

Vγ1(ω)1− VK(ω)

. (5.38)

Calculating the 2d integrals in Eq. (5.35) and γ1(ω) in Eq. (5.34) we obtain

K(ω) =∆

4π

(1 + ω)2

4ωln

1 + ω

1− ω=:

∆

4πF(ω) (5.39)

and

γ1(ω) =1

16π

((1 + ω)2

2ω2 ln1 + ω

1− ω− 1

ω

)=

116π

2ω

(F(ω)− 1

2

).

(5.40)

Inserting these integrals into the solution (5.37) we can compute theresponse function’s excitonic part

ΠEx = χ− χQP =

c

v

Γcv +

v

c

Γvc (5.41)

and its first-order approximation Π(1)Ex . The results are

ΠEx(ω, q) = − q2

16π∆

aω2

(

F(ω)− 12

)2

1− aF(ω)+

(F∗(−ω)− 1

2

)2

1− aF∗(−ω)

(5.42)

and

Π(1)Ex (ω, q) = − q2

16π∆

aω2

((F(ω)− 1

2

)2+(

F∗(−ω)− 12

)2)

, (5.43)

127


where the function F(ω) is defined in Eq. (5.39).The equation for Λ given in Eq. (5.15) can be solved in a similar fashion,

which allows then to calculate the excitonic part of the exact xc kernel.However, it is easier to obtain f Ex

xc directly from Eq. (5.6) and the exactresponse function χ = χQP + ΠEx with ΠEx from Eq. (5.42):

f Exxc (ω, q) =

χ−1QP(ω, q)ΠEx(ω, q)χ−1

QP(ω, q)

1 + χ−1QP(ω, q)ΠEx(ω, q)

. (5.44)

From here the first-order approximation to f Exxc follows immediately:

f Exxc

(1)(ω, q) = χ−1

QP(ω, q)Π(1)Ex (ω, q)χ−1

QP(ω, q). (5.45)

Inserting this f Exxc

(1) in the equation for χ (5.7b) we arrive at an approxi-mate solution for the response function

χ f (1)(ω, q) =

χQP(ω, q)

1−Π(1)Ex (ω, q)χ−1

QP(ω, q). (5.46)

The excitonic part of this approximate response function is then

Πf (1)

Ex (ω, q) =Π

(1)Ex (ω, q)

1−Π(1)Ex (ω, q)χ−1

QP(ω, q). (5.47)

Note that although f Exxc

(1) is based on Π(1)Ex , this formula does not coincide

with Π(1)Ex . In accordance with the equation for χ (5.7b) it accounts for an

infinite series of diagrams instead. This way, the excitonic pole in ΠEx

[Eq. (5.42)] which has been lost in Π(1)Ex [Eq. (5.43)] reappears in Eq. (5.47).

5.3.2 Results

Having calculated the exact and the approximate expressions for the ex-citonic part of the xc kernel and for the response function we are now inthe position to compare these results. Let us start with the excitonic peak

128


in the absorption spectrum. In both the exact and the approximate re-sponse functions the excitonic peak originates from the divergence of theexcitonic part ΠEx(ω). The exact ΠEx in Eq. (5.42) is expressed throughthe function

F(ω) =(1 + ω)2

4ωln

1 + ω

1− ω. (5.48)

For 0 ≤ ω ≤ 1 the function F(ω) is real valued and goes from 12 at ω = 0

to infinity at ω = 1. For −1 ≤ ω ≤ 0 the function F(ω) varies between 0and 1

2 .For small and positive (attractive) a and 0 ≤ ω ≤ 1 there is only one

pole in ΠEx when the denominator of the first term is zero:

1− aF(ω) = 0. (5.49)

This pole approaches ω = 1 if the scattering length a is small. As aincreases, the pole’s energy decreases reaching ω = 0 for a = 2. For thisinteraction strength the energy gain from exciton formation is as largeas the band gap, which allows a spontaneous creation of excitons in thesystem [excitonic instability (e.g., Halperin and Rice, 1968)]. Since we arenot interested in this phenomenon here, we will consider only the case0 ≤ a ≤ 2. Hence we only have to solve Eq. (5.49) to obtain the positionof the excitonic peak. When the dimensionless exciton binding energyε = 1− ω is small, Eq. (5.49) has an approximate solution

ε0 = 2 exp(−1

a

). (5.50)

Another important parameter is the oscillator strength of the excitonicpeak at ε0. To find it we have to expand χ in the vicinity of this peak,obtaining

χ(ε, q) ≈ − q2

16π∆

(1a− 1

2

)2 ε0

ε− ε0

= − q2

16π∆

(ln

2ε0− 1

2

)2 ε0

ε− ε0.

(5.51)

129


Now, we evaluate the position and the strength of the bound excitonusing the approximate response function’s excitonic part (5.47), which is

based on f Exxc

(1). It has a pole when the equation

1− χ−1QP(ω, q)Π

(1)Ex (ω, q) = 0 (5.52)

is fulfilled. Considering similar to (5.50) the case of a small binding en-ergy ε, this equation can be approximately solved by

ε′0 = 2 exp

(−1

a+

1− a−√

1− 2a− a2

2a

), (5.53)

which differs from the approximate solution of the exact condition inEq. (5.50). However, e.g. at a = 0.2 this is only about 14 % larger thanthe exact solution (5.50), which gives ε0 ≈ 0.013. For a smaller scatteringlength and therefore a smaller binding energy the agreement between

the exact binding energy and the one based on f Exxc

(1) is even better. Forcomparing the oscillator strength, we have to expand the approximateχ f (1)

in the vicinity of ε′0:

χ f (1)(ε, q) ≈ − q2

16π∆

(ln(2/ε′0)− 1)2

2a(ln(2/ε′0)− 1/2

)− 1

ε′0ε− ε′0

. (5.54)

Note that in Eq. (5.54) a and ε′0 are linked through Eq. (5.53), but are bothused here for clarity. The oscillator strength is also in good agreementwith the exact result (5.51). For a = 0.2 the second fraction in Eq. (5.54)is only 1 % smaller than the corresponding term in the exact result. Withweaker interaction this error becomes even smaller going to zero in thelimit a → 0. Hence, the error in the oscillator strength stems almostexclusively from the error in the binding energy ε′0, which appears in thenumerator of the third fraction of Eq. (5.54).

In Fig. 5.3 the real and the imaginary part of the exact χ and χ f (1)

are shown for the scattering length a = 0.25. For comparison, χQP isalso displayed. One clearly observes a very good agreement between the

130


0 0.5 1 1.5 2−10

−5

0

5

10

ω

<χ(ω)q2/16π∆

0 0.5 1 1.5 2−4

−3

−2

−1

0

ω

=χ(ω)q2/16π∆

Figure 5.3: The real and the imaginary parts of the exact χ (full) and χ f (1)(dashes)

for the scattering length a = 0.25. The arrows in the lower panel indicate theposition of the excitonic peak. For comparison, the real and the imaginarypart of χQP are shown by dotted lines.

exact and the approximate response functions. Both main features, theenhancement of the imaginary part (the Sommerfeld factor) and the exci-tonic peak are correctly reproduced. The latter is indicated by the arrowson the plots of the imaginary part. The apparent difference between theexact and the approximate response function is the position of the ex-citonic peak. As mentioned above, the exciton’s oscillator strengths are

131


0 0.5 1 1.5 2

−40

−20

0

20

40

ω

<χ(ω)q2/16π∆

0 0.5 1 1.5 2

−0.3

−0.2

−0.1

0

ω

=χ(ω)q2/16π∆

Figure 5.4: The real and the imaginary parts of the exact χ (full) and χ f (1)(dashes)

for the scattering length a = 0.6. The arrows in the lower panel indicate theposition of the excitonic peak.

also different, which is not, however, reflected in the figure.

From the discussion so far one could conjecture that the approximationfor the response function based on the first-order f Ex

xc is always sufficient.This is, of course, not true, as we can see from Fig. 5.4, where the approx-imate and the exact response function are compared for the larger valueof the scattering length a = 0.6. For this strong interaction, the position ofthe excitonic peak in the approximate response function is clearly wrong.

132


0 0.5 1 1.5 2

−4

−3

−2

−1

0

ω

< f Exxc /a

16π∆/q2

0 0.5 1 1.5 20

0.2

0.4

0.6

ω

= f Exxc /a

16π∆/q2

Figure 5.5: The real and the imaginary part of the exact f Exxc /a for a = 0.05 (dots),

a = 0.1 (dashes), a = 0.25 (long dashes), and a = 0.6 (dot-dashes) compared

with f Exxc

(1)/a (full).

In addition, the imaginary part has the wrong magnitude above the bandgap.

To uncover the reason for the good agreement between the exact andthe approximate response functions for a “weak” interaction, let us com-

pare f Exxc

(1) with the exact f Exxc for different interaction strengths. We note

first that f Exxc

(1) is proportional to the scattering length a. Therefore, it is

133


more adequate to compare f Exxc

(1)/a with f Exxc /a, as done in Fig. 5.5. It is

clearly visible that f Exxc

(1) is a very good approximation to the exact f Exxc

in a frequency range close to the band gap. Actually, for ω = 1 we obtainfrom the exact expressions (5.44) and (5.45)

f Exxc

(1)(ω = 1) = −16π∆

q2 a (5.55)

and

f Exxc (ω = 1) = −16π∆

q2a

1− a2/4. (5.56)

Thus even for a very strong interaction of a = 0.6 where f Exxc

(1) leads tothe rather poor response function of Fig. 5.4, there is only a 10 % error

in f Exxc

(1) at ω = 1. In the static case the errors are larger, as we get forω = 0:

f Exxc

(1)(ω = 0) = −16π∆

q298

a (5.57)

and

f Exxc (ω = 0) = −16π∆

q298

a1 + a

. (5.58)

Here a 10 % error is already reached for a = 0.1, as the relative error ofthe first-order f Ex

xc is of order a, whereas it is of order a2 at ω = 1.It is also interesting to look at the difference δ f Ex

xc between the exact f Exxc

and f Exxc

(1) for different interaction strengths. Since the leading term inδ f Ex

xc is of the order a2, these differences are normalized by a2, when plot-ting them in Fig. 5.6. The general behavior of the different curves is quitesimilar, which indicates that it is mostly the second-order approximationf Exxc

(2) which contributes to δ f Exxc . One may be surprised that the curves

for stronger interaction are closer to 0 for ω < 1, which might indicate

that f Exxc

(1) is a better approximation for stronger interactions. However,

this is not the case. This only tells us that f Exxc

(3) is negative for ω < 1.

134


0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

ω

< δf Exxc /a2

16π∆/q2

0 0.5 1 1.5 2

−1

−0.8

−0.6

−0.4

−0.2

0

ω

= δf Exxc /a2

16π∆/q2

Figure 5.6: The real and the imaginary parts of δ f Exxc /a2 = ( f Ex

xc − f Exxc

(1))/a2

for a = 0.05 (dots), a = 0.1 (dashes), a = 0.25 (long dashes), and a = 0.6(dot-dashes).

5.3.3 Validity of the first-order approximation

From the previous section we can conclude that f Exxc

(1) is a good approxi-mation to f Ex

xc close to the band gap ω = 1 practically for any interactionstrength. Even for a strong interaction, where the response function stem-

ming from f Exxc

(1) is quite wrong, the first-order kernel f Exxc

(1) is still very

accurate close to ω = 1. This is the reason for the success of f Exxc

(1) in de-

135


scribing the bound excitonic state as shown in the previous section. Since

around ω = 1 the first-order kernel f Exxc

(1) is a good approximation, the(exact) bound excitonic state will be correctly described, if this state lieswithin this region. If, however, the binding energy is outside this region,

f Exxc

(1) will fail. Note that as seen from Fig. 5.5 this region gets smaller asthe interaction increases and at the same time the binding energy of theexciton increases. Hence the error in the exciton binding energy increaseswith the increase of the interaction strength.

Can we understand the good agreement between f Exxc

(1) and f Exxc for

ω ≈ 1 in terms of the integral equation for Λ? For this we explicitlycalculate the diagrams involved in the replacement shown in Eq. (5.17).Working with the valence- and the conduction-band Green functions wehave four combinations of the external lines and can split this replace-ment into four parts. At energies close to the band gap, the two-particlepropagator with a conduction-band state in the upper line and a valence-band state in the lower line should dominate. Hence, the most importantpart of this replacement is the diagram with a conduction-band state inthe upper line on both sides of the graphs. This diagram can be translatedinto quantities introduced in the previous section:

v k

c k + q

v k′

c k′ + q

−

v k

c k + q

v k′

c k′ + q

= Vg21(k, k′)− Γ1(ω, q, k)χ−1

QP(ω, q)γ∗(q, k′).

(5.59)

Note that we use the bare interaction V here, as the interaction renormal-ization in these diagrams means simply a replacement V → V .

Inserting g1, Γ1, χQP, and γ in Eq. (5.59) and taking the limit ω → 1 weobtain:

Vg21(k, k′)− Γ1(ω = 1, q, k)χ−1

QP(ω = 1, q)γ∗(q, k′)

136


= V[

u2ku2

k′

(1− ∆

Ek′

)+( k−

kvk

)2( k′+k′

vk′)2

+ 2k−k

vkukuk′vk′k′+k′

+ u2k

( k′+k′

vk′)2( q−

q

)2 ∆

Ek′

]. (5.60)

Not surprisingly, the whole expression (5.60) is proportional to the in-teraction V (or V after renormalization). In the proportionality factor insquare brackets all terms contain (1−∆/Ek′) (or powers thereof). For thefirst term this is directly visible, in the others it is hidden in vk′ . Let uslook at this factor more closely. When the diagram of Eq. (5.59) is insertedinto the equation for Λ (5.15) the integration over k′ has to be performed.The main contribution to this integral comes from small momenta, due tothe small energy denominators in the particle–hole propagator. However,for these small momenta the expression (5.60) is small as ∆ ≈ Ek′ .

We can therefore conclude that close to the band gap the kernel in theequation for Λ is indeed small for those states which mainly contribute tothe integral. Thus we explicitly observe the cancellation we qualitativelydiscussed in section 5.1. This cancellation explains the excellent agree-

ment between f Exxc and f Ex

xc(1) for ω ≈ 1. It is worth mentioning that this

cancellation does not occur at every QP excitation energy, as speculatedin section 5.1, but only close to the parabolic critical point in the jointdensity of states at ω = 1.

A complementary interpretation can be obtained from looking at thediagrammatic expansion of the exact response function χ and comparingit with χ f (1)

, which is based on the first-order f Exxc . For example, the

third-order term in the expansion of the exact χ is given by

, (5.61)

whereas the the third-order term of χ f (1)is

. (5.62)

The difference between the two expressions is similar to the replacements

137


discussed in section 5.1. This remains true in all orders of the perturba-tion theory. From the above calculation of the effect of this replacementfollows that the exact χ and χ f (1)

are almost identical in the vicinity of theband gap, independently of the position of the excitonic peak. Therefore,for weak interaction, where the excitonic features of the response function(the bound state and the Sommerfeld factor) are most prominent close tothe band gap, χ f (1)

is a good approximation to. For stronger interaction,χ f (1)

is still a good approximation close to the band gap. However, theexcitonic features of the response function occur at other energies and arenot well described.

5.3.4 Static long-range exchange-correlation kernel

In section 2.4 it was mentioned that the static long-range kernel suggestedby Reining et al. (2002) has been successfully applied to the descriptionof excitonic effects in a variety of semiconductors. It is interesting toinvestigate how this approximation behaves in the 2d Dirac model. Wetherefore approximate the excitonic xc kernel as

f Exxc (ω, q) ≈ −16π∆

q2 β, (5.63)

where β is an adjustable parameter. Looking at the real part of f Exxc in

Fig. 5.5 and the various exact results in Eqs. (5.55)–(5.58) it is temptingto use β = a as a first approximation. An improved value for β canbe obtained by fitting the position of the excitonic peak. Inserting theapproximate f Ex

xc (5.63) into Eq. (5.7b) we obtain for the response function

χLR(ω, q) = − q2

16π∆

D(ω)1− βD(ω)

, (5.64)

with D(ω) = −16π∆/q2χQP(ω). Similar to the procedure in section 5.3.2expanding D(ω) for small ε = 1− ω gives the position of the excitonicpeak as

ε′′0 = 2 exp(− 1

β− 1)

. (5.65)

138


Expanding χLR in the vicinity of ε′′0 one obtains

χLR(ε, q) ≈ − q2

16π∆

1β2

ε′′0ε− ε′′0

(5.66a)

= − q2

16π∆

(ln

2ε′′0− 1)2 ε′′0

ε− ε′′0. (5.66b)

For a simplistic approximation β = a, the binding energy (5.65) is toosmall by a factor of 1/e ≈ 0.369 compared with Eq. (5.50) independentof the scattering length a. Compared with Eq. (5.51) the second fraction1/β2 in Eq. (5.66a) is too large by a factor 1 + a, which slightly compen-sates the error in the oscillator strength caused by the binding energy ε′′0in the numerator of the third fraction of Eq. (5.66a). For example, fora = 0.2 the oscillator strength in this model is 56 % smaller than the exactresult, while the binding energy is 63 % smaller.

One can improve on these results using a more sophisticated fit for β.We can choose β such that the binding energy (5.65) reproduces the exactresult (5.50), i.e.,

β =a

1− a. (5.67)

In that case the second fraction 1/β2 in Eq. (5.66a) is too large by a factor1− a. However, since the binding energy is exact, the oscillator strengthfor a = 0.2 is only 20 % smaller than the exact result, which is comparableto the accuracy we got from the first order approximation to f Ex

xc .In addition, we can compare the frequency dependence of the response

function χLR to the exact χ. In Fig. 5.7 the real and the imaginary partof χ are shown for a = 0.25 as well as the real and the imaginary partof χLR for β = 0.25, β = 0.33, and β = 0.31. For comparison, χQP isalso displayed. Note that in principle β = 0.33 should reproduce theexact exciton binding energy according to Eq. (5.67). However, with a =0.25 we are reaching the regime where the expansion for small ε used inderiving this formula starts to get problematic. For example, Eq. (5.50)gives ε0 = 0.0366 for a = 0.25, while the divergence in the response

139


0 0.5 1 1.5 2−10

−5

0

5

10

ω

<χ(ω)q2/16π∆

0 0.5 1 1.5 2

−5

−4

−3

−2

−1

0

ω

=χ(ω)q2/16π∆

Figure 5.7: The real and the imaginary parts of the exact χ for the scatteringlength a = 0.25 (full) and χLR based on the static long-range kernel (5.63)for β = 0.25 (long dashes), β = 0.33 (dot-dashes), and β = 0.31 (dashes).The arrows in the lower panel indicate the position of the excitonic peak.For comparison, dotted lines show the real and the imaginary part of χQP.

function occurs about 14 % below that at ε = 0.0361. This causes anovercompensation of the binding energy for β = 0.33 clearly visible fromthe arrows in the lower panel of Fig. 5.7. An exact fit of the excitonicpeak can be obtained by using β = 0.31. One can see from Fig. 5.7that the height of the peak in the Sommerfeld correction just above the

140

5.4 Coulomb interaction

band gap is very sensitive to the precise value of β. For β = 0.31 theexact result is almost reproduced, comparable to the results for the firstorder f Ex

xc displayed in Fig. 5.3. However, independently of the value ofβ there is no Sommerfeld correction for ω > 1.3, where =χLR convergesto =χQP. This is not too surprising, since the real part of the exact f Ex

xcis much larger in that frequency regime as can be seen from Fig. 5.5. Italso fits to the results from Botti et al. (2004) who found that a staticlong-range xc kernel fitted to the optical absorption spectrum just abovethe band gap is not capable of correctly describing electron energy lossspectroscopy occurring at higher energy.

The real part of the response functions is displayed in the upper panelof Fig. 5.7. Below the band gap, the main difference is the position of thedivergence, i.e., the position of the excitonic peak. Above the band gap<χLR for different values of β is very similar and consistently underesti-mating the exact χ. Again, this is related to the real part of the exact f Ex

xcbeing much larger in that frequency regime than the long-range kernelof Eq. (5.63).

Overall we see that the static long-range xc kernel of Eq. (5.63) is capa-ble of giving quite accurate results, if β is fitted properly. This approxima-tion is interesting in situations where one wants to use a good descriptionof excitonic effects to calculate other properties of the system, as it wasdone by Marini and Rubio (2004) as well as Bruneval et al. (2005).


In this section, we consider the 2d Dirac model of Eq. (5.19) with theCoulomb interaction between the particles, i.e.,

VC(r − r′) =g2

|r − r′| , (5.68)

where g is a dimensionless charge, which can include, e.g., the dielectricscreening. Note that Eq. (5.68) is the 3d interaction although the modelsystem is 2d. We will not attempt to solve the BSE with this interaction,

141


but rather focus on the properties of shallow excitons. The weak bindinglimit ε = 1 − ω 1 is, in fact, a “nonrelativistic” limit where the BSEreduces to the two-particle Schrödinger equation (Sham and Rice, 1966).The quasiparticle energy eigenvalues are approximately

Ek ≈ ∆ +k2

2∆, (5.69)

and for the eigenvectors holds

uk ≈ 1 and vk ≈ 0. (5.70)

Solving the BSE thus reduces to solving the positronium problem in 2d

for particles with a mass ∆, i.e., for a reduced mass ∆/2. The responsefunction can be written in the spectral representation as (Mahan, 1990)

χ(ω, q) = |γ(q, 0)|2 ∑n

|φn(r = 0)|2ω − (2∆− εn)

≈ |γ(q, 0)|2 |φ0(r = 0)|2ω − (2∆− ε0)

,

(5.71)

where the φn are the eigenfunctions with eigenvalues −εn of the above-mentioned Schrödinger equation. The approximation in Eq. (5.71) is validclose to the excitonic peak of the 1s ground-state, which we are interestedin here. The other states of the infinite Rydberg series are neglected, as thefirst-order f Ex

xc considered later produces only one additional peak in theresponse function. The 1s wave function required in this approximationis given by

φ0(r) =

√2∆ε0

πexp−

√∆ε0r (5.72)

and the exciton binding energy is

ε0 = g4∆. (5.73)

Introducing dimensionless variables ε and ε0 = ε0/(2∆), we arrive at

χ(ε, q) ≈ − q2

2π∆

ε0

ε− ε0(5.74)

142


for ε close to the excitonic peak of the 1s ground-state.

We now compare the position and the oscillator strength of the exci-

tonic peak of the 1s ground-state with the results obtained from f Exxc

(1).For this we first expand χQP of Eq. (5.25) for small ε

χQP(ε, q) ≈ − q2

16π∆

(ln

2ε− 1)

. (5.75)

In order to obtain f Exxc

(1) we also need an expression for the first-orderapproximation to the excitonic part of the response function Π

(1)Ex in the

limit of small ε. This can be done by using two approximations. The firstapproximation is using Eqs. (5.69) and (5.70). The second approximationreduces the number of diagrams that are considered. In a conventionalbasis, Π

(1)Ex is given by a single graph

Π(1)Ex = , (5.76)

while there are four graphs in the diagonal representation

Π(1)Ex =

c c

v v

+

c v

v c

+

v c

c v

+

v v

c c

. (5.77)

The particle–hole propagators with a conduction-band state in the upperline and a valence-band state in the lower line diverge at ω = 1. Theparticle–hole propagators with a valence-band state in the upper line anda conduction-band state in the lower line diverge at ω = −1. Therefore,in the limit where the energy goes to the band gap (or equivalently ε issmall), the main contribution to Π

(1)Ex comes from the diagram with both

particle–hole propagators of the upper line belonging to the conduction-band state and the lower line to the valence-band state. Neglecting the

143


other contributions is the second approximation in calculating Π(1)Ex :

Π(1)Ex ≈

c c

v v

. (5.78)

With these approximations we obtain

Π(1)Ex (ε, q) ≈ −|γ(q, 0)|2

(2∆)2 ∑k,k′

V(k− k′)(12 (k/∆)2 + ε

) (12 (k′/∆)2 + ε

) , (5.79)

where V(k − k′) = 2πg2/|k − k′| is the 2d Fourier representation ofV(r − r′) from Eq. (5.68). The easiest way to solve this double integral isto look at it in real space, instead of Fourier space, where it becomes asingle integral. The 2d Fourier transformation of the particle–hole propa-

gator(

12 (k/∆)2 + ε

)−1gives the modified Bessel function of the second

kind, K0, so that we can write

Π(1)Ex (ε, q) ≈ − g2q2

8π∆√

2ε

∫ ∞

0dρ K2

0(ρ). (5.80)

The integral over K20 computes to (π/2)2, and we arrive at

Π(1)Ex (ε, q) ≈ − q2

16π∆

π2

2

√ε0

ε. (5.81)

Now we have all ingredients to build f Exxc

(1) = χ−1QPΠ

(1)Ex χ−1

QP and insertit in Eq. (5.7b). This gives the following approximation for the response

function based on f Exxc

(1):

χ f (1)(ε, q) =

χQP(ε, q)

1−Π(1)Ex (ε, q)χ−1

QP(ε, q)

= − q2

16π∆

(ln(2/ε)− 1)1− (ln(2/ε)− 1)−1 (π2/2)

√ε0/ε

.

(5.82)

144


Note that this equation is valid only for small ε, since we derived χQP

and Π(1)Ex only for small values of ε.

From Eq. (5.82) we clearly see that χ f (1)contains an additional pole

where the denominator vanishes. Hence, the binding energy of the exci-ton in this approximation ε′0 is implicitly defined by

1−(

ln2ε′0− 1)−1 π2

2

√ε0

ε′0= 0. (5.83)

Equation (5.83) can be solved for ε′0 by using the Lambert W function(Corless et al., 1996). In this way, one could calculate the approximatebinding energy ε′0 in terms of the exact binding energy ε0 or the inter-action strength g. However, it is more instructive to express g and ε0 interms of the ratio of the binding energies:

g4

2= ε0 = 2

ε0

ε′0exp

−(

π2

2

√ε0

ε′0+ 1

). (5.84)

The approximate binding energy ε′0 and the exact binding energy ε0 areidentical for

g4

2= ε0 = 2 exp

−(

π2

2+ 1)

≈ 1189

. (5.85)

In other words, for the interaction strength which corresponds to an ex-act exciton binding energy of 1/189 of the band gap, the approximation

using f Exxc

(1) gives the exact binding energy. Not only for stronger, butalso for weaker interaction strength there is some error. To be more pre-cise, the error is below 10 % for ε0 between 1/165 and 1/222. It is below20 % for ε0 between 1/148 and 1/271. Note that the approximation isconsistent, since in the range of energies where the approximation givesgood results, ε is a small quantity of order 10−2.

It is interesting to note that for the Coulomb interaction – in contrast toa short-range interaction – one does not get the correct exciton binding

energy from f Exxc

(1) in the limit of the interaction strength going to zero.

145


Note also that while the exact solution of the 2d hydrogen problem givesrise to an infinite series of excitonic peaks in the exact response function,we obtain only one excitonic peak in the approximate approach. Although

f Exxc

(1) is frequency dependent, it does not contain the rapid oscillationsneeded to describe the whole series of excitonic states.

To compare the approximate response function of Eq. (5.82) to the ex-act one from Eq. (5.74) and determine the residual it is best to expandχ f (1)

around ε′0. This can be done by calculating the first-order Taylorexpansion of the denominator in Eq. (5.82):

χ f (1)(ε, q) ≈ − q2

16π∆

(π2/2)√

ε0/ε′0

1/2− (2/π2)√

ε′0/ε0

ε′0ε− ε′0

. (5.86)

In the case where the exciton energy is exactly reproduced, i.e. for ε′0 = ε0,the second fraction in Eq. (5.86) is approximately 16.6. The oscillatorstrength is thus too large by a factor of two in spite of the fact that thebinding energy is exact. The value of the second fraction in Eq. (5.86) isalmost constant for a large range of ratios ε′0/ε0, so that mainly the ε′0in the numerator of the third fraction in Eq. (5.86) produces additionalerrors in the oscillator strength. It is not too surprising that the approx-imated response function has an excitonic peak with a larger oscillatorstrength than the 1s peak in χ, as the excitonic oscillator strength is dis-tributed over more peaks in the exact response function.

Similar to the results for the short-range interaction of the previoussection, we observe that the cancellation effects are also effective for thelong-ranged Coulomb interaction. However, the general structure of thiscancellation is different and cannot be universally characterized. Hence

the accuracy of f Exxc

(1) has to be checked for every particular system. Theintegral equation for Λ (5.15) together with the replacement procedure ofEq. (5.16) or Eq. (5.17) provides an appropriate tool for this task.

146

5.5 Outlook

5.5 Outlook

We have seen that cancellation effects occur for both a general short-rangeinteraction and the long-ranged Coulomb interaction between the QPs.However, the cancellation is effective only close to the band gap, whichconstitutes a parabolic critical point in the joint density of states. Thearguments of section 5.3.3 are mainly concerned with the large amountof the phase space occupied by states close to the band gap. Since this issimilar for other critical points, these arguments may be transferable tothem. It is therefore very interesting to test the predictive power of thisapproach for other systems with more realistic band structures, includinghyperbolic critical points in the joint density of states.

Besides the obvious extension to more realistic systems, one shouldalso keep in mind that the model presented in this chapter is one of thevery few models in this field that allow analytic solutions. The analyticsolution often provides more physical insight than numeric calculationsfor realistic systems. Therefore it should be fully explored, including theproperties of f QP

xc . The quasiparticle part of the xc kernel can also becalculated within the model of Eq. (5.20).

Another interesting question is how well the first-order approximationfor f Ex

xc performs if not all states in the system are delocalized. Thisis relevant, e.g., for excitons interacting with impurities. One could tryto model this by adding an “impurity” in form of a delta potential tothe Hamiltonian. Such a system would be particularly interesting whenthe binding energy to the impurity is larger than the excitonic bindingenergy.

147

6 Summary

The density-functional theory (DFT) is one of the most important ab initiomethods for calculating properties of many-body systems. The successof DFT is based on the Kohn-Sham (KS) method and the approximationsfor the exchange-correlation (xc) effects. These approximations are wellunderstood. However, DFT is first and foremost a theory for the groundstate. It is very difficult to obtain properties of excited states.

The time-dependent density-functional theory (TDDFT) can overcomethis restriction and in the linear-response regime can give access to theproperties of excited states. However, the xc effects, which are containedin the xc kernel fxc, have to be approximated. The approximations forfxc are not well understood up to now.

Green-functions-based conventional many-body theory (MBT) is an al-ternative approach to the many-body problem. Especially when com-bined with diagrammatic techniques, MBT allows to derive physicallymotivated approximations.

This work investigates the use of diagrammatic techniques from MBTfor TDDFT. The aim of the investigations is to contribute to the develop-ment, testing, and understanding of the approximations involved in thetime-dependent case. It is shown that it is possible to expand the xc ker-nel in a diagrammatic series. Explicit diagrammatic rules for constructingfxc in any order of the perturbation theory are given.

The derived diagrammatic rules enable us to study the analytic prop-erties of fxc. It is shown that fxc is finite at KS excitation energies in everyorder of the perturbation theory. If this were not the case, finite-order ap-proximations to fxc would not give sensible results for the kernel whichplays the role of a “mass operator” connecting the KS density–densityresponse function with the exact density–density response function.

149

6 Summary

However, there is one important difference between fxc and the self-energy, the latter playing the role of the mass operator connecting freeand exact Green function. This becomes apparent when the perturbativeexpansion for fxc is used to calculate the correction to the KS excitationenergies. This correction is obtained up to the second-order with respectto the interaction and with respect to one- and two-particle irreducibleelements. It is shown that if a perturbative approximation for fxc is used,the correction to the KS excitation energies has to be calculated using aconsistent approach based on the same order of perturbation theory for bothquantities.

One of the most promising applications of TDDFT in the linear re-sponse regime is the calculation of electronic excitation spectra of solids,where TDDFT could replace the computationally expensive solution ofthe BSE, which is normally used for describing the excitonic correlations.It is shown that the xc kernel can be separated into a quasiparticle and aexcitonic part in an exact and unambiguous way. The diagrammatic rulesderived for fxc apply to these parts separately. This allows us to ex-press the excitonic part in terms of the three-point function Λ, for whichan integral equation is derived. This integral equation is similar to theBSE and gives the exact translation of the BSE into the TDDFT language.There exists the possibility of cancellation effects in the kernel of the inte-gral equation. If these effects are effective, a first-order approximation tothe excitonic fxc is sufficient.

The cancellation effects in the equation for Λ are explicitly studied fora two-band model semiconductor. Using a general short-range interactionbetween the quasiparticles (QPs) all relevant equations are solved ana-lytically. It is shown that the cancellation is most effective directly at theband-gap energy. Consequently, the first-order and the exact excitonic fxc

are very similar close to the band-gap energy for all interaction strengths.For a weak interaction the first-order approximation to the excitonic fxc

describes the different excitonic correlations in the density–density re-sponse function very well. Qualitatively similar results are found whenthe QPs interact via the long-ranged Coulomb interaction.

150

Bibliography

A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods ofQuantum Field Theory in Statistical Physics (Prentice-Hall, EnglewoodCliffs, N. J., 1963). (cited on pages 8, 11, 27, 98, and 126)

S. L. Adler, Phys. Rev. 126, 413 (1962). (cited on pages 46 and 47)

G. Adragna, R. Del Sole, and A. Marini, Phys. Rev. B 68, 165108

(2003). (cited on pages 48 and 113)

H. Appel, E. K. U. Gross, and K. Burke, Phys. Rev. Lett. 90, 043005

(2003), cond-mat/0203027. (cited on pages 100 and 101)

W. G. Aulbur, L. Jönsson, and J. W. Wilkins, Solid State Phys. 54, 1

(2000). (cited on pages 45, 79, and 112)

G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961).(cited on page 78)

P. L. de Boeij, F. Kootstra, J. A. Berger, R. van Leeuwen, and J. G.Snijders, J. Chem. Phys. 115, 1995 (2001). (cited on page 46)

S. Botti, F. Sottile, N. Vast, V. Olevano, L. Reining, H.-C.Weissker, A. Rubio, G. Onida, R. Del Sole, and R. W. Godby, Phys.Rev. B 69, 155112 (2004). (cited on pages 46 and 141)

F. Bruneval, F. Sottile, V. Olevano, R. Del Sole, and L. Reining,Phys. Rev. Lett. 94, 186402 (2005), cond-mat/0504565.(cited on pages 117 and 141)

K. Burke and E. K. U. Gross, in Density functionals: theory andapplications, edited by D. Joubert (Springer, Berlin, 1998), vol. 500 ofLecture notes in physics, pp. 116–146. (cited on page 37)

151

http://arxiv.org/abs/cond-mat/0203027


Bibliography

D. M. Bylander and L. Kleinman, Phys. Rev. Lett. 74, 3660 (1995).(cited on page 26)

M. E. Casida, in Recent Developments and Applications of Modern DensityFunctional theory, edited by J. M. Seminario (Elsevier Science,Amsterdam, 1996), vol. 4 of Theoretical and Computational Chemistry, pp.391–439. (cited on pages 83, 100, and 101)

M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub, J. Chem.Phys. 108, 4439 (1998). (cited on page 42)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E.Knuth, Advances in Computational Mathematics 5, 329 (1996).(cited on page 145)

R. Del Sole, G. Adragna, V. Olevano, and L. Reining, Phys. Rev. B67, 045207 (2003). (cited on page 113)

F. Della Sala and A. Görling, J. Chem. Phys. 115, 5718 (2001).(cited on page 26)

J. F. Dobson, Phys. Rev. Lett. 73, 2244 (1994). (cited on page 42)

J. F. Dobson, in Electronic Density Functional Theory, edited by J. F.Dobson, G. Vignale, and M. P. Das (Plenum Press, New York, 1998),pp. 43–53. (cited on page 37)

R. M. Dreizler and E. K. U. Gross, Density-Functional Theory (Springer,Berlin, 1990). (cited on page 21)

M. van Faassen, P. L. de Boeij, R. van Leeuwen, J. A. Berger, andJ. G. Snijders, Phys. Rev. Lett. 88, 186401 (2002). (cited on page 42)

A. L. Fetter and J. D. Walecka, Quantum Theory of Many-ParticleSystems (McGraw-Hill, Inc., New York, 1971).(cited on pages 8, 27, and 114)

152

Bibliography

C. Filippi, C. J. Umrigar, and X. Gonze, J. Chem. Phys. 107, 9994

(1997). (cited on page 104)

V. I. Gavrilenko and F. Bechstedt, Phys. Rev. B 55, 4343 (1997).(cited on pages 42 and 45)

M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951).(cited on page 13)

P. Ghosez, X. Gonze, and R. W. Godby, Phys. Rev. B 56, 12811 (1997).(cited on page 42)

S. J. A. van Gisbergen, F. Kootstra, P. R. T. Schipper, O. V.Gritsenko, J. G. Snijders, and E. J. Baerends, Phys. Rev. A 57, 2556


R. W. Godby, M. Schlüter, and L. J. Sham, Phys. Rev. B 37, 10159


X. Gonze, P. Ghosez, and R. W. Godby, Phys. Rev. Lett. 74, 4035 (1995).(cited on page 42)

X. Gonze and M. Scheffler, Phys. Rev. Lett. 82, 4416 (1999).(cited on pages 72 and 104)

A. Görling, Phys. Rev. A 54, 3912 (1996a). (cited on pages 90 and 104)

A. Görling, Phys. Rev. B 53, 7024 (1996b), 59, 10370(E) (1999).(cited on page 26)

A. Görling, Phys. Rev. A 57, 3433 (1998a). (cited on pages 43 and 51)

A. Görling, Int. J. Quant. Chem. 69, 265 (1998b).(cited on pages 43 and 51)

A. Görling and M. Levy, Phys. Rev. B 47, 13105 (1993).(cited on page 27)

153

Bibliography

A. Görling and M. Levy, Phys. Rev. A 50, 196 (1994).(cited on page 27)

A. Görling and M. Levy, Int. J. Quant. Chem. Symp. 29, 93 (1995).(cited on page 27)

T. Grabo, M. Petersilka, and E. K. U. Gross, J. Mol. Struct.:THEOCHEM 501–502, 353 (2000). (cited on pages 42 and 80)

E. K. U. Gross, C. A. Ullrich, and U. J. Gossmann, in Densityfunctional theory, edited by E. K. U. Gross and R. M. Dreizler

(Plenum Press, New York, 1995), vol. 337 of NATO ASI series, Series B,Physics, pp. 149–171. (cited on page 37)

B. I. Halperin and T. M. Rice, Solid State Phys. 21, 115 (1968).(cited on page 129)

W. Hanke and L. J. Sham, Phys. Rev. B 21, 4656 (1980).(cited on pages 43, 45, and 103)

L. Hedin, Phys. Rev. 139, A796 (1965). (cited on page 45)

L. Hedin and S. Lundqvist, Solid State Phys. 23, 1 (1969).(cited on page 8)

V. Heine and R. O. Jones, J. Phys. C 2, 719 (1969). (cited on page 43)

S. Hirata, S. Ivanov, I. Grabowski, and R. J. Bartlett, J. Chem. Phys.115, 1635 (2001). (cited on page 26)

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).(cited on page 21)

J. F. Janak, Phys. Rev. B 18, 7165 (1978). (cited on page 24)

R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).(cited on pages 21 and 23)

154

Bibliography

E. O. Kane, in Semiconductors and Semimetals, edited by R. K.Willardson and A. C. Beer (Academic Press, New York, 1966),vol. 1, p. 75. (cited on page 118)

Y.-H. Kim and A. Görling, Phys. Rev. B 66, 035114 (2002a).(cited on pages 43 and 46)

Y.-H. Kim and A. Görling, Phys. Rev. Lett. 89, 096402 (2002b).(cited on pages 43 and 46)

Y.-H. Kim, M. Städele, and A. Görling, Int. J. Quant. Chem. 91, 257


W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).(cited on pages 21 and 23)

F. Kootstra, P. L. de Boeij, and J. G. Snijders, Phys. Rev. B 62, 7071


J. B. Krieger, Y. Li, and G. J. Iafrate, Phys. Rev. A 45, 101 (1992).(cited on page 26)

L. D. Landau, Sov. Phys. JETP 3, 920 (1957a). (cited on page 8)

L. D. Landau, Sov. Phys. JETP 5, 101 (1957b). (cited on page 8)

R. van Leeuwen, Phys. Rev. Lett. 82, 3863 (2000). (cited on page 39)

R. van Leeuwen, Int. J. Mod. Phys. B 15, 1969 (2001). (cited on page 37)

M. Lein, E. K. U. Gross, and J. P. Perdew, Phys. Rev. B 61, 13431 (2000).(cited on pages 42 and 80)

M. Lein and S. Kümmel, Phys. Rev. Lett. 94, 143003 (2005).(cited on page 39)

M. Levy, Phys. Rev. A 26, 1200 (1982). (cited on page 23)

P.-O. Löwdin, J. Chem. Phys. 19, 1396 (1951). (cited on page 85)

155


Bibliography

J. M. Luttinger and P. Nozières, Phys. Rev. 127, 1431 (1962).(cited on page 9)

G. D. Mahan, Many-Particle Physics (Plenum Press, New York, 1990),2nd ed. (cited on pages 8, 41, and 142)

A. Marini, R. Del Sole, and A. Rubio, Phys. Rev. Lett. 91, 256402


A. Marini and A. Rubio, Phys. Rev. B 70, 081103 (2004),cond-mat/0405590. (cited on page 141)

R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem(McGraw-Hill, New York, 1976), 2nd ed. (cited on pages 8 and 75)

P. Nozières, Theory of Interacting Fermi Systems (W. A. Benjamin, Inc.,New York, 1964). (cited on page 8)

P. Nozières and J. M. Luttinger, Phys. Rev. 127, 1423 (1962).(cited on page 9)

G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).(cited on pages 45 and 46)

J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983).(cited on page 24)

M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett.76, 1212 (1996). (cited on pages 40, 42, 43, 80, and 83)

M. Petersilka, E. K. U. Gross, and K. Burke, Excitation energy fromtime-dependent density functional theory using exact and approximatepotentials (2000), cond-mat/0001154. (cited on pages 80 and 99)

J. C. Phillips, Solid State Phys. 18, 55 (1966).(cited on pages 43, 44, and 45)

156




Bibliography

D. Pines and P. Nozières, The Theory of Quantum Liquids, vol. I (W. A.Benjamin, Inc., New York, 1966). (cited on page 8)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes: The Art of Scientific Computing (CambridgeUniversity Press, 1986). (cited on pages 72 and 73)

L. Reining, V. Olevano, A. Rubio, and G. Onida, Phys. Rev. Lett. 88,066404 (2002). (cited on pages 46, 47, 48, 111, and 138)

C. F. Richardson and N. W. Ashcroft, Phys. Rev. B 50, 8170 (1994).(cited on page 80)

E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984).(cited on pages 37 and 38)

L. H. Ryder, Quantum field theory (Cambridge University Press, 1996),2nd ed. (cited on page 118)

V. Sahni, J. Gruenebaum, and J. P. Perdew, Phys. Rev. B 26, 4371


E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951).(cited on page 110)

L. J. Sham, Phys. Rev. B 32, 3876 (1985). (cited on page 31)

L. J. Sham and T. M. Rice, Phys. Rev. 144, 708 (1966).(cited on pages 114 and 142)

L. J. Sham and M. Schlüter, Phys. Rev. Lett. 51, 1888 (1983).(cited on pages 24 and 31)

L. J. Sham and M. Schlüter, Phys. Rev. B 32, 3883 (1985).(cited on page 24)

R. T. Sharp and G. K. Horton, Phys. Rev. 90, 317 (1953).(cited on page 25)

157

Bibliography

F. Sottile, V. Olevano, and L. Reining, Phys. Rev. Lett. 91, 056402

(2003). (cited on pages 44, 48, and 111)

M. Städele, J. A. Majewski, P. Vogl, and A. Görling, Phys. Rev. Lett.79, 2089 (1997). (cited on page 26)

M. Städele, M. Moukara, J. A. Majewski, P. Vogl, and A. Görling,Phys. Rev. B 59, 10031 (1999). (cited on page 26)

R. Stubner, I. V. Tokatly, and O. Pankratov, Phys. Rev. B 70, 245119


K. Sturm and A. Gusarov, Phys. Rev. B 62, 16474 (2000).(cited on page 42)

J. D. Talman and W. F. Shadwick, Phys. Rev. A 14, 36 (1976).(cited on page 25)

K. Tatarczyk, A. Schindlmayr, and M. Scheffler, Phys. Rev. B 63,235106 (2001). (cited on pages 42 and 80)

I. V. Tokatly, Phys. Rev. B 71, 165104 (2005a). (cited on page 43)

I. V. Tokatly, Phys. Rev. B 71, 165105 (2005b). (cited on page 43)

I. V. Tokatly and O. Pankratov, Phys. Rev. Lett. 86, 2078 (2001),cond-mat/0008468. (cited on pages 26, 41, 55, 83, 90, and 104)

I. V. Tokatly, R. Stubner, and O. Pankratov, Phys. Rev. B 65, 113107

(2002), cond-mat/0108012. (cited on pages 5, 49, 66, 83, 101, and 116)

C. A. Ullrich, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett.74, 872 (1995). (cited on page 43)

C. A. Ullrich and G. Vignale, Phys. Rev. B 58, 15756 (1998).(cited on page 43)

G. Vignale, Phys. Rev. Lett. 74, 3233 (1995a). (cited on page 42)

158




Bibliography

G. Vignale, Phys. Lett. A 209, 206 (1995b). (cited on page 42)

G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996).(cited on page 42)

G. Vignale and W. Kohn, in Electronic Density Functional Theory, editedby J. F. Dobson, G. Vignale, and M. P. Das (Plenum Press, NewYork, 1998), pp. 199–216. (cited on page 42)

G. Vignale, C. A. Ulrich, and S. Conti, Phys. Rev. Lett. 79, 4878


J. Werschnik, E. K. U. Gross, and K. Burke, to appear in J. Chem.Phys. (2005), cond-mat/0410362. (cited on page 37)

G. C. Wick, Phys. Rev. 80, 268 (1950). (cited on page 13)

N. Wiser, Phys. Rev. 129, 62 (1963). (cited on pages 46 and 47)

W. Yang and Q. Wu, Phys. Rev. Lett. 89, 143002 (2002).(cited on page 26)

A. Zangwill and P. Soven, Phys. Rev. Lett. 45, 204 (1980a).(cited on page 41)

A. Zangwill and P. Soven, Phys. Rev. A 21, 1561 (1980b).(cited on page 41)

159


Acknowledgments

I would like to thank several people for their contributions:

Oleg Pankratov for initiating this work as well as for the freedom I hadwhen completing it.

Ilya Tokatly for all the helpful discussions about physical meaning as wellas technical details.

Roland Winkler for bringing in a different point of view and for the carefuland critical proofreading.

Alexander Mattausch, Günther Schwarz, Marc Siegmund, Markus Hofmann,Matthias Heid, Michael Strass, Tassilo Dannecker, Steffi Gaile, and ValeryValeyev for the very friendly atmosphere at our institute.

Christine Wiedemann for here invaluable help with all the administrativework.

My final thanks go to Rona Röthig for all here love and support duringthe creation of this work.

161

Curriculum vitae

Persönliche Daten

Ralf StubnerKapellenstraße 12

91056 Erlangen

Geboren am 9.2.1973 in Schwabach

Ausbildung

09/1979 – 07/1983 Christian-Maar-Schule, Schwabach

09/1983 – 07/1992 Adam-Kraft-Gymnasium, Schwabach

09/1992 – 10/1993 Zivildienst beim Landesbund für Vogelschutz inBayern e. V., Hilpoltstein

11/1993 – 07/1995 Grundstudium Physik an derFriedrich-Alexander-Universität Erlangen-Nürnberg

09/1995 – 04/1996 Studienaufenthalt an der University of St Andrews,Scotland

05/1996 – 05/1999 Hauptstudium Physik an derFriedrich-Alexander-Universität Erlangen-Nürnberg

05/1999 Diplom in Physik, Thema der Diplomarbeit:Verallgemeinerung der k·p-Theorie für periodischeStörungen

seit 10/1999 Wissenschaftlicher Mitarbeiter amLehrstuhl für theoretische Festkörperphysik,Friedrich-Alexander-Universität Erlangen-Nürnberg

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