Electronic excitations: density-functional versus many...

REVIEWS OF MODERN PHYSICS, VOLUME 74, APRIL 2002

Electronic excitations: density-functional versus many-bodyGreen’s-function approaches

Giovanni Onida*

Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica dell’ Universita diRoma ‘‘Tor Vergata,’’ Via della Ricerca Scientifica 1, I-00133 Roma, Italy

Lucia Reining†

Laboratoire des Solides Irradies, UMR 7642 CNRS-CEA, Ecole Polytechnique,F-91128 Palaiseau, France

Angel Rubio‡

Departamento de Fısica de Materiales, Facultad de Quımicas, Universidad del Paıs Vasco/Euskal Herriko Unibertsitatea, Centro Mixto CSIC-UPV/EHU and Donostia InternationalPhysics Center (DIPC) 20018 San Sebastian/Donostia, Spainand Laboratoire des Solides Irradies, UMR 7642 CNRS-CEA, Ecole Polytechnique,F-91128 Palaiseau, France

(Published 7 June 2002)

Electronic excitations lie at the origin of most of the commonly measured spectra. However, thefirst-principles computation of excited states requires a larger effort than ground-state calculations,which can be very efficiently carried out within density-functional theory. On the other hand, twotheoretical and computational tools have come to prominence for the description of electronicexcitations. One of them, many-body perturbation theory, is based on a set of Green’s-functionequations, starting with a one-electron propagator and considering the electron-hole Green’s functionfor the response. Key ingredients are the electron’s self-energy S and the electron-hole interaction. Agood approximation for S is obtained with Hedin’s GW approach, using density-functional theory asa zero-order solution. First-principles GW calculations for real systems have been successfully carriedout since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeterequation, via a functional derivative of S. An alternative approach to calculating electronic excitationsis the time-dependent density-functional theory (TDDFT), which offers the important practicaladvantage of a dependence on density rather than on multivariable Green’s functions. This approachleads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than afour-point, interaction kernel. At present, the simple adiabatic local-density approximation has givenpromising results for finite systems, but has significant deficiencies in the description of absorptionspectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, andappreciable distortions of the spectral line shapes. The search for improved TDDFT potentials andkernels is hence a subject of increasing interest. It can be addressed within the framework ofmany-body perturbation theory: in fact, both the Green’s functions and the TDDFT approaches profitfrom mutual insight. This review compares the theoretical and practical aspects of the two approachesand their specific numerical implementations, and presents an overview of accomplishments and workin progress.

CONTENTS

I. Introduction 602A. Motivation from experiments 603B. Theoretical framework 603

1. Overview 6032. Effective Hamiltonians and effective

interactions 604II. TDDFT and Green’s-Function Approaches:

Common Ingredients 608

*Present address: Istituto Nazionale per la Fisica della Mate-ria, Dipartimento di Fisica dell’ Universita di Milano, via Ce-loria 16, I-20133 Milano, Italy. Electronic address:[email protected]

†Electronic address: [email protected]‡Electronic address: [email protected]

0034-6861/2002/74(2)/601(59)/$35.00 601

A. Density-functional theory for the ground state 608B. The inverse dielectric function and local-field

effects 609C. Four-point kernels and effective two-particle

equations 611III. Green’s-Function Theory 613

A. The concept of Green’s functions and the self-energy 6131. Connection to spectroscopic measurements 6142. Dyson’s equation: the self-energy operator

S 615B. Hedin’s equations 617C. The iterative approach 617

1. First iteration step: the GW approximation 6182. Second iteration step: vertex correction in

P and S 619IV. Green’s Functions in Practice 621

A. Calculations of one-particle excited states 6211. GW calculations 621

©2002 The American Physical Society

602 Onida, Reining, and Rubio: Density-functional vs many-body

a. Evaluation of the Green’s function G 621b. Evaluation of the screened Coulomb

interaction W5«21v 622c. Application of the self-energy S 622

2. Comparison of GW and DFT results 624B. Calculations of two-particle excited states 625

1. Independent-quasiparticle approximation 6252. Electron-hole attraction 6253. The effects of the electron-hole interaction 6274. Different levels of sophistication 630

V. Time-Dependent Density-Functional Theory 631A. Formalism 631B. Excitation energies in TDDFT 632C. The exchange-correlation kernel fxc 635

VI. TDDFT Versus Bethe-Salpeter 638A. The equations 638B. The limit of isolated electrons 640C. Relaxation and correlation 643D. Comparison of some applications 643

1. Common ingredient: the bare Coulombinteraction 643

2. Exchange and correlation effects 645a. Some applications to finite systems 645b. Some applications to extended systems 647

VII. Conclusions—Frequently Asked Questions andOpen Questions 647

Acknowledgments 650Appendix A: Exact Properties of fxc 650Appendix B: Derivation of the Equations Common to theTDDFT and Bethe-Salpeter Approaches 652

1. Dyson-like equation for the macroscopicdielectric function 652

2. Effective two-particle equations 653Appendix C: An fxc from the Bethe-Salpeter Approach 654References 654

I. INTRODUCTION

In every spectroscopic experiment one perturbs thesample (by incoming photons, electrons, etc.) and mea-sures the response of the system to this perturbation. Inother words, the system is excited. Therefore it is in gen-eral not sufficient to calculate ground-state properties inorder to interpret or predict results of experiments likephotoemission, electron-energy loss, absorption, etc.

Direct and inverse photoemission and absorption canbe taken as the prototype spectroscopies which onewould like to describe in this context. They are sche-matically depicted in Fig. 1. In the photoemission pro-cess the system absorbs a photon hn , and an electron isejected whose kinetic energy Ek is then measured atsome distance. If one considers this photoelectron to becompletely decoupled from the sample, energy and mo-mentum conservation allow one to deduce the change intotal energy of the sample, which is interpreted as theenergy level of the ‘‘hole,’’ i.e., the level that was for-merly occupied by the photoelectron. Hence as a firstapproach one can state that photoemission measures thedensity of occupied states. By analogy, inverse photo-emission yields information about the density of unoc-cupied states. In absorption experiments, an electron isexcited from an occupied state into a conduction state.This process looks, at first glance, like the sum of a pho-

Rev. Mod. Phys., Vol. 74, No. 2, April 2002

toemission and an inverse photoemission experiment(creation of a hole and an electron). Instead, as shown inFig. 1, in the absorption measurement the excited elec-tron remains inside the system and cannot be supposedto be a free electron decoupled from the others. Hencewhereas direct and inverse photoemission results are of-ten already well described by the density of occupiedand unoccupied states, respectively, one realizes that (i)in absorption, the joint density of occupied and unoccu-pied states must be considered, (ii) even over a smallrange of excitation energies, transition probabilities canvary considerably and must be taken into account, and(iii) the excited electron and the hole cannot be treatedseparately, since the electron feels the presence of thehole. Point (iii) constitutes the main difficulty for a cor-rect description of this type of experiment.

The interpretation of photoemission spectra as a den-sity of occupied states is linked to the picture of inde-pendent electrons which occupy some well-defined en-ergy level in the system. Of course, electrons are notindependent, and it is clear that, for example, an elec-tron that leaves the sample will lead the remaining elec-trons to relax. This relaxation energy and otherquantum-mechanical contributions must be taken intoaccount if energy differences are to be calculated cor-rectly. In other words, in photoemission the single-electron energy levels are renormalized by the presenceof the other electrons. One can then still retain the pic-ture of one-particle energy levels, but these particles arequasielectrons and quasiholes, i.e., they contain the ef-fects of all the other particles (Landau, 1957a, 1957b,1959). They can still be described by a sort of one-particle Schrodinger equation, which does, however,contain rather complicated effective potentials (alsocalled optical potentials) reflecting these interactions.Moreover, it is clear that, in the case of absorption, evena very sophisticated one-quasiparticle Schrodinger equa-tion would be inadequate, since the quasielectron andquasihole must be described simultaneously, requiring

FIG. 1. Schematic representation of the excitations involved indirect photoemission, inverse photoemission, and absorptionspectroscopies. Photoemission can be resolved in angle, spin,and time; absorption can be resolved in polarization and time.This allows a direct probing of the electronic and structuralproperties of bulk and low-dimensional samples including dy-namical effects. DE5Ef2Ef8 , apart from phonons and radia-tive losses.

603Onida, Reining, and Rubio: Density-functional vs many-body

an effective two-particle equation. Progress in the theo-retical description of spectroscopy is in fact very oftenlinked to progress in finding better effective one- or two-particle Hamiltonians.

One important motivation for seeking more precisetheoretical descriptions is, of course, the fact that dis-crepancies arise between experimental spectra and thespectra calculated at some approximate level. This mo-tivation becomes stronger as more precise experimentsbecome available.

A. Motivation from experiments

The excitations considered in the present work con-cern electrons, mainly in the valence energy region.Most of the theoretical tools presented in the followingare general enough to be used also for the study of ex-citations involving core electrons, although in that caseone is frequently interested in more complex phenom-ena such as the Auger effect (Verdozzi et al., 2001),which is not explicitly treated in this review. Traditionaltechniques for the study of valence electron excitationsuse either photons (visible and ultraviolet absorption,transmission and reflectivity spectra), electrons(electron-energy-loss spectra), or both (electron photo-emission and inverse photoemission spectra).

During the last two decades, important improvementsin many of these techniques have been achieved, due tothe ongoing substantial progress in obtaining (a) highspectral and spatial resolution, brightness; (b) shortmeasurement times (scale of femtoseconds); (c) highspatial coherence; and (d) low temperatures (Smith,2001). Large contributions come from the progressmade at synchrotron-radiation sources and with ultrafastlasers. In particular, photon beams in the soft-x-ray en-ergy region (extreme ultraviolet) are increasingly used,due to the availability of high-brilliance sources (Smith,2001). Several striking examples can be found in the re-cent literature. Femtosecond lasers yield informationabout elementary electronic processes occurring at sur-faces on time scales from pico- to femtoseconds, whichare relevant for potential technological applications. Forexample, an electronic excitation is the initial step inmany chemical reactions, and the energetics and lifetimeof this process directly govern the reaction probability.Hence chemical selectivity can be obtained through anactivation of the desired reaction via a femtosecondelectronic excitation (Sundstrom, 1996; see Diau et al.,1998, for an example). Time-resolved femtosecond pho-toemission spectroscopy has been used to gain insightinto electronically induced adsorbate reactions at sur-faces and kinetics of growth, and to monitor in real timeand with atomic resolution the dynamics of electrons(Plummer, 1997) and the motion of atoms at surfaces(Petek et al., 2000). Recent advances in ultrafast elec-tron diffraction have yielded direct imaging of transientstructures in chemical reactions (Ihee et al., 2001). X-raymicroscopy allowed the study of biological matter, likethe internal structure of a cell, with a spatial resolutionof 36 nm using photons in the ‘‘water window’’ energy


range, i.e., between 290 and 530 eV, where carbon ab-sorbs (excitation of the 1s electrons) and water is rela-tively transparent. Photoelectron spectra with high spa-tial resolution are at the basis of photoelectronmicroscopy, where a 20-nm resolution can be obtained(Nolting et al., 2000). The determination of the Fermisurface of very complex materials such as cuprate super-conductors (Zhou et al., 1999), the study of a Fermi gapopening in complex structures (Cepek et al., 2001), andthe measurement of the energy dependence of the line-width associated with photoemission peaks near theFermi energy (Valla et al., 1999a, 1999b) are now pos-sible because of the improved spectral resolution.

Often, it is clear that a simple one-particle picture isintrinsically inadequate for describing the processes oc-curring in the experiments. A good example is resonantphotoemission, where core-valence absorption and va-lence electron Auger emission interfere. Many aspectsof these experiments can in principle be described by thetheoretical tools discussed in this review. In the follow-ing, we define the common framework of the ap-proaches and then consider and compare them in detail.

B. Theoretical framework

1. Overview

This review treats approaches using or leading to thepicture of effective particles, i.e., ‘‘quasiparticles,’’ asoutlined above. There are, of course, other ways to treatthe many-body problem, e.g., the configuration-interaction approach of quantum chemistry (see, for ex-ample, Jensen, 1999; Szabo and Ostlund, 1983; Bonacic-Koutecky et al., 1990 for an application). The latter isbased on the minimization of the energy with respect tothe expansion coefficients of a trial many-body wavefunction, written as a linear combination of determi-nants. A description of configuration-interaction-basedtechniques is clearly beyond the scope of this paper, al-though they can be very efficient (e.g., in the determina-tion of higher excited states for point defects in SiO2 ;Raghavachari et al., 2002). However, the drawback ofthe configuration-interaction method with respect todensity-functional-based or Green’s-function-based‘‘quasiparticle’’ methods is its unfavorable scaling withthe system’s size.1

1In particular, full configuration-interaction calculations in-volve a number of determinants which grows exponentiallywith the size of the system. Simplified methods have been de-vised, such as the truncated configuration interaction, whichreduces the scaling to M6 (where M is the number of basisfunctions) when only determinants with a number of excitedelectrons lower than or equal to 2 are included. Other scalings,namely, M8 or M10, are obtained by also including triply orquadruply excited determinants. The approaches which will bediscussed in this review can exhibit a much better scaling, ac-cording to the system and to the numerical implementation(Benedict et al., 1998a; Bertsch et al., 2000; Hahn et al., 2002;Vasiliev et al., 2002).


The concept of quasiparticles is intimately linked tothat of band structure (or discrete energy levels in afinite system), and the corresponding one-electron–likeSchrodinger equations. Band-structure equations cancontain the complications of electron-electron interac-tions implicitly in empirical or semiempirical Hamilto-nians, as in the tight-binding approach, or they can treatthose effects explicitly. It is this latter point which is ofinterest here. We shall hence discuss the quasiparticleformalism of many-body perturbation theory. In addi-tion, we shall also discuss approaches based on the staticand time-dependent density-functional theory (DFT andTDDFT).

Static DFT (Hohenberg and Kohn, 1964) in the for-mulation of Kohn and Sham (1965) is a ground-statetheory in the form of an effective one-particle Schro-dinger equation (the Kohn-Sham equation). The Kohn-Sham eigenvalues are often interpreted as quasiparticleenergies, and their differences as optical excitation ener-gies, without formal justification. The Kohn-Sham eigen-values and eigenfunctions are also often used as startingpoint for further excited-state calculations. On the otherhand, DFT has been extended to time-dependent DFT(Runge and Gross, 1984), which is in principle an exacttheory for the description of neutral excitations (such asthose involved in absorption). In fact, TDDFT can bewritten in the form of an effective two-particle equation,which can be directly compared to the effective two-quasiparticle equation of many-body perturbationtheory.

In the last few years, several reviews have addresseddifferent aspects of the calculation of electronic excita-tions in both finite and infinite systems using either thequasiparticle picture (see, e.g., Aryasetiawan and Gun-narsson, 1998; Strinati 1988; Aulbur et al., 1999; Farid,1999a; Hedin, 1999; Rohlfing and Louie, 2000) or TD-DFT (see, for example, Gross et al., 1994, 1996; Casida1995, 1996; Dobson, Vignale, and Das, 1997; Rubioet al., 1997; van Leeuwen, 2001; Burke et al., 2002). Itemerges that, although both methods are in principleexact when applied to the appropriate problem, theypresent different drawbacks: in the context of many-body perturbation theory, the effective one- and two-quasiparticle Hamiltonians are essentially well estab-lished, but the methods become numericallyimpracticable for complex systems. TDDFT, on theother hand, could in principle lead to technically simplerequations, but its large-scale application is at presentprevented by the fact that a good general approximationfor the corresponding effective two-particle Hamiltonian(or equivalent quantities) has not yet been found. Thelarge increase of interest in theoretical spectroscopicanalysis from first principles suggests a joint venture inexploring both approaches. The aim is to reach an effec-tive method of handling excitations of many-electronsystems with an effort comparable to that of DFT-basedapproaches used for the calculation of ground-stateproperties.

In the present work we review some of the essentialphysics contained in the various approximations used to


describe spectroscopic measurements from first prin-ciples. We address in detail some frequently asked ques-tions related to the calculation of photoemission and op-tical spectra, including those which remain open andmay be solved in the near future. We attempt to give ageneral perspective of our present understanding of dif-ferent spectroscopies and to compare the many-bodyperturbation theory approaches—in particular thosebased on Hedin’s equations (Hedin, 1965)—with time-dependent density-functional theory. This comparisonwith an eye to practical applications, together with thefocus on the calculation of photon absorption andelectron-energy-loss spectra which involve two-quasiparticles (electron-hole) excitations, characterizesthe present work and differentiates it from other re-cently published reviews on Green’s-function methods,more focused on one-particle Green’s functions andsingle-quasiparticle excitations, such as those involved inphotoemission or inverse photoemission (e.g., Aryase-tiawan and Gunnarsson, 1998; Aulbur et al., 1999; Farid,1999a; Hedin, 1999).

Since this field of research is rather broad, and in or-der to make the discussions more focused and specific,we have restricted the review to the description ofpurely electronic excitations from valence states, fornonmagnetic systems. We have hence excluded the ex-plicit discussion of core-level spectroscopy (Almbladhand Hedin, 1983) and Auger spectroscopy (Verdozziet al., 2001), although the theory presented here remainsvalid in that energy range. We also exclude the problemof strong correlation and the methods that have beendeveloped to treat these problems in particular, e.g., thelocal-density approximation plus an on-site Hubbard re-pulsion method (LDA1U; Anisimov et al., 1997) andthe dynamical mean-field theory (Georges et al., 1996).

Unless otherwise stated, we shall use atomic unitsthroughout the paper (i.e., e25\5me51).

2. Effective Hamiltonians and effective interactions

The earliest attempts to cast the many-body probleminto the form of one particle moving in some mean (or,more generally, effective) field due to the electron-electron interaction are the Hartree (1928) and theHartree-Fock (Fock, 1930) approaches: a product or de-terminant ansatz for the many-body wave function al-lows one to minimize the total energy variationally, lead-ing to an effective one-particle Schrodinger equation. Itis interesting to note that the Hartree-Fock method al-ready yields good results, like total energy differences,in certain systems. However, the interpretation of theHartree-Fock eigenvalues as electron addition and re-moval energies does not lead to satisfactory agreementbetween theory and experiment. As a guideline we maytake the minimum direct quasiparticle gap at G in dia-mond, for which Mauger and Lannoo (1977) obtained avalue of Eg.15 eV in a self-consistent Hartree-Fockcalculation, instead of the experimental quasiparticlegap of 7.3 eV (inferred from optical experiments byRoberts and Walker, 1967).


The need for going beyond Hartree-Fock in electronicstructure calculations was in fact recognized early.Many-body correlation effects have been studied sincethe 1930s (Wigner, 1934); an important milestone is thework of Quinn and Ferrell (1958) on the homogeneouselectron gas and metals. A key role is played by theelectron self-energy, which is the proper exchange-correlation potential acting on an excited electron orhole. A sound basis for modern band-structure calcula-tions was laid by Hedin (Hedin, 1965; Hedin and Lun-dqvist, 1969) with the so-called GW approximation forthe self-energy, which was applied to the electron gas inthat work. The essential improvement offered by theGW one-particle Schrodinger equation over theHartree-Fock equation lies in the explicit description ofthe potential induced by the additional particle. In orderto understand this, one might ask why the eigenvalues ofthe Hartree-Fock equation do not yield a satisfactorydescription of quantities like the band structure, al-though Koopman’s theorem says that it is justified toidentify eigenvalues with total energy differences. Sowhat is missing, and which ingredients should reason-ably appear in a theory going beyond Hartree-Fock andallowing one to use the eigenvalues of some effectiveHamiltonian?

First, Koopman’s theorem supposes that the one-electron orbitals are frozen upon changing the numberof electrons. In order to go beyond this drastic approxi-mation, one has to include the relaxation of the orbitalsas a response to the addition of an electron or a hole tothe system. This effect is contained in the so-called delta-self-consistent-field calculations for finite systems, wherethe difference in total energy between two self-consistent calculations, for N and N61 electrons, is ob-tained explicitly. In other words, total energy differencesare calculated successfully within Hartree-Fock, becausein the calculation for N61 electrons the wave functionsare allowed to relax with respect to those of theN-electron calculation. This response of the system tothe additional electron or hole—the screening of the ad-ditional particle—can be reformulated in linear responsein terms of the dielectric matrix «, defined by the relation

Vtot~r!5E dr8«21~r,r8!Vext~r8! (1.1)

between the external potential Vext and the screened(total) potential Vtot . The total potential is the sum ofthe external potential and the potential due to the po-larization of the system induced by the external pertur-bation. It is the relaxation of the wave functions whichgives rise to this polarization, and one can therefore tryto use the concept of the dielectric matrix in order tocorrect the Hartree-Fock one-particle Schrodinger equa-tion, without calculating the relaxation explicitly as inthe delta-self-consistent-field approach. One has henceto calculate the induced potential Vind5Vtot2Vext dueto an additional electron, and acting on that electronitself. As a first step, one could assume that the addi-tional electron in a state n gives rise to an induced po-tential that is proportional to the perturbing ‘‘charge


density’’ ucn(r)u2 (as in the familiar picture of the imagecharge). However, whereas Hartree relaxation effectsare very important in small systems, in solids in a Blochpicture they are negligible, and hence this effect alonecannot lead to a satisfactory correction of the Hartree-Fock equation. Rather, one should take ucn(r)u2 as theprobability to find the additional electron in some pointr, and then, supposing that the additional electron is at r(i.e., taking correlation into account), calculate the in-duced potential in the same point. Then, in the corre-sponding equations ucnu2 is replaced by a d function, andthe potential which should be added to the Hartree-Fock equation turns out to be the so-called Coulomb-hole term 1

2 v(«2121), v being the bare Coulombinteraction.2

Of course, the change in the charge distribution dueto the additional, perturbing, electron will also affect theexchange term of the original Hartree-Fock equation,which now turns out to be screened by «21. These argu-ments lead to an effective one-particle Schrodingerequation for an additional electron (or hole) whichreads

S 2¹2

21E dr8 r~r8!v~r,r8!

112 E dr8v~r8,r!@«21~r,r8!2d~r,r8!# Dcn~r!

2E dr8W~r,r8!(s51

occ

cs* ~r8!cs~r!cn~r8!5encn~r!,

(1.2)

where W5«21v is the screened Coulomb interaction.This is the equation known as the static COHSEX (Cou-lomb hole plus screened exchange) approximation (He-din, 1965), where the terms added to the kinetic-energyoperator and the Hartree potential constitute the self-energy SCOHSEX. For diamond, Hybertsen and Louie(1986) have found that the energy difference betweenvalence and conduction bands is overestimated in theCOHSEX approximation by about 1 eV, which means,however, that most of the error of the Hartree-Fock re-sult cited above is removed by this correction.

To go further, one must take into account the fact thatthe response of a system to an external perturbation isfrequency dependent, which means that one has to in-troduce the concept of a response in time (dynamicalcorrelation and memory effects). In fact, the COHSEXpotential is nothing other than the static limit of theexchange-correlation self-energy calculated in the so-

2The factor 12 describes the fact that a charge e is taken from

infinity to a point r of the system or, equivalently, built up frome50. The part of the work coming from the induced chargewhich is needed to do this is *0

edqVind(q)5*0edqq(«2121)v

51/2e2* (W2v). This is nothing other than the adiabaticbuilding up of charge density (Hedin, 1965, 1999) of classicalelectrostatics.


called GW approximation (Hedin, 1965), which, aspointed out above, can be considered as the state-of-the-art tool for band-structure calculations today.

After its introduction for metals, the GW approxima-tion was also used early for semiconductors and insula-tors, first in static COHSEX (Brinkman and Goodman,1966; Lipari and Fowler, 1970; Brener, 1975a, 1975b),later using a plasmon-pole approximation in a model«21 (Bennett and Inkson, 1977; Inkson and Bennett,1978) and, starting from a Hartree-Fock calculation, us-ing a realistic frequency- and wave-vector-dependent di-electric matrix, in a basis of localized orbitals (Strinatiet al., 1980, 1982).

It was shown that the GW approach managed to cor-rect the largest part of the band-gap error; in fact, forthe example of the gap at G in diamond, Strinati et al.(1980) found a value of 7.4 eV, which is half of theHartree-Fock gap and very close to the experimentalone.3 The fact that the inclusion of dynamical effects inthe response functions leads to a reduction of the quasi-particle gap with respect to the static COHSEX result isa general finding, and the example of diamond showsthe typical order of magnitude of the dynamical effects(i.e., about 10–20 % of the experimental gap).

On the other hand, GW calculations are cumbersomeas compared, for example, to DFT ones. As pointed outabove, the Kohn-Sham equation also has the form of aneffective one-particle Schrodinger equation, and it istherefore tempting to use its eigenvalues as electron ad-dition and removal energies. However, this yields largeerrors (Lundqvist and March, 1983); in particular onefinds a strong underestimate of the band gap of semicon-ductors and insulators. For diamond, the difference ofKohn-Sham eigenvalues calculated in the local-densityapproximation (LDA) has yielded a gap at G of Eg55.51 eV (Hybertsen and Louie, 1985). The origin ofthis failure, whether it is mainly due to the wrong inter-pretation of the Kohn-Sham equation or to the LDA, isstill discussed today and will be studied in Sec. IV.A.2. Inany case, DFT is a good starting point for further calcu-lations, and starting from the work of Hybertsen andLouie (1985, 1986), and Godby et al. (1986, 1987, 1988),today ab initio GW calculations are mostly performedusing Kohn-Sham results as ingredients (to be precise,the Kohn-Sham eigenvalues and eigenfunctions are usedto construct the self-energy of the GW form). Thesefully ab initio GW calculations generally yield very goodagreement, most often better than 10–15 %, betweenthe experimental and the calculated band structure,apart from the case of strongly correlated systems.Again for the example of diamond, Hybertsen andLouie (1985) and Godby et al. (1987) have performed abinitio GW calculations and obtained a quasiparticle gapat G of Eg57.38 eV and Eg57.26 eV, respectively,which compare well with the experimental value of 7.3eV (Roberts and Walker, 1967).

3To be precise, this calculation also includes a vertex correc-tion beyond the usual GW .


The GW approach thus yields very gratifying resultsconcerning the band structure, but having a good bandstructure is not enough when one is interested in spec-troscopies like absorption, where one creates a neutral,i.e., electron-hole type of excitation. One is in that casetalking about the response of the system to a (time-dependent) external potential, which implies that one isinterested in the dielectric matrix itself, as defined in Eq.(1.1) in its obvious time-dependent extension. This re-sponse, when looked at from the point of view of thesystem in its ground state, implies a redistribution ofcharge (wave functions), in other words, depletion ofcharge (holes) or accumulation of charge (electrons) insome places. This creation of electron-hole pairs can beseen explicitly in the corresponding equations. In fact, indirect (r) space, the dielectric function for an electronsystem can be written as

«~r,r8,v!5d~r2r8!2E dr9v~r2r9!P~r9,r8,v! (1.3)

where v is the bare Coulomb interaction and where apolarizability operator P(r9,r8,v) has been introduced.When P is zero, the system is not polarizable and hencethe total potential is equal to the external one. Other-wise, P is, in general and on average, negative, i.e., act-ing against the external potential. The simplest approxi-mation for P is the independent (quasi)particle form:

PIQP~r,r8,v!ª(ij

~f i2f j!c i~r!c j* ~r!c j~r8!c i* ~r8!

v2v ij1ih.

(1.4)

Here v ij5(e j2e i), f i are Fermi occupation numbers,and (i ,j) label the states of energy e j and e i obtainedfrom some (for the moment, unspecified) equation forone-particle states. The small imaginary number ihleads to an imaginary part of P which is proportional tod(e j2e i2v); in other words, one can see the energyconservation for a photon v promoting an electron fromstate i to state j .

The approximation for the polarizability PIQP in Eq.(1.4) as a sum over independent transitions has the formof the random-phase approximation (RPA; Adler, 1962;Wiser, 1963). In fact, the RPA was originally meant todescribe a calculation performed within the (linearized)time-dependent Hartree approach, or Lindhard approxi-mation (Lindhard 1954), for a homogeneous electrongas. Later, the time-dependent Hartree approach wasshown by Ehrenreich and Cohen (1959) to be equivalentto the diagrammatic bubble expansion for the dielectricfunction in many-body perturbation theory (see alsoPines, 1963, and Pines and Nozieres, 1989). Here, theterm RPA form will in the following indicate that theapproximation (1.4) has been used for P . The ‘‘addi-tional’’ holes i and electrons j can of course be calcu-lated within the framework of the one-particle excita-tions outlined above. It turns out that the dielectricfunction evaluated using Kohn-Sham orbitals c i and ei-genvalues e i from LDA in Eq. (1.4) yields absorptionspectra that are in quantitative, and sometimes even inqualitative, disagreement with experiments (Cohen and


Chelikowsky, 1988). Since the factor d(e j2e i2v) im-plies that absorption occurs at eigenvalue differences,this has been in part attributed to the fact that the Kohn-Sham eigenvalues underestimate the quasiparticle gap,which leads to a redshift of the theoretical spectrum withrespect to the experimental one. In fact, for example, ina small sodium cluster Na4 the first allowed transitionbetween Kohn-Sham orbitals occurs at 1.1 eV (Onidaet al., 1995), whereas the first experimental absorptionpeak is seen at 1.8 eV (Wang et al., 1990).

However, even when the quasiparticle gap problem iscorrected by replacing the Kohn-Sham eigenvalues byquasiparticle energies calculated, for example, in theGW approximation (this approach will be namedGW-RPA in the following), important discrepancieswith experiment are found. In general, the redshift prob-lem becomes a blueshift problem—in the case of Na4 ,the first main absorption peak is shifted to 3.3 eV. Thisproblem is not due to a failure of the GW approach. Infact, for Na4 the ionization potential, i.e., the creation ofa hole, is correctly described within GW (Reining et al.,2000). In the latter case, however, one considers the ad-dition of a hole with respect to the ground state and not,as in the case of absorption, the addition of a hole andthen the addition of an electron in the presence of thathole. An additional correction term to the potential ishence needed in the case of absorption, expressed by thechange of the potential upon presence of the hole, and aself-consistent treatment of hole and electron altogether.One has therefore to expect an effective two-particleequation for the response function, including electron-hole interaction effects. This equation—the Bethe-Salpeter equation—is actually found via the rigoroustreatment of the problem within Green’s-function theory(see Sec. IV.B). It requires the correct calculation of thequasielectron and the quasihole (for example, via GW)and contains an interaction term that mixes the formerlyindependent transitions. The electron-hole interactionhence modifies the expression for the polarizability inEq. (1.4). These excitonic effects have been well knownfor some time, starting from a description based on asuperposition of atomic excitations (Frenkel, 1931a,1931b; Peierls, 1932) and the insertion of the concept of‘‘excitons’’ into the band picture (Wannier, 1937), andextending to detailed work in the 1950s like that ofHeller and Marcus (1951), or of Haken (1956, 1957). Anoverview of the intense activity in the field at that timecan be found in the book of Knox (1963).

Sham and Rice (1966) established the link betweenexciton models and many-body theory, by deriving theeffective mass approximation from the Bethe-Salpeterequation. The first realistic nonmodel calculation for va-lence excitons in a solid based on the Bethe-Salpeterequation was performed by Hanke and Sham, first usingthe time-dependent Hartree-Fock approximation (whichis equivalent to introducing an unscreened electron-holeinteraction; Hanke and Sham 1974, 1975) and later alsointroducing a static electron-hole screening (Hanke andSham, 1980). Their calculation, based on the linear com-bination of atomic orbitals, managed to explain the


qualitative features of the line shape of the spectra ofdiamond and silicon. The Na4 cluster was the first real-istic system to be treated in the ab initio scheme startingfrom DFT and going through GW (Onida et al., 1995),with a particularly strong effect of the electron-hole in-teraction due to the finite size of the cluster: the firstabsorption peak was moved back by 1.5 eV from its GWposition, with a result close to the experimental one. Anincreasing number of such ab initio calculations of exci-tonic effects in finite and extended systems (see, for ex-ample, Albrecht et al., 1997, 1998a, 1998b; Benedictet al., 1998a, 1998b; Rohlfing and Louie, 1998a, 1998b)have appeared since then, generally showing a big im-provement over the results of RPA and GW-RPA cal-culations.

Of course these two-particle electron-hole calcula-tions are relatively cumbersome, and it has always beentempting to stay within the framework of DFT. Onecould in fact consider the DFT exchange-correlation po-tential Vxc as an approximation to the self-energy and,knowing that self-energy and electron-hole interactioneffects partially cancel each other,4 hope that Vxc to-gether with its variation with density (the functional de-rivative dVxc /dr , i.e., the so-called exchange-correlationkernel fxc), could yield improved optical spectra. Thisresembles the theoretically more rigorous TDDFT(Runge and Gross, 1984; Gross and Kohn, 1985), wherethe response of the electrons to a time-dependent exter-nal potential is derived by searching the extrema of thequantum-mechanical action functional, which leads to atime-dependent Kohn-Sham equation. It has turned outthat TDDFT in the adiabatic local-density approxima-tion (TDLDA) often yields good spectra for finite sys-tems. For Na4 and other sodium clusters, Vasiliev et al.(1999) have calculated absorption spectra in excellentagreement with experiment, i.e., reproducing peak posi-tions and relative heights within .10% (0.2 eV). This isnot in contradiction with the fact that, as mentionedabove, the Kohn-Sham eigenvalue differences aresmaller than the absorption energies: TDDFT directlydescribes the evolution of the density under the influ-ence of a time-dependent external potential, whichmeans that both the potential (yielding the Kohn-Shameigenvalues) and variations of the potential (which canshift the excitation energies) are taken into account.TDLDA is also successful in describing electron-energy-loss spectra for bulk metals and semiconductors, but im-proves only very slightly the absorption spectra of solidswith respect to RPA calculations (Gavrilenko and Bech-stedt, 1996). Therefore, in the field of TDDFT the maineffort is directed toward finding better approximationsfor both the exchange-correlation potential and its varia-tion with the density (i.e., fxc). To this end, one has tounderstand the main differences between finite and infi-nite systems, as well as between different spectroscopies.

4See the example of Na4 in Onida et al. (1995), and the re-sults of the jellium model for metal clusters by Pacheco andEkardt (1997; also Ekardt and Pacheco, 1995).


In fact, for example, absorption and electron- energy-loss spectra are derived in different ways from the samedielectric function.

In the following we shall therefore concentrate inmore detail on the dielectric function and its link tospectroscopy. In particular, we shall outline those fea-tures that are common to both TDDFT and Bethe-Salpeter approaches, before treating the two approachesseparately and finally comparing them.

II. TDDFT AND GREEN’S-FUNCTION APPROACHES:COMMON INGREDIENTS

TDDFT is an extension of static ground-state density-functional theory. Also, Green’s-function calculations of-ten start from DFT results. In fact, concerning theground state, calculations based on density-functionaltheory using simple density functionals predominate to-day. Current DFT results have often surpassed thosefrom standard ab initio quantum chemistry techniqueslike Hartree-Fock, configuration interaction, etc., whichmay require heavy compromises in their technical real-ization for systems that are not very small (e.g., fullconfiguration-interaction calculations can hardly bedone for more than five electrons; Jensen, 1999). De-spite the failures of DFT due to the approximationsmade for the exchange-correlation contributions, its usecontinues to increase because of its wide applicabilityand its favorable scaling with the number of atoms.

In the following, we briefly review the basic ingredi-ents of DFT in view of its use as a starting point forspectroscopy calculations. For a more detailed descrip-tion we refer the reader to any of the numerous compi-lations published in recent years, e.g., Lundqvist andMarch (1983); Dreizler and Gross (1990); Seminario(1996).

A. Density-functional theory for the ground state

The ground-state energy of a system of interactingelectrons in an external potential can be written as afunctional of the ground-state electronic density. Com-pared to conventional quantum-chemistry methods thisapproach is particularly appealing, since it does not relyon a complete knowledge of the N-electron wave func-tion but only on the electronic density. Of course, al-though the theory is exact, the energy functional is un-known and has to be approximated in practicalimplementations. Today’s DFT starts with the theoremsof Hohenberg and Kohn (1964) for a search of the elec-tronic ground state of an isolated system of N interact-ing electrons in an external potential Vext(r). The firsttheorem (‘‘the density as the basic variable in the elec-tronic problem’’) establishes that the external potentialVext(r) is a functional of the charge density r(r), withinan additive constant. The second theorem establishesthe ‘‘energy variational principle for the density.’’ Thesetwo theorems show that the problem of solving themany-body Schrodinger equation for the ground state


can be exactly recast into the variational problem ofminimizing the Hohenberg-Kohn functional with respectto the charge density.

In practice, the available approximations for kineticenergy and exchange-correlation density-based func-tionals give moderate quantitative agreement with ex-perimental data (Dreizler and Gross, 1990). A much im-proved strategy has been presented by Kohn and Sham(1965) using orbital variables. The fundamental assump-tion is to introduce a reference system of noninteractingelectrons in an external potential Veff

KS(r) such that theground-state charge density of the physical system coin-cides with that of the reference system.5 With the intro-duction of the noninteracting system, the variationalproblem on r(r) is thus finally reformulated in terms ofthe following set of self-consistent Kohn-Sham equa-tions:

F212

¹21VH~r!1Vxc~r!1V0~r!Gc i~r!5« ic i~r!

(2.1)

with the density given by r(r)5( i51N uc i(r)u2. Thus Veff

KS

5VH1Vxc1V0 , where VH is the Hartree potential,Vxc(r)5 dExc@r#/dr(r) is the exchange-correlation po-tential that contains all many-body effects, and V0 is anexternal potential stemming, for example, from theelectron-ion interaction.

The simplest approximation (still widely used) to Excis the local-density approximation (Kohn and Sham,1965). The approximation is based on using theexchange-correlation energy density exc

hom of the homo-geneous electron gas (Ceperley and Alder, 1980; Ortizet al., 1999), namely, Exc

LDA@r#5*exchom

„r(r)…r(r)dr.Hence one replaces the inhomogeneous electron systemat each point r by a homogeneous electron gas havingthe density of the inhomogeneous system at r. The ra-tionale for this approximation is in the limit of slowlyvarying density. Unexpectedly, the domain of applicabil-ity of the LDA has been found to go much beyond thenearly free-electron gas and accurate results can be ob-tained for very inhomogeneous systems. Improvementsover the LDA have been found, e.g., by the generalized-gradient approximations, in which the exchange-correlation energy density is a function not only of theelectron density, but also of its gradient (see, e.g., Per-dew et al., 1996). Further improvements over thegeneralized-gradient functional should go in the direc-tion of nonlocal functionals, in order to describe betterthe inhomogeneity of the exchange-correlation hole, asin the case of the exact exchange potentials (Gross et al.,1996).

The issue of calculating electronic excitations, how-ever, goes beyond the problem of finding a good ap-proximation to the ground-state exchange-correlation

5This definition is meaningful if the charge density is nonin-teracting v representable, so that one has a unique definitionof the total energy Kohn-Sham functional.


functional. The problem is intimately linked to the factthat the one-particle eigenvalues in the Kohn-Shamtheory have often been used, as well, to discuss the ex-citation spectra of solids, molecules and atoms. Al-though there is no rigorous justification for a direct iden-tification of Kohn-Sham eigenvalues with quasiparticleenergies, the formal resemblance between Kohn-Shamand quasiparticle equations, such as the COHSEXequation [Eq. (1.2)], where S(r,r8,v) replaces d(r2r8)Vxc(r) in Eq. (2.1) explains why the Kohn-Shamequation can be considered at least as a starting point.

As pointed out above, once some one-particle–liketheory such as Kohn-Sham has been established, re-sponse functions and spectra can be constructed, for ex-ample in the approximation (1.4). This is the subject ofthe next section.

B. The inverse dielectric function and local-field effects

As discussed above, information about the electronsand holes, and about their interaction, is contained inthe dielectric function via the polarizability P of Eq.(1.3). However, further considerations are necessary inorder to obtain spectra. In fact, independently of how Pand « are calculated, the latter usually has to be in-verted, since one is concerned with the total potentialVtot5«21Vext for a given external potential Vext , andnot vice versa. Equivalently, the induced charge is givenby r ind5xVext , where the reducible polarizability x and«21 are linked by

«21~r,r8,v!5d~r2r8!1E dr9v~r2r9!x~r9,r8,v!. (2.2)

In the following, it is useful to describe explicitly twotypes of spectra containing neutral excitations, namely,photon absorption and electron-energy-loss spectra(EELS). In electron-energy-loss experiments, an elec-tron impinges on the sample and loses energy by excit-ing electron-hole pairs, plasmons, and other high-ordermultipair excitations. This energy loss is given by theimaginary part of the integral of the potential created bythe electron, and the induced charge. When the poten-tial Vext due to an electron is taken proportional to aplane wave, this leads, for a momentum transfer q, tothe loss function

L~v!}2ImS E drdr8e2iqrx~r,r8,v!eiqr8D , (2.3)

which is hence determined by the Fourier transform ofthe inverse dielectric function 2Im@«21(q,q,v)# .

In a finite system, x also yields the photoabsorptioncross section s, via

s~v!54pv

cIm a~v!, (2.4)

where c is the velocity of light and Im a(v) is the imagi-nary part of the dynamical polarizability,

a~v!52E drdr8Vext~r,v!x~r,r8,v!Vext~r8,v!. (2.5)


In particular, for a dipolar external field Vext along the zdirection, the corresponding component of the s tensorreads

szz~v!524pv

cIm E drdr8zx~r,r8,v!z8, (2.6)

which can also be obtained from the long-wavelengthlimit of «21: s(v)5limq→0 (v/c) Im@v(q)x(q,q,v)#along the field direction. The other components of thetensor can be similarly obtained. Note that in molecular-beam experiments, where the molecules are randomlyoriented, one measures the trace of the tensor, that is,sexp(v)5 1

3(i5(x,y,z)sii(v).In the solid, one has to distinguish between the loss

spectra and the absorption spectra, which are insteaddescribed by the macroscopic dielectric function «M . Itsrelation to the microscopic dielectric function « of peri-odic crystals has been discussed by Adler (1962) andWiser (1963) as well as by Ehrenreich (1966):

«M~v![ limq→0

1

«G50,G85021

~q,v!, (2.7)

where «GG8(q,v)ª«(q1G,q1G8,v) is the Fouriertransform to reciprocal space of «(r,r8), G is a recipro-cal lattice vector, and q belongs to the first Brillouinzone. The optical absorption spectrum is then given bythe imaginary part «2(v) of «M(v). The dielectric con-stant «0 is the value of «M(v) at v50. If « is diagonal inG,G8, «M is just «M5limq→0«G50,G850 , i.e., the spatialaverage of the microscopic dielectric function. This is infact the case in the homogeneous electron gas. Other-wise, the microscopic dielectric function «(r,r8) will de-pend explicitly on the positions r and r8, and not simplyon the distance ur2r8u. This is translated into the factthat the dielectric matrix in reciprocal space is not diag-onal. The fact that all elements of the matrix contributeto one element of its inverse reflects the so-called localfield effects: these effects arise whenever the system un-der study is nonhomogeneous on the microscopic scale.In this case, for example, an external spatially constantperturbing field will induce fluctuations on the scale ofthe interatomic distances in the material, giving rise toadditional internal microscopic fields. These conceptswill be helpful in the next section [see the discussionafter Eq. (2.23)], and in Sec. VI, where TDDFT andBethe-Salpeter methods are compared, outlining theequations and the part of the kernel which are commonto both approaches (see in particular Sec. VI.D.1). It isclear that the local field effects should in principle beincluded in both absorption and loss spectra, since thecrucial quantity which appears is always the inverse di-electric function.

Local field effects can shift peak positions, and theycan be very important, as will be discussed later. How-ever, even when local field effects are taken into ac-count, the above conclusions about the need to go be-yond RPA or GW-RPA in the calculation of P , andhence «, remain valid (for example, the inclusion of localfield effects does not generally lead to a significant im-


provement of the absorption spectra of solids; Louieet al., 1975). It should in fact be stressed that the conceptof local field effects is independent of the approach usedto calculate P and «. It is therefore useful to summarizehere the notations that will be used throughout this re-view in order to characterize the different levels of ap-proximation: First, RPA and GW-RPA refer to a calcu-lation in which P has been approximated by the form(1.4), using DFT and GW eigenvalues, respectively, butthe spectrum has then correctly been constructed from«21. When local field effects are neglected in the inver-sion, in the dipole limit Eq. (1.4) leads to the well-knownEhrenreich and Cohen (1959) expression for the imagi-nary part of the macroscopic dielectric function in termsof the matrix elements of the velocity operator v be-tween valence and conduction states:6

Im$«M~v!%516p

v2 (v ,c

u^cvuvucc&u2d~ec2ev2v!. (2.8)

Independently of the quality of the states c entering Eq.(1.4), the electronic excitations described by Eq. (2.8)are restricted to the generation of noninteractingparticle-hole pairs. In the following this approximationwill be called an independent-particle-random-phase ap-proximation macroscopic dielectric function (Del Soleand Girlanda, 1993).

It is also interesting to point out that, on the otherhand, if the matrix inversion is properly taken into ac-count, the formerly independent transitions mix. Inother words, even if no electron-hole interaction is in-cluded in P , there is an effective electron-hole interac-tion showing up in «M . This is actually an electron-holeexchange term, as will be discussed later.

The inclusion of local field effects is, in principle,straightforward: Inverting « and placing the Fouriertransform of the inverse dielectric function in Eq. (2.7)directly yields the macroscopic dielectric function foreach frequency v. For our purpose, however, it is con-venient to use a different formulation for the macro-scopic dielectric function, which is suitable for a subse-quent inclusion of excitonic effects. Moreover, and moreimportantly here, it will enable us to compare in a con-cise way (i) electron-energy-loss and absorption spectraand, later, (ii) TDDFT and the Bethe-Salpeter approach.In fact, one can show (see Hanke, 1978 and AppendixB.1) that «M can be constructed from a modified re-sponse function P ,

«M~v!512 limq→0

@v~q!0PG5G850~q,v!# , (2.9)

where the matrix PG,G8 satisfies the Dyson-like screen-ing equation

P5P1P vP . (2.10)

6This is none other than Fermi’s golden rule in the dipoleapproximation and the one-electron picture for absorptionprocesses (Pines, 1963; Fetter and Walecka, 1971).


Here we have defined the modified Coulomb interactionv(q)G as follows:

v~q!G5H 0 if G50

v~q!G54p

uq1Gu2 else J .

In other words, the difference between v and v is ‘‘only’’in the G50 component.7 Because of the long range ofthe Coulomb interaction, this term is divergent for van-ishing q, which explains its importance, as will be dis-cussed below. v , on the other hand, does not diverge andplays the role of a correction term, namely, the localfield effects. In fact, from Eq. (2.10) it is clear that put-ting v to zero is equivalent to neglecting those effects.

It is interesting to compare Eq. (2.10) to the corre-sponding Dyson-like equation for x, which follows fromEqs. (2.2) and (1.3):

x5P1Pvx , (2.11)

the only difference is in fact the G50 part of the bareCoulomb interaction. This remark turns out to be ex-tremely important, as will be discussed later in the com-parison of TDDFT and Bethe-Salpeter results for finiteand infinite systems. One can put the common term vinto evidence by writing Eq. (2.11) as

xG,G85PG,G8

1(G9

PG,G9vG9xG9,G81PG,0v0x0,G8 . (2.12)

One can then immediately write down Eq. (2.12) and(2.10) for the case when local field effects are neglected,and take the macroscopic limit; two quantities P005P00and x005P00 /(12P00v0) are obtained, which describe,respectively, the absorption and the electron-energy-lossspectra of the system. Also in this simplified case it isclear that x and P are fundamentally different (x isscreened, but P is not), and that this fact is entirely dueto the seemingly tiny difference of the kernel of theDyson-like equations, i.e., the difference between v andv (see detailed discussions and applications in Sec.VI.D).

Until now, all considerations have been general, i.e., Phas not been specified. Only when talking explicitlyabout P will there be a difference between TDDFT andthe Bethe-Salpeter approach. However, in spite of—orbecause of—the differences that will arise, it is worth-while to anticipate a qualitative discussion of the struc-ture of the equations leading to the determination of P .This allows one to highlight the importance and conse-

7In a finite system, one can still have a wave vector k as a sumof a reciprocal lattice vector G and a vector q lying inside thefirst Brillouin zone, by creating a periodic array of ‘‘unit cells’’of vacuum, each of them containing a copy of the system. Thelimit of infinite volume (isolated systems) for those supercellscorresponds to the zero-volume limit for the first Brillouinzone, i.e., to an infinitely dense G-space.


quences of induced local and nonlocal potentials, with-out masking simple concepts by complicated many-bodyformulations. For the case of Hartree-Fock, many de-tails can be found in the reviews of McLachlan and Ball(1964) and Langhoff et al. (1972).

C. Four-point kernels and effective two-particle equations

An effective self-consistent Hamiltonian, which ismade time dependent through the time dependence ofthe wave functions, in the presence of a time-dependentexternal potential, allows one to calculate the induceddensity matrix to first order (Hedin and Lundqvist,1969):

r ind~r,r8,v!5 (n ,n8

cn~r!

3~fn2fn8!^nuH(1)~v!un8&

en2en82vcn8

* ~r8!,

(2.13)

where en and cn(r)5^run& are the eigenvalues andeigenfunctions of the unperturbed Hamiltonian. Then xis determined through

r ind~r,r!5E dr8x~r,r8!Vext~r8!. (2.14)

The perturbation H(1) not only is given by the externalpotential Vext , but also contains self-consistently the in-duced potential, which is proportional to the induceddensity matrix (schematically Vind5r inddVeff /dr).

A typical effective Hamiltonian will have an effectivepotential Veff(r1 ,r2 ,t) that has some contributions thatare local and depend on the density, i.e.,Vloc$@r( rr, t )# ,r1 ,t%d(r12r2) (like the Hartree potentialor the DFT exchange-correlation potential), and othersthat are nonlocal but directly proportional to the densitymatrix (like the Hartree-Fock exchange potential):r(r1 ,r2 ,t)w(r1 ,r2), where w(r1 ,r2) is some generalizedinteraction. For simplicity here we do not consider morecomplicated time-dependent interactions, but includethe possibility of memory effects in Vloc , i.e., Vloc canbe a functional of the density at all (past) times t . In thiscase, the induced potential is

Vind~r1 ,r2 ,t !5E dr3dr4dt8Fd~r12r2!d~r32r4!

3dVloc~@r~ rr, t !# ,r1 ,t !

dr~r3 ,r3 ,t8!

1d~r12r3!d~r22r4!w~r1 ,r2!d~ t2t8!G3r ind~r3 ,r4 ,t8!

ªE dr3dr4dt8K~r1 ,r2 ,r3 ,r4 ,t ,t8!

3r ind~r3 ,r4 ,t8!, (2.15)


which defines the four-point kernel K . Equations (2.13)and (2.15) can be put into a closed form for the induceddensity matrix, by introducing the four-point function4x0 [see Eq. (B16)] such that Eq. (2.13) becomes

r ind~r1 ,r2 ,v!5E dr3dr4dr5dr6

3~124x0K !21~r1 ,r2 ,r3 ,r4 ,v!

34x0~r3 ,r4 ,r5 ,r6 ,v!

3Vext~r5 ,v!d~r52r6!. (2.16)

Hence from Eq. (2.14)

x~r1 ,r2 ,v!5E dr3dr4~124x0K !21~r1 ,r1 ,r3 ,r4 ,v!

34x0~r3 ,r4 ,r2 ,r2 ,v!. (2.17)

An equation of this kind is Eq. (4.9), derived below forthe particle-hole response function in the Bethe-Salpeter scheme. Another one is the TDDFT equation.Note that in this case, since there are only local poten-tials, the d functions in the kernel (2.15) allow us tocontract all equations immediately and to work withtwo-point functions from the beginning (see, e.g.,Bertsch and Tsai, 1975; Bertsch and Broglia, 1994). Inparticular, the four-point function 4x0 can immediatelybe replaced by the two-point function PIQP(r1 ,r2)54x0(r1 ,r1 ,r2 ,r2) defined in Eq. (1.4), as in the mainequation of TDDFT [Eq. (5.11)] in Sec. V.B. A compari-son of Eq. (2.17) and the four-point generalization ofEq. (2.11) allows one to get the four-point polarizability4P from the equation

4P54x014x04f 4P (2.18)

with4f~r1 ,r2 ,r3 ,r4!5K~r1 ,r2 ,r3 ,r4!24v~r1 ,r2 ,r3 ,r4!,

and8

4v~r1 ,r2 ,r3 ,r4!5d~r12r2!d~r32r4!v~r1 ,r3!.

The two-point P is then

P~r1 ,r2!54P~r1 ,r1 ,r2 ,r2!. (2.19)

Using the Green’s-function formalism which will be in-troduced below (Sec. III), x0 and 4x0 will be expressed,respectively, as 2iG0(12)G0(21) and2iG0(13)G0(42).

In general, Veff will of course contain the Hartree po-tential, and hence K will contain the variation of theHartree potential with respect to the density, dVH /dr ,which is just the contribution v appearing in the Dyson-like equation (2.11). This allows one to see the relationbetween the local field effects (see Sec. II.B) and thedensity variation of the Hartree potential: When localfield effects are neglected, all spatial frequencies of the

8The definition of 4v , which is used in the four-point equa-tions involving P , is analogous to that of 4v .


induced Hartree potential are neglected in the Dyson-like equation apart from the spatial frequency given bythe external potential, i.e., apart from the G50 contri-bution when the external potential is macroscopic.9 Infinite systems, macroscopic potentials do not play a role,and the distinction is meaningless—neglecting local fieldeffects becomes equivalent to completely neglecting theinduced Hartree potential. Therefore Eq. (2.6) is at thesame time proportional to the macroscopic loss functionand the to limit of Eq. (2.9) for the case of an infinitelydense G space.

To summarize, one will in general find a four-pointequation for S54x or S54P :

S54x014x0KS , (2.20)

where K contains (K5v1f ) or not (K5 v1f ) thelong-range term v(G50), depending on whether S54x

or S54P . Any density-matrix-dependent effective po-tential will lead to a four-point equation for S. In orderto get the two-point functions, S has to be contractedlike in the expression for the macroscopic dielectricfunction, given by

«M~v!512 limq→0

3Fv~q!E drdr8e2iq•(r2r8) 4P~r,r,r8,r8;v!G .(2.21)

If no nonlocal terms are present in Veff (e.g., when Veff5VHartree1Vxc

DFT), Eq. (2.20) can be contracted immedi-ately, and one can work with two-point functions only.

Generally, in order to solve Eq. (2.20), one has to in-vert a four-point function for each frequency. This hasbeen done even for solids, for example in the case of aFock term in the effective Hamiltonian, in the earlywork of Hanke and Sham (1974, 1975). However, for acomparison between TDDFT and the Bethe-Salpeterequation method it is more convenient to present theequations in a different way, also often used. Thisscheme, described in the following, has the advantage ofputting the two-particle nature of the problem into evi-dence. In fact, it has been shown (see Appendix B.2)that the four-point equation for S can be transformed toan eigenvalue problem involving the effective two-particle Hamiltonian:

H(n1n2),(n3n4)2p [~en2

2en1!dn1n3

dn2n4

1~fn12fn2

!K(n1n2),(n3n4) . (2.22)

The indices ni refer to the fact that matrix elementshave been taken with respect to four eigenfunctions ofthe starting effective static one-particle Hamiltonian.Hence H2p is diagonalized, and from its eigenvalues El

9The local field effects can also be included through a directcalculation of the variation of the Hartree potential in re-sponse to an electric field as done in density functional pertur-bation theory (Baroni et al., 2001 and references therein).


and eigenstates Aln1n2 the four- and two-point quantities

of interest are constructed. For example, the macro-scopic dielectric function is given by

«M~v!512 limq→0

v~q! (l ,l8

S (n1 ,n2

^n1ue2iq•run2&

3Al

(n1n2)

v2El1ihNl ,l8

21

3 (n3 ,n4

^n4ueiq•r8un3&Al8* (n3n4)

~fn32fn4

!D . (2.23)

Here Nl ,l8 is an overlap matrix (see Appendix B.2).Since it can be shown that the ni always appear as pairsconsisting of one occupied and one empty state (see Ap-pendix B.2), this reflects the very physical approach ofrewriting the problem in terms of mixing between differ-ent independent-particle transitions between occupiedand unoccupied eigenstates of a one-particle Hamil-tonian. Equation (2.23) should be compared to Eq. (2.8);the difference is caused by a change in excitation ener-gies [from (ec2ev) to El], but also, very importantly, bythe coefficients Al which mix the formerly independenttransitions. As pointed out above, even if only the Har-tree potential is used in the derivation of K , i.e., if oneconsiders only local field effects and no furtherexchange-correlation contributions, the electron-holeHamiltonian is not diagonal, and the coefficients Al willmix the transitions. This way of formulating the problemis hence also appealing because the knowledge of the Alallows one to interpret spectra in terms of mixing oftransitions.

One can thus define a general two-particle Hamil-tonian which (i) yields, depending on its interaction po-tential, the inverse dielectric function or the macro-scopic one and (ii) does so for systems that are describedby an effective one-particle Hamiltonian either depend-ing on just the density, or containing more complicatedfunctionals involving the density matrix.

The corresponding equations have been derived start-ing from the time-dependent density matrix in Eq.(2.13). In order to talk now in detail about the possibleeffective one-particle Hamiltonians and derived kernels,one is naturally led to rigorously introduce in the follow-ing a sort of time-dependent density matrix for electronsand holes, which will allow the effective Hamiltonianand its derived response functions to be obtained on thesame footing; one is led to introduce the Green’sfunction.10

10Note that the history of modern band-structure calculationsstarted with the Hartree theory, where the essential ingredientis the electronic density, r(r)5(scs* (r)cs(r). Exchange ef-fects (Hartree-Fock) are then included through the nonlocaldensity matrix r(r,r8)5(scs* (r8)cs(r), whereas to introduce atime dependence, the natural way is to do it in the phase of thewave functions, which leads to a time-dependent density ma-trix: r(r,r8,t82t)5(scs* (r8)cs(r)eies(t2t8). This quantity is ac-tually a sort of one-hole Green’s function, which will show upin the following in the more formal discussion of the problem.


III. GREEN’S-FUNCTION THEORY

The Green’s-function technique is indeed extremelyuseful: it allows one to tackle directly the problems ofcalculating excitation and ionization energies, ground-state energies, transition matrix elements, absorption co-efficients, and dynamical polarizabilities, as well as elas-tic and inelastic electron cross sections. Furthermore,self-consistent perturbation theories can be formulatedin terms of the Green’s function. The technique is fullydiscussed in many textbooks and we refer the reader tothose for details (Abrikosov et al., 1963; Hedin and Lun-dqvist, 1969; Fetter and Walecka, 1971; Landau and Lif-schitz, 1980), but in the following we outline the equa-tions needed in the discussion. For further reading aboutthe subtleties of the mathematical concepts involved, seethe review by Farid (1999a) where this many-body ap-proach is developed in detail together with the approxi-mations to Hedin’s equations discussed below.

A. The concept of Green’s functions and the self-energy

The one-electron Green’s function G is defined (atzero temperature) as an expectation value with respectto the many-electron ground state uN&,

G~xt ,x8t8!52i^NuTc~xt !c†~x8t8!uN&, (3.1)

where c(xt) is the field operator in the Heisenberg pic-ture, x stands for three space coordinates (r) plus onespin coordinate (j), and T is the time-ordering operator.In this equation, c†(x ,t)uN& represents an(N11)-electron state in which an electron has beenadded to the system at point r and time t . When t8,t ,the many-body Green’s function gives the probabilityamplitude to detect an electron at point r and time twhen an electron has been added to the system at pointr8 and time t8. When t8.t , the Green’s function de-scribes the propagation of a many-body state in whichone electron has been removed at point r and time t ,that is, the propagation of a hole.11 G is closely relatedto fundamental properties like charge and spin densityor the total energy, and derived spectra like photoemis-sion or Compton scattering. Thus, for example, thecharge density is obtained through

r~rt !52i limt→01 E G~xt ,xt1t!dj .

Similarly, the total electronic energy can be deducedthrough the Galitskii-Migdal formula (Galitskii andMigdal, 1958)

E512 E dx limt8→t1 F ]

]t2ih~x !GG~x ,t ,x8,t8!x8→x ,

(3.2)

11When time-reversal symmetry holds, the Green’s function issymmetric with respect to the interchange of the spatial coor-dinates. This general statement is also true for the density ma-trix.


where h(x) is the density-independent part of the effec-tive Hamiltonian. By inserting a complete set of (N11)- or (N21)-particle states (depending on the timeordering) between the field operators in Eq. (3.1), oneobtains the Lehmann representation of the Green’sfunction in terms of the amplitudes,

fs~x !5H ^Nuc~x !uN11,s& , es.m

^N21,suc~x !uN&, es,m(3.3)

and in terms of the real energies es5E(N11,s)2E(N) „or @E(N)2E(N21,s)#… for es.m (,m):

G~x ,x8,v!5(s

fs~x !fs* ~x8!

v2@es1ih sgn~m2es!#, (3.4)

where the small imaginary part h is needed for the con-vergence of the Fourier transform to frequency space.The poles of G hence correspond to electron additionand removal energies. In particular, the so-called spec-tral function A(v)ªu(1/p)Im G(v)u becomes

A~x ,x8;v!5(s

fs~x !fs* ~x8!d~v2«s!. (3.5)

In the noninteracting case, where uN&, uN11& and uN21& are simply Slater determinants, only states withuN11,s&5cs

†uN& (c† is a creation operator) lead to non-vanishing Lehmann amplitudes. These amplitudes andthe Lehmann energies are then equal to the eigenfunc-tions and eigenvalues of the corresponding one-electronHamiltonian, and the spectral function consists of a setof d peaks at those eigenvalues. Hence each peak corre-sponds to a particle. Furthermore, it can be easily seenthat the Green’s function reduces to the time-dependentelectron or hole density matrix introduced in Sec. II.C.When the electron-electron interaction is turned on it isno longer possible to write uN21& or uN11& simply interms of pure one-electron Slater determinants. Instead,the many-body wave function is a linear combination ofmany of them. For example, the (N11)-particle wavefunction can be expressed as uN11,s&5( lmalm

s cm† uNl& ,

where uNl& is an excited state of the N-particle system.Hence there will now be more nonvanishing contribu-tions to the spectral function (3.5). If these contribu-tions, merged together, form a clearly identifiable mainstructure, which can be thought to derive from a d peakwhen the interactions are switched off, one can stillwork in a particle-like picture, associating each peakwith a ‘‘quasiparticle.’’12 With respect to anindependent-particle d peak, for the quasiparticle one

12The concept of quasiparticles was introduced by Landau(Landau, 1957a, 1957b, 1959; Landau and Lifschitz, 1980) inthe Fermi-liquid theory of ordinary metals as a one-to-one cor-respondence between low-energy excitations of a free Fermigas and those of the interacting electron liquid. A quasiparticlecan be considered as the combination of a real particle (elec-tron or hole) and a cloud of virtual electron-hole pairs sur-rounding it. Through mutual interactions quasiparticles can de-cay into other quasiparticles, leading to a finite lifetime.


finds a spreading of the oscillator strength. Thereforethe broadening carries valuable information aboutmany-body correlation effects in the interacting system.

Another important quantity for the analysis of the op-tical spectra of interacting electron systems is the two-particle Green’s function, defined as

G~1,2,18,28!52^NuTc~1 !c~2 !c†~28!c†~18!uN& ,(3.6)

where space, time, and spin coordinates are indicated inabbreviated form by numbers, i.e., 15(r1 ,t1 ,j1). Thisfunction provides information about processes involvingtwo-particle transitions and their interaction through thetime ordering of the four times appearing in the equa-tion. For the purpose of this review, only a few specifictime orderings need be considered, in particular fort1 ,t18.t2 ,t28

G~1,2,18,28!52(s

Xs~1,18!Xs~2,28!

in terms of the hole-particle (Bethe-Salpeter) ampli-tudes Xs(1,18)5^NuTc(1)c†(18)uN ,s& and Xs(1,18)5^N ,suTc(1)c†(18)uN&. Also the case t1 ,t18,t2 ,t28 de-scribes particle-hole amplitudes, whereas other time or-derings will lead to particle-particle or hole-hole pair-ings. This point is relevant here because only for hole-particle pairings do the states belong to the N-particlesystem. Therefore one can define a specific hole-particleGreen’s function Ghp considering only those two timeorderings (t1 ,t18:t2 ,t28):

G~1,2,18,28!5Ghp~1,2,18,28!1other orderings.(3.7)

The ‘‘other orderings’’ terms correspond to poles of thetwo-particle Green’s function that differ from6@E(N ,s)2E(N)# and that are important for differenttypes of spectroscopies, e.g., Auger in the case of Ghh

(Cini, 1977, 1979).

1. Connection to spectroscopic measurements

In the Introduction we sketched the main spectro-scopic measurements that can be described by the ap-proaches presented in this review. It is now useful tobuild a bridge between those qualitative considerationsand the abstract mathematics of the previous subsection.

In fact, the building block for the description of anyspectrum involving the interaction of radiation and mat-ter is the probability P for an excitation from the initialstate uC0& (typically the ground state) to a set of finalstates uC f&. According to the measurement, the initial orfinal states contain an additional photon. This process iswell described by Fermi’s golden rule:

P52p(f

u^C fuDuC0&u2d~Ef2E0!, (3.8)

where D describes the perturbation due to the photonfield. For a one-photon process, neglecting quadratic


terms in the vector potential A, D5 (e/2mc) (Ap1pA). High photon fluences (multiphoton processes)are excluded here.

From the transition probability, one can construct thespectra. In the case of photoemission (see Fig. 1), onemeasures the photocurrent, which is proportional to theprobability per unit time of emitting a photoelectron ofmomentum k, when the target is irradiated with photonsof frequency v. Hence one looks explicitly at the photo-electron. As a consequence, the simple approximationsfor photoemission decouple the photoelectron from thetarget, usually treating it as a free propagating electronstate. One then finds that the photoemission spectrumcarries directly accessible information about the targetitself, namely, about its electronic structure, or to be pre-cise, the energy of holes (or electrons, for inverse pho-toemission). The final simple relation between the pho-toemission spectrum and the electronic structure of thesample is based on the classical three-step model (see, forexample, Almbladh and Hedin, 1983, and Kevan, 1992):the photoelectron is excited by the photon, loses energyon its way to the surface, and finally has to pass thesurface to propagate (as a free particle) to the detector.Only the first step, the excitation of the photoelectron, iscontained in the so-called intrinsic spectrum, whereaslosses on the way out of the target are called extrinsic.13

Starting from the transition probability [Eq. (3.8)],one can write down the photoelectron current of photo-emission, using the initial-state energy E05EN1v , i.e.,the sum of the N-particle ground-state energy and thephoton energy. In the intrinsic approximation, the finalstate uk;N21,f& is approximated by ck

†uN21,f& with en-ergy ek1EN21,f , where k and ek are the momentum andthe energy of the photoelectron. The photoelectron cur-rent reads then (Almbladh and Hedin, 1983)

Jk~v!5k

4p2 (f

d~ek2e f2v!

3Z^N21,fuck(ij

D ijc i†cjuN&Z2. (3.9)

Here e f5EN2EN21,f , and ( ijD ijc i†cj is the photon op-

erator in second quantization. Using the argument thatthe high-energy one-electron state k does not change thestate of the (N21)-electron system, and that it makes anegligible contribution to the virtual one-electron exci-tations contained in the ground state N (that is ckuN&50), one can write the photoelectron current as

Jk~v!5k

4p2 (ij

DkiAij~ek2v!D jk , (3.10)

where Aij are the matrix elements of the spectral func-tion (3.5) between one-electron orbitals,

Aij~ek2v!5E dxdx8c i* ~x !c j~x8!A~x ,x8,ek2v!.

(3.11)

13For high-energy photoelectrons this contribution is, in gen-eral, much smaller than the intrinsic part.


If one further assumes that the diagonal elements of Aare dominating (which is exact for independent elec-trons) and that the D are constant, the photoemissionspectrum is proportional to the trace of the interactingspectral function (also neglecting electron-phonon cou-pling and impurity effects).

From Eq. (3.5) one can see that the spectral functionhas a direct link to the density of available states for theaddition and removal of electrons. In fact, for indepen-dent particles the trace of the spectral function is pro-portional to the density of occupied and empty states ofthe system, respectively, and provides a good approxi-mation for the direct and inverse photoemission spectra.As we have seen, the same is true for the interactingelectron case: the trace of A(x ,x8;v) can be accessedexperimentally. In this way, the quasiparticle spectrum isdirectly measured in experiments such as direct, inverseand time-resolved two-photon photoemission (see Fig. 1for a schematic diagram of the first two). Still, final-state,extrinsic, and surface (transport) effects, neglected inthis discussion, can play an important role in some spe-cific experimental configurations.

The absorption coefficient, by contrast, is defined asthe ratio of the energy removed from the incident beamper unit time and per unit volume to the incident flux.This energy is absorbed thanks to transitions from theinitial to any final state, with the sole constraint of en-ergy conservation. There is, in general, no simple as-sumption concerning the final state as there is in photo-emission, and one has to deal with holes and electrons atthe same time. It would not be a good approximation forthe final state to decouple the excited electron from theothers, nor to describe it by a scattering (plane-wave)state. One has therefore to sum over all possibleN-particle excited states in order to obtain the absorp-tion coefficient. The energy differences appearing in thed function are now EN ,f2(EN1v), and the sum over allstates leads to a density-density correlation function (ordensity response function).14 The latter can be obtainedfrom the time-ordered (or retarded) two-particleGreen’s function Ghp(1,2,18,28) [Eq. (3.7)], by contract-ing some of its arguments and by taking suitable aver-ages or integrals of the resulting (microscopically vary-ing) functions. That is, the microscopic polarizability ofEq. (1.3) or the density response function entering inEqs. (2.2) and (2.5) are obtained from a Ghp where thelimit (18)→(11); (28)→(21) has been taken. In the

14Indeed, using the standard Kubo formula for the density-density response function (Abrikosov et al., 1963) x(x ,t ;x8,0)52iu(t)^Nu@r(x ,t),r(x8,0)#uN&, together with the clo-sure relation of N-particle states uN ,f& and the time evolu-tion of the density operator r(x ,t)5eiH0tr(x ,0)e2iH0t,one finds the response function x(x ,t ;x8,0)52iu(t)@( fBf(x)Bf* (x8)ei(EN2EN ,f)t2c.c.#, where Bf(x)5^Nur(x ,t)uN ,f& and EN ,f is the energy of the fth N-particleexcited state. Then its time Fourier transform has poles at thetrue excitation energies of the N-particle system, v56(EN2EN ,f).


RPA approximation, PIQP is derived by taking Ghp asthe product of two one-particle Green’s functions:G(1,2,18,28)5G(1,28)G(2,18), which becomesG(1,2)G(2,1) after the contraction. More details aregiven in Sec. IV.B.2.

2. Dyson’s equation: the self-energy operator S

The full one-particle Green’s function cannot be com-puted exactly, and one therefore needs a method to de-rive a physically sound approximation. In the simplecase of the Hartree-Fock approximation, G is writtendirectly in terms of the Hartree-Fock eigenvalues andeigenfunctions as GHF(x ,x8,v)5(scs(x)cs* (x8)/$v2@es1ih sgn(m2es)#%. The inverse of GHF is hencelinked to the Hartree-Fock Hamiltonian HHF:

~GHF!21~x ,x8,v!5v2HHF~x ,x8!. (3.12)

For the general case, it is therefore reasonable to writethe frequency Fourier transform of G21 as

G21~x ,x8,v!5$d~x2x8!@v2HH~x !#2S~x ,x8,v!%,(3.13)

where HH contains the kinetic-energy operator, the ex-ternal and the Hartree potential, and S is supposed toinclude the Fock exchange and the rest of the Coulombinteraction (correlation). This means that in this Dysonequation for G the exchange (the Fock term) and cor-relation contributions are included in an operator that isin principle unknown, the self-energy S.15 Since theright-hand side of Eq. (3.13) contains v, i.e., the Fouriertransform of a time derivative, implicit expressions for Scan be found through an evaluation of the equation ofmotion for the Green’s function. This scheme gives riseto a set of coupled differential equations in whichhigher-order Green’s functions appear. Those equationscan be exactly decoupled in only a few cases, and onehas to resort to approximations to get a practical, solv-able scheme. However one can sum up the series ofcoupled equations for the Green’s functions in a pertur-bative way and recast all the information in the operatorS, as we shall see later.

Suppose for the moment that S is known. ExpressingG in terms of the Lehmann amplitudes and energies[Eq. (3.3)], one finds, after taking (for a discrete level)the v→«s limit,

E $d~x2x9!@«s2HH~x !#2S~x ,x9,«s!%fs~x9!dx950,

(3.14)

i.e., the Dyson equation for the quasiparticle energies «sand amplitudes fs . The amplitudes fs form a completeset but are nonorthogonal, since the self-energy opera-tor is energy dependent. This also implies that the equa-tion can have more solutions than in the static case. In-

15See Farid (2001) for a detailed discussion of the generalproperties of the dynamical self-energy operator, including ananalysis of the shortcomings of the usually applied approaches.


stead, if the electron-electron interaction is smoothlyswitched off, those amplitudes tend to form the standardorthonormal set of an independent-particle system. Thisleads again to the situation described above, with thedifference between the isolated independent-particlepeaks and the broad quasiparticle ones containing a cer-tain number of nonvanishing Lehmann amplitudes. Dy-son’s equation (3.14) indeed has a one-electron form(single-particle Schrodinger-like) but the operator S hasvery nontrivial correlation effects built in and is far froma mean-field approximation.

In order to deal more easily with the pole structuresof G , S, etc., it is convenient to generalize all quantitiesfrom the real frequency axis v to the complex plane z ,i.e., to make an analytic continuation. The one-particleGreen’s function is then written in terms of left and rightsolutions

G~z !5(s

fsr~z !fs

l* ~z !

z2Es~z !(3.15)

defined by the general Dyson equation to be solved inthe full complex plane,

Es~z !5z ,

where

@H01S~z !#fsr~z !5Es~z !fs

r~z !,(3.16)

@H01S~z !#†fsl* ~z !5Es* ~z !fs

l* ~z !.

The self-energy S here plays the role of an effectivepotential for an electron or hole added to the systemwhich is, in general, complex, nonlocal, and energy de-pendent. This potential arises from the exchange andfrom the response of the rest of the electrons to thepresence of the additional particle. S(v) is real when vis set to an energy value below the first inelastic thresh-old of the system.

Thus one has achieved an exact one-particle picture atthe price of introducing a complicated effective poten-tial. The non-hermiticity and energy dependence of Simply that the quasiparticle wave functions are energydependent and not necessarily orthogonal; however,both the Green’s function and Dyson’s equations admit abiorthogonal spectral representation (in terms of leftand right eigenfunctions).16 The important point to noteis that now one is dealing with the energies Es(z), whichare in principle complex.

The mathematical details of the analytic continuationare nontrivial and have important consequences. For arigorous and complete treatment we refer the reader tothe review paper of Farid (1999a). It is, however, usefulto illustrate the connection between the Lehmann rep-resentation and the analytic continuation by an example:

16The biorthonormal and Lehmann representations are bothexact. Even if they look similar they are not identical, since inthe former case one works with the analytic continuations(Farid, 1999a).


Suppose that some matrix element of a Green’s func-tion is given in the Lehmann representation by

G~v!5(s

g~s !

v2es1ih. (3.17)

For simplicity, let us consider a case in which g(s)5(1/p)E2 /@(s2E1)21E2

2# and es5s . Given this par-ticular form of g(s), taking the thermodynamic limit(i.e., making s a continuous variable) yields a spectralfunction A(v)51/pu Im@G(v)#u with a Lorentzian form:

A~v!51p

E2

~v2E1!21E22 . (3.18)

On the other hand, we can also write

A~v!51p UImS 1

v2~E11iE2! D U; (3.19)

thus the Lorentzian, which in the beginning was given bythe imaginary part of a function containing a series ofinfinitely close-lying poles on the real axis (a branchcut),in this particular case can also be described by the imagi-nary part of another function, defined in the whole com-plex plane, having a single, but complex, pole (E11iE2). The position of the peak of A(v) is given by thereal part E1 , and the imaginary part E2 gives its width.This simple example demonstrates the connection be-tween the Lehmann and the complex-pole representa-tion of the Green’s function, in the case of a single qua-siparticle pole. It also explains why the complexrepresentation is advantageous—in this example it re-places the search for a branchcut by the search for onecomplex pole. Moreover, it illustrates the meaning of E1and E2 : the real part of the quasiparticle energies solu-tion of Dyson’s equation gives the band structure, andthe imaginary part the quasiparticle damping (electrondynamics). The solution can be real valued only for ex-citations with infinite lifetime [ImS(z5«s)50; the dis-crete part of the spectrum]. However, for energy rangeswhere « forms a continuum, S(x ,x8;«) is also complexand Dyson’s equation still has a solution with a complexz . In principle, this non-Hermitian eigenvalue problemcan be solved for each and every value of z in the com-plex plane (with the corresponding set of eigenvaluesand eigenvectors). Finally, this simple example is meantto explain why the existence of complex quasiparticleenergies is not in contradiction with the fact that «s ap-pearing in the Lehmann representation of the Green’sfunction are always real quantities.

In order to extend the quasiparticle picture, one canreturn to the description of a photoelectron spectrum interms of the Green’s function and self-energy operator(in its analytic continuation, i.e., using complex ener-gies), by considering the matrix elements of Eq. (3.15):

Gii~v!51

v2Ei~v!. (3.20)

If one expands this expression for energies v close to thegenerally complex quasiparticle energy Ei , one obtains

Gii~v!'Zi

v2Ei,


where Zi is the complex renormalization factor,

Zi51

12]S ii~v!

]v Uv5Ei

. (3.21)

The ith matrix element of the spectral function for vclose to Ei then reads

Aii~v!51p

uIm Ei Re Zi2~v2Re Ei!Im Ziu

~v2Re Ei!21~Im Ei!

2.

If uIm Eiu is small, Aii(v) has a sharp Fano peak at v5Re Ei of width G i5uIm Eiu and strength uRe Ziu, withthe asymmetry set by Im Zi . Zi is hence a measure ofthe strength of the quasiparticle pole and determines thespecific shape of the corresponding excitation in thespectral function. Even for weakly correlated systemslike sp metals and valence semiconductors uRe Ziu is farfrom 1 and can typically be in the range 0.6–0.9 (Hedinand Lundqvist, 1969; Hybertsen and Louie, 1986; Godbyet al., 1988). The sum of all resonances, i.e., the trace ofA , will finally determine the specific shape of the pho-toelectron spectrum.

B. Hedin’s equations

As discussed in the Introduction (Sec. I.B), when go-ing beyond Hartree-Fock one has to deal with electronrelaxation and correlation effects, which can also be de-scribed as dynamical screening effects. In fact, one canwrite an expansion of the self-energy in terms of ascreened Coulomb potential W , by demanding that thetotal energy be stationary with respect to variations ofthe Green’s function (Hedin, 1965). In this process theself-energy should be seen as a functional of the Green’sfunction (S@G#). The use of W instead of the bare Cou-lomb potential is physically more sound: conventionalmany-body perturbation theory suffers in fact from vari-ous convergence problems, which have to be bypassedwith care (see, e.g., Farid, 1999b).

In fact, interactions in a real many-body system arescreened to a large extent, and therefore W should be amuch better behaved quantity for the development of aperturbation expansion than the bare Coulomb poten-tial. The latter is known to lead to convergence prob-lems in the range of densities of normal metals andsemiconductors (Hedin, 1965, 1999; Hedin and Lun-dqvist, 1969, 1971). This expansion allows one to writethe many-body problem as a closed set of five integralequations introduced by Hedin (1965), which relate theGreen’s function, self-energy, polarization propagator,and vertex function. The equations can be obtained byintroducing a local time-dependent source term in theHamiltonian which directly couples to the particle den-sity. This new source term is set to zero once the finalequations are obtained. In this way Hedin derived a self-consistent scheme written in terms of the following setof coupled equations [here, as before, 15(r1 ,t1 ,j1), andv stands for the bare Coulomb interaction]:


S~12!5iE G~13!G~324!W~41!d~34!, (3.22)

W~12!5v~12!1E v~13!P~34!W~42!d~34!, (3.23)

P~12!52iE G~13!G~41!G~342!d~34!, (3.24)

G~123!5d~12!d~13!

1E dS~12!

dG~45!G~46!G~75!G~673!d~4567!,

(3.25)

where P(12) is the time-ordered polarization operator,W(12) is the dynamical screened interaction, andG(123) is the vertex function. The fifth equation is theDyson equation, which links G and S [see Eq. (3.13)].This way of writing the equations is in fact appealing,since it highlights the important physical ingredients: thepolarization [Eq. (3.24)], which contains the response ofthe system to the additional particle or hole, is built upby the creation of pairs of particles and holes (the twoGreen’s functions). The vertex function G contains theinformation that the hole and the electron interact. G, inturn, is determined by the change in the potential uponexcitation [Eq. (3.25)].

C. The iterative approach

Hedin’s equations together with Dyson’s equationform a set of equations that must in principle be solvedself-consistently for G . This means that the Green’sfunction used to calculate the self-energy should coin-cide with the Green’s function obtained from the Dysonequation with the very same self-energy. It is obviousthat this is a very difficult task, and that one must inpractice find a simplification of Hedin’s equations. Sinceit is not possible to find a straightforward, well-definedand convergent perturbation expansion in some smallparameter, the approach to simplifying the equations issomewhat arbitrary and needs to be analyzed in detail.A self-consistent calculation of the interaction of low-energy electrons with an electron gas was first carriedout by Quinn and Ferrell (1958). They performed a self-energy calculation of electron-electron scattering ratesnear the Fermi surface and derived a formula for theinelastic lifetime of hot electrons which is exact in thehigh-density limit. These free-electron-gas calculationswere extended by Ritchie (1959)17 and Quinn (1962) toinclude, within the first Born and random-phase ap-proximations, energies away from the Fermi surface,and by Adler (1963) and Quinn (1963) to take into ac-count the effects of the presence of a periodic latticeand, in particular, the effect of virtual interband transi-

17The 1/2 factor in front of z2 in the expansion of f1 justbefore Eq. (6.15) of this reference must be replaced by 1/3, asdone in a subsequent paper (Ritchie and Ashley, 1965).


tions (Quinn, 1963). In their original work, Quinn andFerrell (1958) had already given the GW expression forS, but did not recognize its usefulness beyond the caseof an electron gas and only evaluated the results as adensity (rs) expansion. Further studies for quasiparticleexcitations and lifetimes in a homogeneous electron gaswere done by Hedin (1965) and Lundqvist (1967, 1968).

1. First iteration step: the GW approximation

The simplest approximation to G(123) assumes thisoperator to be diagonal in space and time coordinates,G(123)5d(12)d(13): the so-called vertex correctionsare neglected. This implies that the irreducible polariz-ability is given by noninteracting quasielectron-quasihole pairs,

P~12!52iG~12!G~211!. (3.26)

It can be evaluated explicitly by using Eq. (3.4). After aFourier transformation, and taking into account thesmall imaginary parts of the energies, one obtains theformula (1.4) in its time-ordered version; this approxi-mation is in fact again the RPA form.

For the self-energy, this approximation yields the form

S~12!5iG~13!W~31!, (3.27)

the so-called GW approximation as introduced by He-din (1965). Of course, the Dyson equation for theGreen’s function G5G01G0SG still has to be added,and even at the GW level one has to deal with a many-body self-consistent problem. However, the GW ap-proximation is a comparatively simple expression for theself-energy operator, which in principle allows theGreen’s function of an interacting many-electron systemto be computed by starting from the Green’s functionG0 of a hypothetical independent-particle system withan effective one-electron potential, and iterate to self-consistency. The GW extends the well-known Hartree-Fock approximation, in which the self-energy is the ex-change potential Sx , by replacing the bare Coulombpotential v by the dynamically screened potential W ,e.g., Sx5iGv is replaced by SGW5iGW . The self-energy operator now consists of a dynamically screenedexchange potential plus a dynamical Coulomb hole. Infact, the static approximation of the GW self-energy isnone other than the COHSEX formula [Eq. (1.2)] dis-cussed in the introduction. GW has been shown to bephysically well motivated and to be superior to theHartree-Fock approximation, especially for metals,where Hartree-Fock (i.e., using the bare Coulomb po-tential) leads to unphysical results (Hedin and Lund-qvist, 1969). The dynamically screened interaction W in-troduces energy-dependent correlation effects absent inthe one-particle picture. Further, an energy-dependentcorrelation decreases the Hartree-Fock band gap byraising the valence-band energy and lowering theconduction-band energy. There is some empirical evi-dence that supports the idea that even in the first itera-tion (that is, using just the noninteracting Green’s func-tion G0) one obtains quite accurate results for one-


electron properties such as the excitation energy(Hybertsen and Louie, 1985, 1986; Godby et al., 1986,1987, 1988; Aryasetiawan and Gunnarsson, 1998) andthe quasiparticle lifetime (Campillo et al., 1999; Schoneet al., 1999; Campillo, Pitarke, et al., 2000; Echeniqueet al., 2000; Campillo, Rubio, et al., 2000; Campillo,Silkin, et al., 2000; Keyling et al., 2000; Silkin et al., 2001;Spataru et al., 2001). This is important for practical ap-plications of the GW approach since, despite its formalsimplicity, the practical solution of the self-consistentGW equations is a formidable task, which has been car-ried out only recently: self-consistent calculations wereperformed for the homogeneous electron gas (Holm andvon Barth, 1998; Holm and Aryasetiawan, 2000; Garcıa-Gonzalez and Godby, 2001), simple semiconductors, andmetals (Shirley, 1996; Schone and Eguiluz, 1998). Self-consistency modifies the one-electron excitation spec-trum (excitation energies and lifetimes) as well as thecalculated screening properties. The results turn out tobe worse than those of the nonself-consistent G0W0calculations18 or those obtained within the so-calledGW0 , in which only the explicit G , and not W , is up-dated at every iteration, thus achieving partial self-consistency in G (von Barth and Holm, 1996). Improve-ments have been obtained, in particular concerning theposition of plasmon satellites, using a partially self-consistent G , i.e. wave functions of order zero but up-dated quasiparticle energies (Bechstedt et al., 1994). Theproblem of self-consistency in GW calculations has beeninvestigated more deeply in such simple systems as thehomogeneous electron gas or an exactly solvable Hub-bard model. The main outcome of the self-consistentGW calculation for the homogeneous electron gas(Holm and von Barth, 1998; Holm and Aryasetiawan,2000; Garcıa-Gonzalez and Godby, 2001) is that the totalenergy computed with the Galitskii and Migdal (1958)formula turns out to be strikingly close to the total en-ergy calculated using quantum Monte Carlo (Ceperleyand Alder, 1980; Ortiz et al., 1999). Here few sum rulesalready determine most of the energy contributions inthe homogeneous electron gas. Encouraging results arealso obtained for the electron gas in three and two di-mensions, even for those ranges of densities for whichthe GW approach is often supposed to fail (Garcıa-Gonzalez and Godby, 2001) and for inhomogeneous sys-tems such as bulk, surfaces, and interfaces (Sanchez-Friera and Godby, 2000; Garcıa-Gonzalez and Godby,2002), and dimers (Aryasetiawan et al., 2002b). This re-sult may be related to the fact that the self-consistentGW scheme conserves electron number, energy, and to-tal momentum (that is, it fulfills the microscopic conser-vation laws; Martin and Schwinger, 1959, Baym andKadanoff, 1961; Baym, 1962). However, concerningspectroscopic properties a systematic deterioration inthe description of the bandwidth, quasiparticle excita-

18In this scheme G0W0 , the full G has been replaced by G0and the corresponding screened Coulomb potential is calledW0 .


tion and lifetimes, and plasmon satellites is found. Re-sults along the same lines were obtained in the case ofan analytically solvable finite Hubbard cluster recentlyused to compare the performances of several types ofGW calculations (Verdozzi et al., 1995; Pollehn et al.1998; Schindlmayr et al., 1998).

An important conclusion of this subsection is that, ifone is interested in the excitation spectrum, self-consistent GW performs worse than the simpler G0W0scheme; efforts should be directed towards consistentlyincluding vertex corrections in the calculations (see nextsection).

2. Second iteration step: vertex correction in P and S

Having arrived at the GW expression for the self-energy (or any approximation for S other than S50),one can go again through Eq. (3.22)–(3.25), starting withthe latter. Now, for a nonvanishing functional derivativedS/dG , one obtains a correction to the bare vertex G5d ; this is the linear response of the self-energy to achange in the total potential of the system. This correc-tion affects the polarizability in Eq. (3.24)—the twoGreen’s functions in P are no longer decoupled. In fact,vertex corrections account for exchange-correlation ef-fects between an electron and the other electrons in thescreening density cloud. In particular this includes theelectron-hole attraction (excitonic effects) in the dielec-tric response, which we shall look at later.

The improved G and P can then be used, through Eqs.(3.23) and (3.22), in order to construct a new self-energy.This iteration step is still important, since, for example,long-range vertex corrections are needed in order to im-prove the structure of plasmon satellites (Aryasetiawanand Gunnarsson, 1998). Also the fact that self-consistency causes the results of GW spectroscopy cal-culations to deteriorate suggests that vertex correctionsare important (Verdozzi et al., 1995). On the other hand,as noted by DuBois (1959), vertex corrections enhancecorrelation functions as the density-density response,whereas the inclusion of self-energy effects in theGreen’s function leads to a reduction. Similar conclu-sions were reached by Hong and Mahan (1994). There-fore one faces competing effects between self-consistency and vertex corrections: contributions fromvertex corrections and self-consistency tend to cancel toa large extent for the 3D homogeneous electron gas.However, vertex corrections in quasi-2D electron-gassystems (like narrow quantum wells) are more impor-tant for the description of the quasiparticle band-gaprenormalization and lifetimes (Marmorkos and DasSarma, 1991).19 The cancellation of vertex correctionswith self-consistency seems to be a quite general feature.Vertex correction diagrams have also been found by de

19For the case of the semimetallic graphite (quasi-2D system)it has been shown by Spataru et al. (2001) that G0W0 describesthe observed energy dependence of the electron lifetimes butnot the absolute value.


Groot et al. (1996) to partially cancel the self-consistency effects in the case of a quasi–one-dimensional semiconducting wire. Their results showthat, after systematic inclusion of all lowest-order cor-rections both to the RPA polarizability and to the GWself-energy, the band gap is roughly the same as at thefirst iteration G0W0 . Cancellation of the first-order ver-tex and self-consistency corrections, however, has beenfound only to a minor extent for the band gaps of bulksilicon and diamond (Ummels et al., 1998). The authorshave included vertex and self-consistency corrections tofirst order and found that corrections to the polarizabil-ity largely compensate each other, while this is not truefor the gaps which become larger than the GW ones by0.7 and 0.4 eV for diamond and silicon, respectively.20

There are also cancellations between the same vertexcorrection used in different places; in fact, Mahan andSernelius (1989) emphasize that a consistent way of han-dling vertex corrections is to include identical vertex cor-rections in both the polarization and the self-energy op-erator.

The simplest improvement to the GW approximationin Eq. (3.22) consists in introducing a vertex correctionconsistent with the starting LDA calculation of the one-electron orbitals. The DFT exchange-correlation poten-tial is regarded as an approximation to the self-energy(Del Sole et al., 1994). Based on this idea, the vertex Gcan be easily expressed in terms of the functional deriva-tive dVxc /dr . This so-called GWG approximation(Rice, 1965; Hybertsen and Louie, 1986; Mahan and Ser-nelius, 1989; Del Sole et al., 1994; Mahan, 1994) reads incompact notation

G51

12fxcP0,

W5W1[~12fxcP0!v

12~v1fxc!P0, G5G0 (3.28)

and thus

S5iG0 W1 G5iG0v

12~v1fxc!P0. (3.29)

S is hence again of the GW form, but with an effectivescreening W2ªv/@12(v1fxc)P0# which is in fact thetestcharge electron dielectric function of linear-responsetheory (Del Sole et al., 1994; Hedin, 1999). The correc-tions given by fxc hence account for exchange-correlation effects at two levels. The first level concernsthe electrons of the ‘‘screening medium,’’ with anexchange-correlation hole around the electrons; whenan electron is participating in the dielectric screening

20Note that there are also cancellations in the T-matrix ap-proximation where short-range interactions between two local-ized holes (electrons) can be described by a series of ladderdiagrams exactly representing all repeated binary collisions(Kanamori, 1963). Due to the effects of the vertex corrections,the self-consistent version of the T matrix (using dressedpropagators in the ladder) is found to be inferior to the non-self-consistent one (Cini and Verdozzi, 1986).


others are less likely to be found nearby. Second, thepotential induced by the additional electron or hole alsoincludes exchange-correlation interactions between theparticle and the system electrons, and not only the clas-sical Coulomb interaction (since the particles are in factindistinguishable). The vertex correction has to be put inboth P and S in order to obtain this result (a vertexcorrection in P alone yields the testcharge-testcharge di-electric function). Results for silicon (Del Sole et al.,1994) show that the quasiparticle energy gaps obtainedusing the RPA screening (as is done in standard GWcalculations) or W2 are close, whereas using S5iG0W1 , i.e., putting the vertex correction only in P ,alters the results, which gives evidence for the impor-tance of consistent corrections. Support for this sort ofGWG approximation can already be found in the workof Rice (1965). He showed that, using Hubbard’s energyexpression corrected for exchange effects in an electrongas, a functional derivative of the total energy with re-spect to the electron occupation number gave a quasi-particle energy corresponding to S5iG0W2 .

The main drawback of the LDA-based GWG ap-proximation is the fact that results for bandgaps in semi-conductors as well as valence bandwidths are very closeto the standard GW values. Further improvementsmight be obtained by using better approximations to theTDDFT kernel, including nonlocality and memory ef-fects (see Sec. V.C), or it might be necessary to workwith the true nonlocality of the exchange-correlationself-energy, which means using a four-point kernel (seeSec. II.C). An example of such a calculation has beengiven by Shirley and Martin (1993), who performed GWcalculations for atoms starting from Hartree-Fock andusing a vertex correction consistent with the Fock ex-change term. Shirley and Martin found that the resultsfor Group I, II, IV, and VIII elements are similar tothose of ordinary GW without vertex corrections, againsuggesting large cancellations. It should, however, benoted that the choice of the ‘‘reference’’ state was cru-cial. For example, electron removal energies from theneutral state were calculated as electron addition ener-gies to the corresponding singly ionized state. This mightbe seen as a partial inclusion of important vertex andself-consistency corrections already in the referencestate, for calculations both with and without explicit ver-tex corrections.

Systematic vertex corrections can hence be obtainedthrough an iterative solution of Hedin’s equations. Onecan push this scheme further by using the GW approxi-mation for S in the equation for the vertex function, andso on. An explicit formula for the vertex function corre-sponding to the full second cycle of iteration can be ob-tained which mixes certain diagrams of different ordersin the screened interaction. Schindlmayr and Godby(1998) have shown that the vertex to order (n11),

G(n11)~123!5d~12!d~13!1E d~4567!dS(n)~12!

dG(n)~45!

3G(n)~46!G(n)~75!G(n11)~673!,

(3.30)


can be expressed in terms of the lowest-order equiva-lents of all quantities except S(n):

G(n11)~123!5d~12!d~13!1E d~4567!dS(n)~12!

dG(0)~45!

3G(0)~46!G(0)~75!G(1)~673!. (3.31)

The appearance of the lowest order reduces the numeri-cal effort substantially. In particular G(0) can be chosensuch that it contains no satellite spectrum, but only a setof robust quasiparticle excitations. In the self-consistentlimit n→` the previous equation implies a relation be-tween the exact self-energy and vertex function. Nu-merical results for a model of strongly correlated elec-trons have indicated that this method yields improvedexcitation energies, in particular concerning the satellitespectrum. Nevertheless, to our knowledge this approachhas never been applied to real systems. Instead, aslightly different version of the first iteration step iswidely used, namely, the expression

G(n11)~123!5d~12!d~13!1E d~4567!dS(n)~12!

dG(n21)~45!

3G(n)~46!G(n)~75!G(n11)~673!

(3.32)

in its lowest-order version,

G(2)~123!5d~12!d~13!1E d~4567!dS(1)~12!

dG(0)~45!

3G(1)~46!G(1)~75!G(2)~673!. (3.33)

Note that the only difference between Eqs. (3.30) and(3.32) is the version of the Green’s function with respectto which the functional derivative of S is taken. Theconsequences might, however, be important, becausewhereas it is obvious that already the first-iteration so-lution of the integral equation (3.33) will shift the polesof P52iGGG , this is not obvious in the case of Eq.(3.31). Rather, for n51 Eq. (3.31) multiplied by2iG(1)G(1) yields a first-order polarizability with polescoming from the poles of both G(0) and G(1).

In fact, Eq. (3.33) is the starting point for the deriva-tion of the Bethe-Salpeter equation, which correctlyyields features like bound excitons in the absorptionspectra, as we shall show in Sec. IV.B.3.

Obviously, choices like the one between Eqs. (3.30)and (3.32) are crucial only because one typically doesnot want to iterate to infinity, but to find the ‘‘best’’ first-iteration step, which is not well defined. This kind ofproblem regularly shows up when one is dealing withmany-body Coulomb equations.

Another way of partially summing higher-order dia-grams for describing vertex corrections to the valenceelectron Green’s function is given by an exponential ex-pression very similar to the (almost exact) solution ob-tained for the core electron Green’s function, as firstshown by Hedin (1980). This approximation can also bederived as the first term in a cumulant expansion (Ar-


yasetiawan and Gunnarsson, 1998). The spectral func-tion in the cumulant expansion for a state below theFermi level is expressed as

A~k ,v!5nk

2p E2`

`

dteivte2i«kt1C(k ,t), (3.34)

where nk is the occupation number of state k , ek is thesingle-particle eigenvalue, and C(k ,t) is the cumulantoperator, which can be derived in different ways (Hedin,1980).21 The cumulant expansion contains only boson-type diagrams describing emission and reabsorption ofplasmons; it does not contain diagrams corresponding tothe interaction between a hole and particle-hole pairs.For this reason, the cumulant expansion primarily cor-rects the satellite description, whereas the quasiparticleenergies are to a large extent determined by GW . Thisapproximation has been applied successfully to describ-ing the multiple plasmon satellite structure in the spec-tra of alkali metals (Aryasetiawan et al., 1996) as well asto substantially improving the calculated electron mo-mentum spectroscopy of graphite in regions of energywhere no intensity is predicted from the LDA bandstructure (Vos et al., 2001).

IV. GREEN’S FUNCTIONS IN PRACTICE

From the previous subsection it emerges that the wayquasiparticle energies and electron-hole excitationsshould be calculated is not unique. In practice schemeshave been designed which allow one to obtain satisfac-tory results in many applications. An important variablein this strategy is the starting point of the calculations,i.e., one has to find the charge density (and often thelattice parameters and other ground-state properties)and some suitable one-electron eigenvalues and eigen-functions which allow construction of a starting one-particle Green’s function. This task can be successfullycarried out using the Hartree-Fock approximation incases like atoms and molecules where screening is weak,or, most often, starting from static density-functionaltheory. In the following we shall summarize how the cal-culations are done in practice.

A. Calculations of one-particle excited states

First-principles GW calculations are today standardpractice in many solid-state theory groups. Several re-views on quasiparticle calculations and the derivationand analysis of Hedin’s equations have been publishedin recent years (e.g., Aryasetiawan and Gunnarsson,1998; Aulbur et al. 1999; Hedin, 1999) and we refer the

21The simplest way is to identify the cumulant expan-sion $G(k ,t)5ie2i«kt1C(k ,t)5G0(k ,t)@11C(k ,t)1

12 C2(k ,t)

1 ¯ #% with the self-energy expansion of the Green’s function(G5G01G0SG01G0SG0SG01 ¯ ). In practice, using theGW approximation for the self-energy and a G0 determinedfrom a DFT calculation, the cumulant is obtained equatingG0C5G0SG0 with S5SGW2Vxc .


reader to those for further details. The fundamental re-view by Hedin and Lundqvist (1969) focuses on the elec-tron gas and contains a comprehensive introduction toall concepts and physical quantities and the connectionto measurable spectra. In the recent reviews by Aryase-tiawan and Gunnarsson (1998) and Aulbur et al. (1999)the successes and shortcomings of the GW approxima-tion and numerical implementations (plane waves, localorbital basis, real-space/imaginary-time) and tricks toimprove the computational efficiency are described, to-gether with a detailed compilation of results obtainedfor semiconductors, transition-metal oxides, fullerenes,superlattices, interfaces, surfaces, defects, Schottky bar-riers, and atoms and molecules. Apart from their impor-tance on their own, quasiparticle energies also serve asan input to further spectroscopy calculations, in particu-lar for the Bethe-Salpeter equation approach. Thereforein the following we outline the way GW calculations areusually performed and give some illustrative examples.

1. GW calculations

A GW calculation requires in principle solution ofEq. (3.16) for the quasiparticle energies and amplitudes,taking for S the product of the Green’s function G andthe screened Coulomb interaction W . The complexity ofGW leads, in practical applications, to approximationsin the construction of G and W , which we shall examinein the following.

a. Evaluation of the Green’s function G

In practice G is constructed using single-particle or-bitals from a Kohn-Sham (or, less often, from a Hartree-Fock) calculation. One considers the single-particle or-bitals and eigenvalues as a zeroth-order approximationto the quasiparticle amplitudes and energies (see Hedin,1995, for a discussion of this issue. This suggests the pos-sibility of looking for a first-order, perturbative solutionof Eq. (3.16) with respect to (S2Vxc) or to (S2Sx),where Vxc is the exchange-correlation potential ofKohn-Sham (or Sx is the exchange potential of Hartree-Fock). This is the approach followed in many practicalGW calculations for real systems.22 The diagonal matrixelements of (S2Vxc) give the quasiparticle energies as

e iQP5e i

KS1Zi^f iKSuS~e i

KS!2VxcKSuf i

KS&, (4.1)

with Zi21512^f i

KSudS/de ue iKSuf i

KS&. In the case of

simple bulk semiconductors, this approximation turnsout to be very close to the exact result obtained by di-agonalizing HH1SGW (but without any self-consistentupdate of G and W ; Hybertsen and Louie, 1986). Inother cases, such as transition metals and finite and low-dimensional systems, the validity of the perturbative ap-proach in S2Vxc should be checked explicitly. For in-

22The solution of the exact local Kohn-Sham exchange hasalso been used as input for the GW quasiparticle calculation(Aulbur et al., 2000). The quasiparticle corrections seems to besmaller in this case.


stance, Pulci et al. (1999) have shown that for the (110)surface of GaAs, even if quasiparticle energies are notaltered when doing the full diagonalization of the quasi-particle Hamiltonian, the quasiparticle wave functionscorresponding to nearly degenerate states, with differentdegrees of surface/bulk localization, strongly hybridize,making the quasiparticle wave functions significantly dif-ferent from the LDA ones. Differences between LDAand GW wave functions have also been found in thecase of SiH4 for states crossing the level of zero poten-tial (Rohlfing and Louie, 2000). In general, differencesbetween LDA and quasiparticle wave functions can atleast be expected in any nonbulk system, i.e., when thereare regions of space where the density goes to zero. Inthese regions, LDA produces an exchange-correlationpotential with an incorrect asymptotic behavior (e.g.,missing the 21/r tail in the case of atoms and clusters);hence, LDA wave functions may have a wrong spatiallocalization. The GW potential, on the other hand, doeshave the correct asymptotic behavior (Charlesworth,et al., 1993; Garcia-Gonzalez and Godby, 2002).23

b. Evaluation of the screened Coulomb interaction W5«21v

One of the main difficulties is the evaluation of thefull dielectric response of the system. In fact, even withinthe chosen zeroth-order scheme, i.e., within RPA-LDA,the calculation of «21(r,r8,v) remains a difficult taskfrom a numerical point of view (due to the spatial non-locality and frequency dependence of «). For this rea-son, approximated schemes have been developed:

• The model dielectric function (Levine and Louie, 1982;Hybertsen and Louie, 1988; Bechstedt, Del Sole,et al., 1992). These approaches allow one to consider-ably reduce the computational effort or even to avoidcompletely the ab initio calculation of «.

• The plasmon-pole approximation (Hybertsen andLouie, 1986; von der Linden and Horsch, 1988; Ha-mada et al., 1990; Engel and Farid, 1993). Most mod-els are based on the observation that «(G,G8,v)21 isgenerally a peaked function in v that can be approxi-mated by a single-pole function in v. The pole posi-tion and strength are determined by imposing sumrules (Hybertsen and Louie, 1986), or by fitting eachelement «21(G,G8) at two points v along the imagi-nary energy axis (Godby and Needs, 1989). Since theevaluation of S involves an integration over the en-ergy, the fine details of the energy dependence are notcritical, and the plasmon-pole approximation turnsout to work reasonably well. One important drawbackis that quasiparticle lifetimes cannot be computedwithin a plasmon-pole approximation, as Im(S) iszero everywhere except at the pole. The validity ofthe plasmon-pole approximation has not been system-

23One recent GW application touching this point is the workof White et al. (1998) on the image potential at the Al(111)surface.


atically tested, except in a very few cases (see, forexample, Aulbur et al., 1999; Arnaud and Alouani,2000).

When the full energy dependence of the dielectric ma-trix is retained, the integration in v may be performedalong the imaginary axis, where «21 is well behaved(Godby et al. 1988; Schone and Eguiluz, 1998), by pick-ing up all the poles along the real axis (Aryasetiawan,1992; Fleszar and Hanke, 2000), or by using thetransition-space spectral representation of «21, whichallows one to perform the frequency integration analyti-cally (Shirley and Martin, 1993).24

c. Application of the self-energy S

For many applications, the simple prescriptions de-scribed above have yielded results within 10–15 % of theexperimental ones, the typical example being the directquasiparticle gap of diamond, as discussed in the intro-duction. An agreement of the same quality betweenquasiparticle energies and photoemission or inversephotoemission data has also been obtained for morecomplex systems like surfaces, atoms, and nanostruc-tures (Aulbur et al., 1999), and the GW calculations aresystematically used to study quasiparticle excitations inrealistic systems of practical interest. For example, in thecase of an ethylene molecule (C2H4) adsorbed on theSi(001)-(231) surface GW seems to improve the cal-culated tunneling currents measured in scanning tunnel-ing microscopy (Rignanese et al., 2001).25 A recent GWstudy on YH3 illustrates the important consequences ofself-energy corrections (Miyake et al., 2000; vanGelderen et al., 2000). It turns out that GW correctionsremove the band overlap responsible for erroneous me-tallic LDA behavior in YH3 that has a measured opticalgap of 2.8 eV. Hence the gap is of electronic originrather than structural, and not due to strong correlation(Eder et al., 1997; Ng et al., 1997).

Another physical quantity that can be obtained from aknowledge of the full, complex self-energy S is the qua-siparticle lifetime, i.e., the electron-electron scatteringcontribution to the linewidth. The damping rate of anexcited electron in the state c0(r) with energy E is infact obtained as (Echenique et al., 2000)

24Mixed-space (Blase et al., 1995) and real-space/imaginary-time techniques (Rojas et al., 1995), as well as a technique toeliminate the unoccupied state summations (Reining et al.,1997), can also be used to reduce the computational effort (seethe reviews by Aulbur et al., 1999; and Aryasetiawan and Gun-narsson, 1998, for a detailed comparison of different numericalimplementations).

25Similar studies concerning the interpretation of scanningtunneling microscopy images have also been performed in theframework of the LDA1U approach [see, for example, thework by Dudarev et al. (1997) on the NiO(100) surface]. Werefer the reader to Anisimov et al. (1997) for a description ofthe relation between GW and LDA1U.


t21522E drE dr8c0* ~r!Im S~r,r8,E !c0~r8!. (4.2)

In the GW approximation, the imaginary part of S isdetermined by the imaginary part of «21 that containsall decay processes of the initial state. Quasiparticledamping times were calculated for diamond by Strinatiet al. (1980, 1982), who found results consistent withphotoemission data. A treatment of the electron dynam-ics, including band structure and dynamical screening ef-fects, is necessary for quantitative comparisons with ex-periment (see, for instance, Burgi et al., 1999; Valla et al.,1999b). An illustrative example is given by Campilloet al. (1999; Campillo, Pitarke, et al., 2000; Campillo, Ru-bio, et al., 2000) and Gerlach et al. (2001), who evaluatedthe quasiparticle lifetimes of electron and holes in bulkCu within the GW scheme, showing an increase in thelifetime close to the Fermi level as compared to the pre-dictions of a free-electron-gas model of the solid. How-ever, the lifetimes get closer to those of a free-electrongas for hole energies below ;3 eV, whereas the experi-mental data show a distinct asymptotic behavior. Thisdiscrepancy between theory and experiment may be asignature of important departures of the DFT-LDAband structure of Cu, which is used in the calculation ofW , from the actual quasiparticle band structure (see Fig.2; Marini et al., 2002).

GW results for Cu agree with photoemission datawithin 30 meV for the highest d band, correcting 90% ofthe LDA error. The energies of the other d bandsthroughout the Brillouin zone are reproduced within 300meV, and the maximum error (.600 meV) is found forthe bottom valence band at the G point, where only 50%of the LDA error is corrected. This level of agreementfor the d bands cannot be obtained without includingself-energy exchange contributions coming from 3s and

FIG. 2. Comparison of the calculated and experimental bandstructures for copper: solid line, GW quasiparticle excitationenergies (Marini et al., 2002); dashed line, DFT-LDA eigenval-ues; s, experimental data compiled by Courths and Hufner(1984). A comparison to more recent experimental data (Stro-cov et al., 1998, 2001) yields the same agreement.


3p atomic levels,26 demonstrating the importance ofthose contributions. On the other hand, the total band-width is still larger than the measured one. This overes-timate of the GW bandwidth with respect to the experi-mental one seems to be a quite general feature, which isnot yet properly understood. (See Yasuhara et al., 1999,for a discussion about Na, Ku et al., 2000 for a comment,and Yasuhara et al., 2000 for reply.)

Fully self-consistent GW calculations have been per-formed by Schone and Eguiluz (1998) for crystalline sili-con and potassium, with the conclusion that for real ma-terials, self-consistent GW without vertex corrections inS yields quasiparticle energies and bandwidths that dis-agree with experiment; for example, the absolute gap ofSi turns out to be 1.91 eV, compared with the experi-mental 1.17 eV. On the other hand, Massidda et al.(1995) have calculated self-consistently the electronicstructure of MnO, using a frequency-independent modelS derived by Gygi and Baldereschi (1989). An encour-aging agreement was found with experiment concerningthe energy gap, magnetic moment, bandwidth, and spec-tral distribution of Mn 3d states. The fact that self-consistency leads to poorer results in the calculations ofSchone and Eguiluz (1998), but leads to an improvementin the calculations of Massidda et al. (1995) shows thatthe dynamical aspect of the self-energy is very problem-atic in this context.

Few works including vertex corrections exist to dateon real materials. One example is given by the GWGcalculation on silicon, described in Sec. III.C.2, where Gis taken from a DFT-LDA approach. However, the ef-fects of this approximate kernel are rather small. It isdifficult to guess what the effect would be if a better (inparticular more long-range) approximation of the kernelwere used. The work of Ummels et al. (1998), discussedin the same section, suggests that some changes might befound.

The effects of vertex corrections can also explain whyGW calculations have not always been successful in de-scribing self-energy effects, for example for d-electronbands (where the problem might come from strong cor-relation or simply from self-interaction effects, i.e., fromthe strong localization). A critical discussion about theextent to which the GW approximation is capable ofdescribing highly correlated systems such as NiO isgiven by Aryasetiawan and Gunnarsson (1995).

Computationally, the evaluation of S as GW is a hardtask.27 Calculations can scale as badly as N4, where N isthe number of atoms, hence reaching the limit of com-puter power well before DFT-LDA calculations, whichusually scale as N3.

26The relevance of semicore states in the self-energy was alsopointed out by Rohlfing et al. (1995, 1998).

27A typical bottleneck is given by the summations over theempty states (Reining et al., 1997), both for the determinationof the screening and for G . Another difficulty comes from theconvergence of Brillouin zone integrals (i.e., k-point sampling)of functions that have a factor uk1Gu2 in the denominator asthe exchange term (Pulci, 1998).


It turns out that, whereas the ‘‘standard’’ prescriptionfor GW calculations yields good results for many appli-cations, one has to be careful when effects beyond thisperturbative-RPA scheme have to be included. In orderto gain some insight concerning the choices to be made,it is useful to resort to the outcome of model calcula-tions.

2. Comparison of GW and DFT results

As mentioned in Sec. II.A, the Kohn-Sham eigenval-ues, appearing as Lagrange multipliers in the minimiza-tion of the density functional, do not carry a directphysical meaning, even though they can be consideredas well-defined approximations to the excitation ener-gies (Gorling, 1996; Filippi, 1997). In particular, they canbe viewed as a good zero-order starting point for pertur-bative self-energy calculations (see Sec. IV.A.1). In fact,there is no equivalent of Koopman’s theorem for theKohn-Sham eigenvalues, and a direct comparison of theLDA energy gap of a semiconductor or insulator withthe experimental (photoemission and inverse photo-emission) band gap normally yields a severe underesti-mate, of the order of 30–50 % or more (see, for ex-ample, Bechstedt, 1992). This error is corrected in a GWcalculation of quasiparticle energies.

For a more precise and general comparison betweenDFT and GW , one has to distinguish between the ef-fects of the approximated exchange-correlation func-tional, for example LDA (which affects the total energydifferences and the effective potential, hence the eigen-values) and the effect of interpreting Kohn-Sham eigen-values as electron addition and removal energies (see,for example, Jones and Gunnarsson, 1989). Almbladhand von Barth (1985) showed that in exact DFT thehighest occupied eigenvalue (highest occupied molecu-lar orbital level, or HOMO) does have a physical inter-pretation: it corresponds to the ionization potential.Hence, in exact DFT, the ionization potential could bederived directly from a single calculation for the groundstate of an N-electron system. Unfortunately, the use ofapproximated exchange-correlation functionals (likeLDA) quite severely affects the Kohn-Sham highest oc-cupied level. Hence, when comparing excitation ener-gies directly with Kohn-Sham eigenvalues, one must dis-tinguish two cases: the highest occupied level, for whichthe error is due only to the LDA, and the other levels[e.g., the lowest unoccupied molecular orbital (LUMO)eigenvalue], for which, even in exact DFT, a discrepancycould be found. For example, in the Be atom (Jones andGunnarsson, 1989), both errors are large: the LDAHOMO (25.6 eV) misses the ionization potential(29.32 eV) by almost 4 eV, and the HOMO-LUMO gapis 3.5 eV in LDA, 3.62 eV in exact DFT, and larger than9 eV experimentally. Part of the error comes from thespurious self-interaction which is contained in the LDA.Perdew and Zunger (1981) have proposed a self-interaction-corrected approach in order to eliminate thiscontribution. This is also the case in realizations of DFTwhich allow the inclusion of the exact local exchange


potential as an optimized effective potential (OEP) (de-fined as the functional derivative of the exact exchangeenergy Ex with respect to the electron density) andwhich have been introduced with great success (Grosset al., 1996; Gorling et al., 1999).

On the other hand, in a finite system it is possible tocompute quasiparticle energies directly as energy differ-ences, provided one is able to compute the total energyof the system with N electrons in the ground state, andthat of the system with N61 electrons in the groundstate or some excited state. This scheme, known as thedelta-self-consistent-field method, has been successfullyused within DFT to describe bound one-electron excita-tions and the ionization potential of atoms and mol-ecules (Jones and Gunnarsson, 1989).

Hence, DFT- or Hartree-Fock–based delta-self-consistent-field calculations can be directly comparedwith GW results. For atoms in the iron series, Shirleyand Martin (1993) have shown that GW yields results ofa quality similar to, or slightly worse than, those com-puted from Hartree-Fock or local-spin-density total en-ergy differences (Baroni, 1984; Vukajlovic et al., 1991).Small sodium clusters can be taken as another example:Na4 and Na6 vertical and adiabatic ionization potentialsobtained with the delta-self-consistent-field approach(Martins et al., 1985) compare well with DFT1GW re-sults for jellium spheres (Saito et al., 1989) and for thereal clusters (Onida et al., 1995; Reining et al., 2000).Similarly, Ogut et al. (1997) studied the evolution of thequasiparticle and optical gaps of hydrogenated siliconclusters as a function of cluster size. Delta-self-consistent-field LDA calculations were used to estimatethe quasiparticle correction to the HOMO-LUMO gap.Ogut et al. concluded that quantum confinement, as wellas reduction of screening due to finite size, leads to ap-preciable excitonic corrections. This correction turns outto be large and size dependent. The delta-self-consistentfield results point out that for finite systems the Hartreerelaxation of the wave functions when the electron isadded or removed is essential, while it is of no relevancefor delocalized states in bulk solids, as pointed out byGodby and White (1998; see also the reply of Ogut et al.,1998). In infinite systems, the correlation part takes overin relevance (see the discussion in Sec. VI.C).

Finally, another way to overcome the ‘‘gap problem’’without performing a self-energy calculation has beenproposed by Mackrodt et al. (1996): Hartree-Fock calcu-lations on nickel oxide (NiO) and on the lithium-dopedmaterial have allowed the determination of the bandgap of the former, by taking eigenvalue differences be-tween two empty states of the latter. Simply taking thedifference between Hartree-Fock eigenvalues of an oc-cupied and an empty state of NiO would lead to far toolarge a gap.

In conclusion, Kohn-Sham eigenvalues are generallyin disagreement with photoemission and inverse photo-emission energies, both for finite and for extended sys-tems, with errors of 30–50 % or more. On the otherhand, self-energy-corrected quasiparticle energies (ob-tained via the GW method) usually agree well with pho-toemission and inverse photoemission data: the reported


errors are typically in the range of 10–15 % of the bandenergies measured with respect to the Fermi level, hencebeing in absolute value larger for systems with a largergap. In the case of finite systems, the delta-self-consistent-field method also yields accurate excitationenergies, since wave-function localization allows for fi-nite relaxation effects. In infinite systems with delocal-ized wave functions the delta-self-consistent-fieldmethod is inadequate, since Hartree relaxation effectsvanish and correlation must be taken into account. GWcontains both relaxation and correlation and is thereforesuitable to describe electron addition and removal ener-gies.

In the next section we shall discuss the use of Green’s-function theory for the study of two-particle excitedstates, such as the electron-hole pairs that are created inabsorption measurements. For an overview of earlierand some more recent work, see Hanke et al. (1983) andRohlfing and Louie (2000), respectively.

B. Calculations of two-particle excited states

The key quantity in the calculation of two-particle ex-cited states is the polarizability P , as introduced in Eq.(1.3).

1. Independent-quasiparticle approximation

As discussed in Sec. III.B, P is given by Eq. (3.24). InSec. III.C we defined the RPA form P52iGG [Eq.(3.26)], from which the screened interaction in GW cal-culations is usually obtained. In principle, Eq. (3.26)could also be used for the calculation of absorption spec-tra and other spectra that involve the creation ofelectron-hole pairs. P is then just the product of twoone-particle Green’s functions. Using Eq. (3.4) and aftera Fourier transformation from time to frequency space,one obtains for the independent-(quasi)particle P theresult (1.4) in its time-ordered form, from which the re-tarded version can be deduced. As in the case of theone-particle Green’s function, the irreducible polariz-ability P (which is derived from a two-particle Green’sfunction) can be understood in its Lehmann representa-tion or in the complex plane. In the second case, which isthe representation used in practical applications, theelectron and hole energies entering the denominator canbe complex, leading to a finite lifetime of the two-particle excitation (because of the finite lifetime of theelectron and/or the hole). Additional deexcitation chan-nels are included when the electron-hole interaction istaken into account; this will be discussed below.

Formula (1.4) is at present still the standard expres-sion used for many ab initio calculation of optical spec-tra of real materials. Note that, whereas the form of P isprescribed by Eq. (3.26), the iteration approach is againnot uniquely defined concerning the ingredients. Hencethe Green’s functions entering Eq. (3.26) may be consid-ered to stem, for example, from a Kohn-Sham or from aGW calculation. In practice, most often DFT Green’sfunctions are used when the approximation P5PIQP[see Eq. (1.4)] is made. This can be understood as an


approximation to the Green’s-function iteration scheme,but also as an approximation to TDDFT, as will be ex-plained in Sec. V. Sometimes a mixed DFT-GW Green’sfunction is used; the eigenvalues entering G are taken tobe GW ones, whereas the wave functions are Kohn-Sham orbitals. This approach (called GW-RPA in theintroduction) was first tried in order to overcome theredshift of DFT-based absorption spectra with respect toexperiment, but in general it overcorrects, and, more-over, does not improve line shapes.28 Today, GW-RPA ismostly seen as a good approximation to the first-iteration result for PIQP , which is then used to include Gin the second iteration, i.e., it is the starting point of theBethe-Salpeter approach, which will be outlined in thefollowing. We shall concentrate on absorption spectra,i.e., on the calculation of P [Eq. (2.9)], where going be-yond the RPA gives the most visible effects. All resultscan be easily generalized to the case in which one isinstead interested in x, replacing the local field effectscontribution v by v (see Sec. II.B).

2. Electron-hole attraction

The inclusion of vertex corrections [i.e., the inclusionof G in Eq. (3.24) for P] can be achieved through a sec-ond iteration of Hedin’s equations. This means that onehas to go again from Eq. (3.22) to Eq. (3.25), now usingS5iGW in the latter. This yields an integral equationfor G,

G~123!5d~12!d~13!

1iW~112 !E d~67!G~16!G~72!G~673!.

(4.3)

Here, the approximation dS/dG5iW is used. Thismeans that (i) one neglects the term iG (dW/dG),which contains information about the change in screen-ing due to the excitation and is considered to be small,and that (ii) Eq. (3.32), and not Eq. (3.30), has beenused. As discussed above, this latter choice in the itera-tion scheme can be crucial when only one or a few itera-tions are performed.

One can transform Eq. (4.3) to an integral equationfor a generalized 3P , defined as

3P~312![2iE d~67!G~16!G~72!G~673!, (4.4)

by multiplying with 2iG(41)G(25) on the left and in-tegrating over d(12):

3P~345!52iG~43!G~35!

1iE d~12!G~41!G~25!W~112 ! 3P~312!.

(4.5)

28It does, however, yield rather good static dielectric con-stants (Levine and Allan, 1989). For silicon and germanium,the error decreases by one order of magnitude when GW in-stead of LDA eigenvalues are used.


The polarization P can hence in principle be obtainedfrom P(31)53P(311). However, the kernel GGW ofthe integral equation (4.5) is a four-point function.Therefore, as outlined in Sec. II.C, it is equally conve-nient to introduce a four-point 4P , which has the advan-tage that local field effects can be included on the samefooting, i.e., one can directly obtain P (or x). In fact,closely following the prescriptions in Sec. II.C, one canintroduce the four-point dynamically screened interac-tion

4W~1234![W~12!d~13!d~24!. (4.6)

Then one gets for Eq. (4.5) the four-point integral equa-tion

4P54PIQP24P 4W 4PIQP , (4.7)

with the obvious generalization from PIQP to 4PIQP [seeEqs. (1.4) and (B16)]. One can now also write Eq. (2.10)for four-point quantities, namely,

4P~1234!54P~1234!

1E d~5678! 4P~1256!d~56!d~78!v~57!

3 4P~7834!. (4.8)

Putting these equations together, one obtains the Bethe-Salpeter equation for 4P :

4P54PIQP14PIQPK 4P , (4.9)

where the kernel K contains an electron-hole exchangecontribution v and electron-hole attraction 2W :

K~1234!5d~12!d~34!v~13!2d~13!d~24!W~12!.(4.10)

The equation for the corresponding 4x differs only bythe fact that v instead of v appears. The diagrams rep-resenting the Bethe-Salpeter equation for x are shownin Fig. 3. The Green’s-functions lines are dressed, i.e.,they are quasiparticle lines.

One can make an immediate connection to the discus-sion of the four-point equations in Sec. II.C. In fact,

FIG. 3. Feynman diagrams representing the Bethe-Salpeterequation for x.


most often W is taken to be the statically screened Cou-lomb interaction. Equation (4.10) for K correspondsthen to the time-dependent screened Hartree-Fockapproximation.29 In particular, one can follow thescheme of Sec. II.C and Appendix B.2, and derive theeffective two-particle Hamiltonian H2p defined in Eq.(2.22), which is often used in the context of the Bethe-Salpeter equation approach. In fact, when static screen-ing is used in W the effective two-particle equation takesthe particularly simple, energy-independent form,

(n3n4

$~en22en1

!dn1n3dn2n4

1~fn12fn2

!@ v(n1n2)(n3n4)

2W(n1n2)(n3n4)#%Al(n3n4)

5ElAl(n1n2) , (4.11)

where it is understood that matrix elements of the ker-nel (4.10) are taken with respect to four one-particleorbitals n1¯n4 : v(n1n2)(n3n4)5^n1n2uvun3n4& andW(n1n2)(n3n4)5^n1n3uWun2n4&. The solution of thisequation allows one to construct then the absorptionspectrum from Eq. (2.23). If one considers only the reso-nant part of H2p, i.e., the part mixing only transitions ofpositive frequency, the resulting operator is Hermitianand its eigenstates orthogonal. Therefore one obtainsthe simpler formula

«M~v!512 limq→0

v0~q!(l

u(n1n2^n1ue2iq•run2&Al

n1n2u2

v2El1ih.

(4.12)

As discussed below, this turns out to be a very goodapproximation for the calculation of bulk absorptionspectra.

Summarizing this section, we recall that (i) the Bethe-Salpeter approach to the calculation of two-particle ex-cited states is a straightforward extension of the GWapproach for the calculation of one-particle excitedstates; and (ii) it leads to an effective two-particleHamiltonian H2p which is of the general form found intime-dependent problems, as explained in the introduc-tion. The electron-hole interaction that appears in H2p

has two contributions: the first involves the bare Cou-lomb interaction (v or v , depending on whether x or Pare calculated), which connects the electron and holeindexes in an exchangelike manner and is therefore alsocalled electron-hole exchange. In order to avoid confu-sion, however, it should not be forgotten that this con-tribution stems from the density variation of the Hartreeterm and contains just the local field effects in the caseof v . The second contribution involves W, which con-nects the electron and hole indexes like a direct Cou-

29It corresponds only for the kernel K , because the eigenval-ues entering PIQP are in practice calculated in the full (dy-namic) GW scheme, and the eigenfunctions are approximatedwith LDA ones, not screened Hartree-Fock ones.


2lomb interaction and is therefore called the screenedelectron-hole interaction. It comes from the variation ofthe self-energy, in other words, from an exchangeliketerm in the one-quasiparticle Hamiltonian. It is this lastterm that is responsible for the appearance of boundstates, and it can even lead to hydrogenlike spectral fea-tures in insulators.

In practice, the spin sn is most often not consideredexplicitly. One then makes use of the fact that only (v ,c)transitions with sv5sc (i.e., singlet excitons) contributeto the optical spectrum. Since W does not contain a spininteraction, and because of the way the d functions con-nect indexes, the matrix elements W(vc)(v8c8) are diago-nal in spin, i.e., sv5sv8 . Instead, the matrix elementsv(vc)(v8c8) are independent of sv and sv8 . Hence spin isimplicitly included via a factor of 2 for v in Eq. (4.11).

3. The effects of the electron-hole interaction

Exciton calculations for real systems within a fullyfirst-principles scheme, starting from DFT, have recentlybeen performed. However, the importance of the exci-tonic effects has been known for a long time, even incases in which the features are less characteristic than,the appearance of a hydrogenic series of bound states inthe band gap. Excitonic effects in the absorption lineshape of semiconductors above the fundamental gapwere calculated some time ago (Hanke and Sham, 1974,1975, 1980; del Castillo and Sham, 1985). Hanke andSham (1980) solved the Bethe-Salpeter equation [Eq.(4.9)] for the particle-hole response function of bulk sili-con, using a linear combination of atomic orbitals basis,with a semiempirical band structure fitted to optical ex-periments. They found important corrections to theindependent-particle result, in particular a strong in-crease in the lowest main absorption peak (E1), leadingfor the first time to a calculated absorption spectrum ofbulk silicon in qualitative agreement with experiment.

The ab initio approaches used today for solution ofthe Bethe-Salpeter equation (Albrecht et al., 1998a,1998b; Benedict et al., 1998a; Rohlfing and Louie,1998b) mostly follow the scheme introduced by Onidaet al. (1995) for the spectrum of the cluster Na4 and thatof Albrecht et al. (1997) for the optical gap of bulkLi2O. This procedure consists of (i) a ground-state DFTcalculation; (ii) a GW calculation to correct the eigen-values; and (iii) solution of the Bethe-Salpeter equationusing GW eigenvalues, Kohn-Sham orbitals, and staticRPA screening for the electron-hole interaction. It thusincludes several steps and choices, which are schemati-cally represented by the flow diagram in Fig. 4.

This scheme, using the approach of the effective two-particle Hamiltonian, has again been applied to the cal-culation of the absorption spectrum of bulk silicon (Al-brecht et al., 1998a), showing that the Bethe-Salpetermethod used without any adjustable parameter can yield


absorption spectra of continuum excitons in bulk semi-conductors in quantitative agreement with experiment(see Fig. 5).30

On the other hand, as pointed out above, the electron-hole attraction can also lead to bound excitons in insu-

30The present curve has been calculated by Olevano andReining (2000) using an improved Brillouin-zone samplingwith respect to the original publication of Albrecht et al.(1998a). See also the discussions by Cardona et al. (1999) andAlbrecht et al. (1999).

FIG. 4. Flow diagram sketching the solution of the Bethe-Salpeter equation in practice.


lators, where the interaction is only weakly screened.The resulting spectra can also be well described by theab initio Bethe-Salpeter approach, as has been illus-trated by the work of Benedict et al. (1998a). These au-thors use a recursive method (Haydock, 1980) to invertthe Bethe-Salpeter equation (4.9), which has the advan-tage that one can make use of the d functions in the Wcontribution to the kernel in order to make calculationsless cumbersome.31 Figure 6 shows the result for bulkLiF (Shirley, 2001).32 The sharp peak in the absorptionspectrum at about 12 eV is clearly due to excitoniceffects—the spectrum without excitons is essentially fea-tureless in the range of the absorption onset. The 12-eVpeak defines the optical gap, which turns out to be onlyabout 0.5 eV lower than the experimental one, whereasthe independent-quasiparticle result shows a consider-able overestimate.

The same approach has also been used by Benedictet al. (1998b) to obtain the absorption spectra of bulk Si,C, Ge, and GaAs. Moreover, they have studied thewide-gap semiconductor GaN (in its wurtzite and zinc-blende structures), and the insulating CaF2 crystal(Benedict and Shirley, 1999). In all cases, the inclusionof excitonic effects turned out to be crucial for a quan-titative agreement between theory and experiment.Rohlfing and Louie (1998b) have introduced a differentway of extending the applicability of the Bethe-Salpeter

31The calculations are simplified further by the use of a modelelectron-hole screening.

32Shirley’s calculation is the same as that of Benedict et al.(1998a), but with an improved Brillouin-zone sampling andincluding more bands.

FIG. 5. Silicon absorption spectrum @Im(«M)#: d, experiment(Lautenschlager et al., 1987); dash-dotted curve, RPA, includ-ing local field effects; dotted curve, GW-RPA; solid curve,Bethe-Salpeter equation.


approach: they noticed that one of the bottlenecks in thecalculation of bulk spectra is the determination of the(Nv3Nc3Nk)2 matrix elements of H(vk ,ck)(v8k8,c8k8)

2p ,where Nv and Nc are the numbers of valence and con-duction bands that have to be taken into account in theconstruction of the spectrum, and Nk is the number of kpoints at which the direct vk→ck transitions are consid-ered. They hence proposed an interpolation scheme in kfor the H(vk ,ck)(v8k8,c8k8)

2p , which allowed them to calcu-late the absorption spectrum of bulk GaAs including theextremely weakly bound exciton states (some meV ofbinding energy) in the gap. The description of the latterfeature in fact required the use of about 1000 k points ina sphere of 0.015-a.u. radius around k50.

Rohlfing and Louie (1998b) and Benedict and Shirley(1999) also noticed that the excitonic effects above thegap are due to the mixing of transitions given by thecoefficients Al

(n1n2) in Eq. (4.12), in other words, inter-ference effects. This can be shown by looking at the den-sity of excitation energies, which turns out to be almostunchanged by the inclusion of the electron-hole interac-tion (see, for example, Fig. 6 in Benedict and Shirley,1999). This means that the apparent shift of peak posi-tions [about 0.4 eV for the E2 peak in silicon (Albrechtet al., 1998a)] is not due to a negative shift of the transi-tion energies.

Of course, this is not true in the case of bound exci-tons, which, apart from the above-mentioned insulators,includes systems having discrete energy levels, like at-oms (Rohlfing and Louie, 2000), molecules and clusters(Onida et al., 1995; Rohlfing and Louie, 1998a), andother low-dimensional systems. In these cases, excitationenergies are shifted; moreover, the effect can be verystrong, since screening is not efficient and the electronand hole are localized.

A good illustration of excitons in low-dimensional sys-tems is given by the case of conjugated polymers liketrans-polyacetylene and poly-phenylene-vinylene(PPV). The optical absorption spectra of these mol-

FIG. 6. LiF, absorption spectrum @«25Im(«M)#. Continuouscurve with crosses, experimental (Roessler and Walker, 1967).Dashed curve, Eq. (2.8) using GW eigenvalues. Plain continu-ous curve, Bethe-Salpeter equation (Shirley, 2001).


ecules have been studied by Rohlfing and Louie (1999a).The electron-hole interaction gives rise to exciton bind-ing energies of the order of one eV and strongly modi-fies the spectral line shapes. Similar results for polymershave also been found by van der Horst et al. (1999) andby Ruini et al. (2002).

Surfaces can also be considered as low-dimensionalsystems, in particular in certain reconstructions like thequasi–one-dimensional chain structures of Si(111)231and Ge(111)(231). A strong excitonic effect has beenfound for Si(111)231 in semiempirical (Reining andDel Sole, 1991) and ab initio calculations (Rohlfing andLouie, 1999b). The excellent agreement found allowsone to use the theoretical results as reference spectra forthe interpretation of experiments. For example, an abinitio exciton calculation by Rohlfing et al. (2000) forGe(111)(231) has demonstrated how optical differen-tial reflectivity spectra can be used to distinguish be-tween the two possible isomers of the reconstructed sur-face. This distinction was made possible by the fact thata quantitative comparison between the calculated andexperimental spectra is possible when electron-hole ef-fects are treated correctly (see Fig. 7).

Rohlfing and Louie (2000) have recently published anextended paper in which their first-principles exciton ab-sorption calculations are reviewed. They present thetheoretical framework, compare different choices of ba-sis sets, and discuss the applications to both finite sys-tems (namely, He, Ne, and Ar atoms, and SiH4 andlarger SinHm clusters) and infinite crystals (GaAs, Si,LiF, and LiCl).

FIG. 7. Normal-incidence differential reflectivity spectrum ofthe Ge(111)(231) surface: solid lines, the spectrum includingself-energy and excitonic effects; dashed lines, spectra ob-tained using Eq. (2.8) and GW eigenvalues. The upper panel isfor the positively buckled isomer, the lower for the negativelybuckled isomer. Dots are experimental data by Nannaroneet al. (1980).


The importance of excitonic effects can also be seen inelectron-energy-loss spectra, linked to the imaginarypart of the inverse dielectric function. Recently,Soininen and Shirley (2000) applied the ab initio excitonscheme to the calculation of the dynamic structure fac-tor S(q,v) of diamond, LiF, and wurtzite GaN for vari-ous momentum transfers q. As in the case of the macro-scopic dielectric function, the inclusion of the electron-hole interaction in «21 was found to substantiallyredistribute the spectral weight with respect to a GW(independent-quasiparticle) calculation, being crucial ininterpreting the experimental dynamical structure factorover a wide range of energies. Essentially, an overallshift to lower energies and an enhancement of the inten-sities on the low-energy side are found. The latter effectcan be traced back to the nontrivial excitonic effects inthe imaginary part of the dielectric function. Concerningthe overall, essentially rigid shift, one might wish to dis-cuss in detail the extent to which the electron-hole at-traction and self-energy corrections cancel in the dy-namical structure factor. For the plasmon peak of siliconat vanishing momentum transfer, it has in fact beenshown that strong cancellations occur (Olevano andReining, 2001a). In other words, whereas it is clear that,at present, absorption spectra of bulk materials most of-ten necessitate the inclusion of self-energy and electron-hole attraction effects, it is less obvious how much theseeffects improve loss spectra in general and to what ex-tent improvements obtained with respect to using Eq.(2.8) are essentially due to local field effects. This latterpoint will also be discussed in Sec. VI.D.

In conclusion, it is important to note that a large va-riety of problems has by now been studied via the abinitio approach, ranging from absorption to energy-lossspectra; from applications to atoms, molecules, and clus-ters to applications to bulk materials; and showing exci-tonic effects as different as the continuum exciton insilicon, the very weakly bound Wannier excitons inGaAs, or the strongly bound exciton in LiF. Models existwhich allow each of these situations to be described, thetwo best known being the hydrogenic Wannier modelfor weakly bound excitons and the Frenkel model forstrongly bound excitons (see Bassani and Pastori Par-ravicini, 1975 for an overview). In fact, certain featuresin the spectra can be very easily predicted by these mod-els. In particular, the knowledge of the effective masses(band curvatures) at the direct gap, and of the dielectricconstant, often yield a very good estimate of excitonbinding energies and allow one to estimate stronglybound excitons like those in LiF. However, none of thesimple models could ever predict an entire spectrum.Solid argon, for example, exhibits a hydrogenlike seriesof peaks below the continuum, the first of which isFrenkel-like, and the others of which are well describedby the Wannier model. More refined model approachesmanage to include both types of bound excitons (Rescaet al., 1978), but are not meant to predict excitonic ef-fects in higher parts of the spectrum, where other detailsof the band structure become important. Instead, the abinitio approach has been shown to treat all those fea-


tures equally well, and hence to be flexible and predic-tive. Of course the agreement with experiment is stillsomewhat limited by the various approximations thatare commonly used.

In the following section, several possible choices willbe discussed.

4. Different levels of sophistication

One of the main difficulties one encounters when dis-cussing the quality of Bethe-Salpeter results is the factthat, as can be seen from Fig. 4, the ab initio approach isa three-step method. First, a Kohn-Sham ground-statecalculation is performed. Second, GW corrections areadded in order to obtain correct quasiparticle energies.Third, the Kohn-Sham and GW results are used in orderto construct the Bethe-Salpeter equation. At each step,approximations are made and one has to be careful inorder to keep the error bar in the final results small.

In the following, we shall examine a list of the mainapproximations that are often used.

(i) Pseudopotentials. All above-cited ab initio calcu-lations use pseudopotentials. The use of pseudowave functions for the construction of transitionmatrix elements might, however, introduce someerror in the results, even when the nonlocality ofthe pseudopotential is correctly taken intoaccount.33 The influence of strongly nonlocalpseudopotentials on the calculation of transitionmatrix elements has been analyzed by Read andNeeds (1991), who have found differences of upto 10–15 %, depending on the element. See alsoMarini et al. (2001) for an example.Arnaud and Alouani (2000, 2001) have calculatedquasiparticle energies and optical spectra includ-ing self-energy and excitonic effects, using the all-electron full-potential projector-augmented wavemethod of Blochl (1994). For Si, GaAs, and dia-mond their GW band structure agrees with theavailable pseudopotential results to within 1% forGaAs and AlAs, 2% for Si, and 5% for InP (Ar-naud and Alouani, 2000). Optical spectra com-puted within the Bethe-Salpeter scheme (Arnaudand Alouani, 2001) are also in agreement towithin a few percent with pseudopotential results,except for the case of diamond, where they founddifferences in the peak positions of the order of0.5 eV.

(ii) Kohn-Sham-LDA wave functions. In the Bethe-Salpeter approach, PIQP and other matrix ele-ments are most often constructed using GW ei-genvalues but LDA wave functions. This has beenjustified by the fact that quasiparticle and Kohn-Sham orbitals are supposed to be close to each

33Naturally, when using nonlocal pseudopotentials, the cor-rect definition of the velocity operator must be adopted, sincemacroscopically wrong results can be obtained otherwise (VanDyke, 1972; Read and Needs, 1991).


other (Hybertsen and Louie, 1986; Godby et al.,1986, 1988). However, this is not always the case.For example, for finite systems the missing 21/rtail of the Vxc

LDA potential can strongly distort thewave functions, and it may be necessary to correcttheir spatial behavior. This has been done byRohlfing and Louie (2000). For the small clustersstudied, the authors have shown that the wavefunctions of the empty states become clearly moredelocalized with respect to the LDA Kohn-Shamones. This has a sizable effect on the exciton bind-ing energies, which are reduced by as much as20%. Note that it would be much more compli-cated to include exact quasiparticle wave func-tions. In fact, in order to obtain Eq. (4.11), thewave functions used for the construction of thematrix elements have to derive from a static op-erator. Otherwise, they might be nonorthogonalfor different states (energies), the above basistransformation would become more complicated,and the calculation could yield different results.

(iii) Static electron-hole screening. This approximationis often justified, since the plasma frequencies ofthe investigated systems are much larger than theexcitonic binding energies. Moreover, it has beenfound that dynamical effects in the electron-holescreening and in the one-particle Green’s functiontend to cancel each other, at least in simple semi-conductors (Bechstedt et al., 1997), which suggeststhat both should be neglected. This might not betrue in other systems where the exciton bindingenergy is large. Dynamical effects in the Bethe-Salpeter equation have been studied for core ex-citons by Strinati (1982, 1984), who derived themore general effective two-particle equation34

(v8c8

Hvcv8c8~El!Al

v8c8~El!5ElAlvc~El!. (4.13)

Note that, as in the case of the one-particle ex-cited states, it is the dielectric function that intro-duces the energy dependence (and hence finallythe possibility of complex eigenvalues) into theequation: the two-particle excited state may nowhave a finite lifetime, shorter than the electronand hole lifetimes. And again, since this comesthrough «21, energy is conserved, since one exci-tation is decaying by exciting others. Dynamicaleffects have been studied for SiH4 by Rohlfingand Louie (2000); they are found to increase theexciton binding energy by about 0.1 eV.

(iv) RPA electron-hole screening. The kernel W is usu-ally evaluated in the RPA. Since the neglect ofexchange-correlation effects in the screening canlead to an underestimate of the dielectric con-stant, this approximation might overestimate exci-

34Strinati found that dynamical screening effects significantlynarrow the core-excitation spectral width, which goes alongwith an increase in the exciton binding energy.


ton binding energies. Chang et al. (2000) havetried to estimate the effect for a-quartz by rescal-ing the electron-hole screening self-consistently,with the dielectric constant calculated includingexcitonic effects. In fact, they find a shift of themain absorption peak towards higher energies,closer to the experimental one. A quantitative dis-cussion is difficult, since it should include a recal-culation of the dielectric matrix for all momentumtransfers and a consistent inclusion of vertex cor-rections (i.e., an additional vertex, because theBethe-Salpeter equation should now be derivedfrom S5iGWG).

(v) Neglect of resonant-antiresonant coupling. Gener-ally, Eq. (4.12) is used for the calculation of ab-sorption spectra. This supposes that transitions atpositive (resonant part) and negative (antireso-nant part) frequencies do not mix (see AppendixB.2). The effects of the resonant-antiresonantcoupling on the exciton binding energy have beendiscussed by Zimmermann (1970) and Del Soleand Selloni (1984). Neglecting the coupling canbias the results, especially for quantities based onthe real part of epsilon, like the dielectric constantor the electron-energy-loss spectra. This has beenillustrated by Olevano and Reining (2001a) forthe case of the plasmon resonance of silicon.

To summarize, we stress that the Bethe-Salpeter ap-proach has up to now yielded results that agree with themeasured ones within 10% for the peak positions and20% for the peak strengths. The error bar is given by thesum of several contributions, and it has not yet beencompletely elucidated where the main sources of errorlie. Well-generated ab initio pseudopotentials should af-fect the results by less than .0.1 eV.35 Also, as men-tioned in Sec. IV.A.2, GW transition energies may be offby 10–15 %, and one suspects that the quality of the GWresult dominates the agreement between the experimen-tal and the theoretical peak positions. The way thescreening is taken into account (e.g., with a model di-electric function, or in the static RPA) also contributesto the error bar by at least another 0.1 eV. As for presentlimitations of the approach, one should mention the ne-glect of the dynamical screening of the electron-hole in-teraction, which could become relevant when looking atthe dielectric function at frequencies comparable withthe plasma frequency, and the complete neglect of thelattice vibrations. Finally, there are the limitations due tothe computational heaviness of the approach, which is inpart necessarily cumbersome, since it involves four-pointquantities. This point is, among others, a strong motiva-tion to search for alternative approaches, the mostprominent being the TDDFT approach, to be discussedin the next section.

35The choice of the lattice constant—experimental ortheoretical—can also contribute 50–200 meV.


V. TIME-DEPENDENT DENSITY-FUNCTIONAL THEORY

Several reviews on general foundations of TDDFTand some applications have appeared recently, includingthose of Gross et al. (1994, 1996), Casida (1995, 1996),Dobson, Vignale, and Das (1997), Burke et al. (2002),and van Leeuwen (2001). Once more we focus the dis-cussion below on the points that are essential for thecomparisons that are within the scope of this review. Inthe following, we summarize the foundations of TDDFT,and its connections with excited-state properties andwith many-body perturbation theory (a detailed com-parison between TDDFT and the Green’s-function ap-proach will be given in Sec. VI).

A. Formalism

The Hohenberg-Kohn-Sham theory as described inSec. II.A is a ground-state theory, and is hence notmeant for the calculation of electronic excitations. How-ever, one can extend the idea of static DFT by analogywith classical mechanics: the ground state is determinedby the minimum of the total energy. When one asks forthe evolution of the system under the influence of atime-dependent external potential, one should searchthe extrema of the quantum-mechanical action,

A5Et0

t1dt^C~ t !ui

]

]t2H~ t !uC~ t !&, (5.1)

just as in classical mechanics the trajectory of a system isdetermined by the extrema of the action * t0

t1dtL(t),

where L is the Lagrangian. Theorems have now beenestablished for time-dependent DFT which are parallelto those of static DFT. Many-body effects are includedin a local exchange-correlation potential, which is nowtime dependent and, as in the case of static DFT, is un-known (for more details see Gross et al., 1996). The firstapplications of TDDFT response theory were made be-fore the formal development and relied on analogieswith time-dependent Hartree-Fock theory (Stott andZaremba, 1980; Zangwill and Soven, 1980). In the workof Runge and Gross (1984) a theory similar to that ofHohenberg, Kohn, and Sham is developed for time-dependent potentials in terms of the action functional.The first theorem proves a one-to-one mapping betweentime-dependent potentials and time-dependent densities(that are v representable); the second proves thestationary-action principle. The scheme is very similar tothe ground-state Kohn-Sham formalism. The proof ofthe first theorem is based directly on the evolution of thetime-dependent Schrodinger equation from a fixed ini-tial many-particle state C(t0)5C0 under the influenceof a time-dependent potential Vext(t) (required to beexpandable in a Taylor series around t0). The initialstate C0 does not need to be the ground state or anyother eigenstate of the initial potential. As one does notrely on the adiabatic connection in standard zero-temperature many-body perturbation theory, the for-malism is able to handle external perturbations varyingrapidly in time.


By virtue of the first theorem, the time-dependentdensity determines the external potential uniquely up toan additive purely time-dependent function. On theother hand, the potential determines the time-dependent wave function; therefore the expectationvalue of any quantum-mechanical operator is a uniquefunctional of the density. This theorem has been gener-alized by van Leeuwen (1999) by showing that a time-dependent density obtained from an initial many-bodysystem can be, in principle, reproduced by a time-dependent external potential in a many-body systemwith different (and possibly null) two-particle interac-tion. This mapping between densities and time-dependent potentials in TDDFT demonstrates the non-interacting v representability of a general many-bodysystem. That is, if one takes the two-particle interactionof the second system to be zero, this theorem establishesthat for a given initial state having the proper initialdensity and time derivative, there is a unique potential(up to a purely time-dependent function) in a noninter-acting system that reproduces the given time-dependentdensity at all times. This result is important for theKohn-Sham formalism of TDDFT (see below).

Moreover, in addition to their dependence on thedensity, the time-dependent functionals depend on theinitial state C0 .36 The time-dependent particle andcurrent density can be calculated exactly from the con-tinuity equation and the equation of motion of theparamagnetic current-density operator jp(r)5 (1/2i) ( j51

N @¹rjd(r2rj)1d(r2rj)¹rj

# , that is,

]r~r,t !]t

52¹j~r,t !, (5.2)

]j~r,t !]t

52i^C~ t !u@ jp ,H~ t !#uC~ t !&. (5.3)

In particular, the current density j(r,t) following fromthe time-dependent Kohn-Sham orbitals and the true in-teracting current density may differ only by the curl ofan arbitrary function (Dobson, Vignale, and Das, 1997).The second theorem deals with the variational principleof the action functional with the initial condition C(t0)5C0 . From the previous one-to-one mapping betweentime-dependent potentials and densities, the action [Eq.(5.1)] is a functional of the density A@r# , which musthave a stationary point at the correct time-dependentdensity. Thus the Euler equation corresponding to theextrema of A@r# , dA@r#/dr(r,t)50 determines thetime-dependent density, just as in the Hohenberg-Kohnformalism the static ground-state density is given by theminimum of the total energy E„dE@r#/dr(r)50…. Simi-larly, one can define a time-dependent Kohn-Shamscheme by introducing a noninteracting system that re-

36In general there are many wave functions leading to thesame density. Any of them will work properly, because thedependence of the effective time-dependent potential is suchthat the interacting density will be reproduced in each case.


produces the exact interacting density r(r,t). Assumingthe (demonstrated) v representability of the time-dependent densities (van Leeuwen, 1999), one gets thefollowing time-dependent Kohn-Sham equations:

F212

¹21Veff~r,t !Gc i~r,t !5i]

]tc i~r,t ! (5.4)

r~r,t !5(i51

N

uc i~r,t !u2 (5.5)

where Veff(r,t)5VH(r,t)1Vxc(r,t)1Vext(r,t) is the ef-fective time-dependent potential felt by the electrons. Itconsists of the sum of the external time-dependent ap-plied field, the time-dependent Hartree term, plus theexchange-correlation potential (defined via the equiva-lence between the interacting and fictitious noninteract-ing systems). The variational principle yields

Vxc~r,t !5dAxc@r#

dr~r,t !, (5.6)

where Axc@r# is the exchange-correlation part of the ac-tion functional.37

The advantage of the time-dependent Kohn-Sham-scheme lies in its computational simplicity compared toother quantum-chemical methods such as time-dependent Hartree-Fock or configuration interaction(Langhoff et al., 1972; Jensen, 1999). Up to here, theformalism presented deals with the quantum nature ofthe electrons only. The equations can be generalized totreat the quantum-mechanical coupling of the nuclearand electronic motions (Kreibich, 2000; Kreibich andGross, 2001), the quantum nature of the electromagneticfield, and even superconductors (Gross et al., 1996;Kurth et al., 1999).

B. Excitation energies in TDDFT

Formally, TDDFT allows the calculation of the(bound and unbound) excited-state energies and transi-tion probabilities of a many-body system, based on in-formation gleaned from an ordinary DFT self-consistentcalculation. In the time-dependent approach, one stud-ies how the system behaves when subject to a time-dependent external perturbation. In this case, the sys-tem’s response is directly related to the N-particleexcited states of an N-particle system, in a manner simi-lar to the way the one-particle Green’s function is re-

37As with the total energy in the static case, here the actionfunctional has been decomposed as

A@r#5( i51N * t0

t1dt^c i(t)ui (]/]t) 112 ¹22Vext(t)uc i(t)&

2AH@r#2Axc@r# ,

where c i(t) are the wave functions of the fictitious noninter-acting Kohn-Sham system, AH is the time-dependent Hartreecontribution 1

2 **drdtr(r,t)VH(r,t), and Axc includes all ex-change and dynamical correlation effects due to the many-body interacting system. Axc is not known and has to be ap-proximated.


lated to the (N11)- and (N21)-particle excited statesof the same system (see Sec. III.A).

Petersilka et al. (1996) have developed a formalismfor calculating the neutral excitations in finite systems.The key formula is a Dyson-like representation of theexact linear density response x of an interacting many-electron system in terms of the noninteracting Kohn-Sham response x0 . To prove this relation, one can startfrom the physical definition of the retarded linear-response function,

x~r,t ,r8,t8!5dr~r,t !

dVext~r8,t8!U

Vext50

, (5.7)

which measures the degree to which the density re-sponds to first order in the external potential. Equiva-lently, the linear response x0 of the fictitious Kohn-Shamsystem that can be described in terms of the Kohn-Shamorbitals is given by

x0~r,t ,r8,t8!5dr~r,t !

dVeff~r8,t8!U

Veff50

, (5.8)

where Veff5Vext1VH1Vxc . Using dr/dVext5(dr/dVeff)3(dVeff /dVext)[x0dVeff /dVext one obtains

dVeff~r,t !dVext~r8,t8!

5d~r2r8!d~ t2t8!

1E Fd~ t2t9!

ur2r9u1fxc~r,t ,r9,t9!G

3x~r9,t9,r8,t8!dr9dt9, (5.9)

where the time-dependent exchange-correlation kernel

fxc~r,t ,r8,t8!5dVxc@r~r,t !#

dr~r8,t8!U

Vext50

(5.10)

has been introduced. Now it is straightforward to get theDyson-like equation

x~r,r8,v!5x0~r,r8,v!

1E dr1dr2x0~r,r1 ,v!K~r1 ,r2 ,v!x~r2 ,r8,v!,

(5.11)

which has to be solved iteratively and where the kernelK has been introduced as

K~r1 ,r2 ,v!51

ur12r2u1fxc~r1 ,r2 ,v!. (5.12)

Equation (5.11) corresponds to Eq. (2.17) of the In-troduction. In fact, Eq. (5.11) and Eq. (5.10) can be ob-tained directly, as discussed in the Introduction, for thecase in which the total potential in the time-dependentHamiltonian is VH1Vxc . As also pointed out there, dueto the density-only dependence of these potentials the dfunctions that appear in the derivation are such that onlytwo-point functions are involved from the beginning.Equation (5.11) is hence the contraction of Eq. (2.20).Of course, the nontrivial point is that the response func-


tion x calculated from the fictitious noninteracting sys-tem using Eq. (5.11) is equal to the true response func-tion of the interacting system.

This scheme provides an exact representation of thefull interacting linear density response as

r1~r,v!5E dr8x0~r,r8;v!Veff~r8,v!

[E dr8x~r,r8;v!Vext~r8,v!.

In other words, the exact linear density response of aninteracting system can be written as the linear densityresponse of a noninteracting system to the effective per-turbation Veff . The exact time-dependent exchange-correlation kernel is of course unknown, and practicalcalculations must rely on some approximation. The mostcommonly used one, due to its simplicity, is the adiabaticlocal-density approximation, also called the time-dependent LDA (TDLDA), in which fxc(r1 ,r2 ,v) is ap-proximated by the (v-independent) functional deriva-tive of the LDA exchange-correlation potential:

fxcTDLDA~r1 ,r2!5d~r12r2!

]VxcLDA

„r~r1!,r1…

]r~r1!. (5.13)

Apart from this approximation for fxc , another approxi-mation has to be made in practical calculations: thestatic Kohn-Sham orbitals and eigenvalues used to con-struct x0 are in fact calculated with an approximateexchange-correlation potential Vxc , typically the sameas the one used in ground-state calculations.

Formally inverting Eq. (5.11), one obtains a compacttwo-point matrix equation,

x~v!5@12x0~v!K~v!#21x0~v!, (5.14)

where the problem of finding the excited-state energiesof an interacting system (poles of x) has been mapped tosearching the values of v for which the operator R(v)512x0(v)K(v) is not invertible. In fact, x has poles atthe true excitation energies v5V j , while x0 has poles atthe Kohn-Sham eigenvalue differences. Hence the sin-gularities of x must be canceled by the zeros of R(v)[and those of x0 by the zeros of R21(v)]. The true ex-citation energies V j can be characterized as those fre-quencies where the eigenvalues of R vanish. In otherwords, the energies of the resulting electron-hole excita-tions will be renormalized with respect to those of thenoninteracting electron-hole pairs. This formalism pro-vides a convenient starting point for calculating the ex-citation spectrum.

Of course, for the practical solution of Eq. (5.14) allpoints discussed in the Introduction apply. In particular,when adding a finite small imaginary part to the fre-quency, the two-point equation can be inverted for eachfrequency in order to obtain the spectrum. Alternatively,Eq. (5.14) can be transformed into an effective eigen-value problem, which implies that one is now workingwith four-point quantities. This second procedure isequivalent to solving the generalized four-point eigen-


value problem 4R(v)ul&50 (see Appendix, B.2, in par-ticular 4R[P21[@124x0K#) and yields the equation

cn1n22 (

n3n4

cn3n4~fn1

2fn2!

K ~n1n2!,~n3n4!~v!

v2~en22en1

!50,

(5.15)

where the kernel K(n1n2),(n3n4)(v)5^Fn1n2

uK(v)uFn3n4&, with Fn1n2

(r)ªcn1(r)cn2

* (r).The index n embodies the spin as well as orbital degreesof freedom.38

Looking at the Coulomb contribution to K , one rec-ognizes the effective exchange interaction between elec-tron and hole (electron-hole exchange), which also ap-pears in the Bethe-Salpeter equation (4.11) (for x and vinstead of P and v). The diagonal element reads

E E dr1dr2Fn1n2* ~r1!

1ur12r2u

Fn1n2~r2!, (5.16)

whereas the fxc contribution within the simple TDLDAintroduces a local and static attractive electron-hole in-teraction:

E E dr1dr2Fn1n2* ~r1!d~r12r2!

]Vxc~r1!

]rFn1n2

~r2!

5E drrn1~r!

]Vxc~r!

]rrn2

~r!. (5.17)

One can see the effects of these two different contribu-tions on the calculated optical spectrum of a finite sys-tem, e.g., a small cluster: the Coulomb part shifts theindependent electron spectrum to high energies,whereas the exchange-correlation brings it partially back(this can to a certain extent be compared with the exci-tonic corrections computed in the framework of themany-body theory, as will be discussed in Sec. VI).These effects are clearly seen in Fig. 8, where the opticalspectrum calculated within TDLDA for three differentclusters is reported.

Equation (5.15) can be simplified by an expansionaround one particular Kohn-Sham transition 1→2, i.e.,by calculating the transition energy v5V in first-orderperturbation theory (single-pole approximation). Fol-lowing Petersilka et al. (1996) and considering explicitlythe spin degrees of freedom,39 one obtains from Eq.(5.15)

V.v121Re@K1↑2↑ ,1↑2↑~v12!6K1↑2↓ ,1↑2↓~v12!# .(5.18)

38The transition-space representation is based on similaritieswith the quantum-chemistry time-dependent Hartree-Fock ap-proach (Langhoff et al., 1972; Casida, 1995). However, a simpletwo-point matrix problem has been transformed into a morecomplex four-point representation. Note also that Eq. (5.15)can be derived from the condition det (H2p2El)50 with H2p

of the form of Eq. (B20), and K of the form defined in Eq.(2.15) without the term involving w .

39In the present section we treat spin effects explicitly, as isnatural when considering atomic transitions.


Note that the noninteracting Kohn-Sham response func-tion is diagonal in the spin variables and exhibits polesat the Kohn-Sham energy differences corresponding tononinteracting electron-hole excitations within the samespin space. The mixing of spin channels comes into playsimply by the spin-dependent exchange-correlation ker-nel, and the magnetization response naturally involvesspin-flip processes. In order to make more explicit thefact that the approximation in Eq. (5.18) embodies thespin-multiplet structure of the excitation spectrum ofotherwise spin-unpolarized ground states, one can re-write the fxc kernel in terms of the two independentcombinations of the spin components of the kernel: fxc

1

FIG. 8. Calculated optical absorption spectrum (given as thedipolar strength function) for several clusters (Marques et al.,2001), compared to experimental data (Itoh et al., 1986; Wanget al., 1990): solid black line, full TDLDA calculations; dashedline, independent Kohn-Sham particle approximation (x0);dotted line, an RPA calculation, i.e., putting fxc50 in Eq.(5.12); gray line, the experimental results. In the inset of Na8are the results for the sodium dimer: heavy solid line, fullTDLDA calculation; gray line, experimental results (Sinhaet al., 1949), but now the dashed line (almost indistinguishablefrom previous one) is for a calculation using an exact-exchangefunctional (Marques et al., 2001). As discussed in the text thekernel includes an effective attractive part (electron-hole at-traction), which reduces the Coulomb term (electron-hole ex-change).


50.5(fxc↑↑1fxc↑↓) and fxc2 50.5(fxc↑↑2fxc↑↓).40 Thus the

singlet and triplet solutions of Eq. (5.18) are (note againthe similarity to the Bethe-Salpeter equation concerningthe contribution of v)

vsinglet5v1212 Re E dr1dr2F12* ~r1!

3S 1ur12r2u

1fxc1 ~r1 ,r2 ,v12! DF12~r2!

and

v triplet5v12

12ReE dr1dr2F12* ~r1!fxc2 ~r1 ,r2 ,v12!F12~r2!.

Promising results are obtained for the lowest excitationenergies of atoms and molecules by using different ap-proximations for Vxc and fxc (Petersilka et al., 1996;March et al., 1999; Casida et al., 2000; Petersilka et al.,2000).

Results for finite systems beyond the single-pole ap-proximation which have been particularly discussed arethose of TDLDA (Rubio et al., 1996; Yabana andBertsch, 1996, 1999a, 1999b; Vasiliev et al., 1999, 2002;Casida et al., 2000; Marques et al., 2001) and the resultsobtained within the OEP scheme41 (Gorling, 1999;Gross et al., 1996; Petersilka et al., 1996, 2000; Graboet al., 2000a, 2000b). The latter approach gives anexchange-correlation potential with the correct 21/r be-havior at long distances (Casida and Salahub, 2000). Theoptimized effective potential yields a uniform shift ofthe energies of transitions to Rydberg states (Petersilkaet al., 2000; Stener et al., 2001) that mimics to some ex-tent the results obtained using the exact Vxc availablefor some atoms (Petersilka et al., 2000). From this workthe importance of a good description of the staticexchange-correlation potential is clear, as is also the factthat only ground-state quantities are needed in the cal-culation of excitations. Vxc is especially important forthe higher atomic excited states, which are almost uni-formly shifted from the true excitation energies (theshift can be related to the difference between the ioniza-tion potential and the highest occupied orbital, which inexact DFT should be zero). Further developments oftime-dependent functionals to treat problems involvingexcited-state dissociation (Cai et al., 2000; Aryasetiawanet al., 2002a) and autoionization resonances (Steneret al., 2001) are needed. In the latter case, even if theatomic photoionization cross sections are well describedusing the exact Vxc and the TDLDA kernel, this is not

40The fxc2 part of the kernel describes exchange-correlation

processes in the Kohn-Sham system related to the linear re-sponse of the frequency-dependent magnetization density,whereas fxc

1 is related to the frequency-dependent density.41The calculations in the OEP scheme handle exchange ex-

actly, and correlations are treated in a local or gradient-corrected adiabatic functional.


the case for the autoionization resonances that turn outto be very sensitive to the particular choice of fxc .

In extended systems with delocalized states one candirectly solve Eq. (5.14) in the complex plane. [Thesecular Eq. (5.15) becomes a four-point integral equa-tion and is computationally not advantageous]. The ze-ros of R provide both the excitations as well as the cor-responding lifetimes. This technique has beensuccessfully used to describe the plasmon dispersion of ahomogeneous electron gas (Tatarczyk et al., 2001). How-ever, the TDLDA has only limited success in describingexcitations in extended systems, and most likely bothnonlocality in space and time are needed in order to getthis renormalization. The successes and failures ofTDLDA in finite and infinite systems, respectively, arenot yet fully understood. However, it appears that im-provements might more easily be found through an im-proved Vxc in the case of finite, and an improved fxc inthe case of infinite, systems. In any case, fxc is a quantityof interest which (i) has been discussed in less detailthan Vxc and (ii) is necessarily more complicated, sinceit is its functional derivative. Therefore we look moreclosely at fxc in the following section.

C. The exchange-correlation kernel fxc

In the previous section we introduced the exchange-correlation kernel fxc to take into account all the dy-namical exchange and correlation effects in the responseof a system to an external perturbing potential. An exactrepresentation of fxc in terms of the response functionsis obtained directly from Eq. (5.11):

fxc~r,t ;r8,t8!5x021~r,t ;r8,t8!2x21~r,t ;r8,t8!

2d~ t2t8!

ur2r8u. (5.19)

Note that for finite systems the frequency-dependent re-sponse operators can be noninvertible at isolated fre-quencies (isolated real poles). However, this is no longerthe case for infinite bulk systems. In Appendix A weprovide a summary of the known exact properties of fxcthat can be used as constraints to build new approxima-tions to the exchange-correlation kernel.

As a consequence of causality, response functionsmust be zero for t8.t . Therefore the fxc kernel cannotbe symmetric under the exchange of (r,t) and (r8,t8).Since, on the other hand, fxc is the functional derivativeof the exchange-correlation potential, one concludesthat either the exact Vxc(r,t) cannot be a functional de-rivative of the Axc functional (i.e., fxc is not a secondfunctional derivative; see Gross et al., 1996; van Leeu-wen, 1998, 1999), or there is a contradiction with thestationary-action principle that we have described as abasic ingredient of time-dependent density-functionaltheory. This problem applies to all twice-differentiableaction functionals defined with respect to the physicaltime. It appears when applying the variational principleto the action in Eq. (5.1). In fact, from the Runge-Gross(1984) theorem, the wave function is determined up to a


time-dependent phase factor; this makes the action notuniquely defined unless one fixes this phase factor (inparticular it can be chosen such that A@C#50). There-fore the action is not a density functional as the densityby itself does not fix the phase of the wave function (vanLeeuwen, 2001). This apparent contradiction has beenformally resolved by van Leeuwen (1998, 2001) by de-fining a new action functional42 that properly incorpo-rates causality effects, and it is not made stationary butrather used as a generating function for the density andresponse functions as in statistical mechanics.

One of the most widely used approximations to time-dependent phenomena is the adiabatic LDA or time-dependent LDA (TDLDA) in which the static LDAfunctional is used for the dynamical properties. Thismeans that the fxc kernel is a contact function in timeand space [Eq. (5.13)]. Thus fxc is not frequency depen-dent at all. TDLDA gives rather accurate results for sys-tems with rapidly varying densities such as atoms, sur-faces, and clusters.43 Gross and Kohn (1985) havepresented an extension of the LDA approximation toinclude dynamical effects in the fxc kernel, and we referthe reader to their work for the details ofparametrization.44 The idea is to use the homogeneouselectron-gas kernel fxc

hom(ur2r8u,v) and make the as-sumption that the linear-induced density is a slowlyvarying function (as in the traditional static LDA). Thisamounts to the approximation fxc(r,r8,v)5 f xc

hom„r(r),v…, where f xc

hom is the q50 Fourier com-ponent of fxc

hom„r(r),ur82r9u….

Other kernels proposed in the literature are the fol-lowing (spin variables are omitted for simplicity):

• Petersilka, Grossman, and Gross (PGG) kernel. Thiswas derived by Petersilka et al. (1996) in the contextof exact exchange time-dependent optimized effectivepotentials and kernels (Gross et al., 1996; Gorling,1998) (x-TDOEP), and it is equivalent to the so-calledSlater approximation in Hartree-Fock calculations(Langhoff et al., 1972). It is a frequency-independentkernel that in real space reads

42The new action is defined using the time contour method ofKeldysh (1965) in which the physical time is parametrized interms of a parameter called pseudotime. By construction, theresponse functions obtained as higher-order derivatives of theaction functional are symmetrical in the Keldysh pseudotime.A back-transformation to the physical time directly providesthe desired causal response functions (see van Leeuwen, 2001for mathematical details of this technique).

43See, for example, (Stott and Zaremba, 1980; Zangwill andSoven, 1980; Dobson et al., 1988; Tsuei et al., 1990; Casida,1995; Petersilka et al., 1996; Rubio et al., 1996, 1997; Liebsch,1997; Vasiliev et al., 1999, 2002).

44Note that the lower the density the larger the frequencydependence of the kernel. The parametrization can be ex-tended to nonvanishing q and to include spin polarization.However, it fails to reproduce some exact relations, such as theharmonic-potential theorem (Dobson, 1994).


fxPGG~r,r8!52

12

1ur2r8u

U(k

fkck~r!ck* ~r8!U2

r~r!r~r8!.

Note that the general expression of the TDOEP ker-nel (similar to time-dependent Hartree-Fock) is fre-quency dependent even at the exchange-only level(Petersilka et al., 1996, 1998), a feature that is not ac-counted for in the PGG approximation. However fora two-electron system at the exchange-only level theTDOEP and PGG kernels are equivalent, explicitlyshowing that the dynamical part of the kernel stemsfrom processes involving multiple valence transitions.

• TDOEP-SIC kernel. This is an attempt to improveover both TDLDA and exact exchange (TDOEP) bycorrecting the TDLDA for the self-interaction error(SIC5self-interaction-corrected). This correctiondoes not affect antiparallel spin contributions. In thiscase, the kernel reads

fxcTDOEP-SIC~r,r8!5fxc

TDLDA~r,r8!

212

(k

fkuck~r!ck* ~r8!u2

r~r!r~r8!

3H ]VxcLDA

„rk~r!,r…

]rk~r8!1

1ur2r8uJ ,

where rk is the density of the orbital k . This expres-sion is exact in the one-electron case and, for moreelectrons, it corrects the spurious self-interaction ofparallel spin, but keeps the antiparallel contributionsas in TDLDA. This functional is nevertheless ill de-fined, since it is not invariant upon a unitary transfor-mation of the Kohn-Sham wave functions.

• BPG kernel. Burke et al. (2002) proposed a hybridmethod to improve the excitation spectra of small at-oms by combining the previous PGG expression forsymmetric spin orientations and the TDLDA for an-tisymmetric spin orientations. For the case of a homo-geneous electron gas it reads fxc

BPG(q)50.5* @fxc ,↑↑

PGG(q)1fxc ,↑↓TDLDA(q)# .

• CDOP kernel. Corradini et al. (1998) gave a param-etrization of the quantum Monte Carlo data of Mo-roni et al. (1995) for the homogeneous electron gasthat satisfies the theoretically known limits for smalland large q. This kernel has an analytical space Fou-rier transform that simplifies its implementation instandard first-principles techniques.

• RA kernel. Richardson and Ashcroft (1994) proposeda dynamical parametrization of the kernel based on asummation of self-energy and fluctuation terms in thediagrammatic expansion of the polarization functionfor the homogeneous electron gas. It is constructed tosatisfy many known exact conditions (static and dy-namic). This parametrization is assumed to be veryclose to the exact dynamical exchange-correlation ker-nel of the homogeneous electron gas. A corrected ver-sion of the RA parametrization can be found in Leinet al. (2000).


• TP kernel. Tokatly and Pankratov (2001), based on amany-body diagrammatic expansion, have derived anexpression for the fxc kernel in terms of the regularpart xr of the Kohn-Sham response function x0 withrespect to a given resonance frequency v ij :fxc~r,r8,v ij!

5D ij

E dr1dr2xr21~r,r1!F ij~r1!F ij* ~r2!xr

21~r2 ,r8!

F E dr1dr2F ij* ~r2!xr21~r2 ,r1!F ij~r1!G2 ,

where D ij5^F ijufxc(v ij)uF ij& [F ij was defined afterEq. (5.15)]. It is shown that the spatial nonlocality isstrongly frequency dependent and diverges for thecase of infinite systems at the excitation energies. Thisresult is similar in form to the correction obtained byGonze and Scheffler (1999) and it is also equivalent tothe Gorling-Levy perturbation theory (Gorling andLevy, 1994).

• RORO kernel. This is a static kernel derived by Rein-ing et al. (2002) from a direct comparison between thetime-dependent density-functional and Bethe-Salpeter equations. In Appendix C we develop thiskernel in greater detail by formally comparing theBethe-Salpeter equation and TDDFT. The kernelconsists of a contribution stemming from the energyshift between Kohn-Sham and GW eigenvalues, and asecond one describing the electron-hole interaction.One can absorb the first, positive contribution into anenergy shift of the starting x0 . Important excitoniceffects are then obtained by using only the static long-range term Dfxc(q,G,G8)52dG,G8a/uq1Gu2.

Some of these kernels have been tested in two limitingcases, for helium and beryllium atoms (Petersilka et al.,2000) and for the correlation energy (Lein et al., 2000)and plasmon dispersion (Tatarczyk et al., 2001) of a ho-mogeneous electron gas. In the case of atoms the de-tailed form of the Vxc potential is the crucial part ingetting the absolute position of most excitation energies,and the TDLDA kernel provides reasonable results.The results are marginally improved using more compli-cated kernels, always keeping the ‘‘exact’’ Vxc fixed. Infact the results obtained by Petersilka et al. (1998, 2000)for the helium and beryllium atom using different ker-nels indicate that the influence of the fxc on the calcu-lated spectra is less important than the choice of a goodVxc potential. However, this is no longer true for thelower excitation energies of beryllium and for singlet-triplet splittings, where the effects coming from Vxc can-cel. Furthermore, Aryasetiawan et al. (2002a) looked atthe singlet excitation of the H2 molecule as a function ofthe internuclear distance. The results are summarized inFig. 9, where it can be seen that the simple TDLDA isgood for intermediate distances only. The inclusion ofspin dependence improves the large-distance results. Bycomparing with exact results of a simple two-site Hub-bard model it was found that, indeed, fxc should be


strongly nonlocal in space and has an important energydependence. The nonlocality has been clearly shown byBaerends (2001). He proposed an orbital-dependentexchange-correlation potential for the H2 molecule thatyields the exact dissociation regime within the Kohn-Sham formalism.

Similar conclusions were obtained by Marques et al.(2001) in a study of the optical absorption spectrum ofsmall sodium and silicon hydrogenated clusters.45 More-over, it is observed that exchange-correlation kernels fit-ted to reproduce atomic properties perform poorly inthe case of an extended electron gas, due mainly to in-correct behavior at long wavelengths. This limit is wellreproduced in the TDLDA but not in the PGG kernel.Furthermore, the static CDOP kernel gives results ofsimilar quality to those of the more elaborated dynami-cal RA kernel. This indicates that the frequency depen-dence of the kernel is of little importance in providinggood total correlation energies and plasmon dispersion.Note that TDLDA does not perform too badly in thislast case (Larson et al., 1996).

Finally, Reining et al. (2002) have tested the above-mentioned long-range contribution to the static ROROkernel by performing a TDDFT calculation for bulk sili-con in the following way. First, they determined theLDA electronic structure. Second, they constructed x0 ,but with the eigenvalues shifted to the GW ones, in or-der to simulate the first part of the kernel as explainedabove. Third, they used Dfxc(r,r8)52a/(4pur2r8u),with the empirical value a50.2. The result of theTDDFT-RORO calculation is shown in Fig. 10. The dotsare the experimental results for the absorption spectrummeasured by Lautenschlager et al. (1987). The dottedand dot-dashed curves are the results of a RPA and a

45All the results for the optical spectrum of small Na clustersare very similar, regardless of the exchange-correlation poten-tial used. In contrast, hydrogenated silicon clusters show that amuch better agreement with diffusion quantum Monte Carlocalculations (Grossman et al., 2001; Porter et al., 2001) andwith experiments can be obtained when the exact exchangepotential is used.

FIG. 9. The excitation energy DE in the hydrogen moleculefor the transition from the 1Sg

1 state to the 1Su1 state as a

function of the nuclear separation d . LDA stands for the ei-genvalue difference in LDA, whereas TDLDA and TDSLDAcorrespond to the calculation of the excitation energy withintime-dependent LDA without and with spin polarization, re-spectively. The full line corresponds to the exact results, pro-vided for comparison (Aryasetiawan et al., 2002).


TDLDA calculation. The well-known discrepancies withexperiment are found. The solid curve is the result of theapproximate TDDFT-RORO calculation, visibly in closeagreement with the Bethe-Salpeter result (dashed) andwith experiment.

Similarly good agreement has been found for the op-tical spectra of other semiconductors (Botti et al., 2002).It turns out that this static long-range contribution to thekernel is sufficient to reproduce strong continuum exci-tonic effects in semiconductors. Promising results werealso obtained for the optical spectrum of some simplesemiconductors by Kim et al. (2002) using the exact ex-change Kohn-Sham formalism together with a TDLDAfxc , and by de Boeij et al. (2001) using a polarization-dependent functional within the current-density func-tional proposed by Vignale and Kohn (1996). This pro-cedure shows the influence of macroscopic electric fieldsin the response function. However, none of these ap-proaches has up to now managed to describe bound ex-citons, and one should certainly at least go beyond thesimple 21/q2 approximation for the RORO kernel inorder to describe such effects.

One can conclude that the use of TDDFT for the cal-culation of neutral excitations is promising: in the low-energy range of the absorption spectra of clusters,TDDFT, even in the adiabatic local-density approxima-tion, significantly corrects peak positions with respect tothose found when Kohn-Sham eigenvalue differencesare interpreted as excitation energies. Typically, the er-ror in excitation energy reduces from an amount of theorder of 50% to an order of magnitude of 10% (often,the main correction comes from v or v , i.e., the variationof the Hartree potential). Also, valence plasmons in sol-

FIG. 10. Optical spectrum @Im(«M)# for silicon: d, experiment(Lautenschlager et al., 1987); dotted curve, RPA; dot-dashedcurve, TDLDA; long-dashed curve, Bethe-Salpeter equation;continuous curve, TDDFT using RORO kernel (Reining et al.,2002; see text for details).


ids are well described in TDLDA for small, momentumexchange, yielding plasmon frequencies that agree withthe experimental ones within 0.5–2 eV, and good lineshapes. However, the optical absorption spectra of solidsare not improved by TDLDA with respect to the sum ofKohn-Sham transitions (see also Sec. VI.D.1). More pre-cisely, calculations performed on many different semi-conductors (Gavrilenko and Bechstedt, 1996; Kootstraet al., 2000) demonstrate that TDLDA spectra are, onaverage, redshifted by about 40% of the gap. Even whenthe gap is corrected with a scissor operator [as is com-monly done for LDA calculations (Levine and Allan,1989)], significant discrepancies with experiment remain:typically, in zinc-blende semiconductors the strength ofthe E1 peak is underestimated by 0–40 %, that of E2 isoverestimated by 30–100 %, and the absorption onset isoverestimated by 0.5 eV (e.g., in GaSb, InAs, ZnTe) oreven by 0.8 eV (in CdTe and InSb). It is clear that betterexchange-correlation potentials Vxc and kernels fxc mustbe found in order to make TDDFT become the ap-proach for the calculation of absorption spectra, just asstatic DFT is today the predominant method for ab ini-tio calculations of ground-state properties. In fact, fxcmust be a strongly nonlocal functional of the density,and, in principle, frequency dependent. It seems to becrucial to include in fxc at least a contribution going as21/q2 for q→0, in order to reproduce continuum exci-tonic effects in the absorption spectra of infinite systems.See also Appendix C.

VI. TDDFT VERSUS BETHE-SALPETER

TDDFT and the Green’s-function approach representtwo complementary ways to calculate a dynamical di-electric matrix and the spectra derived from it. The ulti-mate goal should of course be either to decide whetherone of the two approaches is clearly superior in a givensituation, or whether it is possible to combine the advan-tages of both in order to design a reliable and efficientway to calculate electronic spectra for a broad range ofapplications. It is therefore useful to compare the twoapproaches on different levels: from the mathematicalpoint of view, by trying to work out which effects thedifferent approximations might cause, and by comparingresults for different systems and types of spectroscopies.It should not be forgotten that in principle both ap-proaches are exact, and failures can only be explainedby the unavoidable approximations, or by an applicationthat is in principle not adequate for a particular problem(as in the case of static DFT and the interpretation ofKohn-Sham eigenvalues as electron addition or removalenergies).

A. The equations

Let us first examine the structure of the equations. Itshould be kept in mind that, whether absorption or lossspectra are to be calculated, and whether TDDFT or theBethe-Salpeter approach is used, the equation to besolved is of the form S5PIQP1PIQPKS [see Eqs. (4.10)and (5.12)]. Here S may be either x or P (describing loss


and absorption spectra, respectively), depending onwhether the kernel K contains the full Coulomb interac-tion v or the truncated v . Moreover, K has a contribu-tion F containing the exchange-correlation effects.There are two important differences between theTDDFT and the Bethe-Salpeter approaches. First, PIQPis either the Kohn-Sham (x0) or the quasiparticleindependent-particle response. The second differenceconcerns F . In particular, since the Bethe-Salpeter ap-proach derives from Green’s functions (i.e., density-matrix-like quantities), it naturally leads to a four-pointfunction F , whereas the TDDFT equation, which isdealing with density-only potentials, can be contractedand yields a two-point equation, since the delta func-tions in the v or v part of the kernel [Eq. (4.10)] are suchthat one can contract Eq. (4.9) and get the two-point Swithout solving the four-point equation first [this corre-sponds to using the fxc defined in the previous section,Eq. (5.10)].

The Bethe-Salpeter equation with the full kernel [Eq.(4.10)] cannot be contracted, because of the fact that thed functions of the exchange-correlation part are connect-ing different indices from those of the local field part. Ifone supposes for a moment that the self-energy could beapproximated by the DFT exchange-correlation poten-tial, for the calculation of one-quasiparticle eigenvaluesbut also for the functional derivative leading to the ker-nel, one would obtain

d~r12r2!dVxc~r1!

dG~r3 ,r4!

5d~r12r2!E dr5

dVxc~r1!

dr~r5!

dr~r5!

dG~r3 ,r4!

5fxc~r1 ,r3!d~r12r2!d~r32r4!, (6.1)

the same structure of connecting d functions as in theHartree (i.e., v) part. Again, one can contract the equa-tion and obtain the TDDFT two-point form. This is ofcourse true for any form of the exchange-correlation po-tential that is purely density dependent and local,whereas for the nonlocal and Green’s-function-dependent potentials of many-body perturbation theorythe equations are necessarily four-point ones. Of coursethis reasoning should not cause the misunderstandingthat TDDFT is an approximation to the Bethe-Salpeterequation; besides the fact that it is a true and in principleexact alternative, there are also subtleties linked to theuse of time-ordered and retarded quantities which pro-hibit an easy switching between the two. However, thediscussion of the structure of the equations still remainsvalid.

Equations (4.9) and (5.11) are not the only way towrite the Bethe-Salpeter and TDDFT schemes. In fact,in the previous sections we have already seen that it isoften convenient to transform these equations to an ef-fective eigenvalue equation, namely, Eq. (4.11) for theBethe-Salpeter equation, and Eq. (5.15), multiplied by(v2v jk), for TDDFT. It should be pointed out that inthis case both equations have become four-point equa-


tions, because they have been obtained by a basis trans-formation to a basis of pairs of occupied and emptystates. The two equations now look exactly identical, theonly differences being the meaning of the one-(quasi)particle eigenvalues and the content of the ker-nel. Clearly, this way of rewriting the equations is con-venient if (a) the single diagonalization to be performedis less onerous than the inversion of x for many frequen-cies (which necessarily assumes that the kernel does notdepend on frequency), and/or (b) if the basis of pairs ofstates which must be considered is smaller than a real-space (or reciprocal-space) basis. This is naturally truefor the Bethe-Salpeter equation, where those basis setswould also be quadratic. In the case of TDDFT, the qua-dratic basis of states must be compared to the linearbasis in real or reciprocal space. In that case, the choiceof the four-point form can only be convenient for smallfinite systems with well-spaced energy levels and inwhich only a limited part of the spectrum is needed. Alarger energy range would require a higher number ofstates, increasing considerably the computational effortin this four-point approach. This explains why one findsthis form mostly in applications of quantum chemists.

Writing the equation in the basis of transitions has theadditional drawback that the full Hamiltonian [Eq.(B20)], i.e., that including resonant, antiresonant, andcoupling contributions, is non-Hermitian. It has beenshown that this problem can be avoided by transformingthe equation into a simple quadratic one of the size ofthe resonant contribution only (Casida, 1995; Bauern-schmitt and Ahlrichs, 1996):

~R2C !~R1C !a5v2a , (6.2)

where R and C are the resonant and coupling matrices,respectively, and a is a linear combination of the twocorresponding parts of the full eigenvector Al (see Ap-pendix B.2 for details). In the case of TDDFT and forreal wave functions, Rvcv8c85vvcdvv8dcc81Kvcv8c8 andCvcv8c85Kvcv8c8 . Then R2C is diagonal, and the qua-dratic equation can be simplified; it reads (in its symme-trized form)

@v jk2 d jldkm12Af jkv jkKjk ,lm~v!Af lmv lm#clm5v2cjk ,

(6.3)

where all the spin degrees are embodied in the i ,j in-dexes, and f jk5f j2fk . Here one is explicitly taking intoaccount transitions of the type l→m where l is occupiedand m is unoccupied. In the Bethe-Salpeter equation,the electron-hole attraction term does not show the sym-metry Cvcv8c85Kvcv8c8 , and Eq. (6.2) cannot be simpli-fied. This can be a considerable problem in applicationslike the calculation of electron-energy-loss spectra,where the resonant-antiresonant coupling cannot be ne-glected, since in that case the non-Hermitian Hamil-tonian [or, equivalently, Eq. (6.2)], do not allow the ap-plication of fast inversion techniques like the Haydockrecursion method (Haydock, 1980).

The third main representation of the TDDFT equa-tion is the explicitly time-dependent one. In fact, the


spectra we are interested in are due to the response ofthe system to a time-dependent external field. It is hencepossible to extract the desired information from a calcu-lation of the time evolution of the density when such aperturbation is applied to the system. This correspondsto going one step back in the derivation of the equa-tions, as was done, for example, in the introduction tothis review. Since in TDDFT the density is easily con-structed from the time-dependent wave functions, it issufficient to solve the time-dependent Schrodinger equa-tion for c and construct the density at every step. This isparticularly easy when approximations for the potentialare chosen that do not contain memory effects, whichmeans that not only are the requirements of computertime modest but the storage requirements are low. Thismethod thus uses a direct solution of the time-dependent single-electron Schrodinger equation for theoccupied states,

i]c i~r,t !

]t5HKS~ t !c i~r,t ! ~ i51 ¯occ. !, (6.4)

where HKS is the Kohn-Sham Hamiltonian and r is thetime-dependent density r(r,t)5( i51

occ c i* (r,t)c i(r,t). Thesolution of this equation relies on very simple sparse-matrix-vector multiplications and on a numerical imple-mentation of the unitary time evolution operator (Ya-bana and Bertsch, 1996, 1999a, 1999b; Bertsch, Iwata,et al., 2000; Bertsch, Rubio, and Yabana, 2000). Hence,for example, in the case of the TDLDA approximationfor a cluster, the solution of the matrix equation (5.15)with a given kernel fxc is equivalent to propagating theequation above for some femtoseconds (the number de-pending on the accuracy in energy for the spectrum; Ya-bana and Bertsch, 1996, 1999a) with a given (now time-dependent) Vxc . The real-time method has two majoradvantages: it is nonperturbative and therefore allowsnonlinear effects of large fields to be calculated with thesame effort, and it uses the same energy functional forthe dynamical calculation as for the static calculationused to prepare the ground state, i.e., the potential andits density variation are automatically consistent, with-out the need to calculate the kernel explicitly.

In principle, the Bethe-Salpeter equation can also beput into the same form, since the time-dependent den-sity can be obtained as the diagonal of the time-dependent Green’s function (Kadanoff and Baym,1962). In that case, the time evolution equation for theGreen’s function must be solved. This has already beenproposed by Kwong and Bonitz (2000) and applied tothe model case of plasma oscillations, including dampingeffects on a correlated electron gas. The technique issimilar in nature to solving the time-dependent Hartree-Fock equation (Langhoff et al., 1972); in principle, usingthe proper self-energy as in the GW approximation, thetime evolution propagation is equivalent to solving theBethe-Salpeter equation. An application to real systemshas never been tried, although the idea seems appealing,since instead of solving a four-point equation one needonly perform a sequence of (two-point) matrix-matrixmultiplications, giving the action of S on G . However,


since G is nonlocal, and since the time dependence of S(and hence memory effects) cannot be neglected, it isnot obvious whether such an application would be fea-sible, not least because of storage requirements.

B. The limit of isolated electrons

Before discussing results, it is useful to make a fewremarks using a model system and/or simplified equa-tions. To this end, working with exchange only alreadyallows one to understand many things. Note that, at leastfor finite systems, exchange only should be a much bet-ter approximation to reality for the two-particle excitedstates than for, say, photoemission, since screening is lessimportant for a neutral than for a charged excitation.

This exchange-only approximation has the advantagethat one can work with relatively simple (explicitlyknown) exchange operators, and with the DFT exact-exchange potential (Gross et al., 1996). This latter po-tential (assuming from now on all wave functions to bereal for simplicity) is given by

Vx~r!52(v

(cE dr8E dr1 E dr2cv~r1!

3(v8

cv8~r1!cv8~r2!

ur12r2ucc~r2!

cc~r8!cv~r8!

ev2ecD

3x021~r,r8!, (6.5)

where x0(r,r8) is the static independent particle re-sponse function. To simplify the following consider-ations further, let us suppose that the system has onlyone electron; Vx is then (Taut, 1992)

Vx~@r# ,r!52E dr8r~r8!

ur2r8u. (6.6)

The ‘‘quasiparticle correction’’ ^cvuSxucv&2^cvuVxucv&is hence zero for the occupied state, and for the lowestunoccupied state it becomes

DQP5^ccuSxucc&2^ccuVxucc&

52E drE dr8cc~r!cv~r!cv~r8!cc~r!

ur2r8u

1E drE dr8ucc~r!u2ucv~r8!u2

ur2r8u. (6.7)

Now, when considering only one occupied and the low-est empty state (single-pole approximation), the effectof the Hartree kernel and the electron-hole interactionis a change of the optical gap by the matrix elements ofthe Coulomb interaction v and the electron-hole inter-action (in this case also unscreened) taken with respectto the pair (v ,c). The latter contribution is

De2h52E drE dr8ucc~r!u2ucv~r8!u2

ur2r8u, (6.8)

which exactly cancels the second term 2^ccuVxucc& ofthe quasiparticle correction [Eq. (6.7)], whereas theelectron-hole exchange interaction (the Hartree kernel)


cancels the first term of Eq. (6.7), ^ccuSxucc&. On theother hand, in the TDDFT approach, the total correc-tion is simply the matrix element

DTDDFT5E drE dr8cc* ~r!cv~r!

3S 1ur2r8u

1fx~r,r8! Dcv* ~r8!cc~r8!, (6.9)

which, using fx(r,r8)5dVx /dr(r8)521/ur2r8u with theabove Vx [Eq. (6.5)], yields exactly the same result.Hence in both approaches the net correction to the ei-genvalue gap is zero, as it should be. This is of coursedue to the fact that in this simple model the valenceexchange term is simply the self-interaction correction,and it completely cancels the Hartree term. Neverthe-less, it is instructive to study this case, since it illustrateshow the matrix elements of fx simulate the sum of self-energy corrections and the electron-hole interaction.Moreover, even in more realistic systems, such as mol-ecules or clusters, the self-interaction contributions canbe very strong. In particular, for other approximate ker-nels that do not treat exchange exactly, this cancellationcan be incomplete and the one-electron limit may bewrong.

If one admits more conduction states, it turns outagain that in this exchange-only TDDFT all off-diagonalelements are zero, since the total kernel v1fx is zero.This is not true for the Bethe-Salpeter approach, wherethe contributions to the kernel coming from the Hartreepart and the Fock part do not cancel, since they connectdifferent indices. Therefore, in order to see whether theone-electron limit is correct, one should check the con-tribution of the off-diagonal elements. This is most eas-ily done in perturbation theory. Going beyond the diag-onal (first-order) approximation of the Bethe-Salpeterequation yields a second-order correction to the excita-tion energy which happens to be

El(2)5 (

c8Þc

u2^ccuSxucc8&2^ccuVHucc8&u2

~ecQP2ec8

QP!

, (6.10)

where the first term in the numerator is the matrix ele-ment between particle-hole pairs (v ,c) and (v ,c8) of theelectron-hole exchange, and the second term in the nu-merator is exactly equal to the matrix element of thedirect electron-hole interaction. In the denominator, onehas quasiparticle eigenvalues of empty states. This sug-gests a deviation from the correct one-electron result, bya shift 2uEl

(2)u to lower excitation energies. However,there are further corrections, due to the fact that theabove formulas have been obtained (as is usually thecase in a GW calculation) using Kohn-Sham wave func-tions, i.e., performing the GW calculation in first-orderperturbation theory. One has in fact to calculate the cor-rection obtained by going to second order in the quasi-particle eigenvalues and taking into account the corre-sponding correction to the wave functions in theevaluation of the matrix elements of the exciton Hamil-tonian. It turns out that the change in eigenvalues adds


another 2uEl(2)u to the excitation energy, whereas the

update of the wave functions shifts the optical gap by12uEl

(2)u. Moreover, in this model system the resonant-antiresonant coupling terms are exactly vanishing. Thefinal result is hence again correct.

This suggests the following observation concerningTDDFT in the exact-exchange approximation andBethe-Salpeter calculations for very small systems: It isreasonable to neglect off-diagonal elements in theTDDFT equation (single-pole approximation), as wellas in the Bethe-Salpeter method, if Kohn-Sham wavefunctions are used and GW corrections are calculated tofirst order. However, if off-diagonal elements are in-cluded in the Bethe-Salpeter equation, it may be neces-sary to go beyond first-order perturbation theory in theGW calculation. On the other hand, resonant-antiresonant coupling terms can be neglected for thosesystems.

One can also examine in detail whether the predic-tions made on the basis of these model calculations arevalid. This can be deduced from the results of calcula-tions on SiH4 done by Rohlfing and Louie (2000), inwhich Bethe-Salpeter results are compared to quantumMonte Carlo results (Grossman et al., 2001) and to ex-periment. First, Table III of Rohlfing and Louie (2000)confirms that the resonant-antiresonant coupling is in-deed completely negligible. Second, the same table con-firms that the inclusion of off-diagonal elements of theresonant electron-hole Hamiltonian lowers the excita-tion energy (by 0.41 eV for the first singlet transition).Then, Table II shows an increase of the transition energyby 0.67 eV due to the off-diagonal elements of S2Vxc . This is in agreement with the above qualitativepredictions concerning the cancellation of both effects.It suggests, moreover, as also pointed out by Rohlfingand Louie (2000), that most of what is going on can beunderstood on the basis of Hartree-Fock, self-interaction corrections and long-range potentials. Infact, the (not self-consistent) time-dependent Hartree-Fock results presented in the second column of Table IIare clearly in acceptable agreement with the correspond-ing Bethe-Salpeter results.

On the other hand, these findings should not be ex-trapolated to larger systems: in a solid, in fact, S pre-serves the crystal symmetry and can therefore not mixfunctions of different k points. The exciton Hamil-tonian, on the contrary, does strongly mix those transi-tions which are close in energy and which often comefrom the same band but different k points. Normally,one will find that quasiparticle wave functions are closeto the Kohn-Sham ones, as has been pointed out by Hy-bertsen and Louie (1986), whereas a strong excitonic ef-fect that is entirely due to the mixing of transitions atdifferent k points drastically alters the absorption spec-trum. Note that in a solid first-order perturbation theorywith respect to the electron-hole interaction yields avanishing shift of transition energies, and in fact themodifications in the spectra (even those appearing asshifts of peak positions) are essentially due to the mixingof transition matrix elements. As pointed out in Sec.IV.B, the joint density of states remains virtually un-


changed (Rohlfing and Louie, 1998b; Benedict and Shir-ley, 1999). Of course, this question about wave functionsdoes not arise in the TDDFT approach: since the latternever pass through electron addition and removal ener-gies, only the Kohn-Sham wave functions resulting fromthe ground-state calculation are needed at all steps.

To go further, one can now extend the model and con-sider the case of more than one valence orbital, but inthe approximation that these orbitals are nonoverlap-ping. Again, one can start from the exact-exchange po-tential. Because the orbitals are nonoverlapping, one hasv5v8 in Eq. (6.5), which yields

Vx~r!52(v

(cE dr8 E dr1 E dr2

3ucv~r1!u2

ur12r2ucv8~r2!cc~r2!cc~r8!cv~r8!

ev2ecD

3x021~r8,r!. (6.11)

Now, since the valence orbitals are nonoverlapping,x0(r,r8) is of block form in (r,r8), each of these blocksbeing given by a region Rl which contains both r and r8and having contributions from one valence orbital v lonly. We call these contributions x0v l

(r,r8). The inverseof x0 is then given by the inverse of each block, and theabove formula yields

Vx~r!52 (v l ,v l8

E dr8dr1dr2

3ucv l

~r1!u2

ur12r2ux0v l

~r2 ,r8!x0v l8

21 ~r8,r!. (6.12)

The integral in r8 has nonvanishing contributions of theform d(r22r) when l5l8 and r is in Re, and one gets

Vx~r!52ERr

dr1

r~r1!

ur12ru. (6.13)

The integration is limited to the region Rr where r issituated. The total quasiparticle correction stemmingfrom both occupied and empty states is hence

DQP5^ccuSxucc&

2^cvuSxucv&2^ccuVxucc&1^cvuVxucv& ,

DQP5^ccuSxucc&1E drERr

dr1

rc~r!r~r1!

ur12ru

1(v8

E drE dr1

cv~r!cv8~r!cv8~r8!cv~r8!

ur2r8u

2E drERr

dr1

rv~r!r~r1!

ur12ru. (6.14)

In the diagonal approximation to the Bethe-Salpeterequation, the electron-hole exchange and the electron-hole interaction give, respectively, the contributions

Dxe2h51E drE dr8cc~r!cv~r!cv~r8!cc~r8!/ur2r8u


and

De2h52E drE dr8ucc~r!u2ucv~r8!u2/ur2r8u,

as in the simpler model above. Adding these terms toDQP, one finds that the total correction of the opticalgap to the bare potential (T1V0) gap is

DEgopt5E drE dr1

rc~r!@r~r1!2rv~r1!#

ur12ru

(the Coulomb interaction of the electron with the valencecharge density after excitation),

2E drE dr8 (v8Þv

cc~r!cv8~r!cv8~r8!cc~r!

ur2r8u

(the exchange interaction of the electron with the valencecharge density after excitation),

2E drE dr1

rv~r!r~r1!

ur12ru

(which subtracts the artificial Coulomb interaction of thenow missing valence electron with valence charge density)

1E drE dr8(v8

cv~r!cv8~r!cv~r8!cv~r!

ur2r8u

(which subtracts the artificial exchange interaction of thenow missing valence electron with the valence chargedensity, and, for the nonoverlapping orbitals, exactly can-cels the Coulomb self-interaction term).

This result is very intuitive. In order to get the TDDFTresult, one could now simply take the form (6.13) of Vxderived above for the case of nonoverlapping orbitals,and obtain a kernel

fx~r,r8!5dVx~r!

dr~r8!52

dRr,Rr8

ur2r8u. (6.15)

Hence in the diagonal approximation the optical gapwould turn out to be identical to the Kohn-Sham eigen-value gap, as in the case of only one electron, since fxcancels the Hartree part of the kernel. This is howevernot due to a failure of the exact-exchange approach; infact, the kernel (6.15) has been derived from Vx whichwas approximated for the case of the model of nonover-lapping valence states. Instead, one should first performthe functional derivative and then do the approximation.In that case, one gets additional terms, which shouldfinally yield the correct result for the exchange-onlycase, as has been shown by Gonze and Scheffler (1999).

It is also interesting to note that in the case of non-overlapping valence orbitals, the PGG kernel in the di-agonal approximation yields the same (wrong) result asEq. (6.15).

In summary, one should note that the exchange-correlation kernel has to simulate the result of self-energy corrections to the exact Kohn-Sham eigenvalues,plus the direct electron-hole interaction. In the one-occupied-level limit cancellations occur (for example be-tween the electron-hole interaction and the conduction-


state matrix element of Vxc) which might be used as aninput for the design of kernels in systems with moreelectrons. From the case of several well-localized occu-pied states, one can see the importance of a kernel thatis derived from a Vx that well reproduces not only theground-state density, but also density variations.

C. Relaxation and correlation

It is also worthwhile to analyze and compare the ef-fects of relaxation and correlation. For the N→N11electrons excitation, one has of course a strong relax-ation (Hartree-like for localized states, and through cor-relation for delocalized states). Then, when going overfrom an N→N11 to the N→N* excitation, one has tosubtract the spurious electron-hole interaction and ex-change. Now there is also relaxation due to the presenceof the hole. One can suppose that the relaxation due tothe electron and the hole cancel to first order, so thetotal N→N* problem should have much less relaxationthan the N→N11 electron problem.

The scenario describing exchange and correlation ef-fects in the one- and two-quasiparticle excitation spectraof many-body systems is hence the following: the screen-ing that explicitly appears in the GW self-energy comesfrom the electronic relaxation (classical or through cor-relation, as discussed in the introduction) due to the ad-ditional electron. The diagonalization of the GW Hamil-tonian takes into account the fact that the GW potentialis different from the initial potential, which can be eitherHartree, Hartree-Fock, Vxc , or unknown if one doesGW from scratch. Then, the diagonalization of theBethe-Salpeter equation creates a correlated electron-hole wave function. Before the diagonalization, theelectron-hole wave function is F(rerh)5cv* (rh)cc(re),which is uncorrelated. After the diagonalization, F hasthe correlated form Fl(rerh)5(vcAl

vccv* (rh)cc(re). Ina molecule or cluster one can assume that one would getsome reasonable result even by taking into account onlydiagonal elements; this is an uncorrelated, but interact-ing, electron-hole pair. It might also happen that onlyone occupied state is contributing, and the resultingwave function Fl(rerh)5cv* (rh)@(cAl

c cc(re)# wouldstill be uncorrelated; only the electron has relaxed, i.e.,adjusted itself to the fact that the hole is present. Thesemixing effects on the wave functions can in fact be easilyvisualized in a cluster (see, for example, Albrecht et al.,1998b). In a solid, on the other hand, taking just onevalence Bloch state would mean that one does not in-clude more than one k point, i.e., one would not get anexcitonic effect. As in the case of the one-particle exci-tation, Hartree relaxation alone is not enough to de-scribe the solid: one must consider correlation.

On the other hand, in TDDFT (and in the time-dependent Hartree approach) the off-diagonal elementsimmediately yield the correlated electron-hole pair, andthe same discussion as above holds concerning the dif-ference between an extended crystal and a finite system.


D. Comparison of some applications

In real applications, of course, details of the systemalready at the independent-particle level are often themain ingredient for a first explanation of spectra. Forexample, bandstructure effects turn out to be decisive indetermining not only the existence of interband transi-tions as observed in optical spectroscopy, but also theproperties of plasmonic excitations in simple and noblemetals. Indeed, ab initio calculations of the dynamicalresponse have properly described the plasmon disper-sion in a variety of metals (Quong and Eguiluz, 1993;Aryasetiawan and Karlsson, 1994; Maddocks et al.,1994a, 1994b; Fleszar et al., 1997; Ku and Eguiluz, 1999;Cazalilla et al., 2000). Since, however, the independent-particle approach often fails to yield quantitative, oreven qualitative, agreement with experiment, it is usefulto compare the performance of various approaches ap-plied to these realistic cases.

In the previous section, we have shown or cited sev-eral examples for successful TDDFT calculations of theabsorption spectrum of atoms and clusters. Far fewerexamples exist for TDDFT calculations of the absorp-tion spectrum of bulk materials. The influence of anadiabatic LDA kernel fxc on the absorption spectrum ofsilicon has been calculated by Gavrilenko and Bechstedt(1996), who showed that its effect is negligible. Theseresults have been confirmed by Koostra et al. (2000, al-though no comparison to the RPA result is given in thatpaper), and can also be seen in Fig. 10. In fact, since theindependent-particle response function x0 goes to zeroas q2, any kernel that does not have a 1/q2 divergencedoes not contribute to the head (i.e., the G50, G850element) of the matrix fxcx0 and can at the best yieldvisible effects through off-diagonal elements, i.e., medi-ated by local field effects. Therefore it is often heardthat ‘‘TDLDA works in clusters, but not in solids.’’ Weuse this statement as a guideline for the following dis-cussions and to provide some explanations. In order toallow for a meaningful comparison, it is useful to con-centrate first on the effect of the part of the kernel whichis common to the TDDFT and the Bethe-Salpeter ap-proaches, namely, the v (or v) part. In this way one cansingle out the rest and determine to what extent TDDFTand the Bethe-Salpeter approaches, respectively, actu-ally improve on P—the only quantity that is treated dif-ferently in the two approaches.

1. Common ingredient: the bare Coulomb interaction

As pointed out in Sec. II.B, the G50 element of vdetermines the macroscopic screening 12v0P00 , whichaccounts for the essential difference between absorptionand EELS spectra of solids. The rest of the Coulombinteraction, i.e., v , gives rise to what is called ‘‘local fieldeffects’’ in the language of solid-state physics. Thismeans that it reflects the fact that an inhomogeneoussystem will exhibit an electronic response that is positiondependent (and not only distance dependent). It is intu-itively clear that such an effect will be the stronger themore a system is inhomogeneous. It is also obvious that


an atom, a molecule, or a cluster of whatever size is byitself a strong inhomogeneity in the empty space, andthat the electronic response must depend strongly on thedistance and on the position of the perturbation and theprobe.46 Therefore one can expect that the effect of valone will already be a strong modification of the spectrain a cluster, whereas in a solid it will depend on details ofthe charge density of the latter, with a tendency to havevery weak effects in, say, nearly free electron metals (ho-mogeneous systems) like simple metals. This is in factthe case. In Fig. 8 we have already shown the results forthree different clusters, and discussed the fact that theCoulomb term has a very strong effect and removesmost of the discrepancy between the LDA curves andexperiment.

When moving to solids, a good example is the lossfunction of bulk silicon, in which the localization of thecharge on the bonds gives rise to a non-negligible inho-mogeneity. The results of local field effects on the plas-mon resonance of silicon have already been discussed,for example, by Louie et al. (1975), who found that thelocal field effects are not as pronounced as in the case ofthe cluster, but are sizable. Moreover, as in the case ofthe cluster, agreement with experiment is improvedwhen v is included; in particular, the height of the plas-mon peak is considerably lowered.

When local field effects are included, good-qualityloss spectra are also obtained in transition-metal com-pounds like Ni and NiO (Aryasetiawan, 1994). The goodquality of RPA loss function results and the importanceof local field effects is illustrated for rutile TiO2 in Fig.11 (Vast et al., 2002). One can see that local field effectsare particularly important at higher energies, in the re-gion of transitions from the Ti 3p semicore level, yield-ing good agreement with experiment. This almost seemsto suggest that one could, in a first approach, alwaysneglect the cumbersome exchange-correlation effects.This is of course not true. It is instructive to look, notonly at the loss spectrum, but also at the absorptionspectrum of bulk silicon @Im(«M)#, shown in Fig. 5. Thedot-dashed curve (RPA result) is in fact in poor agree-ment with experiment. Another striking example is theabsorption spectrum of solid argon, shown in Fig. 12.The strong absorption peaks in the experiment (Saile,1976; Saile et al., 1976) on the low-energy side, which areknown to be of excitonic nature, are completely missingin both the RPA and the GW-RPA results (Olevano andReining, 2001b). Note that local field effects are in-cluded in these calculations, but they cannot remove thediscrepancies. The effect of v on Im(«M), in its more orless pronounced form, is in fact essentially to shift oscil-lator strength to higher energy. This is due to the factthat v is positive, being the Coulomb interaction be-

46The essential difference between even a very big cluster andan infinite solid is the presence of a surface between the clusterand vacuum, implying boundary conditions for the electricfield.


tween charge-density waves (Hanke and Sham, 1975). Itnever creates or even significantly enhances structureson the low-energy side.

Hence v , or local field effects, are generally not suffi-cient to obtain quantitative (for «21) or even qualitative(for «M) agreement between theory and absorption ex-periments, and one has to add the exchange-correlationeffects. In other words, one has to use the interacting P(instead of P5PIQP) in Eqs. (2.10) and (2.11). With re-spect to the RPA-plus-local-field result, these effectsshould cause a variety of modifications in the spectra, ifthey are supposed to restore agreement with experi-ment. Essentially, from the above results one can see

FIG. 11. Integrated loss function at q'0.4 Å21 for rutile TiO2(Vast et al., 2002): solid line, experiment; dashed line, RPAresults. The dot-dashed line in the inset shows the results with-out local field effects.

FIG. 12. Optical spectra for argon: d, experiment (Saile,1976); dashed line, RPA; dot-dashed line, TDLDA; solid line,GW-RPA. From Olevano and Reining, 2001b.


that a shift is needed, which can be towards either lowerenergies (as in the case of the loss spectra of both clus-ters and solids) or higher energies (as in the case of theabsorption spectrum of bulk silicon), together with a re-distribution of oscillator strength towards lower ener-gies, which may enhance existing peaks and/or createnew ones, as in the case of solid argon (see Fig. 12).

2. Exchange and correlation effects

Concerning these exchange-correlation effects, theBethe-Salpeter and the TDDFT approaches act in acompletely different way. In fact, the Bethe-Salpeter ap-proach can be seen as a two-step method, where first aGW calculation yields a strong shift of the whole spec-trum towards higher energies, and the subsequent inclu-sion of the electron-hole interaction redistributes oscil-lator strength towards the low-energy peaks, andeventually shifts peaks, or even creates peaks in the qua-siparticle gap. It is clear that there must be a partialcancellation of the self-energy corrections and theelectron-hole interaction. In the small cluster discussedabove, and also for bulk plasmons, where the RPA pluslocal fields calculation already gives very good results,this cancellation is almost perfect. TDDFT, on the otherhand, should describe directly the complete electron-hole excitation, but in an effective way, which may bemore difficult to understand [see also the discussion inAppendix A, after Eq. (A5)]. In this case, too, partialcancellations occur. Figure 13 shows an example of thesecancellations. Fleszar et al. (1995) have calculated thedynamical structure factor of bulk aluminum withinTDDFT. The independent-particle response x0 closelysimulates the measured spectrum, and even the double-

FIG. 13. Dynamical structure factor S(q,v)522\*drdr8e2iq(r2r8) Im x(r,r8,v) of aluminum, calculationsfrom Fleszar et al. (1995). Open circles are experimental datafrom Platzman et al. (1992). In the left panel, they are com-pared with the Kohn-Sham independent particle response (fullline labeled SKS , obtained taking x5x0 in the above expres-sion for S) (the double-bump structure was already obtainedby Maddocks et al., 1994a, 1994b). The dotted line, labeledSLindhard , is the corresponding noninteracting response of jel-lium with rs52.07. In the right panel, the same experimentaldata are compared with S calculated using x from Eqs. (5.11)and (5.12). Full line, TDLDA (fxc) from Eq. (5.13); dashedline, RPA (fxc50).


hump structure is correctly predicted. However, when afull RPA calculation, including the Coulomb kernel, isperformed, the result worsens. Exchange-correlation ef-fects should hence completely cancel the Coulomb re-pulsion in that case (i.e., a kernel fxc521/ur2r8u woulddo the job). Depending on the situation and the system,different aspects of fxc will become important. For ex-ample, Sturm and Gusarov (2000) have shown that inorder to describe the aluminum inelastic x-ray scatteringcross section for large momentum transfer and large v,dynamical correlation effects in fxc are more importantthan band-structure effects (interband transitions are in-sufficient to explain the observed magnitude of the high-energy tail of the spectrum, while multiple particle exci-tations give the correct order of magnitude).

a. Some applications to finite systems

For clusters, the majority of DFT-based calculationstoday rely on the use of the static LDA kernel, since theoptical spectra turn out to be in very good agreementwith experiment even at the TDLDA level (Casida,1995; Rubio et al., 1996; Yabana and Bertsch, 1996; Va-siliev et al., 1999, 2002). Although, as pointed out above,the improvement with respect to a static LDA calcula-tion essentially comes from the Coulomb part of the ker-nel, and not from fxc , these findings have led to an im-portant breakthrough in the calculation of excitationspectra of finite systems and make the TDDFT tech-nique very popular. One example is the benzene mol-ecule (Fig. 14). It is clear that apart from small differ-ences due to the resolution of vibrational modes in theexperimental spectrum, the overall response is very welldescribed by the TDLDA approach.47 Satisfactoryagreement is also obtained for other molecular clustersas shown in Fig. 8. This figure however, also shows theneed for improvements over the simple TDLDA.

A good and widely studied example is the silane mol-ecule SiH4 , for which quantum Monte Carlo (Grossmanet al., 2001), Bethe-Salpeter (Rohlfing and Louie, 1998a,2000; Grossman et al., 2001), and TDDFT results (Ogutet al., 1997; Marques et al., 2001) are available. In par-ticular, TDLDA or gradient corrected functionals pre-dict that bound excitations will be resonances. This arti-fact is clearly corrected by the exact-exchange potentialand a method proposed by van Leeuwen and Baerends(1994) to impose the correct asymptotic behavior on theexchange potential (Marques et al., 2001). Within thisscheme the TDDFT results for SiH4 have the same ac-

47For the chiral C76 fullerene, TDLDA also provides a con-sistent description of the optical absorption spectrum, the cir-cular dichroism spectrum, and the optical rotatory power (ex-cept for an overall shift of the total spectrum; Yabana andBertsch, 1999a, 1999b). Similarly, the optical spectrum can beused to elucidate the ground-state structure of the smallestpossible carbon cage C20 (Castro et al., 2001), which has beenelusive to ground-state calculations in both quantum MonteCarlo (Grossmann et al., 1995) and DFT (Jones and Seifert,1997).


curacy as the GW or quantum Monte Carlocalculations.48 This fact is illustrated in Fig. 15.

TDLDA gives good results for Na4 as well as for smalland large clusters of simple and noble metal atoms (Ru-bio and Serra, 1993; Rubio et al., 1996, 1997; Yabana andBertsch, 1996; Serra and Rubio, 1997; Vasiliev et al.,1999, 2002) and fullerenes (Yabana and Berstch, 1999a,1999b; Castro et al., 2001) but works less well in smallhydrogenated silicon clusters (see Fig. 15).49 This prob-lem might be due to the presence of more localizedbonds. The exact-exchange calculation does help in thiscase and keeps nearly the same accuracy as in the so-dium cluster. Both calculations are in slightly betteragreement with experiment than the Bethe-Salpeter cal-culation (Onida et al., 1995). Note that the two peaks atabout 3 eV in both the exact-exchange and the Bethe-Salpeter calculations are very similar. However, in bothclusters we see an overall tendency of the TDDFT cal-culations with approximate kernels to give larger excita-tion energies (peak positions blueshifted with respect toexperiment). By looking carefully at the different contri-butions coming from the correct asymptotic behavior ofthe potential, as well as at variations of the fxc kernel,Marques et al. (2001) show that there is an inherent

48Very good results are obtained in the three approaches—quantum Monte Carlo, Bethe-Salpeter and TDDFT with exactexchange. In particular, they overestimate the energy of thefirst singlet excitation by 3%, 4.5%, and 1.5%, respectively,whereas the ionization potential (exactly reproduced in quan-tum Monte Carlo) is only 4% overestimated by TDDFT withexact exchange. The quantum Monte Carlo results are equiva-lent to the best quantum-chemical (coupled-cluster and com-plete active space self-consistent-field) calculations (for detailssee Grossmann et al., 2001 and references therein).

49Here it is important to remark that for the optical gaps, thecalculated values within TDLDA for large-size hydrogenatedsilicon clusters (Ogut et al., 1997; Vasiliev et al., 2001) agree towithin .10% with experiment.

FIG. 14. Optical absorption of the benzene molecule, in unitsof eV21. The TDLDA calculation is from Yabana and Bertsch(1999a), whereas the experiment is from Koch and Otto(1972).


source of error in the approximated kernels comingfrom neglect of dynamical (correlation) effects. Thesetend to shift the spectrum to lower energies and are alsoresponsible for improving the excitation energies ofRydberg-like states (where exchange effects are smalldue to the nearly zero overlap between the Rydberg un-occupied state and the other valence states; thus the usu-ally weak polarization effects are going to play an im-portant role here).

For these small molecules, there are thus variousmethods to get good transition energies: quantumMonte Carlo, the Bethe-Salpeter equation, and TDDFTyield satisfactory results. One might of course wonderwhether the calculation of spectra involving empty (ex-tended) states is not questionable in a supercell ap-proach. In fact, high-energy, extended states do repre-sent a problem if they are really needed as the physicalfinal (one-electron) state (e.g., in absorption at higherfrequency, or for a photoemitted electron, if one wantsto go beyond the usual approximation of neglecting thefinal-state effects). Most often, however, these states areused only as intermediate states in summations, i.e., for

FIG. 15. Calculation of the strength function for Na4 and SiH4within TDDFT using different kernels (Marques et al., 2001):solid line, TDLDA; dotted line, exact-exchange. Gray solidline, experimental results; dashed line, the Bethe-Salpeterequation results of Onida et al. (1995) for Na4 . Vertical linesare the lowest excitation energies obtained by Grossman et al.(2001) for SiH4 : solid vertical line, quantum Monte Carlo cal-culations; dashed vertical line, Bethe-Salpeter equation. In theinset: solid line, self-interaction correction; dotted line, vanLeeuwen and Baerends (LB94) prescription.


a closure relation, which is a mathematical property de-riving from the completeness of the basis set. Since thevalidity of the closure relation does not depend on thechosen boundary conditions (even if the actual shape ofthe individual states does), this does not present a prob-lem.

Concerning the Bethe-Salpeter approach for SiH4 , itshould be noted that (i) the results depend on thechoices made for the different steps of this rather com-plicated technique, e.g., the diagonalization of the GWHamiltonian discussed above; and (ii) only transition en-ergies, not line shapes, are shown. However, in TDDFTthe results depend of course on the choice made for thekernel, as discussed above.

b. Some applications to extended systems

In solids one should look at both absorption andelectron-energy-loss spectra. As discussed in the previ-ous sections, TDLDA fails to describe the absorptionspectra of solids. In fact, at least in the case of silicon,the failure of the TDLDA fxc to reproduce the correctlong-range behavior of the kernel explains its bad per-formance, as shown above in Fig. 10. On the other hand,Olevano and Reining (2001a) have calculated theelectron-energy-loss spectrum of silicon including localfield effects, GW corrections, and excitonic effects, forvanishing momentum transfer. The inclusion of excitoniceffects improves the results with respect to both theRPA and, more drastically, the GW calculations (seeFig. 16): GW and excitonic corrections cancel to a largeextent. In contrast to the case of small molecules and toabsorption spectra in bulk, here the correct result couldonly be obtained by taking into account the couplingbetween transitions at positive and negative frequencies[blocks K in Eq. (B20)]. Figure 16 also shows thatTDLDA improves with respect to the RPA calculation(including local field effects), which means that even theTDLDA fxc has a visible effect on the electron-energy-loss spectrum. However, as in the case of clusters, localfield effects are more important than fxc for theelectron-energy loss spectra. This is a very general find-ing in TDLDA electron-energy-loss calculations (see theresults of Waidmann et al., 2000 and references therein).

For nonvanishing momentum transfer, one can com-pare the TDDFT results of Waidmann et al. (2000) ondiamond to the Bethe-Salpeter results of Caliebe et al.(2000). The comparison is limited, however, to the low-energy region well below the plasmon. First of all, thework of Waidmann et al. (2000) shows the increasing im-portance of local field effects with increasing q . Localfield effects start to be visible at q51.0 Å21 (about 1.1GX). At q51.5 Å21 (about 1.7 GX) they are alreadyvery strong. Second, TDLDA gives a reasonable de-scription of the low-energy part of the spectrum. Itshould be pointed out that the authors state that theLDA kernel fxc itself has only a negligible effect on thisresult. On the other hand, in the work of Caliebe et al.(2000) the Bethe-Salpeter method is applied to the cal-culation of the same spectra. Results of comparablequality are obtained in the end. A more detailed com-


parison is difficult, since the Bethe-Salpeter results arecompared to those obtained when GW corrections areadded to the Kohn-Sham eigenvalues in theindependent-particle polarization. Moreover, as is oftenthe case in the Bethe-Salpeter approach, the local fieldeffects are treated as part of the electron-hole interac-tion kernel. In other words, part of the electron-holeeffect shown is due to the local field effects. The netimprovement due to the electron-hole interaction ishence very strong (although not systematic for all spec-tra), but it is difficult to estimate the total improvementdue to self-energy plus electron-hole attraction effects,in comparison to RPA (including local field effects), orbetter, TDLDA, results. A detailed comparison ofTDDFT and Bethe-Salpeter results for the qÞ0 case isin fact still missing in the literature. Bethe-Salpeter re-sults should be superior to TDLDA results in thoseparts of the loss spectra that are dominated by interbandtransitions, i.e., by the structures in the imaginary part ofthe dielectric function, which are not well reproduced inTDLDA.

VII. CONCLUSIONS—FREQUENTLY ASKED QUESTIONSAND OPEN QUESTIONS

Since the very beginning of physics and chemistry, theinteraction of photons and electrons with matter hasbeen a major topic of study. Today, most characteriza-tion tools as well as electro-optical devices are based onour understanding of these interactions. Technologicalapplications are rapidly progressing, but many funda-mental questions concerning theoretical and numericaldescriptions are still open.

FIG. 16. Energy-loss spectra of bulk silicon: d, experiment(Stiebling, 1978); dotted line, Bethe-Salpeter equation withoutresonant-antiresonant coupling; dash-dotted line, GW-RPA;dashed line, RPA; dash–double-dotted line, TDLDA; solidline, Bethe-Salpeter equation with coupling (Olevano andReining, 2001a).


This has motivated us to discuss in the present reviewthe two most widely used techniques for describing elec-tronic excitations in finite and infinite systems, namely,the Green’s-function approach to many-body perturba-tion theory calculations, and time-dependent density-functional theory. We have presented the foundationsand approximations of the two approaches as well as theadvantages and drawbacks of the methods. A special ef-fort has been made to provide a unified picture of theunderlying equations, in order to facilitate the compari-son and to analyze the most frequently used approxima-tions. We have stressed the fact that the two approachesare somewhat complementary and that one should workon both sides in order to optimize the search for system-atic improvements.

We have not presumed to give a complete view of alltheoretical descriptions of spectroscopies, but ratherlooked at the field from a particular point of view, withthe aim of motivating some possible future develop-ments. Some points have been treated in more detailthan others; we have made our choices on the basis ofthe very lively discussions which are now going on in thisfield. As a conclusion, we summarize below a set of fre-quently asked questions, which we have tried to answerin the present paper. Many points are not new, and theywill be obvious for a specialized reader, but we thinkthat it is useful to put them in a common context. Thisshould also be helpful to those who wish to enter thefield.

Other questions do not have an answer yet. They willbe treated at the end of this section—as a motivation forfuture research.

• The delta-self-consistent field approach, i.e., explicitlycalculating total energy differences, is known to workwell. Why can’t we just always calculate total energydifferences in DFT, instead of struggling with the mean-ing of Kohn-Sham eigenvalues? First, DSCF in DFTsupposes that one can simulate the initial and the finalstate of the excitation by occupying selected one-particle orbitals. This excludes excitations that are noteasily described in terms of isolated single-particletransitions (for example, the collective plasmon exci-tations). Second, whereas the choice of the N and N61 states can be straightforward in a small system, itis not necessarily well defined in the bulk; in fact,when Bloch states are chosen to describe the elec-trons, Hartree relaxation effects vanish and hence themain advantage of DSCF is lost. One should then in-clude dynamical correlation effects, which go beyondstatic DFT-DSCF. See discussions in Secs. IV.A.2 andVI.C.

• Does TDDFT give electron removal or addition ener-gies? In Sec. V we described how TDDFT provides anexact framework for getting the excitation energies ofan N-particle system. The theory handles only neutralexcitations, that is, excitations in which the number ofparticles does not change. Concerning N→N61 pro-cesses, TDDFT is only supposed to have the correctionization potential threshold, as this should be guar-


anteed by the exact DFT exchange-correlation poten-tial (Almbladh and von Barth, 1985).

• If quasiparticle energies are electron addition and re-moval energies, i.e., total energy differences, how canthey be complex? In the Lehmann representation ofthe Green’s function the excitation energies involvedare always differences of total energies between the Nand N61 systems; hence they are always real. Theinfinitely close-lying peaks of the spectral function inthe Lehmann representation merge into broad struc-tures that can be identified as quasiparticle peaks.These structures have a finite width and can hence bedescribed by a real part (peak position) and an imagi-nary part (width) of a quasiparticle’s energy. See dis-cussion in Sec. III.A.

• Why can TDDFT yield absorption spectra, when DFTis a ground-state theory? DFT is a ground-state theorybecause it is based on a minimization of the total en-ergy. TDDFT, on the other hand, is derived from theextrema of the quantum-mechanical action, and hencedescribes the evolution of the system instead of itsground state. This difference is equivalent to that inclassical mechanics, where the equilibrium position ofa particle in space can be found by looking for theminimum of the potential, whereas its trajectory is de-rived from the extrema of the classical action. See dis-cussion in Sec. V.

• Can one in principle get excitonic effects in TDDFT?Yes, excitonic effects are in principle exactly containedin the TDDFT equations, if the exact exchange-correlation potential Vxc and kernel @fxc(r,r8,v)# areused. However, the practical implementations usesimple functionals, which lack, for example, theproper spatial nonlocality. Therefore direct electron-hole interaction effects are only partially described infinite systems, and in general still out of reach of to-day’s TDDFT calculations for solids. See Sec. V.C andAppendix C.

• Is the fact that Kohn-Sham eigenvalue differences un-derestimate the gap a problem for the calculation ofabsorption spectra within TDDFT? No. The Kohn-Sham eigenvalue differences are not meant to repro-duce the (optical) gap. The optical gap is determinedby the Kohn-Sham potential (eigenvalues) and byvariations of the Kohn-Sham potential, through thekernel. This latter contribution changes the opticalgap with respect to the Kohn-Sham eigenvalue differ-ence. Of course, results improve when better poten-tials (eigenvalues) are used—but ‘‘better’’ does notmean ‘‘close to quasiparticle energies.’’

• Why does the RPA often yield good results forelectron-energy loss, but not for absorption in solids?This can be traced back to the crucial role played bythe long-range part of the Coulomb potential in thekernel: the exchange-correlation contribution fxc is infact added to the Hartree term v in the case of the lossspectra [see Eqs. (2.3) and (2.11)], but to the differ-ence between the bare v and the long-range part


v(G50) in the case of absorption [see Eqs. (2.9) and(2.10)]. In the latter case, the only long-range termcomes from fxc , which makes it so crucial. See Secs.II.B and VI.D.

• Why does TDLDA work for clusters, but not for theabsorption spectra of solids? First, the main differencebetween the results of TDLDA and those of Eq. (2.8)[the IP-RPA dielectric function] in clusters stems fromthe contribution v in the kernel, which is always exact.Second, for finite systems fxc

TDLDA provides an effec-tive electron-hole attraction, which gives a significantcontribution to the spectra. In infinite nonmetallic sys-tems, the LDA fxc only contributes through local fieldeffects, since the head of the matrix fxcx0 vanishes asq→0. See Secs. V.C, VI.D, and Appendix C.

• What is the link between the Hartree potential, theelectron-hole exchange interaction, and local field ef-fects in a solid? The variation of the Hartree potentialwith respect to the density yields a contribution v tothe kernel of both the TDDFT and the Bethe-Salpeter equations. This contribution has to be takenwith or without (v or v) its long-range (G50) contri-bution in the case of electron-energy loss or absorp-tion, respectively. Neglecting v in the screening equa-tions is equivalent to neglecting local field effects. Thematrix elements of v and v in an electron-hole basisare dipole-dipole like, and one calls this contributiontherefore the ‘‘electron-hole exchange’’ in the Bethe-Salpeter framework. See Secs. II.B and IV.B.2 formore details.

• Is the electron-hole exchange interaction screened orunscreened? The electron-hole exchange interactionstems from the density variation of the Hartree poten-tial and is therefore exactly the unscreened Coulombinteraction (see Secs. II.B, II.C, and IV.B.2). Some-times model calculations can be found in the literaturewhere this interaction is screened. This procedure isjustified when (and only when) an extremely re-stricted space of transitions is explicitly mixed by theexciton; the neglected transitions are then implicitlyput back into the calculation by an effective screeningof the electron-hole exchange.

• To describe two-particle excited states one resorts to thetwo-particle Bethe-Salpeter equation. What aboutthree-, four-, or more-particle excitations? Do we needthree- or more-particle analogies to the Bethe-Salpeterequation? No. In fact, even for the case of two-particleexcited states we do not need the two-particle Bethe-Salpeter equation, since it would be formally sufficientto solve Hedin’s equation for the vertex [Eq. (3.25)].The situation is similar to the case of electron additionor removal energies, where the set of higher-order en-tangled Green’s functions is decoupled by introducingthe concept of the self-energy, which implicitly con-tains all higher-order electron interactions. Similarly,the complex equation for the polarization operator in


terms of higher-order polarization operators can beformally solved by introducing the vertex function.See Sec. IV.B.

• TDDFT equations can be written as a quadratic equa-tion (6.3) or as a Dyson–like equation (5.11): are thetwo forms equivalent, and what are the reasons tochoose one of them? Yes, the two forms are indeedequivalent. The TDDFT equation can be written forfour-point response functions or for two-point (con-tracted) functions [see Eqs. (2.15) and (2.19)]. Thetwo-point formulation (which does not exist for theBethe-Salpeter equation) leads to the Dyson-likeequations for two-index matrices [Eq. (5.11)], with theobvious advantage of a better scaling with the systemsize. On the other hand, the four-point formulationallows one to switch easily to transition space and tocompare directly with the Bethe-Salpeter equation.The transition-space formulation can be written as aneigenvalue equation like Eq. (5.15) or as a quadraticequation for v like Eq. (6.3). It is advantageous forcases in which only a limited number of transitionscontribute to the spectrum. See Secs. V.B, VI.A, andII.C.

• Since most calculations use supercells or finite do-mains, can we trust the higher-energy eigenstates, whichcan ‘‘sense’’ the boundary conditions? If one is explic-itly interested in these states, that can actually be a bigproblem. However, most often (e.g., in the calculationof a Green’s function) they are just virtual states thatare summed over, and one does not care. See also Sec.VI.D.2.a.

One important question now is actually which wayone should go—is the Bethe-Salpeter equation orTDDFT more promising? Since both approaches are ex-act, the answer depends of course on whether—andwhen—the remaining open questions in the two ap-proaches can be solved. Some questions are common toboth. An important point is whether pseudopotentialsare always adequate for describing spectra with the pre-cision one asks for today. For certain applications, oneshould perhaps migrate towards all-electron schemes.Other questions are specific to the Bethe-Salpeter or theTDDFT approach.

The Bethe-Salpeter approach offers a clear physicalpicture and straightforward possibilities for the analysisof results. It seems to work over a wide range of systems.The TDDFT approach, on the other hand, is appealingsince it calculates things in a more direct way (withoutpassing through electron addition and removal energies)and is, in principle, easier to use.

Open problems in the Bethe-Salpeter approach in-clude, of course, the need for technical developments toovercome the bottlenecks linked to the four-point equa-tions. Moreover, just as in GW , not all the ‘‘ingredients’’of the method are uniquely defined: how, for example,should vertex corrections and dynamical screening beincluded consistently? It is clear that the level of ap-proximation which should be used for each ingredient in


the various Green’s-function-based approaches is a deli-cate and nontrivial question. One should wonder why inthe calculation of an electron-energy-loss spectrum it isa good approximation to use «21 constructed from a P5PIQP with DFT eigenvalues, and a much worse onewhen GW eigenvalues are used, whereas in GW calcu-lations, especially for large-gap systems, an update ofthe energies entering in «21 yields improvements for theresulting quasiparticle energies. At first sight, this seemscontradictory. At this point we propose that one answermay be to put all choices on the basis of a consistentiteration scheme. This means that one starts with a GWcalculation and subsequently iterates Hedin’s equations.The iteration first yields P , which contains a vertex cor-rection. This explains why the electron-energy-loss spec-trum using P5PIQP with GW eigenvalues yields poorresults, but using P with GW eigenvalues and a vertexcorrection yields good results. Then one can go furtherand put this P into the equation for S. Now a secondvertex is appearing, and it is has been shown that thereare cancellations between the vertex in P and the ex-plicit one in S. It is known that it is better to consistentlyneglect the vertex in both W and S, which in practicemeans that a simple update of the energy denominator isin general the most consistent thing to do in a GW cal-culation, if one does not want to include both vertices.Note that one quickly runs into trouble when also up-dating the wave functions, because of the dynamical ef-fects, whereas it sometimes turns out to be necessary toupdate their spatial behavior. These are handwaving ar-guments, though essentially these—and the success of anapproach in real calculations—justify the choices thatare made in the iteration of the equations. Moresuccesses—and failures—of the Bethe-Salpeter ap-proach in its actual form are needed in order to sort outthis question.

In TDDFT, calculations are less cumbersome (if thetwo-point representation is chosen) than in the Bethe-Salpeter method at the moment, i.e., using the simplekernels available today, like that of the adiabatic LDA.Better exchange-correlation potentials and better ker-nels have to be found, especially if TDDFT is meant tobecome a method for the calculation of absorption spec-tra in solids. These kernels might well turn out to bevery complicated, and as difficult to treat as the Bethe-Salpeter equation. There is some hope that, at least incertain cases, relatively simple solutions can be found(see Appendix C). The Bethe-Salpeter equation seemsto provide a good starting point for the derivation ofsuch effective kernels, since it contains the essentialphysics in a structure that is actually close to that ofTDDFT.

In conclusion, it seems reasonable to suppose thatprogress will come from a common effort of peopleworking in the fields of the Bethe-Salpeter equation andof TDDFT.

ACKNOWLEDGMENTS

We thank C. O. Almbladh, F. Aryasetiawan, P. Bal-lone, F. Bechstedt, M. Cini, R. Del Sole, J. Dobson, R.


Godby, L. Hedin, C. Hogan, I. Nagy, V. Olevano, F.Sirotti, F. Sottile, N. Vast, and U. von Barth for helpfuldiscussions, collaborations, and critical reading of themanuscript. We are grateful for collaboration and dis-cussions with G. Bertsch, A. Castro, G. Colo, P. M. Ech-enique, W. Ekardt, E. K. U. Gross, O. Gunnarsson, S. G.Louie, M. Marques, A. Schindlmayer, E. Shirley, and K.Yabana. We thank all the participants of the 2000 CE-CAM workshop on Electronic Excitations, where theidea of writing the present review was born. We havealso benefited from insights and discussions provided bythe series of workshops on this subject organized at theEcole Polytechnique, Rome, Valladolid and CECAMsince 1996. We thank A. Castro and V. Olevano for mak-ing some of the figures used in this review. This workwas supported by the RTN program of the EuropeanUnion NANOPHASE (contract HPRN-CT-2000-00167). G.O. acknowledges INFM for financial support(project PRA 1MESS). Computer time has beengranted by IDRIS (CNRS, France) on project 010544.

APPENDIX A: EXACT PROPERTIES OF fxc

Although the development of time-dependent func-tionals is still at a very early stage compared to that ofstatic functionals, some of the known results for a homo-geneous gas can be generalized to the inhomogeneouscase. For example, causality leads to Kramers-Kronigrelations for the real and imaginary parts of fxc(r,r8,v),and the fact that fxc(r,t ;r8,t8) is a real-valued quantityimplies that fxc(r;r8,v)5fxc(r;r8,2v)* . Besides that,the response functions satisfy the spatial symmetry rela-tions x(r,r8,v)5x(r8,r,v), provided that the unper-turbed system has time-reversal symmetry (no magneticfield is applied or generated in the system). Thus onefinds that fxc(r,r8,v)5fxc(r8,r,v).

Further constraints on the potential Vxc and kernel fxccan be deduced from the quantum-mechanical equationof motion applied to the position operator r:„d/dt ^C(t)uruC(t)&5i^C(t)u@H(t), r#uC(t)&…. The finaland rigorous result is (Gross et al., 1996)

E drr~r,t !¹Vxc@r#~r,t !50. (A1)

This relation is also satisfied by the Hartree potential.This is nothing other than a mathematical statement ofthe physical fact that the exchange-correlation potentialdoes not exert a net force on the system. One also ob-tains the corresponding ‘‘zero-torque’’ expression byconsidering the angle operator w and using the rota-tional invariance of the Coulomb interaction, yielding

E drr~r,t !r3¹Vxc@r#~r,t !50. (A2)

Corresponding properties of the exact exchange-correlation kernel are obtained by evaluating the twoprevious expressions at the density r(r,t)5r0(r)1dr(r,t) for arbitrary dr(r,t). In frequency space onegets


E drr0~r!¹rfxc@r0#~r,r8,v!52¹r8Vxc@r0#~r8!, (A3)

E drr0~r!r3¹rfxc@r0#~r,r8,v!52r83¹r8Vxc@r0#~r8!.

(A4)

Concerning the electronic damping mechanism which isincluded in fxc , we should emphasize that the standardstatic approximations for fxc [where fxc(v)ªfxc(v50)is a real quantity] based on the homogeneous gas predictan infinite lifetime for plasmon excitations at small wavevectors. This artifact is inconsistent with the fact that theexact kernel has an imaginary part related to the multi-pair component of the response function xmp(q ,v)(Sturm and Gusarov, 2000), that is,

Im@fxc~q,v!#.2F v

vpG4

v~q!2 Im xmp~q,v!,

where wp is the plasmon frequency. These higher-orderinteractions are beyond the usual description of screen-ing within the RPA or bubble approximation, and in-volve the partial summations of many complex diagramsappearing in the many-body response function.

In the homogeneous electron gas some exact relationsfor the fxc kernel must hold. In particular one has thefollowing relations:

(i) the compressibility sum rule, limq→0fxc(q ,v50)5d2rexc(r)/dr2, where exc(r) denotes theexchange-correlation energy per particle;

(ii) the third frequency moment sum rule:

limq→0

fxc~q ,v5`!5245

r2/3dexc~r!/r2/3

dr

16r1/3dexc~r!/r1/3

dr;

(iii) the static and frequency-dependent short-wavelength behavior: limq→`fxc(q ,v50)}c2 (b/q2) @12g(0)# [the correct values of the con-stants b and c can be found in Moroni et al.(1995) instead of the values used by Gross andKohn (1985)] and limq→`fxc(q ,vÞ0)52 (8p/3q2) @12g(0)# , where g(0) is the pair-correlation function evaluated at zero distance;

(iv) The following relations satisfied in the high-frequency limit by the real and imaginary parts offxc for q,` : limv→`Re fxc(q,v)5fxc(0,`)1c/v3/2

and limv→` Im fxc(q,v)52c/v3/2, with c523p/15in the high-density limit evaluation of the irreduc-ible polarization propagator.

These exact relations were used by Gross and Kohn(1985) to build a Pade approximation to the fxc(q→0,v). Although these results were obtained for thethree-dimensional electron-gas case, similar relationswere obtained for the two-dimensional case (Iwamoto,1984; Holas and Singwi, 1989) and extended to includetwo electron-hole pairs in both the longitudinal and thetransverse response functions (Nifosı et al., 1998).


Higher exact sum rules (up to seventh order) have beenderived by Sturm (1995), and they can be used as a strin-gent test on the quality of the response functions derivedfrom approximated fxc functionals.50

Additional constraints on the matrix elements of theexchange-correlation kernel were obtained by Gonzeand Scheffler (1999) using the Keldysh formalismadopted by van Leeuwen (1998). In particular, at theexact exchange level the following exact relation has tobe satisfied close to the Kohn-Sham resonance energyv ij :

^F ijufxc~v ij!uF ij&5^c iuSx2Vxuc i&

2^c juSx2Vxuc j&2^njuVuni&,

(A5)

where vxHF is the Hartree-Fock nonlocal operator evalu-

ated with the Kohn-Sham wave functions and Vx is theexact exchange potential in DFT. The last term describesan unscreened electron-hole attraction in contrast to theeffective interaction in the TDLDA approximation (asdiscussed above). This is due to the fact that Eq. (A5)for the kernel explicitly shows two parts, namely, thefirst two contributions containing a shift linked toHartree-Fock electron addition and removal energies,and the second one correcting with respect to those en-ergies, whereas the TDLDA kernel never passes explic-itly through electron addition and removal energies. Inthe single-pole approximation, expression (A5) leads toa correction of the excitation energies with respect toKohn-Sham eigenvalue differences, which turns out tobe identical to the correction obtained by Gorling (1996)in a first-order perturbation theory, in the difference be-tween the many-body and the second-quantized Kohn-Sham Hamiltonians. In addition, the inclusion of corre-lation effects leads to dynamically screened matrixelements through the same unscreened exchangeliketerm and an additional screened Coulomb interaction(Gonze and Scheffler, 1999). The similarities of this ap-proach to the GW quasiparticle correction and the in-clusion of electron-hole (excitonic) effects within theBethe-Salpeter equation of many-body theory are de-scribed in Sec. VI.

Another rigorous constraint is known as theharmonic-potential theorem (Kohn, 1961; Dobson,1994). It deals with the motion of an interacting many-electron system confined in a parabolic potential wellunder a spatially uniform time-dependent external field.This system exhibits sharp resonances at the bareharmonic-oscillator frequency, independently of theelectron-electron interaction. This exact result for theresponse of an interacting electron system stems fromthe invariance of the harmonic potential under a trans-formation to a homogeneously accelerated referenceframe (Vignale, 1995). In this case the dynamics of theelectronic center of mass is completely decoupled from

50In particular, the TDLDA and RPA satisfy the first threeodd-frequency sum rules but not the seventh (Sturm, 1995).


that of the internal degrees of freedom. This puts inter-esting constraints on approximate theories of time-dependent many-body physics such as TDDFT. Notethat the TDLDA approximation to TDDFT satisfies theharmonic-potential theorem because the exchange-correlation potential follows the density when the latteris moved (that is, the exchange-correlation potential islocal in both time and space; see below). The Gross andKohn (1985) approximation violates the harmonic-potential theorem as shown by Dobson (1994), but it ispossible to perform a simple modification for frequency-dependent local theories to satisfy this theorem (Vignaleand Kohn, 1996; Dobson, Bunner, and Gross, 1997; Vig-nale et al., 1997). The essential idea is to make the actionfunctional depend on the relative density rrel(r,t)[r„r1RCM(t),t…, where RCM(t) is the time evolution of thecenter of mass of the system. In this spirit an approxi-mate frequency-dependent exchange-correlation func-tional for small displacements from the equilibrium den-sity r0 can be constructed, as was done earlier by Grossand Kohn (1985), but now satisfying the harmonic-potential theorem. Thus the exchange-correlation po-tential reads Vxc(r,t)5Vxc

LDA(r)1* t0

t dt8fxc@r0(r),t2t8#drrel(r,t8), with fxc as given in Gross and Kohn(1985).51

From the previous general relations for fxc one cansee that a local-density approximation for time-dependent linear response in general does not exist, aslong as one keeps on describing dynamical exchange-correlation effects in terms of the density only. Thismeans that fxc is a strongly nonlocal functional of thedensity. This problem can be overcome if the currentdensity is added as a basic variable in a generalizedKohn-Sham scheme (Vignale and Kohn, 1996). In thisway one can derive a local current-density-functionaltheory of both current and density responses valid forslowly varying densities and potentials. This scheme wasgeneralized by Vignale et al. (1997) to include dynamicalcontributions beyond the linear response. In particular,exchange and correlation beyond the TDLDA lead tothe appearance of complex and frequency-dependentviscoelasticity stresses that, in the homogeneouselectron-gas case, provide an additional damping mecha-nism to the decay into electron-hole pairs.

The fact that the kernel is very nonlocal is also con-firmed by the calculation of the correlation energy of the

51A different perspective in functional development isachieved by the fact that the virial theorem for the exchange-correlation potential has been shown to hold for time-dependent electronic systems (Hessler et al., 1999). Moreover,the time dependence of the exchange-correlation energy issolely determined by the exchange-correlation potential@]Exc /]t 5*dr ]r(r,t)/]t Vxc(rt)# . These exact relations haveimportant implications for the construction of approximatetime-dependent functionals. In particular, the TDLDA ap-proximation is shown to fail badly in regions where the time-dependent density differs considerably from its ground-statecounterpart (Hessler et al., 1999).


homogeneous electron gas by Lein et al. (2000). Theyhave shown that the nonzero spatial range of fxc(r,r8,v)cannot be neglected, whereas the frequency dependenceis less important as far as total correlation energies areconcerned. In fact, Gonze et al. (1997) have indicatedthat fxc must have a 1/q2 divergence, for q→0. This con-tribution is supposed to solve the ‘‘metal-insulator para-dox,’’ i.e., the seeming contradiction concerning systemsthat are in reality nonmetallic, but have a vanishingKohn-Sham gap.

APPENDIX B: DERIVATION OF THE EQUATIONSCOMMON TO THE TDDFT AND BETHE-SALPETERAPPROACHES

1. Dyson-like equation for the macroscopic dielectricfunction

Given a matrix of the form

M5S m00 m1T

m2 m D , (B1)

with m00 being a c number, its inverse is

M215S 0 0

0 m21D 11

~m002m1Tm21m2!

3S 1 2m1Tm21

2m21m2 m21m2m1Tm21D . (B2)

If M is the dielectric function «512vP in its matrixform in reciprocal space (G, G8), then

«M51

«0021 5@«002«1

Te21«2# . (B3)

The quantities «00 , «1T , «2 , and e are given by

«00512v0P00 , (B4)

@«1T#G52v0P0G , GÞ0, (B5)

@«2#G852vG8PG80 , G8Þ0, (B6)

@e#GG85dGG82vGPGG8 , G,G8Þ0. (B7)

This yields

«M512v0P002 (G,G8Þ0

v0P0GeGG821 vG8PG80 . (B8)

Now one defines

«GG8[dG,G82 vGPGG8 , (B9)

with

vG[H 0, G50

vG , GÞ0, (B10)

i.e., v indicates that the G50 contribution is to be leftout in the bare Coulomb interaction. In matrix notationthis yields

«GG85S 1 0T

«2 e D . (B11)


Then the inverse of « is given by using Eq. (B2),

«GG821

5S 1 0T

2e21«2 e21D . (B12)

Thus Eq. (B8) leads to Eq. (2.9), if in the latter P isdefined as

PGG8[PGG81 (K ,K8

PGK«KK821 vK8~q !PK8G8 . (B13)

This gives the wanted matrix expression,

P5P1P «21vP , (B14)

and, together with «215(12 vP)21, the Dyson-likeequation (2.10).

2. Effective two-particle equations

In order to solve a Dyson-like equation such as Eq.(2.20), one has to invert a four-point function for eachfrequency. This problem can be reformulated as an ef-fective eigenvalue problem; in fact, the physical pictureof interacting electron-hole pairs suggests using a basisof eigenfunctions cn of the effective one-particle Hamil-tonian, from which the starting density matrix had beenconstructed, expecting that only a limited number ofelectron-hole pairs will contribute to each excitation(see Fetter and Walecka, 1971). These functions are sup-posed to form an orthonormal and complete set. Anyfour-point function S can then be transformed as

S~r1 ,r1 ;r28 ,r28!

5 (n1¯ n4

cn1* ~r1!cn2

~r18!cn3~r2!cn4

* ~r28!S(n1n2)(n3n4) .

(B15)

The four-point independent-quasiparticle polarization4PIQP ,

4PIQP~r1 ,r2 ,r3 ,r4!

5 (n ,n8

~fn2fn8!cn* ~r1!cn8~r2!cn8* ~r4!cn~r3!

en2en82v,

(B16)

is then diagonal in this basis andreads 4PIQP (n1n2)(n3n4) 5 (fn2

2fn1)dn1 ,n3

dn2 ,n4/ (en2

2en12v). With the definition P[@124PIQPK#21, Eq.

(2.20) becomes S5P 4PIQP . In transition space,

P(n1n2)(n3n4)5@~em22em1

2v!dm1 ,m3dm2 ,m4

1~fm12fm2

!K(m1m2)(m3m4)#(n1n2)(n3n4)21

3~en42en3

2v!.

Defining an effective two-particle Hamiltonian,

H(n1n2)(n3n4)2p [~en2

2en1!dn1 ,n3

dn2 ,n4

1~fn12fn2

!K(n1n2)(n3n4) , (B17)


one can then rewrite S as

S(n1n2)(n3n4)5@H2p2Iv#(n1n2)(n3n4)21 ~fn4

2fn3!. (B18)

The one-particle transition energies on the diagonal arethe eigenvalues of the starting one-particle Hamiltonian.Schematically, the effective Hamiltonian [Eq. (B17)] hasthe form

H(n1n2)(n3n4)2p 5S A B

0 D D , (B19)

with

(B20)

(B21)

(B22)

The resonant part is defined as

H(vc)(v8c8)2p ,res [~ec2ev!dv ,v8dc ,c81K(vc)(v8c8) . (B23)

It corresponds to transitions at positive absorption fre-quencies v. Due to the factor (fn3

2fn4) in Eq. (B18),

only the first column of

M(n1n2)(n3n4)21 5@H2p2Iv#(n1n2)(n3n4)

21 , (B24)

i.e., with (n3n4)5$c8v8% and $v8c8%, contributes to thecalculation of S .

Defining Av5A2Iv , it is clear that only Av21 is going

to be relevant, in other words, the part A in Eq. (B20).This means that only pairs containing one filled and oneempty Bloch state contribute, which is physically mean-ingful. It reflects the fact that only the hole-particle part[Eq. (3.7)] of the two-particle Green’s function yields theabsorption and loss spectra, as pointed out in Sec. III.A.

It is this part A in Eq. (B20) which is referred to asthe two-particle Hamiltonian H2p in the present work. Itis in general not Hermitian and can be further separatedinto four blocks: two blocks on the diagonal with thetransition energies and the interaction kernel K , andtwo off-diagonal coupling blocks with contributions onlyfrom the interaction kernel:


H2p5S H2p ,res Hcoupling

2@Hcoupling#* 2@H2p ,res#* D . (B25)

The resonant part is Hermitian, (H2p ,res)*5(H2p ,res)T, while the coupling part alone is symmetric,Hcoupling5(Hcoupling)T. The part in the lower right is de-noted antiresonant. Neglecting the coupling part iscalled the Tamm-Dancoff approximation.

The spectral representation of the inverse two-particleHamiltonian is

@H2p2Iv#(n1n2)(n3n4)21 5 (

l ,l8

Al(n1n2)Nl ,l8

21 Al8* (n3n4)

El2v(B26)

which holds for a system of eigenvectors and eigenval-ues of a general (not necessarily Hermitian) matrix de-fined by H2pAl5ElAl , where Nl ,l8[(n1n2

Al* (n1n2)A

l8

(n1n2)is the overlap matrix of the gen-

erally nonorthogonal eigenstates of H2p. Hence H2p isdiagonalized, and from its eigenvalues El and eigen-states Al

n1n2 the four- and two-point quantities of inter-est are constructed. In particular, one obtains the mac-roscopic dielectric function given in Eq. (2.23), and anequivalent expression for x.

APPENDIX C: AN fxc FROM THE BETHE-SALPETERAPPROACH

In this subsection, we derive a time-dependentdensity-functional formalism for excitation spectra fromthe many-body Bethe-Salpeter equation (Reining et al.,2002). The results obtained using an approximation tothis kernel, denoted as RORO, were already discussedin Sec. V.C.

The main idea is to compare the effective two-particleHamiltonians H2p ,TDDFT and H2p ,BSE derived for TD-DFT and the Bethe-Salpeter equation, respectively. Ifthe quasiparticle and Kohn-Sham eigenfunctions areequal, and if all matrix elements of H2p ,TDDFT andH2p ,BSE involving those transitions which actually con-tribute to the spectrum are equal, than the resultingspectra will be identical. This yields the condition

fxc~q,G,G8!5 (n1n2n3n4

1

~fn12fn2

!F21~n1 ,n2 ;G!

3F(n1n2)(n3n4)~F* !21~n3 ,n4 ;G8!. (B27)

The factor (fn12fn2

) can never be zero for the particle-hole and hole-particle contributions of a nonmetal. Thematrices F are defined as

F~n1 ,n2 ,r!ªcn1~r!cn2

* ~r! (B28)

and the operator F is

F(n1n2)(n3n4)5~en2

QP2en1

QP2en2

DFT1en1

DFT!dn1n3dn2n4

2~fn12fn2

!W(n1n2)(n3n4) . (B29)


It is now clear that, if the transformation from real totransition space xl→cn(xl) was complete in all four in-dices, Eq. (B27) could never be satisfied, since otherwiseH2p ,TDDFT and H2p ,BSE should also be equal in realspace—and they cannot be, due to the way the d func-tions are put. On the other hand, if the two operatorscannot be made equal, then the spectra can be equalonly if at least one of the two operators (in that case, fxc)is energy dependent. However, in practice only a finitenumber of transitions contribute to the optical spectrum.This means that one can use an incomplete basis in tran-sition space. Therefore one can still find a static operatorthat satisfies the required equality in transition space ina particular energy range, even though the real-spaceoperators are not equal. (The invertibility of the matri-ces F may, however, be questionable in some particularcases).

In view of the ongoing discussions about theexchange-correlation kernel it is interesting to examinesome of its features, and in particular its long-range be-havior for solids. To this end, we note thatFv ,c ,k1q,k(G50) goes to zero as q for small q . SinceWvc ,vc behaves as a constant, if F is invertible this im-plies immediately that fxc(q,G5G850) behaves as1/q2. Note that there are two such long-range termscoming from (a) the electron-hole attraction (of nega-tive sign) and (b) the energy shift between the quasipar-ticle and the DFT eigenvalues [of positive sign, as pre-dicted by Gonze et al., 1997, on the basis of their studyof the polarization-dependence of the exchange-correlation energy (Gonze et al., 1995; Resta, 1996), andas verified by a comparison of Kohn-Sham and experi-mental linear and nonlinear susceptibilities by Aulburet al., 1996]. A discussion of how this approximated ker-nel reproduces the optical spectrum of semiconductorsfor the case of bulk silicon is presented in Sec. V.C.

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