Starting from the many-body Bethe-Salpeter equation we derive an
exchange-correlation kernel $f_{xc}$ that reproduces excitonic effects in bulk
materials within time-dependent density functional theory. The resulting
$f_{xc}$ accounts for both self-energy corrections and the electron-hole
interaction. It is {\em static}, {\em non-local} and has a long-range Coulomb
tail. Taking the example of bulk silicon, we show that the $- \alpha / q^2$
divergency is crucial and can, in the case of continuum excitons, even be
sufficient for reproducing the excitonic effects and yielding excellent
agreement between the calculated and the experimental absorption spectrum.Comment: 6 pages, 1 figur

Angel Rubio

D. E. Aspnes

E. K. U. Gross

E. Runge

G. Onida

Giovanni Onida

I. V. Tokatly

L. Hedin

L. X. Benedict

Lucia Reining

M. Rohlfing

P. Hohenberg

P. Lautenschlager

Ph. Ghosez

S. Albrecht

V. I. Gavrilenko

Valerio Olevano

W. Hanke

W. Kohn

X. Gonze

Physical Review Letters

English

arXiv

Starting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel fxc that reproduces excitonic effects in bulk materials within time-dependent density functional theory. The resulting fxc accounts for both self-energy corrections and the electron-hole interaction. It is static, nonlocal, and has a long-range Coulomb tail. Taking the example of bulk silicon, we show that the -α/q2 divergency is crucial and can, in the case of continuum excitons, even be sufficient for reproducing the excitonic effects and yielding excellent agreement between the calculated and the experimental absorption spectrum.This work has been supported in part by the European Community Contract No. HPRN-CT-2000-00167. Computer time has been
granted by IDRIS (Project No. 000544) on the NEC SX5.Peer reviewe

Reining, Lucia

Olevano, Valerio

Rubio, Angel

Onida, Giovanni

Digital.CSIC

VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002Excitonic Effects in Solids Described by Time-Dependent Density-Functional TheoryLucia Reining,1 Valerio Olevano,1 Angel Rubio,1,2 and Giovanni Onida31Laboratoire des Solides Irradiés, CNRS-CEA, École Polytechnique, F-91128 Palaiseau, France2Departamento de Física de Materiales, Facultad de Ciencias Químicas, Universidad del Pais Vasco,and Donostia International Physics Center, E-20018 San Sebastián, Basque Country, Spain3Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica dell’Università di Roma “Tor Vergata”, I-00133 Roma, Italy(Received 23 May 2001; published 25 January 2002)Starting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel fxcthat reproduces excitonic effects in bulk materials within time-dependent density functional theory. Theresulting fxc accounts for both self-energy corrections and the electron-hole interaction. It is static, non-local, and has a long-range Coulomb tail. Taking the example of bulk silicon, we show that the 2aq2divergency is crucial and can, in the case of continuum excitons, even be sufficient for reproducing theexcitonic effects and yielding excellent agreement between the calculated and the experimental absorp-tion spectrum.DOI: 10.1103/PhysRevLett.88.066404 PACS numbers: 71.10.–w, 71.15.Qe, 71.35.–y, 78.20.BhThe calculation of electronic excitations has remaineda major challenge in the field of solid state theory. Infact, whereas ground state properties can be computedtoday with good precision within density functional the-ory (DFT) [1], electronic excitations are accessible onlythrough additional corrections. Within many-body per-turbation theory, Hedin’s GW corrections [2] are usedto get electron addition and removal energies, and theBethe-Salpeter equation (BSE) for neutral excitations asthose measured, for example, in absorption or electron-energy-loss spectroscopy (EELS). In fact, for several yearsab initio BSE calculations have yielded generally goodagreement between the calculated and the experimentalabsorption spectra for both finite [3] and infinite systems[4–6]. However, these calculations are necessarily cum-bersome, which has up to now prevented their applicationto very complex systems.In recent years, an alternative approach has been de-veloped which allows in principle the correct descrip-tion of exchange-correlation effects in the neutral excitedstate, namely time-dependent density functional theory(TDDFT) [7,8]. As in the case of static DFT the mainobstacle resides in finding a good approximation to theunknown exchange-correlation (xc) contribution. For theground state properties of the majority of finite and infi-nite systems, the local density approximation (LDA) hasturned out to yield surprisingly good results. In the caseof the absorption and EELS spectra of finite systems, andof EELS spectra of solids, TDDFT has yielded good re-sults using the adiabatic LDA approximation (TDLDA)for the xc kernel fxc [8]. However, this is not true forthe absorption spectra of solids [9]. In fact, it has turnedout that the results for the latter obtained within TDLDAare extremely close to those obtained in a simple randomphase approximation (RPA) calculation, where the xc ker-nel is completely neglected, and only local field effects (inother words, the contribution coming from the variation ofthe Hartree potential) are taken into account. It should be066404-1 0031-90070288(6)066404(4)$20.00pointed out that this similarity between RPA and TDLDAspectra to some extent also holds in the case of clustersand EELS spectra of solids, but the results are neverthelesssatisfactory because in those cases already RPA spectra,including local field effects, are in good agreement withexperiment.On the other hand, it would be extremely desirable toobtain good absorption spectra of solids within TDDFT,since the equations to be solved are two-point ones, in con-trast to the four-point BSE. The obstacle to be removedto this aim is hence the fact that a good approximation tothe xc kernel must be found. In this context, extensivediscussions can be found in literature about the need to in-clude long-range nonlocal terms and dynamical (memory)effects in the kernel [8,10].In this work, we show how a TDDFT equation for themacroscopic dielectric function can be derived from theBethe-Salpeter equation. The derivation is exact inthe sense that the resulting equation does yield the samespectra as the standard Bethe-Salpeter equation (with itsown various approximations [4–6]). We demonstratethat the resulting xc kernel has a 1q2 contribution, thestrength of which is inversely proportional to the screeningin the system. Finally, we calculate the absorption and therefraction index spectra of bulk silicon within TDDFT,using only this static long range tail as xc kernel, andobtain excellent agreement with experiment.To start with it is useful to put the TDDFT equation andthe Bethe-Salpeter equation for ´21 on the same footing.In fact, both equations can be written schematically in thesame Dyson-like form,S  S0 1 S0KS . (1)Here, S can be either the two-point polarizability x, fromwhich we can obtain the inverse dielectric function ´21 1 1 yx, or S  L, the two-particle correlation function© 2002 The American Physical Society 066404-1VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002which yields x (and then ´21) by contracting two of itsfour indicesxx1, x2  Lx1, x11 ; x2, x12  . (2)Here x stands for space, spin, and time coordinates.Of course, this holds only for what concerns the form,but not the specific details. First, quantities in theTDDFT equation are two-point ones, whereas in theBSE they are four-point ones. Second, in TDDFTS0 is the independent-particle response function x0constructed with the Kohn Sham (KS) orbitals andeigenvalues, whereas in the BSE formalism S0 standsfor the independent quasiparticle response L0, i.e.,it is constructed using quasiparticle eigenvalues andeigenfunctions (e.g., obtained from a GW calculation).Finally, in the TDDFT case, the kernel K is defined asK  y 1 FTDDFT, where y is the Coulomb potential andFTDDFT stands for the fxc kernel. In the case of the BSE,K  y 1 FBSE, where y and FBSE are the four-pointfunctions yx1, x2, x3, x4  dx1,x2dx3,x4yx1, x3and FBSEx1, x2, x3, x4  2dx2,x4dx1, x3Wx1, x2,where W is the screened interaction. In the caseof TDDFT, one can also understand Eq. (1) as afour-point equation, but with FTDDFTx1, x2, x3, x4 dx1,x2dx3,x4fxcx1,x3 which implies that one canimmediately contract the indices by pairs and reduce theequation to a two-point one.If one performs a basis transformation of the form xl !cnxl, where the cnxl are the one-particle orbitals whichdiagonalize the four-point S0, Eq. (1) can be transformedto an effective two-particle Hamiltonian equation,H2pn1n2n3n4  en2 2 en1dn1n3dn2n41 fn1 2 fn2 Kn1n2n3n4. (3)Here the indices ni refer to the fact that matrix elementsinvolve four eigenfunctions of the starting effective one-particle Hamiltonian with eigenvalue eni and occupationfni . Defining the identity operator I  dm1m3dm2m4, S isdirectly obtained from H2p asSn1n2 n3n4  H2p 2 Iv21n1n2 n3n4fn4 2 fn3 . (4)H2p can be diagonalized, and from its eigenvalues El andeigenstates An1n2l the spectral representation of S can beconstructed.Since the cnxl have to diagonalize S0, they must bethe KS orbitals for the TDDFT equation and the QP eigen-functions for the BSE. If these functions are equal, andif the Al and El are equal, the BSE and TDDFT spectrawould be the same. For a static fxc kernel, this impliesthat the matrix elements of the Hamiltonians H2pTDDFT andH2pBSE are equal. In this scenario, we can directly com-pare the BSE and TDDFT approaches. First, in the BSEthe eigenvalues en are quasiparticle energies (as obtained,for example, from a GW calculation), whereas in TDDFTthey are the eigenvalues obtained from the KS equation066404-2with the (in principle exact) xc potential. Second, for theBSE we haveFBSEn1n2n3n4  2Zdr dr0Fn1,n3; r3 Wr 0, rFn2, n4; r0 ,with the matrices F defined according toFn1,n2; r : cn1 rcn2 r , (5)whereas the exchange-correlation contribution to theTDDFT kernel readsFTDDFTn1n2 n3n4 Zdr dr0Fn1,n2; r3 fxcr, r0Fn3, n4; r0 .[The Hartree contribution yn1n2n3n4 is of course equal inboth cases.] A comparison of the two cases does henceimmediately tell us that the BSE and TDDFT equationswould yield the same spectrum if the static fxc satisfiesfn1 2 fn2Zdr dr0Fn1,n2; rfxcr, r0Fn3, n4; r0 Fn1n2n3n4,(6)where [11]Fn1n2n3n4  eQPn2 2 eQPn1 2 eDFTn2 1 eDFTn1 dn1n3dn2n41 fn1 2 fn2FBSEn1n2n3n4. (7)It is clear that, if the transformation xl ! cnxl was com-plete in all four indices, Eq. (6) could never be satisfied.The reason is that the two operators, FTDDFT and F , can-not be equal because of the way the d functions are put inreal space. On the other hand, if the two operators cannotbe made equal, then the spectrum can be equal only if atleast one of the two operators (in that case, fxc) is energydependent [12]. This in principle correct, general state-ment, can be made less restrictive by realizing that in prac-tice only a finite number of transitions contributes to theoptical spectrum. This means that we can use an incom-plete basis in transition space. In this reduced Hilbert spacewe can still find a static operator that satisfies the requiredequality in transition space in a particular energy range,even though the real-space operators are not equal [13]. Inorder to discuss this possibility, we rewrite Eq. (6) asfn1 2 fn2 XG,G0Fn1,n2;Gfxcq,G,G0Fn3, n4;G0 Fn1n2n3n4 ,(8)where q  k2 2 k1  k4 2 k3. Since for the particle-hole and hole-particle contributions of a nonmetal thefactor fn1 2 fn2  can never be zero, we could in prin-ciple obtain the matrix fxc from Eq. (8) by inverting thematrices F:066404-2VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002fxcq,G,G0 Xn1n2n3n41fn1 2 fn2 3 F21n1,n2;GFn1n2n3n43 F21n3,n4;G0. (9)As pointed out above, the sum over the indices ni is nec-essarily limited to some subspace of important transitions,because otherwise the inverse of F does not exist. Incertain cases, a subspace which at the same time allowsthe matrix to be invertible and to reasonably reproducethe exciton spectrum cannot be found. One example isa two-band model with eDFT  eQP consisting of planewave states cbkr  eiGbreikr, with Gb  Gy or Gc. Inthis case the matrices F are k independent, which meansthat all their rows are equal. In other words, the matrixFTDDFT is k independent, whereas the matrix FBSE goesas 1k 2 k02, which is a clear contradiction. On theother hand, if a subspace where F is invertible can be066404-3found, it is clear that the resulting kernel is frequency in-dependent, as none of the quantities implied, FBSE or F,are frequency dependent (if the kernel of the BSE is cho-sen to be frequency independent [12]). Moreover fxc isnecessarily not local, as the expression in Fourier spacedepends separately on G and on G0. In fact, its nonlocal-ity is used as the degree of freedom which is necessary tofulfill equality (6).On the other hand, one can try to look directly at Sinstead of focusing on fxc. To do that we go back to theinitial Eq. (1) which we can write in a symmetric way asS  S0S0 2 S0KS021S0. (10)Since the independent particle polarizability isx0r, r0,v Xn1n2fn1 2 fn2 3Fn1,n2; rFn1,n2; r0eDFTn1 2 eDFTn2 2 v,the equation for the two-point S  x becomesxr, r0,v Zdr00 dr000x 0r,r00 ,v ?∑x0r1, r2,v 2Zdr3 dr4x0r1, r3,vyr3, r4x0r4,r2,v1Xn1n2n3n4Fn1, n2; r1eDFTn2 2 eDFTn1 2 vFn1n2n3n4Fn3,n4; r2fn4 2 fn3 eDFTn4 2 eDFTn3 2 v∏21r00,r000? x 0r000, r0,v , (11)where we have usedFn1, n2, r  Fn2, n1, r, and sub-stituted the term FfxcF using Eq. (6). In other words,for those cases where the equality (6) can be fulfilled, i.e.,in particular when a static kernel can be found, we havesucceeded in writing the two-point TDDFT equation in away that exactly yields the BSE spectrum [14].The explicit knowledge of the kernel is actually notneeded. There is still a four-point quantity appearing,namely F , however only in a matrix product instead of in-versions and diagonalizations, which allows us to changespace in a convenient way as it is also done in recur-sive inversions of BSE [5]. Despite this fact, in view ofthe ongoing discussions about the xc kernel it is interest-ing to examine some of its features, and in particular itslong-range behavior. In fact, for valence y, k and con-duction c,k 1 q states, Fy,k, c,k 1 q;G  0 goesto zero as q for small q. Since Fy,c,y,c in this limitbehaves as a constant, an fxcq,G  G0  0 obtainedfrom Eq. (9) must behave as 1q2. There is in fact a posi-tive long-range contribution stemming from the QP shift ofeigenvalues (as also predicted in Ref. [15]), and a negativeone resulting from the electron-hole interaction, which isthe main point of interest here.Using the above results, one can also understand whythe TDLDA approximation yields much worse results forthe absorption than for the loss spectra of solids. In fact,energy-loss spectra are directly related to the inverse of thedielectric matrix ´21  1 1 yx. For the calculation of x,the kernel fxc is then added to y, which already containsa long-range contribution yG  0. So the presence orabsence of the long range term in fxc does not necessarilyshow up. However, in the case of absorption spectra, onecan show [16] that the macroscopic dielectric function isgiven by´Mv  1 2 limq!0yqx¯GG00q,v , (12)where x¯ has been calculated in a Dyson-like equation suchas (1) using the same fxc, but added to a coulombian y¯which does not contain the long range term, i.e., yG  0is set to zero. Obviously in that case, a neglect of thedivergence in fxc makes an essential difference.We can carry this discussion about the long-rangeelectron-hole interaction term farther by (i) assuming thatwe absorb the first, positive contribution in the energy shiftof our starting x0 (since anyway we do not know theeigenvalues of the exact xc potential which would go alongwith the exact kernel) and (ii) supposing that we have asystem where the long-range term is completely dominat-ing the rest of the xc contribution, namely, where we canapproximately write fxcq,G,G0  2dG,G0ajq 1 Gj2.In other words, the long-range electron-hole attraction partof fxc reduces the Hartree part, which is reasonable sinceit stems from the exchange potential. This approximationworks best for systems with weakly bound excitons.For a demonstration we have therefore performed aTDDFT calculation for bulk silicon, in the followingway: First, we have determined the DFT-LDA electronicstructure. Second, we have constructed x0, but withthe eigenvalues shifted to the GW ones, in order tosimulate the first part of the kernel as explained above.066404-3VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002Third, we have used fxcr, r0  2a4pjr 2 r0j, withthe empirical value a  0.2. The result of the TDDFTcalculation for ´Mv is shown in Fig. 1. The dots arethe experimental results for the absorption spectrumIm´ measured by Lautenschlager et al. [17] and therefraction index Re´ measured by Aspnes and Studna[18]. The dot-dashed curve is the result of a standardTDLDA calculation (i.e., using DFT-LDA eigenvaluesand the static short-range LDA xc kernel). We find thewell-known discrepancies with experiment. The dashedcurve is the BSE result. Finally, the continuous curveis the result of our approximate TDDFT calculation: Itfits almost perfectly all experimental features in both realand imaginary parts of ´. It turns hence out that thisstatic long-range contribution to the kernel is sufficient toreproduce the strong excitonic effect in a material withweakly bound excitons such as silicon. Preliminary resultson other materials such as GaAs or AlAs are showing asimilar quality of agreement with experiment. We refer toa forthcoming manuscript [19] for details.In conclusion, we have derived a TDDFT equation fromthe Bethe-Salpeter equation which should be particularlysuitable for practical applications to the absorption spectra2 3 4 5 6ω [eV]01020304050Im ε(ω)-20-10010203040Re ε(ω)FIG. 1. Silicon, optical absorption (bottom), and refraction in-dex (top panel) spectra. Dots: experiment. Dot-dashed curve:TDLDA result. Dashed curve: result obtained through theBethe-Salpeter method. Continuous curve: TDDFT result us-ing the long-range kernel derived in this work.066404-4of solids. We have demonstrated that the static exchange-correlation kernel has a long-range contribution stemmingfrom the electron-hole interaction. We have explained whythis long-range contribution is particularly important forthe absorption spectra of solids. At the example of bulksilicon, we have shown how a very simple approximationfor the kernel can yield excellent agreement between thecalculated TDDFT absorption spectrum and experiment.We are grateful to P. Ballone and R. Del Sole foruseful discussions and comments. This work has beensupported in part by the European Community ContractNo. HPRN-CT-2000-00167. Computer time has beengranted by IDRIS (Project No. 000544) on the NEC SX5.[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).[2] L. Hedin, Phys. Rev. 139, 796 (1965).[3] G. Onida, L. Reining, R. W. Godby, R. Del Sole, andW. Andreoni, Phys. Rev. Lett. 75, 818 (1995).[4] S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Phys.Rev. Lett. 80, 4510 (1998).[5] L. X. Benedict, E. L. Shirley, and R. B. Bohn, Phys. Rev. B57, R9385 (1998).[6] M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 81, 2312(1998).[7] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997(1984).[8] E. K. U. Gross, F. J. Dobson, and M. Petersilka, DensityFunctional Theory (Springer, New York, 1996).[9] V. I. Gavrilenko and F. Bechstedt, Phys. Rev. B 55, 4343(1997).[10] I. V. Tokatly and O. Pankratov, Phys. Rev. Lett. 86, 2078(2001).[11] In the Hartree Fock limit, the diagonal element of Freduces to Eq. (7) of X. Gonze and M. Scheffler, Phys.Rev. Lett. 82, 4416 (1999) at the resonance energy.[12] The standard and successful way to introduce the electron-hole interaction in the Bethe-Salpeter scheme has beendone so far by the use of a static screened interaction W[3–6] in the kernel of the BSE FBSE.[13] We remark that the spirit of our scheme is to find a staticfxc that mimics the role of the exact frequency dependentfxc in the macroscopic dielectric function, even if it is farfrom the true one.[14] Note that in order to obtain this simple formula, we had toassume that KS orbitals and QP wave functions are equal.Otherwise, we could write, for example, the BSE in theKS basis, and we would get a complicated v dependenceinstead of the simple en1 2 en2 2 v diagonal part inEq. (4).[15] Ph. Ghosez, X. Gonze, and R. W. Godby, Phys. Rev. B 56,12 811 (1997).[16] W. Hanke and L. J. Sham, Phys. Rev. Lett. 43, 387 (1979).[17] P. Lautenschlager et al., Phys. Rev. B 36, 4821 (1987).[18] D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985(1983).[19] S. Botti et al. (to be published).066404-4

Excitonic effects in solids described by time-dependent density-functional theory

6 pages, 1 figureStarting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel $f_{xc}$ that reproduces excitonic effects in bulk materials within time-dependent density functional theory. The resulting $f_{xc}$ accounts for both self-energy corrections and the electron-hole interaction. It is {\em static}, {\em non-local} and has a long-range Coulomb tail. Taking the example of bulk silicon, we show that the $- \alpha / q^2$ divergency is crucial and can, in the case of continuum excitons, even be sufficient for reproducing the excitonic effects and yielding excellent agreement between the calculated and the experimental absorption spectrum

Olevano, Valério

HAL-Polytechnique

Excitonic effects in solids described by time-dependent density functional theory

Crossref

Excitonic Effects in Solids Described by Time-Dependent Density-Functional Theory

Starting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel f(xc) that reproduces excitonic effects in bulk materials within time-dependent density functional theory. The resulting f(xc) accounts for both self-energy corrections and the electron-hole interaction. It is static, non-local. and has a long-range Coulomb tail. Taking the example of bulk silicon, we show that the -alpha/q(2) divergency is crucial and can, in the case of continuum excitons, even be sufficient for reproducing the excitonic effects and yielding excellent agreement between the calculated and the experimental absorption spectrum

L. Reining

V. Olevano

A. Rubio

AIR Universita degli studi di Milano

HAL-CEA

https://hal.archives-ouvertes.fr/hal-00122419

Excitonic effects in solids described by time-dependent density
  functional theory

Excitonic effects in solids described by time-dependent density functional theory

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AIR Universita degli studi di Milano

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