The study of the electronic structure of materials usuallybegins with noninteracting electrons due to the vast numberof particles involved. Many-body methods then provide a hi-erarchy of corrections to account for the Coulomb interactionto various order. Many-body approaches such as GW (Refs. 1and 2) and the Bethe-Salpeter equation (BSE) (Refs. 3 and 4)are frequently and successfully employed in the calculation ofthe electronic structure and excitations of materials. Thoughaccurate and physically sound, these many-body methods canbecome cumbersome and impractical for large systems due tothe steep scaling of the numerical cost versus the system size.
Alternatively, density-functional theory (DFT) and time-dependent density-functional theory (TDDFT) (Refs. 5, 6,and 7) are popular methods for calculating electronic groundstates and excitations, respectively, and are widely usedin chemistry, physics, materials science, and other areas.Density-functional methods solve the many-body problem byconstructing a noninteracting system which reproduces theelectronic density of the interacting, physical system. The fa-vorable balance between accuracy and efficiency makes theresulting DFT and TDDFT schemes unrivaled for large butfinite system sizes.8
Considerable effort has been spent to replicate this suc-cess of TDDFT for periodic solids.9 Generally speaking,TDDFT works very well for simple metallic systems, wherethe excitation spectrum is dominated by collective plas-mon modes. The reason is that common local and semilocalexchange-correlation (xc) functionals are based on the homo-geneous electron liquid as reference system, which is an idealstarting point to describe electrons in metals.
The situation is more complicated in insulators and semi-conductors. The first problem that comes to mind is that of the
bandgap. The exact DFT determines the fundamental gap ofsolids by EKS
g + �xc, where EKSg is the difference between the
lowest unoccupied and highest occupied Kohn-Sham eigen-values (often called “Kohn-Sham gap”), and �xc is the deriva-tive discontinuity.10 Most of the popular approximated xcfunctionals in DFT are based on approximations in terms ofthe local density, and they strongly underestimate the funda-mental gap by having no derivative discontinuity, and under-estimating the Kohn-Sham gap. In principle, TDDFT providesa mechanism to obtain the correct optical bandgap, which isequal to the fundamental gap minus the exciton binding en-ergy for direct-gap solids,11–13 but this puts very strong de-mands on the approximated xc kernel fxc, requiring a strongfrequency dependence to simulate the missing derivative dis-continuity in the approximated ground-state DFT calculation.
The second difficulty is excitonic effects. It is a well-known fact that standard local and semilocal xc functionals donot produce any excitonic binding;4, 9 again, the proper choiceof fxc is crucial. There are many examples in the literatureof successful TDDFT calculations of excitonic effects, usingexact exchange,14 an effective xc kernel engineered from theBSE,15–19 a meta-GGA kernel,20 and a recent “bootstrap” xckernel.21 These kernels all have in common that they have along spatial range; however, it has also been shown that cer-tain excitonic features can be equally well reproduced by sim-ple short-range kernels.9, 16 This calls for further explanation.
Due to the complexity of real solids, the question of thegeneral requirements for excitonic binding in TDDFT hasbeen difficult to analyze. As a first step towards a simpli-fied TDDFT approach for excitons, a two-band model was re-cently developed, which was used to test the performance ofsimple xc kernels for calculating excitonic binding energiesin several III-V and II-VI semiconductors.22, 23 Since other
014513-2 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
low-dimensional model systems for excitons have been usedin the literature with some success,12 we will push this re-ductionist approach further and propose a minimal TDDFTmodel for excitons in this paper.
Our model is one-dimensional (1D) and uses two simpleKronig-Penney-type bands as input. We show that the mini-mal model reproduces and reveals many aspects of excitons.The model is accessible and relatively easy to implement, andit can be used to identify important aspects of the xc func-tional for excitonic effects. It clearly shows that excitons arecollective excitations of the many-body system: the appropri-ate phase-coherent mixing of the single-particle excitations isaccomplished via a coupling matrix featuring fxc. The proper-ties of this coupling matrix are analyzed and compared withits BSE counterpart.
In textbooks, excitons are usually introduced through thetwo-body Wannier equation, which describes an electron anda hole which interact via a screened Coulomb potential. Whilethis arguably constitutes the simplest model for excitons, it isbased on several drastic assumptions which are not fulfilled ingeneral.24, 25 We discuss how and under what circumstancesa Wannier-like equation emerges from our minimal TDDFTmodel.
The paper is structured as follows. We give an introduc-tion to Wannier excitons in Sec. II A and to TDDFT for solidsin Sec. II B, followed by a description of the minimal model inSec. II C. Section III then presents results for the minimal ex-citon model comparing TDDFT with the BSE, and discussesvarious implications. In Sec. IV we show how a Wannier-like equation emerges from TDDFT. Conclusions are given inSec. V. Details of the BSE method are provided in Ap-pendix A. Atomic units (¯= e = me = (4πε0)−1 = 1) are usedthroughout unless otherwise stated, and we will only considerspin-unpolarized systems.
II. BACKGROUND AND MODEL
A. Wannier excitons
The electronic structure of crystalline solids is describedby the Bloch theory, where the electrons move in a peri-odic effective single-particle potential which reflects the crys-tal symmetry. As a result, the electronic states form energybands. In insulators and semiconductors, electronic excita-tions take place between the occupied (valence) and unoc-cupied (conduction) bands. These interband transitions canbe described within a simple independent-particle approachbased on Fermi’s golden rule; one thus obtains a reason-able qualitative account of the optical properties in thesematerials.26
However, experiments reveal that there are importantmodifications to this picture, as illustrated schematically inFig. 1. Above the bandgap, the spectrum appears stronglyenhanced, and below the bandgap one may find discreteabsorption peaks known as bound excitons. The origin ofthese modifications are Coulomb interactions: the simplepicture of independent single-particle excitations is replacedby a more complex scenario where these excitations aredynamically coupled. Excitonic effects are ubiquitous in
ω − ωgap
abso
rpti
on
0
FIG. 1. Schematic optical spectrum near the band edge of a typical 3D direct-gap insulator. Dashed line: independent-particle spectrum. Solid lines: spec-trum including excitonic effects.
nature, and occur in 3D, 2D, and 1D systems alike.27, 28 Thedetails of the excitonic modifications to the noninteractingspectrum depend on the dimensionality of the system.29
The standard textbook explanation of excitons30 is basedon the simple picture of an electron-hole pair held together byCoulomb interactions, see Fig. 2. One thus arrives at a two-body problem similar to the positronium atom, with a center-of-mass momentum k and relative motion described by theWannier equation24, 25, 29[
− ∇2
2mr
+ V (r)
]ψν(r) = Eνψν(r), (1)
where mr is the reduced mass, defined as m−1r = m−1
c − m−1v .
mc and mv are the effective masses of the conduction bandelectrons and the valence band holes. V (r) = 1/εr is theCoulomb interaction between the electron and the hole, di-vided by the static dielectric constant ε of the system. ψν andEν are the excitonic wave function and binding energy, re-spectively.
In 1D systems, the Coulomb interaction is ill-defined andrequires, in general, some parametrized form;31 we will usethe following soft-Coulomb interaction:
V1D(x) = A√x2 + α
, (2)
where A and α are parameters. In the following, we set α
= 0.01 if not mentioned otherwise.The Wannier equation (1) has the form of a hydrogenic
Schrödinger equation, so it possesses a Rydberg series (even
e
h
k
FIG. 2. Illustration of Wannier exciton as an electron-hole pair, extendingover many lattice constants.
014513-3 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
for 1D cases with soft-Coulomb interaction) with infinitelymany eigenvalues below the bandgap, and a continuum abovethe gap.29 The excitons in the Wannier model for 3D and 2Dcases both enhance the optical absorption near the bandgap,while the presence of excitons in 1D systems suppresses theoptical absorption just above the bandgap.
The Wannier picture of excitons appears clear and in-tuitive, but this simplicity is somewhat deceptive. In reality,excitons are a dynamical many-body phenomenon and re-quire a subtle coordination and cooperation of many single-particle transitions between two bands. Though Sham andRice32 showed that the Wannier equation can be justified asan approximation to the BSE, and thus inherently describesthe collective nature of excitons, its real-space representationoften hampers the analysis of excitonic effects, since the col-lective nature of excitons is not explicitly shown. Besides, theterm “Wannier model” is not normally used with such an un-derlying many-body picture in mind, but it is rather treated asa phenomenological model obtained by just considering thescreened interaction within one particle-hole pair, as it wasoriginally proposed.24, 25 In the following, we develop a modelbased on TDDFT which will illustrate the true physical natureof excitons, but which will remain sufficiently transparent toallow a simple interpretation of the collective many-body ef-fects that are responsible for the excitonic binding.
B. TDDFT in finite and periodic systems
TDDFT (Ref. 7) is an in principle exact approach forelectron dynamics, based on the uniqueness of the mappingbetween the time-dependent electronic density and the ex-ternal potential.5 The key equation of TDDFT is the time-dependent Kohn-Sham (TDKS) equation
i∂
∂tφi(r, t)
=[−∇2
2+ vext(r, t) + vH(r, t) + vXC(r, t)
]φi(r, t), (3)
where φi(r, t) are the TDKS orbitals of the noninteractingKohn-Sham system which reproduces the density of the realinteracting system. vext and vH are the external potential ofthe physical system and the Hartree potential, respectively,and the xc potential vXC is the only piece that needs to beapproximated in practice.
The excitation spectrum of a system can be calculated viatime propagation of Eq. (3) following a suitably chosen initialperturbation. Alternatively, one can obtain excitation energiesand optical spectra directly from linear-response TDDFT, us-ing the so-called Casida equation33(
A BB A
)(XY
)= ω
(−1 00 1
) (XY
). (4)
Equation (4) is a generalized eigenvalue equation. One ob-tains the optical transition frequencies from the eigenvaluesω, and the corresponding eigenvectors tell us how the Kohn-Sham single-particle transitions are mixed to form the tran-sitions of the interacting system. A, B, X, Y are objects ofthe transition space, spanned by single-particle transitions of
the Kohn-Sham system. The matrices A and B in Eq. (4) aredefined as
A(ij )(mn)(ω) = δimδjn(εj − εi) + F(ij )(mn)HXC ,
B(ji)(mn)(ω) = F(ji)(mn)HXC ,
(5)
where i, m are labels for occupied ground-state Kohn-Shamorbitals, j,n are for unoccupied ground-state Kohn-Sham or-bitals, and the ε’s are the associated Kohn-Sham orbital ener-gies. F
(ij )(mn)HXC in Eq. (5) is defined as
F(ij )(mn)HXC =2
∫d3r
∫d3r ′ φi(r)φ∗
j (r)fHXC(r, r′, ω)φ∗m(r′)φn(r′),
(6)
where the factor of 2 accounts for the spin, the φ’s are theground-state Kohn-Sham orbitals, and fHXC is the Hartree-exchange-correlation (Hxc) kernel, defined as a Fourier trans-form of
fHXC(r, t, r′, t ′) = δvH(r, t)δn(r′, t ′)
+ δvXC(r, t)δn(r′, t ′)
≡ 1
|r − r′| + fXC(r, t, r′, t ′). (7)
The xc kernel fXC has to be approximated in practice.X and Y make up the eigenvector in Eq. (4), and they
describe excitations and de-excitations, respectively. A com-monly used approximation to Eq. (4), known as Tamm-Dancoff approximation (TDA),34 is to set B = 0, so that theCasida equation reduces to∑
This decouples excitations and de-excitations, and the compu-tational cost is reduced. There are situations, for instance, formolecular excitations of open-shell systems,35 in which theTDA is preferred in practice over the full Casida equation (4).We find that the TDA can also be advantageous for excitons(see Sec. III).
By considering only a single Kohn-Sham transitionin Eq. (4) one arrives at the small-matrix approximation(SMA),36, 37
ω2SMA,ij = ω2
KS,ij + 2ωKS,ijF(ij )(ij )HXC , (9)
where ωKS,ij = εj − εi is the Kohn-Sham transition fre-quency to be corrected. One can further simplify this bymaking the TDA, which yields the single-pole approximation(SPA),
ωSPA,ij = ωKS,ij + F(ij )(ij )HXC . (10)
The SMA and SPA are valid when the considered excitation isfar away from other transitions in the system. Though they arenot usually accurate enough for real calculations, their sim-plicity makes them very useful for theoretical analysis anddevelopment.
In periodic solids, the Kohn-Sham orbitals are labeledwith the band index i and the wavevector k and have the Bloch
014513-4 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
form
φik(r) = eik·ruik(r). (11)
It is in principle possible to adapt the Casida equation (4) forthe case of orbitals of the form (11),38, 39 and use this to cal-culate the excitation spectrum. However, to obtain the full op-tical spectrum in solids it is more convenient to calculate themacroscopic dielectric function4
εM (ω) = limq→0
1
ε−1G=G′=0(q, ω)
= limq→0
1
1 + vG=0(q)χG=G′=0(q, ω), (12)
where the G’s are reciprocal lattice vectors, and χ = δn/δvext
is the linear-response function. εM(ω) can be expressed as4
εM (ω) = 1 − limq→0
vG=0(q)∑
λ
∣∣∑ij 〈i|e−iq·r|j 〉ρ(ij )
λ
∣∣2
ω − ωλ + iη, (13)
where λ labels the solutions of Eq. (8) for an extended system.Beyond linear response, few real-time TDDFT calcula-
tions exist for periodic solids.40–42 Instead of directly solvingthe TDKS equation (3), the TDKS orbitals can be expandedin terms of ground-state Kohn-Sham Bloch functions as
φik(r, t) =∑m
cimk(t)φmk(r). (14)
The time-dependent density matrix is defined as
ρmnik (t) = cimk(t)c∗
ink(t). (15)
The equation of motion of the density matrix is then
i∂
∂tρik(t) = [Hk(t), ρjk(t)], (16)
with the TDKS Hamiltonian matrix Hk(t) defined by
Hmnk (t) =
∫�
d3r φ∗mk(r)HKS(r, t)φnk(r), (17)
where � is the volume of the unit cell, and HKS(r, t) is theTDKS Hamiltonian of Eq. (3). This density-matrix approachhas been used to derive the TDDFT version of the semicon-ductor Bloch equations.22, 23 In this formalism we only con-sider vertical transitions, where the Bloch wavevector k doesnot change during the dynamics. Nonvertical excitations arenot considered, since they involve indirect processes (e.g.,phonon-assisted processes, inelastic x-ray scattering) whichwe ignore here.
C. Minimal model for excitons
Solids are formally described by the many-bodySchrödinger equation. Since exact solutions are not possible,it is instructive to resort to model systems to eliminate unde-sired details of the many-body system and provide clear il-lustrations of the specific features one is interested in. Theoriginal Wannier model for excitons presented in Sec. II A issuch a model.
While intuitive, the Wannier model in its original formassumes the electron-hole interaction as given, so the exci-tonic effects are already built in by default. However, this
does not explain under what conditions one expects to seethe formation of excitons, and the many-body nature of exci-tonic effects is not obvious. We therefore propose a minimalmodel for the study of excitons with a lower abstraction levelthan the Wannier model, and where excitonic effects show upwithout any ad hoc assumptions.
For excitations near the bandgap, a reasonable approx-imation is to use a two-band model, i.e., only to considerthe highest valence band (v) and the lowest conduction band(c). This means that we only need to consider those elementsof the time-dependent density matrix ρmn
ik (t) [Eq. (15)] forwhich mn = cc, cv, vc, vv, and i = v; the latter index willbe dropped in the following.
For the case when a small perturbative electric field isapplied to the system, it is sufficient, to lowest order in theperturbation, to consider only the time evolution of the off-diagonal part of the density matrix,23 ρcv
k . One then obtainsfrom Eq. (16),
i∂
∂tρcv
k (t) = ωcvk ρcv
k (t) + δV cvHXC,k(t), (18)
where ωcvk =εck−εvk, and δV cv
HXC,k(t)=V cvHXC,k(t)−V cv
HXC,k(0).Here, V cv
HXC,k denotes the matrix elements of vH(r, t)+ vXC(r, t), defined similarly to Eq. (17). Fourier transforma-tion of Eq. (18) gives
ρcvk (ω) =
∑k′
{F
(vc)(cv)HXC,k,k′ρ
cv∗k′ (ω) + F
(vc)(vc)HXC,k,k′ρ
cvk′
}ω − ωcv
k, (19)
where
F(ij )(mn)HXC,k,k′
= 2∫
�
d3r
∫�
d3r ′ φik(r)φ∗jk(r)fHXC(r, r′, ω)φ∗
mk′ (r′)φnk′(r′)
≡ 2〈〈ij |fHXC|mn〉〉. (20)
Equation (19) is equivalent to the SMA for finite sys-tems. While the SMA only refers to the transition betweenone individual occupied and one individual unoccupied or-bital, Eq. (19) considers the transitions between the valenceand the conduction band as a whole. Ignoring the couplingbetween excitations and de-excitations by setting F
(vc)(cv)k,k′ = 0
(i.e., making the TDA), one arrives at the solid-state analog ofthe SPA:∑
k′
[ωcv
k′ δk,k′ + F(vc)(vc)HXC,k,k′ (ω)
]ρcv
k′ (ω) = ωρcvk (ω). (21)
An equation of a similar form as Eq. (21) can be derived start-ing from the BSE; the difference is only in the details of theF matrix, which is described in Sec. III B. Equation (21) isthe central equation which we will use to describe excitoniceffects. It requires as input the Kohn-Sham Bloch functionsfor the valence and the conduction band of an insulator orsemiconductor. To keep the model as simple as possible, wewill consider the Kronig-Penney (KP) model30 rather than areal material. The KP model is a 1D noninteracting systemwith a periodic potential of square wells. Within the unit cell[− b, a], the potential is
VKP(x) ={
0 0 < x < a
V0 −b < x < 0 .(22)
014513-5 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
x
V (x)V0
0 a a + b−b−a − b
k− πa+b
πa+b
E
FIG. 3. 1D Kronig-Penney model: potential (lower panel) and band structure(upper panel).
A typical example for the band structure of the KP modelis plotted in Fig. 3. Despite its simple appearance, the KPmodel is very versatile: by varying the values of the latticeconstant a + b, barrier width b, and barrier height V0, a widerange of bandgaps and band curvatures can be achieved. Asquare-well potential of finite depth does not support an infi-nite number of bound states as the Coulomb potential does;but this, in fact, closely reflects the reality of the effective po-tential felt by the valence electrons in a solid, which is rela-tively shallow due to the screening of the bare nuclear chargesby the core electrons. In practice, many solid-state calcula-tions account for this screening by using muffin-tin potentialsor pseudopotentials.43–45 The KP model can be viewed as anelementary version of this approach.
In the following, we always choose the first two bands tobe fully occupied and the higher bands to be empty. We thenmake the two-band approximation for bands 2 and 3, since theshapes of these bands resemble the highest valence band andlowest conduction band in direct-gap materials such as GaAs.The bands in the KP model are sufficiently well separated, sothe two-band approximation is justified.
To establish a connection with TDDFT, we assumethe solution to the noninteracting KP model to be ourground-state Kohn-Sham system. In other words, the poten-tial in Eq. (22) represents the exact Kohn-Sham potentialvext + vH + vxc, which corresponds to a physical systemwhose external potential vext is uniquely determined thanks tothe Hohenberg-Kohn theorem of DFT. For our purpose, it isnot necessary to know what this external potential looks like.The Kohn-Sham Bloch functions can then be determined inan elementary fashion.30
We also carry out BSE calculations in our model sys-tem to check the performance of TDDFT (see Appendix Afor technical details). BSE calculations are typically basedon ground-state quasiparticle states obtained from the GW
method.4 This is because the single-particle gap in GW is usu-ally closer to experiment than the approximate Kohn-Shamgap. However, in our case this distinction is not important be-cause we use the given KP band structure as input for bothBSE and TDDFT. Overall, using the KP model as the groundstate for both BSE and TDDFT is equivalent to ignoring thequasiparticle part of the xc kernel, so that only the excitonicpart of the xc kernel remains.12
Let us also mention that, even though we mainly focus onWannier excitons in this paper, TDDFT (in general, as wellsas within our particular model approach) is also applicable tothe more tightly bound Frenkel-type excitons.
In summary, our minimal TDDFT model for excitonsconsists of the following two ingredients:
(1) A two-band model for the vertical transitions betweenthe highest valence band and the lowest conductionband, see Eq. (21);
(2) the band structure from a 1D KP model.
Of course, the model is not complete without a choice forthe xc kernel fXC. This will be discussed below.
III. RESULTS FROM THE MINIMAL MODEL
A. Bound excitons from the BSE and from TDDFT
The exact xc kernel fXC(r, r′, ω) is unknown and must beapproximated. We restrict ourselves to adiabatic kernels thathave no frequency dependence. The adiabatic local-densityapproximation (ALDA), as well as all semilocal, gradient-corrected xc kernels, are known to be unable to describe ex-citonic effects.4 The exact xc kernel has a long-range decayof 1/|r − r′|, which is absent in all (semi)local xc kernels de-rived from the uniform electron gas. This long-range part isthought to be essential for excitons.4, 7, 18
The long-range behavior of fXC depends on the dimen-sionality, and in our 1D model system we define the follow-ing soft-Coulomb xc kernel as an analog to the long-range xckernel in 3D
f SCXC(x, x ′, ω) = − ASC√
(x − x ′)2 + α. (23)
We also consider an extremely short-range contact xc kernel
f contXC (x, x ′, ω) = −Acontδ(x − x ′). (24)
These model xc kernels depend on the constants ASC andAcont, which we will treat as fitting parameters in the follow-ing. The idea is to tune the parameters in the model xc kernelsso that bound excitons are produced, and to align the lowestbound exciton in the TDDFT spectrum with the lowest boundexciton in the BSE spectrum. A brief discussion of 1D ALDA(Ref. 46) will also be given.
Results for the imaginary part of the dielectric functionare presented in Figs. 4 and 5. We find that both the long-range f SC
XC and the short-range f contXC produce bound excitons,
and thus, strictly speaking, the long-range behavior of the xckernel is not really required for excitonic effects, as previouslyreported for 3D by Sottile et al.16 The BSE results in Figs. 4and 5 show several identifiable bound excitons, in agreement
014513-6 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
0
2
4
6
8
0 0.5 1 1.5 2
Im(ε
M)
ω
BSETDDFT(contact)
TDDFT(soft-Coulomb)
FIG. 4. Imaginary part of the dielectric function (the optical spectrum),Im(εM), calculated with BSE and TDDFT. Parameters of the KP model:a = 2.6, b = 0.4, V0 = 8. For BSE, A = 0.25 in Eq. (2). For TDDFT with thecontact kernel, Eq. (24), Acont = 2.32. For TDDFT with the soft-Coulombkernel, Eq. (23), ASC = 0.898. The BSE produces several bound excitons,but TDDFT only one.
with the Rydberg series predicted by the Wannier model; thenumber of visible bound excitons somewhat depends on thenumerical resolution in momentum space.
For the KP model parameters of Fig. 4, we find that theadiabatic TDDFT can only bind a single excitonic state. Forother KP parameters (specifically those in which the lowestconduction band is above the barrier), TDDFT produces twoexcitons, see Fig. 5, which agree well with the lowest twoexcitons in BSE. There are additional, higher lying bound ex-citons in BSE which are very faint and difficult to resolve nu-merically. For all the KP systems we tested, we never foundmore than two bound excitons with TDDFT. This indicatesthe limitations of the adiabatic xc kernels used here. It hasbeen reported that only one bound exciton is observed for realsolids with adiabatic TDDFT,16 in contrast with our observa-
0
3
6
9
0 0.15 0.3
Im[ε
M]
ω
BSETDDFT(contact)
TDDFT(soft-Coulomb)
FIG. 5. Same as Fig. 4, for KP model parameters a = 3, b = 3, V0 = 1.For BSE, A = 0.14 in Eq. (2). For TDDFT with the contact kernel,Eq. (24), Acont = 3.77. For TDDFT with the soft-Coulomb kernel, Eq. (23),ASC = 0.955. TDDFT produces two bound excitons. Higher lying bound ex-citons exist within BSE but are numerically hard to resolve.
0
0.02
0.04
0.06
-3 -2 -1 0 1 2 3
|ρkcv
|2
k
1st excitonic excitation BSETDDFT
0
0.015
0.03
-3 -2 -1 0 1 2 3
|ρkcv
|2
k
2nd excitonic excitation BSETDDFT
(a)
(b)
FIG. 6. Eigenvectors |ρcvk |2 of the first two excitonic transitions for KP pa-
rameters a = 0.5, b = 0.5, V0 = 20. For BSE, A = 0.25 in Eq. (2). For TDDFT(soft-Coulomb), ASC = 2.39 in Eq. (23).
tion of up to two bound excitons in our model system. Anexplanation is given in Sec. III C.
As mentioned in Sec. II C, the Wannier model in itsreal-space representation does not clearly demonstrate thatexcitons are collective excitations. Since the Wannier modelin its original sense assumes a single electron-hole pair pic-ture, one cannot immediately see that excitonic excitations arecomposed of a coherent superposition of many single-particleexcitations. In our minimal model, we solve the eigenvalueequation (21), and the eigenvectors ρcv
k (which depends onω parametrically) describe how the transitions between non-interacting orbitals form the transitions in the interactingsystem.47 |ρcv
k |2 is the percentage of a noninteracting transi-tion in the transition of the interacting system. Two typicalcases are plotted in Figs. 6 and 7.
Figure 6 clearly shows that excitons are collective exci-tations which are formed by mixing a wide distribution ofsingle-particle transitions. As expected, the lowest excitoneigenfunction is nodeless and the second excitonic eigenfunc-tion has a single node. With purely parabolic bands, the resultsfrom the Wannier model would be recovered, as we will showbelow. In contrast, the transitions in the continuum shown inFig. 7 have a strong single-particle character (the two peaksarise from the ±k degeneracy in the KP model).
Equation (21) is equivalent to the SPA for finite sys-tems, which ignores the coupling between excitations andde-excitations (TDA). We also investigated what happenswhen we do not make the TDA, i.e., when we work instead
014513-7 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
0
0.1
0.2
0.3
-3 -2 -1 0 1 2 3
|ρkcv
|2
k
BSETDDFT
FIG. 7. Eigenvector |ρcvk |2 of a nonexcitonic excitation in the continuum part
of the spectrum. The model parameters are the same as those in Fig. 6.
of Eq. (21) with the full equation for the two-band model,Eq. (19). As long as we describe relatively weakly bound ex-citons that are not too far below the bandgap, we find that thedifference between the two methods is very minor.
However, we find that Eq. (19) can lead to TDDFT exci-tonic binding energies that are purely imaginary when the in-teraction strengths ASC or Acont are too high. In our minimalmodel, such instabilities arise when the interaction strengthA in Eqs. (23) and (24) increases so that the excitonic bind-ing energy becomes greater than the bandgap. This situa-tion is comparable to the well-known triplet instability inTDDFT, for which the TDA generally leads to an overall bet-ter behavior;35 for excitons binding energies in our minimalmodel, we draw similar conclusions. The triplet instabilityfor finite systems can be easily seen with Eqs. (9) and (10).F
(ij )(ij )HXC in these equations is always negative, and it is possible
for the SMA to obtain a purely imaginary solution. This neverhappens for the SPA, which is equivalent to the SMA com-bined with the TDA. Our minimal model is an extension ofthe SMA to periodic systems, and similar instabilities henceexist for excitons. The success of the TDA is known in BSEas well.48
B. Analysis of the coupling matrix
The BSE scheme is commonly implemented within anadiabatic scheme (see Appendix A); it produces a series ofbound excitons. Since we assumed that the KP model is theKohn-Sham ground state in TDDFT and the GW quasiparti-cle ground state in BSE, the difference between TDDFT andBSE becomes easily comparable, since the central equation tobe solved has the same form, Eq. (21). The F(ij)(mn) couplingmatrices for TDDFT and BSE are
where fH is the Hartree kernel (the 1D soft-Coulomb inter-action), and W is the screened interaction. Aside from thechange from fXC to W , the most prominent difference be-tween BSE and TDDFT is the order of the indices for W
-3 -2 -1 0 1 2 3
k
-3
-2
-1
0
1
2
3
k’
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
FIG. 8. Contour plot of the coupling matrix |F (vc)(vc)BSE |. The model parame-
ters are the same as those in Fig. 6.
in Eq. (26). Since the noninteracting ground-state wave func-tions have the Bloch form (11), we can see from Eq. (20) thatF
(ij )(mn)BSE has a strong k − k′ dependence; in Fig. 8 this shows
up as a dominance along the diagonal. By contrast, this k − k′
dependence is clearly absent in F(ij )(mn)TDDFT , as demonstrated in
Fig. 9.〈〈ij |fXC|mn〉〉 and 〈〈im|W |jn〉〉 with only vertical tran-
The xc matrix with only vertical transitions (27) only de-pends on the long-range (q = 0) behavior of its momen-tum space representation fXC(q, G, G′), while the W ma-trix (28) also depends on other q values in its momentumspace representation W (q, G, G′). It is impossible to find an
-3 -2 -1 0 1 2 3
k
-3
-2
-1
0
1
2
3
k’
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
FIG. 9. Contour plot of the coupling matrix |F (vc)(vc)TDDFT |. The model parame-
ters are the same as those in Fig. 6.
014513-8 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
adiabatic fXC that reproduces the BSE coupling matrix asin Fig. 8, since W (q, G, G′) has an extra degree of freedomover fXC(q = 0, G, G′). One can only hope to reproduce aportion of the BSE coupling matrix with adiabatic TDDFT(as pointed out in Ref. 15), or make the xc kernel frequencydependent so that the information from the q-dependencein W (q, G, G′) is mapped into the frequency dependence infXC(q = 0, G, G′, ω).
Considering the nature of the objects involved in thismapping, a highly nontrivial frequency dependence in fXC isrequired to reproduce a series of bound excitons. For exam-ple, one can easily construct an fXC which reproduces a givenseries of bound excitons by using a different contact kernel inthe region ωi − η−
i ≤ ω ≤ ωi + η+i surrounding each exciton
at frequency ωi:
fXC(x, x ′, ω)
= Aiδ(x − x ′)θ [ω − (ωi − η−i )]θ [(ωi + η+
i ) − ω]. (29)
Here, the Ai’s are parameters which are adjusted so that aTDDFT calculation with the frequency-independent kernelf i
XC(x, x ′) = Aiδ(x − x ′) would produce ωi as the lowest ex-citonic binding energy. Such an fXC is of course completelyad hoc, but the fact that the excitonic series can be repro-duced in this way demonstrates that the inclusion of the fre-quency dependence would greatly improve the flexibility ofthe TDDFT scheme.
On the other hand, within the adiabatic approximation thecharacteristics of the F
(ij )(mn)HXC coupling matrix are important
for excitonic effects. To emphasize this, we now show that invery special cases the number of discrete excitonic eigenval-ues can be derived. Consider a real matrix �(0) + F, where �0
is a diagonal real matrix with �0k,k = �
(0)−k,−k = ω
(0)k , and F
has the symmetry Fk,k′ = F−k′,−k = Fk′,k = F−k,k′ = Fk,−k′ .Within second order perturbation theory, there is at most onediscrete eigenvalue of �(0) + F in the limit where k, k′ be-come continuous.49 Though this case does not correspond tothe matrices that would occur in real calculations, it indicatesthe close relationship between the properties of the couplingmatrix and excitonic effects.
It is also possible to derive properties of the discreteeigenvalues if k and k′ are completely decoupled in the xc ker-nel. Owing to the symmetry F
(vc)(vc)k,k′ = F
(vc)(vc)−k′,−k = F
(vc)(vc)∗k′,k
implied by Eq. (20), such separable kernels can only have theform
Fk,k′ = ±A(k)A∗(k′). (30)
For an excitation below the bandgap with frequency ω, wecan show49 that it must satisfy
−∑
k
|A(k)|2ω − ωcv
k
= 1, (31)
where the sum is carried out over the first Brillouin zone(FBZ). Equation (31) shows that Eq. (30) must have the neg-ative sign in order to have bound excitons. The left-hand sideof Eq. (31) is monotonically increasing with ω, so for sepa-rable kernels of the form of Eq. (30), there is only one boundexcitonic solution.
As shown in Fig. 9, TDDFT coupling matrices lack thestrong dependence of k − k′ as in BSE coupling matrices.Expanding the TDDFT coupling matrices into a power seriesof separable matrices and truncating at the first order wouldbe a reasonable approximation, explaining why TDDFT(in adiabatic approximation) produces fewer bound excitons(if any at all) than many-body methods such as BSE.
C. Dimensionality considerations
The contact xc kernel and the soft-Coulomb xc kernel inEqs. (23) and (24) have the following simple form in momen-tum space:
f SCXC(q,G,G′) = −2ASCK0(
√αSC|q + G|)δG,G′ ,
f contXC (q,G,G′) = −AcontδG,G′ , (32)
where q ∈ FBZ, G and G′ are reciprocal lattice vectors, andK0 is a modified Bessel function of the second kind.50 It iscustomary to refer to the matrix elements where G = G′ = 0as “head,” G = 0 or G′ = 0 as “wings,” and G �= 0, G′ �= 0 as“body.”
The 3D Coulomb potential has the form 4π /q2 in momen-tum space. However, in 1D systems there is no real Coulombinteraction which behaves as q−2 for q → 0, and one has touse the soft-Coulomb interaction instead. Though there aremany flavors of the soft-Coulomb interaction, they all havethe same log q behavior for q → 0. However, the linear-response function χ always behaves as q−2 for q → 0 anddoes not depend on the dimensionality. This renders quan-tities like the macroscopic dielectric function (12) ill-definedfor strictly 1D systems. Furthermore, the bootstrap xc kernel21
and other xc kernels that depend on the cancellation of the 3DCoulomb q−2 singularity will not work as designed in strictly1D and 2D systems. Therefore, Im(εM) shown in Figs. 4 and5 are calculated at a small but finite q.
The coupling matrix F(vc)(vc)HXC can be written in momen-
tum space as
F(vc)(vc)HXC,k,k′ = 1
�
∑G,G′
[vG(q = 0)δG,G′ + fXC(q = 0,G,G′)]
×〈c, k|eiGx |v, k〉〈v, k′|e−iG′x |c, k′〉. (33)
For any xc kernel that behaves as q2 for q → 0, one canfurther simplify the calculation by ignoring the so-called lo-cal field effects,51 i.e., instead of summing over G and G′ inEq. (33), only the head is considered. This approximation isvalid except for very strongly bound excitons.16 In 3D sys-tems, a prominent example is the long-range kernel −α/q2,which is obtained as an effective xc kernel with only headmatrix elements from inverting the BSE of contact excitons.16
On the other hand, any xc kernel that diverges moreslowly than q−2 for q → 0 changes the spectrum only throughthe local field effects, i.e., all G and G′ must be summed inEq. (33). In other words, effective xc kernels with only thehead are not feasible in strictly 1D systems due to the asymp-totic behavior of the soft-Coulomb potential discussed above.
014513-9 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
For 1D systems with G = 0, we have
⟨j, kj
∣∣ei(q+G)x∣∣ i, ki
⟩ q→0∼ O(q1),
vG=0(q)q→0∼ O(log q), (34)
and fXC’s with 1D long-range behavior such as the soft-Coulomb kernel also behave as O(log q). ConsideringEq. (33), these asymptotic properties imply that the head andwing contributions to F(ij)(mn) always vanish in strictly 1D sys-tems for physically meaningful xc kernels. Due to these di-mensionality restrictions, the xc kernel changes the strictly1D system only through the local field effects.
In 3D the head contribution to the coupling matrix FHXC
is orders of magnitude bigger than the local field effects,which is the reason that long-range kernels (with nonzerohead) produce excitonic effects much easier than short-rangexc kernels (with vanishing head) such as ALDA. In ourstrictly 1D model system, the head contribution is zero evenfor the BSE due to the soft-Coulomb interaction, and thusthe long-range kernel does not necessarily outperform short-range kernels such as the contact kernel.
Our 1D model shows that short-range xc kernels can pro-duce bound excitons if they are strong enough, and this isalso valid for 3D solids (which is nicely discussed by Sottileet al.16), in which case the body of the short-range xc kernelhas to be comparable in magnitude with the head of long-range xc kernels. It has to be stressed that for an adiabatic xckernel with its strength tuned to produce bound excitons, thereis no guarantee that the continuum part of the spectrum wouldremain reasonable. However, finding adiabatic short-range xckernels that are only devoted to calculations of bound excitonsis possible.
To give the reader an idea how strong the commonlyused short-range xc kernel should be to produce bound exci-tons, we implement the 1D ALDA kernel46 (see Appendix B),with the only tunable parameter being the interaction strengthA of the soft-Coulomb potential Eq. (2). The Kronig-Penneymodel for BSE has the following parameters: a = 0.5,b = 0.5, V0 = 20, A = 0.45, and α = 1. The ALDA by defini-tion has to have the same interaction strength as in BSE (i.e.,A = 0.45), and the resulting spectrum does not have any peaksbelow the bandgap. We then tune the interaction strength ofthe ALDA kernel so that the first bound exciton is alignedwith that of BSE, and we find that this happens for A = 3.88,which is nearly an order of magnitude stronger. For 3D sys-tems due to the previously stated reasons, short-range xc ker-nels have to be even stronger to produce bound excitons.
These peculiarities only occur when one considersstrictly 1D and 2D systems. In a more realistic picture, one en-counters quasi-2D systems52 and quasi-1D systems (such asquantum wires with finite radius or nanotubes53, 54), in whichthe movement of electrons is confined in certain directionssuch that the transverse motion can be averaged in compari-son with the longitudinal motion. Though these systems showlow-dimensional characteristics in various properties due toconfinement, in the limit of q → 0 they eventually differ fromstrictly low-dimensional systems.
In Ref. 16 it was reported that only one bound excitonis obtained using the long-range xc kernel with only the headcontribution. By contrast, we observe two bound excitons inour model system. Considering Eq. (27) with only the head,we see immediately that it has the form of a separable kernelas in Eq. (30); on the other hand, Fk,k′ is never separable in ourmodel system since the sum over G and G′ must be taken, dueto the properties of our strictly 1D geometry. In BSE, evenif only the head is considered, Eq. (28) is never separable.This explains the apparent contradiction, and also suggeststhat while using only the head of the coupling matrix can yieldsatisfying optical spectra, the wings and body contributionshave to be included to obtain more than one bound exciton inadiabatic TDDFT.
IV. THE WANNIER MODEL IN TDDFT
Our minimal model and the Wannier exciton picturecan be connected by considering the Fourier transform ofEq. (21). We define an effective two-body potential Veh viathe Fourier transform of F
(vc)(vc)HXC,k,k′
Veh(R, R′) = a + b
2π
∑k,k′∈FBZ
e−ik·RF(vc)(vc)HXC,k,k′e
ik′ ·R′, (35)
where R is a direct lattice vector. The Fourier transform of thedensity matrix is
ρ(R, ω) ≡ ρcv(R, ω) =∑
k
e−ik·Rρcvk (ω). (36)
Since Wannier excitons extend over many lattice constants,we approximate R as a continuous variable r. Assuming theeffective mass approximation, Eq. (21) becomes
− ∇2
2mr
ρ(r, ω) +∫
d3r ′ Veh(r, r′, ω)ρ(r′, ω) = Eρ(r, ω),
(37)
where E is the excitonic binding energy, and the integrationis carried out over all space. We call Eq. (37) the TDDFTWannier equation, since it has the same form as Eq. (1). Withproper choice of the approximated xc kernel, the nonlocal ef-fective electron-hole interaction potential Veh supports boundexcitonic states.
Due to the similarity between BSE and TDDFT in thetransition space representation, Eq. (21) can also be applied tothe BSE results, and an effective electron-hole interaction canbe derived. Figure 10 shows the effective interaction potentialVeh for TDDFT and BSE.
The TDDFT Wannier equation provides an intuitive wayof describing the effective nonlocal electron-hole interaction,and of explaining why adiabatic TDDFT usually has fewerexcitons than BSE and the Wannier model. However, in mostcases the TDDFT Wannier equation is not suitable for quanti-tative use due to the approximations involved. The approx-imation where we take the lattice vector R as a continu-ous variable assumes that the exciton radius is much largerthan the lattice constant; this works fine in most cases wetested. But the effective mass approximation where ωq is ap-proximated by q2/2mr is only good for transitions near the
014513-10 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
-6 -4 -2 0 2 4 6
x
-6
-4
-2
0
2
4
6
x’
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
-6 -4 -2 0 2 4 6
x
-6
-4
-2
0
2
4
6
x’
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
(a)
(b)
FIG. 10. Contour plots of Veh(x, x′) for (a) BSE and (b) TDDFT with thesoft-Coulomb kernel. The KP system is the same as in Fig. 6.
bandgap, thus requiring these transitions of the noninteract-ing system to dominate the exciton, which is equivalent to theexciton extending over many lattice constants.25 One obtainsthe −∇2/2mr term in Eq. (37) from q2/2mr in the limit wherethe lattice constant a + b → 0, and this approximation is notvalid for most systems.
Although Veh is a nonlocal potential, in most cases wefind that the Veh’s for both TDDFT and BSE are dominated bythe diagonal part, so the exciton problem is in analogy to one-body systems. Figure 11 shows the diagonal part of Veh, whichcan be taken as the effective one-body potential. The Wanniermodel in 1D has the soft-Coulomb potential, which supportsan infinite number of bound excitons29 (the soft-Coulomb in-teraction is fitted so that the binding energy of the first excitonmatches that of the BSE). We find in general that the diagonalparts of Veh for both BSE and TDDFT are much more shallowthan the soft-Coulomb potential, and the TDDFT one is morenarrow than the BSE one. Thus, BSE and TDDFT are not ableto produce a complete excitonic Rydberg series, and TDDFTin general produces fewer bound excitons than the BSE.
One may wonder why the BSE Veh in Fig. 10 is moreshallow than the soft-Coulomb potential; after all, BSE cor-rectly describes the electron-hole interaction within its limits.First, the Wannier model is only an approximation to the BSE,and as a consequence only the diagonal part of Veh is finite. Itis not surprising that the diagonal part of Veh has to be deeperso as to be effectively equivalent to the nonlocal Veh of BSE.
-4
-2
0
-6 -4 -2 0 2 4 6
x’
x
BSETDDFT
Wannier model
FIG. 11. Diagonal part Veh(x, x′) of those shown in Fig. 10.
Second, the BSE is practically limited by the numerical pro-cedure, and we observe that using more k-points in the BSEcalculation in general results in more bound excitons close tothe bandgap.
The TDDFT Wannier equation is not suitable for quanti-tative use for most of our model systems, despite the successof the Wannier model in describing real semiconductors.55, 56
Since the approximations involved in Eq. (37) require that theexciton radius is large compared to the lattice constant, thissuggests that this discrepancy is due to the special nature of1D systems: namely, for similar effective masses the excitonradius in 1D is much smaller than in 3D and 2D.29
V. CONCLUSION
The purpose of this paper is to construct a transpar-ent and accessible minimalist model system that is suitableto describe excitonic effects within the context of TDDFT.TDDFT, in principle, is an exact theory for all types of elec-tronic excitation process, including excitonic effects. In prac-tice, however, the linear-response TDDFT with available adi-abatic xc kernels does not describe the physics of excitonsbetter than BSE. In particular, bound excitons are notoriouslydifficult to obtain within TDDFT. Our model provides a basisfor the development of new techniques for bound excitons inTDDFT, so that one can focus on excitonic effects without thecomplication of real 3D solids.
With our minimal model, we show that adiabatic TDDFTis capable of producing bound excitons through the local fieldeffect even when the xc kernel is local in space, provided thestrength of the kernel is strong enough. This statement is stilltrue in 3D; however, due to the non-vanishing head contri-bution of the exact fXC, we expect that the deviation of theeffective interaction strength of a short-range fXC from thereal, long-range fXC becomes larger than the in our strictly1D model. In this sense the long-range kernel, though veryfavorable, is not a necessary condition for excitonic effects.With only the head of the xc coupling matrix, only one boundexciton can be obtained with an adiabatic fXC; this is becausethe xc coupling matrix then becomes separable. Including thewings and body contributions can lead to additional boundexcitons.
014513-11 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
We show the connection between TDDFT and the Wan-nier model for excitons by deriving the TDDFT Wannierequation, which describes a real-space system featuring anonlocal effective electron-hole interaction. Such a connec-tion intuitively demonstrates how adiabatic TDDFT generallyproduces fewer bound excitons than BSE, and does not havea complete Rydberg series. The eigenvectors of the excitonicexcitations in the minimal model clearly show their collectivenature, which is not obvious from the Wannier model alone.Excitonic instabilities may show up in TDDFT with approx-imate xc kernels, and this suggests that the TDA tends to bemore reliable for excitons than the formally exact method.
The frequency dependence of the exact xc kernel,fXC(r, r′, ω), is usually ignored. Despite the fact that adia-batic xc kernels have met with considerable recent successin producing optical spectra of insulators and semiconduc-tors (see the discussion in the Introduction), they are inca-pable of producing an excitonic Rydberg series. Our modelsystem gives an explanation for why this is the case. This fail-ure of the adiabatic approximation for fXC is quite differentfrom that which is responsible for the inability of adiabaticTDDFT to produce double excitations in finite systems or cer-tain classes of charge-transfer excitations.57, 58 This calls forcontinuing efforts in the search for nonadiabatic xc kernelsfor excitons.
ACKNOWLEDGMENTS
This work is supported by the National Science Founda-tion (NSF) (Grant No. DMR-1005651).
APPENDIX A: THE BETHE-SALPETER EQUATION
Electrons and holes near the Fermi surface are well de-scribed in the quasiparticle picture. The quasiparticle Green’sfunction G is related to that of the noninteracting system, G0,through the use of the self-energy �
G(12) = G0(12) +∫
d(34) G0(13)�(34)G(42), (A1)
where the arguments denote sets of space and time variables.A widely used approximation for the self-energy is the GWapproximation1, 2
�(12) = iG(12)W (12), (A2)
where W is the screened interaction,
W (r, r′, ω) =∫
d3r ′′ ε−1(r, r′′, ω)v(r′′, r′), (A3)
v is the bare Coulomb interaction, and the inverse dielectricfunction ε−1 within the random-phase approximation is ob-tained as
ε−1(r, r′, ω) = δ(r − r′) +∫
d3r ′′ v(r, r′′)χ0(r′′, r′, ω).
(A4)
The linear-response function χ0 can be calculated by itsLehmann representation
χ0(r, r′, ω) =∑ij
ψ∗i (r)ψj (r)ψi(r′)ψ∗
j (r′)
ω − (Ei − Ej ) + iη(fj − fi), (A5)
where ψ i are quasiparticle states, Ei are quasiparticle ener-gies, and fi are occupation numbers. In practice, evaluating χ0
through Eq. (A5) can be quite time consuming, and χ0 is of-ten calculated with the plasmon-pole model.59 However, ourminimal model is simple enough to allow us to use Eq. (A5)directly.
The optical spectrum obtained from a GW calculationdoes not include important dynamical many-body effects,such as the effect of the electron-hole (excitonic) interaction.The two-particle Green’s function includes these effects. TheBSE (Refs. 3 and 4) describes the relation between the four-point polarization function L(1234) of an interacting systemand the corresponding object of the quasiparticle system. Forthe calculation of the optical spectrum, the full L function isnot necessary, and one can work with L̄ defined by
L̄(1234) = L0(1234) +∫
d(5678) L0(1256)K̄(5678)L̄(7834),
(A6)
in which L0 is
L0(1234) = iG(13)G(42), (A7)
and assuming the GW approximation, the kernel K̄ is
Here, v̄ denotes the Coulomb interaction with the long-rangepart removed.4 In practice the BSE is often solved in the tran-sition space, which is spanned by single-particle excitations.A four-point function such as L̄ then becomes
L̄(ij )(mn)(ω)
=∫
dx1 . . . x4 L̄(r1r2r3r4; ω)φi(r1)φ∗j (r2)φ∗
m(r3)φn(r4),
(A9)
where the φ’s can be any complete basis set. Equation (A6) inthe transition space becomes
H (ij )(mn)exc (ω) = (Ej − Ei − ω)δimδjn + (fi − fj )K̄ (ij )(mn)(ω).
(A11)
We make the adiabatic approximation for H(ij )(mn)exc and arrive
at the following eigenvalue problem:∑mn
H (ij )(mn)exc A
(mn)λ = ωexc
λ A(ij )λ , (A12)
and L̄(ij )(mn) can be expressed in terms of these eigenvectorsby
L̄(ij )(mn)(ω) =∑
λ
A(ij )λ A
(mn)∗λ
ωexcλ − ω
. (A13)
014513-12 Yang, Li, and Ullrich J. Chem. Phys. 137, 014513 (2012)
Only the transitions between the valence and conductionbands contribute. Within our two-band model, the excitonicHamiltonian has the following block matrix form:
Hexc =(
Ec − Ev + K̄ (vc)(vc) K̄ (vc)(cv)
−K̄ (vc)(cv)∗ Ev − Ec − K̄ (vc)(vc)∗
).
(A14)
Ignoring the off-diagonal part in Eq. (A14) is equivalent to theTDA.
As shown in Sec. III, it is possible that instabilities showup in the full BSE results when the underlying ground-statecalculation is not exact. Such instabilities in the minimalmodel are an artifact originating from the assumption that thesolution of the KP model constitutes the ground state of themany-body system. However, this is not a matter of great con-cern in practice.
In principle, the transition space spans all possible com-binations of valence and conduction orbitals, including non-vertical transitions connecting different Bloch wavevectors.The kernel K̄ = v̄ − W of the BSE in momentum space,Eq. (A8), has the following ingredients:
Only the excitations with the same momentum transfer q arecoupled due to the δ functions in Eqs. (A15) and (A16), so weonly need to include vertical transitions in the calculations foroptical properties.
APPENDIX B: 1D ALDA KERNEL
The 1D LDA exchange energy density per particle canbe derived similarly as in 3D. The first-order reduced densitymatrix ρ1(x, x′) of a 1D spin-unpolarized uniform electronicgas is
ρ1(u) = 1
2π
∫ kF
−kF
dk exp(−iku) = sin(kF u)
πu, (B1)
where u = |x − x′|, kF = πn/2 is the 1D Fermi energy, and n isthe density of the uniform electronic gas. The LDA exchangehole is then
nx(u) = −2 |ρ1(u)|2n
= − sin2(kF u)
πkF u2, (B2)
where n is the density of the uniform electronic gas. Followingthe derivation of Burke and Perdew,60 the exchange energy
density per particle is
εLDAx (n = 2kF /π ) = 1
2
∫ ∞
−∞du
nx(u)
V1D(u)
= −A
2
∫ ∞
−∞du
sin2(kF u)
πkF u2√
u2 + α. (B3)
The ALDA xc kernel is defined as the functional derivative
f ALDAXC (x, x ′) = δvLDA
XC (x)
δn(x ′), (B4)
where the LDA xc potential is
vLDAXC (x) = δEXC
δn(x)= δ
∫ ∞−∞ dx n(x)εXC[n(x)]
δn(x). (B5)
The exchange kernel is simply obtained as
f ALDAx (x, x ′) = −AK0[n(x)π
√α]δ(x − x ′), (B6)
where K0 is a modified Bessel function of the second kind.50
In a recent work, Helbig et al.46 provided an interpolationformula of the LDA correlation energy density per particleof 1D systems with soft-Coulomb interaction with α = 1 inEq. (2)
εLDAc (rs, ζ ) = −1
2
rs+Er2s
A + Brs+Cr2s + Dr3
s
ln(1 + αrs + βrms ),
(B7)
where A, B, C, D, E, α, β are parameters given inRef. 46 (notice that A and α here have different meaningsfrom previous uses in the paper, see Eq. (2)). The correla-tion kernel is derived by taking the functional derivative as inEqs. (B4) and (B5).
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