Comparison of three quantum chemical ab initio methods for …bezugly/JP_CS_2008.pdf ·...

Comparison of three quantum chemical ab initio methods for

band structure calculations: the hydrogen fluoride chain

Viktor Bezugly1,3

, Martin Albrecht2 and Uwe Birkenheuer

1,4,*

1 Max-Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 2 Theoretical Chemistry, University of Siegen, 57068 Siegen, Germany 3 present: Max-Planck Inst. for Chemical Physics of Solids, 01187 Dresden, Germany 4 present: Inst. of Safety Research, FZ Dresden-Rossendorf, 01328 Dresden, Germany

* corresponding author ([email protected])

Abstract. Three different many-body wave-function-based ab initio methods for the calcu-lation of correlated (or quasi-particle) band structures of periodic systems are presented: the lo-cal Hamiltonian approach, the incremental self-energy method, and the crystal orbital variant of the algebraic diagrammatic construction. All three methods explicitly exploit the local na-ture of electron correlation, and by consequently switching to representations in localized Wannier orbitals O(N) scaling could be achieved in all three cases. These methods were ap-plied to single (HF)2 zigzag chains as found in solid hydrogen fluoride using the same geome-tries and basis sets. Essentially identical quasi-particle band structures were obtained, corrobo-rating the appropriateness of the different concepts pursued in each of the presented quantum chemical correlation methods for band structures of infinite systems.

1. Introduction Sound electronic structure calculations for excited-state properties of extended materials are one of the current challenges to modern many-body theory. In the last decades several schemes have been devel-oped within the framework of density functional theory such as time-dependent density functional theory [1], the GW approximation [2] or the dynamic mean field theory [3]. Significant advances in the description of excited-state wave functions have also been achieved for Quantum Monte-Carlo approaches [4]. And even quantum chemical many-body wave function based methods for ground-state properties of periodic or extended systems became feasible by now, by systematically exploiting the predominantly local character of electron correlation [5-10]. Yet, quantum chemical correlation methods for excited states of extended systems are still rare. There exist some early attempts based on Møller-Plesset perturbation theory (such as the outer valence Green’s function method) [11, 12] but it is only recently that a couple of quantum chemical methods have been developed for excited-state properties which are similar in spirit to the modern local correlation methods for ground states.

The aim of the present study is to compare the performance of three of these modern approaches, the local Hamiltonian approach (LHA) pioneered by Fulde and coworkers [13-17], the incremental self-energy method (ISEM) of Igarashi [18-21], and the crystal-orbital variant of the algebraic dia-grammatic construction (CO-ADC) [22-23] for a given infinite sample system, a single [(HF)2]∞ zig-zag chain as found in solid hydrogen fluoride. Since the electronic properties of this model system have already been discussed in some detail before (e.g. Ref. [24-26] and references therein), we will

Ab initio Simulation of Crystalline Solids: History and Prospects IOP PublishingJournal of Physics: Conference Series 117 (2008) 012006 doi:10.1088/1742-6596/117/1/012006

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focus here on the comparison of the three quantum chemical band structure methods and their ability to deliver reliable correlated band energies of periodic systems, i.e., the quasi-particle band structure.

2. Theory The quantities of interest in this investigation are the correlated valence and conduction band energies of periodic (closed-shell) systems which are defined as

10

val −−= NNEE kk ννε and NN

EE 01cond −= +

kk ννε with 111 ˆ mmm NNNΨHΨE kkk σνσνν = (1)

where NE0 is the correlated ground-state energy of the neutral crystal and 1mN

E kν are the correlated total

energies of the crystal after removal or uptake of an electron, and which should not be confused with

simple one-particle bands. Here, 1mNΨ kσν refer to those eigenstates of the ( 1mN )-electron system with

spin σm and crystal momentum km which can safely be associated with a valence or conduction band index ν , respectively. The band energies can then either be obtained from two separate correlation calculations, one for the neutral system and one for the ( 1mN )-electron system. This is how one pro-ceeds in the local Hamiltonian approach. Alternatively, they can also be extracted directly from the poles of the Fourier transformed one-particle Green’s function

NNΨtctcΨttG 00 ][ )'(ˆ)(ˆTi)',,( +−= kkk σµσννµ (2)

of the periodic system which possesses the Lehmann representation

∑∑ ++

+++

+

−−+

+−−+

−−−=

mNN

m

NNm

Nm

N

nN-n

N

NNn

Nn

N

EE

ΨcΨΨcΨ

EE

ΨcΨΨcΨ

G0i)(

ˆˆ

0i)(

ˆˆ),(

01

011

0

10

011

0

k

kkkk

k

kkkkk

ωωω

σµσσσνσνσσσµ

νµ . (3)

Here n and m run over all eigenstates of the ( 1mN )-electron system with given total spin and crystal

momentum. T in equation (2) is Wick's time order symbol, NΨ 0 the correlated ground state of the crys-

tal, and kσνc are the annihilation operators of Bloch orbitals with spin σ , momentum k and band in-

dex ν . This approach is adopted in the incremental self-energy method and the CO-ADC formalism. The basic idea of all three methods presented here is the use of localized Hartree-Fock orbitals.

This concept is very general and can be applied equally well to molecular and to periodic systems. We focus on the latter case, in which a multi-band (or generalized) Wannier-transformation

∫ ∑

=

1.BZ VBZ

3di )(e k

n-

Unν νσνσ kkR kR and ∑ ∑

=

n nUn )(e *i kRkR

kRνσσν (4)

(applied separately to the occupied and virtual orbitals of interest) mediates between the canonical Bloch orbitals kσν and the desired local Wannier functions σnR with suitably chosen unitary

band mixing matrices )(kU . In terms of these local orbitals, a model (or reference) space 1±NP and

an excitation space 1±NQ are distinguished. For the ( 1−N )-electron states (the case of the ( 1+N )-

electron states being completely analogous) the model space consists of all local single-hole configu-rations HFˆ ΦcΦ aa = which can be constructed from the Hartree-Fock (HF) ground-state reference

determinant HFΦ , while the excitation space is made up by all single, double and higher-order excita-tions on top of these model configurations:

K,ˆˆˆˆ,ˆˆacbrsabr ΦccccΦcc

+++ . (5)

For the sake of simplicity we adopt composite indices K,,, cba and K,,, tsr which represent either

occupied or virtual local Wannier orbitals σnR . The idea of locality now simply translates into a

configuration selection scheme where only configurations with local orbitals from spatially contiguous regions are considered which automatically leads to the desired )(NO scaling of all three methods.


2

2.1. The local Hamiltonian approach

The local Hamiltonian approach (LHA), first formulated in 1993 [13], is well-established by now [14-16]. It is combined here with our recently developed band disentanglement [27] and cluster-in-solid embedding tools [28]. It evaluates the band energies as differences of correlated state energies from separate correlation calculations. Any quantum mechanical program which is able to deliver such en-ergies for finite ( 1±N )-electron systems can be used for that purpose. The basic idea of the LHA is to

focus on those eigenstates 1±NΨ kσν of the ( 1±N )-electron system which are dominated by the single-

particle configurations from the model spaces. Thus, satellites and other states that are not describable in an independent electron picture are explicitly excluded. An effective Hamiltonian is then defined by projecting the full Hamiltonian H of the system onto the subspace of the selected eigenstates. For the ( 1−N )-electron states, for example, this yields

( )∑ −−− −=k kkkσν σνσνσν

110

1effˆ NNNNΨEEΨH . (6)

After identifying the correlated eigenstates 1−NΨ kσν with their non-correlated Hartree-Fock counter parts

HF1 ˆ ΦcΦ

Nkk σνσν =− it is possible to construct a matrix representation

bN

abaab ΨHEΨΨHΨH ˆˆ0

effeff −== (7)

of the effective Hamiltonian in terms of local multi-determinantal ( 1−N )-electron wave functions

∫ ∑

=≡

1.BZ VBZd*i 3

)(e: knna UΨΨΨ

ν νσνσ kkkR

R (8)

where )(kU is the band mixing matrices from the Wannier transformation (4) which was set up to

mediate between the orbitals kσν and σnR . Once, the local representation (7) of the effective

Hamiltonian is found, the correlated valence band energies can be obtained by either diagonalizing the Bloch orbital representation of the effective Hamiltonain:

∑ +=nm mnmn UHUH µννµ )()( effeff kk with ∑=

R R0kR

keff

,ieff e)( σmσnnm HH , (9)

or directly its purely translational symmetry adapted representation )(eff knmH (since )(kU is unitary). In a totally analogue way the ( 1+N )-electron states for the conduction bands are treated.

Because of the predominantly local nature of electron correlation, only for a very limited number of matrix elements electron correlation has to be considered explicitly. The long-range polarization ef-fects are usually small and can, if necessary, very efficiently be estimated in the continuous-media limit [14,15,17]. However, the most important benefit of the local representation is, that the individual matrix elements can actually be extracted from finite (properly embedded) fragments of the infinite periodic system [13-17,28] by using the molecular equivalents of equations (6), (7) and (8).

2.2. The incremental scheme

Within the LHA the problem of finding the correlated band structures of periodic systems is reduced to a series of quantum chemical correlation calculations on finite fragments. Since such multi-state calculations can still be quite demanding, the incremental scheme, originally designed for ground-state energies [5], is used to reduce the computational effort further [13-17]. In that scheme the correlated quantity of interest is determined by performing a sequence of so-called constrained correlation calcu-lations in which the electrons are only allowed to be excited from a few occupied orbitals, and to ex-trapolate from these data to the case of no restrictions. For the local matrix elements (LME) of the ef-fective Hamiltonian this leads to the following incremental expansion:

K+∆+∆+= ∑∑ ≠<≠ badc abbac ababab dcHcHHH,,

effeff ),()(() with (10)

())()( effeffababab HcHcH −=∆ (11a)


3

[ ] .)()(()),(),( effeffdHcHHdcHdcH ababababab ∆+∆+−=∆ (11b)

Here, effabH is the LME of interest, whereas ( )K,,eff

dcH ab are those from a constrained correlation cal-culation with only the orbitals K,, dc being open for correlation in addition to the orbitals a and b .

While the 1-body increments ( )cH ab∆ describe the effect of adding a single occupied orbital to the

correlation space, the 2-body increments ( )dcH ab ,∆ quantify to which extent the 1-body contributions are non-additive. Higher-order increments can be defined in a similar way. Yet, as a matter of experi-ence [13-17], the incremental expansion (10) converges quite rapidly with increasing order such that it can usually be truncated after the 2-body increments.

2.3. The incremental self-energy method

The second approach to correlated band structures presented here is the incremental self-energy method (ISEM). Like the local Hamiltonian approach its computational efficiency mainly stems from the use of an incremental expansion [19-21]. Yet, in contrast to the LHA the incremental self-energy method is entire formulated in a Green’s function framework.

The key quantity in this approach is the frequency dependent self-energy )(ˆ ωΣ which allows to ex-press the desired Green’s function (3) in the form

( ) ( )[ ] 10 ˆˆˆ −−−= ωωω ΣHG (12)

with 0H being the chosen zero-order Hamiltonian of the system (e.g. the Fock-operator F ). The cor-

related band energies are iteratively retrieved as the zeros of the denominator in equation (12). Again, the Bloch orbital representation (cf. equation (9)) or the translation symmetry adapted representation

( ) ∑=R R0

kRk )(e ,

i ωω σσ mnnm Σ,Σ with σωσωσσ mΣnΣ mn R0R0 )(ˆ)(, = (13)

of the self-energy can be used for that purpose. The self-energy itself is evaluated in its local represen-tation )(, ωσσ mnΣ R0 in order to be able to fully exploit locality. Following Igarashi [18] the self-energy

is first decomposed into a pure retarded and a pure advanced part

)()()( AR ωωω pqpqpq ΣΣΣ += (14)

which is correct up to third order in Møller-Plesset perturbation theory. Here p and q are composite indices (like a, b, c, … and r, s, t ,…) which refer to arbitrary local orbitals. To continue, the excitation

spaces 1±NQ are restricted to single excitations and the full Hamiltonian matrices

βααβN

EHH 0R/A ˆ −±= (15)

are diagonalized, yielding eigenvectors R/A,nSα and eigenvalues R/A

nλ with HFˆˆˆ Φcccasr asr++=≡α

for RH and HFˆˆˆ Φcccrba bar

+=≡α for AH . The self-energies )(R/A ωpqΣ are then given by

[ ] ∑∑ +

<

⋅

<

<<

−+

+−

∑∑=+−=

n n

butntub

asrnrsa

butasr

tubrsapq

StuqbSsrpa

qbtuHsrpaΣ0i

0i)(R

,

*R,

*

,

R,

,,

1

,RR

λωωω (16)

and a totally analogous expression for )(A ωpqΣ [18]. Here, lkrijklrijklij1

121

12−− −= is a short-

hand notation for the usual two-electron integrals showing up in perturbation theory of electronic Hamiltonians. In an earlier work [20] it was pointed out that the ordinary Møller-Plesset second order perturbation theory (MP2) expression

∑<

+++−−

⋅=

asr asr

pq

qarssrpaΣ

,

MP2R

0i)(

εεεωω with pFpp

ˆ=ε (17)

is obtained, when the diagonalization of RH is avoided by using its independent-particle counterpart.


4

Otherwise the rather intricate diagonalization is circumvented by using the incremental scheme from Sec. 2.2. In complete analogy to equation (10) one writes

K+∆+∆+= ∑∑ ≠<≠ 0,0

000 )()()()(

ji

j,ipqi

,ipqpqpq ΣΣΣΣ ωωωω (18)

where )(,,0 ωKj,i

pqΣ are the self-energies one obtains when only excitations inside the unit cells 0, iR ,

jR ,… are considered and )(,0 ωK,i

pqΣ∆ are the corresponding increments (cf. equation (11)).

2.4. The crystal orbital ADC formalism

The third band structure method discussed here is the crystal orbital ADC formalism (CO-ADC). It is our recently developed periodic version [22,23] of the molecular algebraic diagrammatic construction (ADC), which is a well-established method in molecular physics to approximate the Feynman-Dyson perturbation series of the self-energy [29]. In particular, ADC is size-consistent (and size-extensive) which is very important for extended systems [30]. Like the incremental self-energy method it sets out from the Dyson equation. Yet, a completly different route is pursued to set up the self-energy and to find the correlated band energies.

The construction starts with a decomposition of the self-energy )(ˆ ωΣ into a static contribution ∞Σ

and two dynamic contributions (retarded and advanced). In Bloch orbital representation this reads

),(),()(),( ωωω νµνµνµνµ kkkk−+∞ ++= MMΣΣ with 0),( =∞→± ωνµ kM (19)

Next, the following fundamental ansatz is made for the retarded and advanced dynamic self-energies,

[ ] ±−±±±± −−= kk kkk µννµ ωω uCKuM1

)()(),( , (20)

which perfectly reproduces the analytical structure of the diagrammatic Feynman-Dyson perturbation

expansion of )(ˆ ω±M . Here, ±

kνu and )(k±C are arbitrary vectors and Hermitian matrices and )(k±

K

are assumed to be diagonal matrices. The n-th order approximation to CO-ADC is obtained by ex-panding equation (20) in powers of the residual interaction from the Møller-Plesset partitioning (up to n-th order) and comparing the resulting expressions with the Feynman-Dyson expansion. The resulting

scheme is denoted as CO-ADC(n). Up to third order each entry in the coupling amplitudes ±kνu is, at

the very end, characterized by a relative two-particle one-hole (2p1h) configuration of total spin τ : )";''',( σσστ mnnI 0RR≡ with τσσσ =−+ "' , (21)

made from two virtual Wannier orbitals σnR and ''' σnR and one occupied orbital ""σn0 from

the reference (or origin) unit cell in the retarded case, and a similarly constructed one-particle two-hole (1p2h) configuration in the advanced case. For more details see Ref. [22].

The particular structure of the algebraic form (20) allows to avoid the tedious iterative pole search of the Green’s function. The correlated band energies are actually given by the eigenvalues of the so-called band structure matrix

+

+

+

=−−−

+++

−+∞

)()(0)(

0)()()(

)()()()(

)(

TT0

kkk

kkk

kkkk

k

CKU

CKU

UUΣH

B (22)

where )(0 kH is the zero-order Hamiltonian in (diagonal) Bloch orbital representation, and TU

± are

the Hermitian adjoints of the amplitude matrices ),,( 21 K±±± = uuU . The diagonalization in the desired

energy window can be carried out very efficiently by a complex band-(or block-)Lanczos algorithm [31]. Like in the two other methods, the individual entries of the band structure matrix B(k) are evalu-ated in the local Wannier representation before they are transformed back to k-space (cf. equations (9) and (13)), for example


5

∑ ++ =m mmII Uuu )()()( * kkk µµ with ∑ ++ =

R R0kR

k σσ mImI uu ,ie)( . (23)

Only local quantities enter the expressions for +σσ mIu R0 , such that cut-off criteria can easily be applied.

As the Fock matrix is no longer diagonal in the local representation, in addition to the two-electron interaction, the off-diagonal elements of the Fock matrix are expanded diagrammatically as well lead-ing to the CO-ADC(m,n) scheme where m is the highest order of diagrams with off-diagonal terms. In CO-ADC(2,2), the variant applied in this study, the local representation of the )(kB matrix reads

0,)(,

.0,)(,

,ˆ)1(,

,,,,

,,,,

,,0

=−+⋅==

=−+⋅==

⋅−=⋅=

−−−

+++

∞

cdsabrrbacdsabrcdsabrpabr

tubrsaasrtubrsatubrsaprsa

qppqpqpqp

CKprabu

CKparsu

qFpΣH

εεεδ

εεεδ

δεδ

(24)

Hence, the off-diagonal matrix elements removed from 0H enter the band structure matrix again as

first-order contributions to the static self-energy. Actually, equation (24) implies that in CO-ADC(2,2) the dynamic self-energy reduces to the standard MP2 expression (17), as it does in the incremental self-energy method when the diagonal approximation is used (which is the case in the present study).

Figure 1. Sketch of the [(HF)2]∞ chain together with contours (in 0.05 au steps) of the local p-type orbitals of σ symmetry with Rn ≡ Sη where S denotes the molecular unit as indicated in the top line. Not shown are the s-type inner valence orbitals (η = 1) and the p-type orbitals of π symmetry (η = 2).

3. Computational details

The geometrical parameters of the [(HF)2]∞ zigzag chain (see figure 1) are adopted from a previous study on the ground-state properties of (HF)∞ [32]. They are 0.92 Å for the molecular HF bond length, 1.58 Å for the H—F hydrogen bond length, 120° for the H–F–H bond angle, and a0 = 4.33 Å for the size of the unit cell. Dunning's correlation-consistent valence-double-zeta basis with polarization func-tions (cc-pVDZ) [33] are employed throughout, i.e. a [3s,2p,1d] basis on F and a [2s, 1p] basis on H.

For the local Hamiltonian approach the occupied local orbitals are generated using the Wannier-Boys localization procedure [34] implemented in CRYSTAL [35] together with our band-disentangle-ment scheme [27] to separate the π band from the σ bands. Applied to the six outermost valence bands of [(HF)2]∞ (excluding the inner s-type valence states at about –40 eV) three almost perfect p-type orbitals are found on each fluorine atom (see figure 1). The other two methods are based on the WANNIER code [36] which directly optimizes the local occupied Wannier orbitals in real space (on a support of nine unit cells) but re-canonicalizes them within the origin unit cells. The local virtual orbi-tals required in these two methods for the spatial excitation space truncation (see equations (16) and (24)) are constructed using the projected atomic orbital (PAO) method [37]. The redundancy problem inherent to this approach for translational symmetric sets of Wannier orbitals was solved by a suitable pre-selection of the atomic basis functions. Spatial separations up to the 2nd-nearest neighbor unit cell turned out to be enough to converge the band energies to less than 0.01 eV (see also Ref. [23,38]).

4. Results and Discussion

The LHA is based on quantum chemical correlation calculations on finite molecular fragments. The multi-reference configuration interaction (MRCI) scheme with single and double excitations (SD) [39]


6

Figure 2. Decay of the increments )(cH aa∆ with increasing distance between the defining orbital aϕ

and the decorating orbital cϕ . For the nomenclature of the orbital indices a ≡ Sη see figure 1.

as implemented in MOLPRO [40] was chosen for that purpose in conjunction with our CRYSTAL-MOLPRO interface [28] which also provides the proper embedding potential for each fragment, an im-portant feature because of the permanent dipole moments of the HF units along the chain axis. With our extension of Pople’s size-consistency correction for MRCI [41] to cationic hole states [17] the LME increments ),,( KdcH ab∆ decay properly with increasing distance of the “decorating” orbitals

c, d, … (as shown exemplarily in figure 2) rather than saturating at –0.01 or 0.01 eV if no or the stan-dard correction [41] is applied. Third-nearest neighbour HF units are enough to achieve total conver-gence within 0.01 eV. The resulting LMEs are summarized in table 1. Correlation contributions have only been evaluated where likely to significantly exceed 0.01 eV. As expected, the major correlation effect is found for the diagonal terms (leading to homogeneous shifts of the bands), but there is also some noticeable impact on the onsite coupling term of the two local orbitals of σ symmetry.

The final quasi-particle bands are presented in figure 3 (with the CO-ADC data been extracted from Ref. [23]). The degeneracy of the bands at the X point is due to the glide plane symmetry of the zigzag chain. The correlation-induced upward shifts are 1.6 eV for the lower band complex, 1.9 eV for the π bands in the upper band complex and 2.0 eV for the σ bands therein. Some minor changes in the dispersion are also discernible. Yet, the main result is, that all three correlation methods, although quite different in nature, yield essentially the same correlated band structure. All deviations remain below 0.1 eV. Only the upper σ bands predicted by CO-ADC are off somewhat more (~0.2 eV).

All three methods have also been applied successfully to 3D bulk systems: diamond, Si and Ge for the LHA [13-15,28] as well as MgO [42] (using the frozen local hole approximation [43,44]), LiH and LiF for the ISEM [19-21] and LiF for the CO-ADC formalism [22], corroborating the principle power of the underlying truncation techniques. The computational effort for the two Green’s function-based methods is determined by the size of the matrices (15) or (22), respectively, which both have to be di-agonalized. In the ISEM this can be done separately for the retarded and the advanced part. Yet, there

Table 1. The local matrix elements of (HF)∞ The Hartree-Fock values HF eff,,baH are extracted from an

[(HF)2]5 cluster, the correlated ones from an [(HF)2]3 cluster. For the nomenclature of the orbital indices a ≡ Sη see figure 1. Elements with magnitudes below 0.1 eV are not shown.

XXXXXXXXXXXXXXX eff02,02H eff

03,03H eff04,04H XXXX eff

04,03H eff12,02H eff

14,03H eff13,04H

aFbH baˆHF eff,

, = –18.18 –19.81 –20.12 –1.57 –0.15 –0.28 –0.11

HF eff,,

eff,

eff, : bababa HHH −=δ 2.01 1.77 1.72 –0.21 — — —

eff,baH –16.17 –18.04 –18.40 –1.78 –0.15 –0.28 –0.11


7

Figure 3. Quasi-particle band structure of [(HF)2]∞ as obtained with the three correlations methods LHA, ISEM and CO-ADC. For comparison the Hartree-Fock bands are shown as well (bold dashed). The right panel is a zoom into the upper band complex (with the π bands being the lower ones at X).

also exists a non-Dyson version of the ADC which has the same feature [45] and it could be shown that a local variant of that formalism can be formulated as well [46]. In addition to configuration selec-tion, the ISEM employs the incremental scheme which allows to reduce the effort for a single calcula-tion drastically (and is sometimes the only way to circumvent other hardware limitations like mem-ory). Hence, the ISEM is a typical divide-and-conquer method like the LHA and their efficiency de-pends on the number of increments that have to be considered. In more complex systems (e.g. trans-polyacetylene [16]) this number can easily reach several hundreds and automatization (as recently ac-complished for ground-state calculations [47]) is vital to handle the data from all the separate calcula-tions. The achievable locality of the Wannier orbitals is another important issue. For valence bands the LHA only requires occupied orbitals which are relatively easy to obtain, while the other two methods need virtual local orbitals in that case as well. Yet, for conduction bands the LHA requires local and energetically low-lying virtual orbitals which is sort of a contradiction in itself [27] and can limit the applicability of the LHA quite noticeable. Also, the LHA has problems when satellites occur (as is the case for the inner s-type valence bands of the [(HF)2]∞ chain studied here).

5. Conclusions

Three rather different approaches to the correlated band structures of extended systems, the local Hamiltonian approach [14,16], the incremental self-energy method [21] and the crystal-orbital ADC formalism [22], have been applied to a periodic [(HF)2]∞ chain using the same structural parameters and Gaussian basis sets. Essentially identical quasi-particle band structures were found, demonstrating that the approximations and concepts underlying these three wave-function-based correlation methods are well-suited to describe excited state properties of extended or even infinite periodic systems.

Acknowledgments

VB and UB gratefully acknowledge the financial support from the German Research Association (DFG) obtained via the priority program SPP 1145.


8

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