Electronic structure of TiSe2 from a quasi-self-consistent G0W0 approach

Maria Hellgren,1 Lucas Baguet,1 Matteo Calandra,2, 3 Francesco Mauri,4 and Ludger Wirtz5

1Sorbonne Universite, Museum National d’Histoire Naturelle, UMR CNRS 7590, Institut de Mineralogie,de Physique des Materiaux et de Cosmochimie (IMPMC), 4 place Jussieu, 75005 Paris, France

2Dipartimento di Fisica, University of Trento, Via Sommarive 14, 38123 Povo, Italy3Sorbonne Universites, CNRS, Institut des Nanosciences de Paris, UMR7588, 75252, Paris, France

4Dipartimento di Fisica, Universita di Roma La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy5Department of Physics and Materials Science, University of Luxembourg,

162a avenue de la Faıencerie, L-1511 Luxembourg, Luxembourg(Dated: December 10, 2020)

In a previous work it was shown that the inclusion of exact exchange is essential for a firstprinciples description of both the electronic- and the vibrational properties of TiSe2 [Phys. Rev.Lett. 119 456325 (2017)]. The GW approximation provides a parameter-free description of screenedexchange but is usually employed perturbatively (G0W0) making results more or less dependent onthe starting point. In this work, we develop a simple quasi-self-consistent extension of G0W0 basedon the random phase approximation (RPA) and the optimized effective potential of hybrid densityfunctional theory. This approach generates an optimal G0W0 starting-point and a hybrid exchangeparameter consistent with the RPA. While self-consistency plays a minor role for systems such asAr, BN and ScN, it is shown to be crucial for TiS2 and TiSe2. We find the high-temperature phaseof TiSe2 to be a semi-metal with a band structure in good agreement with experimental results.Furthermore, the optimized hybrid functional accurately reproduces the low-temperature chargedensity wave phase.

I. INTRODUCTION

TiSe2 is a layered quasi-two-dimensional materialthat undergoes an unconventional charge density wave(CDW) transition below 200 K. The apparent interplaybetween the CDW and superconductivity at finite pres-sure or doping1,2 has lead to numerous studies over thepast years aiming to understand the driving mechanismbehind the CDW. Nevertheless, the relative role playedby excitonic effects and electron-phonon coupling is stilldebated. Experimentally, strong signatures are observedin both vibrational3–6 and angle-resolved photoemissionspectra (ARPES),7–13 and some studies point to soft elec-tronic modes.14

First-principle calculations should be able to explainthe exact mechanism of the CDW transition. How-ever, numerically tractable approaches such as the lo-cal density approximation (LDA) or generalized gradi-ent approximations (GGAs) within density functionaltheory (DFT) fail to give a complete picture.5,15,16 Adramatic improvement is found when including a frac-tion of Hartree-Fock (HF) exchange via the hybridfunctionals.17–19 With a result similar to the DFT+Uapproach,16,20 the Ti-d levels are then well described. Inaddition, the hybrid functionals contain the long-rangeCoulomb interaction which was shown to be crucial toinduce the CDW phase.19 This fact suggests that strongelectron-hole coupling is at play and that an excitonictransition could be of importance.21,22 It was also foundthat the standard medium-range hybrid functional al-ready gives a quantitatively reasonable agreement be-tween theory and experiment. However, the resultsshowed to be strongly dependent on the hybrid parame-ters, making it still uncertain whether a parametrization

optimized on a test-set of standard semi-conductors isadequate.

The GW method is computationally more expensivebut provides a parameter-free and physical descriptionof screened exchange. The bare Coulomb interaction isreplaced by the screened Coulomb interaction, W , whichis determined by the linear density response function ap-proximated at the Hartree level, i.e., the random phaseapproximation (RPA).23,24 The GW approximation forthe self-energy is known to produce accurate band-gapson a wide range of systems.25–28 It is, however, almost al-ways employed perturbatively (G0W0), on top of a DFTKohn-Sham (KS) band structure, assuming that the KSelectronic structure is close enough to the final result.Other variants that bring results closer to self-consistencyhave also been developed.29–32 An alternative to the fullyself-consistent GW scheme is to look for the optimal KSstarting-point via the Sham-Schluter equation.33–35 Theresulting KS potential produces a density similar to theGW density and is known as the RPA potential.36,37 TheKS RPA band structure can be shown to provide a con-sistent starting-point for G0W0.38,39

A high-level calculation of the electronic band struc-ture of TiSe2 in the high-T phase would be valuable.While transport experiments all predict a semi-metallicbehaviour some ARPES measurements have found agap.7,8 The latter scenario was supported by the firstG0W0 calculation and interpreted as an excitonic gap.40

In this work, we will re-examine how G0W0 performson TiSe2 by first showing that it is a case sensitiveto exchange in the starting-point. As a fully self-consistent calculation is out of reach we develop a quasi-self-consistent approach that exploits the local hybrid po-tential as an approximation to the local RPA potential.

arX

iv:2

005.

1460

3v2

[co

nd-m

at.m

trl-

sci]

9 D

ec 2

020

2

In this way, we produce a theoretically justified G0W0 so-lution that approximates the RPA solution. At the sametime we generate an RPA-optimized hybrid functionalthat is used to study the CDW phase.

The paper is organized as follows. In Sec. II we startby reviewing the GW formalism and the RPA as a self-consistent way to do perturbative G0W0. We then in-troduce a hybrid functional approach based on the op-timized effective potential. Using this potential we thendevelop a quasi-self-consistent G0W0 scheme and com-pare it to variants introduced by others. In Sec. IIIwe present numerical results for Ar, BN, ScN, TiS2 andTiSe2. We also use the RPA optimized hybrid functionalto study the CDW phase of TiSe2. Finally, in Sec. IVwe present our conclusions.

II. SCREENED EXCHANGE FROM GW

We will focus on studying the performance of the GWapproximation in describing the band structure of thehigh-T phase of TiSe2. The results turn out to bestrongly dependent on which approximate GW schemeis used. In this section we, therefore, start by review-ing the different ways to solve the GW equations anddiscuss the connections between GW , RPA, COHSEX(COulomb Hole Screened EXchange) and hybrid func-tionals. This will allow us to finally motivate a quasi-self-consistent G0W0 approach based on the local hybridpotential.

A. The GW approximation

We define the self-energy as the nonlocal frequency de-pendent potential Σ that contains all the many-body ef-fect beyond the Hartree (H) approximation. To first or-der, in an expansion in terms of the Green’s function, G,and the Coulomb interaction, v, Σ is just the static butnonlocal Fock term of the HF approximation,

ΣHF = iGv. (1)

By replacing the bare Coulomb interaction in the Fockterm with the dynamically screened Coulomb interaction,W , we obtain the self-energy within the GW approxima-tion

Σ = iGW. (2)

The screened interaction within the GW approxima-tion is approximated at the time-dependent Hartree levelfor which the irreducible polarizability, P , is approxi-mated with P0, i.e., to zeroth order in the explicit de-pendence on the Coulomb interaction. We thus have

W = v + vP0W, P0 = −iGG. (3)

From Dyson’s equation,

G = GH +GHΣ[G]G, (4)

we then have access to the many-body quasi-particlespectrum contained in G.

It can further be shown that the GW approximationis a Φ-derivable approximation41,42 that obeys physicalconservation laws and has an underlying action func-tional. An example of such an action functional is theKlein functional43

YK = −iΦ[G]− UH + iTr [GG−1H − 1 + ln(−G−1)],(5)

where UH is the Hartree energy. With the choice

Φ[G] =1

2Tr{ln[1 + ivGG]} (6)

it is easy to see that YK is stationary when G obeysDyson’s equation (Eq. (4)), and the self-energy is equalto

Σ =δΦ

δG= iGW. (7)

At the stationary point the Klein functional is equalto the GW total energy as obtained from the standardnon-variational Galitskii-Migdal energy expression.44

Instead of using the Hartree approximation as the ze-roth order approximation for G one can start from theDFT KS system. The Dyson’s equation can then be re-written in terms of the single-particle KS Green’s func-tion, Gs, and the exchange-correlation (xc) part of thelocal KS potential

G = Gs +Gs[Σ[G]− vxc]G. (8)

The diagonal of G, i.e. the density, is already exactlydescribed by Gs. In this way, Gs can be assumed tobe ’close’ to G, justifying a perturbative treatment ofEq. (8), and thus circumventing the full solution to thenumerically challenging Dyson’s equation. By writingEq. (8) in the basis of KS orbitals and keeping only thediagonal terms we can write the quasi-particle equationas26,45

Ek = εk + 〈k|Σs(Ek)− vxc|k〉 (9)

where k refers to the Bloch orbital index. The subscripts on the self-energy signifies that it is evaluated with Gs.The energy dependence of Σs can either be treated tozeroth order, i.e. Ek = εk, where εk is the KS eigenvalue,or to first order in a Taylor expansion around εk. Thelatter implies that a renormalization factor

Zk =

[1− ∂<Σs

∂ω

∣∣∣∣ω=εk

]−1

(10)

should be multiplied in the following way

Ek = εk + Zk〈k|Σs(εk)− vxc|k〉. (11)

This G0W0 correction, starting from PBE or LDA, isthe most common GW approach to determine the band

3

structure. The justification of this approach relies, how-ever, on the assumption that PBE or LDA orbitals aresimilar to the true quasi-particle orbitals. The renor-malization factors are usually incorporated but it can beargued that these should be omitted.38 The argumentsare based on the connection between GW and the RPAfor the total energy,36,46,47 as we will now discuss.

Let us go back to the Klein energy functional (Eq. (5))and keep the Φ-functional in the GW approximation.From now on we will add superscripts (ΦGW ,ΣGW ) aswe focus only on this approximation. If we replace theinteracting G, in every term, with a non-interacting Gswe can, after a few manipulations, write Eq. (5) as

Y GWK [Gs] = −iΦGW [Gs] + Ts[n] + UH + Uext (12)

where Ts is the kinetic energy of non-interacting KS elec-trons and Uext is the external potential energy. It is easyto see that ΦGW [Gs] is exactly the same functional asthe xc energy of the RPA energy functional

ERPAxc ≡ −iΦGW [Gs] = − i

2Tr{ln[1 + ivGsGs]}. (13)

Eq. (12) is thus nothing but the RPA total energy, i.e.,Y GWK [Gs] = ERPA.36,47,48

The RPA energy functional can be shown to have aminimum when varied with respect to the total KS po-tential Vs = vext + vH + vxc. Such a variation can becarried out via the functional Gs[Vs]. At the minimumvxc = vRPA

xc obeys the so-called linearized Sham-Schluter(LSS) equation∫

d2χs(1, 2)vRPAxc (2) =

∫d2d3 Λs(3, 2; 1)ΣGWs (2, 3)

(14)where Λs(3, 2; 1) = −iGs(3, 1)Gs(1, 2) and χs(2, 1) =−iGs(2, 1)Gs(1, 2).49 The LSS equation can also be de-rived from the condition that the GW density and theKS RPA density, i.e. the diagonals of G and Gs, arethe same to first order when expanding Dyson’s equation(Eq. (8)).

As the RPA potential is a local KS potential it doesnot reproduce the fundamental gap.33–35 One can, how-ever, still calculate the gap, Eg, corresponding to theRPA functional by taking the derivative of the energyfunctional with respect to particle number N . One finds

Eg = I −A =∂ERPA

∂N

∣∣∣∣+

− ∂ERPA

∂N

∣∣∣∣−. (15)

Evaluating the derivative on the right hand side ’+’, i.e.,the negative of the ionization energy

−I = εv + 〈v|ΣGWs (εv)− vRPAxc |v〉 (16)

and the derivative on the left hand side ’-’, i.e., the neg-ative of the affinity

−A = εc + 〈c|ΣGWs (εc)− vRPAxc |c〉 (17)

we can write

Eg = EKSg + ∆xc (18)

where EKSg is the KS gap and

∆xc = 〈c|ΣGWs (εc)− vRPAxc |c〉 − 〈v|ΣGWs (εv)− vRPA

xc |v〉.(19)

To derive these expressions Eq. (14) has to be used. Thequantity ∆xc equals what is called the derivative discon-tinuity within DFT.50–53

It is now clear that the gap obtained from the RPAfunctional is nothing but theG0W0 correction of Eq. (11),without the Zk factor, evaluated with the RPA potential.The RPA potential can thus be seen as an optimal KSstarting point for G0W0, which then produces a gap equalto the gap extracted from the RPA functional.38 It hasbeen shown on a number of semiconductors39 that usingthe RPA potential for a G0W0 calculation brings gaps incloser agreement with self-consistent GW approaches.30

By expanding the GW quasiparticle energy around thezeroth order RPA KS energy and using Eq. (14) the ex-pressions in Eqs. (16)-(17) are easily extended to thewhole band structure,38

ERPAk = εk + 〈k|ΣGWs (εk)− vRPA

xc |k〉. (20)

To conclude we have reviewed how it is possible tocalculate gaps and even the full band structure from theRPA and that this corresponds to the perturbative G0W0

result evaluated with the local RPA potential. Theseare well-known results that we will base the followingdiscussions on.

B. Hybrid functionals and the COHSEXapproximation

We will now turn to the simpler COHSEX and hy-brid functionals which are often used as cheaper but self-consistent alternatives to the GW approach.

The frequency dependence of the GW self-energy al-lows for the description of many-body effects such asquasi-particle lifetimes and satellite spectra but severelycomplicates a fully self-consistent solution. Often one is,however, only interested in the quasiparticle excitationenergy for which the nonlocality of the self-energy playsthe most important role. It is then motivated to approxi-mate Σ by ignoring dynamical effects in W . This impliessetting

Wstatic = v + vP0(ω = 0)Wstatic, (21)

and results in the so-called COHSEX approximation

ΣCOHSEX = iGWstatic +1

2W dp , (22)

whereW dp = Diag [vP0(ω = 0)Wstatic] is a local Coulomb-

hole potential and the first term is a nonlocal statically

4

screened exchange operator. The COHSEX approxima-tion can easily be solved self-consistently but can stillbe numerically demanding since it requires the genera-tion and summation over all conduction bands. A moredrastic approach that avoids the inclusion of unoccupiedbands is to keep the bare Coulomb interaction as in theHF approximation but scale it down with a constant α.If we then add a compensating fraction of the local PBEexchange and a local PBE correlation term we get theso-called hybrid functionals

ΣHYB,α = αΣHF + (1− α)vPBEx + vPBE

c . (23)

These functionals are structurally similar to COHSEXbut not more demanding than a HF calculation. Oneof the drawbacks is that a free parameter is introduced.A fraction 25% (PBE0) has shown to be reasonable inmany molecular systems. In the HSE06 functional a sec-

ond parameter, µ = 0.2 A−1

, that controls the range ofthe Coulomb interaction is introduced.54 In this way, itis possible to get a good description of many semicon-ductors as well.

Although often used in a DFT context the hybrid func-tionals are almost always treated like the HF approxima-tion, that is, by minimizing the energy with respect to or-bitals that are generated by the nonlocal Fock potential.In this work we will instead use the optimized effectivepotential method55 and minimize the hybrid energy withrespect to a local KS potential. The local KS potentialcorresponding to HF has been evaluated for solids beforeand is know as the exact-exchange (EXX) potential.56,57

The local hybrid potential is given by the sum of the lo-cal potentials derived from the PBE terms and a localexchange potential obtained from an equation similar tothe LSS equation (Eq. (14)) but with ΣGWs replaced bythe scaled HF self-energy. We have

vhyb,αxc = vαx + (1− α)vPBEx + vPBE

c (24)

where∫d2χs(1, 2)vαx (2) = α

∫d2d3 Λs(3, 2; 1)ΣHF

s (2, 3).

(25)The potential vαx can again be seen as the local poten-tial giving a density similar to the density of the fullynonlocal potential, to first order. The gap will, however,differ from the gap of the nonlocal potential, but, whencorrected with the discontinuity

∆xc = 〈c|αΣHFs − vαx |c〉 − 〈v|αΣHF

s − vαx |v〉 (26)

the gap is expected to be close to that of the nonlocal hy-brid functional. Gaps calculated in this way using otherexchange based functionals can be found in Refs. 58 and59.

-3

-2

-1

0

1

2

PBEEXXc

0 1 2 3 4 5PDOS (EXXc)

-3

-2

-1

0

1

2

E-E F [e

V]

Ti 3dSe 4p

MΓ A LK HΓ A

pz

px,y

Figure 1. Band structure and p, d-orbital projected densityof states in the high-T phase of TiSe2. EXXc (full lines)compared to PBE (dashed lines).

C. Optimal G0W0 starting point based on a localhybrid potential

The common crucial ingredient in GW , COHSEX andhybrid functionals is the nonlocal exchange term. Dueto this similarity the hybrids can be used as a way to doapproximate self-consistent GW . Such an approach wasdeveloped in Refs. 32 and 60. By using a hybrid as astarting-point for G0W0 the α parameter is varied untilthe GW correction vanishes. At this value, the GW andhybrid eigenvalues agree

〈k|ΣGWs (εnlk )− ΣHYB,αs |k〉nl = 0⇒ EGWk = εnlk . (27)

Here the matrix elements are evaluated with orbitalsgenerated by the nonlocal ΣHYB,α

s , emphasized by thesub(super)-script nl. This method has been shown toperform well for molecules, improving the ionization en-ergies as compared to standard hybrid functionals andG0W0 based on the PBE starting-point.32 We note, how-ever, that it is not possible to derive an equation similarto Eq. (18) combining the Klein GW energy functionalwith a nonlocal potential. In fact, it has been shown tolack an extremum when varied in a restricted space ofnonlocal but static potentials.61

We will now present a variant that utilizes the RPAenergy and, hence, the optimization with respect to a lo-cal potential. As seen in the previous subsection a G0W0

correction based on the local RPA potential is justifiedvia the GW LSS equation (Eqs. (14) and (20)). Anal-ogously, a hybrid correction based on the local hybridpotential is justified via the hybrid LSS equation. Wehave

EHYB,αk = εk + 〈k|ΣHYB,α

s − vhyb,αxc |k〉. (28)

We will now approximate the RPA potential in Eq. (20)by the local hybrid potential

ERPAk ≈ εk + 〈k|ΣGWs (εk)− vhyb,αxc |k〉 (29)

5

Table I. Band gaps (eV) of Ar, c-BN, ScN, TiS2 and TiSe2. The G0W0 results (without renormalization factors) are obtainedwith G0 of the approximation in the preceding column. The H-G0W0 results are evaluated with the optimized nonlocal hybrid

functional having an equal gap. The EXX results of TiS2 and TiSe2 are obtained with µ = 0.1 A−1

.

Solid PBE G0W0 EXX EXX lit. EXXc G0W0 α lhyb G0W0 α H-G0W0 HSE06 Exp.

Ar 8.65 14.09 9.57 9.61a 9.93 14.30 0.57 9.38 14.19 0.63 14.88 10.32 14.2b

BN 4.54 6.51 5.58 5.57a 5.12 6.78 0.25 4.70 6.61 0.30 6.99 5.85 6.4 ± 0.5c

ScN -0.05 1.01 1.57 1.58d 1.39 0.81 0.17 0.21 0.96 0.24 1.51 0.92 0.9 ± 0.1e

TiS2 -0.10 1.18 1.18 - 1.03 0.30 0.25 0.17 0.82 0.33 1.17 0.57 0.5 ± 0.1f

TiSe2 -0.63 0.37 0.57 - 0.38 -0.85 0.20 -0.45 -0.07 0.32 0.40 -0.15 -0.1-0.1g

a Reference 39.b Reference 62.c Reference 63.d Reference 64.e Reference 65.f Reference 66.g See text.

and optimize α such that the correction in Eq. (28) andEq. (29) are equal. This is equivalent to

〈k|ΣGWs (εk)− ΣHYB,αs |k〉 = 0⇒ ERPA

k = EHYB,αk , (30)

where the self-energy operators are evaluated with or-bitals and eigenvalues from vhyb,αxc (instead of ΣHYB,α

s asin Eq. (27) above). In this way, we optimize α such thatthe DFT hybrid functional behaves like the RPA func-tional when varying the particle number. The differencebetween this approach and the one in Ref. 32 lies in whichtype of reference system is used to evaluate the GW en-ergy. Allowing for nonlocal potentials can have a largeimpact on the energy due to the opening of a large gap.

Due to lack of frequency dependence in ΣHYB,αs the

condition in Eq. (30) is, in general, impossible to ful-fil for all bands at every k, but can be made true forthe difference between the highest occupied level and thelowest unoccupied level. One thus needs to optimize αusing the following condition

〈c|ΣGWs (εc)− ΣHYB,αs |c〉−

〈v|ΣGWs (εv)− ΣHYB,αs |v〉 = 0. (31)

In the next section we will show that self-consistencyhas a very small effect on systems where PBE alreadygives a good description of the orbitals. In contrast, forsystems where exact-exchange plays an important role,self-consistency is necessary and we will show that, viaEq. (31), it is possible to obtain meaningful results with-out leaving the traditional G0W0 scheme. The validity ofthe G0W0-approach is determined by the validity of theRPA for the given system. By approximating the RPApotential with the local hybrid potential we generate, as abi-product, a hybrid functional that can be used to studyother properties such as phonons and lattice instabilities.

III. NUMERICAL RESULTS

In this section we start by introducing TiSe2 and thetechnical aspects of our calculations. We then present

the G0W0 results for TiSe2, TiS2 and a set of well-knownsystems (Ar, c-BN and ScN) for which there already existboth EXX and G0W0 results in the literature. Finally, weinvestigate the performance of the RPA optimized hybridfunctional in capturing the CDW phase of TiSe2.

A. System and computational details

The high-T phase of TiSe2 crystallizes in the spacegroup P 3m1. It belongs to the 1T family of the layeredtransition metal dichalcogenids with the Ti-atom octa-hedrally coordinated by six neighbouring Se-atoms. Asemi-metallic behaviour is found in most experiments.Below 200 K a CDW transition occurs, characterised bya 2 × 2 × 2 superstructure (space group P3c1) and theopening of a small gap. The distortion pattern can beuniquely defined by the displacement δTi and the ra-tio δTi/δSe ≈ 3. Standard DFT functionals predict aphonon instability at the three equivalent L points. Asymmetric combination of these gives the correct CDWpattern, but with a severely underestimated distortionamplitude.16 Hybrid functionals give a better descriptionand have revealed the important role of HF exchangefor the instability. This possibly hints to the presenceof an excitonic instability. Although a weakly screenedelectron-hole interaction is clearly important, no sponta-neous electronic CDW has so far been generated in bulkTiSe2. Hybrid functionals induce the CDW via a strongelectron-phonon coupling combined with the enhancedelectronic susceptibility at the L points. This mecha-nism is given support by the combined accuracy of theelectronic bands, phonons and distortion amplitude.19

In this work we aim for the more sophisticated GWmethod that allows for a physical description of thescreened interaction. Due to the increase in computa-tional cost we have been limited to the high-T phase. Thelow-T CDW phase will be studied with the RPA-basedhybrid functional, optimized according to the proceduredescribed in the previous section.

6

In addition, we have determined the gap of TiS2 whichis structurally identical to TiSe2 but lacks a CDW tran-sition, at least in the bulk. We have only found oneGW study of TiS2 where the gap was determined to 0.75eV,67 which can be compared to the experimental resultof around 0.5 eV.67 We have also looked at solid Ar (avan der Waals bonded large-gap insulator), c-BN (a spbonded insulator) and ScN (a pd bonded semiconductor)in order to illustrate the workings and validity of theequations derived in Sec. II.

The hybrid calculations have been performed withVASP,68–70 Quantum Espresso71 (QE) and the CRYS-TAL program.72 Whenever comparisons could be madethese codes, despite using different pseudopotentials orin the latter case a Gaussian basis-set, agree rather well.For example, pd gaps agree within 0.05 eV. For TiS2

and TiSe2 we have used a range-separation parameter

of µ = 0.1 A−1

in all hybrid calculations. We chose arange twice as large as in HSE06 since our previous workon TiSe2 indicated that HSE06 was somewhat too shortranged.19 The local hybrid potential for solids was gener-ated from an extension of a molecular code for solving theLSS equation.73 The GW self-energy was subsequentlyevaluated using the YAMBO code.74,75 For testing theoptimization scheme in Eq. (27) we switched to the VASPcode which allows the self-energy to be evaluated witha hybrid G0. Agreement between different codes thatuse different numerical techniques and pseudopotentialsis still hard to achieve within GW .76,77 Nevertheless, forAr and c-BN, results on the PBE-G0W0 level agree within0.05 eV. For the pd gapped systems we found variationsup to 0.1 eV (TiS2 and TiSe2) and 0.2 eV (ScN), whichshould be taken into account when comparing the dif-ferent schemes. We used a 12 × 12 × 6 and 10 × 10 × 4k-point grid for TiSe2 and TiS2 respectively. Up to 500empty bands where included in the self-energy.

B. G0W0 results

In Table I we present the band gaps of Ar, c-BN, ScN,TiS2 and TiSe2. The PBE results are presented in thefirst column and the G0W0 results, obtained on top ofPBE orbitals, are presented in the second column. Therenormalization factors are omitted in all G0W0 calcula-tions. In this work, we are using a new implementationof the LSS equation (Eq. (25)) as described in Ref. 73.The gaps obtained within the EXX approximation (thirdcolumn) are, therefore, first compared to values foundin the literature. We find a very good agreement in allcases where results are available. The EXXc results areobtained by adding the PBE correlation potential to theEXX potential. This EXXc potential is then used for ob-taining the G0W0 results in the next column. In this way,we provide both extremes of the α-range: PBE (α = 0)and EXXc (α = 1.0). The α parameter is then optimizedaccording to Eq. (31). The optimized value of α, the KSgap of the corresponding local hybrid potential (lhyb),

HFc Δpd = 3.70 eV

EXXc Δpd = 0.80 eV lhybα=0.2 Δpd = -0.45 eV Ti

Se

HYBα=0.2 Δpd = -0.14 eV

0.38 eV -0.45 eV

-0.14 eV3.70 eV

Figure 2. Iso-surface of a Ti-d-orbital along the Γ−M path.78

To the left EXXc is compared to HFc, and to the right theoptimal hybrid with 20% of nonlocal exchange is compared tothe corresponding local hybrid approximation.

and the final G0W0 gap are presented in the followingcolumns. As a result of Eq. (31), the G0W0 gap has tobe the same as the perturbative gap of the nonlocal hy-brid functional (Eq. (28)) with parameter α. Finally, wepresent the H-G0W0 results which are obtained using theoptimization scheme of Eqs. (27), the HSE06 results andexperimental values.

Looking at the results for Ar and c-BN we immediatelysee that theG0W0 results are not so sensitive to which KSpotential is used. The EXX potential increases the KSgap by around 1 eV but this leads only to a small increaseof 0.2-0.3 eV in the G0W0 gap. By optimizing α we finda gap in between, with α = 0.57 for Ar and α = 0.25 forc-BN. These values are consistent with RPA in a sensethat both RPA and the hybrid functional give the samegap when evaluated with the orbitals of the optimizedlocal hybrid potential. We note that the H-G0W0 resultsgenerally leads to larger values of α.

In ScN, a pd semiconductor, we see a partially differentbehaviour. First, PBE predicts a semi-metallic groundstate with a pd band overlap. Including exact-exchangea KS gap of 1.57 eV opens. This rather large variationproduces again only a small variation at the G0W0 level.However, the behaviour of the correction is opposite ascompared to the correction in Ar and c-BN by givinga smaller G0W0 gap with EXXc than with PBE. Thissomewhat counterintuitive behaviour was noted alreadyin Ref. 79. We now look at TiS2 which also has a pdgap. We see a similar trend but now the GW variation ismuch larger, ranging from 0.3 eV with EXXc to 1.18 eVwith PBE. In this case, the optimization plays a crucialrole. With 25% of exchange the gap optimize to 0.82 eV.After this study we are now ready to turn to TiSe2.

The high-T phase PBE band structure has been pub-lished in several previous works,16 but is repeated here inFig. 1. The PBE results differ strongly from ARPES ex-periments. Similar to ScN and TiS2 the Se-p -Ti-d bandoverlap is severely overestimated and, in this case, eveninverted at Γ, leading to strong pd hybridization.16 Thepz orbitals corresponding to the flattened p-band aroundΓ is pushed above the Fermi level leading to excess d-electron occupation. These large errors invalidate theuse of a standard PBE/LDA starting-point for G0W0. If

7

-3

-2

-1

0

1

2

E-E F [e

V]

Γ ΓM K A L H A

lh-G0W0

HYBα=0.2lhyb-G0W0

Figure 3. Left: Band structure of the optimal hybrid functional with 20% of nonlocal exchange compared to the correspondinglhyb-G0W0 result. Right: Energy gain in the supercell as a function of Ti-distortion keeping the ratio δTi/δSe = 3 fixed.Optimal hybrid (red) is shown together with PBE (green) and the result obtained via the direct optimization scheme (blue,α = 32%).

we use PBE as a starting-point for G0W0, we open a gapof 0.37 eV between Γ and L. The band gap is actuallya bit smaller since we also found ’mexican hat’ featuresaround the Γ point similar to those reported in Ref. 40.Similar features are found already in the HF term butdisappears as soon as the orbitals are updated.19

In the same figure we compare the KS band structureof the EXXc approximation to PBE. The correspondingprojected density of states (PDOS) is shown in the sidepanel. The inclusion of exchange, even with a KS localpotential, corrects the occupations and opens a gap be-tween the Γ and L points. Corrected with a discontinuity(Eq. (26)) we find a gap as large as 3.75 eV in agreementwith the HFc approximation. The d-orbitals of the HFcand EXXc approximations are visualized in Fig. 2. Sincethe EXXc is defined to give a charge density similar toHFc by Eq. (25) the orbitals are also very similar.

If we evaluate G0W0 on top of the EXXc band struc-ture the gap closes and we find a large band overlap (-0.85eV). The magnitude of the variation is very close to theone in TiS2 and it is clear that a self-consistent schemeis necessary. At optimal α = 0.2, we find a small bandoverlap of around 0.1 eV. Most experiments predict anegative gap of around 0.1 eV in the high-T phase. Ourvalue is thus a reasonable prediction and shows that asemi-metallic solution can be found within GW . Theelectron-phonon mechanism found in Ref. 19 did notcrucially depend on the existence of a Fermi surface, buta semi-metallic solution increases the probability for theexistence of a purely electronic CDW.

The band structure along Γ −M and Γ − L is shownin Fig. (3) superimposed on the full band structure ofa hybrid functional with 20% of exchange. We see thatnot only the band overlap around the Fermi-level agreesbut also the band dispersions. Dynamical effects in theself-energy seem important around -3 eV where the pdmixed flat band is shifted downwards in the hybrid func-

tional with respect to G0W0. Experiments place thisband somewhere in between.80 In Fig. (4) we have su-perimposed the same results on an ARPES experimentby Rohwer et al.81 Since spin-orbit coupling (SOC) is notincluded care should be taken when comparing with ex-periment. Previous studies have shown that SOC splitsthe degenerate p-bands at Γ which could have a smalleffect on the comparisons. Overall we see a very goodagreement between theory and experiment noting thatsome of the deviations can be explained by looking atdifferent values for kz (see discussion in Ref. 19).

To the right in Fig. 2 we have compared the d-orbital ofthe local hybrid with the same orbital of the nonlocal hy-brid. Again, the orbitals are very similar despite very dif-ferent underlying gaps. By comparing with EXXc/HFcwe also see the effect of exchange on the d-orbitals whichmight be one explanation for the sensitivity of G0W0 tothe fraction of exchange in the starting-point. The largerα is the more charge localizes between the Ti-atoms. Thecharge on the Se atoms, i.e., the hybridization with Se-p-orbitals, is instead seen to reduce with α. The H-G0W0

yields a gap 0.4 eV at 32% of exchange, which is muchlarger than any experimental value. We also performed aself-consistent COHSEX calculation which gives a morereasonable result of 0.12 eV. We stress that these gaps arenot related to the CDW since the symmetry is preservedin our calculations.

C. RPA optimized hybrid functional

The approach applied above shows that G0W0 predictsa value for the pd band-overlap which is in good agree-ment with most experiment. It also gives us a predictionof α based on the derivative of the RPA functional. Thishybrid functional can now be used to study the CDW in-stability too expensive for an approach like GW or RPA.

8

To the right in Fig. (3) we have used the RPA opti-mized α to calculate the energy gain in the supercell afterdistorting the atoms according to the CDW pattern. The

set of parameters α = 0.2, µ = 0.1 A−1

lies close to thepath used in Ref. 19 and is not so different from theset that agreed best with experiment. We therefore findvery good results for both the energy gain and the δTi-distortion. Note, however, that in contrast to Ref. 19,where the α-parameter was fitted to the band gap and thephonon-frequencies, here it results from a self-consistentcalculation.

If we look at the results for the H-G0W0 optimizedhybrid functional with 32% of exchange the energy gainis quite small and the δTi distortion worsen as comparedto PBE. In this case, we expect the phonon frequenciesto largely deviate from experiment.

IV. CONCLUSIONS

In this work we have applied a self-consistent GWmethod to TiSe2 in order to give an accurate estimateof the much debated band-gap/band-overlap. First ofall, it was found that the standard G0W0 prescriptionbased on a PBE/LDA starting-point is unreliable due toqualitative errors in describing the band structure withinPBE/LDA. To overcome this problem we have developeda simple quasi-self-consistent approach based on the localhybrid potential and the RPA functional. This approachallows for a systematic inclusion of exact-exchange in thestarting-point, which, in the case of TiSe2 has a large im-pact on, e.g., the description of the Ti-d-orbitals. It isshown that G0W0 converges to a semi-metallic ground-state with a band-overlap of 0.1 eV. This is in line withthe majority of experimental results but contradicts firstG0W0 results based on the LDA starting-point.

The G0W0 approach generates a hybrid α-parameterconsistent with RPA. With a motivated choice for µ thishybrid functional produces an electron-phonon coupling

strong enough to induce the CDW transition. Further-more, the potential energy surface lies very close to ourearlier published hybrid results which gave very accuratephonons. While in our previous work, the α-parameterwas chosen as a best fit to both the band gap and thephonon frequencies, here it has been calculated via a self-consistent procedure involving the G0W0 method. In thisway, we provide further support to the proposed mecha-nism of the CDW distortion in TiSe2.

ACKNOWLEDGMENTS

This work was performed using HPC resourcesfrom GENCI-TGCC/CINES/IDRIS (Grant No.A0050907625). M. H. and L. B. acknowledges sup-port from Emergence-Ville de Paris. L. W. and M.C.acknowledges financial support from Agence Nationalede la Recherche (Grant N. ANR-19-CE24-0028) and theFond National de Recherche, Luxembourg via projectINTER/19/ANR/13376969/ACCEPT.

E-E

FM’(L’) Γ(A) M(L)

Figure 4. Band structure of HYBα=0.2 compared to ARPESat 300 K.81 Red lines corresponds to the A−L path and blacklines to the Γ − L path.

1 A. F. Kusmartseva, B. Sipos, H. Berger, L. Forro, andE. Tutis, “Pressure induced superconductivity in pristine1T−TiSe2,” Phys. Rev. Lett. 103, 236401 (2009).

2 E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G.Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong,and R. J. Cava, “Superconductivity in CuxTiSe2,” NaturePhysics 2, 544–550 (2006).

3 F. J. Di Salvo, D. E. Moncton, and J. V. Waszczak, Phys.Rev. B 14, 4321 (1976).

4 J. A. Holy, K. C. Woo, M. V. Klein, and F. C. Brown,“Raman and infrared studies of superlattice formation inTiSe2,” Phys. Rev. B 16, 3628–3637 (1977).

5 F. Weber, S. Rosenkranz, J.-P. Castellan, R. Osborn,G. Karapetrov, R. Hott, R. Heid, K.-P. Bohnen, andA. Alatas, “Electron-phonon coupling and the soft phononmode in TiSe2,” Phys. Rev. Lett. 107, 266401 (2011).

6 C. S. Snow, J. F. Karpus, S. L. Cooper, T. E. Kidd, andT.-C. Chiang, “Quantum melting of the charge-density-wave state in 1T−TiSe2,” Phys. Rev. Lett. 91, 136402(2003).

7 Th. Pillo, J. Hayoz, H. Berger, F. Levy, L. Schlapbach, andP. Aebi, “Photoemission of bands above the fermi level:The excitonic insulator phase transition in 1T − TiSe2,”Phys. Rev. B 61, 16213–16222 (2000).

8 T. E. Kidd, T. Miller, M. Y. Chou, and T.-C. Chiang,“Electron-hole coupling and the charge density wave tran-sition in TiSe2,” Phys. Rev. Lett. 88, 226402 (2002).

9 K. Rossnagel, L. Kipp, and M. Skibowski, “Charge-density-wave phase transition in 1T − TiSe2 : excitonicinsulator versus band-type jahn-teller mechanism,” Phys.Rev. B 65, 235101 (2002).

10 H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. De-spont, M. G. Garnier, H. Beck, P. Aebi, L. Patthey,

9

H. Berger, and L. Forro, “Evidence for an excitonic in-sulator phase in 1T−TiSe2,” Phys. Rev. Lett. 99, 146403(2007).

11 C. Monney, H. Cercellier, F. Clerc, C. Battaglia, E. F.Schwier, C. Didiot, M. G. Garnier, H. Beck, P. Aebi,H. Berger, L. Forro, and L. Patthey, “Spontaneous ex-citon condensation in 1T -TiSe2: Bcs-like approach,” Phys.Rev. B 79, 045116 (2009).

12 P. Chen, Y. H. Chan, X. Y. Fang, Y. Zhang, M Y Chou,S. K. Mo, Z. Hussain, A. V. Fedorov, and T. C. Chiang,“Charge density wave transition in single-layer titaniumdiselenide,” Nature Communications 6, 8943 (2015).

13 M.-L. Mottas, T. Jaouen, B. Hildebrand, M. Rumo,F. Vanini, E. Razzoli, E. Giannini, C. Barreteau, D. R.Bowler, C. Monney, H. Beck, and P. Aebi, “Semimetal-to-semiconductor transition and charge-density-wave sup-pression in 1T − TiSe2−xsx single crystals,” Phys. Rev. B99, 155103 (2019).

14 Anshul Kogar, Melinda S. Rak, Sean Vig, Ali A. Hu-sain, Felix Flicker, Young Il Joe, Luc Venema, Greg J.MacDougall, Tai C. Chiang, Eduardo Fradkin, Jasper vanWezel, and Peter Abbamonte, “Signatures of exciton con-densation in a transition metal dichalcogenide,” Science358, 1314–1317 (2017).

15 Matteo Calandra and Francesco Mauri, “Charge-densitywave and superconducting dome in TiSe2 from electron-phonon interaction,” Phys. Rev. Lett. 106, 196406 (2011).

16 Raffaello Bianco, Matteo Calandra, and Francesco Mauri,“Electronic and vibrational properties of TiSe2 in thecharge-density-wave phase from first principles,” Phys.Rev. B 92, 094107 (2015).

17 P. Chen, Y. H. Chan, X. Y. Fang, Y. Zhang, M Y Chou,S. K. Mo, Z. Hussain, A. V. Fedorov, and T. C. Chiang,“Charge density wave transition in single-layer titaniumdiselenide,” Nature Communications 6, 8943 (2015).

18 P. Chen, Y.-H. Chan, X.-Y. Fang, S.-K. Mo, Z. Hussain,A.-V. Fedorov, M.Y. Chou, and T.-C. Chiang, “Hidden or-der and dimensional crossover of the charge density wavesin TiSe2,” Sci. Rep. 6, 37910 (2016).

19 M. Hellgren, J. Baima, R. Bianco, M. Calandra, F. Mauri,and L. Wirtz, “Critical role of the exchange interaction forthe electronic structure and charge-density-wave formationin TiSe2,” Phys. Rev. Lett. 119, 176401 (2017).

20 Vladimir I. Anisimov, Jan Zaanen, and Ole K. Andersen,“Band theory and mott insulators: Hubbard U instead ofstoner I,” Phys. Rev. B 44, 943–954 (1991).

21 Maria Hellgren, Jacopo Baima, and Anissa Acheche, “Ex-citon peierls mechanism and universal many-body gaps incarbon nanotubes,” Phys. Rev. B 98, 201103 (2018).

22 Diego Pasquier and Oleg V. Yazyev, “Excitonic effects intwo-dimensional TiSe2 from hybrid density functional the-ory,” Phys. Rev. B 98, 235106 (2018).

23 Lars Hedin, “New method for calculating the one-particlegreen’s function with application to the electron-gas prob-lem,” Phys. Rev. 139, A796–A823 (1965).

24 B. Holm and U. von Barth, “Fully self-consistent GW self-energy of the electron gas,” Phys. Rev. B 57, 2108–2117(1998).

25 G. Strinati, H. J. Mattausch, and W. Hanke, “Dynamicalcorrelation effects on the quasiparticle bloch states of acovalent crystal,” Phys. Rev. Lett. 45, 290–294 (1980).

26 Mark S. Hybertsen and Steven G. Louie, “Electron cor-relation in semiconductors and insulators: Band gapsand quasiparticle energies,” Phys. Rev. B 34, 5390–5413

(1986).27 F Aryasetiawan and O Gunnarsson, “The GW method,”

Rep Prog Phys 61, 237 (1998).28 Lucia Reining, “The GW approximation: content, suc-

cesses and limitations,” WIREs Comput Mol Sci , e1344(2017).

29 Fabien Bruneval, Nathalie Vast, and Lucia Reining, “Ef-fect of self-consistency on quasiparticles in solids,” Phys.Rev. B 74, 045102 (2006).

30 M. Shishkin and G. Kresse, “Self-consistent GW calcula-tions for semiconductors and insulators,” Phys. Rev. B 75,235102 (2007).

31 A. Svane, N. E. Christensen, I. Gorczyca, M. van Schil-fgaarde, A. N. Chantis, and T. Kotani, “Quasiparticleself-consistent GW theory of iii-v nitride semiconductors:Bands, gap bowing, and effective masses,” Phys. Rev. B82, 115102 (2010).

32 Viktor Atalla, Mina Yoon, Fabio Caruso, Patrick Rinke,and Matthias Scheffler, “Hybrid density functional theorymeets quasiparticle calculations: A consistent electronicstructure approach,” Phys. Rev. B 88, 165122 (2013).

33 R. W. Godby, M. Schluter, and L. J. Sham, “Trends inself-energy operators and their corresponding exchange-correlation potentials,” Phys. Rev. B 36, 6497–6500(1987).

34 R. W. Godby, M. Schluter, and L. J. Sham, “Self-energyoperators and exchange-correlation potentials in semicon-ductors,” Phys. Rev. B 37, 10159–10175 (1988).

35 Myrta Gruning, Andrea Marini, and Angel Rubio, “Den-sity functionals from many-body perturbation theory: Theband gap for semiconductors and insulators,” The Journalof Chemical Physics 124, 154108 (2006).

36 Maria Hellgren and Ulf von Barth, “Correlation potentialin density functional theory at the GWA level: Sphericalatoms,” Phys. Rev. B 76, 075107 (2007).

37 Maria Hellgren, Daniel R. Rohr, and E. K. U. Gross,“Correlation potentials for molecular bond dissociationwithin the self-consistent random phase approximation,”The Journal of Chemical Physics 136, 034106 (2012).

38 Y. M. Niquet and X. Gonze, “Band-gap energy in therandom-phase approximation to density-functional the-ory,” Phys. Rev. B 70, 245115 (2004).

39 Jiri Klimes and Georg Kresse, “Kohn-sham band gaps andpotentials of solids from the optimised effective potentialmethod within the random phase approximation,” TheJournal of Chemical Physics 140, 054516 (2014).

40 M. Cazzaniga, H. Cercellier, M. Holzmann, C. Monney,P. Aebi, G. Onida, and V. Olevano, “Ab initio many-bodyeffects in TiSe2: A possible excitonic insulator scenariofrom GW band-shape renormalization,” Phys. Rev. B 85,195111 (2012).

41 Gordon Baym and Leo P. Kadanoff, “Conservation lawsand correlation functions,” Phys. Rev. 124, 287–299(1961).

42 Gordon Baym, “Self-consistent approximations in many-body systems,” Phys. Rev. 127, 1391–1401 (1962).

43 Abraham Klein, “Perturbation theory for an infinitemedium of fermions. ii,” Phys. Rev. 121, 950–956 (1961).

44 V Galitskii and A Migdal, Zh. Eksp. Teor. Fiz. 34, 139(1958).

45 G. Strinati, H. J. Mattausch, and W. Hanke, “Dynamicalaspects of correlation corrections in a covalent crystal,”Phys. Rev. B 25, 2867–2888 (1982).

10

46 Mark E. Casida, “Generalization of the optimized-effective-potential model to include electron correlation: Avariational derivation of the sham-schluter equation for theexact exchange-correlation potential,” Phys. Rev. A 51,2005–2013 (1995).

47 Fabio Caruso, Daniel R. Rohr, Maria Hellgren, XinguoRen, Patrick Rinke, Angel Rubio, and Matthias Schef-fler, “Bond Breaking and Bond Formation: How ElectronCorrelation is Captured in Many-Body Perturbation The-ory and Density-Functional Theory,” Phys. Rev. Lett. 110,146403 (2013).

48 Maria Hellgren, Fabio Caruso, Daniel R. Rohr, XinguoRen, Angel Rubio, Matthias Scheffler, and Patrick Rinke,“Static correlation and electron localization in moleculardimers from the self-consistent RPA and GW approxima-tion,” Phys. Rev. B 91, 165110 (2015).

49 L. J. Sham and M. Schluter, “Density-functional theory ofthe energy gap,” Phys. Rev. Lett. 51, 1888–1891 (1983).

50 John P. Perdew, Robert G. Parr, Mel Levy, and Jose L.Balduz, “Density-functional theory for fractional particlenumber: Derivative discontinuities of the energy,” Phys.Rev. Lett. 49, 1691–1694 (1982).

51 O. Gunnarsson and K. Schonhammer, “Density-functionaltreatment of an exactly solvable semiconductor model,”Phys. Rev. Lett. 56, 1968–1971 (1986).

52 Maria Hellgren and E. K. U. Gross, “Discontinuities of theexchange-correlation kernel and charge-transfer excitationsin time-dependent density-functional theory,” Phys. Rev.A 85, 022514 (2012).

53 Maria Hellgren and E. K. U. Gross, “Discontinuousfunctional for linear-response time-dependent density-functional theory: The exact-exchange kernel and approx-imate forms,” Phys. Rev. A 88, 052507 (2013).

54 Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernz-erhof, “Hybrid functionals based on a screened coulombpotential,” The Journal of Chemical Physics 118, 8 (2003).

55 James D. Talman and William F. Shadwick, “Optimizedeffective atomic central potential,” Phys. Rev. A 14, 36–40(1976).

56 M. Stadele, J. A. Majewski, P. Vogl, and A. Gorling,“Exact kohn-sham exchange potential in semiconductors,”Phys. Rev. Lett. 79, 2089–2092 (1997).

57 E. Engel and R. N. Schmid, “Insulating ground states oftransition-metal monoxides from exact exchange,” Phys.Rev. Lett. 103, 036404 (2009).

58 M. Kuisma, J. Ojanen, J. Enkovaara, and T. T. Rantala,“Kohn-sham potential with discontinuity for band gap ma-terials,” Phys. Rev. B 82, 115106 (2010).

59 E. J. Baerends, “From the kohn?sham band gap to thefundamental gap in solids. an integer electron approach,”Phys. Chem. Chem. Phys. 19, 15639 (2017).

60 Chandrima Mitra, “Extracting the hybrid functional mix-ing parameter from a GW quasiparticle approach,” physicastatus solidi (b) 250, 1449–1452 (2013).

61 Sohrab Ismail-Beigi, “Correlation energy functional withinthe GW -RPA: Exact forms, approximate forms, and chal-lenges,” Phys. Rev. B 81, 195126 (2010).

62 N. Schwentner, F. J. Himpsel, V. Saile, M. Skibowski,W. Steinmann, and E. E. Koch, “Photoemission from rare-gas solids: Electron energy distributions from the valencebands,” Phys. Rev. Lett. 34, 528–531 (1975).

63 R.M. Chrenko, “Ultraviolet and infrared spectra of cubicboron nitride,” Solid State Communications 14, 511 – 515(1974).

64 Markus Betzinger, Christoph Friedrich, Andreas Gorling,and Stefan Blugel, “Precise response functions in all-electron methods: Application to the optimized-effective-potential approach,” Phys. Rev. B 85, 245124 (2012).

65 Hamad A. Al-Brithen, Arthur R. Smith, and Daniel Gall,“Surface and bulk electronic structure of ScN(001) investi-gated by scanning tunneling microscopy/spectroscopy andoptical absorption spectroscopy,” Phys. Rev. B 70, 045303(2004).

66 C. Wang, L. Dotson, M. McKelvy, and W. Glaunsinger,“Scanning tunneling spectroscopy investigation of chargetransfer in model intercalation compounds ti1+xs2,” TheJournal of Physical Chemistry 99, 8216–8221 (1995).

67 Adrien Stoliaroff, Stephane Jobic, and Camille Latouche,“Optoelectronic properties of tis2: A never ended storytackled by density functional theory and many-body meth-ods,” Inorganic Chemistry 58, 1949–1957 (2019).

68 G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169(1996).

69 G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15(1996).

70 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).71 P. Giannozzi et al, J. Phys.: Condens. Matter 29, 465901

(2017).72 Roberto Dovesi, Alessandro Erba, Roberto Orlando, Clau-

dio M. Zicovich-Wilson, Bartolomeo Civalleri, LorenzoMaschio, Michel Rerat, Silvia Casassa, Jacopo Baima,Simone Salustro, and Bernard Kirtman, “Quantum-mechanical condensed matter simulations with crystal,”WIREs Computational Molecular Science 8, e1360 (2018).

73 Ngoc Linh Nguyen, Nicola Colonna, and Stefano de Giron-coli, “Ab initio self-consistent total-energy calculationswithin the EXX/RPA formalism,” Phys. Rev. B 90,045138 (2014).

74 A. Marini, C. Hogan, M. Gruning, and D. Varsano, Comp.Phys. Comm. 144, 180 (2009).

75 D. Sangalli et al, J. Phys.: Condens. Matter 31, 325902(2019).

76 Jiri Klimes, Merzuk Kaltak, and Georg Kresse, “Predic-tive GW calculations using plane waves and pseudopoten-tials,” Phys. Rev. B 90, 075125 (2014).

77 Tonatiuh Rangel, Mauro Del Ben, Daniele Varsano,Gabriel Antonius, Fabien Bruneval, Felipe H. da Jor-nada, Michiel J. van Setten, Okan K. Orhan, David D.O?Regan, Andrew Canning, Andrea Ferretti, AndreaMarini, Gian-Marco Rignanese, Jack Deslippe, Steven G.Louie, and Jeffrey B. Neaton, “Reproducibility in g0w0calculations for solids,” Computer Physics Communica-tions 255, 107242 (2020).

78 The iso-surface of the square of the Bloch orbitals are vi-sualized with VESTA (K. Momma and F. Izumi, J. Appl.Crystallogr., 44, 1272 (2011)).

79 Abdallah Qteish, Patrick Rinke, Matthias Scheffler, andJorg Neugebauer, “Exact-exchange-based quasiparticle en-ergy calculations for the band gap, effective masses, anddeformation potentials of scn,” Phys. Rev. B 74, 245208(2006).

80 Z. Vydrova, E. F. Schwier, G. Monney, T. Jaouen, E. Raz-zoli, C. Monney, B. Hildebrand, C. Didiot, H. Berger,T. Schmitt, V. N. Strocov, F. Vanini, and P. Aebi,“Three-dimensional momentum-resolved electronic struc-ture of 1T−TiSe2 : a combined soft-x-ray photoemissionand density functional theory study,” Phys. Rev. B 91,235129 (2015).

11

81 Timm Rohwer, Stefan Hellmann, Martin Wiesenmayer,Christian Sohrt, Ankatrin Stange, Bartosz Slomski, AdraCarr, Yanwei Liu, Luis Miaja Avila, Matthias Kallane,Stefan Mathias, Lutz Kipp, Kai Rossnagel, and Michael

Bauer, “Collapse of long-range charge order tracked bytime-resolved photoemission at high momenta,” Nature471, 490 (2011).