Comparison of codes for many-body perturbation theory calculations

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Comparison of codes for many-body perturbation theory calculations of molecular systems Marco Govoni, and Giulia Galli J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00952 • Publication Date (Web): 03 Feb 2018 Downloaded from http://pubs.acs.org on February 5, 2018

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Journal of Chemical Theory and Computation

Comparison of codes for many-body perturbation theory calculations of molecular systems Marco Govoni∗,†,‡ and Giulia Galli‡,† †Institute for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, United States ‡Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States E-mail: [email protected]

Abstract The reproducibility of calculations carried out within many-body perturbation theory at the G0 W0 level is assessed for 100 closed shell molecules and compared to that of density functional theory. We consider vertical ionization potentials (VIP) and electron affinities (VEA) obtained with five different codes: BerkeleyGW, FHI-aims, TURBOMOLE, VASP, and WEST. We review the approximations and parameters that control the accuracy of G0 W0 results in each code and we discuss in detail the effect of extrapolation techniques for the parameters entering the WEST code. Differences between the VIP and VEA computed with the various codes are within ∼ 60 meV and ∼ 120 meV respectively, which is up to four times larger than in the case of the best results obtained with DFT codes. Vertical ionization potentials are validated against experiment and CCSD(T) quantum chemistry results showing a mean absolute relative error of ∼ 4% for data obtained with WEST. Our analysis of the differences between localized orbitals and plane-wave implementations points out at molecules containing

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Cu, I, Ga, and Xe as major sources of discrepancies, which call for a re-evaluation of the pseudopotentials used for these systems in G0 W0 calculations.

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1

Introduction

Over the last several decades, the validation of molecular properties computed from first principles has been primarily carried out using ab initio quantum chemistry methods, and has led to the definition of numerous sets of molecules used to benchmark thermodynamics and spectroscopic properties, e.g. the G2/97, 1–3 GMTKN30, 4 ISO34, 5,6 and S66 7 sets. Validation of properties based on density functional theory (DFT) has been much less frequent, and only recently papers on reproducibility of DFT results for solids 8–10 have appeared in the literature. Even less common are validation studies involving Green’s function methods, e.g. many-body perturbation theory (MBPT), 11,12 with few papers on molecules published in the last decade. 13–20 However, MBPT has become increasingly popular to investigate complex systems, including heterogeneous solids and complex molecules, thanks to algorithmic advances which have permitted to tackle systems with tens to hundreds of atoms. 16,21,22 Hence the validation of MBPT results obtained with different numerical methods and different codes is an important and pressing task. In this paper we present a comparison of quasi particle energies of molecules, computed using several codes and numerical methods, all implementing perturbative GW calculations (which are denoted G0 W0 ). We present data for vertical ionization potentials (VIP) and electron affinities (VEA) of 100 molecules already studied by other groups, namely the GW100 set. 18 We begin by describing (Sec. 2) validation of results at the DFT level of theory adopted to compute single particle energies and wavefunctions, which are the starting point of G0 W0 calculations. We then introduce in Sec. 3 the algorithms and codes used for the G0 W0 calculations, with focus on WEST, 16 a code which does not require the evaluation of any virtual electronic states, nor direct diagonalization of dielectric matrices. In Sec. 4 we discuss the results obtained with WEST, and in Sec. 5 we compare them to those available in the literature and computed with all-electron and pseudopotential codes. Finally, in Sec. 6 we compare G0 W0 results with experimental values and those obtained using CCSD(T).

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2

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DFT results

Within DFT, the n-th single particle wavefunction ψn and corresponding energy εn of interacting electrons in the external field of ions are obtained by solving the Kohn-Sham (KS) equation: 23–27 ˆ KS |ψn i = εn |ψn i H

(1)

ˆ KS = Tˆ + Vîon + VˆH + Vˆxc is the KS Hamiltonian, Tˆ is the kinetic energy operator, where H and Vîon , VˆH , and Vˆxc are the ionic, Hartree, and exchange-correlation potential operators, respectively. It is well known that in general the eigenvalues of Eq. 1 do not represent electron addition and removal energies. 11,28 In a finite system, the eigenvalue of the highest occupied state would correspond to the ionization energy only if the exact KS potential were used, 29,30 according to Koopmans’ theorem. 31 Therefore, the calculation of VIP and VEA from KS eigenvalues obtained using an approximate Vˆxc usually leads to inaccurate results. 14–16 In the next sessions we discuss the calculation of VIP and VEA, using quasi-particle energies obtained with the G0 W0 method. 11,28,32–39 Since these are perturbative calculations starting from DFT wavefunctions, it is important to first investigate the convergence of VIP and VEA values obtained at the DFT level. We solved Eq. 1 for all molecules belonging to the GW100 set 18 using the pseudopotential, plane-wave method as implemented in the Quantum-ESPRESSO code 40 . 41 Within this formulations, VIP and VEA are defined as:

VIPDF T = Evac − εHOMO ,

(2)

VEADF T = Evac − εLUMO ,

(3)

where Evac is the electrostatic potential averaged on a sphere of diameter a, obtained in a periodically repeated supercell of side a containing the molecule of interest. The size of the cell was chosen so as to minimize the interaction with periodic replica. The eigenvalues εHOMO

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and εLUMO are the energies of the highest occupied (HOMO) and of the lowest unoccupied molecular orbitals (LUMO). We used the PBE exchange and correlation functional 42 and the SG15 Optimized NormConserving Vanderbilt (ONCV) pseudopotentials. 43 The parameters of the SG15 psudopotentials were optimized by Gygi et al. 44 so as to reproduce the results of all-electron calculations with the same functional; details about the pseudopotentials are reported in the SI (Tab. S1). Energy cutoffs (Ecut ) for the plane-wave expansion, yielding a value of the εHOMO of each molecule converged within 10 meV, are reported in the SI (Tab. S2). For both VIP and VEA we determined converged results by independently varying a and Ecut : we used −3

the functions a−3 and Ecut2 to extrapolate results as a function of cell size and energy cutoff, respectively. The linear fits of the functions VIP(a−3 ) and VEA(a−3 ) were performed by computing the VIP and VEA at a = [17, 19, 21, 23, 25] ˚ A and Ecut = 130 Ry. The linear fits −3

−3

A of VIP(Ecut2 ) and VEA(Ecut2 ) were performed by computing the VIP and VEA at a = 25 ˚ and Ecut = [70, 80, 90, 110, 120, 130] Ry. For each molecule the extrapolated DFT result and corresponding absolute error were obtained by taking the average and semi-range 45 between the result of independent extrapolations as a function of a and Ecut . Extrapolated DFT results for VIP and VEA are reported in Tab. S3 and S4, respectively. In Fig. 1 we compare values for the VIP computed at the same level of theory (DFT-PBE) but with different numerical methods: all-electron and localized basis sets (AELO), and pseudopotential plane-wave (PSPW) methods. Results labeled AELO-QZVP and AELOEXTRA were obtained by van Setten et al. 18 using the FHI-aims 46,47 code, with the def2QZVP basis set or with extrapolations to complete basis set values based on calculations with def2-SVP, def2-TZVP, and def2-QZVP basis sets, 18,48 respectively. In Ref. 18 the authors noted that, despite the different treatment of the Gaussian orbitals, the results obtained with the TURBOMOLE 48 code are practically identical to the ones obtained with FHI-aims; hence we refer to both these results as AELO results. Results labeled PSPW-(25 ˚ A-80 Ry) and PSPW-EXTRA were obtained using SG15 ONCV pseudopotentials, with a = 25 ˚ A and

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Ecut = 80 Ry or with extrapolations with respect to a and Ecut , respectively. Throughout the manuscript, each comparison of two sets of results X and Y is carried out P using the mean absolute error M AEXY = N1 N i=1 |Xi − Yi |, and with the maximum absolute error. Fig. 1 shows that the agreement between AELO and PSPW results is very good, and on a par with that obtained for solids in Ref. 9. Our extrapolated PSPW results show a MAE of 1 meV and a maximum absolute error of 4 meV with respect to our non-extrapolated results, which proves that a = 25 ˚ A and Ecut = 80 Ry leads to well converged DFT results. A similar analysis of the MAE between AELO-QZVP and AELO-EXTRA reveals that the effect of extrapolation techniques is one order of magnitude (39 meV) larger in AELO than in PSPW implementations of DFT. Compared to AELO-EXTRA values, our extrapolated results show a MAE of 30 meV, the same as reported by Maggio et al. 19 who used the VASP code with scalar relativistic PAW pseudopotentials (vasp.5.4). The molecules which show the most significant absolute errors between AELO and PSPW extrapolated results are: As2 , Ne, N2 , Ar, CuCN, and SO2 . In addition, similar to the DFT results obtained with VASP, our extrapolated values are in better agreement with the non-extrapolated AELO-QZVP values, supporting the claim of Maggio et al. 19 that the choice of extrapolation by van Setten et al. 18 may have led to a slight overestimate of the complete basis set limit. Finally, Ref. 18 used incorrect geometries for two molecules (namely CH2 CHBr and C6 H5 OH) 49 we used here the corrected geometries reported in Ref. 19. We now turn to the description of G0 W0 results and we will show that the differences between methods and codes can be up to four times larger than for DFT. But first we summarize the methods and algorithms used to carry out G0 W0 calculations.

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Figure 1: Mean absolute error (MAE, upper triangle) and maximum absolute error (lower triangle) of vertical ionization potentials computed using different implementations of DFT. The symbols AELO-QZVP and AELO-EXTRA denote all-electron results from Ref. 18 obtained using local orbitals (AELO). AELO-EXTRA values were extrapolated to complete basis set. 18 The symbols PSPW-(25˚ A-80Ry) and PSPW-EXTRA denote plane-wave results obtained in this work using SG15 ONCV pseudopotentials. 44 PSPW-EXTRA values were extrapolated with respect to cell size and kinetic energy cutoff.

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Implementations of G0W0 calculations

Within many-body perturbation theory (MBPT) electron addition and removal energies can be obtained by computing quasi-particle energies (EnQP ) starting, e.g. from KS eigenvalues and eigenvectors: 11,28,34–39

ˆ QP ) − Vˆxc |ψn i EnQP = εn + hψn | Σ(E n

(4)

where the electron self-energy in the GW approximation 11,28,32,33,38 is defined as:

0

Z

+∞

Σ(r, r ; ω) = i −∞

dω G(r, r0 ; ω + ω 0 )W (r, r0 ; ω 0 ) . 2π

(5)

In Eq. 5, G is the Green’s function, and W is the screened Coulomb interaction. 50,51 Within the G0 W0 approximation the KS eigenvalues and eigenvectors are used to compute G and W in the random-phase approximation (RPA), 11,28,34–39 and corrections to single particle energies (Eq. 4) are used to define VIP and VEA as:

QP VIPQP = Evac − EHOMO ,

(6)

QP VEAQP = Evac − ELUMO ,

(7)

Here the value of Evac is the same as the one computed at the DFT level of theory, and used in Eq.s 2-3, since wavefunctions and hence charge densities are not modified in the G0 W0 scheme. Although quasi-particle energies will be primarily computed using Eq. 4, we also introduce a popular approximation to the quasi-particle energies based on a linear expansion of the electron self-energy: 52,53

ˆ n ) − Vˆxc |ψn i EnQP −lin = εn + Zn hψn | Σ(ε

8


(8)

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where

Zn−1

= 1−

∂ ∂ω

ˆ hψn | Σ(ω) |ψn i

. The validity of such approximation will be discussed ω=n

at the end of Sec. 4. We compare the results obtained with WEST 16 to those computed with four other codes: BerkeleyGW, 54 FHI-aims, 46,47 TURBOMOLE 48 and VASP. 19 The numerical methods and algorithms implemented in these codes differ in many respects: i) the basis set chosen to represent the single particle orbitals (localized orbitals (LO) or PW); ii) the way core-valence partition is treated (i.e. type of pseudopotential (PS): norm-conserving (NC), ONCV or PAW); iii) the inclusion or lack of relativistic effects in the pseudopotentials; iv) the calculation of Green’s functions (and hence of density-density response functions) with explicit summations over empty states or with projector techniques; v) the basis set used to represent response functions; vi) the representation of the self-energy as a function of frequency and its integration; vii) the method used to solve the QP equation (Eq. 4). These technical differences are summarized in Tab. 1, and in Sec. 5 we discuss which ones of them may be majorly responsible for the discrepancies observed between the results of these five codes. Table 1: Comparison between implementations of the G0 W0 methods in five different codes, which have been used to carry out calculations for the VIP and VAE of the GW100 set: BerkeleyGW, 54 FHI-aims, 46,47 TURBOMOLE, 48 VASP, 19 and WEST. 16 a: except for Cu, Ag and Ti. b : except for He. c : the VIP and VEA of 7 molecules were updated in Tab. 2 of Ref. 19, namely Xe, Rb2 , I2 , CH2 CHI, CI4 , AlI3 , and Ag2 . d : “all” means all states given the chosen number of basis functions. Codes basis set for single particle orbitals core-valence electrons relativistic effects virtual states basis set for density-density response function self-energy frequency integration QP solutions

BerkeleyGW

FHI-aims

PW

LO (def2, trapolated)

TURBOMOLE

VASP

WEST

LO (def2)

PW (extrapolated)

PW (extrapolated)

NC PS (PSlib)a

AE

AE

PAW (vasp.5.4)b

ONCV (SG15)

no

noc

no

scalar

scalar

truncated

alld

alld

alld

projected out

PW (truncated)

LO

LO

PW (truncated)

PDEP (extrapolated)

real-axis integration

analytic continuation

fully analytic

analytic continuation

contour mation

graphical section

iterative method

largest weight

Brent’s method

secant method

inter-

ex-

PS

PS

defor-

The solution of the Hedin’s equations implemented in the WEST code includes the fol9



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lowing steps (see Ref. 16 for details about the notation in use): 1. A spectral decomposition of the symmetrized irreducible density-density response function (χ¯0 ) at ω = 0 is obtained by iterative diagonalization using the Davidson algorithm; 55 the calculations of electronic virtual states is avoided by using density functional perturbation theory (DFPT). 56,57 The set of eigenvectors ϕi of χ¯0 , which we call projective dielectric eigenpotentials (PDEP), 58,59 is used as basis set to represent the symmetrized reducible density-density response function (χ) ¯ at finite frequency (ω):

0

χ(r, ¯ r ; ω) =

NX P DEP

ϕi (r)Λij (ω)ϕ∗j (r0 )

(9)

i,j

2. A separable form (low-rank decomposition) for the screened Coulomb interaction W is obtained: NP DEP 1 X ϕ˜i (r)Λij (ω)ϕ˜∗j (r0 ) W (r, r ; ω ) = v(r, r ) + Ω i,j=1 0

0

0

(10)

where v(r, r0 ) is the bare Coulomb potential, and ϕ˜i = v 1/2 ϕi . 3. The Lanczos algorithm 60–62 is employed to compute in parallel W (ω), thus eliminating the need for plasmon pole models 36,37,52,63–67 or any model for the frequency dependence of the density-density response function and of G; the contour deformation 68 technique is used to carry out the frequency integration in Eq. 5, avoiding the shortcomings of analytic continuation. 15,69–71 4. Finally the quasi-particle equation (Eq. 4) is solved self-consistently using a secant method, 16 or by adopting the approximate Eq. 8. In all codes considered in this work, except WEST, the Hamiltonian is explicitly diagonalized and the Green’s function is evaluated by computing occupied and empty states. This strategy does not typically involve any further truncation on the number of empty states included in the calculation, when using LO basis sets. However, in PW basis sets codes summations over virtual states are usually truncated 72,73 due to the large size of the basis set (which may contain up to > 100k elements for the systems investigated in this work), 10


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leading to a prohibitively large number of virtual states. Nevertheless, Maggio et al. 19 fully diagonalized the Hamiltonian for each molecule of the GW100 set by calculating all orbitals spanned by their PW basis set; this procedure is however limited to systems with small unit cells. 74 In codes using PW basis sets, additional energy cutoffs are often introduced, in order to reduce the number of plane-waves necessary to represent density response functions: cutoffs used in BerkeleyGW are listed in Tab. 1 of Ref. 18; in VASP the kinetic energy cutoff for response functions is set to 2/3 of Ecut . 19 The low-rank decomposition of the dielectric matrix used in the WEST code permits a compact representation of density response functions, with a number of basis functions (the number of eigenpotentials NP DEP , see Eq. 9) much smaller than the number of plane waves; in addition, no direct inversion or storage of the matrix is necessary. Note that NP DEP is proportional to the number of electrons in the system, not to the volume of the supercell (contrary to the number of plane-waves). Only the results obtained with FHI-aims (referred to as AELO, as according to Ref. 18 TURBOMOLE gives comparable results), VASP and WEST were extrapolated with respect to complete basis set.

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4

G0W0 results

For each molecule of the GW100 set we computed the VIP and VEA at the G0 W0 level of theory using the WEST code; 16 calculations were performed starting from PBE wavefunctions, as discussed in Sec. 2. We determined extrapolated converged values as a function of four parameters: a, Ecut , NP DEP , and NLanczos . Extrapolations were obtained by linear −3

fits of VIP and VEA as a function of a−3 , Ecut2 , and NP−1DEP , and by an exponential fit as a function of NLanczos . Converged values determined as averages of four independent extrapolations as a function of the 4 parameters are reported in Tab. 2 and Tab. 3. 75 The linear fits of VIP(a−3 ) and VEA(a−3 ) were performed by computing the values of the functions at a = [17, 19, 21, 23, 25] ˚ A and Ecut = 130 Ry, NP DEP = 30 Nel , NLanczos = 30. −3

−3

The linear fits of VIP(Ecut2 ) and VEA(Ecut2 ) were performed by computing the functions at Ecut = [70, 80, 90, 110, 120, 130] Ry and a = 25 ˚ A, NP DEP = 30 Nel , NLanczos = 30. The linear fit as a function of NP−1DEP included calculations for NP DEP = [10, 15, 20, 25, 30] Nel and a = 25 ˚ A, Ecut = 130 Ry, NLanczos = 30. The exponential fit as a function of NLanczos included calculations at NLanczos = [15, 20, 25, 30] and a = 25 ˚ A, Ecut = 130 Ry, NP DEP = 30 Nel . Ne is the number of valence electrons of each molecule, reported in Tab. S2. For each molecule the absolute error associated to the extrapolation was estimated by computing the semi-range between the result of four independent extrapolations as a function of the four parameters (and reported in parentheses). The maximum semi-range found was 73 (9) meV for G0 W0 (DFT) results. We did not report the result of the extrapolation for molecules for which the coefficient of determination of one of the fits was < 0.95, and the deviation from the non-extrapolated result differed by more than 20 meV. Moreover, for CI4 , KBr, NaCl, BN, O3 , BeO, MgO, Cu2 , and CuCN the solution of Eq. 4 is close to one or more poles of the self-energy and multiple solutions of the nonlinear QP equation may be found (see graphical solutions in the SI, reported for all molecules). For these molecules different implementations may converge to different numerical values (not necessarily with maximum QP weight) depending on the algorithm implemented. We note that in general the poles of the self-energy 12


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are found at deeper energy when G0 W0 calculations are performed using hybrid functional instead of PBE wavefunctions, resulting in an improved numerical stability of root finding algorithms employed to solve Eq. 4. In addition, improved accuracy of G0 W0 calculations has been reported when employing hybrid functionals. 14,16,17,76 When using Eq. 8 for the quasiparticle energies, we could perform an extrapolation for the VIP and VAE of all molecules in the set, consistent with the observation reported in Ref. 19 that the linearization used in Eq. 8 yields unique solutions which are better suited for extrapolation than those of the nonlinear Eq. 4.

Figure 2: Energy differences (∆Ei ) between quasi-particle energies computed with Eq. 8 (QPlin) and Eq. 4 (QP, see text) at a = 25 ˚ A, Ecut = 80 Ry, NP DEP = 30 Nel , NLanczos = 30, and their respective extrapolated results. Results for vertical ionization potentials (VIP) and electron affinities (VEA) are reported in Tab. 2 and Tab. 3, respectively. The horizontal lines denote the maximum, mean, and minimum of the distribution of results. The difference between the G0 W0 results obtained with Ecut = 80 Ry and the extrapolated values is reported in Fig. 2. On average the difference with respect to converged values is ∼ −20 meV for VIP and −15 meV for VEA, and the maximum absolute difference is ∼ 90 meV, which is smaller than the basis set correction range reported by Maggio et al. 19 using the VASP code (of the order of 300 − 400 meV). We also note that in Ref. 16 we computed the VIP of 29 molecules of the GW100 set using parameters and PBE norm13



conserving pseudopotentials different from the ones considered in this study, and without implementing extrapolation methods. Compared to the extrapolated results reported in this manuscript, the results of Ref. 16 show a MEA of 69 meV, which is a small difference, on a par with the best agreement reported in the next section. The difference between VIP and VEA and the corresponding extrapolated values as a function of a, Ecut , NP DEP , and NLanczos is reported in Fig. 3. The figure shows that the rate of converge of VIP and VEA with respect to a and Ecut is slower at the G0 W0 level of theory than at the DFT level (compare with Fig. S1). We also observed that NP DEP ' 10Ne is sufficient to converge within 0.1 eV the calculation of both VIP and VEA of all molecules of the set. Finally, Fig. 4 shows that the approximation of Eq. 8 on average performs better for VEA (mean difference = 0.02 eV) than for VIP (mean difference = −0.12 eV). If we exclude three outliers (KH, LiF and LiH), all VIP and VEA computed with the linearization of Eq. 8 differ by < 0.3 eV from the solutions of Eq. 4. The discrepancies between Eq. 4 and Eq. 8 for these molecules are caused by a pole of the self-energy located in close proximity to the QP solution (see graphical solutions in the SI, reported for all molecules). These differences were discussed for a subset of the GW100 molecules in Ref. 16 and observed to be more pronounced for G0 W0 calculations started from PBE than from hybrid DFT wavefunctions.

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Figure 3: Energy differences (∆Ei ) between extrapolated and non extrapolated results (see text) for the vertical ionization potential (VIP, left panels) and electron affinity (VEA, right panels), as a function of the cell size (a), energy cutoff (Ecut ), number of projective dielectric eigenpotentials NP DEP , and number of Lanczos steps (NLanczos ) used in calculations performed with the WEST code. Quasi-particle energies have been computed using Eq. 8. The horizontal lines denote the maximum, mean, and minimum of the distribution of results. 15



Figure 4: Energy differences (∆Ei ) between extrapolated values of the vertical ionization potentials (VIP, left hand side, see Tab. 2) and electron affinity (VEA, right hand side, see Tab. 3), computed using Eq. 8 and Eq. 4 to obtain quasi particle energies. Horizontal lines denote the maximum, mean, and minimum of the distribution of results.

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Table 2: Vertical Ionization Potentials (VIP, eV) computed using G0 W0 @PBE. The symbol QP-lin (QP) denotes results obtained with Eq. 8 (Eq. 4). All theoretical results have been extrapolated. The absolute error associated to the extrapolation (reported in parentheses) was estimated by computing the semi-range between the result of independent extrapolations as a function of the four parameters. QP-lin (Eq. 8) VASP

19

QP (Eq. 4)

WEST

AELO

18

Formula

1

He

23.62

23.648(20)

23.49

23.38

23.417(15)

24.59 77

2

Ne

20.36

20.525(20)

20.33

20.17

20.330(18)

21.56 77

3

Ar

15.42

15.500(15)

15.28

15.32

15.372(12)

15.76 78

4

Kr

14.03

13.867(16)

13.89

13.93

13.764(13)

14.00 79

5

Xe

12.22

13.381(12)

12.22

12.14

13.222(7)

12.13 80

6

H2

16.06

16.031(16)

15.85

15.85

15.842(11)

15.43 81

7

Li2

5.32

5.193(9)

5.05

5.09

5.041(4)

4.73 82

8

Na2

5.06

5.065(8)

4.88

4.93

4.977(5)

4.89 83

9

Na4

4.23

4.282(9)

4.14

4.17

4.241(11)

4.27 84

10

Na6

4.40

4.422(12)

4.34

4.34

4.366(11)

4.12 84

11

K2

4.24

4.214(9)

4.08

4.12

4.136(4)

4.06 83

12

Rb2

4.14

4.078(10)

4.07

4.02

4.015(5)

3.90 83

13

N2

15.06

15.078(19)

15.05

14.93

14.937(17)

15.58 85

14

P2

10.40

10.481(16)

10.38

10.35

10.434(15)

10.62 86

15

As2

9.62

9.581(17)

9.67

9.59

9.551(16)

10.00 87

16

F2

15.08

15.163(33)

15.10

14.93

15.000(30)

15.70 88

17

Cl2

11.40

11.495(22)

11.31

11.32

11.406(20)

11.49 89

18

Br2

10.65

10.517(18)

10.56

10.57

10.435(16)

10.51 89

19

I2

9.59

10.559(16)

9.48

9.52

10.410(14)

9.36 90

20

CH4

14.14

14.105(19)

14.00

14.02

13.991(16)

13.60 91

21

C2 H6

12.58

12.528(21)

12.46

12.50

12.435(19)

11.99 90

22

C3 H8

11.98

11.917(22)

11.89

11.90

11.838(20)

11.51 90

23

C4 H10

11.69

11.479(23)

11.59

11.61

11.406(22)

11.09 90

24

C2 H4

10.50

10.456(17)

10.40

10.42

10.393(15)

10.68 91

25

C2 H2

11.24

11.185(17)

11.09

11.07

11.092(14)

11.49 91

Continued on next page

17


VASP

19

i

WEST

exp.


QP-lin (Eq. 8) i

Formula

VASP

19

Page 18 of 53

QP (Eq. 4)

WEST

AELO

18

VASP

19

WEST

exp.

26

C4

10.97

10.973(20)

10.91

10.89

10.900(18)

12.54 92

27

C3 H6

10.78

10.731(20)

10.65

10.72

10.667(18)

10.54 93

28

C6 H6

9.16

9.127(19)

9.10

9.11

9.082(18)

9.23 94

29

C8 H8

8.24

8.203(20)

8.18

8.19

8.158(20)

8.43 95

30

C5 H6

8.51

8.487(18)

8.45

8.47

8.442(17)

8.53 96

31

CH2 CHF

10.36

10.356(17)

10.32

10.28

10.293(15)

10.63 97

32

CH2 CHCl

10.00

9.997(19)

9.89

9.92

9.936(18)

10.20 98

33

CH2 CHBr

9.83

9.712(19)

9.64

9.75

9.644(18)

9.90 98

34

CH2 CHI

9.36

9.935(17)

9.13

9.27

9.814(13)

9.35 99

35

CF4

15.53

15.653(32)

15.60

15.41

15.509(30)

16.20 100

36

CCl4

11.31

11.410(24)

11.21

11.20

11.292(22)

11.69 90

37

CBr4

10.38

10.225(23)

10.22

10.25

10.106(21)

10.54 101

38

CI4

9.23

10.153(20)

8.97

9.11

—

9.10 102

39

SiH4

12.53

12.551(20)

12.40

12.40

12.423(17)

12.30 103

40

GeH4

12.24

12.443(15)

12.11

12.13

12.325(13)

11.34 104

41

Si2 H6

10.52

10.584(21)

10.41

10.44

10.521(19)

10.53 105

42

Si5 H12

9.19

9.246(23)

9.05

9.13

9.190(23)

9.36 105

43

LiH

7.20

7.195(16)

6.58

6.46

6.625(13)

7.90 106

44

KH

5.37

5.394(18)

4.99

4.97

4.970(14)

8.00 107

45

BH3

13.09

13.080(19)

12.96

12.95

12.953(17)

12.03 108

46

B2 H6

12.04

12.026(21)

11.93

11.94

11.919(19)

11.90 109

47

NH3

10.44

10.403(19)

10.39

10.32

10.184(21)

10.82 110

48

HN3

10.56

10.543(18)

10.55

10.50

10.480(18)

10.72 111

49

PH3

10.45

10.514(16)

10.35

10.35

10.426(13)

10.59 112

50

AsH3

10.36

10.404(18)

10.21

10.26

10.331(17)

10.58 113

51

H2 S

10.30

10.357(18)

10.13

10.11

10.232(18)

10.50 114

52

HF

15.38

15.471(28)

15.37

15.37

15.227(28)

16.12 115

53

HCl

12.51

12.603(18)

12.36

12.45

12.476(15)

12.79 116

54

LiF

10.45

10.537(25)

10.27

10.07

10.112(21)

11.30 117

55

MgF2

12.77

12.842(25)

12.50

12.41

12.460(19)

13.30 118


18


Page 19 of 53 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


QP-lin (Eq. 8) i

Formula

VASP

19

QP (Eq. 4)

WEST

AELO

18

VASP

19

WEST

exp.

56

TiF4

14.22

14.313(26)

14.07

14.01

—

15.30 119

57

AlF3

14.53

14.627(28)

14.48

14.33

14.396(25)

15.45 120

58

BF

10.67

10.710(13)

10.73

10.46

10.564(10)

11.00 121

59

SF4

12.29

12.413(20)

12.38

12.20

12.321(19)

11.69 122

60

BrK

8.04

7.959(24)

7.57

7.80

—

8.82 123

61

GaCl

9.99

10.274(14)

9.74

9.89

10.186(12)

10.07 124

62

NaCl

8.76

8.862(20)

8.43

8.47

—

9.80 123

63

MgCl2

11.41

11.468(19)

11.20

11.19

11.253(17)

11.80 125

64

AlI3

9.69

10.589(20)

9.50

9.58

10.314(73)

9.66 126

65

BN

10.61

11.234(33)

11.15

—

—

66

HCN

13.43

13.409(21)

13.32

13.29

13.218(19)

13.61 127

67

PN

11.41

11.434(20)

11.29

11.24

11.257(17)

11.88 128

68

H2 NNH2

9.45

9.420(20)

9.37

9.33

9.270(18)

8.98 129

69

H2 CO

10.57

10.561(21)

10.46

10.42

10.406(20)

10.88 130

70

CH3 OH

10.72

10.744(22)

10.67

10.61

10.605(22)

10.96 131

71

CH3 CH2 OH

10.33

10.356(23)

10.27

10.21

10.213(21)

10.64 132

72

CH3 CHO

9.80

9.813(23)

9.66

9.63

9.612(18)

10.24 133

73

CH3 CH2 OCH2 CH3

9.52

9.520(26)

9.42

9.43

9.395(23)

9.61 132

74

HCOOH

10.98

11.009(21)

10.87

10.81

10.815(17)

11.50 134

75

HOOH

11.12

11.164(21)

11.10

10.96

11.003(20)

11.70 135

76

H2 O

12.05

12.093(19)

12.05

11.84

11.872(17)

12.62 90

77

CO2

13.44

13.461(20)

13.46

13.36

13.369(19)

13.77 136

78

CS2

10.01

10.104(18)

9.95

9.96

10.046(16)

10.09 137

79

CSO

11.13

11.218(18)

11.11

11.06

11.158(18)

11.19 138

80

CSeO

10.50

10.428(17)

10.43

10.42

10.372(16)

10.37 139

81

CO

13.76

13.791(17)

13.71

13.62

13.663(15)

14.01 138

82

O3

12.07

12.121(19)

11.49

—

—

12.73 140

83

SO2

12.04

12.082(20)

12.06

11.91

11.960(18)

12.50 90

84

BeO

9.50

9.615(16)

8.60

—

—

10.10 141

85

MgO

7.10

7.241(15)

6.75

—

—

8.76 142


19


—


QP-lin (Eq. 8) i

Formula

VASP

19

Page 20 of 53

QP (Eq. 4)

WEST

AELO

18

VASP

19

WEST

exp.

86

C6 H5 CH3

8.79

8.753(20)

8.73

8.75

8.711(19)

8.82 94

87

C8 H10

8.73

8.698(21)

8.66

8.69

8.657(20)

8.77 94

88

C6 F6

9.69

9.701(20)

9.74

9.63

9.649(19)

10.20 100

89

C6 H5 OH

8.43

8.421(20)

8.40

8.38

8.368(19)

8.75 143

90

C6 H5 NH2

7.84

7.801(20)

7.78

7.78

7.733(19)

8.05 144

91

C5 H5 N

9.31

9.283(25)

9.17

9.16

9.131(20)

9.66 145

92

C5 H5 N5 O

7.90

7.865(22)

7.87

7.85

7.816(21)

8.24 146

93

C5 H5 N5

8.18

8.145(21)

8.16

8.12

8.089(20)

8.48 147

94

C4 H5 N3 O

8.50

8.493(22)

8.44

8.40

8.397(21)

8.94 146

95

C5 H6 N2 O2

8.89

8.879(21)

8.87

8.83

8.818(20)

9.20 148

96

C4 H4 N2 O2

9.55

9.265(20)

9.38

9.36

9.193(18)

9.68 149

97

NH2 CONH2

9.59

9.601(23)

9.46

9.35

9.395(20)

9.80 114

98

Ag2

7.95

8.121(8)

7.96

7.83

8.037(7)

7.66 150

99

Cu2

7.40

7.913(65)

7.78

7.19

—

7.46 151

CuCN

9.99

—

9.56

—

—

—

100

20


Page 21 of 53 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3: Vertical Electron Affinity (VEA, eV) computed using G0 W0 @PBE. The symbol QP-lin (QP) denotes results obtained with Eq. 8 (Eq. 4). All theoretical results have been extrapolated. The absolute error associated to the extrapolation (reported in parentheses) was estimated by computing the semi-range between the result of independent extrapolations as a function of the four parameters. QP-lin (Eq. 8) VASP

19

QP (Eq. 4)

WEST

AELO

18

Formula

1

He

—

-0.017(6)

—

—

-0.017(6)

—

3

Ar

—

-0.685(27)

—

—

-0.684(27)

—

4

Kr

—

-0.675(21)

—

—

-0.675(21)

—

5

Xe

-0.70

-0.554(18)

0.07

-0.28

-0.553(18)

—

6

H2

—

-0.052(6)

—

—

-0.052(6)

—

7

Li2

0.54

0.579(11)

0.75

0.61

0.636(11)

—

8

Na2

0.56

0.581(8)

0.66

0.60

0.613(7)

0.54 106

9

Na4

1.03

1.037(14)

1.15

1.07

1.081(15)

0.91 152

10

Na6

1.03

1.001(12)

1.13

1.07

1.035(12)

—

11

K2

0.70

0.719(7)

0.75

0.74

0.745(7)

0.55 106

12

Rb2

0.70

0.709(8)

0.85

0.74

0.731(8)

0.50 152

13

N2

—

-2.191(22)

—

—

-2.145(22)

—

14

P2

0.97

1.083(23)

1.08

0.99

1.096(24)

0.68 153

15

As2

1.06

1.084(22)

1.52

1.07

1.094(23)

0.74 154

16

F2

0.84

1.047(34)

1.23

0.96

1.059(29)

1.24 155

17

Cl2

1.22

1.368(23)

1.40

1.25

1.381(23)

1.02 155

18

Br2

1.97

1.871(24)

1.96

1.99

1.878(25)

1.60 155

19

I2

2.20

3.214(17)

2.28

2.21

3.214(18)

1.70 155

20

CH4

-0.63

-0.762(26)

-2.03

-0.63

-0.761(26)

—

21

C2 H6

—

-0.778(26)

—

—

-0.777(26)

—

22

C3 H8

—

-0.751(26)

—

—

-0.749(26)

—

23

C4 H10

—

-0.738(27)

—

—

-0.736(27)

—

24

C2 H4

—

-1.824(18)

—

—

-1.798(19)

—

25

C2 H2

—

-2.521(18)

—

—

-2.495(18)

—

26

C4

3.08

3.107(21)

3.15

3.09

3.104(22)

3.88 156


21


VASP

19

i

WEST

exp.


QP-lin (Eq. 8) i

Formula

VASP

19

Page 22 of 53

QP (Eq. 4)

WEST

AELO

18

VASP

19

WEST

exp.

27

C3 H6

—

-0.752(31)

—

—

-0.751(31)

—

28

C6 H6

—

-0.949(18)

—

—

-0.930(18)

—

29

C8 H8

0.02

0.047(18)

0.12

0.05

0.068(18)

0.57 157

30

C5 H6

—

-0.918(18)

—

—

-0.897(18)

—

31

CH2 CHF

—

-1.910(17)

—

—

-1.885(17)

—

32

CH2 CHCl

-1.25

-1.238(18)

-1.17

-1.19

-1.212(18)

—

33

CH2 CHBr

—

-1.089(19)

-1.01

—

-1.058(19)

—

34

CH2 CHI

—

-0.238(16)

-0.56

-0.37

-0.218(16)

—

35

CF4

—

-0.833(26)

—

—

-0.832(26)

—

36

CCl4

0.28

0.388(19)

0.54

0.32

0.411(18)

—

37

CBr4

1.44

1.435(21)

1.56

1.47

1.454(21)

—

38

CI4

2.40

3.034(17)

2.47

2.42

3.040(17)

—

39

SiH4

—

-0.765(25)

—

—

-0.762(25)

—

40

GeH4

—

-0.611(13)

—

—

-0.608(13)

—

41

Si2 H6

—

-0.775(18)

—

—

-0.768(18)

—

42

Si5 H12

-0.05

0.050(17)

-0.00

-0.03

0.067(17)

—

43

LiH

0.04

0.051(9)

0.16

0.07

0.073(10)

0.34 158

44

KH

0.22

0.223(9)

0.32

0.25

0.249(8)

—

45

BH3

-0.08

-0.040(15)

-0.03

-0.03

-0.008(16)

0.04 159

46

B2 H6

—

-0.748(18)

—

—

-0.722(18)

—

47

NH3

—

-0.810(12)

—

—

-0.807(12)

—

48

HN3

—

-1.190(21)

—

—

-1.145(20)

—

49

PH3

—

-0.702(16)

—

—

-0.699(16)

—

50

AsH3

—

-0.665(14)

—

—

-0.660(14)

—

51

H2 S

—

-0.781(12)

—

—

-0.775(12)

—

52

HF

—

-1.117(8)

—

—

-1.114(8)

—

53

HCl

—

-1.101(13)

—

—

-1.092(13)

—

54

LiF

-0.17

-0.072(7)

0.01

-0.17

-0.068(7)

—

55

MgF2

0.28

0.317(8)

0.31

0.29

0.329(8)

—

56

TiF4

0.66

0.823(22)

1.06

0.79

0.924(21)

2.50 160


22


Page 23 of 53 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


QP-lin (Eq. 8) i

Formula

VASP

19

QP (Eq. 4)

WEST

AELO

-0.09

0.142(9)

0.23

18

VASP

19

WEST

exp.

-0.08

0.155(9)

—

57

AlF3

58

BF

—

-0.980(14)

—

—

-0.925(13)

—

59

SF4

-0.12

0.049(18)

0.10

-0.07

0.075(18)

1.50 161

60

BrK

0.31

0.388(14)

0.42

0.32

0.402(14)

0.64 162

61

GaCl

0.15

0.417(11)

0.39

0.19

0.438(11)

—

62

NaCl

0.43

0.449(9)

0.42

0.46

0.473(10)

0.73 162

63

MgCl2

0.59

0.684(13)

0.68

0.61

0.700(13)

—

64

AlI3

0.99

1.652(14)

1.18

1.02

1.659(14)

—

65

BN

—

—

—

—

4.077(24)

—

66

HCN

—

-2.281(19)

—

—

-2.250(19)

—

67

PN

—

0.493(21)

—

—

0.515(21)

—

68

H2 NNH2

—

-0.721(8)

—

—

-0.716(8)

—

69

H2 CO

—

-0.805(20)

—

—

-0.764(21)

—

70

CH3 OH

—

-0.911(15)

—

—

-0.909(15)

—

71

CH3 CH2 OH

—

-0.845(15)

—

—

-0.841(15)

—

72

CH3 CHO

-0.87

-0.924(20)

-0.83

-0.87

-0.868(19)

—

73

CH3 CH2 OCH2 CH3

—

-0.715(26)

—

—

-0.712(26)

—

74

HCOOH

-1.72

-1.682(20)

-1.59

-1.64

-1.636(19)

—

75

HOOH

—

-1.836(20)

—

—

-1.796(20)

—

76

H2 O

-1.04

-0.914(9)

-2.01

-1.04

-0.911(9)

—

77

CO2

—

-0.978(11)

—

—

-0.974(11)

—

78

CS2

0.40

0.482(21)

0.55

0.42

0.495(20)

0.55 163

79

CSO

—

-0.974(20)

—

—

-0.944(20)

—

80

CSeO

—

-0.681(20)

—

—

-0.638(20)

—

81

CO

—

-0.483(20)

—

—

-0.438(19)

—

82

O3

2.52

2.605(26)

2.69

2.50

2.558(29)

2.10 164

83

SO2

1.19

1.359(24)

1.49

1.25

1.372(25)

1.11 165

84

BeO

2.37

2.234(17)

2.72

2.73

2.507(8)

—

85

MgO

2.12

2.026(17)

2.13

2.05

1.950(19)

—

86

C6 H5 CH3

—

-0.885(17)

—

—

-0.867(17)

—


23



QP-lin (Eq. 8) i

Formula

VASP

19

Page 24 of 53

QP (Eq. 4)

WEST

AELO

18

VASP

19

WEST

exp.

87

C8 H10

—

-0.923(17)

—

—

-0.904(17)

—

88

C6 F6

-0.27

-0.053(15)

-0.36

-0.24

-0.026(15)

0.70 166

89

C6 H5 OH

—

-0.865(17)

-0.78

—

-0.847(17)

—

90

C6 H5 NH2

—

-1.002(17)

—

—

-0.981(17)

—

91

C5 H5 N

—

-0.380(19)

—

—

-0.361(18)

—

92

C5 H5 N5 O

—

-0.542(17)

—

—

-0.512(17)

—

93

C5 H5 N5

—

-0.318(18)

—

—

-0.290(17)

—

94

C4 H5 N3 O

-0.15

-0.115(18)

-0.01

-0.12

-0.091(18)

0.23 167

95

C5 H6 N2 O2

0.04

0.081(18)

0.18

0.06

0.102(17)

-0.29 168

96

C4 H4 N2 O2

0.09

0.128(18)

0.25

0.11

0.151(17)

-0.22 168

97

NH2 CONH2

—

-0.537(7)

—

—

-0.532(7)

—

98

Ag2

1.31

1.474(10)

1.40

1.35

1.489(10)

1.10 169

99

Cu2

1.21

1.294(10)

1.23

1.24

1.408(4)

0.84 170

CuCN

1.81

1.917(12)

1.85

1.91

1.977(12)

1.47 171

100

24


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5

Verification: comparisons between G0W0 results obtained with different implementations

Fig.s 5-6 show VIPs and VEAs, computed at the G0 W0 @PBE level of theory, as obtained with different codes. 172 The MAE between PSPW codes (WEST and VASP) is 59 meV for VIP and 71 meV for VEA. AELO codes yield a similar agreement for VIP (MAE = 59, 70 compared to VASP and WEST, respectively) but slightly worse for VEA (MAE = 122, 92 compared to VASP and WEST, respectively). The differences observed here are up to four times as large as that found at the DFT level of theory and shown in Fig. 1. We note that in the comparisons reported in Fig.s 5-6 we have excluded molecules with unbound LUMOs, with multiple QP solutions, and eight molecules containing the Cu, I, Ga, and Xe atomic species (namely: Cu2 , CuCN, CI4 , CH2 CHI, I2 , AlI3 , GaCl, and Xe). For VEA energies large discrepancies are observed between codes using localized and extended basis sets for unbound LUMOs. Overall, we note that the maximum differences between PSPW and AELO codes is ∼ 0.2 eV for VIP, and is twice as big for VEA energies. At variance with G0 W0 results, no major differences between DFT results were found in Sec. 2 for molecules containing Cu, I, Ga and Xe elements. This possibly stems from the sensitivity of G0 W0 calculations to the choice of pseudopotentials. All these elements have either partially or fully filled d shells, leading to a non straightforward definition of core and valence partitions. 17 Fig. 7 shows that the eight molecules containing the Cu, I, Ga, and Xe atomic species are responsible for the largest discrepancies between the codes. When these molecules are removed from the comparison the maximum absolute differences are reduced to < 0.3 eV. The only remaining exception occur for the VEA of the As2 molecule where WEST and VASP results give good agreement (24 meV), while the AELO codes yield a result that differs by 426 meV from PSPW results. The As2 molecule yielded the most significant discrepancy between AELO and PSPW codes also at the DFT level of theory (176 meV). Fig.s 5-6 also show a comparison between results obtained with Eq. 4 and Eq. 8. We

25



found that even when the linear approximation is used to determine electron self-energies, the results obtained with VASP and WEST are consistent within 58 meV and 79 meV for VIP and VAE, respectively. By comparing with Ref. 18 we found that results obtained by using the generalized plasmon pole (GPP) model for the frequency dependent dielectric matrix (BerkeleyGW-GPP) show ∼ 0.3 eV and ∼ 0.4 eV average differences from the results obtained with VASP or WEST using Eq. 8 and Eq. 4, respectively. Finally, we note that results reported in Ref. 18 – or previously in the literature –, that suffer from the use of finite basis sets, may be excluded from cross-codes verification purposes.

Figure 5: Mean absolute error (MAE, upper triangle) and maximum absolute error (lower triangle) of vertical ionization potential (VIP) energies computed with different codes, whose names appear on the axes. The comparison does not include eight molecules with Cu, I, Ga and Xe atomic species and the molecules with multiple QP solutions. QP (QP-lin) energies were computed using Eq. 4 (8). All results have been extrapolated. VIP numerical values are reported in Tab. 2.

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Figure 6: Mean absolute error (MAE, upper triangle) and maximum absolute error (lower triangle) of vertical electron affinity (VEA) energies computed with different codes, whose names appear on the axes. The comparison does not include molecules with unbound LUMOs and the eight molecules with Cu, I, Ga and Xe atomic species. QP (QP-lin) energies were computed using Eq. 4 (8). All results have been extrapolated. VEA numerical values are reported in Tab. 3.

27



Figure 7: Energy differences (∆Ei ) between vertical ionization potential (VIP) and vertical electron affinities (VEA) energies computed with the code specified on the abscissa and with the WEST code. The distributions reported on the left (grey) are obtained considering all molecules for which computed VIP or VEA are reported in Tab. 2-3, and removing molecules with unbound LUMOs or with multiple QP solutions. The distributions reported on the right (blue for VIP, and red for VEA) are obtained by further removing eight molecules that include Cu, I, Ga and Xe atomic species. The horizontal lines denote the maximum, mean, and minimum of the distribution of results.

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6

Validation: comparisons with experiments and CCSD(T) quantum chemistry results

Fig. 8 shows that, when compared to CCSD(T) 173 or to experimental values, the results of VASP, WEST and AELO codes give an average deviation of ∼ 0.4 eV, which is comparable to the average deviation of CCSD(T) VIP energies from experiment (∼ 0.3 eV). The coupled cluster results of Krause et al. 173 were computed with the def2-TZVPP basis set; we expect that the estimated accuracy of CCSD(T) may improve if basis set incompleteness errors are included. 174 Overall, even the largest deviations of the VIP (∼ 0.4 eV) are small compared to the values of the VIP considered here, varying in the interval ∼ [4, 23] eV. Due to the unavailability of computed or reference results for some of the molecules of the GW100 set, and due to the exclusion of molecules with unbound LUMOs (for which large discrepancies are observed between codes using localized and extended basis sets), some of the MAE reported in Fig. 8 were obtained considering less than 1/3 of the total number of molecules. We focused the present reproducibility study on G0 W0 @PBE as a substantial amount of computed (and converged) data are available, however several papers in the literature have reported improved accuracy of G0 W0 calculations when hybrid functionals are employed. 14,16,17,76,175–180 We note that, as shown in Ref. 18, the use of generalized plasmon pole models to represent the frequency dependent screening can lead to fortuitous improved agreements with experiments.

29



Figure 8: Mean absolute error (MEA) of vertical ionization potentials (VIP) and electron affinities (VEA), as obtained with the codes listed on the abscissa at the G0 W0 @PBE level of theory (Eq. 4), compared to experimental values and the results of CCSD(T) calculations. CCSD(T) results (obtained at the ∆CCSD(T)/def2-TZVPP level) are from Ref. 173. The comparison is obtained considering molecules for which VIP and VEA are reported in Tab. 23. The comparison does not include molecules with unbound LUMOs or with multiple QP solutions.

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7

Conclusion

In summary, we reported results for the VIP and VEA of the molecules belonging to the GW100 set, as obtained with the WEST code 16 and we compared them to those computed with four other codes: two plane-wave codes (BerkeleyGW 54 and VASP 19 ) and two codes using localized orbitals (FHI-aims 46,47 and TURBOMOLE 48 ). The WEST results were extrapolated with respect to all parameters employed in the calculation (cell size, plane-wave cutoff, number of eigenpotentials to represent the dielectric matrix, and number of Lanczos steps) and they represent converged values which may serve as reference data. Differences between the VIP and VEA computed with different codes vary between 60 and 120 meV, with the best agreement found between WEST and VASP. These differences are up to four times larger than in the case of the best results obtained with DFT codes. Major differences were identified as coming from the choice of frequency integrator (Eq. 5) and the solver used to find numerical solutions of Eq. 4, which is nonlinear and presented multiple solutions for nine molecules (CI4 , KBr, NaCl, BN, O3 , BeO, MgO, Cu2 , and CuCN). However, we also note that even the largest average deviations are relatively small compared to the values of the VIP considered here. The VIP results computed at the G0 W0 @PBE level of theory by the codes considered here show a mean absolute relative error ∼ 4% when compared to experiment and CCSD(T) quantum chemistry results. The verification and validation analysis conducted for VEA energies suffered from the lack of available reference data (both experimental and theoretical). Our analysis of the differences between all-electron and pseudopotential implementations pointed at molecules containing Cu, I, Ga and Xe, as major sources of the differences, and in particular at the pseudopotential used. Further investigations are required to understand the effect of core-valence partition choice for these species. We note that the comparisons presented here included both non-relativistic (BerkeleyGW, FHI-aims, TURBOMOLE) and scalar relativistic (VASP, WEST) calculations, with no sizable discrepancies ascribable to relativistic effects. Irrespective of the basis set used, the best agreement between codes was found for extrap31



olated results. This highlights the importance of extrapolation with respect to all parameters of the calculation when comparing different implementations of the same method. In this paper we performed ' 2000 G0 W0 calculations to obtain extrapolated single particle energies for all 100 molecular systems of the GW100 set; both the accuracy and the feasibility of this verification and validation analysis greatly benefited from the algorithms implemented in WEST. The results reported in this work are available online for further scrutiny in the form Data Collections at www.west-code.org/database.

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Acknowledgement We acknowledge fruitful discussions with Carl Adorf, Timothy Berkelbach, Federico Giberti, Matteo Gerosa, and David Gygi. The data space operations were performed with the Signac framework. 181 This work was supported by MICCoM, as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC02-06CH11357.

Supporting Information Available The Supporting Information contains details of the calculations, convergence studies, and an analysis of the frequency dependence of the G0 W0 self-energy for each molecule.

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Comparison of codes for many-body perturbation theory calculations

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