# GW100: A Slater Type Orbital Perspective

Arno Förster<EMAIL_ADDRESS>Theoretical Chemistry, Vrije Universiteit,
De Boelelaan 1083, NL-1081 HV, Amsterdam, The Netherlands  Lucas Visscher
Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV,
Amsterdam, The Netherlands

###### Abstract

We calculate complete basis set (CBS) limit extrapolated ionization potentials
(IP) and electron affinities (EA) with Slater Type Basis sets for the
molecules in the GW100 database. To this end, we present two new Slater Type
orbital (STO) basis sets of triple- (TZ) and quadruple-$\zeta$ (QZ) quality
whose polarization is adequate for correlated-electron methods and which
contain extra diffuse functions to be able to correctly calculate electron
affinities of molecules with a positive Lowest Unoccupied Molecular Orbital
(LUMO). We demonstrate, that going from TZ to QZ quality consistently reduces
the basis set error of our computed IPs and EAs and we conclude that a good
estimate of these quantities at the CBS limit can be obtained by
extrapolation. With MADs from 70 to 85 meV, our CBS limit extrapolated
ionization potentials are in good agreement with results from FHI-AIMS,
TURBOMOLE, VASP and WEST while they differ by more than 130 meV on average
from nanoGW. With a MAD of 160 meV, our electron affinities are also in good
agreement with the WEST code. Especially for systems with positive LUMOs, the
agreement is excellent. With respect to other codes, the STO type basis sets
generally underestimate EAs of small molecules with strongly bound LUMOs. With
62 meV for IPs and 93 meV for EAs, we find much better agreement to CBS limit
extrapolated results from FHI-AIMS for a set of 250 medium to large organic
molecules.

###### keywords:

GW, GW100, Benchmark, STO, Basis Set

##  1 Introduction

Over the last years, the $GW$ approximation (GWA)1 has increasingly been
applied to calculate charged excitations in finite systems.2, 3, 4, 5, 6, 7,
8, 9 While formally more rigorous, fully self-consistent $GW$ (sc$GW$)
calculations are relatively expensive and not necessarily very accurate for
the calculation of ionization potentials (IP) and electron affinities (EA) for
molecular systems.10, 8 Instead, the perturbative $G_{0}W_{0}$ approach is
often the preferred alternative since it is computationally less demanding
than sc$GW$ and can give significantly more accurate quasiparticle (QP)
energies, provided that it is based on a suitable reference.5, 10, 8

Ideally, the result of a $G_{0}W_{0}$ calculation should be independent of the
particular implementation of the method. In practice, however, choices
regarding the numerical representation of the involved quantities must be
made, including the choice of a single-particle basis as well as a
discretization of frequency and/or time-variables. The choice of the single-
particle basis also entail a choice regarding the representation of the core
electrons as well as a treatment of virtual states. Both factors are decisive
since it is known that core correlation plays a major role in $G_{0}W_{0}$
calculations7 but also since $GW$ QP energies converge very slowly to the
complete basis set (CBS) limit.7, 9, 11, 12 For these reasons, achieving
consensus between different $G_{0}W_{0}$ codes is challenging and requires
careful convergence of a calculation with respect to all technical parameters.
Due to limited resources and/or time constraints, it might not always be
possible in applications to only work with converged parameters. In that case,
one would like to know how a certain technical parameter affects the final
result.

For these reasons, comparison between different codes through systematic
benchmarks is highly desirable. First, it allows to verify that the results
from these codes agree within a reasonable margin of error. For $G_{0}W_{0}$
calculations, one usually aims for an accuracy of 100 meV. Second, such
benchmarks are crucial in order to quantify the influence of the various
technical parameters on the QP energies. Significant efforts in that direction
have been initiated by van Setten et al.7 in 2015 with the publication of the
GW100 database for finite systems. In their work, van Setten et al. compared
the IPs and EAs of 100 small and medium-sized molecules on the
$G_{0}W_{0}$@PBE level of theory, calculated with three different codes, the
Gaussian type orbital (GTO) based all-electron code TURBOMOLE6, 13, the
numerical atomic orbital (NAO) based all-electron code FHI-AIMS14, 15, 4, and
the plane wave (PW) code BerkeleyGW.16, 17 Later, benchmarks for many more
codes followed, including the PW implementations in VASP18, 19, 20, 21 in
201722 and WEST23 in 2018,24 and the real-space finite-element (RSFE)
implementation in nanoGW25 in 2019.26 Also the accuracy of many low-order
scaling implementations of the $G_{0}W_{0}$ method were benchmarked against
the GW100 database.27, 28, 29, 30

These studies established the choice of single-particle basis as a crucial
factor causing major differences between different implementations. For
instance, the results from TURBOMOLE, FHI-AIMS and MOLGW,31 but also from the
low-scaling implementations by Wilhelm et al.28 in CP2K32 and by Duchemin and
Blase30, all using the same def2-GTO type basis sets, agree within a few ten
meV on average for GW100, even though these implementations differ in
frequency treatment as well as calculation of four-center integrals. The
differences between codes using different basis sets are considerably larger.
The discrepancy between the TURBOMOLE and BerkeleyGW results of nearly 300 meV
on average reported in ref. 7 for the Highest Occupied Molecular Orbital
(HOMO) were not necessarily insightful since the BerkeleyGW results were not
CBS limit extrapolated. With only around 60 meV on average, the agreement
between the CBS limit extrapolated PW results obtained with VASP and TURBOMOLE
was found to be significantly better.22 However, for EAs the disagreement
between different codes is considerable larger and differences for systems
with a positive LUMO can easily exceed several eV. It has also been pointed
out in ref. 22 that the type of GTO-type basis set has a major influence on
these EAs and that Dunning’s correlation consistent basis sets are more
suitable than the def2-series which has been used in ref. 7. Beside the choice
of the basis set, the treatment of core electrons (pseudo-potentials vs. all-
electron) also plays a decisive role for many systems.24

Against this background, it seems that further benchmark results for GW100
using different basis set types are useful to advance the current
understanding of the dependence of $GW$ QP energies on the basis set type.
Recently, we presented the first production-level implementation of the
$G_{0}W_{0}$ method with Slater type orbitals (STOs) in the Amsterdam density
functional (ADF) module of the Amsterdam modelling suite (AMS).33 The
$G_{0}W_{0}$ implementation in ADF is detailed in ref. 29 and is based on the
space-time formulation of the $GW$ method first proposed by Godby and
coworkers.34, 35 It uses non-uniform imaginary time and imaginary frequency
grids tailored to the system under investigation36, 37, 21 and treats 4-point
correlation functions with the pair-atomic density fitting (PADF)
approximation, resulting in a low-order implementation with a very low
prefactor.28 To demonstrate the correctness of our implementation, we already
presented IPs and EAs for the GW100 database in ref. 29. However, our results
did not allow a meaningful comparison to implementations with other basis set
types. Since it is not always straightforward to obtain converged minimax
grids,36, 27, 29 we used rather small imaginary frequency and time grids which
were often not converged. Most importantly, in ref. 29 we used standard Slater
type basis sets optimized for independent-electron methods38 which do not
allow a systematic extrapolation to the CBS limit for correlated methods.39,
40

To be able to obtain accurate, CBS limit extrapolated IPs and EAs using STOS,
we report here improvements over our original implementation regarding both
parameters. For once, we implemented improved imaginary time and frequency
grids which allows a systematic convergence to the limit of an infinite number
of grid points. Second, and most importantly, we designed two new Slater type
basis sets for all elements of the periodic table which we call (aug)-TZ3P and
(aug)-QZ6P. They contain extra diffuse functions to be able to obtain accurate
EAs for systems with LUMO above the vacuum level and in the choice of
polarization functions we follow the requirements of correlated-electron
methods41, 40 as closely as possible: ADF only supports basis functions with
angular momenta up to $l=3$ which is a clear restriction since already for
second row atoms, a consistent polarization on the quadruple-$\zeta$ (QZ)
level requires angular momenta up to $l=4$ and higher angular momenta
functions are necessary for heavier elements.40 For simplicity, we will refer
to these basis sets as correlation consistent. However, we emphasize that they
are not correlation consistent in a strict sense but rather as correlation
consistent as possible in our current implementation.

Despite these restrictions, as we will demonstrate in the following by
comparison to results from other codes, the new basis sets allow for a
reliable extrapolation to the CBS limit. Consequently, we present here CBS
limit extrapolated IPs and EAs for the GW100 database with STOs. These results
are the focus of the current work and are meant to complement the previous
studies on GW100 using GTOs, PWs, and RSFEs. Since $G_{0}W_{0}$@PBE generally
do not give accurate QP energies,8, 10, 42 we will only focus on numerical
aspects and abstain from comparison to experimental or high-level quantum
chemistry reference values.43, 44 To complement our results, we also calculate
IPs and EAs of 250 molecules from the GW5000 database.11 This work is
organized as follows: In section 2 we shortly outline the $G_{0}W_{0}$
implementation in ADF and describe our new basis sets. We then present our IPs
and EAs for GW100 and point out similarities and differences to other codes in
section 3. Finally, section 4 summarizes and concludes this work.

##  2 Theory

We start this section by reviewing shortly the $G_{0}W_{0}$ approximation (in
the following, we refer to this approximation as GWA for simplicity) and
describe the main features of the implementation in ADF. For more details we
refer to ref. 29. Here, we mostly focus on the two factors which are most
decisive for this work: The newly designed basis sets and the treatment of
imaginary frequency and imaginary time.

###  2.1 The $GW$ Approximation

The $GW$ approximation is an approximation to the self-energy $\Sigma$ which
appears in the Dyson equation.45 It is defined as the difference between a
dynamical one-body Hamiltonian $H$ and a static reference Hamiltonian $h$46
corresponding to a non-interacting Fermi system (in practice, $h$ will be the
Hamiltonian of generalized47 KS48, 49, 50 problem),

$\Sigma(1,2)=H(1,2)-h(1,2)\;.$ (1)

$H$ is defined by $H(\omega)=\omega-G(\omega)^{-1}$, where $G$ is the Green’s
function of the interacting Fermi-system.
$1=(\bm{r}_{1},\sigma_{1},\omega_{1})$ collects space-, spin- and frequency
coordinates of a particle but, as we do not consider spin-orbit coupling in
the current work, we will omit spin for simplicity in the following. If one
approximates $\Sigma$ as diagonal and real and transforms to a molecular
orbital representation, the Dyson equation becomes

$\omega_{n}=\braket{n}{h}{n}+\braket{n}{\operatorname{Re}\left(\Sigma(\omega_{n})\right)}{n}\;,$
(2)

where $n$ labels a single-particle state. In case $n$ refers to the HOMO
(LUMO) level, $-\omega_{n}$ is equal to the IP (EA) of the system. In the GWA,

$\Sigma(1,2)=iG_{0}(1,2)*W_{0}(1,2)-v_{xc}(1,2)\;,$ (3)

where $*$ denotes convolution, $v_{xc}$ is the exchange-correlation potential
of generalized KS-DFT, $W_{0}$ is the dynamically screened Coulomb interaction
which is related to the bare Coulomb interaction $V$ via a Dyson equation,

$W_{0}(1,2)=V(1,2)+V(1,3)P_{0}(3,4)W_{0}(4,2)\;,$ (4)

and

$G_{0}(\omega)=(\omega-h)^{-1}$ (5)

is the one-body Green’s function corresponding to the non-interacting
reference system. Integration over repeated indices is implied. The kernel of
this Dyson equation, the polarizability $P_{0}$, is in turn calculated from
the non-interacting Green’s function in the random phase approximation
(RPA)46,

$P_{0}(1,2)=-iG_{0}(1,2)*G_{0}(2,1)\;.$ (6)

In principle, the RPA is distinct from the GWA but when referring to the $GW$
method in a quantum chemistry context it is usually implied that the RPA is
made for $P_{0}$.

The $G_{0}W_{0}$ method in ADF is implemented in the space-time formalism
introduced by Godby and coworkers.34, 35 This means, eq. (4) is solved in
imaginary frequency, while $P$ and $\Sigma$ are calculated in imaginary time.
From the imaginary time domain, $\Sigma$ is brought to the imaginary frequency
axis. Since $\Sigma$ is analytic in the upper half plane, it can be
analytically continued to the real axis. In practice this requires discrete
grids to sample the imaginary time and imaginary frequency axes. In the
absence of numerical noise we can use our knowledge of the self-energy on
$N_{\omega}$ points on the imaginary frequency axis to interpolate the input
data with a rational function using the Padé technique as outlined by Vidberg
et al.51 The crucial assumption in this approach is that knowledge of $\Sigma$
at a small number of points on the imaginary frequency axis allows for an
accurate interpolation. In case the single QP picure is valid, this assumption
is reasonable for states in the vicinity to HOMO and LUMO energy since the
interacting Greens-function has qualitatively the same structure as the non-
interacting one (See for example ref. 52 or ref. 53) and thus the self-energy
(by definition) will be smooth.54 The $G_{0}W_{0}$ implementation in ADF
follows the work flow shown in figure 1.

Figure 1: Schematic work flow for the calculation of the self-energy in the
whole complex plane in ADF. FT denotes Fourier transform and AC denotes
analytical continuation.

###  2.2 Imaginary Time and Frequency Grids

To discretize imaginary frequency and imaginary time in a suitable way, we
follow Kresse and coworkers (building on earlier work by Almlöf55 and
Hackbusch and coworkers56) and use non-uniform grids for both domains and
switch between them using non-unifom Fourier transforms (FT).36, 37, 21 The
grid points are selected at run-time and are tailored to the system under
investigation. Our implementation of these grids closely follows Kresse and
coworkers36, 37, 21 and is outlined in appendix A.

Similar grids in space-time method implementations have been used by other
researchers as well27, 28, 30 to implement the $G_{0}W_{0}$ method for finite
systems. Recently, Wilhelm et. al. benchmarked the convergence of QP energies
in the GW100 database with respect to the grid sizes. In our older work,29 our
frequency grids were restricted to not more than 19 grid points and ref. 28
clearly shows such grids to be insufficient to convergence all IPs and EAs in
GW100. Consequently, we calculated new frequency grids which allow us to use
up to 40 imaginary frequency points which ensures that the results are
converged with respect to this parameter.28 In ref. 28, Wilhelm et al. could
also reproduce the results obtained by van Setten et al.7 with the TURBOMOLE
code with an accuracy of a few meV. Since TURBOMOLE performs the frequency
integration fully analytically, we conclude that the frequency integration is
a numerical parameter which is well under control in our updated
implementation.

###  2.3 Discretization of Real Space

To discretize real space, we use a basis of $N_{AO}$ STOs,
$\mathcal{S}=\left\\{\chi_{\mu}\right\\}_{\mu=1,\dots\;,N_{AO}}$, which are
related to the molecular orbitals (MO) $\phi$ by

$\phi_{n}=\sum_{\mu}b_{n\mu}\chi_{\mu}(\bm{r})\;,$ (7)

All quantities which occur in (4) are 4-point correlation functions and to
this end, solution of (4) would scale as $N^{6}$ when expanding these
quantities in terms of $\mathcal{S}$. In the density-fitting (DF)
approximation,57, 58, 59, 60, 61, 62, 63 one approximates the product basis of
atomic orbitals $\mathcal{P}=\mathcal{S}\bigotimes\mathcal{S}$ with an
auxiliary basis
$\mathcal{A}=\left\\{f_{\alpha}\right\\}_{\alpha=1,\dots\;,N_{aux}}$, where
$N_{aux}$ grows linearly with system size, and introduces a basis
transformation,

$\mathcal{C}:\mathcal{P}\mapsto\mathcal{A}\;,\quad\chi_{\mu}(\bm{r})\chi_{\nu}(\bm{r})=\sum_{p}c_{\mu\nu
p}f_{p}(\bm{r})\;.$ (8)

The equations which need to be solved now are

$\displaystyle P_{pq,\tau}=$ $\displaystyle c_{\mu\nu
p}P_{\mu\nu\kappa\lambda,\tau}c_{\kappa\lambda q}=-ic_{\mu\nu
p}\left[G_{0}\right]_{\mu\kappa,\tau}\left[G_{0}\right]_{\nu\lambda,\tau}c_{\kappa\lambda
q}$ (9) $\displaystyle\left[W_{0}\right]_{pq,\omega}=$ $\displaystyle
V_{pq}+V_{pr}\left[P_{0}\right]_{rs,\omega}\left[W_{0}\right]_{sq,\omega}$
(10) $\displaystyle\left[\Sigma_{xc}\right]_{\mu\nu,\tau}=$ $\displaystyle
i\sum_{\kappa\lambda}\sum_{pq}\left[G_{0}\right]_{\kappa\lambda,\tau}c_{\mu\kappa
q}\left[W_{0}\right]_{pq,\tau}c_{\nu\lambda q}\;.$ (11)

The computational bottlenecks in these equations are now the basis
transformations in eqs. (9) and (11) which both scale as $N^{4}$ with system
size. However, when the map $\mathcal{C}$ from eq. (8) is constrained such
that the number of non-zero-elements in $\mathcal{C}$ only grows linearly with
system size, both equations only require $N^{2}$ operations.27, 29 PADF is a
way to achieve this goal by way of restricting the density fit to pairwise
sums only and by subsequently introducing distance cut-offs. Also other
schemes to introduce sparsity into $\mathcal{C}$ have been applied
successfully to achieve low-order scaling $G_{0}W_{0}$ implementations27, 28
(we refer to ref. 28 for a detailed discussion of the pros and cons of
different techniques), but PADF is arguably the best practical method to make
the map $\mathcal{C}$ as sparse as possible. Constructing $\mathcal{C}$ in
that way only requires evaluation of a small number of integrals compared to
other approaches58 which makes this approach especially suitable for localized
basis functions for which Coulomb integrals can not be calculated
analytically, such as STOs or numerical atomic orbitals. In fact, the compute
time spent on evaluating the remaining integrals in our implementation is
negligible compared to the time needed to complete a $G_{0}W_{0}$ calculation.

In PADF, only the fit functions centred on atoms $A$ and $B$ are used to
expand a product of AOs which are centred on atoms $A$ and $B$. This
translates into a matrix $\mathbf{c}$ for which $c_{\mu\nu p}\neq 0$ only if
$\mu,\nu,p\in(A,B)$. One further introduces a cut-off so that for 2 AOs $\mu$,
$\nu$ centred on atoms far away from each other, all corresponding matrix
elements $c_{\mu\nu p}$ are set to zero. Employing PADF, standard auxiliary
basis sets designed for global density fitting are not suitable (they are
simply too small) and dedicated fit sets are needed. For an in-depth
discussion of challenges in the design of these fit sets for correlated-
electron methods we refer the reader to ref. 29 and for a more explicit
description of the fit sets, a list of exponents of the fit functions and more
technical details to ref. 64.

###  2.4 Slater Type Orbital Basis Sets

In eq. 7, $\mu$ is a composite index, collecting the five defining parameters
of a Slater type function

$\chi_{\mu}(r,\theta,\phi)=\chi_{A,\alpha,n,l,m}(r,\theta,\phi)=C(\alpha,n)r^{n-1}e^{-\alpha
r}Z_{lm}(\theta,\phi)\;,\quad r=|\bm{r}-\bm{R}_{A}|\;,$ (12)

the exponent $\alpha$ and the quantum numbers $(n,l,m)$, as well as the
nucleus $A$ on which the function is centred. In (12), $C(\alpha,n)$ is a
normalization constant and $Z_{lm}$ denotes a real spherical harmonic. The
main difference to Gaussian type functions is in the dependence on $r$ in the
exponential instead of $r^{2}$ which mainly results in a different behavior
close to the atomic nuclei and a slower decay for large $r$. While STO-type
basis sets are well developed for independent-electron methods65, 38, 66, 67,
68 we are not aware of any attempt to construct general Slater type basis sets
which are consistent with the requirements of correlated-electrons methods.
Here, we make a first attempt to present such basis sets for the whole
periodic table. The concept of correlation-consistency has also been applied
to NAO basis sets for RPA total energy calculations.69

####  2.4.1 Correlation Consistent Atomic Basis Sets

In the construction of our basis sets we make two assumptions: First, we
assume that the principles guiding the construction of GTO-type basis sets for
correlated-electron methods which have been developed over the last decades
should be applicable to STO-type basis sets as well. This can be justified as
follows: Based on numerical experiments on correlation consistent GTO-type
basis sets, Helgaker et al.70 established the relation

$E^{corr}_{\infty}-E^{corr}_{X}=aX^{-3}$ (13)

between the correlation energy $E^{corr}_{\infty}$ at the CBS limit and the
correlation energy $E^{corr}_{X}$ calculated with a given correlation
consistent basis set with cardinal number $X=l_{max}+1$. Conceptually, their
formula is based on earlier work by Schwartz71 and the mathematical more
rigorous work by Hill72 on the convergence of the ground state of the Helium
atom in a full configuration interaction calculation with respect to the
single-particle basis. Later on, Kutzelnigg and Morgan73 generalized that
result to arbitrary $n$-electron systems for MP2 calculations.

In principle, (13) is only valid in the limit of large $X$, however, there is
numerical evidence that it is already a good approximation for $X=3$ and
$X=4$.70, 39 Since (13) is independent of the type of localized basis
functions71, 72, 73, this should also be the case for STO-type basis sets
provided that they are also constructed in a correlation consistent fashion as
first defined by Dunning;41 such that the total basis set incompleteness error
is distributed equally between the different angular momenta functions. This
requires that polarization functions are added in well-defined sequences.40
There is numerical evidence40 that the consistent polarization for correlated
methods is $1(l_{occ}+1)$ on the double-$\zeta$ (DZ),
$2(l_{occ}+1)1(l_{occ}+2)$ on the TZ and $3(l_{occ}+1)2(l_{occ}+2)(l_{occ}+3)$
on the QZ level, where $l_{occ}$ denotes the angular momentum of the highest
occupied orbital in an atom. This requirement is usually only followed
strictly for the first three rows of the periodic table40 and the design of
correlation consistent basis sets for heavier elements might follow different
principles.74, 75, 76, 77, 78

####  2.4.2 Construction of Correlation Consistent Slater-Type Basis sets

Based on the considerations above, we construct correlation consistent basis
sets of TZ and QZ quality. We name these basis sets TZ3P and QZ6P,
respectively. We also augment these basis sets with additional diffuse
functions and we name these basis sets aug-TZ3P and aug-QZ6P, respectively.
The acronym $xP$ refers to the number of polarization functions we use for the
elements of the first three rows of the periodic table. This choice is
consistent with the requirements for correlation consistent basis sets stated
above.

At this point, we introduce our second assumption: Since the ADF code only
supports basis functions up to $l=3$, for second- and third-row elements we
chose the polarization $2d1f$ and $3d3f$ for TZ and QZ respectively, and for
consistency also $2p1d$ and $pd3d$ for Hydrogen and Helium. A good
justification for the validity of this approximation can not be given and as
we will see later, our results indeed suggest that the replacement of a $g$
with another $f$ functions negatively affects the QP energies for small
molecules. Generalization of ADF to accommodate use of higher angular momenta
functions is therefore desirable.

The TZ2P and QZ4P basis sets which have been described in detail in ref. 38
serve as starting points for our new basis sets. TZ2P (QZ4P) is of DZ (TZ)
quality in the core region and of TZ (QZ) quality in the valence region. For
the TZ3P and QZ6P basis sets, we chose not to optimize exponents but rather to
add additional polarization functions in an ad-hoc manner. This approach can
be justified by the observation that precise values of exponents become less
important as a basis set approaches completeness.65, 40 This is especially
true for molecules as opposed to isolated atoms. The latter case generally
requires larger and more optimized basis sets than the molecular case.
Instead, we rather focus on choosing the exponents in a way that their overlap
is small and linear dependency problems are more likely to be avoided.

TZ3P is simply obtained by augmenting the TZ2P basis set by another
$l_{occ}+1$-function for all elements. The exponent is chosen so that it is
twice as large as the exponent of the $l_{occ}+1$-function in the TZ2P basis
set. This is due to the fact that the exponents of the polarization functions
in the TZ2P and QZ4P basis sets are chosen in a way that the basis sets become
more accurate in the valence region, which is favorable for the calculation of
bonding energies. The calculation of IPs also requires the accurate
representation of the electron density closer to the core, and TZ3P should
yield a major improvement over TZ2P in that respect. The same reasoning has
also been followed in ref. 79. In complete analogy, the QZ6P basis set is
obtained by adding an additional tight $l_{occ}+1$-function and an
$l_{occ}+2$-function for all elements. For Fluorine and Chlorine we added an
additional shell of $d$-function since we found exceptionally slow convergence
of QP energies to the CBS limit. This will be discussed in more detail in
section 3.

The exponents $\alpha_{1},\alpha_{2}$ of the polarization functions in the
QZ4P basis set fulfill $\frac{\alpha_{1}}{\alpha_{2}}\approx 2$ for each $l$.
Thus, the polarization functions in the TZ3P and QZ6P basis sets loosely
follow an even-tempering scheme80,

$\alpha_{i}=\alpha_{1}\beta^{i-1}\;,\quad i=2,\dots,M\;,\quad\beta=2\;,$ (14)

with M being equal to two (three) for $l_{occ}+1$ in TZ3P (QZ6P) and one
(three) for $l_{occ}+2$. The value of $\beta$ is rather large to avoid linear
dependency problems, and in conjunction with a rather small $\alpha_{1}$
ensures that the exponents span a rather wide range of values in the QZ6P
case.81 Ren et al. recently used a similar reasoning to chose the exponents of
Slater type functions for $G_{0}W_{0}$ calculations for periodic systems.82

####  2.4.3 Adding Diffuse Functions

It is known from electron scattering experiments that molecular electron
affinities are sometimes negative, i.e. their anion state at the geometry of
the neutral molecule is unstable.83, 84, 85 This corresponds to a positive
LUMO QP energy which formally corresponds to a non-normalizable continuum
orbital. However, as an artefact of working with a finite basis, the orbital
will always be constrained to be normalizable.86 Very diffuse functions are
then needed to mimic the continuum state, and for this reason we augment our
basis sets with additional diffuse functions (See also ref. 87 and ref. 88).
The resulting basis sets are denoted as aug-TZ3P and aug-QZ6P and are obtained
from TZ3P and QZ6P, respectively by adding diffuse $s$, $p$ and $d$-functions
for all element types, with the important exceptions of first row atoms where
we only add diffuse $s$ and $p$ functions.This is slightly different from
usual augmentation approaches encountered for Gaussian type basis sets.
Dunning and coworkers defined the prefix ”-aug” to mean adding an additional
diffuse functions for all angular momenta already present in a basis set for
every atom type.87 On the contrary, in a process which they denoted as minimal
augmentation, Truhlar and coworkers only added diffuse $s$ and $p$ functions
to all elements heavier than Hydrogen.88 Our approach is can be seen as a
compromise between both. We decided to chose the exponents of the diffuse
functions in line with (14), except for a small shift. More precisely, the
exponent of the most diffuse function for a given angular momentum is
$\alpha_{l,min}$, the exponents of the diffuse function is chosen to be

$\alpha_{l,diffuse}=\alpha_{l,min}/2+0.05\alpha_{l,min}$ (15)

The exceptions are the elements Hydrogen to Beryllium for which we chose the
already optimized exponents by Chong et al.90 These exponents are very close
to fulfilling (15). Our choice of exponents is a compromise between two
requirements: We would like the additional functions to be as diffuse as
possible but we still want to be able to fit them accurately with a large
auxiliary basis set. As discussed for instance in ref. 29, this is already
challenging. In practice, this means that $2\alpha_{l,diffuse}$ should be at
least equal or preferably slightly larger (for this reason we added the shift
in (15)) than the exponent of the most diffuse function in the auxiliary basis
set.

##  3 Results

###  3.1 Computational Details

All calculations have been performed with a locally modified development
version of ADF202091 using the implementation as described in ref. 29 and
using the updated imaginary frequency grids as described in this work.The code
to generate the imaginary time grids can be found on Github.
(https://github.com/bhelmichparis/laplace-minimax). Please also see ref. 56
and 114.

####  3.1.1 GW100

We follow the protocol for GW100 as described by van Setten et al.7 and
perform non-relativistic $G_{0}W_{0}$@PBE93, 94 calculations. We use the
structures as available on the webpage for the GW100 database and also use the
updated structures for Phenol and Vynilbromide.Data downloaded from the
webiste of the GW100 project by Van Setten et al., https://gw100.wordpress.com
For consistency with other benchmarks for GW100, we always use the QP energy
obtained from the KS LUMO energy, which is usually, but not always, the
energetically lowest virtual QP energy. For a detailed discussion of the
effect of orbital reordering we refer the reader to ref. 26.

We performed calculations with the augmented versions of the basis sets
described in this work and extrapolated them to the CBS limit as described for
instance in ref. 6: We calculate the QP energies $\epsilon_{n}$ with the aug-
TZ3P and aug-QZ6P basis sets and estimate the CBS limit as

$\epsilon_{n}^{CBS}=\epsilon_{n}^{QZ}-\frac{1}{N^{QZ}_{bas}}\frac{\epsilon_{n}^{QZ}-\epsilon_{n}^{TZ}}{\frac{1}{N^{QZ}_{bas}}-\frac{1}{N^{TZ}_{bas}}}\;,$
(16)

where $\epsilon_{n}^{QZ}$ ($\epsilon_{n}^{TZ}$) denotes the value of the QP
energy using aug-QZ6P (aug-TZ3P) and $N^{QZ}_{bas}$ and $N^{TZ}_{bas}$ denote
the respective numbers of basis functions. Since we work with spherical
harmonics, there are 5 $d$ and 7 $f$ functions. In all calculations, we set
the numericalQuality key to Good and used 32 imaginary time and 32 imaginary
frequency points each. As explained in section A, this should be understood as
the upper bound for the number of grid points. We report the number of grid
points actually used in each calculation in the supporting information. Per
default we also used the Good auxiliary basis set. However, for all systems in
GW100 containing fourth- or fifth-row elements and for all systems smaller
than 4 atoms we used the Normal auxiliary basis set to prevent issues with
over-fitting (see ref. 64). For systems with a positive LUMO we used the
veryGood fit set. This fit set contains additional diffuse fit functions which
are necessary to accurately represent the diffuse electron densities of these
systems. These fit sets are discussed in ref. 64. Also for Guanine, Uracil,
Pentasilane and Ethoxyethane we used the VeryGood fit set since we observed
inconsistencies with the Good fit set. I all calculations, we set Dependency
Bas=5e-4 in the AMS input as described in ref. 29.

####  3.1.2 GW5000

We also performed $G_{0}W_{0}$@PBE096, 97 calculations for a subset of of 250
molecules from the GW5000 database11 using the zeroth order regular
approximation (ZORA).98, 99, 100, 101 Calculations are performed with the non-
augmented TZ3P and QZ6P basis sets. Eq. (16) is used for CBS limit
extrapolation. We use 24 points in imaginary frequency and imaginary time
each, numericalQuality Good and the Normal auxiliary basis set which is
sufficient for non-augmented basis sets.29 We set Dependency Bas=5e-4.

###  3.2 Basis Set Errors

Before we discuss in detail the comparison of the STO results to the ones from
other codes, we consider the basis set errors and basis set convergence
properties of aug-TZ3P and aug-QZ6P. Using eq. (16), one implicitly assumes
that when going to a larger basis set, each additional basis function reduces
the basis set error on average by the same amount. In other words, one assumes
uniform convergence of the basis set expansion, a natural property of finite
elements in real- or reciprocal space. It is clear that such an assumption
will only be justified for rather large basis sets and, for the same reason,
extrapolation is generally more reliable for larger systems. Usually one would
also like to use three or even more data points instead of using (16). It has,
however, been pointed out39 that including a calculation with a basis set of
quality lower than TZ will detoriate the quality of the fit. Therefore, we
calculate QP energies at the CBS limit from TZ and QZ results only.

Figure 2: Basis set errors with respect to the CBS limit extrapolated IPs in
the GW100 database using the aug-TZ3P and aug-QZ6P basis sets. The univariate
plots show the distributions of errors with respect to the CBS limit
extrapolated values. All values are in eV. Figure 3: Basis set errors with
respect to the CBS limit extrapolated EAs in the GW100 database using the aug-
TZ3P and aug-QZ6P basis sets. The univariate plots show the distributions of
errors with respect to the CBS limit extrapolated values. All values are in
eV.

To demonstrate the accuracy of an extrapolation scheme, one would ideally
compare the extrapolated result to one obtained with a very large basis which
already gives a result very close to the CBS limit. Due to the limitations of
our basis sets to angular momenta $\leq 3$, this is not possible for us. As a
rule of thumb, extrapolation with basis sets of cardinality $X$ and $X-1$ can
provide the accuracy of a calculation using a basis set of accuracy $X+1$.40
Recently, Bruneval et al.12 found basis set errors of about 60 meV for the IPs
of a large set of small to medium organic molecules with the cc-pV5Z basis
set. This is of the same order as the typical accuracy in a photo-ionization
experiment12 and considerably lower than the 150 meV for IPs which are usually
found using the cc-pVQZ basis set.11, 12 For EAs, one can usually expect
errors of the same order of magnitude than IPs when augmented basis sets are
used.102 It is thus reasonable to use this number as an estimate of the
average error in our extrapolation.

Figure 4: Correlation between differences between the aug-TZ3P and aug-QZ6P
results $\Delta_{T-Q}$ for EAs (y-axis) and IPs (x-axis). The univariate plots
show the corresponding distributions. The blue straight line is a linear fit
and the dotted line is defined by $\Delta_{IP}=\Delta_{EA}$. All values are in
eV.

The distributions of basis set errors with respect to the CBS limit
extrapolated IPs and EAs (excluding noble gases and the hydrogen molecule in
the latter) for the GW100 database are shown in figures 2 and 3, respectively.
The average basis set error reduces from 300 (290) meV to 170 (170) meV for
IPs (EAs), i.e. there seems to be no qualitative difference in the convergence
to the CBS limit for IP and EA as one would expect for augmented basis sets.
We also notice that with one exception of the IP of BeO, the basis set error
on the TZ level is always lower than the one on the QZ level. In both plots,
we also highlight some systems for which the convergence to the CBS limit
seems to be rather slow, ie. The difference between QP energies on the TZ and
QZ level are very large. For these systems, CBS limit extrapolation will be
less accurate. Without exception, the problematic systems are composed of only
a few atoms.

Also a good correlation between the basis set errors for IPs and EAs is
desirable since it implies a fast convergence of the HOMO-LUMO QP gap to the
CBS limit. Fast convergence of this quantity with basis sets augmented with
diffuse functions has been demonstrated before.102, 29 At first glance, the
distributions of IP and EA BSEs appear rather similar, suggesting that such an
error cancellation might also be found for our basis sets. To investigate this
further, we plot all pairs $(\Delta(QZ-TZ)_{IP_{i}},\Delta(QZ-TZ)_{EA_{i}})$
(bivariate plot) together with the corresponding error distributions
(univariate plots) in figure 4 (omitting again all noble gases and H2). The
blue solid line is a linear fit, the dotted black line is defined by the
equation $\Delta(QZ-TZ)_{IP_{i}}=\Delta(QZ-TZ)_{EA_{i}}$, and Gaussian kernels
are fitted to the univariate distributions. A few molecules with large BSE for
the IP but small basis set error for the EA (Cu2) or vice-versa (BN, TiF4, F2,
AlI3) aside, most systems cluster around the dotted line in the grey shaded
area in which IP and EA BSEs should cancel each other to a large extent.

###  3.3 Comparison to GTO-Type Basis Sets

Figure 5: Distribution of deviations of the ADF IPs to the def2-GTO IPs for
the GW100 database for TZ and QZ basis sets ($\Delta_{X,i}\;,X=\text{TZ,QZ}$).
The univariate plots show the distribution of the $\Delta_{TZ,i}$ (upper
histogram) and the $\Delta_{QZ,i}$ (left histogram). The bivariate plots shows
the pairs $(\Delta_{TZ,i},\Delta_{QZ,i})$ and the blue line is a linear fit.
The dotted line is defined by $\Delta_{T}=\Delta_{Q}$. Systems containing 5th
row elements are highlighted in red. All values are in eV.

Additional insight into the convergence properties of the STO type basis sets
is provided when comparing them to GTO type ones of the same cardinality. Such
a comparison is made in figure 5 for results on the TZ and and for the QZ
level. The univariate plot on top of figure 5 shows the deviation of the aug-
TZ3P IPs to the ones obtained with def2-TZVP and the univariate plot on the
right side of figure 5 shows the same for aug-QZ6P and def2-QZVP (here and in
the remainder of this paper, we use the results calculated with the TURBOMOLE
code whenever we refer to def2-GTO basis sets. We could have equally well used
results obtained with other codes like FHI-AIMS or MOLGW). Again, Gaussian
kernels are fitted to the univariate distributions. The bivariate plot shows
the individual pairs $(\Delta_{T,i},\Delta_{Q,i})$, with $\Delta_{T,i}$
($\Delta_{Q,i}$) being the differences between the IPs calculated with aug-
TZ3P and def2-TZVP (aug-QZ6P and def2-QZVP) for the $i$th datapoint in the
GW100 database. The dotted line is defined by $\Delta_{T,i}=\Delta_{Q,i}$. The
systems for which the QP equations (2) can have multiple solutions (CI4, KBr,
NaCl, BN, O3, BeO, MgO, Cu2, and CuCN) are excluded from this comparison.24
Also Ag2 is not shown since the deviations on the TZ and QZ level are
exceptionally large.

We observe, that the maximum of the Gaussian kernel function is close to zero
eV for the TZ and closer to 0.1 eV for the QZ basis sets. In other words, the
aug-QZ6P IPs are consistently smaller than the def2-QZVP IPs (with a mean
deviation (MD) of 120 eV), while the aug-TZ3P ones are with a MD of 60 meV
only slightly smaller than the def2-TZVP IPs on average. The missing
$l_{occ}+3$ function in the aug-QZ6P basis set might be a reason for this
discrepancy. We will see in the next subsection that the CBS limit
extrapolated IPs calculated with our basis sets are on average lower than the
def2-GTO results. This section clearly shows that this discrepancy is mostly
caused by the differences on the QZ level.

The deviations of the STO-type to the respective def2- basis sets are strongly
correlated. In cases in which the aug-TZ3P IPs are considerably smaller than
their def2-TZVP counterparts, also the aug-QZ6P IPs will be much smaller than
the def2-QZVP ones. Good examples are the five molecules represented by the
points in the upper right corner of the bivariate plot. Since it is known that
the GTO-type basis sets allow for a reliable CBS limit extrapolation, this
fact is highly important since it guarantees that the same CBS limit
extrapolation can be performed using our STO-type basis sets.

We also shortly comment on the systems containing 5th-row elements,
highlighted in red. With the exception of one of them (Rb2), the agreement on
the TZ is rather good, while they agreement on the QZ level is significantly
worse. This is again due to the inconsistent polarization for the heavier
elements mentioned above; approaching the CBS limit for heavier elements
becomes difficult without Slater functions with angular momentum larger than
$l=3$.

To summarize the key points of this section, our STO type basis sets seem to
behave qualitatively similar to the GTO-type basis sets, although the
improvement when going from TZ to QZ is smaller for the STO- than for the GTO-
type basis sets. Together with the good correlation of deviations on the TZ
and QZ level, this indicates that our basis sets allow indeed for a meaningful
CBS limit extrapolation. However, the CBS limit extrapolated IPs from the STO-
type basis sets will on average be lower than their counterparts calculated
using GTOs.

###  3.4 Comparison to Other Codes

####  3.4.1 KS Eigenvalues

Before we dive into our comparison of the $GW$ QP energies, we shorty compare
our KS eigenvalues for the systems in GW100 to the ones from other codes. Here
and in the following, we do not include Phenol and Vynilbromide in the
statistical analysis since different structures have been used in the past for
both systems.22 Our results (see supporting information for the raw data for
all 100 systems) only confirm what is already well known; For KS eigenvalues,
the agreement between different codes is generally excellent. For example, the
CBS limit extrapolated KS HOMO energies from the WEST code agree with the ones
obtained from def-GTO calculations within 30 meV on average, with a maximum
deviations of 176 meV.24 These figures reduce to 24 meV and 92 meV when the
plane-wave results are compared to def2-QZVP.24 With only 19 meV, the
agreement from VASP to def2-QZVP is even better.22 One should keep in mind
that the extrapolation schemes for correlated-electron methods and localized
basis functions are not necessarily useful to extrapolate KS eigenvalues as
has already been pointed out in ref. 22 and in ref. 24. Such a comparison
should rather be based on non-extrapolated results.103, 104

Our KS HOMO eigenvalues calculated on the aug-QZ6P level of theory show a MAD
of 26 meV to the ones on the def2-QZVP level and of 22 meV to the CBS limit
extrapolated values calculated with the WEST code. Our LUMO eigenvalues only
differ to the ones from WEST by 35 meV on average. Major deviations of more
than 150 meV are only found for Helium (340 meV), H2 (280 meV) and Ag2 (340
meV). With deviations of more than 420 meV to WEST and 490 meV to def-QZVP,
the latter system is also the only outlier for IPs. However, when the ZORA is
made, the deviations to WEST reduce to 15 meV for the HOMO and 17 meV for the
LUMO, respectively. Also, the deviation to the def2-QZVP IP reduces to an
acceptable value of 50 meV which is rather strange, given that the latter has
been calculated without relativistic corrections.

####  3.4.2 Ionization Potentials

We now turn our attention to the IPs calculated within the GWA. In addition to
Phenol and Vynilbromide, we again exclude the systems for which the QP
equations (2) can have multiple solutions (CI4, KBr, NaCl, BN, O3, BeO, MgO,
Cu2, and CuCN)24 from the following statistical comparison, but also TiF4 and
OCS for which no IPs from the WEST code are available. Due to large
discrepancies between WEST and VASP, we also exclude the systems containing
Iodine, Gallium and Xenon. Finally, we also exclude all remaining systems
containing 5th-row elements from our analysis since for these systems
(especially the ones containing Iodine and Ag2) the different treatment of
relativistic effects have been shown to significantly affects QP energies.22
This leaves us with a set of 81 molecules whose IPs we include in the
statistical analysis in this section.

Table 1: $G_{0}W_{0}$@PBE ionization potentials (IP) for the GW100 database (third column) Columns four to seven denote deviations of the ADF IPs to the ones from reference $X$, $\Delta_{X}=IP_{X}-IP_{ADF}$. All values are in eV. | Name | ADF | $\Delta_{def2-GTO}$ | $\Delta_{VASP}$ | $\Delta_{WEST}$ | $\Delta_{nanoGW}$
---|---|---|---|---|---|---
1 | Helium | $23.31$ | $0.18$ | $0.07$ | $0.11$ | $-0.11$
2 | Neon | $20.06$ | $0.27$ | $0.11$ | $0.27$ | $0.39$
3 | Argon | $15.26$ | $0.02$ | $0.06$ | $0.11$ | $0.19$
4 | Krypton | $13.71$ | $0.18$ | $0.22$ | $0.05$ | $-0.06$
5 | Xenon | $11.88$ | $0.45$ | $0.26$ | $1.24$ | $0.23$
6 | Hydrogen | $15.88$ | $-0.03$ | $-0.03$ | $-0.04$ | $-0.13$
7 | Lithium dimer | $5.08$ | $-0.02$ | $0.01$ | $-0.04$ | $-0.04$
8 | Sodium dimer | $4.89$ | $0.02$ | $0.04$ | $0.09$ | $0.05$
9 | Sodium tetramer | $4.25$ | $-0.04$ | $-0.08$ | $-0.01$ | $-0.03$
10 | Sodium hexamer | $4.30$ | $0.04$ | $0.04$ | $0.07$ | $0.05$
11 | Potassium dimer | $3.99$ | $0.09$ | $0.13$ | $0.15$ | $0.07$
12 | Rubidium dimer | $3.78$ | $0.10$ | $0.24$ | $0.23$ | $0.08$
13 | Nitrogen | $14.79$ | $0.26$ | $0.14$ | $0.15$ | $0.04$
14 | Phosphorus dimer | $10.26$ | $0.12$ | $0.09$ | $0.17$ | $0.06$
15 | Arsenic dimer | $9.58$ | $0.08$ | $0.01$ | $-0.03$ | $-0.09$
16 | Fluorine | $14.99$ | $0.12$ | $-0.06$ | $0.01$ | $0.03$
17 | Chlorine | $11.18$ | $0.13$ | $0.14$ | $0.23$ | $0.13$
18 | Bromine | $10.46$ | $0.10$ | $0.11$ | $-0.02$ | $-0.17$
19 | Iodine | $9.04$ | $0.54$ | $0.48$ | $1.37$ | $0.23$
20 | Methane | $13.89$ | $0.11$ | $0.13$ | $0.10$ | $0.26$
21 | Ethane | $12.35$ | $0.11$ | $0.15$ | $0.09$ | $-0.19$
22 | Propane | $11.85$ | $0.04$ | $0.05$ | $-0.01$ | $-0.32$
23 | Butane | $11.52$ | $0.07$ | $0.09$ | $-0.11$ | $-0.27$
24 | Ethylene | $10.28$ | $0.12$ | $0.14$ | $0.11$ | $-0.01$
25 | Acetylene | $11.14$ | $-0.05$ | $-0.07$ | $-0.05$ | $-0.21$
26 | Tetracarbon | $10.74$ | $0.17$ | $0.15$ | $0.16$ | $0.06$
27 | Cyclopropane | $10.63$ | $0.02$ | $0.09$ | $0.04$ | $-0.15$
28 | Benzene | $9.07$ | $0.03$ | $0.04$ | $0.01$ | $-0.11$
29 | Cyclooctatetraene | $8.16$ | $0.02$ | $0.03$ | $0.00$ | $-0.14$
30 | Cyclopentadiene | $8.43$ | $0.02$ | $0.04$ | $0.01$ | $-0.13$
31 | Vinyl fluoride | $10.24$ | $0.08$ | $0.04$ | $0.05$ | $-0.06$
32 | Vinyl chloride | $9.82$ | $0.08$ | $0.10$ | $0.12$ | $-0.01$
33 | Vinyl bromide | $9.03$ | $0.11$ | $0.72$ | $0.61$ | $0.49$
34 | Vinyl iodide | $8.93$ | $0.26$ | $0.34$ | $0.88$ | $0.09$
35 | Tetrafluoromethane | $15.43$ | $0.17$ | $-0.02$ | $0.08$ | $-0.04$
36 | Tetrachloromethane | $11.24$ | $-0.03$ | $-0.04$ | $0.05$ | $-0.00$
37 | Tetrabromomethane | $10.05$ | $0.17$ | $0.20$ | $0.06$ | $-0.01$
38 | Tetraiodomethane | $8.74$ | $0.31$ | $0.37$ |  | $0.07$
39 | Silane | $12.37$ | $0.02$ | $0.03$ | $0.05$ | $-0.12$
40 | Germane | $12.09$ | $0.03$ | $0.04$ | $0.23$ | $-0.18$
41 | Disilane | $10.46$ | $-0.05$ | $-0.02$ | $0.06$ | $-0.14$
42 | Pentasilane | $9.07$ | $-0.02$ | $0.06$ | $0.12$ | $-0.08$
43 | Lithium hydride | $6.52$ | $0.07$ | $-0.06$ | $0.11$ | $0.15$
44 | Potassium hydride | $4.88$ | $0.09$ | $0.09$ | $0.09$ | $-0.20$
45 | Borane | $12.93$ | $0.03$ | $0.02$ | $0.02$ | $-0.15$
46 | Diborane(6) | $11.92$ | $0.01$ | $0.02$ | $-0.00$ | $-0.25$
47 | Amonia | $10.27$ | $0.12$ | $0.05$ | $-0.09$ | $-0.20$
48 | Hydrazoic acid | $10.44$ | $0.11$ | $0.06$ | $0.04$ | $-0.26$
49 | Phosphine | $10.27$ | $0.08$ | $0.08$ | $0.16$ | $0.04$
50 | Arsine | $10.31$ | $-0.11$ | $-0.05$ | $0.02$ | $-0.11$
51 | Hydrogen sulfide | $10.12$ | $0.01$ | $-0.01$ | $0.11$ | $0.07$
52 | Hydrogen fluoride | $15.14$ | $0.23$ | $0.23$ | $0.09$ | $0.12$
53 | Hydrogen chloride | $12.36$ | $0.00$ | $0.09$ | $0.12$ | $0.11$
54 | Lithium fluoride | $10.03$ | $0.24$ | $0.04$ | $0.08$ | $0.03$
55 | Magnesium fluoride | $12.44$ | $0.07$ | $-0.03$ | $0.02$ | $-0.14$
56 | Titanium tetrafluoride | $13.93$ | $0.15$ | $0.08$ |  | $0.00$
57 | Aluminum fluoride | $14.33$ | $0.15$ | $0.00$ | $0.07$ | $-0.13$
58 | Boron monofluoride | $10.59$ | $0.14$ | $-0.13$ | $-0.03$ | $-0.17$
59 | Sulfur tetrafluoride | $12.34$ | $0.05$ | $-0.14$ | $-0.02$ | $-0.18$
60 | Potassium bromide | $7.85$ | $-0.24$ | $-0.05$ |  | $-0.74$
61 | Gallium monochloride | $9.81$ | $-0.07$ | $0.08$ | $0.36$ | $-0.06$
62 | Sodium chloride | $8.32$ | $0.03$ | $0.15$ |  | $-0.14$
63 | Magnesium chloride | $11.07$ | $0.12$ | $0.12$ | $0.18$ | $0.04$
64 | Aluminum iodide | $9.38$ | $0.20$ | $0.20$ | $0.93$ | $-0.07$
65 | Boron nitride | $10.94$ | $0.15$ |  |  | $0.25$
66 | Hydrogen cyanide | $13.10$ | $0.22$ | $0.19$ | $0.12$ | $0.12$
67 | Phosphorus mononitrid | $11.10$ | $0.20$ | $0.14$ | $0.16$ | $0.11$
68 | Hydrazine | $9.40$ | $-0.03$ | $-0.07$ | $-0.13$ | $-0.30$
69 | Formaldehyde | $10.46$ | $0.00$ | $-0.04$ | $-0.05$ | $-0.25$
70 | Methanol | $10.54$ | $0.13$ | $0.07$ | $0.07$ | $-0.23$
71 | Ethanol | $10.17$ | $0.10$ | $0.04$ | $0.04$ | $-0.20$
72 | Acetaldehyde | $9.61$ | $0.05$ | $0.02$ | $0.00$ | $-0.22$
73 | Ethoxy ethane | $9.41$ | $0.02$ | $0.02$ | $-0.02$ | $-0.30$
74 | Formic acid | $10.84$ | $0.03$ | $-0.03$ | $-0.03$ | $-0.13$
75 | Hydrogen peroxide | $10.96$ | $0.14$ | $-0.00$ | $0.04$ | $0.02$
76 | Water | $11.94$ | $0.11$ | $-0.10$ | $-0.07$ | $-0.12$
77 | Carbon dioxide | $13.37$ | $0.09$ | $-0.01$ | $0.00$ | $-0.25$
78 | Carbon disulfide | $9.80$ | $0.15$ | $0.16$ | $0.25$ | $0.09$
79 | Carbon oxide sulfide | $10.90$ | $0.21$ | $0.16$ | $0.26$ | $0.08$
80 | Carbon oxide selenide | $10.40$ | $0.03$ | $0.02$ | $-0.03$ | $-0.22$
81 | Carbon monoxide | $13.66$ | $0.05$ | $-0.04$ | $-0.00$ | $-0.18$
82 | Ozone | $11.74$ | $-0.23$ |  |  | $0.33$
83 | Sulfur dioxide | $11.86$ | $0.20$ | $0.05$ | $0.10$ | $0.05$
84 | Beryllium monoxide | $8.99$ | $-0.48$ |  |  | $0.76$
85 | Magnesium monoxide | $6.82$ | $-0.07$ |  |  | $0.24$
86 | Toluene | $8.72$ | $0.00$ | $0.03$ | $-0.01$ | $-0.14$
87 | Ethylbenzene | $8.65$ | $0.01$ | $0.04$ | $0.01$ | $-0.16$
88 | Hexafluorobenzene | $9.70$ | $0.04$ | $-0.07$ | $-0.05$ | $-0.13$
89 | Phenol | $8.47$ | $0.04$ | $-0.09$ |  | $-0.24$
90 | Aniline | $7.72$ | $0.07$ | $0.06$ | $0.01$ | $-0.14$
91 | Pyridine | $9.18$ | $0.00$ | $-0.02$ | $-0.05$ | $-0.19$
92 | Guanine | $7.77$ | $0.10$ | $0.08$ | $0.05$ | $-0.12$
93 | Adenine | $8.14$ | $0.01$ | $-0.02$ | $-0.05$ | $-0.20$
94 | Cytosine | $8.44$ | $0.01$ | $-0.04$ | $-0.04$ | $-0.19$
95 | Thymine | $8.86$ | $0.01$ | $-0.03$ | $-0.04$ | $-0.22$
96 | Uracil | $9.26$ | $0.12$ | $0.10$ | $-0.07$ | $-0.02$
97 | Urea | $9.25$ | $0.21$ | $0.10$ | $0.14$ | $-0.03$
98 | Silver dimer | $7.06$ | $0.91$ | $0.77$ | $0.98$ | $0.81$
99 | Copper dimer | $8.20$ | $-1.57$ | $-1.01$ |  | $-0.33$
100 | Copper cyanide | $10.25$ | $-0.73$ |  |  | $0.40$

Table LABEL:tab::IPs shows the results for all 100 IPs in the GW100 database
obtained with our code next to the deviations to def2-GTO, VASP, WEST and
nanoGW, if available. To facilitate a discussion of the data, figures 6 shows
MADs and maximum absolute errors (MAE) between all codes, while figure 7
visualizes the distribution of the deviations of the ADF IPs to the ones from
other codes. The IPs from ADF, def2-GTO, VASP and WEST are all in good
agreement with each other, with MADs between 56 and 86 meV, while the
deviations to nanoGW are about twice as large. Figure 7 also shows, that the
deviations of the ADF IPs to the ones from other codes (again, with the
exception of nanoGW) show a small spread and no outliers can be found. We note
again, that we assume the CBS limit extrapolation error to be of the order of
at least 60 meV on average and all the values reported and compared here
should only be interpreted with these error bars. For the plane-wave codes,
the CBS limit extrapolation error is likely smaller than for the localized
basis sets, but there are additional sources of error, most notably pseudo-
potentials and box-size effects. In light of these uncertainties, the
agreement between all four codes can be considered as excellent.

Figure 6: MADs, (upper triangle) and mean deviations (MD) (lower triangle) of
the CBS limit extrapolated IPs in the GW100 database (18 molecules have been
excluded from the comparison, see explanations above) computed with different
codes. All values are in meV. Figure 7: Deviation of IPs calculated with ADF
to different codes. Black dots denote the individual data points. The
horizonal line in each box denotes the median deviation, the box contains all
data points between the first quartile (Q1) and third quartile (Q2) and the
whiskers are at $Q1\pm\frac{3}{2}|Q1-Q3|$ (in case of a normal distribution,
the whiskers include 99.3 % of all data points). All values are in eV.

Looking at the mean deviations to the other codes in the lower triangle of
figure 6 as well as at the boxplots in figure 7, we see that ADF IPs are
generally smaller than the ones from def2-GTO, VASP and WEST. The reasons for
the discrepancy between ADF and def2-GTO have already been discussed above. We
also see that the nanoGW IPs are on average much smaller than the ones from
all others codes. This is in line with the fact that the nanoGW results were
apparently obtained without basis set extrapolation,26 although the numerical
parameters determining the convergence to the CBS limit (grid spacing, chosen
cut-off for virtual states and radius of the sphere around a given finite
systems) where tested for convergence separately. Still, our data analysis
suggests that the nanoGW IPs are not as well converged as the ones from the
other codes.

Systems containing Fluorine and Nitrogen generally show rather pronounced
disagreements between different codes (For example, consider the following
deviations from ADF to def2-GTO: N2: 260 meV, HCN: 220 meV, TiF4: 150 meV,
AlF3: 150 meV, LiF: 240 meV, HF: 230 meV). For Fluorine, this has already been
observed by Maggio and Kresse in ref. 22, who pointed at the default pseudo-
potentials in VASP as a potential source of these discrepancies. Another
possible explanation might be found in a a recent study by Bruneval et al.12
which suggests, that molecules predominantly composed of Carbon and Hydrogen
converge to the CBS limit rather quickly, whereas the convergence is
considerably slower for systems to a large part composed of Fluorine and
Nitrogen. For this reason, we also used stronger polarized basis sets for
Fluorine and Chlorine (For the latter, the basis set convergence is also
rather slow, but less pronounced than for Fluorine).

Finally, we shortly comment on some systems which we have excluded from the
statistical comparisons in figures 6 and figures 7. We find large differences
for system containing 5th row elements, e.g. Xe, ceAg2, I2, or CI4. Here, the
ADF IPs are considerably lower than the ones from other codes, indicating that
the ADF results are not properly converged to the CBS limit which is due to
the missing basis functions with angular momentum higher than $l=3$.

With 1.57 eV, the by far largest deviation reported in table LABEL:tab::IPs
can be found between ADF (8.20 eV) and TURBOMOLE (6.63 eV) for Cu2. FHI-AIMS
gives an IP of 7.78 eV for this system,7 which is in considerably better
agreement with ADF. FHI-AIMS relies on an analytical continuation from the
imaginary to the real frequency axis with 16 sampling points (AIMS-P16), not
much different from the procedure in ADF. We can conclude, that for this
particular system, the large deviation of ADF and FHI-AIMS to TURBOMOLE are
caused by inaccuracies in the frequency grids, which are not converged. The
ADF IP reported in table LABEL:tab::IPs has been obtained from aug-TZ3P and
aug-QZ6P calculation with 24 and 27 imaginary frequency points, respectively.
When 24 imaginary frequency points are used for the aug-QZ6P calculation as
well, the IP of Cu2 reduces to 8.05 eV, which is already in reasonable
agreement with FHI-AIMS. Similar conclusions can be drawn for CuCN. Another
interesting case is BeO, with deviation of 0.48 eV to def-GTO. Again, the
large deviation is due to non-converged frequency grids. the ADF IP is with
8.99 eV very close to the 9.07 eV obtained by AIMS-P16. With 128 imaginary
frequency points (AIMS-P128), FHI-AIMS gives an IP of 9.63 eV which is then in
perfect agreement with TURBOMOLE. Furthermore, in ref. 7, three solutions are
reported for BeO, while ADF only recovers one of them. These three examples
show, that the current frequency treatment in ADF can not properly describe
the IPs of systems for which the single QP picture breaks down in the valence
region.

####  3.4.3 Electron Affinities

We now turn our attention to the EAs. As for the IPs, table LABEL:tab::EAs
shows the EAs calculated with ADF and the differences to the other four codes
excluding all noble gases and H2. However, it is known, that the def2-GTO
basis sets sometimes severely overestimate positive LUMO energies which then
deviate from results from plane-wave codes by more than 1 eV. Furthermore,
since EAs converge slower to the CBS limit than IPs when non-augmented basis
sets are used,12, 29 also basis set extrapolation errors are larger for the
remaining systems. On the other hand, plane-wave calculations require very
large box sizes for these systems which makes it harder to converge the EAs
with respect to this parameter and for this reason results from VASP are often
not available.22

Table 2: $G_{0}W_{0}$@PBE electron affinities (EA) for the GW100 database (third column) Columns four to seven denote deviations of the ADF EAs to the ones from reference $X$, $\Delta_{X}=EA_{X}-EA_{ADF}$. All values are in eV. | Name | ADF | $\Delta_{def2-GTO}$ | $\Delta_{VASP}$ | $\Delta_{WEST}$ | $\Delta_{nanoGW}$
---|---|---|---|---|---|---
7 | Lithium dimer | $0.52$ | $0.23$ | $0.09$ | $0.12$ | $0.14$
8 | Sodium dimer | $0.64$ | $0.02$ | $-0.04$ | $-0.03$ | $-0.00$
9 | Sodium tetramer | $0.92$ | $0.23$ | $0.15$ | $0.16$ | $0.20$
10 | Sodium hexamer | $0.95$ | $0.18$ | $0.12$ | $0.09$ | $0.11$
11 | Potassium dimer | $0.59$ | $0.16$ | $0.15$ | $0.16$ | $0.16$
12 | Rubidium dimer | $0.67$ |  | $0.07$ | $0.06$ | $0.07$
13 | Nitrogen | $-2.40$ | $0.28$ |  | $0.25$ | $0.17$
14 | Phosphorus dimer | $0.64$ | $0.44$ | $0.35$ | $0.45$ | $0.38$
15 | Arsenic dimer | $1.08$ | $0.44$ | $-0.01$ | $0.01$ | $-0.04$
16 | Fluorine | $0.54$ | $0.69$ |  | $0.52$ | $0.58$
17 | Chlorine | $0.83$ | $0.57$ | $0.42$ | $0.55$ | $0.52$
18 | Bromine | $1.40$ | $0.56$ | $0.59$ | $0.48$ | $0.29$
19 | Iodine | $1.56$ |  | $0.65$ | $1.65$ |
20 | Methane | $-0.78$ | $-1.25$ | $0.15$ | $0.02$ | $-0.10$
21 | Ethane | $-0.77$ | $-1.16$ |  | $-0.01$ | $-0.11$
22 | Propane | $-0.72$ | $-1.15$ |  | $-0.03$ | $-0.11$
23 | Butane | $-0.70$ | $-1.13$ |  | $-0.04$ | $-0.13$
24 | Ethylene | $-1.91$ | $0.09$ |  | $0.11$ | $-0.04$
25 | Acetylene | $-2.48$ | $-0.08$ |  | $-0.02$ | $-0.20$
26 | Tetracarbon | $2.62$ | $0.53$ | $0.47$ | $0.48$ | $0.52$
27 | Cyclopropane | $-0.73$ | $-1.23$ |  | $-0.02$ | $-0.14$
28 | Benzene | $-0.96$ | $0.07$ |  | $0.03$ | $-0.10$
29 | Cyclooctatetraene | $0.03$ | $0.09$ | $0.02$ | $0.04$ | $-0.10$
30 | Cyclopentadiene | $-0.91$ | $0.06$ |  | $0.01$ | $-0.15$
31 | Vinyl fluoride | $-1.92$ | $0.04$ |  | $0.03$ | $-0.09$
32 | Vinyl chloride | $-1.31$ | $0.14$ | $0.12$ | $0.10$ | $0.00$
33 | Vinyl bromide | $-1.23$ | $0.12$ |  | $0.17$ | $0.09$
34 | Vinyl iodide | $-0.77$ |  | $0.40$ | $0.55$ | $0.44$
35 | Tetrafluoromethane | $-0.88$ | $-3.00$ |  | $0.05$ | $-0.10$
36 | Tetrachloromethane | $0.04$ | $0.50$ | $0.28$ | $0.37$ | $0.41$
37 | Tetrabromomethane | $0.99$ | $0.57$ | $0.48$ | $0.46$ | $0.41$
38 | Tetraiodomethane | $2.16$ |  | $0.28$ | $0.88$ | $0.32$
39 | Silane | $-0.72$ | $-1.54$ |  | $-0.04$ | $-0.11$
40 | Germane | $-0.47$ | $-1.38$ |  | $-0.14$ | $-0.36$
41 | Disilane | $-0.75$ | $-0.76$ |  | $-0.02$ | $-0.83$
42 | Pentasilane | $-0.08$ | $0.08$ | $0.05$ | $0.15$ | $-0.02$
43 | Lithium hydride | $0.05$ | $0.11$ | $0.02$ | $0.02$ | $0.13$
44 | Potassium hydride | $0.17$ | $0.15$ | $0.08$ | $0.08$ | $0.35$
45 | Borane | $-0.26$ | $0.23$ | $0.23$ | $0.25$ | $0.21$
46 | Diborane(6) | $-0.87$ | $0.13$ |  | $0.15$ | $-0.00$
47 | Amonia | $-0.76$ | $-1.24$ |  | $-0.05$ | $-0.09$
48 | Hydrazoic acid | $-1.40$ | $0.30$ |  | $0.25$ | $0.23$
49 | Phosphine | $-0.67$ | $-1.59$ |  | $-0.03$ | $-0.06$
50 | Arsine | $-0.58$ | $-1.36$ |  | $-0.08$ | $-0.16$
51 | Hydrogen sulfide | $-0.73$ | $-1.52$ |  | $-0.05$ | $-0.11$
52 | Hydrogen fluoride | $-1.06$ | $-0.98$ |  | $-0.05$ | $-0.12$
53 | Hydrogen chloride | $-1.19$ | $-0.34$ |  | $0.10$ | $-0.03$
54 | Lithium fluoride | $-0.04$ | $0.05$ | $-0.13$ | $-0.03$ | $0.11$
55 | Magnesium fluoride | $0.26$ | $0.05$ | $0.03$ | $0.07$ | $0.18$
56 | itanium tetrafluorid | $0.09$ | $0.97$ |  | $0.83$ | $1.14$
57 | Aluminum fluoride | $0.06$ | $0.17$ | $-0.14$ | $0.10$ | $0.01$
58 | Boron monofluoride | $-1.21$ | $0.16$ |  | $0.28$ | $0.24$
59 | Sulfur tetrafluoride | $-0.29$ | $0.39$ | $0.22$ | $0.37$ | $0.35$
60 | Potassium bromide | $0.34$ | $0.08$ | $-0.02$ | $0.06$ | $0.59$
61 | Gallium monochloride | $0.02$ | $0.37$ | $0.17$ | $0.42$ | $0.20$
62 | Sodium chloride | $0.42$ | $-0.00$ | $0.04$ | $0.05$ | $0.61$
63 | Magnesium chloride | $0.68$ | $-0.00$ | $-0.07$ | $0.02$ | $0.05$
64 | Aluminum iodide | $1.18$ |  | $-0.16$ | $0.48$ | $0.02$
65 | Boron nitride | $4.05$ | $-0.10$ |  | $0.03$ | $0.05$
66 | Hydrogen cyanide | $-2.31$ | $0.09$ |  | $0.06$ | $-0.06$
67 | hosphorus mononitrid | $0.12$ | $0.47$ |  | $0.40$ | $0.35$
68 | Hydrazine | $-0.70$ | $-0.98$ |  | $-0.02$ | $-0.08$
69 | Formaldehyde | $-1.06$ | $0.35$ |  | $0.30$ | $0.15$
70 | Methanol | $-0.81$ | $-1.00$ |  | $-0.10$ | $-0.19$
71 | Ethanol | $-0.73$ | $-0.94$ |  | $-0.11$ | $-0.19$
72 | Acetaldehyde | $-1.16$ | $0.33$ | $0.29$ | $0.29$ | $0.25$
73 | Ethoxy ethane | $-0.62$ | $-1.08$ |  | $-0.09$ | $-0.18$
74 | Formic acid | $-1.82$ | $0.23$ | $0.18$ | $0.18$ | $0.04$
75 | Hydrogen peroxide | $-2.06$ | $0.11$ |  | $0.26$ | $0.28$
76 | Water | $-0.88$ | $-1.13$ | $-0.16$ | $-0.03$ | $-0.08$
77 | Carbon dioxide | $-1.03$ | $0.10$ |  | $0.06$ | $-0.03$
78 | Carbon disulfide | $0.10$ | $0.45$ | $0.32$ | $0.40$ | $0.32$
79 | Carbon oxide sulfide | $-1.22$ | $0.39$ |  | $0.28$ | $0.20$
80 | arbon oxide selenide | $-0.93$ | $0.41$ |  | $0.29$ | $0.22$
81 | Carbon monoxide | $-0.84$ | $0.47$ |  | $0.40$ | $0.38$
82 | Ozone | $2.03$ | $0.66$ | $0.47$ | $0.53$ | $0.52$
83 | Sulfur dioxide | $0.86$ | $0.63$ | $0.39$ | $0.51$ | $0.36$
84 | Beryllium monoxide | $1.99$ | $0.73$ | $0.74$ | $0.52$ | $0.54$
85 | Magnesium monoxide | $1.74$ | $0.39$ | $0.31$ | $0.21$ | $0.52$
86 | Toluene | $-0.91$ | $0.08$ |  | $0.04$ | $-0.09$
87 | Ethylbenzene | $-0.87$ | $0.00$ |  | $-0.03$ | $-0.18$
88 | Hexafluorobenzene | $-0.03$ | $-0.33$ |  | $0.00$ | $-0.08$
89 | Phenol | $-0.78$ | $0.04$ |  | $-0.07$ | $-0.18$
90 | Aniline | $-0.91$ | $-0.03$ |  | $-0.07$ | $-0.19$
91 | Pyridine | $-0.44$ | $0.14$ |  | $0.08$ | $-0.05$
92 | Guanine | $-0.48$ | $0.02$ |  | $-0.03$ | $-0.15$
93 | Adenine | $-0.28$ | $0.07$ |  | $-0.01$ | $-0.13$
94 | Cytosine | $-0.18$ | $0.17$ | $0.06$ | $0.09$ | $-0.03$
95 | Thymine | $0.02$ | $0.16$ | $0.04$ | $0.08$ | $-0.06$
96 | Uracil | $0.05$ | $0.20$ | $0.06$ | $0.10$ | $-0.02$
97 | Urea | $-0.49$ | $-0.68$ |  | $-0.04$ | $-0.12$
98 | Silver dimer | $0.91$ |  | $0.44$ | $0.58$ | $0.41$
99 | Copper dimer | $1.00$ | $0.23$ | $0.24$ | $0.41$ | $0.25$
100 | Copper cyanide | $1.47$ | $0.38$ | $0.44$ | $0.51$ | $0.34$
MAD |  |  | $0.48$ | $0.21$ | $0.16$ | $0.21$

Thus, for the full database, only comparison to WEST and nanoGW is possible.
Excluding again all compounds containing Iodine, Copper, Gallium and Xenon as
well as remaining systems containing 5th row elements, we find a MAD of 160
meV to the former and of 210 meV to the latter code. These MADs are about
twice as large as for IPs but in light of the difficulties mentioned above
certainly not surprising and in line with the previous benchmark studies on
GW100.7, 22, 24, 26 Figure 8 shows that the ADF EAs, as for the IPs, are on
average smaller than the ones from WEST while no trend in that direction can
be observed when comparing to nanoGW.

Figure 8: Deviation of EAs calculated with ADF to different codes. Black dots denote the individual data points. The horizonal line in each box denotes the median deviation, the box contains all data points between the first quartile (Q1) and third quartile (Q2) and the whiskers are at $Q1\pm\frac{3}{2}|Q1-Q3|$ (in case of a normal distribution, the whiskers include 99.3 % of all data points). All values are in eV. The deviations to VASP have been excluded due to the lack of reference values for too many systems in GW100. |  | aug. | non.aug. |  |  |
---|---|---|---|---|---|---
Index |  | T | Q | Ex. | $\Delta_{TQ}$ | T | Q | Ex. | $\Delta_{TQ}$ | $\Delta_{TT}$ | $\Delta_{QQ}$ | $\Delta_{EE}$
13 |  N2 | -2.65 | -2.54 | -2.40 | -0.11 | -3.00 | -2.58 | -2.15 | -0.42 | 0.36 | 0.04 | -0.25
20 |  CH4 | -0.97 | -0.89 | -0.78 | -0.08 | -2.20 | -1.59 | -0.95 | -0.61 | 1.23 | 0.70 | 0.17
21 |  C2H6 | -0.96 | -0.88 | -0.77 | -0.08 | -2.13 | -1.52 | -0.87 | -0.62 | 1.17 | 0.64 | 0.10
22 |  C3H8 | -0.92 | -0.83 | -0.72 | -0.08 | -2.06 | -1.44 | -0.78 | -0.62 | 1.14 | 0.61 | 0.06
23 |  C4H10 | -0.89 | -0.81 | -0.70 | -0.08 | -2.04 | -1.41 | -0.74 | -0.63 | 1.15 | 0.60 | 0.04
24 |  C2H4 | -2.12 | -2.03 | -1.91 | -0.09 | -2.40 | -2.12 | -1.81 | -0.28 | 0.28 | 0.09 | -0.10
25 |  C2H2 | -2.76 | -2.65 | -2.48 | -0.11 | -3.24 | -2.87 | -2.44 | -0.37 | 0.48 | 0.22 | -0.04
27 |  C3H3 | -0.98 | -0.88 | -0.73 | -0.10 | -2.29 | -1.61 | -0.87 | -0.68 | 1.31 | 0.73 | 0.14
31 |  C2H3F | -2.21 | -2.09 | -1.92 | -0.12 | -2.50 | -2.22 | -1.91 | -0.28 | 0.29 | 0.13 | -0.01
39 |  SiH4 | -0.92 | -0.83 | -0.72 | -0.09 | -1.75 | -1.42 | -1.08 | -0.33 | 0.83 | 0.59 | 0.36
47 |  NH3 | -0.93 | -0.85 | -0.76 | -0.08 | -1.85 | -1.35 | -0.85 | -0.51 | 0.93 | 0.50 | 0.09
66 |  HCN | -2.59 | -2.48 | -2.31 | -0.12 | -2.96 | -2.61 | -2.23 | -0.35 | 0.37 | 0.14 | -0.08
70 |  CH4O | -1.05 | -0.95 | -0.81 | -0.10 | -2.00 | -1.49 | -0.93 | -0.52 | 0.95 | 0.54 | 0.12
71 |  C2H6O | -0.97 | -0.87 | -0.73 | -0.10 | -1.90 | -1.39 | -0.84 | -0.51 | 0.93 | 0.52 | 0.11
76 |  H2O | -1.02 | -0.96 | -0.88 | -0.06 | -1.75 | -1.34 | -0.89 | -0.41 | 0.73 | 0.38 | 0.01
| MD |  |  |  | 0.09 |  |  |  | 0.48 | 0.81 | 0.43 | 0.05
Table 3: Comparison of EAs for selected molecules from the GW100 database
calculated with ADF with and without diffuse functions. All values are in eV.

One furthermore clearly sees that for a group of molecules with significant
deviations from ADF to def2-GTO of up to several eVs. These are the already
mentioned systems with positive LUMO. Contrariwise, the raw data in table
LABEL:tab::EAs shows that the agreement with WEST is especially good for
systems with positive LUMO. As an example, consider the series of linear
alkane chains, $\text{C}_{n}\text{H}_{2n+1}$ for $n=1,\dots,4$. With
deviations from 10 to 40 meV, the agreement with WEST is excellent, while
def2-GTO overestimates the EAs of these systems by more than 1 eV. In this
context, it is interesting to investigate the effect of the diffuse functions.
This is shown in table 3 for some systems with LUMO well above the vacuum
level. Only comparing the basis set extrapolated values, the effect of the
diffuse functions seem to be rather small; for N2 the EA calculated from the
basis set without the diffuse functions is with 250 meV even higher than the
augmented basis sets, and overall, the average difference is only 50 meV which
is well within the expected error range from the CBS limit extrapolation.
However, comparing the results form the finite basis sets, the differences are
exorbitant. Especially, on the TZ level, the addition of diffuse functions
results in a lowering of the EAs by nearly 1 eV on average. For the non-
augmented basis sets, the average difference between TZ and QZ basis set is
480 meV, resulting in differences of sometimes more than 1 eV between the EAs
on the TZ and on the extrapolated level. In light of these differences the
good agreement between the CBS limit extrapolated EAs is remarkable. Despite
this good agreement, the augmented basis sets should be the preferred choice
to calculate EAs of systems with unbound LUMOs since the extrapolation
procedure is generally less reliable when the differences between the results
for the finite basis sets are larger.

Figure 9: MADs, (upper triangle) and mean deviations (MD) (lower triangle) of
the CBS limit extrapolated EAs of the subset of systems with bound LUMO and
medium sized organic molecules (in total 48 data points) in the GW100 dataset
computed with different codes. All values are in eV.

Finally, we compare our EAs for systems with a bound LUMO to WEST, nanoGW, and
def-GTO results. The MADs and MDs in figure 9 show that def2-GTO, nanoGW and
and WEST are generally in good agreement for these systems while the MADs to
ADF are large. As the MDs show, ADF significantly overestimates these EAs
compared to the other codes, which indicates that the results are not entirely
converged to the CBS limit. This interpretation is also in line with the raw
data in table LABEL:tab::EAs showing that the deviations are generally largest
for di- and triatomic systems as well as molecules containing Fluorine and
Chlorine, while the agreement for the medium organic molecules like the
nucleo-bases is satisfactory. Adding additional diffuse or tight functions
would possible not result in an improved description of the EAs of the former
systems. Instead, reaching the CBS limit is most likely only possible using
basis functions with higher angular momenta than $l=3$ which are not available
to us. However, we can try to simulate the effect of these functions by adding
additional off-center Slater functions to the basis set which can be achieved
conveniently by adding ghost atoms. This approach is reminiscent of bond-
centred basis functions which have for example been used by Dunlap and
coworkers.105

system | T | Q | ex. | ex. (no gh.) | def2-GTO | WEST
---|---|---|---|---|---|---
F2 | 0.20 | 0.51 | 0.90 | 0.54 | 1.23 | 1.06
Cl2 | 0.74 | 0.95 | 1.21 | 0.83 | 1.40 | 1.38
Br2 | 1.27 | 1.45 | 1.69 | 1.40 | 1.96 | 1.88
TiF4 | -0.33 | -0.10 | 0.19 | 0.09 | 1.06 | 0.92
SO2 | 0.84 | 1.07 | 1.38 | 0.86 | 1.49 | 1.37
Table 4: Effect of the addition of additional off-center Slater functions via
Ne ghost atoms on the EAs of selected systems form GW100. All values are in
eV.

As examples, we consider the dihalogens F2, Cl2 and Br2, SO2, as well as TiF4
and we augment these structures with Ne ghost atoms for which we use the same
basis set than for the real atoms. For each atom $A$ in a systems, we place
two ghost atoms $G,G^{\prime}$ on a straight line defined by the position of
$A$ and every neighboring atom $B$ so that the distance between $A$ and $G$
($G^{\prime}$) equals one third (minus one third) of the distance between $A$
and $B$. The results of this augmentation is shown in table 4 and we clearly
see that it reduces the basis set error considerably, as compared to the other
codes. Of course, such an augmentation should not be seen as a practical
solution but it shows that agreement between ADF and the other codes can in
principle be reached also for these systems.

###  3.5 Ionization Potentials and Electron Affinities for a Subset of GW5000

For the GW100 database, discrepancies to other codes are more pronounced for
smaller systems while the agreement for medium systems is significantly
better. To confirm this observation for a larger number of systems, we also
calculated IPs and EAs of a subset of 250 medium to large organic molecules
from the GW5000 database11 for which CBS limit extrapolated references values
calculated with FHI-AIMS using the def2-GTO basis sets are available. We have
already considered this subset in ref 29. We used here the TZ3P and QZ6P basis
sets without diffuse functions since the LUMOs of the considered systems are
negative. As expected, we find better agreement with FHI-AIMS than for GW100:
For IPs, ADF deviated to FHI-AIMS by 62 meV on average as opposed to the 86
meV for GW100. For EAs, the MAD is with 93 meV slightly worse but the
agreement is much better than for GW100.

##  4 Conclusions

The GW100 benchmark by van Setten et al.7 has become the most important
resource for comparing different implementations of the $GW$ method for finite
systems. IPs and EAs calculated with implementations using def2 GTO-type basis
sets,7 plane waves,22, 24 and real-space finite elements26 have been compared
in the recent past and good agreement between CBS limit extrapolated results
has been found for IPs and EAs of molecules with bound LUMO. In this work, we
extended the list of available results and calculated the IPs and EAs in the
GW100 dataset using Slater type orbitals. For this purpose we have developed
new Slater type basis sets which allow us to extrapolate our results to the
CBS limit. These basis sets are available online.The basis set files can be
downloaded from http://www.scm.com/wp-content/uploads/Correlation-basis-
sets.zip

Our study confirms once more that it is possible, though difficult, to reach
consensus between different implementations. All implementations compared in
this work do not only implement the GWA with different basis sets, but they
also differ in other technical parameters like frequency treatment,
description of core electrons, the algorithm used to solve the QP equations
and the numerical treatment of 4-point correlation functions. In light of
these differences, the observed agreement between 55 to 85 meV on average for
IPs between STO- def2-GTO- and plane-wave results is excellent. Reaching the
CBS limit is more difficult for EAs than for IPs. Still, EAs calculated with
ADF are in excellent agreement with the place-wave results from the WEST code
for systems with positive LUMOs, with an overall MAD between both codes of 160
meV. These deviations mostly stem from large basis set errors for the EAs of
small molecules with bound LUMO with the ADF code. For larger, organic
molecules, agreement to def2-GTO results is with a MAD of 93 meV significantly
better.

We observe that ADF tends to give lower EAs and IPs than the def2-GTO and
plane-wave results. This is an indicator that the results are not entirely
converged with respect to the CBS limit. Our STO-type basis sets are
restricted to angular momenta smaller or equal to $l=3$ which sets a limit to
the accuracy currently attainable with STOs. It would be desirable to obtain
results for the GW100 database using even larger STO type basis sets
containing also functions with higher angular momenta. This would enable a
more accurate estimate of the CBS limit and might hopefully lead to an even
better agreement with other implementations. On the other hand, the best IPs
for GW100 in our earlier work showed a MAD of more than 300 meV to def-GTO and
nanoGW.29 Thus, the improvements which are due to our improved basis sets and
imaginary time and frequency grids are already immense.

The close agreement between different codes is highly important in practice
since it allows researchers to interpret the results of their $GW$
calculations, without worrying that they might be skewed by technical aspects.
The slow convergence of $GW$ calculations to the CBS limit remains an obstacle
in practical applications. Especially for $GW$ calculations for finite systems
with hundreds of atoms, which have become the focus of much research in the
last years27, 28, 29, 30, calculations at (or even close to) the CBS limit for
individual QP energies are currently often out of reach and will most likely
not become routine anytime soon. To overcome this issue, finite basis set
corrections107, 108, 109, 12 hold much promise and it is to hope that these
techniques will be further developed and become more widespread available in
the near future. For differences of QP energies, the situation is already much
better. When augmented basis sets are used, basis set errors usually cancel to
a large extent and already on the DZ level fundamental gaps are often
sufficiently converged to the CBS limit.2, 110, 111, 112, 113, 102, 29

##  Appendix A Implementation of Imaginary Time and Frequency Grids

In this appendix we state the formulas we use to calculate imaginary time and
imaginary frequency grids. For derivations, we refer to ref. 36. Since we want
to switch between imaginary time and imaginary frequency, we write the
discrete Fourier transform $f(i\omega_{k})$ of an arbitrary function
$f(i\tau_{j})$ as

$f(i\omega_{k})=-i\sum^{N_{\tau}}_{j}\left\\{\gamma^{(c)}_{kj}\cos(\omega_{k}\tau_{j})\left(f(i\tau_{j})+f(-i\tau_{j})\right)-i\gamma^{(s)}_{kj}\sin(\omega_{k}\tau_{j})\left(f(i\tau_{j})-f(-i\tau_{j})\right)\right\\}\;.$
(17)

Fourier transform from imaginary frequency to imaginary time can be done by
inverting the matrices with the elements
$\gamma^{(c)}_{kj}\cos(\omega_{k}\tau_{j})$ and
$\gamma^{(s)}_{kj}\sin(\omega_{k}\tau_{j})$ (In case $N_{\tau}\neq
N_{\omega}$, a pseudo-inverse can be calculated). At the beginning of a GW
calculation, we find optimal sets of imaginary frequency
$\left\\{\omega_{k})\right\\}_{k=1,\dots N_{\omega}}$ and imaginary time
points $\left\\{\tau_{j})\right\\}_{j=1,\dots N_{\tau}}$. For imaginary time,
we use the algorithm described in ref. 114 to minimize

$\left\|\eta^{(\tau)}(x;\alpha,\tau)\right\|_{\infty}\;,\quad\eta^{(\tau)}(x;\alpha,\tau)=\frac{1}{2x}-\sum^{N_{\tau}}_{j}\alpha_{j}e^{-2x\tau_{j}}.\;,$
(18)

and for imaginary frequency, we first minimize

$\left\|\eta^{(\omega)}(x;\sigma,\omega)\right\|_{2}\;,\quad\eta^{(\omega)}(x;\sigma,\omega)=\frac{1}{x}-\frac{1}{\pi}\sum^{N_{\omega}}_{k}\sigma_{k}\left(\frac{2x}{x^{2}+\omega^{2}_{k}}\right)^{2}\;,$
(19)

using a Levenberg–Marquardt algorithm (LMA). In both minimizations, $x$
denotes the electron-hole pair transition energy Both, $\eta^{(\tau)}$ and
$\eta^{(\omega)}$ are minimized in the interval $[1,x_{min}/x_{max}]$, with
$x_{min}$ ($x_{max}$) being the smallest (largest) considered electron-hole
transition energy. In both optimizations, the interval $[x_{min},x_{max}]$ is
represented on a logarithmic grid. After minimizing the $L_{2}$ norm on the
logarithmic grid, we minimize the $L_{2}$ norm for the positions of the minima
and maximima and repeat this procedure until self-consistency is reached. The
coefficients $\left\\{\sigma_{k})\right\\}_{k=1,\dots N_{\omega}}$ and
$\left\\{\alpha_{j})\right\\}_{j=1,\dots N_{\tau}}$ are byproducts of this
optimization but they can be used to evaluate MP2 or RPA correlation energies.

$\eta^{(\omega)}$ has multiple minima and especially for larger $N_{\omega}$
the LMA only converges to a useful minimum when it is initialized with
$\sigma$ and $\omega$ which are already sufficiently close to the parameters
which minimize $\eta^{(\omega)}$. In practice, we found that at a useful (but
not necessarily global) minimum, $\eta^{(\omega)}$ has $2N_{\omega}$ or
$2N_{\omega}-1$ extrema. This behaviour can be exploited to find good starting
points for the LMA for different ratios $x_{min}/x_{max}$ and $N_{\omega}$. We
always start the LMA from pretabulated values.The pretabulated values are
available on Github (https://github.com/ArnoFoerster/Imaginary-Frequency-
Grids-GW-and-RPA) These values have been obtained by a simple metropolis
algorithm for several $x_{min}/x_{max}$ and for $N_{\omega}$ between 1 and 40
and are chosen so that $\eta^{(\omega)}$ has $2N_{\omega}$ or $2N_{\omega}-1$
extrema for a given $N_{\omega}$. In a GWA calculation, the LMA is then
initialized with the pretabulated parameters which are closest to the
$x_{min}/x_{max}$ of the calculation.

To avoid unnecessarily large grids, the $N_{\omega}$ and $N_{\tau}$ are
determined at run-time and the user only specifies an upper limit of points
for both, imaginary time and frequency grids. More precisely, for small
$N_{\omega}$, grid points and weights are calculated and the $L_{2}$ norm of
$\eta^{\omega}$ is calculated. Then we increase the imaginary frequency grid
until the least square error is smaller than $\epsilon_{\omega}=1e^{-10}$. The
number of points which are required to reach that accuracy strongly depends on
$x_{min}/x_{max}$. In our experience, the QP energies converge faster with
respect to the imaginary time grid than with respect to the $\omega$-grid,
Since we find it convenient to work with grids of the same size (although this
is not necessary), we set $N_{\tau}=N_{\omega}$. For example, for a Hydrogen-
molecule in a triple-$\zeta$ (TZ) basis, $N_{\omega}=7$ will already be
sufficient to reach the desired accuracy, while for the Iodine molecule in a
QZ basis $N_{\omega}=31$ will be necessary.

To calculate the matrices $\gamma^{(c)}$ we minimize

$\left\|\eta^{(c)}\left(x;\gamma^{(c)}\right)\right\|_{2}\;,\quad\eta^{(c)}\left(x,\gamma^{(c)}\right)=\frac{2x}{x^{2}+\omega^{2}_{k}}-\sum^{N_{\tau}}_{j=1}\gamma^{(c)}_{kj}\cos(\omega_{k}\tau_{j})e^{-x\tau_{j}}\;,$
(20)

for all $\omega_{k}$ and for $\gamma^{(s)}$, we minimize

$\left\|\eta^{(s)}\left(x;\gamma^{(s)}\right)\right\|_{2}\;,\quad\eta^{(s)}\left(x,\gamma^{(s)}\right)=\frac{2\omega_{k}}{x^{2}+\omega^{2}_{k}}-\sum^{N_{\tau}}_{j=1}\gamma^{(c)}_{kj}\sin(\omega_{k}\tau_{j})e^{-x\tau_{j}}\;,$
(21)

with a LMA. The procedure is the same as described in ref. 21. In all cases,
the algorithm converges smoothly from arbitrary starting values.

This research received funding from the Netherlands Organisation for
Scientific Research (NWO) in the framework of the Innovation Fund for
Chemistry and from the Ministry of Economic Affairs in the framework of the “
_TKI/PPS-Toeslagregeling_ ”. The authors thank Erik van Lenthe for fruitful
discussions.

KS HOMO and LUMO eigenvalues, QP energies for finite basis sets, raw data for
the subset of 250 molecules from GW5000 and technical parameters of all
calculations for GW100.

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  * 89 This is slightly different from usual augmentation approaches encountered for Gaussian type basis sets. Dunning and coworkers defined the prefix ”-aug” to mean adding an additional diffuse functions for all angular momenta already present in a basis set for every atom type.87 On the contrary, in a process which they denoted as minimal augmentation, Truhlar and coworkers only added diffuse $s$ and $p$ functions to all elements heavier than Hydrogen.88 Our approach is can be seen as a compromise between both.
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  * 91 Baerends, E.; Ziegler, T.; Atkins, A.; Autschbach, J.; Baseggio, O.; Bashford, D.; Bérces, A.; Bickelhaupt, F.; Bo, C.; Boerrigter, P.; Cavallo, L.; Daul, C.; Chong, D.; Chulhai, D.; Deng, L.; Dickson, R.; Dieterich, J.; Ellis, D.; van Faassen, M.; Fan, L.; Fischer, T.; Förster, A.; Guerra, C. F.; Franchini, M.; Ghysels, A.; Giammona, A.; van Gisbergen, S.; Goez, A.; Götz, A.; Groeneveld, J.; Gritsenko, O.; Grüning, M.; Gusarov, S.; Harris, F.; van den Hoek, P.; Hu, Z.; Jacob, C.; Jacobsen, H.; Jensen, L.; Joubert, L.; Kaminski, J.; van Kessel, G.; König, C.; Kootstra, F.; Kovalenko, A.; Krykunov, M.; van Lenthe, E.; McCormack, D.; Michalak, A.; Mitoraj, M.; Morton, S.; Neugebauer, J.; Nicu, V.; Noodleman, L.; Osinga, V.; Patchkovskii, S.; Pavanello, M.; Peeples, C.; Philipsen, P.; Post, D.; Pye, C.; Ramanantoanina, H.; Ramos, P.; Ravenek, W.; Reimann, M.; Rodríguez, J.; Ros, P.; Rüger, R.; Schipper, P.; Schlüns, D.; van Schoot, H.; Schreckenbach, G.; Seldenthuis, J.; Seth, M.; Snijders, J.; Solà, M.; Stener, M.; Swart, M.; Swerhone, D.; Tognetti, V.; te Velde, G.; Vernooijs, P.; Versluis, L.; Visscher, L.; Visser, O.; Wang, F.; Wesolowski, T.; van Wezenbeek, E.; Wiesenekker, G.; Wolff, S.; Woo, T.; Yakovlev, A. ADF2020, locally modified development version
  * 92 The code to generate the imaginary time grids can be found on Github. (https://github.com/bhelmichparis/laplace-minimax). Please also see ref. 56 and 114.
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  * 95 Data downloaded from the webiste of the GW100 project by Van Setten et al., https://gw100.wordpress.com
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  * 106 The basis set files can be downloaded from http://www.scm.com/wp-content/uploads/Correlation-basis-sets.zip
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  * Jacquemin et al. 2015 Jacquemin, D.; Duchemin, I.; Blase, X. Benchmarking the Bethe-Salpeter Formalism on a Standard Organic Molecular Set. _J. Chem. Theory Comput._ 2015, _11_ , 3290–3304
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  * 115 The pretabulated values are available on Github (https://github.com/ArnoFoerster/Imaginary-Frequency-Grids-GW-and-RPA)

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