#  $GW$100: a plane wave perspective for small molecules

Emanuele Maggio  University of Vienna, Faculty of Physics and Center for
Computational Materials Science, Sensengasse 8/12, A-1090 Vienna, Austria
Peitao Liu  University of Vienna, Faculty of Physics and Center for
Computational Materials Science, Sensengasse 8/12, A-1090 Vienna, Austria
Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016,
China  Michiel J. van Setten  Nanoscopic Physics, Institute of Condensed
Matter and Nanosciences, Université Catholique de Louvain, 1348 Louvain-la-
Neuve, Belgium  Georg Kresse<EMAIL_ADDRESS>University of Vienna,
Faculty of Physics and Center for Computational Materials Science, Sensengasse
8/12, A-1090 Vienna, Austria

###### Abstract

In a recent work, van Setten and coworkers have presented a carefully
converged $G_{0}W_{0}$ study of 100 closed shell molecules [J. Chem. Theory
Comput. 11, 5665 (2015)]. For two different codes they found excellent
agreement to within few 10 meV if identical Gaussian basis sets were used. We
inspect the same set of molecules using the projector augmented wave method
and the Vienna ab initio simulation package (VASP). For the ionization
potential, the basis set extrapolated plane wave results agree very well with
the Gaussian basis sets, often reaching better than 50 meV agreement. In order
to achieve this agreement, we correct for finite basis set errors as well as
errors introduced by periodically repeated images. For electron affinities
below the vacuum level differences between Gaussian basis sets and VASP are
slightly larger. We attribute this to larger basis set extrapolation errors
for the Gaussian basis sets. For quasi particle (QP) resonances above the
vacuum level, differences between VASP and Gaussian basis sets are, however,
found to be substantial. This is tentatively explained by insufficient basis
set convergence of the Gaussian type orbital calculations as exemplified for
selected test cases.

###### pacs:

71.15.-m, 71.15.Nc., 71.15.Dx, 71.55 Gs

##  I Introduction

The $GW$ approximation suggested by Lars Hedin Hedin (1965) has a long history
in solid state physics. First practical applications were already published in
the 1980s by Hanke and coworkers soon followed by the often quoted study of
Hybertsen and Louie Hanke and Sham (1979); Strinati  _et al._ (1980, 1982);
Hybertsen and Louie (1986). For solids, it is generally found that even the
simplest approximation $G_{0}W_{0}$ yields reasonably accurate quasiparticle
(QP) energies and band gaps in good agreement with experiment van Schilfgaarde
_et al._ (2006); M.Shishkin and G.Kresse (2007); Fuchs  _et al._ (2007). The
results often improve if the Green’s function is iterated to self-consistency,
either updating the QP-energies only or even the one-electron orbitals Faleev
_et al._ (2004); van Schilfgaarde  _et al._ (2006); M.Shishkin and G.Kresse
(2007); Caruso  _et al._ (2012, 2013); Kaplan  _et al._ (2015, 2016); Lischner
_et al._ (2014); Koval  _et al._ (2014). Applications of the $GW$
approximation to molecules, however, have been comparatively rare, since codes
based on local orbitals, which are by construction particularly well suited to
treat molecules, did not incorporate the $GW$ approximation until recently.
This has changed, with many local basis set codes, such as FHI-aims, MOLGW,
Turbomole, and CP2K now supporting $GW$ calculations Ren  _et al._ (2012);
Blase  _et al._ (2011); Bruneval (2012); van Setten  _et al._ (2013); Wilhelm
_et al._ (2016); Foerster  _et al._ (2011); Ke (2011). Also, efficient plane
wave codes using a Sternheimer approach, such as ABINIT and West Laflamme
Janssen  _et al._ (2015); Govoni and Galli (2015), are becoming available. As
for solids, carefully converged QP calculations are, however, still
comparatively scarce Klimeš  _et al._ (2014).

To fill this gap, Bruneval recently performed systematic studies for about 30
molecules Bruneval and Marques (2013). van Setten and coworkers went one step
further and evaluated basis set extrapolated $GW$ QP energies for 100 closed
shell molecules using several codes van Setten  _et al._ (2015). They found
that the $GW$ QP energies of the highest occupied orbital (HOMO) and lowest
unoccupied orbital (LUMO) of two local basis set codes, FHI-aims and
Turbomole, virtually agree, if identical basis sets are used. In many respects
this is not astonishing, since two codes ought to yield the same results, if
the computational parameters are identical. The two codes are, however,
technically quite different. For instance, they introduce auxiliary basis sets
to avoid storing the two-electron four orbital integrals. Furthermore, FHI-
aims uses a numerical representation of the Gaussians and calculates the self-
energy along the imaginary axis (Wick rotation) requiring an analytic
continuation to the real axis. All these factors can introduce small
uncertainties. Clearly, the study impressively demonstrates that all these
intricacies are well under control, and technically well converged results can
be obtained using both codes.

The paper by van Setten et al.van Setten  _et al._ (2015) also reports results
using the $GW$ Berkeley plane wave code Deslippe  _et al._ (2012). Although
agreement of that code with experiment is very good if the plasmon-pole model
is used, comparison of the fully frequency dependent $G_{0}W_{0}$ HOMO and
LUMO with Gaussian basis set results is less satisfactory. For the considered
molecules, the mean absolute difference between Gaussian type orbitals (GTO)
and plane waves (PWs) is about 200 meV for the HOMO. We note on passing that
the agreement between GTO and other plane wave studies is seemingly superior
Govoni and Galli (2015); Laflamme Janssen  _et al._ (2015), although, this
could be related to the fact that these studies only considered a subset of
the $GW$100 set. The disagreement between the Berkeley $GW$ PW code and GTO
codes is certainly slightly disconcerting, since it puts decades of studies
using PW based $GW$ calculations into question. Remarkably, on the level of
DFT, the reported one-electron energies of the HOMO agree to within few 10
meV. So how can one understand the much larger discrepancies for $GW$ QP
energies?

A partial answer is given by the observation that QP energies converge very
slowly with respect to the basis set size, as well established for Gaussian
type orbitals Bruneval and Marques (2013); van Setten  _et al._ (2015);
Wilhelm  _et al._ (2016). van Setten et al. obtained basis set converged QP
energies by extrapolating against the basis set size or against $1/C_{n}^{3}$,
where $C_{n}$ is the basis set cardinal number van Setten  _et al._ (2015).
Extrapolation was based on def2-SVP, def2-TZVP and def2-QZVP, but even though
def2-QZVP constitutes a fairly complete set, the extrapolated values can
differ by more than 300 meV from the values at the largest considered basis
set. Astonishingly, the reported PW results were not extrapolated to the basis
set limit, although a recent work of Klimes et al. shows that the $GW$ QP
energies converge like one over the number of plane waves Klimeš  _et al._
(2014) and this behaviour is also confirmed by purely analytical arguments
Schindlmayr (2013). Early evidence of this slow convergence using PWs exist
aplenty Shih  _et al._ (2010); Friedrich  _et al._ (2011); Bruneval and Gonze
(2008). In view of this slow convergence, a brute force approach to predict QP
energies seems elusive considering that most codes scale cubically with
respect to the number of basis functions. The present work tries to rectify
this issue by reporting QP energies using the plane wave code VASP, carefully
correcting for basis set incompleteness errors, as detailed in section II.

Another point that we briefly mention in section III is that the calculation
of the poles of the $G_{0}W_{0}$ Green’s function can be unphysical, if the
initial Green’s function yields too small excitation energies. In this case,
first linearizing the $G_{0}W_{0}$ self-energy and then determining the poles
of the Green’s function yields more robust QP energies. We, finally, finish
with discussions and our conclusions.

##  II Theory and Computational method

###  II.1 Theory

$GW$ is a well established perturbative approach to calculate QP energies
Hedin (1965). In the $GW$ approximation, one initiates the calculations using
a groundstate DFT calculation to obtain the DFT one-electron orbitals
$\phi_{n}$ and the corresponding one-electron energies $\epsilon_{n}$.

The first step in a $GW$ calculation is to determine the DFT Green’s
functions,

$G_{0}({\bf r}^{\prime},{\bf r},\omega)=\sum_{n}\frac{\phi_{n}({\bf
r}^{\prime})\phi^{*}_{n}({\bf r})}{\omega-\epsilon_{n}-i\eta\,{\rm
sign}(\mu-\epsilon_{n})},$ (1)

where $\mu$ is the chemical potential of the electrons, and $\eta$ a positive
infinitesimal. From the Green’s function the independent particle
polarizability

$\chi({\bf r},{\bf r}^{\prime},t)=-\mathrm{i}G_{0}({\bf r},{\bf
r}^{\prime},t)G_{0}({\bf r}^{\prime},{\bf r},-t),$ (2)

and the corresponding screened interaction

$W({\bf r},{\bf r}^{\prime},\omega)=v({\bf r},{\bf r}^{\prime})+v({\bf r},{\bf
s})\chi({\bf s},{\bf s}^{\prime},\omega)W({\bf s}^{\prime},{\bf
r}^{\prime},\omega)$ (3)

can be determined. Here $v$ is the Coulomb kernel, and integration over
repeated spatial coordinates ($\bf s$ and ${\bf s}^{\prime}$) is assumed.
Furthermore, the Green’s functions and polarizabilities in frequency and time
domain are related by a Fourier transformation. The final step is to calculate
the interacting Green’s function

$G({\bf r},{\bf r}^{\prime},\omega)=\frac{1}{\omega-T-V^{\rm H}({\bf
r})\delta({\bf r}-{\bf r}^{\prime})-\Sigma({\bf r},{\bf r}^{\prime},\omega)},$
(4)

where $T$ is the kinetic energy operator, $V^{\rm H}$ is the Hartree-
potential, and $\Sigma({\bf r},{\bf r}^{\prime},t)$ is the self-energy in the
$GW$ approximation:

$\Sigma({\bf r},{\bf r}^{\prime},t)=\mathrm{i}G_{0}({\bf r},{\bf
r}^{\prime},t)W({\bf r},{\bf r}^{\prime},t).$ (5)

The poles of the Green’s function then determine the QP energies. In
principle, this cycle can be continued by evaluating $\chi$ in step (2) using
the updated Green’s function and iterated to self-consistency. It is also
possible to obtain partial self-consistency, for instance, by calculating $W$
once and forever using the DFT orbitals and one electron energies and
iterating only the Green’s function until it is self-consistent [i.e.
iterating only Eqs. (4) and (5)].

The most common approximation is, however, the $G_{0}W_{0}$ approximation e.g.
used by Hybertsen and Louie Hybertsen and Louie (1986). Instead of the poles
of the Green’s function, this approximation calculates the nodes of the
denominator in Eq. (4)

$E^{\rm QP}_{n}=\text{Re}\left[\langle\phi_{n}|T+V^{H}({\bf r})+\Sigma(E^{\rm
QP}_{n})|\phi_{n}\rangle\right]$ (6)

in the basis of the DFT orbitals. Since this involves only the diagonal
elements of the self-energy, solutions of this equation are cheaper to
determine than poles of the fully interacting Green’s function. Obviously this
is a good approximation, if the self-energy is diagonally dominant in the
basis of the DFT orbitals. As already pointed out by Hybertsen and Louie this
is generally the case, although there is some evidence that iterating the DFT
orbitals is important Faleev  _et al._ (2004); van Schilfgaarde  _et al._
(2006); Bruneval  _et al._ (2006). This is particularly so for atoms or
molecules, since the KS potential and, as a result, the KS orbitals do not
decay properly at large distances from the molecule.

The solutions obtained by solving Eq. (6) are labeled as $G_{0}W_{0}$ in the
present work. Furthermore, a commonly used approximation is to linearize the
the energy dependence in the self-energy in Eq. (6) at the DFT one-electron
energy and determine the nodes of the linearized equation. This yields the
following approximate position for the nodes Hybertsen and Louie (1986);
Shishkin and Kresse (2006):

$E^{\rm QP}_{n}-\epsilon^{\rm
DFT}_{n}=Z_{n}\text{Re}\left[\langle\phi_{n}|T+V^{\rm H}({\bf
r})+\Sigma(\epsilon^{\rm DFT}_{n})-\epsilon^{\rm
DFT}_{n}|\phi_{n}\rangle\right]$ (7)

were $Z_{n}$ is related to the derivative of the self-energy at $\epsilon^{\rm
DFT}_{n}$

$Z_{n}=\Big{(}1-\left.\frac{\partial\text{Re}[\langle\phi_{n}|\Sigma(\omega)|\phi_{n}\rangle]}{\partial\omega}\right|_{\epsilon^{\rm
DFT}_{n}}\Big{)}^{-1}.$

The correlation factor $Z_{n}$ can be also related to the amplitude of the
corresponding QP peak and is a measure of the degree of correlation. For the
HOMO and LUMO of molecules, $Z$ is commonly between 0.7-0.9, corresponding to
a low to very low degree of correlation. Solutions of the linearized equations
will be labeled as lin-$G_{0}W_{0}$ in the present work, and the first
derivative is evaluated using central difference with the step size of
$\Delta=\pm 0.1$ eV.

###  II.2 Technical details

As in the $GW$100 paper of van Setten et al. van Setten  _et al._ (2015), we
use the PBE functional for the DFT starting point. However, all calculations
include scalar relativistic effects, in contrast to the calculations of van
Setten et al. that are based on non-relativistic potentials. The potentials
used in the present work are the $GW$ potentials distributed with the latest
release of VASP (vasp.5.4), and we followed the recommendations in the VASP
manual on which version to use. Generally this means that lower lying semi-
core states were not correlated in the calculations, except for the alkali and
alkali-earth metals, as well as Ti and Ga. For He, we found issues with the
originally distributed potential. The He_GW potential failed to converge in
DFT calculations when the plane wave cutoff was increased, because a ghost
state was introduced as the basis set size increased. The potential was
slightly modified to remove this problem and will be distributed with the next
release. Furthermore, for boron to fluorine the potentials B$\\_$GW$\\_$new,
…, F$\\_$GW$\\_$new were used (also already distributed with vasp.5.4). These
potentials include $d$ partial waves, whereas the standard $GW$ potentials
choose the $d$ potential as the local potential.

The potentials used in this work are not the most accurate $GW$ potentials yet
available for VASP. Specifically, we have recently shown that norm-conserving
(NC) $GW$ potentials are necessary to predict very accurate QP energies for
$3d$, $4d$ and $5d$ elements Klimeš  _et al._ (2014) with the NC potentials
generally increasing the QP binding energies. In our experience, such highly
accurate potentials are, however, not required in the present case for the
following reasons. For $s$ and $p$ elements the standard potentials conserve
the norm very well to within about 70 %, often even 90 %. Furthermore, errors
introduced by violating the norm-conservation can only occur at very high
scattering energies, since the standard $GW$-PAW potentials predict the
scattering properties correctly up to about 400 eV. Beyond that energy, the
PAW projectors become incomplete. For the elements considered here, we expect
that the combination of these two effects means that the results for the HOMO
and LUMO will be accurate even though we do not use NC potentials. The only
exceptions are copper, neon, fluorine, oxygen and possibly nitrogen. These
elements possess strongly localized $3d$ and $2p$ orbitals. We will return to
this point later.

In the calculations presented here we calculate the Green’s function, the
screened interaction $W$ as well as the self-energy in imaginary time and
frequency. This has several advantages compared to the full real frequency
implementation also available in VASP. The fully frequency dependent version
along the real axis requires at least 100, but for molecules with their sharp
resonances often even several hundred frequency points to converge. Since the
boxes considered in this work are quite large, we also need several thousands
of plane waves to describe the frequency dependent screened interaction and
Green’s function accurately. This becomes very quickly prohibitive. In the
imaginary frequency, on the other hand, only relatively few frequency points
are required. In the calculations presented here, 16 frequency points and the
time and frequency grids discussed by Kaltak et al. are used Kaltak  _et al._
(2014a, b). These 16 points were found to be sufficient to converge the QP
energies of the HOMO and LUMO to about 10 meV Liu  _et al._ (2016). The
downside of working in the imaginary frequency domain is that the results
along the imaginary frequency axis need to be continued to the real axis. This
was done using a (16 point) Padé fit following Thiele’s reciprocal difference
method based on continued fractions Baker (1975). We note that the reported
FHI-aims results in Ref. van Setten  _et al._ , 2015 were— with few
problematic exceptions —also obtained using 16 parameter Padé fits. These
exceptions are BN, O3, BeO, MgO and CuCN where many more points were required.
For the other molecules, the 16 parameter Padé fits yielded excellent
agreement with Turbomole, which calculates the exact $GW$ self-energy along
the real axis. Details of our implementation are reported elsewhere Liu  _et
al._ (2016).

The other crucial issues are basis set extrapolation and convergence with
respect to the box size. To obtain basis set converged results, we used a
relatively small box, but one that still faithfully reproduces the character
of the HOMO and LUMO. For this box, we performed calculations for the default
cutoff as specified by the VASP potentials, and calculations for three
additional plane wave cutoffs, with the largest calculation corresponding to
twice the number of plane waves used in the default setup. These four data
points are fitted assuming that the QP energies as a function of the number of
plane waves $N_{\rm pw}$ converge like

$E^{\rm QP}(N_{\rm pw})=E^{\rm QP}(\infty)+\frac{C}{N_{\rm pw}},$ (8)

where $N_{\rm pw}$ is the number of plane waves in the basis set Harl and
Kresse (2008); Shepherd  _et al._ (2012); Björkman  _et al._ (2012); Klimeš
_et al._ (2014). A four point fit and a two point fit with the largest and
smallest PW basis set yielded a maximum difference of 10 meV in the QP
energies. To illustrate that the basis set dependence is indeed following a
$1/{N_{\rm pw}}$ behavior to great accuracy, we will show data for selected
molecules in Sec. III.2. The only subtlety impeding an accurate and automatic
extrapolation is the use of the Padé fit. The slope of the self-energy can
vary somewhat between different calculations causing some variations in the
predicted QP energies. Extrapolation from these “noisy” data is difficult and
error prone. To circumvent this issue, we perform the extrapolation for the
self-energy evaluated at the DFT one-electron energies, specifically on
$\Delta E=\text{Re}\left[\langle\phi|T+V^{\rm H}+\Sigma(\epsilon^{\rm
DFT})|\phi\rangle\right]-\epsilon^{\rm DFT}$ instead of $\Delta E=E^{\rm
QP}-\epsilon^{\rm DFT}$, and scale the correction by the $Z$-factor at the
smallest, i.e. default, PW cutoff. In this way, we neglect variations of the
$Z$-factor between different basis sets, but these variations are small and
dominated by noise.

A few final comments are in place here. In the calculations presented herein,
we calculate all orbitals spanned by the PW basis set. This implies that the
number of orbitals also increases as the number of plane waves increases.
Second, the kinetic energy cutoff for the response function (ENCUTGW in VASP)
is set to 2/3 of the cutoff used for the plane wave basis of the orbitals
(ENCUT in VASP). Whenever the PW cutoff for the orbitals is increased, the PW
cutoff for the basis set of the response function is increased accordingly.
This means that a single parameter, the PW cutoff for the orbitals (ENCUT),
entirely controls the accuracy of the calculations (at least with respect to
the basis sets). Since all the intermediate control parameters are set
automatically by VASP, and since the QP energy corrections converge like one
over the number of plane waves and orbitals Klimeš  _et al._ (2014),
extrapolation to the infinite basis set limit is straightforward and robust.

Let us now comment on the second point, convergence with respect to the cell
size. In plane wave codes, it is common practice to truncate the Coulomb
kernel at a certain distance $r_{c}$, say half the box size, so that the
periodically repeated orbitals can not screen the central atom. The downside
of this approach is that it modifies the Coulomb kernel to becomeRozzi  _et
al._ (2006)

$\frac{4\pi e^{2}}{|{\bf g}|^{2}}(1-\cos(|{\bf g}|r_{c})),$

where $\bf g$ is a plane wave vector. Obviously, this modifies the Coulomb
kernel at large reciprocal lattice vectors $\bf g$. In test calculations we
found that this spoils the previously mentioned basis set extrapolation (8):
as one increases the plane wave cutoff, one moves through maxima and minima of
the truncated Coulomb kernel, causing superimposed oscillations in the QP
energies. Basis set extrapolation becomes then uncontrolled. To deal with the
repeated images, we instead resort to the standard trick used in periodic
codes: $k\cdot p$ perturbation theory Baldereschi and Tosatti (1978). We
calculate the first order change of the orbitals with respect to $k$ Gajdoš
_et al._ (2006), and accordingly the head and wings of the polarizability and
a correction to the ${\bf g}\to 0$ component of the self-energy. This term
corrects the leading monopole-monopole interaction between repeated images,
but leaves the monopole-dipole and dipole-dipole interactions uncorrected.
These two terms fall off like $1/V$ and $1/V^{2}$, where $V$ is the cell size
volume Makov and Payne (1995). To deal with this, we perform four calculations
at different volumes, with the box size progressively increased by 1 Å and fit
the data to

$a_{0}+a_{1}/V+a_{2}/V^{2}.$ (9)

For most molecules the corrections are small and only of the order of 10-20
meV, whereas for the alkali dimers and some polar molecules the corrections
can be as large as 100-200 meV. In these cases the correction is very well
described by the theoretical equation. We hope to find a better solution in
future work, for instance, an explicit subtraction of monopole and dipole
interactions between periodic images. In terms of compute time, however, the
additional calculations for smaller boxes only require a modest amount of
time: since the total compute time scales quadratic to cubic with respect to
the number of plane waves, the calculations scale also quadratic to cubic in
the volume. Typically we need 12 Å large boxes to obtain results converged to
20 meV with respect to the box size. The additional smaller volumes used for
the extrapolation require only half of the compute time of the largest final
box.

The final QP energies reported in the next section were obtained by
calculating the PBE one-electron HOMO and LUMO for a 25 Å box at an energy
cutoff that is 30 % increased compared to the VASP default values. The vacuum
level, evaluated as the Hartree plus ionic potential, was evaluated at the
position furthest from the center of the molecule and subtracted from the PBE
one-electron energies. We checked that the DFT one-electron energies are
converged to a few meV with this setup. To the DFT one-electron energies, the
shift of the QP energies $E^{\rm QP}-\epsilon^{\rm DFT}$ for the largest
considered box, box size corrections, and basis set corrections as described
above are added. It goes without saying that this procedure is rather involved
and since errors are expected to accumulate, we estimate that the present
predictions are only accurate to about $\pm 50$ meV, where convergence with
cell size is the main source of errors and difficult to estimate precisely.

To give a feeling for the required compute time and computational effort, we
need to stress that our plane wave code is mainly designed for solids.
Nevertheless, a calculations for C6H6 in a 10 Å box at the default cutoff
takes about 4 hours on a single node with 16 Xeon v2 cores. The compute time
stays roughly constant if the box size is increased by 1 Å and the number of
cores is simultaneously doubled. Furthermore, the compute time is mostly
independent of the number of atoms in the box, but increases cubically with
the box size as the total number of plane waves increases linearly with the
box size. By comparison Turbomole, using the def2-TZVP basis and the
resolution of the identity method, takes 30 minutes on a 12 core AMD opteron
6174 for the response and $GW$ part of the calculation (the time spent for the
DFT part is negligible in comparison).

##  III Results

###  III.1 HOMO for $GW$100

Table 1:  Ionization potential (IP, negative of HOMO QP energies) for 100 molecules using $G_{0}W_{0}$ and linearized lin-$G_{0}W_{0}$ method. For comparison the basis set extrapolated values of Ref. van Setten  _et al._ , 2015 and the experimental IPs are given (vertical IPs are in italics). If basis set extrapolated values are not specified in Ref. van Setten  _et al._ , 2015, the AIMS-P16 values are shown in the column GTO (marked by ∗). Last column shows the differences between GTO and PW results. The ∗ indicates differences to non basis set extrapolated values.  |  | $G_{0}W_{0}$ | $G_{0}W_{0}$ | lin-$G_{0}W_{0}$ | EXP | $\Delta$
---|---|---|---|---|---|---
|  | GTOvan Setten  _et al._ (2015) | PW | PW |  | PW-GTO
1 | He | 23.49(0.03) | 23.38 | 23.62 |  24.59 Kelly (1987) | -0.11
2 | Ne | 20.33(0.01) | 20.17 | 20.36 |  21.56 Kelly (1987) | -0.16
3 | Ar | 15.28(0.03) | 15.32 | 15.42 |  15.76 Weitzel  _et al._ (1994) | 0.04
4 | Kr | 13.89(0.16) | 13.93 | 14.03 | 14.00 Wetzel  _et al._ (1987) | 0.04
5 | Xe | 12.02∗ | 12.14 | 12.22 | 12.13 Schäfer and Rabeneck (1987) | 0.12∗
6 | ${\rm H_{2}}$ | 15.85(0.09) | 15.85 | 16.06 | 15.43 McCormack  _et al._ (1989) | 0.00
7 | ${\rm Li_{2}}$ | 5.05(0.02) | 5.09 | 5.32 | 4.73 Dugourd  _et al._ (1992) | 0.04
8 | ${\rm Na_{2}}$ | 4.88(0.03) | 4.93 | 5.06 | 4.89 Kappes  _et al._ (1985) | 0.05
9 | ${\rm Na_{4}}$ | 4.14(0.03) | 4.17 | 4.23 | 4.27 Herrmann  _et al._ (1978) | 0.03
10 | ${\rm Na_{6}}$ | 4.34(0.06) | 4.34 | 4.40 | 4.12 Herrmann  _et al._ (1978) | 0.00
11 | ${\rm K_{2}}$ | 4.08(0.04) | 4.12 | 4.24 | 4.06 Kappes  _et al._ (1985) | 0.04
12 | ${\rm Rb_{2}}$ | 3.79∗ | 4.02 | 4.14 | 3.90 Kappes  _et al._ (1985) | 0.23∗
13 | ${\rm N_{2}}$ | 15.05(0.04) | 14.93 | 15.06 | 15.58 Trickl  _et al._ (1989) | -0.12
14 | ${\rm P_{2}}$ | 10.38(0.04) | 10.35 | 10.40 | 10.62 Bulgin  _et al._ (1976) | -0.03
15 | ${\rm As_{2}}$ | 9.67(0.10) | 9.59 | 9.62 | 10.0 Lau  _et al._ (1982) | -0.08
16 | ${\rm F_{2}}$ | 15.10(0.04) | 14.93 | 15.08 | 15.70 Lonkhuyzen and Lange (1984) | -0.17
17 | ${\rm Cl_{2}}$ | 11.31(0.05) | 11.32 | 11.40 |  11.49 Dyke  _et al._ (1984a) | 0.01
18 | ${\rm Br_{2}}$ | 10.56(0.18) | 10.57 | 10.65 |  10.51 Dyke  _et al._ (1984a) | 0.01
19 | ${\rm I_{2}}$ | 9.23∗ | 9.52 | 9.59 |  9.36 Kimura  _et al._ (1981) | 0.29∗
20 | ${\rm CH_{4}}$ | 14.00(0.06) | 14.02 | 14.14 |  13.6 Bieri and Åsbrink (1980) | 0.02
21 | ${\rm C_{2}H_{6}}$ | 12.46(0.06) | 12.50 | 12.58 |  11.99 Kimura  _et al._ (1981) | 0.04
22 | ${\rm C_{3}H_{8}}$ | 11.89(0.06) | 11.90 | 11.98 |  11.51 Kimura  _et al._ (1981) | 0.01
23 | ${\rm C_{4}H_{10}}$ | 11.59(0.05) | 11.61 | 11.69 |  11.09 Kimura  _et al._ (1981) | 0.02
24 | ${\rm C_{2}H_{4}}$ | 10.40(0.03) | 10.42 | 10.50 |  10.68 Bieri and Åsbrink (1980) | 0.02
25 | ${\rm C_{2}H_{2}}$ | 11.09(0.01) | 11.07 | 11.24 |  11.49 Bieri and Åsbrink (1980) | -0.02
26 | ${\rm C_{4}}$ | 10.91(0.03) | 10.89 | 10.97 | 12.54 Ramanathan  _et al._ (1993) | -0.02
27 | ${\rm C_{3}H_{6}}$ | 10.65(0.04) | 10.72 | 10.78 |  10.54 Plemenkov  _et al._ (1981) | 0.07
28 | ${\rm C_{6}H_{6}}$ | 9.10(0.01) | 9.11 | 9.16 |  9.23 Howell  _et al._ (1984) | 0.01
29 | ${\rm C_{8}H_{8}}$ | 8.18(0.02) | 8.19 | 8.24 |  8.43 Fu and Dunbar (1978) | 0.01
30 | ${\rm C_{5}H_{6}}$ | 8.45(0.02) | 8.47 | 8.51 |  8.53 Kiselev  _et al._ (1992) | 0.02
31 | ${\rm CH_{2}CHF}$ | 10.32(0.02) | 10.28 | 10.36 |  10.63 Bieri  _et al._ (1981a) | -0.04
32 | ${\rm CH_{2}CHCl}$ | 9.89(0.02) | 9.92 | 10.00 |  10.20 Cambi  _et al._ (1983) | 0.03
33 | ${\rm CH_{2}CHBr}$ | 9.14(0.01) | 9.75 | 9.83 |  9.90 Cambi  _et al._ (1983) | 0.61
34 | ${\rm CH_{2}CHI}$ | 9.01∗ | 9.27 | 9.36 |  9.35 Wittel  _et al._ (1974) | 0.26∗
35 | ${\rm CF_{4}}$ | 15.60(0.06) | 15.41 | 15.53 |  16.20 Bieri  _et al._ (1981b) | -0.19
36 | ${\rm CCl_{4}}$ | 11.21(0.06) | 11.20 | 11.31 |  11.69 Kimura  _et al._ (1981) | -0.01
37 | ${\rm CBr_{4}}$ | 10.22(0.16) | 10.25 | 10.38 |  10.54 Dixon  _et al._ (1971) | 0.03
38 | ${\rm CI_{4}}$ | 8.71∗ | 9.11 | 9.23 |  9.10 Jonkers  _et al._ (1982) | 0.40∗
39 | ${\rm SiH_{4}}$ | 12.40(0.06) | 12.40 | 12.53 |  12.3 Roberge  _et al._ (1978) | 0.00
40 | ${\rm GeH_{4}}$ | 12.11(0.04) | 12.13 | 12.24 |  11.34 Potts and Price (1972) | 0.02
41 | ${\rm H_{6}Si_{2}}$ | 10.41(0.06) | 10.44 | 10.52 |  10.53 Bock  _et al._ (1976) | 0.03
42 | ${\rm H_{12}Si_{5}}$ | 9.05(0.05) | 9.13 | 9.19 |  9.36 Bock  _et al._ (1976) | 0.08
43 | LiH | 6.58(0.04) | 6.46 | 7.20 | 7.90 NIS (2015) | -0.12
44 | KH | 4.99(0.01) | 4.97 | 5.37 | 8.00 Farber  _et al._ (1982) | -0.02
45 | ${\rm BH_{3}}$ | 12.96(0.06) | 12.95 | 13.09 | 12.03 Ruščić  _et al._ (1988) | -0.01
46 | ${\rm B_{2}H_{6}}$ | 11.93(0.06) | 11.94 | 12.04 |  11.90 Åsbrink  _et al._ (1981) | 0.01
47 | ${\rm NH_{3}}$ | 10.39(0.05) | 10.32 | 10.44 |  10.82 Baumgaertel  _et al._ (1989) | -0.07
48 | ${\rm HN_{3}}$ | 10.55(0.02) | 10.50 | 10.56 |  10.72 Cvitaš and Klasinc (1976) | -0.05
|  | $G_{0}W_{0}$ | $G_{0}W_{0}$ | lin-$G_{0}W_{0}$ | EXP | $\Delta$
---|---|---|---|---|---|---
|  | GTO | PW | PW |  | PW-GTO
49 | ${\rm PH_{3}}$ | 10.35(0.05) | 10.35 | 10.45 |  10.59 Cowley  _et al._ (1982) | 0.00
50 | ${\rm AsH_{3}}$ | 10.21(0.02) | 10.26 | 10.36 |  10.58 Demuth (1977) | 0.05
51 | ${\rm H_{2}S}$ | 10.13(0.04) | 10.11 | 10.30 |  10.50 Bieri  _et al._ (1982) | -0.02
52 | HF | 15.37(0.01) | 15.37 | 15.38 |  16.12 Banna and Shirley (1975) | 0.00
53 | HCl | 12.36(0.01) | 12.45 | 12.51 | 12.79 Wang  _et al._ (1984) | 0.09
54 | LiF | 10.27(0.03) | 10.07 | 10.45 | 11.30 Berkowitz  _et al._ (1962) | -0.20
55 | ${\rm MgF_{2}}$ | 12.50(0.06) | 12.41 | 12.77 | 13.30 Hildenbrand (1968) | -0.09
56 | ${\rm TiF_{4}}$ | 14.07(0.05) | 14.01 | 14.22 |  15.30 Dyke  _et al._ (1997) | -0.06
57 | ${\rm AlF_{3}}$ | 14.48(0.06) | 14.33 | 14.53 | 15.45 Dyke  _et al._ (1984b) | -0.15
58 | BF | 10.73(0.05) | 10.46 | 10.67 | 11.00 Farber and Srivastava (1984) | -0.27
59 | ${\rm SF_{4}}$ | 12.38(0.07) | 12.20 | 12.29 | 11.69 Fisher  _et al._ (1992) | -0.18
60 | KBr | 7.57(0.13) | 7.80 | 8.04 |  8.82 Potts and Price (1977) | 0.23
61 | GaCl | 9.74(0.07) | 9.89 | 9.99 |  10.07 Grabandt  _et al._ (1990) | 0.15
62 | NaCl | 8.43(0.14) | 8.47 | 8.76 |  9.80 Potts and Price (1977) | 0.04
63 | ${\rm MgCl_{2}}$ | 11.20(0.07) | 11.19 | 11.41 |  11.80 Lee and Potts (1979) | -0.01
64 | ${\rm AlI_{3}}$ | 9.30∗ | 9.58 | 9.69 |  9.66 Barker  _et al._ (1975) | 0.28∗
65 | BN | 11.15(0.03) | - | 10.61 | 11.50 | -
66 | HCN | 13.32(0.01) | 13.29 | 13.43 |  13.61 Kreile  _et al._ (1982) | -0.03
67 | PN | 11.29(0.04) | 11.24 | 11.41 | 11.88 Bulgin  _et al._ (1977) | -0.05
68 | ${\rm N_{2}H_{4}}$ | 9.37(0.04) | 9.33 | 9.45 |  8.98 Vovna  _et al._ (1975) | -0.04
69 | ${\rm H_{2}CO}$ | 10.46(0.02) | 10.42 | 10.57 | 10.88 Ohno  _et al._ (1995) | -0.04
70 | ${\rm CH_{3}OH}$ | 10.67(0.05) | 10.61 | 10.72 |  10.96 Vorob’ev  _et al._ (1989) | -0.06
71 | ${\rm CH_{3}CH_{2}OH}$ | 10.27(0.05) | 10.21 | 10.33 |  10.64 Ohno  _et al._ (1985) | -0.06
72 | ${\rm CH_{3}CHO}$ | 9.66(0.03) | 9.63 | 9.80 |  10.24 Johnson  _et al._ (1982) | -0.03
73 | ${\rm CH_{3}CH_{2}OCH_{2}CH_{3}}$ | 9.42(0.05) | 9.43 | 9.52 |  9.61 Ohno  _et al._ (1985) | 0.01
74 | HCOOH | 10.87(0.01) | 10.81 | 10.98 |  11.50 von Niessen  _et al._ (1980) | -0.06
75 | ${\rm H_{2}O_{2}}$ | 11.10(0.01) | 10.96 | 11.12 |  11.70 Ashmore and Burgess (1977) | -0.14
76 | ${\rm H_{2}O}$ | 12.05(0.03) | 11.84 | 12.05 |  12.62 Kimura  _et al._ (1981) | -0.21
77 | ${\rm CO_{2}}$ | 13.46(0.06) | 13.36 | 13.44 |  13.77 Eland and Berkowitz (1977) | -0.10
78 | ${\rm CS_{2}}$ | 9.95(0.05) | 9.96 | 10.01 |  10.09 Schweig and Thiel (1974) | 0.01
79 | CSO | 11.11(0.05) | 11.06 | 11.13 |  11.19 Potts and Williams (1974) | -0.05
80 | COSe | 10.43(0.09) | 10.42 | 10.50 | 10.37 Cradock and Duncan (1975) | -0.01
81 | CO | 13.71(0.04) | 13.62 | 13.76 |  14.01 Potts and Williams (1974) | -0.09
82 | ${\rm O_{3}}$ | 11.49(0.03) | - | 12.07 |  12.73 Katsumata  _et al._ (1984) | -
83 | ${\rm SO_{2}}$ | 12.06(0.06) | 11.91 | 12.04 |  12.50 Kimura  _et al._ (1981) | -0.15
84 | BeO | 8.60(0.01) | - | 9.50 | 10.10 Theard and Hildenbrand (1964) | -
85 | MgO | 6.75(0.03) | - | 7.10 | 8.76 Dalleska and Armentrout (1994) | -
86 | ${\rm C_{6}H_{5}CH_{3}}$ | 8.73(0.02) | 8.75 | 8.79 |  8.82 Howell  _et al._ (1984) | 0.02
87 | ${\rm C_{8}H_{10}}$ | 8.66(0.02) | 8.69 | 8.73 |  8.77 Howell  _et al._ (1984) | 0.03
88 | ${\rm C_{6}F_{6}}$ | 9.74(0.07) | 9.63 | 9.69 |  10.20 Bieri  _et al._ (1981b) | -0.11
89 | ${\rm C_{6}H_{5}OH}$ | 8.51(0.01) | 8.38 | 8.43 | 8.75 Ballard  _et al._ (1987) | -0.13
90 | ${\rm C_{6}H_{5}NH_{2}}$ | 7.78(0.01) | 7.78 | 7.84 |  8.05 Furin  _et al._ (1987) | 0.00
91 | ${\rm C_{5}H_{5}N}$ | 9.17(0.01) | 9.16 | 9.31 |  9.66 Kobayashi and Nagakura (1974) | -0.01
92 | Guanine | 7.87(0.01) | 7.85 | 7.90 |  8.24 Hush and Cheung (1975) | -0.02
93 | Adenine | 8.16(0.01) | 8.12 | 8.18 |  8.48 Lin  _et al._ (1980) | -0.04
94 | Cytosine | 8.44(0.01) | 8.40 | 8.50 |  8.94 Hush and Cheung (1975) | -0.04
95 | Thymine | 8.87(0.01) | 8.83 | 8.89 |  9.20 Dougherty  _et al._ (1976) | -0.04
96 | Uracil | 9.38(0.01) | 9.36 | 9.55 |  9.68 Palmer  _et al._ (1980) | -0.02
97 | ${\rm NH_{2}CONH_{2}}$ | 9.46(0.02) | 9.35 | 9.59 |  9.80 Bieri  _et al._ (1982) | -0.11
98 | ${\rm Ag_{2}}$ | 7.08∗ | 7.83 | 7.95 | 7.66 Beutel  _et al._ (1993) | 0.75∗
99 | ${\rm Cu_{2}}$ | 7.78(0.06) | 7.19 | 7.40 | 7.46 Franzreb  _et al._ (1990) | -0.59
100 | CuCN | 9.56(0.04) | - | 9.99 | - | -

Let us first note on the agreement at the level of DFT (not shown). In
general, our values agree exceedingly well with the PBE values reported in the
supplementary material of Ref. van Setten  _et al._ , 2015. In most cases, our
PBE HOMO is located between the basis set extrapolated values and the values
obtained with the best basis sets used in the GTO calculations (def2-QZVP). On
average, our PW HOMOs agree better with the non basis set extrapolated values
with a mean deviation (MD) of 7 meV and a mean absolute deviation (MAD) of 19
meV. Compared to the GTO basis set extrapolated values, the MD and MAD are -25
meV and 30 meV (in both cases, CH2CHBr was excluded, see below).

van Setten et al. van Setten  _et al._ (2015) extrapolated the DFT eigenvalues
using a cubic polynomial in the inverse of the basis set cardinal number
($C_{n}^{-3}$); we believe that this is not appropriate and will overestimate
the basis set corrections. It is commonly agreed that DFT calculations
converge exponentially with the cardinal number, whereas any correlated wave
function calculation converges with the inverse of the basis set size
(corresponding roughly to $C_{n}^{-3}$) Bruneval and Marques (2013); Klimeš
_et al._ (2014); van Setten  _et al._ (2015). This is a result of Kato’s cusp
condition Kato (1957) causing a kink in the many-body wave function as two
coordinates approach each other. We have shown that this problem carries over
to $GW$ calculations Klimeš  _et al._ (2014). As a one-electron theory,
density functional theory does not suffer from this slow convergence. We hence
believe that van Setten overestimated the basis set corrections for DFT. This
is supported by the observation that our PBE results tend to be closer to the
non-extrapolated Gaussian results at the level of def2-QZVP.

We now turn to the QP energies predicted at the level of $G_{0}W_{0}$ shown in
Table 1. The agreement between the VASP PW and the GTO results is generally
very good. We note that the $G_{0}W_{0}$ approximation used here is identical
to the one applied by van Setten et al. van Setten  _et al._ (2015).
Specifically, van Setten determined the nodes of Eq. (6), and we do exactly
the same in the present work. Linearization of the QP equation (7) yields
generally somewhat larger QP energies and often improves agreement with
experiment slightly (column lin-$G_{0}W_{0}$). This trend has also been
observed in a recent benchmark for an unrelated set of molecules Scherpelz
_et al._ (2016). In agreement with van Setten van Setten  _et al._ (2015), we
have found poles in the self-energy close to the predicted QP energies for BN,
O3, BeO, MgO and CuCN. Since analytic continuation has difficulties to resolve
the precise pole structure of the Green’s function, we only report the values
obtained from the linearized self-energy.

Our discussion starts with the molecules that show large discrepancies between
VASP and GTO’s. A large out-liner is seemingly CH2CHBr. However, for this
molecule, as well as C6H5OH, we found large forces in the preparatory PBE
calculations. Double checking the original literature Weast and Astle (2011)
suggests that the $GW$100 paper used incorrect geometries. Since the ultimate
purpose is certainly to compare with experiment, we decided to update the
geometries to the correct literature values.

Table 2:  IP (negative HOMO $G_{0}W_{0}$ QP energies) and $G_{0}W_{0}$ LUMO for selected molecules. The GTO values have been calculated using frozen core potentials, scalar relativistic corrections, and are extrapolated to the infinite basis set limit.  |  | IP | IP | LUMO | LUMO
---|---|---|---|---|---
|  | GTO | PW | GTO | PW
5 | Xe | 12.22 | 12.14 | -0.07 | 0.28
12 | ${\rm Rb_{2}}$ | 4.07 | 4.02 | -0.85 | -0.74
19 | ${\rm I_{2}}$ | 9.48 | 9.52 | -2.28 | -2.21
34 | ${\rm CH_{2}CHI}$ | 9.13 | 9.27 | 0.56 | 0.37
38 | ${\rm CI_{4}}$ | 8.97 | 9.11 | -2.47 | -2.42
64 | ${\rm AlI_{3}}$ | 9.50 | 9.58 | -1.18 | -1.02
98 | ${\rm Ag_{2}}$ | 7.96 | 7.83 | -1.40 | -1.35

Among the remaining molecules, errors are large for compounds containing
iodine, rubidium, and silver with a maximum deviation of 400 meV for CI4 and
Rb2, and 750 meV for Ag2. However, in Ref. van Setten  _et al._ , 2015 no
basis set extrapolation was performed for these molecules. From CCl4 to CBr4,
the basis set corrections increase from 300 meV to 350 meV, suggesting a basis
set error of 400 meV for CI4 using GTOs. Similarly, for Rb2 the GTO results
were not basis set corrected, and estimating the basis set error from Na2 and
K2 again suggests that the VASP results are accurate. For Ag2, the difference
between VASP and GTO seem on first sight to be too large to be ascribed to
basis set errors alone. To resolve the issue, one of us (MvS) repeated the Xe,
Rb2, I2, ${\rm CH_{2}CHI}$, CI4, AlI3, and Ag2 calculations using scalar
relativistic corrections and frozen core SVP, TZVP and QZVP basis sets. This
yielded basis set extrapolated values summarized in Tab. 2 certainly now in
good to very good agreement with the VASP values.

Table 3:  IP (negative HOMO $G_{0}W_{0}$ QP energies) for selected molecules calculated for a 9 Å box for the standard $GW$ potentials and normconserving $GW$ potentials. Results differ from the previous table, since calculations in Table 1 have been performed for larger boxes and include a correction for the box size error. The column $\Delta$ reports the difference between the standard PAW and NC PAW potential.  |  |  $GW$ PAW | NC $GW$ PAW | $\Delta$
---|---|---|---|---
13 | ${\rm N_{2}}$ | 14.98 | 15.02 | -0.04
16 | ${\rm F_{2}}$ | 14.97 | 15.13 | -0.15
35 | ${\rm CF_{4}}$ | 15.42 | 15.58 | -0.16
52 | HF | 15.39 | 15.32 | 0.06
58 | BF | 10.42 | 10.46 | -0.04
59 | ${\rm SF_{4}}$ | 12.19 | 12.26 | -0.07
76 | ${\rm H_{2}O}$ | 11.86 | 11.94 | -0.09
99 | ${\rm Cu_{2}}$ | 7.03 | 7.53 | -0.50

For the remaining molecules, the mean absolute deviation between the two codes
and thus two completely different basis sets is only 60 meV, if we also
exclude Cu2. For Cu2, the fluorine containing compounds, H2O, as well as Ne
the ionization potentials (IPs) are smaller in VASP, which we will now show to
be related to slight deficiencies in the PAW potentials. Copper, neon, and
fluorine and, to a lesser extent, oxygen are particularly difficult to
describe using a plane wave based approach, since the $3d$ and $2p$ electrons
are strongly localized. To cope with this, the Cu, F and Ne potentials are
already the three smallest core and hardest potentials used in the present
work. But still, the partial waves do not conserve the norm exactly, which
results in errors, if an electron is scattered into a plane wave with very
high kinetic energy Klimeš  _et al._ (2014). To determine this error, we
performed calculations with norm-conserving (or almost norm-conserving) $GW$
potentials for the molecules Cu2, N2, F2, CF4, HF, BF, SF4 and H2O reported in
Tab. 3 using the potentials Cu$\\_$sv$\\_$GW$\\_$nc,
N$\\_$h$\\_$GW,…,F$\\_$h$\\_$GW. Except for HF, the QP energies are clearly
shifted towards higher binding energies in these calculations, and the
discrepancies to the GTO calculations are reduced to an acceptable level of
100 meV. We also note that the PAW error increases from nitrogen, over oxygen
to fluorine. HF and BF are exceptions, since the HOMOs possess predominantly
hydrogen and boron character and, therefore, do not depend strongly on the F
potential (we note that the HF results were already accurate using the
standard potentials). Finally, the standard carbon and boron potentials used
here are already almost norm conserving, and hence negligible changes are
found for carbon based compounds with harder potentials (not shown).

The final case worthwhile mentioning is KBr. Here the $GW$100 paper van Setten
_et al._ (2015) reports relatively large extrapolation errors of 130 meV,
indicating that in this case the GTO based extrapolation might be inaccurate.

For the remaining systems, we find the agreement to be excellent.
Specifically, for all considered organic molecules the absolute differences
are typically below 50 meV, with very few out-liners. This clearly
demonstrates that plane wave codes can be competitive in terms of precision
with GTOs. Certainly the agreement between GTOs and PWs is better than
originally reported in the $GW$100 paper, a point discussed in more detail in
the next section.

Figure 1:  (color online) Convergence of QP HOMO with respect to the employed
cutoff for various materials. For N2, CF4 and H2O, results are shown for two
potentials, the standard $GW$ potentials, as well as NC potentials. The slopes
are steeper for the NC potentials, which is particularly obvious for CF4. For
H2O results for two box sizes (8 and 9 Å) are shown (see text).

###  III.2 Basis set convergence and comparison to other PW calculations

In Fig. 1, we show the convergence of the HOMO with respect to the plane wave
cutoff for the orbitals. This cutoff also determines the total number of
orbitals as well as the cutoff for the response function. The number of plane
waves and total orbitals is proportional to one over the cutoff to the power
of 3/2. It is clearly visible that the curves follow almost exactly a straight
line. In few cases, out-liners are visible. For instance for H2O, we have
included results for two box sizes 8 and 9 Å. The 9 Å box results have a
slight jump, which is not present for the 8 Å box. However, this out-liner is
small (about 10 meV), and changes extrapolated results only by less than 10
meV. Usually the out-liners could be dealt with by just changing the box size
slightly. We believe that they are related to ”shell”-effects, i.e. a sudden
increase in the number of plane waves when the cutoff is changed through
certain values. Furthermore, the analytic continuation is not always entirely
well behaved and can cause changes of the order of 20 meV. Overall, the plot
demonstrates that extrapolation with respect to the energy-cutoff is very well
under control and can be done with great accuracy.

As noted before, the differences between the GTO and the Berkeley $GW$
calculations reported in Ref. van Setten  _et al._ , 2015 are more sizable. If
we exclude the difficult multipole cases, BN, O3, MgO and BeO, the mean
absolute deviation between Berkeley $GW$ and basis set extrapolated GTOs was
0.2 eV in Ref. van Setten  _et al._ , 2015, whereas it is reduced to 0.05 eV
for VASP PAW potentials (for the same subset). We speculate that this is
mostly related to neglecting basis set extrapolation errors or— less likely
—to an inaccurate treatment of the core-valence interaction. Typically our
basis set corrections are of the order of 300-400 meV at the default cutoff
and therefore very sizable. Even doubling the number of basis functions and
therefore increasing the compute time by a factor of about 8 (cubic scaling),
reduces the error only by a factor 2, to about 150-200 meV. Hence,
calculations without basis set corrections are hardly affordable or
practicable, and it is certainly advisable to perform an extrapolation
whenever possible.

Table 4:  LUMO QP energies for selected molecules using $G_{0}W_{0}$ and the linearized lin-$G_{0}W_{0}$. For comparison, the non basis set extrapolated values (AIMS-P16), the basis set extrapolated values from Ref. van Setten  _et al._ , 2015, and the negative of the experimental electron affinities are shown (vertical attachment energies are in italics). Differences between PW and GTO are shown in last column. The + indicates energies above the vacuum level, and ∗ indicates differences to non basis set extrapolated values.  |  | $G_{0}W_{0}$ | $G_{0}W_{0}$ | $G_{0}W_{0}$ | lin-$G_{0}W_{0}$ | EXP | $\Delta$
---|---|---|---|---|---|---|---
|  | AIMS-P16 | GTO-EXTRA | PW | PW |  | PW-GTO
2 | Ne | 11.64 | - | 0.40 | 0.40 | - | -11.24+
5 | Xe | 4.28 | - | 0.70 | 0.70 | - | -3.58+
6 | ${\rm H_{2}}$ | 3.50 | 3.30(0.52) | 0.07 | 0.07 | - | -3.23+
7 | ${\rm Li_{2}}$ | -0.63 | -0.75(0.04) | -0.61 | -0.54 | - | 0.14
8 | ${\rm Na_{2}}$ | -0.55 | -0.66(0.70) | -0.60 | -0.56 | -0.54 | 0.06
9 | ${\rm Na_{4}}$ | -1.01 | -1.15(0.90) | -1.07 | -1.03 | -0.91 McHugh  _et al._ (1989) | 0.08
10 | ${\rm Na_{6}}$ | -0.97 | -1.13(0.10) | -1.07 | -1.03 | - | 0.06
11 | ${\rm K_{2}}$ | -0.65 | -0.75(0.05) | -0.74 | -0.70 | -0.50 | 0.01
12 | ${\rm Rb_{2}}$ | -0.62 | - | -0.74 | -0.70 | -0.50 McHugh  _et al._ (1989) | -0.12∗
14 | ${\rm P_{2}}$ | -0.72 | -1.08(0.08) | -0.99 | -0.97 | -0.68 Jones  _et al._ (1995) | 0.09
15 | ${\rm As_{2}}$ | -0.85 | -1.52(0.35) | -1.07 | -1.06 | -0.74 Lippa  _et al._ (1998) | 0.45
16 | ${\rm F_{2}}$ | -0.70 | -1.23(0.14) | -0.96 | -0.84 |  -1.24 Ayala  _et al._ (1981) | 0.27
17 | ${\rm Cl_{2}}$ | -0.89 | -1.40(0.12) | -1.25 | -1.22 |  -1.02 Ayala  _et al._ (1981) | 0.15
18 | ${\rm Br_{2}}$ | -1.40 | -1.96(0.29) | -1.99 | -1.97 |  -1.60 Ayala  _et al._ (1981) | -0.03
19 | ${\rm I_{2}}$ | -1.68 | - | -2.21 | -2.20 |  -1.70 Ayala  _et al._ (1981) | -0.53∗
20 | ${\rm CH_{4}}$ | 2.45 | 2.03(0.35) | 0.63 | 0.63 | - | -1.40+
26 | ${\rm C_{4}}$ | -2.94 | -3.15(0.06) | -3.09 | -3.08 | -3.88 Arnold  _et al._ (1991) | 0.06
29 | ${\rm C_{8}H_{8}}$ | 0.06 | -0.12(0.02) | -0.05 | -0.02 | -0.57 Miller  _et al._ (2002) | 0.07
32 | ${\rm CH_{2}CHCl}$ | 1.42 | 1.17(0.03) | 1.19 | 1.25 | - | 0.02+
36 | ${\rm CCl_{4}}$ | -0.01 | -0.54(0.13) | -0.32 | -0.28 | - | 0.22
37 | ${\rm CBr_{4}}$ | -1.08 | -1.56(0.29) | -1.47 | -1.44 | - | 0.09
38 | ${\rm CI_{4}}$ | -2.14 | - | -2.42 | -2.40 | - | -0.28∗
42 | ${\rm H_{12}Si_{5}}$ | 0.16 | 0.00(0.07) | 0.03 | 0.05 | - | 0.03
43 | LiH | -0.07 | -0.16(0.09) | -0.07 | -0.04 | -0.34 Sarkas  _et al._ (1994) | 0.09
44 | KH | -0.18 | -0.32(0.01) | -0.25 | -0.22 | - | 0.07
45 | ${\rm BH_{3}}$ | 0.12 | 0.03(0.05) | 0.03 | 0.08 | -0.04 Wickham-Jones  _et al._ (1989) | 0.00
54 | LiF | 0.09 | -0.01(0.01) | 0.17 | 0.17 | - | 0.18
55 | ${\rm MgF_{2}}$ | -0.14 | -0.31(0.06) | -0.29 | -0.28 | - | 0.02
56 | ${\rm TiF_{4}}$ | -0.60 | -1.06(0.13) | -0.79 | -0.66 | -2.50 Boltalina  _et al._ (1991) | 0.27
57 | ${\rm AlF_{3}}$ | 0.16 | -0.23(0.10) | 0.08 | 0.09 | - | 0.31
59 | ${\rm SF_{4}}$ | 0.38 | -0.10(0.13) | 0.07 | 0.12 | -1.50 Miller  _et al._ (1995) | 0.17
60 | KBr | -0.31 | -0.42(0.06) | -0.32 | -0.31 | -0.64 Miller  _et al._ (1986) | 0.10
61 | GaCl | -0.02 | -0.39(0.15) | -0.19 | -0.15 | - | 0.20
62 | NaCl | -0.39 | -0.42(0.01) | -0.46 | -0.43 | -0.73 Miller  _et al._ (1986) | -0.04
63 | ${\rm MgCl_{2}}$ | -0.43 | -0.68(0.08) | -0.61 | -0.59 | - | 0.07
64 | ${\rm AlI_{3}}$ | -0.80 | - | -1.02 | -0.99 | - | -0.22∗
72 | ${\rm CH_{3}CHO}$ | 1.05 | 0.83(0.05) | 0.87 | 0.87 | - | 0.04+
74 | HCOOH | 1.91 | 1.59(0.00) | 1.64 | 1.72 | - | 0.05+
76 | ${\rm H_{2}O}$ | 2.37 | 2.01(0.16) | 1.04 | 1.04 | - | -0.97+
78 | ${\rm CS_{2}}$ | -0.20 | -0.55(0.09) | -0.42 | -0.40 | -0.55 Cavanagh  _et al._ (2012) | 0.13
82 | ${\rm O_{3}}$ | -2.30 | -2.69(0.11) | -2.50 | -2.52 | -2.10 Arnold  _et al._ (1994) | 0.19
83 | ${\rm SO_{2}}$ | -1.00 | -1.49(0.12) | -1.25 | -1.19 | -1.11 Nimlos and Ellison (1986) | 0.24
84 | BeO | -2.56 | -2.72(0.04) | -2.73 | -2.37 | - | -0.01
85 | MgO | -1.89 | -2.13(0.09) | -2.05 | -2.12 | - | 0.08
88 | ${\rm C_{6}F_{6}}$ | 0.66 | 0.36(0.08) | 0.24 | 0.27 | -0.70 Eustis  _et al._ (2007) | -0.12+
94 | Cytosine | 0.26 | 0.01(0.01) | 0.12 | 0.15 | -0.23 Schiedt  _et al._ (1998) | 0.11
95 | Thymine | 0.06 | -0.18(0.01) | -0.06 | -0.04 |  0.29 Aflatooni  _et al._ (1998) | 0.12
96 | Uracil | 0.01 | -0.25(0.01) | -0.11 | -0.09 |  0.22 Aflatooni  _et al._ (1998) | 0.14
98 | ${\rm Ag_{2}}$ | -1.05 | - | -1.35 | -1.31 |  -1.10 Handschuh  _et al._ (1995) | -0.30∗
99 | ${\rm Cu_{2}}$ | -0.92 | -1.23(0.08) | -1.24 | -1.21 | -0.84 Taylor  _et al._ (1992) | -0.01
100 | CuCN | -1.65 | -1.85(0.05) | -1.91 | -1.81 |  -1.47 Wu  _et al._ (2010) | -0.06

For the core-valence interaction, we emphasize that VASP always evaluates the
interaction at the level of Hartree-Fock if correlated calculations are
performed. More precisely, VASP calculates the PBE core orbitals on the fly
and then recalculates the action of the PBE core states on the valence states
using the Hartree-Fock approximation. Not doing so can have a sizable effect
on the QP energies for heavier atoms Shishkin and Kresse (2006). We are not
aware of other pseudopotential codes following a similar route. This might be
responsible for a small part of the errors in the reported Berkeley $GW$
calculations of Ref. van Setten  _et al._ , 2015, if heavier atoms are
involved.

Calculations for another fairly large set of molecules have been reported by
Govoni and Galli using the West codeGovoni and Galli (2015). 29 molecules are
identical to the $GW$100 set considered here. The mean absolute difference
between the basis set extrapolated GTO results and the VASP results for this
subset is 60 meV, whereas the difference between the West results and the
basis set extrapolated GTO results is about twice as large 120 meV (mean
absolute difference between VASP and West is 90 meV). In many cases, the West
IPs are too small indicating again basis set incompleteness errors. Anyhow,
the West results are closer to the basis set converged values than the $GW$
Berkeley results.

###  III.3 Comparison to experiment

When comparing the present results against the experimental ionisation
energies, a mean absolute error of 0.5 eV is observed. This large discrepancy
is not unexpected given that in this computational approach self-consistency,
vertex corrections and finite temperature effects are omitted. However we can
comment on the biggest outliers in the set. A first example is
$\text{C}_{\text{4}}$: it is well known that the smaller $\text{C}_{\text{2}}$
molecule is particularly challenging to describe, owing to strong electron
correlation Curtiss  _et al._ (2007). For the larger cluster we expect similar
effects, hence the inclusion of the vertex should improve the agreement with
the experiment. We have a similar expectation for the case of
$\text{F}_{\text{2}}$. Our conjecture is substantiated by previous electron
propagator calculations Golab  _et al._ (1986), where the poles of the Green’s
function in the Lehmann representation were located to give the IP, and where
a comparable mismatch to experiment was ascribed to the poor description of
dynamic correlation. For $\text{AlF}_{\text{3}}$, LiF and KH we have to bear
in mind that the experimental value for a vertical transition was not
available, therefore geometry relaxations may explain the mismatch. This is
only partially true for KH, where the inclusion of adiabatic effects in the
perturbative calculations still leaves a sizable disagreement ($\sim$2 eV)
Rayne and Forest (2011); in this case it is not completely unreasonable to
call for a further assessment of the experimental value.

###  III.4 Linearized QP-HOMO for $GW$100

We now turn to results obtained by first linearizing the self-energy and then
determining the QP energy from this linearized equation. This procedure is in
our experience more ”robust” and better behaved than seeking the poles in the
non-linearized equation. The main issue of the latter approach is that, in the
$G_{0}W_{0}$ approximation, the first pole in the self-energy is approximately
located at the energy of the DFT HOMO minus the first excitation energy in the
DFT (LUMO$-$HOMO):

$\epsilon_{\rm HOMO}-(\epsilon_{\rm LUMO}-\epsilon_{\rm HOMO}).$

This is a simple Auger like excitation, where the hole has sufficient energy,
i.e. is sufficiently below the HOMO to be able to excite an electron-hole
pair. As discussed by van Setten, such poles lead to multiple solutions for
the QP energy van Setten  _et al._ (2015) and make the determination of the QP
energies difficult for molecules with small excitation energies. These poles
are, however, an artifact of the $G_{0}W_{0}$ approximation. If the $GW$
procedure were done self-consistently, the first pole in the self-energy would
move to approximately

$E^{\rm QP}_{\rm HOMO}-(E^{\rm QP}_{\rm LUMO}-E^{\rm QP}_{\rm HOMO}).$

In other words, at the valence band edge (HOMO) and conduction band edge
(LUMO) the self-energy never possesses poles. However, in a single shot
procedure and when starting from much too small band gaps, the quasiparticle
energy $E^{\rm QP}$ might move into regions where the self-energy evaluated
from DFT orbitals has a pole. Linearization at the DFT eigenenergies resolves
this issue, as the $G_{0}W_{0}$ self-energy has no poles in the direct
vicinity of the DFT HOMO. The problem is also less severe, if the calculations
are done selfconsistently or when starting from a prescription that yields
larger HOMO-LUMO Kohn-Sham gaps, as shown in a recent evaluation of the
difference between the quasi-particle orbital energies and their linearized
counterparts by Govoni et al. Govoni and Galli (2015). Therein it is shown
that, for a wide range of molecules, this difference is substantially more
pronounced for $GW$ calculations on a PBE reference state than if a hybrid
functional with non-local exchange is used.

In summary, we feel that for code benchmarking as well as for a comparison
with experiment determining the poles of the linearized equation is
preferable, at least, if a PBE reference state is employed. However, it also
needs to be emphasized that for comparison with the already published $GW$100
data, it is of paramount importance to follow exactly the procedures laid out
in the the initial $GW$100 paper.

###  III.5 LUMO for $GW$100

The calculated LUMOs are shown in Table 4. A few important comments are in
place here. First, the table reports the QP energy of the lowest unoccupied
orbital in the preceeding DFT calculations to maintain compatibility with the
previous publication. In some cases (Xe, H2O, CH2CHCl, CH3CHO, and HCOOH) PW
calculations predict at the DFT level a very weakly bound LUMO+1 state (just
below the vacuum level) whose $G_{0}W_{0}$ QP energy is below the QP state
corresponding to the DFT LUMO level. These energy levels are not shown in Tab.
4.

If we consider the $G_{0}W_{0}$ values corresponding to the DFT LUMOs, the
agreement between the $GW$100 reference GTO data and plane waves is
reasonable, although not quite as good as for the HOMO. Specifically
troublesome is the observation that the GTO calculations sometimes predict
much too positive LUMOs several eV above the PW results. Admittedly, box size
convergence can be troublesome for QP energies above the vacuum level, and we
therefore only show few selected positive LUMOs— those where we are confident
that convergence to 50 meV was attained for the cell sizes considered in our
calculations. All positive (unbound) $G_{0}W_{0}$ LUMOs are marked by a
superscript $"+"$ sign in the last column. The differences are particularly
striking for Ne, Xe, H2, H2O and CH4 reaching 11 eV for Ne.

To investigate this issue, we compared the DFT-LUMOs of the PW and GTO
calculations (the latter are available upon request to MvS), and found that
the deviations between PWs and GTOs are much larger than for the DFT HOMOs on
average, and especially large for some of the problematic cases, e.g. Ne, Xe,
H2, or H2O which are the largest outliners in the subsequent QP calculations.
Specifically, for Ne and H2 the PW DFT calculations predict very shallow bound
LUMOs, a few 10 meV below the vacuum level. These can not be reproduced with
any of the available GTO basis sets.

For the other cases with larger discrepencies, we now show that the GTO basis
sets are often not sufficiently flexible to describe unoccupied orbitals. This
is supported by several observations. (i) Basis set corrections using GTOs are
much larger for the LUMO than for the HOMO, as for instance exemplified for
As2, F2 or Cl2. To make this very clear, we have included in Table 4 both, the
basis set extrapolated values (with estimated error bars), as well as the
values at the largest considered GTO basis set. (ii) Non basis set
extrapolated GTO values deviate markedly from PW results. As before, these are
marked by a star superscript in the last column. GTO basis set extrapolated
values are tabulated in Tab. 2 and clearly improve the agreement with the PW
results. For Xe, where the discrepancy was previously 3.6 eV, the error is
reduced to about 0.3 eV. Furthermore, we recalculated the QP energies of H2O
and CH4 using Dunning correlation consistent basis sets and found basis set
extrapolated $G_{0}W_{0}$ QP energies of 1.00 eV and 0.89 eV, now in excellent
and reasonable agreement with the PW results. All in all, we therefore
conclude that the Gaussian basis set results for unoccupied states need to be
considered with some caution, and Dunning correlation consistent basis sets
are seemingly better suited to predict accurate values.

If we restrict the comparison between PWs and GTOs to states below the vacuum
level, we find the agreement to be generally, as for the HOMO, rather
satisfactory. Differences are about a factor two larger than for the HUMO, but
considering the previous discussion on the possible issues with the Gaussian
basis sets for unoccupied orbitals, this is certainly not astonishing.

Finally, concerning the agreement with the experiment, we find a similar
absolute deviation as for the first ionization energies (compare with Tab. 1).
To make the comparison between the LUMO energies and the experiment more
immediate, the second last column reports the negative of the experimental
electron affinities, which is overall in quite satisfactory agreement with
experiment.

##  IV Discussions and Conclusions

The main purpose of the present work is a careful comparison of $GW$ QP
energies obtained using Gaussian type orbitals and plane waves. One important
motivation was that the values reported for the Berkeley $GW$ code were
typically 200 meV smaller than the basis set extrapolated GTO results.
However, the Gaussian basis set extrapolation also often increased the
predicted QP energies by some 100 meV. Since basis set extrapolation using
GTOs is not necessarily accurate, and since the Berkeley $GW$ calculations are
often closer to the uncorrected values than the basis set extrapolated values,
we felt that it is important to bring in a third independent set of
calculations, hopefully confirming one or the other of the previous values.

The main outcome of our work is that our VASP predicted HOMOs are in excellent
agreement with the basis set extrapolated GTO results. We believe this
establishes beyond doubt that the values reported in Ref. van Setten  _et al._
, 2015 are very reliable and can be used as a rigorous benchmark for future
implementations. In the few cases (iodine compounds, Br2 and Ag2), where the
GTO calculations were not extrapolated to the basis set limit, we find— not
unexpectedly —that the non basis set extrapolated GTO values underestimate the
IP by about 300-400 meV. The present work also reports basis set extrapolated
GTO values for these molecules finding good agreement with VASP PW results.

Although the mean absolute deviation between our PAW PW results and the GTO
results is only 60 meV, we found larger discrepancies for molecules containing
copper, fluorine and nitrogen. We traced these differences back to the use of
non-normconserving PAW potentials: using normconserving PAW potentials the
agreement between PW calculations and GTOs improves further.

For the LUMO, results are slightly less satisfactory. Agreement between GTOs
and PWs is good for QP energies below the vacuum level, although even for
those there are more out-liners and the average deviation is larger. For
instance, differences are sizable for some seemingly simple dimers. We
attribute this to very large basis set corrections for GTOs for some molecules
(e.g. 700 meV for As2).

If the predicted QP LUMOs are above the vacuum level, the differences between
the PW and GTO results can be very large and can reach 11 eV (Ne). Our
explanation for this behavior is that the GTO basis sets employed in Ref. van
Setten  _et al._ , 2015 are not always sufficiently flexible to model
unoccupied states. This is particularly true for atoms and small dimers, where
the LUMO has a character that is very different from a linear combination of
atomic like orbitals. In most cases, these basis set issues lead to small but
noticeable errors on the level of DFT, but they are dramatically amplified at
the level of $G_{0}W_{0}$. For Xe, H2O and CH4, GTO calculations with improved
basis sets have been reported finding very good to good agreement with the PW
results.

If we disregard the slightly disconcerting propagation of errors in going from
DFT to $G_{0}W_{0}$ for LUMOs, we are satisfied by the agreement between plane
waves and Gaussian type orbitals. As already stated, for the HOMO the mean
absolute deviation is only 60 meV, which is excellent if one considers that
the computational details are so different. Furthermore, our results have been
obtained using the $GW$ PAW potentials distributed with vasp.5.4, so that
similar calculations e.g. for molecules adsorbed on surfaces can be readily
performed using the projector augmented wave method.

###### Acknowledgements.

This work was supported by the Austrian Science Fund (FWF) within the
_Spezialforschungsbereich Vienna Computational Materials Laboratory_ (SFB
ViCoM, F41) and the _Deutsche Forschungsgruppe Research Unit_ FOR 1346. P. Liu
is grateful to the China Scholarship Council (CSC)-FWF Scholarship Program.
Computational resources were provided by the Vienna Scientific Cluster (VSC)
and supercomputing facilities of the Université catholique de Louvain
(CISM/UCL).

## References

  * Hedin (1965) L. Hedin, Phys. Rev. 139, A796 (1965).
  * Hanke and Sham (1979) W. Hanke and L. J. Sham, Phys. Rev. Lett. 43, 387 (1979).
  * Strinati  _et al._ (1980) G. Strinati, H. J. Mattausch, and W. Hanke, Phys. Rev. Lett. 45, 290 (1980).
  * Strinati  _et al._ (1982) G. Strinati, H. J. Mattausch, and W. Hanke, Phys. Rev. B 25, 2867 (1982).
  * Hybertsen and Louie (1986) M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986).
  * van Schilfgaarde  _et al._ (2006) M. van Schilfgaarde, T. Kotani, and S. Faleev, Phys. Rev. Lett. 96, 226402 (2006).
  * M.Shishkin and G.Kresse (2007) M.Shishkin and G.Kresse, Phys. Rev. B 75, 235102 (2007).
  * Fuchs  _et al._ (2007) F. Fuchs, J. Furthmüller, F. Bechstedt, M. Shishkin, and G. Kresse, Phys. Rev. B 76, 115109 (2007).
  * Faleev  _et al._ (2004) S. V. Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett. 93, 126406 (2004).
  * Caruso  _et al._ (2012) F. Caruso, P. Rinke, X. Ren, M. Scheffler, and A. Rubio, Phys. Rev. B 86, 081102 (2012), arXiv:1202.3547 .
  * Caruso  _et al._ (2013) F. Caruso, P. Rinke, X. Ren, A. Rubio, and M. Scheffler, Phys. Rev. B 88, 075105 (2013).
  * Kaplan  _et al._ (2015) F. Kaplan, F. Weigend, F. Evers, and M. J. van Setten, J. Chem. Theory Comput. 11, 5152 (2015).
  * Kaplan  _et al._ (2016) F. Kaplan, M. E. Harding, C. Seiler, F. Weigend, F. Evers, and M. J. van Setten, J. Chem. Theory Comput. 12, 2528 (2016).
  * Lischner  _et al._ (2014) J. Lischner, S. Sharifzadeh, J. R. Deslippe, J. B. Neaton, and S. G. Louie, Phys. Rev. B 90, 115130 (2014), arXiv:arXiv:1409.2901v1 .
  * Koval  _et al._ (2014) P. Koval, D. Foerster, and D. Sánchez-Portal, Phys. Rev. B 89, 155417 (2014), arXiv:1404.1715 .
  * Ren  _et al._ (2012) X. Ren, P. Rinke, V. Blum, J. Wieferink, A. Tkatchenko, A. Sanfilippo, K. Reuter, and M. Scheffler, New J. Phys. 14, 053020 (2012).
  * Blase  _et al._ (2011) X. Blase, C. Attaccalite, and V. Olevano, Phys. Rev. B 83, 115103 (2011).
  * Bruneval (2012) F. Bruneval, J. Chem. Phys. 136, 194107 (2012).
  * van Setten  _et al._ (2013) M. J. van Setten, F. Weigend, and F. Evers, J. Chem. Theory Comput. 9, 232 (2013).
  * Wilhelm  _et al._ (2016) J. Wilhelm, M. Del Ben, and J. Hutter, J. Chem. Theory Comput. 12, 3623 (2016).
  * Foerster  _et al._ (2011) D. Foerster, P. Koval, and D. Sánchez-Portal, J. Chem. Phys. 135, 074105 (2011).
  * Ke (2011) S. H. Ke, Phys. Rev. B - Condens. Matter Mater. Phys. 84, 205415 (2011), arXiv:1012.1084 .
  * Laflamme Janssen  _et al._ (2015) J. Laflamme Janssen, B. Rousseau, and M. Cote, Phys. Rev. B 91, 125120 (2015).
  * Govoni and Galli (2015) M. Govoni and G. Galli, J. Chem. Theory Comput. 11, 2680 (2015), http://dx.doi.org/10.1021/ct500958p .
  * Klimeš  _et al._ (2014) J. Klimeš, M. Kaltak, and G. Kresse, Phys. Rev. B 90, 075125 (2014), arXiv:arXiv:1404.3101v1 .
  * Bruneval and Marques (2013) F. Bruneval and M. A. L. Marques, J. Chem. Theory Comput. 9, 324 (2013).
  * van Setten  _et al._ (2015) M. J. van Setten, F. Caruso, S. Sharifzadeh, X. Ren, M. Scheffler, F. Liu, J. Lischner, L. Lin, J. R. Deslippe, S. G. Louie, C. Yang, F. Weigend, J. B. Neaton, F. Evers, and P. Rinke, J. Chem. Theory Comput. 11, 5665 (2015), pMID: 26642984.
  * Deslippe  _et al._ (2012) J. Deslippe, G. Samsonidze, D. a. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012), arXiv:1111.4429 .
  * Schindlmayr (2013) A. Schindlmayr, Phys. Rev. B 87, 075104 (2013), arXiv:arXiv:1302.6368v1 .
  * Shih  _et al._ (2010) B.-C. Shih, Y. Xue, P. Zhang, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 105, 146401 (2010).
  * Friedrich  _et al._ (2011) C. Friedrich, M. C. Müller, and S. Blügel, Phys. Rev. B 83, 081101 (2011), $ibid$ 84, 039906(E) (2011).
  * Bruneval and Gonze (2008) F. Bruneval and X. Gonze, Phys. Rev. B - Condens. Matter Mater. Phys. 78, 1 (2008).
  * Bruneval  _et al._ (2006) F. Bruneval, N. Vast, and L. Reining, Phys. Rev. B 74, 045102 (2006).
  * Shishkin and Kresse (2006) M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).
  * Kaltak  _et al._ (2014a) M. Kaltak, J. Klimeš, and G. Kresse, J. Chem. Theory Comput. 10, 2498 (2014a).
  * Kaltak  _et al._ (2014b) M. Kaltak, J. Klimeš, and G. Kresse, Phys. Rev. B 90, 054115 (2014b).
  * Liu  _et al._ (2016) P. Liu, M. Kaltak, J. Klimeš, and G. Kresse, Phys. Rev. B 94, 165109 (2016), arXiv:1607.02859 .
  * Baker (1975) G. A. J. Baker,  _Essentials of Padé Approximants_ (Academic Press, New York, 1975) Chap. 18.
  * Harl and Kresse (2008) J. Harl and G. Kresse, Phys. Rev. B 77, 045136 (2008).
  * Shepherd  _et al._ (2012) J. J. Shepherd, A. Grüneis, G. H. Booth, G. Kresse, and A. Alavi, Phys. Rev. B 86, 035111 (2012).
  * Björkman  _et al._ (2012) T. Björkman, A. Gulans, A. V. Krasheninnikov, and R. M. Nieminen, Phys. Rev. Lett. 108, 235502 (2012).
  * Rozzi  _et al._ (2006) C. A. Rozzi, D. Varsano, A. Marini, E. K. U. Gross, and A. Rubio, Phys. Rev. B 73, 205119 (2006).
  * Baldereschi and Tosatti (1978) A. Baldereschi and E. Tosatti, Phys. Rev. B 17, 4710 (1978).
  * Gajdoš  _et al._ (2006) M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 73, 045112 (2006).
  * Makov and Payne (1995) G. Makov and M. Payne, Phys. Rev. B 51, 4014 (1995).
  * Kelly (1987) R. L. Kelly, J. Phys. Chem. Ref. Data 16, Supplement 1 (1987).
  * Weitzel  _et al._ (1994) K.-M. Weitzel, J. Mähnert, and M. Penno, Chem. Phys. Lett. 224, 371 (1994).
  * Wetzel  _et al._ (1987) R. C. Wetzel, F. A. Baiocchi, T. R. Hayes, and R. S. Freund, Phys. Rev. A 35, 559 (1987).
  * Schäfer and Rabeneck (1987) H. Schäfer and H. Rabeneck, Zeitschrift für Anorg. und Allg. Chemie 545, 224 (1987).
  * McCormack  _et al._ (1989) E. McCormack, J. M. Gilligan, C. Cornaggia, and E. E. Eyler, Phys. Rev. A 39, 2260 (1989).
  * Dugourd  _et al._ (1992) P. Dugourd, D. Rayane, P. Labastie, B. Vezin, J. Chevaleyre, and M. Broyer, Chem. Phys. Lett. 197, 433 (1992).
  * Kappes  _et al._ (1985) M. M. Kappes, P. Radi, M. Schär, and E. Schumacher, Chem. Phys. Lett. 113, 243 (1985).
  * Herrmann  _et al._ (1978) A. Herrmann, S. Leutwyler, E. Schumacher, and L. Wöste, Helv. Chim. Acta 61, 453 (1978).
  * Trickl  _et al._ (1989) T. Trickl, E. F. Cromwell, Y. T. Lee, and A. H. Kung, J. Chem. Phys. 91 (1989).
  * Bulgin  _et al._ (1976) D. K. Bulgin, J. M. Dyke, and A. Morris, J. Chem. Soc. Faraday Trans. 2 72, 2225 (1976).
  * Lau  _et al._ (1982) K. H. Lau, R. D. Brittain, and D. L. Hildenbrand, J. Phys. Chem. 86, 4429 (1982).
  * Lonkhuyzen and Lange (1984) H. V. Lonkhuyzen and C. A. D. Lange, Chem. Phys. 89, 313 (1984).
  * Dyke  _et al._ (1984a) J. M. Dyke, G. D. Josland, J. G. Snijders, and P. M. Boerrigter, Chem. Phys. 91, 419 (1984a).
  * Kimura  _et al._ (1981) K. Kimura, S. Katsumata, Y. Achiba, T. Yamazaki, and S. Iwata, in  _Handb. HeI Photoelectron Spectra Fundam. Org. Compd._ (Japan Scientific Soc. Press, Tokyo, 1981).
  * Bieri and Åsbrink (1980) G. Bieri and L. Åsbrink, J. Electron Spectros. Relat. Phenomena 20, 149 (1980).
  * Ramanathan  _et al._ (1993) R. Ramanathan, J. A. Zimmerman, and J. R. Eyler, J. Chem. Phys. 98, 7838 (1993).
  * Plemenkov  _et al._ (1981) V. Plemenkov, Y. Villem, N. Villem, I. Bolesov, L. Surmina, N. Yakushkina, and A. Formanovskii, Zh. Obs. Khim. 51, 2076 (1981).
  * Howell  _et al._ (1984) J. O. Howell, J. M. Goncalves, C. Amatore, L. Klasinc, R. M. Wightman, and J. K. Kochi, J. Am. Chem. Soc. 106, 3968 (1984).
  * Fu and Dunbar (1978) E. W. Fu and R. C. Dunbar, J. Am. Chem. Soc. 100, 2283 (1978).
  * Kiselev  _et al._ (1992) V. Kiselev, A. Sakhabutdinov, I. Shakirov, V. Zverev, and A. Konovalov, Zh. Org. Khim. 28, 2244 (1992).
  * Bieri  _et al._ (1981a) G. Bieri, W. von Niessen, L. Åsbrink, and A. Svensson, Chem. Phys. 60, 61 (1981a).
  * Cambi  _et al._ (1983) R. Cambi, G. Ciullo, A. Sgamellotti, F. Tarantelli, R. Fantoni, A. Giardini-guidoni, I. E. McCarthy, and V. di Martino, Chem. Phys. Lett. 101, 477 (1983).
  * Wittel  _et al._ (1974) K. Wittel, H. Bock, and R. Manne, Tetrahedron 30, 651 (1974).
  * Bieri  _et al._ (1981b) G. Bieri, L. Åsbrink, and W. V. Niessen, J. Electron Spectros. Relat. Phenomena 23, 281 (1981b).
  * Dixon  _et al._ (1971) R. N. Dixon, J. N. Murrell, and B. Narayan, Mol. Phys. 20, 611 (1971).
  * Jonkers  _et al._ (1982) G. Jonkers, C. Lange, and J. G. Snijders, Chem. Phys. 69, 109 (1982).
  * Roberge  _et al._ (1978) R. Roberge, C. Sandorfy, J. I. Matthews, and O. P. Strausz, J. Chem. Phys. 69, 5105 (1978).
  * Potts and Price (1972) A. W. Potts and W. C. Price, Proc. R. Soc. London A Math. Phys. Eng. Sci. 326, 165 (1972).
  * Bock  _et al._ (1976) H. Bock, W. Ensslin, F. Feher, and R. Freund, J. Am. Chem. Soc. 98, 668 (1976).
  * NIS (2015) “NIST Chemistry WebBook,”  (2015).
  * Farber  _et al._ (1982) M. Farber, R. D. Srivastava, and J. W. Moyer, J. Chem. Thermodyn. 14, 1103 (1982).
  * Ruščić  _et al._ (1988) B. Ruščić, C. A. Mayhew, and J. Berkowitz, J. Chem. Phys. 88, 5580 (1988).
  * Åsbrink  _et al._ (1981) L. Åsbrink, A. Svensson, W. von Niessen, and G. Bieri, J. Electron Spectros. Relat. Phenomena 24, 293 (1981).
  * Baumgaertel  _et al._ (1989) H. Baumgaertel, H. W. Jochims, E. Ruehl, H. Bock, R. Dammel, J. Minkwitz, and R. Nass, Inorg. Chem. 28, 943 (1989).
  * Cvitaš and Klasinc (1976) T. Cvitaš and L. Klasinc, J. Chem. Soc. Faraday Trans. 2 72, 1240 (1976).
  * Cowley  _et al._ (1982) A. H. Cowley, R. A. Kemp, M. Lattman, and M. L. McKee, Inorg. Chem. 21, 85 (1982).
  * Demuth (1977) R. Demuth, Zeitschrift für Naturforsch. B 32, 1252 (1977).
  * Bieri  _et al._ (1982) G. Bieri, L. Åsbrink, and W. von Niessen, J. Electron Spectros. Relat. Phenomena 27, 129 (1982).
  * Banna and Shirley (1975) M. S. Banna and D. A. Shirley, J. Chem. Phys. 63, 4759 (1975).
  * Wang  _et al._ (1984) R. Wang, M. A. Dillon, and D. Spence, J. Chem. Phys. 80, 63 (1984).
  * Berkowitz  _et al._ (1962) J. Berkowitz, H. A. Tasman, and W. A. Chupka, J. Chem. Phys. 36, 2170 (1962).
  * Hildenbrand (1968) D. L. Hildenbrand, J. Chem. Phys. 48, 3657 (1968).
  * Dyke  _et al._ (1997) J. M. Dyke, D. Haggerston, A. Wright, A. Morris, E. van Lenthe, and J. Snijders, J. Electron Spectros. Relat. Phenomena 85, 23 (1997).
  * Dyke  _et al._ (1984b) J. M. Dyke, C. Kirby, A. Morris, B. Gravenor, R. Klein, and P. Rosmus, Chem. Phys. 88, 289 (1984b).
  * Farber and Srivastava (1984) M. Farber and R. D. Srivastava, J. Chem. Phys. 81, 241 (1984).
  * Fisher  _et al._ (1992) E. R. Fisher, B. L. Kickel, and P. Armentrout, J. Chem. Phys. 97, 4859 (1992).
  * Potts and Price (1977) A. W. Potts and W. C. Price, Phys. Scr. 16, 191 (1977).
  * Grabandt  _et al._ (1990) O. Grabandt, R. Mooyman, and C. De Lange, Chem. Phys. 143, 227 (1990).
  * Lee and Potts (1979) E. P. F. Lee and A. W. Potts, Proc. R. Soc. A Math. Phys. Eng. Sci. 365, 395 (1979).
  * Barker  _et al._ (1975) G. K. Barker, M. F. Lappert, J. B. Pedley, G. J. Sharp, and N. P. C. Westwood, J. Chem. Soc. Dalt. Trans. , 1765 (1975).
  * Kreile  _et al._ (1982) J. Kreile, A. Schweig, and W. Theil, Chem. Phys. Lett. 87, 473 (1982).
  * Bulgin  _et al._ (1977) D. K. Bulgin, J. M. Dyke, and A. Morris, J. Chem. Soc., Faraday Trans. 2 73, 983 (1977).
  * Vovna  _et al._ (1975) V. Vovna, F. Vilesov, and S. Lopatin, Opt. Spectrosc. 38, 143 (1975).
  * Ohno  _et al._ (1995) K. Ohno, K. Okamura, H. Yamakado, S. Hoshino, T. Takami, and M. Yamauchi, J. Phys. Chem. 99, 14247 (1995).
  * Vorob’ev  _et al._ (1989) A. S. Vorob’ev, I. I. Furlei, A. S. Sultanov, V. I. Khvostenko, G. V. Leplyanin, A. R. Derzhinskii, and G. A. Tolstikov, Bull. Acad. Sci. USSR Div. Chem. Sci. 38, 1388 (1989).
  * Ohno  _et al._ (1985) K. Ohno, K. Imai, and Y. Harada, J. Am. Chem. Soc. 107, 8078 (1985).
  * Johnson  _et al._ (1982) K. Johnson, I. Powis, and C. Danby, Chem. Phys. 70, 329 (1982).
  * von Niessen  _et al._ (1980) W. von Niessen, G. Bieri, and L. Åsbrink, J. Electron Spectros. Relat. Phenomena 21, 175 (1980).
  * Ashmore and Burgess (1977) F. S. Ashmore and A. R. Burgess, J. Chem. Soc. Faraday Trans. 2 73, 1247 (1977).
  * Eland and Berkowitz (1977) J. H. D. Eland and J. Berkowitz, J. Chem. Phys. 67, 5034 (1977).
  * Schweig and Thiel (1974) A. Schweig and W. Thiel, Mol. Phys. 27, 265 (1974).
  * Potts and Williams (1974) A. Potts and T. Williams, J. Electron Spectros. Relat. Phenomena 3, 3 (1974).
  * Cradock and Duncan (1975) S. Cradock and W. Duncan, J. Chem. Soc. Faraday Trans. 2 71, 1262 (1975).
  * Katsumata  _et al._ (1984) S. Katsumata, H. Shiromaru, and T. Kimura, Bull. Chem. Soc. Jpn. 57, 1784 (1984).
  * Theard and Hildenbrand (1964) L. P. Theard and D. L. Hildenbrand, J. Chem. Phys. 41, 3416 (1964).
  * Dalleska and Armentrout (1994) N. Dalleska and P. Armentrout, Int. J. Mass Spectrom. Ion Process. 134, 203 (1994).
  * Ballard  _et al._ (1987) R. Ballard, J. Jones, D. Read, A. Inchley, and M. Cranmer, Chem. Phys. Lett. 137, 125 (1987).
  * Furin  _et al._ (1987) G. Furin, A. Sultanov, and I. I. Furlei, Dokl. Phys. Chem. 3, 530 (1987).
  * Kobayashi and Nagakura (1974) T. Kobayashi and S. Nagakura, J. Electron Spectros. Relat. Phenomena 4, 207 (1974).
  * Hush and Cheung (1975) N. Hush and A. S. Cheung, Chem. Phys. Lett. 34, 11 (1975).
  * Lin  _et al._ (1980) J. Lin, C. Yu, S. Peng, I. Akiyama, K. Li, L. K. Lee, and P. R. LeBreton, J. Am. Chem. Soc. 102, 4627 (1980).
  * Dougherty  _et al._ (1976) D. Dougherty, K. Wittel, J. Meeks, and S. P. McGlynn, J. Am. Chem. Soc. 98, 3815 (1976).
  * Palmer  _et al._ (1980) M. H. Palmer, I. Simpson, and R. J. Platenkamp, J. Mol. Struct. 66, 243 (1980).
  * Beutel  _et al._ (1993) V. Beutel, H.-G. Krämer, G. L. Bhale, M. Kuhn, K. Weyers, and W. Demtröder, J. Chem. Phys. 98, 2699 (1993).
  * Franzreb  _et al._ (1990) K. Franzreb, A. Wucher, and H. Oechsner, Zeitschrift f r Phys. D Atoms, Mol. Clust. 17, 51 (1990).
  * Kato (1957) T. Kato, Comm. Pure Appl. Math. 10, 151 (1957).
  * Scherpelz  _et al._ (2016) P. Scherpelz, M. Govoni, I. Hamada, and G. Galli, J. Chem. Theory Comput. 12, 3523 (2016).
  * Weast and Astle (2011) R. C. Weast and M. J. Astle,  _CRC Handbook of Chemistry and Physics, 92nd ed._ (CRC Press, New York, 2011).
  * McHugh  _et al._ (1989) K. M. McHugh, J. G. Eaton, G. H. Lee, H. W. Sarkas, L. H. Kidder, J. T. Snodgrass, M. R. Manaa, and K. H. Bowen, J. Chem. Phys. 91 (1989).
  * Jones  _et al._ (1995) R. O. Jones, G. Ganteför, S. Hunsicker, and P. Pieperhoff, J. Chem. Phys. 103, 9549 (1995).
  * Lippa  _et al._ (1998) T. P. Lippa, S.-J. Xu, S. A. Lyapustina, J. M. Nilles, and K. H. Bowen, J. Chem. Phys. 109 (1998).
  * Ayala  _et al._ (1981) J. A. Ayala, W. E. Wentworth, and E. C. M. Chen, J. Phys. Chem. 85, 768 (1981).
  * Arnold  _et al._ (1991) D. W. Arnold, S. E. Bradforth, T. N. Kitsopoulos, and D. M. Neumark, J. Chem. Phys. 95, 8753 (1991).
  * Miller  _et al._ (2002) T. M. Miller, A. A. Viggiano, and A. E. S. Miller, J. Phys. Chem. A 106, 10200 (2002).
  * Sarkas  _et al._ (1994) H. W. Sarkas, J. H. Hendricks, S. T. Arnold, and K. H. Bowen, J. Chem. Phys. 100, 1884 (1994).
  * Wickham-Jones  _et al._ (1989) C. T. Wickham-Jones, S. Moran, and G. B. Ellison, J. Chem. Phys. 90, 795 (1989).
  * Boltalina  _et al._ (1991) O. Boltalina, A. Borshchevskii, L. Sidorov, and V. Chepurnykh, Zh. Fiz. Khim. SSSR 65, 928 (1991).
  * Miller  _et al._ (1995) A. E. S. Miller, T. M. Miller, A. A. Viggiano, R. A. Morris, J. M. Van Doren, S. T. Arnold, and J. F. Paulson, J. Chem. Phys. 102, 8865 (1995).
  * Miller  _et al._ (1986) T. M. Miller, D. G. Leopold, K. K. Murray, and W. C. Lineberger, J. Chem. Phys. 85, 2368 (1986).
  * Cavanagh  _et al._ (2012) S. J. Cavanagh, S. T. Gibson, and B. R. Lewis, J. Chem. Phys. 137, 144304 (2012).
  * Arnold  _et al._ (1994) D. W. Arnold, C. Xu, E. H. Kim, and D. M. Neumark, J. Chem. Phys. 101, 912 (1994).
  * Nimlos and Ellison (1986) M. R. Nimlos and G. B. Ellison, J. Phys. Chem. 90, 2574 (1986).
  * Eustis  _et al._ (2007) S. N. Eustis, D. Wang, K. H. Bowen, and G. Naresh Patwari, J. Chem. Phys. 127, 114312 (2007).
  * Schiedt  _et al._ (1998) J. Schiedt, R. Weinkauf, D. M. Neumark, and E. Schlag, Chem. Phys. 239, 511 (1998).
  * Aflatooni  _et al._ (1998) K. Aflatooni, G. A. Gallup, and P. D. Burrow, J. Phys. Chem. A 102, 6205 (1998).
  * Handschuh  _et al._ (1995) H. Handschuh, C. Cha, P. S. Bechthold, G. Ganteför, and W. Eberhardt, J. Chem. Phys. 102, 6406 (1995).
  * Taylor  _et al._ (1992) K. J. Taylor, C. L. Pettiette-Hall, O. Cheshnovsky, and R. E. Smalley, J. Chem. Phys. 96, 3319 (1992).
  * Wu  _et al._ (2010) X. Wu, Z. Qin, H. Xie, R. Cong, X. Wu, Z. Tang, and H. Fan, J. Phys. Chem. A 114, 12839 (2010).
  * Curtiss  _et al._ (2007) L. a. Curtiss, P. C. Redfern, and K. Raghavachari, J. Chem. Phys. 126, 084108 (2007).
  * Golab  _et al._ (1986) J. T. Golab, B. S. Thies, D. L. Yeager, and J. a. Nichols, J. Chem. Phys. 84, 284 (1986).
  * Rayne and Forest (2011) S. Rayne and K. Forest, Comput. Theor. Chem. 974, 163 (2011).