# An Optimally-Tuned Starting Point for Single-Shot $GW$ Calculations of
Solids

Stephen E. Gant  Department of Physics, University of California Berkeley,
Berkeley, California 94720, USA  Jonah B. Haber  Department of Physics,
University of California Berkeley, Berkeley, California 94720, USA  Marina R.
Filip  Department of Physics, University of Oxford, Clarendon Laboratory,
Oxford OX1 3PU, United Kingdom  Francisca Sagredo  Department of Physics,
University of California Berkeley, Berkeley, California 94720, USA  Dahvyd
Wing  Department of Molecular Chemistry and Materials Science, Weizmann
Institute of Science, Rehovoth 76100, Israel  Guy Ohad  Department of
Molecular Chemistry and Materials Science, Weizmann Institute of Science,
Rehovoth 76100, Israel  Leeor Kronik  Department of Molecular Chemistry and
Materials Science, Weizmann Institute of Science, Rehovoth 76100, Israel
Jeffrey B. Neaton<EMAIL_ADDRESS>Department of Physics, University of
California Berkeley, Berkeley, California 94720, USA  Molecular Foundry,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  Kavli
Energy NanoScience Institute at Berkeley, Berkeley, California 94720, USA

###### Abstract

The dependence of _ab initio_ many-body perturbation theory within the $GW$
approximation on the eigensystem used in calculating quasiparticle corrections
limits this method’s predictive power. Here, we investigate the accuracy of
the recently developed Wannier-localized optimally tuned screened range-
separated hybrid (WOT-SRSH) functional as a generalized Kohn-Sham starting
point for single-shot $GW$ ($G_{0}W_{0}$) calculations for a range of
semiconductors and insulators. Comparison to calculations based on well-
established functionals, namely PBE, PBE0, and HSE, as well as to self-
consistent $GW$ schemes and to experiment, shows that band gaps computed via
$G_{0}W_{0}$@WOT-SRSH have a level of precision and accuracy that is
comparable to that of more advanced methods such as quasiparticle self-
consistent $GW$ (QS$GW$) and eigenvalue self-consistent $GW$ (ev$GW$). We also
find that $G_{0}W_{0}$@WOT-SRSH improves the description of states deeper in
the valence band manifold. Finally, we show that $G_{0}W_{0}$@WOT-SRSH
significantly reduces the sensitivity of computed band gaps to ambiguities in
the underlying WOT-SRSH tuning procedure.

##  I Introduction

Ab initio many-body perturbation theory (MBPT) within the $GW$ approximation
is a state-of-the-art approach for calculations of the quasiparticle (QP) band
structures of crystalline solids [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14]. In the $GW$ approximation, the self-energy $\Sigma$ is given by the
convolution $\Sigma=iGW$, where $G$ is the single particle Green’s function
and $W$ is the dynamically screened Coulomb interaction. The $GW$ self-energy
is normally first constructed from a (generalized) Kohn-Sham (G)KS [15]
“starting point”, an eigensystem computed from density functional theory
(DFT). While semi-local functionals, such as the local density approximation
(LDA) [16] or generalized gradient approximations (GGAs) like PBE [17], have
historically been the standard choice for constructing this starting point
eigensystem [3, 6, 7, 9], hybrid functionals are increasingly used [18, 19,
20, 21, 22, 23, 24, 25, 14, 26].

In practice, there are a variety of choices regarding how $GW$ calculations
are carried out, with significant consequences for accuracy [27, 14]. Once the
$GW$ self-energy has been constructed, the quasiparticle energies can be
computed via first-order correction to the (G)KS eigenvalues, the so-called
single-shot $GW$ ($G_{0}W_{0}$) approach [3], or $G$ and/or $W$ can be
iterated to self-consistency [28, 29, 30, 31, 32, 33, 34, 35, 36, 19, 37, 38,
20, 39, 40, 41, 23, 42, 43, 22, 44, 24, 45, 46, 47, 26, 48]. The $G_{0}W_{0}$
method is the computationally least expensive approach, and, as has been well-
established, the quasiparticle band structures computed with $G_{0}W_{0}$
approaches typically substantially improve agreement with experiment compared
to those obtained directly from the (G)KS eigenvalues of their underlying DFT
starting points [18, 49, 42, 47, 14]. For example, QP band gap data from an
analysis of $G_{0}W_{0}$ calculations for various semiconductors and
insulators by Grumet et al. [47] exhibited a mean absolute error (MAE) of
$0.2$ eV compared to an MAE of $1.2$ eV for the underlying DFT functionals
used. However, $G_{0}W_{0}$ results exhibit a starting-point dependence, where
results can depend considerably on the DFT functional used to construct the
starting eigensystem [18, 10, 50, 14, 25]. For molecules, a range of about $1$
eV in $G_{0}W_{0}$ calculations of highest occupied molecular orbital (HOMO)
energies has been reported [51, 52]. Likewise, $G_{0}W_{0}$ results for solids
(Si, InN, ZnO, ZnS, CdS, and GaN) [18, 53] have shown a similar starting point
dependence of up to $2$ eV in computed band gaps. As such, it is common
practice to differentiate $G_{0}W_{0}$ calculations by the functional used in
their starting point, denoted by $G_{0}W_{0}$@(…). Relatedly, the accuracy of
$G_{0}W_{0}$ calculations based on semi-local DFT functionals is known to
depend on a fortuitous and sometimes unreliable cancellation of error between
the lack of consideration of vertex corrections, which tends to cause under-
screening in $W_{0}$, and the systematic underestimation of band gaps computed
from semi-local functionals, which tends to cause over-screening [33, 34].

One way to address the issue of starting point dependence is to construct the
self-energy in a more self-consistent manner, leading to the development of
methods like eigenvalue self-consistent $GW$ (ev$GW$) [4, 30, 33] and
quasiparticle self-consistent $GW$ (QS$GW$) [31, 32, 34]. In ev$GW$, the
eigenvalues used to construct $G$ and $W$ are iterated to self-consistency.
Though ev$GW$ is noticeably less dependent on the starting point used, the
wavefunctions used in constructing $G$ and $W$ are not updated in this
appraoch, leading to a residual modest starting point dependence ($0.4$ eV in
the case of azabenzenes [51]). On the other hand, QS$GW$ seeks to
variationally minimize the difference between the self-energy and a static
nonlocal potential by updating both the wavefunctions and eigenvalues used to
construct $G$ and $W$ and has been shown to be independent of the starting
point used [54] (though there do exist questions as to whether this holds true
for some metal oxides [55, 56]). While iterating on $G$ and/or $W$ provides
more consistent results, it also requires greater computational resources.
Additionally, while the self-consistent correction of the QP eigenvalues
accounts for the error due to DFT band gap underestimation in these methods,
it does not account for the lack of vertex corrections, leading to under-
screening and larger QP band gaps [35, 33]. For example, Grumet et al. report
that ev$GW$ and QS$GW$ overestimate QP gaps by $1.0$ eV and $0.8$ eV on
average, respectively [47].

While $GW$ self-consistency schemes can reduce the starting-point dependence
of $G_{0}W_{0}$, the increased cost of going beyond $G_{0}W_{0}$ has
incentivized the development of starting points for $G_{0}W_{0}$ calculations
which do not suffer from the same level of starting-point dependence [53, 18,
19, 20, 22, 23, 25, 26]. In particular, hybrid DFT functionals, which include
exact exchange, are an appealing candidate for improved $G_{0}W_{0}$ starting
points for a few reasons. First, the GKS band gaps computed with these
functionals vary with the amount of exact exchange present, and therefore can
be used to remedy the over-screening due to band gap underestimation that is
present in semi-local functionals [14]. Moreover, hybrid functionals can
better address the starting point dependence associated with more localized
$d$ states [53, 33, 57], where self-interaction errors (SIEs) present in semi-
local functionals are more pronounced [58] and lead to spurious orbital energy
ordering that can propagate to the $GW$ eigenspectrum [14]. In such cases, the
presence of exact exchange can help to reduce this error [53, 59, 26, 51, 19,
60].

The use of hybrid functionals like PBE0 [61] and HSE [62] as starting points
for $G_{0}W_{0}$ calculations has been shown to generally improve agreement
with experiment [18, 25]. Moreover, some hybrid functionals can be tuned [63]
to satisfy the ionization potential (IP) theorem [64, 65], suggesting the
possibility of a more physically accurate and consistent starting point
eigensystem. Specifically, Wing et al. [66] developed a procedure for
parameterizing a class of screened range-separated hybrid (SRSH) functionals
capable of accurately predicting the band gaps of solid state materials
without empirical parameters, directly from density functional theory. The
parameterization is arrived at by capturing the asymptotic limit of the
screened exchange potential and by using an ansatz based on the IP theorem
which applies to localized Wannier functions in systems with periodic boundary
conditions [67]. This class of Wannier-localized optimally tuned screened
range-separated hybrid (WOT-SRSH) functionals has been recently used to
calculate the fundamental band gaps of semiconductors and insulators, leading
to excellent agreement with experiment, with an MAE of 0.1 eV [66].

For molecules, the use of optimally tuned range-separated hybrid (RSH)
functionals which enforce the IP theorem as a starting point for $G_{0}W_{0}$
as suggested in [21], has been shown to be successful [40, 41, 23, 68, 45,
69]. However, as of yet, there has not been an analogous exploration of these
non-empirical WOT-SRSH starting points which approximately satisfy the IP
theorem for $G_{0}W_{0}$ calculations of solid-state systems.

Here, we undertake such an exploration and analyze the performance of single-
shot $G_{0}W_{0}$@WOT-SRSH calculations. For a series of 15 semiconductor and
insulators, we construct $G_{0}$ and $W_{0}$ using WOT-SRSH and compute band-
gaps as well as properties associated with states deeper in the valence band
manifold such as valence bandwidths and $d$ band positions. We then compare
results with experiments and calculations from other DFT starting points. We
also discuss how $G_{0}W_{0}$ corrections affect the sensitivity of computed
bands gaps to ambiguities in the WOT-SRSH tuning procedure. Overall, our
calculations demonstrate that a $G_{0}W_{0}$@WOT-SRSH approach provides
accurate quasiparticle properties for a broad range of materials, opening the
door to predictive single-shot $G_{0}W_{0}$ calculations for chemically
complex solids.

##  II Theory

###  II.1 DFT

The starting point for our $GW$ calculations are (G)KS orbitals
$\phi_{n\bm{k}}$ and eigenenergies $\epsilon^{0}_{n\bm{k}}$, where $n$ is the
band index and $\bm{k}$ the wave vector. Here, we primarily focus on the SRSH
functional scheme [70, 37, 71, 72]. This class of functionals is formulated by
partitioning the exchange portion of the Coulomb potential into

$\frac{1}{r}=\frac{\alpha+\beta\text{erf}\left(\gamma
r\right)}{r}+\frac{1-\left[\alpha+\beta\text{erf}\left(\gamma
r\right)\right]}{r}.$ (1)

This partition introduces three parameters $\alpha$, $\beta$, and $\gamma$,
the physical and computational significance of which is discussed shortly.
When implemented in the hybrid functional, the first term of Eq. (1) is
treated explicitly with Fock exchange, whereas the second term is replaced
with an approximate semi-local exchange functional [70, 37]. In this
framework, $\alpha$ regulates the amount of exact exchange in the short range,
$\alpha+\beta$ regulates the amount of exact exchange in the long range, and
$\gamma$ is the length scale for the transition between these two limits. The
correlation component is treated with the same functional used for the semi-
local part of the aforementioned exchange partition. By specifying the values
of $\alpha$, $\alpha+\beta$, and $\gamma$, we can recover various well-known
hybrid functionals (Table 1). For example, if the semi-local exchange is based
on the PBE functional [17], then setting $\gamma=0$ produces a global hybrid
functional, PBE$\alpha$ [73, 74], and if $\alpha=0.25$, PBE0 is obtained. For
$\gamma=0.106$ $a_{0}^{-1}$, setting $\alpha+\beta=0$ and $\alpha=0.25$ yields
the HSE functional [62].

| $\bm{\alpha}$ | $\bm{\alpha+\beta}$ |  $\bm{\gamma}$ ($\bm{a_{0}^{-1}}$)
---|---|---|---
PBE0 | $0.25$ | $0.25$ | 0
HSE06 | $0.25$ | 0 | $0.106$
WOT-SRSH | Variesa | $\varepsilon_{\infty}^{-1}$ | Tuned
Table 1: Hybrid functionals in the SRSH formalism
a: By default $\alpha$ is set to be $0.25$, but in cases where
$\alpha+\beta\sim 0.25$ the value of $\alpha$ is increased slightly until the
IP ansatz can be satisfied. For more details see [66] or the discussions in
sections II.1 and IV.2.

In this work, we focus on the novel WOT-SRSH formulation [66] of the SRSH
functional. Here, the choice $\alpha+\beta=\varepsilon^{-1}_{\infty}$, where
$\varepsilon^{-1}_{\infty}$ is the orientationally-averaged electronic
contribution to the dielectric constant, enforces the asymptotically correct
long-range screening in the Coulomb potential [37, 71, 72]. The range-
separation parameter $\gamma$ is non-empirically selected by enforcing an
ansatz which extends the IP theorem to the removal of an electron from the
highest-energy occupied maximally-localized Wannier function (MLWF) [67]. By
default, we choose $\alpha=0.25$ because, as seen in global hybrids, setting
$\alpha=0.25$ has proven effective for many molecular and solid-state systems
[73, 75, 76, 39]. In cases where setting $\alpha=0.25$ does not yield a unique
choice of $\gamma$ via the IP ansatz, as is often the case when
$\varepsilon_{\infty}^{-1}\sim 0.25$, $\alpha$ is increased slightly until an
optimal value of $\gamma$ that does not approach zero can be found. The need
for a lower bound on the size of $\gamma$ is related to the “$\gamma$ collapse
problem” [77, 78], where small values of $\gamma$ result in an unphysical
effectively PBE$\alpha$ hybrid functional if $\gamma^{-1}$ exceeds the size of
the unit cell of the calculation. As seen in Table 2, the largest value that
$\alpha$ needed to be increased to was $0.35$. With these constraints, WOT-
SRSH functionals are a system-specific but non-empirical class of exchange
correlation (xc) functionals that result in a GKS eigensystem that
consistently and accurately predicts the QP band gaps of solids, compensating
by construction for the derivative discontinuity error present in most density
functionals [79, 80, 81, 15, 9, 82, 83, 84].

###  II.2 $GW$ Method

In the ab initio $GW$ approach, the self-energy $\Sigma=iGW$ of a system is
constructed from a DFT (G)KS eigensystem. As discussed, this (G)KS eigensystem
$\left\\{\phi_{n\bm{k}},\epsilon^{\text{DFT}}_{n\bm{k}}\right\\}$ depends on
the underlying xc functional $V_{xc}$ used to compute it, and by extension the
self-energy computed from this eigensystem is also sensitive to the choice of
$V_{xc}$. Specifically, the single-particle Green’s function $G_{0}$ is
constructed as

$G_{0}(\bm{r},\bm{r}^{\prime};\omega)=\sum_{n\bm{k}}\frac{\phi_{n\bm{k}}(\bm{r})\phi_{n\bm{k}}^{*}(\bm{r}^{\prime})}{\omega-\epsilon_{n\bm{k}}^{\text{DFT}}\pm
i\eta},$ (2)

where $\eta$ is a positive infinitesimal real number, and the $\pm$ in front
of it is $-$ for occupied states and $+$ for empty states. The dynamically
screened Coulomb interaction $W_{0}$ is given by

$W_{0}(\bm{r},\bm{r}^{\prime};\omega)=\int
d\bm{r}^{\prime\prime}\varepsilon^{-1}(\bm{r},\bm{r}^{\prime\prime};\omega)\frac{1}{\left|\bm{r}^{\prime}-\bm{r}^{\prime\prime}\right|},$
(3)

where the dielectric function,

$\displaystyle\varepsilon^{-1}(\bm{r},\bm{r}^{\prime};\omega)=$
$\displaystyle\delta(\bm{r},\bm{r}^{\prime})$ (4) $\displaystyle-\int
d\bm{r}^{\prime\prime}v(\bm{r},\bm{r}^{\prime\prime})\chi_{0}(\bm{r}^{\prime\prime},\bm{r}^{\prime},\omega),$

is computed within the random-phase approximation (RPA) [4] based on the
polarizability $\chi_{0}(\bm{r},\bm{r}^{\prime},\omega)$, given by the Adler-
Wiser expression [85, 86]

$\chi_{0}(\bm{r},\bm{r}^{\prime},\omega)=\sum_{n\bm{k}}^{\text{occ.}}\sum_{n^{\prime}\bm{k}^{\prime}}^{\text{emp.}}\left[\frac{\phi_{n\bm{k}}^{*}(\bm{r})\phi_{n^{\prime}\bm{k}^{\prime}}(\bm{r})\phi_{n\bm{k}}^{*}(\bm{r}^{\prime})\phi_{n^{\prime}\bm{k}^{\prime}}(\bm{r}^{\prime})}{\omega-\left(\epsilon_{n^{\prime}\bm{k}^{\prime}}^{\text{DFT}}-\epsilon_{n\bm{k}}^{\text{DFT}}\right)+i\eta}-\frac{\phi_{n\bm{k}}(\bm{r})\phi_{n^{\prime}\bm{k}^{\prime}}^{*}(\bm{r})\phi_{n\bm{k}}(\bm{r}^{\prime})\phi_{n^{\prime}\bm{k}^{\prime}}^{*}(\bm{r}^{\prime})}{\omega+\left(\epsilon_{n^{\prime}\bm{k}^{\prime}}^{\text{DFT}}-\epsilon_{n\bm{k}}^{\text{DFT}}\right)-i\eta}\right],$
(5)

where the summations are over the occupied and unoccupied bands. In practice,
$\chi_{0}(\bm{r},\bm{r}^{\prime},\omega)$ is often evaluated statically
($\omega=0$), and a simplified model frequency dependence, such as a plasmon
pole model, is used instead [4, 87, 88]. We also note that a consideration of
the denominators in Eq. (5) clarifies why the under- or overestimation of the
band gap can result in over- or under-screening in $W_{0}$, respectively.

With the above quantities, the $G_{0}W_{0}$ self-energy becomes

$\displaystyle\Sigma(\bm{r},\bm{r}^{\prime};\omega)=\frac{i}{2\pi}\int$
$\displaystyle
d\omega^{\prime}G_{0}(\bm{r},\bm{r}^{\prime};\omega+\omega^{\prime})W_{0}(\bm{r},\bm{r}^{\prime};\omega^{\prime})$
(6) $\displaystyle\times e^{i\omega^{\prime}\eta}.$

This $G_{0}W_{0}$ operator can then be used to correct the DFT eigenvalues
perturbatively via

$\epsilon_{n\bm{k}}^{\text{QP}}=\epsilon^{\text{DFT}}_{n\bm{k}}+\braket{n\bm{k}}{\Sigma(\epsilon_{n\bm{k}}^{\text{QP}})-V_{xc}}{n\bm{k}}.$
(7)

Due to the fact that $\epsilon_{n\bm{k}}^{\text{QP}}$ in Eq. (7) depends on
itself, evaluating this expression is non-trivial. However, as is common
practice [89, 90, 91], we expand Eq. (7) to first order about
$\epsilon^{\text{DFT}}_{n\bm{k}}$ to evaluate it efficiently.

##  III Computational Details

###  III.1 DFT Calculations

|  | Lattice
---
Parameters
| WOT-SRSH
---
Parameters${}^{\text{d}}$
| $a$ | $c$ | $u$ | $\alpha$ | $\beta$ |  $\gamma$ ($a_{0}^{-1}$)
InSb | 6.48${}^{\text{a}}$ |  |  | 0.25 | -0.1745 | 0.17
InAs | 6.06${}^{\text{a}}$ |  |  | 0.25 | -0.1623 | 0.16
Ge | 5.66${}^{\text{a}}$ |  |  | 0.25 | -0.1824 | 0.19
GaSb | 6.1${}^{\text{a}}$ |  |  | 0.25 | -0.1733 | 0.19
Si | 5.43${}^{\text{a}}$ |  |  | 0.25 | -0.1611 | 0.24
InP | 5.87${}^{\text{a}}$ |  |  | 0.25 | -0.1373 | 0.23
GaAs | 5.65${}^{\text{a}}$ |  |  | 0.25 | -0.1549 | 0.15
AlSb | 6.14${}^{\text{a}}$ |  |  | 0.25 | -0.1482 | 0.14
AlAs | 5.66${}^{\text{a}}$ |  |  | 0.3 | -0.1779 | 0.18
GaP | 5.45${}^{\text{a}}$ |  |  | 0.25 | -0.1375 | 0.21
AlP | 5.47${}^{\text{a}}$ |  |  | 0.25 | -0.1128 | 0.16
C | 3.57${}^{\text{a}}$ |  |  | 0.3 | -0.1198 | 0.23
AlN | 3.11${}^{\text{a}}$ | 4.98${}^{\text{a}}$ | 0.3821${}^{\text{c}}$ | 0.35 | -0.1073 | 0.26
MgO | 4.22${}^{\text{a}}$ |  |  | 0.25 |  0.0948 | 1.5
LiF | 4.03${}^{\text{b}}$ |  |  | 0.25 |  0.2681 | 1.08
Table 2:  Parameters used in the DFT starting point calculations. Lattice
parameters were taken from experiment, and WOT-SRSH parameters were taken from
prior work [66].
a: [92], b: [93], c: [94]

Our DFT calculations are performed using a modified version of the QUANTUM
ESPRESSO plane-wave code [95, 96, 97] that allows for the use of the SRSH
functional [37] of Eq. (1) with arbitrary $\alpha$, $\beta$, and $\gamma$
parameters. Other modifications also allow for a more efficient calculation of
many hundreds of unoccupied states for GKS systems via what amounts to a non-
self-consistent field calculation once the occupied orbitals and ground state
density have been converged (see SI Section S-I [98] for more details). All
calculations utilize fully relativistic optimized norm-conserving (NC)
Vanderbilt pseudopotentials [99] obtained from the PSEUDO-DOJO repository
[100]. Using these pseudopotentials, the effects of spin-orbit coupling (SOC)
are included self-consistently at the DFT level for all calculated
observables. For Ge, Ga, In, Sb, and As, the electrons within a complete set
of semi-core shells of the same principal quantum number are treated as
valence electrons. For calculations using hybrid functionals and the $GW$
methods, the explicit consideration of these deeper states has been shown to
be necessary for the accurate description of the electronic structure of such
systems [101, 30, 102, 103]. A plane wave energy cutoff of $135$ Ry and
experimental room temperature lattice parameters (summarized in Table 2) are
used for all systems.

For hybrid functionals, the energy cutoff involved in constructing the exact
exchange operator is lowered, without significantly impacting the results at
the DFT or $G_{0}W_{0}$@DFT levels, from its default value of four times the
plane wave energy cutoff to $150$ Ry. In some rare cases where this causes
numerical instability in the self-consistent evaluation of the exchange,
namely computing the PBE0 starting points for Ge and InAs, this cutoff is
raised to the default value of four times the plane wave energy cutoff.

###  III.2 $GW$ Calculations

All our $GW$ calculation are carried out using the BerkeleyGW package [4,
104]. In an effort to minimize the cost of computing many hundreds of
unoccupied states using hybrid functionals, the dielectric function is
constructed using a symmetry-reduced unshifted Monkhorst-Pack $\bm{q}$-grid.
Frequency dependence in the dielectric function is included approximately via
the Godby-Needs plasmon pole model (PPM) [87, 88], which has been shown to
reproduce the computed bands gaps of full-frequency integration at reduced
cost [105]. It should be noted, however, that this comparable level of
accuracy can wane for deeper valence states; previous studies [106, 107]
report that valence bandwidths and $d$ band binding energies computed using
the Godby-Needs PPM are modestly overestimated relative to full-frequency
integration.

The static remainder approximation to $\Sigma$ [108] is used whenever it
yields faster convergence with respect to the number of bands, which is the
case for all materials except AlN, MgO, and LiF. The band gaps of all
materials are converged within (or well-within) $50$ meV with respect to the
number of bands used to construct $\varepsilon$ and $\Sigma$, the energy
cutoff in the construction of $\varepsilon$, and the unshifted $\bm{k}$-grid
being used. For more convergence details, see SI Section S-II [98].

The effects of SOC are computed at the DFT level and added perturbatively at
the $G_{0}W_{0}$ level for all materials, an approximation which has precedent
and justification for the classes of materials under study [109, 110, 111].
While BerkeleyGW does allow for the explicit computation of SOC effects at the
$GW$ level, this would require twice as many bands in the starting point
eigensystem, quadrupling the cost of already expensive calculations. However,
we find the error of including SOC perturbatively to be minimal. For example
we report that for AlSb, a system with a strong SOC band gap renormalization
of $240$ meV, the error in the computed SOC renormalization of the band gap is
only $6$ meV. For systems with weaker renormalizations like GaP, this error is
only $1$ meV.

For band structures with conduction band minima off high symmetry points (as
is the case for Si, C, GaP, and AlSb), eigenvalues are interpolated using the
Wannier90 code [112]. Due to the similarity in orbital character of the states
near the band gap for all the aforementioned systems, only the four highest
occupied and four lowest unoccupied bands about the band edges are
Wannierized, with $sp^{3}$ starting projections being used for all of them.
SOC corrections to the interpolated bands are determined for each eigenenergy
$\epsilon_{n\bm{k}}$ and interpolated using MLWFs, as outlined by Malone et
al. [109].

|  |  $\bm{G}_{0}W_{0}$@
---
PBE
|  $\bm{G}_{0}W_{0}$@
---
PBE0
|  $\bm{G}_{0}W_{0}$@
---
HSE
WOT-SRSH |  |  $\bm{G}_{0}W_{0}$@
---
WOT-SRSH
ev$GW$ | QS$GW$ | Ref | Expt, ZPR
InSb | 0.09 | 0.58 | 0.45 | 0.32 | 0.44 | 0.79${}^{\text{a}}$ | 0.61${}^{\text{a}}$ | 0.19 | 0.17${}^{\text{e}}$, 0.02${}^{\text{j}}$
InAs | 0.13 | 0.68 | 0.50 | 0.42 | 0.48 | — | 0.66${}^{\text{c}}$ | 0.37 | 0.35${}^{\text{e}}$, 0.02${}^{\text{j}}$
Ge | 0.47 | 0.91 | 0.78 | 0.69 | 0.74 | 0.95${}^{\text{b}}$ | 0.95${}^{\text{c}}$ | 0.71 | 0.66${}^{\text{f}}$, 0.05${}^{\text{j}}$
GaSb | 0.46 | 1.00 | 0.88 | 0.69 | 0.86 | — | 1.15${}^{\text{c}}$ | 0.76 | 0.73${}^{\text{e}}$, 0.03${}^{\text{j}}$
Si | 1.18 | 1.57 | 1.42 | 1.14 | 1.40 | 2.18${}^{\text{a}}$ | 1.49${}^{\text{a}}$ | 1.18 | 1.12${}^{\text{f}}$, 0.06${}^{\text{j}}$
InP | 1.41 | 1.96 | 1.81 | 1.56 | 1.80 | 1.97${}^{\text{a}}$ | 1.64${}^{\text{a}}$ | 1.40 | 1.35${}^{\text{e}}$, 0.05${}^{\text{j}}$
GaAs | 1.01 | 1.59 | 1.46 | 1.41 | 1.48 | 1.85${}^{\text{b}}$ | 1.96${}^{\text{c}}$ | 1.47 | 1.42${}^{\text{e}}$, 0.05${}^{\text{j}}$
AlSb | 1.51 | 1.90 | 1.74 | 1.71 | 1.78 | 2.61${}^{\text{a}}$ | 2.22${}^{\text{a}}$ | 1.65 | 1.61${}^{\text{e}}$, 0.04${}^{\text{j}}$
AlAs | 2.04 | 2.49 | 2.33 | 2.25 | 2.41 | 2.98${}^{\text{a}}$ | 2.84${}^{\text{a}}$ | 2.20 | 2.16${}^{\text{e}}$, 0.04${}^{\text{j}}$
GaP | 2.34 | 2.75 | 2.60 | 2.39 | 2.61 | 2.77${}^{\text{a}}$ | 2.67${}^{\text{a}}$ | 2.35 | 2.27${}^{\text{e}}$, 0.08${}^{\text{j}}$
AlP | 2.44 | 2.92 | 2.75 | 2.52 | 2.82 | 3.2${}^{\text{a}}$ | 2.94${}^{\text{a}}$ | 2.51 | 2.49${}^{\text{e}}$, 0.02${}^{\text{j}}$
C | 5.58 | 5.95 | 5.82 | 5.76 | 5.92 | 6.41${}^{\text{a}}$ | 6.43${}^{\text{a}}$ | 5.85 | 5.47${}^{\text{g}}$, 0.38${}^{\text{k}}$
AlN | 5.72 | 6.55 | 6.35 | 6.56 | 6.69 | — | 6.80${}^{\text{c}}$ | 6.52 | 6.14${}^{\text{e}}$, 0.38${}^{\text{k}}$
MgO | 6.96 | 8.07 | 7.99 | 8.16 | 8.62 | 9.53${}^{\text{a}}$ | 9.58${}^{\text{a}}$ | 8.36 | 7.83${}^{\text{h}}$, 0.53${}^{\text{l}}$
LiF | 13.58 | 14.75 | 14.55 | 15.34 | 15.63 | 15.90${}^{\text{b}}$ | 16.63${}^{\text{d}}$ | 15.35 | 14.20${}^{\text{i}}$, 1.15${}^{\text{l}}$
MAE | 0.40 | 0.31 | 0.22 | 0.07 | 0.19 | 0.66${}^{\text{m}}$ | 0.51 |  |
MSE | -0.40 | 0.19 | 0.038 | 0.003 | 0.19 | 0.66${}^{\text{m}}$ | 0.51 |  |
Max Error | -1.77 | -0.60 | -0.80 | -0.20 | 0.40 | 1.17${}^{\text{m}}$ | 1.28 |  |
Table 3:  QP band gaps (in eV) at the WOT-SRSH and $G_{0}W_{0}$@DFT level for
the various compounds and functionals under study. At the bottom of the table
are the MAE (mean absolute error), MSE (mean signed error), and Max Error; all
are in eV and measured relative to the reported reference values, which are
arrived at by incorporating ZPR corrections into experimental band gap data.
Experimental results are arrived at via an analysis of optical absorption
spectroscopy data, where excitonic effects are taken into account to arrive at
the fundamental gap (See [66] for details).
a: [47], b: [35], c: [32], d: [46], e: [113], f: [92], g: [114], h: [115], i:
[116], j: [117], k: [118], l: [119, 120]
m: The MAE, MSE, and Max Error for ev$GW$ were computed using the available
data for 12 out of 15 compounds.  Figure 1:  QP band gaps computed using
$G_{0}W_{0}$@PBE, $G_{0}W_{0}$@PBE0, $G_{0}W_{0}$@HSE, $G_{0}W_{0}$@WOT-SRSH,
ev$GW$, and $QSGW$ in reference to ZPR corrected experimental results. Data
are taken from table 3. The inset in the lower right corner is a zoom-in of
the bottom 3 eV of the data set.

##  IV Results and Discussion

###  IV.1 Quasiparticle Band Structures

####  IV.1.1 Band Gaps

The QP band gaps of 15 semiconductors and insulators calculated from WOT-SRSH,
as well as $G_{0}W_{0}$ using four starting points (PBE, PBE0, HSE06, and WOT-
SRSH), are given in Table 3 and graphed in Figure 1. Additional results from
DFT functionals other than WOT-SRSH can be found in the SI, section S-III.A
[98]. While zero point renormalization (ZPR) effects due to electron-phonon
coupling have a significant impact on the band gaps of many solids [121, 122,
123, 124, 125], they are not addressed computationally in this work. Instead,
computed band gaps are compared to reference band gaps which remove ZPR
effects from the experimental measurements (see [66]). Additionally, excitonic
effects are accounted for in our reference set by adding estimated or
calculated exciton binding energies to the measured optical absorption edge or
by inferring the fundamental band gap position based on the location and
identification of excitonic absorption peaks in experimental data (See [66]
for more details).

In line with what we have reported previously [66], the WOT-SRSH functional
yields an excellent MAE of 0.07 eV and an MSE of 0.00 eV for band gaps—the
highest accuracy of all of the methods under study for this set of solids. As
the MSE indicates, the data are nearly equally spread between over- and
underestimating band gaps. Also, unlike the other functionals, WOT-SRSH has
accuracy that is maintained for wider-band-gap systems and has a much smaller
maximum magnitude error of 0.2 eV.

Performing $G_{0}W_{0}$ based on the WOT-SRSH starting point for this set of
materials yields an MAE of 0.19 eV, with the $G_{0}W_{0}$@WOT-SRSH calculated
band gaps maintaining a similar level of precision with an maximum error of
0.40 eV. Notably, the $G_{0}W_{0}$@WOT-SRSH band gaps are all slightly
overestimated, consistent with the overestimation observed with more
rigorously self-consistent methods such as ev$GW$ and QS$GW$ [47, 35, 32, 46]
(see table 3 and Figure 1). Some of the reported overestimation for these
methods has been attributed to the absence of ZPR effects in the band gap [32,
35], but our reference band gap accounts for ZPR effects and still indicates
some systematic overestimation. However, it is also known that the RPA
dielectric function can under-screen and thus overestimate band gaps. As
previously noted [35, 43, 46, 126, 127, 128], beyond-RPA vertex corrections
for a similar set of semiconductors and insulators can provide an improvement
in the accuracy of the screening and QP band gaps once a consistent starting
point that no longer underestimates the band gap is reached.

Comparing $G_{0}W_{0}$@WOT-SRSH to self-consistent $GW$ approaches in Figure
1, we find excellent agreement and superior performance relative to experiment
for the systems studied here, at a lower computational cost. $G_{0}W_{0}$@WOT-
SRSH also has a similar qualitative performance to these methods, consistently
modestly overestimating band gaps across a broad range of materials.

As is well known and in agreement with prior work [18, 49, 42, 47, 14],
$G_{0}W_{0}$@PBE significantly improves the accuracy of PBE band gaps, in this
case bringing its mean absolute error (MAE) from 1.5 eV to 0.4 eV.
$G_{0}W_{0}$@PBE also corrects major qualitative issues such as the inverted
band gaps of InSb, InAs, and GaSb. Notably, however, band gaps of some
insulators are still underestimated by more than 1 eV (e.g. MgO, LiF) by
$G_{0}W_{0}$@PBE, leading to a substantial max error of -1.77 eV.

Figure 2: Bandstructure of GaAs, including SOC, from WOT-SRSH and
$G_{0}W_{0}$@WOT-SRSH. a) A full plot of the bandstructure, including the
eight highest occupied orbitals and the first few eV of the next eight
unoccupied orbitals. b) A zoomed-in inset—depicted by a gray box in a)—of the
valence and conduction band extrema. For both plots, $E_{F}$ is taken to be
the energy of the valence band maxima

For the insulators studied (C, AlN, MgO, and LiF) we find that well-
established hybrid functionals such as HSE and PBE0 offer a significant
improvement over PBE as starting points for $G_{0}W_{0}$ calculations.
However, these hybrids perform slightly worse than PBE for the remaining
semiconductors. Overall, for $G_{0}W_{0}$@PBE0, we observe a MAE of 0.31 eV,
an MSE of 0.19 eV, and a max error of -0.60 eV. The smaller MSE relative to
the MAE reflects that the band gaps computed from $G_{0}W_{0}$@PBE0 switch
from being overestimated for small-band-gap semiconductors to being
underestimated for wide-band-gap insulators.

In the case of $G_{0}W_{0}$@HSE, calculated QP band gaps have an MAE of 0.22
eV and an MSE of 0.04 eV. This comparatively near-zero MSE reflects that the
band gaps computed from $G_{0}W_{0}$@HSE switch from being overestimated for
small-band-gap semiconductors to being underestimated for wide-band-gap
insulators. Moreover, while the MAE calculated for $G_{0}W_{0}$@HSE is quite
low and comparable to what is seen for $G_{0}W_{0}$@WOT-SRSH, the performance
of $G_{0}W_{0}$@HSE is not consistent. For wide-band-gap insulators such as
LiF, $G_{0}W_{0}$@HSE underestimates the band gap by nearly 1 eV, leading to a
max error of -0.80 eV.

|  |  $\bm{G}_{0}W_{0}$@
---
PBE
|  $\bm{G}_{0}W_{0}$@
---
PBE0
|  $\bm{G}_{0}W_{0}$@
---
HSE
WOT-SRSH |  |  $\bm{G}_{0}W_{0}$@
---
WOT-SRSH
Expt.
InSb | 11.30 | 11.51 | 11.32 | 11.96 | 11.32 | 11.7${}^{\text{a}}$ (XPS), 10.8${}^{\text{a}}$ (ARPES)
InAs | 11.90 | 12.42 | 12.34 | 13.08 | 12.34 | 12.3${}^{\text{a}}$ (XPS)
Ge | 12.82 | 13.52 | 13.30 | 14.04 | 13.26 | 12.6${}^{\text{a}}$ (XPS)
GaSb | 11.74 | 12.36 | 12.16 | 12.83 | 12.13 | 11.6${}^{\text{a}}$ (XPS), 11.64${}^{\text{a}}$ (ARPES)
Si | 11.51 | 12.32 | 12.10 | 13.07 | 12.04 | 12.5${}^{\text{a}}$ (XPS)
InP | 11.22 | 11.93 | 11.72 | 12.60 | 11.72 | 11.0${}^{\text{a}}$ (XPS), 11.4${}^{\text{a}}$ (IPES)
GaAs | 12.77 | 13.45 | 13.26 | 14.09 | 13.29 | 13.8${}^{\text{a}}$ (XPS), 13.1${}^{\text{a}}$ (ARPES)
AlSb | 10.67 | 11.36 | 11.14 | 12.06 | 11.20 | —
AlAs | 11.64 | 12.39 | 12.17 | 13.35 | 12.29 | —
GaP | 12.18 | 12.92 | 12.70 | 13.70 | 12.71 | 12.5${}^{\text{a}}$ (ARPES)
AlP | 11.00 | 11.83 | 11.59 | 12.75 | 11.67 | —
C | 22.23 | 23.23 | 23.04 | 24.02 | 23.25 | 21${}^{\text{a}}$ (XPS)
AlN | 6.55 | 6.73 | 6.69 | 6.65 | 6.75 | —
MgO | 5.09 | 5.19 | 5.18 | 5.07 | 5.26 | 6.5${}^{\text{b}}$ (XPS), 7${}^{\text{b}}$ (XES)
LiF | 3.50 | 3.50 | 3.50 | 3.30 | 3.51 | 3.4${}^{\text{c}}$ (XPS)
MAE | 0.59 | 0.68 | 0.64 | 1.08 | 0.65 |
MSE | -0.24 | 0.31 | 0.16 | 0.81 | 0.18 |
Table 4:  QP valence bandwidths (in eV), at the WOT-SRSH and $G_{0}W_{0}$@DFT
level, for the various compounds and functionals under study. For zinc blende
materials, the valence bandwidth is defined as the maximal energy difference
between the top four (excluding spin degeneracy) valence bands. For the
wurtzite and rock salt compounds, the valence bandwidth is defined as the
maximal energy difference between the top three valence bands for LiF and MgO
and the top six valence bands for AlN. At the bottom of the table are the MAE
and MSE; all are in eV and calculated using the leftmost reported experimental
values. Experimental data are obtained via XPS, angle-resolved photo-emission
spectroscopy (ARPES), and X-ray emission spectroscopy (XES). Due to a lack of
quality data on the contributions of ZPR in these results, we do not attempt
to correct for such effects in our analysis.
a: [129], b: [130], c: [131] |  |  $\bm{G}_{0}W_{0}$@
---
PBE
|  $\bm{G}_{0}W_{0}$@
---
PBE0
|  $\bm{G}_{0}W_{0}$@
---
HSE
WOT-SRSH |  |  $\bm{G}_{0}W_{0}$@
---
WOT-SRSH
Expt.
InSb | 16.18 | 16.74 | 16.57 | 16.24 | 16.55 | 17.1a, 16.98b, 17.41c
InAs | 15.31 | 16.1 | 16.04 | 15.8 | 16.03 | 16.9a, 17.40c, 17.38d
Ge | 26.97 | 28.32 | 28.13 | 27.25 | 28.09 | 29.4f
GaSb | 17.12 | 15.52 | 18.15 | 17.52 | 18.11 | 18.8a, 18.9g
InP | 14.86 | 15.71 | 15.57 | 15.37 | 15.55 | 17.1a
GaAs | 16.81 | 17.98 | 17.81 | 17.14 | 17.83 | 18.7a, 18.7b, 18.82c
AlSb | 29.68 | 30.66 | 30.5 | 30.05 | 30.55 | 31.15e, 31.60d
AlAs | 36.66 | 37.97 | 37.82 | 37.56 | 38.03 | 39h
GaP | 16.03 | 17.2 | 17.03 | 16.64 | 17.02 | 18.6a, 18.7c
MAE | 1.92 | 1.19 | 1.03 | 1.48 | 1.01 |
MSE | -1.92 | -1.19 | -1.03 | -1.48 | -1.01 |
Table 5:  QP highest $d$ band positions, at the WOT-SRSH and $G_{0}W_{0}$@DFT
level, for the various functionals and $d$-electron containing compounds under
study. At the bottom of the table are the MAE and MSE; all are in eV and
measured relative to the leftmost reported experimental values. All
experimental data are obtained via X-ray photo-emission spectroscopy (XPS).
Due to a lack of quality data on the contributions of ZPR in these results, we
do not attempt to correct for such effects in our analysis.
a: [132], b: [133], c: [134], d: [135], e: [136], f: [137], g: [138], h: [139]

####  IV.1.2 Band Structure

In Figure 2 we plot the calculated band structures from WOT-SRSH and
$G_{0}W_{0}$@WOT-SRSH for GaAs. Apart from a small shift, the bands are nearly
identical. Additionally, the similarity of their curvature, especially near
the band gap can be seen in the lower inset plot. $G_{0}W_{0}$@WOT-SRSH
corrections do, however, result in a flattening of the valence bands compared
to those of WOT-SRSH. This can be seen in the top figure, where the lowest
valence band from WOT-SRSH is $\sim 0.5$ eV lower than its $G_{0}W_{0}$@WOT-
SRSH counterpart. This indicates that away from the band gap, there may be
more significant differences between bandstructures of $G_{0}W_{0}$@WOT-SRSH
and WOT-SRSH. In sections IV.1.3 and IV.1.4 we analyze these differences in
greater detail.

####  IV.1.3 Bandwidths

The calculated valence bandwidths for all compounds are reported in Table 4.
Additional results from DFT functionals other than WOT-SRSH can be found in
the SI, section S-III.B [98]. For zinc blende materials, where there is strong
$sp^{3}$ hybridization, the valence bandwidth is defined as the maximal energy
difference between the top four (excluding spin degeneracy) valence bands. For
the wurtzite and rock salt compounds, the valence bandwidth is defined as the
maximal energy difference between the top three valence bands for LiF and MgO
and the top six valence bands for AlN since it has twice as many atoms per
unit cell. For more information on the states under consideration to compute
bandwidths, see the leftmost column in Table 4. Unlike for QP band gaps, the
effects of ZPR are not incorporated when comparing to experiment. Details on
the DFT calculations (excluding WOT-SRSH) can be found in the SI. For WOT-
SRSH, the MAE and MSE are 1.08 eV and 0.81 eV respectively, suggesting the
method tends to overestimate valence bandwidths by $\sim 1$ eV.
$G_{0}W_{0}$@WOT-SRSH has an MAE of 0.65 eV and an MSE of 0.18 eV, showing
that $G_{0}W_{0}$ corrections away from the band gap offer a significant
improvement in accuracy relative to WOT-SRSH. Notably, the valence bandwidths
for the zinc blende compounds are generally overestimated relative to
experiment by both WOT-SRSH and $G_{0}W_{0}$@WOT-SRSH, while for the rock salt
compounds studied, the valence bandwidths are, if anything, underestimated.

Moving to the well-established starting point functionals, $G_{0}W_{0}$@PBE
computes bandwidths quite well, with an MAE of 0.59 eV and an MSE of -0.24 eV.
It also tends to underestimate bandwidths as its MSE suggests. For hybrids,
$G_{0}W_{0}$@PBE0 and $G_{0}W_{0}$@HSE have MAEs of 0.68 and 0.64 eV and MSEs
of 0.31 and 0.16 eV respectively. Interestingly, $G_{0}W_{0}$@HSE and
$G_{0}W_{0}$@WOT-SRSH have comparable levels of accuracy for bandwidths.
Unlike in the case of band gaps, this similar level of accuracy persists for
wide-gap insulators.

####  IV.1.4 $d$ Band Energies

For each semiconductor in our set that has elements for which $d$ orbitals are
explicitly treated as valence states, the $d$ band position, defined as the
highest $d$ orbital eigen-energies relative to the valence band maxima, is
reported in Table 5. Additional results from DFT functionals other than WOT-
SRSH can be found in the SI, section S-III.C [98]. As in the case of
bandwidths, the effects of ZPR are not incorporated when comparing to
experiment. For all calculations, we observe a universal underestimation of
the $d$-orbital locations, making the distinction between the MAE and MSE
meaningless. $G_{0}W_{0}$ corrections offer an improvement in accuracy for all
starting points. For WOT-SRSH, the MSE decreases from 1.48 to 1.01 eV. For
$G_{0}W_{0}$@PBE, it plumets from 3.8 to 1.92 eV. For $G_{0}W_{0}$@PBE0 it
decreases from 1.7 to 1.19 eV, and for $G_{0}W_{0}$@HSE it decreases from 1.47
to 1.03 eV. In total, $G_{0}W_{0}$@HSE and $G_{0}W_{0}$@WOT-SRSH appear to
perform the best and offer a comparable level of accuracy. However, both
methods still deviate from experimental reports by $\sim 1$ eV.

###  IV.2 Parameter Sensitivity of WOT-SRSH and $G_{0}W_{0}$@WOT-SRSH

The IP ansatz used to tune the range-separation parameter in the WOT-SRSH
functional determines $\gamma$ uniquely for a given choice of $\alpha$ and
$\varepsilon_{\infty}$. However, there can be ambiguities in the selection of
$\alpha$ and $\varepsilon_{\infty}$, with consequences for the predictive
power of this WOT-SRSH band gaps. Assuming first that $\varepsilon_{\infty}$
has been computed accurately and that $\beta$ is set to enforce
$\alpha+\beta=\varepsilon_{\infty}^{-1}$, there exists, in principle, a range
of choices of $\alpha$ for each material where one can find an optimal
$\gamma>0$ satisfying the IP ansatz. These optimal $(\alpha,\gamma)$ pairs
produce band gaps which can differ by up to a few hundred meV. Some of the
ambiguity in selecting $\alpha$ is avoided by setting $\alpha=0.25$ by
default, but as discussed in II.1 an optimal $\gamma$ cannot always be found
when $\alpha=0.25$, especially if $\varepsilon_{\infty}^{-1}\sim 0.25$. In
such cases, $\alpha$ must be varied until it becomes possible to find an
optimal $\gamma$ which satisfies the IP ansatz and the constraint
$\gamma>L^{-1}$, where $L$ is the unit cell size used in the calculations.
Additionally, it should be noted that while in principle $\alpha$ can be
increased to be as large as $1$, in practice values approaching unity are
generally considered to be unphysically large for most systems [66]. Thus,
WOT-SRSH predictions are, in practice, more precise than those one would
obtain from considering the full range of $\alpha$ values.

Nonetheless, it is of significant interest to explore the ambiguity in
selecting $\alpha$ in the WOT-SRSH framework further and its consequences for
$G_{0}W_{0}$@WOT-SRSH. To do so, we systematically vary $\alpha$ and $\gamma$
and compute GKS and $G_{0}W_{0}$ QP band gaps for AlN. AlN is a good candidate
for investigation since it has a dielectric constant that is very close to
$0.25$ and its band gap exhibits significant variation, on the order of $100$s
of meV, between optimal $(\alpha,\gamma)$ pairs. The difference $\Delta
E_{g}=E_{g}-E_{g,\text{ref}}$ between computed band gaps, relative to the
chosen reference gaps $E_{g,\text{ref}}$ for SRSH and $G_{0}W_{0}$@SRSH
calculations of AlN, can be seen over a range of $\alpha$ and $\gamma$ in
Figure 3. Note that, as indicated, we are using, strictly speaking, the SRSH
functional, as opposed to WOT-SRSH, meaning the IP anatz is not satisfied for
most of the data shown in Figure 3. The only overall constraint applied here
is $\beta=\varepsilon_{\infty}^{-1}-\alpha$. Pairs of $(\alpha,\gamma)$
satisfying the IP ansatz are marked with black diamonds, and the reference
band gap $E_{g,\text{ref}}$ is chosen to be the band gap obtained with the
WOT-SRSH parameters of prior work [66]. A range of $\pm 100$ meV about this
reference value is specified in white in the colormap.

Overall, $G_{0}W_{0}$ corrections to the SRSH starting point substantially
reduce the sensitivity of the computed band gap to variations in $\alpha$ and
$\gamma$ by about a factor of three. Specifically, at the SRSH level $\Delta
E_{g}$ varies by 6.0 eV for the large ranges of $\alpha$ and $\gamma$
considered, while for $G_{0}W_{0}$@SRSH it varies by only 2.14 eV. This
reduction in sensitivity becomes much more pronounced when only
$(\alpha,\gamma)$ pairs satisfying the IP ansatz are considered. At the WOT-
SRSH level, the $\Delta E_{g}$ values produced by these pairs have a range of
$322$ meV and depart from the white $\pm 100$ meV range about
$E_{g,\text{ref}}$ for the somewhat unphysical larger choices of $\alpha$. In
contrast, at the $G_{0}W_{0}$@WOT-SRSH level, the exhibited range is only $26$
meV. This reduction is by more than an order of magnitude, and substantially
lower than the reduction observed for the overall SRSH functional. A similar
set of trends is also observed for the other materials; see SI Section S-IV.1
[98].

Figure 3: The direct band gap of AlN, relative to a reference value ($\Delta
E_{g}=E_{g}-E_{g,\text{ref}}$) at the SRSH and $G_{0}W_{0}$@SRSH levels,
interpolated over a wide range of $(\gamma,\alpha)$ values. The grid of
performed calculations is represented as gray dots, and the pairs satisfying
the WOT-SRSH constraint are depicted as diamonds, with the reference pair for
$\Delta E_{g}$ in red. A range of $\pm 100$ meV about the reference is shaded
in white. $G_{0}W_{0}$ can be seen to suppress the overall variation at the
SRSH level by about a factor of three. Moreover, the pairs satisfying the WOT-
SRSH constraint can be seen to leave the $\pm 100$ meV range for the somewhat
unphysically large values of $\alpha$ for SRSH, but not for $G_{0}W_{0}$@SRSH.

We also note that there exists some uncertainty as to how to select the long-
range screening $\alpha+\beta=\varepsilon_{\infty}^{-1}$. For example, one
could compute $\varepsilon_{\infty}$ by considering the head of the RPA
dielectric matrix used in $GW$ [3]. However, the response to a finite electric
field can also be employed, leading to a beyond-RPA value of
$\varepsilon_{\infty}$ [140, 141]. For example, the inclusion of local field
effects for such methods can significantly affect the calculated response
[142]. Additionally, the underlying DFT functional affects the computed
$\varepsilon_{\infty}$; and it is even possible to self-consistently update an
SRSH functional so that the dielectric constant it predicts matches the long-
range screening, $\alpha+\beta$, that it employs [39, 143].

Accordingly, we consider the effects of varying the dielectric constant for
AlN. For the WOT-SRSH functional used here, $\varepsilon_{\infty}$ was
computed in prior work [66] via finite electric fields using a PBE0
functional. In lieu of re-computing the optimal $\gamma$ value for different
values of $\varepsilon_{\infty}$, we simply consider the effects of changing
$\varepsilon_{\infty}$ by $\pm 10\%$ while keeping $\alpha$ and $\gamma$
fixed. This choice likely exacerbates the sensitivity of the functional
because the IP ansatz is slightly violated for the values of
$\varepsilon_{\infty}$ which differ from the original one used to tune
$\gamma$. As can be seen in Section S-IV.2 of the SI [98], these perturbations
in $\varepsilon_{\infty}$ result in the band gap changing by 260 meV at the
SRSH level but only 80 meV at the $G_{0}W_{0}$@SRSH level. This behavior is in
line with the approximately threefold band gap range flattening observed
above. A similar trend is also observed for the other materials considered in
the SI, Section S-IV.2 [98].

##  V Conclusions

We have shown that a new WOT-SRSH class of range-separated hybrid functionals,
which is tuned to satisfy an IP ansatz for localized orbitals and to have the
correct asymptotic boundary conditions of the Coulomb potential in solids,
offers an excellent starting point for $G_{0}W_{0}$ calculations of the
electronic structure for a wide range of semiconductors and insulators.
$G_{0}W_{0}$@WOT-SRSH calculations perform at a level of accuracy that is
equal to or better than state-of-the-art ev$GW$ and QS$GW$ calculations at a
lower computational cost. Additionally, $G_{0}W_{0}$@WOT-SRSH tends to improve
the description of states deeper in the valence band manifold, leading to more
accurate valence bandwidths and $d$ band locations. Finally, we have shown
that $G_{0}W_{0}$@WOT-SRSH corrections greatly reduce the sensitivity of
computed bands gaps to variations in the underlying WOT-SRSH parameters that
can arise from ambiguities in the optimal tuning procedure. In total, these
calculations demonstrate that pairing WOT-SRSH with single-shot $G_{0}W_{0}$
methods offers a high-accuracy predictive method for calculating QP properties
of materials with a wide range of band gaps.

## Acknowledgements

This work was supported via US-Israel NSF-Binational Science Foundation (BSF)
Grant DMR-1708892. Computational resources were provided by the Extreme
Science and Engineering Discovery Environment (XSEDE) supercomputer Stampede2
at the Texas Advanced Computing Center (TACC) through the allocation TG-
DMR190070 and additional computational resources were provided by the National
Energy Research Scientific Computing Center (NERSC), DOE Office of Science
User Facilities supported by the Office of Science of the US Department of
Energy under Contract DE-AC02-05CH11231. MRF acknowledges support from the UK
Engineering and Physical Sciences Research Council (EPSRC), grant no.
EP/V010840/1, and LK thanks the Aryeh and Mintzi Katzman Professorial Chair
and the Helen and Martin Kimmel Award for Innovative Investigation.

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