# A pathway towards high throughput Quantum Monte Carlo simulations for
alloys: A case study of two-dimensional (2D) GaSxSe1-x

Daniel Wines  Department of Physics, University of Maryland Baltimore County,
Baltimore MD 21250  Kayahan Saritas  Department of Applied Physics, Yale
University, New Haven CT 06520  Can Ataca<EMAIL_ADDRESS>Department of
Physics, University of Maryland Baltimore County, Baltimore MD 21250

###### Abstract

The study of alloys using computational methods has been a difficult task due
to the usually unknown stoichiometry and local atomic ordering of the
different structures experimentally. In order to combat this, first-principles
methods have been coupled with statistical methods such as the Cluster
Expansion formalism in order to construct the energy hull diagram, which helps
to determine if an alloyed structure can exist in nature. Traditionally,
density functional theory (DFT) has been used in such workflows. In this work
we propose to use chemically accurate many-body variational Monte Carlo (VMC)
and diffusion Monte Carlo (DMC) methods to construct the energy hull diagram
of an alloy system, due to the fact that such methods have a weaker dependence
on the starting wavefunction and density functional, scale similarly to DFT
with the number of electrons, and have had demonstrated success for a variety
of materials. To carry out these simulations in a high-throughput manner, we
propose a method called Jastrow sharing, which involves recycling the
optimized Jastrow parameters between alloys with different stoichiometries. We
show that this eliminates the need for extra VMC Jastrow optimization
calculations and results in a significant computational cost savings (on
average 1/4 savings of total computational time). Since it is a novel post-
transition metal chalcogenide alloy series that has been synthesized in its
few-layer form, we used monolayer GaSxSe1-x as a case study for our workflow.
By extensively testing our Jastrow sharing procedure for monolayer GaSxSe1-x
and quantifying the cost savings, we demonstrate how a pathway towards
chemically accurate high-throughput simulations of alloys can be achieved
using many-body VMC and DMC methods.

††preprint: AIP/123-QED

##  I Introduction

The accurate modeling of alloys with first-principles methods is a difficult
computational effort in the materials science community van de Walle (2009).
The main issue arises from the usually unknown stoichiometry and local atomic
ordering of the alloyed structures. One way to combat this is to manually
create supercells with a specific stoichiometry and local atomic ordering
determined empirically or intuitively Jung  _et al._ (2015); Ersan, Ozaydin,
and Aktürk (2018); Kanli  _et al._ (2019). Another, more systematic option is
to create a large set of alloyed structures (of varying compositions and
stoichiometries) and determine the energy hull diagram of the alloy series
from first-principles coupled with statistical methods van de Walle (2009);
Wines, Ersan, and Ataca (2020); Cai  _et al._ (2018). Popular methods usually
involve the Special Quasirandom (SQS) Zunger  _et al._ (1990) generation of
several structures, evaluation of energies from first-principles, and then
using the first-principles energies as a training set for statistical methods
such as the Cluster Expansion van de Walle (2009) formalism.

Constructing an accurate energy hull diagram is essential for predicting the
energetic stability and favorability of alloys. The accuracy of an energy hull
diagram depends critically on the choice of the method used to generate total
energies. Density functional theory (DFT) has been used for this purpose,
thanks to its balance between accuracy and computational scaling on modern
computer clusters. On the other hand, diffusion Monte Carlo (DMC) Foulkes  _et
al._ (2001); Needs  _et al._ (2009) is another ground state method which can
provide total energy differences with significantly better accuracy than DFT,
with minimal functional and basis set dependence Wines, Saritas, and Ataca
(2020); Shin  _et al._ (2021); Frank  _et al._ (2019); Shulenburger  _et al._
(2015); Annaberdiyev  _et al._ (2021). DMC has a very similar computational
scaling to DFT, $N^{3-4}$, though with nearly a thousand times larger
prefactor, but its embarrassingly parallel algorithm can utilize parallel
computation architectures using larger number of nodes. Recently, DMC has been
used to calculate chemically accurate total energies and electronic properties
for a wide variety of two-dimensional (2D) and bulk structures, demonstrating
the routine applicability of this method Shin  _et al._ (2021); Hao  _et al._
(2018); Mostaani, Drummond, and Fal’ko (2015); Shulenburger  _et al._ (2015);
Kadioglu  _et al._ (2018); Szyniszewski  _et al._ (2017a, b); Mostaani  _et
al._ (2017); Hunt  _et al._ (2020); Saritas  _et al._ (2019a); Frank  _et al._
(2019); Saritas  _et al._ (2018); Luo  _et al._ (2016); Benali  _et al._
(2016); Santana  _et al._ (2015); Shin  _et al._ (2017, 2018); Saritas,
Krogel, and Reboredo (2018); Saritas  _et al._ (2019b, 2020); Nazarov  _et
al._ (2016); Annaberdiyev  _et al._ (2021); Wang, Annaberdiyev, and Mitas
(2020); Bennett  _et al._ (2021). In this work, building on our previous DMC
study of 2D GaSe Wines, Saritas, and Ataca (2020), we propose to use DMC to
reliably calculate the energetics of 2D GaSxSe1-x as a case study. This will
enable accurate simulations of alloy properties for experimental
characterization and validation.

DMC calculations require multiple steps to obtain the total energy of a
structure and the stochastic nature of the method introduces uncertainty in
the observables, in contrast to the deterministic nature of DFT. Constructing
an adequate energy hull diagram requires the high-throughput calculation of
the energy of several structures. Similarly to the development of the decade
old field of high throughput DFT Jain  _et al._ (2011), researchers have
attempted to bridge the gap between high throughput computing and Quantum
Monte Carlo (QMC) and increase the computational efficiency of the QMC
workflow Zen  _et al._ (2018); Saritas  _et al._ (2017). For example, Saritas
et al. Saritas  _et al._ (2017) developed a high throughput procedure to
determine the DMC formation energy of certain materials directly from the ICSD
database. In this work, we propose a new high throughput workflow to determine
the alloy formation energy of 2D GaSxSe1-x with near chemical accuracy. We
went on to test various wavefunction optimization methods, which included
recycling Jastrow parameters (which we call Jastrow sharing) among various
alloy configurations to reduce the overall computational cost and localization
errors.

In addition to 2D GaSxSe1-x being a suitable case study that builds off our
previous QMC work Wines, Saritas, and Ataca (2020), it has been reported that
alloying is a promising route to control the properties of several monolayer
or few layer materials Wang  _et al._ (2018); Kanli  _et al._ (2019); Yang
_et al._ (2019); Ersan, Ozaydin, and Aktürk (2018); Ersan, Gökoğlu, and Aktürk
(2015); Mo  _et al._ (2019); Miyazato  _et al._ (2020); Raffone  _et al._
(2016) (including post transition metal chalcogenides, or PTMCs Almadvari,
Nayeri, and Fotoohi (2020); Wines, Ersan, and Ataca (2020); Nguyen  _et al._
(2020); Cai  _et al._ (2016, 2018)), which makes the study of 2D alloys with
accurate QMC methods useful for theorists and guiding experimentalists in
synthesis and characterization. For example, it has experimentally been
reported that when GaSe nanostructures are alloyed with Te, the material
undergoes a hexagonal to monoclinic transition and there exists an instability
region where the phases compete and two different band gaps can be found at
the same composition Cai  _et al._ (2018). Janus monolayers Ersan and Ataca
(2020); Nguyen  _et al._ (2020) (a class of two-dimensional 2D alloyed
structures) have also gained attention due to their applications for Schottky
contacts, as demonstrated theoretically with Janus GaSSe on top of graphene
Nguyen  _et al._ (2020). In addition, PTMCs such as GaSe and GaS can be
engineered for specific applications by chemical functionalization Zhou  _et
al._ ; Ao  _et al._ (2015); Meng  _et al._ (2015), creating heterostructures
Nguyen  _et al._ (2020); Li  _et al._ (2016); Liu  _et al._ (2018); Yang  _et
al._ (2016); Jappor (2017), and applying strain Longuinhos and Ribeiro-Soares
(2019); Liu  _et al._ (2015); Wang  _et al._ (2016). PTMCs such as GaSe have
also been reported to be suitable substrate material for other 2D structures,
as well Wines  _et al._ (2020); Yang  _et al._ (2018).

In section II we outline our DFT and QMC approaches and convergence criteria.
In section III.1 we present our DFT results in order to initially screen 2D
GaSxSe1-x alloys. In section III.2 we describe our Jastrow sharing methodology
and present a detailed analysis of this methodology at the variational Monte
Carlo (VMC) and DMC level. In section III.3 we present DMC results using
various supercell sizes for alloys and compare extrapolated results using
different wavefunction optimization methods. In section III.4 we give an
analysis of the computational cost savings of our approach. Finally, we
provide concluding remarks and future perspectives in section IV.

##  II Computational Methods and Theory

Benchmarking DFT calculations were performed using the VASP code with
projector augmented wave (PAW) potentials Kresse and Furthmüller (1996);
Kresse and Joubert (1999). For these VASP benchmarking calculations, the
Perdew-Burke-Ernzerhof (PBE)Perdew, Burke, and Ernzerhof (1996) and strongly
constrained and appropriately normed (SCAN)Sun, Ruzsinszky, and Perdew (2015)
meta-GGA functionals were used. In addition, the PBE+D2 Grimme (2006) and
PBE+D3 Grimme  _et al._ (2010) methods of Grimme and the SCAN+rvv10 Peng  _et
al._ (2016) functionals were used to investigate vdW effects at the DFT level.
At least 20 Å of vacuum was given between periodic layers of GaSxSe1-x in the
$c$-direction, a kinetic energy cutoff of 350 eV was used, and a 6x6x1
reciprocal grid was used for the primitive cells of the alloys and the number
of k-points were scaled appropriately with supercell size.

DMC calculations use wavefunctions from other ab-initio methods, in our case
DFT-PBE, as input (trial wavefunction) to obtain an equilibrium that is
computationally tractable. For DFT calculations within the QMC workflow, the
Quantum Espresso (QE) Giannozzi  _et al._ (2009) code was used. For Ga, S, and
Se we used norm-conserving Burkatzki-Filippi-Dolg (BFD) Burkatzki, Filippi,
and Dolg (2007, 2008) pseudopotentials. DMC calculations require norm-
conserving pseudopotentials and BFD pseudopotentials have been thoroughly
tested in DMC Dzubak, Krogel, and Reboredo (2017); Wines, Saritas, and Ataca
(2020). For details of pseudopotential testing, conversion and validation,
refer to the discussion and Table S1 in the Supplementary Information (SI).
For the pseudopotentials, we used a kinetic energy cutoff of 160 Ry (see Fig.
S1), which gives a convergence of less than 1 meV at the DFT level. We used
the same supercell reciprocal twist grid in QE as our benchmarking VASP
calculations which is tested in Fig. S2.

After the trial wavefunction is generated using DFT, VMC and DMC Foulkes  _et
al._ (2001); Needs  _et al._ (2009) calculations were carried out using the
QMCPACK Kim  _et al._ (2018); Kent  _et al._ (2020) code. VMC calculations are
the intermediate steps between the DMC and DFT calculation, where the single
determinant DFT wavefunction is converted into a many-body wavefunction,
through the Jastrow parameters Slater (1929); Jastrow (1955). Jastrow
parameters help model the electron correlation and reduce the uncertainty in
the DMC calculations Umrigar and Filippi (2005); Mitas, Shirley, and Ceperley
(1991). The automated DFT-VMC-DMC workflows were generated using the Nexus
Krogel (2016) software suite. Up to three-body Jastrow Drummond, Towler, and
Needs (2004) correlation functions were included. The linear method Umrigar
_et al._ (2007) was used to minimize the variance and energy respectively of
the VMC total energies . The cost function of the variance optimization is
100$\%$ variance minimization and the cost function of the energy optimization
is split as 95$\%$ energy minimization and 5$\%$ variance minimization, which
has been shown to reduce uncertainty for DMC results Umrigar and Filippi
(2005). The explicit details of how we modified this optimization procedure
for our high-throughput QMC workflow will be given in section III.2.

Being a real-space wavefunction method, DMC calculations need to be performed
at increasing supercell sizes to eliminate finite-size errors. We used
supercell sizes up to 72 atoms and extrapolated to the infinite cell size. In
order to smooth the image interactions at each supercell used in the
extrapolation, we used the optimal_tilematrix function in Nexus, which allows
constructing appropriate supercells with the largest Wigner-Seitz (WS) radius
for a given size. We used Jackknife fitting to obtain a linear fit of DMC data
and extrapolate to the infinite-size limit for total ground state energy of
each alloyed structure. In addition, the locality approximation Mitas,
Shirley, and Ceperley (1991) was used to evaluate the nonlocal
pseudopotentials in DMC. A timestep of 0.01 Ha-1 (convergences tests are shown
in Fig. S3) was used for all DMC simulations.

To initially create and screen these GaSxSe1-x alloyed structures with varying
concentrations, the Special Quasirandom Structure (SQS) method Zunger  _et
al._ (1990) was employed as implemented in the ATAT code van de Walle (2009).
The alloy formation energy in units of eV/formula unit (otherwise known as the
mixing enthalpy) is defined as:

$E_{\textrm{{form}}}=E_{{\textrm{GaS}_{1-x}\textrm{Se}_{x}}}-(1-x)E_{\textrm{{GaS}}}-xE_{\textrm{{GaSe}}}$
(1)

where $E_{{\textrm{GaS}_{1-x}\textrm{Se}_{x}}}$ is the energy of a GaSxSe1-x
cell, $E_{\textrm{{GaS}}}$ is the energy of a GaS cell, and
$E_{\textrm{{GaSe}}}$ is the energy of a GaSe cell, all per formula unit
(f.u.). After the SQS generation of random alloys, first-principles
calculations are done to calculate the formation energy of each structure.
These energies are then used to obtain fitted energies via the Cluster
Expansion formalism van de Walle (2009). Further details and convergence
criteria can be found in the Supplementary Information (SI).

##  III Results and Discussion

###  III.1  DFT Calculations

In contrast to a previous DFT study where the 2D GaSxSe1-x alloys were
constructed manually Jung  _et al._ (2015), we used the SQS method to generate
alloys and DFT to initially optimize the geometries. Lattice parameters of 2D
GaS and GaSe have been determined experimentally Jung  _et al._ (2015),
therefore accurate determination of the geometry is an important step for the
respective 2D alloys, which have unknown experimental lattice constants. Due
to this, we performed geometry and lattice optimization calculations with PBE,
PBE-D2, PBE-D3, SCAN and SCAN+rvv10 DFT functionals to benchmark. This process
also allows us to construct the energy hull diagrams using each functional
(see Fig. 1 and Fig. S4). We find that all the tested functionals are in good
agreement with each other in their formation energies, but lattice parameters
differ slightly for each structure. However, we find that the SCAN+rvv10
functional gives the overall best agreement with the experimental lattice
parameters of 2D GaS and GaSe Jung  _et al._ (2015); Li  _et al._ (2014) (see
Table S2 in SI). This is expected since it has been previously reported that
SCAN yields lattice constants and optimal geometry in closer agreement to
experiment for several 2D materials Buda  _et al._ (2017); Wines, Saritas, and
Ataca (2020). Therefore, we use the geometries from the SCAN+rvv10 functional
as a starting point for DMC calculations. To ensure that the optimal geometry
of these structures are closest to exact and do not bias the energy, we
isotropically scaled the lattice (starting from the geometry and lattice
constants obtained from SCAN+rvv10) and confirmed what the location of the
minimum energy was with respect to lattice constant for the equation of state
(see Fig S5-9). The benchmarking of lattice parameters with different
functionals are done using VASP because the plane wave cutoff of PAW
potentials are more than an order of magnitude smaller than BFD
pseudopotentials.

Figure 1: The atomic structure (top and side view) of selected SQS GaSxSe1-x
alloys from the energy hull diagram: a) Janus GaS0.5Se0.5, b) GaS0.5Se0.5
(lowest E${}_{\textrm{form}}$), c) GaS0.333Se0.667 (lowest
E${}_{\textrm{form}}$), d) GaS0.667Se0.333 (lowest E${}_{\textrm{form}}$), e)
GaS0.875Se0.125 (smallest Wigner-Seitz radius) and f) the energy hull diagram
obtained using DFT (SCAN+rvv10, PAW potentials), where the structures depicted
(a - e) are circled with the corresponding color outline and the band gap is
given in the color axis for each structure.

As seen in Fig. 1 f), all the monolayer GaSxSe1-x structures have positive
formation energy, which implies that these structures are potentially not
energetically favorable at zero temperature and cannot form spontaneously in
vacuum (metastable). In addition, Janus GaS0.5Se0.5 (where S and Se atoms are
on the opposing faces of the 2D layer), has the highest formation energy,
which means that this structure is the least energetically favorable in free-
standing form. This is a similar finding to Ersan et al. Ersan and Ataca
(2020), where it was found that monolayer Janus PtXY (X = S, Se, Te) is the
least favorable alloyed structure in the 2D PtXY alloy series. Even though
these reported Janus structures are thermodynamically stable (as a result of
phonon and molecular dynamics simulations) in vacuum Ersan and Ataca (2020),
they can perhaps be synthesized on a substrate material Zhang  _et al._
(2013); Lu  _et al._ (2017). In free-standing form, molecular dynamics
simulations showed that certain Janus structures can spontaneously curl Xiong
_et al._ (2018). Figure S4 and 1 f) also reflects the band gap tunability
(color axis) of GaSxSe1-x monolayers at the DFT level, opening the door to
various electronic applications. Following the initial geometric optimization
and construction of the energy hull diagram with DFT, we selected certain
structures to further run QMC on to test our Jastrow sharing procedure which
we will discuss in the next subsection.

###  III.2 Jastrow Parameter Sharing

In attempt to reduce the number of wavefunction optimization simulations
required to perform DMC for all the 2D GaSxSe1-x structures involved in the
Cluster Expansion formalism, we propose a systematic method we call Jastrow
sharing. Wavefunction optimization using Jastrow parameters is key to reducing
variance and the localization error of subsequent DMC simulations Nazarov  _et
al._ (2016); Dzubak, Krogel, and Reboredo (2017); Krogel and Kent (2017).
Jastrow parameters are typically optimized using the cheaper VMC calculations,
but these costs can be important considering the large number of materials
simulated in the Cluster Expansion formalism. Computational cost analysis will
further be discussed in Section III.4. Our Jastrow sharing approach involves a
judicious selection of optimized Jastrow parameters from a single compound,
and reusing it in all the rest of the materials studied. In this section we
discuss the various methods we used to test and validate this method for
GaSxSe1-x and the limitations of this method with respect to other material
classes.

In our method, we use a supercell that has the smallest Wigner-Seitz (WS)
radius to obtain the optimized Jastrow parameters that will later be used in
all DMC calculations. The maximum Jastrow cutoff radius possible in a
structure is equal to its WS radius, which is equal to the radius of the
largest inscribing sphere that can fit into the simulation cell. Even though
the Jastrow cutoff can be optimized, this increases the computational scaling
of the VMC optimization Drummond, Towler, and Needs (2004), hence it is more
practical to use the maximum Jastrow cutoff possible for a periodic system (WS
radius), and update each Jastrow parameter at a fixed cost. Therefore, it is
imperative that the smallest WS radius among all the structures is determined
in advance in order to reuse the same set of Jastrow parameters for all the
structures. For three-body Jastrows, a cutoff radius of 2 Å (on the order of
GaSxSe1-x bond lengths) was used. Due to the fact that three-body Jastrows
(energy optimization) usually capture shorter-range correlations, we made this
arbitrary choice to minimize computational cost while attempting to capture
such interactions. The full workflow of our procedure is given in Fig. 5.

We performed DMC calculations on four structures: x = 0.333 (12 atom cell,
Fig. 1 c)), x = 0.5 (16 atom cell, Fig. 1 b)) and x = 0.667 (12 atom cell,
Fig. 1 d)) and the Janus GaS0.5Se0.5 structure (highest formation energy, 16
atom cell, Fig. 1 a)). Except for the Janus structure, all these compounds lie
on the bottom of the DFT hull diagram. However, this selection can be made on
a finite threshold above the DFT hull or more ambitious plans can involve all
the SQS generated structures. The reason we decided to investigate materials
with low and high formation energy is to observe if QMC can correctly capture
these energy differences and to demonstrate that our method can work for the
entire energy hull.

DMC-J3 | Janus | x = 0.5 | x = 0.333 | x = 0.667
---|---|---|---|---
Janus Jastrow | 0 | -6(6) | 2(6) | 3(10)
x = 0.5 Jastrow | 0(4) | 0 | 4(7) | 15(11)
x = 0.333 Jastrow | -1(4) | -2(4) | 0 | 17(10)
x = 0.667 Jastrow | -6(5) | -10(6) | -1(6) | 0
x = 0.875 Jastrow | -2(5) | -15(7) | -2(5) | 9(9)
DMC-J2 | Janus | x = 0.5 | x = 0.333 | x = 0.667
Janus Jastrow | 0 | -3(7) | -9(7) | -6(7)
x = 0.5 Jastrow | 3(4) | 0 | 5(7) | -9(8)
x = 0.333 Jastrow | -7(5) | 4(9) | 0 | 6(9)
x = 0.667 Jastrow | -9(7) | -2(8) | -2(6) | 0
x = 0.875 Jastrow | -8(4) | -5(8) | -2(6) | -3(7)
Table 1: The differences (in meV/f.u.) between total energies (DMC) calculated
with self-Jastrows and shared-Jastrows including up to two-body and up to
three-body terms with the associated error bars in parenthesis. The rows
represent which Jastrows are used and the columns represent each alloyed
structure for which the DMC energies are calculated.

Among the unit cells of the alloyed structures we selected to study, x = 0.875
has the smallest WS radius (see Fig. 1 e)). Therefore, according to our
procedure, we would use GaS0.875Se0.125 to generate the Jastrow parameters to
be used in the remainder of the calculations. However, in order to understand
the sensitivity of our method to this decision, we make a benchmark study
where we optimize Jastrow parameters of all the structures separately as it is
typically done in DMC calculations. For testing, a 16 atom cell was used for x
= 0.5, Janus, 0.875 and a 12 atom cell was used for x = 0.333, 667. To be able
to use all 5 of these sets of Jastrow parameters across all the structures we
study, we still use the smallest WS radius as the cutoff parameter for two-
body Jastrow parameters. The smallest WS radius we have in this case is 3.12
Å. This is longer than 2 Å, hence no modifications are required for 3-body
Jastrows. The testing of our Jastrow sharing procedure is tabulated in Table
1, where we show the energy difference (in meV/f.u.) between the total energy
of each structure using its own optimized Jastrow parameters (we will define
this as a self-Jastrows) and the total energy of each structure using the
optimized Jastrow parameters of a different structure (we will define these as
shared-Jastrows), with the associated error. In order to accept the Jastrow-
sharing procedure as valid, we expect the localization error obtained with
self-Jastrows and shared-Jastrows to be comparable and the total energy
difference (per f.u.) between self-Jastrows and shared-Jastrows to be as
minimal as possible.

In Table 1, the columns represent the different selected structures and the
rows represent the various optimized Jastrow parameters. The upper quadrant
contains the energy difference at the DMC level using up to three-body
Jastrows and the lower quadrant contains the energy difference at the DMC
level using up to two-body Jastrows. A more detailed table with VMC energies,
including two- and three-body Jastrows are given in Table S3 in the SI. From
the data in Table 1, we observe that at the DMC level, the total energy
differences between shared-Jastrows and self-Jastrows are overall smaller,
with differences ranging from 0 - 17 meV/f.u. We also observe that the energy
differences calculated with DMC are nearly identical whether or not two-body
or three-body interactions are included in the Jastrow factor due to the fact
that we don’t have enough resolution to differentiate between the results. The
lack of significant variations in the DMC energies here also suggest that the
localization errors of these pseudopotentials are very small or negligible
since the quality of the Jastrow parameters, with regards to the Jastrow
cutoff, do not affect the DMC calculations.

In Figure 2, we additionally show that DMC-J3 total energies are always lower
than the DMC-J2 energies. This has been observed for a variety of systems
Krogel and Kent (2017). However, in a practical sense, the variation in a set
of Jastrow parameters (J2 or J3) is more important than the separation between
the DMC energies calculated with J2 and J3 Jastrow parameters. This is because
often the user makes one selection regarding the number of Jastrow parameters
and applies it throughout for all similar materials studied in the same
calculation set, such as polymorphs or alloys. Ideally, the quality of the
Jastrow parameters should only change the effort required to get the target
uncertainty in the DMC total energies. However, as observed in here, we often
see that the quality of the Jastrow parameters also change the degree of
localization error Nazarov  _et al._ (2016); Dzubak, Krogel, and Reboredo
(2017); Krogel and Kent (2017) which can lead to inconsistencies when energy
differences are calculated between two structures.

Figure 2: The total DMC energy (eV/f.u.) as a function of total VMC energy
(eV/f.u.) for the Janus, x = 0.5, x = 0.333, and x = 0.667 structures. The
formula for shared-Jastrow sensitivity (per f.u.), which is defined as the
change in DMC energy per change in VMC energy (slope of the graph), is given
in each inset. The data points with a black center represent J3 results while
the other points represent J2 results. Due to their non linearity, the data
points using the Janus Jastrow were excluded (see SI).

In order to quantify the amount of localization error introduced as a function
of the quality of the Jastrow parameter, we can utilize a quantity called
Jastrow sensitivity. The Jastrow sensitivity is defined as the amount of
energy decrease in the DMC energy per unit decrease in the VMC energy and is a
property of the pseudopotential used in the calculation Krogel and Kent
(2017). Ideally, the Jastrow sensitivity should be small for a given
pseudopotential for all of the relevant valence enviroments (i.e. different
phases of a given material). We infer that a low enough Jastrow sensitivity
could be a reason why we were successfully able to transfer different Jastrow
parameters across the 2D GaSxSe1-x series. In theory, if the sensitivity of
the pseudopotentials is low for the material class of study, our Jastrow
sharing procedure should be valid.

To further quantify Jastrow sensitivity in our work, we define a new quantity
called shared-Jastrow sensitivity (SJ,S), which is defined as the DMC energy
per unit decrease in the VMC energy calculated with self-Jastrows and shared-
Jastrows. In order to calculate this for each structure, we plotted the DMC
total energies vs. the VMC total energies (per f.u.) and obtained the slope
for the data (using J2 and J3 parameters, from Table 1), which is depicted in
Fig. 2. Due to the fact that VMC energies obtained with the Janus Jastrow
parameters have a large deviation attributed to cutoff radius, we excluded
these points from the linear fitting. A more detailed discussion related to
the Jastrow parameters of the Janus structure can be found in the SI. From
Fig. 2 we find the shared-Jastrow sensitivity (per f.u.) to be 0.094 (0.047
per atom) for the Janus structure, 0.116 (0.058 per atom) for the x = 0.5
structure, 0.084 (0.042 per atom) for the x = 0.333 structure, and 0.115
(0.058 per atom) for the x = 0.667 structure.

These sensitivities are comparable in value to the lower bounds on the Jastrow
sensitivities reported by Krogel and KentKrogel and Kent (2017) (0.05 for Ce
4+ using the locality approximation) and Dzubak, Krogel, and ReboredoDzubak,
Krogel, and Reboredo (2017) (0.07, 0.05, 0.04 for Mn, Fe, Co respectively). In
addition to our calculated shared-Jastrow sensitivity, we calculated the
standard Jastrow sensitivity of the individual Ga, S and Se atoms to further
prove transferability across different atomic systems. We obtained this by
calculating the DMC energy with no Jastrow parameters and the DMC energy using
up to J2 and J3 Jastrows, then fitting a line to the three data points. We
obtain a Jastrow sensitivity of 0.07, 0.14 and 0.04 for Ga, S and Se
respectively. As a result of these sensitivity calculations, we can quantify
that the Jastrows can be transferable between various stoichiometries in the
GaSxSe1-x system. If one wished to implement this Jastrow-sharing procedure
for another system, calculating the shared-Jastrow sensitivity using VMC and
DMC energies from self-Jastrows and shared-Jastrows and the standard Jastrow
sensitivity of the respective atoms and comparing to tabulated results in
literature could be a viable method to check if Jastrow-sharing is feasible.

###  III.3 Finite-size Effects

| Cohesive Energy (eV/f.u.)
---|---
Structure |  | DFT
---
(PBE)
| DMC
---
self-Jastrow
| DMC
---
WS-Jastrow
GaSe | -7.135 | -7.028(3)Wines, Saritas, and Ataca (2020) | -
GaS | -7.789 | -7.659(4) | -
Janus | -7.360 | -7.231(6) | -7.240(6)
x = 0.5 | -7.456 | -7.35(1) | -7.35(2)
x = 0.333 | -7.347 | -7.259(6) | -7.258(6)
x = 0.667 | -7.564 | -7.461(6) | -7.464(7)
| Formation Energy (eV/f.u.)
Structure |  | DFT
---
(PBE)
| DMC
---
self-Jastrow
| DMC
---
WS-Jastrow
Janus | 0.1019 | 0.106(4) | 0.096(4)
x = 0.5 | 0.0059 | 0.01(1) | 0.00(2)
x = 0.333 | 0.0057 | 0.006(4) | 0.007(4)
x = 0.667 | 0.0064 | 0.013(3) | 0.010(5)
Table 2: The calculated cohesive energy (top half) and alloy formation energy
(bottom half) all in eV/f.u. obtained from DFT (PBE, BFD pseudopotentials) and
DMC (using self-Jastrows and WS-Jastrows) for the Janus, x = 0.5, x = 0.333,
and x = 0.667 structures. Figure 3: The DMC calculated formation energies as a
function of inverse number of atoms (N) in the supercell, extrapolated to the
thermodynamic limit (N$\rightarrow\infty$) for the Janus, x = 0.5, x = 0.333,
and x = 0.667 structures. Red represents DMC energies calculated with self-
Jastrows while blue represents DMC energies calculated with WS-Jastrows.
Figure 4:  The partially constructed energy hull diagram for GaSxSe1-x
calculated with DFT (PBE) in green and DMC (using self-Jastrows in red and WS-
Jastrows in blue), with associated error bars.

To construct our DMC energy hull diagram and eliminate finite-size effects, we
extrapolate all structures to the thermodynamic limit. Therefore, our DMC
calculations are performed at different supercell sizes and then extrapolated
to the infinite sized limit as the energy scales as 1/N, where N is the number
of particles. In addition to this, we use a converged reciprocal grid obeying
periodic boundary conditions for each supercell, which is known as twist
averaging. For an additional benchmark to assess the validity of our approach,
we calculated the DMC cohesive energy (per f.u.) of each structure by
subtracting the appropriate single atom energies of Ga, Se and S (see SI for
single atom calculation details) from the extrapolated total energy. We
calculated the alloy formation energy ($E_{\textrm{{form}}}$) at the DMC level
using equation 1 (see Table 2). The total energy of the GaSe and GaS
supercells with the exact same number of atoms as the specific alloy supercell
was subtracted from the total energy of said alloy. This alloy formation
energy was calculated at multiple supercell sizes and extrapolated to the
thermodynamic limit (see Figure 3).

For the DMC extrapolated cohesive energies and formation energies, self-
Jastrows were used for each respective supercell. In addition, the optimized
Jastrow parameters of the alloy with the smallest WS radius (we will refer to
this as WS-Jastrow, with parameters taken from the 16 atom GaS0.875Se0.125
cell) were used to calculate the total DMC energy for each supercell size. A
comparison of the DMC results (for extrapolated cohesive energies and alloy
formation energies) using self-Jastrows and WS-Jastrows and comparison to DFT
is found in Table 2. In addition, the partial energy hull diagram for the
selected structures using DFT and DMC (with self-Jastrows and WS-Jastrows) is
depicted in Figure 4. From the linear extrapolated results (Fig. 3) depicted
in Fig. 4 and tabulated in Table 2, we clearly see the energies calculated
(and then extrapolated) using self-Jastrows and WS-Jastrows are nearly
indistinguishable.

These extrapolated formation energies demonstrate that an adequate energy hull
diagram can be constructed from DMC using just one set of Jastrow parameters
from the smallest WS cell, in contrast to optimizing the Jastrows for every
structure at every supercell size. Additionally, using a many-body approach
such as QMC for alloys can provide near-chemically accurate confirmation of
stability and other important electronic and energetic properties. 2D
GaSxSe1-x was chosen because it is a convenient system to test this
methodology on. This is convenient because it builds on our previous 2D GaSe
Wines, Saritas, and Ataca (2020) work and since there is relatively good
agreement between DFT functionals for the formation energy hull diagram (Fig.
S4), it allows us to benchmark the resolution of our DMC extrapolated (to meV
resolution) energies with reliable DFT energies.

A full schematic of this workflow is given in Fig. 5. This workflow starts
with an input of an initial lattice, followed by the SQS generation of random
alloys and then the construction of the energy hull diagram with DFT. After
analyzing the DFT hull diagram, the user can choose to run QMC for structures
below a certain energy threshold or discard them. From the SQS structures, the
user must determine which has the smallest WS radius, and then perform J2 and
J3 Jastrow optimization for such. These J2 and J3 parameters are then reused
for DMC calculations of the other selected structures. From this, the DMC
energy diagram can be constructed. A more detailed analysis of the cost
savings of this workflow is given in the following section.

Figure 5: The full high throughput workflow proposed in this work to obtain
the energetics of an alloy system with QMC methods.

###  III.4 Computational Cost Analysis

VMC Computational Time
---
| 1 Supercell | Extrapolation | Energy Hull
Self Jastrow | $755\pm 180$ | $7,124\pm 1,816$ | $24,043\pm 5,881$
WS Jastrow | $134\pm 40$ | $134\pm 40$ | $134\pm 40$
DMC Computational Time
| 1 Supercell | Extrapolation | Energy Hull
Self Jastrow | $2,143\pm 1,071$ | $32,899\pm 16,450$ | $93,066\pm 46,533$
WS Jastrow | $2,109\pm 1,054$ | $33,259\pm 16,630$ | $93,170\pm 46,616$
Table 3: A measure of the computational cost (time in seconds) using Self
Jastrows and WS Jastrows for 1 supercell, one extrapolated result, and the
entire energy hull diagram for VMC (upper half) and DMC (lower half)
simulations. The quantity in the $\pm$ gives the upper and lower bound on cost
(discarding unnecessary QMC steps after target error is reached).

We claim that reusing a fixed single set of Jastrow paramaters across all
structures involved in creating a DMC energy hull diagram is a valid and
computationally efficient approach. Therefore, we compare the efficiency of
our procedure to another more conventional way of performing these DMC
calculations, where the Jastrow parameters are optimized separately for each
structure including the different sized supercells used in the finite size
extrapolations. Otherwise, all the settings in the DMC and VMC calculations
are uniform across all the calculations, including the number of cores and
number of walkers used. In this section we analyze the computational cost
associated with Jastrow sharing. We can see from our results in Table 1 that
the VMC and DMC energies are converging to the appropriate ground states with
comparable localization error using the same VMC and DMC parameters on the
same number of nodes/cores, which demonstrates that Jastrow sharing for
GaSxSe1-x is a valid approach. Therefore, we report the computational time for
these two procedures.

Before proceeding any further, we provide details of our computational
environment. All of the VMC and DMC simulations are conducted on 8 nodes of
our local cluster. Each node is connected via OmniPath networking and contains
2 Intel 20-Core Xeon Gold 6138 CPUs clocked at 2GHz and 384 GB of 2666MHz DDR4
memory. QMCPack version 3.9.0 is compiled using OpenMPI version 3.0.1 and
Intel Fortran, C and C++ (version 19.0.4.243) compiler.

To quantitatively investigate the computational cost savings of Jastrow
sharing, we report the computational time of VMC and DMC (total time including
every twist) in Table 3. We report the computational time obtained with
parameters we used in our QMC simulations (upper bound) and calculate a lower
bound, which is obtained by removing the number of extra unnecessary steps
after a target uncertainty was reached ($\sim$10 meV/f.u. for VMC, $\sim$5
meV/f.u. for DMC). We used the 36 atom supercell of GaS0.333Se0.667 as an
example of a single supercell (second column of Table 3). As seen from Table 3
(second column), the VMC time associated with optimizing the Jastrow
parameters of the smallest WS structure is much lower due to the fact that the
WS structure only has 16 atoms (while the other supercell has 36 atoms).

To report total cost savings for all of the calculations performed in this
paper, we give the VMC and DMC time needed to achieve the extrapolated results
(using x = 0.333 as an example, see third column of Table 3). We also report
the computational time to achieve the extrapolated results for the four
considered structures (Janus, x = 0.5, x = 0.333, and x = 0.667) to get an
estimate of the cost savings for constructing the energy hull diagram (see
Fig. 4). Since the DMC extrapolated results using the WS-Jastrow only requires
one VMC optimization simulation (extra VMC calculations are avoided), the VMC
time for the extrapolation is the same as the single supercell (third column
of Table 3). The substantial savings of constructing the hull diagram (at the
VMC level) is indicated in the fourth column of Table 3, where the VMC
computational time of using WS-Jastrows is two orders of magnitude smaller
than using self-Jastrows.

From Table 3, we observe that using either the self-Jastrows or the WS-Jastrow
does not change the DMC computational time significantly, which means that
Jastrow-sharing does not increase cost at the DMC level in our case.
Specifically, each set of Jastrow parameters are of comparable quality and do
not increase the uncertainty or amount of DMC steps needed to reach the target
uncertainty for each calculation. It is clear from this cost analysis that by
sharing Jastrow parameters, it is possible to achieve substantial savings by
reducing the VMC computational time. The cost of VMC calculations is often
disregarded when compared to DMC cost, but in our work we observe that when
self-Jastrows are used, VMC accounts for roughly 1/4 (on average) of the total
cost. Therefore by using the WS-Jastrow approach, the high-throughput QMC
calculations of alloy systems can be significantly less costly.

##  IV Conclusion

We have outlined a high-throughput procedure to calculate the energy hull
diagram for an alloy system using QMC methods. This involves optimizing the
Jastrow parameters of the alloy with the smallest WS radius, and using these
optimized parameters for all of the other subsequent structures. We tested the
validity of this Jastrow-sharing procedure on monolayer GaSxSe1-x by swapping
the Jastrow parameters of structures with different stoichiometries and
tabulated the Jastrow sensitivities. Finally, we quantified the substantial
computational cost savings obtained from avoiding extra VMC optimization
simulations. We hope this method can be implemented for other systems where
the Jastrow sensitivities of the subsequent pseudopotentials are low, which
can aid in future accurate and high-throughput studies for alloys.

## Supplementary Material

See the supplementary information for additional details and convergence tests
of DFT and DMC calculations, convergence criteria for the SQS and Cluster
Expansion, DFT and DMC details about the optimal geometries of each structure,
testing of Jastrow-sharing at the VMC level, and details about finite-size
extrapolation.

###### Acknowledgements.

This work was supported by the National Science Foundation through the
Division of Materials Research under NSF DMR-1726213.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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