QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion Quantum Monte Carlo

We review recent advances in the capabilities of the open source ab initio Quantum Monte Carlo (QMC) package QMCPACK and the workflow tool Nexus used for greater efficiency and reproducibility. The auxiliary field QMC (AFQMC) implementation has been greatly expanded to include k-point symmetries, tensor-hypercontraction, and accelerated graphical processing unit (GPU) support. These scaling and memory reductions greatly increase the number of orbitals that can practically be included in AFQMC calculations, increasing accuracy. Advances in real space methods include techniques for accurate computation of band gaps and for systematically improving the nodal surface of ground state wavefunctions. Results of these calculations can be used to validate application of more approximate electronic structure methods including GW and density functional based techniques. To provide an improved foundation for these calculations we utilize a new set of correlation-consistent effective core potentials (pseudopotentials) that are more accurate than previous sets; these can also be applied in quantum-chemical and other many-body applications, not only QMC. These advances increase the efficiency, accuracy, and range of properties that can be studied in both molecules and materials with QMC and QMCPACK.

We review recent advances in the capabilities of the open source ab initio Quantum Monte Carlo (QMC) package QMCPACK and the workflow tool Nexus used for greater efficiency and reproducibility. The auxiliary field QMC (AFQMC) implementation has been greatly expanded to include k-point symmetries, tensorhypercontraction, and accelerated graphical processing unit (GPU) support. These scaling and memory reductions greatly increase the number of orbitals that can practically be included in AFQMC calculations, increasing accuracy. Advances in real space methods include techniques for accurate computation of band gaps and for systematically improving the nodal surface of ground state wavefunctions. Results of these calculations can be used to validate application of more approximate electronic structure methods including GW and density functional based techniques.
To provide an improved foundation for these calculations we utilize a new set of correlation-consistent effective core potentials (pseudopotentials) that are more accurate than previous sets; these can also be applied in quantum-chemical and other many-body applications, not only QMC. These advances increase the efficiency, accuracy, and range of properties that can be studied in both molecules and materials with QMC and QMCPACK. a) kentpr@ornl.gov

I. INTRODUCTION
Quantum Monte Carlo (QMC) methods are an attractive approach for accurately computing and analyzing solutions of the Schrödinger equation. [1][2][3] The methods form a general ab initio methodology able to solve the quantum many-body problem, applicable to idealized models such as chains or lattices of atoms through to complex and low-symmetry molecular and condensed matter systems, whether finite or periodic, metallic or insulating, and with weak to strong electronic correlations. Significantly, the methods can naturally treat systems with significant multi-reference character, and are without electron self-interaction error, which challenges many quantum chemical approaches and density functional theory approximations, respectively. The methods continue to be able to take advantage of improvements in computational power, giving reduced time to solution with new generations of computing. Due to these features, usage of QMC methods for first principles and ab initio calculations is growing.
Compared to traditional deterministic approaches, QMC methods are generally distinguished by: (1) use of statistical methodologies to solve the Schrödinger equation. This allows the methods to not only treat problems of high-dimensionality efficiently, but also potentially use basis, wave function, and integral forms that are not amenable to numerical integration. (2) Use of few and well-identified approximations that can potentially be quantified or made systematically convergeable. (3) A low power scaling with system-size, but large computational cost prefactor. (4) High suitability to large scale parallel computing owing to lower communications requirements than conventional electronic structure methods.
Scaling has been demonstrated to millions of compute cores. 4 Modern applications of QMC have expanded to cover many of the same systems studied by density functional theory (DFT) and quantum chemical approaches, and in many cases also at a similar atom and electron count, although at far greater computational cost. Besides those described in below, recent molecular applications of QMC include studies of the nature of the quadruple bond in C 2 5 , acenes 6 , physisorption of water on graphene 7 , binding of transition metal diatomics 8 , and DNA stacking energies 9 . Materials applications include nitrogen defects in ZnO 10 , excitations in Mn doped phosphors 11 , and the singlet-triplet excitation in MgTi 2 O 4 12 . Methodological improvements include: reducing the sensitivity of pseudopotential evaluation 13 , extensions to include linear response 14 , density functional embedding 15 , excited states including geometry optimization 16 , improved twist averaging 17 , and accurate trial wavefunctions via accurate densities 18 . Importantly, for model systems such as the hydrogen chain, the methods can be used to benchmark themselves as well as other many-body approaches 19 . This partial list of developments and applications from the last two years alone indicates that the field is growing and maturing.
In this article we describe recent updates to the QMCPACK code and its ecosystem which also entails major updates to the testing, validation and maintainability. complexity of the overall calculation process. All of these contribute to the effort required by the user to obtain desired results. In essence, higher required effort translates directly into lower productivity of the user base. Lower productivity in turn risks a lower overall adoption rate and thus blunts overall impact of the method. It is therefore important to seek to understand and minimize barriers to the practical use of QMC.
To illustrate the complexity of the QMC calculation process, we describe below a basic but realistic sequence of calculations (a scientific workflow) that is required to obtain a final fixed node DMC total energy per formula unit for a single crystalline solid with QMCPACK.
In this workflow, we suppose that self-consistent (SCF) and non-self-consistent (NSCF) calculations are performed with Quantum Espresso 24 and wavefunction optimization (OPT), Scientific workflow tools make the QMC process more accessible in multiple ways: (1) bringing the constellation of electronic structure codes needed to produce a single QMC result under a single framework, (2) reducing the number of inputs required to request a desired result to a single user-facing input file, (3) reducing overall complexity by abstracting the execution process, (4) minimizing the direct effort required to execute the workflow process by assuming the management of simulation execution and monitoring from the user.
Workflow tools have been applied with significant benefit to related electronic structure methods such as DFT [28][29][30][31] and also to QMC 32,33 .
The Nexus workflow automation system 34 was created to realize these advantages for users of QMCPACK. Nexus is a Python-based, object oriented workflow system that can be run on a range of target architectures. Nexus has been used successfully on simple workstations and laptops, small group or institutional computing clusters, university level high performance computing centers in the U.S. and internationally, and Leadership Computing Facilities supported by the U.S. Department of Energy. Nexus has been used in a growing number of QMC studies involving QMCPACK and its uptake by new users is high.
Nexus abstracts user's interactions with each target simulation code that are components of a desired simulation workflow. Access to each respective code is enabled through single function calls that only require the user to specify a reduced set of important input parameters. Each function call resembles a small input block from a standard input file for an electronic structure code. Taken together, a sequence of these blocks comprises a new metainput file that represents the data flow and execution pattern of the underlying simulation codes as a combined workflow.
Nexus assumes the responsibility of initiating and monitoring the progress of each simulation job in the workflow. Nexus generates expanded input files to each code based on the reduced inputs provided by the user. It also generates job submission files and monitors job execution progress via a lightweight polling mechanism. Apart from direct execution of each workflow step, Nexus also automates some tasks that previously fell to users. One example is that Nexus selects the best wavefunction produced during the non-linear statistical optimization process employed by QMCPACK and automatically passes this wavefunction to other calculations (such as diffusion Monte Carlo), which require it.
In the future, additional productivity gains might be realized with Nexus by further abstracting common workflow patterns. For example, convergence studies for orbital param-eters (k-points, mesh-factors, source DFT functional) often follow similar patterns which could be encapsulated as simple components for users. Additionally, more of the responsibility for obtaining desired results, e.g. total energies to a statistically requested tolerance, could be handled by Nexus through algorithms that create and monitor dynamic workflows.

IV. EFFECTIVE CORE POTENTIALS
A. Introduction  to the other ECPs. In addition, Fig. 1 shows these improvements to be consistent for different elements and varying geometries. Hence, we believe that ccECP accomplishes the best accuracy compromise for atomic spectral and molecular properties. Furthermore, ccECPs are provided with smaller cores than conventionally used ones in some cases where large errors were observed. This includes Na-Ar with [He] core and H-Be with softened/canceled Coulomb singularity at the origin (ccECP(reg)). Selected molecular test results for these are shown in Figure 3.
For reference, we also provide accurate total and kinetic energies for all ccECPs 46 using methods such as CCSDT(Q)/FCI (FCI, full configuration interaction) with DZ-6Z extrapolations to estimate the complete basis set limit. This data, for instance, is useful in the assessment of fixed-node DMC biases. Figure 2b shows the summary of single-reference (HF) fixed-node DMC errors for ccECP pseudo atoms.

C. ccECP Database and Website
In order to facilitate the use ccECPs, we have provided basis sets and a variety of ECP formats available at https://pseudopotentiallibrary.org, shown in    Bond Length (Å) ccECP [He] (b) SiO binding curve discrepancies Molpro, GAMESS, NWChem, and PySCF which uses the NWChem format. We also provide an XML format which can directly be used in QMCPACK.
In addition to the ccECPs themselves, we have also provided basis sets appropriate for

V. AUXILIARY FIELD QUANTUM MONTE CARLO
The latest version of QMCPACK offers a now mature implementation of the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method 20,52 capable of simulating both molecular 53,54 and solid state systems [55][56][57] . AFQMC is usually formulated as an orbital-space approach in which the Hamiltonian is represented in second-quantized form aŝ whereĉ † i andĉ i are the fermionic creation and annihilation operators, h ij and v ijkl are the oneand two-electron matrix elements and E II is the ion-ion repulsion energy. Key to an efficient implementation of AFQMC is the factorization of the 4-index electron-repulsion integral (ERI) tensor v ijkl , which is essential for the Hubbard-Stratonovich (HS) transformation 58,59 .
QMCPACK offers three factorization approaches which are appropriate in different settings. The most generic approach implemented is based on the modified-Cholesky factorization 60-64 of the ERI tensor: where the sum is truncated at N chol = x c M , x c is typically between 5 and 10, M is the number of basis functions and we have assumed that the single-particle orbitals are in general complex. The storage requirement is thus naively O(M 3 ) although sparsity can often be exploited to keep the storage overhead manageable (see Table II). Note that QMCPACK can accept any 3-index tensor of the form of L n ik so that alternative density-fitting based approaches can be used. Although the above approach is efficient for moderately sized molecular and solid-state systems, it is typically best suited to simulating systems with fewer than 2000 basis functions.
To reduce the memory overhead of storing the three-index tensor we recently adapted the tensor-hypercontraction 65-67 (THC) approach for use in AFQMC 56 . Within the THC approach we can approximate the orbital products entering the ERIs as where ϕ i (r) are the one-electron orbitals and r µ are a set of specially selected interpolating points, ζ µ (r) are a set of interpolating vectors and N µ = x µ M . We can then write the ERI tensor as a product of rank-2 tensors where To determine the interpolating points and vectors we use the interpolative separable density fitting (ISDF) approach [68][69][70] . Note that the storage requirement has been reduced to O(M 2 ).
For smaller system sizes the three-index approach is preferred due to the typically larger THC prefactors determined by x µ ≈ 15 for propagation and x µ ≈ 10 for the local energy evaluation. The THC approach is best suited to simulating large supercells, and is also easily ported to GPU architectures due to its smaller memory footprint and use of dense linear algebra. Although the THC-AFQMC approach has so far only been used to simulate periodic systems, it is also readily capable of simulating large molecular systems using the advances from Ref. 71.
Finally, we have implemented an explicitly k-point dependent factorization for periodic systems 72 where now i runs over the number of basis functions (m) for k-point k i in the primitive cell,  Table. II), perhaps the most significant advantage is that it permits the use of batched dense linear algebra and is thus highly efficient on GPU architectures. Note that the THC and k-point symmetric factorization can be combined to simulate larger unit cells and exploit k-point symmetry, however this has not been used to date. We compare the three approaches in Table II and provide guidance for their best use.
In addition to state-of-the-art integral factorization techniques, QMCPACK also permits the use of multi-determinant trial wavefunction expansions of the form   respectively. This approach essentially removes the memory overhead associated with storing the ERIs at the cost of using a potentially very large plane-wave basis set 73,74 . This plane wave approach is not yet available in QMCPACK.
determinants than their orthogonal counterparts to achieve convergence in the AFQMC total energy 53 (see Fig. 5).
QMCPACK also permits the evaluation of expectation values of operators which do not commute with the Hamiltonian using the back propagation method 59,81,82 . In particular, the back-propagated one-particle reduced density matrix (1RDM) as well components or contracted forms of the two-particle reduced density matrix are available. As an example we plot in Fig. 6 the natural orbital occupation numbers computed from the back-propagated phaseless AFQMC 1RDM.
Tools to generate the one-and two-electron integrals and trial wavefunctions for molecular and solid state systems are also provided through the afqmctools package distributed with QMCPACK. To date these tools are mostly dependent on the PySCF software package 23 , however we provide conversion scripts for FCIDUMP formatted integrals, as well as simple calculation, wavefunction conversion, and AFQMC calculation. We are using this integration to perform a study of the relative strengths of AFQMC and real space QMC methods.
Over the next year we plan to extend the list of observables available as well as complete GPU ports for all factorization and wavefunction combinations. In addition we plan to implement the finite temperature AFQMC algorithm [84][85][86][87][88] , and spin-orbit Hamiltonians with non-collinear wavefunctions. We will also release our ISDF-THC factorization tools and our interface to Quantum Espresso 24 . We hope our open-source effort will enable the wider use of AFQMC in a variety of challenging settings.

CALCULATIONS
The key factor in reaching high accuracy using QMC is the choice of trial wave function Reaching systematic convergence of the trial wavefunction and its nodal surface for general systems has been a key challenge for real space QMC methods since their invention.
Besides increasing accuracy in calculated properties, this is also required to remove the start- The most straightforward method to improve the quality of the trial wavefunction nodes in a convergeable manner is to increase its complexity via a multi-Slater determinant (MSD) or configuration-interaction (CI) expansion: where Ψ T is expanded in a weighted (c i ) sum of products of up and down spin determinants D i , and J is the Jastrow correlation factor. In the limit of a full configuration interaction calculation in a sufficiently large and complete basis set, this wavefunction is able to represent the exact wavefunction. However, direct application of configuration interaction is prohibitively costly for all but the smallest systems, because very large numbers of determinants are usually required. To speed application, an efficient selection procedure for the determinants is needed. This can be combined with efficient algorithms for evaluating the wavefunction in QMC. 95 For systems where CIPSI can be fully converged to the FCI limit and reliably extrapolated to the basis set limit, QMC is not required, but for any reasonable number of electrons, QMC can be used to further improve the convergence. The wavefunctions produced from CIPSI can be used either directly, in which case the nodal error is determined by the CIPSI procedure, or used to provide an initial selection of determinants whose coefficients are subsequently reoptimized in presence of a Jastrow function, or used within DMC where the projection procedure will improve on the CIPSI wavefunction. This procedure is equally applicable to solids as well as molecules, provided k-points and their symmetries are fully implemented.
In the following, we illustrate these techniques by application to molecular and solid-state

C. Solid-state Lithium Fluoride
Solid LiF is a face centered cubic material with a large gap, used mainly in electrolysis for his role in facilitating the formation of an Li-C-F interface on the carbon electrodes 107 . The purpose of this example is to demonstrate basis set effect and the systematic convergence of DMC energy with the number of determinants (a paper demonstrating convergence to the thermodynamic limit is in preparation). We simulated a cell of (LiF ) 2 (4 atoms per cell) at the Gamma point using correlation consistent electron-core potentials (ccECP)Bennett For such a small simulation cell, it is possible to convergence the sCI wavefunction to the FCI limit with a reasonable number of determinants, as can be seen in Fig-9 be systematically converged to give in-principle exact results.
As discussed above, convergent wavefunctions and energies can be constructed using sCI techniques. However, even for small primitive cells with relatively uncorrelated electronic structures, this approach quickly requires millions of determinants making it expensive to apply today. We have developed theories, methods and implementations to obtain the band-edge wavefunctions around the fundamental gap and their relative energies efficiently and to a high accuracy. Error cancelation is built into the methodology so that simpler trial wavefunctions are effective and the scheme is substantially more efficient to apply.
Surprisingly, for the systems examined so far, only single and double excitations need be considered to obtain accurate band gaps, even using the simple VMC method. This makes the technique comparatively cheap to apply.
To compute the optical band gaps of insulators and semi-conductors we use the energy difference of optimized wave functions that describe the valence band maximum (VBM) and the conduction band minimum (CBM). Optimizations use the recently developed excited state variational principle, [108][109][110] whose global minimum is not the ground state but the eigenstate with energy immediately above the chosen value ω, which could be placed within the band gap to target the first  Figure 10 shows that the predicted optical gaps are in excellent agreements with experimental values and the mean absolute deviation (MAD) is just 3.5%, compared to MADs more than twice this large for the optical gaps obtained by subtracting the known exciton binding energy from G 0 W 0 and self-consistent GW gaps. These calculations were able to run of departmental level computing and did not require supercomputers due to the use of VMC and moderate sized supercells utilized.
Remaining errors that have yet to be investigated include the size of the CI expansion and the limited correlation energy obtained with VMC, the pseudopotentials, and residual finite size error. DMC may further improve the VMC results.
In order to further show the method's advantage, we performed a thorough analysis of zinc oxide, a material that is particularly challenging for MBPT. 115 As shown in Figure 10, the perturbative nature of G 0 W 0 makes its prediction highly sensitive to the amount of exact exchange included in the DFT reference. As G 0 W 0 assumes a zeroth-order picture, in which electronic excitations are simple particle-hole transitions between the one-particle eigenstates of DFT, such a sensitivity indicates the break down of this assumption and then G 0 W 0 becomes unreliable. On the other hand, our VMC approach is designed to be insensitive to the DFT choice for two reasons: (1) its ability to include approximate orbital relaxation counteracts the shortcomings of the DFT orbitals. (2) unlike G 0 W 0 it does not require orbital energies as input. From Figure 10 we do find that its prediction to optical gap is both accurate and independent to the choice of DFT functionals.

E. Summary
Using DMC as a post-sCI method is very promising to systematically improve molecular and solid-state calculations beyond the single-determinant picture. It converges faster the sCI methods used on their own. From a practical perspective, PySCF, Quantum Package and QMCPACK are fully interfaced with each other through the Nexus workflow automation system. The necessary multi-step workflow to run the above examples is fully implemented.
In the case of solids, Nexus can automatically manage finite-size scaling calculations by setting the size of the super-cells, the number of twists angles, and drive PySCF, Quantum Package, and QMCPACK appropriately and automatically.

VII. APPLICATIONS
To illustrate the application of recent developments in QMCPACK, in Section VII A we give an example of using real space QMC to study non-valence anions, which are particularly challenging systems. Section VII B gives an example of computing the excited states of localized defects, which is a challenge for all electronic structure methods. In Section VII C we give an example of computing the momentum-distribution and Compton profile from real space methods. The necessary estimators have recently been specifically optimized for these tasks.

A. Applications of DMC to Non-valence Anions
In the aug-cc-pVDZ+7s7p basis set is 38meV smaller than the DMC EBE obtained using the trial wave function employing B3LYP orbitals. This suggests that the EBE from EOM calculations may not be fully converged in these large basis sets. Overall, these results demonstrate that DMC is a viable approach for the characterization of NVCB anions.

B. Excitation energies of localized defects
For defects and interfaces, most ab-initio methods can only achieve qualitative agreement on the optical properties. We have recently studied emission energies of Mn 4+ -doped solids using DMC, which is chosen as proof of principle 11 . We show that our approach is applicable to similar systems, provided that the excitation is sufficiently localized. In support of this configuration all on the t 2g orbitals. The ground state is in t ↑↑↑ 2g ( 4 A 2g ) configuration due to Hund's rules, but the excited state is found to be as t ↑↑↓ 2g ( 2 E g ) 135 . Therefore, the emission energy is simply defined as E em = E( 2 E g ) − E( 4 A 2g ). In Fig. 12(b), we show the spin flipped electrons per radial volume n(r) = ρ ↑ ground − ρ ↑ excited which is spherically integrated around the Mn 4+ atom. n(r) approaches to zero with increasing radius indicating that excited electron density is strongly localized on the impurity atom.
DMC and LDA+U n(r) densities are almost identical to each other despite the large difference in their emission energies which underscore the difficulties that needed to be overcome for better DFT functionals 139 .
C. Calculation of the many-body properties: the momentum distribution As full-many body methods, QMC can be used to calculate many-body properties that can not be readily obtained from single-particle or mean-field techniques. We have recently updated and optimized calculation of the momentum distribution.
In general, the differential cross sections of the scattering can be related to the momentum distribution. These experimental techniques are powerful probes for understanding subtle details in the ground state properties of materials, which are manifested in the MDF.
In normal Fermi liquids, the electron MDF has a discontinuity at the Fermi momentum p F . In three-dimensional systems this discontinuity defines the shape of the Fermi surface, which is also related to the screening properties of the electrons 144 . The Fermi surface can be extracted from the p-space MDF via back-folding 145 . This leads to occupation density within the first Brillouin zone from which the Fermi surface topology can be considered 146 .
The magnitude of the discontinuity at the Fermi surface, however, quantifies the strength of a quasiparticle excitation and is generally referred to as the renormalization factor 147,148 .
For strongly coupled systems the renormalization factor tends to zero as the coupling strength increases, and thus, it provides an estimate for the strength of electron correlations. Interestingly, the discontinuity at the Fermi momentum disappears for superfluids or superconducting materials. For insulators the discontinuity is absent and the sharp drop is noticeably broadened, which also holds true for some semi-metals 149,150 . Even small scale charge density oscillations lead to clear signatures in the MDF 151 . Therefore, the momentum distribution function provides complementary and informative knowledge to other characterizations of many-body systems.
In variational Monte Carlo this is expressed as where R = {r 1 + s, . . . , r Ne }. Thus, we get for the MDF: In practice, the Monte Carlo estimate for the MDF with N s samples is given by where R includes the coordinates of all the electrons, and s i j is a displacement vector acting on the j th electron of the i th sample. In diffusion Monte Carlo calculations the mixed distribution replaces |Ψ(R)| 2 , and additional measures must be taken to calculate or estimate the density matrix. Notice that the momentum distribution normalizes to the number of electrons p n(p) = N e = Ω (2π) d dp n(p), in which d refers to dimensionality. In Eq. (12) a finite system and a system at the thermo- term, the evaluation of single particle orbitals at r j + s i j dominates the cost. In fact, its call count can be reduced from N s N e to N s by making where r i is the electron coordinates of sample i. Although the leading cost of is optimized away, its remaining terms still scale as O(N s N 2 e ) and the computational cost should be comparable to the non-local pseudopotential calculation.
Unlike the wavefunction ratios which needs to be computed only once for all the p, phase factors are computed for each p and also take significant portion of time. Similarly, for each p, the number of evaluations can be reduced to N s + N e times by separating indices i and j in two terms like Eq. (13). The calculation of e −ip·r i and e −ip·r j can be efficiently vectorized using the single instruction multiple data (SIMD) unit in modern processors. By applying the above techniques and ensuring vectorization of all operations, the overhead for evaluating the MDF in a 48 atom cell VO 2 was reduced from additional 150% to only 50% cost increase compared to a DMC run without any estimators.
Within the so-called impulse approximation (IA) the Compton profile as well as the dynamical structure factor are proportional to the projection of n(p) onto a scattering vector 149,152 . In this case directional Compton profile in z-direction would be expressed as J(q) = Ω (2π) 3 n(p x , p y , p z = q)dp x dp y .
The IA is especially appropriate for X-ray Compton scattering from electronic systems 142,149 , and thus, it is capable of providing a unique perspective for understanding the electronic structure of materials; bulk properties, in particular.
In Ref. 151 QMCPACK was used in obtaining the MDFs and Compton profiles for VO 2 across its metal insulator transition from metallic rutile (R) phase to insulating monoclinic (M1) phase. There the analysis of the MDF shows signatures of the non-Fermi liquid character of the metallic phase of vanadium dioxide. Moreover, findings therein provide an explanation for the experimentally observed anomalously low electronic thermal conductivity 153 , which manifests as back scattering characteristics within the momentum distribution function. Fig. 13 shows some examples of MDF differences across the phase transition in two planes as well as for a few different directions 151 .

VIII. SUMMARY
We have described recent enhancements to the open source QMCPACK package. Besides increases in capability for both real space and auxiliary field Quantum Monte Carlo (QMC) methods, the surrounding ecosystem has also been improved. These enhancements include the workflow system Nexus, which aims to reduce the complexity of performing research studies and the tens to hundreds of individual calculations that might be entailed. A new set and open database of effective core potentials has also been established at https://pseudopotentiallibrary.org, and we expect that these will be of interest for other quantum chemical and many-body calculations due to their increased accuracy, including for stretched bonds and excited states. We have also described how improvements in open software development have benefitted the project. Besides the activities described in this article, we note that there is substantial ongoing work to enhance the architecture of QMCPACK for GPU accelerated machines and to obtain portable performance from a single code base. Once the new design is proven on diverse GPUs it will be described in a future article.
Overall the applicability of QMC continues to expand, it is becoming easier to apply, and there are many systems and phenomena where the higher accuracy and many-body nature of QMC is both warranted and can now be applied. We hope that this article will help encourage these new applications.

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