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[Kousuke Nakano](https://orcid.org/0000-0001-7756-4355), Sandro Sorella, Dario Alfè, Andrea Zen

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[Beyond Single-Reference Fixed-Node Approximation in <i>Ab Initio</i> Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons](https://mdr.nims.go.jp/datasets/924dfdb1-fd85-4190-8052-c994ba804132)

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Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of ElectronsBeyond Single-Reference Fixed-Node Approximation in Ab InitioDiffusion Monte Carlo Using Antisymmetrized Geminal PowerApplied to Systems with Hundreds of ElectronsKousuke Nakano,* Sandro Sorella, Dario Alfe,̀ and Andrea Zen*Cite This: https://doi.org/10.1021/acs.jctc.4c00139 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: Diffusion Monte Carlo (DMC) is an exact technique toproject out the ground state (GS) of a Hamiltonian. Since the GS is alwaysbosonic, in Fermionic systems, the projection needs to be carried out whileimposing antisymmetric constraints, which is a nondeterministic polynomialhard problem. In practice, therefore, the application of DMC on electronicstructure problems is made by employing the fixed-node (FN) approx-imation, consisting of performing DMC with the constraint of having a fixed,predefined nodal surface. How do we get the nodal surface? The typicalapproach, applied in systems having up to hundreds or even thousands ofelectrons, is to obtain the nodal surface from a preliminary mean-fieldapproach (typically, a density functional theory calculation) used to obtain asingle Slater determinant. This is known as single reference. In this paper, wepropose a new approach, applicable to systems as large as the C60 fullerene,which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz(antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-fieldapproach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing theFN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on severalexamples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicableto large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable andreliable within the proposed scheme.1. INTRODUCTIONAb initio electronic structure calculations, which compute theelectronic structure of materials nonempirically, have becomean essential methodology in the materials science andcondensed matter physics communities. Density functionaltheory (DFT), a mean-field approach which was originallyproposed by Kohn and Hohenberg,1 is the most widely usedmethodology for ab initio electronic structure calculations.DFT has enjoyed widespread success, despite its reliance onthe so-called exchange−correlation (XC) functional, whoseexact form is yet to be discovered. Although many XCs havebeen proposed, no functional that performs universally well forall materials is established.Several methodologies transcend the mean-field paradigm.For example, in the quantum chemistry community, thecoupled cluster method with single, double, and perturbativetriple excitations,2 denoted as CCSD(T), is widely recognizedas the gold-standard approach, balancing accuracy andcomputational efficiency. This technique has been employedas a reference in many benchmark tests, both for isolated andperiodic systems.2−5 CCSD(T) is mostly applied in relativelysmall systems, as it becomes very computationally intensive asthe simulated systems get larger (hundreds of electrons ormore). Moreover, despite the many successes of CCSD(T),there are a few cases where CCSD(T) fails, mostly attributedto the multireference character of a chemical system (strongcorrelation) and where other methods, more expensivecomputationally, are needed.6 A different approach, adoptedby the condensed matter community as the gold standard, isthe diffusion Monte Carlo (DMC) method.7 DMC has goodscaling with the system size and it uses algorithms that can beparallelized with little or no efficiency loss, fully exploitingmodern supercomputers and making relatively large systemstreatable.CCSD(T) and DMC predictions typically show consensusin the computed physical properties, such as heats of formationand binding energies, and good agreement with experi-Received: February 1, 2024Revised: May 5, 2024Accepted: May 9, 2024Articlepubs.acs.org/JCTC© XXXX The Authors. Published byAmerican Chemical SocietyAhttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXThis article is licensed under CC-BY 4.0Downloaded via NATL INST FOR MATLS SCIENCE (NIMS) on May 28, 2024 at 01:39:29 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kousuke+Nakano"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Sandro+Sorella"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Dario+Alfe%CC%80"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Andrea+Zen"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.jctc.4c00139&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=tgr1&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://pubs.acs.org/JCTC?ref=pdfhttps://pubs.acs.org/JCTC?ref=pdfhttps://acsopenscience.org/researchers/open-access/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/ments.5,8−13 It was believed that CCSD(T) and DMC wouldalso agree on extended systems, but recent findings by Al-Hamdani et al.12 have unveiled discrepancies in binding energycalculations between these methods for large systems, such as aC60 buckyball inside a [6]-cycloparaphenyleneacetylene ring(C60@[6]CPPA). It is unclear which approach is to be trustedin these tricky cases. These findings raise a pivotal question:what is the reference approach for noncovalent interactionsbetween large systems? To answer this question, Al-Hamdaniet al.12 discussed possible discrepancy sources coming fromuncontrollable errors existing in both CCSD(T) and DMC.Both approaches employ some approximations and have theirweaknesses, and the debate is still open. To draw a moreconclusive determination, one should develop a scheme whichmitigates the impact of uncontrollable errors in the methods.In this work, we focus on improvements in the DMC approachthat alleviate its largest source of error: the fixed-node (FN)approximation.DMC yields the exact ground state (GS) in bosonic systems.In Fermionic systems (for instance, in electronic structurecalculations), DMC suffers from the so-called negative signproblem, arising from the fact that the Fermionic GS haspositive and negative regions. The negative sign problem in theDMC method for Fermions has been proven to be anondeterministic polynomial hard problem;14 thus, it seemsunrealistic to find a general solution at present. This problem isavoided, in practice, by modifying the DMC projectionalgorithm with the introduction of the FN approximation,where the projected wave function ΦFN is constrained to havethe nodes of a predetermined guiding function ΨT. The FNapproximation keeps the projected wave function ΦFNantisymmetric, but ΦFN is the exact GS Φ0 only if its nodesare exact. A general property of ΦFN is that it is always theclosest function to Φ0 within the FN constraint. For trialfunctions obtained from mean-field approaches, such asHartree−Fock (HF) or DFT, it is generally believed that theerror associated with the FN approximation is small andbenefits from a large error cancellation in the evaluation ofbinding energies.8 However, the FN error is typically notaccessible, as Φ0 is unknown, and this yields an uncontrollableerror in FN-DMC.In standard FN-DMC simulations, the nodal surface is givenby an approximate wave function, which is typically obtainedstarting from a mean-field approach, such as HF or DFT. Thevariational principle can still be applied to FN-DMC,a and soto go beyond the mean-field solution, one should optimize thegiven nodal surface by minimizing the FN-DMC energy EFN(which is the expectation value of ΦFN), going in the directionof the exact wave function Φ0 and the exact energy E0. Thisprocedure is seldomly followed in DMC simulations, especiallyon large systems (say, with hundreds or thousands ofelectrons), as it is hardly affordable computationally and theuncertainty on the optimization of the FN surface could beeasily comparable, if not larger, than the binding energy underconsideration. Thus, the standard approach is to just keep thenodal surface of the Slater determinant (SD) built with theKohn−Sham orbitals obtained from a DFT calculation. Whilethe FN surface from DFT might be suboptimal, this approachtypically yields quite reliable results, especially in theevaluations of noncovalent interactions, due to very favorableerror cancellations.5,8In smaller systems (with say, tens of electrons), it is possibleto improve the nodal surface, and the most standard approachis to use an ansatz that has more degrees of freedom than theinitial SD, such as the antisymmetrized geminal power(AGP),19−21 the Pfaffian,22−24 the complete active space,25,26the valence bond,27,28 the backflow,23,29,30 and multideter-minant expansions,31−38 including methods employing neuralnetworks and machine learning techniques.39−45 The standardapproach here is to optimize the wave function parameters atthe level of theory of variational Monte Carlo (VMC).19,46−50Indeed, optimization at the FN-DMC (FN-opt) level impliesfurther difficulties, as we will discuss below. However,optimization at the VMC (VMC-opt) level has some flaws.In VMC-opt, the object that is optimized is the variationalwave function ΨT, which is obtained from the product of oneof the ansatzes discussed above and the Jastrow factor.b Thecloser ΨT gets to the GS Φ0, the smaller its VMC energy(variational principle) and its VMC variance (zero-varianceproperty) are. VMC-opt explores the parameters’ variationalspace, seeking the set which minimizes the VMC energy or theVMC variance, and it is often done by employing the VMCgradient. It is not guaranteed that the parameters obtainedfrom VMC-opt are those giving the best possible nodal surfaceallowed by the employed ansatz (unless we are in the limit casewhere ΨT yields VMC with zero variance, such that we knowthat ΨT is an eigenstate of the Hamiltonian). Although thisapproach, in practice, gives a better nodal surface than theDFT one, it sometimes gives unreasonable outcomes, e.g., itoverestimates binding energies, as revealed in this work. Itwould be desirable, instead, to implement an optimization atthe FN-DMC level of theory, where the parameters of thefunction ΨT giving the nodal surface are optimized so as tominimize the FN energy. This would guarantee to find the bestnodal surface allowed by the adopted wave function ansatz. Tothe best of our knowledge, the first attempt to directly optimizethe variational parameters included in a trial wave function atthe FN-DMC level was done by Reboredo et al.51 in the abinitio framework. They proposed a way to iteratively generatenew trial wave functions to get a better nodal surface. Theygeneralized the method to excited states52 and finite temper-atures53 and also applied for large systems such as C20.54 Veryrecently, McFarland and Manousakis55 reported successfulenergy minimizations with approximated and exact FNgradients. They proposed to optimize nodes using acombination of FN gradients and the projected gradientdescent (PGD) method. The PGD method works for Be, Li2,and Ne using all-electron DMC calculations,55 while it hasbeen successful only for small molecules.When it comes to optimizing the nodal surface of a largesystem, the main problem is that the number of variationalparameters determining the nodal surface often scales morethan linearly with the size of the system. For instance, thenumber of variational parameters in the determinant part ofthe AGP ansatz scales with O(L2), where L is the number ofbasis functions in a system. It makes the parameter space to beoptimized so complex that the optimization is easily trapped inlocal minima and one cannot find the true GS. Moreover, sincethe optimization algorithms are stochastic, there is always anadditional uncertainty on the optimized parameters, which arenot going to be exact and the corresponding DMC energy hastherefore an optimization bias. The optimization bias increaseswith the system size and with the number of variationalparameters and can be reduced only at the cost of increasingthe statistical sampling (and the computational cost). Theevaluation of binding energy implies the difference betweenJournal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXBpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-astwo or more DMC energies, and it is often a tiny fraction ofthe total energy. Therefore, the optimization uncertainty canoften be comparable to the binding energy, making theevaluation of the interaction energy unreliable. Moreover, weneed to verify that the adopted approach satisfies basic physicalproperties, such as being size-consistent.c At the VMC level,the size consistency is a property of the wave function ansatzemployed, and it depends on the optimization procedure. Atthe FN-DMC level, size consistency might depend on somechoices on the algorithm,56 on the ansatz of the wave functionproviding the FN constraint, and on the optimization.In this paper, we propose a scheme which aims to addressthese issues. In particular, our scheme satisfies the followingpoints: (i) it is systematically more accurate than the standardapproach of employing a single SD, (ii) it is size-consistent,and (iii) it is applicable also to large systems. The ideaunderlying the present work is the combination of the AGPwave function consisting of molecular orbitals (MOs), dubbedAGPn,21 the use of natural orbitals (NOs), and theoptimization of its nodal surface using FN gradients on aselected subset of the AGPn parameters. In particular, weinitialize the orbitals in the AGPn wave function using NOs,which are kept fixed afterward, such that only the coefficientscombining them are optimized to relax the nodal surface. Wecall this scheme the fixed node antisymmetrized geminal poweractive space (FNAGPAS). Since the orbitals are fixed, thisresults in a much smaller number of variational parameters inthe ansatz; thus, one can apply it for larger systems, such as C60fullerene. We show that our scheme gives a better nodalsurface (i.e., a lower energy in the FN-DMC calculation)compared to the typical Slater−Jastrow ansatz, and it reliablydescribes also strongly correlated systems (such as diradicals).We show that the use of FN-opt is important to fulfill the size-consistency property.2. FNAGPAS SCHEMEWe describe here the scheme that we suggest to improve theaccuracy of FN-DMC over the traditional single-determinantSlater−Jastrow ansatz. The key idea is the combination of theAGPn,21 which is the AGP wavefunction constraint with MOs,and the optimization of the ansatz using approximated FNgradients.55 We describe the ansatz in the following section,assuming an unpolarized system for simplicity. The schematicillustration explaining the key concept and its workflow isshown in Figure 1.The real-space quantum Monte Carlo (QMC) typicallyemploys a many-body wave function ansatz Ψ written as theproduct of two terms, ΨQMC = ΦAS × exp J. The term exp J,conventionally dubbed Jastrow factor, is symmetric, and theterm ΦAS is antisymmetric. The Jastrow factor is explicitlydependent on electron−electron distance and often includeselectron−nucleus and electron−electron−nucleus terms.d Thenodal surface of a wave function is determined by theantisymmetric part ΦAS (because exp J ≥ 0). Thus, in FN-DMC, the accuracy of the results depends crucially on thequality of the nodes of ΦAS.The antisymmetric part of a trial wave function is initiallyconstructed from a mean-field self-consistent-field (SCF)approach, such as DFT or HF. The standard QMC setup inlarge systems is to define ΦAS as the single SD obtained fromsuch preliminary SCF calculations. The corresponding ΨQMC isdubbed JSD. Therefore, the nodes of JSD are predefinedbefore any QMC calculation and unrelaxed. Initializing the SDusing different setups for the SCF calculations (e.g., differentexchange−correlation functionals) leads to slightly differenttotal energies, but most of the times, the interaction energies(which are evaluated from energy differences between two ormore systems) are almost unaffected by the details of thepreliminary SCF calculation, especially for weak noncovalentinteractions. This is an indication that there is an almostperfect cancellation of the error induced by the FNFigure 1. Panel a: Schematic illustration of the FNAGPAS scheme. We perform a preliminary mean-field calculation to obtain MOs, followed by acorrelated calculation yielding NOs. The AGPn ansatz corresponds to a multideterminant expansion built on the NOs and depending on thecoefficients λi associated with each orbital i and optimized in order to minimize the FN energy. Panel b: Flowchart illustrating the FNAGPASscheme workflow.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXChttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig1&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asapproximation within the JSD ansatz, provided that the SD isinitialized consistently in all systems.However, changing the setup of the SCF calculation onlyallows the nodes to move within the variational freedom of asingle SD. By contrast, giving ΦAS the variational freedom torelax the nodes beyond the JSD ansatz leads to animprovement of the FN-DMC total energy of the system,58and possibly, also the interaction energies could change. Thechallenge that we take here is to generalize the ansatz in a waythat large systems are still doable.Here, we suggest to use the AGP ansatz as ΦAS. AGP is animplicitly multideterminant ansatz,56,59 which corresponds to aconstrained zero-seniority expansion, as illustrated schemati-cally in Figure 1. The evaluation of an AGP function can bereduced to the computation of a determinant; therefore, theAGP ansatz is computationally comparable to an SD(differently from explicitly multideterminant functions), thusensuring the cubic scaling with the system size of both thevariational and FN algorithms.e The AGP ansatz for a systemof Nel electrons is= [ ]g g gx x x x x x( , ) ( , )... ( , )N NAGP 1 2 3 4 1el el (1)(we are assuming for simplicity an unpolarized system with aneven number of electrons, but the ansatz can be generalized asdiscussed in ref 19), where is the antisymmetrizationoperator and the function g is the geminal function=g fx x r r( , ) ( , )1 2 1 2(1) (2) (1) (2)2, which is a pairing func-tion between two electrons with coordinates x1 and x2 forminga spin singlet. The spatial part f(r1, r2) is symmetric, and it canbe written in terms of a basis set {χμ} for the single-electronorbital space as follows:=f cr r r r( , ) ( ) ( )L L1 2 1 2(2)where μ and ν run over all the L basis orbitals and cμν arevariational parameters. Notice that in general, L ≫ N, and thenumber of variational parameters cμν is equal to L2. Theparameters define a L × L symmetric matrix C (the symmetryof f implies cμν = cνμ), so there is an orthogonal transformationU which diagonalizes C and allows rewriting f as=f r r r r( , ) ( ) ( )L1 2 1 2(3)where ϕμ = ∑νUμνχν. With no loss of generality, we canassume that λs are ranked in a decreasing order of theirabsolute value. Notice that if only the first Nel/2 λs aredifferent from zero, then ΨAGP corresponds to a single SD builton the orbitals , ..., N1 /2eloccupied with both spin-up andspin-down electrons. Since such an SD built on orbitals froman SCF calculation is the standard QMC setup, and it typicallydelivers good results, we tried to relax the nodes by consideringa subset norb (larger than Nel/2 but ≪ L) of the orbitalsobtained from the SCF calculation. This is what we call theAGPn ansatz.For efficient and effective use in QMC, the AGP and AGPnfunctions shall be multiplied by a Jastrow factor, yielding theso-called JAGP and JAGPn functions. The Jastrow factor canhave the same variational form used also in JSD, which allowsfor the JSD, JAGP, and JAGPn functions to satisfy the cuspconditions and to effectively recover the dynamical correla-tions. Indeed, the main improvement of JAGP and JAGPn overJSD is their ability to capture static correlations, yielding toqualitatively different results on systems with an underlyingmultireference character, both at the variational and at the FNlevel of theory.56,59 The optimization of the parameters in theJastrow is usually quite a feasible problem also on largesystems, as their number does not grow uncontrollably withthe size of the system. In practice, every QMC codeimplements a slightly different functional form of the Jastrow,but they share the general features mentioned above. TheQMC code used in this work is TurboRVB,57 an open-sourcepackage. The Jastrow factor implemented in TurboRVB(described in ref 57) has a number of parameters growinglinearly with the size of the system (as shown in the Resultsand Discussion section).In this work, we keep the orbital frozen and optimize thecoefficients λ1, ..., λn dorbof the JAGPn ansatz using FN-DMCgradients. A similar idea, but at the variational level, was alsomentioned in a seminal work by Casula and Sorella to describethe BCS pairing function in iron-based superconductors.60JAGPn dramatically reduces the number of variationalparameters with respect to the JAGP ansatz, such that theoptimization of the JAGPn function is doable even in prettylarge systems, in contrast to JAGP which is affordable only onrelatively small systems. Nevertheless, employing JAGPnsignificantly improves the FN-DMC energy (as well as thevariational QMC energy) over the results within the traditionalJSD function, as we will show in the results section. Of course,the JAGP ansatz has higher variational freedom than JAGPn,so JAGP can in principle improve further over JAGPn.However, in practice, we observe that FN-DMC energiesobtained from the JAGP ansatz are comparable to thoseobtained from JAGPn on small systems (and both JAGP andJAGPn are significantly better that JSD), while, in largesystems, JAGP is unaffordable because the optimization can bestuck at local minima at the variational level and can becomeunstable at the FN level. The latter instability is probably dueto insufficient signal-to-noise ratios61 that the QMCoptimization always suffers from, but the origin of theinstability is yet unclear. On intermediate systems, we noticethat JAGP FN-DMC energy is worse than the JAGPn FN-DMC energy, as a clear indication that despite the highervariational freedom on JAGP, the optimization of that manyparameters is not converging and there is too much noise onthe parameters.The main problem of the AGP ansatz (and AGPn) is that itis not size-consistent at the variational level of theory, butJAGP (JAGPn) is size-consistent if we employ a very flexibleJastrow factor.62,63 Since the FN-DMC corresponds toapplying an infinitely flexible Jastrow factor to the determinantpart, optimizing the AGPn parameters at the FN level ensuresthe size consistency of our approach.A crucial point to make JAGPn almost as accurate as JAGP,despite employing only a small number norb of parameters λs, isto carefully choose the orbitals. We notice that the virtualorbitals obtained from SCF calculations are typically notoptimal, as we need a large number of them (of the order of L)to converge to the best JAGPn FN energy. Moreover, if wecannot afford a systematic test of the convergence of norb foreach system of interest, it is difficult to define a sensiblecriterion to decide which norb to pick. We solved both theJournal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXDpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asproblems by employing NOs for expanding the pairingfunction, instead of using MOs. NOs were constructed fromsecond-order Møller−Plesset (MP2) calculations. This isbecause the MP2 unoccupied orbitals incorporate perturbationeffects and are physically better than those obtained with HFor DFT,64 as shown in the Supporting Information. Morespecifically, we constructed NOs by diagonalizing the densitymatrix obtained by MP2 calculations. We also notice that amethod to construct NOs should be affordable also for largesystems. This is also a reason why we chose MP2 forconstructing NOs in this study. In practice, from the weight ofthe NOs, we can easily define a cutoff value to select norb oneach system, and we notice that we get to converged resultsalready with a value of n that is not much larger than Nel/2(norb = Nel/2 would correspond to a single SD).3. COMPUTATIONAL DETAILSWe applied our scheme to planar and twisted ethylenes, eighthydrocarbons (CH4, C2H4, C2H6, C6H6, C10H8, C14H10,C18H12, and C20H10), the C60 fullerene, and water−methanedimer (see Supporting Information for their coordinates). Thenumber of valence electrons treated in this study is 12, 12, 8,12, 14, 30, 48, 66, 84, 90, 240, and 16, respectively. The MP2calculations (HF and DFT calculations for comparison) togenerate nodal surfaces of trial wave functions were performedusing PYSCF v.2.0.1.65,66 The trial wave functions wereconverted to the TurboRVB wave function format usingTurboGenius67 via TREXIO68 files. We employed the cc-pVQZ basis set accompanied by the ccECP pseudopotentials69for the eight hydrocarbons and C60 fullerene, while the cc-pVTZ basis set accompanied by the ccECP pseudopotentials69for the water−methane and for the torsion calculation ofethylene. We employed [3s], [3s1p], and [3s1p] primitiveJastrow basis for H, C, and O atoms, respectively. The Jastrowfactor and the weights of the NOs in the pairing function (i.e.,the nodal surface of a wave function) were optimized using thestochastic reconfiguration method70 implemented in Tur-boRVB57 with an adaptive hyperparameter.71 The Jastrowfactor was optimized only with VMC gradients, and it was heldfixed during optimization with FN gradients. The FN gradientswere computed from a standard walker distribution usingmixed estimators, which corresponds to method A in ref 55.The lattice-discretized version of the FN-DMC calculations(LRDMC)72,73 was used in this study. The single-shotLRDMC calculations were performed by the single-gridscheme72 with lattice spaces a = 0.30, 0.25, 0.20, and 0.10Bohr, and the energies were extrapolated to a → 0 using f(a2)= k4·a4 + k2·a2 + k0. The LRDMC calculations for computingthose gradients were performed by the single-grid scheme72with lattice space a = 0.20 Bohr. The determinant localityapproximation (DLA)18 was employed for the LRDMCcalculations.f We notice that the LRDMC frameworkguarantees the variational principle even with the presence ofnonlocal pseudopotentials, as proven in the Appendix. Themolecular structures are depicted using VESTA.744. RESULTS AND DISCUSSION4.1. FNAGPAS Captures Strong Correlation. We showthat the proposed FNAGPAS is able to incorporate thecorrelation effect that the JSD ansatz cannot do at all. We applyour scheme for the torsion energy estimation of ethylene(C2H4). The torsion energy is defined as the energy differencebetween the GS ethylene structure (denoted as planarethylene) and the orthogonally rotated ethylene structure(denoted as twisted ethylene), which are both shown in theinset of Figure 2. Here, we consider only the singlet states forboth configurations. It was shown59 that the JSD ansatz cannotdescribe the torsion energy correctly since the ansatz cannotconsider the static electronic correlation of the twistedethylene, which has a diradical character. This is true both atthe variational and at the FN level of theory.59 The lack ofreliability in the FN results based on a JSD ansatz indicatesthat projection schemes cannot recover strong correlation ifthe FN constraints are given from a wave function withqualitative issues, due to the constraint on the projectioncoming from the trial wave function. Thus, the way to improvethe quality of the FN results is to adopt a more general ansatz,able to improve the nodes of the trial wave function andenhance the reliability of FN estimations.The planar ethylene has an electronic structure characterizedby a highest occupied MO (HOMO) of type π and a lowestunoccupied MO (LUMO) of type π*, and the HOMO−LUMO gap is finite. A single SD having two electrons of unlikespin on the HOMO and no electrons on the LUMO capturesqualitatively well the nature of the wave function and there isno static correlation. However, when the molecule is twisted,the HOMO−LUMO gap decreases because the overlapbetween the p orbitals (orthogonal to the plane of the−CH2 atoms) of the two carbons decreases. At a torsionalangle of 90° (i.e., twisted ethylene), the two p orbitals becomeorthogonal and the frontier orbitals become degenerate,forming two singly occupied MOs. We can define threeindependent (orthogonal) wave functions having two electronson two degenerate orbitals forming a spin singlet, a diradical,and two zwitterionic states.75 Their wave functions imply theuse of more than one SD, i.e., their electronic structure showsstrong correlation. Thus, a multireference ansatz is needed tocorrectly describe the diradical character of the orthogonallytwisted ethylene.59Figure 2. Torsion energy of ethylene from FN-DMC with JSD,JAGPn or JAGP wave functions. The values of JAGP and MR-CISD +Q (horizontal broken line) are taken from refs 59 and 76, respectively.The inset shows the structure of the planar and twisted ethylene.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXEhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c00139/suppl_file/ct4c00139_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c00139/suppl_file/ct4c00139_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig2&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asFigure 2 shows the torsion energies of ethylene computedwith the JSD ansatz with an HF nodal surface and the sameenergies computed with the JAGPn ansatz with HF MOs,gwhose weights are optimized using DMC gradients. As acomparison, we also show results obtained with the full JAGPansatz optimized using VMC gradients, which was taken fromref 59. The reference value in Figure 2 is taken from ref 76, andit is computed using MR-CISD + Q.h The JSD ansatz gives133.1(4) kcal/mol for the torsion energy, which is far from thereference value obtained by MR-CISD + Q (i.e., 69.2 kcal/mol76). Our JAGPn ansatz gives an FN energy of 73.0(4) kcal/mol for the torsion energy, which is close to the referencevalues. This result demonstrates that the JAGPn ansatzoptimized using FN gradients correctly describes the diradicalcharacter of the orthogonally twisted ethylene, something thatthe JSD ansatz cannot do.4.2. Application of FNAGPAS to Small and LargeSystems.We now show that the FNAGPAS scheme leads to asystematic improvement over the traditional JSD ansatz inmolecular systems of increasing size, showing an accuracy inline with the full JAGP ansatz (and better on systems wherethe optimization error for the JAGP ansatz is large), whilebeing affordable on much larger systems. We consider the eighthydrocarbons and the C60 fullerene, represented in Figure 3.Figure 4 (top panel) shows the energy gain in the LRDMCtotal energies (a → 0) by the nodal-surface optimizations ofJAGP and JAGPn over the traditional JSD ansatz (with thenodal surface taken from the DFT LDA calculations). Ourproposed FNAGPAS scheme (JAGPn ansatz optimized usingFN gradients) shows positive gains for all molecules, indicatingthat the nodal-surface optimizations improve the nodes of theSD obtained from DFT. Therefore, there is a systematicimprovement in the description of the correlation energy. Theenergy gain scales linearly with the number of electrons in thesystem. The traditional JAGP ansatz (optimized using VMCgradients) was computationally affordable only on the foursmallest systems, due to the rapid increase of the number ofvariational parameters (see the bottom panel in Figure 4),which makes the optimization unstable or not converging. Inaddition, we could only use VMC-opt for the JAGP ansatzbecause FN-opt is not stable. This highlights an additionalcrucial advantage of FNAGPAS over the traditional JAGPapproach. In the four systems where we have both thetraditional JAGP and the FNAGPAS results, the latter isequivalent to the former on ethane, and it recovers morecorrelation energy in methane, ethylene, and benzene. Largersystems were computationally unaffordable with JAGP, whileJAGPn optimization remains feasible both at the variationaland at the FN level. In fact, FNAGPAS has been successfullyperformed up to C60 fullerene. The gain in C60 is ∼2 meV/valence electron, as shown in the inset of Figure 4. This is areasonable value, considering a previous study by Marchi et al.reporting ∼3 meV/valence electron for the finite-size graphenecalculations with the same atoms as the C60.77Let us consider more closely the medium-size molecules.Figure 4 shows that the gains of JAGPn (optimized with FNgradients) are larger than those of JAGP (optimized usingVMC gradients) in spite of the compactness of the AGPnansatz. In fact, the number of variational parameters in thebenzene molecule is 86 for the JAGPn ansatz and is 17,629 forthe JAGP ansatz. Moreover, JAGP is a generalization ofJAGPn. Therefore, one could naively expect that the larger thenumber of variational parameters, the lower the energy. Here,we observe an exception to this expectation. For this point, werecall that the calculations reported in Figure 4 are obtainedwith a quite small Jastrow factor, employing a [3s1p] basis setfor C atoms and a [3s] for H atoms. This is because we targetlarge systems with FNAGPAS, for which the use of largeJastrow factors is unaffordable. It has been reported that anincomplete Jastrow factor leads to misdirection of the nodalsurface within the variational optimization of the JAGP ansatzin the square H4.78 To confirm if this is the case in the presentcalculations, we performed additional calculations with a largerJastrow factor in the JAGP ansatz calculations (i.e., a basis setof [4s3p1d] and [3s1p] for C and H atoms, respectively) andobtained that the larger Jastrow factor leads to a much largerenergy gain than that obtained with the JAGP ansatz with asmall Jastrow [see results in the Supporting Information(Table S-I and Figure S-I)]. The result indicates that the smallJastrow factor leads to misdirection of the nodal surface of theJAGP ansatz also in this study. On the other hand, Figure 4demonstrates that the FNAGPAS scheme works even with asmall Jastrow factor and a minimal number of parameters inFigure 3. Molecular systems considered in this work, whose FN energy has been computed with the traditional JSD ansatz and with the JAGPnansatz (within the FNAGPAS scheme) discussed in this work. The energy gain (i.e., the improvement of the FNAGPAS scheme over the traditionalscheme which employs the JSD ansatz) and the number of variational parameters in the wave function for each system are shown in Figure 4.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXFhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c00139/suppl_file/ct4c00139_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig3&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asthe antisymmetric part, making the approach applicable tolarger systems.As mentioned in the method part, see Section 2, the twomain features over which FNAGPAS is built are (1) the AGPnansatz and (2) the optimization of its nodal surface using FNgradients. To reveal which of the two is more crucial for thesuccess of the method, i.e., the ansatz or the gradient, we triedthe following combinations: (i) JAGPn with VMC-opt; (ii)JAGPn with FN-opt, (iii) JAGP with VMC-opt; and (iv) JAGPwith FN-opt. Note that (ii) corresponds to FNAGPAS. Thescheme (iv), unfortunately, is not possible as the JAGP has toomany parameters and the FN optimization becomes unstable.Results obtained with schemes (i−iii) are reported in theSupporting Information (Table S-I and Figure S-I). Weobserve that scheme (ii) gives the best gains. Scheme (i) givesgains close to (ii), and they both are much better than (iii).Thus, it appears that freezing the orbitals to those obtained bya mean-field approach plays a crucial role in avoidingmisdirection of the node optimization.4.3. FNAGPAS Scheme Is Size-Consistent. We haveshown that the AGPn ansatz is able to gain correlation energiesat the FN level using very few variational parameters. Inaddition to their role in improving the nodal surface, FNgradients also appear to be crucial when calculating bindingenergies of molecules, preserving size consistency. As shown inTable 1 and discussed hereafter for the particular case of thewater−methane dimer, this is not the case when VMCgradients are used. Therefore, when calculating bindingenergies of molecules, the use of VMC gradients in theJAGPn ansatz gives incorrect results, while the use of FNgradients plays a crucial role in it.Table 1 contains the binding energies of the methane−waterdimer computed with the JSD ansatz, with the JAGPn ansatzoptimized using either VMC or FN gradients (the FNAGPASapproach), and with the JAGP ansatz optimized with VMCgradients. The binding energy is evaluated as the energydifference between the dimer and the sum of the energies ofthe two molecules: Eb = Ewater−methane − Ewater − Emethane. Thereference value for the binding energy of the water−methanedimer, −27 meV, was computed by CCSD(T) implemented inthe ORCA79,80 program.i We chose the CCSD(T) value as areference because the bounded water−methane dimer is not astrongly correlated system, thus CCSD(T) should describe thebinding energy correctly. In this system, the JSD ansatz gives abinding energy of −27(2) meV, which is in good agreementwith the CCSD(T) values of −27.0 meV. Thus, a new DMCapproach with nodal-surface optimization should lower thevalue of the total energies but should not affect the energydifferences. The FNAGPAS scheme, which optimizes theJAGPn parameters with the FN gradients, behaves as expected,yielding a binding energy of −29(2) meV, still in goodagreement with the reference value. However, this is not thecase for the JAGPn ansatz optimized with the VMC gradients,Figure 4. Top panel shows the improvement, dubbed energy gain, ofthe JAGP (red) and JAGPn (blue) ansatz with respect to thetraditional JSD ansatz for each of the considered systems, as afunction of the number of valence electrons. The energy gain is thedifference between FN energy of the JSD ansatz and the JAGP (orJAGPn) ansatz. The bottom panel shows the number of parameters inthe Jastrow factor, in the JAGP, and in the JAGPn wave functions.The dashed lines show the linear (gray for JSD and cyan for JAGPn)and quadratic (orange for JAGP) fitting curves.Table 1. FN Binding Energy Eb and Size-ConsistencyEnergy Error ESCE, Computed with LRDMC a → 0, asObtained with the JSD, JAGPn, and JAGP Wave Functionsaansatz nodes opt Eb (meV) ESCE (meV)JSD −27(2) −1(1)JAGPn VMCopt −46(2) 10(2)JAGPn FNopt −29(2) −2(2)JAGP VMCopt −41(3) 11(3)CCSD(T) −27 0aFor JAGPn, we consider both the case of using VMC and FNgradients to optimize the nodal surface. The latter is the schemedubbed FNAGPAS in this work.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXGhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c00139/suppl_file/ct4c00139_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?fig=fig4&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-aswhich gives Eb = −46(2) meV, or for the JAGP ansatz (withVMC optimization), which gives Eb = −41(3) meV.We can interpret the deterioration of the binding energy asfollows: binding energies are computed from relative energiesamong two or more molecules; thus, the accuracy relies on itserror cancellation. The error cancellation in DMC wasreviewed and discussed by Dubecky ́ in 2016.8 Their conclusionis that one can rely on error cancellation as long as one keepsthe constructions and optimizations of the corresponding wavefunctions as systematic as possible. Indeed, this cancellationworks when the nodes are kept at the same systematic accuracyat every step of the trial wave function constructions. In fact,for the water−methane dimer calculations in this study, ourJSD ansatz fully satisfies the size consistency and givessatisfactory binding energy, which means that the errorcancellation works with the DFT nodal surfaces. In thisstudy, we found that error cancellation was deteriorated by thenodal-surface optimizations using the VMC gradients whilerecovered by those using the FN-DMC gradients. When onecomputes the binding energy of a complex system, one usuallyuses the same Jastrow basis sets for each element in thecomplex and the isolated systems. The use of the same Jastrowbasis sets does not guarantee the same contribution to the totalenergy of both the complex and the isolated systems at theVMC level. Indeed, during the nodal-surface optimization atthe VMC level, the incomplete Jastrow factor affects the nodalsurface differently between the complex and isolated systems;thus, the resultant nodal surface gives the incorrect bindingenergy. The recovery should be because FN-DMC is aprojection method to relax the amplitude of the AGPn ansatz,which corresponds to adding an unlimited flexible Jastrowfactor to a given ansatz.The Jastrow incompleteness is also related to thedeterioration of the size consistency for JAGPn and JAGPwith VMC optimization. The size consistency is a propertythat guarantees the consistency of the energy behavior whenthe interaction between the involved molecular system isnullified (e.g., by a long distance). If the size consistency isfulfilled, the energy of the far-away system should be equal tothe sum of the energies of the two isolated molecules. The lastcolumn in Table 1 shows the difference in energies of the far-away water−methane complex (at a distance of ∼11 Å) andthe sum of the isolated molecules, which can be considered thesize-consistency error and is here dubbed ESCE. The JSD ansatzis size-consistent, as expected.81 The table clearly shows thatthe size consistency is deteriorated by the optimization usingVMC gradients, i.e., the difference between the isolated andfar-away energies is finite. In contrast, the size consistency isperfectly retrieved by the optimization using FN gradients.Neuscamman63 pointed out that the deterioration of the sizeconsistency comes from an incomplete Jastrow factor. Morespecifically, the real-space three/four-body Jastrow factor,which was employed in the present study, cannot completelyremove the size-consistency error unless we use unlimitedflexibility in the Jastrow.63 To solve the problem, Goetz andNeuscamman proposed the so-called number-counting Jastrowfactors that can suppress the unfavorable ionic terms and isable to solve the size-consistency problem82,83 within the VMCframework. In this regard, our proposed scheme can beinterpreted as an alternative approach because, again, FN-DMC is a projection method to relax the amplitude of theAGPn ansatz, which corresponds to adding an unlimitedflexible Jastrow factor to a given ansatz.4.4. Discussion. First, we compare our approach withothers that also target to go beyond the single-reference FNapproximation. A well-established strategy is to use themultideterminant ansatz, which has witnessed numeroussuccesses so far.84−92 The multideterminant approach offersthe advantage of systematic improvement by increasing thenumber of SDs. Nonetheless, the number of SDs for acomprehensive representation exponentially scales with systemsize, imposing substantial computational demands for largesystems. Therefore, this method has mainly been applied tosmall molecular systems.84−86 However, there have beensuccessful efforts to reduce the number of requireddeterminants by neglecting less important ones87,88 using, forinstance, the configuration interaction using a perturbativeselection made iteratively (CIPSI), which mitigates theexponential character of the multideterminant approach inpractice.90,92 Recently, Benali et al. successfully applied themultideterminant approach for solids with more than ahundred electrons by combining the CIPSI technique with arestricted active space built using NOs,91 which is a similaridea as we present in this study. Indeed, they demonstratedthat one can go beyond the single-reference nodal surface inlarge systems by the multideterminant approach in practice,though its naive asymptotic scaling is exponential. Themultideterminant approach is becoming as practical andpromising as the single-determinant approach.Concerning the actual computational costs of our proposedmethods, the choice of ansatz (i.e., JSD or AGPn) does notsignificantly affect the cost of wave function optimization,while the choice of gradients does. For instance, for C60,Jastrow optimization with the JSD ansatz and Jastrow + nodal-surface (i.e., weights of NOs) optimization with the JAGPnansatz using VMC gradients require 11.9 and 43.6 core hoursper optimization step with ∼7 mHa accuracy on the totalenergy evaluation at each optimization step, respectively.jHowever, if one uses FN gradients for wave functionoptimization, one needs more computational time. Forinstance, for C60, the nodal-surface (i.e., weights of NOs)optimization with the JAGPn ansatz using FN gradients with a= 0.20 au requires 195.3 core hours per optimization step with∼7 mHa accuracy on the total energy evaluation at eachoptimization step. Thus, our FNAGPAS scheme using FNgradients shows the same scaling of the number of variationalparameters as the single-reference FN DMC with JSD ansatz,while it increases the prefactor of computational cost.Based on the results obtained in this work so far, we finallydiscuss how to improve a Fermionic ansatz in ab initio QMCcalculations, in general. Recently, there have been manysuccessful reports about machine-learning-inspired ansatz witha huge degree of freedom in describing electronic and spinstates, such as deep neural networks,93 restricted Boltzmannmachines,94−96 and transformers,97 which are utilized as ansatzof wave functions to solve the Schrödinger equation withlattice Hamiltonians. Also, in the ab initio community, ansatzesusing deep neural networks have been successfully applied forrealistic problems, such as PauliNet,42 FermiNet,39 andothers.40,41,43−45 In light of the present results, let us considerexploiting an ansatz with a huge degree of freedom (i.e., manyvariational parameters) in ab initio QMC calculations topursue an exact Fermionic GS. If we stop at the VMC level, wemay apply such a flexible ansatz to Jastrow factors, thedeterminant part, or both parts, and it is expected that thelarger the degree of freedom an ansatz has, the larger theJournal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXHpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asenergy gain should be. However, improvements at the VMClevel do not necessarily lead to improvements at the FN level,especially if the determinant part is optimized at the variationallevel. A variational optimization improves the overall shape ofthe trial wave function ΨT, while the nodal surface might notbe as optimized as the ΨT. In this work, indeed, we have seenhow the JAGPn ansatz optimized at the FN level leads to muchbetter results than the JAGP ansatz optimized at the VMClevel, despite the latter having many more variationalparameters and it is much better at the VMC level. Moreover,we also observed how the JAGPn (and JAGP, for that matter)ansatz itself yields a size-consistency error at the FN level if theparameters are optimized at the VMC level, while the sameansatz with parameters optimized at the FN level is notaffected by this issue. Thus, caution should be used whenemploying these new highly flexible machine-learning-basedwave function parameterizations, as it is not guaranteed thatimprovements in the VMC energy are reflected in improve-ments in the FN energy in a consistent way. Basic physicalproperties, which were present in the most standard wavefunctions (such as the JSD), might not appear in the fancierapproaches, similar to the mentioned problem of sizeinconsistency in JAGP and JAGPn.5. CONCLUSIONS AND PERSPECTIVESIn this study, we propose a method for variational optimizationof the AGP wave function expressed in terms of NOs, withpairing coefficients optimized using FN gradients. Within ourscheme, the variational parameter space increases only linearlywith the system size, as opposed to the quadratic scaling of thestandard parameterization of AGP, with the result that ourproposed method allows the optimization of the nodal surfacesfor large systems, which has been difficult to achieve withconventional approaches. In addition to demonstrating thatour scheme can be applied to systems as large as C60, weshowed that our scheme also achieves better (i.e., lower) DMCenergies than the single-reference FN DMC calculations.Moreover, we have shown that our approach is size-consistentand can be used to estimate binding energies.We showed that the Jastrow incompleteness affecting nodal-surface optimizations can be mitigated by using FN gradientscombined with the JAGPn ansatz. However, in this study, wedid not investigate the effect of the basis set incompleteness onthe determinant part (i.e., nodal surface). The basis setincompleteness is believed to be less severe in QMCcalculations than in quantum chemistry methods because theJastrow factor (at the variational level) or the projection (at theFN level) mitigates its error. However, to the best of ourknowledge, no one has seriously investigated the error so far.Considering binding energy calculations done by DMCreported so far,8 the basis set incompleteness should have asmall effect on small molecules, but it should be carefullyconsidered when studying large molecules using DMC donewith localized basis sets. This is one of the intriguing futureworks for applying the single-reference DMC and ourproposed methods to large systems.■ APPENDIXProof for the Variational Principle of the LRDMCOptimization with DLAAs pointed out in seminal works by Casula et al.,72,98 the use ofa pseudopotential that has the so-called nonlocal term inducesan additional sign problem in the standard DMC approachwith the LA; thus the variational principle, which justifies theenergy minimization strategy, is deteriorated. Instead, one ofthe advantages of the LRDMC is that the use ofpseudopotentials does not deteriorate the variational princi-ple;72 thus, the energy minimization is justified. Recently, weimplemented the DLA18 into the TurboRVB package. In thisstudy, we combine the DLA with the LRDMC frameworkimplemented in the TurboRVB package. We prove here thatthe variational principle holds also in the LRDMC with theDLA. This proof is inspired by the proof by Haaf et al.99 thatthe lattice Green’s function Monte Carlo method is variational.In LRDMC calculations with the DLA, the effectiveHamiltonian (i.e., the FN Hamiltonian) reads=+ =>lmooooooooonoooooooooHH V x xx x x H xH x xfor0 for if ( ) ( )0for otherwisex xx x x xx xx x,FN, ,sf,DLAT , T,(4)where | |H x H xx x, , =V D x H D x( ) / ( )x x x x x x,sf,DLA for sfT , T ,by which the original term in the LRDMC approach,=V x H x( ) / ( )x x x x x x,sf for sfT , T , is replaced, and sf meansthat all x′ (≠x) satisfy ΨT(x′)Hx,x′ΨT(x) > 0. Here, we omitthe lattice-space dependency of the Hamiltonian (i.e., H ≡ Ha)because one can extrapolate energies to the a → 0 limit. Noticethat we assume that a trial wave function can be decomposedinto the Jastrow and determinant parts, i.e., ΨT = JTDT. Wealso notice that> >x H x D x H D x( ) ( ) 0 ( ) ( ) 0x x x xT , T T , Tsince the Jastrow factor does not affect the sign of a wavefunction. We define the following notations:= | ||EHMATFNFNT FN (5)= | ||EHFNFNFNFNFN FN (6)= | ||EHFN FNFN FN (7)= | ||EH00 00 0 (8)where |ΦFN⟩ is the FN GS of ĤFN. In the following, we willshow that the following equations hold:=E E E EMA FN 0 (9)The first equality (EMA = EFN) holds because |ΦFN⟩ is the exactGS of HFN (i.e., ĤFN|ΦFN⟩ = EFN|ΦFN⟩). This is also true withthe nonlocal terms of pseudopotentials. Now, we define thedifference between the effective FN energy obtained with theeffective Hamiltonian ĤFN and that obtained with the trueHamiltonian Ĥ:= | |E E E H HFN FNFNFN (10)Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXIpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asWe want to prove that ΔE ≥ 0 for the FN state and that theequality holds for ΦFN = ΨT = Ψ0, where we denote Ψ0 as theexact wave function of the original Hamiltonian Ĥ, i.e.,| = |H E0 0 0 . Hereafter, we will do the same exercisewritten in ref 99. We define the difference between theeffective FN energy obtained with the effective HamiltonianHFN and that obtained with the true Hamiltonian H:= | |= | |E E EH HV HFNFNFNFNFNsf sfFN (11)where we define a truncated Hamiltonian Htr and a spin-flipHamiltonian Hsf by= +H H Htr sf (12)and= +H H VFN tr sf (13)Indeed, the matrix elements are==>lmooooooonooooooHH x xx x x H xH x xfor0 for if ( ) ( ) 0for otherwisex xx xx xx x,tr,T , T, (14)and==>lmooooonoooooHx xH x x x H xx x0 forfor if ( ) ( ) 00 for otherwisex x x x x x,sf, T , T(15)ΔE can be written explicitly in terms of the matrix elements:= * [ | | | | ]E x x V x x x H x x( ) ( ) ( )x xFNsfFNsfFN(16)This can be rewritten as= *ÄÇÅÅÅÅÅÅÅÅÅÅÅ ÉÖÑÑÑÑÑÑÑÑÑÑÑE x HD xD xxH x( )( )( )( )( )x xx xxx xFNsf,TTFNsf, FN(17)where sf means that all x′ (≠ x) satisfy ΨT(x′)Hx,x′ΨT(x) > 0.In this double summation, each pair of configurations (x, x′)appears twice. Therefore, we can combine these terms andrewrite it as a summation over the pairs:= | | + | |* *ÄÇÅÅÅÅÅÅÅÅÅÅ ÉÖÑÑÑÑÑÑÑÑÑÑE HD xD xxD xD xxx x x x( )( )( )( )( )( )( ) ( ) ( ) ( )x xx x( , )sf,TTFN2 TTFN2FN FN FN FN(18)Notice that the Hamiltonian is Hermitian: Hx,x′ = Hx′,x. Sinceall the pairs satisfy >H 0D xD x x x( )( ) ,TT, we obtain= | | = | |HD xD xHD xD xHD xD xHD xD x( )( )( )( )( )( )( )( )x x x x x x x x,TT,TT,TT,TT(19)Then= | | | | + | |* *ÄÇÅÅÅÅÅÅÅÅÅÅÅ ÉÖÑÑÑÑÑÑÑÑÑÑÑÑE HD xD xxD xD xxx x x x x x x x( )( )( )( )( )( )sgn( , ) ( ) ( ) sgn( , ) ( ) ( )x xx x( , )sf,TTFN2 TTFN2FN FN FN FN(20)where sgn(x, x′) denotes the sign of Hx,x′. Finally, we get= | |E HD xD xx x xD xD x(x)( )( )sgn( , ) ( )( )( )x xx x,sf,FNTTFNTT2(21)indicating that ΔE is positive for any wave function ΦFN. Thus,the GS energy of HFN is an upper bound for the GS energy ofthe original Hamiltonian H (i.e., EFN ≥ E). Hereafter, weconsider the case that one uses the true GS Ψ0 for thedeterminant of the trial wave function (i.e., ΨT = JT·Ψ0), toprove that EFN = E holds with ΨT = JT·Ψ0 (i.e., DT = Ψ0): Forall the pairs (x, x′), Ψ0Hx,x′Ψ0 > 0 is satisfied, meaning sgn(x,x′) → + and +xx( )( )00or sgn(x, x′) → − and xx( )( )00.Thus, the above condition is fulfilled when the followingcondition is satisfied:=xxxx( )( )( )( )FNFN00 (22)In the DLA approach, the spin-flip term is composed only ofthe determinant of the trial wave function. Therefore, the FNoutcome with the DLA approach is not affected by thepresence of the Jastrow factor in the trial wave function (in thea → 0 limit). Therefore, one gets ΦFN = Ψ0 with ΨT = JT·Ψ0.Thus, ΔE = 0 is fulfilled with ΨT = JT·Ψ0, and the followingrelations hold:= = = ·E E E E J(with )MA FN 0 T T 0 (23)meaning that the effective Hamiltonian H FNand the trueHamiltonian Ĥ have the same GS energy E0 and the same GSΦFN = Ψ0 with ΨT = JT ·Ψ0, where the final equality E = E0comes from the usual variational principle.In the DLA approach, we can update the trial wave functionΨT such that EMA goes down according to the gradient ∂αEMAor using a more sophisticated optimization scheme. As writtenabove, the equals EFN = E = E0 are met when ΨT = J ·Ψ0. Itimplies that one can look for the true GS energy and wavefunction by variation of the determinant part of the trial wavefunction. Indeed, in the LRDMC calculations with the DLA,one can access the mixed-average energy EMA and its derivativeEMA , where α⃗ is a set of the variational parameters. SinceEMA satisfies the variational principle, i.e., EMA ≥ E0, theequality holds when ΨT = JT·Ψ0; as proven above, one canupdate the determinant part of the trial wave function, DT,such that EMA goes down, then, it is expected that DT finallyreaches DT → Ψ0 and EMA → E0.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXJpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as■ ASSOCIATED CONTENTData Availability StatementThe QMC kernel used in this work, TurboRVB, is availablefrom its GitHub repository [https://github.com/sissaschool/turborvb].*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139.Total energies and the number of variational parametersof hydrocarbons and fullerene, total energies of themethane−water complex and corresponding fragmentsand of the planar and twisted ethylene molecule, anddiscussion on the role of MOs or NOs in the AGPnansatz and the size-consistency error (PDF)Geometries, in xyz format, of all the systems studied(ZIP)■ AUTHOR INFORMATIONCorresponding AuthorsKousuke Nakano − Center for Basic Research on Materials,National Institute for Materials Science (NIMS), Tsukuba,Ibaraki 305-0047, Japan; International School for AdvancedStudies (SISSA), 34136 Trieste, Italy; orcid.org/0000-0001-7756-4355; Email: kousuke_1123@icloud.comAndrea Zen − Dipartimento di Fisica Ettore Pancini,Universita ̀ di Napoli Federico II, 80126 Napoli, Italy;Department of Earth Sciences, University College London,London WC1E 6BT, U.K.; orcid.org/0000-0002-7648-4078; Email: andrea.zen@unina.itAuthorsSandro Sorella − International School for Advanced Studies(SISSA), 34136 Trieste, ItalyDario Alfe ̀ − Dipartimento di Fisica Ettore Pancini, Universita ̀di Napoli Federico II, 80126 Napoli, Italy; Department ofEarth Sciences, University College London, London WC1E6BT, U.K.; Thomas Young Centre and London Centre forNanotechnology, London WC1H 0AH, U.K.; orcid.org/0000-0002-9741-8678Complete contact information is available at:https://pubs.acs.org/10.1021/acs.jctc.4c00139NotesThe authors declare no competing financial interest.■ ACKNOWLEDGMENTSK.N. is grateful for computational resources from theNumerical Materials Simulator at the National Institute forMaterials Science (NIMS). K.N. is grateful for computationalresources of the supercomputer Fugaku provided by RIKENthrough the HPCI System Research Projects (project IDs:hp220060 and hp230030). K.N. acknowledges financialsupport from the JSPS Overseas Research Fellowships, fromGrant-in-Aid for Early Career Scientists (grant no.JP21K17752), from Grant-in-Aid for Scientific Research(grant no. JP21K03400), and from MEXT Leading Initiativef o r E x c e l l e n t Young Re s e a r c h e r s ( g r a n t no .JPMXS0320220025). D.A. and A.Z. acknowledge supportfrom Leverhulme grant no. RPG-2020-038. D.A. and A.Z. alsoacknowledge support from the European Union under theNextGenerat ion EU (projects 20222FXZ33 andP2022MC742). The authors acknowledge the use of theUCL Kathleen High Performance Computing Facility(Kathleen@UCL), and associated support services, in thecompletion of this work. This research used resources of theOak Ridge Leadership Computing Facility at the Oak RidgeNational Laboratory, which is supported by the Office ofScience of the U.S. Department of Energy under contract no.DE-AC05-00OR22725. Calculations were also performedusing the Cambridge Service for Data Driven Discovery(CSD3) operated by the University of Cambridge ResearchComputing Service (www.csd3.cam.ac.uk), provided by DellEMC and Intel using Tier-2 funding from the Engineering andPhysical Sciences Research Council (capital grant EP/T022159/1 and EP/P020259/1). This work also used theARCHER UK National Supercomputing Service (https://www.archer2.ac.uk), the United Kingdom Car−Parrinello(UKCP) consortium (EP/F036884/1). We dedicate thispaper to one of the authors, S.S. (SISSA), who passed awayduring the collaboration. He initially devised an idea to use FNgradients for wave function optimizations, which is one of thekeys for the success of the present work. The QMCcommunity will remember that he is one of the mostinfluential contributors of the past and of the beginning ofthe present century to the community and in particular fordeeply inspiring this work with his development of the ab initioQMC package, TurboRVB.57■ ADDITIONAL NOTESaIn all-electron calculations, FN-DMC is always variational,meaning that the lowest FN energy EFN is obtained when theexact nodal surface is used, otherwise EFN > E0. Whenpseudopotentials are employed, there are also nonlocaloperators in the Hamiltonian. This yields to a problem similarto the sign problem, which requires a further approximation.There are a few alternatives to deal with pseudopotentials inDMC: the LA,15 the T-move (TM),16,17 the DLA, and thedeterminant locality TM (DLTM).18 TM and DLTM arevariational, meaning that their energy (EFN,TM or EFN,DLTM) isan upper bound of the exact GS energy E0.bThe Jastrow factor correlates explicitly all pairs of electrons; itis a symmetric positive function, so it recovers dynamicalcorrelation and it does not change the nodal surface.cAn approach is size-consistent if the energy of a systemconstituted by two or more noninteracting subsystems (e.g.,two molecules far away) is the same of the sum of the energiesof the subsystems.dSee ref 57 for details about the functional form of termsimplemented in the TurboRVB package used for this work.eIt is generally claimed that the cost of FN-DMC scales as thecube of the number of electrons Nel. This is true forsimulations where the antisymmetric part of the wave functioncan be computed as a determinant and Nel up to roughly athousand. For larger systems, the cost for a MC step is N( )el3and therefore the cost of FN-DMC is quartic.fThe use of DLA in LRDMC is equivalent to the DLTM18scheme in standard DMC.gThe HF orbitals obtained with the Fermi−Dirac smearingmethod were used for the occupied and the virtual orbitals ofthe JAGPn ansatz for the twisted ethylene because the HOMOand LUMO should have the same energies. Note, in this case,we did not use the NOs (introduced in the discussion above)because this system is characterized by strong correlationcoming from the two frontier orbitals, which are easily derivedJournal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXKhttps://github.com/sissaschool/turborvbhttps://github.com/sissaschool/turborvbhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c00139/suppl_file/ct4c00139_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c00139/suppl_file/ct4c00139_si_002.ziphttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kousuke+Nakano"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0001-7756-4355https://orcid.org/0000-0001-7756-4355mailto:kousuke_1123@icloud.comhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Andrea+Zen"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-7648-4078https://orcid.org/0000-0002-7648-4078mailto:andrea.zen@unina.ithttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Sandro+Sorella"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Dario+Alfe%CC%80"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-9741-8678https://orcid.org/0000-0002-9741-8678https://pubs.acs.org/doi/10.1021/acs.jctc.4c00139?ref=pdfhttp://www.csd3.cam.ac.ukhttps://www.archer2.ac.ukhttps://www.archer2.ac.ukpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asalready from the HF theory. Moreover, in the twisted ethylenecase, we allowed the optimization of the off-diagonalcoefficient of the AGP matrix that pairs the two frontierorbitals.hThe twisted ethylene is a prototypical example of a systemcharacterized by strong correlation where single-referenceperturbative approaches, such as CCSD(T), fail and multi-reference approaches are needed.iIn particular, we performed canonical CCSD(T) calculationswith the automatic basis set extrapolation implemented inOrca (which assumes an exponential convergence for the HFenergy and a polynomial convergence for the correlationenergy) using Dunning correlation-consistent core-polarizedbasis sets, cc-pCVnZ, with quadruple-ζ (n = Q) and quintuple-ζ (n = 5) basis sets. We performed both estimations with andwithout counterpoise correction, both yielding a bindingenergy of −27.2 meV.jWe measured the computational times on the NumericalMaterials Simulator at the National Institute for MaterialsScience (NIMS) using 1536 cores [32 nodes × Intel XeonPlatinum 8268 (2.9 GHz, 24 cores) × 2 per node].■ REFERENCES(1) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys.Rev. 1964, 136, B864−B871.(2) Bartlett, R. J.; Musiał, M. Coupled-cluster theory in quantumchemistry. Rev. Mod. Phys. 2007, 79, 291−352.(3) Řezác,̌ J.; Hobza, P. Describing Noncovalent Interactionsbeyond the Common Approximations: How Accurate Is the “GoldStandard,” CCSD(T) at the Complete Basis Set Limit? J. Chem.Theory Comput. 2013, 9, 2151−2155.(4) Řezác,̌ J.; Hobza, P. Benchmark Calculations of InteractionEnergies in Noncovalent Complexes and Their Applications. Chem.Rev. 2016, 116, 5038−5071.(5) Al-Hamdani, Y. S.; Tkatchenko, A. Understanding non-covalentinteractions in larger molecular complexes from first principles. J.Chem. Phys. 2019, 150, 010901.(6) Szalay, P. G.; Müller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R.Multiconfiguration Self-Consistent Field and Multireference Config-uration Interaction Methods and Applications. Chem. Rev. 2012, 112,108−181.(7) Foulkes, W. M. C.; Mitas, L.; Needs, R. J.; Rajagopal, G.Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 2001,73, 33−83.(8) Dubecky,́ M.; Mitas, L.; Jurecǩa, P. Noncovalent Interactions byQuantum Monte Carlo. Chem. Rev. 2016, 116, 5188−5215.(9) Al-Hamdani, Y. S.; Rossi, M.; Alfe,̀ D.; Tsatsoulis, T.;Ramberger, B.; Brandenburg, J. G.; Zen, A.; Kresse, G.; Grüneis, A.;Tkatchenko, A.; Michaelides, A. Properties of the water to boronnitride interaction: from zero to two dimensions with benchmarkaccuracy. J. Chem. Phys. 2017, 147, 044710.(10) Brandenburg, J. G.; Zen, A.; Fitzner, M.; Ramberger, B.; Kresse,G.; Tsatsoulis, T.; Grüneis, A.; Michaelides, A.; Alfe,̀ D. Physisorptionof Water on Graphene: Subchemical Accuracy from Many-BodyElectronic Structure Methods. J. Phys. Chem. Lett. 2019, 10, 358−368.(11) Zen, A.; Brandenburg, J. G.; Klimes,̌ J.; Tkatchenko, A.; Alfe,̀D.; Michaelides, A. Fast and accurate quantum Monte Carlo formolecular crystals. Proc. Natl. Acad. Sci. U.S.A. 2018, 115, 1724−1729.(12) Al-Hamdani, Y. S.; Nagy, P. R.; Zen, A.; Barton, D.; Kállay, M.;Brandenburg, J. G.; Tkatchenko, A. Interactions between largemolecules pose a puzzle for reference quantum mechanical methods.Nat. Commun. 2021, 12, 3927.(13) Shi, B. X.; Zen, A.; Kapil, V.; Nagy, P. R.; Grüneis, A.;Michaelides, A. Many-Body Methods for Surface Chemistry Come ofAge: Achieving Consensus with Experiments. J. Am. Chem. Soc. 2023,145, 25372−25381.(14) Troyer, M.; Wiese, U. J. Computational complexity andfundamental limitations to fermionic quantum Monte Carlosimulations. Phys. Rev. Lett. 2005, 94, 170201.(15) Mitas, L.; Shirley, E. L.; Ceperley, D. M. Nonlocalpseudopotentials and diffusion Monte Carlo. J. Chem. Phys. 1991,95, 3467−3475.(16) Casula, M. Beyond the locality approximation in the standarddiffusion Monte Carlo method. Phys. Rev. B: Condens. Matter Mater.Phys. 2006, 74, 161102.(17) Casula, M.; Moroni, S.; Sorella, S.; Filippi, C. Size-consistentvariational approaches to nonlocal pseudopotentials: Standard andlattice regularized diffusion Monte Carlo methods revisited. J. Chem.Phys. 2010, 132, 154113.(18) Zen, A.; Brandenburg, J. G.; Michaelides, A.; Alfe,̀ D. A newscheme for fixed node diffusion quantum Monte Carlo withpseudopotentials: Improving reproducibility and reducing the trial-wave-function bias. J. Chem. Phys. 2019, 151, 134105.(19) Casula, M.; Sorella, S. Geminal wave functions with Jastrowcorrelation: A first application to atoms. J. Chem. Phys. 2003, 119,6500−6511.(20) Casula, M.; Attaccalite, C.; Sorella, S. Correlated geminal wavefunction for molecules: An efficient resonating valence bondapproach. J. Chem. Phys. 2004, 121, 7110−7126.(21) Marchi, M.; Azadi, S.; Casula, M.; Sorella, S. Resonatingvalence bond wave function with molecular orbitals: Application tofirst-row molecules. J. Chem. Phys. 2009, 131, 154116.(22) Bajdich, M.; Mitas, L.; Drobny,́ G.; Wagner, L. K.; Schmidt, K.E. Pfaffian Pairing Wave Functions in Electronic-Structure QuantumMonte Carlo Simulations. Phys. Rev. Lett. 2006, 96, 130201.(23) Bajdich, M.; Mitas, L.; Wagner, L. K.; Schmidt, K. E. Pfaffianpairing and backflow wavefunctions for electronic structure quantumMonte Carlo methods. Phys. Rev. B: Condens. Matter Mater. Phys.2008, 77, 115112.(24) Genovese, C.; Shirakawa, T.; Nakano, K.; Sorella, S. GeneralCorrelated Geminal Ansatz for Electronic Structure Calculations:Exploiting Pfaffians in Place of Determinants. J. Chem. Theory Comput.2020, 16, 6114−6131.(25) Toulouse, J.; Umrigar, C. J. Full optimization of Jastrow−Slaterwave functions with application to the first-row atoms andhomonuclear diatomic molecules. J. Chem. Phys. 2008, 128, 174101.(26) Zimmerman, P. M.; Toulouse, J.; Zhang, Z.; Musgrave, C. B.;Umrigar, C. J. Excited states of methylene from quantum MonteCarlo. J. Chem. Phys. 2009, 131, 124103.(27) Anderson, A. G.; Goddard, W. A. Generalized valence bondwave functions in quantum Monte Carlo. J. Chem. Phys. 2010, 132,164110.(28) Braïda, B.; Toulouse, J.; Caffarel, M.; Umrigar, C. J. QuantumMonte Carlo with Jastrow-valence-bond wave functions. J. Chem.Phys. 2011, 134, 084108.(29) Drummond, N. D.; Ríos, P. L.; Ma, A.; Trail, J. R.; Spink, G. G.;Towler, M. D.; Needs, R. J. Quantum Monte Carlo study of the Neatom and the Ne+ ion. J. Chem. Phys. 2006, 124, 224104.(30) Ríos, P. L.; Ma, A.; Drummond, N. D.; Towler, M. D.; Needs,R. J. Inhomogeneous backflow transformations in quantum MonteCarlo calculations. Phys. Rev. E 2006, 74, 066701.(31) Toulouse, J.; Umrigar, C. J. Optimization of quantum MonteCarlo wave functions by energy minimization. J. Chem. Phys. 2007,126, 084102.(32) Seth, P.; Ríos, P. L.; Needs, R. J. Quantum Monte Carlo studyof the first-row atoms and ions. J. Chem. Phys. 2011, 134, 084105.(33) Clark, B. K.; Morales, M. A.; McMinis, J.; Kim, J.; Scuseria, G.E. Computing the energy of a water molecule using multi-determinants: A simple, efficient algorithm. J. Chem. Phys. 2011,135, 244105.(34) Morales, M. A.; McMinis, J.; Clark, B. K.; Kim, J.; Scuseria, G.E. Multideterminant Wave Functions in Quantum Monte Carlo. J.Chem. Theory Comput. 2012, 8, 2181−2188.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXLhttps://doi.org/10.1103/PhysRev.136.B864https://doi.org/10.1103/RevModPhys.79.291https://doi.org/10.1103/RevModPhys.79.291https://doi.org/10.1021/ct400057w?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct400057w?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct400057w?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.chemrev.5b00526?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.chemrev.5b00526?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/1.5075487https://doi.org/10.1063/1.5075487https://doi.org/10.1021/cr200137a?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/cr200137a?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1103/RevModPhys.73.33https://doi.org/10.1021/acs.chemrev.5b00577?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.chemrev.5b00577?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/1.4985878https://doi.org/10.1063/1.4985878https://doi.org/10.1063/1.4985878https://doi.org/10.1021/acs.jpclett.8b03679?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpclett.8b03679?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpclett.8b03679?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1073/pnas.1715434115https://doi.org/10.1073/pnas.1715434115https://doi.org/10.1038/s41467-021-24119-3https://doi.org/10.1038/s41467-021-24119-3https://doi.org/10.1021/jacs.3c09616?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/jacs.3c09616?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1063/1.460849https://doi.org/10.1063/1.460849https://doi.org/10.1103/PhysRevB.74.161102https://doi.org/10.1103/PhysRevB.74.161102https://doi.org/10.1063/1.3380831https://doi.org/10.1063/1.3380831https://doi.org/10.1063/1.3380831https://doi.org/10.1063/1.5119729https://doi.org/10.1063/1.5119729https://doi.org/10.1063/1.5119729https://doi.org/10.1063/1.5119729https://doi.org/10.1063/1.1604379https://doi.org/10.1063/1.1604379https://doi.org/10.1063/1.1794632https://doi.org/10.1063/1.1794632https://doi.org/10.1063/1.1794632https://doi.org/10.1063/1.3249966https://doi.org/10.1063/1.3249966https://doi.org/10.1063/1.3249966https://doi.org/10.1103/PhysRevLett.96.130201https://doi.org/10.1103/PhysRevLett.96.130201https://doi.org/10.1103/PhysRevB.77.115112https://doi.org/10.1103/PhysRevB.77.115112https://doi.org/10.1103/PhysRevB.77.115112https://doi.org/10.1021/acs.jctc.0c00165?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.0c00165?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.0c00165?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/1.2908237https://doi.org/10.1063/1.2908237https://doi.org/10.1063/1.2908237https://doi.org/10.1063/1.3220671https://doi.org/10.1063/1.3220671https://doi.org/10.1063/1.3377091https://doi.org/10.1063/1.3377091https://doi.org/10.1063/1.3555821https://doi.org/10.1063/1.3555821https://doi.org/10.1063/1.2204600https://doi.org/10.1063/1.2204600https://doi.org/10.1103/PhysRevE.74.066701https://doi.org/10.1103/PhysRevE.74.066701https://doi.org/10.1063/1.2437215https://doi.org/10.1063/1.2437215https://doi.org/10.1063/1.3554625https://doi.org/10.1063/1.3554625https://doi.org/10.1063/1.3665391https://doi.org/10.1063/1.3665391https://doi.org/10.1021/ct3003404?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-aspubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as(35) Filippi, C.; Assaraf, R.; Moroni, S. Simple formalism for efficientderivatives and multi-determinant expansions in quantum MonteCarlo. J. Chem. Phys. 2016, 144, 194105.(36) Scemama, A.; Applencourt, T.; Giner, E.; Caffarel, M. QuantumMonte Carlo with very large multideterminant wavefunctions. J.Comput. Chem. 2016, 37, 1866−1875.(37) Dash, M.; Moroni, S.; Scemama, A.; Filippi, C. PerturbativelySelected Configuration-Interaction Wave Functions for EfficientGeometry Optimization in Quantum Monte Carlo. J. Chem. TheoryComput. 2018, 14, 4176−4182.(38) Scemama, A.; Giner, E.; Benali, A.; Loos, P.-F. Taming thefixed-node error in diffusion Monte Carlo via range separation. J.Chem. Phys. 2020, 153, 174107.(39) Pfau, D.; Spencer, J. S.; Matthews, A. G. D. G.; Foulkes, W. M.C. Ab initio solution of the many-electron Schrödinger equation withdeep neural networks. Phys. Rev. Res. 2020, 2, 033429.(40) Li, X.; Fan, C.; Ren, W.; Chen, J. Fermionic neural networkwith effective core potential. Phys. Rev. Res. 2022, 4, 013021.(41) Ren, W.; Fu, W.; Wu, X.; Chen, J. Towards the ground state ofmolecules via diffusion Monte Carlo on neural networks. Nat.Commun. 2023, 14, 1860.(42) Hermann, J.; Schätzle, Z.; Noé, F. Deep-neural-networksolution of the electronic Schrödinger equation. Nat. Chem. 2020,12, 891−897.(43) Choo, K.; Mezzacapo, A.; Carleo, G. Fermionic neural-networkstates for ab-initio electronic structure. Nat. Commun. 2020, 11, 2368.(44) Li, R.; Ye, H.; Jiang, D.; Wen, X.; Wang, C.; Li, Z.; Li, X.; He,D.; Chen, J.; Ren, W. others Forward Laplacian: A New Computa-tional Framework for Neural Network-based Variational Monte Carlo.arXiv 2023, arXiv:2307.08214. preprint(45) von Glehn, I.; Spencer, J. S.; Pfau, D. A self-attention ansatz forab-initio quantum chemistry. arXiv 2022, arXiv:2211.13672. preprint(46) Sorella, S.; Seki, K.; Brovko, O. O.; Shirakawa, T.; Miyakoshi,S.; Yunoki, S.; Tosatti, E. Correlation-Driven Dimerization andTopological Gap Opening in Isotropically Strained Graphene. Phys.Rev. Lett. 2018, 121, 066402.(47) Nakano, K.; Maezono, R.; Sorella, S. All-Electron QuantumMonte Carlo with Jastrow Single Determinant Ansatz: Application tothe Sodium Dimer. J. Chem. Theory Comput. 2019, 15, 4044−4055.(48) Genovese, C.; Shirakawa, T.; Nakano, K.; Sorella, S. GeneralCorrelated Geminal Ansatz for Electronic Structure Calculations:Exploiting Pfaffians in Place of Determinants. J. Chem. Theory Comput.2020, 16, 6114−6131.(49) Ammar, A.; Giner, E.; Scemama, A. Optimization of LargeDeterminant Expansions in Quantum Monte Carlo. J. Chem. TheoryComput. 2022, 18, 5325−5336.(50) Raghav, A.; Maezono, R.; Hongo, K.; Sorella, S.; Nakano, K.Toward Chemical Accuracy Using the Jastrow Correlated Anti-symmetrized Geminal Power Ansatz. J. Chem. Theory Comput. 2023,19, 2222−2229.(51) Reboredo, F. A.; Hood, R. Q.; Kent, P. R. C. Self-healingdiffusion quantum Monte Carlo algorithms: Direct reduction of thefermion sign error in electronic structure calculations. Phys. Rev. B:Condens. Matter Mater. Phys. 2009, 79, 195117.(52) Reboredo, F. A. Systematic reduction of sign errors in many-body problems: Generalization of self-healing diffusion Monte Carloto excited states. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80,125110.(53) Reboredo, F. A.; Kim, J. Generalizing the self-healing diffusionMonte Carlo approach to finite temperature: A path for theoptimization of low-energy many-body bases. J. Chem. Phys. 2014,140, 074103.(54) Bajdich, M.; Tiago, M. L.; Hood, R. Q.; Kent, P. R.; Reboredo,F. A. Systematic reduction of sign errors in many-body calculations ofatoms and molecules. Phys. Rev. Lett. 2010, 104, 193001.(55) McFarland, J.; Manousakis, E. Gradient-descent optimization offermion nodes in the diffusion Monte Carlo technique. Phys. Rev. A2022, 105, 032815.(56) Zen, A.; Coccia, E.; Gozem, S.; Olivucci, M.; Guidoni, L.Quantum Monte Carlo Treatment of the Charge Transfer andDiradical Electronic Character in a Retinal Chromophore MinimalModel. J. Chem. Theory Comput. 2015, 11, 992−1005.(57) Nakano, K.; Attaccalite, C.; Barborini, M.; Capriotti, L.; Casula,M.; Coccia, E.; Dagrada, M.; Genovese, C.; Luo, Y.; Mazzola, G.; Zen,A.; Sorella, S. TurboRVB: A many-body toolkit for ab initio electronicsimulations by quantum Monte Carlo. J. Chem. Phys. 2020, 152,204121.(58) Annaberdiyev, A.; Melton, C. A.; Bennett, M. C.; Wang, G.;Mitas, L. Accurate Atomic Correlation and Total Energies forCorrelation Consistent Effective Core Potentials. J. Chem. TheoryComput. 2020, 16, 1482−1502.(59) Zen, A.; Coccia, E.; Luo, Y.; Sorella, S.; Guidoni, L. Static anddynamical correlation in diradical molecules by quantum monte carlousing the jastrow antisymmetrized geminal power ansatz. J. Chem.Theory Comput. 2014, 10, 1048−1061.(60) Casula, M.; Sorella, S. Improper s-wave symmetry of theelectronic pairing in iron-based superconductors by first-principlescalculations. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88,155125.(61) Becca, F.; Sorella, S. Quantum Monte Carlo Approaches forCorrelated Systems; Cambridge University Press, 2017.(62) Sorella, S.; Casula, M.; Rocca, D. Weak binding between twoaromatic rings: Feeling the van der Waals attraction by quantumMonte Carlo methods. J. Chem. Phys. 2007, 127, 014105.(63) Neuscamman, E. Size Consistency Error in the AntisymmetricGeminal Power Wave Function can be Completely Removed. Phys.Rev. Lett. 2012, 109, 203001.(64) Grüneis, A.; Booth, G. H.; Marsman, M.; Spencer, J.; Alavi, A.;Kresse, G. Natural orbitals for wave function based correlatedcalculations using a plane wave basis set. J. Chem. Theory Comput.2011, 7, 2780−2785.(65) Sun, Q.; Berkelbach, T. C.; Blunt, N. S.; Booth, G. H.; Guo, S.;Li, Z.; Liu, J.; McClain, J. D.; Sayfutyarova, E. R.; Sharma, S.; et al.PySCF: the Python-based simulations of chemistry framework. WileyInterdiscip. Rev.: Comput. Mol. Sci. 2018, 8, No. e1340.(66) Sun, Q.; Zhang, X.; Banerjee, S.; Bao, P.; Barbry, M.; Blunt, N.S.; Bogdanov, N. A.; Booth, G. H.; Chen, J.; Cui, Z. H.; Eriksen, J. J.;Gao, Y.; Guo, S.; Hermann, J.; Hermes, M. R.; Koh, K.; Koval, P.;Lehtola, S.; Li, Z.; Liu, J.; Mardirossian, N.; McClain, J. D.; Motta, M.;Mussard, B.; Pham, H. Q.; Pulkin, A.; Purwanto, W.; Robinson, P. J.;Ronca, E.; Sayfutyarova, E. R.; Scheurer, M.; Schurkus, H. F.; Smith, J.E.; Sun, C.; Sun, S. N.; Upadhyay, S.; Wagner, L. K.; Wang, X.; White,A.; Whitfield, J. D.; Williamson, M. J.; Wouters, S.; Yang, J.; Yu, J. M.;Zhu, T.; Berkelbach, T. C.; Sharma, S.; Sokolov, A. Y.; Chan, G. K. L.Recent developments in the PySCF program package. J. Chem. Phys.2020, 153, 024109.(67) Nakano, K.; Kohulák, O.; Raghav, A.; Casula, M.; Sorella, S.TurboGenius: Python suite for high-throughput calculations of abinitio quantum Monte Carlo methods. J. Chem. Phys. 2023, 159,224801.(68) Posenitskiy, E.; Chilkuri, V. G.; Ammar, A.; Hapka, M.; Pernal,K.; Shinde, R.; Borda, E. J. L.; Filippi, C.; Nakano, K.; Kohulák, O.;Sorella, S.; de Oliveira Castro, P.; Jalby, W.; Ríos, P. L.; Alavi, A.;Scemama, A. TREXIO: A file format and library for quantumchemistry. J. Chem. Phys. 2023, 158, 174801.(69) Bennett, M. C.; Melton, C. A.; Annaberdiyev, A.; Wang, G.;Shulenburger, L.; Mitas, L. A new generation of effective corepotentials for correlated calculations. J. Chem. Phys. 2017, 147,224106.(70) Sorella, S. Green function Monte Carlo with stochasticreconfiguration. Phys. Rev. Lett. 1998, 80, 4558−4561.(71) Sorella, S.; Casula, M.; Rocca, D. Weak binding between twoaromatic rings: Feeling the van der Waals attraction by quantumMonte Carlo methods. J. Chem. Phys. 2007, 127, 014105.(72) Casula, M.; Filippi, C.; Sorella, S. Diffusion Monte Carlomethod with lattice regularization. Phys. Rev. Lett. 2005, 95, 100201.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXMhttps://doi.org/10.1063/1.4948778https://doi.org/10.1063/1.4948778https://doi.org/10.1063/1.4948778https://doi.org/10.1002/jcc.24382https://doi.org/10.1002/jcc.24382https://doi.org/10.1021/acs.jctc.8b00393?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.8b00393?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.8b00393?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/5.0026324https://doi.org/10.1063/5.0026324https://doi.org/10.1103/PhysRevResearch.2.033429https://doi.org/10.1103/PhysRevResearch.2.033429https://doi.org/10.1103/PhysRevResearch.4.013021https://doi.org/10.1103/PhysRevResearch.4.013021https://doi.org/10.1038/s41467-023-37609-3https://doi.org/10.1038/s41467-023-37609-3https://doi.org/10.1038/s41557-020-0544-yhttps://doi.org/10.1038/s41557-020-0544-yhttps://doi.org/10.1038/s41467-020-15724-9https://doi.org/10.1038/s41467-020-15724-9https://doi.org/10.1103/PhysRevLett.121.066402https://doi.org/10.1103/PhysRevLett.121.066402https://doi.org/10.1021/acs.jctc.9b00295?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.9b00295?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.9b00295?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.0c00165?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.0c00165?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.0c00165?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.2c00556?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.2c00556?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.2c01141?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.2c01141?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1103/PhysRevB.79.195117https://doi.org/10.1103/PhysRevB.79.195117https://doi.org/10.1103/PhysRevB.79.195117https://doi.org/10.1103/PhysRevB.80.125110https://doi.org/10.1103/PhysRevB.80.125110https://doi.org/10.1103/PhysRevB.80.125110https://doi.org/10.1063/1.4861222https://doi.org/10.1063/1.4861222https://doi.org/10.1063/1.4861222https://doi.org/10.1103/physrevlett.104.193001https://doi.org/10.1103/physrevlett.104.193001https://doi.org/10.1103/PhysRevA.105.032815https://doi.org/10.1103/PhysRevA.105.032815https://doi.org/10.1021/ct501122z?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct501122z?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct501122z?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/5.0005037https://doi.org/10.1063/5.0005037https://doi.org/10.1021/acs.jctc.9b00962?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.9b00962?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct401008s?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct401008s?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct401008s?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1103/PhysRevB.88.155125https://doi.org/10.1103/PhysRevB.88.155125https://doi.org/10.1103/PhysRevB.88.155125https://doi.org/10.1063/1.2746035https://doi.org/10.1063/1.2746035https://doi.org/10.1063/1.2746035https://doi.org/10.1103/PhysRevLett.109.203001https://doi.org/10.1103/PhysRevLett.109.203001https://doi.org/10.1021/ct200263g?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/ct200263g?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1002/wcms.1340https://doi.org/10.1063/5.0006074https://doi.org/10.1063/5.0179003https://doi.org/10.1063/5.0179003https://doi.org/10.1063/5.0148161https://doi.org/10.1063/5.0148161https://doi.org/10.1063/1.4995643https://doi.org/10.1063/1.4995643https://doi.org/10.1103/PhysRevLett.80.4558https://doi.org/10.1103/PhysRevLett.80.4558https://doi.org/10.1063/1.2746035https://doi.org/10.1063/1.2746035https://doi.org/10.1063/1.2746035https://doi.org/10.1103/PhysRevLett.95.100201https://doi.org/10.1103/PhysRevLett.95.100201pubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as(73) Nakano, K.; Maezono, R.; Sorella, S. Speeding up ab initiodiffusion Monte Carlo simulations by a smart lattice regularization.Phys. Rev. B 2020, 101, 155106.(74) Momma, K.; Izumi, F. VESTA3 for three-dimensionalvisualization of crystal, volumetric and morphology data. J. Appl.Crystallogr. 2011, 44, 1272−1276.(75) Salem, L.; Rowland, C. The Electronic Properties of Diradicals.Angew Chem. Int. Ed. Engl. 1972, 11, 92−111.(76) Barbatti, M.; Paier, J.; Lischka, H. Photochemistry of ethylene:A multireference configuration interaction investigation of theexcited-state energy surfaces. J. Chem. Phys. 2004, 121, 11614−11624.(77) Marchi, M.; Azadi, S.; Sorella, S. Fate of the resonating valencebond in graphene. Phys. Rev. Lett. 2011, 107, 086807.(78) Genovese, C.; Meninno, A.; Sorella, S. Assessing the accuracyof the Jastrow antisymmetrized geminal power in the H4 modelsystem. J. Chem. Phys. 2019, 150, 084102.(79) Neese, F. The ORCA program system. Wiley Interdiscip. Rev.:Comput. Mol. Sci. 2012, 2, 73−78.(80) Neese, F. Software update: the ORCA program system, version4.0. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018, 8, 4−9.(81) Zen, A.; Sorella, S.; Gillan, M. J.; Michaelides, A.; Alfe,̀ D.Boosting the accuracy and speed of quantum Monte Carlo: Sizeconsistency and time step. Phys. Rev. B 2016, 93, 241118.(82) Van Der Goetz, B. W.; Neuscamman, E. Suppressing IonicTerms with Number-Counting Jastrow Factors in Real Space. J. Chem.Theory Comput. 2017, 13, 2035−2042.(83) Van Der Goetz, B. W.; Otis, L.; Neuscamman, E. Clean andConvenient Tessellations for Number Counting Jastrow Factors. J.Chem. Theory Comput. 2019, 15, 1102−1121.(84) Booth, G. H.; Alavi, A. Approaching chemical accuracy usingfull configuration-interaction quantum Monte Carlo: A study ofionization potentials. J. Chem. Phys. 2010, 132, 174104.(85) Petruzielo, F. R.; Toulouse, J.; Umrigar, C. J. Approachingchemical accuracy with quantum Monte Carlo. J. Chem. Phys. 2012,136, 124116.(86) Morales, M. A.; McMinis, J.; Clark, B. K.; Kim, J.; Scuseria, G.E. Multideterminant wave functions in quantum Monte Carlo. J.Chem. Theor. Comput. 2012, 8, 2181−2188.(87) Giner, E.; Scemama, A.; Caffarel, M. Fixed-node diffusionMonte Carlo potential energy curve of the fluorine molecule F2 usingselected configuration interaction trial wavefunctions. J. Chem. Phys.2015, 142, 044115.(88) Caffarel, M.; Applencourt, T.; Giner, E.; Scemama, A.Communication: Toward an improved control of the fixed-nodeerror in quantum Monte Carlo: The case of the water molecule. J.Chem. Phys. 2016, 144, 151103.(89) Yao, Y.; Giner, E.; Li, J.; Toulouse, J.; Umrigar, C. J. Almostexact energies for the Gaussian-2 set with the semistochastic heat-bathconfiguration interaction method. J. Chem. Phys. 2020, 153, 124117.(90) Scemama, A.; Giner, E.; Benali, A.; Loos, P.-F. Taming thefixed-node error in diffusion Monte Carlo via range separation. J.Chem. Phys. 2020, 153, 174107.(91) Benali, A.; Gasperich, K.; Jordan, K. D.; Applencourt, T.; Luo,Y.; Bennett, M. C.; Krogel, J. T.; Shulenburger, L.; Kent, P. R.; Loos,P.-F.; et al. Toward a systematic improvement of the fixed-nodeapproximation in diffusion Monte Carlo for solids�A case study indiamond. J. Chem. Phys. 2020, 153, 184111.(92) Malone, F. D.; Benali, A.; Morales, M. A.; Caffarel, M.; Kent, P.R. C.; Shulenburger, L. Systematic comparison and cross-validation offixed-node diffusion Monte Carlo and phaseless auxiliary-fieldquantum Monte Carlo in solids. Phys. Rev. B 2020, 102, 161104.(93) Carleo, G.; Nomura, Y.; Imada, M. Constructing exactrepresentations of quantum many-body systems with deep neuralnetworks. Nat. Commun. 2018, 9, 5322.(94) Carleo, G.; Troyer, M. Solving the quantum many-bodyproblem with artificial neural networks. Science 2017, 355, 602−606.(95) Nomura, Y.; Darmawan, A. S.; Yamaji, Y.; Imada, M. RestrictedBoltzmann machine learning for solving strongly correlated quantumsystems. Phys. Rev. B 2017, 96, 205152.(96) Nomura, Y.; Imada, M. Dirac-Type Nodal Spin LiquidRevealed by Refined Quantum Many-Body Solver Using Neural-Network Wave Function, Correlation Ratio, and Level Spectroscopy.Phys. Rev. X 2021, 11, 031034.(97) Viteritti, L. L.; Rende, R.; Becca, F. Transformer VariationalWave Functions for Frustrated Quantum Spin Systems. Phys. Rev.Lett. 2023, 130, 236401.(98) Casula, M. Beyond the locality approximation in the standarddiffusion Monte Carlo method. Phys. Rev. B: Condens. Matter Mater.Phys. 2006, 74, 161102.(99) Ten Haaf, D. F. B.; Bemmel, H. J. M. V.; Eeuwen, J. M. J. V. I.;Saarloos, W. V.; Ceperley, D. M. Proof for an upper bound in fixed-node Monte Carlo for lattice fermions. Phys. Rev. 1995, 51, 13039.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c00139J. Chem. Theory Comput. XXXX, XXX, XXX−XXXNhttps://doi.org/10.1103/PhysRevB.101.155106https://doi.org/10.1103/PhysRevB.101.155106https://doi.org/10.1107/S0021889811038970https://doi.org/10.1107/S0021889811038970https://doi.org/10.1002/anie.197200921https://doi.org/10.1063/1.1807378https://doi.org/10.1063/1.1807378https://doi.org/10.1063/1.1807378https://doi.org/10.1103/PhysRevLett.107.086807https://doi.org/10.1103/PhysRevLett.107.086807https://doi.org/10.1063/1.5081933https://doi.org/10.1063/1.5081933https://doi.org/10.1063/1.5081933https://doi.org/10.1002/wcms.81https://doi.org/10.1002/wcms.1327https://doi.org/10.1002/wcms.1327https://doi.org/10.1103/physrevb.93.241118https://doi.org/10.1103/physrevb.93.241118https://doi.org/10.1021/acs.jctc.7b00158?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.7b00158?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.8b01139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jctc.8b01139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/1.3407895https://doi.org/10.1063/1.3407895https://doi.org/10.1063/1.3407895https://doi.org/10.1063/1.3697846https://doi.org/10.1063/1.3697846https://doi.org/10.1021/ct3003404?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1063/1.4905528https://doi.org/10.1063/1.4905528https://doi.org/10.1063/1.4905528https://doi.org/10.1063/1.4947093https://doi.org/10.1063/1.4947093https://doi.org/10.1063/5.0018577https://doi.org/10.1063/5.0018577https://doi.org/10.1063/5.0018577https://doi.org/10.1063/5.0026324https://doi.org/10.1063/5.0026324https://doi.org/10.1063/5.0021036https://doi.org/10.1063/5.0021036https://doi.org/10.1063/5.0021036https://doi.org/10.1103/PhysRevB.102.161104https://doi.org/10.1103/PhysRevB.102.161104https://doi.org/10.1103/PhysRevB.102.161104https://doi.org/10.1038/s41467-018-07520-3https://doi.org/10.1038/s41467-018-07520-3https://doi.org/10.1038/s41467-018-07520-3https://doi.org/10.1126/science.aag2302https://doi.org/10.1126/science.aag2302https://doi.org/10.1103/PhysRevB.96.205152https://doi.org/10.1103/PhysRevB.96.205152https://doi.org/10.1103/PhysRevB.96.205152https://doi.org/10.1103/PhysRevX.11.031034https://doi.org/10.1103/PhysRevX.11.031034https://doi.org/10.1103/PhysRevX.11.031034https://doi.org/10.1103/PhysRevLett.130.236401https://doi.org/10.1103/PhysRevLett.130.236401https://doi.org/10.1103/PhysRevB.74.161102https://doi.org/10.1103/PhysRevB.74.161102https://doi.org/10.1103/PhysRevB.51.13039https://doi.org/10.1103/PhysRevB.51.13039pubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c00139?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as