{"id":"2087d9e1-2367-48cb-989d-e65b1f2b6f9f","arxiv_id":"2505.19909","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Using local pseudo-Hamiltonians makes neural-network quantum Monte Carlo about ten times faster and enables simulations of iron-sulfur clusters with over 250 electrons.","lead":"Neural-network quantum Monte Carlo gets a speed and accuracy boost by replacing all-electron and semilocal pseudopotential treatments with local pseudo-Hamiltonians, allowing simulations of iron-sulfur clusters with hundreds of electrons. The result is a practical scaling advance for accurate electronic-structure calculations in chemistry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.4's Forward-Laplacian identity (Eq. 15) treats the position-dependent change of coordinates v_i = Q_i^{-1} r_i as if Q_i were constant; as written this exact identity does not hold, and the 'negligible overhead' claim rests on it.","rationale":"I read the paper as a genuine engineering advance: coupling local pseudo-Hamiltonians with neural-network QMC is plausible, the open-source code is a real asset, and the reported speedups on sulfur systems are credible if the Hamiltonian is evaluated correctly. My main reservation is not the fit-transferability issue emphasized by the reader, although that is also valid and I would keep it as a condition. The sharper concern is internal to Section 4.4: Eq. (15) looks like an illegitimate use of a coordinate transformation whose Jacobian actually depends on the coordinate. If Q_i is constant, then A(r_i) cannot vary as required; if Q_i varies, the claimed identity is not exact and the missing first-order terms are not accounted for. This is the type of defect that would invalidate the central claim, but it is also directly testable because the authors have released code. I therefore do not change the reader's CONDITIONAL verdict: the paper should be accepted only after the Eq. (15) identity is either proven with the correct chain rule or validated numerically against a direct implementation of Eq. (8). I also note the secondary issues the reader raised: no statistical error bars in the main accuracy figures, some validation targets overlap the sulfur-PH fitting set, and the AE baselines are unconverged; these are further reasons why the current evidence is conditional rather than definitive.","tokens_in":20828,"tokens_out":21945,"duration_ms":265773,"concrete_test":"Cross-check the JaQMC PH implementation against a direct reference implementation of Eq. (8) on a small system where both can be run with the same LapNet wavefunction checkpoint, e.g., the S atom or S2 at a fixed geometry. Compute local energies on a few hundred sampled configurations using the official Forward-Laplacian path and using standard automatic differentiation or finite differences applied to the literal PH Hamiltonian. If the two sets of local energies agree to numerical precision, Eq. (15) is being implemented correctly and the concern is resolved. If they disagree beyond machine/large-tolerance, the Forward-Laplacian treatment is not evaluating the stated pseudo-Hamiltonian, and the headline efficiency and accuracy claims need to be re-examined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central efficiency advantage of the pseudo-Hamiltonian is that the L^2 term adds almost no cost under the Forward Laplacian framework. The key step is Eq. (15): sum_alpha partial^2_{v_{i,alpha}} psi equals sum_{alpha,beta} A_{alpha,beta}(r_i) partial_{r_{i,alpha}} partial_{r_{i,beta}} psi, with v_i = Q_i^{-1} r_i and A(r_i) = Q_i^T Q_i. This identity is exact only when Q_i is independent of the coordinate being differentiated. But A(r_i) is position dependent through V_{L2,I}(|r_i - R_I|) (Supplementary Eq. 22), so Q_i must depend on r_i. The chain rule then produces additional first-order terms involving derivatives of Q_i, and those terms are not present in Eq. (15) or in the explicit b_alpha formula of Supplementary Eq. (23). If the implementation treats Q_i as a detached constant when computing derivatives, the local energy is not the pseudo-Hamiltonian of Eq. (8), so both the reported PH energies and the over-tenfold speedup would not correspond to the stated Hamiltonian. This is the most load-bearing point because it sits underneath every accuracy and efficiency comparison. The reader's transferability concern is legitimate but secondary: even a perfectly transferable sulfur PH must be evaluated with the correct differential operator.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to use local pseudo-Hamiltonians (PHs) inside neural-network quantum Monte Carlo (NNQMC) as a cheaper and more accurate alternative to all-electron (AE) and semilocal effective-core-potential (ECP) Hamiltonians. The authors construct a sulfur PH by fitting to ccECP reference data, combine it with existing transition-metal PHs, and report over-tenfold speedups, near-chemical-accuracy benchmarks on sulfur molecules and transition-metal sulfides, and a demonstration on Fe2S2(SCH3)4 and Fe4S4(SCH3)4. The core technical claim is that the PH's L^2 term can be evaluated at almost no extra cost through a Forward-Laplacian coordinate transformation.","tokens_in":21191,"tokens_out":5279,"duration_ms":60683,"significance":"If the implementation is correct, the manuscript would address a real bottleneck: nonlocal ECP projections are expensive in NNQMC, and local PHs plus the Forward Laplacian could substantially extend NNQMC to transition-metal systems. The paper has several genuine strengths: the PH construction procedure in Section 4.3 and Supplementary Note 4 is detailed, the code is open-sourced, and the iron-sulfur cluster applications are ambitious. However, the central efficiency derivation in Section 4.4 appears mathematically inconsistent as written, and the accuracy claims in Fig. 3 are partly tested against the same ccECP data used to fit the sulfur PH. The promise of the approach is clear, but the current manuscript does not yet establish the stated Hamiltonian, speedups, or transferability at the claimed level.","major_comments":[{"comment":"The Forward-Laplacian identity is not valid as written for the PH operator. Equation (14) defines v_i = Q_i^{-1} r_i with Q_i an 'invertible constant matrix' for the decomposition A(r_i) = Q_i^T Q_i, but A_{alpha beta}(r_i) in Supplementary Eq. (22) depends on the electron position through V_{L2,I}(|r_i - R_I|). A position-dependent A cannot be represented by a single constant Q_i per electron and geometry. The exact chain rule for an r_i-dependent Q_i contains additional first-order derivative terms involving partial derivatives of Q_i, and these terms are absent from Eq. (15) and from the explicit b_alpha formula in Supplementary Eq. (23). If the implementation evaluates the Laplacian by treating Q_i as a detached constant, then the local energy computed is not the pseudo-Hamiltonian of Eq. (8). This issue sits under both the claimed 'negligible overhead' for the L^2 term and every PH energy reported in Figs. 3-4. Please provide the corrected identities including the Q_i-derivative contributions, or state explicitly which operator was actually implemented and demonstrate that the omitted terms vanish by construction or are negligible.","section":null},{"comment":"The sulfur PH is fitted directly to ccECP reference data: atomic eigenenergies, norm conservation, excitation energies, the S2 binding curve, and the FeS binding curve (cost function in Supplementary Note 4, Eqs. (4)-(13), and step 2 of Section 4.3). The close PH/ECP agreement for the S2 potential energy curve in Fig. 3c and for FeS in Fig. 3d is therefore partially a consistency check of the fitting procedure rather than an independent validation. To support the transferability claim, the authors should explicitly separate training systems from validation systems, or benchmark the sulfur PH on sulfur environments (and molecules) that were not used in the optimization, and then reassess the wording of the accuracy conclusions.","section":null},{"comment":"No statistical uncertainties are reported for the energy differences in Fig. 3 or for the timing comparisons in Fig. 2. NNQMC energies are stochastic; the text reports values like -467.8407(3) in Supplementary Table 2, so error bars are available for some quantities, but the central benchmark plots omit them. Statements such as 'PH achieves chemical accuracy for most species' (Fig. 3b), 'within the experimental uncertainties' (Fig. 3d), and 'over tenfold' acceleration (Section 2.2) cannot be assessed without error bars or repeated-run estimates. Please add uncertainties to all plotted quantities and to the efficiency ratios, and qualify conclusions accordingly.","section":null}],"minor_comments":[{"comment":"Typo: 'psedo-Hamiltonian' should be 'pseudo-Hamiltonian'.","section":null},{"comment":"Typo: 'Dissociatino Energy Error' should be 'Dissociation Energy Error'.","section":null},{"comment":"The symbol i is used both as an electron index and as the imaginary unit; this creates confusion. Use \\mathrm{i} for the imaginary unit or rename the electron index.","section":null},{"comment":"The inset magnifying the PH and ECP convergence curves is too small to read; please enlarge it or show the same data in a separate panel.","section":null},{"comment":"The statement that the coordinate transformation 'can be easily integrated with Forward Laplacian by modifying the input' is too brief for a reader wishing to reproduce the implementation; give explicit formulas for how the modified input and the derivative pattern are constructed in the code.","section":null},{"comment":"The magnetic exchange coupling constant J is quoted without an uncertainty. Since both E_BS, E_HS and the spin-squared expectation values are stochastic, please quote a statistical error for J.","section":null}],"recommendation":"major_revision","confidential_remarks":"The Forward-Laplacian issue in Section 4.4 is the main technical barrier; it is potentially fixable, but the authors need to either derive the correct differential operator or clearly specify the implemented operator and rebenchmark. The transferability concern is also substantive but could be addressed by reframing the claims and adding independent validation systems. I would not reject because the open-source code and detailed methodology should allow verification, and the proposed PH+NNQMC direction is valuable if the derivation is corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: the paper is a serious attempt to make NNQMC practical for transition-metal clusters by combining local pseudo-Hamiltonians with LapNet and the Forward Laplacian. The efficiency gains they report (10x+) would be a big deal if correct. But the derivation of the Forward Laplacian integration in Section 4.4 does not hold as written, and that is load-bearing.\n\nThe identity in Eq. (15) treats the matrix Q_i as constant in the coordinate transformation v_i = Q_i^{-1} r_i. But A(r_i), defined in Eq. (22) and used in Eq. (14), depends on position through V_L2(|r_i - R_I|). If Q_i depends on r_i, the chain rule produces extra first-order terms involving derivatives of Q_i. Those terms are not in Eq. (15) or in b_alpha in Eq. (23). So either the local energy implemented is not the pseudo-Hamiltonian of Eq. (8), or the derivation hides a real approximation. Either way, the claim that the L^2 term adds negligible cost needs a correct derivation.\n\nThis is not a minor typo. Every PH energy, every speedup number, and the Fe4S4 cluster result rest on this. If the implemented operator is not the pseudo-Hamiltonian, the benchmarks don't mean what the paper says.\n\nThere is real substance here. The sulfur PH construction is careful: they optimize to reproduce ccECP atomic eigenenergies, norms, excitations, and S2 binding curves, then refine with Bayesian optimization and CCSD(T)-level corrections. That is a substantial piece of parameterization work, building sensibly on earlier DMC pseudo-Hamiltonians. The iron-sulfur cluster demonstrations are nice showcases. But the validation is weaker than it looks: the sulfur PH is fitted to ccECP, and several benchmarks in Figure 3 are effectively tests of the fit. No statistical error bars appear in the main accuracy figures, which is unusual for QMC. The AE comparison uses runs that are far from converged, which makes the 'PH beats AE' claim less clean.\n\nThe paper is worth refereeing, but the referee should demand a corrected derivation of the Forward Laplacian step and a clear statement of what operator is actually implemented. If the identity is wrong, the authors need to show the correct coordinate transformation or the additional terms, and explain why the reported energies are still valid. Right now, the central efficiency claim is not established.\n\nMy take: this is a paper for the NNQMC community; they'll care a lot. If the math gets fixed, it's a strong contribution. As written, the load-bearing flaw keeps me from trusting the numbers. Send it to review—a serious referee is needed—but expect major revision.","headline":"Promising integration of local pseudo-Hamiltonians with NNQMC, but the Forward Laplacian derivation in Eq. (15) wrongly treats the position-dependent Q_i as constant, undercutting the reported efficiency and accuracy claims until fixed.","tokens_in":21750,"tokens_out":5243,"would_cite":false,"duration_ms":54897,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A fully local pseudo-Hamiltonian, not all-electron or semilocal effective-core forms, is the efficient, accurate Hamiltonian choice for neural-network quantum Monte Carlo, taking it to 268-electron iron-sulfur clusters.","keywords":["neural network quantum Monte Carlo","local pseudopotential","pseudo-Hamiltonian","effective core potential","iron-sulfur cluster","Forward Laplacian","variational Monte Carlo","diffusion Monte Carlo"],"falsifier":"Take the fitted sulfur pseudo-Hamiltonian and compute the dissociation curve of a sulfur-containing system outside the fitting set, for instance SO$_2$ at a highly stretched geometry or a periodic sulfur crystal, using NNQMC with PH, and compare against high-level CCSD(T) at the complete-basis limit. A systematic error growing beyond chemical accuracy with geometry would show that the ccECP-fitted local potential does not transfer beyond its training data.","tokens_in":20605,"feed_emoji":"⚛️","tokens_out":10130,"duration_ms":76872,"temperature":0.7,"pith_summary":"This paper argues that neural-network quantum Monte Carlo (NNQMC) becomes far more practical when the Hamiltonian is a local pseudo-Hamiltonian (PH) rather than the all-electron Coulomb form or the standard semilocal effective-core potential. The authors construct a PH for sulfur by fitting it to ccECP reference data, and combine it with the Forward Laplacian derivative scheme so that the PH's angular-momentum-squared term adds almost no cost. They report over tenfold faster training, accuracy equal to or better than all-electron and ECP calculations, and the first NNQMC treatment of a 268-electron iron-sulfur cubane cluster. If correct, this removes the main scaling bottleneck of NNQMC and extends the method to transition-metal chemistry.","feed_headline":"Neural quantum Monte Carlo gets 10x speedup from local pseudopotentials","feed_subtitle":"Fully local pseudo-Hamiltonians remove the core-electron bottleneck, pushing NNQMC to 268-electron iron-sulfur clusters.","key_machinery":"The load-bearing object is the pseudo-Hamiltonian $h^{\\mathrm{PH}}(r) = -\\tfrac{1}{2}\\, \\hat{p}\\, a(r)\\, \\hat{p} + V^{\\mathrm{PH}}_{\\mathrm{local}}(r) + V_{L2}(r)\\, \\hat{L}^2$, with $a(r)=0$ in this work so the kinetic operator keeps its standard form. Unlike a semilocal effective core potential, this form has no $|\\ell m\\rangle\\langle\\ell m|$ projectors; the angular dependence is carried by $\\hat{L}^2$, and inside the Forward Laplacian scheme the required second derivatives are evaluated through a per-electron coordinate scaling $v_i = Q_i^{-1} r_i$, turning them into ordinary Laplacians with negligible extra cost. For sulfur, the potential functions are fitted so that their channel matrix elements reproduce the ccECP reference values (Eq. 11), under the linear-dependency constraint $2V_0(r)-3V_1(r)=0$ and the positivity bound $1+2r^2 V_{L2}(r)>0$, using atomic eigenenergies, norm-conservation outside the cutoff, excitation energies, and the $S_2$/FeS binding curves as reference data. This single construction carries both the efficiency gain (fewer electrons, no spherical-harmonic integrals) and the accuracy gain (smooth core region without nuclear Coulomb singularities).","core_discovery":"The paper's central claim is that the pseudo-Hamiltonian (PH), a fully local pseudopotential that replaces the nonlocal spherical-harmonic projectors of an effective core potential with the angular-momentum-squared operator $\\hat{L}^2$ and a radial effective-mass term, is the preferred Hamiltonian choice inside NNQMC. Setting $a(r)=0$ keeps the kinetic term standard, and the $\\hat{L}^2$ term is absorbed into a Forward Laplacian coordinate transformation so its second derivatives are nearly free. Across sulfur chains, sulfur-containing molecules, transition-metal sulfides, and two iron-sulfur clusters, the authors find PH over tenfold faster than both all-electron and ECP treatments, accuracy matching or beating all-electron results, and agreement with ECP, MRCC, and experiment. They further obtain a magnetic exchange coupling constant $J=221\\,\\mathrm{cm}^{-1}$ for $[\\mathrm{Fe}_2\\mathrm{S}_2(\\mathrm{SCH}_3)_4]^{2-}$, consistent with experiments and high-level theory, and a physically meaningful spin distribution for the cubane cluster $[\\mathrm{Fe}_4\\mathrm{S}_4(\\mathrm{SCH}_3)_4]$. The paper concludes that PH generally enhances NNQMC's computational efficiency by over tenfold and enables reliable treatment of systems that were previously beyond the method's reach.","pith_inferences":["If the same fitting recipe transfers element by element, a library of local pseudo-Hamiltonians would make NNQMC routine for transition-metal and period-3 chemistry, not just the sulfur and iron cases demonstrated here.","The Forward-Laplacian-plus-$\\hat{L}^2$ trick should carry over to periodic solids through solid-state neural ansatz, potentially opening d- and f-electron solid-state problems to NNQMC.","A decisive test is systematic transferability benchmarking across oxidation states and coordination environments: the paper notes that PH construction still requires system-specific tuning, so the practical payoff depends on how much re-fitting each new element needs."],"forward_implications":["NNQMC can now handle transition-metal clusters with hundreds of electrons, such as the 268-electron cubane $[\\mathrm{Fe}_4\\mathrm{S}_4(\\mathrm{SCH}_3)_4]$, with over twentyfold acceleration.","The pseudo-Hamiltonian is strictly more efficient than semilocal ECPs inside NNQMC, because the $\\hat{L}^2$ term costs almost nothing under the Forward Laplacian scheme while ECP's spherical-harmonic integrals dominate the runtime.","Removing core electrons improves convergence as well as cost: PH reaches chemical accuracy after about 50,000 training iterations, whereas all-electron training is still far from convergence after 150,000.","PH-based NNQMC yields quantitatively reliable magnetic observables, including $J=221\\,\\mathrm{cm}^{-1}$ for the dinuclear iron-sulfur cluster, positioning the method as a practical tool for strongly correlated bioinorganic systems."],"supporting_citations":[{"why":"Supplies the neural-network ansatz and the Forward Laplacian scheme that makes the PH's extra second derivatives nearly free.","marker":"[10]"},{"why":"The ccECP reference data for sulfur and transition metals that the sulfur PH is fitted to reproduce.","marker":"[26]"},{"why":"Provides the locality-error-free pseudo-Hamiltonians for 3d transition metals used for iron in the clusters.","marker":"[29]"},{"why":"Provides the high-accuracy transition-metal effective-core/PH construction method extended to sulfur.","marker":"[30]"},{"why":"Introduces the original pseudo-Hamiltonian idea with a position-dependent mass and $\\hat{L}^2$ term.","marker":"[27]"},{"why":"Documents the computational overhead of nonlocal ECP terms in neural-network wavefunctions, motivating the fully local PH.","marker":"[33]"},{"why":"W4 high-confidence benchmark dataset used to judge the atomization-energy accuracy of PH, ECP, and AE.","marker":"[45]"},{"why":"Provides experimental and MRCC benchmark dissociation energies for the transition-metal sulfides.","marker":"[47]"},{"why":"Supplies the iron-sulfur cluster structures and high-level quantum-chemistry results that PH results are compared with.","marker":"[49]"}],"fun_headline_variants":["Pseudo-Hamiltonian gives neural quantum Monte Carlo 10x speed boost","Local pseudopotentials unlock neural quantum Monte Carlo for large systems","Neural QMC scales up with fully local pseudopotentials, tames Fe4S4","Fully local pseudopotentials accelerate neural quantum Monte Carlo 10-fold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sulfur pseudo-Hamiltonian is fitted to ccECP reference data for a few atomic and dimer properties, and the paper's broad accuracy claims depend on that fitted potential transferring reliably to every sulfur environment, molecule, and NNQMC calculation tested; if the fit only works near its training set, the general accuracy result does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Pseudo-Hamiltonian gives neural quantum Monte Carlo 10x speed boost","Local pseudopotentials unlock neural quantum Monte Carlo for large systems","Neural QMC scales up with fully local pseudopotentials, tames Fe4S4","Fully local pseudopotentials accelerate neural quantum Monte Carlo 10-fold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1565,"prompt_tokens":1012,"completion_tokens":553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":470}},"tokens_in":628,"tokens_out":553,"duration_ms":4979,"temperature":1.0,"reasoning_tokens":470,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:04:22.016398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the fitted sulfur pseudo-Hamiltonian and compute the dissociation curve of a sulfur-containing system outside the fitting set, for instance SO$_2$ at a highly stretched geometry or a periodic sulfur crystal, using NNQMC with PH, and compare against high-level CCSD(T) at the complete-basis limit. A systematic error growing beyond chemical accuracy with geometry would show that the ccECP-fitted local potential does not transfer beyond its training data.","supporting_citations":[{"cited_title":"Knowles, Frederick R","cited_arxiv_id":null,"evidence_quote":"Supplies the neural-network ansatz and the Forward Laplacian scheme that makes the PH's extra second derivatives nearly free."}],"review_version":1}