{"id":"ab61d349-0e32-417e-8f66-6d2851074d1f","arxiv_id":"2505.20187","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Optimizing the Jastrow factor against CASSCF or FCIQMC references removes single-reference artifacts and yields chemically accurate transcorrelated energies for strongly correlated molecules.","lead":"This paper fixes a known failure of the transcorrelated quantum chemistry method by tuning its Jastrow factor against multireference wavefunctions instead of a single Hartree-Fock determinant. The corrected workflow reproduces the nitrogen binding curve to chemical accuracy and gives fast-converging excitation energies for N2, CO, and ammonia.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The xTC approximation replaces multireference 2RDMs by antisymmetrized 1RDM products without quantifying the neglected cumulant; if this error exceeds ~1.6 mHa, the claimed chemical accuracy is coincidental.","rationale":"The paper's central advance is the use of multireference references for Jastrow optimization and for building the xTC Hamiltonian. All reported energies inherit whatever error the xTC approximation makes when applied to multireference references. The original xTC derivation is explicitly single-determinant; extending it to CASSCF or FCIQMC 1RDMs is an unvalidated assumption. This is more load-bearing than the acknowledged stochastic noise in the FCIQMC reference snapshot, because that noise is reducible by averaging, whereas the xTC cumulant neglect is a systematic bias in the effective Hamiltonian. The reader's conditional verdict is appropriate: the method is promising and well demonstrated, but the xTC error must be quantified before the chemical-accuracy claim can be taken as systematic. Since my concern matches the reader's primary weakest assumption, the verdict should remain unchanged.","tokens_in":14362,"tokens_out":3604,"duration_ms":37900,"concrete_test":"Compute TC-FCIQMC energies for N2 at r = 4.5, 6.0, and 8.0 bohr using the full transcorrelated Hamiltonian (keeping explicit three-body terms, i.e., no xTC approximation) and compare these with the xTC results reported in the paper. Deviations larger than 1-2 mHa would demonstrate that the xTC approximation is not controlled for multireference references. As a complementary check, evaluate the magnitude of the contracted 2RDM cumulant term in Eqs. (7a)-(7b) using the exact FCIQMC 2RDM and compare it with the antisymmetrized-product approximation used in xTC.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.1 derives the xTC Hamiltonian by normal-ordering with respect to a single determinant and then approximating higher-order density matrices as antisymmetrized products of the 1RDM (Eqs. 6-8), which is exact only for single-determinant references. In Section 2.2 the same expressions are used with CASSCF or FCIQMC multireference 1RDMs, silently neglecting the connected (cumulant) part of the 2RDM. This cumulant encodes static correlation, so the resulting xTC Hamiltonian may misrepresent the effective interaction in strongly multireference regions such as stretched N2 or excited states. The paper never estimates this error, yet all reported energies are computed with this approximate Hamiltonian. If the neglected cumulant contributes more than the claimed chemical-accuracy threshold (~1.6 mHa), the agreement with experiment and benchmark FCI results is not systematic but fortuitous.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a multireference extension of the transcorrelated workflow by optimizing the Jastrow factor in the presence of a multireference reference wavefunction (CASSCF or a truncated, unconverged FCIQMC CI vector) and using the corresponding multireference 1RDM in the xTC approximation. The central demonstrations are the N2 binding curve with aug-cc-pVTZ compared against the experimental potential of Le Roy et al., and vertical excitation energies for N2, CO, and NH3 compared against extrapolated FCI references and experiment. The authors report that the multireference Jastrow optimization removes the unphysical dip and large size-consistency error seen with an RHF-based Jastrow, yielding dissociation energies close to experiment (CASSCF-Jastrow 361.2 mHa, FCIQMC-Jastrow 364.2 mHa vs. experimental 363.7 mHa).","tokens_in":14538,"tokens_out":4366,"duration_ms":47696,"significance":"If the results hold, the paper provides a practical and general workflow for applying transcorrelated methods to strongly multireference problems, which is an important step for the field. The strengths are that the energies are benchmarked against external experimental and high-level theoretical data (LeRoy N2 potential, HEAT, extrapolated FCI from reference [43]), and the Jastrow parameters are variance-minimized rather than fitted to the target energies. The clean contrast between the RHF-Jastrow baseline (size-consistency error 12.7 mHa, non-parallelity error 8.0 mHa) and the two multireference variants (errors of -1.5/0.5 mHa and 3.1/4.6 mHa, respectively) directly supports the main methodological claim. However, the reported energies are all computed from the xTC Hamiltonian, whose multireference generalization still lacks a quantified error bound, and the FCIQMC-based Jastrow reference introduces uncontrolled stochastic noise.","major_comments":[{"comment":"The xTC approximation replaces higher-order density matrices by antisymmetrized products of the 1RDM, which the paper itself states is exact only for single-determinant references. In Section 2.2 the same approximation is then used with CASSCF and FCIQMC multireference 1RDMs, silently neglecting the connected (cumulant) part of the 2RDM. Since all reported TC-FCIQMC energies are obtained from this approximate Hamiltonian, the authors must quantify the error introduced by this neglect, for example by comparing xTC results against a full TC treatment that retains the three-body operator or explicitly includes the 2RDM cumulant, at representative geometries (e.g., N2 at equilibrium and at 10 bohr, and at least one excited state). Without such a numerical bound, the claimed chemical accuracy could be coincidental rather than systematic.","section":"Section 2.1-2.2, Eqs. (6)-(8)"},{"comment":"The FCIQMC-based Jastrow reference is described as a single imaginary-time snapshot from a 3×10^7-walker calculation, truncated to the 100 most important determinants, and the paper notes in Section 3.2 that this is a source of noise. No statistical error bars or convergence tests are reported for the resulting Jastrow parameters or for the final TC-FCIQMC energies, such as the FCIQMC-Jastrow dissociation energy of 364.2 mHa in Table 1. The authors should provide error bars from independent repetitions or show that the optimized Jastrow and final energies are stable with respect to snapshot length and determinant cutoff, particularly for the stretched geometries where the method is most stochastic.","section":"Section 3.1 and Section 3.2"},{"comment":"The abstract claims chemical accuracy across the entire binding curve, but the non-parallelity errors reported in Table 1 are 3.1 mHa (CASSCF-Jastrow) and 4.6 mHa (FCIQMC-Jastrow), both larger than the ±1.6 mHa chemical-accuracy window. A non-parallelity error larger than 3.2 mHa necessarily means at least one point lies outside that window, so the paper should report the maximum absolute deviation from experiment for each curve, or qualify the chemical-accuracy claim accordingly. This does not undermine the main methodological improvement, but it affects the precision of the abstract's central claim.","section":"Section 3.2, Table 1 and Figure 4"}],"minor_comments":[{"comment":"Equation (6) states that the Hamiltonian is normal ordered with respect to Φ_SD, while Section 2.2 uses the same xTC expressions with multireference 1RDMs; the notation should be revised to make clear that the reference in the normal-ordering formalism is not restricted to a single determinant.","section":"Section 2.1, Eq. (6)"},{"comment":"The pytchint library is cited as 'to be released'; if the code is not yet public, the authors should state its availability or provide a stable reference, since the reproducibility of the xTC integrals is relevant to the reader.","section":"Reference [38]"},{"comment":"The horizontal axis in Figures 5-7 is labeled with expressions such as '1 avdz' and '1 CBS' rather than a clear axis title; the figures should explicitly label the abscissa as 1/n_orb and define the basis-set abbreviations in the captions.","section":"Figures 5-7"},{"comment":"For ammonia, the paper states that the excitations were treated as vertical, which is likely the cause of the large discrepancy with experiment, but the corresponding experimental values should be identified as adiabatic (or the comparison should be explicitly stated as vertical-theory versus adiabatic-experiment) to avoid ambiguity in the figure.","section":"Section 3.3"},{"comment":"The phrase 'state-averaged CASSCF Jastrow ansatzes' is potentially confusing because the Jastrow factors are optimized state-specifically; the text should clarify that the orbitals come from state-averaged CASSCF while the CI vector and Jastrow are optimized for each target state individually.","section":"Section 3.3"},{"comment":"The initiator parameter n_add = 3 is mentioned but not defined; for reproducibility, the authors should briefly define the initiator approximation parameter or cite the relevant equation in reference [18].","section":"Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"The main barrier to acceptance is the unquantified xTC cumulant error for multireference references. If the authors can provide a numerical bound at representative geometries and excited states showing the error is below the claimed chemical-accuracy threshold, the paper would be a solid contribution. The FCIQMC-snapshot noise issue is secondary but should be addressed with error bars or convergence evidence. I would not recommend rejection because the central methodological comparison is sound and the external benchmarking is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is real: instead of optimizing the Jastrow factor against a single Slater determinant, they use CASSCF or FCIQMC references, and feed the matching 1RDM into the xTC approximation. That fixes the unphysical dip and the 12.7 mHa size-consistency error in N2, with the FCIQMC-Jastrow dissociation energy landing at 364.2 mHa versus 363.7 mHa experimental. The excited-state calculations also look good, reaching near-basis-set-limit accuracy with modest basis sets. This is a solid methodological step for the transcorrelated/FCIQMC community, not just a tweak.\n\nWhat the paper does well is the direct comparison: RHF-Jastrow versus the two multireference variants on the same system and basis. The evidence supports the central claim that the reference choice matters for Jastrow optimization. The benchmarks are against experiment and independent extrapolated FCI, so the numbers are not circular.\n\nThe soft spots are real but not fatal. First, the xTC approximation replaces higher-order density matrices with antisymmetrized products of the 1RDM, which is exact only for single determinants. With CASSCF or FCIQMC references, the neglected 2RDM cumulant is unquantified, and all reported energies ride on that approximation. The good agreement with experiment suggests the error is small in these systems, but that is a hope, not a demonstrated bound. Second, pytchint is not released, so the workflow is not fully reproducible as described. Third, the figures have no error bars, and several curves look noisy in the stretched region, which matters when claiming chemical accuracy across the whole curve. The CASSCF-Jastrow dissociation energy is off by about 2.5 mHa and the non-parallelity errors are 3-5 mHa, so the \"chemical accuracy\" phrasing is a little generous. None of this breaks the main finding, but it should be tightened in the final version.\n\nThis paper deserves a serious referee. I'd send it out, asking for an estimate of the cumulant error, release of the code or at least a stable version, and error bars on the figures. The central argument holds up; the presentation just needs to be more careful about what is demonstrated versus what is expected.","headline":"Genuine advance in transcorrelated methods: multireference Jastrow references fix the N2 curve, but the unquantified xTC cumulant error and unreleased code keep this from being fully settled.","tokens_in":15189,"tokens_out":1568,"would_cite":true,"duration_ms":17608,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that optimising the Jastrow factor against a multireference wavefunction, rather than a single Hartree-Fock determinant, removes the transcorrelated method's failure on strongly correlated systems, giving chemically…","keywords":["transcorrelated method","Jastrow factor","multireference wavefunction","full configuration interaction quantum Monte Carlo","xTC approximation","nitrogen binding curve","excitation energies","chemical accuracy"],"falsifier":"Perform a TC-FCIQMC calculation at a strongly stretched N-N distance (e.g. 6 bohr) with the full three-body transcorrelated Hamiltonian instead of the xTC approximation, using the same Jastrow factor and reference; if the energy shifts by more than about 1.6 mHa relative to the xTC value, the reported chemical accuracy is an artefact of the approximation rather than a systematic result.","tokens_in":14125,"feed_emoji":"⚛️","tokens_out":7303,"duration_ms":68198,"temperature":0.7,"pith_summary":"The paper claims that the transcorrelated workflow fails on multireference problems because the Jastrow factor is optimised against a single determinant. The fix is to optimise J against a multireference reference wavefunction, either from a small FCIQMC run or from CASSCF, and to feed the same reference's one-body density matrix into the xTC approximation. With this change, TC-FCIQMC reproduces the experimental N$_2$ binding curve to chemical accuracy across the full range of bond lengths, removes the unphysical dip and size-consistency error seen with the Hartree-Fock reference, and yields vertical excitation energies for N$_2$, CO and NH$_3$ that match much larger-basis non-transcorrelated calculations. The authors conclude that the main obstacle in the transcorrelated workflow is resolved.","feed_headline":"Multireference Jastrow fix hits chemical accuracy for N2","feed_subtitle":"Optimising the correlator against CASSCF or FCIQMC references removes the transcorrelated method's bond-stretching artefacts.","key_machinery":"The load-bearing object is the multireference Jastrow optimisation: the variance objective $\\sigma^2_{\\mathrm{ref}}$ is built from the similarity-transformed Hamiltonian acting on the expansion $\\Phi_0 = \\sum_I c_I |D_I\\rangle$, and the same $\\Phi_0$ supplies the one-body reduced density matrix used by the xTC approximation to fold three-body terms into one- and two-body operators. The Jastrow factor is a polynomial in electron-electron, electron-nuclear, and electron-electron-nuclear distances with optimised coefficients, truncated at 100 determinants from the reference CI vector. Together these choices define a TC Hamiltonian whose FCIQMC solution is evaluated with a multi-determinant trial wavefunction to control stochastic noise.","core_discovery":"On the paper's own terms, the central discovery is that the choice of reference wavefunction during Jastrow optimisation is the decisive ingredient for transcorrelated calculations of strongly multireference systems. Beginning from the Jastrow ansatz $\\Psi = e^{J}\\Phi$, the authors show that when $\\Phi$ is a single Slater determinant the optimised Jastrow carries a single-reference bias: the resulting TC-FCIQMC binding curve of N$_2$ acquires an unphysical dip near 6 bohr and a long-distance asymptote about 10 mHa below twice the atomic energy. Replacing $\\Phi$ with a multireference expansion—a truncated FCIQMC wavefunction or a CASSCF(10e,8o) wavefunction—and using the corresponding one-body reduced density matrix in the xTC approximation removes these artefacts. The FCIQMC-Jastrow route gives a dissociation energy of 364.2 mHa against the experimental 363.7 mHa, with size-consistency error 0.5 mHa and non-parallelity error 4.6 mHa; the CASSCF-Jastrow route is slightly less accurate but deterministic. The workflow extends to excited states by optimising a state-specific Jastrow for each target, with accurate vertical excitation energies for N$_2$, CO and NH$_3$ using small basis sets.","pith_inferences":["The same workflow is a testable candidate for other strongly correlated curves (e.g. O$_2$, C$_2$, or stretched CH bonds), where the single-reference dip should reappear with an RHF Jastrow and disappear with a multireference one.","Because the FCIQMC reference is a single imaginary-time snapshot, averaging the CI vector and 1RDM over a period of imaginary time would likely reduce the remaining noise in the binding curve; the authors note this option but do not implement it.","The apparent success raises the question of whether the xTC approximation's neglect of the 2RDM cumulant is benign for multireference references; a cumulant-corrected or exact-three-body TC calculation at one stretched geometry would settle whether the chemical accuracy is systematic.","Iterating the workflow self-consistently—using the TC-FCIQMC output as the next reference for Jastrow optimisation—points toward a fully correlated TC-MCSCF method, an outlook the authors sketch."],"forward_implications":["TC-FCIQMC with a multireference Jastrow reproduces the full experimental N$_2$ binding curve within chemical accuracy (about $\\pm 1.6$ mHa), including the strongly stretched regime where single-reference methods fail.","The size-consistency error drops from 12.7 mHa with a Hartree-Fock reference to 0.5 mHa with the FCIQMC-Jastrow and $-$1.5 mHa with the CASSCF-Jastrow, restoring the correct long-distance asymptote.","Vertical excitation energies computed at aug-cc-pVDZ are comparable to extrapolated FCI at aug-cc-pVQZ, implying much faster basis-set convergence for excited states.","A state-specific Jastrow, aided by a spin-penalty term, lets the method target individual singlet excited states without collapsing to lower triplets.","The CASSCF-Jastrow variant is deterministic and available from standard quantum chemistry tools, making the workflow practical as a default choice."],"supporting_citations":[{"why":"Supplies the variance-optimised Jastrow workflow and the polynomial Jastrow form that this paper modifies.","marker":"[20]"},{"why":"Introduces the xTC approximation that folds three-body terms into one- and two-body operators using the 1RDM.","marker":"[22]"},{"why":"Provides the experimental N2 potential curve used as the ground truth for the binding curve comparison.","marker":"[27]"},{"why":"Establishes that FCIQMC can handle the non-Hermitian transcorrelated Hamiltonian, the solver used throughout.","marker":"[19]"},{"why":"Defines the initiator approximation used in all FCIQMC calculations of this paper.","marker":"[18]"},{"why":"The earlier TC-DMRG study that found a similar dip artefact, motivating the multireference reference.","marker":"[28]"},{"why":"Supplies the extrapolated FCI reference data that the excitation energies are compared against.","marker":"[43]"},{"why":"Provides the replica method used to compute the 1RDM from FCIQMC.","marker":"[30]"}],"fun_headline_variants":["Multireference Jastrow fixes transcorrelated artefacts","Reference wavefunction choice cures transcorrelated bias","Multireference Jastrow yields chemical accuracy for N2","CASSCF or FCIQMC references remove Jastrow artefacts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper relies on the xTC approximation—replacing the higher-order density matrices by antisymmetrised products of the one-body density matrix—being accurate for genuinely multireference references, yet the error from the dropped cumulant is never quantified.","fun_headline_variants_meta":{"raw":{"variants":["Multireference Jastrow fixes transcorrelated artefacts","Reference wavefunction choice cures transcorrelated bias","Multireference Jastrow yields chemical accuracy for N2","CASSCF or FCIQMC references remove Jastrow artefacts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000652,"raw_usage":{"total_tokens":3002,"prompt_tokens":973,"completion_tokens":2029,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":1959}},"tokens_in":589,"tokens_out":2029,"duration_ms":17210,"temperature":1.0,"reasoning_tokens":1959,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:58:22.756824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform a TC-FCIQMC calculation at a strongly stretched N-N distance (e.g. 6 bohr) with the full three-body transcorrelated Hamiltonian instead of the xTC approximation, using the same Jastrow factor and reference; if the energy shifts by more than about 1.6 mHa relative to the xTC value, the reported chemical accuracy is an artefact of the approximation rather than a systematic result.","supporting_citations":[{"cited_title":"Le Roy, Yiye Huang, and Calvin Jary","cited_arxiv_id":null,"evidence_quote":"Provides the experimental N2 potential curve used as the ground truth for the binding curve comparison."},{"cited_title":"Combining the Transcorrelated method with Full Configuration Interaction Quantum Monte Carlo: application to the homogeneous electron gas","cited_arxiv_id":"1712.07524","evidence_quote":"Establishes that FCIQMC can handle the non-Hermitian transcorrelated Hamiltonian, the solver used throughout."},{"cited_title":"A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks.Journal of Chemical Theory and Computation, 14(8):4360–4379, 2018","cited_arxiv_id":null,"evidence_quote":"Supplies the extrapolated FCI reference data that the excitation energies are compared against."},{"cited_title":"Unbiased reduced density matrices and electronic properties from full configuration interaction quan- tum Monte Carlo.The Journal of Chemical Physics, 141:244117, 2014","cited_arxiv_id":null,"evidence_quote":"Provides the replica method used to compute the 1RDM from FCIQMC."}],"review_version":1}