{"id":"ddbcd210-046a-48d3-9543-a42fe112e778","arxiv_id":"2506.10429","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Transcorrelated pseudopotential calculations yield ionization and excitation energies for Sc-Zn that are mostly within 1 kcal/mol of experiment using aug-cc-pVQZ, without complete basis set extrapolation.","lead":"This paper tests a transcorrelated Hamiltonian with pseudopotentials on all ten 3d transition metal atoms, from scandium to zinc, calculating ionization and 4s-to-3d excitation energies. It reports chemical accuracy in modest Gaussian basis sets without basis set extrapolation or explicit relativistic corrections.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Eq. 12 2-RDM reconstruction may bias the xTC Hamiltonian itself; CC/FCIQMC agreement cannot detect this shared bias.","rationale":"The reader's weakest-assumption analysis identifies exactly the load-bearing point: the xTC Hamiltonian is built using a 2-RDM reconstructed from the 1-RDM (Eq. 12), and the paper states that explicit 2-RDMs are under active investigation. This matters because the similarity-transformed Hamiltonian is the object shared by both the coupled-cluster and FCIQMC calculations; consistency between the two solvers therefore tests the solver, not the Hamiltonian. If the reconstruction is inaccurate for open-shell 3d states, every reported energy inherits the same bias. The strongest claim in the abstract — chemical accuracy for all atoms and properties without CBS extrapolation or DK — rests on this unvalidated approximation plus the empirical assertion of all errors below 1 kcal/mol. The latter is contradicted by the V and Co xTC-CCSDT AVQZ ionization potentials in Table 3, so an adjustment of the claim is needed regardless. A concrete, finite validation is feasible: compute exact FCIQMC 2-RDMs for a few representative atoms and rebuild the Hamiltonian, checking whether energy differences move outside the chemical-accuracy band. This directly tests whether the Eq. 12 approximation is benign or load-bearing. The paper has genuine strengths — the workflow is concrete, the results are reproducible in principle, and the consistency between CC and FCIQMC is meaningful for solver robustness — but the central methodological assumption is not yet independently verified.","tokens_in":16314,"tokens_out":5415,"duration_ms":66935,"concrete_test":"For V and Co in aug-cc-pVTZ (or aug-cc-pVQZ), compute the unbiased two-body RDM with FCIQMC (following Ref. 30) for the neutral ground state, the ion, and the relevant excited state, then rebuild the xTC Hamiltonian of Eqs. 3-5 using this explicit 2-RDM in place of the Eq. 12 reconstruction and rerun xTC-FCIQMC (or xTC-CCSDT). If the resulting IPs or excitation energies shift by more than about 0.3-0.5 kcal/mol relative to Table 3, the reconstructed 2-RDM is load-bearing and the 'chemical accuracy for all atoms' claim is not yet established; if the shifts are below 0.3 kcal/mol, the approximation is validated for these systems.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing assumption is Eq. 12: the antisymmetrized two-body RDM used to contract the three-body L-integrals in Eq. 4 is reconstructed as an antisymmetrized product of the 1-RDM. The resulting xTC Hamiltonian is therefore state-dependent and approximate for strongly correlated open-shell 3d states, and the paper explicitly notes that explicit 2-RDMs are 'under active investigation' (Section 4.5). Both xTC-CCSDT and xTC-FCIQMC solve this same approximate Hamiltonian, so their mutual agreement in total energies and the compactification metrics in Fig. 2 cannot validate the Hamiltonian itself. The good agreement with experiment for IPs and excitations could in principle arise from cancellation between the ground and ionized/excited states of the same biased Hamiltonian. This methodological concern is distinct from a second, more empirical issue: Table 3 shows that two of the ten xTC-CCSDT AVQZ ionization potentials (V: -1.42 kcal/mol, Co: -1.54 kcal/mol) fall outside the 1 kcal/mol band, so the abstract's 'chemical accuracy for all atoms and properties' is not supported even on the paper's own reported numbers.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper benchmarks transcorrelated (TC) theory with pseudopotentials for the 3d transition metal atoms Sc–Zn, computing 4s→3d excitation energies and first ionization potentials. The workflow combines orbital generation (HF/DFT/SA-CASSCF), FCIQMC-based trial wavefunctions, VMC-optimized Jastrow factors, and the xTC approximation of the transcorrelated Hamiltonian, with energies obtained from xTC-CCSD(T), xTC-CCSDT, and xTC-FCIQMC. The authors report mean absolute errors of 0.39 kcal/mol for excitations (AVQZ) and 0.55 kcal/mol for ionization potentials (xTC-CCSDT, AVQZ), and claim chemical accuracy for all atoms and properties without complete-basis-set extrapolation or Douglas–Kroll–Hess Hamiltonians.","tokens_in":16530,"tokens_out":4475,"duration_ms":50055,"significance":"If the results hold, this is a practically valuable demonstration that transcorrelation with pseudopotentials can reach near-experimental accuracy for a notoriously difficult row of atoms with modest basis sets and without explicit relativistic corrections. The explicit comparison to external experimental benchmarks and to prior coupled-cluster results [5] is a strength. The main advertised claim—chemical accuracy for every atom and property—is not supported by the paper's own data for the ionization potentials of V and Co. In addition, the xTC Hamiltonian is constructed using a 2-RDM reconstructed from the 1-RDM (Eq. 12), an approximation whose accuracy is not assessed for the strongly correlated open-shell states; this is a load-bearing modeling assumption because both correlated solvers share the same approximate Hamiltonian.","major_comments":[{"comment":"The statement 'chemical accuracy for all atoms and properties' is contradicted by Table 3: the xTC-CCSDT AVQZ ionization potentials for V and Co are -1.42 and -1.54 kcal/mol, respectively, which are outside the ±1 kcal/mol band. The MAE of 0.55 kcal/mol reflects average accuracy, not per-system chemical accuracy. The abstract, Section 6.5, and the Conclusions should be revised to state that chemical accuracy is achieved on average (MAE < 1 kcal/mol) but with individual outliers for V and Co IPs.","section":"Abstract; Section 6.5; Section 7"},{"comment":"The xTC Hamiltonian is built from a two-body density matrix reconstructed as the antisymmetrized product of the one-body density matrix (Eq. 12), with the note that explicit 2-RDMs are 'under active investigation.' For strongly correlated open-shell 3d states, this reconstruction may introduce a systematic bias in the Hamiltonian itself. Because both xTC-CCSDT and xTC-FCIQMC solve the same approximate Hamiltonian, their mutual agreement (Fig. 3) cannot validate the Hamiltonian. The authors should provide quantitative evidence for the accuracy of Eq. 12 (e.g., by comparing against explicit 2-RDMs for a subset of atoms, or by showing that results are insensitive to the reconstruction), or discuss the potential error and its possible cancellation in energy differences.","section":"Section 4.5, Eq. (12)"},{"comment":"The excited-state FCIQMC procedure orthogonalizes right eigenvectors (Eqs. 14–15), which is not formally correct for the non-Hermitian xTC Hamiltonian, as the right eigenvectors are not orthogonal. The authors justify this by citing a Hubbard model study [17], but the convergence to the correct excited-state energy for the molecular TC Hamiltonian is not guaranteed. The paper should provide additional validation for the excited-state energies, for example by comparing a few cases against a separate multireference method or against exact diagonalization in a small active space, to rule out a systematic bias in the 4s→3d excitation energies.","section":"Section 4.6, Eqs. (14)–(15)"}],"minor_comments":[{"comment":"There are several typographical errors in the expanded basis-set labels, such as 'A VTZ', 'A VZT', and 'A VQZ' instead of 'AVTZ' and 'AVQZ'.","section":"Abstract; Section 6.3; Section 7"},{"comment":"The notation 'spin quantum numbers Sz = m−2 and Sz = m' appears to use 2S_z rather than S_z (e.g., for Sc, m−2 = 1 corresponds to a doublet and m = 3 to a quartet). This should be defined explicitly to avoid confusion.","section":"Section 2"},{"comment":"The ionization potential entries for Ni are given to inconsistent precision (e.g., -2.196 and -0.705 for AVTZ and AVQZ), while other entries use two decimals. Uniform precision would improve readability.","section":"Table 3"},{"comment":"The statement that 'The table shows that in a quadruple-zeta basis set the xTC-FCIQMC and the xTC-CCSDT methods deliver chemical accuracy for all of the transition-metal atoms studied' is inconsistent with the V and Co IPs in the same table; this sentence should be amended for consistency.","section":"Section 6.5"},{"comment":"The conclusions repeat the 'chemical accuracy for atoms Sc–Zn' claim for IPs; this should be qualified as above, and the typo 'A VZT' should be corrected.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript makes a strong and interesting methodological case, but the overstatement of per-system chemical accuracy for IPs is a clear issue that must be fixed. The more substantive concern is the unvalidated 2-RDM reconstruction in Eq. (12); if the authors can show insensitivity or provide benchmark tests, the paper would be considerably stronger. The excited-state FCIQMC issue is also worth addressing, though the good agreement with experiment suggests it may be benign in practice. I would encourage the editor to request a revision rather than reject, as the core methodology and the reported MAEs are likely of interest to the quantum chemistry community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a worthwhile benchmark study: it shows that the transcorrelated Hamiltonian with pseudopotentials (xTC-PP) gets excitation energies and ionization potentials for Sc–Zn in AVTZ/AVQZ basis sets to within a few tenths of a kcal/mol of experiment, without CBS extrapolation or DKH. The MAEs (0.39 kcal/mol for excitations, 0.55 for IPs with CCSDT/AVQZ) are close to the best prior coupled-cluster results and better than most practical methods. That is a real, useful result, and the careful reporting (walker counts, Jastrow parameters, separate comparisons against experiment) is a plus.\n\nThe main soft spot is the abstract's claim of 'chemical accuracy for all atoms and properties'. Their own table shows V and Co IPs at -1.42 and -1.54 kcal/mol, both outside the ±1 kcal/mol band. 'Average chemical accuracy' is defensible; 'all atoms' is not. That is a straightforward fix.\n\nThe more substantive concern is Eq. 12: they reconstruct the antisymmetrized 2-RDM from the 1-RDM and use it to contract the three-body integrals. The paper notes explicit 2-RDMs are under investigation. This means the xTC Hamiltonian itself is approximate, and because both xTC-CCSDT and xTC-FCIQMC use the same Hamiltonian, their mutual agreement does not validate the Hamiltonian. The external agreement with experiment is the real evidence, and it is good, but in principle errors could cancel between ground and ionized/excited states. I don't think this is fatal—the comparison to experiment is not circular—but I would want to see the 2-RDM reconstruction tested (or at least its effect on the final properties estimated) before taking the workflow as a general route to heavier transition metals.\n\nMinor: the Jastrow is optimized per system and per basis set, so there are tunable parameters, but they are not fitted to the target values; the central comparison to experiment stays independent.\n\nVerdict: the paper deserves a serious referee. It is a clean demonstration of a practical workflow with new benchmark numbers, and the caveats are identifiable and addressable. I would send it to peer review and ask the authors to soften the abstract and discuss the 2-RDM approximation more openly.","headline":"Useful benchmark study of xTC-PP on Sc-Zn with honest reporting; abstract overclaims 'chemical accuracy for all atoms', and the 1-RDM-based 2-RDM in Eq. 12 needs scrutiny, but the work deserves peer review.","tokens_in":17102,"tokens_out":2794,"would_cite":true,"duration_ms":31288,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Transcorrelated theory brings all ten 3d transition-metal atoms to chemical accuracy.","keywords":["transcorrelated methods","transition metal atoms","pseudopotentials","chemical accuracy","ionization potentials","4s-3d excitations","coupled cluster","FCIQMC"],"falsifier":"Compute the same Sc–Zn excitation energies and ionization potentials with an explicit two-body density matrix from FCIQMC, instead of the antisymmetrized product, and compare the results; if the differences exceed roughly 1 kcal/mol for any atom, or if a strongly correlated case such as Cr or Fe changes by that amount, the approximation is load-bearing and the claimed accuracy is not intrinsic to the transcorrelated Hamiltonian.","tokens_in":16075,"feed_emoji":"⚛️","tokens_out":6826,"duration_ms":73710,"temperature":0.7,"pith_summary":"Transition-metal atoms are notoriously hard for quantum chemistry because the open 3d shell mixes static and dynamic correlation, relativistic effects cannot be ignored, and basis-set extrapolation is usually required for benchmark accuracy. This paper claims that a transcorrelated Hamiltonian, built with the xTC approximation and combined with scalar-relativistic pseudopotentials, removes those bottlenecks: treating the 3s and 3p semicore as valence and freezing only 1s–2p, it reproduces experimental ionization potentials and $4s\\to 3d$ excitation energies for all ten atoms Sc–Zn within $\\pm 1$ kcal/mol using aug-cc-pVTZ or aug-cc-pVQZ basis sets. The claim matters because it would make benchmark-quality transition-metal chemistry available in modest basis sets without complete-basis-set extrapolation or separate scalar-relativistic Hamiltonian corrections, and it points toward a practical route for transcorrelated methods on heavier transition-metal molecules and solids.","feed_headline":"All ten 3d transition metals reach chemical accuracy","feed_subtitle":"Ionization and 4s-to-3d excitation energies match experiment in modest basis sets, with no extrapolation or relativistic corrections.","key_machinery":"The load-bearing object is the second-quantized transcorrelated Hamiltonian $\\hat{H}_{\\mathrm{TC}} = e^{J} \\hat{H} e^{-J}$, evaluated in the xTC approximation so that only one- and two-body operators survive, with extra commutator terms accounting for the electron–nucleus pseudopotential. The Jastrow factor $J$ is optimized by variance minimization on a trial wave function from FCIQMC, and the one-body density matrix from FCIQMC is used to approximate the two-body density matrix as an antisymmetrized product (Eq. 12). This Hamiltonian moves correlation into the operator, compactifies the wave function, and suppresses basis-set error, which is why coupled cluster and FCIQMC both reach chemical accuracy in triple- or quadruple-zeta bases.","core_discovery":"The central discovery is that the transcorrelated workflow with pseudopotentials is consistently chemically accurate for transition-metal atoms in affordable basis sets. With the full semicore (3s, 3p) included in the valence and only 1s–2p frozen, xTC-FCIQMC gives $4s\\to 3d$ excitation energies with a mean absolute error of 0.39 kcal/mol in aug-cc-pVQZ across Sc–Cu, and xTC-CCSDT gives ionization potentials with a mean absolute error of 0.55 kcal/mol across Sc–Zn; in most cases aug-cc-pVTZ already suffices. The paper further finds that total energies from xTC-CCSD(T), xTC-CCSDT, and xTC-FCIQMC agree to within a few millihartree even when using different orbital sets (Hartree–Fock, DFT, or state-averaged CASSCF), which it reads as evidence that the transcorrelated Hamiltonian is robust to the choice of reference orbitals and that wave-function compactification (typical $\\xi \\approx 0.3$) is what makes low-rank methods accurate.","pith_inferences":["A direct test of the paper's route is to apply the same xTC-plus-pseudopotential workflow to 4d and 5d transition metals (for example Mo or W), where scalar-relativistic and core-correlation effects are stronger; the paper's logic predicts chemical accuracy in modest basis sets without complete-basis-set extrapolation.","Because the Jastrow factor is optimized separately per state and basis, the robustness claim may weaken in molecules where a single Jastrow cannot be optimized to the same variance; a transition-metal diatomic test would expose the limits.","If explicit two-body density matrices replace the antisymmetrized-product reconstruction, the agreement between xTC-CCSDT and xTC-FCIQMC might change for the most strongly correlated atoms; observing whether total-energy agreement persists would separate Hamiltonian bias from solver accuracy."],"forward_implications":["Benchmark-quality ionization potentials and $4s\\to 3d$ excitation energies for all ten 3d transition metals can be obtained without complete-basis-set extrapolation and without separate scalar-relativistic all-electron Hamiltonians.","xTC-CCSDT, used alone without FCI corrections, reaches chemical accuracy for Sc–Zn ionization potentials in aug-cc-pVQZ.","xTC-FCIQMC in aug-cc-pVQZ gives a $4s\\to 3d$ excitation mean absolute error of 0.39 kcal/mol for Sc–Cu, matching the accuracy of previous coupled-cluster benchmark results.","Wave-function compactification of about $\\xi \\approx 0.3$ means lower excitation levels suffice with a transcorrelated Hamiltonian, making both coupled-cluster and FCIQMC calculations cheaper."],"supporting_citations":[{"why":"Provides the xTC approximation that reduces the three-body transcorrelated interaction to one- and two-body terms, making the current calculations feasible.","marker":"[20]"},{"why":"Extends transcorrelated theory to pseudopotentials, establishing the pseudopotential capability this work relies on for j-averaged transition-metal energies.","marker":"[21]"},{"why":"Supplies the eCEPP pseudopotentials used for Sc through Fe.","marker":"[22]"},{"why":"Supplies the ccECP pseudopotentials used for Co through Zn and reports pseudopotential-to-all-electron accuracy of 3–16 meV, the stated basis for targeting chemical accuracy.","marker":"[23]"},{"why":"Provides the all-electron coupled-cluster benchmark excitation energies, ionization potentials, and electron affinities that this work compares against.","marker":"[5]"},{"why":"Provides previous FCIQMC ionization potentials for transition-metal atoms, the accuracy baseline the present coupled-cluster and FCIQMC results are compared with.","marker":"[4]"},{"why":"Establishes the Jastrow-factor variance-minimization procedure that the transcorrelated Hamiltonian construction depends on.","marker":"[15]"},{"why":"Describes the multireference transcorrelated workflow (FCIQMC trial wave function, one-body density matrix, Jastrow optimization) used here to build the transcorrelated Hamiltonian.","marker":"[31]"},{"why":"Demonstrates excited-state energy estimation with a non-Hermitian transcorrelated Hamiltonian via shift-based propagation, used for the spin-conserving excitations.","marker":"[17]"}],"fun_headline_variants":["Transcorrelation brings transition metals to chemical accuracy","All 3d metals reach chemical accuracy without huge basis sets","TC theory beats basis set limits for Sc-Zn atoms","Ionization and excitation energies exact for 3d block","Transcorrelated Hamiltonian tames transition metal spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The two-body part of the transcorrelated Hamiltonian is built from the one-body density matrix through the antisymmetrized-product approximation of Eq. (12); if that reconstruction is inaccurate for strongly correlated 3d states, the Hamiltonian is biased even though the final energies agree with experiment, because both coupled cluster and FCIQMC use the same approximate Hamiltonian.","fun_headline_variants_meta":{"raw":{"variants":["Transcorrelation brings transition metals to chemical accuracy","All 3d metals reach chemical accuracy without huge basis sets","TC theory beats basis set limits for Sc-Zn atoms","Ionization and excitation energies exact for 3d block","Transcorrelated Hamiltonian tames transition metal spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1419,"prompt_tokens":942,"completion_tokens":477,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":398}},"tokens_in":558,"tokens_out":477,"duration_ms":5295,"temperature":1.0,"reasoning_tokens":398,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:27:05.825614+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same Sc–Zn excitation energies and ionization potentials with an explicit two-body density matrix from FCIQMC, instead of the antisymmetrized product, and compare the results; if the differences exceed roughly 1 kcal/mol for any atom, or if a strongly correlated case such as Cr or Fe changes by that amount, the approximation is load-bearing and the claimed accuracy is not intrinsic to the transcorrelated Hamiltonian.","supporting_citations":[{"cited_title":"xtc: An efficient treatment of three-body interactions in transcorrelated methods","cited_arxiv_id":null,"evidence_quote":"Provides the xTC approximation that reduces the three-body transcorrelated interaction to one- and two-body terms, making the current calculations feasible."},{"cited_title":"Philip Haupt, Daniel Kats, Pablo Lopez-Rios, and Ali Alavi","cited_arxiv_id":null,"evidence_quote":"Extends transcorrelated theory to pseudopotentials, establishing the pseudopotential capability this work relies on for j-averaged transition-metal energies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the eCEPP pseudopotentials used for Sc through Fe."},{"cited_title":"Melton, M","cited_arxiv_id":null,"evidence_quote":"Supplies the ccECP pseudopotentials used for Co through Zn and reports pseudopotential-to-all-electron accuracy of 3–16 meV, the stated basis for targeting chemical accuracy."},{"cited_title":"Balabanov and Kirk A","cited_arxiv_id":null,"evidence_quote":"Provides the all-electron coupled-cluster benchmark excitation energies, ionization potentials, and electron affinities that this work compares against."},{"cited_title":"Thomas, George H","cited_arxiv_id":null,"evidence_quote":"Provides previous FCIQMC ionization potentials for transition-metal atoms, the accuracy baseline the present coupled-cluster and FCIQMC results are compared with."},{"cited_title":"Optimizing jastrow factors for the transcorrelated method.The Journal of Chemical Physics, 158(22), 2023","cited_arxiv_id":null,"evidence_quote":"Establishes the Jastrow-factor variance-minimization procedure that the transcorrelated Hamiltonian construction depends on."},{"cited_title":"Transcorrelated Methods for Multireference Problems","cited_arxiv_id":"2505.20187","evidence_quote":"Describes the multireference transcorrelated workflow (FCIQMC trial wave function, one-body density matrix, Jastrow optimization) used here to build the transcorrelated Hamiltonian."},{"cited_title":"Compact numerical solutions to the two- dimensional repulsive hubbard model obtained via nonunitary similarity transformations.Phys","cited_arxiv_id":null,"evidence_quote":"Demonstrates excited-state energy estimation with a non-Hermitian transcorrelated Hamiltonian via shift-based propagation, used for the spin-conserving excitations."}],"review_version":1}