{"id":"bbc3fbce-60bb-4f50-8b8c-5d4f4f483959","arxiv_id":"2412.05885","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Transcorrelated calculations with pseudopotentials become chemically accurate when two nonlocal pseudopotential commutator corrections are included, and Jastrow optimization variance drops by roughly an order of magnitude.","lead":"This paper extends the transcorrelated electronic structure method to use pseudopotentials, which replace core electrons with an effective potential, and derives the new correction terms this introduces. The resulting method reaches chemical accuracy for ionization energies, atomization energies, and dissociation curves of first-row systems while reducing the variance that makes Jastrow optimization expensive.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PP-2 chemical accuracy rests on an unquantified truncation of three-body PP commutator terms (Eq. 9); a direct numerical bound on the dropped Γ terms is needed.","rationale":"The reader's weakest assumption identifies exactly the same point: the PP commutator three-body truncation is unquantified. I agree. The paper is otherwise well-supported: benchmarks are extensive, the variance-reduction claim is backed by tabulated data, and the PP-2 versus PP-3/4 comparison shows convergence in the two-body commutator order. However, that comparison is orthogonal to the three-body truncation. Since the method's headline result is chemical accuracy at the 1-2 mHa level, a missing estimate of a potentially comparable error source leaves the central claim conditional. The proposed test is a natural extension of tools already in TCHINT/xTC and would settle the matter. Thus I leave the reader's CONDITIONAL verdict unchanged.","tokens_in":20546,"tokens_out":5141,"duration_ms":52081,"concrete_test":"Implement the dropped three-body PP commutator terms (the i>j>m sums in Eq. 9 for H_PP_en) using the paper's existing numerical quadrature for Eq. (5), and estimate their effect: (i) compute the Frobenius norm of the resulting three-body matrix elements in an AVTZ basis for Be and B with both eCEPP and ccECP; (ii) incorporate the terms into the xTC generalized normal-ordering approximation (as done for the kinetic-energy L tensor) and recompute xTC-CCSD(T) ionization energies and atomization energies. If the energy shift exceeds 1 mHa (the chemical-accuracy threshold used in the paper), the PP-2 truncation is not established as reliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central benchmark claim, that xTC-CCSD(T)(PP-2) reaches chemical accuracy for first-row ionization energies, atomization energies, and N2/F2 dissociation curves, depends on the Hamiltonian in Eq. (10). For the pseudopotential contribution, this Hamiltonian drops all three-body terms in the second commutator, i.e., the sums over i>j>m in Eq. (9). The paper states (Section: Pseudopotentials in the Transcorrelated Hamiltonian) that this is done 'under the assumption that the 3-body interactions of the valence electrons close to the core are negligible,' and explicitly leaves their treatment to future work. No numerical estimate of these terms is provided. The convergence shown in Fig. 3 and Table 4 across PP-n (n=0-4) does not test this truncation, because all n drop the same three-body terms; it only probes convergence in the two-body part of the commutator series. For first-row atoms the valence electrons have non-negligible probability inside the nonlocal-core region, so the dropped terms could be on the order of the 1-2 mHa accuracy target and could be partially compensated by the xTC approximation applied to the kinetic-energy three-body terms. In all-electron TC, the BCH series terminates at the second commutator, but with nonlocal PPs it does not, so the all-electron analogy does not justify the truncation. Without a bound on the dropped three-body PP terms, the chemical-accuracy results are conditional on an unverified approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents an extension of the transcorrelated (TC) method to calculations with nonlocal pseudopotentials (PPs). The similarity transformation of the Hamiltonian with a Jastrow factor is derived for the PP operator, and the resulting commutator terms are evaluated numerically in a modified TCHINT code. The method is tested on first-row atoms and molecules, with ionization potentials, atomization energies, and N2/F2 dissociation curves computed using xTC-CCSD(T), xTC-CCSDT, and xTC-FCIQMC. The central claims are that the PP-2 approximation (including the first two commutators of the PP with the Jastrow factor) is necessary and sufficient to reach chemical accuracy for the studied properties, and that the use of PPs substantially reduces the variance of the Jastrow-optimization VMC calculation.","tokens_in":20865,"tokens_out":8748,"duration_ms":84680,"significance":"If the central claims hold, this work is an important step toward applying transcorrelated methods to heavier elements and periodic systems, where the all-electron approach is costly. The paper provides a clear account of the commutator integrals, a practical numerical integration scheme, and benchmark data across several properties. The demonstrated variance reduction and the availability of optimized Jastrow parameters in the Supplementary Material are valuable. However, the chemical-accuracy claims rest on an unquantified truncation of the PP commutator series, and the combined Jastrow treatment introduces a size-consistency error that is on the order of the accuracy target. These issues make the presented results conditional rather than definitive.","major_comments":[{"comment":"The truncation of the PP commutator series by dropping all three-body terms (the sums over i>j>m in Eq. 9) is not numerically justified. The paper correctly notes that for nonlocal PPs the BCH series does not terminate at the second commutator, but then it removes the three-body contributions under an assumption of negligible valence-electron density near the core. No estimate of these terms is provided. The convergence test in Fig. 3 and Table 4 (varying the number of retained commutators, n=0-4) does not probe this truncation because all n drop the same three-body terms; it only tests convergence of the two-body part. For first-row atoms, valence electrons have significant probability inside the nonlocal core region, so the omitted terms may be comparable to the 1-2 mHa chemical-accuracy target. The authors should provide a numerical estimate of the dropped three-body PP commutator terms for a representative system (e.g., Be or B), ideally by computing them explicitly or by comparing PP-2 results with an implementation that includes them, and show that they are below the accuracy target.","section":"Pseudopotentials in the Transcorrelated Hamiltonian, Eqs. (9)-(10)"},{"comment":"The combined Jastrow treatment used for the dissociation curves introduces a size-consistency error of -1.5(2) mHa for N2 at the largest bond length, as reported in the text. This error is at the same scale as the chemical-accuracy threshold (1.6 mHa), and the paper does not quantify how it affects the error curves in Figs. 9 and 10. Since dissociation curves are one of the three principal benchmark classes, the authors should either recompute key points with the separate Jastrow treatment (which they state eliminates the error) or provide a quantitative estimate of the size-consistency error along the computed curves to confirm that the chemical-accuracy claims are unaffected.","section":"Dissociation energies"}],"minor_comments":[{"comment":"The phrase 'cusps, 1 in the wave function' should have a space before the citation, and the reference formatting should be consistent throughout.","section":"Introduction"},{"comment":"In the sentence 'for to simplify presentation', the word 'the' is missing; it should read 'to simplify the presentation'.","section":"Pseudopotential approximation"},{"comment":"In Eq. (13), the summation index 'i' is reused for both the electron index and the grid-point index, which is confusing; a different index (e.g., 'k') should be used for the spherical grid points.","section":"Evaluation of transcorrelated Hamiltonian with pseudopotentials"},{"comment":"The acronym 'NECP' is used in the text but never defined; presumably it denotes the number of grid points under pseudopotential influence, but it should be spelled out at first use.","section":"Evaluation of transcorrelated Hamiltonian with pseudopotentials"},{"comment":"The notation 'xTC-CCSD(T)(PP-0)' is ambiguous because the parentheses could be misread as subtraction; using a clearer notation such as 'xTC-CCSD(T)-PP0' would improve readability.","section":"Notation"},{"comment":"The phrase 'a feature than can provide useful' should be 'a feature that can be useful'.","section":"Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to the development of transcorrelated methods with pseudopotentials, and the benchmark data are useful. The main obstacle to acceptance is the unquantified truncation of three-body PP commutator terms, which is explicitly acknowledged by the authors. This is a fixable issue if the authors provide a numerical estimate of the omitted terms for a representative system or implement them (possibly within the xTC approximation). The size-consistency error in the combined Jastrow treatment is also worth quantifying. The manuscript is otherwise well organized, though the reference list has some incomplete entries that should be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real advance, and the benchmarks look credible within their stated scope. The thing to watch is the three-body pseudopotential commutator truncation. The paper acknowledges it openly but gives no numerical bound on the dropped terms, so the chemical-accuracy claims are conditional until someone estimates them.\n\nWhat is actually new: the derivation of the nonlocal PP commutator contributions to the transcorrelated Hamiltonian (Eqs. 11-12), and the first systematic PP-based xTC benchmarks. The math up to that point is sound; the issue is what happens after. The paper shows cleanly that the second commutator (PP-2) is necessary for chemical accuracy, and that PP-3 and PP-4 do not change results — that convergence check is real, within the two-body sector. The variance reduction is quantified and behaves sensibly, roughly 1/Nv scaling. Benchmark coverage is solid: ionization energies of Be-F, atomization energies against HEAT, and N2/F2 dissociation curves with FCIQMC. No target quantity is fitted — Jastrow parameters are variance-optimized on HF wave functions, PPs are literature values, accuracy is judged against experiment and HEAT — so the circularity burden is low. The citation pattern leans heavily on the authors' own TC series, which is appropriate since those are the enabling works. I also credit them for flagging the 1-2 mHa size-inconsistency from the combined Jastrow treatment and the odd ccECP AVQZ basis-set behaviour rather than smoothing either over.\n\nThe soft spots, in order of weight. First, the truncation concern in the stress-test note lands. With nonlocal PPs the similarity transformation does not terminate at the second commutator, and the paper drops the i>j>m sums in Eq. 9 'under the assumption that the 3-body interactions of the valence electrons close to the core are negligible.' The PP-n convergence study cannot detect a problem here because every n drops the same three-body terms. For first-row atoms, valence electrons do penetrate the nonlocal-core region, so a 1-2 mHa contribution is not implausible. Not a demonstrated flaw, but an unverified approximation under the central accuracy claim. Second, the released artifacts are thin (Be FCIDUMP files only, no TCHINT), limiting independent verification. Third, the cost-reduction claim is qualitative: variance reduction is shown but no timings are given.\n\nThe CONDITIONAL verdict and the stress-test are both fair. I would not escalate to a refutation. The paper is explicit about what it drops, the P tensor contributions are empirically small corrections, and the xTC machinery has a track record of making similar approximations work. But the burden is on the authors to bound the dropped terms.\n\nThis paper is for people working in transcorrelated methods, QMC, and pseudopotential development, plus anyone planning TC for solids or transition metals. It deserves a serious referee. Send it to review, with the specific request that the authors either estimate the dropped three-body PP terms or state clearly the regime in which the approximation is controlled.","headline":"A genuine, honest methods advance — the first nonlocal pseudopotential treatment in transcorrelated theory — whose chemical-accuracy claims rest on an unquantified three-body commutator truncation that the authors flag but do not bound.","tokens_in":21388,"tokens_out":6818,"would_cite":true,"duration_ms":57911,"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":"First two pseudopotential commutators give transcorrelated theory chemical accuracy.","keywords":["transcorrelated methods","Jastrow factor","pseudopotentials","effective core potentials","explicitly correlated methods","coupled cluster","FCIQMC","chemical accuracy"],"falsifier":"Compute the dropped three-body pseudopotential commutator terms (the $\\sum_{i>j>m}$ part of Eq. (9)) for a first-row atom or for N$_2$ and show that they change the energy by more than the claimed sub-milliHartree level; alternatively, run the full commutator series with the three-body terms retained and compare against PP-2.","tokens_in":20365,"feed_emoji":"⚛️","tokens_out":13459,"duration_ms":104339,"temperature":0.7,"pith_summary":"Transcorrelated methods improve basis-set convergence by applying a similarity transformation to the Hamiltonian with a Jastrow factor, a function of interelectronic distances, and their practical reach has been limited by the cost of optimizing that factor. This paper shows that replacing core electrons with pseudopotentials cuts the variational Monte Carlo variance to roughly 8–30% of the all-electron value, making the optimization much cheaper. The catch is that nonlocal pseudopotentials do not commute with the Jastrow factor, so the similarity transformation no longer stops after the second commutator. The paper derives the required commutator terms, truncates at the first two non-zero ones, and shows that xTC-CCSD(T)(PP-2) reaches chemical accuracy for first-row ionization energies, atomization energies, and N2 and F2 dissociation curves. This opens a route toward transcorrelated calculations on larger molecules, transition metals, and solids.","feed_headline":"Two commutators make transcorrelated theory work with pseudopotentials","feed_subtitle":"Including the first two nonlocal-potential commutators gives accurate first-row energies and dissociation curves.","key_machinery":"The machinery is the nested commutator expansion $\\hat H_{\\mathrm{TC}} = e^{-J}\\hat H e^{J} = \\hat H + [\\hat H,J] + \\frac{1}{2!}[[\\hat H,J],J] + \\cdots$, applied to the nonlocal pseudopotential $\\hat V_{\\mathrm{eff}}(r) = V_{l_{\\max}}(r) + \\sum_{l=0}^{l_{\\max}-1} V_l(r)\\sum_{m=-l}^{l} |Y_{lm}\\rangle\\langle Y_{lm}|$. The pseudopotential's angular projection operator means it does not commute with the Drummond–Towler–Needs Jastrow factor (a sum of one-, two-, and three-body correlation functions), so the paper must compute the one- and two-body commutator terms $\\Pi$ and $\\Gamma$ of Eqs. (11)–(12), evaluating the spherical projection numerically on an icosahedral grid with $N_s = 12$ points. Terms involving three distinct electrons ($i > j > m$) from the pseudopotential commutators are dropped, while the three-body kinetic-energy commutator terms are handled by the xTC approximation. The result is a second-quantized transcorrelated Hamiltonian with one-, two-, and three-body pieces whose pseudopotential correction is the tensor $P^{pq}_{rs}$ entering alongside the kinetic-energy corrections $K$ and $L$.","core_discovery":"The central result is that a pseudopotential-based transcorrelated Hamiltonian is reliable only if the nonlocal pseudopotential's non-commutation with the Jastrow factor is corrected through at least the first two non-zero commutators, the PP-2 level. In the all-electron case the similarity transformation terminates exactly at the second commutator; with a nonlocal effective potential it does not, and the authors truncate the series by dropping the three-body $i > j > m$ terms under the assumption that valence electrons near the core make them negligible. With that approximation, the corrected transcorrelated Hamiltonian makes xTC-CCSD(T)(PP-2) chemically accurate for the first-row ionization energies Be–F with both eCEPP and ccECP pseudopotentials, for atomization energies of CN, CO, CF, N2, O2, F2, H2O, and CO2 (eCEPP at AVQZ and ccECP at AVTZ), and for xTC-FCIQMC dissociation curves of N2 and F2 at AVQZ. Higher commutator orders change energies by about 1 mHa or less, which the authors take as evidence that PP-2 is sufficient.","pith_inferences":["The PP-2 truncation should be re-tested for transition metals and heavier elements, where pseudopotentials have more angular-momentum channels and the nonlocal commutators are likely larger.","The non-monotonic basis-set behavior of ccECPs (better at AVTZ than AVQZ for both xTC and F12) suggests the pseudopotential and basis set are not fully converged together; joint reoptimization could remove the artifact.","One could estimate the dropped three-body pseudopotential commutator terms by comparing PP-2 against a version that keeps them within the xTC treatment, giving a direct numerical check of the core assumption without all-electron calculations.","For strongly correlated dissociation like N2, the need for a multideterminant trial wave function in Jastrow optimization hints that single-reference VMC may be the next bottleneck when applying this method to bond breaking."],"forward_implications":["With PP-2, xTC-CCSD(T) reaches chemical accuracy for first-row ionization energies (Be–F) in triple- and quadruple-zeta bases with both eCEPP and ccECP pseudopotentials.","Atomization energies of the eight-molecule test set reach chemical accuracy with xTC-CCSDT(PP-2) using eCEPPs in AVQZ and with ccECPs in AVTZ.","xTC-FCIQMC(PP-2) reproduces the N2 and F2 dissociation curves to chemical accuracy in AVQZ, apart from the most compressed N2 bond lengths.","Because pseudopotentials cut the VMC variance to roughly 8–30% of the all-electron value, Jastrow optimization becomes cheaper, making larger molecules, transition metals, and solid-state systems more accessible.","The small size-inconsistency of the combined Jastrow treatment (~1–2 mHa) is removed by the separated Jastrow treatment, which the paper verifies gives identical equilibrium results."],"supporting_citations":[{"why":"Establishes the Jastrow similarity transformation and the commutator framework that this paper extends to pseudopotentials.","marker":"[7]"},{"why":"Introduces the xTC approximation for three-body kinetic-energy commutator terms on which the pseudopotential implementation relies.","marker":"[15]"},{"why":"Supplies the eCEPP family of effective core potentials used in the calculations.","marker":"[22]"},{"why":"Supplies the ccECP family of effective core potentials; comparing the two families drives the accuracy analysis.","marker":"[23]"},{"why":"Defines the Drummond–Towler–Needs Jastrow factor form whose optimized parameters enter the transcorrelated Hamiltonian.","marker":"[25]"},{"why":"Provides the reference atomization energies against which chemical accuracy is measured.","marker":"[27]"},{"why":"Provides the variational Monte Carlo code used for Jastrow factor optimization; its variance reduction is a central motivation.","marker":"[29]"},{"why":"Gives the biorthogonal coupled-cluster treatment needed because the transcorrelated Hamiltonian is non-Hermitian.","marker":"[31]"},{"why":"Supplies the initiator FCIQMC approach used for the N2 and F2 dissociation curves.","marker":"[36]"}],"fun_headline_variants":["Two commutators make pseudopotential transcorrelated theory accurate","Just two nonlocal commutators unlock transcorrelated pseudopotentials","PP-2 commutators give chemically accurate transcorrelated results","Two commutators suffice for transcorrelated pseudopotentials","Pseudopotential transcorrelated theory: just two commutators required"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that the three-body pseudopotential commutator terms dropped from the series are negligible for valence electrons near the core; the paper provides no numerical estimate of their size.","fun_headline_variants_meta":{"raw":{"variants":["Two commutators make pseudopotential transcorrelated theory accurate","Just two nonlocal commutators unlock transcorrelated pseudopotentials","PP-2 commutators give chemically accurate transcorrelated results","Two commutators suffice for transcorrelated pseudopotentials","Pseudopotential transcorrelated theory: just two commutators required"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000771,"raw_usage":{"total_tokens":3412,"prompt_tokens":941,"completion_tokens":2471,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":2384}},"tokens_in":557,"tokens_out":2471,"duration_ms":16410,"temperature":1.0,"reasoning_tokens":2384,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:14:06.709602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the dropped three-body pseudopotential commutator terms (the $\\sum_{i>j>m}$ part of Eq. (9)) for a first-row atom or for N$_2$ and show that they change the energy by more than the claimed sub-milliHartree level; alternatively, run the full commutator series with the three-body terms retained and compare against PP-2.","supporting_citations":[{"cited_title":"J.; Luo, H.; Guther, K.; Dobrautz, W.; Tew, D","cited_arxiv_id":null,"evidence_quote":"Establishes the Jastrow similarity transformation and the commutator framework that this paper extends to pseudopotentials."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the xTC approximation for three-body kinetic-energy commutator terms on which the pseudopotential implementation relies."},{"cited_title":"R.; Needs, R","cited_arxiv_id":null,"evidence_quote":"Supplies the eCEPP family of effective core potentials used in the calculations."},{"cited_title":"C.; Melton, C","cited_arxiv_id":null,"evidence_quote":"Supplies the ccECP family of effective core potentials; comparing the two families drives the accuracy analysis."},{"cited_title":"D.; Towler, M","cited_arxiv_id":null,"evidence_quote":"Defines the Drummond–Towler–Needs Jastrow factor form whose optimized parameters enter the transcorrelated Hamiltonian."},{"cited_title":"G.; Császár, A","cited_arxiv_id":null,"evidence_quote":"Provides the reference atomization energies against which chemical accuracy is measured."},{"cited_title":"J.; Towler, M","cited_arxiv_id":null,"evidence_quote":"Provides the variational Monte Carlo code used for Jastrow factor optimization; its variance reduction is a central motivation."},{"cited_title":"M.; Schraivogel, T.; Alavi, A","cited_arxiv_id":null,"evidence_quote":"Gives the biorthogonal coupled-cluster treatment needed because the transcorrelated Hamiltonian is non-Hermitian."},{"cited_title":"H.; Alavi, A","cited_arxiv_id":null,"evidence_quote":"Supplies the initiator FCIQMC approach used for the N2 and F2 dissociation curves."}],"review_version":1}