{"id":"a1f000cf-b410-4c82-9278-70f79dc9a4a0","arxiv_id":"1908.02528","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A fully numerical atomic program now handles fractional occupations and range-separated exchange, producing microhartree-accurate energies and HF/HFS ground-state reference tables for all elements up to Z=118.","lead":"Lehtola extends a finite element atomic structure code to handle fractionally occupied orbitals, range-separated exchange, and effective atomic potentials for SCF initial guesses. The result is microhartree-level total energies for atoms up to Z=118 and a benchmark dataset useful for density functional development.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence to sub-microhartree accuracy for all Z up to 118 is asserted from a single fixed mesh but never demonstrated; heavy-atom energies in Tables IV and V could carry unrecognized basis-set error.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the fixed 139-basis-function mesh is asserted to give microhartree accuracy but no per-element convergence studies are shown, especially for heavy atoms. My review confirms that this is the principal soft spot. The paper's implementation details (range-separated exchange, fractional occupations, SAP potentials) are internally consistent and validated where independent data exist: the LC-BLYP comparison to Gaussian-basis results in Table III is convincing, and the noble-gas comparison to Saito is a good external check, though its precision is not stated. The potential errors are in the unvalidated extrapolation of the fixed mesh to all elements, which is central to the claimed basis-set-limit benchmarks and to the 'literature deviates by up to 10 µEh' conclusion. If the concern lands, the remedy is straightforward: add convergence tests, report noble-gas differences quantitatively, and soften the abstract's ground-state wording to acknowledge unconverged cases. These are addressable, so CONDITIONAL remains the appropriate verdict; my analysis does not change it.","tokens_in":27543,"tokens_out":3649,"duration_ms":45976,"concrete_test":"Recompute the Hartree-Fock total energies for the noble gases He, Ne, Ar, Kr, Xe, Rn, and Og, and for representative heavy open-shell atoms (Hg, Ra, Pu, Cf), using a refined finite element basis, e.g., fifteen 20-node elements and r∞=80 a0, and also with r∞=40 a0, and compare against the ten-element results in Tables IV-V. If any total energy changes by more than 1 µEh, the 'beyond microhartree' claim is false for those systems. In addition, tabulate the signed differences between the present noble-gas HF energies and Saito's B-spline values to 0.01 µEh, so that 'perfect agreement' is quantified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central accuracy claim rests on the assertion in Section III that ten 15-node radial finite elements with r∞=40 a0 and 139 basis functions yield energies 'converged beyond microhartree accuracy' for every atom and cation studied, including Z up to 118. No convergence study is presented, so the claim is supported only by comparisons to literature: Kraisler et al. for Z≤86 (Tables I-II) and Saito's B-spline Hartree-Fock values for noble gases (Section III, described only as 'perfect'). This is load-bearing because Tables I-II show differences of up to ~10 µEh for heavy atoms, which the paper attributes to reference error. That attribution is valid only if the present basis is already converged to well below 10 µEh. Without a per-element convergence test, the apparent 'literature deviation' could partly be the authors' own basis error. The concern is acute for heavy atoms: the 1s orbital at Z=118 has a decay scale of order 1/118 a0, while the 7s/7p tails extend to tens of bohr. A fixed ten-element mesh must resolve both extremes simultaneously, and nothing in the manuscript demonstrates that this is achieved. The abstract's 'ground states' claim is also weakened by the italicized entries in Tables IV-V where a lower-lying configuration failed to converge, but the convergence issue is the more fundamental problem for the benchmark quality of the reported energies.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the HelFEM fully numerical finite element program to atomic calculations with spherically symmetric densities via fractionally occupied orbitals. Specialized implementations are described for LDA and GGA functionals, Hartree-Fock exchange, and range-separated exchange with Yukawa and erfc kernels, together with the generation of radial potentials for the superposition-of-atomic-potentials initial guess. Results include VWN and PBE total energies for neutral atoms and cations with 1 ≤ Z ≤ 86 compared to the fully numerical reference of Kraisler et al., LC-BLYP energies for first- and second-row atoms compared to Gaussian-basis calculations, and spin-restricted HF and HFS energies and configurations for all elements 1 ≤ Z ≤ 118. The central claims are that the approach reaches beyond microhartree accuracy with only 139 radial basis functions, that the literature values deviate by up to 10 μEh from the complete basis set limit, and that the erfc range-separation implementation is validated against large Gaussian-basis results.","tokens_in":27834,"tokens_out":3040,"duration_ms":37970,"significance":"If the numerical accuracy claims are substantiated, this is a valuable contribution: it provides a fully numerical, open, and fast route to complete-basis benchmarks of modern range-separated hybrid functionals on atoms, and the fractional-occupation machinery is practically important for generating atomic densities, potentials, and initial guesses. The work contains no fitting or circular validation: the erfc expansion is taken from an independent reference, the range-separation parameters belong to functional definitions, and comparisons are made to literature values from independent groups. The specialized angular reduction for LDA/GGA, exact exchange, and range-separated kernels is a useful methodological development, and the reported periodic-table energy tables will likely be used by the community as reference data.","major_comments":[{"comment":"The central accuracy claim that the ten 15-node element mesh with r∞ = 40 a0 and 139 radial basis functions gives energies \"converged beyond microhartree accuracy\" for every atom and cation up to Z = 118 is asserted but not demonstrated. No convergence study is presented, so the only evidence is agreement with literature values. This is load-bearing because Tables I and II report differences of up to about 10 μEh for heavy atoms and attribute those differences to error in the reference data; that attribution is valid only if the present basis is itself converged to well below 10 μEh. The concern is acute for heavy atoms, where the 1s orbital is very compact while the valence tails are diffuse, and a fixed ten-element mesh must resolve both. I request explicit convergence tests: for representative elements across the periodic table, including heavy atoms and cations, report the total energy as a function of element subdivision, polynomial order, and r∞, or provide an equivalent per-element error analysis.","section":"Section III, first paragraph"},{"comment":"The abstract and Section IV state that spin-restricted ground states are reported for HF and HFS for all Z = 1 to 118, but the tables themselves contain italicized entries (Pr in HF; Cf, Es, Fm in HFS) for which a lower-lying configuration was identified but failed to converge. For these elements the reported states are not established ground states. The text is transparent about this, but the central periodic-table claim is stronger than the evidence. Please either revise the wording to \"converged spin-restricted states\" with the caveats made explicit in the abstract, or extend the calculations so that the lower-lying configurations are converged and the ground-state statement is accurate.","section":"Section III, Tables IV and V; Abstract"},{"comment":"The claim that agreement with Saito's B-spline Hartree-Fock values for noble gases is \"perfect\" is stated without any numerical comparison. Since the noble-gas comparison is the only external benchmark that extends to Z > 86 and is used to underwrite the accuracy of the heavy-atom HF and HFS tables, please provide a table or listing of the actual differences for the noble gases He through Og, or at minimum the largest absolute deviation. Without numerical values, the reader cannot assess whether the agreement is at the claimed microhartree level or only at a coarser level.","section":"Section III, paragraph on Saito comparison"}],"minor_comments":[{"comment":"The displayed equation for the exchange matrix element appears to contain a typographical error: the Gaunt-type coefficient is written twice identically in the numerator, and the notation is not defined. Please correct the expression and define the angular momentum coupling symbol used.","section":"Section II B, Eq. (24)"},{"comment":"The Introduction states that the reference calculations of Kraisler et al. employed 16 000 point grids, while the Results section says 10 000 radial grid points were used in ref. 82. This inconsistency should be resolved.","section":"Section I vs Section III"},{"comment":"There are several typographical errors and artifacts: \"accucacy\" in Section IV, \"occu pations\" in the title line, and some font inconsistencies in the equations. A careful proofreading pass is recommended.","section":"Throughout"},{"comment":"The table reports five decimals for the finite element values and fewer for the Gaussian basis values. Please state explicitly in the caption or text that all tabulated digits are significant and specify the total energy unit (Eh).","section":"Table III"},{"comment":"The statement that the aug-pc-∞ Gaussian basis truncation error is under 1 μEh for light atoms and tens of μEh for heavier atoms is important but is only referenced indirectly. Since this underpins the choice of ref. 15 as the validation target, a one-sentence summary of how that error was established would improve the presentation.","section":"Section III, erfc validation paragraph"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is methodologically sound and the implementation appears to be a genuinely useful contribution, with no circularity or fitting concerns. The main issue is purely evidentiary: the beyond-microhartree convergence claim and the periodic-table ground-state claim are stronger than the presented data support. Both are fixable within the scope of the paper through additional convergence studies and careful wording, so I do not see a reason to reject. The journal fit is appropriate for physics.comp-ph, and the open-source nature of HelFEM makes the validation reproducible in principle."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing worth knowing: this paper delivers the first fully numerical atomic implementation of erfc range-separated exchange and a complete set of spin-restricted HF/HFS fractional-occupation energies for Z=1–118. That is a genuinely useful resource for validating functionals and building initial guesses. The erfc scheme works—the LC-BLYP results match Gaussian-basis numbers to the published digits for light atoms, and the paper catches a typo in the Ángyán expansion. The Kraisler comparisons are also sensible: differences appear where the reference used cross-shell fractional charge, and grow slowly with Z, which is consistent with the reference being less converged.\n\nThe soft spot is the convergence claim. The text says ten 15-node elements with r∞=40 a0 give 'beyond microhartree accuracy' for every atom and cation, but no convergence study is shown. That matters because Tables I–II show deviations up to ~10 µEh for heavy atoms, and the paper interprets those as reference error. Without a per-element convergence test, part of that could be the authors' own basis error. The 1s orbital at Z=118 and the outer s/p tails put very different length scales in one fixed mesh; I'm not saying the energies are wrong, just that the evidence as presented is assertion, not demonstration. A few convergence curves for, say, U and Og would settle it.\n\nThe abstract also says 'ground states' for Z=1–118, but the tables italicize cases where a lower-lying configuration was found but failed to converge. The text is honest about this; the abstract overstates it.\n\nMinor: the erfc validation stops at Ar, so the short-range Taylor-regime handling for heavy atoms is untested, and the 'perfect' agreement with Saito for noble gases is not quantified. Neither is fatal.\n\nOverall: solid methods paper, worth refereeing seriously, but it needs a convergence appendix and a more careful abstract. The data will be cited if the convergence is confirmed.","headline":"A useful new erfc range-separated implementation and complete HF/HFS reference tables for Z=1–118, but the microhartree convergence claim is asserted without a per-element convergence study and the abstract overstates 'ground states.'","tokens_in":28317,"tokens_out":2576,"would_cite":true,"duration_ms":29960,"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":"Fractional occupations let finite-element atoms reach sub-microhartree accuracy with only 139 radial basis functions.","keywords":["finite element method","fractional occupations","Hartree-Fock","density functional theory","range-separated exchange","erfc kernel","Yukawa kernel","atomic total energies"],"falsifier":"Rerun a heavy closed-shell case such as Rn or Og with a second mesh of more elements, a larger practical infinity, and tighter quadrature; if the Hartree–Fock or Hartree–Fock–Slater energy shifts by more than about one microhartree, the single-mesh convergence claim fails for that part of the table.","tokens_in":1647,"feed_emoji":"⚛️","tokens_out":2256,"duration_ms":70633,"temperature":0.7,"pith_summary":"The paper extends a fully numerical finite element atomic structure program to atoms whose density is made spherically symmetric by splitting electron occupations fractionally across degenerate shells. It derives specialized Hartree–Fock, local density, and generalized gradient approximation equations for this setting, and adds range-separated exchange with both Yukawa and complementary-error-function kernels. The central numerical claim is that a single radial mesh of ten 15-node elements, only 139 radial basis functions, converges atomic total energies past the microhartree level, while reproducing published LDA and GGA values for neutral atoms and cations through Z=86. The paper also reports new spin-restricted non-relativistic Hartree–Fock and Hartree–Fock–Slater ground states for all elements through Z=118. If true, this gives chemistry a simple, parameter-free route to complete-basis-set atomic reference energies, including for functionals that contain exact exchange.","feed_headline":"139 basis functions push atomic energies to sub-microhartree accuracy","feed_subtitle":"Spherically symmetric finite-element atoms reach basis-set-limit energies for every element, enabling accurate hybrid benchmarks.","key_machinery":"The load-bearing object is the radial finite element basis χ_{nlm}=$r^{{-1}}$B_n(r)Y_l^m(θ,φ) combined with fractional shell occupations f_{nl}, so that the density n(r)=Σ_{nl} f_{nl}(2l+1)R_{nl}^2/4π is spherically symmetric by construction. This makes the exact-exchange matrix depend on orbital angular momenta only through the coupled angular momentum L between incoming and outgoing shells, reducing the self-consistent field calculation to radial quadrature. The range-separated kernels enter through their Green's functions: the Coulomb form r_<^L/r_>^{L+1}, the Yukawa form (2L+1)λ i_L(λ r_<)k_L(λ r_>), and the erfc kernel through the scaled Φ_n(Ξ,ξ) spherical-harmonic expansion with a Taylor stabilization for short ranges. This machinery removes all angular quadrature, which is what lets microhartree-level atomic calculations run with only 139 radial functions instead of dense numerical grids.","core_discovery":"The discovery is that the spherically averaged atomic problem becomes nearly trivial when occupations are shared equally over each (n,l) shell: the density matrix is diagonal in l and m, the Coulomb term collapses to a single L=0 component, and exact exchange, including range-separated exchange, reduces to one-dimensional radial integrals with a finite set of coupled angular momenta. With this reduction, ten 15-node radial finite elements reaching r∞=40 a0 are claimed to give energies converged beyond 1 µEh for atoms and cations with 1≤Z≤86. The paper also validates the erfc range-separation implementation against large Gaussian-basis calculations for spherically symmetric atoms, and argues that earlier literature values deviate from the complete basis set limit by as much as 10 µEh for heavier atoms.","pith_inferences":["The convergence claim is stated for a single fixed mesh in Section III but is not shown element by element; testing heavier elements such as Og with additional radial elements would settle whether 139 functions truly deliver microhartree accuracy across the whole periodic table.","The same spherical-averaging construction could naturally extend to fractional occupations within a shell rather than only across all 2l+1 orbitals, which the paper itself notes as future work.","Because the Hartree–Fock and Hartree–Fock–Slater tables cover all elements to Z=118, they could serve as a uniform non-relativistic baseline against which relativistic and quantum-electrodynamic corrections in superheavy elements could be assessed.","A natural next test is to apply the erfc Green's function machinery to open-shell and excited-state atoms with fractional occupations, since the paper validates it mainly on spherically symmetric closed-shell species."],"forward_implications":["Complete basis set benchmarks for LDA, GGA, hybrid, and range-separated hybrid functionals on atoms become routine, exposing basis-set truncation errors in earlier literature.","The reported Hartree–Fock and Hartree–Fock–Slater ground states for Z=1 to 118 provide a consistent non-relativistic, spin-restricted reference set for functional development and for generating pseudopotentials and numerical basis sets.","Fractional-occupation calculations resolve negative-gap and shell-degeneracy situations, so ground and low-lying excited atomic states can be compared at the basis set limit.","The erfc range-separation implementation makes it possible to benchmark modern screened hybrid functionals at the complete basis set limit, something Gaussian basis sets can only approximate.","The computed radial effective potentials give a superposition-of-atomic-potentials initial guess for molecular calculations with documented sub-microhartree accuracy."],"supporting_citations":[{"why":"Supplies the underlying finite element machinery for atomic Hartree–Fock and density functional calculations that this work extends.","marker":"[14]"},{"why":"Provides the reference LDA and PBE total energies for neutral atoms and cations through Z=86 that the new results reproduce and compare against.","marker":"[82]"},{"why":"Supplies the spherical-harmonic expansion of the erfc-screened Coulomb Green's function that the range-separation implementation is built on.","marker":"[100]"},{"why":"Provides large Gaussian-basis LC-BLYP energies used to validate the erfc range-separation scheme against a well-established basis set.","marker":"[15]"},{"why":"Provides high-accuracy B-spline Hartree–Fock energies for noble gases, used to check the heavy-element Hartree–Fock results.","marker":"[132]"},{"why":"Motivates the superposition-of-atomic-potentials initial-guess approach whose radial effective potentials the new routines compute.","marker":"[78]"}],"fun_headline_variants":["Fractional occupations slash atomic basis to 139 functions","Sub-microhartree atoms via fractionally occupied orbitals","Range-separated exchange becomes 1D for fractional occupations","Accurate atomic energies from 139 basis functions","139 functions solve all atoms to sub-microhartree"],"cache_read_input_tokens":30464,"weakest_assumption_plain":"The numerical premise that one fixed radial finite element mesh, ten 15-node elements ending at r∞=40 a0, is already converged to microhartree accuracy for every atom and cation in the tables, including heavy elements past Z=86.","fun_headline_variants_meta":{"raw":{"variants":["Fractional occupations slash atomic basis to 139 functions","Sub-microhartree atoms via fractionally occupied orbitals","Range-separated exchange becomes 1D for fractional occupations","Accurate atomic energies from 139 basis functions","139 functions solve all atoms to sub-microhartree"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000993,"raw_usage":{"total_tokens":4249,"prompt_tokens":1032,"completion_tokens":3217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":3150}},"tokens_in":648,"tokens_out":3217,"duration_ms":23611,"temperature":1.0,"reasoning_tokens":3150,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:40:54.573849+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun a heavy closed-shell case such as Rn or Og with a second mesh of more elements, a larger practical infinity, and tighter quadrature; if the Hartree–Fock or Hartree–Fock–Slater energy shifts by more than about one microhartree, the single-mesh convergence claim fails for that part of the table.","supporting_citations":[{"cited_title":"Optimization of Gaussian basis sets for density-functional calcu- lations,","cited_arxiv_id":null,"evidence_quote":"Provides the reference LDA and PBE total energies for neutral atoms and cations through Z=86 that the new results reproduce and compare against."},{"cited_title":"The QTP family of consistent functionals and poten- tials in Kohn-Sham density functional theory,","cited_arxiv_id":null,"evidence_quote":"Supplies the spherical-harmonic expansion of the erfc-screened Coulomb Green's function that the range-separation implementation is built on."},{"cited_title":"A Variational Ap- proach to the Unipotential Many-Electron Prob- lem,","cited_arxiv_id":null,"evidence_quote":"Provides high-accuracy B-spline Hartree–Fock energies for noble gases, used to check the heavy-element Hartree–Fock results."},{"cited_title":"Numerical atomic basis orbitals from H to Kr,","cited_arxiv_id":null,"evidence_quote":"Motivates the superposition-of-atomic-potentials initial-guess approach whose radial effective potentials the new routines compute."}],"review_version":1}