{
  "model": "HEG3D",
  "lattice": "sc",
  "n_sites": 54,
  "boundary": "P",
  "params": {
    "N_up": 27,
    "N_dn": 27,
    "rs": 0.5,
    "twist": "Gamma"
  },
  "instance_id": "HEG3D/sc_54_P_27_27_0.5",
  "rows": [
    {
      "energy": 173.946204,
      "sigma": 0.000108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "DeepSolid",
      "method_detail": "1 determinant, 3 layers 256/32",
      "method_as_published": "DeepSolid (1 determinant)",
      "family": "other NQS",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "X. Li, Z. Li & J. Chen, Ab initio calculation of real solids via neural network ansatz, Nat. Commun. 13, 7895 (2022), arXiv:2203.15472, doi:10.1038/s41467-022-35627-1",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2203.15472 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.221226(2)",
        "note": "Supplementary Table 11, row rs=0.5, column 'Net'; sources/2203.15472.txt line 1013; printed in Ha per electron; version read: arXiv:2203.15472v2 text, Supplementary Table 11; Nature Communications copy not read | Ha per electron, 54-electron closed-shell gas in a simple cubic cell (27 + 27); QMBL's HF check: at rs = 0.5 the energy 3.221226 sits 0.044 Ha below QMBL's HF energy 3.2653295.",
        "secondary_of": null
      }
    },
    {
      "energy": 173.92644,
      "sigma": 0.00108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "FCIQMC",
      "method_detail": "initiator (n_add = 3), 2838 spin orbitals",
      "method_as_published": "i-FCIQMC (2M = 2838 spin orbitals)",
      "family": "FCIQMC / CC",
      "bound_type": "projected",
      "bound_type_reason": "projection (GFMC/fixed-node) on a trial state; not a strict variational bound (assigned during source reading)",
      "reference": "J. J. Shepherd, G. H. Booth, A. Grüneis & A. Alavi, Full configuration interaction perspective on the homogeneous electron gas, Phys. Rev. B 85, 081103(R) (2012), arXiv:1109.2635, doi:10.1103/PhysRevB.85.081103",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:1109.2635 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22086(2)",
        "note": "Table I, rs=0.5 row, column \"FCI 2M = 2838\"; sources/1109.2635.txt line 123; printed in Ha per electron; version read: arXiv text; APS copy not read | Hartree per electron, 54 electrons in a simple cubic cell, plane waves at Gamma, H = T + Ewald + N v_M / 2 (Madelung term included). At rs = 1 the complete-basis value 0.5300(3) agrees with arXiv:0801.0518's backflow DMC 0.52989(1).",
        "secondary_of": null
      }
    },
    {
      "energy": 173.8908,
      "sigma": 0.0108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "FCIQMC",
      "method_detail": "initiator (n_add = 3); extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "i-FCIQMC, 1/M extrapolation",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "J. J. Shepherd, G. H. Booth, A. Grüneis & A. Alavi, Full configuration interaction perspective on the homogeneous electron gas, Phys. Rev. B 85, 081103(R) (2012), arXiv:1109.2635, doi:10.1103/PhysRevB.85.081103",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:1109.2635 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.2202(2)",
        "note": "Table I, rs=0.5 row, column \"FCI M = infinity\"; sources/1109.2635.txt line 123; printed in Ha per electron; version read: arXiv text; APS copy not read | Hartree per electron, 54 electrons in a simple cubic cell, plane waves at Gamma, H = T + Ewald + N v_M / 2 (Madelung term included). At rs = 1 the complete-basis value 0.5300(3) agrees with arXiv:0801.0518's backflow DMC 0.52989(1).",
        "secondary_of": null
      }
    },
    {
      "energy": 176.3586,
      "sigma": 0.0378,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Hartree-Fock",
      "method_detail": "plane-wave determinant, VMC estimate",
      "method_as_published": "VMC of the Hartree-Fock determinant",
      "family": "mean field",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler & R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006), arXiv:0801.0518, doi:10.1103/PhysRevE.74.066701",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:0801.0518 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.2659(7)",
        "note": "Table I, rs=0.5, row HF; sources/0801.0518.txt line 243; printed in Ha per electron; version read: arXiv text, Table I; APS copy not read | Ha per electron, 54 electrons (27 + 27) in a simple cubic cell. The paper's Hartree-Fock energies (VMC of the plane-wave determinant) match QMBL's HF energy within their error bars at all six rs, e.g. 0.5689(4) against 0.569245 at rs = 1 (checks/heg-hf/convention-check-2026-10-17.json).",
        "secondary_of": null
      }
    },
    {
      "energy": 174.0744,
      "sigma": 0.0108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow",
      "method_detail": "",
      "method_as_published": "VMC, SJ",
      "family": "classic VMC",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler & R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006), arXiv:0801.0518, doi:10.1103/PhysRevE.74.066701",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:0801.0518 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.2236(2)",
        "note": "Table I, rs=0.5, row SJ; sources/0801.0518.txt line 244; printed in Ha per electron; version read: arXiv text, Table I; APS copy not read | Ha per electron, 54 electrons (27 + 27) in a simple cubic cell. The paper's Hartree-Fock energies (VMC of the plane-wave determinant) match QMBL's HF energy within their error bars at all six rs, e.g. 0.5689(4) against 0.569245 at rs = 1 (checks/heg-hf/convention-check-2026-10-17.json).",
        "secondary_of": null
      }
    },
    {
      "energy": 174.0123,
      "sigma": 0.00486,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow; extrapolated: Δτ → 0",
      "method_as_published": "DMC, SJ nodes",
      "family": "DMC",
      "bound_type": "projected",
      "bound_type_reason": "projection (GFMC/fixed-node) on a trial state; not a strict variational bound (assigned during source reading)",
      "reference": "P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler & R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006), arXiv:0801.0518, doi:10.1103/PhysRevE.74.066701",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:0801.0518 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22245(9)",
        "note": "Table I, rs=0.5, row SJ, column ED; sources/0801.0518.txt line 244; printed in Ha per electron; version read: arXiv text, Table I; APS copy not read | Ha per electron, 54 electrons (27 + 27) in a simple cubic cell. The paper's Hartree-Fock energies (VMC of the plane-wave determinant) match QMBL's HF energy within their error bars at all six rs, e.g. 0.5689(4) against 0.569245 at rs = 1 (checks/heg-hf/convention-check-2026-10-17.json).",
        "secondary_of": null
      }
    },
    {
      "energy": 174.0582,
      "sigma": 0.0108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow",
      "method_detail": "three-electron Jastrow",
      "method_as_published": "VMC, SJ3",
      "family": "classic VMC",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler & R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006), arXiv:0801.0518, doi:10.1103/PhysRevE.74.066701",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:0801.0518 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.2233(2)",
        "note": "Table I, rs=0.5, row SJ3; sources/0801.0518.txt line 245; printed in Ha per electron; version read: arXiv text, Table I; APS copy not read | Ha per electron, 54 electrons (27 + 27) in a simple cubic cell. The paper's Hartree-Fock energies (VMC of the plane-wave determinant) match QMBL's HF energy within their error bars at all six rs, e.g. 0.5689(4) against 0.569245 at rs = 1 (checks/heg-hf/convention-check-2026-10-17.json).",
        "secondary_of": null
      }
    },
    {
      "energy": 173.95128,
      "sigma": 0.00378,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "homogeneous backflow",
      "method_as_published": "VMC, BF",
      "family": "classic VMC",
      "bound_type": "variational",
      "bound_type_reason": "variational ansatz at a stated size; energy is an upper bound (assigned during source reading)",
      "reference": "P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler & R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006), arXiv:0801.0518, doi:10.1103/PhysRevE.74.066701",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:0801.0518 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22132(7)",
        "note": "Table I, rs=0.5, row BF; sources/0801.0518.txt line 246; printed in Ha per electron; version read: arXiv text, Table I; APS copy not read | Ha per electron, 54 electrons (27 + 27) in a simple cubic cell. The paper's Hartree-Fock energies (VMC of the plane-wave determinant) match QMBL's HF energy within their error bars at all six rs, e.g. 0.5689(4) against 0.569245 at rs = 1 (checks/heg-hf/convention-check-2026-10-17.json).",
        "secondary_of": null
      }
    },
    {
      "energy": 173.94048,
      "sigma": 0.00216,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow-backflow; extrapolated: Δτ → 0",
      "method_as_published": "DMC, BF nodes",
      "family": "DMC",
      "bound_type": "projected",
      "bound_type_reason": "projection (GFMC/fixed-node) on a trial state; not a strict variational bound (assigned during source reading)",
      "reference": "P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler & R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006), arXiv:0801.0518, doi:10.1103/PhysRevE.74.066701",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:0801.0518 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22112(4)",
        "note": "Table I, rs=0.5, row BF, column ED; sources/0801.0518.txt line 246; printed in Ha per electron; version read: arXiv text, Table I; APS copy not read | Ha per electron, 54 electrons (27 + 27) in a simple cubic cell. The paper's Hartree-Fock energies (VMC of the plane-wave determinant) match QMBL's HF energy within their error bars at all six rs, e.g. 0.5689(4) against 0.569245 at rs = 1 (checks/heg-hf/convention-check-2026-10-17.json).",
        "secondary_of": null
      }
    },
    {
      "energy": 173.92266,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "CCD",
      "method_detail": "extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "CCD, complete-basis limit",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "K. Liao, T. Schraivogel, H. Luo, D. Kats & A. Alavi, Towards efficient and accurate ab initio solutions to periodic systems via transcorrelation and coupled cluster theory, Phys. Rev. Research 3, 033072 (2021), arXiv:2103.03176, doi:10.1103/PhysRevResearch.3.033072",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:2103.03176 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22079",
        "note": "Table II, row rs=0.5, column CCD; sources/2103.03176.txt line 389; printed in Ha per electron; version read: arXiv text, Tables I and II; APS copy not read | 'Total energy (a.u./electron), including the Madelung constant' (Table captions), 14 and 54 electrons in a simple cubic cell; the CC energies are extrapolated to the complete basis (linear in 1/M, Sec. IV). At N = 54, rs = 1 the DCD value agrees with arXiv:0801.0518's backflow DMC to 1e-4 Ha.",
        "secondary_of": null
      }
    },
    {
      "energy": 173.90808,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "DCD",
      "method_detail": "extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "DCD, complete-basis limit",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "K. Liao, T. Schraivogel, H. Luo, D. Kats & A. Alavi, Towards efficient and accurate ab initio solutions to periodic systems via transcorrelation and coupled cluster theory, Phys. Rev. Research 3, 033072 (2021), arXiv:2103.03176, doi:10.1103/PhysRevResearch.3.033072",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:2103.03176 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22052",
        "note": "Table II, row rs=0.5, column DCD; sources/2103.03176.txt line 389; printed in Ha per electron; version read: arXiv text, Tables I and II; APS copy not read | 'Total energy (a.u./electron), including the Madelung constant' (Table captions), 14 and 54 electrons in a simple cubic cell; the CC energies are extrapolated to the complete basis (linear in 1/M, Sec. IV). At N = 54, rs = 1 the DCD value agrees with arXiv:0801.0518's backflow DMC to 1e-4 Ha.",
        "secondary_of": null
      }
    },
    {
      "energy": 173.92158,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "TC-CCD",
      "method_detail": "correlator cutoff k_c = √2 × 2π/L; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "TC-CCD (k_c = √2), complete-basis limit",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "K. Liao, T. Schraivogel, H. Luo, D. Kats & A. Alavi, Towards efficient and accurate ab initio solutions to periodic systems via transcorrelation and coupled cluster theory, Phys. Rev. Research 3, 033072 (2021), arXiv:2103.03176, doi:10.1103/PhysRevResearch.3.033072",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:2103.03176 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22077",
        "note": "Table II, row rs=0.5, column TC-CCD; sources/2103.03176.txt line 389; printed in Ha per electron; version read: arXiv text, Tables I and II; APS copy not read | 'Total energy (a.u./electron), including the Madelung constant' (Table captions), 14 and 54 electrons in a simple cubic cell; the CC energies are extrapolated to the complete basis (linear in 1/M, Sec. IV). At N = 54, rs = 1 the DCD value agrees with arXiv:0801.0518's backflow DMC to 1e-4 Ha.",
        "secondary_of": null
      }
    },
    {
      "energy": 173.91834,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "TC-DCD",
      "method_detail": "correlator cutoff k_c = √2 × 2π/L; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "TC-DCD (k_c = √2), complete-basis limit",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "K. Liao, T. Schraivogel, H. Luo, D. Kats & A. Alavi, Towards efficient and accurate ab initio solutions to periodic systems via transcorrelation and coupled cluster theory, Phys. Rev. Research 3, 033072 (2021), arXiv:2103.03176, doi:10.1103/PhysRevResearch.3.033072",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:2103.03176 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22071",
        "note": "Table II, row rs=0.5, column TC-DCD; sources/2103.03176.txt line 389; printed in Ha per electron; version read: arXiv text, Tables I and II; APS copy not read | 'Total energy (a.u./electron), including the Madelung constant' (Table captions), 14 and 54 electrons in a simple cubic cell; the CC energies are extrapolated to the complete basis (linear in 1/M, Sec. IV). At N = 54, rs = 1 the DCD value agrees with arXiv:0801.0518's backflow DMC to 1e-4 Ha.",
        "secondary_of": null
      }
    },
    {
      "energy": 173.892790783,
      "sigma": 0.00702,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "FCIQMC",
      "method_detail": "initiator (n_add = 3), run at 922 spin orbitals; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "i-FCIQMC, single-point basis-limit estimate (run at M = 922)",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "J. J. Shepherd, G. H. Booth & A. Alavi, Investigation of the full configuration interaction quantum Monte Carlo method using homogeneous electron gas models, J. Chem. Phys. 136, 244101 (2012), arXiv:1201.4691, doi:10.1063/1.4720076",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:1201.4691 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "E_c = -2.435(7) (total, Ha)",
        "note": "Table II, rs=0.5, N=54, column Ecorr (a.u.); sources/1201.4691.txt line 21000; printed correlation energy -2.435(7) Ha for 54 electrons; E = E_HF + E_c with E_HF = 3.26532945895 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text, Tables I and II; AIP copy not read | Correlation energy E_c = E - E_HF, total for the cell, Hartree; 14, 38 or 54 electrons in a simple cubic cell at the Gamma point, a closed shell, so E_HF is the unique plane-wave determinant's. Stored E = E_HF + E_c with E_HF from QMBL (checks/heg-hf/hf.mjs; RULES.md 12.3, 2026-10-19).",
        "secondary_of": null
      }
    },
    {
      "energy": 173.92698,
      "sigma": 0.00108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "ph-AFQMC",
      "method_detail": "RHF trial, time step 0.005 a.u., 2880 walkers; basis M=1419 (Table V caption: 'Both ph-AFQMC and i-FCIQMC numbers are obtained from M = 1419",
      "method_as_published": "RHF+ph-AFQMC [RHF trial, time step 0.005 a.u., 2880 walkers; basis M=1419 (Table V caption: 'Both ph-AFQMC and i-FCIQMC numbers are obtained from M = 1419] (arXiv:1905.04361)",
      "family": "AFQMC / GFMC",
      "bound_type": "projected",
      "bound_type_reason": "projection (GFMC/fixed-node) on a trial state; not a strict variational bound (assigned during source reading)",
      "reference": "Joonho Lee, Fionn D. Malone, Miguel A. Morales, An Auxiliary-Field Quantum Monte Carlo Perspective on the Ground State of the Dense Uniform Electron Gas: An Investigation with Hartree-Fock Trial Wavefunctions, J. Chem. Phys. 151, 064122, arXiv:1905.04361, doi:10.1063/1.5109572",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:1905.04361 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "3.22087(2)",
        "note": "Table V, rs=0.5, column ph-AFQMC; sources/1905.04361.txt line 675; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | Eq. (19) (txt line 210): H = T + Vee + E_M; txt lines 232-237: 'the Madelung energy E_M is included to account for the self-interaction of the Ewald sum under periodic boundary conditions. For simplicity we use the formula proposed by Schoof and co-workers' E_M ~ -2.837297 (3/4pi)^(1/3) N^(2/3)/rs (Eq. 22, txt line 238-239). So the Madelung term is included, but through the approximate closed form (Ref 76: T. Schoof, S. Groth, J. Vorberger, M. Bonitz, PRL 115, 130402 (2015)), not through a stated Ewald sum. No Hartree-Fock energy is printed. Reader's own check (python, scratch-R8/hf.py, NOT QM",
        "secondary_of": null
      }
    },
    {
      "energy": 173.955790783,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "CCD",
      "method_detail": "full CCD (driver, ladders, rings, crossed-rings, mosaics); equals CCSD in the HEG; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "CCD [full CCD (driver, ladders, rings, crossed-rings, mosaics); equals CCSD in the HEG; extrapolated: 1/M → 0 (complete basis)] (arXiv:1310.6806)",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "James J. Shepherd, Thomas M. Henderson, and Gustavo E. Scuseria, Coupled Cluster Channels in the Homogeneous Electron Gas, J. Chem. Phys. 140, 124102, arXiv:1310.6806, doi:10.1063/1.4867783",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:1310.6806 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "E_c = -2.372",
        "note": "Table II, rs=0.5, column CCD; sources/1310.6806.txt line 460; printed in Ha, total; printed correlation energy -2.372, E = E_HF + E_c with E_HF = 3.26532945895 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text in sources/; journal copy not read | Sec. II (txt line 64-81): HF eigenvalues with the zero-momentum integral given by 'vM, the Madelung constant ... defined uniquely for a cell geometry and box length, and here we calculate it to be vM L ~ 2.8372'. No Ewald-sum statement beyond that. The paper prints correlation energies only, which do not depend on the Madelung term, and no Hartree-Fock energy.",
        "secondary_of": null
      }
    },
    {
      "energy": 173.902790783,
      "sigma": 0.001026,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Transcorrelated FCIQMC",
      "method_detail": "FCIQMC on the transcorrelated Hamiltonian H_TC = exp(-tau) H exp(tau) with the basis-adapted correlation factor of Eq. (22), three-body term; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "FCIQMC-TC [FCIQMC on the transcorrelated Hamiltonian H_TC = exp(-tau) H exp(tau) with the basis-adapted correlation factor of Eq. (22), three-body term; extrapolated: 1/M → 0 (complete basis)] (arXiv:1712.07524)",
      "family": "FCIQMC / CC",
      "bound_type": "extrapolated",
      "bound_type_reason": "zero-variance or bond-dimension extrapolation; not an upper bound (assigned during source reading)",
      "reference": "Hongjun Luo and Ali Alavi, Combining the Transcorrelated method with Full Configuration Interaction Quantum Monte Carlo: application to the homogeneous electron gas, J. Chem. Theory Comput. 14, 1403-1411, arXiv:1712.07524, doi:10.1021/acs.jctc.7b01257",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:1712.07524 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading",
        "reported_as": "E_c = -2.425(1)",
        "note": "Table 1, N=54, rs=0.5, column FCIQMC-TC; sources/1712.07524.txt line 792; printed in Ha, total; printed correlation energy -2.425(1), E = E_HF + E_c with E_HF = 3.26532945895 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text in sources/; journal copy not read | Not stated: grep for Madelung and Ewald finds nothing. Only correlation energies are printed (Table 1), so the convention is moot for the numbers, but the HF energy of the same systems is not printed either. Interaction Fourier transform w(k) = 4 pi / k^2 for k != 0, 0 for k = 0 (Eq. 28, txt line 595-598).",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/HEG3D/sc_54_P_27_27_0.5/",
  "per_site_divisor": 54,
  "per_site_label": "E/N (Ha)",
  "record": {
    "energy": 173.8908,
    "sigma": 0.0108,
    "method": "FCIQMC",
    "method_detail": "initiator (n_add = 3); extrapolated: 1/M → 0 (complete basis)",
    "reference": "J. J. Shepherd, G. H. Booth, A. Grüneis & A. Alavi, Full configuration interaction perspective on the homogeneous electron gas, Phys. Rev. B 85, 081103(R) (2012), arXiv:1109.2635, doi:10.1103/PhysRevB.85.081103",
    "bound_type": "extrapolated",
    "energy_per_site": 3.2202
  },
  "no_record_reason": null
}