{
  "model": "HEG3D",
  "lattice": "sc",
  "n_sites": 54,
  "boundary": "P",
  "params": {
    "N_up": 27,
    "N_dn": 27,
    "rs": 5,
    "twist": "Gamma"
  },
  "instance_id": "HEG3D/sc_54_P_27_27_5",
  "rows": [
    {
      "energy": -4.27086,
      "sigma": 0.000108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "MP-NQS",
      "method_detail": "plane-wave orbitals",
      "method_as_published": "MP-NQS (plane-wave orbitals)",
      "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": "G. Pescia, J. Nys, J. Kim, A. Lovato & G. Carleo, Message-passing neural quantum states for the homogeneous electron gas, Phys. Rev. B 110, 035108 (2024), arXiv:2305.07240, doi:10.1103/PhysRevB.110.035108",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2305.07240 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": "−0.079090(2)",
        "note": "Table II (Appendix G), N=54 unpolarized, rs=5, column MP-NQS; sources/2305.07240.txt line 680; printed in Ha per electron; version read: arXiv text, Appendix G Tables I-V; APS copy not read | Ha per electron, simple cubic cell; Eq. 13 gives the Ewald sum and 'const.'. The WAP-net values it quotes equal arXiv:2202.04622's Table III (Madelung included) / 2 to 5e-7, and its own N = 14, rs = 1 energy sits 0.0376 Ha below QMBL's HF energy, as theirs does.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.269456,
      "sigma": 0.000216,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "MP-NQS",
      "method_detail": "Gaussian orbitals on bcc sites",
      "method_as_published": "MP-NQS (Gaussian orbitals on bcc sites)",
      "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": "G. Pescia, J. Nys, J. Kim, A. Lovato & G. Carleo, Message-passing neural quantum states for the homogeneous electron gas, Phys. Rev. B 110, 035108 (2024), arXiv:2305.07240, doi:10.1103/PhysRevB.110.035108",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2305.07240 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": "−0.079064(4)",
        "note": "Table II (Appendix G), N=54 unpolarized, rs=5, column MP-NQS (Gauss); sources/2305.07240.txt line 680; printed in Ha per electron; version read: arXiv text, Appendix G Tables I-V; APS copy not read | Ha per electron, simple cubic cell; Eq. 13 gives the Ewald sum and 'const.'. The WAP-net values it quotes equal arXiv:2202.04622's Table III (Madelung included) / 2 to 5e-7, and its own N = 14, rs = 1 energy sits 0.0376 Ha below QMBL's HF energy, as theirs does.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.24818,
      "sigma": 0.00054,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow",
      "method_as_published": "FN-DMC (Slater-Jastrow trial state)",
      "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": "G. Pescia, J. Nys, J. Kim, A. Lovato & G. Carleo, Message-passing neural quantum states for the homogeneous electron gas, Phys. Rev. B 110, 035108 (2024), arXiv:2305.07240, doi:10.1103/PhysRevB.110.035108",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2305.07240 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": "−0.07867(1)",
        "note": "Table II (Appendix G), N=54 unpolarized, rs=5, column FN-DMC; sources/2305.07240.txt line 680; printed in Ha per electron; version read: arXiv text, Appendix G Tables I-V; APS copy not read | Ha per electron, simple cubic cell; Eq. 13 gives the Ewald sum and 'const.'. The WAP-net values it quotes equal arXiv:2202.04622's Table III (Madelung included) / 2 to 5e-7, and its own N = 14, rs = 1 energy sits 0.0376 Ha below QMBL's HF energy, as theirs does.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.2571116,
      "sigma": 0.0000108,
      "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": "-0.0788354(2)",
        "note": "Supplementary Table 11, row rs=5, column 'Net'; sources/2203.15472.txt line 1016; 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": -3.0375,
      "sigma": 0.00378,
      "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": "−0.05625(7)",
        "note": "Table I, rs=5.0, row HF; sources/0801.0518.txt line 258; 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": -4.2201,
      "sigma": 0.00054,
      "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": "−0.07815(1)",
        "note": "Table I, rs=5.0, row SJ; sources/0801.0518.txt line 259; 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": -4.227336,
      "sigma": 0.000486,
      "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": "−0.078284(9)",
        "note": "Table I, rs=5.0, row SJ3; sources/0801.0518.txt line 260; 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": -4.247046,
      "sigma": 0.000378,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow + three-electron Jastrow; extrapolated: Δτ → 0",
      "method_as_published": "DMC, SJ3 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": "−0.078649(7)",
        "note": "Table I, rs=5.0, row SJ3, column ED; sources/0801.0518.txt line 260; 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": -4.263894,
      "sigma": 0.00027,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "homogeneous backflow + three-electron Jastrow",
      "method_as_published": "VMC, BF3",
      "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": "−0.078961(5)",
        "note": "Table I, rs=5.0, row BF3; sources/0801.0518.txt line 261; 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": -4.267944,
      "sigma": 0.000162,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow-backflow + three-electron Jastrow; extrapolated: Δτ → 0",
      "method_as_published": "DMC, BF3 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": "−0.079036(3)",
        "note": "Table I, rs=5.0, row BF3, column ED; sources/0801.0518.txt line 261; 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": -4.20066,
      "sigma": 0.00189,
      "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": "Y. Kwon, D. M. Ceperley & R. M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study, Phys. Rev. B 58, 6800 (1998), arXiv:cond-mat/9803092, doi:10.1103/PhysRevB.58.6800",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:cond-mat/9803092 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": "-0.15558(7)",
        "note": "Table II, row E^SJ_VMC, rs=5.0; sources/cond-mat-9803092.txt line 618; printed in Ry per electron; version read: arXiv text (sources/cond-mat-9803092.txt), Tables II and III; APS copy not read | Ry per electron (converted × 1/2), N electrons in a cube with periodic boundary conditions (simple cubic). Checked against arXiv:0801.0518 on the same instance: the fixed-node DMC energy with backflow nodes, 1.0601(2) Ry = 0.53005 Ha at N = 54, rs = 1, agrees with its 0.52989(1) Ha, 0.0376 Ha below QMBL's HF energy 0.569245.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.20363,
      "sigma": 0.00135,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow",
      "method_detail": "three-body correlation",
      "method_as_published": "VMC SJ+3BD",
      "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": "Y. Kwon, D. M. Ceperley & R. M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study, Phys. Rev. B 58, 6800 (1998), arXiv:cond-mat/9803092, doi:10.1103/PhysRevB.58.6800",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:cond-mat/9803092 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": "-0.15569(5)",
        "note": "Table II, row E^SJ+3BD_VMC, rs=5.0; sources/cond-mat-9803092.txt line 619; printed in Ry per electron; version read: arXiv text (sources/cond-mat-9803092.txt), Tables II and III; APS copy not read | Ry per electron (converted × 1/2), N electrons in a cube with periodic boundary conditions (simple cubic). Checked against arXiv:0801.0518 on the same instance: the fixed-node DMC energy with backflow nodes, 1.0601(2) Ry = 0.53005 Ha at N = 54, rs = 1, agrees with its 0.52989(1) Ha, 0.0376 Ha below QMBL's HF energy 0.569245.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.24683,
      "sigma": 0.00135,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "",
      "method_as_published": "VMC SJ+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": "Y. Kwon, D. M. Ceperley & R. M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study, Phys. Rev. B 58, 6800 (1998), arXiv:cond-mat/9803092, doi:10.1103/PhysRevB.58.6800",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:cond-mat/9803092 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": "-0.15729(5)",
        "note": "Table II, row E^SJ+BF_VMC, rs=5.0; sources/cond-mat-9803092.txt line 620; printed in Ry per electron; version read: arXiv text (sources/cond-mat-9803092.txt), Tables II and III; APS copy not read | Ry per electron (converted × 1/2), N electrons in a cube with periodic boundary conditions (simple cubic). Checked against arXiv:0801.0518 on the same instance: the fixed-node DMC energy with backflow nodes, 1.0601(2) Ry = 0.53005 Ha at N = 54, rs = 1, agrees with its 0.52989(1) Ha, 0.0376 Ha below QMBL's HF energy 0.569245.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.24845,
      "sigma": 0.00135,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "three-body correlation",
      "method_as_published": "VMC SJ+3BD+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": "Y. Kwon, D. M. Ceperley & R. M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study, Phys. Rev. B 58, 6800 (1998), arXiv:cond-mat/9803092, doi:10.1103/PhysRevB.58.6800",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:cond-mat/9803092 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": "-0.15735(5)",
        "note": "Table II, row E^SJ+3BD+BF_VMC, rs=5.0; sources/cond-mat-9803092.txt line 621; printed in Ry per electron; version read: arXiv text (sources/cond-mat-9803092.txt), Tables II and III; APS copy not read | Ry per electron (converted × 1/2), N electrons in a cube with periodic boundary conditions (simple cubic). Checked against arXiv:0801.0518 on the same instance: the fixed-node DMC energy with backflow nodes, 1.0601(2) Ry = 0.53005 Ha at N = 54, rs = 1, agrees with its 0.52989(1) Ha, 0.0376 Ha below QMBL's HF energy 0.569245.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.24818,
      "sigma": 0.00081,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow",
      "method_as_published": "DMC SJ",
      "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": "Y. Kwon, D. M. Ceperley & R. M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study, Phys. Rev. B 58, 6800 (1998), arXiv:cond-mat/9803092, doi:10.1103/PhysRevB.58.6800",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:cond-mat/9803092 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": "-0.15734(3)",
        "note": "Table II, row E^SJ_FN, rs=5.0; sources/cond-mat-9803092.txt line 623; printed in Ry per electron; version read: arXiv text (sources/cond-mat-9803092.txt), Tables II and III; APS copy not read | Ry per electron (converted × 1/2), N electrons in a cube with periodic boundary conditions (simple cubic). Checked against arXiv:0801.0518 on the same instance: the fixed-node DMC energy with backflow nodes, 1.0601(2) Ry = 0.53005 Ha at N = 54, rs = 1, agrees with its 0.52989(1) Ha, 0.0376 Ha below QMBL's HF energy 0.569245.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.26546,
      "sigma": 0.00108,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow-backflow + three-body",
      "method_as_published": "DMC SJ+3BD+BF",
      "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": "Y. Kwon, D. M. Ceperley & R. M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study, Phys. Rev. B 58, 6800 (1998), arXiv:cond-mat/9803092, doi:10.1103/PhysRevB.58.6800",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-17",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-17",
        "method": "source text of arXiv:cond-mat/9803092 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": "-0.15798(4)",
        "note": "Table II, row E^SJ+3BD+BF_FN, rs=5.0; sources/cond-mat-9803092.txt line 624; printed in Ry per electron; version read: arXiv text (sources/cond-mat-9803092.txt), Tables II and III; APS copy not read | Ry per electron (converted × 1/2), N electrons in a cube with periodic boundary conditions (simple cubic). Checked against arXiv:0801.0518 on the same instance: the fixed-node DMC energy with backflow nodes, 1.0601(2) Ry = 0.53005 Ha at N = 54, rs = 1, agrees with its 0.52989(1) Ha, 0.0376 Ha below QMBL's HF energy 0.569245.",
        "secondary_of": null
      }
    },
    {
      "energy": -4.04568,
      "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": "-0.07492",
        "note": "Table II, row rs=5.0, column CCD; sources/2103.03176.txt line 392; 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": -4.1337,
      "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": "-0.07655",
        "note": "Table II, row rs=5.0, column DCD; sources/2103.03176.txt line 392; 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": -4.185,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "TC-CCD",
      "method_detail": "correlator cutoff k_c = √5 × 2π/L; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "TC-CCD (k_c = √5), 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": "-0.07750",
        "note": "Table II, row rs=5.0, column TC-CCD; sources/2103.03176.txt line 392; 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": -4.23198,
      "sigma": null,
      "energy_variance": null,
      "dof": 54,
      "einf": null,
      "v_score": null,
      "method": "TC-DCD",
      "method_detail": "correlator cutoff k_c = √5 × 2π/L; extrapolated: 1/M → 0 (complete basis)",
      "method_as_published": "TC-DCD (k_c = √5), 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": "-0.07837",
        "note": "Table II, row rs=5.0, column TC-DCD; sources/2103.03176.txt line 392; 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": -4.09806,
      "sigma": 0.0027,
      "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": "-0.07589(5)",
        "note": "Table V, rs=5.0, column ph-AFQMC; sources/1905.04361.txt line 678; 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": -4.03510572305,
      "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 = -0.995",
        "note": "Table II, rs=5.0, column CCD; sources/1310.6806.txt line 465; printed in Ha, total; printed correlation energy -0.995, E = E_HF + E_c with E_HF = -0.0562982541305 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
      }
    }
  ],
  "url": "https://qmbl.org/i/HEG3D/sc_54_P_27_27_5/",
  "per_site_divisor": 54,
  "per_site_label": "E/N (Ha)",
  "record": {
    "energy": -4.27086,
    "sigma": 0.000108,
    "method": "MP-NQS",
    "method_detail": "plane-wave orbitals",
    "reference": "G. Pescia, J. Nys, J. Kim, A. Lovato & G. Carleo, Message-passing neural quantum states for the homogeneous electron gas, Phys. Rev. B 110, 035108 (2024), arXiv:2305.07240, doi:10.1103/PhysRevB.110.035108",
    "bound_type": "variational",
    "energy_per_site": -0.07909
  },
  "no_record_reason": null
}