{
  "model": "HEG3D",
  "lattice": "bcc",
  "n_sites": 27,
  "boundary": "P",
  "params": {
    "N_up": 27,
    "N_dn": 0,
    "rs": 90,
    "twist": "Gamma"
  },
  "instance_id": "HEG3D/bcc_27_P_27_0_90",
  "rows": [
    {
      "energy": -0.229392,
      "sigma": 0.000027,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "FermiNet",
      "method_detail": "16 dense determinants, 4 layers 512/64",
      "method_as_published": "FermiNet (16 dense determinants)",
      "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. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer & W. M. C. Foulkes, Discovering Quantum Phase Transitions with Fermionic Neural Networks, Phys. Rev. Lett. 130, 036401 (2023), arXiv:2202.05183, doi:10.1103/PhysRevLett.130.036401",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2202.05183 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": "−0.008496(1)",
        "note": "Table IV (SI), rs=90.0, column 'FermiNet'; sources/2202.05183.txt line 714; printed in Ha per electron; version read: arXiv text, Supplementary Table IV; APS copy not read | Hartree per electron, spin-polarized N = 27 in a body-centred cubic cell (text and caption). QMBL's HF check fixes the cell: at rs = 0.2 the FermiNet energy 44.18067 is 0.047 Ha below QMBL's bcc HF energy (open shell), against 5.1 and 2.6 Ha above it for sc and fcc cells.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.2275479,
      "sigma": 0.0000027,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "plane-wave determinant",
      "method_as_published": "SJB-VMC (gas)",
      "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": "G. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer & W. M. C. Foulkes, Discovering Quantum Phase Transitions with Fermionic Neural Networks, Phys. Rev. Lett. 130, 036401 (2023), arXiv:2202.05183, doi:10.1103/PhysRevLett.130.036401",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2202.05183 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": "−0.0084277(1)",
        "note": "Table IV (SI), rs=90.0, column 'SJB-VMC Gas'; sources/2202.05183.txt line 714; printed in Ha per electron; version read: arXiv text, Supplementary Table IV; APS copy not read | Hartree per electron, spin-polarized N = 27 in a body-centred cubic cell (text and caption). QMBL's HF check fixes the cell: at rs = 0.2 the FermiNet energy 44.18067 is 0.047 Ha below QMBL's bcc HF energy (open shell), against 5.1 and 2.6 Ha above it for sc and fcc cells.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.2291274,
      "sigma": 0.0000027,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow-backflow, plane-wave determinant",
      "method_as_published": "SJB-DMC (gas 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. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer & W. M. C. Foulkes, Discovering Quantum Phase Transitions with Fermionic Neural Networks, Phys. Rev. Lett. 130, 036401 (2023), arXiv:2202.05183, doi:10.1103/PhysRevLett.130.036401",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2202.05183 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": "−0.0084862(1)",
        "note": "Table IV (SI), rs=90.0, column 'SJB-DMC Gas'; sources/2202.05183.txt line 714; printed in Ha per electron; version read: arXiv text, Supplementary Table IV; APS copy not read | Hartree per electron, spin-polarized N = 27 in a body-centred cubic cell (text and caption). QMBL's HF check fixes the cell: at rs = 0.2 the FermiNet energy 44.18067 is 0.047 Ha below QMBL's bcc HF energy (open shell), against 5.1 and 2.6 Ha above it for sc and fcc cells.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.22930398,
      "sigma": 0.00000189,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Gaussian-orbital (crystal) determinant",
      "method_as_published": "SJB-VMC (crystal)",
      "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": "G. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer & W. M. C. Foulkes, Discovering Quantum Phase Transitions with Fermionic Neural Networks, Phys. Rev. Lett. 130, 036401 (2023), arXiv:2202.05183, doi:10.1103/PhysRevLett.130.036401",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2202.05183 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": "−0.00849274(7)",
        "note": "Table IV (SI), rs=90.0, column 'SJB-VMC Crystal'; sources/2202.05183.txt line 714; printed in Ha per electron; version read: arXiv text, Supplementary Table IV; APS copy not read | Hartree per electron, spin-polarized N = 27 in a body-centred cubic cell (text and caption). QMBL's HF check fixes the cell: at rs = 0.2 the FermiNet energy 44.18067 is 0.047 Ha below QMBL's bcc HF energy (open shell), against 5.1 and 2.6 Ha above it for sc and fcc cells.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.2298699,
      "sigma": 0.0000054,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Fixed-node DMC",
      "method_detail": "trial state: Slater-Jastrow-backflow, Gaussian-orbital (crystal) determinant",
      "method_as_published": "SJB-DMC (crystal 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. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer & W. M. C. Foulkes, Discovering Quantum Phase Transitions with Fermionic Neural Networks, Phys. Rev. Lett. 130, 036401 (2023), arXiv:2202.05183, doi:10.1103/PhysRevLett.130.036401",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-16",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-16",
        "method": "source text of arXiv:2202.05183 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": "−0.0085137(2)",
        "note": "Table IV (SI), rs=90.0, column 'SJB-DMC Crystal'; sources/2202.05183.txt line 714; printed in Ha per electron; version read: arXiv text, Supplementary Table IV; APS copy not read | Hartree per electron, spin-polarized N = 27 in a body-centred cubic cell (text and caption). QMBL's HF check fixes the cell: at rs = 0.2 the FermiNet energy 44.18067 is 0.047 Ha below QMBL's bcc HF energy (open shell), against 5.1 and 2.6 Ha above it for sc and fcc cells.",
        "secondary_of": null
      }
    },
    {
      "energy": -0.2274048,
      "sigma": 0.0000081,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=1 keeps 8 terms i",
      "method_as_published": "SJB-VMC [Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=1 keeps 8 terms i] (arXiv:2201.08743)",
      "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": "Sam Azadi and N. D. Drummond, Low-density Phase Diagram of the Three-Dimensional Electron Gas, Phys. Rev. B 105, 245135, arXiv:2201.08743, doi:10.1103/PhysRevB.105.245135",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:2201.08743 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": "−0.0084224(3)",
        "note": "Table I, g=1; sources/2201.08743.txt line 204; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | The text names the Ewald interaction ('the interaction ... is of Ewald form', txt line ~287, 'the use of the Ewald interaction rather than 1/r', line ~173) but never the Madelung term. The paper prints no Hartree-Fock energy, so no HF check is possible. Fig. 15 caption (txt line 711): 'The Madelung energy of the bcc lattice has been subtracted off' from the DMC energies, i.e. the energies as printed contain the Madelung term. Indirect check: ferromagnetic bcc crystal at rs=90, N=27 has E = -0.0084925 Ha/el. (Table I), close to the bcc Madelung lattice energy -0.89593/rs = -0.009955 plus the ze",
        "secondary_of": null
      }
    },
    {
      "energy": -0.22932747,
      "sigma": 0.00000189,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=2 keeps 24 terms ",
      "method_as_published": "SJB-VMC [Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=2 keeps 24 terms ] (arXiv:2201.08743)",
      "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": "Sam Azadi and N. D. Drummond, Low-density Phase Diagram of the Three-Dimensional Electron Gas, Phys. Rev. B 105, 245135, arXiv:2201.08743, doi:10.1103/PhysRevB.105.245135",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:2201.08743 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": "−0.00849361(7)",
        "note": "Table I, g=2; sources/2201.08743.txt line 205; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | The text names the Ewald interaction ('the interaction ... is of Ewald form', txt line ~287, 'the use of the Ewald interaction rather than 1/r', line ~173) but never the Madelung term. The paper prints no Hartree-Fock energy, so no HF check is possible. Fig. 15 caption (txt line 711): 'The Madelung energy of the bcc lattice has been subtracted off' from the DMC energies, i.e. the energies as printed contain the Madelung term. Indirect check: ferromagnetic bcc crystal at rs=90, N=27 has E = -0.0084925 Ha/el. (Table I), close to the bcc Madelung lattice energy -0.89593/rs = -0.009955 plus the ze",
        "secondary_of": null
      }
    },
    {
      "energy": -0.22930236,
      "sigma": 0.00000189,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=3 keeps 27 terms ",
      "method_as_published": "SJB-VMC [Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=3 keeps 27 terms ] (arXiv:2201.08743)",
      "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": "Sam Azadi and N. D. Drummond, Low-density Phase Diagram of the Three-Dimensional Electron Gas, Phys. Rev. B 105, 245135, arXiv:2201.08743, doi:10.1103/PhysRevB.105.245135",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:2201.08743 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": "−0.00849268(7)",
        "note": "Table I, g=3; sources/2201.08743.txt line 206; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | The text names the Ewald interaction ('the interaction ... is of Ewald form', txt line ~287, 'the use of the Ewald interaction rather than 1/r', line ~173) but never the Madelung term. The paper prints no Hartree-Fock energy, so no HF check is possible. Fig. 15 caption (txt line 711): 'The Madelung energy of the bcc lattice has been subtracted off' from the DMC energies, i.e. the energies as printed contain the Madelung term. Indirect check: ferromagnetic bcc crystal at rs=90, N=27 has E = -0.0084925 Ha/el. (Table I), close to the bcc Madelung lattice energy -0.89593/rs = -0.009955 plus the ze",
        "secondary_of": null
      }
    },
    {
      "energy": -0.22930101,
      "sigma": 0.00000189,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=5 keeps 27 terms ",
      "method_as_published": "SJB-VMC [Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=5 keeps 27 terms ] (arXiv:2201.08743)",
      "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": "Sam Azadi and N. D. Drummond, Low-density Phase Diagram of the Three-Dimensional Electron Gas, Phys. Rev. B 105, 245135, arXiv:2201.08743, doi:10.1103/PhysRevB.105.245135",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:2201.08743 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": "−0.00849263(7)",
        "note": "Table I, g=5; sources/2201.08743.txt line 207; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | The text names the Ewald interaction ('the interaction ... is of Ewald form', txt line ~287, 'the use of the Ewald interaction rather than 1/r', line ~173) but never the Madelung term. The paper prints no Hartree-Fock energy, so no HF check is possible. Fig. 15 caption (txt line 711): 'The Madelung energy of the bcc lattice has been subtracted off' from the DMC energies, i.e. the energies as printed contain the Madelung term. Indirect check: ferromagnetic bcc crystal at rs=90, N=27 has E = -0.0084925 Ha/el. (Table I), close to the bcc Madelung lattice energy -0.89593/rs = -0.009955 plus the ze",
        "secondary_of": null
      }
    },
    {
      "energy": -0.22929831,
      "sigma": 0.00000189,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=7 keeps 27 terms ",
      "method_as_published": "SJB-VMC [Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=7 keeps 27 terms ] (arXiv:2201.08743)",
      "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": "Sam Azadi and N. D. Drummond, Low-density Phase Diagram of the Three-Dimensional Electron Gas, Phys. Rev. B 105, 245135, arXiv:2201.08743, doi:10.1103/PhysRevB.105.245135",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:2201.08743 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": "−0.00849253(7)",
        "note": "Table I, g=7; sources/2201.08743.txt line 208; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | The text names the Ewald interaction ('the interaction ... is of Ewald form', txt line ~287, 'the use of the Ewald interaction rather than 1/r', line ~173) but never the Madelung term. The paper prints no Hartree-Fock energy, so no HF check is possible. Fig. 15 caption (txt line 711): 'The Madelung energy of the bcc lattice has been subtracted off' from the DMC energies, i.e. the energies as printed contain the Madelung term. Indirect check: ferromagnetic bcc crystal at rs=90, N=27 has E = -0.0084925 Ha/el. (Table I), close to the bcc Madelung lattice energy -0.89593/rs = -0.009955 plus the ze",
        "secondary_of": null
      }
    },
    {
      "energy": -0.22929858,
      "sigma": 0.00000189,
      "energy_variance": null,
      "dof": 27,
      "einf": null,
      "v_score": null,
      "method": "Slater-Jastrow-backflow",
      "method_detail": "Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=15 keeps 46 terms",
      "method_as_published": "SJB-VMC [Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=15 keeps 46 terms] (arXiv:2201.08743)",
      "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": "Sam Azadi and N. D. Drummond, Low-density Phase Diagram of the Three-Dimensional Electron Gas, Phys. Rev. B 105, 245135, arXiv:2201.08743, doi:10.1103/PhysRevB.105.245135",
      "peer_reviewed": true,
      "source": "sweep-allresults-2026-10-19",
      "provenance": "primary",
      "verified": {
        "checked_on": "2026-10-19",
        "method": "source text of arXiv:2201.08743 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": "−0.00849254(7)",
        "note": "Table I, g=15; sources/2201.08743.txt line 209; printed in Ha per electron; version read: arXiv text in sources/; journal copy not read | The text names the Ewald interaction ('the interaction ... is of Ewald form', txt line ~287, 'the use of the Ewald interaction rather than 1/r', line ~173) but never the Madelung term. The paper prints no Hartree-Fock energy, so no HF check is possible. Fig. 15 caption (txt line 711): 'The Madelung energy of the bcc lattice has been subtracted off' from the DMC energies, i.e. the energies as printed contain the Madelung term. Indirect check: ferromagnetic bcc crystal at rs=90, N=27 has E = -0.0084925 Ha/el. (Table I), close to the bcc Madelung lattice energy -0.89593/rs = -0.009955 plus the ze",
        "secondary_of": null
      }
    }
  ],
  "url": "https://qmbl.org/i/HEG3D/bcc_27_P_27_0_90/",
  "per_site_divisor": 27,
  "per_site_label": "E/N (Ha)",
  "record": {
    "energy": -0.2298699,
    "sigma": 0.0000054,
    "method": "Fixed-node DMC",
    "method_detail": "trial state: Slater-Jastrow-backflow, Gaussian-orbital (crystal) determinant",
    "reference": "G. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer & W. M. C. Foulkes, Discovering Quantum Phase Transitions with Fermionic Neural Networks, Phys. Rev. Lett. 130, 036401 (2023), arXiv:2202.05183, doi:10.1103/PhysRevLett.130.036401",
    "bound_type": "projected",
    "energy_per_site": -0.008513699999999999
  },
  "no_record_reason": null
}