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All instances / Electron gas, 3D

Electron gas, 3D bcc cell, N = 27, rs = 90, polarized

Record

-0.0085137(2) E/N (Ha)

Fixed-node DMC (trial state: Slater-Jastrow-backflow, Gaussian-orbital (crystal) determinant)

10.1103/PhysRevLett.130.036401 · lowest unflagged energy published, a projected estimate (rules §6)

The Hamiltonian

H = −½ ∑i ∇i2 + ∑i<j vE(ri − rj) + Nξ/2

Homogeneous electron gas in 3D: N↑ = 27, N↓ = 0 in a periodic bcc cell at rs = 90 bohr, Γ point. vE is the Ewald-summed Coulomb interaction with a uniform neutralizing background, ξ the Madelung term. Hartree atomic units; the energy per electron is E/N.

cell
bcc
electrons
27
boundary
periodic
N_up
27
N_dn
0
rs
90
twist
Gamma
energies
11

Stored as a total energy in Hartree (rules §12); quoted here as E/N (Ha), the energy per electron (how the conversion works).

Every published energy

Grouped by what the number is. Where the instance is not solved, the lowest unflagged energy of any group is the record.

Strict variational upper bounds (9)

Where the instance is not solved, the record is the lowest unflagged energy in this group or among the projected and extrapolated ones, with or without an error bar; where it is solved, the exact energy is the record and the lowest of these is the closest challenger.

E/N (Ha)distance to recordσVar(E)V-score methodsourceyear
-0.008496 1.8e-5 1.0e-6 n/a n/a FermiNet (16 dense determinants, 4 layers 512/64)
peer reviewed
 10.1103/PhysRevLett.130.036401 2023
-0.00849361 2.0e-5 7.0e-8 n/a n/a Slater-Jastrow-backflow (Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=2 keeps 24 terms )
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.00849274 2.1e-5 7.0e-8 n/a n/a Slater-Jastrow-backflow (Gaussian-orbital (crystal) determinant)
peer reviewed
 10.1103/PhysRevLett.130.036401 2023
-0.00849268 2.1e-5 7.0e-8 n/a n/a Slater-Jastrow-backflow (Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=3 keeps 27 terms )
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.00849263 2.1e-5 7.0e-8 n/a n/a Slater-Jastrow-backflow (Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=5 keeps 27 terms )
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.00849254 2.1e-5 7.0e-8 n/a n/a Slater-Jastrow-backflow (Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=15 keeps 46 terms)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.00849253 2.1e-5 7.0e-8 n/a n/a Slater-Jastrow-backflow (Wigner-crystal Slater determinant of Gaussian orbitals (C = 0.000148 a.u.), SJB optimised with g=7; truncation parameter g=7 keeps 27 terms )
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.0084277 8.6e-5 1.0e-7 n/a n/a Slater-Jastrow-backflow (plane-wave determinant)
peer reviewed
 10.1103/PhysRevLett.130.036401 2023
-0.0084224 9.1e-5 3.0e-7 n/a n/a Slater-Jastrow-backflow (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)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a

Projected or fixed-node estimates (2)

Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Node- or constraint-dependent, so not cleanly comparable to each other or to strict bounds. Like any unflagged energy, one holds the record where it is the lowest on an unsolved instance.

E/N (Ha)distance to recordσVar(E)V-score methodsourceyear
-0.0085137record 0 2.0e-7 n/a n/a Fixed-node DMC (trial state: Slater-Jastrow-backflow, Gaussian-orbital (crystal) determinant)
peer reviewed
 10.1103/PhysRevLett.130.036401 2023
-0.0084862 2.7e-5 1.0e-7 n/a n/a Fixed-node DMC (trial state: Slater-Jastrow-backflow, plane-wave determinant)
peer reviewed
 10.1103/PhysRevLett.130.036401 2023

How these numbers were read

Where each number below was read, and how it was checked.

-0.0085137(2) — Fixed-node DMC (trial state: Slater-Jastrow-backflow...

Checked 2026-10-16 · 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)

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.

-0.008496(1) — FermiNet (16 dense determinants, 4 layers 512/64)

Checked 2026-10-16 · 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)

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.

-0.00849361(7) — Slater-Jastrow-backflow (Wigner-crystal Slater determinant...

Checked 2026-10-19 · 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)

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

-0.00849274(7) — Slater-Jastrow-backflow (Gaussian-orbital (crystal)...

Checked 2026-10-16 · 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)

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.

-0.00849268(7) — Slater-Jastrow-backflow (Wigner-crystal Slater determinant...

Checked 2026-10-19 · 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)

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

-0.00849263(7) — Slater-Jastrow-backflow (Wigner-crystal Slater determinant...

Checked 2026-10-19 · 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)

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

-0.00849254(7) — Slater-Jastrow-backflow (Wigner-crystal Slater determinant...

Checked 2026-10-19 · 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)

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

-0.00849253(7) — Slater-Jastrow-backflow (Wigner-crystal Slater determinant...

Checked 2026-10-19 · 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)

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

-0.0084862(1) — Fixed-node DMC (trial state: Slater-Jastrow-backflow...

Checked 2026-10-16 · 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)

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.

-0.0084277(1) — Slater-Jastrow-backflow (plane-wave determinant)

Checked 2026-10-16 · 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)

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.

-0.0084224(3) — Slater-Jastrow-backflow (Wigner-crystal Slater determinant...

Checked 2026-10-19 · 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)

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

Coverage

Newest published number here: 2023.

This instance as data

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