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All instances / Electron gas, 3D

Electron gas, 3D sc cell, N = 54, rs = 50, unpolarized

Record

-0.0145015(1) E/N (Ha)

MP-NQS (plane-wave orbitals)

10.1103/PhysRevB.110.035108 · lowest unflagged energy published, a strict variational bound (rules §6)

The Hamiltonian

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

Homogeneous electron gas in 3D: N↑ = 27, N↓ = 27 in a periodic sc cell at rs = 50 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
sc
electrons
54
boundary
periodic
N_up
27
N_dn
27
rs
50
twist
Gamma
energies
7

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 (2)

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
-0.0145015record 0 1.0e-7 n/a n/a MP-NQS (plane-wave orbitals)
peer reviewed
10.1103/PhysRevB.110.035108 2024
-0.0144999 1.6e-6 1.0e-7 n/a n/a MP-NQS (Gaussian orbitals on bcc sites)
peer reviewed
10.1103/PhysRevB.110.035108 2024

Projected or fixed-node estimates (1)

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
-0.01387 6.3e-4 2.0e-5 n/a n/a Fixed-node DMC (trial state: Slater-Jastrow)
peer reviewed
10.1103/PhysRevB.110.035108 2024

Extrapolations (4)

Zero-variance, bond-dimension or Trotter-error extrapolations. Not a bound: no ansatz reached the number itself. 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
-0.01281 1.7e-3 n/a n/a n/a TC-DCD (correlator cutoff k_c = 4 × 2π/L; extrapolated: 1/M → 0 (complete basis))
peer reviewed
10.1103/PhysRevResearch.3.033072 2021
-0.01267 1.8e-3 n/a n/a n/a DCD (extrapolated: 1/M → 0 (complete basis))
peer reviewed
10.1103/PhysRevResearch.3.033072 2021
-0.01263 1.9e-3 n/a n/a n/a TC-CCD (correlator cutoff k_c = 4 × 2π/L; extrapolated: 1/M → 0 (complete basis))
peer reviewed
10.1103/PhysRevResearch.3.033072 2021
-0.01223 2.3e-3 n/a n/a n/a CCD (extrapolated: 1/M → 0 (complete basis))
peer reviewed
10.1103/PhysRevResearch.3.033072 2021

How these numbers were read

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

-0.0145015(1) — MP-NQS (plane-wave orbitals)

Checked 2026-10-16 · 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. −0.0145015(1)

Table II (Appendix G), N=54 unpolarized, rs=50, column MP-NQS; sources/2305.07240.txt line 683; 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.

-0.0144999(1) — MP-NQS (Gaussian orbitals on bcc sites)

Checked 2026-10-16 · 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. −0.0144999(1)

Table II (Appendix G), N=54 unpolarized, rs=50, column MP-NQS (Gauss); sources/2305.07240.txt line 683; 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.

-0.01387(2) — Fixed-node DMC (trial state: Slater-Jastrow)

Checked 2026-10-16 · 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. −0.01387(2)

Table II (Appendix G), N=54 unpolarized, rs=50, column FN-DMC; sources/2305.07240.txt line 683; 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.

-0.01281 — TC-DCD (correlator cutoff k_c = 4 × 2π/L; extrapolated: 1/M...

Checked 2026-10-17 · 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. -0.01281

Table II, row rs=50.0, column TC-DCD; sources/2103.03176.txt line 395; 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.

-0.01267 — DCD (extrapolated: 1/M → 0 (complete basis))

Checked 2026-10-17 · 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. -0.01267

Table II, row rs=50.0, column DCD; sources/2103.03176.txt line 395; 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.

-0.01263 — TC-CCD (correlator cutoff k_c = 4 × 2π/L; extrapolated: 1/M...

Checked 2026-10-17 · 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. -0.01263

Table II, row rs=50.0, column TC-CCD; sources/2103.03176.txt line 395; 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.

-0.01223 — CCD (extrapolated: 1/M → 0 (complete basis))

Checked 2026-10-17 · 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. -0.01223

Table II, row rs=50.0, column CCD; sources/2103.03176.txt line 395; 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.

Coverage

Newest published number here: 2024.

This instance as data

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