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All instances / Electron gas, 3D

Electron gas, 3D bcc cell, N = 64, rs = 100, polarized

Record

-0.00769619(9) E/N (Ha)

Fixed-node DMC (Gaussian exponent C = 0.00011 [Eq. (7)]; VMC variance (Ha^2) in the same row: 0.00000252)

10.1103/PhysRevB.105.245135 · 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↑ = 64, N↓ = 0 in a periodic bcc cell at rs = 100 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
64
boundary
periodic
N_up
64
N_dn
0
rs
100
twist
Gamma
energies
8

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 (4)

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
-0.0076803 1.6e-5 2.0e-7 n/a n/a Slater-Jastrow-backflow (Gaussian exponent C = 0.0001503 (opt.); VMC variance (Ha^2) in the same row: 0.00000201)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.0076786 1.8e-5 2.0e-7 n/a n/a Slater-Jastrow-backflow (Gaussian exponent C = 0.00011 [Eq. (7)]; VMC variance (Ha^2) in the same row: 0.00000252)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.0076737 2.2e-5 2.0e-7 n/a n/a Slater-Jastrow (Gaussian exponent C = 0.0001553 (opt.); VMC variance (Ha^2) in the same row: 0.00000274)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.0076719 2.4e-5 3.0e-7 n/a n/a Slater-Jastrow (Gaussian exponent C = 0.00011 [Eq. (7)]; VMC variance (Ha^2) in the same row: 0.00000329)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a

Projected or fixed-node estimates (4)

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
-0.00769619record 0 9.0e-8 n/a n/a Fixed-node DMC (Gaussian exponent C = 0.00011 [Eq. (7)]; VMC variance (Ha^2) in the same row: 0.00000252)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.00769616 3.0e-8 8.0e-8 n/a n/a Fixed-node DMC (Gaussian exponent C = 0.00011 [Eq. (7)]; VMC variance (Ha^2) in the same row: 0.00000329)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.0076960 1.9e-7 1.0e-7 n/a n/a Fixed-node DMC (Gaussian exponent C = 0.0001503 (opt.); VMC variance (Ha^2) in the same row: 0.00000201)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a
-0.0076957 4.9e-7 2.0e-7 n/a n/a Fixed-node DMC (Gaussian exponent C = 0.0001553 (opt.); VMC variance (Ha^2) in the same row: 0.00000274)
peer reviewed
 10.1103/PhysRevB.105.245135 n/a

How these numbers were read

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

-0.00769619(9) — Fixed-node DMC (Gaussian exponent C = 0.00011 [Eq. (7)];...

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. −0.00769619(9)

Table II, SJB, C=0.00011 [Eq. (7)], column DMC energy; sources/2201.08743.txt line 309; 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

-0.00769616(8) — Fixed-node DMC (Gaussian exponent C = 0.00011 [Eq. (7)];...

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. −0.00769616(8)

Table II, SJ, C=0.00011 [Eq. (7)], column DMC energy; sources/2201.08743.txt line 307; 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

-0.0076960(1) — Fixed-node DMC (Gaussian exponent C = 0.0001503 (opt.); VMC...

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. −0.0076960(1)

Table II, SJB, C=0.0001503 (opt.), column DMC energy; sources/2201.08743.txt line 310; 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

-0.0076957(2) — Fixed-node DMC (Gaussian exponent C = 0.0001553 (opt.); VMC...

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. −0.0076957(2)

Table II, SJ, C=0.0001553 (opt.), column DMC energy; sources/2201.08743.txt line 308; 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

-0.0076803(2) — Slater-Jastrow-backflow (Gaussian exponent C = 0.0001503...

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. −0.0076803(2)

Table II, SJB, C=0.0001503 (opt.), column VMC energy; sources/2201.08743.txt line 310; 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

-0.0076786(2) — Slater-Jastrow-backflow (Gaussian exponent C = 0.00011 [Eq....

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. −0.0076786(2)

Table II, SJB, C=0.00011 [Eq. (7)], column VMC energy; sources/2201.08743.txt line 309; 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

-0.0076737(2) — Slater-Jastrow (Gaussian exponent C = 0.0001553 (opt.); VMC...

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. −0.0076737(2)

Table II, SJ, C=0.0001553 (opt.), column VMC energy; sources/2201.08743.txt line 308; 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

-0.0076719(3) — Slater-Jastrow (Gaussian exponent C = 0.00011 [Eq. (7)];...

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. −0.0076719(3)

Table II, SJ, C=0.00011 [Eq. (7)], column VMC energy; sources/2201.08743.txt line 307; 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

No row here carries a publication year.

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