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All instances / Electron gas, 3D

Electron gas, 3D sc cell, N = 14, rs = 3, unpolarized

Record

-0.07432(2) E/N (Ha)

CCSDTQ5 (coupled cluster Monte Carlo (HANDE), non-initiator; level CCSDTQ5; ion: CCSD fitted in 1/M with chosen polynomial order, higher levels by in; extrapolated: 1/M → 0 (complete basis))

10.1063/1.5003794 · lowest unflagged energy published, an extrapolation, not a bound (rules §6)

The Hamiltonian

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

Homogeneous electron gas in 3D: N↑ = 7, N↓ = 7 in a periodic sc cell at rs = 3 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
14
boundary
periodic
N_up
7
N_dn
7
rs
3
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.

Projected or fixed-node estimates (3)

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
-0.07352 8.0e-4 3.6e-5 n/a n/a ph-AFQMC (phaseless AFQMC with a single-determinant RHF trial, time step 0.005 a.u., 2880 walkers (txt line 325-327))
peer reviewed
 10.1063/1.5109572 n/a
-0.07351 8.1e-4 2.9e-5 n/a n/a ph-AFQMC (phaseless AFQMC with a single-determinant RHF trial, time step 0.005 a.u., 2880 walkers (txt line 325-327))
peer reviewed
 10.1063/1.5109572 n/a
-0.07256 1.8e-3 1.4e-5 n/a n/a ph-AFQMC (phaseless AFQMC with a single-determinant RHF trial, time step 0.005 a.u., 2880 walkers (txt line 325-327))
peer reviewed
 10.1063/1.5109572 n/a

Extrapolations (5)

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
-0.07432record 0 2.1e-5 n/a n/a CCSDTQ5 (coupled cluster Monte Carlo (HANDE), non-initiator; level CCSDTQ5; ion: CCSD fitted in 1/M with chosen polynomial order, higher levels by in; extrapolated: 1/M → 0 (complete basis))
peer reviewed
 10.1063/1.5003794 n/a
-0.07429 2.9e-5 2.1e-5 n/a n/a CCSDTQ (coupled cluster Monte Carlo (HANDE), non-initiator; level CCSDTQ; ion: CCSD fitted in 1/M with chosen polynomial order, higher levels by ing; extrapolated: 1/M → 0 (complete basis))
peer reviewed
 10.1063/1.5003794 n/a
-0.07391 4.1e-4 2.1e-5 n/a n/a CCSDT (coupled cluster Monte Carlo (HANDE), non-initiator; level CCSDT; ion: CCSD fitted in 1/M with chosen polynomial order, higher levels by ing ; extrapolated: 1/M → 0 (complete basis))
peer reviewed
 10.1063/1.5003794 n/a
-0.071164 3.2e-3 7.1e-6 n/a n/a CCSD (coupled cluster Monte Carlo (HANDE), non-initiator; level CCSD; ion: CCSD fitted in 1/M with chosen polynomial order, higher levels by ing l; extrapolated: 1/M → 0 (complete basis))
peer reviewed
 10.1063/1.5003794 n/a
-0.0711279 3.2e-3 n/a n/a n/a CCD (full CCD (driver, ladders, rings, crossed-rings, mosaics); equals CCSD in the HEG; extrapolated: 1/M → 0 (complete basis))
peer reviewed
 10.1063/1.4867783 n/a

How these numbers were read

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

-0.07432(2) — CCSDTQ5 (coupled cluster Monte Carlo (HANDE)...

Checked 2026-10-19 · source text of arXiv:1706.09923 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 = -0.3837(3)

Table I, row CCSDTQ5, rs=3.0 a0 (own value); sources/1706.09923.txt line 309; printed in Ha, total; printed correlation energy -0.3837(3), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text in sources/; journal copy not read | Eq. (7) (txt line 154-159): 'H = sum_k 1/2 (k^2 + V_M) c+c + ...', with 'V_M the Madelung constant that does not affect the correlation energy'; cubic box of length L; no Ewald-sum statement; no Hartree-Fock energy printed. Only correlation energies are printed, so the convention cannot be checked against a printed HF value.

-0.07429(2) — CCSDTQ (coupled cluster Monte Carlo (HANDE), non-initiator;...

Checked 2026-10-19 · source text of arXiv:1706.09923 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 = -0.3833(3)

Table I, row CCSDTQ, rs=3.0 a0 (own value); sources/1706.09923.txt line 308; printed in Ha, total; printed correlation energy -0.3833(3), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text in sources/; journal copy not read | Eq. (7) (txt line 154-159): 'H = sum_k 1/2 (k^2 + V_M) c+c + ...', with 'V_M the Madelung constant that does not affect the correlation energy'; cubic box of length L; no Ewald-sum statement; no Hartree-Fock energy printed. Only correlation energies are printed, so the convention cannot be checked against a printed HF value.

-0.07391(2) — CCSDT (coupled cluster Monte Carlo (HANDE), non-initiator;...

Checked 2026-10-19 · source text of arXiv:1706.09923 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 = -0.3780(3)

Table I, row CCSDT, rs=3.0 a0 (own value); sources/1706.09923.txt line 307; printed in Ha, total; printed correlation energy -0.3780(3), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text in sources/; journal copy not read | Eq. (7) (txt line 154-159): 'H = sum_k 1/2 (k^2 + V_M) c+c + ...', with 'V_M the Madelung constant that does not affect the correlation energy'; cubic box of length L; no Ewald-sum statement; no Hartree-Fock energy printed. Only correlation energies are printed, so the convention cannot be checked against a printed HF value.

-0.07352(4) — ph-AFQMC (phaseless AFQMC with a single-determinant RHF...

Checked 2026-10-19 · 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. E_c = -0.3725(5)

Table III, rs=3.0, M=925, column ph-AFQMC; sources/1905.04361.txt line 502; printed in Ha, total; printed correlation energy -0.3725(5), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); 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

-0.07351(3) — ph-AFQMC (phaseless AFQMC with a single-determinant RHF...

Checked 2026-10-19 · 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. E_c = -0.3723(4)

Table III, rs=3.0, M=179, column ph-AFQMC; sources/1905.04361.txt line 501; printed in Ha, total; printed correlation energy -0.3723(4), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); 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

-0.07256(1) — ph-AFQMC (phaseless AFQMC with a single-determinant RHF...

Checked 2026-10-19 · 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. E_c = -0.3590(2)

Table III, rs=3.0, M=81, column ph-AFQMC; sources/1905.04361.txt line 500; printed in Ha, total; printed correlation energy -0.3590(2), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); 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

-0.071164(7) — CCSD (coupled cluster Monte Carlo (HANDE), non-initiator;...

Checked 2026-10-19 · source text of arXiv:1706.09923 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 = -0.3395(1)

Table I, row CCSD, rs=3.0 a0 (own value); sources/1706.09923.txt line 306; printed in Ha, total; printed correlation energy -0.3395(1), E = E_HF + E_c with E_HF = -0.0469136387818 Ha per electron (QMBL, checks/heg-hf/hf.mjs); version read: arXiv text in sources/; journal copy not read | Eq. (7) (txt line 154-159): 'H = sum_k 1/2 (k^2 + V_M) c+c + ...', with 'V_M the Madelung constant that does not affect the correlation energy'; cubic box of length L; no Ewald-sum statement; no Hartree-Fock energy printed. Only correlation energies are printed, so the convention cannot be checked against a printed HF value.

-0.0711279 — CCD (full CCD (driver, ladders, rings, crossed-rings...

Checked 2026-10-19 · 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 = -0.339

Table III, rs=3.0, column CCD; sources/1310.6806.txt line 495; printed in Ha, total; printed correlation energy -0.339, E = E_HF + E_c with E_HF = -0.0469136387818 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.

Coverage

No row here carries a publication year.

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