The Hamiltonian
H = −½ ∑i ∇i2 + ∑i<j vE(ri − rj) + Nξ/2
Homogeneous electron gas in 3D: N↑ = 19, N↓ = 0 in a periodic sc cell at rs = 5 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
- 19
- boundary
- periodic
- N_up
- 19
- N_dn
- 0
- rs
- 5
- twist
- Gamma
- energies
- 5
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.
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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.0672540(5) — Fixed-node DMC (trial state: iterative backflow)
Checked 2026-10-16 · source text of arXiv:2202.04622 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. -0.134508(1)
Table III column EIBDMC, block Ne = 19 (spin-polarised), row rs = 5.0; sources/2202.04622.txt line 531; printed in Ry per electron; version read: arXiv:2202.04622v3 text (title 'Wave function Ansatz (but Periodic) Networks and the Homogeneous Electron Gas'); APS copy not read | Table III in Ry per electron (converted × 1/2); Ewald sum with the Madelung term N v_M added (text); simple cubic cell. QMBL's HF check: at rs = 1, N = 14 the energies sit 0.0376 Ha per electron below QMBL's HF energy 0.6065343, a correlation energy; without the Madelung term they would sit 0.365 Ha higher.
-0.0672510(2) — WAP-net (n_d = 8)
Checked 2026-10-16 · source text of arXiv:2202.04622 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. -0.1345021(5)
Table III column 3 (WAP-Net nd=8), block Ne = 19 (spin-polarised), row rs = 5.0; sources/2202.04622.txt line 531; printed in Ry per electron; version read: arXiv:2202.04622v3 text (title 'Wave function Ansatz (but Periodic) Networks and the Homogeneous Electron Gas'); APS copy not read | Table III in Ry per electron (converted × 1/2); Ewald sum with the Madelung term N v_M added (text); simple cubic cell. QMBL's HF check: at rs = 1, N = 14 the energies sit 0.0376 Ha per electron below QMBL's HF energy 0.6065343, a correlation energy; without the Madelung term they would sit 0.365 Ha higher.
-0.0672500(3) — WAP-net (n_d = 1)
Checked 2026-10-16 · source text of arXiv:2202.04622 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. -0.1345001(7)
Table III column 2 (WAP-Net nd=1), block Ne = 19 (spin-polarised), row rs = 5.0; sources/2202.04622.txt line 531; printed in Ry per electron; version read: arXiv:2202.04622v3 text (title 'Wave function Ansatz (but Periodic) Networks and the Homogeneous Electron Gas'); APS copy not read | Table III in Ry per electron (converted × 1/2); Ewald sum with the Madelung term N v_M added (text); simple cubic cell. QMBL's HF check: at rs = 1, N = 14 the energies sit 0.0376 Ha per electron below QMBL's HF energy 0.6065343, a correlation energy; without the Madelung term they would sit 0.365 Ha higher.
-0.0672489(3) — 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. −0.0672489(3)
Table IV (Appendix G), N=19 polarized, rs=5, column MP-NQS; sources/2305.07240.txt line 708; 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.
-0.0672400(5) — Iterative backflow (single determinant)
Checked 2026-10-16 · source text of arXiv:2202.04622 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. -0.134480(1)
Table III column EIBVMC, block Ne = 19 (spin-polarised), row rs = 5.0; sources/2202.04622.txt line 531; printed in Ry per electron; version read: arXiv:2202.04622v3 text (title 'Wave function Ansatz (but Periodic) Networks and the Homogeneous Electron Gas'); APS copy not read | Table III in Ry per electron (converted × 1/2); Ewald sum with the Madelung term N v_M added (text); simple cubic cell. QMBL's HF check: at rs = 1, N = 14 the energies sit 0.0376 Ha per electron below QMBL's HF energy 0.6065343, a correlation energy; without the Madelung term they would sit 0.365 Ha higher.