The Hamiltonian
H = −½ ∑i ∇i2 + ∑i<j vE(ri − rj) + Nξ/2
Homogeneous electron gas in 2D: N↑ = 91, N↓ = 0 in a periodic triangular cell at rs = 30 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
- triangular
- electrons
- 91
- boundary
- periodic
- N_up
- 91
- N_dn
- 0
- rs
- 30
- twist
- Gamma
- energies
- 3
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.
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.
-0.03193763(3) — Self-attention NQS (spatially weighted particle attention)
Checked 2026-10-16 · source text of arXiv:2607.08616 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. −0.03193763(3)
Table I (main text), N_e=91 rs=30, row NN-VMC, column E; sources/2607.08616.txt line 66; printed in Ha per electron; version read: arXiv text with its Supplemental Material; no journal version (Crossref) | Ha per electron, triangular cell, Gamma point ('we did not apply twist averaging'). The paper's E minus its E_corr gives HF = -0.0273098 Ha per electron at N = 91, rs = 30, which QMBL's HF energy reproduces to all seven digits (Madelung term included).
-0.0319370(1) — Fixed-node DMC (trial state: Slater-Jastrow-backflow;...
Checked 2026-10-16 · source text of arXiv:2607.08616 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. −0.0319370(1)
Table I (main text), N_e=91 rs=30, row SJB-DMC, column E; sources/2607.08616.txt line 65; printed in Ha per electron; version read: arXiv text with its Supplemental Material; no journal version (Crossref) | Ha per electron, triangular cell, Gamma point ('we did not apply twist averaging'). The paper's E minus its E_corr gives HF = -0.0273098 Ha per electron at N = 91, rs = 30, which QMBL's HF energy reproduces to all seven digits (Madelung term included).
-0.0318792(2) — Slater-Jastrow-backflow
Checked 2026-10-16 · source text of arXiv:2607.08616 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. −0.0318792(2)
Table I (main text), N_e=91 rs=30, row SJB-VMC, column E; sources/2607.08616.txt line 64; printed in Ha per electron; version read: arXiv text with its Supplemental Material; no journal version (Crossref) | Ha per electron, triangular cell, Gamma point ('we did not apply twist averaging'). The paper's E minus its E_corr gives HF = -0.0273098 Ha per electron at N = 91, rs = 30, which QMBL's HF energy reproduces to all seven digits (Madelung term included).