The Hamiltonian
H = −½ ∑i ∇i2 + ∑i<j vE(ri − rj) + Nξ/2
Homogeneous electron gas in 2D: N↑ = 57, N↓ = 0 in a periodic square cell at rs = 40 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
- square
- electrons
- 57
- boundary
- periodic
- N_up
- 57
- N_dn
- 0
- rs
- 40
- twist
- Gamma
- energies
- 2
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. On a solved instance the exact energy is the record; otherwise the lowest unflagged energy of any group holds it (rules §6).
Strict variational upper bounds (1)
Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.
Projected or fixed-node estimates (1)
Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Each depends on its node or constraint, so two of them are not cleanly comparable to each other or to strict bounds.
How these numbers were read
Where each number below was read, and how it was checked.
-0.024420(1), Fixed-node GFMC (trial state: Slater-Jastrow)
Checked 2026-10-22 · doi:10.1103/PhysRevB.39.5005: text layer of the publisher PDF (1989 scan), every number checked against the page image at 300 dpi; no LLM transcription of numbers; single reading + page-image check
Reported as. −0.048841(2)
Table II (fully polarized fluid), row E_57^GFMC, column rs = 40; dl/tanatar-ceperley-1989.txt line 225; printed in Ry per electron; matches the page image | Energies in Ry per electron ('Energies in this paper are in units of rydbergs/electron'), converted × 1/2. Ewald sums with each electron's own images ('an electron must also interact with all of its own images which form a square lattice'): the Madelung term is included. Square simulation cell, periodic boundary conditions, plane-wave orbitals filling closed shells (Gamma). QMBL's HF check (checks/heg-hf/hf.mjs, square cell): E - E_HF per electron is -0.099 Ha at N = 26, rs = 1, -0.046 at N = 26, rs = 5, -0.0078 at N = 114, rs = 50, -0.0112 at N = 37 polarized, rs = 5 and -0.0023 at N = 113 polarized, rs = 75, close to the paper's E_c(rs) (Tables I, II); without the Madelung term the N = 26, rs = 1 energy would sit 0.216 Ha per electron higher. Every number checked against the page image (300 dpi).
-0.024360(5), Slater-Jastrow (RPA (Gaskell) pseudopotential)
Checked 2026-10-22 · doi:10.1103/PhysRevB.39.5005: text layer of the publisher PDF (1989 scan), every number checked against the page image at 300 dpi; no LLM transcription of numbers; single reading + page-image check
Reported as. −0.04872(1)
Table II (fully polarized fluid), row N = 57, column rs = 40; dl/tanatar-ceperley-1989.txt line 217; printed in Ry per electron; matches the page image | Energies in Ry per electron ('Energies in this paper are in units of rydbergs/electron'), converted × 1/2. Ewald sums with each electron's own images ('an electron must also interact with all of its own images which form a square lattice'): the Madelung term is included. Square simulation cell, periodic boundary conditions, plane-wave orbitals filling closed shells (Gamma). QMBL's HF check (checks/heg-hf/hf.mjs, square cell): E - E_HF per electron is -0.099 Ha at N = 26, rs = 1, -0.046 at N = 26, rs = 5, -0.0078 at N = 114, rs = 50, -0.0112 at N = 37 polarized, rs = 5 and -0.0023 at N = 113 polarized, rs = 75, close to the paper's E_c(rs) (Tables I, II); without the Madelung term the N = 26, rs = 1 energy would sit 0.216 Ha per electron higher. Every number checked against the page image (300 dpi).