Electron gas, 2D square cell, N = 21, rs = 20, polarized

Record

-0.04634(2) E/N (Ha)

Slater-Jastrow (RPA (Gaskell) pseudopotential)

10.1103/PhysRevB.39.5005 · lowest unflagged energy published, a strict variational bound (rules §6)

The Hamiltonian

H = −½ ∑_i ∇_i² + ∑_i<j v_E(r_i − r_j) + Nξ/2

Homogeneous electron gas in 2D: N_↑ = 21, N_↓ = 0 in a periodic square cell at r_s = 20 bohr, Γ point. v_E 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: 21
boundary: periodic
N_up: 21
N_dn: 0
rs: 20
twist: Gamma
energies: 1

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.

E/N (Ha)	distance to record	σ	Var(E)	V-score	method	source	year
-0.04634record	0	1.5e-5	n/a	n/a	Slater-Jastrow (RPA (Gaskell) pseudopotential) peer reviewed	10.1103/PhysRevB.39.5005	1989

How these numbers were read

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

-0.04634(2), 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. −0.09267(3)

Table II (fully polarized fluid), row N = 21, column rs = 20; dl/tanatar-ceperley-1989.txt line 215; 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).

Coverage

Newest published number here: 1989.

This instance as data

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