The Hamiltonian
H = −½ ∑i ∇i2 − ∑i,I 1/|ri − RI| + ∑i<j 1/|ri − rj| + ∑I<J 1/|RI − RJ|
Hydrogen sheet: 4 protons on a 2 × 2 square grid, spacing R = 2.4 bohr, open (a finite molecule); N↑ = 2, N↓ = 2. Basis-free: the continuum Born–Oppenheimer Hamiltonian with point protons. Hartree atomic units, nuclear repulsion included; the energy per atom is E/N.
- geometry
- square
- atoms
- 4
- boundary
- open
- N_up
- 2
- N_dn
- 2
- R
- 2.4
- basis
- none
- energies
- 4
Stored as a total energy in Hartree, nuclear repulsion included (rules §13); quoted here as
E/N (Ha), the energy per atom
(how the conversion works).
Every published energy
Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).
Strict variational upper bounds (3)
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.
-0.52812(5), LR-DMC (JAGP)
Checked 2026-10-20 · source text of arXiv:1905.06971 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. −2.1125± 0.0002 Ha per total
Table II; sources/1905.06971.txt line 243; version read: arXiv text; journal copy not reached (publisher returned HTTP 403 or a login page, 2026-10-21); read again when reachable (RULES.md §8) | Table I caption (line 210): 'All the energies are expressed in Hartree'; Fig. 1 caption (line 134-135): ry fixed at 2.4 a.u., so rx = 2.40 is the square (2x2) geometry; energies ~ -2.1 are totals for H4 | Table II caption says 'Difference between the energies calculated with the DMC...' but entries are absolute energies in Ha. Assumes ry = 2.4 a.u. as in Fig. 1; rx = 2.400 is the square row. Nodes from JAGP(cc-pVDZ) (Fig. 4 caption), continuum DMC. | reader confidence medium
-0.52710(7), VMC (JAGP)
Checked 2026-10-20 · source text of arXiv:1905.06971 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. −2.1084± 0.0003 Ha per total
Table I; sources/1905.06971.txt line 214; version read: arXiv text; journal copy not reached (publisher returned HTTP 403 or a login page, 2026-10-21); read again when reachable (RULES.md §8) | Table I caption (line 210): 'All the energies are expressed in Hartree'; Fig. 1 caption (line 134-135): ry fixed at 2.4 a.u., so rx = 2.40 is the square (2x2) geometry; energies ~ -2.1 are totals for H4 | Only the rx = ry = 2.4 a.u. row is a square; rx=1.8 and 3.0 rows are rectangles (not rows). Row/column alignment checked against the header; a.u. = bohr. Basis is that of the AGP/SD determinant; Jastrow-correlated VMC is a continuum Hamiltonian, so the mapping to basis instance (cc-pVDZ vs none) is a ruling. | reader confidence medium
-0.5269(1), VMC (JAGP)
Checked 2026-10-20 · source text of arXiv:1905.06971 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. −2.1075± 0.0004 Ha per total
Table I; sources/1905.06971.txt line 214; version read: arXiv text; journal copy not reached (publisher returned HTTP 403 or a login page, 2026-10-21); read again when reachable (RULES.md §8) | Table I caption (line 210): 'All the energies are expressed in Hartree'; Fig. 1 caption (line 134-135): ry fixed at 2.4 a.u., so rx = 2.40 is the square (2x2) geometry; energies ~ -2.1 are totals for H4 | Only the rx = ry = 2.4 a.u. row is a square; rx=1.8 and 3.0 rows are rectangles (not rows). Row/column alignment checked against the header; a.u. = bohr. Basis is that of the AGP/SD determinant; Jastrow-correlated VMC is a continuum Hamiltonian, so the mapping to basis instance (cc-pVDZ vs none) is a ruling. | reader confidence medium
-0.5173(1), VMC (SJ)
Checked 2026-10-20 · source text of arXiv:1905.06971 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. −2.0694± 0.0004 Ha per total
Table I; sources/1905.06971.txt line 214; version read: arXiv text; journal copy not reached (publisher returned HTTP 403 or a login page, 2026-10-21); read again when reachable (RULES.md §8) | Table I caption (line 210): 'All the energies are expressed in Hartree'; Fig. 1 caption (line 134-135): ry fixed at 2.4 a.u., so rx = 2.40 is the square (2x2) geometry; energies ~ -2.1 are totals for H4 | Only the rx = ry = 2.4 a.u. row is a square; rx=1.8 and 3.0 rows are rectangles (not rows). Row/column alignment checked against the header; a.u. = bohr. Basis is that of the AGP/SD determinant; Jastrow-correlated VMC is a continuum Hamiltonian, so the mapping to basis instance (cc-pVDZ vs none) is a ruling. Printed as '-2.0694± 0.0004' (glued minus). | reader confidence medium