The Hamiltonian
H = −½ ∑i ∇i2 − ∑i,I 1/|ri − RI| + ∑i<j 1/|ri − rj| + ∑I<J 1/|RI − RJ|
Hydrogen chain: 4 protons equally spaced on a line, spacing R = 1.8 bohr, open (a finite molecule); N↑ = 2, N↓ = 2. Projected onto the STO-6G basis set; full CI in it is this instance's exact energy. Hartree atomic units, nuclear repulsion included; the energy per atom is E/N.
- geometry
- chain
- atoms
- 4
- boundary
- open
- N_up
- 2
- N_dn
- 2
- R
- 1.8
- basis
- STO-6G
- 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).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (2)
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.
Approximate methods (no bound) (1)
Methods that converge to a fixed approximation and not to the exact energy: RPA, finite-order perturbation theory, coupled cluster restricted to some of its terms. Their error has no known sign, so they are compared only with each other (rules §4).
How these numbers were read
Where each number below was read, and how it was checked.
-0.5476025, CCSD(T)
Checked 2026-10-20 · source text of arXiv:2305.12776 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. −2.19041 Ha per total
Table S2; sources/2305.12776.txt line 1150; 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) | line 1143: 'ground-state energies'; no unit printed; H10 FCI 1.8 bohr -5.42439 = -0.5424/atom, matches Motta 2017 total Ha incl. nuclear repulsion | Unit Ha and nuclear repulsion inferred from agreement with the known H10 STO-6G value, not stated. CMO/LMO = NQS in canonical/localized MO basis. | reader confidence medium
-0.547595, VMC (RBM)
Checked 2026-10-20 · source text of arXiv:2305.12776 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. −2.19038 Ha per total
Table S2; sources/2305.12776.txt line 1150; 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) | line 1143: 'ground-state energies'; no unit printed; H10 FCI 1.8 bohr -5.42439 = -0.5424/atom, matches Motta 2017 total Ha incl. nuclear repulsion | Unit Ha and nuclear repulsion inferred from agreement with the known H10 STO-6G value, not stated. CMO/LMO = NQS in canonical/localized MO basis. | reader confidence medium
-0.547595, VMC (RBM)
Checked 2026-10-20 · source text of arXiv:2305.12776 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. −2.19038 Ha per total
Table S2; sources/2305.12776.txt line 1150; 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) | line 1143: 'ground-state energies'; no unit printed; H10 FCI 1.8 bohr -5.42439 = -0.5424/atom, matches Motta 2017 total Ha incl. nuclear repulsion | Unit Ha and nuclear repulsion inferred from agreement with the known H10 STO-6G value, not stated. CMO/LMO = NQS in canonical/localized MO basis. | Variational (ruled 2026-10-24). The paper does not state whether the printed LMO energy is the optimisation-time estimator of eq. 12 or the variational energy of the truncated state of eq. 13 at the selection cutoff 1e-3 used for these runs; it calls the two equal only for cutoffs up to 1e-5. | reader confidence medium
-0.547595, FCI
Checked 2026-10-20 · source text of arXiv:2305.12776 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. −2.19038 Ha per total
Table S2; sources/2305.12776.txt line 1150; 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) | line 1143: 'ground-state energies'; no unit printed; H10 FCI 1.8 bohr -5.42439 = -0.5424/atom, matches Motta 2017 total Ha incl. nuclear repulsion | Unit Ha and nuclear repulsion inferred from agreement with the known H10 STO-6G value, not stated. CMO/LMO = NQS in canonical/localized MO basis. | reader confidence medium