The Hamiltonian
H = −½ ∑i ∇i2 − ∑i,I 1/|ri − RI| + ∑i<j 1/|ri − rj| + ∑I<J 1/|RI − RJ|
Hydrogen chain: 10 protons equally spaced on a line, spacing R = 1 bohr, open (a finite molecule); N↑ = 5, N↓ = 5. Projected onto the cc-pV5Z 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
- 10
- boundary
- open
- N_up
- 5
- N_dn
- 5
- R
- 1
- basis
- cc-pV5Z
- energies
- 5
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.
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.
-0.44236(8), ph-AFQMC
Checked 2026-10-20 · dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0; source text of arXiv:1705.01608 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. -4.4236 0.0008 (total for N = 10; second number the stated uncertainty)
dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_10_OBC/R_1.0/AFQMC_basis-5Z in sources/1705.01608-data-b99a183.tsv; number first printed in the dataset (RULES.md §8), the paper is the method reference | Hartree, lengths in bohr (Appendix B). Total energies with nuclear repulsion: FCI/STO-6G H10 at R = 1.8 is -5.42438538 in the dataset and -0.542439 per atom in Table II, and the paper's FCI E_inf = -0.471039 is the STO-6G hydrogen-atom energy per atom.
-0.43927, MP2
Checked 2026-10-20 · source text of arXiv:1911.01618 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. HF -0.41550 + MP2 correlation -0.02377 per atom
SI Table III, row "5Z" (NAO 550), HF and MP2 columns; sources/1911.01618-anc-si.txt line 1272; version read: APS Supplemental Material (si.pdf, sources/1911.01618-PhysRevX.10.031058-si.txt) and the arXiv ancillary SI; identical | Hartree per atom, total energy with nuclear repulsion: the cc-pVDZ HF value -0.40101 matches RHF/cc-pVDZ of the 2017 data set (-4.010066006349 for H10).
-0.4155, HF
Checked 2026-10-20 · source text of arXiv:1911.01618 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. -0.41550 per atom (HF column)
SI Table III, row "5Z" (NAO 550), HF column; sources/1911.01618-anc-si.txt line 1272; version read: APS Supplemental Material (si.pdf, sources/1911.01618-PhysRevX.10.031058-si.txt) and the arXiv ancillary SI; identical | Hartree per atom, total energy with nuclear repulsion: the cc-pVDZ HF value -0.40101 matches RHF/cc-pVDZ of the 2017 data set (-4.010066006349 for H10).
-0.4154885, HF
Checked 2026-10-20 · dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0; source text of arXiv:1705.01608 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. -4.154885002392 (total for N = 10)
dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_10_OBC/R_1.0/RHF_basis-5Z in sources/1705.01608-data-b99a183.tsv; number first printed in the dataset (RULES.md §8), the paper is the method reference | Hartree, lengths in bohr (Appendix B). Total energies with nuclear repulsion: FCI/STO-6G H10 at R = 1.8 is -5.42438538 in the dataset and -0.542439 per atom in Table II, and the paper's FCI E_inf = -0.471039 is the STO-6G hydrogen-atom energy per atom.
-0.4154884, HF
Checked 2026-10-20 · dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0; source text of arXiv:1705.01608 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. -4.154883934306 (total for N = 10)
dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_10_OBC/R_1.0/UHF_basis-5Z in sources/1705.01608-data-b99a183.tsv; number first printed in the dataset (RULES.md §8), the paper is the method reference | Hartree, lengths in bohr (Appendix B). Total energies with nuclear repulsion: FCI/STO-6G H10 at R = 1.8 is -5.42438538 in the dataset and -0.542439 per atom in Table II, and the paper's FCI E_inf = -0.471039 is the STO-6G hydrogen-atom energy per atom.