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All instances / Hydrogen chain

Hydrogen chain H30 chain, open, R = 1.6, cc-pVTZ

Best values (rules §6)

best variational -0.5334553 HF
10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected -0.55942(3) ph-AFQMC
10.1103/PhysRevX.7.031059 · the lowest unflagged projected energy, a projected estimate, which depends on its node or constraint
best approximate -0.559162 RCCSD(T)
10.1103/PhysRevX.7.031059 · the lowest unflagged approximate energy, an approximate method, whose error has no known sign

Energies as E/N (Ha). Classes are different claims and are not ranked against each other.

The Hamiltonian

H = −½ ∑i ∇i2 − ∑i,I 1/|ri − RI| + ∑i<j 1/|ri − rj| + ∑I<J 1/|RI − RJ|

Hydrogen chain: 30 protons equally spaced on a line, spacing R = 1.6 bohr, open (a finite molecule); N↑ = 15, N↓ = 15. Projected onto the cc-pVTZ 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
30
boundary
open
N_up
15
N_dn
15
R
1.6
basis
cc-pVTZ
energies
7

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 (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.

E/N (Ha)distance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.5334553best variational 0 n/a n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5326362 8.2e-4 n/a n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Projected or fixed-node estimates (4)

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.

E/N (Ha)distance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.55942best projected 0 3.3e-5 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5582387 1.2e-3 n/a n/a n/a RCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.557894 1.5e-3 n/a n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5571134 2.3e-3 n/a n/a n/a DMET
peer reviewed
 10.1103/PhysRevX.7.031059 2017

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).

E/N (Ha)distance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.559162best approximate 0 n/a n/a n/a RCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017

How these numbers were read

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

-0.55942(3), 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. -16.7827 0.0010 (total for N = 30; 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_30_OBC/R_1.6/AFQMC_basis-TZ 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.

-0.559162, RCCSD(T)

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. -16.77486 (total for N = 30)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_30_OBC/R_1.6/RCCSD(T)_basis-TZ 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.

-0.5582387, RCCSD

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. -16.74716 (total for N = 30)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_30_OBC/R_1.6/RCCSD_basis-TZ 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.

-0.557894, UCCSD

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. -16.73682 (total for N = 30)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_30_OBC/R_1.6/UCCSD_basis-TZ 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.

-0.5571134, DMET

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. -16.713401 (total for N = 30)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_30_OBC/R_1.6/DMET_2_basis-TZ 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.

-0.5334553, 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. -16.003660 (total for N = 30)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_30_OBC/R_1.6/UHF_basis-TZ 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.

-0.5326362, 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. -15.9790855 (total for N = 30)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_30_OBC/R_1.6/RHF_basis-TZ 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.

Coverage

Newest published number here: 2017.

This instance as data

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