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All instances / Hydrogen chain

Hydrogen chain H10 chain, open, R = 2, basis-free

Best values (rules §6)

best variational -0.563249(4) VMC (JAGP)
10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected -0.563852(7) LR-DMC (LDA determinant)
10.1103/PhysRevX.7.031059 · the lowest unflagged projected energy, a projected estimate, which depends on its node or constraint
best extrapolated -0.5643(1) SBDMRG
10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate -0.56391(1) 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: 10 protons equally spaced on a line, spacing R = 2 bohr, open (a finite molecule); N↑ = 5, N↓ = 5. Basis-free: the continuum Born–Oppenheimer Hamiltonian with point protons. 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
2
basis
none
energies
17

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
-0.563249best variational 0 4.0e-6 n/a n/a VMC (JAGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.562776 4.7e-4 6.0e-6 n/a n/a VMC (SJ)
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Projected or fixed-node estimates (2)

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
-0.563852best projected 0 7.0e-6 n/a n/a LR-DMC (LDA determinant)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.563831tied with best projected at 2σ 2.1e-5 5.0e-6 n/a n/a LR-DMC (AGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Extrapolations (9)

Zero-variance or bond-dimension extrapolations. These are not bounds: no ansatz reached the number.

E/N (Ha)distance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.5643best extrapolated 0 1.4e-4 n/a n/a SBDMRG
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56412tied with best extrapolated at 2σ 1.7e-4 3.0e-5 n/a n/a MRCI+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56410tied with best extrapolated at 2σ 1.9e-4 3.0e-5 n/a n/a MRCI-F12+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56396tied with best extrapolated at 2σ 3.3e-4 6.0e-5 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5632 1.1e-3 1.2e-4 n/a n/a DMET
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56311 1.2e-3 1.0e-5 n/a n/a RCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56239 1.9e-3 1.0e-5 n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.53822 2.6e-2 2.0e-5 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.53663 2.8e-2 1.3e-5 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Approximate methods (no bound) (4)

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
-0.56391best approximate 0 1.0e-5 n/a n/a RCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56341 5.0e-4 1.0e-5 n/a n/a UCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56114 2.8e-3 7.0e-5 n/a n/a PC-NEVPT2
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.56100 2.9e-3 8.0e-5 n/a n/a SC-NEVPT2
peer reviewed
 10.1103/PhysRevX.7.031059 2017

How these numbers were read

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

-0.5643(1), SBDMRG

Checked 2026-10-20 · 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. -0.56428(14) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/SBDMRG_CBS: -5.64288 0.00140648 | 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.

-0.56412(3), MRCI+Q

Checked 2026-10-20 · 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. -0.56411(3) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/MRCI+Q_CBS: -5.641160 0.000300 | 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.

-0.56410(3), MRCI-F12+Q

Checked 2026-10-20 · 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. -0.56410(3) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/MRCI+Q+F12_CBS: -5.640990 0.000329 | 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.

-0.56396(6), ph-AFQMC

Checked 2026-10-20 · 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. -0.56396(6) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/AFQMC_CBS: -5.63964 0.00059032 | 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.

-0.56391(1), RCCSD(T)

Checked 2026-10-20 · 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. -0.56391(1) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/RCCSD(T)_CBS: -5.63906 0.000135677 | 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.

-0.563852(7), LR-DMC (LDA determinant)

Checked 2026-10-20 · 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. -0.563852(7) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/LRDMC_LDA_basis-TZ: -5.63851669863051 7.794640032341178E-005 | 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.

-0.563831(5), LR-DMC (AGP)

Checked 2026-10-20 · 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. -0.563831(5) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/LRDMC_AGP_basis-TZ: -5.63831003131921 5.187775442283965E-005 | 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.

-0.56341(1), UCCSD(T)

Checked 2026-10-20 · 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. -0.56341(1) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/UCCSD(T)_CBS: -5.63408 0.000135332 | 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.

-0.563249(4), VMC (JAGP)

Checked 2026-10-20 · 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. -0.563249(4) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/VMC_AGP_basis-TZ: -5.63248627404995 4.050485374458314E-005 | 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.

-0.5632(1), DMET

Checked 2026-10-20 · 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. -0.56321(12) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/DMET_2_CBS: -5.63212 0.00123559 | 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.

-0.56311(1), RCCSD

Checked 2026-10-20 · 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. -0.56311(1) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/RCCSD_CBS: -5.63106 0.000150005 | 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.

-0.562776(6), VMC (SJ)

Checked 2026-10-20 · 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. -0.562776(6) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/VMC_LDA_basis-TZ: -5.62775908497710 5.631746367726193E-005 | 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.

-0.56239(1), UCCSD

Checked 2026-10-20 · 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. -0.56239(1) per atom

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/UCCSD_CBS: -5.62393 0.000149013 | 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.

-0.56114(7), PC-NEVPT2

Checked 2026-10-20 · 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. -0.56113(7) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/PC-NEVPT2_CBS: -5.61136 0.000802421 | 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.

-0.56100(8), SC-NEVPT2

Checked 2026-10-20 · 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. -0.56099(8) per atom

Table III (first block), R = 2 row; sources/1705.01608.txt line 1272; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/SC-NEVPT2_CBS: -5.60995 0.000809322 | 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.

-0.53822(2), HF

Checked 2026-10-20 · 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. -5.38220(2) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 2 row; sources/1705.01608.txt line 1285; version read: journal PDF (PRX 7, 031059, APS, sources/1705.01608-PhysRevX.7.031059.txt) and arXiv text; the table values agree digit for digit. Dataset file N_10_OBC/R_2.0/UHF_CBS: -5.3821994 0.0001349 | 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.

-0.53663(1), 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. -5.3663015 0.0001326 (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_2.0/RHF_CBS 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.

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