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All instances / Hydrogen chain

Hydrogen chain H10 chain, open, R = 3.6, basis-free

Best values (rules §6)

best variational -0.514968(3) VMC (JAGP)
10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected -0.51558(1) 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 -0.51612(9) DMET
10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate -0.5143(1) UCCSD(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 = 3.6 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
3.6
basis
none
energies
15

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
-0.514968best variational 0 3.0e-6 n/a n/a VMC (JAGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.513869 1.1e-3 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
-0.51558best projected 0 1.1e-5 n/a n/a LR-DMC (LDA determinant)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.515571tied with best projected at 2σ 1.1e-5 7.0e-6 n/a n/a LR-DMC (AGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Extrapolations (8)

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
-0.51612best extrapolated 0 9.0e-5 n/a n/a DMET
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.51569 4.3e-4 9.0e-5 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5157tied with best extrapolated at 2σ 4.4e-4 1.4e-4 n/a n/a SBDMRG
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.51554 5.8e-4 5.0e-5 n/a n/a MRCI-F12+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.51549 6.3e-4 8.0e-5 n/a n/a MRCI+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5141 2.1e-3 1.0e-4 n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.505039 1.1e-2 7.0e-6 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.47109 4.5e-2 1.3e-5 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Approximate methods (no bound) (3)

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
-0.5143best approximate 0 1.0e-4 n/a n/a UCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.51406tied with best approximate at 2σ 2.3e-4 8.0e-5 n/a n/a PC-NEVPT2
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.51404tied with best approximate at 2σ 2.4e-4 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.

-0.51612(9), 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. -0.51612(9) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/DMET_2_CBS: -5.16118 0.000955347 | 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.

-0.51569(9), 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. -0.51568(9) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/AFQMC_CBS: -5.15686 0.000895438 | 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.

-0.5157(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. -0.51568(14) per atom

Table III (second block), R = 3.6 row; sources/1705.01608.txt line 1289; 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_3.6/SBDMRG_CBS: -5.15681 0.00140194 | 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.

-0.51558(1), 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. -0.515582(11) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/LRDMC_LDA_basis-TZ: -5.15581697074419 1.146206152758427E-004 | 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.

-0.515571(7), 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. -0.515571(7) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/LRDMC_AGP_basis-TZ: -5.15570628627727 7.421204351732657E-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.

-0.51554(5), 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. -0.51553(5) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/MRCI+Q+F12_CBS: -5.155350 0.000524 | 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.

-0.51549(8), 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. -0.51549(8) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/MRCI+Q_CBS: -5.154930 0.000838 | 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.

-0.514968(3), 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. -0.514968(3) per atom

Table III (second block), R = 3.6 row; sources/1705.01608.txt line 1289; 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_3.6/VMC_AGP_basis-TZ: -5.14968170284009 2.876923009610780E-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.

-0.5143(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. -0.51428(10) per atom

Table III (second block), R = 3.6 row; sources/1705.01608.txt line 1289; 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_3.6/UCCSD(T)_CBS: -5.14283 0.00101089 | 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.

-0.5141(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. -0.51407(10) per atom

Table III (second block), R = 3.6 row; sources/1705.01608.txt line 1289; 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_3.6/UCCSD_CBS: -5.14071 0.000977084 | 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.

-0.51406(8), 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. -0.51405(8) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/PC-NEVPT2_CBS: -5.14055 0.00080757 | 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.

-0.51404(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. -0.51403(8) per atom

Table III (first block), R = 3.6 row; sources/1705.01608.txt line 1276; 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_3.6/SC-NEVPT2_CBS: -5.14037 0.000803288 | 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.

-0.513869(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. -0.513869(6) per atom

Table III (second block), R = 3.6 row; sources/1705.01608.txt line 1289; 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_3.6/VMC_LDA_basis-TZ: -5.13868932027577 5.568477608828433E-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.

-0.505039(7), 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. -5.050387(7) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 3.6 row; sources/1705.01608.txt line 1289; 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_3.6/UHF_CBS: -5.0503872 0.0000737 | 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.

-0.47109(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. -4.7109433 0.0001271 (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_3.6/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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