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All instances / Hydrogen chain

Hydrogen chain H10 chain, open, R = 1, basis-free

Best values (rules §6)

best variational -0.441499(9) VMC (JAGP)
10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected -0.442430(7) LR-DMC (AGP)
10.1103/PhysRevX.7.031059 · the lowest unflagged projected energy, a projected estimate, which depends on its node or constraint
best extrapolated -0.4432(3) MRCI+Q
10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate -0.4427(2) 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 = 1 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
1
basis
none
energies
14

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
-0.441499best variational 0 9.0e-6 n/a n/a VMC (JAGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.44061 8.9e-4 1.0e-5 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
-0.442430best projected 0 7.0e-6 n/a n/a LR-DMC (AGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.442093 3.4e-4 9.0e-6 n/a n/a LR-DMC (LDA determinant)
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
-0.4432best extrapolated 0 3.2e-4 n/a n/a MRCI+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4430tied with best extrapolated at 2σ 2.3e-4 3.1e-4 n/a n/a MRCI-F12+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4428tied with best extrapolated at 2σ 4.0e-4 2.4e-4 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4421 1.1e-3 2.1e-4 n/a n/a SBDMRG
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4421 1.2e-3 2.6e-4 n/a n/a RCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4421 1.2e-3 2.6e-4 n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4154 2.8e-2 1.8e-4 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4154 2.8e-2 1.6e-4 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Approximate methods (no bound) (2)

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
-0.4427best approximate 0 2.1e-4 n/a n/a RCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.4427tied with best approximate at 2σ <1e-4 2.1e-4 n/a n/a UCCSD(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.

-0.4432(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. -0.44324(32) per atom

Table III (first block), R = 1 row; sources/1705.01608.txt line 1267; 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_1.0/MRCI+Q_CBS: -4.432420 0.003256 | 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.

-0.4430(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. -0.44301(31) per atom

Table III (first block), R = 1 row; sources/1705.01608.txt line 1267; 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_1.0/MRCI+Q+F12_CBS: -4.430120 0.003094 | 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.

-0.4428(2), 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. -0.44284(24) per atom

Table III (first block), R = 1 row; sources/1705.01608.txt line 1267; 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_1.0/AFQMC_CBS: -4.42843 0.00250894 | 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.

-0.4427(2), 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. -0.44268(21) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/RCCSD(T)_CBS: -4.42676 0.00277214 | 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.

-0.4427(2), 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. -0.44268(21) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/UCCSD(T)_CBS: -4.42676 0.00277214 | 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.

-0.442430(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. -0.442430(7) per atom

Table III (first block), R = 1 row; sources/1705.01608.txt line 1267; 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_1.0/LRDMC_AGP_basis-TZ: -4.42430119748770 6.907982901638674E-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.

-0.4421(2), 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. -0.44209(21) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/SBDMRG_CBS: -4.42098 0.00228272 | 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.

-0.442093(9), 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. -0.442093(9) per atom

Table III (first block), R = 1 row; sources/1705.01608.txt line 1267; 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_1.0/LRDMC_LDA_basis-TZ: -4.42093061454202 9.113339676805278E-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.

-0.4421(3), 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. -0.44207(26) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/RCCSD_CBS: -4.42071 0.00275895 | 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.

-0.4421(3), 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. -0.44207(26) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/UCCSD_CBS: -4.42071 0.00275895 | 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.

-0.441499(9), 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. -0.441499(9) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/VMC_AGP_basis-TZ: -4.41499134293753 8.925990089022460E-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.

-0.44061(1), 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. -0.44061(1) per atom

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/VMC_LDA_basis-TZ: -4.40614443735753 1.047289398018219E-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.

-0.4154(2), 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. -4.1536332 0.0018035 (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/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. | The complete-basis RHF value lies 0.0012518 Ha (total) above the paper's own RHF/cc-pV5Z energy -4.154885002392 on the same geometry, although a finite-basis Hartree-Fock energy bounds the complete-basis limit from above: a property of the fit, noted only, no flag (ruled 2026-10-24).

-0.4154(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. -4.15363(16) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 1 row; sources/1705.01608.txt line 1280; 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_1.0/UHF_CBS: -4.1536325 0.0018030 | 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. | The complete-basis UHF value lies 0.001254 Ha (total) above the paper's own UHF/cc-pV5Z energy -4.154883934306 on the same geometry, although a finite-basis Hartree-Fock energy bounds the complete-basis limit from above: a property of the fit, noted only, no flag (ruled 2026-10-24).

Coverage

Newest published number here: 2017.

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