Hydrogen chain H10 chain, open, R = 1.4, basis-free

Best values (rules §6)

best variational	-0.548046(7)	VMC (JAGP) 10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected	-0.548748(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	-0.5493(2)	MRCI+Q 10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate	-0.5491(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

Hydrogen chain: 10 protons equally spaced on a line, spacing R = 1.4 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: 1.4
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 best	σ	Var(E)	V-score	method	source	code	year	comment
-0.548046best variational	0	7.0e-6	n/a	n/a	VMC (JAGP) peer reviewed	10.1103/PhysRevX.7.031059		2017
-0.547650	4.0e-4	8.0e-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 best	σ	Var(E)	V-score	method	source	code	year	comment
-0.548748best projected	0	7.0e-6	n/a	n/a	LR-DMC (LDA determinant) peer reviewed	10.1103/PhysRevX.7.031059		2017
-0.548742tied with best projected at 2σ	6.0e-6	6.0e-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 best	σ	Var(E)	V-score	method	source	year
-0.5493best extrapolated	0	1.5e-4	n/a	n/a	MRCI+Q peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5492tied with best extrapolated at 2σ	<1e-4	1.2e-4	n/a	n/a	MRCI-F12+Q peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5491tied with best extrapolated at 2σ	1.2e-4	1.1e-4	n/a	n/a	ph-AFQMC peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5491tied with best extrapolated at 2σ	2.1e-4	2.8e-4	n/a	n/a	DMET peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5486	6.6e-4	1.5e-4	n/a	n/a	SBDMRG peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5485	8.0e-4	1.4e-4	n/a	n/a	RCCSD peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5485	8.0e-4	1.4e-4	n/a	n/a	UCCSD peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.52285	2.6e-2	3.5e-5	n/a	n/a	HF peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.52285	2.6e-2	4.0e-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 best	σ	Var(E)	V-score	method	source	year
-0.5491best approximate	0	1.4e-4	n/a	n/a	RCCSD(T) peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5491tied with best approximate at 2σ	<1e-4	1.4e-4	n/a	n/a	UCCSD(T) peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.54594	3.1e-3	3.0e-5	n/a	n/a	PC-NEVPT2 peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.54569	3.4e-3	3.0e-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.

-0.5493(2), 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. -0.54926(15) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/MRCI+Q_CBS: -5.492550 0.001542 | 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.

-0.5492(1), 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. -0.54917(12) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/MRCI+Q+F12_CBS: -5.491740 0.001162 | 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.

-0.5491(1), 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. -0.54914(11) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/AFQMC_CBS: -5.49141 0.00109055 | 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.

-0.5491(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. -0.54907(14) per atom

-0.5491(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. -0.54907(14) per atom

-0.5491(3), 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. -0.54905(28) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/DMET_2_CBS: -5.49049 0.00276097 | 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.

-0.548748(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. -0.548748(7) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/LRDMC_LDA_basis-TZ: -5.48748347990741 7.319702771584984E-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.

-0.548742(6), 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. -0.548742(6) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/LRDMC_AGP_basis-TZ: -5.48742444630286 5.923118418134931E-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.

-0.5486(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. -0.54859(15) per atom

-0.5485(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. -0.54846(14) per atom

-0.5485(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. -0.54846(14) per atom

-0.548046(7), 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. -0.548046(7) per atom

-0.547650(8), 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. -0.547650(8) per atom

-0.54594(3), 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. -0.54593(3) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/PC-NEVPT2_CBS: -5.45939 0.000477537 | 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.

-0.54569(3), 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. -0.54569(3) per atom

Table III (first block), R = 1.4 row; sources/1705.01608.txt line 1269; 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_1.4/SC-NEVPT2_CBS: -5.45694 0.000477335 | 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.

-0.52285(4), 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. -5.2284533 0.0003531 (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_1.4/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 0.0002624 Ha (total) above the paper's own RHF/cc-pV5Z energy -5.228715729826 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).

-0.52285(4), 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. -5.22845(4) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 1.4 row; sources/1705.01608.txt line 1282; 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_1.4/UHF_CBS: -5.2284533 0.0003531 | 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 0.0002657 Ha (total) above the paper's own UHF/cc-pV5Z energy -5.228715717303 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.

This instance as data

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