Hydrogen chain H10 chain, open, R = 1.6, basis-free

Best values (rules §6)

best variational	-0.562237(5)	VMC (JAGP) 10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected	-0.562900(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	-0.56325(8)	MRCI+Q 10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate	-0.56311(9)	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 = 1.6 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: 1.6
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 best	σ	Var(E)	V-score	method	source	code	year	comment
-0.562237best variational	0	5.0e-6	n/a	n/a	VMC (JAGP) peer reviewed	10.1103/PhysRevX.7.031059		2017
-0.561823	4.1e-4	7.0e-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
-0.562900best projected	0	7.0e-6	n/a	n/a	LR-DMC (LDA determinant) peer reviewed	10.1103/PhysRevX.7.031059		2017
-0.562878tied with best projected at 2σ	2.2e-5	6.0e-6	n/a	n/a	LR-DMC (AGP) peer reviewed	10.1103/PhysRevX.7.031059		2017

Extrapolations (7)

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
-0.56325best extrapolated	0	8.0e-5	n/a	n/a	MRCI+Q peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.56321tied with best extrapolated at 2σ	3.6e-5	7.0e-5	n/a	n/a	MRCI-F12+Q peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5632tied with best extrapolated at 2σ	9.6e-5	1.2e-4	n/a	n/a	ph-AFQMC peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5631tied with best extrapolated at 2σ	1.3e-4	1.4e-4	n/a	n/a	SBDMRG peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.5626tied with best extrapolated at 2σ	6.7e-4	3.0e-4	n/a	n/a	DMET peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.53682	2.6e-2	2.6e-5	n/a	n/a	HF peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.53682	2.6e-2	3.0e-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
-0.56311best approximate	0	9.0e-5	n/a	n/a	RCCSD(T) peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.56311tied with best approximate at 2σ	<1e-5	9.0e-5	n/a	n/a	UCCSD(T) peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.56011	3.0e-3	1.0e-5	n/a	n/a	PC-NEVPT2 peer reviewed	10.1103/PhysRevX.7.031059	2017
-0.55991	3.2e-3	1.0e-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.

-0.56325(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. -0.56324(8) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/MRCI+Q_CBS: -5.632460 0.000831 | 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.

-0.56321(7), 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. -0.56321(7) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/MRCI+Q+F12_CBS: -5.632120 0.000737 | 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), 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. -0.56315(12) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/AFQMC_CBS: -5.63153 0.00115245 | 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.

-0.5631(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. -0.56311(14) per atom

-0.56311(9), 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. -0.56311(9) per atom

-0.56311(9), 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. -0.56311(9) per atom

-0.562900(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. -0.562900(7) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/LRDMC_LDA_basis-TZ: -5.62900079058741 7.088643856917804E-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.

-0.562878(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. -0.562878(6) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/LRDMC_AGP_basis-TZ: -5.62877909685974 5.462790894697366E-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.

-0.5626(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. -0.56258(30) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/DMET_2_CBS: -5.6258 0.00300588 | 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.

-0.562237(5), 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. -0.562237(5) per atom

-0.561823(7), 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. -0.561823(7) per atom

-0.56011(1), 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. -0.56011(1) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/PC-NEVPT2_CBS: -5.60113 0.000276705 | 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.

-0.55991(1), 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. -0.55991(1) per atom

Table III (first block), R = 1.6 row; sources/1705.01608.txt line 1270; 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_1.6/SC-NEVPT2_CBS: -5.59911 0.000276668 | 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.

-0.53682(3), 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. -5.3682046 0.0002567 (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_1.6/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. | The complete-basis RHF value lies 0.0001543 Ha (total) above the paper's own RHF/cc-pV5Z energy -5.368358892993 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).

-0.53682(3), 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. -5.36820(3) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 1.6 row; sources/1705.01608.txt line 1283; 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_1.6/UHF_CBS: -5.3682046 0.0002567 | 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 0.0001589 Ha (total) above the paper's own UHF/cc-pV5Z energy -5.368358893359 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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