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All instances / Hydrogen chain

Hydrogen chain H10 chain, open, R = 1.2, basis-free

Best values (rules §6)

best variational -0.513646(9) VMC (JAGP)
10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected -0.514432(6) 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 -0.5153(2) MRCI+Q
10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate -0.5149(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 = 1.2 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
1.2
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
-0.513646best variational 0 9.0e-6 n/a n/a VMC (JAGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.513225 4.2e-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 bestσVar(E)V-score methodsourcecodeyearcomment
-0.514432best projected 0 6.0e-6 n/a n/a LR-DMC (AGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.514418tied with best projected at 2σ 1.4e-5 7.0e-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
-0.5153best extrapolated 0 1.5e-4 n/a n/a MRCI+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5151tied with best extrapolated at 2σ 2.3e-4 1.2e-4 n/a n/a MRCI-F12+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5149tied with best extrapolated at 2σ 4.0e-4 1.4e-4 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5143 9.7e-4 1.9e-4 n/a n/a RCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5143 9.7e-4 1.9e-4 n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5140 1.3e-3 1.5e-4 n/a n/a SBDMRG
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.48837 2.7e-2 5.0e-5 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.48837 2.7e-2 5.4e-5 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
-0.5149best approximate 0 1.9e-4 n/a n/a RCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5149tied with best approximate at 2σ <1e-4 1.9e-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.

-0.5153(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. -0.51529(15) per atom

Table III (first block), R = 1.2 row; sources/1705.01608.txt line 1268; 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_1.2/MRCI+Q_CBS: -5.152930 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.

-0.5151(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. -0.51506(12) per atom

Table III (first block), R = 1.2 row; sources/1705.01608.txt line 1268; 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_1.2/MRCI+Q+F12_CBS: -5.150630 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.

-0.5149(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. -0.51490(19) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/RCCSD(T)_CBS: -5.14902 0.00196811 | 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.

-0.5149(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. -0.51490(19) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/UCCSD(T)_CBS: -5.14902 0.00196811 | 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.

-0.5149(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. -0.51489(14) per atom

Table III (first block), R = 1.2 row; sources/1705.01608.txt line 1268; 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_1.2/AFQMC_CBS: -5.14894 0.00138821 | 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.

-0.514432(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. -0.514432(6) per atom

Table III (first block), R = 1.2 row; sources/1705.01608.txt line 1268; 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_1.2/LRDMC_AGP_basis-TZ: -5.14432069019419 6.114219359422410E-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.

-0.514418(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. -0.514418(7) per atom

Table III (first block), R = 1.2 row; sources/1705.01608.txt line 1268; 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_1.2/LRDMC_LDA_basis-TZ: -5.14417513945647 7.558616328967620E-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(2), 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. -0.51432(19) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/RCCSD_CBS: -5.14316 0.00192456 | 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(2), 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. -0.51432(19) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/UCCSD_CBS: -5.14316 0.00192456 | 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.

-0.5140(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. -0.51399(15) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/SBDMRG_CBS: -5.13994 0.00149949 | 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.

-0.513646(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. -0.513646(9) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/VMC_AGP_basis-TZ: -5.13645939668728 8.684042872338628E-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.

-0.513225(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. -0.513225(8) per atom

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/VMC_LDA_basis-TZ: -5.13224718956311 8.510167831527644E-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.

-0.48837(5), 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. -4.88368(5) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 1.2 row; sources/1705.01608.txt line 1281; 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_1.2/UHF_CBS: -4.8836782 0.0005371 | 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 0.0004114 Ha (total) above the paper's own UHF/cc-pV5Z energy -4.884091362602 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).

-0.48837(5), 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. -4.8836797 0.0005354 (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_1.2/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 0.0004117 Ha (total) above the paper's own RHF/cc-pV5Z energy -4.884091359616 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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