QMBLthe quantum many-body leaderboard

All instances / Hydrogen chain

Hydrogen chain H10 chain, open, R = 3.2, basis-free

Best values (rules §6)

best variational -0.524373(4) VMC (JAGP)
10.1103/PhysRevX.7.031059 · the lowest unflagged variational energy, a strict variational bound
best projected -0.524987(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 -0.52512(6) DMET
10.1103/PhysRevX.7.031059 · the lowest unflagged extrapolated energy, an extrapolation, not a bound
best approximate -0.5257(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

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 = 3.2 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
3.2
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 bestσVar(E)V-score methodsourcecodeyearcomment
-0.524373best variational 0 4.0e-6 n/a n/a VMC (JAGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.523557 8.2e-4 5.0e-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
-0.524987best projected 0 7.0e-6 n/a n/a LR-DMC (AGP)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.524974tied with best projected at 2σ 1.3e-5 9.0e-6 n/a n/a LR-DMC (LDA determinant)
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 bestσVar(E)V-score methodsourcecodeyearcomment
-0.52512best extrapolated 0 6.0e-5 n/a n/a DMET
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.5251tied with best extrapolated at 2σ <1e-5 1.4e-4 n/a n/a SBDMRG
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52500tied with best extrapolated at 2σ 1.2e-4 5.0e-5 n/a n/a MRCI-F12+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52500tied with best extrapolated at 2σ 1.2e-4 7.0e-5 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52499tied with best extrapolated at 2σ 1.2e-4 6.0e-5 n/a n/a MRCI+Q
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52419 9.3e-4 8.0e-5 n/a n/a RCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52303 2.1e-3 8.0e-5 n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.509670 1.5e-2 6.0e-6 n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.486951 3.8e-2 5.3e-6 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 bestσVar(E)V-score methodsourcecodeyearcomment
-0.5257best approximate 0 1.0e-4 n/a n/a RCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52340 2.3e-3 8.0e-5 n/a n/a UCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52302 2.7e-3 9.0e-5 n/a n/a PC-NEVPT2
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.52299 2.7e-3 9.0e-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.

-0.5257(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. -0.52574(10) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/RCCSD(T)_CBS: -5.25742 0.000972659 | 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.

-0.52512(6), 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. -0.52512(6) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/DMET_2_CBS: -5.25123 0.000606248 | 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.

-0.5251(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. -0.52511(14) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/SBDMRG_CBS: -5.25114 0.00140136 | 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.

-0.52500(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. -0.52500(5) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/MRCI+Q+F12_CBS: -5.250000 0.000458 | 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.

-0.52500(7), 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. -0.52499(7) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/AFQMC_CBS: -5.24997 0.000706798 | 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.

-0.52499(6), 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. -0.52499(6) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/MRCI+Q_CBS: -5.249950 0.000619 | 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.

-0.524987(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. -0.524987(7) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/LRDMC_AGP_basis-TZ: -5.24987250108512 6.840897035041683E-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.

-0.524974(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. -0.524974(9) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/LRDMC_LDA_basis-TZ: -5.24973919682442 9.869193738861033E-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.

-0.524373(4), 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. -0.524373(4) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/VMC_AGP_basis-TZ: -5.24372765577335 3.384158361392282E-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.

-0.52419(8), 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. -0.52419(8) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/RCCSD_CBS: -5.2419 0.000786923 | 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.

-0.523557(5), 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. -0.523557(5) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/VMC_LDA_basis-TZ: -5.23556666449395 4.717928839959921E-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.

-0.52340(8), 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. -0.52340(8) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/UCCSD(T)_CBS: -5.23399 0.000832091 | 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.

-0.52303(8), 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. -0.52303(8) per atom

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/UCCSD_CBS: -5.23034 0.000798288 | 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.

-0.52302(9), 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. -0.52302(9) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/PC-NEVPT2_CBS: -5.23025 0.000913586 | 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.

-0.52299(9), 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. -0.52299(9) per atom

Table III (first block), R = 3.2 row; sources/1705.01608.txt line 1275; 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_3.2/SC-NEVPT2_CBS: -5.2299 0.000912688 | 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.

-0.509670(6), 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. -5.096702(6) (printed as the H10 total, in a column of per-atom values)

Table III (second block), R = 3.2 row; sources/1705.01608.txt line 1288; 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_3.2/UHF_CBS: -5.0967023 0.0000617 | 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.

-0.486951(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. -4.8695081 0.0000531 (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_3.2/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.

Coverage

Newest published number here: 2017.

This instance as data

JSON · source file on GitHub · add a result · open an issue about this instance