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All instances / Hydrogen chain

Hydrogen chain H50 chain, open, R = 1, STO-6G

Best values (rules §6)

best variational -0.3456814 LDMRG
10.1063/1.2345196 · the lowest unflagged variational energy, a strict variational bound
best projected -0.3466732 DMET
10.1103/PhysRevX.7.031059 · the lowest unflagged projected energy, a projected estimate, which depends on its node or constraint
best approximate -0.345728 GW
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: 50 protons equally spaced on a line, spacing R = 1 bohr, open (a finite molecule); N↑ = 25, N↓ = 25. Projected onto the STO-6G basis set; full CI in it is this instance's exact energy. Hartree atomic units, nuclear repulsion included; the energy per atom is E/N.

geometry
chain
atoms
50
boundary
open
N_up
25
N_dn
25
R
1
basis
STO-6G
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 (7)

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
-0.3456814best variational 0 n/a n/a n/a LDMRG
peer reviewed
 10.1063/1.2345196 2006
-0.3456812 2.0e-7 n/a n/a n/a DMRG
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.3456804 1.0e-6 n/a n/a n/a LDMRG
peer reviewed
 10.1063/1.2345196 2006
-0.345643 3.8e-5 n/a n/a n/a LDMRG
peer reviewed
 10.1063/1.2345196 2006
-0.345352 3.3e-4 n/a n/a n/a LDMRG
peer reviewed
 10.1063/1.2345196 2006
-0.3372976 8.4e-3 n/a n/a n/a HF
peer reviewed
 10.1063/1.2345196 2006
-0.3372975 8.4e-3 n/a n/a n/a HF
peer reviewed
 10.1103/PhysRevX.7.031059 2017

Projected or fixed-node estimates (6)

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
-0.3466732best projected 0 n/a n/a n/a DMET
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.345704 9.7e-4 2.0e-6 n/a n/a ph-AFQMC
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.3454434 1.2e-3 n/a n/a n/a CCSD
peer reviewed
 10.1063/1.2345196 2006
-0.3454434 1.2e-3 n/a n/a n/a UCCSD
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.3449205 1.8e-3 n/a n/a n/a SEET
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.3431425 3.5e-3 n/a n/a n/a DMET
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
-0.345728best approximate 0 n/a n/a n/a GW
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.3456454 8.3e-5 n/a n/a n/a CCSD(T)
peer reviewed
 10.1063/1.2345196 2006
-0.3456452 8.3e-5 n/a n/a n/a UCCSD(T)
peer reviewed
 10.1103/PhysRevX.7.031059 2017
-0.3445266 1.2e-3 n/a n/a n/a MP2
peer reviewed
 10.1063/1.2345196 2006

How these numbers were read

Where each number below was read, and how it was checked.

-0.3466732, DMET

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. -17.333661 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/DMET_5_basis-STO 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.

-0.345728, GW

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. -17.2864 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/SC-GW_basis-STO 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.

-0.345704(2), ph-AFQMC

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. -17.2852 0.0001 (total for N = 50; 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_50_OBC/R_1.0/AFQMC_basis-STO 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.

-0.3456814, LDMRG

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.41919 = -17.28407 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3456812, DMRG

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. -17.28406 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/DMRG_basis-STO 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. | Read as the variational energy of the final sweep at bond dimension D = 2000 (Appendix A runs Block with the standard settings of Ref. [66], whose schedule ends at the maximum M; the paper names no extrapolation in the bond dimension). Medium confidence: Ref. [66] also describes extrapolation to zero discarded weight, and the paper does not say whether these energies were extrapolated (ruled 2026-10-24). | reader confidence medium

-0.3456804, LDMRG

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.41914 = -17.28402 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3456454, CCSD(T)

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.41739 = -17.28227 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3456452, UCCSD(T)

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. -17.282262 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/UCCSD(T)_basis-STO 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.

-0.345643, LDMRG

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.41727 = -17.28215 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3454434, CCSD

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.40729 = -17.27217 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3454434, UCCSD

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. -17.272169 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/UCCSD_basis-STO 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.

-0.345352, LDMRG

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.40272 = -17.26760 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3449205, SEET

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. -17.24602616 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/SEET_CI-GF2_mix(6,SAO)_basis-STO 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.

-0.3445266, MP2

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.86488 + E_corr -0.36145 = -17.22633 (totals)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3431425, DMET

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. -17.157125 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/DMET_2_basis-STO 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.

-0.3372976, HF

Checked 2026-10-20 · source text of arXiv:cond-mat/0606115 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading

Reported as. E_RHF -16.864 88 (total)

Table VI, R = 1 row; sources/cond-mat-0606115.txt line 826; version read: arXiv v3 (journal copy not reached, AIP HTTP 403) | Hartree totals: E_RHF (total, with nuclear repulsion) and E_RHF,el printed side by side; at R = 1.8 bohr E_RHF + E_corr[LDMRG(500)] = -27.03865 against the 2017 STO-6G DMRG -27.03864 for H50.

-0.3372975, 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. -16.8648759170 (total for N = 50)

dataset github.com/simonsfoundation/hydrogen-benchmark-PRX, commit b99a183 (2017-10-09, uploaded by the authors), CC BY 4.0, file N_50_OBC/R_1.0/UHF_basis-STO 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.

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