The Hamiltonian
H = −½ ∑i ∇i2 − ∑i,I 1/|ri − RI| + ∑i<j 1/|ri − rj| + ∑I<J 1/|RI − RJ|
Hydrogen chain: 8 protons equally spaced on a line, spacing R = 3 bohr, open (a finite molecule); N↑ = 4, N↓ = 4. Projected onto the STO-3G 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
- 8
- boundary
- open
- N_up
- 4
- N_dn
- 4
- R
- 3
- basis
- STO-3G
- energies
- 7
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).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (3)
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.
Projected or fixed-node estimates (3)
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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.4973, CCSDT
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.9784 Ha per total
Table I; sources/2305.05168.txt line 225; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. | reader confidence medium
-0.4965875, CCSD
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.9727 Ha per total
Table I; sources/2305.05168.txt line 224; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. | reader confidence medium
-0.4930875, ED
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.9447 Ha per total
Table I; sources/2305.05168.txt line 228; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. | reader confidence medium
-0.4929875, CCSDTQ
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.9439 Ha per total
Table I; sources/2305.05168.txt line 226; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. | reader confidence medium
-0.491525, QFlow
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.9322 Ha per total
Table I; sources/2305.05168.txt line 227; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. QFlow value is from the primary active space (footnote a). | reader confidence medium
-0.4582, CASCI
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.6656 Ha per total
Table I; sources/2305.05168.txt line 223; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. | reader confidence medium
-0.4465375, HF
Checked 2026-10-20 · source text of arXiv:2305.05168 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading (Sonnet reader)
Reported as. -3.5723 Ha per total
Table I; sources/2305.05168.txt line 222; version read: arXiv text; no journal version found | Table I caption line 216: 'energies (in Hartree)'; R in 'a.u.' (R_H-H=2.0 a.u.); magnitude consistent with total energy incl. nuclear repulsion | Boundary not stated; 'linear chains' of H atoms, taken as open. Column header gives (2.0 a.u)/(3.0 a.u). Nuclear repulsion not stated; value magnitude suggests included. | reader confidence medium