The Hamiltonian
H = −½ ∑i ∇i2 − ∑i,I 1/|ri − RI| + ∑i<j 1/|ri − rj| + ∑I<J 1/|RI − RJ|
Hydrogen chain: 6 protons equally spaced on a line, spacing R = 3 bohr, open (a finite molecule); N↑ = 3, N↓ = 3. Projected onto the cc-pVDZ 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
- 6
- boundary
- open
- N_up
- 3
- N_dn
- 3
- R
- 3
- basis
- cc-pVDZ
- energies
- 6
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.
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.
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).
How these numbers were read
Where each number below was read, and how it was checked.
-0.5269833, 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. -3.1619 Ha per total
Table II; sources/2305.05168.txt line 260; version read: arXiv text; no journal version found | Table II caption line 254: 'energies (in Hartree)', R_H-H=3 a.u. | Boundary not stated, open assumed. | reader confidence medium
-0.5265167, 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. -3.1591 Ha per total
Table II; sources/2305.05168.txt line 261; version read: arXiv text; no journal version found | Table II caption line 254: 'energies (in Hartree)', R_H-H=3 a.u. | Boundary not stated, open assumed. | reader confidence medium
-0.5265167, 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. -3.1591 Ha per total
Table II; sources/2305.05168.txt line 265; version read: arXiv text; no journal version found | Table II caption line 254: 'energies (in Hartree)', R_H-H=3 a.u. | Boundary not stated, open assumed. | reader confidence medium
-0.5264833, DUCC-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. -3.1589 Ha per total
Table II; sources/2305.05168.txt line 264; version read: arXiv text; no journal version found | Table II caption line 254: 'energies (in Hartree)', R_H-H=3 a.u. | Boundary not stated, open assumed. | reader confidence medium
-0.5261833, 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. -3.1571 Ha per total
Table II; sources/2305.05168.txt line 259; version read: arXiv text; no journal version found | Table II caption line 254: 'energies (in Hartree)', R_H-H=3 a.u. | Boundary not stated, open assumed. | reader confidence medium
-0.5261667, DUCC-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. -3.1570 Ha per total
Table II; sources/2305.05168.txt line 263; version read: arXiv text; no journal version found | Table II caption line 254: 'energies (in Hartree)', R_H-H=3 a.u. | Boundary not stated, open assumed. | reader confidence medium