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 bohr, open (a finite molecule); N↑ = 5, N↓ = 5. Projected onto the aug-cc-pVQZ 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
- 10
- boundary
- open
- N_up
- 5
- N_dn
- 5
- R
- 1
- basis
- aug-cc-pVQZ
- energies
- 2
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 (1)
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.
Approximate methods (no bound) (1)
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.
-0.43918, MP2
Checked 2026-10-20 · source text of arXiv:1911.01618 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. HF -0.41566 + MP2 correlation -0.02352 per atom
SI Table III, row "aug-QZ" (NAO 460), HF and MP2 columns; sources/1911.01618-anc-si.txt line 1275; version read: APS Supplemental Material (si.pdf, sources/1911.01618-PhysRevX.10.031058-si.txt) and the arXiv ancillary SI; identical | Hartree per atom, total energy with nuclear repulsion: the cc-pVDZ HF value -0.40101 matches RHF/cc-pVDZ of the 2017 data set (-4.010066006349 for H10).
-0.41566, HF
Checked 2026-10-20 · source text of arXiv:1911.01618 read locally (pypdf layout text; the generator finds every value again on its line), no LLM transcription of numbers; single reading
Reported as. -0.41566 per atom (HF column)
SI Table III, row "aug-QZ" (NAO 460), HF column; sources/1911.01618-anc-si.txt line 1275; version read: APS Supplemental Material (si.pdf, sources/1911.01618-PhysRevX.10.031058-si.txt) and the arXiv ancillary SI; identical | Hartree per atom, total energy with nuclear repulsion: the cc-pVDZ HF value -0.40101 matches RHF/cc-pVDZ of the 2017 data set (-4.010066006349 for H10).