aiccm2026dev-a vs aiccm2026dev-b — a comparative analysis of two independent AICCM formulations¶
Analysis prepared by the aiccm2026dev-a line, 2026-06-22, after reading the
aiccm2026dev-b manuscript in full.
Sources.
-a:
docs/manuscripts/aiccm2026dev_a_theory.md(this line) and the algorithm referenceAICCM_ALGORITHM.md§§12–13.-b:
docs/manuscripts/aiccm_comparison/aiccm_comparison.tex(1347 lines, “The Ab Initio Cyclic Cluster Model Reconsidered…”) + itsdata/*.json.baseline: Peintinger & Bredow, J. Comput. Chem. 35, 839 (2014) — the 2014 AICCM both lines re-examine.
Methodological note. Until this analysis the two lines were derived blind to each other (the value of an independent cross-check). This document is written after reading -b’s manuscript; -a’s derivation predates and did not use it. The empirical head-to-head (running both on the shared 28-system set) is still the decisive test and is unaffected.
Erratum / update (round-2 debate, 2026-06-22). -b’s critical reply corrected three claims below; this note supersedes them (the inline text is left as the record of the exchange):
Q1 is settled, and it revises §3/§4. -b ran the exact 8×12×12-bohr H₂ (2,1,1) case: A
run_ccm_rhf_gdf= B RI-RHF = −1.1213722182230756 Ha/cell, Δ = 0.0. So A’s production RI is the multi-k BvK-Madelung-corrected GDF — i.e. the character route — and is bit-identical to B. The 4.64 mHa was A’s legacy Γ-supercell helper (exxdiv=None) vs the productionexxdiv=ewald— a finite-mesh exchange-q=0 convention difference, not an A/B disagreement. Consequence: -a’s “stays real-Γ, avoids the round-trip” (§4 Argument 2) is retracted — production is character-native and coincides with -b.The Madelung result is narrower than stated (§1, §4, §7, §8).
ξ·S⊗Sis the leading molecular-limit term, not exactly rank-1 for a finite solid; and the background is block-diagonal in occ/virt, so it leaves the correlation numerator unchanged only at a fixed reference — it shifts the orbital- energy denominators, so independent bare-vs-neutral SCF→correlation do not agree and the neutral reference is required. “Provably Madelung-robust correlation” is therefore a fixed-reference diagnostic, not an all-post-HF theorem.
ω^symis a model tensor, not a demonstrated ERI (its AO-centre fold is basis-label-dependent; a physical kernel is a functional of the pair density). It stays a historical diagnostic, out of the production hierarchy.The corrected statements are now in
aiccm2026dev_a_theory.md(§§6.1, 7).
1. Executive summary¶
Two teams set out, independently and without reading each other, to put the 2014 ab-initio Cyclic Cluster Model on a rigorous footing. They converged on the same physics. Both identify the cyclic cluster as a finite Born–von-Kármán (BvK) torus; both prove the Γ-supercell calculation equals the full unreduced Γ-centred character mesh exactly (not as a quadrature approximation); both adopt the charge-neutral periodic Green’s function (G = 0 dropped) with a single gauge for all Coulomb channels; both reinterpret the Wigner–Seitz fractions as a partition of unity over tied minimum-image representatives rather than weights multiplied into integrals; both diagnose the 2014 four-centre factor as bra–ket-asymmetric and therefore not the derivative of a scalar energy; and both build the same downstream stack (HF/KS, RI, RIJCOSX, MP2, CCSD(T), Wannier/IAO localization, PAO/PNO, space-group action on the torus).
That independent agreement on the foundations is the single most important result of the comparison: it is strong evidence the foundations are right.
The lines differ in four substantive ways and several presentational ones:
Default representation. -a is real-space, supercell-Γ native (CCM ≡ SCM-Γ); -b is reciprocal/character native and adds an exact inverse transform to a real torus for local correlation.
The four-centre object. -a keeps two — a symmetric weighted bare-1/r four-centre (
ω^sym, for small-cluster/1-D analysis) and the neutral periodised tensorg_eff = ΣLL(production). -b keeps one — the periodised neutral tensor only — and argues that any weight on a non-periodised kernel, even a symmetric one, is the wrong kind of object.The Madelung treatment. -a characterises the bare→neutral difference analytically as a rank-1 background
ξ·S⊗Sand proves correlation is Madelung-robust; -b is neutral by construction and never builds a bare tensor to compare.Scope and validation emphasis. -a is broader (open-shell, a 28-system all-Bravais test set) and makes “resolved” claims; -b is narrower (closed-shell only) but carries the single strongest external check (3-D RI-MP2 vs PySCF KRHF/KMP2 below one nanohartree per cell) and is markedly more conservative, disabling features until parity is proved.
Crucially, the two production paths are essentially the same object (the neutral periodised tensor / its RI factorisation), so on neutral systems with matched cutoffs the head-to-head energies should agree closely. The interesting divergences will be on the regimes both papers flag as open: non-orthorhombic ≥2-D, ionic 3-D, and the low-dimensional long-range gauge.
3. The central theoretical divergence: a symmetric weight, or no weight¶
This is the sharpest and most interesting difference, and -b names it directly. -b describes the -a line (their §”Term-by-term comparison”) as one that
“replaces the historical bridge by an explicitly symmetric four-centre bridge while retaining the weighted-cluster construction.”
That is accurate. The two lines part company on what the corrected four-centre object should be:
-a keeps a four-centre weight, but makes it symmetric. Its
ω^sym(§6.1) is a symmetric intra-pair × inter-electron min-image partition,ω^sym = ω_{μν}·ω_{ρσ}·ω^↔, with ρ and σ folded independently to their own minimum images. This is exactly 8-fold symmetric (≤ 3×10⁻¹⁸) on 1-D, 2-D-hex, and 2-D-oblique lattices — a genuine repair of eq-18 that preserves the weighted-cluster lineage. -a presents this as a headline result and uses it as the small-cluster / 1-D / post-HF analysis tool.-b abandons four-centre weights entirely. It periodises the kernel first (the neutral torus ERI, Eq. periodic_eri), which is symmetric by construction, and uses the WS fractions only to pick tied representatives — “no second, centre-dependent multiplier is applied.” -b argues (their §”Double counting”) that the unproved step in the whole AICCM tradition is precisely “turning a representative count into a centre-pair multiplier on a non-periodised four-centre kernel” — which would make even a symmetric weight the wrong kind of object, because it modifies a kernel that was never periodised.
The reconciliation. -a does not actually use ω^sym (bare-1/r) for
production. -a’s own paper (§7) shows the bare symmetric four-centre over-binds
ionic 3-D by the Madelung background and is non-separable, and routes production
HF/KS through the neutral periodised tensor g_eff = ΣLL via GDF — which is
the same object -b uses. So:
On the production path, -a and -b compute with the same neutral periodised tensor (or its RI). The divergence is presentational/conceptual: -a presents a hierarchy (symmetric-weighted bare → neutral periodised) with explicit continuity to 2014; -b presents a single periodised object and argues the weight idea should be dropped at the root.
Both views are defensible. -a’s hierarchy makes the relationship to the 2014
construction (and the precise nature of its error) maximally explicit and gives a
cheap small-cluster tool; -b’s single-object view is conceptually cleaner and
avoids ever constructing a non-separable tensor. -b’s caution — that the 1-D
equality of the weighted constructions “is not evidence of equality for general
3-D boundary multiplicities” — applies to -a’s ω^sym only insofar as -a were to
use it in 3-D, which it does not.
4. Point-by-point differences¶
Axis |
|
|
|---|---|---|
Native representation |
real-space supercell-Γ (CCM ≡ SCM-Γ) |
reciprocal/character; exact inverse transform to real torus for local correlation |
Four-centre object(s) |
symmetric-weighted bare |
neutral periodised tensor only (+ its RI factorisation) |
Stance on WS as a 4c weight |
retains a (now-symmetric) weight as a central object |
rejects 4c weights; WS only selects tied representatives |
Madelung / ionic |
characterises bare−neutral = |
neutral by construction; no bare tensor built; notes a BvK-supercell exchange Madelung term must enter K |
RI three-centre |
neutral cderi |
pair-resolved periodic DF `(μν |
Open shell |
UHF / UKS / UMP2 |
closed-shell only (explicitly not claimed) |
Localization spread |
Pipek–Mezey / Boys; Resta position operator flagged as TODO |
implements the circular (Resta-type) operator `⟨w |
IAO |
reuse molecular IAO route (planned) |
minimal-reference IAO, evaluated in an explicit supercell (not yet periodised across the boundary) |
Space-group exploitation |
reports pair-orbit reduction (MgO(2,2,2) 256→32, 8×); union-V breaks symmetry → use neutral V |
enumerates k/pair/quartet orbits; scattering disabled until Fock/energy parity proven |
Historical-related-work audit |
eq-18 only (deMon2k in the algorithm reference, not the paper) |
separate deMon2k audit; notes its 3-centre already symmetrises both orderings ⇒ not the eq-18 defect |
Units |
atomic / Hartree (Ha/atom) |
SI primary (kJ/mol), Hartree in parentheses |
Test breadth |
28 systems, all 7 crystal systems + cubic centerings; CRYSTAL23 refs (harness ready) |
compact H₂/H₄/LiH/graphene controls + 18-system A/B fleet inputs; 31 runnable jobs |
Strongest external check |
CCM ≡ SCM-Γ == KRHF settling test (production HF) |
3-D RI-MP2 vs PySCF KRHF/KMP2 < 1 nanohartree/cell (executed) |
Tone |
“resolved” (Madelung, RI-consistency) |
conservative; pervasive “open / not claimed / fail closed” |
5. How each treats the 2014 AICCM (Peintinger & Bredow)¶
Both lines treat the 2014 paper as the baseline to be diagnosed and superseded, not discarded, and both reproduce its 1-D numbers.
Shared diagnosis. The eq-18 factor
x_{lm}·½(x_{lq}+x_{mq})·x_{qr}is bra–ket-asymmetric; a converged SCF can hide this because DIIS solves a fixed-point equation without checking it is Euler–Lagrange for a scalar energy, and the boundary factors differ from unity only on a few short-ranged terms (both papers make this “convergence conceals the defect” argument).-a’s continuity is tighter: it keeps eq-18 (
union12) as the documented 1-D tool, identifies eq-18 as the WS-symmetry-hidden special case of the symmetric folding, and buildsω^symas the minimal symmetric repair in the same weighted-cluster language.-b’s break is cleaner: it reconstructs the 2014 equations faithfully (their §”Historical AICCM formulation”), proves the defect as the explicit condition
x_{mq}=x_{rl}, and then leaves the weighted-tensor idea behind entirely. -b also extends the audit to the 2008 deMon2k KS-ADFT CCM and is careful not to tar it with the eq-18 brush, since deMon2k’s three-centre term already averages both AO-centre orderings.
Net: -a frames the new line as “2014, repaired and generalised”; -b frames it as “2014, re-derived from the finite group, with the weighting tradition diagnosed as unnecessary.”
6. Numerical evidence — what each has actually shown¶
The two evidence bases are complementary rather than overlapping.
-b’s strongest results (all executed, conservative provenance):
3-D RI-MP2 vs out-of-process PySCF KRHF/KMP2: total agreement +0.945×10⁻⁶ kJ/mol per cell (sub-nanohartree); max imaginary residue 8.7×10⁻¹⁹ Ha. This is a genuine external cross-code validation and the single most convincing number in either paper.
Real-torus = character local MP2 to 6.9×10⁻¹⁸ Ha/cell; LiH no-truncation local CCSD/(T) = canonical finite-torus to ~10⁻¹² Ha — i.e., the inverse transform and the local-correlation machinery are exact in the no-truncation limit.
H₂ backend control: direct / RI / RIJCOSX within 0.115 kJ/mol (RHF and PBE0).
Openly flagged gaps: vacuum-padded H₄ 2-cell direct-vs-RI = 58.296 kJ/mol (unresolved truncation/embedding); lower-dimensional 4c/RI offsets of hundreds of kJ/mol (unmatched long-range gauges); dense ionic ladders (LiH/MgO/NaCl/ diamond) and an external periodic-CC benchmark not yet run.
-a’s strongest results:
The symmetric four-centre is exactly 8-fold symmetric (≤ 3×10⁻¹⁸) on 1-D, 2-D-hex, 2-D-oblique — versus eq-18’s 1.6×10⁻³ → 2.1×10⁻²; the energetic footprint (hex −0.400456 vs union12 −0.399044 Ha/atom) is shown.
Full-stack agreement with the molecular kernels in the isolated limit (HF == padded to µHa; KS == molecular < 10⁻⁶; MP2/CCSD == molecular; RIJCOSX to ~5 µHa); RI(GDF) == four-centre to ≲ 0.1 mHa/atom in 1-D and 3-D.
The Madelung gap resolved analytically:
g_eff = bare − ξ·S⊗S,ξ →the cell Madelung constant (ratio 0.975 → 0.991 over L = 12 → 20 bohr), and the background contributes exactly zero to correlation (machine ε, any ξ).Localization validated (density residual 2.4×10⁻¹⁶, unitarity 6.7×10⁻¹⁶); the 28-system Bravais-spanning test set + CRYSTAL23 reference harness in place.
A concrete head-to-head data point already visible. -b reports that its
RI-RHF differs from “the older Γ-supercell HF helper” by 12.185 kJ/mol
(4.64 mHa) on the compact two-cell H₂ control, and rejects that legacy helper as
a post-HF reference. -a’s production RI route (run_ccm_rhf_gdf) is a GDF
Γ-on-supercell calculation of exactly that family. Whether -b’s “legacy helper”
is literally -a’s GDF path or a different one, a ~4–5 mHa discrepancy on a tiny
vacuum-padded cell is precisely the kind of long-range-gauge / cutoff difference
the head-to-head must reconcile — and both papers independently flag the
small-cell low-dimensional gauge as the leading open issue.
7. Predicted head-to-head outcome¶
Because both production paths reduce to the neutral periodised tensor (or its RI) on the same finite torus, the prediction is:
Neutral, gapped, ≥2 cells, matched cutoffs: close agreement, of the order of -b’s RI-MP2-vs-PySCF result, limited by auxiliary-basis / grid / cutoff choices rather than by the formulations.
Compact / vacuum-padded / low-dimensional cells: disagreement at the mHa-to-tens-of-kJ/mol level, from the different long-range gauges each line currently uses below 3-D (both flag this).
Ionic 3-D: agreement iff both use their neutral routes; -a’s bare
ω^symwould diverge by theξ·S⊗SMadelung background, but -a does not use it for production there.Correlation energies: expected to agree well (both are Madelung-robust — -a by the
ξ·I⊗Icancellation proof, -b by neutral construction), and -b’s exact-limit local-correlation controls give a clean target.
Discriminating systems (both papers concur): non-orthorhombic ≥2-D crystals and ionic 3-D solids, evaluated at ≥2 cells with matched cutoffs. These are well covered by -a’s 28-system Bravais set (corundum, wurtzite, black-P, fluorite, rock-salts, CsCl, …), which is the natural arena for the run.
8. Honest assessment of relative strengths and gaps¶
-b is ahead on: rigor of the finite-group/character derivation; the single strongest external validation (sub-nanohartree 3-D RI-MP2 vs PySCF); exact-limit local-correlation proofs; the circular (Resta-type) spread operator; the scholarly deMon2k audit; conservative, fully-provenanced reporting; SI discipline.
-a is ahead on: scope (open-shell; the full 4c→RI→RIJCOSX hierarchy as shipped methods); the analytic Madelung characterisation and Madelung-robust correlation proof; explicit continuity to (and minimal repair of) the 2014 weighting; the breadth of the validation surface (all-Bravais 28-system set + CRYSTAL23 harness); and an already-validated production HF path tied to CCM ≡ SCM-Γ.
Shared open items (neither has closed): a scalable dense-3-D path beyond small clusters / RI; the low-dimensional long-range Coulomb gauge (the leading source of inter-line disagreement); turning the space-group machinery from diagnostic into an actual integral/Fock reduction with proven parity; external periodic dense-SCF and periodic-CC benchmarks on real ionic solids.
A fair reading is that -b is the more rigorously proven core on a narrow base, and -a is the more broadly built and validated-against-molecular-limits method with the Madelung physics worked out. They are complementary, and their independent agreement on the foundations is mutually corroborating.
9. Conclusion¶
The two lines are not rival theories of different physics; they are two independent, largely-agreeing derivations of the same corrected cyclic cluster model, differing mainly in (i) which representation is treated as primary (real-space-Γ for -a, reciprocal/character for -b), (ii) whether a symmetric four-centre weight is retained as a central object (-a) or dropped in favour of the periodised tensor alone (-b), (iii) how explicitly the Madelung background is characterised (-a quantifies it; -b avoids it by construction), and (iv) breadth versus conservatism (open-shell + all-Bravais coverage with “resolved” claims for -a; closed-shell with a sub-nanohartree external check and pervasive caution for -b).
Their production objects coincide (the neutral periodised tensor / its RI), so the
empirical head-to-head should show close agreement on neutral gapped systems and
localise any disagreement to the low-dimensional gauge and small-cell embedding —
exactly the open frontier both papers name. The right next step is the run the
maintainer planned: both methods on the shared 28-system Bravais set with matched
cutoffs, compare.py --vs, plus the permutation-symmetry, per-atom-energy, and
nrep/k-mesh-convergence diagnostics.
That two from-scratch derivations land on the same finite-torus / neutral-kernel / partition-of-unity foundation — while a 12-year-old published weighting is shown, twice and by different arguments, to lack the symmetry it needs — is the strongest statement either paper makes.