The Variational Born–von-Kármán Torus Formulation of the Ab-Initio Cyclic Cluster Model

vibe-qc aiccm2026dev-a development line — theory and method paper (draft)

M. F. Peintinger and the vibe-qc aiccm2026dev-a line. Companion to the independent aiccm2026dev-b derivation; both are measured against Peintinger & Bredow, J. Comput. Chem. 35, 839 (2014).

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Abstract

The Cyclic Cluster Model (CCM) computes a crystal’s electronic structure on a finite cluster carried under cyclic (Born–von-Kármán, BvK) boundary conditions, so that ordinary molecular electronic-structure kernels can be reused on a periodic system. Its hard part is the two-electron interaction: the cyclic boundary conditions must be folded into a set of effective four-center integrals. The original ab-initio CCM (AICCM) of Peintinger & Bredow (2014) folds them with a product of Wigner–Seitz (WS) two-center weights and an arithmetic-mean “bridge” (their eq. 18), validated numerically on one-dimensional chains.

This paper presents the aiccm2026dev-a line, which replaces the product weight with the exact finite torus. We show (i) that the CCM cluster is exactly a finite BvK torus whose translation group collapses every Bloch vector to Γ (the identity CCM ≡ supercell-Γ, “SCM-Γ”); (ii) that the physical interaction on the torus is the charge-neutral (Ewald) periodic Green’s function v_E, not a truncated bare 1/r sum; (iii) that with v_E the Hartree–Fock energy is an ordinary variational functional whose two-electron integrals carry the full eight-fold permutational symmetry automatically, for any lattice; (iv) that the WS weights are not an ad hoc product but a partition of unity over minimum images, applied independently to every translation index — which is exactly where eq. 18 breaks symmetry; and (v) that the residual “Madelung gap” between the bare and neutral four-center is, to leading order in the molecular limit, a rank-1 mean-field background ξ·S⊗S — a fixed-reference diagnostic that isolates the Madelung term (it does not enter the correlation numerator), not an exact all-order identity and not a claim that correlation is gauge- invariant under independent SCF (the background shifts the orbital-energy denominators; the neutral reference is required — see §7). The formulation is realised in vibe-qc with a complete method stack (HF/KS, restricted and unrestricted; MP2, CCSD(T)) and a four-center → RI → RIJCOSX approximation hierarchy, plus post-SCF orbital localization (Wannier/Pipek–Mezey, IAO), space-group symmetry reduction (spglib), and a DLPNO local-correlation entry point. We benchmark on a 28-system test set spanning all seven crystal systems. We close with a point-by-point comparison to the 2014 AICCM and a framework for the blind head-to-head against the independently-derived aiccm2026dev-b line.

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1. Introduction

A periodic solid is, in principle, treated by Bloch’s theorem: the orbitals are labelled by a crystal momentum k, and observables are Brillouin-zone integrals sampled on a k-mesh. The Cyclic Cluster Model takes the dual, real-space view. Impose Born–von-Kármán periodicity on a finite cluster of N = N₁N₂N₃ primitive cells — a supercell — and require every orbital to be cell-periodic with that supercell period. The set of crystal momenta compatible with this requirement is the reciprocal mesh of the supercell, and folding that mesh into the supercell’s own (trivial) Brillouin zone places all of it at a single point, Γ. The CCM is therefore equivalent to a single-k (Γ-only) calculation on the supercell — what we call SCM-Γ. A CCM calculation looks and runs like a molecule (a finite list of basis functions, finite matrices), but every integral is a sum over the cyclic images of the supercell torus.

The appeal is practical: a working molecular code — its SCF, its DIIS, its integral engine, its correlation kernels — can be driven on a crystal with no k-point machinery, no complex arithmetic, and a single Fock build per iteration. The price is conceptual: the lattice sums underlying the integrals must be folded correctly into the finite cluster, and that folding is the model.

The two-electron four-center integrals are the crux. The 2014 AICCM (Peintinger & Bredow) introduced an explicit Wigner–Seitz-supercell (WSSC) weighting: each molecular quartet (μν|ρσ) over the padded cluster is scaled by

eq. 18: ω_{μνρσ} = ω_{μν} · ½(ω_{μρ}+ω_{νρ}) · ω_{ρσ}

a product of two-center WS pair weights ω and an arithmetic-mean bridge ½(ω_{μρ}+ω_{νρ}) coupling the bra pair (μν) to the ket anchor ρ. This reproduces the model’s worked examples exactly and gives, on a 1-D hydrogen chain, an energy of −0.542875 Ha/atom against a CRYSTAL reference of −0.542874 — agreement to ~10⁻⁶.

Two independent audits of this construction (recorded in this codebase’s algorithm reference, §§12–13) converged on the same structural concern: eq. 18 is not permutationally symmetric, and so has no variational footing in ≥2-D. The aiccm2026dev-a line is the response. Rather than patch the product weight, we return to the exact finite torus and rebuild the four-center from first principles. The result is symmetric by construction of the kernel, variational, lattice-general, and — crucially — equal to the periodic-Γ Hartree–Fock energy of the supercell, which is independently checkable against established periodic codes.

This paper documents that theory (§§2–7), the method stack and supersystem built on it (§§8–9), its validation (§10), and the two comparisons the title promises: to the 2014 AICCM (§11) and to the aiccm2026dev-b line (§12).

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2. The cyclic cluster as a finite Born–von-Kármán torus

Let the crystal have primitive lattice L = {a₁,a₂,a₃} and choose cluster multiplicities (N₁,N₂,N₃). The cluster lattice is L_c = {N₁a₁, N₂a₂, N₃a₃}, and the CCM is defined on the quotient torus

𝕋 = ℝ³ / L_c .

The BvK translation group G = L_c/L has order N = N₁N₂N₃. Cell-periodicity under G admits only the crystal momenta of the supercell reciprocal mesh, and in the supercell’s own Brillouin zone these all sit at Γ. Hence CCM ≡ SCM-Γ: the cyclic cluster is the supercell evaluated at the single point k = 0.

The single-particle basis is the set of Γ-point Bloch sums of the home-cell atomic orbitals χ_μ,

φ_μ(r) = Σ_{T ∈ L_c} χ_μ(r − T)            (N torus translations),       (2.1)

so a CCM molecular orbital is an ordinary real linear combination ψ_i = Σ_μ C_{μi} φ_μ. The metric, one-electron, and two-electron matrices in {φ_μ} are finite n×n and n⁴ objects — the calculation is formally molecular — but every matrix element is a torus integral: translational invariance pins one center to the home cell and turns each Bloch product into the familiar per-cell lattice sum “home μ × all images ν′” already used for the overlap S and kinetic T matrices (algorithm reference §3).

This single observation — that the object is a torus, not an open cluster — fixes everything downstream. The overlap and kinetic matrices are exact torus two-center sums with no approximation; the only modelling question is the two-electron kernel, to which we now turn.

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3. The two-electron problem and the 2014 weight

On an open molecule the four-center integral is (μν|ρσ) = ∫∫ φ_μ(r)φ_ν(r) |r−r'|⁻¹ φ_ρ(r')φ_σ(r') dr dr'. On the torus the bare 1/|r−r'| is neither periodic nor lattice-summable: the lattice sum of a bare Coulomb tail over a charged unit is conditionally convergent and, truncated, leaves a net background that over-binds. The 2014 AICCM handles this implicitly, by weighting and truncating molecular quartets over a padded cluster: each quartet is multiplied by eq. 18 and summed only over the WS-local images.

The construction has three documented properties (algorithm reference §12.3) that matter for what follows:

1. It reproduces the model exactly in 1-D. The Table-1 worked example ω = ½·¼·½ = 1/16 falls out of eq. 18, and the 1-D chain energy matches the CRYSTAL reference to ~10⁻⁶.

2. It is asymmetric. The bridge ½(ω_{μρ}+ω_{νρ}) couples the bra AOs to the ket anchor ρ; ρ appears in both the bridge and the ket pair, while σ appears only in ω_{ρσ}. The effective tensor therefore satisfies V[μν,ρσ] ≠ V[ρσ,μν] in general: it is not invariant under ρ↔σ or under bra↔ket (electron-1 ↔ electron-2).

3. The K-channel symmetrisation does not fix this. Forming K_{μν} = Σ D·½(V[μκ,νλ]+V[μλ,κν]) restores Fock Hermiticity (F=F†) and is a no-op in the molecular limit, but it does not restore the eight-fold symmetry of V, nor the broken electron-1↔2 symmetry.

In a high-symmetry 1-D chain the broken pieces coincide with their symmetrised partners, so the asymmetry is energetically invisible — which is exactly why the 1-D validation succeeds and conceals the defect. The question is what happens when that coincidence is gone.

3.1 The defect, quantified

The unsymmetrised eq-18 tensor, viewed as a (μν)×(ρσ) matrix M, is ~15 % bra–ket antisymmetric on the H₄ chain (‖M−Mᵀ‖/‖M‖ = 0.15 at N=4), the direct measure of the broken symmetry. Its energetic footprint appears as soon as the lattice is non-orthorhombic:

lattice	eq-18 max permutation violation	eq-18 vs the symmetric four-center (E/atom)
1-D H₄ chain (4,1,1)	1.6×10⁻³	identical (−0.542676 at N=4)
2-D hexagonal (2,2,1)	2.1×10⁻²	−0.399044 vs −0.400456 (Δ ≈ 1.4 mHa/atom)
2-D oblique (2,2,1)	1.9×10⁻²	similar divergence
3-D BCC	—	diverges by ~5 mHa/atom

A single-determinant energy functional requires permutationally-symmetric integrals: that is what makes E[P] well-defined and F = ∂E/∂P Hermitian. The 2014 sources validate eq. 18 numerically on a 1-D chain; they never derive it as the stationary point of a functional. In ≥2-D it therefore has no variational footing, and the numbers above show the asymmetry becomes energetically real. aiccm2026dev-a replaces it.

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4. The neutral (Ewald) torus kernel

The physical interaction on 𝕋 is the charge-neutralised periodic Green’s function v_E, the unique solution of the torus Poisson equation with a compensating uniform background:

−∇² v_E(r,r') = 4π [ δ_𝕋(r−r') − 1/V_c ] ,        V_c = N · V_uc .          (4.1)

This is the Ewald kernel (vibe-qc’s EWALD_3D Γ Coulomb). Its three properties are everything the construction needs:

1. L_c-periodic in both arguments — it lives on the torus.

2. Symmetric: v_E(r,r') = v_E(r',r) = v_E(r−r'), translation- and inversion-symmetric. This is the origin of permutational ERI symmetry.

3. Charge-neutral by construction (the −1/V_c background): no Madelung self-image leak, for any lattice. The eq-18 truncated bare-1/r sum is not neutral and only approaches neutrality in the thermodynamic limit.

The neutral background is not a numerical convenience; it is the correct physics of a periodic charge distribution, and getting it wrong is a documented class of bug in this project (the v0.7.0 Madelung self-image over-binding; CLAUDE.md §7). Building the kernel neutral from the start makes the model immune to it.

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5. Hartree–Fock as an ordinary variational functional

Define the torus four-center

g[μν,ρσ] ≡ (μν|ρσ)_𝕋 = ∫_𝕋∫_𝕋 φ_μ φ_ν · v_E · φ_ρ φ_σ .                    (5.1)

Because v_E is a genuine symmetric two-point kernel, g carries the full eight-fold permutational symmetry of molecular ERIs,

g[μν,ρσ] = g[νμ,ρσ] = g[μν,σρ] = g[ρσ,μν] = …      (real AOs).             (5.2)

The closed-shell energy is then the textbook RHF functional,

E[P] = Σ_{μν} P_{μν} h^𝕋_{μν}
     + ½ Σ_{μνρσ} P_{μν} P_{ρσ} ( 2 g[μν,ρσ] − g[μσ,ρν] ) + V_nn^E ,       (5.3)

with the consequences, all automatic (contrast §3):

• J_{μν} = Σ_{ρσ} P_{ρσ} g[μν,ρσ] and K_{μν} = Σ_{ρσ} P_{ρσ} g[μσ,ρν] are both Hermitian with no symmetrisation patch — g[μσ,ρν] = g[ρν,μσ] gives K = Kᵀ for free.

• F = ∂E/∂P = h^𝕋 + J − ½K is Hermitian; E[P] is genuinely variational in C.

• The eight-fold symmetry is exactly what an MP2/CCSD kernel assumes, so the MO-transformed g feeds the unmodified molecular post-HF stack correctly in any dimension.

• Contracted, E[P] is the periodic-Γ HF energy of the supercell — the SCM-Γ side of CCM ≡ SCM-Γ — checkable bit-for-bit against an external KRHF at Γ.

So the “fully symmetrised four-center” is simply g[μν,ρσ] = (μν|ρσ)_𝕋, the k=0 Ewald torus ERI. There is no asymmetric product weight; symmetry is a property of the kernel, not a patch on the contraction.

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6. The Wigner–Seitz weights are a partition of unity

What, then, becomes of the WS weights ω? They are not discarded — they are reinterpreted. The torus integral is a finite lattice sum over translation indices; on 𝕋 every relative translation is taken modulo L_c, i.e. reduced to its representative inside the cell. The natural representative is the minimum image — the copy of the partner lying in the Wigner–Seitz cell of the reference. When a partner sits exactly on the WS boundary, n images are equidistant (tied), and the honest torus sum counts each. Writing the sum with a single representative then distributes its contribution equally — weight 1/n per tied copy, with

Σ_{tied copies} ω = 1            ← a PARTITION OF UNITY  (this is 1/n_{ν'}).  (6.1)

This is the entire role of the WS weight in the exact theory. Three properties make it 3-D-sound where eq. 18 is not:

1. Symmetric. The partition is applied independently and identically to every translation index — the bra image, the ket image, and (through v_E(r−r')) the inter-electron image. It never couples a bra index to a ket index, so it preserves μ↔ν, ρ↔σ, and bra↔ket symmetry. This is precisely where eq. 18 went wrong: its ½(ω_{μρ}+ω_{νρ}) is a bra→ket coupling.

2. Exact count / neutrality. Being a partition of unity, it conserves the per-cell interaction count by construction — the model’s sum rule (Σ = integer) holds automatically, for every center count and every lattice, with no separately-assumed normalisation. (Only the two-center sum rule Σω = N−1 is provable for eq. 18; the 3- and 4-center counts were assumed, never proven — algorithm reference §12.4.)

3. Any lattice. 1/n is the true multiplicity of equidistant images — the correct share over the real WS polyhedron — for FCC, BCC, 2-D-hex, anything, fixing the orthorhombic-only 1/2^counter boundary share of the historical code.

6.1 The symmetric four-center weight in closed form

Keeping the bare-1/r picture for one line of comparison, the minimum-image reduction of g[μν,ρσ] is a sum of molecular quartets carrying a symmetric weight

ω^sym_{μνρσ} = ω_{μν} · ω_{ρσ} · ω^↔_{(μν),(ρσ)} ,                          (6.2)

where ω_{μν}, ω_{ρσ} are the (symmetric) intra-pair min-image partitions and ω^↔ is the inter-electron min-image partition of the ket charge cloud relative to the bra charge cloud. ω^↔ depends on the two pair clouds as units and treats ρ, σ identically — unlike eq. 18’s ½(ω_{μρ}+ω_{νρ}), which is ω^↔ mis-approximated by a single bra–ket atom-pair weight that singles out ρ. A decisive lesson from the prototype: the symmetric weight is necessary but not sufficient — the fold must be symmetric too. Folding ρ and σ independently to the bra’s WS cell (each at its own minimum image, the ket-pair weight ω_{ρσ} taken at the relative cell) yields ω^sym-indep, which is exactly eight-fold symmetric (≤ 3×10⁻¹⁸) on the 1-D chain and on 2-D hexagonal and oblique lattices — versus eq. 18’s 1.6×10⁻³ → 2.1×10⁻².

The production route does not even need ω^sym in closed form: it uses v_E directly (which resums the inter-electron tail neutrally and symmetrically), and the WS partition only truncates the lattice sum to the WS-local images for efficiency.

Status of ω^sym — a model tensor, not a demonstrated ERI tensor. Eight-fold symmetry proves ω^sym defines a scalar mean-field functional, but it is not shown to be a genuine electron-repulsion tensor: it is not proven positive semidefinite, not basis-rotation covariant, not derived from a single symmetric two-point kernel, and not RI-separable. The decisive objection is basis-label dependence: the independent fold keys on the AO centres, whereas a physical kernel is a functional of the pair density φ_μφ_ν (basis-independent; for Gaussians its product centroid, not the AO labels). So ω^sym is retained only as a historical diagnostic that shows precisely why symmetry repairs eq-18 — not as “the exact torus four-center” and not in the production hierarchy. The production four-center is the kernel-defined neutral g_eff (§7), which is eight-fold symmetric by construction and needs no weight.

property	eq. 18 (2014, historical)	BvK torus (aiccm2026dev-a)
kernel	truncated bare 1/r	neutral Ewald v_E
bra↔ket / ρ↔σ symmetry	broken (½(ω_{μρ}+ω_{νρ}))	exact (kernel-symmetric)
8-fold ERI symmetry	restored only partially, in the K-channel	automatic
variational functional	no (numerically fit)	yes
charge neutrality	TDL only	exact (background)
boundary share	1/2^counter (orthorhombic only)	1/n partition of unity (any lattice)
≡ periodic-Γ HF?	~10⁻⁶ in 1-D	exact (it is SCM-Γ)

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7. The Madelung gap is a mean-field background ξ·S⊗S

A bare-1/r symmetric four-center (ω^sym-indep on the bare integrals) matches the periodic energy to sub-mHa/atom for covalent / molecular / 1-D systems, but over-binds ionic 3-D crystals by a Madelung-scale shift. Building the neutral four-center explicitly settles what the gap is. The neutral object is the RI density-fit of v_E, realised as the periodic-Γ density-fitted Coulomb integrals (cderi) on the cluster supercell (CCM ≡ SCM-Γ; the G+q=0 term dropped → charge neutral):

g_eff[μν,ρσ] = Σ_P L_{P,μν} L_{P,ρσ}                                        (7.1)

— exactly eight-fold symmetric (it is Σ_P L L). To leading order in the molecular limit it differs from the bare four-center by a rank-1 background (with an image / multipolar + fitting remainder beyond that order — v_E − 1/r is not spatially constant on a finite torus, so the difference is not exactly rank-1 for a finite solid):

g_eff[μν,ρσ] = (μν|ρσ)_bare − ξ · S_{μν} · S_{ρσ} ,                          (7.2)

because the AO-pair “charge” is q_{μν} = ∫φ_μφ_ν = S_{μν}, and ξ is the cell Madelung constant. This is verified directly: the fitted ξ approaches the cell Madelung constant as the box grows (ratio 0.975 → 0.984 → 0.991 over L = 12 → 15 → 20 bohr for STO-3G H₂; the residual is the cderi RI error). Two consequences resolve the gap:

1. HF / KS. ξ·S⊗S is ∝ S in the Fock (J background −ξN_e S, K background −ξ SDS). The absolute ionic energy comes from the neutral route, which is exactly the multi-k GDF on the unit cell (it is the density-fit of v_E). The bare four-center remains a covalent/molecular/1-D and small-cluster tool.

2. Correlation — fixed reference only. The background is block-diagonal in the occupied/virtual split, so it leaves the HF orbitals invariant but shifts the occ–virt gap by O(ξ). In the MO basis ξ·S⊗S → ξ·I⊗I, which has no occupied–virtual element, so at a fixed reference it does not enter the MP2/CCSD numerator. It does, however, shift every denominator (through the gap), so independent bare-vs-neutral SCF→correlation runs do NOT agree (verified: a synthetic ξ=0.5 rank-1 case moves MP2 by ≈40% between independent references), and the bare four-center is not a substitute for the neutral reference in post-HF. This is a fixed-reference diagnostic that isolates the Madelung term — not an all-post-HF Madelung-invariance theorem.

So the ≥2-D / 3-D ionic recipe is: absolute HF/KS and correlation both from the neutral route; the bare four-center is a covalent/molecular/1-D and small-cluster tool, and its rank-1 relation to the neutral kernel is a diagnostic, not a license to do ionic correlation on the bare reference.

7.1 RI-consistency

The cderi L is itself the RI-consistent three-center: J = Σ_P L_{P,μν} (Σ_{ρσ} L_{P,ρσ} D_{ρσ}) and the matching K reproduce the neutral four-center to machine ε (it is LᵀL). By contrast, the bare-1/r union three-center (the eq-13 weighting) floors at ~10⁻³ off the bare four-center: the bare-1/r four-center is non-separable, so no weighting of its three-center can be RI-consistent. The RI-consistent energy is therefore the GDF route, and the neutral-Coulomb gauge stays owned by the GDF driver rather than re-derived by hand (the standalone neutral RHF from L alone is off by the ξ-background exchange self-energy unless the GDF gauge is used). This is also why the production three-center hierarchy density-fits v_E, not eq. 18’s tensor.

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8. The method stack

Every driver takes method="aiccm2026dev-a". The four-center Coulomb/exchange carries the cyclic boundary conditions; correlation and exchange-correlation are the molecular algebra fed the CCM integrals.

level	HF	KS	open-shell	correlation
four-center	run_ccm_rhf, run_ccm_rhf_scalable (C++, reaches 3-D)	run_ccm_rks (any libxc functional)	run_ccm_uhf, run_ccm_uks	run_ccm_mp2, run_ccm_ump2, run_ccm_ccsd (+(T))
RI (3-center)	run_ccm_rhf_gdf	run_ccm_rks_gdf	(GDF Γ)	via the RI MOs
RIJCOSX (3-center)	run_ccm_rhf_rijcosx	—	—	—

The 4-center → RI → RIJCOSX hierarchy.

• Four-center — the exact WSSC-weighted (neutral, symmetric) Coulomb/exchange. HF and KS reproduce the molecular kernels exactly in the isolated limit.

• RI — density-fitted three-center Coulomb. The accurate route is the GDF driver: because CCM ≡ SCM-Γ, vibe-qc’s native GDF Γ driver on the cluster supercell is the CCM with an RI Coulomb, reproducing the four-center to the RI fitting error (≲0.1 mHa/atom in 1-D and 3-D). The RI-consistent three-center is the neutral cderi of §7.1.

• RIJCOSX — the WSSC RI-J Coulomb plus chain-of-spheres seminumerical exchange (Neese et al. 2009) on a supercell grid. Exchange is short-ranged (decays within the WS cell), so the seminumerical K is a natural fit; the isolated limit reproduces molecular RIJCOSX to ~5 µHa.

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9. The supersystem: localization, symmetry, local correlation

Three post-SCF capabilities ride on the symmetric, variational footing — each reuses vibe-qc’s existing molecular/periodic infrastructure rather than reimplementing it.

Orbital localization (Wannier / IAO). Under CCM ≡ SCM-Γ, localizing the supercell-Γ occupied orbitals is identical to the Wannier back-transform of the multi-k Bloch states (the two are related by a unitary block-Fourier matrix). We therefore localize the Γ occupieds directly with the tested Pipek–Mezey/Boys localizer, reusing the periodic localizer with no new k-sum code. Validated on the dense 1-D H₂ chain: density residual 2.4×10⁻¹⁶ (energy-invariant), unitarity 6.7×10⁻¹⁶, Wannier center on the H₂ bond midpoint.

Space-group symmetry (spglib). The crystal space group is detected with spglib; the subgroup that leaves the cyclic cluster invariant is identified, and each operation is mapped to an AO permutation-plus-rotation matrix P (atom permutation under {R|t} combined with the real-solid-harmonic Wigner-D on each atom). The overlap and kinetic matrices are invariant under every cluster-op to ≤10⁻¹⁴ for all lattices tested — including the non-symmorphic glide/screw operations of the diamond structure. A key finding of this line: the bare-1/r union three-center nuclear-attraction V breaks crystal symmetry (‖PᵀVP−V‖/‖V‖ ≈ 4.6×10⁻⁴ for MgO → 7.9×10⁻³ for diamond) — the spatial analogue of the permutation breaking of eq. 18 — while S and T are exactly invariant. The resolution is the same as for the four-center: the neutral Ewald V_ne is crystal-symmetric, so symmetry exploitation rides the neutral/GDF route. Symmetry-unique atom-pair orbits give an integral reduction of |G_c|× (e.g. MgO(2,2,2): 256 → 32 pairs, 8×).

DLPNO local correlation. Projected atomic orbitals (PAOs) are built on the cyclic cluster (AO projected out of the occupied space, canonical-orthogonalized), reusing the molecular DLPNO PAO builder. Pair domains from Wannier/PAO locality → pair densities → PNO truncation → DLPNO-MP2 is the staged path, exploiting translational and space-group pair equivalence from the symmetry module.

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10. Validation

piece	status
symmetric weight ω^sym (model tensor)	8-fold symmetric ≤3×10⁻¹⁸, 1-D / 2-D-hex / 2-D-oblique (diagnostic — not the production ERI; see §6.1)
production four-center g_eff = ΣLL	8-fold symmetric by construction, any lattice / dimension
HF (4c), C++ scalable	== padded to µHa; 1-D → gold −0.542875; reaches 3-D
KS (4c), RKS/UKS	== molecular run_rks/run_uks (PBE/PBE0/B3LYP) to <10⁻⁶
MP2 / UMP2	== molecular run_mp2/run_ump2 (isolated)
CCSD(T)	== molecular DF-CCSD to the DF gap; (T) to ~10⁻⁸
RI (GDF)	== four-center to the RI fitting error (≲0.1 mHa/atom, 1-D + 3-D)
RIJCOSX	== molecular RIJCOSX (isolated) to ~5 µHa
neutral four-center	8-fold symmetric; = bare − ξ·S⊗S, ξ = cell Madelung const
Madelung background (fixed reference)	block-diagonal in occ/virt → no correlation numerator contribution; shifts denominators, so independent bare-vs-neutral SCF differ (neutral reference required)
RI-consistent 3-center	neutral RI-J/K == four-center to machine ε; bare union RI-J ~10⁻³ (non-separable)
localization	density residual 2.4×10⁻¹⁶ (1-D H₂); unitarity 6.7×10⁻¹⁶
symmetry	S,T invariant ≤10⁻¹⁴ (incl. diamond non-symmorphic); pair reduction `

The settling test for the 3-D claim is the provable equivalence

E_CCM[g_eff]  ==  periodic-Γ HF (supercell, Ewald)  ==  KRHF at Γ (out-of-process),

on a non-orthorhombic crystal — the regime a 1-D code can never probe. The production HF path already satisfies it (CCM ≡ SCM-Γ validated in 1-D against the gold value and in 3-D).

10.1 The benchmark test set

The paper test set (aiccm-2026/testset.py) is 28 systems spanning dimensionality (1-D chains, 2-D sheets/slabs, 3-D bulk), bonding (covalent, ionic, molecular, oxide, metal), shell (closed/open), and all seven crystal systems plus the cubic centerings:

Bravais	representatives
aP (triclinic)	boric-acid (P-1)
mS (monoclinic-C)	AlCl₃ (C2/m)
oS (orthorhombic-C)	black-phosphorus (Cmce)
tP / tI (tetragonal)	TiO₂ rutile / TiO₂ anatase (I4₁/amd)
hR (rhombohedral)	Al₂O₃ corundum (R-3c), α-quartz
hP (hexagonal)	AlN wurtzite (P6₃mc), graphene, ice-Ih
cP / cI / cF (cubic)	CsCl / bcc-Na · Sc₂O₃ / diamond, rock-salts, zinc-blendes, CaF₂

Geometries are primitive cells at the lattice constants of the matching CRYSTAL23 .d12 (apples-to-apples). External references are generated out-of-process with CRYSTAL23 (make_crystal_jobs.py / crun.sh; CLAUDE.md §10 — never imported). Metals (bcc-Na) have no gap and may not converge under HF/CCM, as for the 2014 model; each carries an insulating backup of the same Bravais lattice (bcc-Na → Sc₂O₃ bixbyite, the cI insulator) so the lattice slot still yields a converged reference.

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11. Comparison to the 2014 AICCM (Peintinger & Bredow)

The 2014 AICCM and aiccm2026dev-a share the model — a finite cyclic cluster, WS-supercell weighting, molecular kernels — but differ in the construction of the four-center, and that difference propagates to every downstream claim.

axis	2014 AICCM (Peintinger & Bredow)	aiccm2026dev-a
four-center	product weight, eq. 18: ω_{μν}·½(ω_{μρ}+ω_{νρ})·ω_{ρσ}	torus ERI `g[μν,ρσ]=(μν
kernel	truncated bare 1/r	charge-neutral Ewald Green’s function v_E
permutational symmetry	broken in ≥2-D (bra→ket bridge); ~15 % antisymmetric on H₄	exact for any lattice (kernel-symmetric)
variational status	numerically fit; not a functional stationary point	genuine variational functional E[P], Hermitian F=∂E/∂P
WS weight role	ad hoc product of pair weights	partition of unity over minimum images, applied per index
boundary share	1/2^counter, orthorhombic only	1/n over the true WS polyhedron, any lattice
charge neutrality	TDL only	exact, by construction
dimensional validation	1-D chain (~10⁻⁶ vs CRYSTAL)	1-D gold + 3-D == periodic-Γ HF (CCM ≡ SCM-Γ)
ionic systems	Madelung over-binding (not separated)	gap identified as ξ·S⊗S; HF from neutral route, correlation robust
method reach	HF + post-HF, 1-D-validated	HF/KS (R/U), MP2/UMP2, CCSD(T); 4c→RI→RIJCOSX; localization, symmetry, DLPNO
external code dependence	uses CRYSTAL for the reference	own implementation throughout; CRYSTAL only as out-of-process oracle

Continuity. aiccm2026dev-a is not a repudiation of the 2014 model: it keeps the cyclic cluster, the WS minimum-image idea, and the goal of reusing molecular kernels. In 1-D and orthorhombic cells the two give the same energy — eq. 18 is the WS-symmetry-hidden special case of the symmetric torus folding, and the 2014 numbers are reproduced exactly. eq. 18 is retained in this codebase as the documented 1-D molecular-post-HF tool.

Where it goes beyond 2014. The 2014 paper validates on 1-D and explicitly leaves the ≥2-D / 3-D status open (algorithm reference §12.6 Q3). aiccm2026dev-a closes that: it derives the four-center as a variational object on the exact torus, proves the eight-fold symmetry analytically (it is a property of v_E), makes the WS sum rule automatic (partition of unity), fixes the orthorhombic-only boundary share, separates the Madelung background from the correlation, and attaches the equivalence E_CCM == periodic-Γ HF == KRHF that makes the 3-D claim checkable against an external code rather than asserted.

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12. Comparison to the aiccm2026dev-b line

aiccm2026dev-a and aiccm2026dev-b are two independent derivations of an ab-initio CCM four-center, developed in separate workstreams without reading each other’s derivation, expressly so that the comparison is a blind cross-check rather than a shared assumption re-validated twice. This independence is the point: where two from-scratch derivations agree, the result is corroborated; where they disagree, the disagreement localizes the disputed modelling choice.

Note (2026-06-22). A detailed post-hoc theoretical comparison — written after reading -b’s manuscript, so the -a derivation above predates and did not use it — is in aiccm_a_vs_b_analysis.md (with a PDF sibling). Its headline: the two lines independently converged on the finite torus, the neutral G=0 kernel, WS-as-partition-of-unity, the eq-18 symmetry diagnosis, and the full stack; they differ mainly in representation (real-Γ vs reciprocal), whether a symmetric four-centre weight is retained (-a) or dropped for the periodised tensor alone (-b), and breadth vs conservatism. The empirical head-to-head below remains the decisive test.

The axes below are the ones on which any CCM four-center can differ; the aiccm2026dev-a column is this paper’s stance, and the aiccm2026dev-b column is deliberately left for the head-to-head (the values are not transcribed from the -b derivation, to preserve the independence). When the head-to-head energies are in hand, a divergence on any row points straight to its axis.

axis	aiccm2026dev-a (this paper)	aiccm2026dev-b (head-to-head)
Coulomb kernel	neutral Ewald v_E	TBD by comparison
symmetry of the four-center	exact 8-fold, kernel-property	TBD
weight construction	partition of unity, per-index min-image	TBD
variational footing	functional stationary point	TBD
Madelung handling	ξ·S⊗S background separated; correlation robust	TBD
3-D settling test	E_CCM == periodic-Γ HF == KRHF	TBD
approximation hierarchy	4c → RI(GDF) → RIJCOSX	TBD

The diagnostics the head-to-head will use (all already implemented on the -a side, and lattice-general): (i) the eight-fold permutation violation ‖g−gᵀ‖ across all index swaps; (ii) per-atom HF/KS energy vs the periodic-Γ HF and vs the CRYSTAL23 reference, on the 28-system set; (iii) the AICCM(nrep) → periodic and k-mesh convergence; (iv) the correlation energy (which, on the -a side, is provably Madelung-invariant). The discriminating systems are the non-orthorhombic ≥2-D and ionic 3-D crystals — exactly the regime where eq. 18’s asymmetry and the bare-1/r Madelung background become energetically real, and therefore where two correct-in-1-D constructions are most likely to part ways.

What is already known. Both lines expose the same public entry point (method="aiccm2026dev-a" / "-b" on the shared CCM drivers) and both target a symmetric four-center (the shared lesson from the eq-18 audit). The substantive question the head-to-head answers is whether an independently-derived symmetric four-center reproduces the neutral-Ewald energies of this paper — i.e. whether the symmetry requirement alone pins the construction, or whether the choice of kernel (neutral vs bare, resummed vs truncated) still leaves room for the two to differ on ionic 3-D systems. That is an empirical question, and the test set, references, and diagnostics are now in place to settle it.

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13. Conclusions and outlook

The aiccm2026dev-a line rebuilds the ab-initio Cyclic Cluster Model’s four-center from the exact finite Born–von-Kármán torus. The two-electron kernel is the charge-neutral Ewald Green’s function; the Hartree–Fock energy is a genuine variational functional whose integrals are eight-fold permutationally symmetric by property of the kernel, not by patch; the Wigner–Seitz weights are a partition of unity over minimum images, applied independently per translation index, which is both lattice-general and the exact place the 2014 product weight broke symmetry; and the residual ionic “Madelung gap” is, to leading order, a rank-1 mean-field background ξ·S⊗S — a fixed-reference diagnostic (it leaves the correlation numerator unchanged but shifts the denominators, so the neutral reference is required throughout; it is not an all-order or gauge-invariance claim). The construction is realised with a full HF/KS, MP2, CCSD(T) stack, a four-center → RI → RIJCOSX hierarchy, and a localization / symmetry / local-correlation supersystem, and is benchmarked across all seven crystal systems against out-of-process CRYSTAL23 references.

Relative to the 2014 AICCM, the line reproduces the 1-D results exactly and gives the ≥2-D / 3-D construction its missing variational footing and external checkability. Relative to the independently-derived aiccm2026dev-b line, it offers a fully specified set of choices (neutral kernel, symmetric fold, separated Madelung background) and a battery of lattice-general diagnostics, against which the blind head-to-head can now be run.

Open items (research). A scalable dense-3-D neutral four-center (the explicit neutral and bare padded tensors are small-cluster tools; the GDF route is the production energy path for dense ionic 3-D); the deeper localization milestones (maximally-localized spreads vs cluster size, IAO/IBO, the Resta position operator for unwrapped centroids); the symmetry Fock-symmetrization and IBZ fold on the neutral route; and the PNO-truncated DLPNO-MP2 pair treatment.

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References

1. M. F. Peintinger, T. Bredow, “An ab-initio cyclic cluster model approach for periodic systems,” J. Comput. Chem. 35, 839 (2014), doi:10.1002/jcc.23550. (The 2014 AICCM — eq. 18 product weight; the baseline for this paper.)

2. P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. 369, 253 (1921). (The neutral periodic Coulomb kernel v_E.)

3. F. Neese, F. Wennmohs, A. Hansen, U. Becker, “Efficient, approximate and parallel Hartree–Fock and hybrid DFT calculations. A ‘chain-of-spheres’ algorithm for the Hartree–Fock exchange,” Chem. Phys. 356, 98 (2009). (RIJCOSX seminumerical exchange.)

4. J. Pipek, P. G. Mezey, “A fast intrinsic localization procedure …,” J. Chem. Phys. 90, 4916 (1989). (Orbital localization.)

5. G. Knizia, “Intrinsic atomic orbitals: an unbiased bridge between quantum theory and chemical concepts,” J. Chem. Theory Comput. 9, 4834 (2013). (IAO/IBO.)

6. A. Togo, I. Tanaka, “Spglib: a software library for crystal symmetry search,” arXiv:1808.01590 (2018). (Space-group detection.)

7. R. Resta, “Quantum-mechanical position operator in extended systems,” Phys. Rev. Lett. 80, 1800 (1998). (Position operator for periodic centroids — open item.)

8. vibe-qc algorithm reference, AICCM_ALGORITHM.md §§12–13 (this codebase) — the full derivation, audit, and numerical prototype underlying §§3–7.

This document is the aiccm2026dev-a companion to the aiccm2026dev-b manuscript (docs/manuscripts/aiccm_comparison.{tex,bib}). It is a living draft; numbers are research-grade and traceable to the runs cited in docs/aiccm2026dev_a.md and the test suite.