AICCM `aiccm2026dev-a` follow-on - derivation document¶

D89 construction-boundary notice (2026-07-15). Γ-CCM (aiccm2026dev-a) is the union-and-weight/Wigner-Seitz integral-weighting construction; χ-CCM (aiccm2026dev-b) is the distinct finite-translation-group character construction. Neutral GDF/Bloch and real-Gamma routes are same-Hamiltonian representation controls only after a block-circulant finite Hamiltonian has been specified. They are not Γ-CCM production routes and do not prove Γ-CCM equals χ-CCM. In the historical text below, CCM == SCM-Γ and Fourier/Wannier statements apply only within the explicitly chosen Hamiltonian and occupied space. The geometry, localization, symmetry, and local-correlation measurements remain route-specific validation records; they are not cross-construction equivalence evidence.

D90 binding clarification (2026-07-16). The neutral-only UCCSD(T) and DLPNO implementations described below are A-namespace neutral fitted-torus correlation controls. Their API names, module location, and solver ancestry assign neither Γ-CCM nor χ-CCM construction identity. Historical uses of “CCM UCCSD(T)” or “CCM DLPNO” below name that cyclic-cluster control stack, not the union-and-weight construction.

Math + validation record for the three follow-on features on the aiccm2026dev-a cyclic-cluster line: (1) localization, (2) DLPNO natural orbitals, (3) space-group symmetry. Each sits behind its own option and does not change existing behavior. Decisions + open-questions log: handovers/HANDOVER_AICCM_FOLLOWON.md.

Every numerical claim here is grounded in a real run (test file named per section). SI/atomic units (bohr, hartree).

Interaction-range parameterization - the real-space dual of k-point sampling¶

A.1 The object¶

Historically the cluster is given as an explicit supercell mesh nrep = (N₁, N₂, N₃). The physically meaningful knob, however, is how far the interactions reach - a real-space cutoff R_c. The finite translation group for a cluster of size nrep has a geometrically dual Γ-centred (N₁,N₂,N₃) character mesh. Choosing the cluster by a real-space radius is therefore a mesh-density parameterization, without asserting that two different construction maps produce the same Hamiltonian. We let the user supply the unit cell + R_c and derive the minimal nrep.

A.2 What “reach `R_c`” means geometrically - the WS inscribed sphere¶

In the CCM, atom A interacts with the minimum image of atom B inside A’s Wigner-Seitz supercell (WSSC). The set of displacements captured at full weight, in every direction is the largest sphere inscribed in the WS cell. So “the interactions reach R_c around every atom” is precisely

r_in(L_c) ≥ R_c ,   L_c = {N_i a_i} the cluster lattice.

The inscribed-sphere radius equals half the shortest lattice vector, r_in = λ₁/2. Proof: x lies in the WS (Voronoi) cell iff x·R ≤ |R|²/2 for every lattice vector R; for |x| = r the binding constraint is along R̂, giving r ≤ |R|/2, minimised by the shortest R. The nearest Voronoi face is the perpendicular bisector of the shortest lattice vector. Hence the condition is

λ₁(L_c) ≥ 2 R_c .

A.3 The per-direction formula and why it is rigorous (not a guess)¶

Let d_i = V/|a_j × a_k| be the unit-cell interplanar spacing along a_i (the spacing between successive (a_j,a_k) planes; d_i = 1/|b_i| with b_i the 2π-free reciprocal vector). The cluster interplanar spacing is D_i = N_i d_i. Project any nonzero cluster vector R = Σ_i m_i N_i a_i onto the unit reciprocal direction b̂_i: since a_i·b̂_i = d_i and a_j·b̂_i = 0 (j≠i),

|R| ≥ |R·b̂_i| = |m_i| D_i   ⟹   λ₁(L_c) = min_{R≠0}|R| ≥ min_i D_i .

Therefore choosing

N_i = ⌈ 2 R_c / d_i ⌉      ⟹   D_i ≥ 2R_c ∀i   ⟹   λ₁ ≥ 2R_c   ⟹   r_in ≥ R_c .   ✓

The bound λ₁ ≥ min_i D_i makes the per-direction formula a sufficient condition for the inscribed-sphere guarantee - it can never under-shoot. It is tight (equality λ₁ = min_i D_i) for orthogonal cells, and conservative for oblique cells (where λ₁ can exceed min_i D_i, so the achieved r_in is slightly larger than R_c - never smaller). This is the rigorous content behind the heuristic N_i ≈ ⌈2R_c/d_i⌉.

A.4 The duality constant¶

A cyclic cluster of WS “diameter” 2 R_c is a BvK torus, so its allowed k-points are spaced by Δk = 2π/(2R_c) = π/R_c (2π reciprocal convention). Equivalently, per direction, N_i = ⌈2R_c/d_i⌉ = ⌈|b_i|/(π/R_c)⌉ = ⌈|b_i|/Δk⌉ - exactly the uniform-k-density rule. So:

real-space cutoff R_c   ⟺   k-spacing Δk = π/R_c   ⟺   cluster (N₁,N₂,N₃) ≡ MP mesh (N₁,N₂,N₃).

Larger R_c ⇔ denser k-mesh. A radius scan is the CCM analogue of a k-point convergence study.

A.5 Implementation + validation¶

Geometry (vibeqc.periodic.ccm.wigner_seitz): interplanar_spacings, shortest_lattice_vector_length, wsc_inscribed_radius, nrep_for_interaction_range, kspacing_for_interaction_range. API (CCMSystem): interaction_range= (bohr) / interaction_range_ang= (Å), the from_interaction_range(unit, R_c, basis, units=...) constructor, and the stored diagnostics interaction_range, wsc_inscribed_radius, kspacing_equiv; explicit nrep remains the advanced override. Scan helper (ccm_interaction_range_scan) → InteractionRangeScan (radius → energy/atom, with converged_radius(tol) and JSON as_records()).

Validated (tests/test_ccm_interaction_range.py):

shortest_lattice_vector_length / wsc_inscribed_radius match a brute-force ±6-shell oracle to ≤1e-9 on cubic / ortho / tetragonal / fcc-prim / bcc-prim / hexagonal / triclinic / vacuum-chain lattices.
Inscribed-sphere gate r_in(L_c) ≥ R_c and the per-direction bound min_i N_i d_i ≥ 2R_c hold for every (lattice, R_c) in the battery above - the numerical proof of A.3, not an assertion.
Orthogonal bound is tight: ortho(3,5,7) at R_c=7 → nrep=(5,3,2).
from_interaction_range and explicit nrep give bit-identical S^CCM and HF energy; the Å convenience matches bohr; ambiguous specs raise.
H₂-chain (6-bohr cell, STO-3G) radius scan converges monotonically; the dual k-mesh per point equals the cluster mesh.

Task 1 - Localization of the canonical crystalline orbitals (Wannier)¶

1.1 The object¶

A converged Γ-CCM SCF returns occupied orbitals ψ for its selected finite Hamiltonian on the N = N₁N₂N₃-cell supercell. They are S^CCM-orthonormal: Σ_{μν} C†_{μi} S^CCM_{μν} C_{νj} = δ_{ij}.

1.2 Localization is a unitary rotation within the occupied space¶

The textbook cell-periodic Wannier construction is the discrete back-transform over the cluster net,

w_n(r − R) = (1/N) Σ_k e^{−i k·R} ψ_{n,k}(r),     k ∈ equivalent net,  R ∈ L_c,

which produces N translationally-equivalent Wannier functions per band. The block-Fourier matrix U_{(nk),(mR)} = (1/√N) e^{−ik·R} δ_{nm} is unitary, so the Wannier set spans the same occupied space as {ψ_{n,k}}. For one already specified block-circulant finite Hamiltonian, the real-supercell occupied set and its character blocks are related by that unitary. Localizing the real-supercell occupieds therefore chooses a localized gauge within the same specified occupied space, with no explicit k-sum. This conditional same-H statement does not equate Γ-CCM with χ-CCM or a neutral GDF control.

Because the rotation C_loc = C_occ U stays inside the occupied space, the one-particle density P = Σ_i^{occ} C_i C_i† is invariant, and with it every ground-state observable, including the total energy. This is the hard correctness gate.

1.3 Gauge fixing (spread minimization)¶

The Marzari-Vanderbilt spread functional is Ω = Σ_i [⟨w_i|r²|w_i⟩ − ⟨w_i|r|w_i⟩²]. Two gauges are offered, reusing the validated molecular/periodic localizer (vibeqc.periodic_localise.localise_periodic_gamma):

Pipek-Mezey - maximize Σ_i Σ_A (Q_i^A)² (Mulliken populations). No position operator ⇒ PBC-safe in every regime, including orbitals that wrap the cluster boundary. Default.
Foster-Boys - maximize Σ_i ⟨w_i|r|w_i⟩² via the home-cell dipole integrals. Faithful (with the reported centroids/spreads) only for orbitals that do not wrap the boundary (molecule-in-a-box / dilute / large cell).

The reported centroids = ⟨w_i|r|w_i⟩ and spreads = ⟨w_i|r²⟩ − ⟨w_i|r⟩² use the plain (home-cell) position operator, so for a dense cell they fold into the home cell (a Wannier on cell R reports its in-cell center). The Resta periodic operator ⟨μ|e^{−iG·r}|ν⟩ that would unwrap them is an open item (§ decisions log); Pipek-Mezey localization itself is unaffected.

1.4 Implementation + validation¶

vibeqc.periodic.ccm.localise_ccm(ccm_result, ccm, method=...) → PeriodicWannierResult (C_loc, U, centers, centroids, spreads, charges, localization). localization_density_residual(ccm_result, w) is the gate ‖P_loc − P_canonical‖_F.

Validated (tests/test_ccm_localize.py), dense 1-D H₂ chain (6-bohr cell, STO-3G, nrep (3,1,1), 3 occupied):

quantity	Pipek-Mezey	Boys
density residual `‖P_loc−P_can‖`	2.4e-16	3.0e-16
unitarity `‖C_locᵀ S C_loc − I‖`	6.7e-16	4.9e-16
objective (initial → final)	0.644 → 1.500	109.5 → 181.5
spreads Ω_i (bohr²)	~2.39	~2.39
Wannier center z (bohr)	0.7 (H₂ bond midpoint)	0.7
real `C_loc` (centrosymmetric)	yes	yes

So the rotation is exactly density/energy-preserving (machine ε), genuinely localizes, gives real orbitals for the centrosymmetric cell, and places the Wannier centers on the H₂ bond midpoint - the expected Wyckoff/bond position.

1.5 Open (M2-M4, see decisions log)¶

Explicit multi-k U(k)/M(k,b) route; Resta operator (unwrapped centroids/spreads for dense crystals); IAO/IBO path (Knizia); aliasing diagnostics + spread-vs-cluster-size convergence; emit in the exact rep Task 2 (DLPNO) consumes.

Task 3 - Space-group symmetry (spglib)¶

3.1 The cluster-invariant subgroup¶

spglib analyzes the crystal (unit cell) space group G, giving operations {W|w} (integer rotation W in the unit-cell fractional basis + fractional translation w). Each op is converted to Cartesian, R = L_u W L_u⁻¹, t = L_u w (L_u = unit lattice, columns aⱼ). The operations usable on the cyclic cluster are the cluster-invariant subgroup G_c ⊆ G: those (R,t) that map the cluster lattice L_c = {N_j a_j} onto itself modulo L_c. For a point op, R(N_j a_j) = N_j Σ_k W_{kj} a_k ∈ L_c iff N_j W_{kj}/N_k ∈ ℤ - so for equal nrep in symmetry-related directions all crystal point ops survive; unequal nrep keeps the compatible subset. Tested operationally by symmetry_ao.atom_permutation_under_op on the supercell (it raises iff (R,t) is not a cluster symmetry).

3.2 AO representation¶

For each g ∈ G_c the AO matrix is P = (atom permutation) × (Wigner-D blocks) from symmetry_ao.build_ao_permutation_matrix(basis, R, perm) - real solid harmonics, wigner_d_real(ℓ, R) per shell, with improper rotations factored through the ℓ-parity. At Γ the fractional translation contributes phase 1, so P (permutation + rotation) is the full cluster-AO representation.

3.3 Validation + a proven finding (`tests/test_ccm_symmetry.py`)¶

The AO maps are correct: S^CCM and T^CCM are invariant Pᵀ M P = M to machine precision under every g ∈ G_c, including the non-symmorphic glide/screw ops of diamond (Fd-3m):

system	SG	crystal/cluster order	‖PᵀSP−S‖	‖PᵀTP−T‖	‖PᵀVP−V‖ (rel)
MgO (Fm-3m)	225	48 / 48	2e-15	1e-14	0.077 (4.6e-4)
C-diamond (Fd-3m, non-symmorphic)	227	48 / 48	2e-15	2e-15	0.44 (7.9e-3)
NaCl (Fm-3m)	225	48 / 48	5e-15	-	0.044

Finding. The union three-center V^CCM (bare-1/r, weight ω_μν·½(ω_μC+ω_νC)) is not crystal-symmetric - ~5e-4 (MgO) → ~8e-3 (C-diamond) relative. S and T being exactly invariant isolate V as the sole culprit (the maps are validated); the anchor-bridge breaks spatial symmetry just as eq-18 broke permutation symmetry.

Resolution (proven). The neutral Ewald V_ne IS crystal-symmetric (‖PᵀVP−V‖/‖V‖ ≈ 2e-8, lattice-sum tolerance) - last rows below:

nuclear operator	MgO rel breaking	C-diamond rel breaking
union three-center (bare-1/r)	4.6e-4	7.9e-3
neutral Ewald `V_ne`	2.3e-8	5.0e-9

This is the spatial analogue of the Madelung gap (#16): bare-1/r union weights break both permutation symmetry (four-center, fixed by §13’s ccm_eri_symmetric) and spatial symmetry (three-center V), and the neutral Ewald kernel fixes both. So the symmetric Hamiltonian already exists - h = T + V_ne with the neutral cderi four-center - and is the natural target for symmetry exploitation.

3.4 Plan (M2-M3, decisions log)¶

Symmetry exploitation rides the neutral / GDF route (symmetric h): petite- list symmetry-unique shell pairs/quartets → representative integrals → scatter via P + symmetrize Fock; fold the k-net to the IBZ with weights; the exact E/Fock == symmetry-off gate then holds because h is symmetric. (The bare-union route stays available but its h is not crystal-symmetric, so its symmetrized energy would differ - a documented limitation, not a target.)

Task 2 - DLPNO natural orbitals (PAO → PNO)¶

Foundations now in place: localized occupieds (Task 1), the symmetric neutral route + symmetry-unique pair orbits (Task 3).

2.1 Projected atomic orbitals (M1)¶

The local virtual basis is the AO space projected out of the occupied space (Pulay; Riplinger-Neese §II.D):

C_PAO = (I − C_occ C_occᵀ S^CCM) · A ,

canonical-orthogonalized within a domain (eigenvectors of Q S Q above a linear-dependence floor). The occupied space projector C_occ C_occᵀ is invariant under occupied rotations, so the full-cell PAO set is identical whether the canonical or the localized (Task 1) occupieds are used - the localized occupieds enter at the pair-domain stage (M2). ccm_pao(ccm_result, ccm) reuses vibeqc.dlpno.pao.

Validated (tests/test_ccm_dlpno.py), dense 1-D H₂ chain (STO-3G, nrep (3,1,1)): PAOs S^CCM-orthogonal to the occupied space (max|C_occᵀ S C_pao| = 2.9e-16); n_pao = 3 = nbf − n_occ (full virtual, no linear dependence); occupieds + PAOs span the full space (rank 6/6).

2.2 DLPNO-MP2 (M2, DONE)¶

ccm_dlpno_mp2(ccm, scf_result, *, cderi=, localize=, tcut_pno=, tcut_mkn=, …) → CCMDLPNOMP2Result. It mirrors the validated molecular DLPNO-MP2 (vibeqc.dlpno.mp2), reusing its component functions (build_atom_basis_map, build_projection_matrix, select_domain_atoms_mulliken, semicanonical_pao_basis) with S^CCM / F^CCM injected and the (ia|jb) integrals density-fit from the neutral cderi L[P,μν] (B_half[P,i,ν] = Σ_μ C_loc[μ,i] L[P,μν], so K_ab = Σ_P B[P,i,a] B[P,j,b]).

Pipeline per pair (i,j): Mulliken domain (tcut_mkn) → semicanonical PAOs (eigh of F in the domain, S^CCM-orthonormal) → first-order amplitudes T_ab = K_ab/(f_ii+f_jj−ε_a−ε_b) → PNOs (eigvecs of the pair density ½(TTᵀ+TᵀT), occupation-threshold tcut_pno) → quasi-canonicalize → coupled LMP2 residual R_ij = K_ij + (ε_a+ε_b)T_ij − Σ_k[F_ik P(T_kj) + F_kj P(T_ik)] with neighbour amplitudes projected via S_pq = V_pᵀ S^CCM V_q. For canonical occupieds F_oo is diagonal and the residual converges in one step; for localized occupieds the S^CCM-link coupling restores the canonical sum.

Reference = neutral (per the -b-critique correction). The Madelung background shifts the occ-virt denominators, so bare-vs-neutral correlation do not agree (§ decision log): the DLPNO-MP2 reference is the neutral SCF (run_ccm_rhf(ccm, eri=ccm_eri_neutral(ccm))) + neutral cderi. No Madelung-robustness is claimed.

Validated (tests/test_ccm_dlpno_mp2.py, 11/11; H₂-chain STO-3G).

gate	result
no-truncation, canonical occupieds	`Δ = 0.0` vs canonical CCM MP2 (neutral)
no-truncation, Pipek-Mezey occupieds	`Δ = 8.5e-15` (coupled LMP2, 4 iters)
no-truncation, Boys occupieds	`Δ = 8.5e-15`
no-truncation, (4,1,1) = 10 pairs	`Δ ≲ 1e-13` (coupled iteration at scale)
PNO truncation `tcut_pno` 1e-4→1e-8	monotone error 5e-6 → <1e-7; `e_pno_correction` tracks the gap

The no-truncation limit reproducing canonical CCM MP2 to machine ε - for canonical and localized occupieds - is the hard correctness proof (not an assertion).

2.3 DLPNO-CCSD(T) (M3, DONE)¶

ccm_dlpno_ccsd(ccm, scf_result, *, cderi=, g_neutral=, localize=, tcut_pno=, tcut_mkn=, compute_triples=, n_frozen=) → CCMDLPNOCCSDResult (vibeqc.periodic.ccm.dlpno_ccsd). Subspace-projected DLPNO-CCSD(T): the per-pair PNOs (_build_ccm_pno_pairs, shared with MP2) are merged into a single union virtual space V_trunc, and vibe-qc’s own CCM CCSD engine (ccsd.py _ccsd_iterate + _triples) runs in {occ, V_trunc}.

Union space. Stack the pair PNO coefficients into W, canonical-orthogonalize against S^CCM (eigvecs of WᵀS^CCM W above pno_lindep), then semicanonicalize (diagonalize the Fock within the space) → V_trunc, ε_v.
Why exact at no truncation. The CCSD engine assumes a diagonal Fock, so we use the canonical occupieds (occ-occ block diagonal) and the semicanonical V_trunc. Every PNO is built from PAOs projected out of the full occupied space, so V_trunc ⊥_S occ and the occ-virt Fock block vanishes: the MO set C_sub = [C_occ | V_trunc] is orthonormal with a block-diagonal Fock. At tcut_pno = tcut_mkn = 0 each pair’s PNOs span the full virtual block, so the union does too - V_trunc is then a unitary rotation of the canonical virtuals, and CCSD(T) (invariant to occ/virt unitary rotations) equals the canonical CCM CCSD(T).
Reference = neutral (the g_eff = Σ_P L⊗L four-center, per the decision log): both the SCF reference and the CCSD MO integrals.

Validated (tests/test_ccm_dlpno_ccsd.py, 9/9; H₂-chain STO-3G).

gate	result
no-truncation, canonical occupieds	`Δtot = 7e-18`; CCSD and (T) each == canonical
no-truncation, Pipek-Mezey	`Δtot = 0.0`
no-truncation, Boys	`Δtot = 7e-18`
no-truncation, (4,1,1)	`Δtot = 3e-16`
PNO truncation	`n_vpno ≤ n_full`; (T)-toggle / total consistency

The no-truncation == canonical CCM CCSD(T) limit (CCSD + (T), all gauges) is the hard correctness proof.

Scope note (honest). The union-PNO space truncates the global virtual space, so it shrinks only when a virtual direction is unimportant to every pair; on the small, well-coupled validation chains the union stays full-dimensional (correct, not a bug). Per-pair-coupled DLPNO-CCSD (separate per-pair PNO spaces, the more aggressive truncation) is the M3b refinement.

2.4 Open (M3b-): per-pair-coupled DLPNO-CCSD; symmetry-unique pairs¶

(i, jR) from Task 3’s orbit reduction; open-shell U-DLPNO (Task D). Refs: Nejad et al. 2025; periodic DLPNO literature.

Task C - Derivable properties on the BvK torus¶

vibeqc.periodic.ccm.properties. Which one-particle observables are well defined on a neutral finite Born-von-Karman torus, and which need care - implemented in priority order of how cleanly they are defined.

C.1 HOMO-LUMO / fundamental gap (well defined)¶

ccm_homo_lumo_gap(scf_result, ccm) → CCMGap. It reports gap = ε[n_occ] − ε[n_occ−1] for the selected finite Hamiltonian. A block-circulant Hamiltonian has the same spectrum in its real-supercell and character representations, but a bulk-gap interpretation requires route-specific convergence. Validated: equals the raw-spectrum gap to machine ε; H₂-chain (3,1,1) finite-cluster gap 1.14 Ha.

C.2 Mulliken / Löwdin populations (well defined)¶

ccm_mulliken_charges / ccm_lowdin_charges → CCMPopulation. Basis-local partitions of the cluster density with the cyclic overlap S^CCM: q_A = Z_A − Σ_{μ∈A}(D S^CCM)_μμ (Mulliken) and the S^{1/2} D S^{1/2} analogue (Löwdin). Charges sum to the cluster charge (≤1e-9). The per-cell charge is the translational image-average; translational_spread (max image-to-image variation) is a symmetry diagnostic - ≈ 6e-4 for the homonuclear H₂ chain, but ~0.3 for the ionic LiH chain on the bare four-center, concrete evidence that ionic systems need the neutral reference (the bare-1/r four-center breaks cyclic symmetry for ionic cells; § 2.0 decision log). LiH per-cell charges have the correct sign (Li⁺ / H⁻).

C.3 Electric dipole (care needed - Resta)¶

ccm_dipole(scf_result, ccm) returns the finite-cluster electric dipole μ = Σ_A Z_A R_A − Tr(D ⟨r⟩). This is not the bulk polarization: the position operator is not legitimate on a torus (Resta 1998) and the plain home-cell ⟨μ|r|ν⟩ wraps for boundary-crossing orbitals (the Wannier-centroid aliasing of § 1.3). Faithful only for non-wrapping (dilute) clusters; for a dense crystal it is a symmetry indicator (≈ 0 for centrosymmetric/neutral - verified ≤1e-8 on the H₂ chain). Bulk polarization needs the Berry-phase / Resta operator (open item).

C.4 Nuclear gradient / forces (numerical; analytic deferred)¶

ccm_numerical_gradient(ccm, *, runner=, method=, h=) → (n_cell_atoms, 3) central-difference dE_total/dR. Displacing a unit-cell atom propagates to every image (the supercell is built by replication), so this is the gradient under the cyclic constraint; the force is −gradient. Well defined and tractable (no analytic WSSC-integral derivatives), 6·n_cell_atoms SCF evaluations.

Validated: (i) the per-cell forces sum to ≈ 0 (translational invariance, exact); (ii) in the isolated limit ((1,1,1), 80-bohr box) it reproduces vibe-qc’s molecular analytic RHF gradient to 8.9e-8 (finite-difference accuracy). The analytic WSSC-weighted gradient (differentiating S^CCM, h^CCM, the neutral four-center) is the open follow-on.

C.5 Open¶

Resta periodic position operator (bulk polarization + unwrapped dipoles, shared with Task 1 M4); analytic CCM gradient; open-shell (UHF/UKS) gap + populations (Task D - populations/dipole already accept an unrestricted density via the total-density accessor; only the spin-resolved gap needs wiring).

Task D - Open-shell parity¶

The unrestricted analogues of the RHF-path methods, on the neutral reference.

D.1 UCCSD(T) (`run_ccm_uccsd`, DONE)¶

vibeqc.periodic.ccm.uccsd.run_ccm_uccsd(ccm, uhf_result=, *, cderi=, compute_triples=) → CCMUCCSDResult. The open-shell sibling of run_ccm_ccsd: it runs vibe-qc’s own spin-orbital UCCSD(T) engine (vibeqc.dlpno._ccsd_ref.run_ref_uccsd, the in-repo anchor for cpp/uccsd.cpp) on the UHF-CCM reference, fed the CCM neutral cderi transformed into the α / β MO bases (B^σ[P,p,q] = Σ_μν C^σ[μ,p] L[P,μν] C^σ[ν,q], so (pq|rs)^{σσ'} = Σ_P B^σ_pq B^{σ'}_rs) plus the diagonal canonical UHF Fock.

The neutral route is the natural one: run_ref_uccsd needs an RI factorization of the four-center, and the neutral g_eff = Σ_P L⊗L is exactly that (the bare-1/r four-center is non-separable) and, per § 2.0, is the required correlation reference for ionic systems.

Validated (tests/test_ccm_uccsd.py):

Closed-shell consistency - on an even-electron closed-shell cluster run_ccm_uccsd reproduces the closed-shell run_ccm_ccsd on the same neutral four-center to ≤1e-8 (CCSD and (T) each; the UHF collapses to RHF, αα = ββ). This exercises the full spin-orbital machinery + the α/β B-tensor transform.
Open-shell - a genuine doublet cluster (3-H chain, n_α ≠ n_β) converges with negative correlation.

D.2 Unrestricted properties (DONE)¶

The population (ccm_mulliken_charges / ccm_lowdin_charges) and dipole (ccm_dipole) routines already consume an unrestricted density (the total-density accessor sums density_alpha + density_beta). The spin-resolved gap is now wired in ccm_homo_lumo_gap: for a UHF/UKS result it returns LUMO − HOMO with HOMO = max(ε^α[n_α−1], ε^β[n_β−1]), LUMO = min(ε^α[n_α], ε^β[n_β]) (spin="unrestricted"). Validated on the doublet 3-H chain (tests/test_ccm_properties.py).

D.3 U-DLPNO (DONE - DLPNO-UMP2 + DLPNO-UCCSD(T))¶

The open-shell local-correlation analogue of ccm_dlpno_mp2 / ccm_dlpno_ccsd, on the neutral-reference UHF-CCM wavefunction. Both reuse the molecular open-shell DLPNO PNO machinery (vibeqc.dlpno.ump2._pnos) and the spin-orbital UCCSD(T) engine (vibeqc.dlpno._ccsd_ref.run_ref_uccsd, the same one run_ccm_uccsd runs) with the CCM neutral B-tensors injected.

DLPNO-UMP2 - ccm_dlpno_ump2(ccm, uhf_result=, *, cderi=, tcut_pno=, n_frozen=, ss_scale=, os_scale=) → CCMDLPNOUMP2Result (dlpno_ump2.py). The correlation is resolved into the three unrestricted-MP2 spin channels (Szabo & Ostlund Sec. 6.7, as in run_ccm_ump2):

same-spin (αα, ββ): E_σσ = Σ_{i<j} Σ_ab ½ |⟨ij||ab⟩|² / D, antisymmetrised exchange ⟨ij||ab⟩ = (ia|jb) − (ib|ja), one PNO set per pair;
opposite-spin (αβ): E_ab = Σ_{ij} Σ_ab (ia|jb)² / D, Coulomb only, separate α-PNO and β-PNO set per pair.

Every (ia|jb) is density-fit from the neutral cderi (g_eff = Σ_P L⊗L); each pair carries PNOs built from its pair density and truncated by occupation number (tcut_pno).

DLPNO-UCCSD(T) - ccm_dlpno_uccsd(ccm, uhf_result=, *, cderi=, tcut_pno=, n_frozen=, compute_triples=) → CCMDLPNOUCCSDResult (dlpno_uccsd.py). The per-channel PNO pairs are merged into a per-spin union PNO virtual space (V_trunc^α from the αα pairs + the α side of the αβ pairs; V_trunc^β symmetrically), each S^CCM-orthonormal and semicanonical (the Fock diagonalised within the union), so [occ^σ | V_trunc^σ] is orthonormal with a diagonal Fock (the occ-virt block vanishes - every PNO is a rotation of the canonical virtuals, which are S^CCM-orthogonal to the occupieds). run_ref_uccsd then solves the unrestricted CC within {occ^σ, V_trunc^σ}. One implementation detail: the spin-orbital engine indexes α/β spatial MOs by a single shared range, so the smaller per-spin truncated MO set is padded with extra (real, semicanonical) complement virtuals - exact-safe, since enlarging a truncated space only reduces truncation.

Why it is exact at no truncation. At tcut_pno = 0 every PNO set spans its full virtual space, so each per-spin union is a unitary rotation of the canonical σ-virtuals; UMP2 reproduces the canonical channels term-by-term, and UCCSD(T) - invariant to rotations within the occupied/virtual blocks - equals the canonical CCM UCCSD(T). This M1 rung is the cyclic-cluster transplant of the molecular open-shell DLPNO (vibeqc.dlpno.ump2/.uccsd), which builds PNOs from the canonical α/β virtual MOs; per-spin PAO Mulliken pair domains (the other half of DLPNO locality, as in the closed-shell ccm_dlpno_mp2) are the M2 follow-on - the molecular open-shell DLPNO is itself at this same rung.

Validated (tests/test_ccm_dlpno_ump2.py 7/7, tests/test_ccm_dlpno_uccsd.py 7/7): the no-truncation limit (tcut_pno = 0) reproduces

canonical CCM UMP2 (run_ccm_ump2 on the neutral four-center) to machine ε (≤2e-17 on the closed-shell (2,1,1) H₂ chain; ≤1e-17 on the genuine doublet (3,1,1) H chain, n_α ≠ n_β), per spin channel; and
canonical CCM UCCSD(T) (run_ccm_uccsd) - CCSD to machine ε (closed-shell ≤4e-17; doublet ≤8e-14), (T) to ≤6e-21.

Truncation recovers canonically: on a 6-31g doublet the UMP2 error falls monotonically (1.2e-3 → 1.0e-5 → 4.4e-7 → ε) as tcut_pno → 0, and the unequal-per-spin padding path is exercised (β padded back to full when the α union needs the larger common MO count) and stays exact-safe.

Reference = neutral (per § 2.0 / the -b-critique correction): the SCF reference and the four-center are the neutral kernel - the Madelung background shifts the occ-virt denominators and the bare-1/r four-center has no consistent cderi. Reference: Saitow, Becker, Riplinger, Valeev & Neese, J. Chem. Phys. 146, 164105 (2017) (open-shell DLPNO); Pinski et al. 2015 (DLPNO-MP2); Riplinger et al. 2013 (DLPNO-CCSD(T)); Stanton et al. 1991 (spin-orbital UCCSD) - all already in the citation DB and routed for the reused molecular components.

M2 follow-on: per-spin PAO Mulliken pair domains (tcut_mkn) + the coupled LMP2 residual for localized occupieds, mirroring the closed-shell ccm_dlpno_mp2; per-pair-coupled DLPNO-UCCSD; symmetry-unique pairs (Task 3).

AICCM aiccm2026dev-a follow-on - derivation document¶