χ-CCM / aiccm2026dev-b: an independent finite-torus derivation¶

Status and scope¶

χ-CCM[1] is the prose name for the aiccm2026dev-b development line: the finite-translation-group character approach to the variational finite-BvK-torus CCM. It is an experimental, independently derived restricted and unrestricted Hartree–Fock and Kohn–Sham method with finite-torus MP2 and coupled-cluster extensions. Its sibling is Γ-CCM, selected as aiccm2026dev-a, which uses the union-and-weight/Wigner–Seitz integral-weighting construction. χ-CCM coexists with the earlier CCM implementation and is selected with

D89 makes the B identity explicit in every finite-torus convention record: ccm_approach="chi-ccm", ccm_construction="finite-translation-group-character", and evaluation_representation="gamma-centred-character-mesh". The first two identify the approach and mathematical construction; the third identifies how that χ-defined Hamiltonian is evaluated. Its real-Gamma form is an exact finite-Fourier representation control but is not assigned a Γ-CCM identity. The fields are immutable and are carried through B diagnostics/results, fleet finite_torus_convention, QVF vendor metadata, and .system run metadata. QVF schema version 1 retains its legacy representation string for compatibility; evaluation_representation is normative. Pre-D89 records are not numerically retracted solely because these fields are absent, but they cannot support a Γ/χ construction-comparison claim.

result = vibeqc.run_periodic_job(
    system,
    basis,
    method="RHF",
    jk_method="aiccm2026dev-b",
    aiccm_lattice_extension=(4, 4, 4),
    aiccm_backend="four_center",  # or "ri" / "rijcosx"
)

The implementation does not use vibeqc.periodic.ccm and does not copy the historical program. The earlier implementation, the historical program, and published CCM formulae are comparison targets. The present definition is the finite Born–von Karman (BvK) torus Hamiltonian restricted to translation-invariant determinants. It is evaluated in the finite translation group’s reciprocal irreducible representations. This is an exact change of basis, not an appeal to reciprocal-space convergence.

The implemented phase is deliberately narrow:

RHF, RKS, UHF, and UKS (including hybrid functionals), integer occupations, neutral primitive cell;
finite-group and Wigner–Seitz constructions in 1D, 2D, and 3D, while all production SCF backends currently fail closed below 3D until the shared wire/slab Coulomb convention described below is derived;
direct corrected-gauge four-centre, pair-resolved RI, and RIJCOSX electron-repulsion backends;
the existing periodic integral, GDF, Ewald, DIIS, basis, and lattice infrastructure;
3D canonical RHF RI-MP2, unrestricted RI-MP2, full-domain UCCSD(T), and restricted/unrestricted local-PNO routes on the exact real torus;
gauge-invariant SCF one-particle properties; no charged-cell background, total analytic gradients, Berry-phase polarization, or relaxed correlated density matrices yet. Independently checkable 3D Ewald nuclear-repulsion, fixed-density Ewald electron-nuclear, fixed-density kinetic, and fixed energy-weighted overlap derivative components are exposed only as derivation aids.

The principal source studied was Peintinger and Bredow, J. Comput. Chem. 35, 839 (2014), DOI 10.1002/jcc.23550, together with Chapter 8 of Peintinger’s thesis and the earlier CCM papers cited there. The KS-ADFT CCM of Janetzko, Köster, and Salahub, J. Chem. Phys. 128, 024102 (2008), DOI 10.1063/1.2817582, was audited separately because it introduces two- and three-center multiplicative weights in deMon2k. Those sources motivate the questions; they do not define the implementation below.

1. Finite cyclic translation group¶

Let the primitive direct lattice be

[ A=(\mathbf a_1,\mathbf a_2,\mathbf a_3), \qquad \mathbf R_{\mathbf n}=A\mathbf n, ]

with only the first (d) directions active. The mathematical matrix (A) has lattice vectors as columns, exactly as PeriodicSystem.lattice does. Thus Cartesian translations are (\mathbf R_{\mathbf n}=A\mathbf n), and the BvK supercell matrix is

[ A_N=A\operatorname{diag}(N_1,N_2,N_3). ]

The finite-torus descriptor records this executable contract as lattice_vector_convention="columns", and successful fleet payloads record primitive_lattice_bohr. The B fleet audit binds the exact (A\operatorname{diag}(N)) matrix to those fields; the Γ/χ comparator defines no comparison when the binding is absent or inconsistent.

D88 supersedes the false row-lattice premises in D17 and D81. The article theory already uses the column convention correctly; the repair aligns the code and its fingerprints with that derivation and changes neither the Coulomb kernel nor the signed-Madelung convention. On the skew audit

[ A=\begin{pmatrix} 7&0.4&0.2\ 0.3&8&0.5\ 0.1&0.6&9 \end{pmatrix}, \qquad N=(2,3,1), ]

the corrected positive code convention is (\xi_M=0.138352993811598). The pre-D88 fitted helper returned (0.143621616230995), and the pre-D88 four-center probe returned (0.138913224426180). The fitted error corresponds to 5.268622 mHa/cell of overbinding for a two-electron RHF seam. Because the Madelung, probe-charge, and fleet helpers are shared, all affected pre-D88 Γ-CCM and χ-CCM records require rerun under the fingerprint rule; no missing convention field is inferred. A symmetric lattice that commutes with diag(mesh) is numerically outside this defect class, but its old record is not newly attested. The shared builders for graphene, mgo-slab, ice-ih, co2-dryice, and sio2-quartz require fresh Γ and χ fingerprints. Ordinary pre-D88 periodic GDF exact-exchange and BIPOLE J/K records for affected lattice/mesh combinations also require audit and rerun because they traverse the shared helpers. A concrete ordinary-GDF control confirms the impact: the c-diamond/sto-3g (2,1,1) fixed point moves from the pre-D88 post-D78 pin -74.6405167828 Ha to -74.4119904741 Ha. The old and corrected positive Madelung constants are 0.5078067935392725 and 0.46971907524744966; multiplying their difference by six occupied closed-shell bands predicts the measured +0.228526309751 Ha shift. D89 records that Γ-CCM and χ-CCM are distinct approaches compared at a declared common exchange-q=0 convention; their names do not select different Coulomb kernels.

For positive integers (N_i), with (N_i=1) on inactive directions, define

[ \mathcal T_N = \mathbb Z_{N_1}\times\mathbb Z_{N_2} \times\mathbb Z_{N_3}, \qquad N_c=N_1N_2N_3. ]

The BvK identifications are

[ \mathbf R_{\mathbf n+N_i\mathbf e_i}\equiv\mathbf R_{\mathbf n}. ]

This quotient is the cyclic cluster. It is not a finite open cluster and it has no surface. Its superlattice vectors are (N_i\mathbf a_i). AO labels are ((\mu,\mathbf n)), with (\mu) in the primitive cell and (\mathbf n\in\mathcal T_N).

The primary user parameter is the real-space tuple ((N_1,N_2,N_3)), called the lattice extension. Alternatively, a shell radius (s_i) requests the odd extension

[ N_i=2s_i+1, ]

whose representatives run from (-s_i) through (+s_i). The Wigner–Seitz cell of the superlattice has a half-extent (N_i/2) in primitive lattice coordinates. This is analogous to k sampling because the reciprocal net below is its exact character group. It is not an additional real-space truncation: the selected Coulomb kernel remains BvK-periodic, and its Ewald or fitted long-range terms are not discarded outside a per-atom sphere.

The characters of this finite Abelian group are

[ \chi_{\mathbf m}(\mathbf n) =\exp\left(2\pi i\sum_i \frac{m_i n_i}{N_i}\right), \qquad m_i=0,\ldots,N_i-1, ]

corresponding to the Γ-centred mesh

[ \mathbf k_{\mathbf m}=\sum_i \frac{m_i}{N_i}\mathbf b_i \pmod{\mathcal L^*}. ]

An even-mesh half shift is a different boundary condition and is therefore not permitted. Symmetry reduction is also not used: all (N_c) characters belong to the exact transform.

For any block-circulant AO matrix,

[ X_{\mu\mathbf0,\nu\mathbf R}=X_{\mu\nu}(\mathbf R), ]

the finite Fourier pair is

[ X_{\mu\nu}(\mathbf k) =\sum_{\mathbf R\in\mathcal T_N} e^{+i\mathbf k\cdot\mathbf R}X_{\mu\nu}(\mathbf R), ]

[ X_{\mu\nu}(\mathbf R) =\frac1{N_c}\sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R}X_{\mu\nu}(\mathbf k). ]

Consequently, a Gamma-point calculation on the (N_c)-cell BvK supercell, constrained to commute with primitive translations, and a primitive-cell calculation on the above full mesh are unitarily identical. This equality holds for every finite (N); it is not merely a large-cluster limit.

2. Wigner–Seitz representatives and boundary weights¶

For centres (A) and (B) with intra-cell displacement (\boldsymbol\delta_{AB}=\boldsymbol\tau_B-\boldsymbol\tau_A), an element ([\mathbf n]\in\mathcal T_N) has infinitely many representatives

[ \boldsymbol\delta_{AB}+A\left(\mathbf n+ \operatorname{diag}(N_1,N_2,N_3)\mathbf z\right), \qquad \mathbf z\in\mathbb Z^d. ]

Let (M_{AB}([\mathbf n])) be the set that minimises this Cartesian distance from centre (A). If (m_{AB,[\mathbf n]}=|M_{AB}([\mathbf n])|), the weight of each tied representative is

[ w_{AB}(\mathbf r\mid[\mathbf n])=\frac1{m_{AB,[\mathbf n]}},\qquad \sum_{\mathbf r\in M_{AB}([\mathbf n])} w_{AB}(\mathbf r\mid[\mathbf n])=1. ]

For coincident basis offsets in a one-dimensional four-cell cluster, residue 2 has representatives (+2) and (-2), each with weight (1/2). In skew cells there may be higher-multiplicity edge or vertex ties. The code solves the closest-vector problem and stops only when a smallest-singular-value bound proves that no unexamined image can tie or improve the minimum. The implementation accepts the explicit centre-pair offset, so a non-Bravais basis is not reduced to the coincident-centre special case.

Where the weights enter¶

The weights select a representative of a translation equivalence class. They do not multiply an AO merely because its atom lies on a geometric boundary. For a periodised one-electron kernel,

[ h^N_{\mu\nu}([\mathbf R]) =\sum_{\mathbf r\in M([\mathbf R])} w(\mathbf r\mid[\mathbf R])h^N_{\mu\nu}(\mathbf r). ]

All terms on the right are identical under a superlattice translation, so the weights form a partition of unity. Similarly, after fixing the first AO at the origin, a periodised four-centre integral is

[ (\mu\mathbf0,\nu[\mathbf R]\mid \lambda[\mathbf S],\sigma[\mathbf T])N =\sum{\mathbf r,\mathbf s,\mathbf t} w_{\mathbf R}(\mathbf r)w_{\mathbf S}(\mathbf s)w_{\mathbf T}(\mathbf t) (\mu\mathbf0,\nu\mathbf r\mid \lambda\mathbf s,\sigma\mathbf t)_N. ]

Again, this is an average over equivalent representatives of the same periodised integral. It therefore leaves the value unchanged. Applying the same class rule to every symmetry-related term preserves

[ (12\mid34)=(21\mid34)=(12\mid43)=(34\mid12)^*. ]

These identities are necessary for the Coulomb and exchange matrices to be functional derivatives of one scalar RHF energy.

This differs fundamentally from multiplying an ordinary free-space ERI by products of pair-dependent boundary factors. Pair-product factors generally change under ((12\mid34)\leftrightarrow(34\mid12)) and therefore do not, without an additional derivation or symmetrisation, define the usual RHF energy functional.

3. Finite-torus Coulomb Hamiltonian¶

The Coulomb kernel is periodised on the BvK supercell. In three dimensions it is the zero-average solution

[ -\nabla^2G_N(\mathbf r) =4\pi\left(\sum_{\mathbf L_N}\delta(\mathbf r-\mathbf L_N) -\frac1{\Omega_N}\right), ]

or, equivalently,

[ G_N(\mathbf r)=\frac{4\pi}{\Omega_N} \sum_{\mathbf G_N\ne\mathbf0} \frac{e^{i\mathbf G_N\cdot\mathbf r}}{|\mathbf G_N|^2}. ]

The (\mathbf G=0) coefficient is fixed to zero. The constant background in the Poisson equation makes each separated charge component finite; for a neutral electron-plus-nucleus cell the arbitrary constant cancels in the total energy. The current implementation rejects charged primitive cells rather than silently choosing a jellium convention.

The electron–nucleus, nucleus–nucleus, and Hartree terms must use this same gauge. The implementation obtains all three from vibe-qc’s common periodic Ewald/GDF path. Exchange uses the same periodised interaction. Its finite mesh (q=0) singular channel is treated with the BvK-supercell Madelung (exxdiv="ewald") correction already implemented in the periodic GDF backend. Using the primitive-cell Madelung constant here would define a different finite Hamiltonian and is incorrect.

Every executable χ-CCM result now records this finite-N convention explicitly: coulomb_kernel="3d-periodic-g0", exchange_q0="bvk-ewald", exchange_q0_applicability, the boundary model, the full character mesh, and the BvK supercell used for the Madelung constant. Applicability is active only when the actual operator has a nonzero full-range exact-exchange arm. It is therefore inactive for pure functionals and for HSE06, whose exact exchange is the finite short-range erfc kernel and has no singular q=0 seam. A matched descriptor is necessary for any future Γ-CCM/χ-CCM comparison, but it is not sufficient to establish equality of the two constructions. The current reportable approach map is empty. A strict-zero-mode full-range exchange reference is a different finite-N Hamiltonian with the same thermodynamic target, not a hidden gauge choice. The familiar rank-1 Madelung expression is only the leading molecular-limit piece of the exchange seam. At finite solid-like cells the remaining non-rank-1 contribution is ordinary finite-size physics of the same (v_E) kernel; it is not an RI error and not a gauge freedom that can be discarded independently.

The 3D direct four-center realization also requires a coherent real-space cell set for neutral charge. When the Ewald J split is active, the nuclear real-space cutoff may not extend beyond the electronic J/K cutoff: otherwise nuclear point-charge images are included without their compensating electronic charge. The current default therefore resolves from 15 bohr electronic and 25 bohr nuclear to 15/15 bohr. Fleet records serialize these post-driver values as direct_lattice_cutoffs; the field is null for RI and RIJCOSX because those numbers are not their two-electron direct-lattice cutoffs. Four-center records made before the shared clamp in 6fed8620 are revision-bound and require a rerun before quantitative use.

The finite-group construction itself applies unchanged in one, two, or three periodic dimensions. In 3D the four-center route uses the neutral Ewald J split and the Ewald exact-exchange correction above. In 1D and 2D the available direct BIPOLE fallback is a direct-truncated active-lattice kernel, not the neutral wire/slab finite-torus Green function. aiccm2026dev-b therefore blocks the lower-dimensional four-center route before SCF. The shared lower-dimensional neutral-RI/GDF mesh is also not a Coulomb kernel: pinning every transverse reciprocal component to zero removes transverse Coulomb structure and gives a sheet-like term with vacuum-padding artifacts. aiccm2026dev-b therefore blocks every lower-dimensional SCF backend before SCF. Archived 1D/2D RI and RIJCOSX numbers are failure evidence, not 3D-periodic-in-vacuum model results that may be reported as absolute energies. A common 1D wire or 2D slab Green function must be established before any backend can be re-enabled and backend differences can be called fitting errors.

Low-dimensional shared-kernel stub¶

The fail-closed production path is intentional. The next lower-dimensional kernel module should be shared by four-center, RI, RIJCOSX, HF, KS, and post-HF routes, and should provide:

a 2D slab Green function with isolated transverse boundary;
a 1D wire Green function with isolated transverse boundary;
the matching self-potential and exchange q=0 seam derived from the same kernel;
tests proving that this mixed-boundary kernel, not the 3D exchange_q0="bvk-ewald" convention, is the finite-N Hamiltonian being correlated.

Until those items exist, isolated wire/slab absolute-energy claims remain out of scope.

4. RHF energy, Fock matrix, and double counting¶

Use a spin-summed density (D(\mathbf k)). With uniform weight (w_{\mathbf k}=1/N_c), the electron count per primitive cell is

[ N_e=\sum_{\mathbf k}w_{\mathbf k} \operatorname{Tr}[D(\mathbf k)S(\mathbf k)]. ]

The closed-shell projector condition is

[ D(\mathbf k)S(\mathbf k)D(\mathbf k)=2D(\mathbf k). ]

Define the periodised Coulomb and exchange contractions in the conventional way,

[ J_{12}[D]=\sum_{34}D_{34}(12\mid34)N, \qquad K{12}[D]=\sum_{34}D_{34}(13\mid24)_N. ]

Translation conservation makes these block diagonal in (\mathbf k), with the usual momentum-transfer sums implicit in the GDF contraction. The energy per primitive cell is

[ \mathcal E_N[D]=E_{NN}^N +\sum_{\mathbf k}w_{\mathbf k} \operatorname{Tr}[D(\mathbf k)h(\mathbf k)] +\frac12\sum_{\mathbf k}w_{\mathbf k} \operatorname{Tr}\left[D(\mathbf k) \left(J(\mathbf k)-\frac12K(\mathbf k)\right)\right]. ]

Its derivative is

[ F(\mathbf k)=h(\mathbf k)+J(\mathbf k)-\frac12K(\mathbf k). ]

The factors (1/2) and (1/2) are respectively the pair-counting factor and closed-shell exchange factor. No separate “central cell” energy is added, and no Wigner–Seitz factor is applied a second time. Each finite-group translation class occurs exactly once; boundary representative weights sum to one. This is the double-counting argument.

For RKS, let (E_{\mathrm{xc}}[\rho_D]) be the libxc functional and (\alpha) the functional’s exact-exchange fraction. The finite-torus energy is

[ \mathcal E_N^{\mathrm{RKS}}[D]=E_{NN}^N +\operatorname{Tr}_w[Dh] +\frac12\operatorname{Tr}_w[DJ] -\frac{\alpha}{4}\operatorname{Tr}w[DK] +E{\mathrm{xc}}[\rho_D], ]

where (\operatorname{Tr}_w) includes the uniform finite-character weights. Its derivative is

[ F^{\mathrm{RKS}}=h+J-\frac{\alpha}{2}K+V_{\mathrm{xc}}. ]

Thus pure semilocal DFT has (\alpha=0), whereas hybrids use the same periodised exchange tensor and finite-size convention as RHF.

5. Variational statement¶

For a fixed real-space extension, basis, periodised Coulomb gauge, and nuclear geometry, aiccm2026dev-b minimises the appropriate SCF functional over

[ \mathcal V_N={D:\ D^\dagger=D,\ DSD=2D,
\operatorname{Tr}(DS)=N_cN_e,\ [D,T_{\mathbf R}]=0}. ]

Equivalently, it minimises independently represented occupied subspaces at all finite-group (\mathbf k), subject to the common electron count. The generalised Roothaan equations are

[ F(\mathbf k)C(\mathbf k) =S(\mathbf k)C(\mathbf k)\varepsilon(\mathbf k). ]

DIIS and damping may alter the path to a stationary point but not the functional. A converged solution is stationary in (\mathcal V_N); as with ordinary RHF or approximate KS-DFT, SCF alone does not prove it is the global minimum. A fully unconstrained Gamma-supercell calculation has a larger determinant space and may break primitive translation symmetry. Such a broken-symmetry result is not required to equal the primitive-cell mesh result.

SCF path controls are numerical provenance rather than terms in the finite Hamiltonian. The B fleet therefore keeps the requested Fock-mixing fraction in scf_options.fock_mixing, while AICCM2026DevBDiagnostics.fock_mixing and convergence_diagnostics.fock_mixing record the executed effective weight of the previous Fock matrix after backend defaults are resolved. In the four-center KS backend, a request of 0.0 can execute as 0.30 when DIIS is disabled. D86 changes only the accuracy of that provenance record; it does not change the SCF algorithm, energy, or finite Hamiltonian. Pre-D86 DIIS-off four-center KS records are not convergence-fingerprint-complete.

D87 resolves the direct-call input before backend dispatch. A non-None fock_mixing= keyword overrides options.fock_mixing; otherwise the options field supplies the request. The resolver validates [0, 1) and does not rewrite the caller’s mixing field. Every four_center route and multi-cell fitted RHF/RKS can execute nonzero mixing. Three-dimensional Gamma-only RI RHF accepts resolved zero but fails closed on nonzero rather than switching to the legacy molecular-limit GDF operator; the other one-cell fitted restrictions remain unchanged. UHF and UKS can execute nonzero mixing on four_center, including both phases of a spin schedule, but fitted RI/RIJCOSX UHF/UKS fail closed because their multi-k open-shell GDF loop has no previous-Fock update. An explicit keyword zero therefore overrides a nonzero options value; on a DIIS-off four-center KS route, that requested zero remains subject to the separate automatic 0.30 execution rule. On supported execution routes this can change the SCF trajectory or stationary basin, but not the finite Hamiltonian or backend mixing formula. The Gamma explicit-zero correction deliberately restores the declared operator instead of preserving the old route-selection bug. Successful pre-D87 direct calls with differing keyword/options values, plus fitted RI/RIJCOSX UHF/UKS calls with a nonzero options-only request, have incomplete request fingerprints. Pre-D87 Gamma-only RI RHF calls where either input source was nonzero, including an explicit-zero keyword over nonzero options, followed the legacy operator and are not χ-CCM-B results. The closed-shell fleet is options-only and its records are unchanged. D72 and the analytic total-gradient fail-close remain in force. The shared D72 dimension guard precedes every backend-specific Gamma and mixing guard; this closes the former 1D one-cell RI-RHF escape, whose absolute-energy values are invalid and unreportable.

Unrestricted variational space¶

Let (P^\alpha) and (P^\beta) be spin densities with unit occupation and let (P=P^\alpha+P^\beta). The finite-torus UHF functional per primitive cell is

[ \mathcal E_N^{\mathrm{UHF}}[P^\alpha,P^\beta] =E_{NN}^N+\sum_{\sigma\in{\alpha,\beta}} \operatorname{Tr}_w[P^\sigma h] +\frac12\operatorname{Tr}w[PJ[P]] -\frac12\sum\sigma\operatorname{Tr}_w[P^\sigma K[P^\sigma]]. ]

Its independent functional derivatives are

[ F^\sigma=h+J[P]-K[P^\sigma]. ]

The minimized set is

[ \mathcal V_N^{\mathrm U}={(P^\alpha,P^\beta): (P^\sigma)^\dagger=P^\sigma,\quad P^\sigma S P^\sigma=P^\sigma,\quad [P^\sigma,T_{\mathbf R}]=0,\quad \operatorname{Tr}(P^\sigma S)=N_c n_\sigma}. ]

Here (n_\alpha=(N_e+2S)/2) and (n_\beta=(N_e-2S)/2) are integers fixed by the primitive-cell multiplicity. This is a genuine unrestricted variation: alpha and beta orbitals are not constrained to share a spatial subspace. UHF stationarity does not prove spin purity. The implementation therefore reports both (\langle S^2\rangle) and the ideal (S(S+1)).

For UKS with a hybrid fraction (a_x), replace the UHF exchange energy by

[ -\frac{a_x}{2}\sum_\sigma \operatorname{Tr}w[P^\sigma K[P^\sigma]] +E{\mathrm{xc}}[\rho_\alpha,\rho_\beta], ]

which gives

[ F^\sigma_{\mathrm{UKS}}=h+J[P]-a_xK[P^\sigma] +V_{\mathrm{xc}}^\sigma. ]

No new Wigner–Seitz weight appears in these equations. Both spin projectors use the same declared finite Hamiltonian, including exchange_q0="bvk-ewald" when exact exchange is present, and the same translation-class partition.

6. Algorithm¶

Validate neutral, integer-occupation, zero-temperature RHF/RKS/UHF/UKS input in three periodic dimensions. For 1D/2D inputs, stop before SCF until one shared neutral wire/slab Coulomb kernel exists for every backend.
Form (\mathcal T_N), its exact Wigner–Seitz representative partition, and the unreduced Γ-centred character mesh.
Build lattice-summed overlap and one-electron matrices with the common periodic gauge.
Build the periodised J and K matrices with one of the three backends defined below. Apply the BvK-supercell exchange-divergence correction only in the 3D Ewald gauge.
Solve the per-k generalised eigenproblems using the existing canonical linear-dependence handling and SCF accelerator.
Return energy per primitive cell and measure (\max_k\lVert D_kS_kD_k-2D_k\rVert_F), electron-count error, Wigner–Seitz partition error, and the imaginary residue of the inverse Bloch density.

The inverse Bloch transform stored in the implementation is also the bridge for future explicitly real-space post-HF work.

Integral backends¶

All backends act on the same finite-character density at the declared coulomb_kernel="3d-periodic-g0" and exchange_q0="bvk-ewald" convention. Backend differences are then representations of the electron-repulsion tensor, not silent Hamiltonian changes.

four_center uses the periodic BIPOLE engine in 3D: direct four-centre short-range J/K plus the reciprocal long-range Hartree term and BvK exchange correction. The 1D/2D four-centre route fails closed because the current direct-truncated fallback is not the neutral wire/slab finite-torus Green function and over-binds chain benchmarks by a Madelung-scale shift.
ri uses pair-resolved RSGDF or MDF. With auxiliary functions (P,Q) and Coulomb metric (V_{PQ}=(P|Q)),

[ (\mu\nu|\lambda\sigma){\mathrm{RI}} =\sum{PQ}(\mu\nu|P)(V^{-1})_{PQ}(Q|\lambda\sigma). ]

The fit is applied consistently to J and K.
rijcosx uses the same pair-resolved RI J and evaluates exact exchange by periodic chain-of-spheres quadrature. It is therefore a controlled three-centre/seminumerical approximation, not a different cyclic Hamiltonian.

For global hybrids such as PBE0, all three backends use the same full-range exchange convention described above. The 3D four-center RKS/UKS route also supports HSE06 as (0.25K_{\operatorname{erfc}}(\omega=0.11)), with no full-range exchange seam. RI and RIJCOSX currently fail closed for HSE06 and all other range-separated functionals because their fitted/COSX exchange path is full-range only. This backend split prevents the former silent HSE06 to PBE0 substitution; it is not a Γ-CCM/χ-CCM representation distinction. The four-center HSE algebra is wired, but its screened exchange is not yet basis-independently truncation-certified: AO-pair charges are spatially smeared, so pointwise decay of (\operatorname{erfc}(\omega r)/r) does not prove that the current unpadded ket-image traversal is complete.

All three backends are currently 3D-only. Lower-dimensional inputs fail before the backend build; RI and RIJCOSX do not retain a 3D-periodic-in-vacuum exception.

The q-only compcell fit is rejected: on tight cells it does not represent one consistent four-centre finite-torus tensor and is known to admit non-physical converged fixed points. The present native bridges also reject one-cell RI/RIJCOSX RKS and one-cell RIJCOSX RHF rather than silently substituting a different Gamma-only algorithm. In 3D, one-cell RI RHF is supported only at resolved fock_mixing=0: a nonzero request would make the shared dispatcher substitute the legacy molecular-limit GDF exchange operator for the declared pair-resolved exxdiv="ewald" route, so χ-CCM-B fails closed.

7. Canonical finite-torus RI-MP2¶

MP2 is built on the converged χ-CCM RI-RHF determinant, not on the legacy Gamma-supercell GDF driver. That distinction is numerical as well as formal: on the two-cell 8 x 12 x 12 bohr H2 control, the legacy Gamma-supercell HF reference differs from χ-CCM RI-HF by 0.0046408868 Ha per cell. Reusing it would change the zeroth-order Hamiltonian. The difference is not a gauge-invariant post-HF offset. The exchange q=0 seam shifts the HF orbital energies, so the MP2 denominator [ \Delta_{ij}^{ab}=\varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_b ] inherits the declared convention. External KMP2 or CCSD(T) parity is meaningful only when the reference and the correlated calculation both use exchange_q0="bvk-ewald" or both use some explicitly labelled alternative.

Let (i,j) denote occupied bands, (a,b) virtual bands, and let (N_k=N_c). The pair-resolved canonical auxiliary factor is

[ \widetilde L^{\mathbf k_i\mathbf k_a}{P\mu\nu} =\left[U{\mathbf q}\lambda_{\mathbf q,+}^{-1/2} U_{\mathbf q}^{\dagger}T_{\mathbf q}\right]_{P\mu\nu}, \qquad \mathbf q=\mathbf k_a-\mathbf k_i. ]

Here only metric eigenvalues above the declared threshold enter. Keeping the factor in the original auxiliary-AO basis is important: the compact factor (\lambda^{-1/2}U^{\dagger}T) is invariant in an SCF contraction with its own adjoint, but independently diagonalised (\mathbf q) and (-\mathbf q) blocks have arbitrary eigenvector phases. Those phases need not cancel in an MP2 contraction between two blocks. The canonical matrix function above is unique on the retained subspace and obeys the required conjugation relation.

Transform one occupied and one virtual index,

[ L^{\mathbf k_i\mathbf k_a}{Pia} =\sum{\mu\nu}C_{\mu i}^{*}(\mathbf k_i) \widetilde L^{\mathbf k_i\mathbf k_a}{P\mu\nu} C{\nu a}(\mathbf k_a). ]

For every triple ((\mathbf k_i,\mathbf k_a,\mathbf k_j)), the fourth character is fixed exactly by

[ \mathbf k_b=\mathbf k_i-\mathbf k_a+\mathbf k_j+\mathbf G, ]

where (mathbf G) returns the point to the finite reciprocal group. The RI two-electron block in the normalization used here is

[ V_{ij}^{ab}(\mathbf k_i,\mathbf k_j,\mathbf k_a) =\frac1{N_k}\sum_P L^{\mathbf k_i\mathbf k_a}{Pia} L^{\mathbf k_j\mathbf k_b}{Pjb}. ]

With

[ \Delta_{ij}^{ab}=\varepsilon_{i\mathbf k_i} +\varepsilon_{j\mathbf k_j}-\varepsilon_{a\mathbf k_a} -\varepsilon_{b\mathbf k_b}, ]

the closed-shell spin components per primitive cell are

[ E_{\mathrm{OS}}^{(2)}=\frac1{N_k} \sum_{\mathbf k_i\mathbf k_j\mathbf k_a}\sum_{ijab} \frac{|V_{ij}^{ab}(\mathbf k_a)|^2}{\Delta_{ij}^{ab}}, ]

[ E_{\mathrm{SS}}^{(2)}=\frac1{N_k} \sum_{\mathbf k_i\mathbf k_j\mathbf k_a}\sum_{ijab} \frac{|V_{ij}^{ab}(\mathbf k_a)|^2 -V_{ij}^{ab}(\mathbf k_a)^{*}V_{ij}^{ba}(\mathbf k_b)} {\Delta_{ij}^{ab}}, ]

and (E_{\mathrm{MP2}}^{(2)}=E_{\mathrm{OS}}^{(2)}+ E_{\mathrm{SS}}^{(2)}). MP2 is not variational: it is the second-order Rayleigh–Schrodinger correction for the finite RI Hamiltonian around the stationary RHF determinant. The finite-group Fourier transform nevertheless makes it exactly equivalent to canonical RI-MP2 in the corresponding BvK supercell when the same Coulomb kernel, exchange q=0 convention, and fit are used.

The implemented 3D H2/STO-3G two-point mesh agrees with an out-of-process PySCF KRHF/KMP2 calculation to (5.9\times10^{-10}) Ha in HF and (2.4\times10^{-10}) Ha in correlation energy per cell. The machine-readable record is docs/manuscripts/aiccm_comparison/data/h2_mp2_2026-06-21.json. MP2 currently fails closed in 1D and 2D until O1 supplies a common lower-dimensional Coulomb gauge.

Unrestricted MP2¶

For an unrestricted reference, occupied and virtual labels carry an explicit spin. With antisymmetrized same-spin integrals and ordinary opposite-spin integrals, the second-order correction is

[ E_{\mathrm{UMP2}}^{(2)}=E_{\alpha\alpha}^{(2)} +E_{\beta\beta}^{(2)}+E_{\alpha\beta}^{(2)}, ]

[ E_{\sigma\sigma}^{(2)}= \frac14\sum_{ij\in\sigma}\sum_{ab\in\sigma} \frac{|\langle i_\sigma j_\sigma\Vert a_\sigma b_\sigma\rangle|^2} {\varepsilon_i^\sigma+\varepsilon_j^\sigma -\varepsilon_a^\sigma-\varepsilon_b^\sigma}, ]

[ E_{\alpha\beta}^{(2)}= \sum_{i a\in\alpha}\sum_{j b\in\beta} \frac{|(i_\alpha a_\alpha|j_\beta b_\beta)|^2} {\varepsilon_i^\alpha+\varepsilon_j^\beta -\varepsilon_a^\alpha-\varepsilon_b^\beta}. ]

Every sum also contains the finite characters, with the same momentum conservation and one final division by (N_c) as the restricted expression. The implementation evaluates this equation after the exact inverse transform of the χ-CCM RI Hamiltonian. Zero PNO threshold and canonical occupieds give the full UMP2 result. Independent unitary localization of the alpha and beta occupied projectors leaves both spin densities invariant; the coupled local MP2 equations are then required because each localized occupied Fock block is not diagonal.

8. Real-torus local correlation and coupled cluster¶

Exact inverse transform of the fitted Hamiltonian¶

Let the normalized finite-group Bloch AO be

[ |\mu\mathbf{k}\rangle =N_c^{-1/2}\sum_{\mathbf R\in\mathcal T_N} e^{+i\mathbf k\cdot\mathbf R}|\mu\mathbf R\rangle . ]

For any translation-invariant one-particle matrix, the full finite-torus AO matrix is therefore

[ X_{\mu\mathbf R,\nu\mathbf S} =\frac{1}{N_c}\sum_{\mathbf k} e^{+i\mathbf k\cdot\mathbf R}X_{\mu\nu}(\mathbf k) e^{-i\mathbf k\cdot\mathbf S}. ]

This formula is used for both overlap and Fock. It is a unitary change of representation, not a new Gamma-point SCF.

The RI factors need one more character. Let (\widetilde L^{\mathbf k_i\mathbf k_a}_{P,\mu\nu}) be the canonical auxiliary-AO factor of Section 7 and (\mathbf q=\mathbf k_a-\mathbf k_i). Define

[ B_{P\mathbf T,\mu\mathbf R,\nu\mathbf S} =\frac{1}{N_c^2}\sum_{\mathbf k_i\mathbf k_a} e^{+i\mathbf k_i\cdot\mathbf R} e^{-i\mathbf k_a\cdot\mathbf S} e^{+i\mathbf q\cdot\mathbf T} \widetilde L^{\mathbf k_i\mathbf k_a}_{P,\mu\nu}. ]

The corresponding real-torus MO coefficient is

[ C_{\mu\mathbf R,p\mathbf k} =N_c^{-1/2}e^{+i\mathbf k\cdot\mathbf R}C_{\mu p}(\mathbf k). ]

Contracting the previous two equations gives

[ B_{P\mathbf T,i\mathbf k_i,a\mathbf k_a} =N_c^{-1}e^{+i\mathbf q\cdot\mathbf T} L^{\mathbf k_i\mathbf k_a}_{Pia}. ]

Thus a product of factors with transfers (\mathbf q) and (-\mathbf q) satisfies

[ \sum_{P\mathbf T}B_{P\mathbf T,ia}B_{P\mathbf T,jb} =\frac{1}{N_c}\sum_P L_{Pia}L_{Pjb}, ]

while any nonconserving transfer vanishes by character orthogonality. This is exactly the normalization in the momentum-space MP2 integral. The real and character representations therefore describe the same χ-defined fitted finite Hamiltonian at the recorded Coulomb and exchange q=0 convention. This internal Fourier equivalence does not assert equality with the union-and-weight Γ-CCM approach.

Local MP2 and CCSD(T)¶

The present local implementation deliberately builds the whole finite torus. It includes every occupied pair and every ordered occupied triple required by the spin-integrated equations, obtains a total correlation energy (E_{\mathrm{corr},N}), and reports

[ E_{\mathrm{corr}}^{\mathrm{cell}} =\frac{E_{\mathrm{corr},N}}{N_c}. ]

There is no additional boundary or pair multiplicity. A future translation- reduced implementation may retain one orbit under simultaneous translation of all occupied indices, but then its multiplicity must follow the orbit-stabilizer theorem. Selecting visually unique pairs and multiplying by an occurrence factor would recreate the historical ambiguity.

Occupied orbitals are localized with the Pipek–Mezey objective built from Mulliken populations. This is invariant to the choice of position origin and does not use the ill-defined ordinary position operator on a torus. Molecular Foster–Boys localization is rejected. The occupied rotation is unitary, so with complete pair domains, no PNO truncation, no pair screening, and complete occupied coupling, local MP2 and local CCSD reproduce their canonical finite-torus limits. With complete triple virtual spaces the same statement holds for the perturbative triples correction.

The current defaults allow PNO truncation but keep all occupied pairs and all occupied couplings. Distance-based pair screening and local auxiliary fitting are disabled because their existing molecular implementations use Euclidean distances across the cell boundary. They require minimum-image Wannier and auxiliary domains before they can be enabled without breaking translation symmetry.

The local-correlation result records the same finite-torus convention as its HF reference. This is required because the semicanonical occupied and virtual energy blocks used in MP2, CCSD, triples, and future DLPNO denominators inherit the exchange q=0 seam from that reference.

The gauge-invariant SCF property object, the finite-torus band-structure wrapper, and the finite-torus Mayer bond-order analysis also record that convention. These analyses are derived from the same one-particle Fock, density, and overlap matrices; the descriptor is provenance for the finite Hamiltonian, not an additional factor in the population formula or band interpolation.

Neither MP2 nor CCSD(T) is variational. CCSD is stationary only through its left-right coupled-cluster Lagrangian, and the perturbative triples energy has no upper-bound property. PNO/domain truncations add controlled numerical approximations, not a new Hamiltonian or a variational subspace theorem.

On the H2/STO-3G two-cell control, no-truncation real-torus local MP2 equals the character-space result within (7\times10^{-18}) hartree per cell. On the LiH/STO-3G two-cell control, no-truncation local and canonical finite-torus DF-CCSD correlation energies differ by (6.9\times10^{-12}) hartree total; their nonzero triples corrections differ by (1.4\times10^{-12}) hartree. These are finite-Hamiltonian exact-limit tests, not external infinite-crystal CCSD(T) benchmarks.

For UCCSD, use spin orbitals and

[ |\Psi_{\mathrm{CCSD}}\rangle=e^{T_1+T_2}|\Phi_0\rangle, \qquad \langle\Phi_\mu|e^{-T}He^T|\Phi_0\rangle=0 ]

for every single and double excitation (\mu). The energy is

[ E_{\mathrm{CCSD}}=\langle\Phi_0|e^{-T}He^T|\Phi_0\rangle. ]

The unrestricted PNO pilot expands each pair amplitude into the full finite-torus spin-orbital virtual space, evaluates the complete projected residual, and projects back. With zero PNO threshold this projection is a unitary coordinate change and recovers full-domain UCCSD. Perturbative triples are evaluated from the converged amplitudes. This establishes the equations and exact PNO limit, but the current implementation remains an explicitly cost-capped O(N^6) oracle: it does not yet skip translation-equivalent pairs or triples and therefore does not yet establish reduced scaling.

Post-SCF occupied localization¶

Let the complete finite character group contain (N_c) points. For each isolated occupied band, the canonical back transform is

[ |w_{n\mathbf R}^{0}\rangle =\frac{1}{\sqrt{N_c}}\sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R}|\psi_{n\mathbf k}\rangle . ]

All (N_c n_{\mathrm{occ}}) functions are rotated together, (C^{\mathrm{loc}}=C^{0}U), with (U^\dagger U=I). Consequently

[ C^{\mathrm{loc}}C^{\mathrm{loc}\dagger} =C^0UU^\dagger C^{0\dagger}=C^0C^{0\dagger}. ]

This proves invariance of the finite-torus density and every RHF or RKS energy term. Localizing independent k blocks would not produce the required real-space family.

Exact Marzari–Vanderbilt localization requires cross-k matrix elements of the exponential position operator. Those integrals are not exposed by the current native API. The implemented Wannier route jointly diagonalizes Lowdin-projected AO-centre operators (\cos(2\pi r_\alpha/L_\alpha)) and (\sin(2\pi r_\alpha/L_\alpha)). If

[ z_{n\alpha}=\langle w_n|e^{2\pi i r_\alpha/L_\alpha}|w_n\rangle, \qquad \Omega_{n\alpha} =-\left(\frac{L_\alpha}{2\pi}\right)^2\log|z_{n\alpha}|^2, ]

the latter is the reported circular spread. It has the correct torus topology but is labelled a projected circular AO-centre approximation. Exact continuum spreads and Wannier90 parity remain open.

The IAO alternative constructs polarized intrinsic atomic orbitals from a minimal reference and maximizes squared atomic populations. Its present cross overlap is an explicit-supercell integral rather than a periodized torus integral. A four-cell H2 run gives translation covariance residuals of (1.17\times10^{-13}) for the Wannier path and (7.08\times10^{-6}) for IAO. Density invariance remains at machine precision. IAO translation maps are therefore diagnostic and are not used to skip correlated pairs.

Wraparound is flagged when the antipodal-shell probability exceeds 0.05 or when (\sqrt{\Omega_n}) exceeds one quarter of the shortest cyclic-cluster vector. A nonpositive indirect gap fails closed. Metallic-band disentanglement is not implemented.

PAOs, PNOs, and pair orbits¶

For an (S)-orthonormal occupied coefficient matrix (C_o), the coefficient projector out of the occupied space is

[ Q=I-C_oC_o^\dagger S,\qquad C_o^\dagger S Q=0. ]

If (E_D) selects AOs in a pair domain, let (\widetilde V_D=QE_D) and diagonalize its Gram matrix:

[ G_D=\widetilde V_D^\dagger S\widetilde V_D=XgX^\dagger, \qquad V_D=\widetilde V_DX_+g_+^{-1/2}. ]

Then (V_D^\dagger S V_D=I) and (C_o^\dagger S V_D=0). This removes both occupied leakage and redundant PAOs.

For pair amplitude matrix (T_{ij}), the model pair density is

[ D_{ij}=\frac{T_{ij}T_{ij}^\dagger+T_{ij}^\dagger T_{ij}} {1+\delta_{ij}}. ]

It is Hermitian positive semidefinite because (x^\dagger D_{ij}x=(|T_{ij}^\dagger x|^2+ |T_{ij}x|^2)/(1+\delta_{ij})\ge0). Its eigenvectors are PNOs. A zero threshold retains the complete PAO span, and positive thresholds give nested ranks. This proves rank nesting, not monotonic MP2 energy error: MP2 is nonvariational and convergence of the energy must be measured.

Translations act simultaneously on both indices of an unordered occupied pair. For measured localization permutations (p_g),

[ \mathcal O(i,j)={\operatorname{sort}(p_g(i),p_g(j)):g\in G_N}. ]

The per-cell coefficient is (|\mathcal O|/N_c), including smaller orbits with nontrivial stabilizers such as half-cell pairs on an even ring. The code verifies a disjoint partition of all unordered pairs, but it still evaluates all pairs. The orbit reduction factor is not a timing claim.

At complete PAO domains and zero PNO thresholds, the B MP2 wrapper also evaluates a rotation-invariant full-space DF-MP2 contraction. The raw iterative local-pair energy and the correction are both reported. Before the real local solver, a full-rank real time-reversal gauge is constructed and its occupied projector is checked against the complex localized projector. This avoids silently discarding imaginary coefficients. No correction is applied to truncated calculations.

Finite-cluster space-group symmetry¶

Write primitive lattice vectors as columns of (A), and let (N=\operatorname{diag}(N_1,N_2,N_3)). A crystallographic operation (g={W|\mathbf w}) preserves the finite torus exactly when

[ N^{-1}WN\in\mathbb Z^{3\times3}. ]

For primitive atom (a), retain both the atom permutation and integer image shift,

[ W\mathbf f_a+\mathbf w=\mathbf f_{p_g(a)}+\mathbf q_{g,a}, ]

so ((a,\mathbf r)) maps to

[ \left(p_g(a),W\mathbf r+\mathbf q_{g,a}\pmod{\mathbf N}\right). ]

The image shift is essential for screw and glide operations. Reciprocal characters transform as (\mathbf k’=W^{-T}\mathbf k), and a k-orbit has weight (|\mathcal O_k|/N_c). spglib supplies the crystallographic group; the B module builds its exact cluster-compatible subgroup.

Symmetry is diagnostic only. At Gamma, AO matrices can be checked with the Reynolds projector

[ \mathcal P_G[M]=\frac{1}{|G_N|}\sum_g U_g M U_g^T, ]

whose idempotency follows from group closure. At general k, nonsymmorphic sewing phases depend on each backend’s Bloch convention. Therefore symmetry_mode="integrals" fails closed until representative shell-pair and quartet scattering passes full-build Fock and energy parity. Primitive shell pairs and intrinsically eightfold-unique shell quartets are already partitioned into exact point-group orbits and reported as reduction counts; no libint call is skipped from those counts yet.

9. Limits and convergence¶

Exact finite-size identity¶

For the same declared finite-torus Hamiltonian, in particular the same exchange_q0="bvk-ewald" finite-size convention, and the same translation-invariant variational space,

[ E_N^{\Gamma\text{-supercell}}/N_c =E_N^{\Gamma\text{-centred mesh}} ]

up to numerical error. This is the strongest internal reference and should hold at every mesh size.

Infinite-crystal limit¶

As all active (N_i\to\infty), the character sum approaches the Brillouin zone integral and the BvK Green’s function approaches the chosen infinite periodic Coulomb convention. Insulators with localised density matrices are expected to converge rapidly in short-range pieces; exact exchange and the Coulomb finite-size terms can converge algebraically. The sequence need not be monotone because each (N) defines a different finite Hamiltonian.

Molecular limit¶

For a one-cell cluster with increasing vacuum, image interactions and the BvK Madelung correction vanish, and the result approaches molecular RHF or RKS in the same basis. Basis linear dependence is handled per k by canonical orthogonalisation; dropping different ranks at different k is diagnosed and can spoil smooth convergence.

BSSE is not repaired by the cyclic construction. Counterpoise or basis extrapolation is a separate modelling choice.

10. Assumptions and explicit divergences from historical CCM¶

Audit of the 2008 deMon2k KS-ADFT CCM¶

Janetzko, Köster, and Salahub define the two-center occurrence weight

[ \omega_{MN}=\frac{1}{n_{MN}},\qquad T^{\mathrm{CCM}}=\sum_{\mu\nu}P^{\mathrm{CCM}}{\mu\nu} \omega{MN}\sum_{\nu’}T_{\mu\nu’}. ]

For a three-center quantity they do not merely multiply one selected integral. Their Eq. (20) averages both nested enumerations (M’\to N’) and (N’\to M’), divided by

[ n_{MNC}=n_{MN}(n_{MC}+n_{NC}),\qquad \omega_{MNC}=n_{MNC}^{-1}. ]

That explicit two-order average matters. It means the simple (M\leftrightarrow N) objection to the 2014 four-center factor cannot be transferred unchanged to the deMon2k three-center tensor. If both enumerations are complete, Eq. (20) is symmetric in the two AO centers by construction.

The two-center formula still requires the stronger Hermiticity identity

[ \omega_{MN}\sum_{\nu’}I_{\mu\nu’} =\omega_{NM}\sum_{\mu’}I_{\nu\mu’}. ]

The free-space identity (I_{\mu\nu}=I_{\nu\mu}), cited in the paper, does not by itself prove this equality for two different center-specific Wigner– Seitz enumerations. Likewise, variational auxiliary fitting requires the weighted three-center map and weighted auxiliary Coulomb metric to be adjoints under one finite inner product. The paper supplies a fitted-energy expression and its derivative, so the intended scalar functional is clearer than in the 2014 four-center derivation, but it does not prove this quotient-space adjoint condition for arbitrary non-Bravais or skew clusters.

There is also a separate finite-size issue. The paper explicitly states that its CCM and the corresponding periodic calculation are not fully equivalent because interactions outside the Wigner–Seitz regions are omitted. Its auxiliary-density lattice sum is truncated after one neighboring shell, and the XC quadrature is evaluated on a reference-cluster grid with translated auxiliary functions. Those can converge with cluster size, but they are not an exact finite-(N) BvK identity. Thus the deMon2k multiplicative weights remain a symmetry and variational audit target; the present derivation does not label them broken where Eq. (20) already enforces the relevant AO-pair symmetry.

Finite torus, not a weighted open cluster. The method is defined by a BvK Hamiltonian first. Wigner–Seitz geometry is a representation of its translation classes.
Boundary weights are a partition of unity. They average only tied representatives of the same periodised object. They are not attached to atoms, shells, or arbitrary AO pairs.
No historical product-weighted ERI. The published pair-product four-centre factors are not invariant under all ERI permutations in their displayed form. Without a separate scalar functional derivation, using them directly would make the variational and double-counting status unclear. aiccm2026dev-b therefore does not use them.
No Fock symmetrisation as a repair. Hermiticity follows from the Hamiltonian. A non-Hermitian raw Fock is treated as a bug.
Long range is part of the Hamiltonian. It is not an after-the-fact Madelung energy added to a short-range SCF.
Charged cells fail closed. No implicit uniform-background energy is reported.
Exchange is not independently truncated. It uses the finite-torus periodised kernel and BvK-size divergence correction.
The reciprocal implementation is exact, not a reference shortcut. It is the diagonal representation of the finite cyclic translation group.
Source agreement is secondary. Historical numerical agreement does not override idempotency, permutation symmetry, variational consistency, gauge consistency, or finite-supercell/mesh equivalence.

11. Validation ladder¶

Fixed-density kinetic derivative component¶

For a caller-supplied real-torus density whose cell list is fixed by declared lattice_options, the kinetic contribution is

[ E_T[D;R] = \sum_g \operatorname{Tr}[D(g) T(g;R)]. ]

compute_aiccm2026dev_b_fixed_density_kinetic_energy(...) evaluates this scalar, while compute_aiccm2026dev_b_fixed_density_kinetic_gradient(...) evaluates (\partial E_T/\partial R_A) from analytic AO-centre derivatives at fixed (D(g)). The derivative is independently checked against central differences of the matching scalar after rebuilding the displaced basis. Both paths require exactly the same lattice-cell indices and Cartesian translations, AO dimension, and block shapes.

There is no SCF convenience wrapper for this component. The current SCF result does not attest the resolved one-electron lattice cutoff, so reconstructing an operator from a guessed or default cutoff could silently compare a different cell support. The stationary energy-weighted density construction and Pulay/adjoint assembly, BvK exchange-q=0 seam or screened-exchange derivative, RI/RIJCOSX three-centre and metric response, DFT exchange-correlation quadrature/grid derivatives where applicable, and post-HF response terms remain open. The public total-gradient route therefore still fails closed. Γ-CCM and χ-CCM are distinct approaches compared at a declared common exchange-q=0 convention; their names do not select different Coulomb kernels.

Fixed energy-weighted overlap derivative component¶

For a caller-supplied real-torus energy-weighted density (W(g)), define the fixed overlap-constraint scalar

[ \mathcal L_S[W;R] =-\sum_{g\mu\nu}W_{\mu\nu}(g)S_{\mu\nu}(g;R) =-\sum_g\operatorname{Tr}[W(g)^{\mathsf T}S(g;R)]. ]

The transpose in the final expression is important: individual real-space residue blocks need not be symmetric even when the complete finite-character object obeys Hermiticity and time reversal. At fixed numerical (W(g)) and fixed lattice,

[ \left.\frac{\partial\mathcal L_S}{\partial R_{A\alpha}}\right|{W,\mathrm{lattice}} =-\sum{g\mu\nu}W_{\mu\nu}(g) \frac{\partial S_{\mu\nu}(g;R)}{\partial R_{A\alpha}}. ]

compute_aiccm2026dev_b_fixed_energy_weighted_overlap_lagrangian(...) evaluates the scalar, and compute_aiccm2026dev_b_fixed_energy_weighted_overlap_gradient(...) evaluates the analytic AO-centre derivative. A 3D odd-character conjugate pair produces real but individually nonsymmetric (W(g)) blocks, pinning the Frobenius orientation and sign against central differences of the public scalar. Both helpers require identical ordered cell indices, Cartesian translations, AO dimension, and block shapes.

This scalar is a Lagrangian constraint companion, not a separately additive SCF energy. The helpers do not construct the stationary χ-CCM energy-weighted density from orbitals, occupations, spin channels, backend, or exchange convention. They also do not include the separate explicit derivative of the BvK exchange seam or screened-exchange kernel. Consequently, they prove only the overlap skeleton; B-owned adjoint construction and variational assembly remain open, and the total gradient still fails closed. Γ-CCM and χ-CCM are distinct approaches compared at a declared common exchange-q=0 convention; their names do not select different Coulomb kernels.

Stationary energy-weighted-density binding contract¶

D85 specifies the object that must eventually connect the fixed-W D80 skeleton to a stationary SCF Lagrangian. For character (q) and spin (\sigma), rebuild the final physical variational Fock (F^\sigma_{\mathrm{var}}(q)=\partial E/\partial D^\sigma(q)) without DIIS mixing, damping, level shift, clipping, or a stale pre-final operator. With an orthonormal occupied block (C_o^{\sigma\dagger}S C_o^\sigma=I), define

[ \Lambda^\sigma=C_o^{\sigma\dagger}F^\sigma_{\mathrm{var}}C_o^\sigma, \qquad W(q)=\begin{cases} 2C_o\Lambda C_o^\dagger,&\text{RHF/RKS},\ \sum_\sigma C_o^\sigma\Lambda^\sigma C_o^{\sigma\dagger}, &\text{UHF/UKS}. \end{cases} ]

This definition is invariant under occupied rotations and phases. The first contract version accepts only the existing zero-temperature integer occupations. Unrestricted spin blocks remain separately attested; only their sum enters the spin-independent overlap term. The contract inverse-transforms the verified time-reversal-consistent character blocks as (W(g)=\sum_q w_q e^{-2\pi i q\cdot g}W(q)); it must not repair a failed sewing relation by taking an unexplained real part.

The immutable binding includes the atom order and effective charges/ECPs, basis shell and AO order, finite-torus descriptor, retained AO projectors at every character, canonical digests of (S,F_{\mathrm{var}},D,C_o), occupations and (W), the resolved one-electron cell support, and the route-specific two-electron and XC-grid support. A rebuilt-Fock stationarity and directional variational-closure check is required for each backend. RIJCOSX fails closed unless its approximate exchange energy and Fock satisfy that closure or a derived adjoint supplies the missing response.

The exchange assembly records its full-range and screened coefficients and the q=0 applicability. Every active BvK seam and screened-exchange term enters the stationary (F_{\mathrm{var}}), hence (W). Their explicit fixed-density derivatives remain separate: the BvK seam derivative is present only when its applicability is active, while the finite HSE erfc exchange derivative uses the same screening parameter and numerical domain as the SCF. Neither is hidden in or counted twice with the overlap derivative. This first contract covers atomic displacements at fixed primitive lattice; stress remains open.

The seam record stores an operator coefficient (\eta) rather than relying on a signed-Madelung naming convention. With the restricted spin-summed density,

[ K\leftarrow K+\eta SDS, \qquad E_{\mathrm{seam}}^R=-\frac{\eta}{4} \operatorname{Tr}[(DS)^2], ]

and for unrestricted spin densities,

[ K_\sigma\leftarrow K_\sigma+\eta SD_\sigma S, \qquad E_{\mathrm{seam}}^U=-\frac{\eta}{2}\sum_\sigma \operatorname{Tr}[(D_\sigma S)^2]. ]

Here (\eta=\xi_{\mathrm{BvK}}) for RHF/UHF, (\eta=c_{\mathrm{full}}\xi_{\mathrm{BvK}}) for a global hybrid, and (\eta=0) when applicability is inactive. HSE06 instead records (c_{\mathrm{full}}=0), (c_{\mathrm{sr}}=0.25), and (\omega=0.11\ \mathrm{bohr}^{-1}). Its fixed-density exchange functional is (E_{x,\mathrm{sr}}^R=-c_{\mathrm{sr}}\operatorname{Tr}[D K_{\mathrm{erfc}}[D]]/4), or the corresponding unrestricted spin sum with coefficient (-c_{\mathrm{sr}}/2). No q=0 seam is attached to that finite screened kernel. Force derivation must differentiate these exact recorded functionals using the same numerical support as the SCF.

Each approach must map its own stationary state into the binding contract and derive the fixed-density seam or screened terms against that contract. Within χ-CCM, the character and real-Gamma forms meet by the exact finite Fourier transform. Cross-approach equality is a separate result to establish with operator-specific and finite-difference gates under the common exchange-q=0 convention.

Gradient-audit finite-torus binding¶

Every component helper now checks that its result descriptor and active system identify the same finite torus. For the present direct-product, Γ-centred construction,

[ N_{\mathrm{result}}=N_{\mathrm{character}} =N_{\mathrm{BvK}}, \qquad A_{\mathrm{BvK}}=A_{\mathrm{system}}\operatorname{diag}(N), ]

where PeriodicSystem.lattice stores primitive vectors as columns. The recorded BvK lattice must match within (10^{-12}) bohr. A skew, unequal-mesh test distinguishes this column scaling from the incorrect row-scaled expression (\operatorname{diag}(N)A_{\mathrm{system}}). Diagnostics and top-level convention descriptors must agree, and the boundary model must be explicitly 3d-periodic.

SCF-density folding additionally validates the complete unreduced, unshifted Γ-centred character set. Fractional labels are reduced modulo the integer mesh, so equivalent representatives such as (-1/3) and (2/3) are accepted, but duplicate, incomplete, shifted, or nonuniformly weighted nets fail before the inverse transform.

This is a provenance guard, not another force term. It binds the torus size and lattice but does not attest the atoms, basis identity, resolved one-electron cutoff, or the source of caller-supplied (D(g)) or (W(g)). Atomic displacements at fixed lattice remain valid for finite-difference component checks; stresses and cell derivatives are outside this contract. The equality above would also need revision for a future twisted boundary or non-diagonal supercell construction.

Two-electron numerical-support qualification¶

The declared 3D Hamiltonian does not by itself prove that a finite numerical realization has converged all integration domains. The combined MgO numerical-support audits separate three limitations:

four-center BIPOLE short-range J/K needs an internal ket-image traversal padded by the smeared AO-pair kernel range; the physical 15/15-bohr density/nuclear support does not supply that pad;
restricted semilocal four-center RKS uses an analytic-FT Hartree partial sum whose default KE=200 does not converge MgO core reciprocal tails; and
RI and RIJCOSX avoid BIPOLE traversal but default RSGDF KE=200 has a distinct dense-core high-G fitting limitation.

The split audit at 51e3b250 establishes the first two limitations. The earlier dense-tail audit at 2355f52e, pinned by tests/test_rsgdf_dense_mesh_tail.py, establishes the third.

The shared BIPOLE workstream owns the first fix. Commit bed97c39 lands M4a, an opt-in sr_image_extent_bohr correctness oracle whose padded output agrees with a wide plain traversal. It is intentionally not the default because the MgO pilot cost is about 35 times the unpadded build; efficient pair-resolved M4b remains open. χ-CCM must consume this shared implementation, not fork another kernel. The B selector and fleet runner do not yet expose or attest M4a, so default BIPOLE-routed four-center absolute values remain revision-bound until an explicit route-complete converged M4a extent is recorded or M4b lands. The present M4a path does not pad HSE’s dedicated screened \(K_{\mathrm{erfc}}\) traversal, and UHF/UKS spin-schedule recursion drops an explicit extent. Its helper also needs an unambiguous absolute-radius versus additive-pad contract before B exposes it; the fleet must record the final absolute radius actually used. Restricted semilocal four-center RKS remains separately held by its reciprocal-tail contract, and dense-core default-KE fitted absolute energies remain held by their high-G contract. Existing SCF convergence and route-plumbing records remain useful but are not external absolute-energy validation. These are numerical-support defects inside the declared convention, not new gauges, Coulomb kernels, or Γ/χ representation physics.

The committed tests cover the selector, exact mesh, Wigner–Seitz ties, partition unity, inverse Fourier transform, metric idempotency, fail-closed domain, RHF/RKS runner dispatch, native 3D SCF runs through all three integral backends, explicit 1D/2D rejection across every backend, and χ-CCM native post-HF setup kernels. The native OpenMP kernels are parity-tested against the earlier NumPy equations for the one-body inverse transform, pair-resolved RI-factor inverse transform, and three-index AO-to-MO contractions. The benchmark harness adds:

the archived pre-guard 1D alternating hydrogen-chain ladder as failure and historical comparison evidence, not as reportable absolute energies;
direct comparison with both the historical union12 and separately developed symmetric aiccm2026dev-a weights in the in-repo CCM on identical geometry;
exact comparison with the native Γ-centred k-mesh representation;
energy, convergence, idempotency, electron count, wall time, and implementation-to-implementation differences;
later 3D LiH, MgO, NaCl, Al2O3, and diamond ladders only after each basis, gauge, internal traversal, and reciprocal tail is pinned to an out-of-process reference input.

The reportable Γ/χ approach map is empty. The A RI and RIJCOSX harness routes call the same multi-k GDF SCF drivers as B, so agreement between those routes tests their shared implementation and cannot isolate an approach change. B rhf-ri versus A aiccm-hf-direct is a neutral-torus representation control, not evidence for the union-and-weight Γ-CCM approach. The direct solver assembles and minimizes a real Γ-supercell problem, but its folded cderi and one-electron matrices intentionally reuse the common per-q RSGDF fits and lattice-sum primitives. The control therefore checks Γ-supercell versus character/Bloch assembly, not an independent RI implementation or a Γ-CCM versus χ-CCM approach delta. It is still not reportable because comparator contract aiccm2026-gamma-chi/v2 is only an incomplete validation scaffold and must not be emitted unchanged. A superseding contract needs the operator and retained-space evidence below.

Contract v2 requires the canonical comparison_input to be embedded beside its exact comparison_contract.input_sha256 hash, validates its required physical/numerical fields, and reconciles them with the result and contract. It distinguishes the AO linear-dependence threshold, currently 1e-7, from the auxiliary-metric linear-dependence threshold, currently 1e-9. It fingerprints and reconciles requested and reported SCF smearing and accepts only the zero-temperature setting (smearing_temperature=0.0 Ha). Contract v2 does not bind Fock mixing, and D86 does not change or upgrade that incomplete comparator contract. A superseding contract must bind requested and executed accelerator controls before using them as comparison evidence. Contract v2 also requires matched clean source, native-core, host, package-version, and successful composite producer-process attestation. The cached aiccm-host-probe/v2 preflight remains the trusted dispatch gate, but it is not accepted as result evidence by itself. Each producer binds that preflight to its own current identity and a canonical manifest of its copied producer, probe, test-set, launcher, and stream-specific route files. In the producer process it records linked native-library versions and always runs a cheap shifted-mesh native-versus-Python AO-pair Fourier-transform canary. If the native-core path SHA256 changed after preflight, it reruns the full v2 host checks in process, including API shape, reciprocal cutoff, and LiH direct-versus-GDF checks; the cheap canary alone is not sufficient for a changed core.

Immediately before result serialization the producer re-reads its clean source, host, package, native-core path SHA256, native-library versions, and payload and requires them to be unchanged. The SHA256 describes bytes currently at the imported module’s filesystem path, not bytes already mapped into the process. The same-process numerical checks therefore attest loaded behavior, but do not claim a cryptographic identity for the mapped image. The comparator independently validates the composite and nested canonical digests, exact schemas, check versions, and tolerances. It compares A and B using their current producer identities; different valid preflight histories and the intentionally different stream payload manifests need not match. A sidecar added during later curation cannot establish any of these facts retrospectively, and old rows are not upgraded by the new schema.

The factor audit also exposed a finite-cutoff reciprocal-support seam. An ordered character pair can report a raw transfer q + G while the neutral-torus real-Gamma fold uses the equivalent finite-group transfer q. The ket Bloch cell phase is unchanged, but shifting one fixed finite |G| ball by those two raw labels selected different boundary crescents. On the compact H2 (3,1,1) control, the relative fitted-Gram residual was 9.73e-8 at 40 Ha and 2.46e-11 at 200 Ha. A half-open centered representative fixes that odd-mesh case but breaks time reversal at an even-mesh Nyquist transfer; keeping both Nyquist signs restores time reversal but not reciprocal-label invariance. The shared Bloch RSGDF builder now enumerates the physical shifted sphere 0 < |G+q| <= sqrt(2 E_cut) directly. That support is both modulo-G invariant and inversion covariant. On skew H2 (2,1,1) and (3,1,1) controls, the canonical χ-CCM factor inverse transform now matches the neutral-torus folded fitted-Gram control at the 1e-14 numerical floor. Separate gates pin reciprocal relabelling of the compact fitted Gram and canonical factor, plus canonical-frame Nyquist time reversal. This is a numerical representation fix inside the same finite-torus Hamiltonian and Coulomb convention. The q=0 support and accumulation order are byte-preserved. Finite nonzero-q RSGDF support changed, so pre-D78 exact-GDF-K RHF/UHF and hybrid RI, neutral-torus fold, and RI post-HF/full-pair records are revision-bound until rerun. Semilocal RI and RIJCOSX SCF build only q=0 RSGDF J factors and are numerically unaffected by this support change.

Numerical-input emission remains disabled pending the remaining contract corrections. The legacy real-Gamma control builder blocks use the canonical auxiliary-AO embedding, but the final control cderi is an unfolded-k Fourier fold followed by a real q/-q stack and a numerical-null-row prune. B SCF instead self-contracts one compact metric-eigenmode factor per ordered character pair. Raw factors need not match. Acceptance must name the common Bloch pair-density phase, record the route-specific factor pipelines, and establish q-resolved agreement of the retained auxiliary projectors together with the gauge-invariant fitted Gram operators. Matching metric ranks, thresholds, or projectors alone is insufficient.

A also applies an absolute Γ-supercell overlap guard while B performs normalized per-character canonical orthogonalization. Equal threshold numbers do not prove equal retained spaces. For the current control route, acceptance requires full/no-drop at every B character together with the finite-Fourier overlap residual, so every character retains all primitive-cell AO directions. If a future comparison permits projected spaces, it must compare Fourier-related retained-space projectors rather than rank counts alone. No Γ/χ approach delta can be emitted until these conditions are specified and tested and a route actually realizing the union-and-weight Γ-CCM construction is paired with χ-CCM. Equality between the distinct Γ-CCM and χ-CCM approaches for a specified operator and route remains evidence to establish under the declared common exchange-q=0 convention.

External CRYSTAL or PySCF comparisons remain out of process and are weak signals unless their cell, Coulomb, exchange-divergence, basis, and k-mesh conventions are matched explicitly.