Comparing Γ-CCM and χ-CCM¶

Γ-CCM (aiccm2026dev-a) and χ-CCM[1] (aiccm2026dev-b) are distinct approaches to the ab-initio Cyclic Cluster Model. Γ-CCM uses union-and-weight/Wigner–Seitz integral weighting; χ-CCM uses a finite-translation-group character construction. They are developed in separate workstreams, share only the geometry registry (testset.py), and are compared head-to-head to determine, rather than presume, where specified finite operators and routes agree.

Important

The reportable Γ/χ approach-comparison map is currently empty. The existing aiccm-hf-direct versus B rhf-ri pair is a neutral-torus real-Gamma versus character representation control, not evidence for the union-and-weight Γ-CCM approach. Within the χ-defined Hamiltonian those two representations are related by an exact finite Fourier transform. Cross-approach equality remains evidence to establish at a declared common exchange-q=0 convention.

Running the same system through both approaches¶

Via the test-set runners¶

The aiccm-2026/ directory has separate runners that share only testset.py for the geometry:

cd aiccm-2026

# Γ-CCM: four-center HF on diamond
python run_case.py c-diamond aiccm-hf --out results-a/

# χ-CCM: RHF with four-center backend on diamond
python run_case_b.py c-diamond rhf-4c --out results-b/

# Inspect the two records. The current comparator reports the approach
# comparison as not-defined; it does not form an energy delta.
python compare_b.py results-b/

compare.py still produces a single-line route table. Its old --vs head-to-head mode is disabled because route names alone cannot prove a matched Γ-CCM/χ-CCM operator. compare_b.py reports the gamma_ccm_chi_ccm_approach_comparison_status explicitly as not-defined.

Running the neutral-torus Fourier control¶

The one retained cross-run control is not an approach comparison. It compares B rhf-ri, evaluated on the character mesh, with aiccm-hf-direct, evaluated in a real-Gamma supercell, for a separately attested neutral fitted-torus Hamiltonian:

python run_case.py c-diamond aiccm-hf-direct --out results-control/
python run_case_b.py c-diamond rhf-ri --out results-b/

python compare_b.py results-b/ \
    --real-gamma-control-results results-control/

When the complete control contract is present, its result is written only to the real_gamma_control_* and character_minus_real_gamma_control_mha_per_atom fields. It is never labelled as a Γ-CCM/χ-CCM delta. The legacy --a-results spelling is accepted only as an alias for --real-gamma-control-results and should not be used in new commands.

Fleet-scale comparison¶

Generate both batches and submit:

# Γ-CCM full matrix
python make_jobs.py | sh

# χ-CCM SCF matrix (all systems, all routes)
python make_jobs_b.py --profile scf | sh

# After results land, inspect the B records. No approach delta is formed.
python compare_b.py results-b/ --csv comparison.csv

# Optionally add the separately generated neutral-torus real-Gamma control.
python compare_b.py results-b/ \
    --real-gamma-control-results results-control/ --csv control.csv

Adding a CRYSTAL23 reference¶

Populate the crystal_refs_b.json template with per-system per-atom energies from actual CRYSTAL23 runs, then:

python compare_b.py results-b/ --crystal-refs crystal_refs_b.json

The template schema is:

{
  "c-diamond": {"rhf": -38.07736, "pbe": -38.07736}
}

Values are energy per primitive-cell atom in Hartree. Only 3-D systems belong in this B reference template while the all-backend 1-D/2-D fail-close is active.

What to expect¶

A matched convention is necessary, not sufficient¶

A meaningful approach comparison must hold the Coulomb operator and exchange-q=0 convention fixed. For current 3-D χ-CCM-B records those labels are coulomb_kernel="3d-periodic-g0" and exchange_q0="bvk-ewald". Matching them does not prove that the union-and-weight and finite-character constructions produced the same finite Hamiltonian, variational space, or approximation. Those bindings require a route-specific derivation and fingerprint.

No such cross-approach route is currently reportable. In particular, run_ccm_rhf_gdf and aiccm-hf-direct are neutral fitted-torus controls; they are not the union-and-weight Γ-CCM construction. Their agreement with B rhf-ri tests a specified fitted operator or its Fourier evaluation, not Γ-CCM/χ-CCM approach equality.

Current comparison boundaries¶

axis	current rule	reason
Approach construction	report `not-defined`	No route map currently binds a union-and-weight Γ-CCM result to a finite-character χ-CCM result for one fully attested operator.
Neutral-torus Fourier control	use only `real_gamma_control_*` fields	Character and real-Gamma evaluations of one specified block-circulant Hamiltonian are Fourier related. This is an internal representation theorem, not a construction identity.
Exchange seam	require the same active `exchange_q0` convention	Different seams define different finite-size operators. A matching label is still only one part of the contract.
Low dimensions	do not run or quote χ-CCM-B absolute energies	All 1-D/2-D B SCF backends fail closed until a shared wire/slab Coulomb convention is derived.
Cluster size	study each approach separately	No monotone cross-approach delta or common finite-size remainder is assumed. A thermodynamic-limit claim requires its own convergence evidence.

The non-orthorhombic and ionic 3-D systems remain useful discriminating tests. Agreement on them would be evidence only after both records pass a complete construction, operator, numerical-input, and producer fingerprint.

Interpreting a disagreement¶

Check the construction identity. A B record must carry the exact ccm_approach="chi-ccm", ccm_construction="finite-translation-group-character", and evaluation_representation="gamma-centred-character-mesh" fields. A future Γ record needs its own union-and-weight identity. A real-Gamma control must not be relabelled as Γ-CCM.
Check the convention descriptor. Both results must carry coulomb_kernel="3d-periodic-g0" and exchange_q0="bvk-ewald". If not, the disagreement is a convention mismatch. If they do match, further operator and construction checks are still required.
Check what is being compared. If the pair is B rhf-ri versus aiccm-hf-direct, interpret it only as the neutral-torus character versus real-Gamma control. For an actual Γ-CCM/χ-CCM pair, stop unless a current route-specific comparison contract exists.
Check every numerical fingerprint. Basis, structure, mesh, AO space, finite operator, auxiliary fit, thresholds, smearing, source, native core, and producer attestation must match the applicable contract. Do not infer a missing field from a filename or route name.
Check dimensional support. A 1-D/2-D χ-CCM-B absolute energy is unreportable regardless of backend. Keep such rows as explicit unsupported coverage until the shared mixed-boundary kernel is derived.
If a future complete approach contract passes and a delta remains, the result is evidence of a construction, discretization, or implementation difference to localize. It is not automatically a bug in either approach.

Cross-stream property comparison¶

property	Γ-CCM	χ-CCM	current cross-approach status
HF/KS energy per atom	`energy_per_atom`	`energy / n_atoms`	`not-defined`; needs a matched construction/operator contract
Correlation energy	`e_correlation`	route-specific correlation fields	`not-defined`; reference and correlated spaces are not yet cross-bound
HOMO/LUMO gap	`ccm_homo_lumo_gap`	`fundamental_gap` in SCF record	`not-defined`; eigenvalue conventions and finite operators must be bound
Mulliken charges	`ccm_mulliken_charges`	`mulliken_charges` in properties	`not-defined`; overlap and density contracts must be established
Dipole moment	`ccm_dipole` (finite cluster)	not emitted	unavailable and gauge dependent
Forces	Γ route-specific force surface	analytic total gradient fails closed	unavailable
Density idempotency	not emitted	`density_idempotency_error`	N/A