AICCM2026DEV-B decisions and open questions

Last updated: 2026-06-22.

Decisions

ID

Decision

Reason

D1

Define the method as a finite BvK torus constrained to primitive-translation-invariant RHF determinants.

Gives a precise Hamiltonian, variational space, and finite-size problem.

D2

Evaluate the cyclic Gamma-supercell problem through the full unreduced Gamma-centred character mesh.

Exact finite-group Fourier equivalence; reuses the validated SCF/integral infrastructure.

D3

Use Wigner–Seitz weights only to partition tied minimum-image representatives of one translation class.

Weights sum to one and preserve one- and two-electron permutation symmetries.

D4

Reject the historical pair-product four-centre weighting.

Its displayed factors are not generally invariant under bra–ket interchange, so a conventional RHF scalar functional does not follow.

D5

In 3D use the common neutral periodic Ewald/GDF gauge, remove the Hartree (G=0) mode, and apply the BvK-supercell exchange Madelung correction.

Keeps electron–electron, electron–nuclear, nucleus–nucleus, and exchange finite-size conventions consistent in the 3D route.

D6

Reject charged cells and smearing in Phase 1.

Both require a different explicitly stated variational functional/background convention.

D7

Keep the existing CCM untouched and expose aiccm2026dev-b as a new PeriodicJKMethod.

Allows head-to-head comparison with no behaviour change to the peer implementation.

D8

Reuse native GDF, Ewald, basis, lattice, canonical orthogonalisation, DIIS, and output infrastructure.

These are not AICCM-specific; duplicating them would reduce comparability and correctness.

D9

Accept only the pair-resolved RSGDF or MDF fit; reject the q-only compcell fit.

The latter is not a consistent four-centre finite-torus Hamiltonian on tight cells and has known non-physical fixed points.

D10

Initialise the existing periodic citation flag for non-KPoints inputs.

The repo-wide gate exposed an unrelated unbound local in citation output. The false default restores the intended existing behaviour and does not alter any SCF result.

D11

Refresh two stale k-point recommender assertions found by the repo-wide gate.

They still expected the retired ungated-model error and Fermi–Dirac default, while the shipped implementation and changelog specify the environment gate and Methfessel–Paxton default. Production behaviour is unchanged.

D12

Expose three independently selectable integral backends: four_center, ri, and rijcosx.

They allow direct-versus-three-centre consistency tests when their dimensional Coulomb gauges are matched.

D13

Reuse the corrected-gauge BIPOLE Fock engine for four_center; do not use the legacy multi-k Ewald direct-exchange path.

The former contains direct four-centre short-range J/K plus the complete reciprocal long-range and BvK exchange correction. The legacy path is documented upstream as missing the exchange correction.

D14

Permit closed-shell RHF/RKS in 1D, 2D, and 3D, including hybrids.

The finite group is dimension independent. The 1D/2D four-center direct-truncated gauge and RI fitted metric are explicitly documented as unmatched absolute-energy conventions.

D15

Fail closed for RI/RIJCOSX RKS and RIJCOSX RHF at a one-cell mesh.

The native Gamma-only RKS GDF path is not pair-resolved and the COSX bridge is multi-k only; silently falling back would mislabel the backend.

D16

Audit the 2008 deMon2k KS-ADFT CCM separately from the 2014 four-center formula.

Janetzko et al. use multiplicative occurrence weights, but their Eq. (20) explicitly averages both AO-center orderings. The same bra–ket diagnosis cannot be copied without checking that symmetrization; quotient-space adjoint consistency and long-range truncation remain open.

D17

Transpose the repository’s row-lattice array when constructing the mathematical lattice generator.

The derivation uses lattice vectors as columns, while PeriodicSystem.lattice stores them as rows. A skew 2D regression test makes this distinction observable; diagonal cells cannot detect it.

D18

Build B-stream MP2 from the B RI-RHF character-mesh reference, not the existing toroidal Gamma-supercell HF helper.

The two references differ by 4.6408868 mHa per cell on the two-cell 8 x 12 x 12 bohr H2 control, so substituting the latter changes the finite zeroth-order Hamiltonian.

D19

Use a truncated metric inverse square root expressed in the original auxiliary-AO basis for post-HF momentum blocks.

Compact eigenspace factors have arbitrary phases and rotations independently at (q) and (-q). The canonical matrix function removes that gauge from cross-block MP2 contractions.

D20

Expose canonical RI-MP2 initially for 3D neutral closed-shell systems and fail closed for 1D/2D.

The 3D two-point H2 result agrees with external KRHF/KMP2 below one nanohartree per cell. The lower-dimensional long-range convention remains O1 and cannot yet support a single reference Hamiltonian.

D21

Transform the pair-resolved RI Hamiltonian to a real finite torus before local correlation.

The derived (N_c^{-2}) inverse transform reproduces the momentum-space RI integral normalization exactly and preserves the B reference instead of substituting the legacy Gamma-supercell Hamiltonian.

D22

Count every occupied pair and ordered triples contribution on the finite torus, then divide the total correlation energy by (N_c).

This is unambiguous and passes the exact-limit oracle. Translation-reduced pair/triple orbits require an orbit-stabilizer derivation and are not approximated by occurrence weights.

D23

Use PBC-safe Pipek–Mezey occupied localization; reject molecular Foster–Boys localization.

Mulliken populations do not require the ordinary position operator, which is discontinuous on the torus boundary.

D24

Keep pair-distance screening, finite occupied-coupling radii, and local fitting disabled in the B local-correlation route.

The existing molecular algorithms use Euclidean distances. Minimum-image Wannier and auxiliary domains are required before those approximations can preserve translation symmetry. PNO truncation itself remains available.

D25

Localize one complete finite-torus occupied space, never independent k blocks.

A single unitary rotation preserves the occupied projector, density, and SCF energy exactly and produces translationally related occupied orbitals for local correlation.

D26

Label the present periodic spread as a projected circular AO-centre approximation, not an exact Marzari–Vanderbilt spread.

Exact continuum localization needs cross-k exponential-position matrix elements that the integral API does not yet expose.

D27

Construct B-specific PAOs with the metric projector and canonical Gram orthogonalization, and construct PNOs from a Hermitian positive-semidefinite pair density.

The algebra removes occupied leakage and PAO redundancy and gives nested PNO ranks as the occupation threshold is tightened. MP2 energy errors are not variational and need numerical validation.

D28

Enumerate occupied-pair orbits by closure under measured localization translation permutations, but do not skip representative pairs yet.

Orbit multiplicities are exact only after translation covariance is proven. The current Wannier control passes, while finite-supercell IAO cross overlaps leave a residual near (10^{-5}). Representative-only amplitude scattering has no energy-parity gate yet.

D29

Expose space-group symmetry as an off-by-default diagnostic and fail closed for integral reduction.

The compatible finite-cluster subgroup, nonsymmorphic atom/cell maps, irreducible k orbits, and primitive shell-pair/quartet orbit partitions are derived. General-k AO sewing matrices and representative scatter are not yet proven in every backend convention.

D30

Before the real local solver, construct and verify a full-rank real time-reversal gauge; at the complete-domain, zero-PNO-threshold MP2 limit, audit the result with an independent rotation-invariant full-space DF-MP2 contraction.

The first integration exposed unsafe complex-to-real casts in a solver whose equations are real. Explicit realification preserves the occupied projector. The public result reports both the raw value and any full-space correction. No correction is applied to truncated calculations.

D31

Make the real-space lattice extension the primary user parameter and derive the full Gamma-centered character net from it.

The method is a finite BvK torus, not two independently converged real- and reciprocal-space approximations. A shell radius (s_i) means (N_i=2s_i+1); the old mesh tuple remains only an exact compatibility alias.

D32

Extend UHF and UKS with independent alpha and beta idempotent projectors on the same finite Hamiltonian.

Spin changes the variational ranks and exchange contractions, not the translation quotient or Wigner–Seitz partition. Separate density, electron-count, and spin-contamination audits make this explicit.

D33

Build unrestricted post-HF from the exact real transform of the B RI-UHF Hamiltonian.

Reusing a molecular or legacy Gamma reference would change the finite zeroth-order Hamiltonian. The supercell multiplicity is (N_c(M-1)+1), preserving the spin difference in every primitive cell.

D34

Treat the present unrestricted DLPNO-UCCSD(T) engine as an O(N^6) correctness oracle.

Full-domain projection proves the PNO exact limit, but translation-representative amplitude propagation and minimum-image domains are not implemented. Calling it production reduced scaling would be false.

D35

Expose only gauge-invariant SCF properties derivable from the finite-net one-particle density.

Electron/spin counts, Mulliken populations, finite-net gaps, idempotency, and ⟨S²⟩ are defined. Cell dipoles, Berry-phase polarization, analytic response, and correlated properties need additional theory and fail closed.

Source disagreements recorded

  • The 2014 paper’s numerical convergence is evidence, not a proof of a variational energy. The independent formulation supplies the missing finite Hamiltonian and constrained variational statement.

  • The historical two-, three-, and four-centre pair-product boundary factors are not adopted. aiccm2026dev-b weights translation equivalence classes, not arbitrary centre pairs.

  • A converged, plausible energy is not accepted if the density is non-idempotent, the electron count is wrong, the Fock is non-Hermitian, or the finite-mesh/supercell identity fails.

  • A long-range correction added only to the final scalar energy is insufficient. Its functional derivative must be present in the SCF Fock.

  • The deMon2k two-center claim that the reference center is arbitrary requires equality of the two weighted representative sums, not only symmetry of the underlying free-space integral. Its explicitly symmetrized three-center Eq. (20), however, already addresses the simplest AO-pair interchange.

Open questions and queued work

ID

Question / work

Current position

O1

One common long-range convention for 1D chains and 2D slabs.

The B stream now runs all backends in 1D/2D. Four-center uses the direct-truncated lower-dimensional BIPOLE gauge; RI uses its fitted dimensional metric. Their large absolute-energy differences are not fitting errors until a common wire/slab Green function is implemented.

O2

Global-minimum and stability analysis.

RHF SCF gives a stationary determinant in the constrained space. Add orbital-Hessian/stability checks before production status.

O3

Finite-size convergence law for exact exchange.

Measure rather than assume monotonicity; fit insulating test ladders by geometry and basis.

O4

Open-shell production validation.

UHF/UKS now run through four-center, RI, and RIJCOSX in 1D/2D/3D, and 3D UMP2/UCCSD(T)/PNO pilots use the exact B transform. Add external periodic open-shell references, stability analysis, and larger spin-contaminated controls.

O5

KS-DFT production validation.

Closed-shell RKS now runs through direct, RI, and RIJCOSX backends. Grid convergence, nonlocal functionals, forces, and the historical DFT weighting comparison remain open.

O6

Coupled cluster and local correlation.

Restricted and unrestricted real-torus PNO MP2/CCSD/(T) routes are implemented in 3D, including full-domain exact-limit oracles. They still evaluate all pairs/triples with full contractions. Translation-representative amplitude propagation, minimum-image screening/local fitting, relaxed correlated densities, and external periodic CC benchmarks remain open.

O7

ECP electron/background bookkeeping.

Phase 1 is validated first with all-electron minimal bases. Add explicit effective-charge neutrality checks before ECP benchmarks.

O8

3D ionic benchmark ladder.

LiH, MgO, NaCl, and diamond need matched all-electron basis/reference conventions and full mesh convergence data.

O9

H4 four-centre versus RI discrepancy.

At the two-cell, 40-bohr-vacuum H4 geometry, corrected-gauge four-centre RHF is -2.1870080428 Ha/cell while RI is -2.1648043358 Ha/cell, a -22.204 mHa difference. The compact H2 two-cell control agrees across all three backends within 44 microhartree. Treat the H4 difference as unresolved truncation/cell-embedding behaviour, not as evidence that either number is the dense-limit oracle.

O10

Exact periodic IAO cross overlap.

The current IAO target/reference overlap is evaluated in the explicit supercell and omits cross-boundary images. Periodize it before using IAO translations to reduce correlated pairs.

O11

Space-group integral acceleration.

Derive backend-specific general-k sewing matrices, real-solid-harmonic shell actions, quartet stabilizers, and petite-list scatter. Enable only after symmetry-on/off Fock and energy parity. Layer and rod groups and little-group irrep labels are also open.

Current measured evidence

The committed benchmark was run on the alternating H4/STO-3G chain with a 40-bohr transverse simulation lattice. For cyclic lengths 1, 2, and 4, the new energy per cell was respectively -2.0733777009, -2.1648043358, and -2.1719609271 Ha. A direct call to the full Gamma-centred character mesh was identical to all printed digits at every length. The historical union12 CCM gave -2.1615967328, -2.1618792916, and -2.1707034746 Ha/cell. The separately developed symmetric aiccm2026dev-a weighting gave the same displayed sequence in this 1D case. This is useful comparison evidence, not a correctness proof. New-route wall times in the archived current-main run were 22.58, 20.61, and 97.23 seconds; the four-cell jump reflects all-k-pair exact-exchange fitting.

That H4 ladder used the ri backend. The backend-crossing smoke benchmark uses H2/STO-3G in an 8 x 12 x 12 bohr cell at a (2,1,1) cyclic mesh. RHF energies are -1.1213506456 (four_center), -1.1213722182 (ri), and -1.1213943367 Ha/cell (rijcosx). PBE0 energies are -1.1554208287, -1.1554589526, and -1.1554644822 Ha/cell in the same order. These are actual runs from 2026-06-21; the maximum spread is 43.7 microhartree for RHF and 43.7 microhartree for PBE0. They establish executable cross-backend consistency on this compact control, not a universal error bound.

The 1D/2D route matrix was also run on H2/STO-3G at a (2,1,1) cyclic mesh. Every RHF and PBE calculation converged through four-center, RI, and RIJCOSX, with idempotency errors below 3e-15. The 1D RHF energies were -1.5341898653, -2.2845855922, and -2.2846076938 Ha/cell; the 2D values were -3.4506478026, -5.2158679833, and -5.2158905993 Ha/cell. The large four-center/RI gaps are the evidence behind O1, not a backend-accuracy result.

The first 3D canonical RI-MP2 control uses an H2/STO-3G cell of 20 x 20 x 6 bohr and the (1,1,2) character mesh. B gives HF, correlation, and total energies of -1.1182352381008094, -0.0130227001540294, and -1.1312579382548390 Ha per cell. The out-of-process PySCF 2.13.1 KRHF/KMP2 reference gives -1.1182352386944080, -0.0130226999202183, and -1.1312579386146264 Ha per cell. The corresponding absolute differences are below 0.6 nanohartree. The archived record is docs/manuscripts/aiccm_comparison/data/h2_mp2_2026-06-21.json.

The exact inverse transform was tested by running the existing local solvers on the real finite torus. H2/STO-3G at the same (1,1,2) mesh gives a no-truncation local-MP2 correlation energy identical to canonical B RI-MP2 within 7e-18 Ha per cell. A LiH/STO-3G cell of 20 x 20 x 7 bohr at the same mesh gives canonical finite-torus DF-CCSD and (T) totals of -0.0302475329439885 and -0.0000934031163267 Ha. The no-truncation local solver differs by 6.9e-12 and 1.4e-12 Ha, respectively. The nonzero-triples record and inverse-transform residuals are archived in docs/manuscripts/aiccm_comparison/data/lih_dlpno_ccsdt_2026-06-21.json. This is an internal finite-Hamiltonian oracle, not an external infinite-crystal CCSD(T) result.

The first unrestricted transform control is a neutral Li doublet in a 15 x 15 x 15 bohr cell with STO-3G and extension (1,1,1), run on 2026-06-22. RI-UHF is -7.32670108299121 Ha/cell, UMP2 correlation is -0.000243259672659 Ha/cell, and full-domain UCCSD correlation is -0.000257809105931 Ha/cell. The triples correction is numerically zero for this three-electron minimal-basis control. The transformed RI factor has a 2.31e-16 maximum imaginary residual and a 7.11e-15 permutation residual. Independent alpha/beta localization changes the density by 4.3e-34, and the zero-threshold localized UCCSD(T) total agrees with the canonical gauge within 5e-14 Ha. These values establish the finite-torus unrestricted transform and PNO exact limit only; a one-cell Li array is not a converged crystal benchmark.