Internal OMx Gaussian pair operands

The private native semiempirical.nddo._evaluate_omx_gaussian_pair accessor accepts two _OMxGaussianContraction records with explicit exponents, s coefficients, optional p coefficients and positive zeta. Coefficients multiply normalized Cartesian primitives. Exponents in bohr^-2 are multiplied by zeta squared and contractions are explicitly normalized. raw_norms_a/b record squared norms before this normalization. No published basis tables are bundled by this interface.

Atom-local AO order is s, px, py, pz (or s only), with positive Cartesian p phases. Origins are bohr. Overlap is dimensionless. ERIs, attractions and core energy are Hartree. eri_unscaled and eri_scaled are pair matrices with row mu*n_a+nu and column lambda*n_b+sigma, representing (mu_A nu_A | lambda_B sigma_B). All density-product channels are retained. The supplied gamma_ss sets f_ko = gamma_ss / eri_unscaled[0,0]. Attractions include the negative sign and the other core’s supplied charge; core energy is f_ko * core_charge_a * core_charge_b / R.

coulomb_a(P_b) and coulomb_b(P_a) contract the complete scaled tensor with a symmetric spin-summed atom-local density, without an extra spin factor. They return Coulomb contributions only. _evaluate_omx_legacy_pair_operands provides the existing two-atom overlap and F(P)-F(0) for algebraic baseline comparisons without running SCF.

These are internal development operands, following Dral et al., JCTC 12, 1082 (2016), DOI 10.1021/acs.jctc.5b01046, section 2.3 and equations 18-20. The current OM2/OM3 driver and parameter tables are unchanged. The pair API is not a complete published OMx Hamiltonian: consistent overlap, penetration, valence-core, orthogonalization and derivative integration still require implementation and independent numerical validation. Operand agreement does not establish an SCF state, minimum, external gradient tolerance or molecular energy agreement. Refs #496.