Finite-character RI-MP2 in Paper 2¶
The Paper 1 finite-torus RHF proofs are indexed in formal_aiccm.md. For Paper 2, VibeQCAICCM/ChiMP2.lean starts from supplied, whitened primitive factors and canonical closed-shell orbital energies. It does not prove a second SCF, fit or coupled-cluster implementation.
Eqs. 17, 19 and 20. For each fixed auxiliary and primitive AO pair, chi_inverse_entries expands W times the phase-weighted block times W adjoint. Each orbital column supplies 1/sqrt(N), and chiLeg carries 1/N, so the cell tensor has 1/N^2. chi_project_leg applies the two unitary inverses. The primitive occupied and virtual coefficients contract within their respective modes. Conjugation of the resulting cell entry is trivial only under a reality condition; chi_real_cell proves sufficiency of conjugate partners under simultaneous momentum inversion in the real AO frame.
Eqs. 11 to 13 and 21. chi_fourth_iff gives b=i-a+j from opposite transfers. Character orthogonality in the auxiliary-cell sum, not in a pointwise pair product, gives the momentum delta and a factor N. chi_transfer_sum and chi_fitted_integral leave a bilinear product of the opposite-transfer columns divided by N. chi_cell_eri composes this result with the transform of each of the two cell matrices. CanonicalFactorAdapter.integral instead pairs (i,a) with (b,j) at the same transfer and conjugates the reversed second leg. chi_reverse_mo proves equivalence under reversal by adjunction in a common retained auxiliary frame. Time reversal and pair adjunction are distinct.
Eqs. 9 and 14 to 16. For fixed occupied bands and momenta, label a virtual choice by (a,A,B). Interchanging the virtuals sends it to (i-a+j,B,A). This permutation is its own inverse; chi_den_swap shows it preserves the real energy denominator. Pair a term with its swap: the sum of the two complex exchange cross terms is twice their real part. chi_same_spin converts the complete same-spin sum to one half of the squared antisymmetrized integral divided by the denominator. No individual term needs to be real. Identical alpha and beta channels supply one quarter each. chi_mesh_scale and chi_mp2_energy give the combined OS plus SS energy with N^-3 on raw primitive contractions: two powers from the squared integral, one per cell. Keep the complete swap-closed virtual space, fixed active occupations and nonzero denominators. The code rejects nonnegative denominators and negative ones within its tolerance. It uses the same reference seam in the eigenvalues, without adding a separate MP2 seam energy.
Run python3 formal/check_proofs.py with the pinned Lake. Submit formal/check_chi_mp2.py as a single file via vq on a qualified run-python. It builds a native three-character closed-shell H2 reference and calls run_chi_mp2. A separate calculation Fourier-transforms the actual AO factors to cells, rotates complex MO phases and sums all four character indices without the production conservation table. It checks OS, SS, the forbidden integrals and the total, with wrong conjugation, exchange and normalization controls. This is a same-fit comparison, not external validation or an error bound on density fitting. No production code, toolchain or dependency change is needed.
Pair density before PNO truncation (Paper 2, Eq. 22)¶
In the real finite-torus local-MP2 driver, take one placed occupied pair and its full supplied pair virtual space. Remove linear dependencies and use overlap-orthonormal, semicanonical coordinates. This space may already be smaller than the torus virtual space. The seed is the square real matrix T_ab = K_ab / (F_ii + F_jj - eps_a - eps_b). The numerator and orbital energies are finite, and each denominator must be nonzero. No assumption about its sign is required for this identity. T, D and the occupations are dimensionless. Let w = 1/(1 + delta_ij), where equality compares the complete placed occupied labels, and H = w (4T - 2T^t). The density passed to diagonalization is D = H T^t + H^t T, with no further pair or cell factor. This is the closed-shell spin-adapted density of Riplinger and Neese (J. Chem. Phys. 138, 034106, Eqs. 23-24).
chi_pair_gram: put C = T + T^t and A = T - T^t. Expanding gives D = w (C C^t + 3 A A^t). Thus each real test vector has nonnegative quadratic form (chi_pair_pos), and all occupations are nonnegative (chi_pair_occupations). Taking the trace sums the squared entries; chi_pair_trace and chi_pair_total_occupation identify this weighted sum of squares with the total occupation.
chi_pair_reverse: replacing T by its transpose changes only the sign of A. This gives pair-order independence when reversal transposes the seed in the same frame. For a diagonal pair with symmetric seed, A vanishes and w = 1/2: D = 2 T T^t (chi_pair_diagonal).
chi_pair_bounds: for B = w (T T^t + T^t T), the differences D - 2B and 6B - D are respectively 2w A A^t and 2w C C^t. The bare density B is not generally proportional to the spin-adapted density.
chi_pair_embed: for real U with U^t U = I, the density of U T U^t equals U D U^t by contraction of the inner U^t U. This also covers a rectangular embedding in orthonormal coordinates, not a raw array of nonorthogonal AO coefficients. chi_pair_complete: if D V = V n with diagonal n and V V^t = I, then V n V^t = D, including any zero occupations. This identifies the density, not individual eigenvectors in a degenerate eigenspace.
The production entry point periodic.chi.local_mp2.run_chi_local_mp2 requests pno_norm=”mp2”. dlpno.translation_builder.build_translation_pair_pnos forms the seed, calls dlpno.pno_density.pair_density and sorts eigenvalues in descending order. Zero cutoff retains the full input space. A positive cutoff is strict, with a leading-vector fallback if none qualify. The subsequent Fock diagonalization rotates within the retained span. Separately, periodic.chi.pno.build_pair_natural_orbitals uses the bare Hermitian density with adjoints. The real spin-adapted result does not justify substituting adjoints in that spin formula in a complex frame.
Run formal/check_chi_pair_density.py via the same qualified runtime as the canonical MP2 check. On asymmetric real input and on the seeds from the native three-cell hydrogen calculation, the reference stacks the columns of C once and of A three times, scales by sqrt(w), and uses an SVD rather than the production density contraction and eigensolver. It checks the real driver, a complex bare helper, pair order, diagonal weight, zero occupations and a rectangular isometry. Deliberate errors include the best scalar multiple of the bare density and omission of two odd-spin channels. This check does not certify a selected PAO domain, PNO truncation error, floating-point equivalence or a CCSD(T) energy.