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Projected-AGP reference

CAS source

Let |Psi_CAS> be a state-specific singlet CASSCF solution. The spin-summed spatial one-particle RDM is

gamma[p,q] = <Psi_CAS| sum_sigma a[q,sigma]^dagger a[p,sigma] |Psi_CAS>.

Within each spatial-symmetry block, active orbitals are rotated so that gamma[p,q] = n[p] delta[p,q]. The active 2-RDM is transformed by the same rotation. In the spin-free convention used by the implementation, the fit target is the symmetrized pair-transfer block

Pi[p,q] = 1/2 (Gamma[pq,pq] + Gamma[qp,qp]).

This exact index convention must be recorded with every reference artifact.

Fixed-number AGP fit

For n = N_active / 2 active electron pairs and pair creator P[p]^dagger = a[p,alpha]^dagger a[p,beta]^dagger, the fitted state is

|AGP_Nactive> proportional to
P_Nactive product_p (1 + g[p] P[p]^dagger) |vacuum>.

For an n-orbital subset I, define C_I = product_(p in I) g[p] and Z_n = sum_(|I|=n) C_I^2. Exact subset sums give the AGP occupations and pair transfers:

n_AGP[p] = (2/Z_n) sum_(I contains p) C_I^2

Pi_AGP[p,q] = (2/Z_n) sum_(I contains q, p not in I) C_(I-q+p) C_I,
Pi_AGP[p,p] = n_AGP[p].

Relative signed geminals minimize the equal-weight least-squares differences between n_AGP and n_CAS, and between Pi_AGP and Pi_CAS. Initial magnitudes follow sqrt(n[p]/(2-n[p])); relative signs come from the dominant eigenvector of the target pair-transfer matrix. Logarithmic magnitude ratios remove the irrelevant common scale of fixed-number AGP.

No energy, FCI result, CC result, or later geometry is used to select the fit.

Unprojected Bogoliubov vacuum

The common AGP scale cancels in the fixed-number state but matters in the unprojected quasiparticle vacuum. A positive scale lambda is therefore chosen to satisfy

sum_(p in active) 2 (lambda g[p])^2 / (1 + (lambda g[p])^2) = N_active.

Then

u[p] = 1 / sqrt(1 + (lambda g[p])^2)
v[p] = lambda g[p] u[p].

Inactive correlated orbitals use (u,v)=(0,1) and external orbitals use (u,v)=(1,0). The resulting matrices must satisfy

U^dagger U + V^dagger V = I
U^T V + V^T U = 0.

The Gaussian Bogoliubov vacuum used in Wick contractions and its target-number AGP component are related, but they are not the same object.

What information is and is not retained

The reference uses the diagonal active 1-RDM and one pair-transfer block of the active 2-RDM. It does not use every element of the active 2-RDM, nor the active 3- or 4-RDM. A generic CAS wavefunction cannot be reconstructed from this information. The compression error is a reference-model limitation, not a QPCCSD convergence error, and is never converted into an energy correction.

Each result must retain at least:

  • CASSCF convergence and energy as provenance only;
  • active electron and orbital counts;
  • frozen, active, and external orbital identities;
  • natural occupations and orbital-symmetry labels;
  • signed relative geminals and the global number-setting scale;
  • maximum occupation-fit and pair-transfer-fit errors;
  • normalized pair-subspace fidelity;
  • target mean active number; and
  • both Bogoliubov canonical defects.

The production serializer retains these compactly as active natural occupations, relative and number-scaled signed geminals, and the explicit global number-setting scale. It also records deterministic SHA-256 fingerprints of the active 1-RDM, active 2-RDM, and correlated MO coefficient array without embedding those full arrays in result JSON.

The source-RDM record is fail-closed. It must state exactly the spin-summed spatial 1-RDM convention, the PySCF spin-free 2-RDM convention, the active_rdm2[p,q,p,q] pair-transfer indexing, the versioned projected-AGP fit protocol, and complete_active_rdm2_reconstructed = false. A checkpoint that omits any of these fields is not inferred or upgraded during serialization.

The serialized production result also retains compact SHA-256 fingerprints of the active 1-RDM, active 2-RDM, and correlated molecular-orbital coefficients. These fingerprints identify the numerical reference inputs without expanding the result JSON with full dense arrays.

For a potential-energy curve, orbital/block alignment, CI phase continuity, and signed-geminal continuity should also be recorded. They diagnose branch continuity but do not replace the fit-error fields.