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Production method

Scope

The production route is one named scientific method:

projected-AGP-referenced, direct full-coordinate QPCCSD followed by fixed-amplitude particle-number PAV with the Ser2/W2 closure.

The reference model, coordinate definition, energy bookkeeping, and projection schedule are invariants in the production API. They cannot be changed through a configuration file.

Quasiparticle reference

A state-specific singlet CASSCF calculation supplies the active one-particle RDM and active two-particle RDM. Within each spatial-symmetry block, the active orbitals are rotated to diagonalize the spin-summed 1-RDM. A signed projected AGP is then fitted to:

  • the resulting natural occupations; and
  • the symmetrized pair-transfer block of the active 2-RDM.

The relative geminal signs retain pair-phase information that natural occupations alone cannot provide. A final common scale sets the mean particle number of the unprojected paired Bogoliubov vacuum. Inactive correlated orbitals are occupied and external orbitals are empty in that vacuum.

The fit does not reconstruct the complete CAS 2-RDM. See reference.md for the definition and mandatory diagnostics.

Hamiltonian and cluster operator

The molecular Hamiltonian is transformed over every correlated spin orbital and normal ordered with respect to the canonical Bogoliubov vacuum |Phi>. QPCCSD uses the pure-creation cluster operator

T = T1 + T2
T1 = (1/2!) t[pq] beta[p]^dagger beta[q]^dagger
T2 = (1/4!) t[pqrs] beta[p]^dagger beta[q]^dagger
                   beta[r]^dagger beta[s]^dagger.

The connected Baker-Campbell-Hausdorff expansion terminates after the fourth nested commutator for a two-body Hamiltonian. Generated, antisymmetry-reduced Wick contractions evaluate the scalar, two-creation, and four-creation components of exp(-T) H exp(T).

Full symmetry-allowed coordinate space

The residual equations are solved in the total-singlet, totally symmetric coordinate space. All allowed indices span the complete correlated orbital space, including pair and quadruple coordinates wholly internal to the active orbitals. There is no active-space excitation mask.

“Full” therefore means all coordinates allowed by spin and molecular symmetry. It does not discard those symmetries. The frozen-core N2 CAS(6,6) regression counts are 8 pair plus 57 quadruple coordinates in STO-3G and 32 pair plus 1641 quadruple coordinates in 6-311G.

Direct energy contract

For the converged unprojected amplitudes T*,

E_raw = <Phi| exp(-T*) H exp(T*) |Phi>.

This scalar is the reported QPCCSD total energy. The CASSCF RDMs influence the reference, but the CASSCF energy is not added. The following expressions define different, experimental methods and are prohibited in production:

E_CASSCF + E_QP(T*)
E_CASSCF + E_QP(T*) - E_QP(0)
E_CASSCF + Delta_QPCCSD.

The result record stores the literal energy convention direct together with the production method identifiers, so downstream plotting cannot confuse the methods.

Unprojected optimization and rescue

The production solver combines a diagonal quasiparticle preconditioner, limited-history secant updates, an analytic Jacobian-vector product, and a right-preconditioned Newton-Krylov hookstep when ordinary iterations stall. A denominator floor may regularize only the preconditioner used to propose a step. It never shifts the Hamiltonian, the residual equations, or the energy.

Certification always reevaluates the original unshifted equations from the saved terminal amplitudes. For difficult potential-energy curves, geometry continuation and forward/reverse closure are separate campaign operations; they do not change the single-point method.

Particle-number projection after variation

For target correlated electron number N,

P_N = (1/2 pi) integral d(phi) exp(i phi (Nhat - N)).

The PAV energy is evaluated at the fixed unprojected amplitudes:

E_PAV^N = <Phi| P_N H exp(T*) |Phi>
          --------------------------------.
          <Phi| P_N   exp(T*) |Phi>

No projected residual is solved. The amplitudes and orbitals are not relaxed in the presence of the projector. This is a coupled-cluster projective energy, not a Hermitian variational quotient; it need not be lower than the unprojected value and is not an upper bound to FCI.

Production evaluates the U(1) integral on a shifted midpoint grid. At each angle, the rotated correlated state is represented by a scalar correlated norm and disentangled W1 and W2 clusters. The Ser2/W2 closure retains all components of W1 and W2, including pure-active components, and sets W3 = 0. “Ser2” denotes this closure, not a projected reoptimization.

The base grid supplies the reported energy. A second calculation on the doubled grid is a validation, not a Richardson extrapolation. The N2 defaults are 9/18 grid points for STO-3G and 25/50 for 6-311G.

Scaling and limitations

With M quasiparticle spin orbitals and L gauge points, the dominant rank-four contractions scale as O(L M^6). The implementation does not form state vectors or runtime tensors above rank four.

The main controlled approximations are the compression of a general CAS state into a paired Gaussian reference, the QPCCSD cluster truncation, and the Ser2 condition W3 = 0. Particle-number PAV removes wrong-number components but does not repair all three approximations. All reference-fit and projection diagnostics must therefore accompany energies used in a scientific claim.