Experimental chi same-source RHF/UHF CCSD and triples

The chi CC route consumes the sealed stationary classic RI RHF or UHF state and its retained canonical factors. The public runner selects it with method="aiccm", variant="chi", correlation="ccsd", scf_reference="rhf" or scf_reference="uhf", aiccm_backend="ri", fit_source="classic", and the same explicit classic physical-pair controls required by chi MP2. It computes CCSD plus the precisely defined canonical spin-orbital triples correction below by default. Pass chi_cc_options=ChiCCOptions(compute_triples=False) for CCSD alone; ChiCCOptions lives in vibeqc.periodic.chi.canonical_cc. The library entry point is vibeqc.periodic.chi.correlation.run_chi_ccsd(reference, ...).

The source, support, kernel, metric rank and HF orbitals are never rebuilt or substituted. Integer gapped explicit-electron occupations, complete unshifted characters and the existing tight stationary-HF acceptance are required. chi_frozen_core counts lowest occupied orbitals per character in each spin. An integer applies to both spins; UHF also accepts (ncore_alpha, ncore_beta). The remaining alpha/beta occupied and virtual dimensions may differ. All-electron and classic scalar ECP references are admitted. Frozen spaces contain only explicit occupied orbitals, after ECP core removal. SR CC, ROHF-CC, KS/hybrid, spin-orbit ECPs and fractional occupations are unsupported in this increment. The accepted SR RHF MP2 route is unchanged; original SR10 negative metric modes and SR32 complex UHF operator failures remain gates. Chi is experimental; D93 fitted/post-HF is not-qualified. R5.3 is an unrestricted source implementation. Restricted R5.2 scientific acceptance remains separate, and neither it nor accepted scoped UMP2 results provide CC acceptance by inheritance.

Equations and conventions

The implementation uses the complex spin-orbital form of Stanton et al. (1991), section II, equations 1-13, with t1[a,i], t2[a,b,i,j] and antisymmetrized g[p,q,r,s] = <pq||rs>. Amplitudes multiply excitation operators and are not conjugated in the CCSD energy or residual equations. Bra orbitals in integrals are conjugated. The factor product is (pq|rs)=L[pq] L[sr]^H, using one shared transfer metric, including at complex characters.

The occupied and virtual lists carry (character, canonical MO, spin) labels. Each UHF spin channel uses its own sealed coefficients, orbital energies, density, occupations and Fock matrix. The consumer checks orbital orthonormality, F_sigma C_sigma = S C_sigma epsilon_sigma, and the occupied projector against that spin’s density. It makes no semicanonical rotation and does not identify alpha and beta spatial orbitals. Freezing occurs after the accepted canonical frame is established, without further core/valence mixing. Source seals bind both spin channels and the original source, kernel, q=0 convention, support, metric rank and factors.

T1 = sum_ai t1[a,i] a^dagger_a a_i and T2 = (1/4) sum_abij t2[a,b,i,j] a^dagger_a a^dagger_b a_j a_i. The full antisymmetric spin-orbital amplitudes contain independent alpha/beta singles and aa/ab/bb doubles. The energy is sum_ia f[i,a] t1[a,i] + (1/4) sum_ijab g[i,j,a,b] t2[a,b,i,j] + (1/2) sum_ijab g[i,j,a,b] t1[a,i] t1[b,j]. For spin-integrated doubles, the same-spin contributions therefore have weight 1/4 and the mixed-spin contribution has weight one. No extra closed-shell factor of two is applied. The residuals are projections of exp(-T) H exp(T) and retain all spin sectors allowed by the determinant.

Each excitation conserves crystal momentum modulo a reciprocal vector. As in McClain et al. (2017), section 3, equations 26-28, the reference is a finite complete-character Hamiltonian. Primitive-cell factors give supercell integrals after division by Nk; the resulting supercell correlation energy is divided by Nk. Dense spin amplitudes currently retain zero forbidden sectors. This costs more memory than fully momentum-compressed amplitudes without changing the mathematics. There is no CCM construction or molecular-source substitution.

The retained factors omit zero momentum. HF already includes the BvK Madelung exchange term -xi S D_sigma S. ChiCCOptions.coulomb_zero_mode explicitly selects the finite-mesh CC convention:

  • "hf-energy-only" preserves the historical default: corrected HF energies and bare retained ERIs. This is an energy-only finite-size approximation; it is not a constant-kernel many-electron Hamiltonian.

  • "matched-eri" also adds Nk*xi to a primitive-cell Coulomb ERI when both orbital pairs are diagonal in character and canonical MO. After division by Nk, its spin-orbital correction is xi*(delta_pr delta_qs - delta_ps delta_qr).

Both use the actual physical HF Focks and unchanged fitted factors. The executed scalar is recovered from both spin Focks and checked against every matrix element without rebuilding native integrals. Output identifies the convention and madelung_hartree. Library calls and the public runner use the same ChiCCOptions; pass chi_cc_options to run_periodic_job.

The matched treatment follows Xing and Lin (2024), section II B, equations 7-11; their signed constant is -xi here. A constant interaction contributes only a constant on a fixed-electron Hilbert space after the matching uniform Hartree/background convention. Its occupied-energy and general-ERI contributions therefore cancel in the converged CCSD residual. Independent fixed-electron determinant tests check this cancellation, also for complex amplitudes. The energy-only approximation does not obey that invariance; the two conventions must not be compared as the same finite Hamiltonian. Their common thermodynamic limit does not establish accuracy at a tested finite mesh. Changing the historical default remains a separate maintainer decision, consistent with the periodic source design record.

Frozen spaces retain the same scalar; ECP-removed electrons never enter the active labels. Occupied-virtual MP2 numerators have no diagonal zero-mode contribution, so MP2 is unchanged. The defined perturbative triples formula retains its HF denominators in both CC modes; constant-kernel invariance of converged CCSD is not a claim of invariance of finite-order triples.

The missing explicit CC convention, unavailable matched-ERI treatment and absence of a convention-specific restart boundary are repaired here. HF archives remain reusable, and old CC v1 archives remain readable only as energy-only calculations. Changing modes requires fresh correlation from the retained HF, as documented in restart. The finite-size diagnostic separates this choice from the remaining HF cell/character and fitted-cutoff errors.

The triples definition is canonical-spin-orbital-hermitian-(T)/v1, based on Raghavachari et al. (1989), section 5, equation 14. With the normal-ordered fluctuation operator V, define C=<triple|(V T2)_connected|reference> and D=<triple|V T1|reference>. For i<j<k, a<b<c, the correction is sum C.conj()*(C+D)/Delta, with Delta = epsilon_i+epsilon_j+epsilon_k-epsilon_a-epsilon_b-epsilon_c. More explicitly, with A_ijk A_abc the signed sums over all 36 permutations,

C_ijk^abc = (1/4) A_ijk A_abc [sum_e t_ij^ae g_bc,ek
                              - sum_m t_im^ab g_mc,jk]
D_ijk^abc = (1/4) A_ijk A_abc [t_i^a g_bc,jk]

The factor 1/4 accounts for each seed’s already antisymmetric pairs; unique triples replace the fully ordered energy sum’s 1/36. All allowed aaa, aab, abb and bbb spin sectors are included, with spin and momentum conserved between holes and particles. The singles coefficient is one, as in equation 14; this is neither doubles-only CCSD+T(CCSD) nor QCISD(T).

For a real canonical UHF determinant this is the conventional spin-orbital CCSD(T) expression. For complex orbitals the stated conjugation is an explicit Hermitian extension; the real-orbital paper alone does not establish its accuracy for every complex Hamiltonian. The implementation refuses an imaginary correction exceeding 1e-9 Ha in the supercell sum instead of silently taking its real part. Independent orbital-phase and determinant checks establish the stated finite-Hamiltonian contraction, not a general open-shell triples qualification. Result aliases are UCCSD / UCCSD(T) with reference_method="UHF"; triples_variant records the definition and citation. RHF aliases remain CCSD / CCSD(T).

The formula requires canonical HF with zero occupied-virtual Fock elements. It does not implement additional general-reference Fock terms, ROHF semicanonical choices, spin-adapted open-shell CC, Lambda-CCSD(T), or CCSD-T. Those names must not be used interchangeably with this UHF route. Two independent HF calculations are not a same-root closed-shell reduction test; that test requires identical alpha/beta orbitals, energies and occupied spaces.

Resource and convergence contract

The virtual ladder streams one external virtual index. Non-virtual integral blocks are built from cached MO factor panels; a full four-index spin integral tensor and a six-index triples tensor are not required. The conservative complex array workspace estimate includes cached blocks, intermediates, amplitudes and eight DIIS vectors/errors, plus the MO pair factor cache and transform workspace. It excludes the borrowed HF factors and external BLAS/interpreter overhead. memory_byte_cap refuses an excessive estimate before MO blocks or amplitudes are allocated. No contributions are dropped to fit the budget. Triples still have canonical seventh-order cost.

Both the maximum projected residual and energy change must converge. max_iter, energy_tolerance, residual_tolerance and denominator_tolerance are explicit ChiCCOptions controls. Capped or nonfinite iterations raise an error. Single, double and triple denominators use the summed occupied minus virtual canonical energies. Nonnegative or small denominators in allowed sectors are refused, never clamped. Forbidden spin/momentum sectors and repeated antisymmetric indices do not require denominators. DIIS affects iteration only, not the target equations.

Output separates HF energy, CCSD correlation, triples, total correlation and correlated total, all per primitive cell. The nuclear contribution and correlated electronic energy (total minus nuclear repulsion) are separate manifest fields, with electronic energy also printed in .out. The .out and .system records include source/reference seals, active counts, retained ranks, convergence, amplitude maxima, residuals, workspace estimate and actual citation route. Active counts and canonical residual diagnostics are resolved by spin. The wrapper’s energy remains the HF energy, matching existing periodic correlated runners; use e_total_correlated for the correlated total.

Independent tests construct a determinant Hamiltonian and evaluate exp(-T) H exp(T) at arbitrary complex amplitudes without calling the production residual builder. A separate determinant contraction checks a nonzero triples correction. These algebra checks do not establish physical basis, auxiliary-space or thermodynamic convergence or production qualification.

Persistent continuation

run_chi_ccsd and evaluate_same_source_cc accept checkpoint_path, resume_from and checkpoint_interval. The classic source, canonical/frozen spaces and equation controls must match. Increasing ChiCCOptions.max_iter continues an iteration-limited solve with its amplitudes, cumulative trace and compatible DIIS history. A triples checkpoint retains the converged CC state and verifies complete-prefix coverage before continuing its partial sum. Neither an iteration-limited CC solve nor incomplete triples is a completed result. See checkpoint and restart for the .vqr format, public runner options, atomic publication and exact compatibility policy.

Validator cases and comparison boundary

Submit fresh R5.3 calculations through vq using the delivered runtime that contains this source. Preserve existing restricted jobs and their runtime. Record the source and imported Python/native runtime identities, complete input, basis and auxiliary basis, geometry/cell, spin root, source descriptor, frozen orbitals, SCF acceptance, CC thresholds and separated energies once on the tracking issue. These are proposed cases, not measured acceptance results.

A reproducible small unrestricted starting case is neutral LiH2, multiplicity two, with Cartesian positions in bohr Li=(0,0,0), H=(0,0,3), H=(2.5,0,-1), in an orthorhombic 24 x 26 x 28 bohr three-dimensional cell. Use STO-3G, the explicit def2-SVP-JKFIT auxiliary basis, classic physical-pair support, no frozen core, and full unshifted character meshes (1,1,1) then (3,1,1). This gives three alpha and two beta electrons per character with nonempty virtual spaces and admits nonzero triples; measure the correction rather than inferring it from electron count. Fix initial support at 12 bohr for one-electron and pair radii, 20 Ha reciprocal cutoff, and nuclear Ewald alpha=0.4 inverse bohr, 12 bohr real radius and 8 inverse bohr reciprocal radius. Use zero fitting screening and record the retained metric ranks. These values define a finite test input, not converged physical cutoffs. Refine each support axis independently under the fitted-support contract. Use CC energy tolerance 1e-10 Ha, residual tolerance 1e-8, denominator tolerance 1e-12 Ha, and an explicit adequate memory cap.

For this same finite Hamiltonian, independently form the complete spin-orbital integrals from the retained alpha/beta factors and compare the CCSD residuals and converged energy with a determinant-space calculation where feasible. Evaluate the stated triples correction by independent determinant contractions, including a nonzero aab or abb sector. This compares exactly the executed Hamiltonian but cannot verify the underlying fitted operator’s physical accuracy. Additional cases should cover a same-root closed-shell reference, a two-electron reference (CCSD equals finite-space FCI and triples vanish), unequal frozen counts (1,0), and allowed orbital phases/permutations.

For a molecular-limit comparison, use the same LiH2 geometry and STO-3G orbital basis in canonical ORCA UHF-CCSD(T), all electrons correlated:

! UHF CCSD(T) STO-3G VeryTightSCF NoFrozenCore Bohrs
%mdci
  Include_4thOrder_DT_in_Triples false
end
* xyz 0 2
Li  0.0  0.0  0.0
H   0.0  0.0  3.0
H   2.5  0.0 -1.0
*

The ORCA 6.1 manual, section 3.10.2, documents canonical RHF/UHF MDCI and the additional open-shell fourth-order doubles-triples switch. Record this switch and verify the matched canonical UHF root; QRO, ROHF, DLPNO and KS orbitals are different references. First compare CCSD and triples separately. The ORCA molecular four-center Hamiltonian differs from a finite periodic fitted chi Hamiltonian, so direct equality is not an acceptance criterion. Increase the cell and physical supports, refine auxiliary fitting, and quantify the remaining image, fitting and q=0 differences before using ORCA as a molecular-limit comparison. CRYSTAL HF agreement can test matched periodic reference conventions; it does not supply unrestricted CCSD or this triples correction by inference. If no external program reproduces the finite chi Hamiltonian, report the determinant comparison and its physical limitation explicitly, leaving native scientific acceptance pending.

Classic ROHF and Knowles RMP2 use a separate shared-orbital reference and perturbation consumer. They do not admit ROHF-CCSD or triples through the unrestricted CC solver.