Coupled cluster

"RI-CC" here means the two-electron integrals in the amplitude equations come from density fitting (three-centre B tensors from an RI auxiliary basis), not from exact four-centre integrals. Everything below needs an orbital basis and an RI auxiliary basis. Formal cost is O(N⁶) for CCSD and O(N⁷) for (T).

CCSD and CCSD(T)

What it is. RI-CCSD and the perturbative triples correction (T), in two implementations:

• Spin-adapted, closed shell (ccsd_closed_shell, ccsd_t_closed_shell): amplitudes over spatial orbitals, the algorithm PySCF's cc.CCSD uses. This is what both the CLI and Python run for an RHF reference. Its VVVV block is 16× smaller than the spin-orbital one; measured about 8–10× faster than the spin-orbital CCSD at cc-pVDZ, and the (T) step 9.6–42× faster.
• Spin-orbital (ccsd, ccsd_t): built from the same RHF spatial orbitals and kept as the cross-check of the spin-adapted solvers, so it is closed shell only as well. Its (T) streams one occupied triple at a time, so memory is O(n_o·n_v³)-class rather than the dense six-index tensor.

No open-shell CCD, CCSD or CCSD(T) exists in ferric: every CC solver reads restricted orbitals and refuses an unrestricted reference. The only open-shell coupled-cluster-type method is LinLCCD(hh): the CLI linlccd kind is closed shell, and its open-shell (UHF) version is library-only (ferric_cc::linlccd_u::u_linlccd).

Run it.

• CCSD: method.kind = "ccsd" (examples/water-ccsd.toml, water/cc-pVDZ); Python ferric.run_ccsd(mol, bs, aux).
• CCSD(T): method.kind = "ccsd(t)" (examples/water-ccsd-t.toml); Python ferric.run_ccsd_t(mol, bs, aux).
• CCD: method.kind = "ccd" (examples/water-ccd.toml); Python ferric.run_ccd(mol, bs, aux).
• All three kinds are closed shell only and refuse multiplicity > 1.

Aux basis. The RI error is not negligible at CC accuracy: on water / cc-pVDZ with cc-pvdz-ri, RI-CCSD differs from exact-integral CCSD by about 1.3e-4 Ha. Pick the aux for the accuracy you need, not by habit.

Accuracy. ccsd is Proven.

Limits. Closed-shell entry points only in the CLI and Python. The spin-adapted (T) rejects spin-orbital amplitudes with a typed error rather than mixing conventions.

LinLCCD and ωB97X-L-V

LinLCCD(hh) is linearized coupled-cluster doubles with the hole–hole ladder kept to all orders, closed shell only. The ladder keeps the correlation energy finite as the HOMO–LUMO gap closes, where MP2 diverges. method.kind = "linlccd" (examples/water-linlccd.toml). Proven (narrow): no other quantum chemistry code implements LinLCCD(hh), so its energy is checked against an independent numpy solve of the same equations on PySCF density-fitted integrals (H2O and NH3 through the closed-shell path, and UHF OH through the library-only open-shell path; agreement ≤1.2e-12 Ha). With the ladder off it reduces exactly to RI-MP2, and with exact integrals its driver terms reproduce canonical MP2.

[mp2] linlccd_variant (Python run_linlccd(variant=...)) selects the method's ladder terms: hh (default), drivers-only (no ladder, equal to RI-MP2) or full (hole–hole plus particle–particle, with CCD-like VVVV memory). Every variant is computed exactly by default. All three variants match an independent numpy solve on the same density-fitted integrals (H2O and NH3, 6-31G and cc-pVDZ, ≤ 4.4e-13 Ha); drivers-only also matches PySCF DFMP2.

Local LinLCCD. With [local] scheme = "amplitude-threshold" and eps (examples/water-linlccd-local.toml; Python run_linlccd(..., local="amplitude-threshold", eps=1e-4)) the same variant is solved in the Boys-localized basis with pair amplitudes at or below eps dropped (see Exact and local correlation). eps has no default and is printed and logged with the kept fraction; eps = 0 reproduces the exact LinLCCD of the same variant and the numpy reference (≤ 4.4e-13 Ha, every variant), which is how the local path is anchored. [local] reference = true also runs the exact LinLCCD and prints the local error. Closed shell and energy only.

ωB97X-L-V is a double-hybrid functional that uses short-range LinLCCD(hh) instead of MP2 for its correlation term. It converges its own ωB97X-L Kohn–Sham reference (a non-converged reference is an error), then adds the LinLCCD(hh) correction on those orbitals. method.kind = "wb97x-l-v" (examples/water-wb97xlv.toml). [dft] lambda and omega override the published 0.6 and 0.1 Bohr⁻¹; omitting them gives the published values. λ enters the amplitude equations as well as the energy (the paper's eqn 22), so the correlation term is quadratic in λ at leading order. Validated against PySCF with the published parameters plus a numpy LinLCCD(hh) on water and OH, and against the paper's Be₂ bond energy (2.3 kcal/mol; ferric 2.299).

MP2-based double hybrids

B2PLYP and DSD-PBEP86: a KS reference with weighted exchange and correlation components, plus scaled (SCS-)RI-MP2 correlation. method.kind = "b2plyp" / "dsd-pbep86" (examples/water-b2plyp.toml); Python ferric.run_double_hybrid(mol, bs, aux, kind="b2plyp"). Spike: no comparison to a reference code yet.

Implementation

All contractions go through einsum!, a macro that maps tensor contractions onto BLAS3 GEMMs. The permutation copies that feed those GEMMs are parallelized, because for a strided permutation the copy can dominate the contraction it feeds: measured at 47% at n_v = 40 and 70% at n_v = 80. The copies are bit-identical regardless of thread count, since a permutation writes each output element exactly once; a test pins this and has been checked to fail when deliberately broken.

Memory

The amplitude tensors dominate and grow as \( n_o^2 n_v^2 \), or \( (2n_o)^2 (2n_v)^2 \) in the spin-orbital drivers. Memory budgets are enforced: an oversized job is refused with a breakdown naming the dominant term instead of being OOM-killed partway through.

Cite

CCSD: Scuseria, Janssen & Schaefer 1988; spin-adapted closed-shell CCSD equations: Hirata et al. 2004. (T): Raghavachari et al. 1989; closed-shell (T) algorithm: Rendell, Lee & Komornicki 1991. LinLCCD(hh): Carter-Fenk 2025. ωB97X-L-V: Ransford & Carter-Fenk 2026. B2PLYP: Grimme 2006. DSD-PBEP86: Kozuch & Martin 2011. Review: Bartlett & Musiał 2007. Full entries in References.