Coupled-cluster pairing models for radicals with strong correlations

The pairing hierarchy of perfect pairing (PP), perfect quadruples (PQ) and perfect hextuples (PH) are sparsified coupled cluster models that are exact in a pairing active space for 2, 4, and 6 electron clusters, respectively. We describe and implement three extensions for radicals. First is the trivial generalization that does not correlate radical orbitals. The second model (PQr, PHr) includes terms that entangle pair indices and radical indices such that their maximum total number is 2 for PQ and 3 for PH (like their closed-shell versions). The third family of extended radical models (PPxr, PQxr, and PHxr) include cluster amplitudes that entangle up to 1, 2, and 3 pair indices with up to 1, 2, and 3 radical indices. Notably, PPxr and PQxr are exact for (3e,3o) and (5e,5o), respectively, while still having only $O(N)$ and $O(N^{2}$) amplitudes like their parent models (for $N$ paired electrons). Orbital optimization is considered for PPxr. A series of large-scale numerical tests of these models are presented for spin gaps, and ionization energies of polyenes and polyenyl radicals, ranging in size from ethene and allyl radical up to C$_{22}$H$_{24}$ in full-valence active spaces up to (122e,122o). The xR models perform best.