Hf₃ Cluster Is Triply (σ-, π-, and δ-) Aromatic in the Lowest D_3h, ¹A₁‘ State

The extensive search for the global minimum structure of Hf₃ at the B3LYP/LANL2DZ level of theory revealed that D_3h ³A₂‘ (1a₁‘²1a₂‘ ‘²1e‘⁴2a₁‘²1e‘ ‘²) and D_3h ¹A₁‘ (1a₁‘²2a₁‘²1e‘⁴1a₂‘ ‘²3a₁‘²) are the lowest triplet and singlet states, respectively, with the triplet state being the lowest one. However, at the CASSCF(10,14)/Stuttgart+2f1g level of theory these two states are degenerate, indicating that at the higher level of theory the singlet state could be in fact the global minimum structure. The triplet D_3h ³A₂‘ (1a₁‘²1a₂‘ ‘²1e‘⁴2a₁‘²1e‘ ‘²) structure is doubly (σ- and π-) aromatic and the singlet D_3h ¹A₁‘ (1a₁‘²2a₁‘²1e‘⁴1a₂‘ ‘²3a₁‘²) structure is the first reported triply (σ-, π-, and δ-) aromatic system.