Hyper Open-Shell Excited Spin States of Transition-Metal Compounds: FeF₂, FeF₂···Ethane, and FeF₂···Ethylene

Spin-state energetics are important for understanding properties that involve more than one spin state, for example, catalysis occurring on two or more potential energy surfaces corresponding to different electronic spins. Very often, multiple-spin processes involve transition-metal compounds, and therefore, it is important to understand the electronic structure and energetics of such compounds in different spin states. In this work, we benchmark relative spin-state energies of FeF₂ with respect to the quintet ground spin state using both single-configurational and multiconfigurational methods, and we examine how they are affected by the binding of ethane and ethylene to the iron center. We also benchmark the binding energies of the complexes. The single-configurational methods used in this work are the Hartree-Fock method, 32 exchange-correlation functionals, and the CCSD(T) coupled-cluster method in both restricted and unrestricted formalisms. The multiconfigurational methods that have been used are CASSCF, CASPT2, CASPT3, MRCI, MRCI+Q, and MR-ACPF. The spin-state splitting energies depend on the functional chosen, and of the 32 exchange-correlation functionals investigated here, we find that for the septet and spin-projected triplet states of FeF₂ the M06 functional is the best when compared to our best estimates from multireference calculations. If all nine excitation energies are considered, where there are three excited spin states (singlet, triplet, and septet) for each of the three systems (FeF₂, FeF₂···ethane, and FeF₂···ethylene), the three best-performing functionals are HLE16, SOGGA11-X, and M06-2X. We find that the binding of ethane perturbs the relative spin-state energy of FeF₂ by only a small amount, but the stronger binding of ethylene has a larger effect. For the spin-state splitting energies of FeF₂ using single-reference CCSD(T), we find that the predicted results depend very strongly on precisely how the calculations are done, in particular, on the spin-restricted or spin-unrestricted character of the SCF reference state, which can differ even by around 50 kcal/mol for the SCF reference state and the subsequent CCSD(T) calculations. Upon analyzing the wave functions of both the spin-restricted or spin-unrestricted formalisms, we find that the lowest-energy singlet and triplet states of the complexes, just like FeF₂ in isolation, can have more unpaired electrons than they are usually assumed to have, i.e., they can be hyper open-shell electronic configurations, and this can significantly lower the energy.