Improving the Accuracy in the Prediction of Transition-Metal Spin-State Energetics Using a Robust Variation-Based Approach: Density Functional Theory, CASPT2 and MC-PDFT Applied to the Case Study of Tris-Diimine Fe(II) Complexes

Designing ligands for transition metal complexes with a specified low-spin, high-spin or spin-crossover behavior is challenging. A major advance was recently made by Phan et al. [J. Am. Chem. Soc. 2017, 139, 6437–6447] who showed that the spin state of a homoleptic tris-diimine Fe(II) complex can be predicted from the N–N distance in the free diimine. They could thus predict the change in magnetic behavior on passing from the complexes of 2,2′-bipyridine (bpy), 2,2′-biimidazole (bim) and 2,2′-bis-2-imidazoline (bimz) ligands to those obtained with the modified analogs 4,5-diazafluoren-9-one (dafo), 1,1′-(α,α′-o-xylyl)-2,2′-bisimidazole (xbim) and 2,3,5,6,8,9-hexahydrodiimidazo[1,2-a:2′, 1′-c]pyrazine (etbimz), respectively. Theoretically, the challenge lies in the accurate determination of the HS–LS zero-point energy difference ΔE_HL^°. The issue can be circumvented by using a variation-based approach, wherein ΔE_HL^° is not directly evaluated but obtained from the estimate of its variation Δ(ΔE_HL^°) in series of related systems, which include one whose ΔE_HL^° is accurately known [Phys. Chem. Chem. Phys. 2013, 15, 3752–3763; J. Phys. Chem. A 2022, 126, 6221–6235]. In this study, density functional theory (DFT), second-order multireference perturbation theory in its CASPT2 formulation, multiconfigurational pair DFT (MC-PDFT) and its hybrid formulation (HMC-PDFT) have been applied to the determination of Δ(ΔE_HL^°) in the pairs of complexes ([Fe(bpy)3]2+, [Fe(dafo)3]2+), ([Fe(bim)3]2+, [Fe(xbim)3]2+) and ([Fe(bimz)3]2+, [Fe(etbimz)3]2+). In DFT, we used several semilocal functionals and their global hybrids, as well as their D2, D3, D3BJ and D4 dispersion-corrected forms; and in MC-PDFT, different translated and fully translated functionals. The results are consistent with one another and in very good agreement with experiments. They show small to vanishing dependence on key details of the methods used: namely, the exact-exchange contribution to global hybrids; the ionization potential-electron affinity shift and basis sets used in the CASPT2 calculations; and the active spaces employed for the CASSCF wave functions used in the MC-PDFT and HMC-PDFT calculations. Insights into the change in the spin-state energetics accompanying the ligand exchanges were gained through a complexation energy analysis. Using the accurate CCSD(T) estimate of the HS–LS adiabatic energy difference in [Fe(NCH)6]2+ [J. Chem. Theory Comput. 2012, 8, 4216–4231], the Δ(ΔE_HL^°)-approach has been applied to the determination of ΔE_HL^° in the diimine complexes. The CASPT2 and DFT-D2 methods only give results in agreement with experiments. This suggests for the other methods a limitation in their treatment of dispersion which prevents them from accurately describing the spin-state energetics change accompanying the passing from [Fe(NCH)6]2+ with the tetragonal arrangement of its nitrile ligands to the tris-diimine complexes with the trigonal packing of their bulkier ligands.