Performance of quantum chemistry methods for a benchmark set of spin-state energetics derived from experimental data of 17 transition metal complexes (SSE17)

Abstract

Accurate prediction of spin-state energetics for transition metal (TM) complexes is a compelling problem in applied quantum chemistry, with enormous implications for modeling catalytic reaction mechanisms and computational discovery of materials. Computed spin-state energetics are strongly method-dependent and credible reference data are scarce, making it difficult to conduct conclusive computational studies of open-shell TM systems. Here, we present a novel benchmark set of first-row TM spin-state energetics, which is derived from experimental data of 17 complexes containing Fe^II, Fe^III, Co^II, Co^III, Mn^II, and Ni^II with chemically diverse ligands. The estimates of adiabatic or vertical spin-state splittings, which are obtained from spin crossover enthalpies or energies of spin-forbidden absorption bands, suitably back-corrected for the vibrational and environmental effects, are employed as reference values for benchmarking density functional theory (DFT) and wave function methods. The results demonstrate a high accuracy of the coupled-cluster CCSD(T) method, which features the mean absolute error (MAE) of 1.5 kcal mol^-1 and maximum error of -3.5 kcal mol^-1, and outperforms all the tested multireference methods: CASPT2, MRCI+Q, CASPT2/CC and CASPT2+δMRCI. Switching from Hartree-Fock to Kohn-Sham orbitals is not found to consistently improve the CCSD(T) accuracy. The best performing DFT methods are double-hybrids (PWPB95-D3(BJ), B2PLYP-D3(BJ)) with the MAEs below 3 kcal mol^-1 and maximum errors within 6 kcal mol^-1, whereas the DFT methods so far recommended for spin states (e.g., B3LYP*-D3(BJ) and TPSSh-D3(BJ)) are found to perform much worse with the MAEs of 5-7 kcal mol^-1 and maximum errors beyond 10 kcal mol^-1. This work is the first such extensive benchmark study of quantum chemistry methods for TM spin-state energetics making use of experimental reference data. The results are relevant for the proper choice of methods to characterize TM systems in computational catalysis and (bio)inorganic chemistry, and may also stimulate new developments in quantum-chemical or machine learning approaches.

Fig. 1. Molecular structures of 17 complexes studied in this work (hydrogens omitted for clarity): A1–A9 SCO complexes, B1–B4 complexes with LS ground state, C1–C4 complexes with HS ground state. Multiplicities of the considered spin states are given in the superscript. Ligand abbreviations: acac₂trien = dianion of Schiff base obtained from the 2 : 1 condensation of acetylacetone with triethylenetetramine; HB(pz)₃ = hydrotris(pyrazol-1-yl)borate; HB(tz)₃ = hydrotris(1,2,4-triazol-1-yl)borate; tacn = 1,4,7-triazacyclononane; 1-bpp = 2,6-di(pyrazol-1-yl)pyridine; tpp = tetraphenylporphyrin; ipimpy = 2,6-bis(isopropyliminomethyl)pyridine; acac = acetylacetonate; Cp = cyclopentadienyl; Cp* = pentamethylcyclopentadienyl; en = ethylenediamine; ox = oxalate.

Fig. 2. Distribution of errors in the SSE17 spin-state splittings calculated using selected WFT methods (box-plot) and the resulting MAE of each method (point-plot). Each box represents 50% of the population (with the median marked in the middle) and the whiskers extend from the minimum to the maximum of the population. Individual data are shown as points. To guide the eye, error ranges ±1 kcal mol⁻¹ (“chemical accuracy”) and ±3 kcal mol⁻¹ (“TM chemical accuracy”) are colored in green and yellow, respectively.

Fig. 3. Distribution of errors in the SSE17 spin-state splittings calculated using selected DFT methods (box-plot) and the resulting MAE of each method (point-plot). The boxes are colored by functional type: gradient (G), meta-gradient (mG), hybrid (H), meta-hybrid (mH), range-separated hybrid (RSH), local hybrid (LH), double-hybrid (DH).

Fig. 4. Mean signed errors (kcal mol⁻¹) of selected methods for different classes of complexes.