Accurate Spin-State Energies for Iron Complexes

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

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 FeII, FeIII, CoII, CoIII, MnII, and NiII 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.

Electronic and magnetic properties of porphyrin nanoribbons with chelated metals

Advances in surface-assisted synthesis routes now allow for precise control in the preparation and modification of low-dimensional structures. The choice of molecular precursors plays a fundamental role in these processes since the structural details and properties of the resulting nanostructures directly depend on the molecular block used. From this perspective, units based on porphyrins have proven to be promising candidates for the construction of nanosystems with nontrivial geometry. In particular, efforts have been made to synthesize different arrangements of π-conjugated porphyrins. With this motivation, we use computational simulations to investigate the electronic and magnetic properties of nanoribbons constructed from the concatenation of π-extended porphyrins hosting transition metal atoms. We show that the binding energy of these systems and the specific way the electrons populate the d-shells are strongly influenced by the type of the transition metal. Furthermore, it was observed that most systems with chelated metals (except Ni and Zn) feature magnetic properties. The systems considered in this work have analogs in finite structures recently synthesized in the laboratory so the nanomaterials proposed here have a high potential to be produced in the near future.

A comparative study on the interaction of M3+(H2O)5-6 ions (M = Fe, Ru, and Os) with the hydroxy group

CONTEXT

The heavy metal ions are the typical carcinogenic agents. Up to now, the interaction mechanism of toxic metal ions with the biomolecules such as carbohyrate have not been elucidated and reported in the detail. In this research work, the adjacent dissociation Gibbs energy (EAB) of M3+(H2O)5(R-OH) complexes depended significantly on the molecular volume of primary alcohols and the inductive effects of substituent R in primary alcohols (R = CH3, CH3CH2, CH3CH2CH2, CH3CH2CH2CH2, Cl-CH2, F-CH2) as well as the length of linear cellulose. The affinity of M3+(H2O)5 ions with the sixth water molecule in gas phase reduced in the order as follows: Fe3+  > Ru3+  > Os3+, which were determined by the EAB values and bond lengths of M-O. The water solvent made the EAB values of Fe3+(H2O)6 ions and Fe3+(H2O)5(CH3OH) ions changed completely in the case of the polarizable continuum model, while the Onsager model gave the good agreement with the gas phase model.

METHODS

The nature of interaction between hydrated Fe3+(H2O)5 ion and the hydroxy groups of primary alcohols were investigated using density functional theory method at the B3LYP/6-311 + G** level, the PBEPBE/6-311 + G** level. The influence of water solvent was evaluated using the Onsager model and the polarizable continuum model. The two-layer ONIOM approach and the local softness analysis were employed for the hydroxy groups of linear cellulose at the B3LYP/6-311 + G**:HF/6-31G* level. The affinity of M3+(H2O)5 ions (M = Fe, Ru, and Os) with the sixth water molecule were probed at the B3LYP/QZVPP/6-311 + G** level (QZVPP basis set for the metal atoms).

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 ΔEHL°. The issue can be circumvented by using a variation-based approach, wherein ΔEHL° is not directly evaluated but obtained from the estimate of its variation Δ(ΔEHL°) in series of related systems, which include one whose ΔEHL° 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(ΔEHL°) in the pairs of complexes ( [ F e ( b p y ) 3 ] 2 + , [ F e ( d a f o ) 3 ] 2 + ) , ( [ F e ( b i m ) 3 ] 2 + , [ F e ( x b i m ) 3 ] 2 + ) and ( [ F e ( b i m z ) 3 ] 2 + , [ F e ( e t b i m z ) 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 [ F e ( N C H ) 6 ] 2 + [J. Chem. Theory Comput. 2012, 8, 4216-4231], the Δ(ΔEHL°)-approach has been applied to the determination of ΔEHL° 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 [ F e ( N C H ) 6 ] 2 + with the tetragonal arrangement of its nitrile ligands to the tris-diimine complexes with the trigonal packing of their bulkier ligands.

Tuning the spin-crossover properties of [Fe2] metal-organic cages

A computational study on the interplay between ligand functionalization and guest effects on the transition temperature (T1/2) in the [Fe2(L1R)3]@X (L1 = 1,3-bis-(3-(pyridin-2-yl)-1H-pyrazol-5-yl)benzene, X = H-, F-, Cl-, Br-, I- and [BF4]-, R = H, F, or CH3) family of metal-organic cages (MOCs) is presented. Our results indicate that ligand functionalization with electron-donating or electron-withdrawing groups can significantly impact the T1/2 as expected, while the guest effect in lowering the T1/2 has a linear correlation with the increasing guest size. More importantly, small guests can move away from the center of the cavity, thus enhancing the two-step characteristic of the transition. All the data can be understood by analyzing the underlying electronic structure of the studied systems in terms of the relevant d-based molecular orbitals. These results can help in the rational design of new MOCs that can operate as sensors at specific temperatures, thus accelerating the discovery of new SCO devices with tailored properties.

Automated prediction of ground state spin for transition metal complexes

Exploiting crystallographic data repositories for large-scale quantum chemical computations requires the rapid and accurate extraction of the molecular structure, charge and spin from the crystallographic information file. Here, we develop a general approach to assign the ground state spin of transition metal complexes, in complement to our previous efforts on determining metal oxidation states and bond order within the cell2mol software. Starting from a database of 31k transition metal complexes extracted from the Cambridge Structural Database with cell2mol, we construct the TM-GSspin dataset, which contains 2063 mononuclear first row transition metal complexes and their computed ground state spins. TM-GSspin is highly diverse in terms of metals, metal oxidation states, coordination geometries, and coordination sphere compositions. Based on TM-GSspin, we identify correlations between structural and electronic features of the complexes and their ground state spins to develop a rule-based spin state assignment model. Leveraging this knowledge, we construct interpretable descriptors and build a statistical model achieving 98% cross-validated accuracy in predicting the ground state spin across the board. Our approach provides a practical way to determine the ground state spin of transition metal complexes directly from crystal structures without additional computations, thus enabling the automated use of crystallographic data for large-scale computations involving transition metal complexes.

Benchmarking Periodic Density Functional Theory Calculations for Spin-State Energies in Spin-Crossover Systems

Spin energetics is one of the biggest challenges associated with energy calculations for electronic structure methods. The energy differences of the spin states in spin-crossover compounds are very small, making them one of the most difficult systems to calculate. Few methods provide accurate results for calculating these energy differences. In addition, studies have usually focused on calculating energetics of single molecules, while spin-crossover properties are usually experimentally studied in the solid phase. In this paper, we have used periodic boundary conditions employing methods based on density functional theory to calculate the high- and low-spin energy differences for a test case of 20 extended systems. Compounds with different metals and ligands have been selected, and the results indicate that a semiquantitative description of the energy differences can be obtained with the combination of geometry optimization using the PBE functional including many-body dispersion approach and the use of meta-GGA functionals, such as r2SCAN but especially KTBM24, for the energy calculation. Other hybrid functionals, such as TPSSh, give generally good results, but the calculation of the exact exchange with periodic boundary conditions involves a huge increase in computer time and computational resources. It makes the proposed nonhybrid functional approach (KTBM24//PBE+MB) a great advantage for the study of periodic systems.

Predicting spin states of iron porphyrins with DFT methods including crystal packing effects and thermodynamic corrections

Accurate computational treatment of spin states for transition metal complexes, exemplified by iron porphyrins, lies at the heart of quantum bioinorganic chemistry, but at the same time represents a great challenge for approximate density functional theory (DFT) methods, which are predominantly used. Here, the accuracy of DFT methods for spin-state splittings in iron porphyrin is assessed by probing the ability to correctly predict the ground states for six FeIII or FeII complexes experimentally characterized in solid state. For each case, molecular and periodic DFT calculations are employed to quantify the effect of porphyrin side substituents and the crystal packing effect (CPE) on the spin-state splitting. It is proposed to partition the total CPE into additive components, the direct and structural one, the importance of which is shown to significantly vary from case to case. By knowing the substituent effect, the CPE, and the Gibbs free energy thermodynamic correction from calculations, one can employ the experimental ground-state information in order to derive a quantitative constraint on the electronic energy difference for a simplified (porphin) model of the experimentally characterized metalloporphyrin. The constraints derived in such a way-in the form of single or double inequalities-are used to assess the accuracy of dispersion-corrected DFT methods for 6 spin-state splittings of [FeIII(P)(2-MeIm)2]+, [FeIII(P)(2-MeIm)]+, [FeII(P)(THF)2] and [FeII(P)] models (where P is porphin, 2-MeIm is 2-methylimidazole, THF is tetrahydrofuran). These data constitute the new benchmark set of spin states for crystalline iron porphyrins (SSCIP6). The highest accuracy is obtained in the case of double-hybrid functionals (B2PLYP-D3, DSD-PBEB95-D3), whereas hybrid functionals, especially those with reduced admixture of the exact exchange (B3LYP*-D3, TPSSh-D3), are found to considerably overstabilize the intermediate spin state, leading to incorrect ground-state prediction in FeIII porphyrins. The present approach, which can be generalized to other transition metal complexes, is not only useful in method benchmarking, but also sheds light on the interpretations of experimental data for metalloporphyrins, which are important models to understand the electronic properties of heme proteins.

Benchmarks for transition metal spin-state energetics: why and how to employ experimental reference data?

Accurate prediction of energy differences between alternative spin states of transition metal complexes is essential in computational (bio)inorganic chemistry-for example, in characterization of spin crossover materials and in the theoretical modeling of open-shell reaction mechanisms-but it remains one of the most compelling problems for quantum chemistry methods. A part of this challenge is to obtain reliable reference data for benchmark studies, as even the highest-level applicable methods are known to give divergent results. This Perspective discusses two possible approaches to method benchmarking for spin-state energetics: using either theoretically computed or experiment-derived reference data. With the focus on the latter approach, an extensive general review is provided for the available experimental data of spin-state energetics and their interpretations in the context of benchmark studies, targeting the possibility of back-correcting the vibrational effects and the influence of solvents or crystalline environments. With a growing amount of experience, these effects can be now not only qualitatively understood, but also quantitatively modeled, providing the way to derive nearly chemically accurate estimates of the electronic spin-state gaps to be used as benchmarks and advancing our understanding of the phenomena related to spin states in condensed phases.

Artificial Neural Network-Based Density Functional Approach for Adiabatic Energy Differences in Transition Metal Complexes

During the past decades, approximate Kohn-Sham density functional theory schemes have garnered many successes in computational chemistry and physics, yet the performance in the prediction of spin state energetics is often unsatisfactory. By means of a machine learning approach, an enhanced exchange and correlation functional is developed to describe adiabatic energy differences in transition metal complexes. The functional is based on the computationally efficient revision of the regularized, strongly constrained, and appropriately normed functional and improved by an artificial neural network correction trained over a small data set of electronic densities, atomization energies, and/or spin state energetics. The training process, performed using a bioinspired nongradient-based approach adapted for this work from the particle swarm optimization, is analyzed and discussed extensively. The resulting machine learned meta-generalized gradient approximation functional is shown to outperform most known density functionals in the prediction of adiabatic energy differences for a diverse set of transition metal complexes with varying local coordinations and metal choices.

Adaptive σ aromaticity in the rhenacyclopropene rings

Species generally exhibit one-state aromaticity either in the lowest singlet state (S0 ) or the lowest triplet state (T1 ) according to the Hückel's and Baird's rules. Hence, it is rare for species exhibit two-state aromaticity in both the S0 and T1 states (termed as adaptive aromaticity), let alone adaptive σ aromaticity. Here, we report adaptive σ aromaticity in unsaturated rhenacyclopropene rings via density functional theory (DFT) calculations. Various aromaticity indices including NICS, ACID, EDDB together with isodesmic reactions support the adaptive σ aromaticity in these rhenacyclopropene rings. As the T1 state of these species is formed by the ππ* excitation, the σ-aromaticity of these three-membered rings in the S0 state could hold in the T1 state. In addition, the aromaticity effect of the fused rings is also examined. Our findings expand the family of adaptive σ aromaticity, enriching the metallaaromatic chemistry.

Spin state and magnetic coupling in polynuclear Ni(II) complexes from density functional theory: is there an optimal amount of Fock exchange?

Reliable prediction of the ground-state spin and magnetic coupling constants in transition-metal complexes is a well-known challenge for density functional theory (DFT). One popular strategy for addressing this long-standing issue involves the modification of the fraction of Fock exchange in a hybrid functional. Here we explore the viability of this approach using three polynuclear metal-organic complexes based on a Ni4O4 cubane motif, having different ground state spin values (S = 0, 2, 4) owing to the use of different ligands. We systematically search for an optimum fraction of Fock exchange, across various global, range-separated, and double hybrid functionals. We find that for all functionals tested, at best there only exists a very narrow range of Fock exchange fractions which results in a correct prediction of the ground-state spin for all three complexes. The useful range is functional dependent, but general trends can be identified. Typically, at least two similar systems must be used in order to determine both an upper and lower limit of the optimal range. This is likely owing to conflicting demands of minimizing delocalization errors, which typically requires a higher percentage of Fock exchange, and addressing static correlation, which typically requires a lower one. Furthermore, we find that within the optimal range of Fock exchange, the sign and relative magnitude of Ni-Ni magnetic coupling constants are reasonably well reproduced, but there is still room for quantitative improvement in the prediction. Thus, the prediction of spin state and magnetic coupling in polynuclear complexes remains an ongoing challenge for DFT.

Controlling Spin Crossover in a Family of Dinuclear Fe(III) Complexes via the Bis(catecholate) Bridging Ligand

Spin crossover (SCO) complexes can reversibly switch between low spin (LS) and high spin (HS) states, affording possible applications in sensing, displays, and molecular electronics. Dinuclear SCO complexes with access to [LS-LS], [LS-HS], and [HS-HS] states may offer increased levels of functionality. The nature of the SCO interconversion in dinuclear complexes is influenced by the local electronic environment. We report the synthesis and characterization of [{FeIII(tpa)}2spiro](PF6)2 (1), [{FeIII(tpa)}2Br4spiro](PF6)2 (2), and [{FeIII(tpa)}2thea](PF6)2 (3) (tpa = tris(2-pyridylmethyl)amine, spiroH4 = 3,3,3',3'-tetramethyl-1,1'-spirobi(indan)-5,5',6,6'-tetraol, Br4spiroH4 = 3,3,3',3'-tetramethyl-1,1'-spirobi(indan)-4,4',7,7'-tetrabromo-5,5',6,6'-tetraol, theaH4 = 2,3,6,7-tetrahydroxy-9,10-dimethyl-9,10-dihydro-9,10-ethanoanthracene), utilizing non-conjugated bis(catecholate) bridging ligands. In the solid state, magnetic and structural analysis shows that 1 remains in the [HS-HS] state, while 2 and 3 undergo a partial SCO interconversion upon cooling from room temperature involving the mixed [LS-HS] state. In solution, all complexes undergo SCO from [HS-HS] at room temperature, via [LS-HS] to mixtures including [LS-LS] at 77 K, with the extent of SCO increasing in the order 1 < 2 < 3. Gas phase density functional theory calculations suggest a [LS-LS] ground state for all complexes, with the [LS-HS] and [HS-HS] states successively destabilized. The relative energy separations indicate that ligand field strength increases following spiro4- < Br4spiro4- < thea4-, consistent with solid-state magnetic and EPR behavior. All three complexes show stabilization of the [LS-HS] state in relation to the midpoint energy between [LS-LS] and [HS-HS]. The relative stability of the [LS-HS] state increases with increasing ligand field strength of the bis(catecholate) bridging ligand in the order 1 < 2 < 3. The bromo substituents of Br4spiro4- increase the ligand field strength relative to spiro4-, while the stronger ligand field provided by thea4- arises from extension of the overlapping π-orbital system across the two catecholate units. This study highlights how SCO behavior in dinuclear complexes can be modulated by the bridging ligand, providing useful insights for the design of molecules that can be interconverted between more than two states.

Solvothermal synthesis of cobalt PCP pincer complexes from [Co2(CO)8]

Treatment of [Co2(CO)8] with the ipso-substituted P(C-X)PY ligands (X = Br, Cl; R = iPr, tBu) bearing Y = NH and CH2 linkers under solvothermal conditions affords the five-coordinate Co(I) and Co(III) complexes [CoI(PCPY-R)(CO)2] and [CoIII(PCPY-R)X2]. The later are paramagnetic exhibiting a solution magnetic moment in the range of 3.0-3.3 μB which is consistent with a d6 intermediate spin system corresponding to two unpaired electrons. In the case of P(C-X)PY ligands (X = Br, Cl; R = tBu; Y = NH) the formation of the square planar Co(II) complex [Co(PCPNH-tBu)X] was favored. This complex gives rise to a magnetic moment of 1.8 μB being consistent with a d7 low spin system corresponding to one unpaired electron. All complexes are characterized by means of spectroscopic techniques (NMR, IR), HR-MS. Representative complexes were also characterized by X-ray crystallography.

Graphical abstract

Supplementary Information

The online version contains supplementary material available at 10.1007/s00706-023-03123-x.

Spin-state energetics and magnetic anisotropy in penta-coordinated Fe(III) complexes with different axial and equatorial ligand environments

The penta-coordinated trigonal-bi-pyramidal (TBP) Fe(III) complex (PMe₂Ph)₂FeCl₃ shows a reduced magnetic anisotropy in its intermediate-spin (IS) state as compared to its methyl-analog (PMe₃)₂Fe(III)Cl₃. In this work, the ligand environment in (PMe₂Ph)₂FeCl₃ is systematically altered by replacing the axial -P with -N and -As, the equatorial -Cl with other halides, and the axial methyl group with an acetyl group. This has resulted in a series of Fe(III) TBP complexes modelled in their IS and high-spin (HS) states. Lighter ligands -N and -F stabilize the complex in the HS state, while the magnetically anisotropic IS state is stabilized by -P and -As at the axial site, and -Cl, -Br, and -I at the equatorial site. Larger magnetic anisotropies appear for complexes with nearly degenerate ground electronic states that are well separated from the higher excited states. This requirement, largely controlled by the d-orbital splitting pattern due to the changing ligand field, is achieved with a certain combination of axial and equatorial ligands, such as -P and -Br, -As and -Br, and -As and -I. In most cases, the acetyl group at the axial site enhances the magnetic anisotropy compared to its methyl counterpart. In contrast, the presence of -I at the equatorial site compromises the uniaxial type of anisotropy of the Fe(III) complex leading to an enhanced rate of quantum tunneling of magnetization.

Photodissociation Spectroscopy and Photofragment Imaging of the Fe+(Acetylene) Complex

Tunable laser photodissociation spectroscopy in the 700-400 nm region and photofragment imaging experiments are employed to investigate the Fe⁺(acetylene) ion-molecule complex. At energies above a threshold at 679 nm, continuous dissociation is detected throughout the visible wavelength region, with regions of broad structure. Comparison to the spectrum predicted by time-dependent density functional theory (TD-DFT) indicates that the complex has a quartet ground state. The dissociation threshold for Fe⁺(acetylene) at 679 nm provides the dissociation energy on the quartet potential energy surface. Correction for the atomic quartet-sextet spin state energy difference provides an adiabatic dissociation energy of 36.8 ± 0.2 kcal/mol. Photofragment imaging of the Fe⁺ photoproduct produced at 603.5 nm produces significant kinetic energy release (KER). The photon energy and the maximum value of the KER provide an upper limit on the dissociation energy of D₀ ≤ 34.6 ± 3.2 kcal/mol. The dissociation energies determined from the spectroscopy and photofragment imaging experiments agree nicely with the value determined previously by collision-induced dissociation (38.0 ± 2.6 kcal/mol). However, both values are significantly lower than those produced by computational chemistry at the DFT level using different functionals recommended for transition-metal chemistry.

Spin-state gaps and self-interaction-corrected density functional approximations: Octahedral Fe(II) complexes as case study

Accurate prediction of a spin-state energy difference is crucial for understanding the spin crossover phenomena and is very challenging for density functional approximations, especially for local and semi-local approximations due to delocalization errors. Here, we investigate the effect of the self-interaction error removal from the local spin density approximation (LSDA) and Perdew-Burke-Ernzerhof generalized gradient approximation on the spin-state gaps of Fe(II) complexes with various ligands using recently developed locally scaled self-interaction correction (LSIC) by Zope et al. [J. Chem. Phys. 151, 214108 (2019)]. The LSIC method is exact for one-electron density, recovers the uniform electron gas limit of the underlying functional, and approaches the well-known Perdew-Zunger self-interaction correction (PZSIC) as a particular case when the scaling factor is set to unity. Our results, when compared with reference diffusion Monte Carlo results, show that the PZSIC method significantly overestimates spin-state gaps favoring low spin states for all ligands and does not improve upon density functional approximations. The perturbative LSIC-LSDA using PZSIC densities significantly improves the gaps with a mean absolute error of 0.51 eV but slightly overcorrects for the stronger CO ligands. The quasi-self-consistent LSIC-LSDA, such as coupled-cluster single double and perturbative triple [CCSD(T)], gives a correct sign of spin-state gaps for all ligands with a mean absolute error of 0.56 eV, comparable to that of CCSD(T) (0.49 eV).

One-electron self-interaction error and its relationship to geometry and higher orbital occupation

Density Functional Theory (DFT) sees prominent use in computational chemistry and physics; however, problems due to the self-interaction error (SIE) pose additional challenges to obtaining qualitatively correct results. As an unphysical energy an electron exerts on itself, the SIE impacts most practical DFT calculations. We conduct an in-depth analysis of the one-electron SIE in which we replicate delocalization effects for simple geometries. We present a simple visualization of such effects, which may help in future qualitative analysis of the one-electron SIE. By increasing the number of nuclei in a linear arrangement, the SIE increases dramatically. We also show how molecular shape impacts the SIE. Two- and three-dimensional shapes show an even greater SIE stemming mainly from the exchange functional with some error compensation from the one-electron error, which we previously defined [D. R. Lonsdale and L. Goerigk, Phys. Chem. Chem. Phys. 22, 15805 (2020)]. Most tested geometries are affected by the functional error, while some suffer from the density error. For the latter, we establish a potential connection with electrons being unequally delocalized by the DFT methods. We also show how the SIE increases if electrons occupy higher-lying atomic orbitals; seemingly one-electron SIE free methods in a ground are no longer SIE free in excited states, which is an important insight for some popular, non-empirical density functional approximations (DFAs). We conclude that the erratic behavior of the SIE in even the simplest geometries shows that robust DFAs are needed. Our test systems can be used as a future benchmark or contribute toward DFT development.

Coordination and Spin States in Fe+(C2H2)n Complexes Studied with Selected-Ion Infrared Spectroscopy

Fe⁺(acetylene)_n ion-molecule complexes are produced in a supersonic molecular beam with pulsed laser vaporization. These ions are mass selected and studied with infrared photodissociation spectroscopy in the C-H stretching region, complemented by computational chemistry calculations. All C-H stretch vibrations are shifted to frequencies lower than the vibrations of isolated acetylene because of the charge transfer that occurs between the metal ion and the molecules. Complexes in the size range of n = 1-4 are found to have structures with individual acetylene molecules bound to the core metal ion via cation-π interactions. The coordination is completed with four ligands in a structure close to a distorted tetrahedron. Larger complexes in the range of n = 5-8 have external acetylene molecules solvating this n = 4 core ion via CH-π bonding to inner-shell ligands. DFT computations predict that quartet spin states are more stable for all complex sizes, but infrared spectra for quartet and doublet spin states are quite similar, precluding definitive determination of the spin states. There is no evidence for any of these complexes having acetylenes coupled into reacted structures. This is consistent with computed thermochemistry, which finds significant activation barriers to such reactions.

Iron-Catalyzed Olefin Metathesis: Recent Theoretical and Experimental Advances

The "metathesis reaction" is a straightforward and often metal-catalyzed chemical reaction that transforms two hydrocarbon molecules to two new hydrocarbons by exchange of molecular fragments. Alkane, alkene and alkyne metathesis have become an important tool in synthetic chemistry and have provided access to complex organic structures. Since the discovery of industrial olefin metathesis in the 1960s, many modifications have been reported; thus, increasing scope and improving reaction selectivity. Olefin metathesis catalysts based on high-valent group six elements or Ru(IV) have been developed and improved through ligand modifications. In addition, significant effort was invested to realize olefin metathesis with a non-toxic, bio-compatible and one of the most abundant elements in the earth's crust; namely, iron. First evidences suggest that low-valent Fe(II) complexes are active in olefin metathesis. Although the latter has not been unambiguously established, this review summarizes the key advances in the field and aims to guide through the challenges.

On the free radical scavenging and metallic ion chelating activities of pyridoxal - Could the pro-oxidant risk be competitive?

Primary and secondary antioxidant activities of pyridoxal have been investigated by using density functional theory (DFT) at the M05-2X level combined with 6-311++G(d,p) basis set for non-metallic atoms and LanL2DZ for metallic ions. The former has been examined by its free radical scavenging activity towards HOO^●, HO^●, and NO₂^●via different mechanisms including formal hydrogen transfer (FHT), proton transfer (PT), single electron transfer (SET), and radical adduct formation (RAF). The latter has been accomplished through its transition metal-chelating ability with Fe(III)/Fe(II) and Cu(II)/Cu(I) ions. The results show that pyridoxal illustrates as an efficient radical scavenger, especially, for HO^● and NO₂^● in water. The overall rate constants (k_overall) for the reactions with HOO^●, HO^●, and NO₂^● radicals are 1.30 × 10⁴, 5.76 × 10⁹, and 1.43 × 10⁹ M^-1s^-1, respectively. The SET from the anionic state is the most dominant for the HOO^● and NO₂^● scavenging reactions, while both RAF and SET contribute largely to the reaction with highly reactive HO^● radicals. Moreover, the anionic form of pyridoxal demonstrates a better role as a metal chelator than the neutral. However, the pro-oxidant risks of the formed complexes could be observed if there are superoxide radical anion (O₂^●-) and ascorbate (Asc^-) in aqueous media.

FCIQMC-Tailored Distinguishable Cluster Approach: Open-Shell Systems

A recently proposed tailored approach based on the distinguishable cluster method and the stochastic FCI solver, FCIQMC [J. Chem. Theory Comput. 2020, 16, 5621], is extended to open-shell molecular systems. The method is employed to calculate spin gaps of various Fe(II) complexes, including a Fe(II) porphyrin model system. Both distinguishable cluster and fully relaxed CASSCF natural orbitals were used in this work as reference for the subsequent tailored distinguishable cluster calculations. The distinguishable cluster natural orbitals occupation numbers were also used as an aid to the selection of the active space. The effect of the active space sizes and of the explicit correlation correction (F12) onto the predicted spin gaps is investigated. The tailored distinguishable cluster with singles and doubles yields consistently more accurate results compared to the tailored coupled cluster with singles and doubles.

Computational Investigation of the Metal and Ligand Substitution Effects on the Structure and Electronic States of the Phosphoranimide Tetramer Complexes of Cu(I), Ni(I), Co(I), and Fe(I)

The structurally unique saddle-shaped paramagnetic tetrametallic clusters of Co(I) and Ni(I) with phosphoranimide ligands have been synthesized and proposed as catalyst precursors. The analogous Cu(I) nanocluster is planar and diamagnetic. These notable variations in geometry and ground electronic states indicate that the effect of metal and ligand substituents on the structure and electronic properties of these complexes requires investigation. We present a computational study of a series of these novel homoleptic complexes containing Co(I), Ni(I), and Cu(I) as well as Fe(I) coordinated to phosphoranimides with electron-donating and withdrawing substituents, conducted at the relativistic density functional theory level using ZORA-PBE/TZP. The optimized structures of the saddle-shaped Co(I) and Ni(I) and planar Cu(I) tetramers with linear N-M-N coordination are validated with respect to X-ray diffraction determinations. The ground-state analysis indicates that Cu(I) complexes are diamagnetic, whereas Ni(I) and Co(I) complexes are in high-spin states, in agreement with magnetic susceptibility measurements. The computational results show that Fe(I) complexes are saddle shaped and high spin. The Co(I) complex is stabilized by a metal macrocycle distortion from square to diamond, as elucidated from its Walsh diagram. The effects of metals and ligand substituents on the ground electronic state, metal center coordination environment, and energy of the complexes are investigated. The bulky tertiary butyl substituent causes the largest saddle-shape distortion of the tetramer marcocycle, which partially offsets its electron-donating effect. Macrocycle distortions with N-M-N site angles ranging from obtuse to alternating obtuse reflex are correlated with the increasing number of unpaired electrons. The phenyl-substituted complexes are expected to have the highest reactivity toward electrophiles. Understanding the interplay between structural and electronic parameters is intended to guide the development of synthetic cooperative systems for multielectron redox reactions, models of biological systems, and molecular magnets.

CASPT2 molecular geometries of Fe(II) spin-crossover complexes

Using fully internally contracted (FIC)-CASPT2 analytical gradients, geometry optimizations of spin-crossover complexes are reported. This approach is tested on a series of Fe(II) complexes with different sizes, ranging from 13 to 61 atoms. A combination of active space and basis set choices are employed to investigate their role in determining reliable molecular geometries. The reported strategy demonstrates that a wave function-based level of theory can be used to optimize the geometries of metal complexes in reasonable times and enables one to treat the molecular geometry and electronic structure of the complexes using the same level of theory. For a series of smaller Fe(II) SCO complexes, strong field ligands in the LS state result in geometries with the largest differences between DFT and CASPT2; however, good agreement overall is observed between DFT and CASPT2. For the larger complexes, moderate sized basis sets yield geometries that compare well with DFT and available experimental data. We recommend using the (10e,12o) active space since convergence to a minimum structure was more efficient than with truncated active spaces despite having similar Fe-ligand bond distances.

Effect of Oriented Electric Fields on Biologically Relevant Iron-Sulfur Clusters: Tuning Redox Reactivity for Catalysis

Enzyme-based iron-sulfur clusters, exemplified in families such as hydrogenases, nitrogenases, and radical S-adenosylmethionine enzymes, feature in many essential biological processes. The functionality of biological iron-sulfur clusters extends beyond simple electron transfer, relying primarily on the redox activity of the clusters, with a remarkable diversity for different enzymes. The active-site structure and the electrostatic environment in which the cluster resides direct this redox reactivity. Oriented electric fields in enzymatic active sites can be significantly strong, and understanding the extent of their effect on iron-sulfur cluster reactivity can inform first steps toward rationally engineering their reactivity. An extensive systematic density functional theory-based screening approach using OPBE/TZP has afforded a simple electric field-effect representation. The results demonstrate that the orientation of an external electric field of strength 28.8 MV cm^-1 at the center of the cluster can have a significant effect on its relative stability in the order of 35 kJ mol^-1. This shows clear implications for the reactivity of iron-sulfur clusters in enzymes. The results also demonstrate that the orientation of the electric field can alter the most stable broken-symmetry state, which further has implications on the directionality of initiated electron-transfer reactions. These insights open the path for manipulating the enzymatic redox reactivity of iron-sulfur cluster-containing enzymes by rationally engineering oriented electric fields within the enzymes.

A radical mechanism for C–H bond cross-coupling and N2 activation catalysed by β-diketiminate iron complexes

Density functional theory calculations and electronic structure analyses reveal a radical mechanism with spin-crossovers for C–H bond cross-coupling and N2 activation catalysed by β-diketiminate iron complexes.

Antiaromaticity-Promoted Radical Anion stability in α-vinyl Heterocyclics

As an electron-rich species, radical anions have a wide range of applications in organic synthesis. In addition, aromaticity is an essential concept in chemistry that has attracted considerable attention from...

Catalytic properties of the ferryl ion in the solid state: a computational review

This review summarises the last findings in the emerging field of heterogeneous catalytic oxidation of light alkanes by ferryl species supported on solid-state systems such as the conversion of methane into methanol by FeO-MOF74.

Role of Magnetization on Catalytic Pathways of Non-Oxidative Methane Activation on Neutral Iron Carbide Clusters

Methane has emerged as a promising fuel due to its abundance and clean combustion properties. It is also a raw material for various value added chemicals. However, the conversion of...

Stereodivergent atom-transfer radical cyclization by engineered cytochromes P450

Naturally occurring enzymes can be a source of unnatural reactivity that can be molded by directed evolution to generate efficient biocatalysts with valuable activities. Owing to the lack of exploitable stereocontrol elements in synthetic systems, steering the absolute and relative stereochemistry of free-radical processes is notoriously difficult in asymmetric catalysis. Inspired by the innate redox properties of first-row transition-metal cofactors, we repurposed cytochromes P450 to catalyze stereoselective atom-transfer radical cyclization. A set of metalloenzymes was engineered to impose substantial stereocontrol over the radical addition step and the halogen rebound step in these unnatural processes, allowing enantio- and diastereodivergent radical catalysis. This evolvable metalloenzyme platform represents a promising solution to tame fleeting radical intermediates for asymmetric catalysis.

Accurate calculation of spin-state energy gaps in Fe(III) spin-crossover systems using density functional methods

Fe(III) complexes are receiving ever-increasing attention as spin crossover (SCO) systems because they are usually air stable, as opposed to Fe(II) complexes, which are prone to oxidation. Here, we present the first systematic study exclusively devoted to assess the accuracy of several exchange-correlation functionals when it comes to predicting the energy gap between the high-spin (S = 5/2) and the low-spin (S = 1/2) states of Fe(III) complexes. Using a dataset of 24 different Fe(III) hexacoordinated complexes, it is demonstrated that the B3LYP* functional is an excellent choice not only for predicting spin-state energy gaps for Fe(III) complexes undergoing spin-transitions but also for discriminating Fe(III) complexes that are either low- or high-spin in the whole range of temperatures. Our benchmark study has led to the identification of a very versatile Fe(III) compound whose SCO properties can be engineered upon changing a single axial ligand. Overall, this work demonstrates that B3LYP* is a reliable functional for screening new spin-crossover systems with tailored properties.

Structural and Electronic Properties of Iron(0) PNP Pincer Complexes

In the present work we have prepared and fully characterized several Fe(0) complexes of the type [Fe(PNP)(CO)2] treating Fe(II) complexes [Fe(PNP)(Cl)2] with KC8 in the presence of carbon monoxide. While complexes [Fe(PNPNMe-iPr)(CO)2], [Fe(PNPNEt-iPr)(CO)2] adopt a trigonal bipyramidal geometry, the bulkier and more electron rich [Fe(PNPNH-tBu)(CO)2] is closer to a square pyramidal geometry. Mössbauer spectra showed isomer shifts very close to 0 and similar to those reported for Fe(I) systems. Quadrupole splitting values range between 2.2 and 2.7 mm s-1 both in experiments and DFT calculations, while those of Fe(I) complexes are much smaller (∼0.6 mm s-1).

Infrared photodissociation spectroscopy of heteronuclear group 15 metal-iron carbonyl cluster anions AmFe(CO)n- (A = Sb, Bi; m, n = 2, 3)

Heteronuclear group 15 metal-iron carbonyl cluster complexes of A_mFe(CO)_n^- (A = Sb, Bi; m, n = 2-3) were generated in the gas phase and studied by infrared photodissociation spectroscopy in the carbonyl stretching region. Their structures were determined by comparing the experimental spectra with predicted spectra derived from DFT calculations at the B3LYP and BP86 levels. All of the A_mFe(CO)_n^- cluster anions were determined to have Fe(CO)_n^- fragments with all of the CO ligands terminally bonded to the iron center, and they can be regarded as being formed via the interactions of the neutral group 15 metal clusters with the Fe(CO)_n^- fragments. Bonding analyses indicated that each A₂Fe(CO)_n^- (n = 2, 3) cluster anion contained two A-Fe single bonds and one A-A double bond. Each A₃Fe(CO)_n^- (n = 2, 3) cluster anion involved three A-Fe single bonds and three A-A single bonds. There is an isolobal relationship between the Fe(CO)₃^- group and the group 15 atoms. The substitution of an Fe(CO)₃^- group in place of one A atom in the tetrahedral A₄ molecule resulted in an A₃Fe(CO)₃^- cluster anion with the closed-shell electronic configuration for all the group 15 metals and iron atoms.

Improved Spin-State Energy Differences of Fe(II) Molecular and Crystalline Complexes via the Hubbard U-Corrected Density

We recently showed that the DFT+U approach with a linear-response U yields adiabatic energy differences biased toward high spin [Mariano et al. J. Chem. Theory Comput. 2020, 16, 6755-6762]. Such bias is removed here by employing a density-corrected DFT approach where the PBE functional is evaluated on the Hubbard U-corrected density. The adiabatic energy differences of six Fe(II) molecular complexes computed using this approach, named PBE[U] here, are in excellent agreement with coupled cluster-corrected CASPT2 values for both weak- and strong-field ligands resulting in a mean absolute error (MAE) of 0.44 eV, smaller than that of the recently proposed Hartree-Fock density-corrected DFT (1.22 eV) and any other tested functional, including the best performer TPSSh (0.49 eV). We take advantage of the computational efficiency of this approach and compute the adiabatic energy differences of five molecular crystals using PBE[U] with periodic boundary conditions. The results show, again, an excellent agreement (MAE = 0.07 eV) with experimentally extracted values and a superior performance compared with the best performers M06-L (MAE = 0.08 eV) and TPSSh (MAE = 0.31 eV) computed on molecular fragments.

First-Principles Investigations of Magnetic Anisotropy and Spin-Crossover Behavior of Fe(III)-TBP Complexes

With the ongoing effort to obtain mononuclear 3d-transition-metal complexes that manifest slow relaxation of magnetization and, hence, can behave as single-molecule magnets (SMMs), we have modeled 14 Fe(III) complexes based on an experimentally synthesized (PMe₃)₂FeCl₃ complex [J. Am. Chem. Soc. 2017, 139 (46), 16474-16477], by varying the axial ligands with group XV elements (N, P, and As) and equatorial halide ligands from F, Cl, Br, and I. Out of these, nine complexes possess large zero field splitting (ZFS) parameter D in the range of -40 to -60 cm^-1. The first-principles investigation of the ground-spin state applying density functional theory (DFT) and wave function-based multiconfigurations methods, e.g., SA-CASSCF/NEVPT2, are found to be quite consistent except for few delicate cases with near-degenerate spin states. In such cases, the hybrid B3LYP functional is found to be biased toward high-spin (HS) state. Altering the percentage of exact exchange admixed in the B3LYP functional leads to intermediate-spin (IS) ground state consistent with the multireference calculations. The origin of large zero field splitting (ZFS) in the Fe(III)-based trigonal bipyramidal (TBP) complexes is investigated. Furthermore, a number of complexes are identified with very small ΔG_HS-IS^adia. values indicating the possible spin-crossover phenomenon between the bistable spin states.

Nickel complexes based on BIAN ligands: transformation and catalysis on ethylene polymerization

Treatment of bis(arylimino)acenaphthene (ArBIAN) with Ni(COD)2 in toluene afforded dmpBIANNi(COD) (2a, dmp = 2,6-Me2C6H3) and dippBIANNi(COD) (2b, dipp = 2,6-iPrC6H3), respectively, in moderate yields. Complexes 2a and 2b can be oxidized by a small amount of oxygen at low temperature leading to oxygen-bridged dinuclear Ni(ii) complexes (dmpBIANNi)2(μ-O)2 (4a) and (dippBIANNi)2(μ-O)2 (4b), respectively, as a purple powder. The reaction of ArBIAN with 0.5 equiv of Ni(COD)2 or Ni(Ph3P)4 gave bisligated complexes (dmpBIAN)2Ni (3a) and (dippBIAN)2Ni (3b), which can be considered as Ni(0) complexes supported by two neutral BIAN ligands. Oxidation of the bisligated nickel complexes 3a and 3b with [Cp2Fe][B(C6F5)4] afforded cationic Ni(i) complexes [(dmpBIAN)2Ni][B(C6F5)4] (5a) and [(dippBIAN)2Ni][B(C6F5)4] (5b), respectively, in which the Ni(i) centre is chelated by two neutral Ar-BIAN ligands. These complexes were characterized by NMR and IR spectroscopy and DFT calculation, and the molecular structures of 3b, 4b, and 5b were well established by X-ray diffraction analysis. These complexes were evaluated as catalysts for ethylene polymerization in which 2b showed high activity in the presence of AlMe3. 13C NMR analysis of polymers showed that the 2b/AlMe3 catalytic system gave less-branched polymers when compared to that obtained with dippBIANNiBr2 under the same conditions.

A Theoretical Assessment of Spin and Charge States in Binuclear Cobalt-Ruthenium Complexes: Implications for a Creutz-Taube Model Ion Separated by a C60-Derivative Bridging Ligand

We investigate the spin-state energetics and the role of ionic charges in the electronic configuration of binuclear complexes of the form [(NH₃)₅Co(py)-X-(py)Ru(NH₃)₅]^q+. In these compounds with q = 4-6, py = pyridine, and X = C≡C and C₆₀, the Co-Ru distance varies from ∼1.4 to ∼2.1 nm. We carry out a systematic electronic structure calculation using different exchange-correlation (xc) approaches within spin-density functional theory, which are largely employed to investigate the properties of a variety of coordination complexes. To evaluate the effects of spin states and type of spacer in the bridging ligand on the valence tautomerism between Co^2+/3+ and Ru^2+/3+, we examine in more detail the case of Creutz-Taube-type ions [(NH₃)₅Co(py)-X-(py)Ru(NH₃)₅]⁵⁺. Our analysis shows that the stabilization of low- and high-spin states critically depends on the total charge of the complex, type of X-bridged ligand, and employed xc approach to calculate the electron spin density. Importantly, the C₆₀-bridged group may result in a blockage of the valence tautomerism of the Creutz-Taube complex, inducing bistable charge configurations. Overall, our results also show that an adiabatic description in terms of the frontier molecular spin-orbitals for analyzing the distinct spin-charge states of these complexes may dramatically depend on the density-functional description.

Spin-state dependence of exchange-correlation holes

Applications of density-functional theory (DFT) in computational chemistry rely on an approximate exchange-correlation (xc) functional. However, existing approximations can fail dramatically for open-shell molecules, in particular for transition-metal complexes or radicals. Most importantly, predicting energy differences between different spin-states with approximate exchange-correlation functionals remains extremely challenging. Formally, it is known that the exact xc functional should be spin-state dependent, but none of the available approximations feature such an explicit spin-state dependence [C. R. Jacob and M. Reiher, Int. J. Quantum Chem., 2012, 112, 3661-3684]. Thus, to find novel approximations for the xc functional for open-shell systems, the development of spin-state dependent xc functionals appears to be a promising avenue. Here, we set out to shed light on the spin-state dependence of the xc functional by investigating the underlying xc holes, which we extract from configuration interaction calculations for model systems. We analyze the similarities and differences between the xc holes of the lowest-energy singlet and triplet states of the dihydrogen molecule, the helium atom, and the lithium dimer. To shed further light on the spin-state dependence of these xc holes we also discuss exact conditions that can be derived from the spin structure of the reduced two-electron density matrix. Altogether, our results suggest several possible routes towards the construction of explicitly spin-state dependent approximations for the xc functional.

Probing the Origin of Adaptive Aromaticity in 16-Valence-Electron Metallapentalenes

Species with adaptive aromaticity are aromatic in the ground and lowest-lying triplet excited states and they have normally intermediate singlet-triplet gaps. Few examples of compounds with adaptive aromaticity are known to date, including 16-valence-electron (16e) metallapentalenes. A sweeping search could be conducted to discover new members of this group, but efficient designs with an explicit strategy would facilitate the quest for new members of this elusive family. Density functional theory calculations and aromaticity evaluations have been performed to reveal the nature of triplet-state aromaticity in 16e metallapentalenes. Our results show that coordination of strong σ- or π-donor ligands helps achieving adaptive aromaticity of 16e metallapentalenes by means of a spin delocalization mechanism. These results have important implications for understanding the unusual properties of the organometallic adaptive aromatics, leading the way to efficient design of new compounds with tunable singlet-triplet gaps.

A DFT Study on FeI/FeII/FeIII Mechanism of the Cross-Coupling between Haloalkane and Aryl Grignard Reagent Catalyzed by Iron-SciOPP Complexes

To explore plausible reaction pathways of the cross-coupling reaction between a haloalkane and an aryl metal reagent catalyzed by an iron–phosphine complex, we examine the reaction of FeBrPh(SciOPP) 1 and bromocycloheptane employing density functional theory (DFT) calculations. Besides the cross-coupling, we also examined the competitive pathways of β-hydrogen elimination to give the corresponding alkene byproduct. The DFT study on the reaction pathways explains the cross-coupling selectivity over the elimination in terms of Fe^I/Fe^II/Fe^III mechanism which involves the generation of alkyl radical intermediates and their propagation in a chain reaction manner. The present study gives insight into the detailed molecular mechanic of the cross-coupling reaction and revises the Fe^II/Fe^II mechanisms previously proposed by us and others.

Spin Splitting Energy of Transition Metals: A New, More Affordable Wave Function Benchmark Method and Its Use to Test Density Functional Theory

Accurately predicting the spin splitting energy of chemical species is important for understanding their reactivity and magnetic properties, but it is very challenging, especially for molecules containing transition metals. One impediment to progress is the scarcity of accurate benchmark data. Here we report a set of calculations designed to yield reliable benchmarks for simple transition-metal complexes that can be used to test density functional methods that are affordable for large systems of more practical interest. Various wave function methods are tested against experiment for Fe²⁺, Fe³⁺, and Co³⁺, including CASSCF, CASPT2, CASPT3, MRCISD, MRCISD+Q, ACPF, AQCC, CCSD(T), and CASPT2/CCSD(T) and also a new method called CASPT2.5, which is performed by taking the average of the CASPT2 and CASPT3 energies. We find that MRCISD+Q, ACPF, and AQCC require smaller active spaces for good accuracy than are required by CASPT2 and CASPT3, and this aspect may be important for calculations on larger molecules; here we find that CASPT2.5 extrapolated to a complete basis set is the most suitable method-in terms of computational cost and in terms of accuracy on monatomic systems-and therefore we chose this method for molecular benchmarks. Then Kohn-Sham density functional calculations with 60 exchange-correlation functionals are tested for FeF₂, FeCl₂, and CoF₂. We find that MN15-L, M06-SX, and revM06 have very good agreement with CASPT2.5 benchmarks in terms of both the spin splitting energy and the optimized geometry for each spin state. In addition, we recommend def2-TZVP as the most suitable basis set to perform density functional calculations for molecular spin splitting energies; extra polarization functions in the basis set do not help to increase the accuracy of the spin splitting energy in KS calculations.

Using electronegativity and hardness to test density functionals

Density functional theory (DFT) is used in thousands of papers each year, yet lack of universality reduces DFT's predictive capacity, and functionals may produce energy-density imbalances. The absolute electronegativity (χ) and hardness (η) directly reflect the energy-density relationship via the chemical potential ∂E/∂N and we thus hypothesized that they probe universality. We studied χ and η for atoms Z = 1-36 using 50 diverse functionals covering all major classes. Very few functionals describe both χ and η well. η benefits from error cancellation, whereas χ is marred by error propagation from IP and EA; thus, almost all standard GGA and hybrid functionals display a plateau in the MAE at ∼0.2 eV-0.3 eV for η. In contrast, variable performance for χ indicates problems in describing the chemical potential by DFT. The accuracy and precision of a functional is far from linearly related, yet for a universal functional, we expect linearity. Popular functionals such as B3LYP, PBE, and revPBE perform poorly for both properties. Density sensitivity calculations indicate large density-derived errors as occupation of degenerate p- and d-orbitals causes "non-universality" and large dependency on exact exchange. Thus, we argue that performance for χ for the same systems is a hallmark of an important aspect of universality by probing ∂E/∂N. With this metric, B98, B97-1, PW6B95D3, MN-15, rev-TPSS, HSE06, and APFD are the most "universal" among the tested functionals. B98 and B97-1 are accurate for very diverse metal-ligand bonds, supporting that a balanced description of ∂E/∂N and ∂E²/∂N², via χ and η, is probably a first simple probe of universality.