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.

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.

CH4 activation by PtX+ (X = F, Cl, Br, I)

Reactions of PtX⁺ (X = F, Cl, Br, I) with methane have been investigated at the density functional theory (DFT) level. These reactions take place more easily along the low-spin potential energy surface. For HX (X = F, Cl, Br, I) elimination, the formal oxidation state of the metal ion appears to be conserved, and the importance of this reaction channel decreases in going as the sequence: X = F, Cl, Br, I. A reversed trend is observed in the loss of H₂ for X = F, Cl, Br, while it is not favorable for PtI⁺ in the loss of either HI or H₂. For HX eliminations, the transfer form of H is from proton to atom, last to hydride, and the mechanisms are from PCET to HAT, last to HT for the sequence of X = F, Cl, Br, I. One reason is mainly due to the electronegativity of halogens. Otherwise, the mechanisms of HX eliminations also can be explained by the analysis of Frontier Molecular Orbitals. While for the loss of H₂, the transfer of H is in the form of hydride for all the X ligands. Noncovalent interactions analysis also can be explained the reaction mechanisms.

Environmental Control of the Magnetic Behavior of Transition Metal Complexes: Density Functional Theory Study of Zeolite Y Embedded Complexes [M(bpy)3]2+@Y (M = Fe2+, Co2+)

Using the supramolecular approach developed for the study of the guest-host interactions in the zeolite Y encapsulated [Fe(bpy)₃]²⁺ compound: [Fe(bpy)₃]²⁺@Y (bpy = 2,2'-bipyridine) [Vargas et al., J. Chem. Theory Comput. 2009, 5, 97-115], we apply density functional theory (DFT) to the study of the influence of zeolite Y encapsulation on the structural and energetic properties of [Co(bpy)₃]²⁺ in the low-spin (LS) and high-spin (HS) states, while revisiting [Fe(bpy)₃]²⁺@Y. Although the accurate prediction of the HS-LS energy difference ΔE_HL^el remains challenging for current DFT methods, they give accurate estimates of its variation Δ(ΔE_HL^el) in a series of complexes of a given transition metal ion. Therefore, denoting [M(bpy)₃]²⁺@Y_SM as the supramolecular model of the inclusion compounds, the values of ΔE_HL^el for the bpy complexes in the gas phase and in the supercage of zeolite Y were determined by combining the DFT estimates of Δ(ΔE_HL^el) in the series {[M(NCH)₆]²⁺, [M(bpy)₃]²⁺, and [M(bpy)₃]²⁺@Y_SM}, with accurate CCSD(T) estimates of ΔE_HL^el in the benchmark complexes [M(NCH)₆]²⁺ (M = Fe, Co) [Lawson Daku et al., J. Chem. Theory Comput., 2012, 8, 4216-4231]. Generalized gradient approximations as well as global and range-separated hybrids were employed. In order to better account for the key role of dispersion, they were also augmented with the semiempirical D2, D3BJ, and D3BJM dispersion corrections when available. The use of the D3BJ and D3BJM corrections led to similar results, and this is only with the use of the D2 scheme that (i) the free and encapsulated [Fe(bpy)₃]²⁺ are correctly predicted as LS species and that (ii) the encapsulation of both complexes translates into a destabilization of their HS state with respect to their LS state. The increase of the HS-LS energy difference is smaller for [Co(bpy)₃]²⁺ than [Fe(bpy)₃]²⁺ because the HS-LS molecular volume difference ΔV_HL in [Co(bpy)₃]²⁺ is ∼50% smaller than in [Fe(bpy)₃]²⁺. Periodic DFT calculations performed on crystalline [M(bpy)₃]²⁺@Y show that the employed [M(bpy)₃]²⁺@Y_SM supramolecular model allows the influence of encapsulation on the geometry and the spin-state energetics of [M(bpy)₃]²⁺ (M = Fe, Co) to be quantitatively captured.

Reconciling Local Coupled Cluster with Multireference Approaches for Transition Metal Spin-State Energetics

Spin-state energetics of transition metal complexes remain one of the most challenging targets for electronic structure methods. Among single-reference wave function approaches, local correlation approximations to coupled cluster theory, most notably the domain-based local pair natural orbital (DLPNO) approach, hold the promise of bringing the accuracy of coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T), to molecular systems of realistic size with acceptable computational cost. However, recent studies on spin-state energetics of iron-containing systems raised doubts about the ability of the DLPNO approach to adequately and systematically approximate energetics obtained by the reference-quality complete active space second-order perturbation theory with coupled-cluster semicore correlation, CASPT2/CC. Here, we revisit this problem using a diverse set of iron complexes and examine several aspects of the application of the DLPNO approach. We show that DLPNO-CCSD(T) can accurately reproduce both CASPT2/CC and canonical CCSD(T) results if two basic principles are followed. These include the consistent use of the improved iterative (T₁) versus the semicanonical perturbative triple corrections and, most importantly, a simple two-point extrapolation to the PNO space limit. The latter practically eliminates errors arising from the default truncation of electron-pair correlation spaces and should be viewed as standard practice in applications of the method to transition metal spin-state energetics. Our results show that reference-quality results can be readily achieved with DLPNO-CCSD(T) if these principles are followed. This is important also in view of the applicability of the method to larger single-reference systems and multinuclear clusters, whose treatment of dynamic correlation would be challenging for multireference-based approaches.

Halogenation affects driving forces, reorganization energies and "rocking" motions in strained [Fe(tpy)2]2+ complexes

Controlling the energetics of spin crossover (SCO) in Fe(II)-polypyridine complexes is critical for designing new multifunctional materials or tuning the excited-state lifetimes of iron-based photosensitizers. It is well established that the Fe-N "breathing" mode is important for intersystem crossing from the singlet to the quintet state, but this does not preclude other, less obvious, structural distortions from affecting SCO. Previous work has shown that halogenation at the 6 and 6'' positions of tpy (tpy = 2,2';6',2''-terpyridine) in [Fe(tpy)₂]²⁺ dramatically increased the lifetime of the excited MLCT state and also had a large impact on the ground state spin-state energetics. To gain insight into the origins of these effects, we used density functional theory calculations to explore how halogenation impacts spin-state energetics and molecular structure in this system. Based on previous work we focused on the ligand "rocking" motion associated with SCO in [Fe(tpy)₂]²⁺ by constructing one-dimensional potential energy surfaces (PESs) along the tpy rocking angle for various spin states. It was found that halogenation has a clear and predictable impact on ligand rocking and spin-state energetics. The rocking is correlated to numerous other geometrical distortions, all of which likely affect the reorganization energies for spin-state changes. We have quantified trends in reorganization energy and also driving force for various spin-state changes and used them to interpret the experimentally measured excited-state lifetimes.

Exploring the light-induced dynamics in solvated metallogrid complexes with femtosecond pulses across the electromagnetic spectrum

Oligonuclear complexes of d⁴-d⁷ transition metal ion centers that undergo spin-switching have long been developed for their practical role in molecular electronics. Recently, they also have appeared as promising photochemical reactants demonstrating improved stability. However, the lack of knowledge about their photophysical properties in the solution phase compared to mononuclear complexes is currently hampering their inclusion into advanced light-driven reactions. In the present study, the ultrafast photoinduced dynamics in a solvated [2 × 2] iron(II) metallogrid complex are characterized by combining measurements with transient optical-infrared absorption and x-ray emission spectroscopy on the femtosecond time scale. The analysis is supported by density functional theory calculations. The photocycle can be described in terms of intra-site transitions, where the Fe^II centers in the low-spin state are independently photoexcited. The Franck-Condon state decays via the formation of a vibrationally hot high-spin (HS) state that displays coherent behavior within a few picoseconds and thermalizes within tens of picoseconds to yield a metastable HS state living for several hundreds of nanoseconds. Systematic comparison with the closely related mononuclear complex [Fe(terpy)₂]²⁺ reveals that nuclearity has a profound impact on the photoinduced dynamics. More generally, this work provides guidelines for expanding the integration of oligonuclear complexes into new photoconversion schemes that may be triggered by ultrafast spin-switching.

Benchmarking quantum chemistry methods for spin-state energetics of iron complexes against quantitative experimental data

The accuracy of relative spin-state energetics predicted by selected quantum chemistry methods: coupled cluster theory at the CCSD(T) level, multiconfigurational perturbation theory (CASPT2, NEVPT2), multireference configuration interaction at the MRCISD+Q level, and a number of DFT methods, is quantitatively evaluated by comparison with the experimental data of four octahedral iron complexes. The available experimental data, either spin-forbidden transition energies or spin crossover enthalpies, are corrected for relevant environmental effects in order to derive the quantitative benchmark set of iron spin-state energetics. Comparison of theory predictions with the resulting reference data: (1) validates the high accuracy of the CCSD(T) method, particularly when based on Kohn-Sham orbitals, giving the maximum error below 2 kcal mol-1 and the mean absolute error (MAE) below 1 kcal mol-1; (2) corroborates the tendency of CASPT2 to systematically overstabilize higher-spin states by up to 5.5 kcal mol-1; (3) confirms that the latter problem is partly remedied by the recently proposed CASPT2/CC approach [Phung et al., J. Chem. Theory Comput., 2018, 14, 2446-2455]; (4) demonstrates that NEVPT2 performs worse than CASPT2, by giving errors up to 7 kcal mol-1; (5) shows that the accuracy of MRCISD+Q spin-state energetics strongly depends on the size-consistency correction: the Davidson-Silver and Pople corrections perform best (MAE < 3 kcal mol-1), whereas the standard Davidson correction is not recommended (MAE of 7 kcal mol-1). Only a few DFT methods (including the best performing ones identified in this study: B2PLYP-D3 and OPBE) are able to provide a balanced description of the spin-state energetics for all four studied iron complexes simultaneously, corroborating the non-universality problem of approximate density functionals.

Ligand Substituent Effects on the Spin-Crossover Behaviors of Dinuclear Iron(II) Compounds

Six analogue compounds with the general formula [Fe₂( xL)₅(NCS)₄]· yMeOH ( x = o-Cl, y = 3 for compound 1; x = m-Cl, y = 5 for 2; x = p-Cl, y = 1 for 3; x = o-Me, y = 2 for 4; x = m-Me, y = 2 for 5; x = p-Me, y = 3 for 6; L = N-phenylmethylene-4-amino-1,2,4-triazole) were synthesized. The two Fe(II) ions are triply bridged by the triazole groups of three xL ligands and each Fe(II) is further capped with two NCS^- groups and one more xL ligand. These compounds show regular patterns in their magnetic properties that depend on the positions the substituent groups (-Cl or -Me) ride, i.e., ortho-substituted compounds 1 and 4 undergo complete one-step spin crossover (SCO), while meta-substituted compounds 2 and 5 display incomplete one-step SCO with lower transition temperatures, and para-substituted compounds 3 and 6 are in the high-spin states in all temperature ranges. Structural analyses reveal that the molecular geometry and intermolecular interactions of these compounds, which should account for the differences in magnetic properties, are obviously depend on the positions of substituent groups (steric effect), despite them being electron-withdrawing chlorine or electron-donating methyl, whereas theoretical calculations confirm that the electronic effects of substituent groups exert no effect on the magnetic properties.

A Simple Approach for Predicting the Spin State of Homoleptic Fe(II) Tris-diimine Complexes

We propose a simple method for predicting the spin state of homoleptic complexes of the Fe(II) d⁶ ion with chelating diimine ligands. The approach is based on the analysis of a single metric parameter within a free (noncoordinated) ligand: the interatomic separation between the N-donor metal-binding sites. An extensive analysis of existing complexes allows the determination of critical N···N distances that dictate the regions of stability for the high-spin and low-spin complexes, as well as the intermediate range in which the magnetic bistability (spin crossover) can be observed. The prediction has been tested on several complexes that demonstrate the validity of our method.

Femtosecond X-Ray Scattering Study of Ultrafast Photoinduced Structural Dynamics in Solvated [Co(terpy)_{2}]^{2+}

We study the structural dynamics of photoexcited [Co(terpy)_{2}]^{2+} in an aqueous solution with ultrafast x-ray diffuse scattering experiments conducted at the Linac Coherent Light Source. Through direct comparisons with density functional theory calculations, our analysis shows that the photoexcitation event leads to elongation of the Co-N bonds, followed by coherent Co-N bond length oscillations arising from the impulsive excitation of a vibrational mode dominated by the symmetrical stretch of all six Co-N bonds. This mode has a period of 0.33 ps and decays on a subpicosecond time scale. We find that the equilibrium bond-elongated structure of the high spin state is established on a single-picosecond time scale and that this state has a lifetime of ∼7  ps.

Ammonia-modified Co(II) sites in zeolites: spin and electron density redistribution through the Co(II)-NO bond

Electronic factors essential for the bonding of a non-innocent NO ligand to ammonia-modified Co(2+) sites in cobalt-exchanged zeolites are examined for small cluster models using DFT and advanced correlated wave function calculations. The analysis of charge transfer processes between the NO ligand and the cobalt center involves two protocols: valence-bond expansion of the multiconfiguration CASSCF wave function (in terms of fragment-localized active orbitals) and spin-resolved natural orbitals for chemical valence (SR-NOCV). Applicability of SR-NOCV analysis to transition metal complexes involving non-innocent fragments is critically assessed and the approach based on the CASSCF wave function turns out to be much more robust and systematic for all studied models. It is shown that the character and direction of electron density redistribution through the Co-N-O bond, quantified by relative share of the Co(II)-NO(0), Co(III)-NO(-), and Co(I)-NO(+) resonance structures in the total wave function, fully rationalize the activation of the N-O bond upon NH3 co-ligation (evidenced by calculated and measured red-shift of the NO stretching frequency and commonly ascribed to enhanced backdonation). The huge red-shift of νN-O is attributed to an effective electron transfer between the ammonia-modified Co(ii) centers and the NO antibonding π*-orbitals (related to the increased share of the Co(III)-NO(-) form). Unexpectedly, the effect is stronger for the singlet complex with three NH3 ligands than for that with five NH3 ligands bound to the cobalt center. Our results also indicate that high-efficiency electron transfers between the Co(ii) center and the NO ligand may be enabled for the selected spin state and disabled for the other spin state of the adduct. This illustrates how the cobalt center may serve to fine-tune the electronic communication between the NO ligand and its binding site.

Abrupt versus Gradual Spin-Crossover in Fe(II)(phen)2(NCS)2 and Fe(III)(dedtc)3 Compared by X-ray Absorption and Emission Spectroscopy and Quantum-Chemical Calculations

Molecular spin-crossover (SCO) compounds are attractive for information storage and photovoltaic technologies. We compared two prototypic SCO compounds with Fe(II)N6 (1, [Fe(phen)2(NCS)2], with phen = 1,10-phenanthroline) or Fe(III)S6 (2, [Fe(dedtc)3], with dedtc = N,N'-diethyldithiocarbamate) centers, which show abrupt (1) or gradual (2) thermally induced SCO, using K-edge X-ray absorption and Kβ emission spectroscopy (XAS/XES) in a 8-315 K temperature range, single-crystal X-ray diffraction (XRD), and density functional theory (DFT). Core-to-valence and valence-to-core electronic transitions in the XAS/XES spectra and bond lengths change from XRD provided benchmark data, verifying the adequacy of the TPSSh/TZVP DFT approach for the description of low-spin (LS) and high-spin (HS) species. Determination of the spin densities, charge distributions, bonding descriptors, and valence-level configurations, as well as similar experimental and calculated enthalpy changes (ΔH), suggested that the varying metal-ligand bonding properties and deviating electronic structures converge to similar enthalpic contributions to the free-energy change (ΔG) and thus presumably are not decisive for the differing SCO behavior of 1 and 2. Rather, SCO seems to be governed by vibrational contributions to the entropy changes (ΔS) in both complexes. Intra- and intermolecular interactions in crystals of 1 and 2 were identified by atoms-in-molecules analysis. Thermal excitation of individual dedtc ligand vibrations accompanies the gradual SCO in 2. In contrast, extensive inter- and intramolecular phen/NCS vibrational mode coupling may be an important factor in the cooperative SCO behavior of 1.

Predicting the spin state of paramagnetic iron complexes by DFT calculation of proton NMR spectra

Many transition-metal complexes easily change their spin state S in response to external perturbations (spin crossover). Determining such states and their dynamics can play a central role in the understanding of useful properties such as molecular magnetism or catalytic behavior, but is often far from straightforward. In this work we demonstrate that, at a moderate computational cost, density functional calculations can predict the correct ground spin state of Fe(ii) and Fe(iii) complexes and can then be used to determine the (1)H NMR spectra of all spin states. Since the spectral features are remarkably different according to the spin state, calculated (1)H NMR resonances can be used to infer the correct spin state, along with supporting the structure elucidation of numerous paramagnetic complexes.

Valence tautomeric dinuclear adducts of Co(ii) diketonates with redox-active diquinones for the design of spin qubits: computational modeling

DFT calculations performed on dinuclear adducts of Co(ii) diketonates with redox-active diquinones showed that with the proper structural tuning of the ligands, these paramagnetic complexes possess the properties required in a 2-qubit molecular system.

Spin crossover with thermal hysteresis: practicalities and lessons learnt

Memory applications of spin crossover require bistability: magnetic data must be appropriately collected and reported, and consideration given to lifetimes.