Various aspects related to the use of DFT method for the study of magnetic, geometry and energetic properties of transition metal complexes with redox-active ligands are considered. Particular attention is given to the correct choice of model compounds and methodology of the calculations.
Transition metal complexes with redox-active (non-innocent) ligands play a key role in many biologically important enzyme-driven ,  and catalytic , , ,  reactions and are implemented as the magnetically responsive molecular switches in various molecular electronic and spintronic devices , , . Because of the closeness of energy levels of the valence d-shell electrons of a metal and frontier MOs of the ligand these complexes have highly flexible electronic structure susceptible to low energy barrier intramolecular electron transfers within the molecule and providing for the formation of electronic isomers (electromers ) with two (and more than two in the dinuclear complexes) migrating paramagnetic centers. Magnetic behavior of the complexes with redox-active ligands, kinetic and thermodynamic parameters of rearrangements of the electromers are highly sensitive with respect to even tiny structural modifications and effects of solvation and nonspecific intermolecular interactions. Moreover, definitions of some of the widely used notions of coordination chemistry, such as oxidation number become to be ambiguous when applied to complexes with redox-active ligands and are required to be refined or, at least, viewed under a new angle.
The proper understanding and interpretation of the peculiar properties and reactivity of these compounds must, obviously, be based on the application of methods of modern quantum chemistry, which is the goal of the present microreview largely addressed to chemists employing quantum chemical calculations for prediction of novel and description of properties of experimentally studied transition metal complexes with redox-active ligands. A distinctive feature of the adopted approach is that careful attention is given to the precise account taken for of all structural peculiarities of the molecular systems, underestimation of which in calculations of model systems often leads to wrong results and may disorient experimentalists. The data reported in the paper are based on the experience accumulated by the authors mainly in the course of their computational studies of CoII/CoIII complexes correlated with results of the experimental investigation, but the conclusions made are not limited by this group of the complexes.
Methodology and computational details
All the calculations were performed with the density functional theory (DFT) method which remains to be the most general current tool of theoretical analysis and prediction of properties of substances and materials. It has been shown , , ,  that in spite of the single determinant character of DFT and that the exact energy functional is unknown, with the properly chosen exchange-correlation (XC) functional and all-electronic basis set one can reach the accuracy of correlated methods and get reliable predictions on energy, geometry and electronic structures of molecular systems and solids. According to a recent detailed analysis  of the use of DFT method for calculations of properties and mechanisms of reactions of metal coordination compounds, “in coordination chemistry, the DFT calculations have to be validated by rigorous ab initio or by accurate experimental data for each new problem or new system”. Since application of ab initio methods to polyatomic transition metal complexes can now hardly be regarded as the realistic approach, the principal strategy employed in the present work for testing selected DFT protocols is based on correlation of results of calculations with the experimental data available for the compounds structurally identical or most similar to those under study.
The isomers of transition metal complexes with redox-active ligands have differing spin states, which imposes certain restrictions on the choice of an appropriate functional that is capable to correctly describe spin states of magnetically active complexes . Thus, the use of the widely employed hybrid B3LYP functional for calculations of relative energies of the truncated structures neglecting the tert-butyl groups of the valence tautomeric (VT ) cobalt complex 1 with 9-hydroxy-2,4,6,8-tetra-(tert-butyl)phenoxazin-1-one ligands (Fig. 1) leads to a wrong conclusion on the energy preference of the high-spin (HS) electromer 1′ Cat-N-Q–HSCoII–Cat-N-Q (Q – neutral quinone form, Cat – dianionic catecholate form of the redox-active ligand). The correct result corresponding to the experimental data  on the energy favored low-spin (LS) electromer 1′ Cat-N-Q–LSCoIII–Cat-N-SQ (SQ – radical-anion semiquinone form of the redox-active ligand) was obtained with the modified B3LYP* functional , in which the Hartree-Fock (H-F) part of the XC functional is reduced from 20 (in B3LYP) to 15%. The B3LYP*/6-311++G(d,p) approach well reproduces not only thermodynamic (Table 1), but also kinetic (energy barriers) parameters of the VT rearrangements of complexes 1 and their analogs , . Good agreement between computational and experimental data was also achieved with the use of another hybrid functional – TPSSh  (the H-F contribution is equal to 10%) employed for the study of transition metal complexes with open-shell electronic systems (Table 1).
aHereafter apostrophe indicates the truncated structures stripped off the tert-butyl groups.
At the same time, application of B3LYP* functional to the study of magnetic behavior of a cationic CoII complex based on o-benzoquinone and a di-tert-butyl derivative of 2,11-diaza[3.3](2,6)pyridinophane 2  led to the incorrect conclusion on the energy preference of the low-spin electromeric form SQ–LSCoII, whereas magnetic measurements point to the catecholate form of the quinone moiety of the complex . According to the B3LYP*/6-311++G(d,p) calculations, the structure Cat–LSCoIII corresponding to this spin state is by 6.9 kcal mol−1 destabilized relative to the SQ–LSCoII form (Fig. 2). The coincidence of the theoretical and experimental assignment of the ground state of complex 2 was attained with the use of the TPSSh/6-311++G(d,p) approximation, which also predicted a possibility of occurrence of the centered on CoII ion spin-crossover rearrangement .
The considered examples of computational modeling spin states of transition metal complexes clearly show that in the absence of systematic way of choosing a proper DFT functional each new problem or new system must be treated individually based on careful comparison with experimental data.
Special attention has to be given to the choice of a basis set, the size and completeness of which may render drastic influence on the principal conclusions and accuracy of the performed calculations. At the initial stage of many DFT studies of transition metal complexes, it seems expedient to employ a simple double-zeta Pople’s 6-31G(d,p) basis set, the calculations with which can be done with the help of rather modest computational resources and generally provide for good reproduction of energy and structural characteristics , . Less successful is the application of this basis set for the study of magnetic properties of metal complexes. The B3LYP*/6-31G(d,p) calculations performed on the adducts of bischelate cobalt complexes with redox-active phenoxazin-1-one predicted the low-spin SQ–LSCoIII ground state electromeric forms for the adducts of CoII(bis-N-phenylsalicylaldiminate) 3 and CoII(bis-hexafluoroacetylacetonate) 4 (Fig. 3), whereas results of X-ray and magnetochemical studies ,  showed that the structure of the latter must be described as a high-spin Q–HSCoII molecular complex. Moreover, the predicted energy difference between the isomers 3, SQ–LSCoIII and 3, Q–HSCoII (16.5 kcal mol−1) excludes the possibility of thermally initiated redox-process, which is not in consistent with magnetochemical data . To reach agreement with the experimental findings , , it was necessary to use the extended 6-311++G(d,p) basis set including diffuse and polarization functions on both heavy and light atomic centers. A good alternative to this basis is given by the Ahlrich’s valence triple-zeta def2-TZVP basis set .
Dispersion interactions and solvation effects
In solution and in the solid state, transition metal complexes with organic ligands tend to self-association (oligomerization). This process can significantly affect magnetic properties, structural parameters of molecules of the complexes and even alter stereochemistry of the metal coordination site. The crystal structures of the majority of transition metal complexes are represented by polynuclear architectures, magnetic properties of which differ from those of the isolated molecules. To account for that part of noncovalent interactions of molecules leading to their association and brought about by the dispersion forces, an approach has been advanced allowing evaluation of this effect through complementing the functional with the so-called dispersion corrections. The most extensively employed methods of accounting for dispersion corrections were elaborated by Yanai (CAM ) and Grimme (D3BJ ). In the case of polymolecular systems with closed electron shells this methodology usually helps to markedly improve agreement between the theory and experiment , , . At the same time, implementation of CAM and D3BJ schemes for estimation of the role of dispersion interactions in the formation of oligomers of transition metal diketonates  with the open electron shell structures leads to significant overestimation of contribution of dispersion energy to the total stabilization energies of these aggregates and, thus, should be done with great caution. This conclusion is illustrated by the data of Table 2. Whereas the low value (0.8 kcal mol−1) of energy of stabilization (Estab) of the high-spin electromeric form Q–HSCoII of the adduct 3 obtained by the B3LYP*/6-311++G(d,p) calculations  well agrees with the observed dissociation of this mixed-ligand complex under heating its toluene solution, the calculations performed with inclusion of the D3BJ correction lead to the highly overestimated value of Estab.
|Electromeric form||B3LYP*||B3LYP* D3BJ||B3LYP*+Solv|
In coordination chemistry, the most widely used methods of estimation of solvation are represented by the various versions of the general polarized continuum model (PCMs), the principal descriptors of which are given by dielectric constant, refractive index, acidity and basicity parameters of solvent , , . Account for contribution of the solvation effects was found to be critically important for the correct theoretical description of mechanisms of some of reactions, e.g. hydroamination , catalyzed by transition metal complexes, but it is usually less significant for estimation of thermodynamic characteristics of intramolecular processes, particularly those occurring in low polarity solvents. Thus, the relative energies of the redox-isomers of mixed-ligand complex 3 and those of complexes formed by addition of pyrene-4,5-dione to CoII diketonates  calculated in the gas phase and in toluene solution (in the B3LYP*/6-311++G(d,p)+IEFPCM approximation) differ by no more than 2.5 kcal mol−1.
A comprehensive overview of recent progress in application of DFT methods in coordination chemistry  contains a number of useful instructions for successful application of DFT addressed to practitioners of the method. A particularly underlined is the recommendation of calibration of calculations on model molecules similar to the real objects of intended study. When choosing the model molecules one usually strips the principal structure from alkyl groups and the substituents positioned apart from the reaction center and, seemingly, render no direct influence on reaction mechanism or the studied properties. This widely used approach can, however, be liable to break down in the case of compounds bearing strong electron releasing or withdrawing groups and/or voluminous substituents affecting reaction pathway or reactive conformation of the compounds.
Many usable redox-active ligands contain bulky alkyl substituents and theoretical calculations of transition metal complexes with these ligands are often performed on the truncated structures omitting the substituents. For the case of an interesting bis((3,5-di-tert-butylcatechol-N-(3,5-di-tert-butyl)biquinone)imine-N,O)-zinc(II) complex 5 , paramagnetic properties of the high-spin electromeric form of which are due to the transition of a ligand to its excited triplet state, the geometries calculated for the structure with four tert-butyl groups and that stripped off these groups as well as the energy gaps between the low- and high-spin electromers (16.0 kcal mol−1 for the complex comprising tert-butyl groups and 16.9 kcal mol−1 for that omitting the substituents) practically coincide (Fig. 4).
In contrast with ZnII complex 5, in which the bulky tert-butyl groups do not practically affect its geometry and magnetic properties, NiII complex 6 with a stereochemically rigid phenoxazine-1-one redox-active ligand manifests high sensitivity of the coordination site with respect to tert-butyl substituents in the 2, 8 positions of the heterocycle . According to the B3LYP*/6-311++G(d,p) calculations carried out on the model complex 6′ stripped off the bulky substituents, the ground state is represented by the low-spin electromer 6′ trans-LSNiII with trans-positioned phenoxazine-1-one ligands (Fig. 5). The low-spin cis-isomer 6′ cis-LSNiII and the high-spin isomer with the tetrahedral nickel center 6′ HSNiII are energy disfavored relative to 6′ trans-LSNiII by 9.6 and 5.7 kcal mol−1, respectively. The conclusion on the structure of the energy preferred form of 6′ trans-LSNiII disagrees with data of X-ray crystallography according to which the angle between the planes of the two heterocyclic ligands is equal to 72°. Good agreement with the experimental findings was achieved when calculations were performed on the molecules containing four tert-butyl groups which sterically hinder formation of the trans-isomer. The high-spin structure 6HSNiII with the tetrahedral nickel center becomes to be the ground state isomer of the complex and the computed angle (65°) between the planes of the ligands is close to the experimentally determined value.
It is commonly supposed that calculations of the salt-like coordination compounds can be restricted by consideration of the metal-containing parts of the molecules, while the counter-ions can be disregarded to save the time for computation and improve iterative convergence. Since small counter-ions are capable of entering into the first coordination sphere of the complexes and changing the coordination number of central metal , , this approach should be used rather carefully. In the case of compounds with redox-active ligands it may lead to even incorrect predictions.
A number of papers have been earlier published , ,  devoted to studying the structure and magnetic properties of zinc(II) bischelates with redox-active diazabutadiene (DAD) ligands 7, oxidation states of which are changed within the range of charges between 0 and −2. To investigate the electronic structure of these radical-containing complexes we have performed calculations on neutral, monoanionic and monocationic forms of these compounds and reveal that the geometry optimized anionic and cationic structures incorrectly reproduce the redox-states of the ligands. According to the X-ray structural determinations of the charged complexes , various states of DAD are easily identified in crystal, while the DFT calculations result in the symmetrical structures 7′− and 7′+ (Fig. 6), in which the unpaired electron is delocalized over the ligand system. To find the reason for this disagreement, we have performed calculations on the structures including the corresponding counter-ions and found out that just as in the experimentally detected structure, potassium ion solvated by a tetrahydrofuran molecule presented in the crystal is oriented above the redox-active ligand, which leads to distortion of the chelate cycle. The calculated C–N and C–C bond lengths of the ligand (1.40 and 1.37 Å) in 7−, SQ–ZnII–Cat are in a good agreement with dianionic form of DAD observed in the crystalline state. The unpaired electron is localized on the other ligand, geometry characteristics of which correspond to radical-anion state. The calculations on the complex with trifluoromethanesulfonate anion led to the structure 7+, SQ–ZnII–Q, in which, according to the intraligand bond lengths (Fig. 6), one of the ligands exists in the electrically neutral and another one in the radical-anion semiquinonate form. The example of complexes 7 manifests importance of inclusion of counter-ions into the computational scheme when studying complexes with redox-active ligands.
Several local minima on the PESs
In specific cases analysis of intramolecular rearrangements of transition metal complexes with redox-active ligands is complicated by the existence of two or more electromers of the same multiplicity on the intersecting potential energy surfaces (PESs). An example of this situation is given by the VT system of CoII/CoIII complex 1 (Fig. 1), for which the doublet PES along with the ground state LSCoIII electromer contains also the LSCoII electromer and the quartet PES has minima of HSCoII and also ISCoIII (IS – the intermediate spin state) electromeric forms (Fig. 7).
This means that theoretically predicted and experimentally detected Cat-N-Q–LSCoIII–Cat-N-SQ→Cat-N-Q–HSCoII–Cat-N-Q rearrangement characteristic of this complex can, in principle, be complicated with the parallel spin-forbidden transitions between the additional electromeric forms on the doublet and quartet energy surfaces. In reality, feasibility of these transitions is controlled by the energy gaps between the electromers themselves and energy barriers (MECPs, minimal energy crossing points) for their rearrangements. For the considered complex the calculations  showed that the additional trajectories for rearrangements of the electromers are prohibited because of the energy abundance of Cat-N-Q–ISCoIII–Cat-N-SQ isomer.
Relative stabilities of the electromers with the same spin states can be controlled by redox-active ligand properties. The results of the DFT B3LYP*/6-311++G(d,p) calculations of the adduct 8 of N,N′-ethylene-bis-(acetylacetoiminato)cobalt(II) with o-quinone are in a good agreement with experimental data, according to which its formation is accompanied by electron transfer from the metal to the ligand . As a consequence, the ground state of this complex is presented by the low-spin structure comprising the metal ion and the ligand in the LSCoIII and SQ forms (Fig. 8). The calculation of the high-spin isomers predicts the presence of two minima on the quartet PES, the most stable of which corresponds to 8, SQ-ISCoIII electromer with the trivalent cobalt ion in the intermediate spin state. The isomer 8, Q-HSCoII expected as a consequence of VT rearrangements is by 5.9 kcal mol−1 destabilized relative to the 8, SQ-ISCoIII counterpart. Transition to the complex 9 with o-quinone diimine ligand does not lead to significant changes in the HS-LS energy differences, but is accompanied by inversion in relative energies of the isomers on the quartet PES. In this adduct, the structure 9, Q-HSCoII is by 3.1 kcal mol−1 energy preferred as compared to the 9, SQ-ISCoIII electromer. Analogous results have been obtained in quantum chemical modeling of adducts of CoII aminovinylketonates with redox-active ligands .
Oxidation states of metal centers and ligands
An important characteristic of a metal complex is the formulation of the oxidation state of its metal center, which is usually defined according to the charge accumulated at this center. For transition metal complexes with redox-active ligands in which valence electrons readily circulate between metal and ligand, assignment of the metal to a certain oxidation state based on the calculated electric charge becomes to be ambiguous and disagreeing with the conventional description. An illustrative example is given by the valence tautomeric rearrangement of cobalt complex of 9-hydroxy-2,4,6,8-tetra-(tert-butyl)phenoxazin-1-one 1 (Fig. 1) .
The effective magnetic moment determined for the ground state low-spin form of the complex is equal to 1.89 μВ and ESR spectrum registers the dianion-radical form of the redox-active ligand. These data are indicative of the low-spin structure Cat-N-Q–LSCoIII–Cat-N-SQ of the complex comprising cobalt in the three-charged oxidation state. Heating toluene solution of 1 is accompanied by increase in the magnetic moment of the complex to 5.07 μВ and leads to extinction of the ESR signals. These transformations evidence the formation of the high-spin electromeric form Cat-N-Q–HSCoII–Cat-N-Q of 1, with cobalt in the two-charged oxidation state. Figure 1 presents a typical description of the VT rearrangement. Neither of the commonly used quantum chemical methods of calculation of electronic charges reveals a clear correspondence between the computed charges of the 1′ and the traditionally defined oxidation numbers (Table 3).
Analysis of spin density distribution over the molecules of transition metal complexes with redox-active ligands comprises a better alternative way to estimation of oxidation states of metal centers. Thus, the absence of spin density at the cobalt ion in the low-spin electromer 1′ Cat-N-Q–LSCoIII–Cat-N-SQ (Fig. 9) unequivocally points to the trivalent state of the metal, whereas three unpaired electrons on the cobalt center of the high-spin electromer indicate its HSCoII state.
To judge on the oxidation state of redox-active ligands, at which these are coordinated to the metal, it is usually sufficient to consider the lengths of the bonds within coordination site of the complex which are particularly sensitive to oxidation states of the ligands. In some rare cases, however, no unequivocal assignment can be done on the basis of only geometric arguments and it becomes necessary to refer to distribution of spin density. However, the presence at a metal center of a large number of unpaired electrons may complicate using this approach. Figure 10 demonstrates results of the calculations performed on structurally similar adducts of ruthenium , , iron  and cobalt  diketonates with o-benzoquinone 10 (M=Ru, Fe, Co), in which C–C and C–O bond lengths in the redox-active ligand have similar values. The calculation of the mixed-ligand complex of iron bis-acetylacetonate with o-benzoquinone 10 (M=Fe) reveal concentration of an unpaired electron on the o-quinone fragment pointing to the semiquinonate form of the redox-active ligand and also spin density at the metal center. These findings are compatible with the conclusion on the ferrous high-spin state of the metal bearing four unpaired electrons. A detailed analysis of the electronic structure of this electromer of 10 (M=Fe) indicates the presence of a significant amount of spin density on the oxygen atoms of diketone ligands (Fig. 10). In sum with this yields the value 4.70, which testifies realization of the SQ–HSFeIII electromeric form. Analogous results have been obtained when studying iron bis-acetylacetonate complex with o-diiminobenzoquinone . On the other hand, in the cobalt complex 10 (M=Co), the metal center bears three unpaired electrons and there is no spin density on the ligand This character of spin density distribution clearly points to the unreduced quinone form of the ligand.
Transition metal complexes with redox-active ligands possess flexible valence electronic shell formed by conjugation of d-AO of a metal and valence MO of the ligands. To ensure correct description of magnetic, geometry and energetic properties of these compounds the applied computational protocol must be validated by way of comparison of results of calculations with the experimental data available for the model compounds structurally similar to objects of the study and carefully tested for stability of the calculated wave functions at the localized stationary points. When selecting the appropriate model compounds it is necessary to take an account for the effects which might be rendered by the electron-active, bulky substituents and counter-ions. Interpretation of results of calculations performed on transition metal complexes with redox-active ligands susceptible to VT rearrangements may be complicated when the PESs of the compounds contain several local minima of the isomeric (electromeric) forms of the same multiplicity or when strong antiferromagnetic interactions significantly affect magnetic parameters of the electromers. Finally, it may be noted that the specific electronic structure of the complexes with redox-active ligands and the unusual spin-forbidden character of rearrangements between their isomers impose a certain impression on the meanings of some of the widely accepted in coordination chemistry terms, such as oxidation numbers and energy barriers, and even lead to the appearance of new ones, such as electromerism.
A collection of invited papers based on presentations at 8th International IUPAC Symposium on Macro- and Supramolecular Architectures and Materials: Multifunctional Materials and Structures (MAM-17) held in Sochi, Russia, 6–10 June 2017.
A.G. Starikov thanks the Ministry of Education and Science of the Russian Federation (state assignment no. 4.1774.2017/4.6) and V.I. Minkin thanks Russian Federation President Council on Grants (no. NSh-8201.2016.3) for financial support.
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