Magnetostructural Correlation for Rational Design of Mn(II) Hybrid

Magnetostructural Correlation for Rational Design of Mn(II) Hybrid-Spin Complexes ... Publication Date (Web): December 20, 2012. Copyright © 2012 Ame...
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Magnetostructural Correlation for Rational Design of Mn(II) HybridSpin Complexes Anela Ivanova,*,†,‡ Julia Romanova,† Alia Tadjer,† and Martin Baumgarten‡ †

Department of Physical Chemistry, Faculty of Chemistry and Pharmacy, University of Sofia, 1 James Bourchier Avenue, 1164 Sofia, Bulgaria ‡ Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany S Supporting Information *

ABSTRACT: The magnetic properties of a series of manganese(II) diacetylacetonate and dihexafluoroacetylacetonate hybrid-spin complexes with neutral pyridine-based organic radicals were characterized theoretically by DFT calculations. Three stable radicals, in which a radical group is bound in either para or meta position with respect to the pyridine nitrogen atom, were considered. The correct stable structures and multiplets of the complexes were obtained by full geometry optimization starting from an ideal structure. A total of three important geometry descriptors of the complexes were monitored and related to their magnetic characteristics. These structural parameters are (i) the torsion angle governing the conjugation of the organic radical m-PyNO (anti versus gauche), (ii) the coordination geometry of the acetyl acetonate ligands around the metal ion (square versus rhombic), and (iii) the relative orientation of the organic radical with respect to the acetyl acetonate plane (parallel versus perpendicular). It was found that the magnetic properties are not sensitive to the orientation of the radicals with respect to the equatorial plane but do depend on the conformation of the organic radicals. Even a spin switch between the ferromagnetic (S = 7/2) and antiferromagnetic (S = 3/2) ground state was found to be feasible for one of the complexes upon variation of the organic radical geometry, namely, the dihedral angle between the organic radical moiety and the pyridine ring. The pattern of molecular orbital overlap was determined to be the key factor governing the exchange coupling in the modeled systems. Bonding π-type overlap provides antiferromagnetic coupling in all complexes of the para radicals. In the meta analogues, the spins are coupled through the σ orbitals. A low-spin ground state occurs whenever a continuous σ-overlap pathway is present in the complex. Ferromagnetic interaction requires σ−π orthogonality of the pyridine atomic orbitals and/or π-antibonding Mn− pyridine natural orbital overlap. Using an estimate of the donor−acceptor energy stabilization, the affinity of a given Mn(II) dorbital to mix with the sp2 orbital from pyridine can be predicted.

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INTRODUCTION Complexes of stable organic radicals with transition-metal ions bearing unpaired electrons are often used as building blocks of so-called hybrid-spin materials.1,2 The latter are advantageous because they avoid the extremely low spin-ordering temperatures of organic molecular magnets1a,c and at the same type preserve a low weight and easy handling through methods of organic synthesis. Several ligating sites in the organic moieties enable the formation of three-dimensional spin networks through the binding of separate complexes into crystals. Moreover, the magnetic interactions in the solid state can be tailored by crystal engineering, thus adjusting the features of the material to some predefined values. A metal ion frequently included in hybrid-spin complexes is manganese(II). Because of its five d electrons, Mn(II) offers a broad variety of possible spin states leading to overall antiferromagnetic (AFM), ferrimagnetic, or ferromagnetic (FM) interactions3,4 with the organic radicals. The reasons for the stabilization of high-spin or low-spin states are still unclear: they could be related to varying ligation sites of the organic © 2012 American Chemical Society

radicals, to structural distortions, or to intermolecular interactions in the crystal. In addition, complexes with weak spin−spin coupling such as those of Mn(II) are interesting for applications in quantum computing, for example. The possibility for a spin switch reported here is a novelty opening new perspectives for the utilization of similar molecules as magnetic sensors as well. Some theoretical calculations on Mn(II) complexes have been reported previously.5−10 Mainly density functional theory (DFT) within the broken-symmetry framework has been applied, but sometimes, multireference methods have also been used for comparison. However, the majority of the computations5−8 contain models with more than one metallic center, and the analysis has been aimed primarily at estimating the metal− metal exchange interaction. Most of the ligands studied have been relatively simple spin-inactive inorganic or organic Received: December 13, 2012 Revised: December 20, 2012 Published: December 20, 2012 670

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molecules. Several magnetostructural correlations specific for the investigated systems have been outlined. Most often, the variation of the exchange interaction strength with changing distance between the metal ions or between a metal ion and the atoms ligated directly to it has been discussed. There have also been simulations of complexes containing a single Mn(II) center,9 with mostly porphyrin- or phthalocyanine-based ligands. The primary goal of these studies was elucidation of the groundstate multiplicity of the complexes. In some works,10 spin transfer from Mn to organic conjugated ligands through ligand reduction has been addressed. There are only a few instances11 in which the interaction between the spin of manganese and that of the organic radicals has been investigated. No geometry optimization of the complexes was performed in these studies, but rather some essential magnetic properties, such as spin-exchange constants and character of singly occupied molecular orbitals, of structures yielded by X-ray crystallography have been determined. Therefore, more detailed theoretical studies with not just comparative but also partially predictive nature are needed. Because magnetic properties are highly sensitive to the quality of the molecular structure, the geometry of several model Mn(II) hybrid-spin complexes is optimized within this study and compared to experimental data. Then, the magnetic characteristics of these structures are rationalized in terms of molecular structure, spin density distribution, and nature of singly occupied molecular orbitals. The reasons for selecting this particular oxidation state of the metal ion are that we wished to make a direct comparison to our previous results for Cu(II)12,13 and that the precursor Mn(acac)2 [or Mn(hfac)2] is experimentally known and has been characterized, the latter rendering hybridspin complexes based on it feasible. Thus, the broader applicability of our recently proposed computational protocol12 can be checked. The second goal of this systematic study was to elucidate the relationship between structural parameters, orbital overlap, and character of spin coupling at the molecular level in such Mn(II) complexes.

Figure 1. Scheme of the studied Mn(II) complexes Mn-p-PyNS (left), Mn-p-PyNO (middle), and Mn-m-PyNO (right); notations of key atoms and structural parameters are provided.

(quartet, sextet, and octet) are energetically accessible (see the Multiplicity of the complexes subsection in the Supporting Information), so only these geometries will be discussed. The functional used was UB3LYP16 combined with the basis set 631G*17 for all atoms, with the core electrons on Mn replaced by an RSC Stuttgart−Dresden effective core potential.18 To comply with the experimental data 14,15 and with our previous findings,12,13 in most of the calculations (when not explicitly specified otherwise), the symmetry group was fixed to Ci. However, the effect of symmetry was checked by also performing symmetry-unrestrained optimizations for some of the structures (see below). All computations were done with the program package Gaussian 09.19 Two geometric isomers, namely, rhombic (Rh) and square (Sq)20 (Figure 2), are possible for such type of distorted

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MOLECULAR MODELS AND COMPUTATIONAL PROTOCOL The structures of the calculated complexes are shown in Figure 1. The metal ion is Mn(II), and the diamagnetic ligands are acetylacetonates (acac). The organic radicals are para-pyridyl (pPyNO) and meta-pyridyl (m-PyNO) nitroxides and a parapyridyl thyazyl (p-PyNS) derivative. Their complexes were selected because the magnetic characteristics and two of the crystal structures are experimentally available.14,15 This renders the three complexes suitable test cases for validation of our computational scheme for a different metal ion. So far, we have successfully applied this scheme for the simulation of Cu(II) hybrid-spin complexes.12,13 In the calculated Mn-p-PyNS complex, the hexafluoroacetylacetonate (hfac) ligands (with which the experiments were conducted) were replaced by acac for computational efficiency, after it was confirmed that the trifluoromethyl substituents had a very minor influence on the spin density of manganese. (See Table S1 and Figure S1 of the Supporting Information for estimates of the effect of this substitution.) With five unpaired electrons from Mn(II) and two from the organic radicals, a total of four multiplicities are possible for the studied complexes: doublet (D), quartet (Q), sextet (S), and octet (O). The geometries of all these multiplets were optimized for each complex, but it was found that only three of them

Figure 2. Rhombic (A) and square (B) coordination geometry of Mn(II); the directions of the Cartesian axes used throughout the Ci calculations are shown.

octahedral complexes21 and were thus optimized for each complex because this was found to be essential for proper description of the magnetic properties of the analogous Cu(II) systems.12 It should be noted that the radical m-PyNO has several stable conformers differing in the torsion angle θ (Figure 1).12 The two with dissimilar magnetic characteristics are those with the nitroxide group positioned anti (Figure 3C) and gauche (Figure 3D) with respect to the single bond connecting the radical 671

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Figure 3. Optimized geometry of the most stable spin state of Mn-p-PyNS (A), Mn-p-PyNO (B), anti-m-PyNO (C), and gauche-m-PyNO (D) with perpendicular (left in each panel) and parallel (right in each panel) orientation of the organic radicals with respect to acac; side (top in each panel) and top (bottom in each panel) view is provided; hydrogen atoms are omitted in the top view for clarity.

moiety to the pyridine ring.22 The complexes of both conformers were investigated. The exchange integral (J) (Table 1) characterizing the type and magnitude of spin−spin interaction in the studied complexes was calculated as a Boltzmann-weighted average from the three energy gaps: quartet−octet (ΔEQO), quartet−sextet (ΔEQS), and sextet−octet (ΔESO). Thus, the multireference character of the low-spin state was implicitly taken into account. J is related to each energy gap as follows: ΔEQO = 12J, ΔEQS = 6J, ΔESO = 6J. These relationships were obtained by finding the difference between the relevant eigenvalues of a Heisenberg spin Hamiltonian for a linear three-spin system with neglected through-space spin−spin interaction (because of the long radical−radical distance) between the two organic radicals ̂ ·SMn ̂ + 2SMn ̂ ·SR2 ̂ ) Ĥ = −J(2SR1

Table 1. Exchange Integrals (J/kB) and Types of Structures (Rhombic or Square) Obtained after Geometry Optimization of the Complexes in Ci Symmetrya complex

structure

J/kB (K)

Mn-p-PyNS (⊥) Mn-p-PyNS (∥) Mn-p-PyNO (⊥) Mn-p-PyNO (∥) Mn-m-PyNO (⊥)

Rh Sq Rh Sq Sq Rhb Rh Sqb

−2.8 −3.4 −9.8 −12.5 4.1 −5.6b 3.4 −3.6b

Mn-m-PyNO (∥) a

Additional data are provided in Table S4 of the Supporting Information. bResults for gauche-Mn-m-PyNO.

(1)

̂ are the spin operators of the radicals and the where ŜR and SMn metal ion, respectively. The exact derivation of the eigenvalues and energy differences is provided in the Supporting

Information. Because the quartet states turned out to be spincontaminated (Table S3, Supporting Information), as could be expected from the hybrid nature of the DFT functional used, the 672

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square isomer to a rhombic one (Figure 2) and vice versa. This is true for all Mn(II) complexes studied. Thus, control of this degree of freedom could be used for trapping the systems in the desired minimum. The magnetic characteristics of the complexes can be rationalized in terms of their structures. (See also Tables S6− S8 and Figure S2 of the Supporting Information for more detailed description of the geometry.) In all cases, the perpendicular alignment is more stable than the parallel one (Table S5 of the Supporting Information). However, the energy differences between the two orientations are not very large, indicating that coexistence of the two types of complexes is possible. anti-Mn-m-PyNO is more stable than gauche-Mn-mPyNO in both alignments (Table S4, Supporting Information). The energy separation between the respective ground states is 6.1 kcal/mol for the perpendicular complexes and 5.0 kcal/mol for the parallel complexes. This still does not rule out the presence of the gauche structure in a powder sample of this complex, which might also be the reason for the peculiar magnetic behavior observed experimentally (see above). The complexes of the two para-pyridyl radicals (Figure 3A,B) are oriented in a very similar manner with respect to the acac plane, which indicates that this alignment is governed mainly by the coordination of the pyridine ring and not by the nature of the spin-bearing unit. In the perpendicular gauche-Mn-m-PyNO complex, there is a noticeable tilt of the two organic radicals with respect to the acac plane that was not observed in the other structures. Moreover, this is the only system in which the two pyridine nitrogen atoms are not equidistant from the two acetylacetonates. This structural dissimilarity of Mn-m-PyNO implies different interactions between the organic radicals and the metal ion in this complex compared to the para structures and to the anti conformer. The optimized bond lengths of the studied systems are in good agreement with the lengths of similar bond types reported in experimental X-ray studies9a,21c,25−31 of Mn(II) complexes. Comparison of the calculated bond lengths to the respective values reported for the X-ray structures of the two para complexes shows that, qualitatively, their structures are well reproduced.14,15 In both cases, the bond lengths and the torsion angles of the perpendicular complexes are closer to the experimental values, which coincides with the relative stability of the two structures and can be considered as an additional validation of the computational scheme. Thus, this scheme can be used for reliable prediction of the geometries of unknown hybrid-spin complexes. In all systems modeled, the Mn−R• spin−spin interaction is not very strong, and hence, the exchange integral does not exceed 10 K. This is because, even in the rhombic structures, the Mn−N bond remains the longest among all bonds in the coordination sphere of Mn(II). However, it is not generally true for the studied compounds that the shorter the bond, the stronger the spin−spin interaction. Obviously, additional factors influence the spin exchange. Therefore, the spin density distribution is analyzed next. Spin Density. The atomic spin densities of the parts of the complexes illustrating the metal−organic radical interaction are presented in the Supporting Information (Figures S3−S8), where a more detailed discussion of the spin density distribution is also provided (in the section Spin density). In summary, because spin delocalization is comparable in all cases, it cannot be responsible for the nature of the ground state of the complexes. This leaves as the only alternative the type of molecular orbital

effective exchange integrals of all complexes were also estimated by the spin-projected formula of Yamaguchi et al.23 However, no qualitative or significant quantitative differences were witnessed [Tables 1 and S4 (Supporting Information)]. Therefore, the direct mapping of the spin states to the Heisenberg Hamiltonian is preserved throughout the discussion to match as close as possible the interpretation of the experimental data. Further computational details are provided in the Supporting Information (Computational protocol details section).

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RESULTS AND DISCUSSION Energy and Geometry. Illustrative optimized structures are shown in Figure 3.24 Two minima with respect to the torsion angle γ (Figure 1) were found for each complex upon geometry optimization. In one of the structures (Figure 3, left) the pyridine rings are almost perpendicular to the C−O bonds of acac, whereas in the other (Figure 3, right), they are parallel to each other. Later in this work, the former will be denoted as “perpendicular” (⊥) and the latter − as “parallel” (∥). The experimental value of J/kB for Mn-p-PyNS (with hfac instead of acac ligands), extracted from a fit of the temperature dependence of the magnetic susceptibility using the same Hamiltonian, is Jexp/kB = −4.2 K.15 The effective exchange constant for Mn-p-PyNO (prepared with acac ligands) is Jexp/kB = −10.2 ± 0.05 K.14 These two experiments were carried out on single crystals of the complexes. Mn-m-PyNO, however, could not be crystallized, and the magnetic measurements were made for powder samples. The magnetic susceptibility of Mn-m-PyNO turned out to be almost temperature-independent throughout the entire temperature range,14 which was attributed by the authors to cancellation of the organic spins through intermolecular interactions with neighboring complexes. The theoretical energy gaps between the various multiplicities of a given complex are smaller than the thermal energy at room temperature (300 K). This indicates possible coexistence of the three multiplets at high temperature, which is in accordance with the experimental data.14,15 Nevertheless, the most stable multiplet should be reflective of the ground state of each complex. For the two para-pyridyl radicals, this was found to be the quartet resulting from antiferromagnetic coupling between the organic radicals and manganese, also in line with the experimental findings.14,15 The magnitude of the exchange coupling is close to the experimental values as well, especially for the perpendicular structure of Mn-p-PyNO. The relative strength of the exchange in the entire series of complexes is reproduced correctly, as well. A very interesting magnetic phenomenon was observed for Mn-m-PyNO. In both the perpendicular and parallel alignments, the complex of the anti radical has the most stable octet state, whereas in that of the gauche radical, the quartet state has the lowest energy. This means that, in the former system, the organic radicals are ferromagnetically coupled to Mn(II) and in the latter, they are antiferromagnetically coupled. The most significant fact is that this spin switch can be accomplished by a relatively mild structural change of the complex, which can, in principle, occur in solution or in the solid state. The alignment of the organic radicals with respect to the acac plane seems to be important from a structural aspect. For the two para complexes, the change from perpendicular to parallel orientation has nearly no influence on the spin exchange or on the nature of the ground state. However, the variation of the angle γ, which drives the complexes between the perpendicular and parallel orientations, always triggers a transition from a 673

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Figure 4. SOMOs illustrating the spin−spin interaction within the perpendicular (left) and parallel (right) complexes Mn-p-PyNS optimized with symmetry constraints; all singly occupied orbitals are given in Figure S10 of the Supporting Information; occupation numbers different from one are given in brackets.

Figure 5. SOMOs illustrating the spin−spin interaction within the perpendicular complexes Mn-p-PyNO (left) and Mn-m-PyNO (middle and right) optimized with symmetry constraints; all singly occupied orbitals are given in Figure S11 of the Supporting Information; occupation numbers different from one are given in brackets.

overlap to be the factor responsible for the overall FM or AFM exchange between the organic radicals and the metal ion. Natural Orbitals. The natural singly occupied molecular orbitals (SOMOs) of all modeled Ci complexes are presented in Figures 4, 5, and S9−S11 (Supporting Information).32 As for the

other properties, the two para complexes have much in common in the SOMO overlap patterns. The most distinct feature is related to the quartets, for which only two of the seven SOMOs (SOMO1 and SOMO7) have contributions from the three spinbearing units, that is, are responsible for the spin exchange 674

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Table 2. Energy of Donor−Acceptor Stabilization (Eq 2) for the Ci Complexes of Mn-m-PyNO with Lowest and Highest Multiplicities (Ms)a complex

Ms

LPMn

ΔE(2) DA (kcal/mol)

complex

Ms

LPMn

ΔE(2) DA (kcal/mol)

gauche-Mn-m-PyNO (⊥)

Q

dxz dx2−y2, dz2 dyz, dz2, dxy dxy, dx2−y2 dyz, dxy dxz dx2−y2 dz2, dyz dxy dyz, dz2 dxz, dx2−y2 dz2, dxz dyz, dx2−y2, dz2 dyz, dz2, dxy dxy, dyz dxz, dx2−y2 dz2, dxz dyz, dx2−y2 dyz, dz2 dxy

−0.184 −10.247 −34.737b −26.769b −8.229 −0.003 −49.594b −22.453b −16.176 −5.875 −8.313 −1.047 −111.276b −2.040 −1.247 −7.602 −0.487 −114.734b −2.376 −1.352

gauche-Mn-m-PyNO ( ∥ )

Q

dxz dz2, dx2−y2 dx2−y2, dz2 dyz, dxy dxy, dyz dxz dz2 dx2−y2 dyz dxy dxz dz2, dx2−y2 dx2−y2, dz2 dyz, dxy dxy, dyz dxz dz2, dx2−y2 dx2−y2, dz2 dyz, dxy dxy, dyz

−6.029 −0.008 −91.818b −14.477 −0.340 −15.216 −0.166 −83.990b −0.131 −0.030 −16.279 −0.936 −79.483b −0.266 −0.030 −17.034 −44.953b −9.073 −2.488 −0.094

O

anti-Mn-m-PyNO (⊥)

Q

O

a

O

anti-Mn-m-PyNO ( ∥ )

Q

O

d AOs of Mn with large coefficients in each of its LPs are listed. bMost significant ΔEDA value(s) for the given structure.

between the unpaired electrons.33 In both of these SOMOs, the interaction takes place through π-type overlap between the pz AO of N* (Figure 1) and the dyz AO of Mn(II). The interaction is strongly bonding (covalent-like) in SOMO1 and antibonding in SOMO7. The former corresponds to significant antiferromagnetic coupling between the organic and inorganic spins. Moreover, SOMO1 always has an occupation number larger than 1, which makes its weight in the spin exchange predominant. This indicates that all hybrid-spin complexes of Mn(II) with para radicals with large coefficients for the pz AO on N* will have a substantial propensity for AFM spin coupling due to this p−d πbonding overlap. Unlike the quartets, the octets of the para complexes are characterized by different types of N*−Mn overlap. None of them, however, turns out to provide an as-efficient ferromagnetic interaction route (see the Supporting Information), and hence, all para complexes have quartet ground states. The mechanism of spin exchange in the meta complexes is completely different [Figures 5 and S11 (Supporting Information)]. The spin is transferred entirely through σ pathways in both the parallel and perpendicular structures. This is similar to the analogous copper complexes.12 The only common feature of the meta complexes of Mn with their isomeric para compounds is that, in all quartets, only SOMO1 and SOMO7 are responsible for the coupling, again being in-phase and out-of-phase linear combinations of the same organic and inorganic orbitals. However, the orbitals involved are completely different (see the Supporting Information). The overlap between the sp2 AO of N* and the d AO of Mn is σ-antibonding in all quartet meta complexes. There is a significant difference, however, in the overlap of the sp2 AO of N* with the AOs on the other atoms of the pyridine ring. In the two anti complexes, the pyridine carbon atoms participate in SOMO1 and SOMO7 with their pz AOs, which are orthogonal to the N* AO [Figures 5 and S11 (Supporting Information)]. Thus, the spin from Mn(II) cannot be transferred efficiently to the N−O group, and the quartet is destabilized. In the gauche

complexes, however, the pyridine carbon atoms also participate with sp2 AOs in the two SOMOs. Moreover, in this case, the p AOs of the N−O group close a significant dihedral angle with the pyridine ring, which allows for efficient σ overlap. Thus, an uninterrupted σ pathway for spin coupling between the organic and inorganic radicals exists therein, which leads to stabilization of the gauche quartets. This is one of the reasons for the switch of the ground state from ferromagnetic to antiferromagnetic between the gauche and anti complexes of m-PyNO. The second factor is rooted in the nature of the natural-orbital overlap in the respective octets, where the MO overlap contributes to stabilization of the octet state of anti-Mn-m-PyNO and destabilization of the high-spin state of gauche-Mn-m-PyNO (see the Supporting Information). Thus, the orbital analysis explains the nature of the ground states of the studied complexes and the magnitude of the spin exchange therein. Driving Force for Spin Coupling through Orbital Interaction. As is evident from the discussion in the previous section, a multitude of orbital overlaps is possible between the singly occupied d AOs of the manganese ion and the pyridine nitrogen atom of the organic radicals. The type of this overlap is the most essential factor in the magnetic behavior of the hybridspin complexes, which, for Mn, does not obey strictly the topological rules34 and thus cannot be predicted therefrom. Consequently, the ability to determine the pattern of overlap beforehand (i.e., before preparing the complex) would be extremely valuable from a predictive perspective. This means that one should be able to estimate the affinity of the AOs on N* to mix with each of the d AOs of Mn.35 A plausible driving force for orbital mixing would be the strength of the donor−acceptor interaction between the ligands and the metal ion. This could be estimated by natural bond orbital (NBO) analysis of the relevant orbitals of the separate building blocks of the complexes if one assumes that the N* lone pair (LP) is the donor and the Mn d AO is the acceptor. Then, the donor−acceptor energy lowering 675

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removing the symmetry constraints did not materially change the magnetic characteristics of the complexes.

(ΔEDA) due to the interaction between each such pair of NBOs is given as a second-order perturbative term36 (2) ΔE DA = −λD

⟨D|F |A⟩2 εA − εD

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CONCLUSIONS Quantum chemical DFT calculations were performed for a series of distorted octahedral hybrid-spin complexes of manganese(II) diacetylacetonate with three different pyridine-based neutral organic radicals. Geometry optimization of the structures revealed two stable orientations of the organic radicals with respect to the C−O bonds of the acetylacetonates: one parallel and one perpendicular. The torsion angle changing upon transition between them was found to be responsible for the interconversion between square and rhombic geometric isomers of the complexes. Even though structurally dissimilar, the two alignments were found to result in comparable magnetic characteristics. The magnetic properties of the studied complexes do not depend essentially on the symmetry of the structure, on the distance from Mn(II) to pyridine, or on the extent of spin delocalization and spin polarization. The governing factor defining the ground spin state of the systems is the orbital overlap between the organic and metal subunits. All studied para-substituted pyridine radicals are antiferromagnetically coupled to the metal ion, which is attributed to the strong π-bonding orbital overlap between the singly occupied orbitals of the organic radicals and Mn(II). The estimated exchange coupling constants are in good agreement with experimental data from magnetic susceptibility measurements. 3-(N-Oxyl-tert-butylamino)pyridine binds in a different way to the metal ion than its para analogue. Most important, even a switch of the ground spin state from high-spin to low-spin is possible in its complex upon variation of the torsion angle of the nitroxide group with respect to the pyridine ring. This ferromagnetic-to-antiferromagnetic spin switch depends on the particular Mn(II) orbitals coupling with the organic radical and on the type of MO overlap. To achieve ferromagnetic ground state, the σ-type spin-transfer path between the metal and the organic radical in the low-spin state has to be disconnected by σ−π orthogonal orbitals on the pyridine ring. In addition, no overlap or π-antibonding Mn−pyridine overlap should exist in the high-spin state. Finally, donor−acceptor (pyridine nitrogen lone pair as the donor and manganese singly occupied orbitals as the acceptors) NBO energy stabilization is proposed as a criterion for a priori determination of the spin-active Mn(II) atomic orbitals in a given hybrid-spin complex.

(2)

where λD is the occupation number of the donor NBO, ⟨D|F|A⟩ is an off-diagonal element of the Fock matrix involving the donor and acceptor orbitals, and εA and εD are the corresponding NBO energies. Equation 2 differs slightly from the original definition36 in the sense that the acceptor orbital is not an NBO antibond but an orbital of lone-pair character. Nevertheless, the physical meaning of the modified expression should be similar. NBO analysis of the isolated radicals shows that, in all of these radicals, the lone pair on N* has exclusively sp2 character, in accordance with its formal electron configuration. This renders the donor−acceptor analysis meaningful only for the meta complexes, because in these complexes, an sp2 AO of N* is involved in the SOMO overlap, namely, in the spin−spin interaction. Estimating the donor−acceptor energy stabilization is nevertheless valuable because knowing the likelihood of MO mixing in advance would allow one to predict whether the interaction in a meta system would be ferro- or antiferromagnetic, whereas for a para complex, the low-spin state is more or less predefined by the strong bonding π overlap (see above). Table 2 contains the values of ΔEDA for the high- and low-spin states of the parallel and perpendicular Mn-m-PyNO complexes.37 A fairly straightforward correlation is observed between the values of ΔEDA and the type of d AOs of Mn that actually participate in the SOMOs of the complexes, if one takes into account the directions of the interacting orbitals and not the weights of the specific d AO in each linear combination. In most of the quartets, there is a definite preference for interaction with one of the d AOs, and it is the one present in the complexes. Only for one quartet, namely, perpendicular gauche-Mn-m-PyNO, are the interactions in the yz and xy planes competitive. Consequently, a different d AO participates in its SOMOs. There is an analogous competing interaction also in the octet of the perpendicular gauche-Mn-m-PyNO complex. In this case, the orbital with the second-highest magnitude of ΔEDA is also observed in the complex SOMOs. In the remaining octets, the orbital stabilized most by the donor−acceptor interaction actually combines with the natural orbital of the organic radical in the complex. The data in Table 2 confirm the hypothesis that the values of ΔEDA can be used as a general criterion for determining the type of overlap between the sp2 AO of the pyridine nitrogen and the d AOs of Mn(II) and, hence, for anticipating the nature of the ground state (ferromagnetic or antiferromagnetic) of meta complexes. Unfortunately, the highest values of ΔEDA alone are not instructive for the magnitude of the relative stability of the spin states of a given complex. Other factors such as spin polarization or π-type overlap contribute to this as well. Effect of Symmetry. All of the results discussed so far were obtained for complexes for which the symmetry was fixed to the Ci group. This was done to comply as closely as possible with the experimental conditions. However, not all hybrid-spin complexes are bound to crystallize in this symmetry group. Therefore, it was checked whether relieving the symmetry constraints would influence the magnetic properties of the studied complexes (Tables S9 and S10 and Figures S12 and S13 in the Effect of symmetry section of the Supporting Information). Overall,

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ASSOCIATED CONTENT

S Supporting Information *

Derivation of the relationship between energy gaps and the exchange integral for a linear three-spin system. Magnetic characteristics of Mn(hfac)2 and Mn(acac)2 (Table S1, Figure S1). Energies of the various spin states of Mn-p-PyNO (Table S2). Expectation values of the squared spin operator of the studied complexes (Table S3). Energies, spin configurations, and exchange integrals obtained by the spin-projected formula of Yamaguchi et al.23 (Table S4). Energy difference between the optimized perpendicular and parallel structures of each complex (Table S5). Optimized bond lengths (Table S6) and torsion angles (Table S8, Figure S2) of the various multiplets of the Ci complexes. Differences in bond lengths between optimized and experimental structures (Table S7). Atomic spin densities of the studied compounds (Figures S3−S8). Singly occupied natural orbitals of the parallel complexes of m-PyNO and p-PyNO 676

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(Figure S9). All singly occupied natural orbitals of the parallel and perpendicular complexes of p-PyNS (Figure S10); all singly occupied natural orbitals of the perpendicular complexes of mPyNO and p-PyNO (Figure S11). Energies (Table S9), structural parameters (Table S10), spin density distributions (Figure S12), and singly occupied natural orbitals (Figure S13) of the complexes optimized without symmetry restraints. This material is available free of charge via the Internet at http://pubs. acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]fia.bg. Phone: ++35928161374. Fax: ++35929625438. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was performed during a research stay (of A.I.) funded by the Alexander von Humboldt Foundation. Dr. Galia Madjarova is acknowledged for helpful discussions.

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REFERENCES

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