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Increasing Magnetic Coupling through Oxidation of a Ferrocene Bridge Suranjan Shil† and Carmen Herrmann* Institute for Inorganic and Applied Chemistry, University of Hamburg, Martin-Luther-King-Platz 6, 20146 Hamburg, Germany S Supporting Information *

ABSTRACT: Ferrocene is an interesting coupler for designing magnetic molecules because of its rich chemistry and controllable oxidation state. In this work we have calculated the exchange spin coupling of a ferrocene-coupled nitronyl nitroxide diradical in its neutral and oxidized state (in which an additional spin center is introduced on the metallocene subunit). We do so by carrying out spin-unrestricted Kohn−Sham density functional theory (KS-DFT) calculations with different approximate exchangecorrelation functionals and basis sets. We find that the neutral complex is weakly ferromagnetically coupled (in contrast to experimental results on single crystals), whereas the spin centers in the cationic complex are strongly antiferromagnetically coupled, resulting in an overall ferrimagnetic arrangement of the spins. Our calculations suggest that the magnetic exchange occurs through a spin alternation mechanism and that the lowest unoccupied molecular orbital (LUMO) plays an important role. The ferromagnetic behavior of the neutral complex is very sensitive to rotating one Cp ring versus the other. In the case of the cationic complex, the magnetic coupling is nearly independent of such structural changes. Thus, oxidation allows for switching between a weakly coupled and a strongly coupled, robust overall ferrimagnetic spin arrangement.

1. INTRODUCTION The choice of the linker is important for the properties of diradical-based magnetic molecules.1 Therefore, the study and design of new coupling units is essential. Depending upon the nature of the coupling unit, ferromagnetically and antiferromagnetically coupled magnetic molecules are obtained. In recent times metallocenes are being used as couplers to design magnetic molecules because of their rich chemistry and frequent stability in two oxidation states. Among the metallocenes, ferrocene is most popular in this context.2−6 Jürgens et al. have synthesized a ferrocene-coupled nitronyl nitroxide diradical and found that it is antiferromagnetically coupled.3 They proposed that the magnetic interaction occurs through spin polarization and through-space direct interaction between singly occupied molecular orbitals (SOMOs) depending on the rotation around the cyclopentadienes’ main symmetry axis. Ferrocene can also act as a ferromagnetic coupler when the radical units have a suitable spin density distribution:2 Perchlorotriphenylmethyl radicals coupled through ferrocene give a ferromagnetically coupled complex.2,4 The reasons behind this ferromagnetic and antiferromagnetic behavior of ferrocene-coupled diradicals are not yet entirely clear, and a © XXXX American Chemical Society

comprehensive study on the magneto-structural correlation is still missing.6 We know that in nitronyl nitroxide the unpaired electron resides mainly on the N and O atoms which are located at the outside of the molecule; i.e., intermolecular interaction may play a role in the crystal. In the perchlorotriphenylmethyl radical the unpaired electron resides on an inner carbon of the radical moeity which is sterically shielded by the three aromatic substituents. Intermolecular interactions are more unlikely. This is one possible reason for the different behavior. Another important aspect is the difference in the spatial distribution of the radical’s magnetic orbitals: for NNO, there are no magnetic orbital coefficients on the carbon atom adjacent to the ferrocene bridge in the Hückel picture, while for the triphenyl radical, the carbon atom does carry coefficients.2,4,6 Furthermore, the energy barrier for rotation around the ferrocene’s main symmetry axis is low, which further complicates elucidating magneto-structural correlations experimentally. Received: July 28, 2015

A

DOI: 10.1021/acs.inorgchem.5b01707 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry

Table 1. Distance between Two Cp Rings (d) and Cp Rotational Angle (A) between Two Cp Rings and Their Deviations from Experiment Δd (|dexpt − dcalc|) and ΔA (|Aexpt − Acalc|) for Different Functionals and Basis Sets in the High-Spin State d/Å

A/deg

neutral functional

basis set

BP86 LANL2DZ BP86 6-31G(d,p) B3LYP LANL2DZ B3LYP 6-31G(d,p) B3LYP (solvent THF) 6-31G(d,p) B3LYP 6-31+G(d,p) M06 LANL2DZ M06-HF LANL2DZ exptl (ref 6)

% of HF exchange in the functional 0 20

27 100

neutral

calcd

Δd

cation

calcd

ΔA

cation

3.376 3.251 3.507 3.316 3.317 3.355 3.337 3.970 3.352

0.024 0.101 0.155 0.036 0.035 0.003 0.015 0.618 0

3.524 3.366 3.550 3.358 3.358 3.405 3.460 3.991

138.65 175.74 172 177.81 177.97 138.75 178.18 174.99 180

41.35 4.26 8 2.19 2.03 41.25 1.82 5.01 0

177.031 174.761 172.252 176.95 177.58 172.230 179.08 176.819

Figure 1. Scheme of the study (according to our DFT calculations, the complex is switched from a weakly ferromagnetically coupled to a strongly antiferromagnetically coupled state upon oxidation).

Another aspect that has so far not been addressed is the possibility to switch the magnitude and possibly the sign of the magnetic coupling by oxidizing the closed-shell ferrocene bridge to ferrocenium, which introduces one unpaired electron on the bridge. Radical bridges have been shown to significantly increase magnetic coupling.7 Spin-state switching through oxidation of the coupling unit has been reported in combined experimental and theoretical studies for para-phenylenediamine-linked nitronyl nitroxide groups8 and for nitroxide radicals linked by a quinone-hydroquinone switch.9 Redox switching of magnetic coupling has also been demonstrated using a semiquinone bridge.10 We suggest the use of a metallocene bridge for redox-induced switching of the magnetic coupling which allows for exploiting the rich chemistry of these compounds for fine-tuning of the resulting systems’ properties. We focus on NNO−ferrocene−NNO systems, for which experimentally an antiferromagnetically coupled singlet ground state has been found (magnetic exchange coupling constant −5 cm−1).6 Our goal is to investigate whether oxidizing the bridge would lead to a more strongly and possibly ferromagnetically coupled state, thus switching between molecules without and with a magnetic moment. We do so by carrying out spinunrestricted Kohn−Sham density functional theory (KS-DFT) calculations with different approximate exchange-correlation functionals and basis sets. We will show that the DFT ground state for an isolated molecule is not a singlet, but a weakly coupled triplet for a range of exchange-correlation functionals and basis sets. This difference in the experiment may be due to intermolecular interactions in the crystal or to a systematic shortcoming in present-day standard exchange-correlation functionals. According to our calculations the effect of oxidizing the bridge is to

switch to a strongly coupled state, whose coupling is much less dependent on the rotation of the cyclopentadienyl rings around their main symmetry axis than in the neutral state. Thus, even though the effect is less pronounced than anticipated, the switch from a weakly coupled to a strongly coupled system, both spin-polarized, could potentially be interesting for molecular magnetic materials.

2. COMPUTATIONAL METHODOLOGY All calculations were carried out using spin-unrestricted KS-DFT employing the Gaussian 0911 program package, modeling the low-spin states by KS determinants of broken spin symmetry.12 We optimized all molecular structures in each spin state using different combinations of exchange-correlation functionals (B3LYP,13,14 M06,15 M06HF,16,17 and BP8613,18) and basis sets (LANL2DZ,19 6-31G(d,p),20,21 and 631+G(d,p)20,21) (Table 1). Additional single point calculations were done using B3LYP/6-311++G(d,p)22 on the structures optimized with B3LYP/LANL2DZ. In the neutral state, the molecule has two spin centers. Therefore, we have generated initial guess molecular orbitals (MOs) for the broken-symmetry determinant by using the converged MOs from high-spin calculations and mixing them as implemented in Gaussian 09 (convergence criterion 10−6 a.u. for energies and 10−3 a.u. for molecular geometry). In the cationic state, the molecule has three spin centers; therefore, there are two low-spin determinants possible (↑↑↓ and ↑↓↑). There is no clear preference for either the ↑↑↓ or ↑↓↑ doublet state. We assume that Fe is in the low-spin state (as experimentally known for the ground state of ferrocene). We have determined initial guess MOs for the low-spin state by localizing the spin on the respective center and flipping it as implemented in Gaussian 09. We use the stable = opt keyword for the calculation of the low-spin states to achieve convergence to the lowest-energy broken-symmetry state. We use the Yamaguchi23 formula for the evaluation of the magnetic exchange coupling constant J for the diradical, which is given by B


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Inorganic Chemistry

Figure 2. Optimized high-spin structure of the ferrocene-coupled diradical (B3LYP/6-31G(d,p)) from the side and from above [d is the distance between the centers of the two Cp rings, ϕ1,2 are the twist angles between the Cp ring and the NNO radical (dihedral angle between the N−C bond of the NNO radical and the C−C bond of the Cp ring), and A is the rotational angle between the two Cp rings (dihedral angle between the connecting carbon atoms of the NNO and the Fc of one side of the complex with the other side)].

Table 2. Distance between Two Cp Rings (d) and Cp Rotational Angle (A) between Two Cp Rings and Their Deviations from Experiment Δd (|dexpt − dcalc|) and ΔA (|Aexpt − Acalc|) for Different Functionals and Basis Sets in the Low-Spin BS Determinant d/Å

A/deg

neutral functional

basis set

BP86 LANL2DZ BP86 6-31G(d,p) B3LYP LANL2DZ B3LYP 6-31G(d,p) B3LYP (solvent THF) 6-31G(d,p) B3LYP 6-31+G(d,p) M06 LANL2DZ M06-HF LANL2DZ exptl (ref 6)

J=

E BS − E HS ⟨S ⟩HS − ⟨S2̂ ⟩BS 2̂

2̂

% of HF exchange in the functional 0 20

27 100

neutral

calcd

Δd

cation

calcd

ΔA

cation

3.376 3.240 3.486 3.312 3.313 3.357 3.338 3.980 3.352

0.024 0.028 0.134 0.04 0.039 0.005 0.014 0.628 0

3.552 3.389 3.550 3.444 3.318 3.362 3.474 4.017

138.27 175.74 136.850 177.15 177.92 138.75 114.188 173.08 180

41.73 4.26 43.15 2.85 2.08 41.25 65.812 6.92 0

174.62 176.968 172.252 176.658 175.49 137.91 179.43 179.585

computed value of the oxidation potential E(0) for ferrocene in acetonitrile with respect to a standard hydrogen electrode (SHE) is 0.44 V. The calculated standard oxidation potential from ref 24 is 0.51 V. The deviation from our results may be due to the LANL08 basis set on Fe used in ref 24. Both values are quite close to the experimental value of 0.65 ± 0.01 V/SHE. 3.1. Molecular Structures. The calculated structural parameters like the distance between the two Cp rings (d) and the Cp rotational angle (A) between two nitronyl nitroxide radicals of the neutral complex (as shown in Figure 2) and its cation in the high-spin state are given in Table 1. We have optimized the structures of the complex with different density functionals and basis sets using a transoid structure as an initial guess as was experimentally found for the crystal structure in ref 6. From Table 1 we can see that for all functionals the radical subunits are in transoid form compared to each other, except for B3LYP/6-31+G(d,p) and BP86/LANL2DZ. It is also clear from the table that B3LYP/6-31+G(d,p) gives the smallest deviation in the distance between the two Cp rings but the largest deviation in the Cp rotational angle (A) in the neutral state (about 40°), compared to experiment, which is also true for BP86/LANL2DZ. All other methods give comparable results with the experiment. In the oxidized state, all the methods lead to transoid configuration when starting from a transoid guess. The geometrical parameters d and A for the BS determinants are given in Table 2. We can see that M06/ LANL2DZ gives the largest structural deviation from the experiment in the neutral state. BP86/LANL2DZ, B3LYP/ LANL2DZ, and B3LYP/6-31+G(d,p) give a comparable deviation in A as for the high-spin structure. Since it gives

(1) ̂2

where EBS, EHS, and ⟨S ⟩BS, ⟨S ⟩HS are the energy and average spin square values for corresponding BS and high-spin states. As the relation between magnetic exchange coupling constants J and spinstate energetics for more than two spin centers is not always unique, we only report spin-state energy gaps for the cationic complex.

3. RESULTS AND DISCUSSION In ferrocene, iron has oxidation state (II) with formally no spin on it, while oxidizing will introduce one unpaired electron. In this work we have chosen a ferrocene-coupled nitronyl nitroxide diradical, and we have studied the magnetic properties of that complex in its neutral and oxidized state as shown in Figure 1. We have calculated the oxidation energy (Ecation − Eneutral) of the nitronyl nitroxide (NNO) and ferrocene using the B3LYP functional in combination with a 6-31G(d,p) basis set (molecular structures optimized in their neutral and cationic state). The oxidation energy of NNO is 6.87 eV, and that of ferrocene is 6.90 eV, which suggest that the NNO and ferrocene have comparable oxidation energies. Of course the oxidation energy will vary with the solvent and may change considerably between the isolated subsystems for which the calculations were carried out, and the same subsystems as part of a larger molecular structure. To provide a measure for the reliability of our oxidation energies, we have calculated the oxidation potential of ferrocene using the methodology discussed in ref 24 in combination with the same functional and basis set as above (B3LYP/6-31G(d,p)), and compared it to both experimental and theoretical literature values. Our C


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Inorganic Chemistry

Table 3. Spin-State Energy Gaps in kJ/mol, Magnetic Exchange Coupling Constant J in cm−1, and Cp Rotational Angle (A) between the Radicals for the Complex Calculated with Different Functionals and Basis Sets neutral basis set

ΔE↑↓−↑↑

J (cm )

A/deg

ΔE↑↓↑−↑↑↑

ΔE↑↓↑−↑↑↑ (cm )

ΔE↑↑↓−↑↑↑

ΔE↑↑↓−↑↑↑(cm−1)

A/deg

BP86 BP86 B3LYP B3LYP B3LYP (solvent THF) B3LYP B3LYP M06 M06-HF

LANL2DZ 6-31G(d,p) LANL2DZ 6-31G(d,p) 6-31G(d,p) 6-31+G(d,p) 6-311++G(d,p)c LANL2DZ LANL2DZ

0.034 0.105 4.44 0.94 0.89 −0.55b 2.76 1.52 6.64

2.19 8.78 370.91 76.70 71.75 −45.18 230.72 123.59 426.16

138.7 175.7 172 177.8 178 138.8 172 178.2 175

−14.62 −35.41 −97.67 −69.79 −68.26 −47.89 d −4.59 −13.96

−1222.13 −2960.03 −8164.53 −5833.96 −5706.06 −4003.27

−7.56 −37.54 −97.01 −72.54 −70.20 −47.94 −80.08 −7.12 −20.14

−631.96 −3138.08 −8109.36 −6063.84 −5868.23 −4007.45 −6694.13 −595.18 −1683.56

177 174.8 172.3 177 177.6 172.2 172.3 179.1 176.2

a

−1

cation

functional

a

−1

−383.69 −1166.96

Calculated using eq 1. bFor discussion see section 3.1 cSingle point on the optimized B3LYP/LANL2DZ structure. dSCF did not converge.

the overall smallest deviation from experiment in d and A for both spin states, we rely on B3LYP/6-31G(d,p) to calculate the other properties of these complexes. One of us has also seen previously that this method may be reliable for the calculation of magnetic behavior.25−27 The twist angles ϕ1 and ϕ2 between the Cp rings and the NNO radicals are presented in Table S1 in the Supporting Information. The experimental structure has a twist angle of 13.4°. However, in our calculations we find that the twist angle is always smaller than that from the experiment. To confirm that our calculated structure is not a transition state, we have done a vibrational frequency calculation and find that there are no imaginary frequencies. Oxidizing the complex leads to an increase in the distance between the Cp rings of 0.05 Å or more. 3.2. Exchange Spin Coupling. In two-spin systems, the exchange interaction between magnetic sites is proportional to the energy difference between the antiferromagnetically (AF) and the ferromagnetically (F) coupled states of the complexes.12 A negative sign of the energy difference (EAF − EF) indicates an antiferromagnetic interaction in the complex, whereas a positive sign indicates ferromagnetic interaction. The spin-state energy gaps of the complexes in their neutral and cationic states are listed in Table 3. For clarity, the orientations of the local spins on the first NNO, the ferrocene (for the cation), and the second NNO are indicated by arrows. All exchange-correlation functionals and basis sets give positive AF−F gaps in the neutral two spin system, i.e., the complex exhibits a weak ferromagnetic interaction which is in contrast to experiment, where it was reported that the complex is antiferromagnetically coupled and is very sensitive to molecular conformation.6 This sensitivity we have found for B3LYP/631+G(d,p), where due to the smaller dihedral angle of 138.75° the neutral complex shows antiferromagnetic behavior (we will discuss the dihedral angle dependence in detail later). We also calculated the AF−F gap taking the nuclear coordinates from the X-ray crystallographic structure.6 In this case also we have found that the complex is ferromagnetically coupled. The experimental findings that the complex is antiferromagnetically coupled might be due to the intermolecular interaction between the radicals. Of course we cannot exclude that there is a general flaw that affects several of today’s standard exchange-correlation functionals and leads to wrong AF−F gaps. Note, however, that we have checked the dependence on the parameter which is known to systematically influence spinstate energy gaps, the exact-exchange admixture, by comparing pure functionals (BP86) and hybrid functionals (B3LYP), and also checked meta-GGA functional (M06, M06HF), and found

ferromagnetic exchange coupling even for pure functionals which tend to favor low-spin states compared with hybrid ones. As iron has a flexible oxidation state, we have studied the magnetic nature of the diradical complex in its cationic state where iron is in its Fe(III) oxidation state and a new spin center is generated on Fe. Thus, in the cationic state the complex has three magnetic centers. We have calculated the energies of the ferromagnetically coupled quartet state and two possible broken-symmetry determinants modeling the doublet state, and taken their energy differences. We have found that the complex in its cationic state shows considerable negative energy gaps between the ↑↓↑ doublet and the ↑↑↑ quartet determinants; i.e., the complex shows antiferromagnetic coupling between nearest-neighbor spin centers for all functionals. The ↑↑↓ determinant, which in the brokensymmetry framework, also models a doublet, and is energetically close to the ↑↓↑ determinant. Due to this and because spin densities in broken-symmetry DFT are artificial, we do not suggest to take these results as showing uniquely antiferromagnetic coupling between ferrocenium and NNO, but rather pointing to a ferrimagnetic arrangement of the spins. For a more detailed discussion of the exchange interaction between NNO and ferrocenium, see the next section. To take solvent effects into account, we also calculated the spin-state energy gaps taking tetrahydrofuran (THF) as a solvent in combination with B3LYP/6-31G(d,p). We used the polarizable continuum (PCM) model for the consideration of solvent effects. In this case also we have found that the complex shows a transition from weak ferromagnetic to a strongly coupled ferrimagnetic spin structure. 3.3. Spin Density Analysis. DFT can give acceptable spin densities for radical species.28 In addition to the experimental work discussed in ref 6, the authors have calculated the spin density based on the B3LYP functional and an EPR-II basis set. They concluded that the low spin density on the metallocene bridge leads to an antiferromagnetic coupling. In this work we have computed the spin density using the B3LYP functional and a 6-31G(d,p) basis set taking the experimental structure and found that the spin alternation rule29,30 is a possible explanation for the ferromagnetic coupling (Figure 3a). Only one carbon atom from each Cp ring participates in the spin delocalization whereas the other carbon atoms have nearly zero spin density in the triplet state. This suggests that the magnetic exchange occurs through a spin alternation mechanism resulting in a ferromagnetic exchange interaction. In the case of the cation, when there is an unpaired electron on Fe, to follow the spin alternation Fe must also have a down spin D


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Inorganic Chemistry

Figure 3. Spin density alternation of the ferrocene-coupled diradical in the (a) neutral triplet state and (b) cationic doublet state (B3LYP/631G(d,p)) (single point on experimental structure). The arrows represent the Mulliken local spin density (1 unpaired electron ∼ local spin density 1). Big blue arrows represent the total spin on two nitrogen and two oxygen atoms of the radical.

following the qualitative pattern in Figure 3a, but with a larger magnitude spin on Fe. This results in an overall ferrimagnetic arrangement of the spins (see Figure 3b). The Mulliken local spin densities on the individual radical units and on ferrocene on which Figure 3 is based are given in the Supporting Information (Tables S2 and S3). It has been shown that local spins are much less dependent on the local partitioning scheme than partial charges, so we do not expect these results to change significantly when employing a different local partioning scheme.31 The spin density plots of the optimized complex in the neutral and cationic states are shown in Figure S1 in the Supporting Information. To further study the coupling between NNO and ferrocenium, we have blocked one radical by H resulting in two spin centers, one on Fe and one on the radical (Figure S2 in the Supporting Information). The calculated AF−F gap indicates that this complex is an antiferromagnetically coupled one. This suggests that in the cationic state the nearestneighbor spin centers interact antiferromagnetically, so the ↑↓↑ determinant may be a better representation of the cation’s doublet ground state than the ↑↑↓ one. 3.4. Molecular Orbital Analysis. In addition to the local spin densities previously discussed, molecular orbital analysis is very important to understand the magnetic interactions in a molecule. The energies and shapes of molecular orbitals can give us insight into the magnetic coupling. While the Kohn− Sham orbitals are auxiliary quantities, describing the wave function of a nonphysical noninteracting Fermion system, their analysis may lead to valuable qualitative insight. All effectively singly occupied orbitals (SOMOs) (that is, occupied orbitals that do not have a counterpart of similar shape occupied by an electron of opposite spin) will be termed “singly occupied orbitals” in general, and “magnetic orbitals” in the BS determinant. The energies and shapes of the molecular orbitals of the neutral complex are listed in Figure 4 and for the cationic complex in Figure S3 in the Supporting Information. From Figure 4 we can see that the energy difference between singly occupied orbitals (HOMO − 1α and HOMOα) is very low; i.e., they are nearly degenerate. According to Hoffmann,32 if the energy difference between two consecutive singly occupied molecular orbitals (ΔESS) is less than 1.5 eV, then to minimize the electrostatic repulsion between two degenerate orbitals, parallel orientation of spins occurs. Constantinides et al.33 showed that when ΔESS > 1.3 eV, antiparallel orientation of spins results; i.e., the singlet state is the ground state. Zhang et al.34 have shown that the critical value of ΔESS is different in different cases. In our case we find that the spin orientation follows Hoffmann’s and Constantinides’ arguments in the

Figure 4. Molecular orbitals of the complex (NNO−Fc−NNO) in the neutral triplet state (B3LYP/6-31G(d,p), iso surface value 0.02).

neutral state; that is, the complex shows ferromagnetic behavior. Not only the energy of the orbitals but also the shapes are important for analyzing the magnetic nature of molecular systems. From Figure 6 we can see that the high-spin singly occupied orbitals HOMOα and HOMO − 1α are on the radicals. Both are π orbitals. If we look at the high-spin LUMOα in Figure 6, we can see this orbital has large coefficients not only on the ferrocene, but also on the radical. Therefore, there may be an important role of the LUMO in the magnetic interaction. In a simple MO picture exchange spin coupling occurs due to the interaction between magnetic orbitals. In our case the magnetic orbitals are taken to be the broken-symmetry SOMOs whose positive and negative linear combinations usually resemble the triplet SOMOs (although strictly speaking a more rigorous definition of magnetic orbitals is necessary).35 In the MO diagram of the neutral complex in the triplet state (Figure 4), we can see that the magnetic orbitals are on the radicals, and they need a linker MO through which they can interact. The LUMO can play such a role between the magnetic orbitals. In support of this argument we have calculated the natural orbital occupation and find that the LUMO has an occupation number of 0.04 in the neutral system and 0.03 for the cation in their ferromagnetically coupled state. This observation clearly suggests that there is a role of the LUMO in magnetic exchange. 3.5. Magneto-Structural Correlation. In section 3.2 we have seen that the neutral complex shows ferromagnetic coupling with B3LYP/6-31G(d,p) and antiferromagnetic coupling when we add a diffuse function to the basis set, i.e., B3LYP/6-31+G(d,p). We have also found that the Cp rotational angles (A) between two radicals are very different for these two basis sets. Therefore, the Cp rotational angle plays a major role in the magnetic nature as also suggested by the experiment.6 Now, we study the energetics and magnetic nature of the complexes as a function of the Cp rotational angle (A) E


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Inorganic Chemistry

Figure 5. Relative energy of the complex as a function of Cp rotational angle (A) in the neutral triplet and cationic doublet state (↑↓↑).

and twist angle ϕ (see Figure 2 for definitions). To study the magneto-structural correlation we do single point calculations on the modified coordinates of optimized structures of each spin state. The structures were optimized in each spin state with each functional, and then twisted around the Cp main symmetry axis. 3.5.1. Potential Energy Surfaces for Spin States. We have plotted the relative total energy in the high-spin state of the neutral complex (Figure 5, left) and low-spin state (Figure S4 in the Supporting Information) with the B3LYP functional and two different basis sets, namely, 6-31G(d,p) and 6-31+G(d,p). For both basis sets we have found similar overall trends of the relative energy, but for 6-31+G(d,p) there is a local minimum around a Cp rotational angle of 140° rather than at 180°. In case of the neutral complex we find that the complex is most stable at a Cp rotational angle of 60°. The reason behind this stability comes from the geometrical orientation of the radicals. At 60° Cp rotational angle the N and O atoms of the two radicals come close to each other (the inter-radical O−N distances are 3.22 and 3.27 Å, respectively), and there may be a covalent-type interaction which stabilizes this conformation as illustrated by the MO plots (Figure 6). Indeed, such a structure

mol and more) and has therefore not been included in Figure 5. The relative energies for varying twist angle between Cp and NNO planes are represented in Figure S5 in the Supporting Information for the neutral complex and in Figure S6 in the Supporting Information for the cationic one. The complex is most stable at low twist angle. 3.5.2. Spin-State Energy Gaps. The spin-state energy gaps of the complexes are plotted in Figure 7 as a function of the Cp rotational angle. We can see that the complex in the neutral state (left-hand side) shows ferro- or antiferromagnetic interaction depending upon the dihedral angle. Dihedral angles of 100−140° and below 70° show antiferromagnetic interaction; in the other angles, the complex is ferromagnetic in nature. The radical−Cp twist angle ϕ1 only affects the values of the gap, not the sign (Figures S7 and S8 in the Supporting Information), but note that the coupling can reach close to zero for ϕ1 = 70°. Therefore, the coupling nature is not sensitive to the twist angle ϕ1 whereas it is sensitive to the dihedral angle (A). For the cationic complex (right-hand side of Figure 7), while the absolute values of the energy gap between the ↑↓↑ and ↑↑↑ determinants depend even more strongly on the angle A than for the neutral system, this results in no qualitative changes and a comparatively small relative variation due to the much larger magnitude of the spin-state energy splittings in the cation. 3.5.3. Molecular Orbitals. We have plotted the α (=majority spin) molecular orbital energy gaps of the complex in the neutral state in Figure 8 as a function of Cp rotational angle. From Figure 8 we can see that the energy gaps between SOMOs (HOMO − 1α and HOMOα) are dependent on the Cp rotational angle, but there is no clear trend to explain the magnetic nature of the complex. We can only say that, at 120°, the HOMO − 1α and HOMOα are degenerate. However, in Figure 8b, we can see a resemblace with Figure 7a; i.e., there is a close relationship between the AF−F gap and the HOMOα− LUMOα gap. A low HOMOα−LUMOα gap correlates with antiferromagnetic coupling whereas a larger HOMOα−LUMOα gap correlates with ferromagnetic coupling in this complex. One can argue that at 60° and 90 °Cp rotational angles the HOMOα−LUMOα gaps are equivalent although they correlate with opposite magnetic natures. At 60° Cp rotational angle the two radicals of the complex come close to each other so there might be some covalent-type interaction between two radicals,36 which is also clear from the MO plots (see Figure 6). From the MO diagrams (Figure S10.0,1) in the Supporting Information we also notice that for all Cp rotational angles the SOMOs (HOMOα and HOMO − 1α) are placed on the radical

Figure 6. Molecular orbitals of the complex at the Cp rotational angle (A) 60° in the neutral triplet state (B3LYP/6-31G(d,p), isosurface value 0.02).

has been reported as a crystal structure for the neutral system in ref 6. We are focusing here on the transoid structure as reported as a crystal structure in ref 6. Note that focusing a twisted structure would not change our overall conclusion that oxidation leads to a switch from weak magnetic coupling to a strongly coupled ferrimagnetic arrangement of the spins. Also, the rotation around the Cps’ main symmetry axis could in principle be controlled synthetically by introducing chains linking the two Cp rings. In the cationic state the conformation with 80° Cp rotational angle is the most stable one. For angles below 60° for the neutral complex and below 70° for the cationic complex, the relative energy rises considerably (77 kJ/ F


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Inorganic Chemistry

Figure 7. Energy gap between antiferromagnetically coupled low-spin (LS) and ferromagnetically coupled high-spin (HS) determinants (ELS − EHS) as a function of Cp rotational angle (A) in the neutral and cationic complex (B3LYP/6-31G(d,p)) (see Figure 2 for definition of A).

Figure 8. Energy gaps between (a) HOMO − 1α (SOMO1) and HOMOα (SOMO2), and (b) HOMOα − LUMOα of the complex with respect to the Cp rotational angle (A) in the neutral triplet state (B3LYP/6-31G(d,p)).

nearest-neighbor interactions determine the strong antiferromagnetic coupling irrespective of the dihedral angle. Even though the complex could not be switched from a nonmagnetic, antiferromagnetically coupled form to a ferromagnetically coupled one, as originally hoped for, it can be switched from a weakly coupled to a strongly coupled ferrimagnetic spin arrangement by oxidation, which could be interesting for magnetic materials. Other metallocene bridges might lead to molecules that can be switched between a magnetic and a nonmagnetic form. Note also that the structural dependence of the coupling in the neutral complex could be controlled, in principle, by introducing covalent bridging units between the Cp rings.

centers and the LUMO is always placed on the ferrocene moiety. As discussed above, the fact that the SOMOs do not share coefficients on the bridge confirms that the bridgecentered LUMO may play a significant role in the exchange interaction. To summarize, our data suggest that the LUMOα has an important role in the magnetic interactions in the complex under study.

4. CONCLUSIONS In this work we have studied the magnetic behavior of a ferrocene-coupled nitronyl nitroxide diradical in its neutral and oxidized state. Different exchange-correlation functionals and basis sets are employed. All functionals and basis sets used for the neutral complex show weak ferromagnetic coupling, and antiferromagnetic nearest-neighbor interaction leading to an overall ferrimagnetic arrangement of the spins in the oxidized form. The magnetic nature of the neutral complex depends upon the rotation around the cyclopentadienyl rings’ main symmetry axis, confirming previous results, whereas the oxidized form shows antiferromagnetic behavior irrespective of this structural parameter. The weak ferromagnetic nature of the neutral complex is consistent with a spin alternation mechanism. Also, the LUMOα plays an important role for the magnetic coupling. At smaller dihedral angles when the radicals are close enough, the complex shows antiferromagnetic interaction in the neutral state. Therefore, through-space interaction between radicals may also play a crucial role to determine the magnetic behavior. In the cationic state, the

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b01707. Twist angle ϕ between the Cp rings and NNO radical and their deviations from experiment; energies, ⟨Ŝ2⟩ values and Mulliken spin densities in the neutral and cationic state, spin density plots, molecular orbitals of the complex at cationic quartet state, relative energy of the complex as a function of Cp rotational angle (A), twist angle ϕ in the neutral and cationic state, spin-state energy gaps of the complex with respect to twist angles ϕ, molecular orbitals of the complex as a function of the G


Article

Inorganic Chemistry

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(25) Shil, S.; Misra, A. J. Phys. Chem. A 2009, 113, 1259−1267. (26) Bhattacharya, D.; Shil, S.; Panda, A.; Misra, A. J. Phys. Chem. A 2010, 114, 11833−11841. (27) Bhattacharya, D.; Shil, S.; Klein, D. J.; Misra, A. Theor. Chem. Acc. 2010, 127, 57−67. (28) Yamanaka, S.; Kawakami, T.; Yamada, S.; Nagao, H.; Nakano, M.; Yamaguchi, K. Chem. Phys. Lett. 1995, 240, 268−277. (29) Trindle, C.; Datta, S. N. Int. J. Quantum Chem. 1996, 57, 781− 799. (30) Trindle, C.; Datta, S. N.; Mallik, B. J. Am. Chem. Soc. 1997, 119, 12947−12951. (31) Herrmann, C.; Reiher, M.; Hess, B. A. J. Chem. Phys. 2005, 122, 034102−10. (32) Hoffmann, R.; Zeiss, G. D.; Van Dine, G. W. J. Am. Chem. Soc. 1968, 90, 1485−1499. (33) Constantinides, C. P.; Koutentis, P. A.; Schatz, J. J. Am. Chem. Soc. 2004, 126, 16232−16241. (34) Zhang, G.; Li, S.; Jiang, Y. J. Phys. Chem. A 2003, 107, 5573− 5582. (35) Neese, F. J. Phys. Chem. Solids 2004, 65, 781−785. (36) Shil, S.; Paul, S.; Misra, A. J. Phys. Chem. C 2013, 117, 2016− 2023.

Cp rotational angle, energy gap between molecular orbitals, and optimized Cartesian coordinates of the complexes (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. † E-mail: [email protected].

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ACKNOWLEDGMENTS We thank the Sonderforschungsbereich 668 (project B17) and the Fonds der Chemischen Industrie for funding, and the Hamburg regional computing center (RRZ) for computational resources. We further thank Jürgen Heck, University of Hamburg, for helpful comments.

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