Simple but Useful Scheme toward Understanding of Intramolecular

Article pubs.acs.org/JPCA

Simple but Useful Scheme toward Understanding of Intramolecular Magnetic Interactions: Benzene-Bridged Oxoverdazyl Diradicals Kyoung Chul Ko, Young Geun Park, Daeheum Cho, and Jin Yong Lee* Department of Chemistry, Sungkyunkwan University, Suwon 440-746, Korea S Supporting Information *

ABSTRACT: It has recently been shown that the types of intramolecular magnetic interactions of diradical systems can be changed by the types of radical group: syn-group (or α-group) and anti-group (or β-group). The aim of this study is to establish a useful scheme to understand and explain the intramolecular magnetic interactions in diradical systems regardless of radical groups and the topology of a coupler. We investigated the intramolecular magnetic coupling constant (J) of six oxoverdazyl diradicals (i−vi) coupled with a benzene ring based on the unrestricted DFT calculations. On the basis of our results, we devised a simple but useful scheme by combining the spin alternation rule and the concept of radical group classification. Consequently, it was found that the calculated J values and plots of spin density distributions were consistent with our proposed scheme. In addition, we discussed the closed-shell singlet (CS) state and the dihedral angle effect on J values in detail to comprehensively understand the magnetic interactions of diradical systems. Our scheme can provide the basic framework to design future organic high-spin molecules and organic magnetic materials.

1. INTRODUCTION Organic molecular magnets have generated considerable recent interest due to their magnetic property,1 superconductivity,2,3 spintronic property,4,5 photomagnetic behavior,6,7 and so on. In particular, organic molecular magnets for spintronic applications have been extensively investigated.8−11 In order to induce spin−spin interactions in molecular magnetic materials, a variety of organic radicals bearing unpaired electrons have been utilized as a building block.12 Obviously these organic radicals must have enough stability to be isolated, handled, and stored under experimental conditions which contain oxygen and moisture. From this point of view, oxoverdazyl radical is undoubtedly a good candidate as a spin source for generating magnetic interactions in organic magnetic materials due to the outstanding stability13 by resonance forms, as depicted in Figure 1. Generally, overall magnetic properties of solid states are controlled by intramolecular as well as intermolecular magnetic interactions.14 For diradical-based magnets, estimating the intramolecular magnetic interactions is a prerequisite task for designing a potential magnetic material that possesses appropriate spin−spin interactions.15 For the cases of oxoverdazyl-based diradicals, the magnetic properties of direct coupled oxoverdazyl diradical were characterized by Brook et al.16 In addition, Gilroy et al. synthesized oxoverdazyl diradicals connected by a benzene coupler and studied their intramolecular magnetic couplings.17 Misra and co-workers investigated the intramolecular magnetic interactions for © 2014 American Chemical Society

diverse oxoverdazyl diradicals coupled with 5-membered and 6-membered aromatic rings and polyacenes based on density functional theory (DFT) calculations.18−20 On the other hand, Polo et al. attempted to describe the intramolecular magnetic interactions for a series of tetrathiafulvalene and oxoverdazyl diradical cations connected by diverse π-conjugated couplers.21 Moreover, various coupled diradicals, constructed from oxoverdazyl and nitronyl nitroxide, have been extensively studied by Datta and co-workers.22 Recently, there is growing interest in the photomagnetic properties of diradicals which contain oxoverdazyl radicals.23−25 It is worth noting that finding diradical molecules possessing strong intramolecular magnetic interactions is a major issue. Recently, we reported a systematic way of designing strongly coupled diradicals using DFT calculations.26,27 In this previous study we successfully predicted the overall trends in the strength of intramolecular magnetic interactions according to different monoradicals based on proposed simple diradical model systems.28 We also found that those monoradicals can be classified into two groups (α-group and β-group) depending on the spin density of connecting atoms to the coupler.26,27 Consequently, we found that both radical groups clearly show the opposite type of magnetic couplings for every diradical model system investigated. In addition, we proposed a scheme Received: July 18, 2014 Revised: September 13, 2014 Published: September 15, 2014 9596

dx.doi.org/10.1021/jp5072007 | J. Phys. Chem. A 2014, 118, 9596−9606

The Journal of Physical Chemistry A

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Figure 1. Structures of oxoverdazyl radicals (OVER(C) and OVER(N)), their resonance structures, and Mulliken atomic spin density distributions. Filled and empty circles denote the positive (α) and negative (β) spin densities, respectively.

diradicals experimentally as well as theoretically.16−25 Datta and co-workers first suggested OVER(N) form to induce different connectivity of oxoverdazyl radical for designing strongly coupled diradicals in their theoretical study.22 For the above two types of oxverdazyl radicals, the total number of is exactly the same but the difference between them is the atomic connectivity; the carbon/nitrogen atom of OVER(C)/OVER(N) was connected to the coupler. Additionally, the relationship of the atomic spin density between the connecting atom and the radical dot atom is a crucial point. As seen in Figure 1 (also in Figure S1, Supporting Information), for OVER(C), the connecting atom (carbon) has negative (β) spin density whereas the radical dot atom (nitrogen) has positive (α) spin density. On the other hand, for OVER(N), both the connecting atom (nitrogen) and the radical dot atom (nitrogen) have positive (α) spin density. Thus, on the basis of our proposed classification for radical groups (Scheme 1) OVER(C) can be classified as anti-group and OVER(N) as syn-group. Interestingly, the largest positive spin density was located at the radical dot atom for both OVER(C) and OVER(N). In this study, we investigated the magnetic interactions of benzene-bridged diradical systems (i−vi in Figure 2) in which OVER(C) and OVER(N) were used as spin sources by DFT calculations. It should be noted again that OVER(C)-based diradicals coupled with a benzene ring already have been studied in detail in previous experimental as well as theoretical works.16−21 Herein, we further extend the area of interest by including the OVER(N) radical as another type of spin source to benzene-bridged diradical systems. The main aim of this study is to provide a useful scheme to understand the intramolecular magnetic interactions of diradical systems by applying the concept of our newly proposed classification regarding radical groups. In addition, we discussed the validation of this scheme considering the closed-shell singlet (CS) state of diradicals. Lastly, we also investigated the dihedral angle effect on magnetic couplings for these systems to check the influence of dihedral angles on our scheme. We do believe that our study provides the basic framework for future studies related with the diradical systems composed of diverse radicals belonging to the syn-group or anti-group and diverse couplers.

to standardize the coupler by unrestricted density functional theory calculations27 as well as radical sources.26 The schematic diagram of two radical groups first devised by us is shown in Scheme 1. In this study, we tried to change the Scheme 1. Schematic Diagram of Radical Groups Classified into Syn-Group (or α-group) and Anti-Group (or β-group)

name of the radical groups from α-group/β-group to syngroup/anti-group for better understanding. As seen in Scheme 1, for the radicals belonging to the syn-group, the atomic spin density of the connecting atom has the same sign as that of the atom denoted by a radical dot which is called the radical dot atom throughout this paper. On the other hand, for the antigroup, the signs of spin densities of the connecting atom and radical dot atom are opposite. The connecting atom is defined as the atom directly connected to the coupler. Generally, most spin densities were localized at the radical dot atom in a monoradical as reported previously.26,27 Thus, the sign of the spin density at radical dot atoms of both radicals in a diradical is crucial to determine the ferromagnetism or antiferromagnetism of diradicals. We focused on the relationship of atomic spin densities between the connecting atom and the radical dot atom to classify the radical groups. Oxoverdazyl is a very interesting monoradical because this radical can be included in both radical groups according to the choice of the connecting atom. Figure 1 shows those two types of oxoverdazyl radicals denoted as OVER(C) and OVER(N). A number of studies have been conducted on OVER(C)-based 9597



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E(S = 1) − E(S = 0) = −2J

(2)

where E(S = 0) and E(S = 1) are the energies of the singlet and triplet states of a diradical, respectively. Normally unrestricted density functional theory (UDFT) methods have been employed to present solutions of both spin states for diradicals. However, in spite of using the unrestricted formalism based on a single-determinant wave function, we cannot truly describe the pure open-shell singlet (OS) state of diradicals.29 Thus, the broken symmetry (BS) formalism proposed by Noodleman was adapted to describe OS state diradicals as a compromise approach.30,31 Essentially, the BS solution does not represent the pure eigenstate of an OS diradical but the mixed state of singlet and triplet spin states. Hence, in order to evaluate the reliable magnetic coupling constant J, the energy value needs to be refined by the spin projection method to eliminate spin contamination. Diverse spin-projected methods have been applied to obtain the J values according to the degree of overlap between the magnetically active orbitals.32−37 We used the following formalism proposed by Yamaguchi and co-workers38,39

J=

(E BS − E T) ⟨S2⟩T − ⟨S2⟩BS

(3)

where EBS and ET are the energies of the BS and triplet states and ⟨S2⟩BS and ⟨S2⟩T denote the average spin square values for those spin states, respectively. In order to evaluate the intramolecular J values, DFT calculations were performed using a suite of Gaussian 09 programs.40 Geometry optimizations were carried out at unrestricted spin-polarized density functional theory, the UB3LYP/6-311++G(d,p) level. Herein, the B3LYP functional, which has been widely used to compute the J values, was adopted. For basis sets, normally adding more polarization and diffuse functions to the basis sets produces more accurate results. Thus, in this study, we took into account our computational resources and time and choose an adequately large basis set (6-311++G(d,p)). First, the triplet state diradicals were optimized. Then, the BS state diradicals were optimized on the basis of those geometries. (See details of BS calculations in the Supporting Information.) Finally intramolecular J values were evaluated using the obtained molecular energies of BS states and triplet states. All molecules of BS states and triplet states have been fully optimized. Subsequently, frequency calculations have been performed to confirm the minima on the potential energy surfaces.

Figure 2. Investigated oxoverdayl (OVER(C) and OVER(N)) diradical systems (i−vi) coupled with a benzene ring and dihedral angles (DA1−DA7).

3. RESULTS AND DISCUSSION Figure 2 shows the diradical systems (i−vi) in which OVER(C) and OVER(N) are used as spin sources and benzene was used as a coupler. We designed six oxoverdazyl diradicals (i−vi) by taking into account the topology of the benzene coupler, i.e., meta-geometrical topology (i, iii, and v) and para-geometrical topology (ii, iv, and vi). In addition, the magnetic couplings between two oxoverdazyl radicals via the benzene ring were prepared by considering various composition. Two OVER(C) classified as anti-group radical were used as spin sources in i and ii. On the other hand, for v and vi two OVER(N) belonging to the syn-group radical were connected by the benzene coupler. In addition, iii and iv were designed as mixed radical groups, namely, one OVER(C) radical (antigroup) and one OVER(N) (syn-group) radical. In order to

2. COMPUTATIONAL DETAILS The magnetic exchange interactions between two magnetic sites 1 and 2 can be expressed by the phenomenological Heisenberg spin Hamiltonian Ĥ = −2JS1̂ S2̂ (1) where J is the magnetic exchange coupling constant between two magnetic centers and S1̂ and Ŝ2 are the respective spin angular momentum operators. The J value can be estimated from the energy difference between high-spin and low-spin states for the magnetic systems. For a diradical system, the coupling constant J value of eq 1 is calculated as 9598



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reduce the computational costs, −R groups of oxoverdazyl diradicals were replaced with −H atoms. 3.1. Optimized Structures and Calculated J Values. To find the most stable structure for the BS and triplet states of i− vi we performed a series of calculations of molecular optimization by changing the dihedral angles between a radical and a benzene coupler, from 0° to 180°, with every 10° step. Especially for iii, v, and vi, the directional arrangement of the carboxyl group (CO) in OVER(N) can induce the differences in molecular geometry and symmetry. Consequently, we obtained the most stable geometries of i−vi, as illustrated in Figure 2, and the calculated dihedral angles of OVER(C) and OVER(N) are listed in the Supporting Information (Table S1). OVER(C) showed almost planar geometry with a small dihedral angle (0−3.82°) with respect to the benzene plane, whereas OVER(N) showed broken planarity of diradical systems with a dihedral angle of around 30° (28.53−33.32°). It appears that for OVER(C) there is no steric repulsion between OVER(C) and the benzene coupler, but for OVER(N) the carboxyl groups apparently induce steric repulsion with the hydrogen atoms of the benzene and resulted in a distorted geometry. These points indicate that OVER(C) is a good spin source for designing the π-conjugated magnetic molecules without significant molecular distortions. Details of the dihedral angle effect on magnetic couplings will be discussed in section 3.4. The calculated magnetic coupling constant (J) values for i−vi are listed in Table 1 (see also Table S2, Supporting

overestimation of calculated J values was found. This overestimation of calculated J values is perhaps not surprising because DFT methods use an approximate exchangecorrelation functional instead of an exact exchange-correlation functional. In addition, DFT calculations with an approximate functional suffer from systematic errors, such as self-interaction error (SIE) and static correlation error, as discussed by Cohen et al.41,42 In our previous study, in order to reduce the discrepancy between calculated J values obtained from DFT calculations and experimentally obtained J values, we proposed a scaling factor (0.380) for the diradical systems coupled with a 5membered or 6-membered aromatic ring for the first time.43 The remarkable linear relationships were found between the calculated and the experimental J values for 9 diradicals, and a scaling factor can be used to estimate reliable J values based on DFT calculations. By applying this scaling factor we obtained the scaled J values as listed in Table 1. The scaled J values of i (20.5 cm−1) and ii (−33.7 cm−1) seemed to be quantitatively quite reliable compared to the experimentally obtained J values. The experimental J values for iii−vi have not been reported yet. On the basis of our scaling approach the intramolecular J values were predicted to be −23.7 (iii), 54.3 (iv), and 20.8 (v) cm−1. Among i−v diradical systems, ii and iv have the strongest antiferromagnetic and ferromagnetic interactions, respectively. Interestingly, both ii and iv contain the para-substituted benzene as a coupler. This result is in line with our previous finding that the intramolecular magnetic interaction through the benzene coupler in the para-position is two times stronger than through the benzene coupler in the meta-position,26,27 even though the signs of J values are opposite each other. For i, iii, and v containing meta-substituted benzene as a coupler we found that the meta-substituted benzene coupler can lead to both ferromagnetic and antiferromagnetic interactions depending on the spin sources. The metasubstituted benzene coupler gave a ferromagnetic coupling when the two radicals belong to the same group of spin sources (i and v) while antiferromagnetic coupling when they belong to the different group (iii). The para-substituted benzene coupler (ii and iv) showed opposite types of magnetic coupling compared to the meta-substituted benzene coupler (i and iii). 3.2. Simple but Useful Scheme for Determination of the Type of Magnetic Interactions. Various experimental and theoretical studies invoked that meta- and para-substitution can determine the types of magnetic interactions in benzenebridged diradical systems.17−22,44−46 For homodiradical systems that are composed of two identical radicals as spin sources it is well known that the meta-substituted benzene coupler and para-substituted benzene coupler resulted in ferromagnetic and antiferromagnetic coupling, respectively.15 However, this rule might not be applicable for hetero diradical systems (i.e., iii and iv) that are composed of two radicals belonging to different radical groups. Thus, we intended to suggest a simple but useful scheme to promote comprehensive understanding for this phenomenon. First, it is necessary to understand the spin polarization47 pattern of the benzene coupler in diradical systems via πconjugated bonds. Scheme 2 shows the schematic representation of spin polarization in a π-conjugated system. For the carbon atom denoted by C1, π and σ electrons have parallel spin alignments because the parallel spin alignment is much easier by favorable exchange than the antiparallel spin alignment. In the σ-orbital connecting two carbon atoms

Table 1. Calculated and Scaled J Values for Oxoverdazyl Diradicals (i−vi)a diradical OVER(C)-m-benzeneOVER(C) (i) OVER(C)-p-benzeneOVER(C) (ii) OVER(C)-m-benzeneOVER(N) (iii) OVER(C)-p-benzeneOVER(N) (iv) OVER(N)-m-benzeneOVER(N) (v) OVER(N)-p-benzeneOVER(N) (vi)

our calcd J value (cm−1)

scaled (by 0.380) J value (cm−1)

53.9

20.5

−88.6

−33.7

−62.4

−23.7

143.0

54.3

54.8

20.8

N/A

N/A

a

N/A: Not available due to the convergence error for the BS state optimization.

Information). It was found that the calculated J values of i, iv, and v are positive, causing ferromagnetic (F) coupling. On the other hand, for ii and iii, the calculated J values are negative, causing antiferromagnetic (AF) coupling. In other words, the energetically favorable spin state for i, iv, and v is the triplet state, while that for ii and iii is the BS state. Unfortunately, for vi, the BS state optimization failed due to the convergence error; thus, the calculated J value of vi was omitted. We tried to explain the underlying reason for this convergence error for the BS state optimization of vi, and that will be discussed in section 3.3. For i and ii, experimental J values were obtained to be 19.3 and −29.7 cm−1, respectively, as studied by Gilroy et al.17 Considering these experimentally obtained J values, our calculated J values of i (53.9 cm−1) and ii (−88.6 cm−1) show identical types of magnetic interaction, even though the 9599



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Scheme 2. Schematic Representation of the Spin Polarization Mechanism for π-Conjugated Systems (a), and Proposed Spin Alignments Based on the Spin Alternation Rule for Meta-Substituted (b) and Para-Substituted (c) Benzene-Based Diradicals

Scheme 3. Our Proposed Scheme of the Spin Alignments for i−vi Based on Both Spin Alternation Rule and Classification of Radical Groups

denoted by C1 and C2, the paired electrons should have an opposite spin alignment according to Pauli’s exclusion principle. As a result, for the C2 atom which is adjacent to C1 atom the spin alignment of the π electron is antiparallel to that of C1. Generally, the adjacent atomic centers in π-conjugated systems have opposite spins, that is, α-spin and β-spin alternatively. This is also consistent with spin polarization and spin alternation in a π-conjugated system. By extending the concept of the above spin alternation rule,48,49 we can draw the overall spin polarization patterns for meta-substituted and para-substituted benzene couplers, as depicted in Scheme 2. Interestingly, for the meta-substituted benzene coupler, the positions of two connecting atoms denoted by a square have an identical sign of spin (or ferromagnetic coupling). On the other hand, opposite spin alignment (or antiferromagnetic coupling) is preferred for parasubstituted benzene coupler. For this reason, these spin alternation patterns for the benzene couplers have been used to explain that the meta-substituted and para-substituted benzene coupler can be utilized as ferromagnetic and antiferromagnetic coupling units, respectively.15 However, as seen in our calculated results, the spin alternation rule of the benzene coupler is not enough to explain the calculated J values for iii and iv. We tried to extend the spin pattern of diradical systems by adopting our concept of radical group classification. Scheme 3 shows our proposed scheme of the spin alignments for i−vi. In this scheme the spin alternation pattern of a benzene coupler is maintained based on the spin alternation rule. However, the main difference is that the relationship of spin alignments between the connecting atom and the radical dot atoms is involved. Since the OVER(C) radical is one of the anti-group radicals, the connecting atom and the radical dot atom have an opposite sign of spin alignment. On the other hand, OVER(N), which is one of the syn-group radicals, has an identical sign of spin alignment between them. We simply applied these spin relationships of radical groups to the spin alternation pattern of benzene coupler as illustrated in Scheme 3. Since the most significant spin density of the monoradical was distributed at the radical dot atom, the energetically favorable spin state of diradical systems can be determined by relative spin orientations of the two radical dot atoms (circled arrow). The spin densities for all of the atoms except the radical dot atoms were negligible because of the cancelation according to spin alternation. Therefore, the total spin of ground state can be simply determined by spin densities at radical dot atoms. For

example, for iii, the appreciable negative (β) spin density is located at the radical dot atom of the antigroup radical (OVER(C)), and substantial positive (α) spin density is distributed at the radical dot atom of the syn-group radical (OVER(N)). Thus, the energetically favorable spin state would be the singlet state (S = 0). On the other hand, for iv, since two negative (β) spin densities are located at radical dot atoms, the triplet (S = 1) state would be the ground spin state. Consequently, according to our scheme shown in Scheme 3, ferromagnetic couplings can be constructed in the ground state for i, iv, and v diradical systems while antiferromagnetic couplings for ii, iii, and vi. It is worth noting that the above predictions about the types of magnetic interaction based on our scheme are fully consistent with our calculated J values. In particular, using our scheme it is possible to explain the magnetic interactions of hetero diradical systems (iii and iv). In addition, the plots of spin density distributions for i−vi shown in Figure S2, Supporting Information, can support that the spin alignment patterns in our scheme (Scheme 3) would be correct for the ground spin states. For energetically favorable spin states of i−vi in Figure S2, Supporting Information, the spin polarization patterns, including the relationships of atomic 9600



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state optimization at the RB3LYP/6-311++G(d,p) level based on the triplet state geometries. The calculated dihedral angles of the optimized CS state for i−vi are listed in Table S1, Supporting Information. All of the dihedral angles of OVER(C) remained as almost 0° regardless of spin states for the entire diradical systems (i−vi). Thus, for i and ii, planar geometries were kept for CS states, similar to those of BS or triplet states. OVER(N) showed a dihedral angle of around 30° for BS and triplet states. Interestingly, for iii and vi the dihedral angles of OVER(N) decreased to about 16° at CS solutions. The most marked geometrical changes were expected in the CS states of v and vi, which have the largest (101.6%) and smallest (93.3%) diradical character obtained from our CASSCF calculations, respectively. For the CS state of vi, an entirely planar molecular geometry was obtained. Considering the CS states of iii, iv, and vi have more planar molecular geometries than the BS and triplet states, it was quite abnormal that geometrical distortion for the CS state of v was more prominent than those of BS and triplet states. It reveals that the diradical character of the singlet state might have hidden relationships with geometrical optimization of the CS state. Generally, the change of molecular geometry accompanies the changes of total energy as well as orbital energies. The relative molecular energies for triplet, BS, and CS states are listed in Table S4, Supporting Information. Calculated results reveal that the OS state was energetically more stable than the CS state for all systems (i−v). Considering the energies of the BS and triplet states were very close to each other, we can assume that the BS state has almost the same SCF energy as the triplet state (EBS ≈ ET). From this assumption it is expected that the energy difference (11.09 kcal/mol) between the BS and the CS state of vi might be significantly lower than those for i−v (23.04−28.03 kcal/mol) by a factor of 1/2 (Figure S4, Supporting Information). This indicates that a dramatic change of the molecular structure for vi, especially the planarity, caused a small energy variation between the BS and the CS state compared to those of i−v. Calculated HOMO and LUMO energies and HOMO− LUMO gaps of the CS states for i−vi are listed in Table S5, Supporting Information. The HOMO−LUMO gap of vi was found to be the largest (1.18 eV). From the CS solution, openshell singlet states might be achieved by promoting an electron from the HOMO into the LUMO as the easiest way. Therefore, the HOMO−LUMO gaps of CS solutions will be proportional to the promotion energy for generating the diradical character of singlet states. In this regard, among i−vi, vi would be the most difficult to induce diradical character to the CS state. The relationship between the diradical character of the singlet state and the HOMO−LUMO energy gaps of CS solutions is clearly shown in Figure 3. The HOMO−LUMO gap of CS solution was inversely proportional to the diradical character of the singlet state. Thus, this plot helps to understand that the largest HOMO−LUMO gaps of CS solution and the smallest diradical character of the singlet state were simultaneously obtained for vi. Interestingly, in Figure 3 the point of vi denoted by a dotted circle is apparently apart from other systems (i−v) denoted by a gray circle. Considering the BS state optimization of vi failed among i−vi, intuitively it seems that both the diradical character of the singlet state and the HOMO−LUMO gap of the CS state can be used as indicators for expecting the fate of BS state optimization. Calculated bond orders of chemical bonds and NBO charges of atoms for the CS state of vi are listed in Table 2. On the

spin densities between the connecting atoms and the radical dot atoms, are in good agreement with our scheme. For the energetically unfavorable spin state for i−vi, the spin polarization through the benzene coupler was blocked with mismatching of the spin alternation. 3.3. Diradical Character (y) and Closed-Shell Singlet (CS) States. On the basis of our scheme, the energetically favorable spin state of vi was predicted to be the BS state. However, during optimization in the BS state of vi the ⟨S2⟩ value of BS solution changed to 0.0000 at the second optimization step, and at the same time the SCF energy was abnormally increased, as shown in Figure S3, Supporting Information. Then the SCF energy showed a regularly fluctuating curve, and eventually the BS state optimization could not be converged. In this section, we tried to find the clues of the underlying reason why this unexpected convergence error was observed only for the case of vi and not for the other cases of i−v. Moreover, we intend to check the validation of our proposed scheme for vi. As seen in Figure S3, Supporting Information, the convergence failure of vi came along with the significant change of ⟨S2⟩ value from 1.0167 to 0.0000. Considering the ideal ⟨S2⟩ value of BS solution is 1.0000, indicating an equal mixture of singlet and triplet states, the 0.0000 value of ⟨S2⟩ reveals that the BS solution of vi is approximately changed to CS singlet solution in the progress of BS state optimization. It should be noted that a CS solution does not take into account spin polarization and does not possess any diradical character. In this context, we sought to investigate the diradical character of singlet states and the relationship between BS and CS solutions for i−vi. First, the CASSCF (2,2) calculations were performed on singlet state diradicals of i−vi to obtain the orbital occupation numbers, which was used as an index of diradical character (y),50,51 based on the optimized structures of the triplet state for i−vi. In principle, in order to calculate the diradical character (y) of OS diradicals, it requires one to do calculations related with diradical character based on the optimized structures of OS states. In broken symmetry approaches based on the DFT method, one can use the optimized structures of the BS state instead of OS states as a compromise. In this research, owing to the convergence error of BS state optimization for vi, inevitably we used the optimized structures of the triplet state to obtain the diradical character (y) of the singlet state with the purpose of performing overall calculations through the same way for all diradical systems (i−vi) and facilitating comparison. We assumed that the geometrical differences between BS and triplet states might not be serious to influence the diradical character. The calculated occupation numbers of the HOMO and LUMO and diradical character (y) for the singlet spin states of i−vi are listed in Table S3, Supporting Information. The occupation numbers of i−vi were calculated to be almost 1.0 for both HOMO and LUMO, implying that the singlet spin states possessed significant diradical character. The LUMO occupation numbers could be directly converted to the percentage of diradical character (%) by simply multiplying by 100. Among i−vi systems, vi gave the smallest diradical character (93.3%), and the other systems showed diradical character over 98%. The relatively lower value of diradical character of vi might be related with the convergence error of BS state optimization. Second, in order to investigate the changes from BS solutions to CS solutions for i−vi, we also carried out calculations of CS 9601



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shown in Figure 4b. In addition, comparing the NBO charge of N1 (−0.216), N2 (−0.343), N3 (−0.401), and N4 (−0.305), N1 and N4 can be considered as sp2-hybridized nitrogen while N2 and N3 as sp3 nitrogen. As a result, we could complete the molecular structure for the CS state of vi, as illustrated in Figure 4c, eventually resulting in a closed-shell singlet (CS) state. The open-shell singlet state has a diradical molecular structure (Figure 5a), whereas the closed-shell singlet state

Figure 3. Correlation between the diradical character (y) of the singlet state and the HOMO−LUMO gaps of the CS state for i−vi.

Table 2. Calculated Bond Orders of Chemical Bonds and NBO Charges of Atoms for the CS State of vi bond

bond order

atom

NBO charge

C1−C2 C2−C3 C3−C4 C4−N1 N1−C5 C5−O1 C5−N3 N3−N4 C6−N4 C6−N2 N1−N2

1.245 1.569 1.251 1.154 0.941 1.652 1.166 1.116 1.503 1.399 1.136

C1 C2 C3 C4 C5 C6 N1 N2 N3 N4 O1

0.157 −0.197 −0.193 0.157 0.787 0.207 −0.216 −0.343 −0.401 −0.305 −0.611

Figure 5. Resonance structures of the (a) open-shell diradical and (b) closed-shell quinoid structures for the singlet state of vi.

has a quinoid molecular structure (Figure 5b). We could expect that the open-shell singlet diradical of vi might have about 30° dihedral angles between OVER(N) and the benzene coupler considering the optimized structures for the triplet state of vi. However, for the CS state, the molecular geometry changed to be planar. From bond order analysis and NBO charges, the CS state of vi might have the closed-shell doubly zwitterionic structure, as depicted in Figures 4 and 5b. It is clear that BS state optimization is not feasible to represent the OS state for vi, in which the resonance structures between the OS state and the CS state can be formed. In this

basis of the calculated results for Wiberg bond order and natural bond orbital (NBO) charge, we drew the molecular structure for the CS state of vi starting from a simple schematic in Figure 4a. From bond order analysis and molecular symmetry, we could draw the molecular structure of vi as

Figure 4. Schematic figure (a) and the determination process of drawing the molecular structure (b and c) for the closed-shell singlet (CS) state of vi. 9602



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Figure 6. Calculated J values vs dihedral angles DA1−7.

angles from 0° to 90° with a 10° step. Plots of calculated J values as a function of dihedral angle are shown in Figure 6 (see also Table S7, Supporting Information, for details). It was found that for DA1−2 and DA4−6 the absolute J value decreased as dihedral angle increased, as shown in Figure 6a and 6b. This indicates that the magnetic interaction tends to decrease with geometrical distortion. In geometrical distortion, the exchange coupling might be diminished regardless of the types of magnetic interactions because the conjugation of the diradical systems decreases. More dramatic changes of J values could be found in Figure 6c and 6d. In these two cases, the maximum (DA7) and minimum (DA3) absolute J values were observed at dihedral angles of 30° and 60°, respectively, as listed in Table S7, Supporting Information. For both DA7 of v and DA3 of iii, the meta-substituted benzene was used as a coupler and the OVER(N) was used as a fixed stationary radical with around 30° of dihedral angle according to the optimized structure. It implies that the optimal dihedral angle might exist for generating the strongest/weakest magnetic interaction for the diradical systems containing a meta-substituted benzene coupler as well as a distorted radical. In particular, for DA7, it was found that the spin cross over took place at around a 75° dihedral angle as shown in Figure 6c. In other words, the ground spin state changed from the triplet to thr singlet state in the course of twisting. Such a spin cross over was also found for DA4 and DA6, but the absolute J values for the dihedral angle of 90° were very small (1.5 cm−1, see Table S7, Supporting Information).

case, using the single-point calculation of the BS state based on the molecular geometry of the triplet state might be a compromise to represent the OS state of diradicals. The J value of vi shown in Table S6, Supporting Information, was obtained based on single-point calculation of the BS state. The J value of vi was obtained to be −231.0 cm−1, which was scaled to −87.8 cm−1. Consequently, the negative sign of the J value was obtained for vi, indicating the antiferromagnetic interaction, which is consistent with our proposed scheme (Scheme 3). Our calculations also reveal that in the case of vi the CS state should be considered carefully when it has a resonance relationship with the OS state. Although the OS state is more stable than the CS state, practically it would be possible to observe both the CS and the OS states depending on experimental conditions, as reported in Chichibabin’s hydrocarbon derivatives.52 3.4. Dihedral Angle Effect on J Values. We tried to check the validation of our proposed scheme under the influence of dihedral angles between the radical and the benzene coupler. Scheme 1 shows the seven systems investigated with dihedral angles denoted as DA1−7 in i−v systems. The system vi was excluded in this part due to the complexity related with the resonance relationship between the OS and the CS state and the convergence problem of the BS state optimization. To examine the effect of dihedral angles on the magnetic interactions, one dihedral angle was fixed at the optimized structures when the other was not and vice versa. In addition, J values were calculated by changing the dihedral 9603



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In summary, the plots of calculated J values depending on the dihedral angle can be classified into three types in this study: (1) decreased J value as the dihedral angle increased (Figure 6a and 6b); (2) J value has the maximum point (Figure 6c); (3) J value has the minimum point (Figure 6d). Our devised scheme in Scheme 3 might work well for most of the dihedral angles (DA1−6) for predicting the types of magnetic interactions for i−vi. However, the spin cross over arising from the distortion of DA7 would be an exceptional case.

interactions in delocalized diradical systems, regardless of radical groups and the geometrical topology of couplers. Our results will be helpful in designing organic high-spin molecules and organic magnetic materials.

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ASSOCIATED CONTENT

S Supporting Information *

Computational details of BS state calculations, atomic spin density of oxoverdazyl radical, calculated dihedral angles, intramolecular magnetic coupling constants, spin density distributions, diradical character of the OS state, HOMO− LUMO gaps of the CS state, Weiberg bond orders, NBO charges, J values depending on dihedral angles, and log files of the calculations for the triplet, BS, and CS states. This material is available free of charge via the Internet at http://pubs.acs.org.

4. CONCLUSIONS We propose that organic monoradicals in diradical systems can be classified into two groups according to the relationship of atomic spin densities between the connecting atom and the radical dot atom: syn-group and anti-group. The oxoverdazyl radical can be included in both groups depending on the choice of the connecting atom (OVER(C) or OVER(N)). In this study, we investigated the intramolecular magnetic coupling constant (J) of six oxoverdazyl diradicals (i−vi) coupled with the benzene ring based on the unrestricted DFT calculations. The diradical systems were carefully designed to analyze various magnetic interactions induced by the topology of the benzene coupler as well as the type of radical groups for oxoverdazyl radical. For BS and triplet states, it was found that OVER(C) and OVER(N) have dihedral angles of around 0° and 30°, respectively. It was predicted that antiferromagnetic couplings were preferred for i, iv, and v, whereas ferromagnetic couplings were preferred for ii and iii. We tried to explain the types of magnetic interactions for i−vi. The spin alternation rule was not enough to understand the magnetic couplings for hetero diradical systems, iii and v. Thus, we devised a new and simple but useful scheme (Figure 4) to understand the overall magnetic interactions of i−v via combination of the spin alternation rule and our new concept of radical group classification. Most notably, our proposed scheme was in good agreement with the calculated J values and the spin density distributions for all of i−v. We found the underlying reasons of the convergence error during calculation of BS state optimization for vi. The diradical character of the OS state, the HOMO−LUMO gap of the CS state, and the change of molecular geometry as well as the energy depending on spin states were analyzed. It was found that the resonance structures between the open-shell diradical and the closed-shell doubly zwitterionic quinoid form could exist for vi. Thus, during the BS state optimization, the BS solution of vi might be changed to CS solution owing to the influence of strong characteristics of the CS state that might be originated from the resonance structures. In such a case of vi, single-point calculation of the BS state based on the molecular geometry of the triplet state might be a compromise to present the OS state. The result of vi based on single-point calculation of the BS state was also consistent with our scheme. We verified the validation of our scheme in the environment of the variation of the dihedral angle (DA1−DA7) between the oxoverdazyl radical and the benzene coupler. In most cases, J values decreased as the dihedral angle increased without a change on the types of magnetic interactions. However, our scheme should be carefully applied to explain the magnetic interactions for severely distorted diradical systems, such as a case having spin cross over in the course of twisting of DA7. Nevertheless, our devised scheme would be able to provide insight into the interpretation and prediction of the magnetic

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AUTHOR INFORMATION

Corresponding Author

*Phone: +82-31-299-4560. Fax: +82-31-290-7075. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by National Research Foundation (NRF) grants funded by the Korean government (MEST; 2007-0056343 and 2013R1A1A2062901).

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REFERENCES

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