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A theoretical study of zero-field splitting of organic biradicals
Polyhedron 24 (2005) 2708–2715 www.elsevier.com/locate/poly
A theoretical study of zero-field splitting of organic biradicals Mitsuo Shoji *, Kenichi Koizumi, Tomohiro Hamamoto, Takeshi Taniguchi, Ryo Takeda, Yasutaka Kitagawa, Takashi Kawakami, Mitsutaka Okumura, Shusuke Yamanaka, Kizashi Yamaguchi Department of Chemistry, Graduate School of Science, Osaka University, Machikaneyama-cho 1-1, Toyonaka, Osaka 560-0043, Japan Received 6 October 2004; accepted 1 December 2004 Available online 20 July 2005
Abstract A theoretical method for ab initio calculations of the zero-field splitting parameters from electron spin–spin interaction contributions is described. This methodology is applied to typical organic biradicals within the density-functional theory. It is shown that this approach can provide results in good agreement with experimental values for typical organic biradicals at moderate computational cost. Magnetic properties are discussed in detail for typical organic biradicals, such as carbene, vinylmethylene, phenylcarbene, and diphenylcarbene. Their spin orbitals, magnetic axes, and effects of conformational changes are investigated. Moreover, the relationship between those three factors and the zero-filed splitting parameters (D- and E) including their signs is revealed. This method has been proved to be quite valid in studying organic ferromagnets. 2005 Elsevier Ltd. All rights reserved. Keywords: Ab initio calculation; DFT; Spin–spin dipolar coupling; Zero-field splitting parameters; Organic biradicals; Carbene; Vinylmethylene; Phenylcarbene; Diphenylcarbene
1. Introduction Recently, molecular magnets have attracted considerable interest because many new types of magnets have been discovered [1–3]. For instance, those are ferro-, photo-induced, low dimensional, and single molecular magnets. In particular, single-molecule magnets (SMMs) have given us a lot of attention. These new molecular materials have characteristic magnetic properties because of the quantum tunnelling and show magnetization hysteresis like bulk magnets. These molecules will provide high efficient and ultra-compact molecularbased circuit if used as devices in near the future. These magnetic properties are explained by magnetic anisotropy. Many systematic attempts have been made to real*
Corresponding author. Tel.: +81 6 6850 5405; fax: +81 6 6850 5550. E-mail address: [email protected] (M. Shoji). 0277-5387/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.poly.2005.03.135
ize the SMMs that have large spin multiplicities and large anisotropy energies. The anisotropy originates in the spin–orbit coupling (SOC) and the spin–spin couplings (SSC) interactions. Theoretical and synthetic studies are in an early phase of the development and have just begun to be applied. Though these magnetic properties were considered to be inherent in metal complex systems, organic systems and hybrid systems have been synthesized [4–9] in these days. Also for the studies of spin systems, the zero-field splitting parameter (ZFS) of the electron paramagnetic resonance (EPR) spectroscopy is one of the most important experimental information [10]. Since the ZFS for organic radicals is due to the direct SSC interactions between unpaired electrons, it is expected to provide significant information reflecting the spin-orbital distributions and the molecular structures. Particularly, it is more essential to most of the triplet states of organic biradicals owing to their high reactivities or instabilities.
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It is necessary for the accurate investigations of such compounds to calculate and explain such interactions as SSC interactions within molecular orbital levels, since most of triplet carbenes were determined by the usage of EPR. Especially, high level ab initio studies can be significantly useful not just for inquiring into the origin of magnetic anisotropy but also to the flexible and systematic molecular designs, as well as the interpretation of spectra and the determination of molecular structures. It was difficult to study ZFSs theoretically, because the electron correlation problems are deeply involved and even more so in excited spin states. Only ab initio studies of ZFSs have been carried out for such small molecules as CH2, SiH2, O2, H2CO [11–14]. Although semi-empirical methods have often been used and ab initio calculations are still difficult due to the much computing costs for aromatic and heterocyclic hydrocarbons. In this study, theoretical studies of zero-field splitting energies from the spin–spin coupling contributions for ground triplet molecules [15] are reported. An efficient method for ZFS by the hybrid-density-functional theory (HDFT) is proposed, which can be applied up to relatively large molecules. First, this procedure is discussed for typical organic biradicals. Next, its detailed applications for traditional problems of such carbenes as vinylmethylene, phenylcarbene, and diphenylcarbenes are discussed from the viewpoint of the calculated ZFSs parameters, magnetic axes and spin orbitals.
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ropy is an ellipsoid with two different axes. The spin ellipsoid looks like a disc if D > 0, and the spin ellipsoid resembles a bar or a rugby ball if D < 0. 2.2. The origin of anisotropy in real Hamiltonian The SOC and SSC expressions for the magnetic anisotropy are obtained under relativistic correction of the Breit–Pauli Hamiltonian [16–20]. The spin–orbit term is ! SO e2 h X SðiÞ Mn ðiÞ X 2SðiÞ Mi ðjÞ þ SðiÞ Mj ðiÞ b H ¼ 2 2 ; Zn 2m c r3ni r3ij n;i i6¼j
ð3Þ n
where S(i) is a spin angular momentum operator. M (i) is a gauge-invariant angular momentum operator of ith electron defined as Mn ðiÞ ¼ rin pi .
ð4Þ
The first term in Eq. (3) is the sum over the one-electron SOCs in the Coulomb field of nuclei n. The second term is the interaction of electron spin i with the magnetic field induced by the orbital current of jth electron. In most of organic free radicals, the orbital angular momentum is quenched and SOC is much smaller than SSC. The spin–spin term is 2 2 X r2 ðSðiÞ SðjÞÞ 3ðr SðiÞÞðr SðjÞÞ ij ij ij b SS ¼ e h H ; m2 c2 i
ð5Þ 2. Theoretical backgrounds 2.1. Anisotropy in spin hamiltonian An anisotropic term in spin-Hamiltonian is constructed from the total spin operators as the form X H AS ¼ Dij S i S j ; ði; j ¼ x; y; zÞ; ð1Þ i;j
where Dij is a traceless symmetric rank-two tensor. Diagonalization of the D tensor can represent Eq. (1) in a principal axis (X, Y, Z) coordinate system as 1 H AS ¼ D S 2Z S 2 þ EðS 2X S 2Y Þ; ð2Þ 3 where D and E are the independent parameters: D = 3/2DZZ and E = 1/2(DYY DXX). The Z is the main axis. D is the principle value and E < |D/3|, which represents anisotropy within XY plane. D and E values for organic radical are experimentally determined by EPR fine structure around the large signal g = 2.0. The magnetic axis is rarely determined by EPR study in solid crystals. The axes direction and the sign of E value change depending on the orientation definitions. The labels of the axes of EPR are conventionally chosen so as to make DE < 0. The energy surface of spin anisot-
which means the spin–spin dipolar coupling. This term is the main interaction of the ZFSs in organic radicals and the g-factors is near to the value for a free electron (ge = 2.0023). 2.3. Spin orbitals The unrestricted wave function method is employed in this HDFT study. These orbitals are spin-polarized and negative spin density is found over atoms. The spin orbitals are obtained from natural (spin) orbitals with occupation number 1.0, so-called singly occupied natural orbital (SONO). Natural orbitals (NOs) are obtained as an eigenvector by the diagonalization of all electron (or charge) density as X qðr0 ; rÞ ¼ ni ui ðr0 Þui ðrÞ. ð6Þ i
Each eigenvalue of NOs corresponds to occupation number ni (0 6 ni 6 2). Similar procedure to Eq. (6) provides spin natural orbital (SNO) from all electron spin density. Each eigenvalue of SNOs corresponds to spin occupation number nSNO (in [0, 1]) where a spin is dei fined to be positive. The SONOs and HOMOs of a orbitals are almost the same, but this NO procedure
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theoretically means the spin projection into pure high spin states from spin-contaminated states. These nonSONO NOs (i.e., HONOs) and its occupation numbers can analyse spin polarizations and magnetic interactions quantitatively [21,22]. The total spin orbitals are derived by the antisymmetrization of spin orbitals. An expectation value of an b for magnetic properties is obtained as: operator O b E ¼ hUj OjUi. ð7Þ For the triplet states of organic biradicals, only two SONOs affect for their magnetic properties. The total spin orbitals are 1 j3 Ui ¼ jui uj i ¼ pffiffiffi ðui uj uj ui Þ; ð8Þ 2 and the anisotropic D tensor is obtained by the SSC contribution to the ZFS, as _ SS
D ¼ HSS ¼ h3 UjH j3 Ui.
ð9Þ
3. Computational procedures The present study was started with coding a new computer programs called ‘‘GSO-ZFS’’, in order to calculate SSC. The two electron integrals in Eq. (1) were calculated by differentiating two-electron integrals based on the Obara–Saika methodology [23–26]. This computational procedure is described as below. First, ab initio MO calculations are performed for the molecular structures of the ground triplet state and spin orbitals with GAUSSIAN 98 program packages [27] using UB3LYP//631G method [28,29]. The D of SSC interaction is then evaluated by GSO-ZFS. ZFS parameters D, E and magnetic axis are provided by the diagonalization of D. the g-factor is fixed to be 2.0. In the triplet states, two SONOs were selected for the SSC calculation. In general 2S spin orbitals are necessary for the SSC calculation in the highest 2S + 1 spin multiplicity state. The SNOs are necessary for evaluating SCC from unrestricted orbital calculations because of the separation from the slight spin contaminations by spin polarization. The SSC interactions between these spin orbitals were evaluated by the effective total spin wave functions of the antisymmetrized spin orbitals. If the spin state is of the excited one, multi-reference configuration interaction (MR-CI) and MR-SCF calculations will be necessary for the SSC calculations. 4. Calculated results and discussions 4.1. Applications to typical organic biradicals D and E for typical organic biradicals depicted in Fig. 1 [5,16] were calculated by the method described above.
Fig. 1. Molecular structures of typical organic biradicals (1: carbene, 2: nitrene, 3: cyclopentadienylidene, 4: phenylcarbene, 5: diphenylcarbene, 6: vinylcarbene (s-E), 7: vinylcarbene (s-Z), 8: p-tolylcarbene, 9: pyridylcarbene, 10: iminocyclohexadienylidene, 11: 4-oxo-2,6-cyclohexadienylidene, 12: ferrocene cation, 13: triphenylene, 14: coronene dication, 15: bis[9-(10-phenyl)anthryl]carbene).
The results are summarized in Table 1 with their experimental values, indicating these calculated D and E values were in good agreements with their experimental values. All of these organic biradicals have positive D values because their spin orbitals distribute over one ring. In contrast of such spin distributions, if spins stay away each other, the D value is expected to have negative sign by the application of the point dipole approximation. These D values are often used for the experimental measurement between two spin sites. The largest D value was obtained for 2 because the spin orbitals are the closest to each other. Also the D of carbene (1) was large for the same reason. Among carbene derivatives, the smallest D was obtained for 15 because each spin orbital at carbene centre is delocalized over the anthryl moieties and spins are distant each other [3,30,31]. E value shows how bending the carbene centre is. Such linear biradicals as 15 and 2 have characteristically 0 values of E, so do such molecules with more than C3 symmetry as 12 and 13. Cyclopropenyl cation 13 has a 4p-electron cyclic system. It has the triplet ground state in order to keep C5 symmetry of the molecular structure [32–35]. Coronene 14 takes thermally accessible triplet state in dication state [36], which is extremely stable at ambient room temperature, decomposing above 50–70 C. The molec-
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Table 1 D and E for the triplet ground states of organic molecules No.
Molecules
Ab initio calculationa 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 14 0 15 a
carbene NH C5H4 HCPh diphenylcarbene HCCHCH2 (s-E) HCCHCH2 (s-Z) p-MePhCH HCC5NH4 HNC6H4 C6H4O C5 H5 þ tripen2 coronen2+ coronen2 PhAtCAtPh
Exp. 1
Ref. 1
1
D (cm )
E (cm )
E/D
|D| (cm )
|E| (cm )
|E/D|
0.8104 1.698 0.4412 0.5193 0.3570 0.3762 0.4355 0.512 0.5443 0.0847 0.2057 0.1269 0.0485 0.1145 0.0394 0.0391
0.0576 0.0000 0.0164 0.0346 0.0125 0.0296 0.0219 0.0343 0.0358 0.0048 0.0036 0.0000 0.0000 0.0000 0.0000 0.0000
0.0711 0.0000 0.0373 0.0667 0.0335 0.0787 0.0504 0.0670 0.0658 0.0567 0.0174 0.0000 0.0000 0.0000 0.0000 0.0000
0.7567 1.86 0.4089 0.5150 0.4055 0.4093 0.4578 0.516 0.537 0.1704 0.3179 0.1868 0.0460 0.0591 0.0542 0.105
0.0461 0.0000 0.0120 0.024 0.0194 0.0224 0.0193 0.0240 0.027 0.0025 0.0055 0.0000 0.0088 0.0000 0.0000 0.0004
0.0609 0.0000 0.0293 0.0487 0.0478 0.0547 0.0422 0.0465 0.0503 0.0150 0.017 0.0000 0.0000 0.0000 0.0000 0.0000
[44] [49] [48] [42] [42] [40] [40] [45] [46] [47] [7] [6] [32] [36] [37] [3]
Spin–spin coupling contributions.
ular structure of this dication spin state has D6h symmetry, but the structure of the ground state with the singlet spins has D2h symmetry under the influence of its Jahn– Teller distortions. We implemented geometry optimization for 14 to evaluate the structural distortion effect and the S-T gas. UB3LYP//6-31G showed that the total energy of the singlet state with D2h was more stable than the one of the triplet spin state with D6h by 147.5 kcal mol1. The Jahn–Teller distortions had small contributions to molecular stability for the singlet state of 14, since the energy gap between the singlet structure and that of the triplet was only 8.4 kcal mol1. Hence, it is found that the observed EPR signals for 142+ are arise from its thermally excited triplet spin state. The temperature dependency in EPR signal intensity experimentally estimated S–T gap to be 1.4 kcal mol1. Also 142 takes the thermally excited biradical state with 23 kcal mol1 S–T gap [37]. In contrast, our UB3LYP//6-31G calculations evaluated the triplet spin state of 142 to be slightly more stable by 2.1 kcal mol1 (UB3LYP//631 + G* also did by 1.6 kcal mol1). Such phenomena as triplet stability in dianion and dication states resulted from the doubly degeneration of HOMO and LUMO orbitals in the neutral state of coronene. Thermally excited triplet spin states were also observed in dianion state on strong Jahn–Teller distortions in hexabenzocoronene (HBC) derivatives that are more conjugated polycyclic aromatic hydrocarbons [38]. The HBCs have some characteristic features that HOMOs and LUMOs are nearly triple degenerated, and take such high-oxidized state as tetra anion. EPR observation then indicated that HBCs took quartet spin state, not triplet. These biradicals have p-conjugated spin orbitals and their magnetic hard axis (Z) is vertical to the aromatic rings from the result of our calculations. As
the number of aromatic ring increases, D values decreases, simply due to the spin orbitals delocalized over the aromatic ring and increase in average spin-spin distances. EPR observed D and E value from HBC dianions are so small: 0.00145 and 0.00005, respectively. Since the molecular structure of coronene (14) has C3 symmetry and the observed E value is characteristic zero value. Only bis[9-(10-phenyl)anthryl]carbene 15 [3,39]was calculated by UB3LYP//4-31G because it is a relatively large molecule. The optimized bond angle at the carbene centre (h) was evaluated to be 179.7. The carbene was slightly bent and had small E value. The spin orbitals were much delocalized to each side of anthryl moieties. It is concluded that the weak carbene nature makes the carbene centre almost linear geometry. More detailed discussions for triplet carbene, vinylmethylene and diphenylcarbene are given below. 4.2. Triplet carbene Bond angles at the carbene centre (h) have been experimentally estimated based on E/D values [39,40], which is also important for the delocalization nature of the spin orbitals at the carbene centre. The relationship of the ZFS values to h was investigated by the calculations using UHF, UBHandHLYP, UB3LYP, and ˚ . The reUBLYP. CH bond length was fixed to 1.10 A sults are displayed in Fig. 2. At the linear structure, D = 0.8 and E = 0. When h decreases from 180, D decreases while E increases. Therefore, E/D values decrease with the bond angle h (Fig. 3). The decrease in E/D values is that spin orbitals are inhomogeneous between YZ-plane and ZX-plane where carbene locates because of the presence of hydrogen. The differences in
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Fig. 4. Triplet carbene. The optimized molecular structure with the magnetic axes (A) and spin orbitals (B, C).
Fig. 2. D (solid symbols) and E (open symbols) values dependence of carbene as a function of the bond angle h of carbene centre. D values are UHF (d), UBHandHLYP (n), UB3LYP (¤) and UBLYP (m). E values are UHF (s), UBHandHLYP (h), UB3LYP (}) and UBLYP (n).
DFT (BLYP). Magnetic hard axis (Z) was parallel to the line connecting the two hydrogen atoms, while Magnetic easy plane (XY-plane) was vertical to the Z-axis. Therefore, two spin orbitals of carbenes are consisted of px and py orbitals, respectively (Fig. 4). The ZFS values D/cm1 and E/cm1 in the optimized structure were 0.8104 and 0.0576, respectively, as shown in Table 1. These values are slightly larger than experimental values: 0.7567 and 0.0461, respectively. Havlas et al. have reported D = 0.8043 cm1 and E = 0.0428 cm1 for carbene by CISDTQ/cc-pVDZ methods [11], which is comparable to our values. The SOC interaction causes a slightly smaller contributions than SSC by D 0 = D + 0.0023 cm1 and E 0 = E + 0.0001 cm1 [11,20]. 4.3. Isomers of vinylmethylene
Fig. 3. E/D values dependences of carbene as a function of bond angle h of carbene centre calculated by UHF (s), UBHandHLYP (h), UB3LYP (}) and UBLYP (n).
computational methods of HF, BHandHLYP, B3LYP, BLYP are small, indicating that the differences in the shape of spin orbitals are almost the same as the ones for the SSC interactions. The total spin orbitals are then antisymmetrized by the Slater determinants in the process of the SSC calculation. Consequently, the spin correlations exist for all the spin orbitals, even by the pure
There are two isomers of vinylmethylene: s-E and s-Z, and EPR signals for each of these isomers have been observed. It is interest that whether we can theoretically distinguish the difference in fine structures of ZFS for those isomers, or not. Geometry optimizations were performed for each isomer. The h s were 136.0 and 136.6 for s-E and s-Z, respectively. Fig. 5A displays that the spin orbital perpendicular to the methylene plane is delocalized over the p-orbitals for each s-E and s-Z isomers. The spin orbital populations on the C2 atom were almost zero at the node of the spin orbitals. On the contrary, the other spin orbitals (r type spin orbital) parallel to methylene plane localized at carbene centre (see Fig. 5C). The calculated D value of s-Z was larger than that of s-E, while E values of s-Z were smaller than that of s-E (Table 1). These results were consistent with experimental ones. Magnetic axis and spin orbitals are shown in Fig. 5A. As is already pointed by R.S. Hutton et al. [40], spin orbitals spread into Z-axis of the s-E isomer and the corresponding spin orbitals spread over XZ direction of the s-Z isomers. These spin orbitals clearly shows that the difference in delocalization direction affects the D value. The UB3LYP/6-31G calculated the s-E isomer to be more stable than s-Z by 0.207 kcal mol1. The signal rates of s-Z/s-E in EPR are
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Fig. 5. Vinylcarbene of s-E (1) and s-Z (2) isomers. Each optimized molecular structures with their magnetic axes (A) and two spin orbitals (B, C).
0.05–0.65 depending on temperature and solvents. This experimental result agrees with our calculation result, showing s-E is thermodynamically more stable. 4.4. Phenylcarbene The ground state structure of phenylcarbene is investigated by ZFSs. The molecular structure of phenylcarbene is considered to be either: the phenyl group is parallel or vertical to carbene C–H plane (denoted by q0 and q90, respectively). Each of these structures wass
theoretically optimized. It was revealed that q0 was more stable than q90 by 4.335 kcal mol1. The stable optimized structure q0 was in agreement with other ab initio calculation results by Geise et al. [41]. For isomers q0 and q90, (D, E) = (0.5193,0.0346) cm1 and (0.6005, 0.0308) cm1, respectively. The D for q0 was closer to the experimental value |D| = 0.5150 cm1, while the experimental value |E| = 0.024 cm1 was smaller than the calculated results. These spin orbitals for each isomer resembled each other and ZFS values were relatively equal (Fig. 6).
Fig. 6. Phenylcarbene of q90 (1) and q0 (2). Each optimized molecular structures with their magnetic axes (A) and two spin orbitals (B, C).
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Fig. 7. Structures of diphenylcarbene: a1 (1), a2 (2) and b (3). Each optimized molecular structures with their magnetic axes (A) and two spin orbitals (B, C).
4.5. Diphenylcarbene The molecular structure of diphenylcarbene is classical problem in EPR studies [5,17,18,31,43]. There are three conceivable structures. The first one (a1) is that phenylcarbenes are parallel each other and carbene centre is liner, i.e., h = 0 (Fig. 7-2). The second one (a2) is that a1 is bent at carbene centre in the phenyl ring plane so as the whole structure is within one plane (Fig. 7-1). The third one (b) is that the phenyl ring is vertical each other and carbene centre is bent (Fig. 7-3). The experimental ZFS values proposed that ground state is a2. (The most stable structure seems to b, because two spin orbitals at carbene centre can spread over each phenyl rings) We then decide the structure theoretically, especially using the calculated ZFS values. In optimizing the forth geometry a1 0 , which is bent at carbene centre in the vertical direction, a1 0 fell into the geometry a1 and was revealed to be unstable. The angle h optimized to be 147.0 and 149.0 for a2 and b, respectively. The most stable structure of these three structures was decided to be a2 by UB3LYP//6-31G. Energy gap between a2 and a1 was 85.58 kcal mol1, and that between a2 and b were 81.28 kcal mol 1. In a1 and a2, one spin orbital vertical to the phenyl ring was delocalized onto p-orbitals over both phenyl rings and the other spin orbital was localized at the carbene centre. The carbene centre was in line and E 0 at the structure a1. ZFS values were as (D, E) = (0.3740, 0.00185) cm1 and (0.3774, 0.01657) cm1 for a1 and a2, respectively. On the other hand, the spin orbital at the carbene centre was delocalized over the next phenyl rings each other at the structure b. And the D value is rather smaller because average distances between spins are relatively
longer than a1 and a2. ZFS values for b were evaluated to be (D, E) = (0.3569, 0.01254) cm1, while Experimental ZFS values are |D| = 0.4055 cm1 and |E| = 0.0194 cm1 [39]. Spin orbitals and magnetic axis are shown in Fig. 7. The most stable structure of phenylcarbene was estimated as a2, which reproduces experimental values. It is proved that molecular structure can be determined from the ab initio calculation of ZFS values.
5. Conclusion The results of this approach were in good agreement with experimental values at moderate computational cost. By using this method, theoretical applications to organic biradicals are further examined for carbene structures, isomers of vinylcarbenes and diphenylcarbenes. We showed this calculation procedure was accurate enough for these the ZFS calculations and effective for the theoretical investigations for organic radicals. This means that high spin states in organic radicals are close to that of the single particle picture. It was also found that the SSC between spin orbitals are the dominant factor for the ZFS and the contributions from the SOC to the ZFS will be vanishingly small. This method has been quit valid in studies of organic ferromagnets.
Acknowledgement This work has been supported by the 21 Century COE program entitled ÔThe Creation of Integrated EcoChemistryÕ in Osaka University and Grants-in-Aid for
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Scientific Research on Priority Areas (Nos. 16750049 and 14204061) from Ministry of Education, Culture, Sports, Science and Technology, Japan.
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A theoretical study of zero-field splitting of organic biradicals Mitsuo Shoji *, Kenichi Koizumi, Tomohiro Hamamoto, Takeshi Taniguchi, Ryo Takeda, Yasutaka Kitagawa, Takashi Kawakami, Mitsutaka Okumura, Shusuke Yamanaka, Kizashi Yamaguchi Department of Chemistry, Graduate School of Science, Osaka University, Machikaneyama-cho 1-1, Toyonaka, Osaka 560-0043, Japan Received 6 October 2004; accepted 1 December 2004 Available online 20 July 2005
Abstract A theoretical method for ab initio calculations of the zero-field splitting parameters from electron spin–spin interaction contributions is described. This methodology is applied to typical organic biradicals within the density-functional theory. It is shown that this approach can provide results in good agreement with experimental values for typical organic biradicals at moderate computational cost. Magnetic properties are discussed in detail for typical organic biradicals, such as carbene, vinylmethylene, phenylcarbene, and diphenylcarbene. Their spin orbitals, magnetic axes, and effects of conformational changes are investigated. Moreover, the relationship between those three factors and the zero-filed splitting parameters (D- and E) including their signs is revealed. This method has been proved to be quite valid in studying organic ferromagnets. 2005 Elsevier Ltd. All rights reserved. Keywords: Ab initio calculation; DFT; Spin–spin dipolar coupling; Zero-field splitting parameters; Organic biradicals; Carbene; Vinylmethylene; Phenylcarbene; Diphenylcarbene
1. Introduction Recently, molecular magnets have attracted considerable interest because many new types of magnets have been discovered [1–3]. For instance, those are ferro-, photo-induced, low dimensional, and single molecular magnets. In particular, single-molecule magnets (SMMs) have given us a lot of attention. These new molecular materials have characteristic magnetic properties because of the quantum tunnelling and show magnetization hysteresis like bulk magnets. These molecules will provide high efficient and ultra-compact molecularbased circuit if used as devices in near the future. These magnetic properties are explained by magnetic anisotropy. Many systematic attempts have been made to real*
Corresponding author. Tel.: +81 6 6850 5405; fax: +81 6 6850 5550. E-mail address: [email protected] (M. Shoji). 0277-5387/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.poly.2005.03.135
ize the SMMs that have large spin multiplicities and large anisotropy energies. The anisotropy originates in the spin–orbit coupling (SOC) and the spin–spin couplings (SSC) interactions. Theoretical and synthetic studies are in an early phase of the development and have just begun to be applied. Though these magnetic properties were considered to be inherent in metal complex systems, organic systems and hybrid systems have been synthesized [4–9] in these days. Also for the studies of spin systems, the zero-field splitting parameter (ZFS) of the electron paramagnetic resonance (EPR) spectroscopy is one of the most important experimental information [10]. Since the ZFS for organic radicals is due to the direct SSC interactions between unpaired electrons, it is expected to provide significant information reflecting the spin-orbital distributions and the molecular structures. Particularly, it is more essential to most of the triplet states of organic biradicals owing to their high reactivities or instabilities.
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It is necessary for the accurate investigations of such compounds to calculate and explain such interactions as SSC interactions within molecular orbital levels, since most of triplet carbenes were determined by the usage of EPR. Especially, high level ab initio studies can be significantly useful not just for inquiring into the origin of magnetic anisotropy but also to the flexible and systematic molecular designs, as well as the interpretation of spectra and the determination of molecular structures. It was difficult to study ZFSs theoretically, because the electron correlation problems are deeply involved and even more so in excited spin states. Only ab initio studies of ZFSs have been carried out for such small molecules as CH2, SiH2, O2, H2CO [11–14]. Although semi-empirical methods have often been used and ab initio calculations are still difficult due to the much computing costs for aromatic and heterocyclic hydrocarbons. In this study, theoretical studies of zero-field splitting energies from the spin–spin coupling contributions for ground triplet molecules [15] are reported. An efficient method for ZFS by the hybrid-density-functional theory (HDFT) is proposed, which can be applied up to relatively large molecules. First, this procedure is discussed for typical organic biradicals. Next, its detailed applications for traditional problems of such carbenes as vinylmethylene, phenylcarbene, and diphenylcarbenes are discussed from the viewpoint of the calculated ZFSs parameters, magnetic axes and spin orbitals.
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ropy is an ellipsoid with two different axes. The spin ellipsoid looks like a disc if D > 0, and the spin ellipsoid resembles a bar or a rugby ball if D < 0. 2.2. The origin of anisotropy in real Hamiltonian The SOC and SSC expressions for the magnetic anisotropy are obtained under relativistic correction of the Breit–Pauli Hamiltonian [16–20]. The spin–orbit term is ! SO e2 h X SðiÞ Mn ðiÞ X 2SðiÞ Mi ðjÞ þ SðiÞ Mj ðiÞ b H ¼ 2 2 ; Zn 2m c r3ni r3ij n;i i6¼j
ð3Þ n
where S(i) is a spin angular momentum operator. M (i) is a gauge-invariant angular momentum operator of ith electron defined as Mn ðiÞ ¼ rin pi .
ð4Þ
The first term in Eq. (3) is the sum over the one-electron SOCs in the Coulomb field of nuclei n. The second term is the interaction of electron spin i with the magnetic field induced by the orbital current of jth electron. In most of organic free radicals, the orbital angular momentum is quenched and SOC is much smaller than SSC. The spin–spin term is 2 2 X r2 ðSðiÞ SðjÞÞ 3ðr SðiÞÞðr SðjÞÞ ij ij ij b SS ¼ e h H ; m2 c2 i
ð5Þ 2. Theoretical backgrounds 2.1. Anisotropy in spin hamiltonian An anisotropic term in spin-Hamiltonian is constructed from the total spin operators as the form X H AS ¼ Dij S i S j ; ði; j ¼ x; y; zÞ; ð1Þ i;j
where Dij is a traceless symmetric rank-two tensor. Diagonalization of the D tensor can represent Eq. (1) in a principal axis (X, Y, Z) coordinate system as 1 H AS ¼ D S 2Z S 2 þ EðS 2X S 2Y Þ; ð2Þ 3 where D and E are the independent parameters: D = 3/2DZZ and E = 1/2(DYY DXX). The Z is the main axis. D is the principle value and E < |D/3|, which represents anisotropy within XY plane. D and E values for organic radical are experimentally determined by EPR fine structure around the large signal g = 2.0. The magnetic axis is rarely determined by EPR study in solid crystals. The axes direction and the sign of E value change depending on the orientation definitions. The labels of the axes of EPR are conventionally chosen so as to make DE < 0. The energy surface of spin anisot-
which means the spin–spin dipolar coupling. This term is the main interaction of the ZFSs in organic radicals and the g-factors is near to the value for a free electron (ge = 2.0023). 2.3. Spin orbitals The unrestricted wave function method is employed in this HDFT study. These orbitals are spin-polarized and negative spin density is found over atoms. The spin orbitals are obtained from natural (spin) orbitals with occupation number 1.0, so-called singly occupied natural orbital (SONO). Natural orbitals (NOs) are obtained as an eigenvector by the diagonalization of all electron (or charge) density as X qðr0 ; rÞ ¼ ni ui ðr0 Þui ðrÞ. ð6Þ i
Each eigenvalue of NOs corresponds to occupation number ni (0 6 ni 6 2). Similar procedure to Eq. (6) provides spin natural orbital (SNO) from all electron spin density. Each eigenvalue of SNOs corresponds to spin occupation number nSNO (in [0, 1]) where a spin is dei fined to be positive. The SONOs and HOMOs of a orbitals are almost the same, but this NO procedure
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theoretically means the spin projection into pure high spin states from spin-contaminated states. These nonSONO NOs (i.e., HONOs) and its occupation numbers can analyse spin polarizations and magnetic interactions quantitatively [21,22]. The total spin orbitals are derived by the antisymmetrization of spin orbitals. An expectation value of an b for magnetic properties is obtained as: operator O b E ¼ hUj OjUi. ð7Þ For the triplet states of organic biradicals, only two SONOs affect for their magnetic properties. The total spin orbitals are 1 j3 Ui ¼ jui uj i ¼ pffiffiffi ðui uj uj ui Þ; ð8Þ 2 and the anisotropic D tensor is obtained by the SSC contribution to the ZFS, as _ SS
D ¼ HSS ¼ h3 UjH j3 Ui.
ð9Þ
3. Computational procedures The present study was started with coding a new computer programs called ‘‘GSO-ZFS’’, in order to calculate SSC. The two electron integrals in Eq. (1) were calculated by differentiating two-electron integrals based on the Obara–Saika methodology [23–26]. This computational procedure is described as below. First, ab initio MO calculations are performed for the molecular structures of the ground triplet state and spin orbitals with GAUSSIAN 98 program packages [27] using UB3LYP//631G method [28,29]. The D of SSC interaction is then evaluated by GSO-ZFS. ZFS parameters D, E and magnetic axis are provided by the diagonalization of D. the g-factor is fixed to be 2.0. In the triplet states, two SONOs were selected for the SSC calculation. In general 2S spin orbitals are necessary for the SSC calculation in the highest 2S + 1 spin multiplicity state. The SNOs are necessary for evaluating SCC from unrestricted orbital calculations because of the separation from the slight spin contaminations by spin polarization. The SSC interactions between these spin orbitals were evaluated by the effective total spin wave functions of the antisymmetrized spin orbitals. If the spin state is of the excited one, multi-reference configuration interaction (MR-CI) and MR-SCF calculations will be necessary for the SSC calculations. 4. Calculated results and discussions 4.1. Applications to typical organic biradicals D and E for typical organic biradicals depicted in Fig. 1 [5,16] were calculated by the method described above.
Fig. 1. Molecular structures of typical organic biradicals (1: carbene, 2: nitrene, 3: cyclopentadienylidene, 4: phenylcarbene, 5: diphenylcarbene, 6: vinylcarbene (s-E), 7: vinylcarbene (s-Z), 8: p-tolylcarbene, 9: pyridylcarbene, 10: iminocyclohexadienylidene, 11: 4-oxo-2,6-cyclohexadienylidene, 12: ferrocene cation, 13: triphenylene, 14: coronene dication, 15: bis[9-(10-phenyl)anthryl]carbene).
The results are summarized in Table 1 with their experimental values, indicating these calculated D and E values were in good agreements with their experimental values. All of these organic biradicals have positive D values because their spin orbitals distribute over one ring. In contrast of such spin distributions, if spins stay away each other, the D value is expected to have negative sign by the application of the point dipole approximation. These D values are often used for the experimental measurement between two spin sites. The largest D value was obtained for 2 because the spin orbitals are the closest to each other. Also the D of carbene (1) was large for the same reason. Among carbene derivatives, the smallest D was obtained for 15 because each spin orbital at carbene centre is delocalized over the anthryl moieties and spins are distant each other [3,30,31]. E value shows how bending the carbene centre is. Such linear biradicals as 15 and 2 have characteristically 0 values of E, so do such molecules with more than C3 symmetry as 12 and 13. Cyclopropenyl cation 13 has a 4p-electron cyclic system. It has the triplet ground state in order to keep C5 symmetry of the molecular structure [32–35]. Coronene 14 takes thermally accessible triplet state in dication state [36], which is extremely stable at ambient room temperature, decomposing above 50–70 C. The molec-
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Table 1 D and E for the triplet ground states of organic molecules No.
Molecules
Ab initio calculationa 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 14 0 15 a
carbene NH C5H4 HCPh diphenylcarbene HCCHCH2 (s-E) HCCHCH2 (s-Z) p-MePhCH HCC5NH4 HNC6H4 C6H4O C5 H5 þ tripen2 coronen2+ coronen2 PhAtCAtPh
Exp. 1
Ref. 1
1
D (cm )
E (cm )
E/D
|D| (cm )
|E| (cm )
|E/D|
0.8104 1.698 0.4412 0.5193 0.3570 0.3762 0.4355 0.512 0.5443 0.0847 0.2057 0.1269 0.0485 0.1145 0.0394 0.0391
0.0576 0.0000 0.0164 0.0346 0.0125 0.0296 0.0219 0.0343 0.0358 0.0048 0.0036 0.0000 0.0000 0.0000 0.0000 0.0000
0.0711 0.0000 0.0373 0.0667 0.0335 0.0787 0.0504 0.0670 0.0658 0.0567 0.0174 0.0000 0.0000 0.0000 0.0000 0.0000
0.7567 1.86 0.4089 0.5150 0.4055 0.4093 0.4578 0.516 0.537 0.1704 0.3179 0.1868 0.0460 0.0591 0.0542 0.105
0.0461 0.0000 0.0120 0.024 0.0194 0.0224 0.0193 0.0240 0.027 0.0025 0.0055 0.0000 0.0088 0.0000 0.0000 0.0004
0.0609 0.0000 0.0293 0.0487 0.0478 0.0547 0.0422 0.0465 0.0503 0.0150 0.017 0.0000 0.0000 0.0000 0.0000 0.0000
[44] [49] [48] [42] [42] [40] [40] [45] [46] [47] [7] [6] [32] [36] [37] [3]
Spin–spin coupling contributions.
ular structure of this dication spin state has D6h symmetry, but the structure of the ground state with the singlet spins has D2h symmetry under the influence of its Jahn– Teller distortions. We implemented geometry optimization for 14 to evaluate the structural distortion effect and the S-T gas. UB3LYP//6-31G showed that the total energy of the singlet state with D2h was more stable than the one of the triplet spin state with D6h by 147.5 kcal mol1. The Jahn–Teller distortions had small contributions to molecular stability for the singlet state of 14, since the energy gap between the singlet structure and that of the triplet was only 8.4 kcal mol1. Hence, it is found that the observed EPR signals for 142+ are arise from its thermally excited triplet spin state. The temperature dependency in EPR signal intensity experimentally estimated S–T gap to be 1.4 kcal mol1. Also 142 takes the thermally excited biradical state with 23 kcal mol1 S–T gap [37]. In contrast, our UB3LYP//6-31G calculations evaluated the triplet spin state of 142 to be slightly more stable by 2.1 kcal mol1 (UB3LYP//631 + G* also did by 1.6 kcal mol1). Such phenomena as triplet stability in dianion and dication states resulted from the doubly degeneration of HOMO and LUMO orbitals in the neutral state of coronene. Thermally excited triplet spin states were also observed in dianion state on strong Jahn–Teller distortions in hexabenzocoronene (HBC) derivatives that are more conjugated polycyclic aromatic hydrocarbons [38]. The HBCs have some characteristic features that HOMOs and LUMOs are nearly triple degenerated, and take such high-oxidized state as tetra anion. EPR observation then indicated that HBCs took quartet spin state, not triplet. These biradicals have p-conjugated spin orbitals and their magnetic hard axis (Z) is vertical to the aromatic rings from the result of our calculations. As
the number of aromatic ring increases, D values decreases, simply due to the spin orbitals delocalized over the aromatic ring and increase in average spin-spin distances. EPR observed D and E value from HBC dianions are so small: 0.00145 and 0.00005, respectively. Since the molecular structure of coronene (14) has C3 symmetry and the observed E value is characteristic zero value. Only bis[9-(10-phenyl)anthryl]carbene 15 [3,39]was calculated by UB3LYP//4-31G because it is a relatively large molecule. The optimized bond angle at the carbene centre (h) was evaluated to be 179.7. The carbene was slightly bent and had small E value. The spin orbitals were much delocalized to each side of anthryl moieties. It is concluded that the weak carbene nature makes the carbene centre almost linear geometry. More detailed discussions for triplet carbene, vinylmethylene and diphenylcarbene are given below. 4.2. Triplet carbene Bond angles at the carbene centre (h) have been experimentally estimated based on E/D values [39,40], which is also important for the delocalization nature of the spin orbitals at the carbene centre. The relationship of the ZFS values to h was investigated by the calculations using UHF, UBHandHLYP, UB3LYP, and ˚ . The reUBLYP. CH bond length was fixed to 1.10 A sults are displayed in Fig. 2. At the linear structure, D = 0.8 and E = 0. When h decreases from 180, D decreases while E increases. Therefore, E/D values decrease with the bond angle h (Fig. 3). The decrease in E/D values is that spin orbitals are inhomogeneous between YZ-plane and ZX-plane where carbene locates because of the presence of hydrogen. The differences in
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Fig. 4. Triplet carbene. The optimized molecular structure with the magnetic axes (A) and spin orbitals (B, C).
Fig. 2. D (solid symbols) and E (open symbols) values dependence of carbene as a function of the bond angle h of carbene centre. D values are UHF (d), UBHandHLYP (n), UB3LYP (¤) and UBLYP (m). E values are UHF (s), UBHandHLYP (h), UB3LYP (}) and UBLYP (n).
DFT (BLYP). Magnetic hard axis (Z) was parallel to the line connecting the two hydrogen atoms, while Magnetic easy plane (XY-plane) was vertical to the Z-axis. Therefore, two spin orbitals of carbenes are consisted of px and py orbitals, respectively (Fig. 4). The ZFS values D/cm1 and E/cm1 in the optimized structure were 0.8104 and 0.0576, respectively, as shown in Table 1. These values are slightly larger than experimental values: 0.7567 and 0.0461, respectively. Havlas et al. have reported D = 0.8043 cm1 and E = 0.0428 cm1 for carbene by CISDTQ/cc-pVDZ methods [11], which is comparable to our values. The SOC interaction causes a slightly smaller contributions than SSC by D 0 = D + 0.0023 cm1 and E 0 = E + 0.0001 cm1 [11,20]. 4.3. Isomers of vinylmethylene
Fig. 3. E/D values dependences of carbene as a function of bond angle h of carbene centre calculated by UHF (s), UBHandHLYP (h), UB3LYP (}) and UBLYP (n).
computational methods of HF, BHandHLYP, B3LYP, BLYP are small, indicating that the differences in the shape of spin orbitals are almost the same as the ones for the SSC interactions. The total spin orbitals are then antisymmetrized by the Slater determinants in the process of the SSC calculation. Consequently, the spin correlations exist for all the spin orbitals, even by the pure
There are two isomers of vinylmethylene: s-E and s-Z, and EPR signals for each of these isomers have been observed. It is interest that whether we can theoretically distinguish the difference in fine structures of ZFS for those isomers, or not. Geometry optimizations were performed for each isomer. The h s were 136.0 and 136.6 for s-E and s-Z, respectively. Fig. 5A displays that the spin orbital perpendicular to the methylene plane is delocalized over the p-orbitals for each s-E and s-Z isomers. The spin orbital populations on the C2 atom were almost zero at the node of the spin orbitals. On the contrary, the other spin orbitals (r type spin orbital) parallel to methylene plane localized at carbene centre (see Fig. 5C). The calculated D value of s-Z was larger than that of s-E, while E values of s-Z were smaller than that of s-E (Table 1). These results were consistent with experimental ones. Magnetic axis and spin orbitals are shown in Fig. 5A. As is already pointed by R.S. Hutton et al. [40], spin orbitals spread into Z-axis of the s-E isomer and the corresponding spin orbitals spread over XZ direction of the s-Z isomers. These spin orbitals clearly shows that the difference in delocalization direction affects the D value. The UB3LYP/6-31G calculated the s-E isomer to be more stable than s-Z by 0.207 kcal mol1. The signal rates of s-Z/s-E in EPR are
M. Shoji et al. / Polyhedron 24 (2005) 2708–2715
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Fig. 5. Vinylcarbene of s-E (1) and s-Z (2) isomers. Each optimized molecular structures with their magnetic axes (A) and two spin orbitals (B, C).
0.05–0.65 depending on temperature and solvents. This experimental result agrees with our calculation result, showing s-E is thermodynamically more stable. 4.4. Phenylcarbene The ground state structure of phenylcarbene is investigated by ZFSs. The molecular structure of phenylcarbene is considered to be either: the phenyl group is parallel or vertical to carbene C–H plane (denoted by q0 and q90, respectively). Each of these structures wass
theoretically optimized. It was revealed that q0 was more stable than q90 by 4.335 kcal mol1. The stable optimized structure q0 was in agreement with other ab initio calculation results by Geise et al. [41]. For isomers q0 and q90, (D, E) = (0.5193,0.0346) cm1 and (0.6005, 0.0308) cm1, respectively. The D for q0 was closer to the experimental value |D| = 0.5150 cm1, while the experimental value |E| = 0.024 cm1 was smaller than the calculated results. These spin orbitals for each isomer resembled each other and ZFS values were relatively equal (Fig. 6).
Fig. 6. Phenylcarbene of q90 (1) and q0 (2). Each optimized molecular structures with their magnetic axes (A) and two spin orbitals (B, C).
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Fig. 7. Structures of diphenylcarbene: a1 (1), a2 (2) and b (3). Each optimized molecular structures with their magnetic axes (A) and two spin orbitals (B, C).
4.5. Diphenylcarbene The molecular structure of diphenylcarbene is classical problem in EPR studies [5,17,18,31,43]. There are three conceivable structures. The first one (a1) is that phenylcarbenes are parallel each other and carbene centre is liner, i.e., h = 0 (Fig. 7-2). The second one (a2) is that a1 is bent at carbene centre in the phenyl ring plane so as the whole structure is within one plane (Fig. 7-1). The third one (b) is that the phenyl ring is vertical each other and carbene centre is bent (Fig. 7-3). The experimental ZFS values proposed that ground state is a2. (The most stable structure seems to b, because two spin orbitals at carbene centre can spread over each phenyl rings) We then decide the structure theoretically, especially using the calculated ZFS values. In optimizing the forth geometry a1 0 , which is bent at carbene centre in the vertical direction, a1 0 fell into the geometry a1 and was revealed to be unstable. The angle h optimized to be 147.0 and 149.0 for a2 and b, respectively. The most stable structure of these three structures was decided to be a2 by UB3LYP//6-31G. Energy gap between a2 and a1 was 85.58 kcal mol1, and that between a2 and b were 81.28 kcal mol 1. In a1 and a2, one spin orbital vertical to the phenyl ring was delocalized onto p-orbitals over both phenyl rings and the other spin orbital was localized at the carbene centre. The carbene centre was in line and E 0 at the structure a1. ZFS values were as (D, E) = (0.3740, 0.00185) cm1 and (0.3774, 0.01657) cm1 for a1 and a2, respectively. On the other hand, the spin orbital at the carbene centre was delocalized over the next phenyl rings each other at the structure b. And the D value is rather smaller because average distances between spins are relatively
longer than a1 and a2. ZFS values for b were evaluated to be (D, E) = (0.3569, 0.01254) cm1, while Experimental ZFS values are |D| = 0.4055 cm1 and |E| = 0.0194 cm1 [39]. Spin orbitals and magnetic axis are shown in Fig. 7. The most stable structure of phenylcarbene was estimated as a2, which reproduces experimental values. It is proved that molecular structure can be determined from the ab initio calculation of ZFS values.
5. Conclusion The results of this approach were in good agreement with experimental values at moderate computational cost. By using this method, theoretical applications to organic biradicals are further examined for carbene structures, isomers of vinylcarbenes and diphenylcarbenes. We showed this calculation procedure was accurate enough for these the ZFS calculations and effective for the theoretical investigations for organic radicals. This means that high spin states in organic radicals are close to that of the single particle picture. It was also found that the SSC between spin orbitals are the dominant factor for the ZFS and the contributions from the SOC to the ZFS will be vanishingly small. This method has been quit valid in studies of organic ferromagnets.
Acknowledgement This work has been supported by the 21 Century COE program entitled ÔThe Creation of Integrated EcoChemistryÕ in Osaka University and Grants-in-Aid for
M. Shoji et al. / Polyhedron 24 (2005) 2708–2715
Scientific Research on Priority Areas (Nos. 16750049 and 14204061) from Ministry of Education, Culture, Sports, Science and Technology, Japan.
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