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Theoretical study of magnetic interaction in pyrazole-bridged dinuclear Cu(II) complex
Polyhedron xxx (2017) xxx–xxx
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Theoretical study of magnetic interaction in pyrazole-bridged dinuclear Cu(II) complex Koji Miyagi a, Yasutaka Kitagawa a,b,⇑, Mizuki Asaoka a, Rena Teramoto a, Yoshiki Natori a, Toru Saito c, Masayoshi Nakano a,b,⇑ a b c
Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan Department of Biomedical Information Sciences, Graduate School of Information Sciences, Hiroshima City University, Asa-Minami-Ku, Hiroshima 731-3194, Japan
a r t i c l e
i n f o
Article history: Received 31 December 2016 Accepted 14 March 2017 Available online xxxx Keywords: Pyrazole-bridged dinuclear Cu(II) complex Effective exchange integral (J) values Density functional theory (DFT) Approximate spin projection (AP) method Orbital complementarity
a b s t r a c t A difference in a magnetic behavior of two different types of pyrazole-bridged dinuclear Cu(II) complexes are explained by a concept of the orbital complementarity and the density functional theory (DFT) calculations. The DFT calculations reproduce the experimental J values, and the estimated orbital overlap between Cu(II) ions are quite small, indicating that the spins are localized on the Cu(II) ions. The quasi-degenerate frontier orbital energies indicate that the ferromagnetic interaction of the acetate complex can be explained by the orbital complementarity. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The magnetic interactions between metal ions in the polynuclear metal complexes are usually characterized by metal and ligand species [1]. It is therefore possible to change the magnetic properties of the metal complexes by substituting the metal and ligand species. In other words, it is possible to design the magnetic property of the complex according to a guiding principle if one finds a relationship between chemical species, molecular structures, electronic structures and magnetic properties. Many experimental studies have been devoted to find the relationship [2,3]. For example, it has been known that some heterogeneous bridging ligands between two Cu(II) ions make different contributions to the magnetic exchange interactions [4]. In the late 1980s, Okawa et al. reported that dinuclear pyrazoleCu(II) complexes composed of azide or acetate bridging-ligands shown in Fig. 1 exhibit ferromagnetic (FM) and anti-ferromagnetic (AFM) behavior, respectively [4]. Here, those azide- and acetatebridged pyrazole-Cu(II) complexes are referred to as complex 1 and 2, respectively. Okawa and co-workers explained this phenomenon by an orbital complementarity [4,5] that originates in ⇑ Corresponding authors at: Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 5608531, Japan. E-mail addresses: [email protected] (Y. Kitagawa), mnaka@ cheng.es.osaka-u.ac.jp (M. Nakano).
the orbital interaction between Cu(II) and bridging-ligands. Although the concept of the orbital complementarity, which is summarized below, explains the magnetic behavior of the dinuclear pyrazole-Cu(II) complexes well, it has still been a working hypothesis. In the last two decades, on the other hand, there has been a drastic progress in theoretical calculations of the effective exchange integrals (J) especially for the real metal complexes [6,7]. It is found that the J values of huge metal complexes calculated by the brokensymmetry (BS) density functional theory (DFT) method without any structural simplifications well reproduce the experimental results, and that the obtained information helps us to understand their mechanisms in terms of the molecular orbital interaction. In this study, therefore, we examine a relationship between the orbital complementarity and the magnetic exchange interactions of those dinuclear Cu(II) complexes by BS-DFT calculations. At first, the molecular structures of those complexes are optimized, and then the electronic structures and magnetic interactions are discussed in detail to verify the above hypothesis.
2. Theoretical background 2.1. Orbital complementarity The concept of the orbital complementarity is briefly explained by using the illustration in Fig. 2 [4]. At first, two electrons (spins)
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Fig. 1. Dinuclear Cu(II) complexes bridged by (A) azide (complex 1) and (B) acetate (complex 2) ligands, respectively.
on each Cu(II) ions form quasi-degenerate symmetric bonding (us) and anti-symmetric anti-bonding (ua) orbitals as illustrated in Fig. 2(A). And the bonding and anti-bonding Cu(II) dimer orbitals interact with the bridging-ligand orbitals. In the case of complex 1, both pyrazole (upz) and azide (uaz) orbitals can interact with ua (Fig. 2(B)), and the out-of-phase orbital (u⁄a-(pz+az)) becomes unstable, while the bonding-orbital us does not interact with ligand orbitals. As a result, two spins belong to the lower us rather than u⁄a(pz+az) and thus the AFM state is predicted to become the ground state. On the other hand, in the case of complex 2, us and ua interact with pyrazole (upz) and acetate (uac), respectively. Consequently, formed two out-of-phase orbitals, i.e., u⁄a-pz and u⁄a-ac, tend to be quasi-degenerate. Therefore, the FM state is predicted to become the ground state of complex 2. 2.2. Computational details For the complexes 1 and 2, we employ BS-B3LYP/6-31G⁄ calculations to obtain their molecular structures because X-ray structural data have not been reported. The molecular structures are optimized for both the ferromagnetic (FM) and anti-ferromagnetic (AFM) states, and it is confirmed that the optimized geometry does not have any imaginary frequencies. Because the BS method often suffers from the spin contamination error, the coordination of Cu (II) ions is optimized by the approximate spin-projection (AP) optimization method to eliminate the error [8]. In this study, we used a numerical sampling method for the first-order differentiation of hS2iAFM values that are originally proposed in our previous paper [8]. Therefore, in this paper, the coordination of Cu(II) ions is partly re-optimized by the AP optimization method, starting from the BS (AFM) structure under the assumption that the spin contamination error is small in ligands. As mentioned in the calculated results, the difference in structures between the AFM and FM structures is small. As a consequence, therefore, it is considered that the assumption is suitable for these systems. To evaluate the magnetic interactions of Cu(II) complexes, we use effective exchange integrals (J) calculated by Yamaguchi equation,
J¼
EAFM EFM hS2 iFM hS2 iAFM
;
ð1Þ
where EX and hS2iX denote total nergies and hS2i values of spin state X (X = FM and AFM states). An overlap between two Cu(II) ions is also estimated by a natural orbital (NO) analysis in order to interpret the calculated J values [9]. The overlap Ti between spin-polarized a and b orbital-pair i can be calculated by its occupation number ni of the AFM state as
T i ¼ ni 1
ð2Þ
All DFT calculations were performed using Gaussian09 [10], and the AP optimization method is performed by an own-coding program [8].
3. Results and discussion The Cu(II)-Cu(II) distances in each optimized geometry are summarized in Table 1. The optimized Cartesian coordinate is also summarized in Tables S1 and S2 in Supporting Information. The complex 1 shows a slightly larger difference in Cu(II)-Cu(II) distance between the AFM(BS) and the FM states than that of complex 2, suggesting an existence of spin contamination error in the geometry in complex 1. In order to elucidate the error, we also optimized the geometry by the AP method. It is found that the Cu(II)-Cu(II) distance becomes slightly shorter by eliminating the error. The optimized Cu(II)-Cu(II) distances are comparable to similar pyrazole-bridged complexes such as pyrazole-azide complex (3) (4.147 Å) [11] and pyrazole-carbonate complex (4) (4.089 Å) [12] although they have different lateral ligands as illustrated in Fig. S1 in Supporting Information. Those distances that are much longer than the usual Cu(II)-Cu(II) direct bond (c.a. 2.6 Å [13]) suggest that the magnetic exchange interaction between Cu(II) ions is mainly mediated by bridging-ligands [14]. Next, J values of each complex are calculated at the optimized structures (see Table 2). The complex 1 shows a negative J value, indicating that the ground state is the AFM state. The result is semi-quantitatively reproduces the experimental value, however it slightly overshoots the experimental one. The overestimation (c.a. 17%) is considered to originate in B3LYP functional set as already reported [15]. In order to elucidate the effect of the functional set, we also estimate J values with BHandHLYP functional set, which consists of a larger amount of the Hartree–Fock (HF) exchange term (50%). In fact, BHandHLYP shows smaller absolute values even at the same structures, clearly indicating the importance of the HF ratio for the estimation of the J value. On the other hand, a calculated positive J value of the complex 2, which is con-
Table 1 Optimized Cu(II)-Cu(II) distance [Å] for complexes 1 and 2. Complex
Spin state for geometry optimization AFMa
1 2 a
FM
BS
AP
4.350 4.153
4.346 4.153
Both results of the BS and AP methods are summarized.
Please cite this article in press as: K. Miyagi et al., Polyhedron (2017), http://dx.doi.org/10.1016/j.poly.2017.03.017
4.366 4.155
3
K. Miyagi et al. / Polyhedron xxx (2017) xxx–xxx
Fig. 2. Illustration for the concept of the orbital complementarity [4]. (A) Orbital interaction diagram of complex 1. (B) Orbital interaction between anti-bonding Cu(II) dimer (ua) and bridging-ligands (upz and uaz) of complex 1. (C) Orbital interaction diagram of complex 2. (D) Orbital interaction between Cu(II) dimer (ua) and pyrazole ligand (upz) orbital of complex 2. (E) Orbital interaction between Cu(II) dimer (us) and acetate ligand (uac) of complex 2.
Table 2 Calculated J values [cm1] at the optimized structures at B3LYP/6-31G* level of theory. Complex 1
2
a b c
Spin state for geometry optimization AFM AFM FM AFM AFM FM
c
(BS) (AP)c c
(BS) (AP)c
Calc.a 436 (271) 439 (274) 364 (238) 13.5 (10.6) 13.6 (10.5) 23.0 (10.9)
BHandHLYP/6-31G* values are written in parentheses. Experimental values in Ref. [4] Both J values at the BS and AP structures are summarized.
Table 3 Calculated occupation numbers (n) and overlap (T) between Cu(II) ions estimated by Eq. (2).
Exptl.b
Complex
Spin state for geometry optimization
n
T
371
1
AFM (BS, AP) FM AFM (BS, AP) FM
1.188 1.183 1.002 1.002
0.188 0.183 0.002 0.002
2 >8.9
sistent with the experimental result shows that the ground state is the FM state. In order to elucidate the mechanism of the FM and AFM behavior of those complexes, the natural orbital analysis is performed.
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The NOs of the magnetic orbitals in the AFM state are shown in Fig. S2 in Supporting Information. The occupation numbers of the highest occupied natural orbitals (HONOs) and the overlap between Cu(II) ions calculated by Eq. (2) are summarized in Table 3. Both complexes show small overlap (T) values, indicating that spins of Cu(II) ions are almost localized. Due to a negligible difference in geometry, the BS and AP methods show almost the same occupation numbers. The larger overlap of complex 1 than that of complex 2 indicates the stronger anti-ferromagnetic interaction in complex 1 in spite of the longer Cu(II)-Cu(II) distance. However, the difference in overlap values between complexes 1 and 2 (ca. 0.18) seems to be too small to explain the drastic change, i.e., the AFM and FM states, so that we must consider another possibility for the explanation; the orbital complementarity. Finally, we consider the orbital complementarity. In order to explain the magnetism of complexes 1 and 2 by the orbital complementarity, the orbital energy differences between two orbitals must be estimated as illustrated in Fig. 2. Here we define the differences in orbital energies, DE(complex 1) and DE(complex 2) as follows,
DEðcomplex 1Þ ¼ Eð/aðpzþazÞ Þ Eð/s Þ
ð3Þ
DEðcomplex 2Þ ¼ Eð/apz Þ Eð/sac Þ
ð4Þ
If the DE is negligibly small, two magnetic orbitals shown in Fig. 2 are quasi-degenerated. Calculated frontier orbitals and their energies used here are shown in Fig. S3 in Supporting Information. The estimated DE(complex 1) and DE(complex 2) values are 1.33 and 0.60 eV, respectively, showing that the energy gap (DE) of complex 1 is twice as large as that of complex 2. Therefore, it is actually feasible that one electron moves from u⁄s-ac to u⁄a-pz due to a strong on-site Coulomb interaction [16] in complex 2, and then those two spins are aligned to be mutually parallel by the exchange interaction. As a result, the ground state of complex 2 is expected to be ferromagnetic. 4. Conclusion In this study, the difference in the magnetic behavior of the dinuclear pyrazole-Cu(II) complexes are explained by the concept of the orbital complementarity and DFT calculations. To the best of our knowledge, this is the first computational verification for the orbital complementarity in regard to the magnetism of the dinuclear Cu(II) complexes. The orbital complementarity can explain the spin-alignment of the complexes in a simple manner. In this sense, it is a useful idea for a description of the magnetic interactions as well as for a construction of the molecular design guidelines by combining with electronic structure calculations. Acknowledgements This work has been supported by Grant-in-Aid for Scientific Research (KAKENHI) (C) (No. JP26410093, JP15KT0143) from Japan Society for the Promotion of Science (JSPS). It is also partly supported by Grant-in-Aid for Scientific Research (A) (No.
JP25248007) from Japan Society for the Promotion of Science (JSPS), a Grant-in-Aid for Scientific Research on Innovative Areas ‘‘Stimuli-Responsive Chemical Species” (No. JP24109002), ‘‘p-System Figuration” (JP15H00999), ‘‘Photosynergetics” (No. JP26107004), MEXT, Japan. This is also partly supported by King Khalid University through a grant RCAMS/KKU/001-16 under the Research Center for Advanced Materials Science at King Khalid University, Kingdom of Saudi Arabia. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.poly.2017.03.017. References [1] H. Nishihara, Chem. Lett. 43 (2014) 388–395. [2] F.A. Cotton, R.A. Walton, Multiple Bonds between Metal Atoms, Clarendon Press, Oxford, 1993, and references therein. [3] M.H. Chisholm, Acc. Chem. Res. 33 (2000) 53. [4] T. Kamiusuki, H. Okawa, E. Kitaura, M. Koikawa, N. Matsumoto, S. Kida, H. Oshio, J. Chem. Soc., Dalton Trans. (1989) 2077. [5] Y. Nishida, S. Kida, Inorg. Chem. 27 (1988) 447. [6] (a) P. Kiehl, M.-M. Rohmer, M. Bénard, Inorg. Chem. 43 (2004) 3151; (b) M. Bénard, J.F. Berry, F.A. Cotton, C. Gaudin, X. López, C.A. Murillo, M.-M. Rohmer, Inorg. Chem. 45 (2006) 3932; (c) X. López, M. Bénard, M.-M. Rohmer, Inorg. Chem. 46 (2007) 5; (d) N.A.G. Banderia, Z. Tabookht, C. de Graaf, X. López, Compt. Rend. Chim. 15 (2012) 163; (e) N. Gallego-Planas, A. Martín-Rodríguez, E. Ruiz, Dalton Trans. 45 (2016) 18867. [7] (a) Y. Kitagawa, M. Shoji, K. Koizumi, T. Kawakami, M. Okumura, K. Yamaguchi, Polyhedron 24 (2005) 2751; (b) Y. Kitagawa, T. Matsui, Y. Nakanishi, Y. Shigeta, T. Kawakami, M. Okumura, K. Yamaguchi, Dalton Trans. 42 (2013) 16200. [8] Y. Kitagawa, T. Saito, M. Ito, M. Shoji, K. Koizumi, S. Yamanaka, T. Kawakami, M. Okumura, K. Yamaguchi, Chem. Phys. Lett. 442 (2007) 445–450. [9] K. Yamaguchi, F. Jensen, A. Dorigo, K.N. Houk, Chem. Phys. Lett. 149 (1988) 537. [10] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G.A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H.P. Hratchian, A.F. Izmaylov, J. Bloino, G. Zheng, J.L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J.A. Montgomery Jr., J.E. Peralta, F. Ogliaro, M. Bearpark, J.J. Heyd, E. Brothers, K.N. Kudin, V.N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J.C. Burant, S.S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J.M. Millam, M. Klene, J.E. Knox, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, R.L. Martin, K. Morokuma, V.G. Zakrzewski, G.A. Voth, P. Salvador, J.J. Dannenberg, S. Dapprich, A.D. Daniels, Ö. Farkas, J.B. Foresman, J.V. Ortiz, J. Cioslowski, D.J. Fox, Gaussian 09, Revision C.01, Gaussian Inc., Wallingford, CT, 2009. [11] L. Behle, M. Neuberger, M. Zehndar, T.A. Kaden, Helv. Chim. Acta 78 (1995) 693. [12] J. Ackermann, F. Meyer, E. Kaifer, H. Pritzkow, Chem. Eur. J. 8 (2002) 247. [13] J.E. Weder, T.W. Hambley, B.J. Kennedy, P.A. Lay, D. MacLachlan, R. Bramley, C. D. Delfs, K.S. Murray, B. Moubaraki, B. Warwick, J.R. Biffin, H.L. Regtop, Inorg. Chem. 38 (1999) 1736. [14] Y. Kitagawa, T. Kawakami, Y. Yoshioka, K. Yamaguchi, Polyhedron 20 (2001) 1189. [15] (a) T. Soda, Y. Kitagawa, T. Onishi, Y. Takano, Y. Shigeta, H. Nagao, Y. Yoshioka, K. Yamaguchi, Chem. Phys. Lett. 319 (2000) 223; (b) Y. Kitagawa, T. Soda, Y. Shigeta, S. Yamanaka, Y. Yoshioka, K. Yamaguchi, Int. J. Quant. Chem. 84 (2001) 592; (c) Y. Kitagawa, T. Kawakami, K. Yamaguchi, Mol. Phys. 2002 (1829) 100. [16] P. Rivero, C. Loschen, I.-P. Moreira, F. Illas, J. Comput. Chem. 30 (2009) 2316.
Please cite this article in press as: K. Miyagi et al., Polyhedron (2017), http://dx.doi.org/10.1016/j.poly.2017.03.017
Contents lists available at ScienceDirect
Polyhedron journal homepage: www.elsevier.com/locate/poly
Theoretical study of magnetic interaction in pyrazole-bridged dinuclear Cu(II) complex Koji Miyagi a, Yasutaka Kitagawa a,b,⇑, Mizuki Asaoka a, Rena Teramoto a, Yoshiki Natori a, Toru Saito c, Masayoshi Nakano a,b,⇑ a b c
Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan Department of Biomedical Information Sciences, Graduate School of Information Sciences, Hiroshima City University, Asa-Minami-Ku, Hiroshima 731-3194, Japan
a r t i c l e
i n f o
Article history: Received 31 December 2016 Accepted 14 March 2017 Available online xxxx Keywords: Pyrazole-bridged dinuclear Cu(II) complex Effective exchange integral (J) values Density functional theory (DFT) Approximate spin projection (AP) method Orbital complementarity
a b s t r a c t A difference in a magnetic behavior of two different types of pyrazole-bridged dinuclear Cu(II) complexes are explained by a concept of the orbital complementarity and the density functional theory (DFT) calculations. The DFT calculations reproduce the experimental J values, and the estimated orbital overlap between Cu(II) ions are quite small, indicating that the spins are localized on the Cu(II) ions. The quasi-degenerate frontier orbital energies indicate that the ferromagnetic interaction of the acetate complex can be explained by the orbital complementarity. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The magnetic interactions between metal ions in the polynuclear metal complexes are usually characterized by metal and ligand species [1]. It is therefore possible to change the magnetic properties of the metal complexes by substituting the metal and ligand species. In other words, it is possible to design the magnetic property of the complex according to a guiding principle if one finds a relationship between chemical species, molecular structures, electronic structures and magnetic properties. Many experimental studies have been devoted to find the relationship [2,3]. For example, it has been known that some heterogeneous bridging ligands between two Cu(II) ions make different contributions to the magnetic exchange interactions [4]. In the late 1980s, Okawa et al. reported that dinuclear pyrazoleCu(II) complexes composed of azide or acetate bridging-ligands shown in Fig. 1 exhibit ferromagnetic (FM) and anti-ferromagnetic (AFM) behavior, respectively [4]. Here, those azide- and acetatebridged pyrazole-Cu(II) complexes are referred to as complex 1 and 2, respectively. Okawa and co-workers explained this phenomenon by an orbital complementarity [4,5] that originates in ⇑ Corresponding authors at: Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 5608531, Japan. E-mail addresses: [email protected] (Y. Kitagawa), mnaka@ cheng.es.osaka-u.ac.jp (M. Nakano).
the orbital interaction between Cu(II) and bridging-ligands. Although the concept of the orbital complementarity, which is summarized below, explains the magnetic behavior of the dinuclear pyrazole-Cu(II) complexes well, it has still been a working hypothesis. In the last two decades, on the other hand, there has been a drastic progress in theoretical calculations of the effective exchange integrals (J) especially for the real metal complexes [6,7]. It is found that the J values of huge metal complexes calculated by the brokensymmetry (BS) density functional theory (DFT) method without any structural simplifications well reproduce the experimental results, and that the obtained information helps us to understand their mechanisms in terms of the molecular orbital interaction. In this study, therefore, we examine a relationship between the orbital complementarity and the magnetic exchange interactions of those dinuclear Cu(II) complexes by BS-DFT calculations. At first, the molecular structures of those complexes are optimized, and then the electronic structures and magnetic interactions are discussed in detail to verify the above hypothesis.
2. Theoretical background 2.1. Orbital complementarity The concept of the orbital complementarity is briefly explained by using the illustration in Fig. 2 [4]. At first, two electrons (spins)
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Fig. 1. Dinuclear Cu(II) complexes bridged by (A) azide (complex 1) and (B) acetate (complex 2) ligands, respectively.
on each Cu(II) ions form quasi-degenerate symmetric bonding (us) and anti-symmetric anti-bonding (ua) orbitals as illustrated in Fig. 2(A). And the bonding and anti-bonding Cu(II) dimer orbitals interact with the bridging-ligand orbitals. In the case of complex 1, both pyrazole (upz) and azide (uaz) orbitals can interact with ua (Fig. 2(B)), and the out-of-phase orbital (u⁄a-(pz+az)) becomes unstable, while the bonding-orbital us does not interact with ligand orbitals. As a result, two spins belong to the lower us rather than u⁄a(pz+az) and thus the AFM state is predicted to become the ground state. On the other hand, in the case of complex 2, us and ua interact with pyrazole (upz) and acetate (uac), respectively. Consequently, formed two out-of-phase orbitals, i.e., u⁄a-pz and u⁄a-ac, tend to be quasi-degenerate. Therefore, the FM state is predicted to become the ground state of complex 2. 2.2. Computational details For the complexes 1 and 2, we employ BS-B3LYP/6-31G⁄ calculations to obtain their molecular structures because X-ray structural data have not been reported. The molecular structures are optimized for both the ferromagnetic (FM) and anti-ferromagnetic (AFM) states, and it is confirmed that the optimized geometry does not have any imaginary frequencies. Because the BS method often suffers from the spin contamination error, the coordination of Cu (II) ions is optimized by the approximate spin-projection (AP) optimization method to eliminate the error [8]. In this study, we used a numerical sampling method for the first-order differentiation of hS2iAFM values that are originally proposed in our previous paper [8]. Therefore, in this paper, the coordination of Cu(II) ions is partly re-optimized by the AP optimization method, starting from the BS (AFM) structure under the assumption that the spin contamination error is small in ligands. As mentioned in the calculated results, the difference in structures between the AFM and FM structures is small. As a consequence, therefore, it is considered that the assumption is suitable for these systems. To evaluate the magnetic interactions of Cu(II) complexes, we use effective exchange integrals (J) calculated by Yamaguchi equation,
J¼
EAFM EFM hS2 iFM hS2 iAFM
;
ð1Þ
where EX and hS2iX denote total nergies and hS2i values of spin state X (X = FM and AFM states). An overlap between two Cu(II) ions is also estimated by a natural orbital (NO) analysis in order to interpret the calculated J values [9]. The overlap Ti between spin-polarized a and b orbital-pair i can be calculated by its occupation number ni of the AFM state as
T i ¼ ni 1
ð2Þ
All DFT calculations were performed using Gaussian09 [10], and the AP optimization method is performed by an own-coding program [8].
3. Results and discussion The Cu(II)-Cu(II) distances in each optimized geometry are summarized in Table 1. The optimized Cartesian coordinate is also summarized in Tables S1 and S2 in Supporting Information. The complex 1 shows a slightly larger difference in Cu(II)-Cu(II) distance between the AFM(BS) and the FM states than that of complex 2, suggesting an existence of spin contamination error in the geometry in complex 1. In order to elucidate the error, we also optimized the geometry by the AP method. It is found that the Cu(II)-Cu(II) distance becomes slightly shorter by eliminating the error. The optimized Cu(II)-Cu(II) distances are comparable to similar pyrazole-bridged complexes such as pyrazole-azide complex (3) (4.147 Å) [11] and pyrazole-carbonate complex (4) (4.089 Å) [12] although they have different lateral ligands as illustrated in Fig. S1 in Supporting Information. Those distances that are much longer than the usual Cu(II)-Cu(II) direct bond (c.a. 2.6 Å [13]) suggest that the magnetic exchange interaction between Cu(II) ions is mainly mediated by bridging-ligands [14]. Next, J values of each complex are calculated at the optimized structures (see Table 2). The complex 1 shows a negative J value, indicating that the ground state is the AFM state. The result is semi-quantitatively reproduces the experimental value, however it slightly overshoots the experimental one. The overestimation (c.a. 17%) is considered to originate in B3LYP functional set as already reported [15]. In order to elucidate the effect of the functional set, we also estimate J values with BHandHLYP functional set, which consists of a larger amount of the Hartree–Fock (HF) exchange term (50%). In fact, BHandHLYP shows smaller absolute values even at the same structures, clearly indicating the importance of the HF ratio for the estimation of the J value. On the other hand, a calculated positive J value of the complex 2, which is con-
Table 1 Optimized Cu(II)-Cu(II) distance [Å] for complexes 1 and 2. Complex
Spin state for geometry optimization AFMa
1 2 a
FM
BS
AP
4.350 4.153
4.346 4.153
Both results of the BS and AP methods are summarized.
Please cite this article in press as: K. Miyagi et al., Polyhedron (2017), http://dx.doi.org/10.1016/j.poly.2017.03.017
4.366 4.155
3
K. Miyagi et al. / Polyhedron xxx (2017) xxx–xxx
Fig. 2. Illustration for the concept of the orbital complementarity [4]. (A) Orbital interaction diagram of complex 1. (B) Orbital interaction between anti-bonding Cu(II) dimer (ua) and bridging-ligands (upz and uaz) of complex 1. (C) Orbital interaction diagram of complex 2. (D) Orbital interaction between Cu(II) dimer (ua) and pyrazole ligand (upz) orbital of complex 2. (E) Orbital interaction between Cu(II) dimer (us) and acetate ligand (uac) of complex 2.
Table 2 Calculated J values [cm1] at the optimized structures at B3LYP/6-31G* level of theory. Complex 1
2
a b c
Spin state for geometry optimization AFM AFM FM AFM AFM FM
c
(BS) (AP)c c
(BS) (AP)c
Calc.a 436 (271) 439 (274) 364 (238) 13.5 (10.6) 13.6 (10.5) 23.0 (10.9)
BHandHLYP/6-31G* values are written in parentheses. Experimental values in Ref. [4] Both J values at the BS and AP structures are summarized.
Table 3 Calculated occupation numbers (n) and overlap (T) between Cu(II) ions estimated by Eq. (2).
Exptl.b
Complex
Spin state for geometry optimization
n
T
371
1
AFM (BS, AP) FM AFM (BS, AP) FM
1.188 1.183 1.002 1.002
0.188 0.183 0.002 0.002
2 >8.9
sistent with the experimental result shows that the ground state is the FM state. In order to elucidate the mechanism of the FM and AFM behavior of those complexes, the natural orbital analysis is performed.
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K. Miyagi et al. / Polyhedron xxx (2017) xxx–xxx
The NOs of the magnetic orbitals in the AFM state are shown in Fig. S2 in Supporting Information. The occupation numbers of the highest occupied natural orbitals (HONOs) and the overlap between Cu(II) ions calculated by Eq. (2) are summarized in Table 3. Both complexes show small overlap (T) values, indicating that spins of Cu(II) ions are almost localized. Due to a negligible difference in geometry, the BS and AP methods show almost the same occupation numbers. The larger overlap of complex 1 than that of complex 2 indicates the stronger anti-ferromagnetic interaction in complex 1 in spite of the longer Cu(II)-Cu(II) distance. However, the difference in overlap values between complexes 1 and 2 (ca. 0.18) seems to be too small to explain the drastic change, i.e., the AFM and FM states, so that we must consider another possibility for the explanation; the orbital complementarity. Finally, we consider the orbital complementarity. In order to explain the magnetism of complexes 1 and 2 by the orbital complementarity, the orbital energy differences between two orbitals must be estimated as illustrated in Fig. 2. Here we define the differences in orbital energies, DE(complex 1) and DE(complex 2) as follows,
DEðcomplex 1Þ ¼ Eð/aðpzþazÞ Þ Eð/s Þ
ð3Þ
DEðcomplex 2Þ ¼ Eð/apz Þ Eð/sac Þ
ð4Þ
If the DE is negligibly small, two magnetic orbitals shown in Fig. 2 are quasi-degenerated. Calculated frontier orbitals and their energies used here are shown in Fig. S3 in Supporting Information. The estimated DE(complex 1) and DE(complex 2) values are 1.33 and 0.60 eV, respectively, showing that the energy gap (DE) of complex 1 is twice as large as that of complex 2. Therefore, it is actually feasible that one electron moves from u⁄s-ac to u⁄a-pz due to a strong on-site Coulomb interaction [16] in complex 2, and then those two spins are aligned to be mutually parallel by the exchange interaction. As a result, the ground state of complex 2 is expected to be ferromagnetic. 4. Conclusion In this study, the difference in the magnetic behavior of the dinuclear pyrazole-Cu(II) complexes are explained by the concept of the orbital complementarity and DFT calculations. To the best of our knowledge, this is the first computational verification for the orbital complementarity in regard to the magnetism of the dinuclear Cu(II) complexes. The orbital complementarity can explain the spin-alignment of the complexes in a simple manner. In this sense, it is a useful idea for a description of the magnetic interactions as well as for a construction of the molecular design guidelines by combining with electronic structure calculations. Acknowledgements This work has been supported by Grant-in-Aid for Scientific Research (KAKENHI) (C) (No. JP26410093, JP15KT0143) from Japan Society for the Promotion of Science (JSPS). It is also partly supported by Grant-in-Aid for Scientific Research (A) (No.
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