- Email: [email protected]
The density functional calculations on the structural stability, electronic properties, and static linear polarizability of the endohedral metallofullerene Ba@C74
Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115
Contents lists available at ScienceDirect
Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
The density functional calculations on the structural stability, electronic properties, and static linear polarizability of the endohedral metallofullerene Ba@C74 Chunmei Tang a, Shiyou Fu b, Kaiming Deng a,*, Yongbo Yuan a, Weishi Tan a, Decai Huang a, Xin Wang a a b
Department of Applied Physics, Nanjing University of Science and Technology, Xiaolinwei 200, Nanjing, Jiangsu 210094, PR China Harbin Institute of Technology at Weihai, Weihai, Shandong 264209, PR China
a r t i c l e
i n f o
Article history: Received 29 August 2007 Received in revised form 27 July 2008 Accepted 29 July 2008 Available online 7 August 2008 Keywords: Endohedral fullerene Ba@C74 Static linear polarizability Density functional theory
a b s t r a c t The generalized gradient approximation based on density functional theory is applied to study the structural stability, electronic properties, and static linear polarizability of Ba@C74. It is found that the most favorable endohedral site for a Ba atom is off-center under a [6, 6] double bond along the C2 axis on the rh plane, which is denoted as Ba@C74-2. This result can be explained by the electrostatic potential map of the C74 2 on the rh plane. Of particular interest is the static linear polarizability of Ba@C74-2. The calculated polarizability component azz of Ba@C74-2 687.3 Å3 is nine times that of C60 77.4 Å3, while axx and ayy are zero. This large anisotropy may be due to its lower C2v symmetry compared to C60 with Ih-symmetry, as well as the transference of about two valence electrons from the Ba atom to the carbon cage. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The C74 fullerene has only a unique isomer satisfying the isolated pentagon rule (IPR) [1] with D3h symmetry. It is interesting that C74 has been known as an unusual fullerene because it exists in the raw soot produced by arc discharge [1] but is insoluble in CS2 and toluene, the common organic solvents for other classical fullerenes such as C60. This unusual feature is attributed to its very small highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) energy gap (Eg), resulting in high reactivity [2]. However, mass analysis of the soot has shown the high-yield production of C74 [3]. Upon exohedral reduction [4] by means of electrochemical methods, the resulting dianion C74 2 has been found as a stable closed shell molecule. Along another direction, endohedral reduction by an incorporated divalent metal atom to form M2þ @C2 74 also leads to stable compounds [5]. For example, encapsulation of a Ca [6] atom or a Eu [7] atom inside D3h–C74 have already been reported. Experimentally, Shiga et al. [8] and Peters et al. [9] successfully synthesized the endohedral metallofullerene Ba@C74 using the radio frequency method. Haufe et al. [10] reported the isolation of Ba@C74 by means of multi-step High Performance Liquid Chromatography. Reich et al. [5] also reported the successful synthesis * Corresponding author. Tel./fax: +86 25 84315882. E-mail addresses: [email protected] (K. Deng), [email protected] (D. Huang). 0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.07.032
of [email protected] (OEP)2C6H6, implying the existence of the stable Ba@C74 molecule. Theoretically Slanina et al. calculated six isomers of Ba@C74 [11] and Yb@C74 [12] at DFT/B3LYP level, and found that both the endohedral Ba@C74 and Yb@C74 derived from the sole D3h–C74 cage as the major isomer from the Gibbs energy view, however, they did not discuss the most stable position of the encaged metal atom. Therefore, to find out the most stable position of a Ba atom in the D3h–C74 cage as well as to discuss the electronic structure of the ground state of Ba@C74 is desired. On the other hand, the static linear polarizability a represents one of the most important observations to the understanding of the electronic properties of molecule, since it is very sensitive to the delocalization of valence electrons as well as the structure and shape [13]. In the literature, there are two different ways to get the static linear polarizability. One is sum-over-states (SOS) [14] and the other is the finite-field (FF) approach [15,16]. SOS treatment seems to be out of reach for ab initio methods since they necessitate the calculations of many excited states. The FF method was developed by Kurtz et al. [15] and Williams [16] separately. For a considerable number of molecules, the FF approach has been proven to be acceptable [17]. It has been verified that the FF approach based on density functional theory can produce reliable static polarizability since even the local density approximation for exchange and correlation produces an accurate electron screening response to the application of an external field to the system [18]. There are two aims of this paper. One is to find out the most stable position of a Ba atom in Ba@C74 and to explore its electronic
112
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115
properties. The other is to calculate the static linear polarizability of Ba@C74. The paper is organized as follows. We describe our computational details in Section 2 and present our results and discussion in Section 3. Finally the conclusion is given in Section 4.
(BFGS) algorithm [26] with a convergence criterion of 103 a.u. on the displacement and 105 a.u. on the energy. The mean polarizability hai is calculated from the polarizability components as [13]
2. Computational details
hai ¼
The majority of the computations was performed with the DMOL3 package [19]. The density functional theory (DFT) approach, due to its significant character of high accuracy and low cost, has become one of the most popular calculation routines for large systems [20]. It has been found that the generalized gradient approximation (GGA) is more accurate than the local density approximation (LDA) in the calculations of the bond length and binding energy [21]. Therefore, the Becke-Lee-Yang-Parr (BLYP) correlation exchange functional at the generalized gradient approximation (GGA) [22] level, and DNP basis set are adopted in this paper. BLYP functional is the combination of the exchange functional exploited by Becke [23] and the correlation functional exploited by Lee, Yang and Parr [24]. The DNP basis sets, comparable to Gaussian 6-31G** sets, are the double numerical atomic orbitals augmented by polarization functions. The calculations are of all-electron type. Mulliken population analysis is made to obtain the effective charge and net spin populations on each atom. The electronic structure is obtained by solving the Kohn-Sham(KS) equations [25] self-consistently using the spin unrestricted scheme. Self-consistent field procedures are done with a convergence criterion of 106 a.u. on the energy and electron density. Geometry optimizations with only symmetry restriction and without any other constrained parameter (fixed atom, distance, angle etc) are performed using the Broyden-Fletcher-Goldfarb-Shanno
The static linear polarizability anisotropy Da is defined as [13,27,28]
Da ¼
1 ½axx þ ayy þ azz 3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ðaxx ayy Þ2 þ ðaxx azz Þ2 þ ðayy azz Þ2 2
ð1Þ
ð2Þ
The FF method is used in our calculations. Electronic fields with the strength of 0.005 a.u., similar to previous calculations, [28] are added along the three different Cartesian axes, respectively. 3. Results and discussion 3.1. Structural stability Recently, Kovalenko et al. [29] studied the geometric structure of D3h–C74 and predicted that the possible stable position for an atom inside the cage should be on the rh plane. Thus, in order to find out the most stable position of a Ba atom inside the C74, the Ba atom was placed at four different possible sites, respectively, and most of them are on the rh plane. They are: (1) off-center along the C3 principal axis; (2) off-center under a [6, 6] double bond along the C2 axis on the rh plane; (3) off-center under the hexagon center along the same C2 axis of (2) on the rh plane; and (4) in the cage center. Then these four structures were optimized, yielding Ba@C74-1, Ba@C74-2, Ba@C74-3, and Ba@C74-4, respectively, as
Fig. 1. The geometric structures of the four Ba@C74 (big ball stand for Ba).
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115
shown in Fig. 1. In order to check the reliably of the results, we reoptimized the four configurations of the Ba@C74 at the B3LYP/ Lanl2dz level using the Gaussian 03 package. The calculated minimal vibrational frequencies for Ba@C74-1, Ba@C74-2, and Ba@C74-3 are 34.34, 68.42, and 50.42 cm1, so they are do the local minimum of the related energy surface. However, Ba@C74-4 has three imagery frequencies, i.e., 41.60, 41.59, and 33.52 cm1, therefore, the center of the C74 cage is not the stable position for the Ba atom. Table 1 lists the energy of the HOMO, the energy of the LUMO, energy gap (Eg) between HOMO and LUMO, binding energy (BE), the magnetic moment of the Ba atom (MMA), and the magnetic moment of the molecule (MMM). Commonly, the thermodynamic stability of a fullerene is determined by its BE [30], which is defined as the difference between the energy sum of all free atoms constituting the molecule and the total energy of the molecule. Obviously, similar to the cases of a Ca [6] and a Eu [7] encaged inside C74, the structure with the Ba atom off-center under a [6, 6] double bond along the C2 axis on the rh plane (Ba@C74-2) has the largest BE of 528.09 eV, therefore, Ba@C74-2 is considered as the most stable isomer. This result can be explained as follows. The Ba atom with the largest BE of Ba@C74 should fall into the bottom of the electrostatic potential valley [31]. Fig. 2 shows the electrostatic potential map of the C74 2 on the rh plane, where the data give the magnitude of the electrostatic potential at the minimum points with the unit of kcal/mol. It is found that the electrostatic potentials obviously are negative and suitable for accommodating cationic species. To our expectation, there are 3-fold degenerate minima at the off-center sites under the [6, 6] double bond along the C2 axis on the rh plane. This result also be confirmed by the Car–Parrinello molecular dynamics calculations [32] for the movement of the Ba atom into the C74 cage. Furthermore, it is clear from Table 1, that the magnetic moment of the Ba atom inside the cages and those of the four Ba@C74 molecules are all zero, indicating their
Table 1 The HOMO, LUMO, Eg, BE with the unit in eV, and the magnetic moment of the Ba atom (MMA), the magnetic moment of the molecule (MMM) with the unit in lB for the C74, four Ba@C74 isomers and C74 2 Molecule
HOMO
LUMO
Eg
BE
MMA
MMM
C74 Ba@C74-1 Ba@C74-2 Ba@C74-3 Ba@C74-4 C74 2
5.09 5.07 5.17 5.19 5.18 0.70
5.00 4.45 4.61 4.55 4.47 1.43
0.09 0.62 0.56 0.64 0.71 0.73
524.96 528.01 528.09 527.98 527.46 528.09
– 0.00 0.00 0.00 0.00 –
– 0.00 0.00 0.00 0.00 0.00
Fig. 2. The electrostatic potential map of C74 2 on the rh plane.
113
Fig. 3. The orbital levels of C74, Ba@C74-2, and C74 2 .
closed-shell electronic structures. Meanwhile, with the fact of about 2e effective charge of the Ba atom from the Mulliken population analysis, we inferred that the electronic structure of Ba@C74-2 can be formally described as Ba2þ @C2 74 . Moreover, the computed nearest Ba–C distance 2.98 Åof Ba@C74-2 is longer than the sum of the radius of a Ba atom and a C atom of 2.75 Å, therefore, the Ba–C bond should be ionic. 3.2. Electronic properties of Ba@C74 Fig. 3 presents the orbital levels of Ba@C74-2 and C74 2 , together with those of C74 for comparison. The molecular orbitals whose energies agree with each other within a difference of 0.05 eV are regarded as degenerate, and the length of the horizontal bar shows the orbital degeneracy. Here, the level with one electron occupation is considered as one degeneracy because of the spin unrestricted scheme we used. The broken lines stand for the unoccupied orbital and the solid lines stand for the occupied orbital. The Fermi level (Ef) is defined as the halfway between HOMO and LUMO. We first compared the Egs of all the considered isomers, because they are often correlated with the kinetic stabilities [33,34]. The Eg calculated for the hollow cage C74 is 0.09 eV, much smaller than those of other fullerenes [35]. The Eg of Ba@C74-2, 0.56 eV, is much larger than that of the hollow C74 cage, indicating that incorporating a Ba atom into the C74 cage can strikingly enhance the kinetic stability of the cage. Furthermore, it is noteworthy that C74 2 , which is considered to be formed by the way that two electrons directly occupy on the LUMO of C74, [36] has a larger 0.73 eV energy gap, so the Eg of the C74 cage can also be obviously widened by the exohedral reduction. The contributions from the Ba atom states to the energy levels near the Fermi energy are marked in terms of percentage of population beside the levels. Clearly, the Ba has no contribution to LUMO and HOMO, revealing that both HOMO and LUMO of Ba@C74-2 are carbon like. Therefore, both the electronic detachment and electronic attachment, similar to the case of Si@C74 [35], should occur at the carbon sites of the cage. Fig. 4 shows the total density of states (TDOS) of Ba@C74-2 and C74, and the partial density of states (PDOS) of the Ba in Ba@C74-2. The DOS are obtained by a Lorentzian extension of the discrete energy levels, with weights being the orbital populations in the levels and a summation over them. The broadening width parameter is chosen to be 0.15 eV and the Ef is taken as zero. We explore the
114
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115 Table 3 Diagonal components and average values of a, and the static linear polarizability anisotropies Da for C60 and Ba@C74-2 with the unit in Å3
C60 Ba@C74-2
Fig. 4. The A-TDOS of Ba@C74-2, B-TDOS of C74, and C-PDOS of Ba.
mixture of the orbitals by analyzing TDOS of Ba@C74-2 and PDOS of the Ba in Ba@C74-2. It is obvious from the figure that there are almost no peaks in the PDOS of the Ba from 14 eV to Ef, showing the much weak mixture between the atomic orbitals of the Ba and those of the nearest neighboring carbon atoms of the C74. This result further confirms the ionic character of the Ba–C bond in Ba@C74-2. We also calculate the ionization potentials (VIP) and the vertical electron affinities (VEA) of Ba@C74-2 and C74. The VIP is the energy difference between the positively charged and neutral clusters. The VEA is evaluated by adding one electron to the neutral cluster in its equilibrium geometry and taking the difference between their total energies. The calculated VIP and VEA of the C74 are 6.55 and 3.53 eV, in agreement with the previously calculated value of 6.93 eV by Lu et al. [17] and the experimentally observed 3.28 ± 0.07 eV by Boltalina et al. [36], respectively. The computed VIP and VEA of Ba@C74-2 are 6.57 and 3.16 eV. We hope the further experiments will confirm our results. 3.3. Static linear polarizability of Ba@C74 In order to check the reliability of our method as well as to get the results of pure C60 cluster for comparison, we first calculated the mean static linear polarizability hai of C60. Our value and the other theoretical results are listed in Table 2. It can be found that our average value of a (77.4 Å3) is almost the same as most of other calculations with the same basis set, for example, Yang et al. [27] got the values of 77.6 and 76.1 Å3, using the same DNP basis sets but with the perdew-Burke-Ernzerhof (PBE) functional at the GGA level and the Vosko-Wilk-Nusair (VWN) functional at the LDA level, respectively. Weiss et al. [37], using the random phase approximation (RPA) and the basis set of 6-31G+sd, got a value of 78.8 Å3. It is worthy to note these results are in good agreement with the experimental value of 76.5 ± 8.0 Å3 obtained by Antoine et al. [38], using the molecular beam deflection method (MBD). Therefore our calculation method and the basis sets are both reliable. Table 2 The calculated and experimental average values for the static linear polarizability hai of C60 with the unit in Å3 Methods
BLYP/ DNP
PBE/DNP [27]
VWN/DNP [27]
PRA/6-31G+sd [37]
MBD [38]
hai
77.4
77.6
76.1
78.8
76.5 ± 8.0
axx
ayy
azz
hai
Da
77.4 0.0
77.4 0.0
77.4 687.3
77.4 229.1
0.0 687.3
The calculated results of a tensors, the average linear static polarizability hai, and the static linear polarizability anisotropy Da for Ba@C74-2 as well as their counterparts of C60 for comparison are presented in Table 3. Ba@C74-2 is oriented with its permanent dipole moment along the z axis as indicated in Fig. 1. For Ba@C74-2, the components of a in the x and y directions are zero while that in the z direction 687.3 Å3 is about nine times that for C60. Such large anisotropy of polarizability for Ba@C74-2 may be attribute to two reasons. One is the lower C2v symmetry structure of Ba@C74-2 compared with the Ih symmetry structure of C60. The other is that there are about two electron transference from the Ba atom to its neighboring carbons of the cage. Therefore, the anisotropy of polarizability is very sensitive to the structure or shape as well as the delocalization of the valence electrons [13]. 4. Conclusion The generalized gradient approximation based on density functional theory is applied to study the structural stability, electronic properties, and static linear polarizability of endohedral metallofullerene Ba@C74. It is found that among the four possible isomers of Ba@C74, the most favorable endohedral site for the Ba atom is off-center under a [6, 6] double bond along the C2 axis on the rh plane. This result can be explained by the analysis of the electrostatic potential map of the C74 2 on the rh plane. Inferred from its longer bond length as well as the transference of about two electrons from the Ba to the cage, the nearest Ba–C bond is mainly ionic. From the effective charges of about two electrons for the Ba atom in the cage, we infer that the electronic state of the molecule can be formally described as Ba2þ @C2 74 . Finally, the computed polarizability components, i.e., the azz of Ba@C74-2 687.3 Å3 is about nine times that of C60, while axx and ayy of it are both zero. This large anisotropy anisotropic result may be due to its lower C2v symmetric structure in contrast to C60 isotropic polarizability with Ih-symmetric structure as well as the transference of about two valence electrons from the Ba atom to the neighboring carbons of the cage. Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant No. 10174039), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK2002099 and BK2006204), the NJUST Young Scholar Research Fund (Grant No. 200705), and the Graduate Cultivation Foundation of Jiangsu Province (2006). We also thank Wuxi Supercomputing Center for computing resources. References [1] H.W. Kroto, Nature 329 (1987) 529. [2] H. Shinohara, H. Yamaguchi, N. Hayashi, H. Sato, M. Ohkohchi, Y. Saito, J. Phys. Chem. 97 (1993) 4259. [3] R. Hatakeyama, T. Hirata, H. Ishida, N. Sato, Appl. Phys. Lett. 73 (1998) 888. [4] A.A. Goryunkov, V.Y. Markov, I.N. Ioffe, R.D. Bolskar, M.D. Diener, I.V. Kuvychko, S.H. Strauss, O.V. Boltalina, Angew. Chem. Int. Ed. 43 (2004) 997. [5] A. Reich, M. Ranthöfer, H. Modrow, U. Wedig, M. Jansen, J. Am. Chem. Soc. 126 (2004) 14428. [6] T. Kodama, R. Fujii, Y. Miyake, S. Suzuki, H. Nishikawa, I. Ikemoto, K. Kikuchi, Y. Achiba, Chem. Phys. Lett. 399 (2004) 94.
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115 [7] H. Matsuoka, N. Ozawa, T. Kodama, H. Nishikawa, I. Ikemoto, K. Kikuchi, K. Furukawa, K. Sato, D. Shiomi, T. Takui, T. Kato, J. Phys. Chem. B 108 (2004) 13972. [8] K. Shiga, K. Ohno, Y. Kawazoe, Y. Maruyama, T. Hirata, R. Hatakeyama, N. Sato, Mater. Sci. Eng. A 290 (2002) 6. [9] G. Peters, M. Jansen, Angew. Chem. Int. Ed. Engl. 31 (1992) 223. [10] O. Haufe, A. Reich, C. Möschel, M.Z. Anorg, Allg. Chem. 627 (2001) 23. [11] Z. Slanina, S. Nagase, Chem. Phys. Lett. 422 (2006) 133. [12] Z. Slanina, F. Uhll´k, S. Nagase, J. Phys. Chem. A 110 (2006) 12860. [13] H.Y. Wang, X.B. Li, Y.J. Tang, H.P. Mao, Z.H. Zhu, Chin. J. Chem. Phys. 18 (2005) 50. [14] A.H. Göller, S. Erhardt, U.W. Grummt, J. Mol. Struct. (THEOCHEM) 585 (2002) 143. [15] H.A. Kurtz, J.J.P. Stewart, K.M. Dieter, J. Comput. Chem. 11 (1990) 82. [16] G.R.J. Williams, J. Mol. Struct. (THEOCHEM) 151 (1987) 215. [17] J. Lu, X. Zhang, X. Zhao, Appl. Phys. A 70 (2000) 461. [18] E. Westin, A.R. Rosen, G.T. Valde, E.J. Baerends, J. Phys. B 29 (1996) 5087. [19] DMOL version, vol. 960, Biosym Technologies, San Diego, CA, 1996. [20] G.E. Scuseria, Science 271 (1996) 942. [21] C. Filippi, C.J. Umrigar, M. Taut, J. Chem. Phys. 100 (1994) 1290. [22] C.G. Ding, J.L. Yang, Q.X. Li, K.L. Wang, F. Toigo, Phys. Lett. A 256 (1998) 417.
115
[23] A.D. Becke, J. Chem. Phys. 88 (1988) 1053. [24] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [25] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) 864; W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [26] R. Fletcher, Practical Methods of Optimization, vol. 1, Wiley, New York, 1980. [27] Y. Yang, F.H. Wang, Y.S. Zhou, Phys. Rev. A 71 (2005) 13202. [28] P. Calaminici, K. Jug, A.M. Köster, J. Chem. Phys. 109 (1998) 7756. [29] V.I. Kovalenko, A.R. Khamatgalimov, Chem. Phys. Lett. 377 (2003) 263. [30] G.Y. Sun, M.C. Nicklaus, R.H. Xie, J. Phys. Chem. A 109 (2005) 4617. [31] K. Kobayashi, S. Nagase, T. Akasak, Chem. Phys. Lett. 261 (1996) 502. [32] R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. [33] J.I. Aihara, Theo. Chem. Acc. 102 (1999) 134. [34] J.I. Aihara, Chem. Phys. Lett. 343 (2001) 465. [35] C.M. Tang, Y.B. Yuan, K.M. Deng, Y.Z. Liu, X.Y. Li, J.L. Yang, X. Wang, J. Chem. Phys. 125 (2006) 104307. [36] O.V. Boltalina, I.N. Ioffe, I.D. Sorokin, L.N. Sidorov, J. Phys. Chem. A 101 (1997) 9561. [37] H. Weiss, R. Ahlrichs, M. Häser, J. Chem. Phys. 99 (1993) 1262. [38] R. Antoine, P. Dugourd, D. Rayanc, E. Benichou, M. Broyer, F. Chandezon, C. Guet, J. Chem. Phys. 110 (1999) 9771.
Contents lists available at ScienceDirect
Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
The density functional calculations on the structural stability, electronic properties, and static linear polarizability of the endohedral metallofullerene Ba@C74 Chunmei Tang a, Shiyou Fu b, Kaiming Deng a,*, Yongbo Yuan a, Weishi Tan a, Decai Huang a, Xin Wang a a b
Department of Applied Physics, Nanjing University of Science and Technology, Xiaolinwei 200, Nanjing, Jiangsu 210094, PR China Harbin Institute of Technology at Weihai, Weihai, Shandong 264209, PR China
a r t i c l e
i n f o
Article history: Received 29 August 2007 Received in revised form 27 July 2008 Accepted 29 July 2008 Available online 7 August 2008 Keywords: Endohedral fullerene Ba@C74 Static linear polarizability Density functional theory
a b s t r a c t The generalized gradient approximation based on density functional theory is applied to study the structural stability, electronic properties, and static linear polarizability of Ba@C74. It is found that the most favorable endohedral site for a Ba atom is off-center under a [6, 6] double bond along the C2 axis on the rh plane, which is denoted as Ba@C74-2. This result can be explained by the electrostatic potential map of the C74 2 on the rh plane. Of particular interest is the static linear polarizability of Ba@C74-2. The calculated polarizability component azz of Ba@C74-2 687.3 Å3 is nine times that of C60 77.4 Å3, while axx and ayy are zero. This large anisotropy may be due to its lower C2v symmetry compared to C60 with Ih-symmetry, as well as the transference of about two valence electrons from the Ba atom to the carbon cage. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The C74 fullerene has only a unique isomer satisfying the isolated pentagon rule (IPR) [1] with D3h symmetry. It is interesting that C74 has been known as an unusual fullerene because it exists in the raw soot produced by arc discharge [1] but is insoluble in CS2 and toluene, the common organic solvents for other classical fullerenes such as C60. This unusual feature is attributed to its very small highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) energy gap (Eg), resulting in high reactivity [2]. However, mass analysis of the soot has shown the high-yield production of C74 [3]. Upon exohedral reduction [4] by means of electrochemical methods, the resulting dianion C74 2 has been found as a stable closed shell molecule. Along another direction, endohedral reduction by an incorporated divalent metal atom to form M2þ @C2 74 also leads to stable compounds [5]. For example, encapsulation of a Ca [6] atom or a Eu [7] atom inside D3h–C74 have already been reported. Experimentally, Shiga et al. [8] and Peters et al. [9] successfully synthesized the endohedral metallofullerene Ba@C74 using the radio frequency method. Haufe et al. [10] reported the isolation of Ba@C74 by means of multi-step High Performance Liquid Chromatography. Reich et al. [5] also reported the successful synthesis * Corresponding author. Tel./fax: +86 25 84315882. E-mail addresses: [email protected] (K. Deng), [email protected] (D. Huang). 0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.07.032
of [email protected] (OEP)2C6H6, implying the existence of the stable Ba@C74 molecule. Theoretically Slanina et al. calculated six isomers of Ba@C74 [11] and Yb@C74 [12] at DFT/B3LYP level, and found that both the endohedral Ba@C74 and Yb@C74 derived from the sole D3h–C74 cage as the major isomer from the Gibbs energy view, however, they did not discuss the most stable position of the encaged metal atom. Therefore, to find out the most stable position of a Ba atom in the D3h–C74 cage as well as to discuss the electronic structure of the ground state of Ba@C74 is desired. On the other hand, the static linear polarizability a represents one of the most important observations to the understanding of the electronic properties of molecule, since it is very sensitive to the delocalization of valence electrons as well as the structure and shape [13]. In the literature, there are two different ways to get the static linear polarizability. One is sum-over-states (SOS) [14] and the other is the finite-field (FF) approach [15,16]. SOS treatment seems to be out of reach for ab initio methods since they necessitate the calculations of many excited states. The FF method was developed by Kurtz et al. [15] and Williams [16] separately. For a considerable number of molecules, the FF approach has been proven to be acceptable [17]. It has been verified that the FF approach based on density functional theory can produce reliable static polarizability since even the local density approximation for exchange and correlation produces an accurate electron screening response to the application of an external field to the system [18]. There are two aims of this paper. One is to find out the most stable position of a Ba atom in Ba@C74 and to explore its electronic
112
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115
properties. The other is to calculate the static linear polarizability of Ba@C74. The paper is organized as follows. We describe our computational details in Section 2 and present our results and discussion in Section 3. Finally the conclusion is given in Section 4.
(BFGS) algorithm [26] with a convergence criterion of 103 a.u. on the displacement and 105 a.u. on the energy. The mean polarizability hai is calculated from the polarizability components as [13]
2. Computational details
hai ¼
The majority of the computations was performed with the DMOL3 package [19]. The density functional theory (DFT) approach, due to its significant character of high accuracy and low cost, has become one of the most popular calculation routines for large systems [20]. It has been found that the generalized gradient approximation (GGA) is more accurate than the local density approximation (LDA) in the calculations of the bond length and binding energy [21]. Therefore, the Becke-Lee-Yang-Parr (BLYP) correlation exchange functional at the generalized gradient approximation (GGA) [22] level, and DNP basis set are adopted in this paper. BLYP functional is the combination of the exchange functional exploited by Becke [23] and the correlation functional exploited by Lee, Yang and Parr [24]. The DNP basis sets, comparable to Gaussian 6-31G** sets, are the double numerical atomic orbitals augmented by polarization functions. The calculations are of all-electron type. Mulliken population analysis is made to obtain the effective charge and net spin populations on each atom. The electronic structure is obtained by solving the Kohn-Sham(KS) equations [25] self-consistently using the spin unrestricted scheme. Self-consistent field procedures are done with a convergence criterion of 106 a.u. on the energy and electron density. Geometry optimizations with only symmetry restriction and without any other constrained parameter (fixed atom, distance, angle etc) are performed using the Broyden-Fletcher-Goldfarb-Shanno
The static linear polarizability anisotropy Da is defined as [13,27,28]
Da ¼
1 ½axx þ ayy þ azz 3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ðaxx ayy Þ2 þ ðaxx azz Þ2 þ ðayy azz Þ2 2
ð1Þ
ð2Þ
The FF method is used in our calculations. Electronic fields with the strength of 0.005 a.u., similar to previous calculations, [28] are added along the three different Cartesian axes, respectively. 3. Results and discussion 3.1. Structural stability Recently, Kovalenko et al. [29] studied the geometric structure of D3h–C74 and predicted that the possible stable position for an atom inside the cage should be on the rh plane. Thus, in order to find out the most stable position of a Ba atom inside the C74, the Ba atom was placed at four different possible sites, respectively, and most of them are on the rh plane. They are: (1) off-center along the C3 principal axis; (2) off-center under a [6, 6] double bond along the C2 axis on the rh plane; (3) off-center under the hexagon center along the same C2 axis of (2) on the rh plane; and (4) in the cage center. Then these four structures were optimized, yielding Ba@C74-1, Ba@C74-2, Ba@C74-3, and Ba@C74-4, respectively, as
Fig. 1. The geometric structures of the four Ba@C74 (big ball stand for Ba).
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115
shown in Fig. 1. In order to check the reliably of the results, we reoptimized the four configurations of the Ba@C74 at the B3LYP/ Lanl2dz level using the Gaussian 03 package. The calculated minimal vibrational frequencies for Ba@C74-1, Ba@C74-2, and Ba@C74-3 are 34.34, 68.42, and 50.42 cm1, so they are do the local minimum of the related energy surface. However, Ba@C74-4 has three imagery frequencies, i.e., 41.60, 41.59, and 33.52 cm1, therefore, the center of the C74 cage is not the stable position for the Ba atom. Table 1 lists the energy of the HOMO, the energy of the LUMO, energy gap (Eg) between HOMO and LUMO, binding energy (BE), the magnetic moment of the Ba atom (MMA), and the magnetic moment of the molecule (MMM). Commonly, the thermodynamic stability of a fullerene is determined by its BE [30], which is defined as the difference between the energy sum of all free atoms constituting the molecule and the total energy of the molecule. Obviously, similar to the cases of a Ca [6] and a Eu [7] encaged inside C74, the structure with the Ba atom off-center under a [6, 6] double bond along the C2 axis on the rh plane (Ba@C74-2) has the largest BE of 528.09 eV, therefore, Ba@C74-2 is considered as the most stable isomer. This result can be explained as follows. The Ba atom with the largest BE of Ba@C74 should fall into the bottom of the electrostatic potential valley [31]. Fig. 2 shows the electrostatic potential map of the C74 2 on the rh plane, where the data give the magnitude of the electrostatic potential at the minimum points with the unit of kcal/mol. It is found that the electrostatic potentials obviously are negative and suitable for accommodating cationic species. To our expectation, there are 3-fold degenerate minima at the off-center sites under the [6, 6] double bond along the C2 axis on the rh plane. This result also be confirmed by the Car–Parrinello molecular dynamics calculations [32] for the movement of the Ba atom into the C74 cage. Furthermore, it is clear from Table 1, that the magnetic moment of the Ba atom inside the cages and those of the four Ba@C74 molecules are all zero, indicating their
Table 1 The HOMO, LUMO, Eg, BE with the unit in eV, and the magnetic moment of the Ba atom (MMA), the magnetic moment of the molecule (MMM) with the unit in lB for the C74, four Ba@C74 isomers and C74 2 Molecule
HOMO
LUMO
Eg
BE
MMA
MMM
C74 Ba@C74-1 Ba@C74-2 Ba@C74-3 Ba@C74-4 C74 2
5.09 5.07 5.17 5.19 5.18 0.70
5.00 4.45 4.61 4.55 4.47 1.43
0.09 0.62 0.56 0.64 0.71 0.73
524.96 528.01 528.09 527.98 527.46 528.09
– 0.00 0.00 0.00 0.00 –
– 0.00 0.00 0.00 0.00 0.00
Fig. 2. The electrostatic potential map of C74 2 on the rh plane.
113
Fig. 3. The orbital levels of C74, Ba@C74-2, and C74 2 .
closed-shell electronic structures. Meanwhile, with the fact of about 2e effective charge of the Ba atom from the Mulliken population analysis, we inferred that the electronic structure of Ba@C74-2 can be formally described as Ba2þ @C2 74 . Moreover, the computed nearest Ba–C distance 2.98 Åof Ba@C74-2 is longer than the sum of the radius of a Ba atom and a C atom of 2.75 Å, therefore, the Ba–C bond should be ionic. 3.2. Electronic properties of Ba@C74 Fig. 3 presents the orbital levels of Ba@C74-2 and C74 2 , together with those of C74 for comparison. The molecular orbitals whose energies agree with each other within a difference of 0.05 eV are regarded as degenerate, and the length of the horizontal bar shows the orbital degeneracy. Here, the level with one electron occupation is considered as one degeneracy because of the spin unrestricted scheme we used. The broken lines stand for the unoccupied orbital and the solid lines stand for the occupied orbital. The Fermi level (Ef) is defined as the halfway between HOMO and LUMO. We first compared the Egs of all the considered isomers, because they are often correlated with the kinetic stabilities [33,34]. The Eg calculated for the hollow cage C74 is 0.09 eV, much smaller than those of other fullerenes [35]. The Eg of Ba@C74-2, 0.56 eV, is much larger than that of the hollow C74 cage, indicating that incorporating a Ba atom into the C74 cage can strikingly enhance the kinetic stability of the cage. Furthermore, it is noteworthy that C74 2 , which is considered to be formed by the way that two electrons directly occupy on the LUMO of C74, [36] has a larger 0.73 eV energy gap, so the Eg of the C74 cage can also be obviously widened by the exohedral reduction. The contributions from the Ba atom states to the energy levels near the Fermi energy are marked in terms of percentage of population beside the levels. Clearly, the Ba has no contribution to LUMO and HOMO, revealing that both HOMO and LUMO of Ba@C74-2 are carbon like. Therefore, both the electronic detachment and electronic attachment, similar to the case of Si@C74 [35], should occur at the carbon sites of the cage. Fig. 4 shows the total density of states (TDOS) of Ba@C74-2 and C74, and the partial density of states (PDOS) of the Ba in Ba@C74-2. The DOS are obtained by a Lorentzian extension of the discrete energy levels, with weights being the orbital populations in the levels and a summation over them. The broadening width parameter is chosen to be 0.15 eV and the Ef is taken as zero. We explore the
114
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115 Table 3 Diagonal components and average values of a, and the static linear polarizability anisotropies Da for C60 and Ba@C74-2 with the unit in Å3
C60 Ba@C74-2
Fig. 4. The A-TDOS of Ba@C74-2, B-TDOS of C74, and C-PDOS of Ba.
mixture of the orbitals by analyzing TDOS of Ba@C74-2 and PDOS of the Ba in Ba@C74-2. It is obvious from the figure that there are almost no peaks in the PDOS of the Ba from 14 eV to Ef, showing the much weak mixture between the atomic orbitals of the Ba and those of the nearest neighboring carbon atoms of the C74. This result further confirms the ionic character of the Ba–C bond in Ba@C74-2. We also calculate the ionization potentials (VIP) and the vertical electron affinities (VEA) of Ba@C74-2 and C74. The VIP is the energy difference between the positively charged and neutral clusters. The VEA is evaluated by adding one electron to the neutral cluster in its equilibrium geometry and taking the difference between their total energies. The calculated VIP and VEA of the C74 are 6.55 and 3.53 eV, in agreement with the previously calculated value of 6.93 eV by Lu et al. [17] and the experimentally observed 3.28 ± 0.07 eV by Boltalina et al. [36], respectively. The computed VIP and VEA of Ba@C74-2 are 6.57 and 3.16 eV. We hope the further experiments will confirm our results. 3.3. Static linear polarizability of Ba@C74 In order to check the reliability of our method as well as to get the results of pure C60 cluster for comparison, we first calculated the mean static linear polarizability hai of C60. Our value and the other theoretical results are listed in Table 2. It can be found that our average value of a (77.4 Å3) is almost the same as most of other calculations with the same basis set, for example, Yang et al. [27] got the values of 77.6 and 76.1 Å3, using the same DNP basis sets but with the perdew-Burke-Ernzerhof (PBE) functional at the GGA level and the Vosko-Wilk-Nusair (VWN) functional at the LDA level, respectively. Weiss et al. [37], using the random phase approximation (RPA) and the basis set of 6-31G+sd, got a value of 78.8 Å3. It is worthy to note these results are in good agreement with the experimental value of 76.5 ± 8.0 Å3 obtained by Antoine et al. [38], using the molecular beam deflection method (MBD). Therefore our calculation method and the basis sets are both reliable. Table 2 The calculated and experimental average values for the static linear polarizability hai of C60 with the unit in Å3 Methods
BLYP/ DNP
PBE/DNP [27]
VWN/DNP [27]
PRA/6-31G+sd [37]
MBD [38]
hai
77.4
77.6
76.1
78.8
76.5 ± 8.0
axx
ayy
azz
hai
Da
77.4 0.0
77.4 0.0
77.4 687.3
77.4 229.1
0.0 687.3
The calculated results of a tensors, the average linear static polarizability hai, and the static linear polarizability anisotropy Da for Ba@C74-2 as well as their counterparts of C60 for comparison are presented in Table 3. Ba@C74-2 is oriented with its permanent dipole moment along the z axis as indicated in Fig. 1. For Ba@C74-2, the components of a in the x and y directions are zero while that in the z direction 687.3 Å3 is about nine times that for C60. Such large anisotropy of polarizability for Ba@C74-2 may be attribute to two reasons. One is the lower C2v symmetry structure of Ba@C74-2 compared with the Ih symmetry structure of C60. The other is that there are about two electron transference from the Ba atom to its neighboring carbons of the cage. Therefore, the anisotropy of polarizability is very sensitive to the structure or shape as well as the delocalization of the valence electrons [13]. 4. Conclusion The generalized gradient approximation based on density functional theory is applied to study the structural stability, electronic properties, and static linear polarizability of endohedral metallofullerene Ba@C74. It is found that among the four possible isomers of Ba@C74, the most favorable endohedral site for the Ba atom is off-center under a [6, 6] double bond along the C2 axis on the rh plane. This result can be explained by the analysis of the electrostatic potential map of the C74 2 on the rh plane. Inferred from its longer bond length as well as the transference of about two electrons from the Ba to the cage, the nearest Ba–C bond is mainly ionic. From the effective charges of about two electrons for the Ba atom in the cage, we infer that the electronic state of the molecule can be formally described as Ba2þ @C2 74 . Finally, the computed polarizability components, i.e., the azz of Ba@C74-2 687.3 Å3 is about nine times that of C60, while axx and ayy of it are both zero. This large anisotropy anisotropic result may be due to its lower C2v symmetric structure in contrast to C60 isotropic polarizability with Ih-symmetric structure as well as the transference of about two valence electrons from the Ba atom to the neighboring carbons of the cage. Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant No. 10174039), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK2002099 and BK2006204), the NJUST Young Scholar Research Fund (Grant No. 200705), and the Graduate Cultivation Foundation of Jiangsu Province (2006). We also thank Wuxi Supercomputing Center for computing resources. References [1] H.W. Kroto, Nature 329 (1987) 529. [2] H. Shinohara, H. Yamaguchi, N. Hayashi, H. Sato, M. Ohkohchi, Y. Saito, J. Phys. Chem. 97 (1993) 4259. [3] R. Hatakeyama, T. Hirata, H. Ishida, N. Sato, Appl. Phys. Lett. 73 (1998) 888. [4] A.A. Goryunkov, V.Y. Markov, I.N. Ioffe, R.D. Bolskar, M.D. Diener, I.V. Kuvychko, S.H. Strauss, O.V. Boltalina, Angew. Chem. Int. Ed. 43 (2004) 997. [5] A. Reich, M. Ranthöfer, H. Modrow, U. Wedig, M. Jansen, J. Am. Chem. Soc. 126 (2004) 14428. [6] T. Kodama, R. Fujii, Y. Miyake, S. Suzuki, H. Nishikawa, I. Ikemoto, K. Kikuchi, Y. Achiba, Chem. Phys. Lett. 399 (2004) 94.
C. Tang et al. / Journal of Molecular Structure: THEOCHEM 867 (2008) 111–115 [7] H. Matsuoka, N. Ozawa, T. Kodama, H. Nishikawa, I. Ikemoto, K. Kikuchi, K. Furukawa, K. Sato, D. Shiomi, T. Takui, T. Kato, J. Phys. Chem. B 108 (2004) 13972. [8] K. Shiga, K. Ohno, Y. Kawazoe, Y. Maruyama, T. Hirata, R. Hatakeyama, N. Sato, Mater. Sci. Eng. A 290 (2002) 6. [9] G. Peters, M. Jansen, Angew. Chem. Int. Ed. Engl. 31 (1992) 223. [10] O. Haufe, A. Reich, C. Möschel, M.Z. Anorg, Allg. Chem. 627 (2001) 23. [11] Z. Slanina, S. Nagase, Chem. Phys. Lett. 422 (2006) 133. [12] Z. Slanina, F. Uhll´k, S. Nagase, J. Phys. Chem. A 110 (2006) 12860. [13] H.Y. Wang, X.B. Li, Y.J. Tang, H.P. Mao, Z.H. Zhu, Chin. J. Chem. Phys. 18 (2005) 50. [14] A.H. Göller, S. Erhardt, U.W. Grummt, J. Mol. Struct. (THEOCHEM) 585 (2002) 143. [15] H.A. Kurtz, J.J.P. Stewart, K.M. Dieter, J. Comput. Chem. 11 (1990) 82. [16] G.R.J. Williams, J. Mol. Struct. (THEOCHEM) 151 (1987) 215. [17] J. Lu, X. Zhang, X. Zhao, Appl. Phys. A 70 (2000) 461. [18] E. Westin, A.R. Rosen, G.T. Valde, E.J. Baerends, J. Phys. B 29 (1996) 5087. [19] DMOL version, vol. 960, Biosym Technologies, San Diego, CA, 1996. [20] G.E. Scuseria, Science 271 (1996) 942. [21] C. Filippi, C.J. Umrigar, M. Taut, J. Chem. Phys. 100 (1994) 1290. [22] C.G. Ding, J.L. Yang, Q.X. Li, K.L. Wang, F. Toigo, Phys. Lett. A 256 (1998) 417.
115
[23] A.D. Becke, J. Chem. Phys. 88 (1988) 1053. [24] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [25] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) 864; W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [26] R. Fletcher, Practical Methods of Optimization, vol. 1, Wiley, New York, 1980. [27] Y. Yang, F.H. Wang, Y.S. Zhou, Phys. Rev. A 71 (2005) 13202. [28] P. Calaminici, K. Jug, A.M. Köster, J. Chem. Phys. 109 (1998) 7756. [29] V.I. Kovalenko, A.R. Khamatgalimov, Chem. Phys. Lett. 377 (2003) 263. [30] G.Y. Sun, M.C. Nicklaus, R.H. Xie, J. Phys. Chem. A 109 (2005) 4617. [31] K. Kobayashi, S. Nagase, T. Akasak, Chem. Phys. Lett. 261 (1996) 502. [32] R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. [33] J.I. Aihara, Theo. Chem. Acc. 102 (1999) 134. [34] J.I. Aihara, Chem. Phys. Lett. 343 (2001) 465. [35] C.M. Tang, Y.B. Yuan, K.M. Deng, Y.Z. Liu, X.Y. Li, J.L. Yang, X. Wang, J. Chem. Phys. 125 (2006) 104307. [36] O.V. Boltalina, I.N. Ioffe, I.D. Sorokin, L.N. Sidorov, J. Phys. Chem. A 101 (1997) 9561. [37] H. Weiss, R. Ahlrichs, M. Häser, J. Chem. Phys. 99 (1993) 1262. [38] R. Antoine, P. Dugourd, D. Rayanc, E. Benichou, M. Broyer, F. Chandezon, C. Guet, J. Chem. Phys. 110 (1999) 9771.