Journal of Structural Chemistry. Vol. 47, No. 6, pp. 1016-1021, 2006 Original Russian Text Copyright © 2006 by M. A. Mileev, S. M. Kuzmin, and V. I. Parfenyuk
AB INITIO CALCULATIONS OF STRUCTURE AND STABILITY OF SMALL BORON NITRIDE CLUSTERS M. A. Mileev, S. M. Kuzmin, and V. I. Parfenyuk
UDC 544.13+544.15
Clusters of boron nitride BxNx (x = 1-4, 12, 15, 30) were investigated by the Hartree–Fock and density functional methods using the 6-31G* basis. It was found that linear, cyclic, and shell structures are stable against minor deformations of the BxNx cluster. Inclusion of electron correlation in calculation markedly changes the electron density distribution and the structure of the clusters. Keywords: structure, stability, boron nitride, isomers.
INTRODUCTION Investigation of fullerenes started in 1985 when Kroto et al. [1, 2] synthesized the carbon fullerene molecule C60. In 1991 Iijima [3] reported on the discovery of carbon nanotubes. Later it was predicted that similar structures of non-carbon atoms can also exist in nature [4, 5]. Successful attempts to synthesize nanosized structures from boron nitride molecules [6-11] and the development of procedures for the synthesis of pure boron-nitride nanotubes [13] have stirred the growing interest in calculation of BN structures: B12N12 [14], B12+3nN12+3n (n = 0-3), and B22N22 [15], B12N12 and B16N16 [16, 17], B36N36 [18], B12N12, B24N24 and B60N60 [19], BnNn+4 (n = 8-33) [20], and B30N30 as a formal analog of Buckminster fullerene. Jensen and Toftlund [14] carried out ab initio quantum chemical calculations on B12N12 with various geometries. They found that B12N12 fullerenes are characterized by higher stability than C24 provided that they have four- and sixmembered rings, but no so-called “erroneous” bonds BB and NN. Blasé and co-authors [21] established that the total energy of the B24N24 molecule increases by about 1.6 eV if the molecule contains “erroneous” bonds. They also showed that rings with odd numbers of atoms at the growing end of BN nanotubes are unstable. Stability of B12N12 was also confirmed in a theoretical work [22] using density functional theory. The authors also predicted that other fullerenelike BxNx structures would be especially stable for x = 16 and 29 and, to a smaller extent, for x = 18, 22, and 25. Silaghi-Dumitrescu et al. [15] performed calculations on low-molecular (B12N12 and B15N15) spherical clusters of boron nitride. Ab initio (Hartree–Fock) calculations with a small standard basis set 3-21G did not employ polarization or diffuse functions. These authors also carried out AM1 semiempirical calculations for B12N12, B15N15, B18N18, B21N21, and B22N22 clusters. They calculated formation enthalpies for these clusters and even recalculated the enthalpies per BN bond for several clusters. Calculations were fulfilled using Molek_9000 and Spartan 4.0 quantum chemical programs. D. M. Sheichenko, A. V. Pokropivnyi, and V. V. Pokropivnyi [23] evaluated the energies and performed structure calculations on B12N12, B24N24 and other related molecules and nanosized clusters using HyperChem 5.0 computer
Institute of Solution Chemistry, Russian Academy of Sciences, Ivanovo;
[email protected]. Translated from Zhurnal Strukturnoi Khimii, Vol. 47, No. 6, pp. 1028-1033, November-December, 2006. Original article submitted January 12, 2006; revised March 22, 2006. 1016
0022-4766/06/4706-1016 © 2006 Springer Science+Business Media, Inc.
simulation programs. M. Zandler et al. [24] carried out theoretical investigation of the B12N12 boron nitride cluster using the GAUSSIAN-92 quantum mechanical program. The Hartree–Fock method was employed without including polarization and diffuse functions (3-21G basis set) and neglecting electron correlation. Ten isomers of the B32N32 molecule varying in structure and energy were considered in [25]. The authors investigated the structure and stability of the clusters at a level of density functional theory and used an extensive basis set 631G including the polarization function. For calculations they employed Becke’s three-parameter hybrid functional with the LYP electron correlation functional. The list of structures analyzed by theoretical methods is obviously not exhausted. Energy competition between different types of structure remains an open question. This competition should obviously be most pronounced for small clusters (x < 10). Therefore the aim of this work is theoretical investigation of BxNx clusters (linear and cyclic isomers for x = 1, 2, 3, 30; shell structure and linear and cyclic isomers for x = 4, 12, 15).
DETAILS OF CALCULATIONS Theoretical investigation of molecules was carried out using the GAUSSIAN-03 quantum chemical program [26]. All calculations were performed by the Hartree–Fock method using the standard wide basis set 6-31G complemented by a d type polarization function. For linear and shell type isomers, electron correlation was included in calculations by using density functional theory and a composite BVWN potential. The potential has two components, namely, the electron exchange and electron correlation terms. For the exchange (ȼ) term we used the exchange functional developed by Becke in 1988; it includes Slater’s electron exchange between atoms and a correction for the electron density gradient. For the correlation (VWN) term of the functional we employed the correlation functional developed by Vosko, Wilk, and Nusair in 1980, better known as LSD (Local Spin Density) correlation functional 3 [27]. RESULTS AND DISCUSSION According to our calculations, both linear and cyclic isomers, as well as shell type clusters, are stable against minor deformations of the conformation of BxNx. The calculated total energies of the clusters are listed in Table 1. For small clusters (x < 10), the total energy of the molecule decreases from linear to cyclic conformation. For small clusters, the cyclic conformation is conceivably a global energy minimum, the energy of the shell structure exceeding the energy of the cycle. For clusters with more than 24 atoms, shell type clusters (fullborenes) are most stable. The energy per bond is higher for linear isomers compared with cyclic isomers. As the cluster size increases, the energy per bond increases for cyclic isomers, but decreases for linear isomers. Accordingly, the difference between the cyclic and linear structures decreases from the viewpoint of energy. For shell type clusters, variation of energy per bond is significant (12-13 au) and slightly increases with the number of atoms. Inclusion of electron correlation decreases the total energy of the molecule. The decrease was from 0.237 au to 1.2 au per bond. For shell type clusters, this correction is virtually independent of the cluster size and amounts to 0.35r0.01 au. Table 2 presents the results of Mulliken’s atomic charge calculations and the dipole moments for linear isomers. According to our calculations, the cyclic isomers and the shell type cluster have a zero dipole moment, which agrees with the high symmetry of the molecule. Note that inclusion of electron correlation markedly changes the picture of electron density distribution. For linear isomers, electron density is distributed along the chain in a rather complex manner. The charge varies from –0.08 to 0.55 on boron atoms and from –0.18 to –0.48 on nitrogen atoms. Note that the charge on shell type clusters differs widely between structures of different symmetries. Thus B12N12 is an ionic cluster, while B15N15 approaches a covalent cluster according to the type of bond. The differences are presumably determined by the structural differences between clusters because both the linear and cyclic isomers of this composition differ insignificantly. Table 3 lists the equilibrium geometrical parameters of the molecules. For cyclic isomers, Dxh was used for 1017
TABLE 1. Total Energies for Clusters No. of atoms
Linear isomer
Cyclic isomer
Shell type cluster
1 2
–78.883* (–80.097)* –158.086 –52.695* (3 bonds) –237.288 –47.458* (5 bonds) –316.491 –45.213* (7 bonds) –950.157 –41.311* (23 bonds) –1188.135 –40.970* (29 bonds) –2375.944 –40.270 (59 bonds)
Nonexistent –158.092 (–159.039) –39.523* (–39.760)* (4 bonds) –237.455 (–240.650) –39.576* (–40.108)* (6 bonds) –316.726 (–320.970) –39.591* (–40.121)* (8 bonds) –950.497 –39.604* (24 bonds) –1188.137 –39.604* (30 bonds) –2376.306 –39.605 (60 bonds)
Nonexistent
3
4
12
15
30
Nonexistent –237.101 –29.628* (8 bonds) –316.468 (–320.739) –26.372* (–26.728)* (12 bonds) –950.020 (–962.829) –26.389* (–26.745)* (36 bonds) –1188.312 (–1204.080) –26.407* (–26.757)* (45 bonds) No data available
Note. The values in parentheses are the total energies of the clusters obtained by the electron density theory method. *Energy per BN bond. TABLE 2. Dipole Moments and Mulliken’s Charges for Isomers No. of atoms
Linear isomer
Cyclic isomer
Shell type cluster
1
B –0.31 (–0.21); N 0.31 (0.21); P = 4.4 (2.6) B 0.29; N –0.29; P = 3.3 B 0.32; N –0.32; P = 6.9 B –0.32; N –0.32; P = 11.2 B 0.33; N –0.33; P = 55.9 B 0.33; N –0.33; P = 74.2 B 0.33; N –0.33; P = 167.0
Nonexistent
Nonexistent
B 0.54 (0.39); N –0.54 (–0.39);
Nonexistent
B 0.51 (0.35); N –0.51 (–0.35);
B 0.52; N –0.52
B 0.52 (0.35); N –0.52 (–0.35);
B 0.66 (0.47); N –0.66 (–0.47);
B –0.31 (–0.21); N 0.31 (0.21);
B 1.02 (0.61); N –1.02 (–0.61);
B –0.31 (–0.21); N 0.31 (0.21);
B –0.31 (–0.21); N 0.31 (0.21);
B –0.31 (–0.21); N 0.31 (0.21);
No data available
2
3
4
12
15
30
1018
TABLE 3. Geometrical Parameters of Molecules Molecule
BN bond length, nm
Angle
Dihedral angle
BN Linear isomer B2N2 Linear isomer B3N3 Linear isomer B4N4 Linear isomer B12N12
180 180 180 180 BNB = NBN = 180
— 0 0 0 BNBN = NBNB = 0
BNB = NBN = 180
BNBN = NBNB = 0
BNB = NBN = 180
BNBN = NBNB = 0
Cyclic isomer B2N2
0.124 (0.128) 0.138 0.133 0.132 r = 0.130 l = 3.0 r = 0.130 l = 3.78 r = 0.130 l = 7.68 0.139 (0.140)
0 (0)
Cyclic isomer B3N3
0.135 (0.137)
Cyclic isomer B4N4
0.133 (0.135)
Cyclic isomer B12N12
r = 0.130 d = 0.98 r = 0.130 d = 1.25 r = 0.130 d = 2.48 r44 = 0.149 (0.152)
BNB = 65.1 (64.2) NBN = 114.9 (115.8) BNB = 90.8 (87.9) NBN = 149.2 (152.1) BNB = 105.4 NBN = 164.6 BNB = 150.5 NBN = 179.5 BNB = 156.2 NBN = 179.8 BNB = 168 NBN = 180 BNB = 75.3 (75.4) NBN = 103.0 (102.9) BNB4 = 84 (83) NBN4 = 96 (96) BNB6 = 115 (114) NBN6 = 124 (124.5) BNB4 = 80 (80) NBN4 = 98.5 (98.7) NBN6 = 115-127 (119-127) BNB6 = 109-115 (109-117) NBN6center = 111
Linear isomer B15N15 Linear isomer B30N30
Cyclic isomer B15N15 Cyclic isomer B30N30 Shell type cluster B4N4 Shell type cluster B12N12
r64 = 0.150 (0.152) r66 = 0.143 (0.145)
Shell type cluster B15N15
r64 = 0.146-0.148 (0.148-0.150) r66 = 0.143-0.145 (0.144-0.146) r66center = 0.149 (0.151)
0 (0) 0 (0) 0 0 0 BNBN = NBNB = 14.2 (14.1) BNBN4 = NBNB4 = 7.4 (8.4) BNBN6 = NBNB6 = 16 (18.5)
BNBN4 = NBNB4 = 14 (14) BNBN6 = NBNB6 = 11-19 (10-18)
molecular symmetry, where x is the number of atoms of each element. In particular, B12N12 has D12h symmetry (thus the B12N12 molecule coincides with itself when rotated through an angle of 180q/15 = 12q). Accordingly, D15h was employed for B15N15 (180q/12 = 15q) and D30h for B30N30 (180q/6 = 30q). Linear isomers (Fig. 1Ⱥ) always have ɋv symmetry. For cyclic isomers (Fig. 1B, C), there is a clear tendency for atoms to unite in groups each including two or three nitrogen atoms and one boron atom. In this case, cyclic structures resemble polyhedra whose face is a linear NBN group (the angle is approximately 180q). The BNB angle grows with the number of atoms in the molecule, namely, from 65q for B2N2 to 168q for B30N30. All linear and cyclic structures are characterized by similar BN bond lengths (about 0.130 nm). The B12N12 cluster with a shell type structure (Fig. 2Ⱥ) has D3 symmetry; B15N15 has ɋ3 symmetry. The B12N12 molecule includes eight six-membered and six four-membered cycles; the B15N15 has eleven hexagons and six tetragons. Calculations showed that for the shell of B12N12, the angles in tetragons and hexagons vary from 84q to 96q and from 115q to
1019
Fig. 2. Structures of fullborenes: Ⱥ — shell type cluster B12N12, B — shell type cluster B15N15.
Fig. 1. Structures of clusters: A — linear isomer of B12N12, B — cyclic isomer of B4N4, C — cyclic isomer of B12N12.
124q, respectively. Furthermore, the cycles of the shell proved to be nonplanar. The dihedral angle is 7-8q for four-membered cycles and 16-18q for six-membered cycles, respectively. The case is similar for the B15N15 cluster (Fig. 2B). According to our estimations, the ring diameter of B12N12 measured based on boron atoms is 0.974 nm, while similar measurement using nitrogen atoms yielded 1.007 nm. The diameter of the B12N12 shell type cluster was 0.450 nm in measurement using boron atoms and 0.476 nm in measurement using nitrogen atoms. The displacement of boron atoms relative to nitrogen atoms toward the center of the cluster (Fig. 1B, 2B) proved to be a distinction independent of the structure of the clusters under study. The BN bond length in shell type structures is from 0.143 nm to 0.150 nm. Inclusion of electron correlation increases this value by ~0.003 nm. The angles in the four- and six-membered cycles, as well as dihedral angles, also change.
CONCLUSIONS Thus we have carried out theoretical investigation of BxNx boron nitride clusters (x = 1-4, 12, 15, 30) using the Hartree–Fock and density functional methods and a wide basis set. The linear, as well as the cyclic and shell type, structures were found to be stable against minor deformations of the conformations of the BxNx cluster. Evidence has been found in support of the fact that inclusion of electron correlation in calculations changes considerably both the electron density distribution and the structure of the clusters.
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