The nomenclature can characterize different isomers very well, for different isomers at a given size generally have different nomenclatures. For example, the three isomers of (BN)₁₀ have [3, 3, 6] (a, b, c in [a, b, c] denotes N₄₀₄, N₄₁₄, N₄₂₄ respectively), [4, 0, 4], [6, 0, 0] nomenclature respectively. Additionally, the nomenclature can easily describe the f₄ adjacencies. For example, the smallest boron nitride cluster with all f₄ isolated, i.e. the isomer (BN)₁₂: 3 in T_h symmetry has nomenclature as [0, 12, 0], which N₄₀₄=0 denote the 6 f₄ are isolated. Compare with the all f₄ completely isolated nomenclature [0, 0, 0] ( (BN)₂₈: 23, (BN)₃₁: 21, (BN)₃₄: 35 ), (BN)₁₂: 3 still have 12 B–N bonds that bridged f₄, and (BN)₁₆: 8 with [0, 0, 24] have 24 B–N–B or N–B–N linkers that bridged two isolated f₄.
3.2 Stability
It is the main purpose of this paper to present optimized structures for a large class of clusters, correlate structural data with relative stabilities and to check the new N_4x4 (x = 0, 1 and 2) topological rule. In this situation, a systematic theoretical study would seem to be valuable if it can provide reliable quantitative estimates for the structures, relative stabilities. So a systematic calculation of energies and topological properties has been carried out to all boron nitride alternant structures from (BN)₁₀ to (BN)₃₅. The total electronic energies and the N_4x4 numbers of all structures are summarized in the supporting Information. The most stable structures of each given size are shown in Fig. 2. The relative energies of them are defined as zero.

Fig. 3 Variation of relative energy with number of N₄₀₄ for (BN)_n (n=10-35)
The scale in the vertical coordinate axis is only giving their relative difference at each given size, no relationship between energies of different nuclearity.

    From the calculation up to 70 atoms in size we carried out, the most striking feature is that, the most stable structure has the minimal value of N₄₀₄, i.e., the number of shared f₄ bonds, while the least stable isomer has the maximum N₄₀₄ value. The result is well satisfied with the simple rule of thumb ISR ^[], namely, the most stable structure at any given n is one that minimizes the number of square adjacencies.
    The correlations between N₄₀₄ and relative energies of (BN)_n isomers are showed in Fig.3. It is found that with increased number of N₄₀₄, the relative energies increase steadily and considerably. It shows a general increasing tendency within the rang of (BN)_n we tested. Examination of the full set of curves shows that the gradients of (BN)_n (n=10-35) are very close, especially for the small cages. For large-sized (BN)_n, the curves distort sharply when N₄₀₄ number over 3. As a whole, the penalty caused by N₄₀₄ is significant and similar to different sized (BN)_n. To be mentioned, it is not claimed that all less value N₄₀₄ isomers have lower energies than all higher value N₄₀₄ isomers. As the energy range spanned when the number of isomers grow, some overlaps occur. However, most of them obey this rule and the most stable isomers can be surely included and selected.
    Additionally, we studied the other two merits N₄₁₄ and N₄₂₄ in detail. For they can provide more detailed information when isomers have the same N₄₀₄ value. Examination of computed energies of the full set of optimized geometries shows that, the most stable structure of a given size not only has the minimal value of N₄₀₄, but also preferred to has the minimal N₄₁₄ number, even N₄₂₄ number. For example, there are 3 IPR (N₄₀₄=0) isomers for (BN)₂₂, isomer (BN)₂₂: 14 ([0, 0, 12]) in C₃ symmetry that has the minimal N₄₁₄ number is the most stable, the other isomers (BN)₂₂: 13 ([0, 3, 0]), (BN)₂₂: 12 ([0, 2, 4]) are higher in energy 3.8 kcal/mol, 10.5 kcal/mol respectively. For (BN)₂₈, there are 5 IPR isomers with N₄₁₄=0, and the most stable one (BN)₂₈: 23 ([0, 0, 0]) in T symmetry has the minimal N₄₂₄ number.

Fig.4 (BN)₁₈ IPR isomers

    Exceptionally，isomer (BN)₁₈: 6 ([0, 4, 4]) that has the least number of N₄₁₄ is higher in energy than isomer (BN)₁₈: 5 ([0, 6, 0]) 15.0 kcal/mol, but is lower in energy than another isomer (BN)₁₈: 7 ([0, 6, 0]) 72.2 kcal/mol (Table 1). Fig. 4 gives their structures. We can see that (BN)₁₈: 5 is tube-like, while (BN)₁₈: 6 and (BN)₁₈: 7 are both oblate. As we know, generally, the tube-like isomer is more stable than the oblate one for the strain reason. So (BN)₁₈: 5 that has higher N₄₁₄ number but tube-like is more stable. As the same reason, isomers (BN)₂₈: 18, (BN)₂₈: 19 that both with [0, 0, 12] nomenclature and C_3v symmetry is more stable than (BN)₂₈: 20 with [0, 0, 8] nomenclature that has lower N₄₁₄ number. We can deem that the stability difference between isomers is also related to the isomer's shape and symmetry. These energetic differences are very similar to those of large sized fullerenes obeying the isolated pentagons rule (IPR) ^[], and the relative stabilities of them reflect the intimate interplay between strains of individual rings, conjugations and curvatures of the cages.

Table 1 B3LYP/6-31G* energies E_T (a.u.), E_rel relative energies (kcal/mol) and their N_4x4
(x = 0, 1, 2) number of some (BN)_n isomers

    The other exceptional cases come up to (BN)₂₆ and (BN)₃₅. Isomer (BN)₂₆: 13 ([0, 1, 4]) that has the least number of N₄₁₄ is higher in energy than the two [0, 2, 0] isomers (BN)₂₆: 10, and (BN)₂₆: 12. Similarly, isomer (BN)₃₅: 21 ([0, 0, 4]) is also less stable than the two [0, 1, 0] isomers (BN)₃₅: 20 and (BN)₃₅: 15. By comparing their N_4x4 numbers, it is found that their N₄₁₄ number is very close but the N₄₂₄ number differs sharply. We can get the conclusion that the high N₄₂₄ number may compensate the decrease of energy because of lower N₄₁₄ number. In other words, an isomer with lower N₄₁₄ number but higher N₄₂₄ number may be comparative in energy with the one with higher N₄₁₄ number but lower N₄₂₄ number. As shown in Table 1, (BN)₃₂: 19 ([0, 0, 4]) is very close in energy with (BN)₃₂: 7 ([0, 1, 0]) (within 1 kcal/mol), and structures 16, 17, 20 of (BN)₂₈ also validate this conclusion. This case can also be found in other sized (BN)_n.
    Based on the discussion above, it can be concluded that, the most stable isomer has the minimal number of N₄₀₄ while the least stable one has the maximum N₄₀₄ number, and with increased number of N₄₀₄, the relative energies increase steadily and considerably. Furthermore, the most stable isomer also preferred to has the minimal N₄₁₄ number, even N₄₂₄ number, and the high N₄₂₄ number may compensate the decrease of energy because of lower N₄₁₄ number. Comparing the penalty caused by N_4x4 (x=0, 1, 2), the deviation between different N₄₀₄ isomers is very large than those caused by N₄₁₄ and N₄₂₄, i.e., N₄₀₄ is dominant in effecting the relative energies of isomers. Additionally, the stability also relates to the isomer's shape and symmetry. As a filtering principle, the simple N_4x4 (x = 0, 1, 2) topological rule characterizing the number of bonds between f₄ is very helpful to narrow the range of likely candidates.

4 CONCLUSION
In summary, the topological properties and stabilities of all (BN)_n (n=10-35) alternate cages have been investigated at the B3LYP level systemically. It is found that in general, the most stable isomer of (BN)_n has the least number of N_4x4 (x = 0, 1, 2), and the relative energies show an increasing tendency with N₄₀₄ number. N₄₀₄ is always dominant in effecting the relative energies of isomers, but the influences of the N₄₁₄ and N₄₂₄ are also significant. The present study suggests a simple topological criterion that can be used to sort an approximate stability order of all the isomers. The simple filtering scheme is ideally suited for prescreening the thermodynamically viable structures of boron nitride clusters. It may also give us a guideline for designing and synthesis of the BN clusters.

REFERENCES
[1] Kroto H. W.，Heath J. R.， OBrien, S. C.， Curl R. F.， Smalley R. E. Nature (London), 1985, 318: 162.
[2] Iijima S. Nature, 1991, 354: 56.
[3] Oku T., Kuno M., Kitahara H., Nartia I. Int. J. Inorg. Mater., 2001, 3: 597
[4] (a) Saito Y., Maida M. J. Phys. Chem. A, 1999, 103 (10): 1291 (b) Kongsted J., Osted A., Jensen L., ?strand P.-O., Mikkelsen. K. V. J. Phys. Chem. B, 2001, 105: 10243 (c) Ma R., Bando Y., Sato T., Kurashima K. Chem. Mater., 2001, 13: 2965 (d) Wu H. S., Xu X. H., Zhang F. Q., Jiao H. J. J. Phys. Chem. A, 2003, 107: 6609 (e) Xu H., Ma J., Chen X., Hu Zh., Huo K.-F., Chen Y. J. Phys. Chem. B, 2004, 108: 4024
[5] (a) Alexandre S. S., Nunes R. W., Chacham H. Phys. Rev. B, 2002, 66: 085406 (b) Oku T., Kuno M., Nartia I., Diamond Relat. Mater., 2002, 11: 940 (c) Oku T., Nishiwaki A., Nartia I., Physica B, 2004, 351: 184 (d) Zope R. R., Baruah T., Pederson M. R., Dunlap B. I. Chem. Phys. Lett., 2004, 393: 300
[6] Stéphan O., Bando Y., Loiseau A., Willaime F., Shramchenko N., Tamiya T., Sato T. Appl. Phys. A., 1998, 67: 107
[7] (a) Oku T., Nishiwaki A., Nartia I., Gonda M. Chem. Phys. Lett., 2003, 380: 620 (b) Oku T., Nishiwaki A., Nartia I., Solid State Comm., 2004, 130: 171
[8] (a) Martin J. M. L., El-Yazal J., Fran?ois J.-P., Gijbels R. Chem. Phys. Lett., 1995, 232: 289 (b) Martin J. M. L., El-Yazal J. Chem. Phys. Lett., 1996, 248: 95 (c) Strout D. L. J. Phys. Chem. A, 2001, 105: 261 (d) Jia J. F., Wu H. S., Jiao H. J. Acta Chim. Sin.(Huaxue Xuebao), 2003 , 61(5): 653
[9] (a) Sun M. L., Slanina Z, Lee S. L. Chem. Phys. Lett., 1995, 233: 279 (b) Slanina Z., Sun M. L., Lee S. L. NanoStruct. Mater., 1997, 8: 623
[10] Rogers K. M., Fowler P. W., Seifert G. Chem. Phys. Lett., 2000, 332: 43
[11] Strout D. L. Chem. Phys. Lett., 2004, 383: 95
[12] (a) Wu H. S., Jiao H. J. Chem. Phys. Lett., 2004, 386: 369 (b) Wu H. S., Xu X. H., Jia J. F. Acta Chim. Sin.(Huaxue Xuebao), 2004, 62(1): 28. (c) Cui X. Y., Wu H. S. Chin. J. Chem.( zhongguohuaxue), 2005, 23: 117
[13] (a) Wu H. S., Cui X. Y., Qin X. F., Jiao H. J. J. Mol. Struct.(THEOCHEM), 2004, 714: 153 (b) Wu H. S., Cui X. Y., Xu X. H. J. Mol. Struct. (THEOCHEM), 2005,717: 107
[14] (a) Kroto H. W. Nature (London), 1987, 329: 529. (b) Schmalz T. G., Seitz W. A., Klein D. J., Hite G.E. J. Am. Chem. Soc., 1988,110: 1113.
[15] Fowler P. W., Heine T., Mitchell D., Schmidt R., Seifert G. J. Chem. Soc. Faraday Trans., 1996, 92(12): 2197
[16] Fowler P. W., Rogers K. M., Seifert G., Terrones M., Terrones. H. Chem. Phys. Lett., 1999, 299: 359
[17] Wu H. S., Xu X. H., Jiao H. J. J. Phys. Chem. A. 2004, 108:3813
[18] M. J. Frisch, G. W. Trucks, H. B. Schlegel,et al. Gaussian 03, Revision A.1, Gaussian, Inc., Pittsburgh PA, 2003.
[19] Manolopoulos D. E., May J. C., Down S. E. Chem. Phys. Lett. 1991, 181: 105
[20] Fowler P. W., Manolopoulos D. E., An Atlas of Fullerenes, Oxford: Oxford University Press, 1995
[21] Cioslowski J. Rao N. Moncrieff D. J. Am. Chem. Soc., 2000, 122: 8265


(BN)_n笼的稳定性及笼中四元环间键连类型对笼稳定性的影响
武海顺 田欣欣
（山西师范大学化学与材料科学学院 临汾 041004）
摘要 采用密度泛函B3LYP/6-31G*方法对(BN)_n (n=10-35) 四六结构的拓扑学性质和稳定性进行了研究。结果发现它们的稳定性顺序遵循一个简单的用来描述四元环间键连类型的拓扑学规则N_4x4 (x = 0，1 ，2)。计算表明，各个异构体笼的相对能量随N₄₀₄值的增大而增大，且最稳定构型具有最小的N_4x4 (x = 0, 1，2)值。这一简单的拓扑学标准可用来预测各异构体大概的相对稳定性顺序并能预先挑选出热力学稳定的氮化硼笼状结构。
关键词 氮化硼 稳定性 拓扑学 命名法