Chemical Physics Letters 482 (2009) 121–124

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Chemical Physics Letters

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Structure and stability of B13N13 polyhedrons with octagon(s)

Rui Li a, Li-Hua Gan a,*, Qian Li b, Jie An a

a School of Chemistry and Chemical Engineering, Southwest University, Chongqing 400715, Chinab Precision Driving Laboratory, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

a r t i c l e i n f o

Article history:Received 27 August 2009In final form 28 September 2009Available online 1 October 2009

0009-2614/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cplett.2009.09.094

* Corresponding author.E-mail address: ganlh@swu.edu.cn (L.-H. Gan).

a b s t r a c t

The structures and stabilities of B13N13 polyhedrons with alternant B and N atoms formed by squares,hexagons and octagons are studied with DFT method. It is found that the isomers with octagon(s) alsosatisfy the square adjacency penalty rule, and their relative energies markedly increase with the numberof octagons generally. However, an isomer with one octagon in C1 symmetry is thermodynamically morestable than other isomers and it has approximate sphericity and fewest B44 bonds. These findings suggestthat the isomers with octagon(s) should be considered during the search for the lowest energy isomer ofðBNÞn polyhedrons.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Carbon fullerenes and nanotubes have been the subject ofextensive experimental and theoretical investigation. Boron nitrideðBNÞn, the iso-electronic analogues of the fullerenes, has also re-ceived extensive attentions. A number of studies were undertakento synthesize and/or characterize boron nitride cages [1,2] andnanotubes [3–6]. Theoretical studies demonstrated that (BN)n

polyhedrons composed of squares and hexagons (F4F6-(BN)n poly-hedrons) with alternant B and N atoms are more stable than thefullerene-like cages with pentagons and hexagons, such as B12N12

[7], B12N12, B13N13�B14N14 and B16N16 [8]. Moreover, these F4F6-(BN)n polyhedrons obey the isolated-square rule and the squareadjacency penalty rule [9,10], which are the counterparts of thepowerful isolated pentagon rule and the pentagon adjacencypenalty rule for all-carbon fullerenes [11,12].

However, recent studies indicate that some (BN)n polyhedronswith octagons are more stable than those structures only com-posed of squares and hexagons [13–15]. For example, density func-tional theory study on B24N24 (B3LYP/6-31G*) reveals that theisomer with 2 octagons, 16 hexagons and 8 squares in S8 symmetryis more stable than the lowest energy F4F6 isomer with 20 hexa-gons and 6 squares in S4 symmetry [13,14]. Moreover, B24N24 havebeen synthesized by arc-melting method and detected by laserdesorption time-of-flight mass spectrometry, and its most stablestructure judged from molecular orbital calculations (PM5) is com-prised of 12 tetragonal, 8 hexagonal and 6 octagonal BN rings sat-isfying the isolated-square rule [16]. For B13N13 polyhedrons, themost stable structure consisting of squares and hexagons has beenreported by Strout et al. [8]. However, no other possible isomerswith octagon(s) are included.

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A DFT study is performed on all possible isomers of B13N13

formed by square (F4), hexagon (F6) and octagon (F8) to gain insightinto the structures and stability of B13N13 polyhedrons. The influ-ence of square–square bonds (for simplification, hereafter namedB44 bonds), asphericity (AS), pyramidalization angle (PA), sphericalaromaticity and the enthalpy–entropy interplay on the stability ofB13N13 polyhedrons are investigated in detail.

2. Computational details

According to the Euler’s theorem, the B13N13 polyhedrons sat-isfy the following equations:

V ¼ 2n; ð1ÞE ¼ 3n; ð2Þn4 � n8 ¼ 6; ð3Þn4 þ n6 þ n8 ¼ nþ 2; ð4Þ

where n, V and E denote the number of BN, vertices and edges; n4, n6

and n8 denote the number of squares, hexagons and octagons,respectively.

The coordinates of all possible F4F6F8 polyhedrons of B13N13

were constructed by a revised version of CAGE software [17] andafterward the carbon atoms were replaced with B and N atoms.There are 26 isomers, two of them are made from six squaresand nine hexagons (for simplification, hereafter named (F4F6-0F8-(BN)13), five from seven squares, seven hexagons and one octagon(F4F6-1F8-(BN)13), nine from eight squares, five hexagons and twooctagons (F4F6-2F8-(BN)13) and the remaining ten from ninesquares, three hexagons and three octagons (F4F6-3F8-(BN)13).

All the isomers are optimized by DFT method at the B3LYP/6-31G* [18] level of theory, which has been demonstrated to be reli-able for the description of the structures and properties of (BN)n

polyhedron recently [7,13]. Harmonic vibration frequencies are

Table 1B3LYP/6-31G* relative energy (RE, kcal mol�1), symmetry, HOMO–LUMO Gap (eV),asphericity (AS), the number of B44 bonds (E44), nucleus-independent chemical shift(NICS) as well as the average PA (�) of B (PAB) and N (PAN) atoms of B13N13

polyhedrons.

Isomer Sym. RE Gap AS E44 PAB PAN NICS

F4F6-0F8-1 C1 0.75 5.75 0.68 2 11.41 26.08 �5.01F4F6-0F8-2 C3V 44.73 6.19 0.78 3 11.77 26.52 �5.68F4F6-1F8-4 C1 0.00 6.04 0.32 2 10.63 25.65 �4.13F4F6-1F8-3 C1 28.36 5.99 0.77 3 11.18 26.20 �4.59F4F6-1F8-2 C1 37.90 5.87 1.13 3 11.29 26.25 �4.86F4F6-1F8-5 CS 54.81 5.82 0.51 3 11.34 26.30 �4.03F4F6-1F8-1 C1 79.22 5.69 1.37 4 12.15 27.23 �4.50F4F6-2F8-7 C1 38.65 6.20 0.41 4 10.91 25.48 �4.49F4F6-2F8-9 C1 55.04 6.15 1.05 4 10.66 25.79 �4.99F4F6-2F8-1 C1 63.18 5.82 1.10 4 11.06 26.11 �5.19F4F6-2F8-6 C1 66.89 5.65 0.78 4 11.28 26.96 �4.78F4F6-2F8-3 C1 81.01 5.57 0.37 4 11.58 26.97 �4.34F4F6-2F8-8 C1 89.14 4.62 0.94 5 11.75 26.27 �3.73F4F6-2F8-4 C1 125.49 5.59 1.15 5 12.77 26.37 �4.75F4F6-2F8-5 C1 141.84 5.05 1.34 6 12.46 28.45 �4.89F4F6-2F8-2 C1 182.10 4.29 2.45 6 13.44 29.43 �4.22F4F6-3F8-1 C1 69.53 5.99 0.48 5 10.97 25.92 �4.38F4F6-3F8-10 C3 90.78 4.44 0.36 6 11.03 26.08 �5.01F4F6-3F8-4 C1 91.29 5.83 0.84 6 11.13 26.25 �5.08F4F6-3F8-7 C1 118.71 5.17 0.87 6 11.45 27.69 �5.16F4F6-3F8-8 C1 120.89 5.06 1.16 7 11.80 27.79 �4.91F4F6-3F8-3 C1 131.67 5.05 1.40 7 11.83 27.69 �3.90F4F6-3F8-2 C1 145.78 4.32 2.11 7 11.91 27.71 �4.96F4F6-3F8-9 CS 148.70 5.16 0.75 7 11.96 28.34 �5.74F4F6-3F8-6 C3V 195.75 4.30 1.32 9 14.65 26.89 �7.92F4F6-3F8-5 CS 218.37 3.97 1.39 9 15.67 28.01 �2.25

122 R. Li et al. / Chemical Physics Letters 482 (2009) 121–124

also calculated to confirm that the optimized structures of the iso-mers with the lowest energy are minima on the potential energysurface. Rotational–vibrational partition functions are constructedfrom the optimized structures and vibration data with rigid rotatorand harmonic oscillator approximations, and no frequency scalingis performed. All the calculations are carried out with GAUSSIAN 03software package [19].

The AS [20] and PA [21] of all the considered isomers based onthe optimized structures are defined as in Eq. (5) and (6),respectively:

AS ¼X

i

ðri � r0Þ2

r20

; ð5Þ

hp ¼ hrp � 90�; ð6Þ

where A, B and C are the rotational constants, and hrp is the anglebetween p-orbit and its three adjacent B–N bonds.

Relative concentration (mole fractions) xi of the ith isomeramong the m isomers can be expressed through the partition func-tion qi and the ground-state energies DH�0;i by a compact formula[22]:

xi ¼qi exp½�DHo

0;i=ðRTÞ�Pm

j¼1qj exp½�DH�0;j=ðRTÞ�ð7Þ

where R is the gas constant and T is the absolute temperature. DH�0;iis the relative ground-state energies.

3. Results and discussion

3.1. Structures

The optimized structures of the lowest energy isomers with dif-ferent number of octagons are shown in Fig. 1 and the computedresults are given in Table 1. Each isomer has 39 B–N bonds, andthe bond lengths of different kinds of B–N bonds for the lowest en-ergy isomers are listed in Table 2.

As seen in Table 1, isomer F4F6-0F8-1 with two B44 bonds is thelowest energy isomer of F4F6-0F8-(BN)13 polyhedrons (as illus-trated in Fig. 1), which has been reported by Strout [8]. IsomerF4F6-1F8-4 with two B44 bonds, in C1 symmetry, the lowest energyisomer among F4F6-1F8-ðBNÞ13 polyhedrons is also the most stableone among all isomers of B13N13 polyhedrons. Isomer F4F6-2F8-7 isthe lowest energy isomer of F4F6-2F8-ðBNÞ13 polyhedrons with fourB44 bonds, in C1 symmetry, and isomer F4F6-3F8-1 is the lowest en-ergy isomer of F4F6-3F8-ðBNÞ13 polyhedrons with five B44 bonds, inC1 symmetry.

In order to further investigate the structure of B13N13 polyhe-drons, we examine the bond lengths of the lowest energy isomersbased on B3LYP/6-31G* level. Table 2 shows that the average bondlengths of the lowest energy isomers are all shorter than the B–Nsingle bond in H3B�NH3 (1.668 Å), and the average bond lengthsof B44 and B46 in F4F6-0F8-1 are close to those bond lengths of the

F4F6-0F8-1 F4F6-1F8-4

Fig. 1. The four most stable isomers of B13N13

B12N12 structure with Th symmetry [23]; but longer than that ofthe double bond in H2B ¼ NH2 (1.393 Å) at the same level of theory[7]. The order of the average bond lengths of all different kinds ofbonds is that B44 � B46 > B48 > B66 > B68 > B88.

3.2. Stability

As shown in Table 1, the most stable isomer is F4F6-1F8-4; fol-lowed by F4F6-0F8-1; which is 0.75 kcal mol�1 higher in energythan the former. The relative energies of the most stable isomersof F4F6-2F8-ðBNÞ13 and F4F6-3F8-ðBNÞ13 isomers are 38.65 and69.53 kcal mol�1, respectively. The frequency calculations at theB3LYP/6-31G* level demonstrate that the four most stable isomersare minima on the potential energy surface. The stability of ðBNÞnisomers is related to their HOMO–LUMO gaps [24]. In most cases,the HOMO–LUMO gaps of the most stable isomers of B13N13 are in-deed the largest ones among each kind of isomers.

In order to test the reliability of employed methods, BHandH-LYP/6-31G*, B3LYP/6-311+G* and MP2/6-31G* methods are alsocarried out for the four most stable isomers. The results are listedin Table 3. Table 3 shows that the stability order of B13N13 isomersbased on different methods are the same with each other.

It is known that electronic energetics itself cannot predict rela-tive stabilities in an isomeric system, especially at high tempera-

F4F6-2F8-7 F4F6-3F8-1

polyhedrons based on B3LYP/6-31G* level.

Table 2The B3LYP/6–31Ga average bond lengths (Å) and the range of average bond lengths of the lowest energy isomers of B13N13 polyhedrons.

Isomer B44 B46 B48 B66 B68 B88

F4F6-0F8-1 1.475 1.480 – 1.466 – –– 1.449/1.508 – 1.431/1.514 – –

F4F6-1F8-4 1.482 1.484 1.479 1.457 1.432 –1.480/1.484 1.438/1.523 1.453/1.491 1.421/1.512 1.411/1.443 –

F4F6-2F8-7 1.496 1.486 1.469 1.459 1.444 –1.462/1.522 1.460/1.511 1.439/1.501 1.443/1.476 1.423/1.466 –

F4F6-3F8-1 1.483 1.488 1.473 1.441 1.443 1.4261.465/1.492 1.454/1.538 1.420/1.506 – 1.433/1.451 1.417/1.454

Average 1.484 1.485 1.474 1.456 1.440 1.426

a Subscripts denote the bonds shared by corresponding square, hexagon and octagon.

Table 3The relative energies (kcal mol�1) of the most stable isomers with different methods.

Isomer BH and HLYP B3LYP MP2

6-31G* 6-31G* 6-311+G* 6-31G*

F4F6-0F8-1 0.07 0.75 1.02 1.24F4F6-1F8-4 0.00 0.00 0.00 0.00F4F6-2F8-7 42.04 38.65 38.98 45.04F4F6-3F8-1 75.04 69.53 69.90 79.11

R. Li et al. / Chemical Physics Letters 482 (2009) 121–124 123

tures, as stability interchange induced by the enthalpy–entropyinterplay is possible. As this situation is particularly pertinent tofullerene [22], the same situation can also be expected in ðBNÞnpolyhedrons. Consequently, we include entropic effects and evalu-ate the relative concentrations through the Gibbs free energy at theB3LYP/6-31G* level of theory. Considering the high computationalcost of these calculations, only five most stable isomers arecounted for equilibrium statistical thermodynamic analyses inthe present work. In Fig. 2, the temperature evolution of the equi-librium concentrations of B13N13 isomers has been evaluated. Itturns out that F4F6-1F8 � 4 prevails in the whole temperatureinterval considered. The second one is F4F6 � 0F8-1 over a widerange of temperatures. These results show that F4F6-1F8-4 not onlypossesses the lowest energy but also is thermodynamically moststable; meanwhile, F4F6-0F8-1 may be important component dur-ing the formation of B13N13 polyhedrons.

As shown in Table 1, the relative energies of different kinds ofisomers is that F4F6 � 0F8 � ðBNÞ13 < F4F6-1F8 � ðBNÞ13 F4F6�2F8 � ðBNÞ13 F4F6 � 3F8 � ðBNÞ13 as a whole. The average relative

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

50

60

70

80

90

100

X i (

%)

T(K)

F4F6-2F8-7

F4F6-1F8-2

F4F6-1F8-3

F4F6-0F8-1

F4F6-1F8-4

Fig. 2. B3LYP/6-31G* relative concentrations of the five lowest energy isomers ofB13N13 polyhedrons.

energies of them are 22.67, 40.06, 93.71 and 133.15 kcal mol�1,respectively. This demonstrates that the relative energies ofB13N13 isomers increase with the number of octagons. The morestable isomer has fewer B44 bonds and the relative energies of allisomers increases with the number of B44 bonds. For the structurescontaining octagons, according to Eq. (3), their squares increasewith the number of octagons, and so do the number of B44 bonds.This is the reason why the stability of isomers decreases withincreasing number of octagons. For the isomer F4F6-1F8-4, com-pared with the most stable isomer of F4F6ðBNÞ13 polyhedrons(F4F6-0F8-1), its number of B44 bonds is the same to that ofF4F6-0F8-1 although it contains an additional square. However,the average PAs of isomer F4F6-1F8-4 (PAB 10.63�, PAN 25.65�) islower than that of isomer F4F6-0F8-1 (PAB 11.41�, PAN 26.08�) lead-ing to lower local curvature, which is attributed to the introductionof one octagon. But, as a whole, the number of octagons in ðBNÞnpolyhedrons should be strictly limited, or else it can results in moresquares and B44 bonds.

For each kind of isomers, their relative energies also increaseswith the number of B44 bonds. The average energy penalty of aB44 bond in F4F6-0F8-ðBNÞ13, F4F6-1F8-ðBNÞ13, F4F6-2F8-ðBNÞ13 andF4F6-3F8-ðBNÞ13 isomers are 44.12, 39.98, 55.73 and 32.90kcal mol�1, respectively, which is similar to the role of pentagon–pentagon bonds in fullerenes [25]. This demonstrates that theB13N13 polyhedrons with octagon(s) also obey the square adjacencypenalty rule.

In order to get insight into the relationship between the stabil-ity and geometrical structures of B13N13 polyhedrons, we calcu-lated the AS of all B13N13 isomers based on the B3LYP/6-31G*geometries. The lower the value of AS, the more spherical the cor-responding ðBNÞn structures. The calculated AS results are listed inTable 1. The lowest energy isomer has lowest AS value for eachkind of isomers and the AS value of F4F6-1F8-4 is one of the lowesttwo. This demonstrates that the isomers with approximate sphe-ricity are more stable than other isomers.

It is known that spherical aromaticity is an important indicatorof chemical stability for a polycyclic p-electron system, and aro-matic molecules are chemically more stable than those less aro-matic or antiaromatic molecules. Maximum spherical aromaticityoccurs in icosahedral fullerenes when the valence p-shells arecompletely filled with 2(N + 1)2 electrons. This rule can also be ap-plied to less symmetrical small fullerenes [26,27]. The nucleus-independent chemical shift (NICS) has been widely accepted as areliable parameter of aromaticity [28,29]. Hence, we calculate theNICS values of all considered isomers to evaluate the stability. Asshown in Table 1, the calculated results demonstrate that the trendof NICS values is not in agreement with that of relative energies ofdifferent isomers. These results suggest that spherical aromaticitymay not be used to explain/predict the chemical stability ofðBNÞn polyhedrons.

124 R. Li et al. / Chemical Physics Letters 482 (2009) 121–124

For carbon fullerenes, a pyramidalization angle (PA) can beintroduced to measure the deviation of a sp2-hybridized carbonatom from the plane of three adjacent carbon atoms [21]. For a tri-valent carbon cage molecule, the atoms are in sp2 hybridization, andthere should be as little curvature as possible to achieve the nearlyideal sp2 geometry for r-skeleton and the overlapping of adjacentp-like orbits as large as possible [30]. To investigate if there is a sim-ilar rule in ðBNÞ13 polyhedrons with octagon(s), we calculate thePAs of all B and N atoms for each considered isomer. The averagePA of B (PAB) and N (PAN) atoms of each isomer based on theB3LYP/6-31G* optimized geometries are also listed in Table 1.

As shown in Table 1, the PAB are about 15� smaller than PAN forall isomers, suggesting that the surface of ðBNÞn polyhedrons withalternant B and N atoms is crinkly, as also seen from Fig. 1. Theseresults are evidently different from the cases in carbon fullerenes,in which the carbon atoms tend to form spherical surface and thedifference between the PA of adjacent atoms are generally slight.

Table 1 shows that the PA (PAB and PAN) of the four most stableisomers are the lowest ones in their isomers, and the lowest energyisomer F4F6-1F8-4 has a smaller PAB (10.63�) and PAN (25.65�) thanthe other three most stable isomers. It is known that the valencestates of B and N atoms are 2s22p1 and 2s22p3, respectively. ForðBNÞn polyhedrons, B atoms prefer the planar geometry (sp2

hybridization) and N atoms prefer the pyramidalization to accom-modate a lone pair of electrons [12]. It is obvious that the B and Natoms of the lowest energy isomers are closest to ideal hybridiza-tion. Accordingly, the PAs of B and N atoms determine the stabilityof B13N13 polyhedrons.

4. Conclusions

A DFT study is performed on all isomers of B13N13 polyhedronswith alternant B and N atoms. The results demonstrate that thenew proposed isomer with one octagon in C1 symmetry is the moststable one in all isomers. The isomers with octagon(s) also satisfythe square adjacency penalty rule, and their relative energiesmarkedly increase with the number of octagon(s). The thermody-namically most stable isomers have large HOMO–LUMO gaps,and lower sphericity value than other isomers. The pyramidaliza-tion of B and N atoms determine the stability of B13N13

polyhedrons.

Acknowledgements

Financial support from Southwest University, China (No.SWNUB2005002) and Key Laboratory of the Three Gorges Reser-voir Region’s Eco-Environment within Ministry of Education,Chongqing University (Grant No. KLVF-2007-5) is gratefullyacknowledged.

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