Structure and electronic properties of C₃₆-rings

Jin Yajuan, Huang Yuanhe*, Li Yuxue, Liu Ruozhuang
(Department of Chemistry, Beijing Normal University, Beijing 100875, China)

Fig. 1 Reference frame of C₃₆-rings with D_nh symmetry for the case of belt A.

                            belt A                                                belt B
Fig.2 Two D_nh bonding-belts as shown by bold line area on the C₃₆ cages. The Greek alphabets denote atom indices.

Table 1. Energy E (Kcal/mol), standard deviation s (in Å)and intermolecular bond length (Å) of the C₃₆-rings

    From a view of energy, theoretical calculations here give that forming rings are favorable to the stabilization of the systems. This effect is similar to the previous results but quite different from the situation of the C₆₀-rings, among them more than half become unstable^[14].
    It can be seen that at least one of the intermolecular bond lengths is smaller than 1.65Å. Although 1.65Å is somewhat longer than 1.54Å of usual single C-C bond, it is still in the range of covalent lengths. Like the C₆₀-rings and the C₃₆-rings studied in ref. [13] and [14] therefore, these C₃₆-rings can be considered as covalent C₃₆ oligomers according to the bond lengths between neighbor C₃₆ cages listed in Table 1.
    The stability of C₃₆-rings may be affected by several factors in which the change of the strain energy play an important role. Formation of covalent bond between neighbor C₃₆ cages will cause the distortion of the C₃₆ cage, which results in the increase of the strain energy.
    The parameter s has been used to estimate the strain change of C₃₆ cage distortion for the rings^[13]. We choose D_6h C₃₆ cage optimized at AM1 level as a benchmark for the distortion of C₃₆ cage in the rings. As shown in Fig.1 a_n is the tangent plane of the C₃₆ cage perpendicular to line OO', P_n is the tangent point and T is the position of a possible bonding atom on the bonding-belt. TQ is the vertical distance from the possible bonding atom to a_n and concerned with the relative distance of possible bonding atom pair, such as T, H. The parameter s then can be obtained by

    where m is the number of TQ including reference point P_n (TQ =0 at P_n) for a certain linking pattern, such as m=5 for "aabb". The calculation of s is dependent only on the structure of a single undistorted C₃₆ cage, the linking pattern and the ring size. The value of s(Å) for every linking pattern and ring size are listed in Table 2. But only the rings with the values expressed by italic boldface type can be obtained by full optimization. It can be seen that smaller value of s is favorable to the formation of C₃₆-rings. The rings with s bigger than 0.21Å   cannot lead to convergent results. From Table 1, it can be inferred that the smaller the  the lower the energies of the rings for the same linking pattern except for the two D_7h rings. This indicates that the distortion has a great influence upon the stability and that parameter s can be used to determine the order of stability for those rings with the same linking pattern. It should be noted, however, that D_7h symmetry is unavailable in the GAMESS program. The two D_7h rings are obtained finally by the point-wise optimization. We think this is the main reason why the stability of D_7h rings does not follow the order of s.

Table 2. The values of the parameter s for all the rings considered here.

    Besides the strain, other factors such as the type of the bonding atoms, the size of retained conventional aromatic domains and the double bonds introduced into pentagons have also important influence on the stability of C₃₆-rings.
    The C₃₆ molecule with D_6h symmetry has three kinds of non-equivalent atoms a, b and g as shown in Fig.3. Bonding between the atoms at different position will cause different degree of distortion and reduction of aromatic domain in the cage.

Fig.3 Denotation of three non-equivalent atoms and the aromatic domains (bracketed by dashed lines) on the D_nh C₃₆ cage.

It was pointed out that large p-orbital axis vector (POAV) angle caused large strain on the sp² carbon atom and had high reactivity^[17]. The sp² carbon atom with large POAV angle turned into sp³ hybridization will lead to smaller distortion of the carbon cage. The POAV angles are 107.03°, 106.13° and 102.82° respectively for a, b and g. There are no convergent structures for those rings with pure g-g intermolecular bonds here. However, Table 1 shows that the rings through pure a-a connection, such as A-D_4h-aaand A-D_5h-aa, are less stable than those with pure b-b bonds, such as A-D_3h-bb and B-D_3h-b. Although a atom has larger POAV angle, formation of a-a bonds destroys the aromaticity of hexagon domains on the polar region of C₃₆ cage. Kroto has proposed that the derivative stability is increased for small fullerenes by maximizing conventional aromatic domains concomitant with alleviating cage strain^[18]. The rings containing C₃₆ cages with larger size of conventional aromatic domains should be relatively more stable. In addition, if a double bond is produced on the shared side of two abutting pentagons, larger strain will be introduced into the C₃₆ cage and decrease the stability of the rings. The shared pentagon-pentagon bonds of isolated D_6h C₃₆ cage can be considered as single bonds (1.511Å calculated by AM1). It is found that the stability of rings increase as the pentagon-pentagon bond tends to be a single bond. For instance the shared pentagon-pentagon bonds of the most stable ring (A-D_3h-aabb) are all longer than 1.511Å while the unstable ring (A-D_4h-bbg) has four of the bonds with the lengths of 1.409Å.
3.2 Electronic structures.
The calculated energy levels and symmetries of the higher occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), as well as the energy gaps between them (E_g.), are listed in Table 3 for the C₃₆-rings. It can be seen that the energy gaps are all bigger than that of C₃₆ cage except A-D_4h-aa. Here the most stable ring (A-D_3h-aabb) has the largest energy gap with the value of 5.92ev. It was reported that the energy gaps are reverent to kinetic stability of the fullerene^[19], larger energy gap results in high kinetic stability. Thus forming of ring structures is favorable to both kinetic and thermodynamic stability of the systems. The rings formed through b-b bonds have relatively large E_g such as A-D_3h-bb and B-D_3h-b while pure a-a bond leads to comparatively small ones such as A-D_4h-aa and A-D_5h-aa.
    It can also be found that doubly degenerated frontier orbitals (HOMOs or LUMOs) occur mainly in the rings with odd number of C₃₆ cages. For the rings with even size (n=2,4,6,8), only HOMO of the ring A-D_4h-bbg is degenerated with E_g symmetry. This is some similar to the cases of the C₆₀-rings, the doubly degenerated frontier orbitals are produced only for those with odd size (n=3,5,7)^[14]. Different distribution of energy levels will give different character of spectrum, which may help us to determine the ring structures.

Table 3. Energy gaps (E_g), HOMO and LUMO levels and symmetry of D_nh C₃₆-rings (in ev)

4. CONCLUSION
The D_nh C₃₆-rings constructed through the two new bonding-belts are investigated using the semi-empirical AM1 molecular orbital method. The strain-associated factor s is used to estimate the stability of the C₃₆-rings, and it can be regarded as an indicator of stability order for the rings with the same linking pattern. Besides strain, several interaction factors are also concerned with the stability of the C₃₆-rings, such as the size of conventional aromatic domains, the position of the bonding atoms, and the number of shared pentagon-pentagon double bonds. The energy gaps between HOMO and LUMO are all bigger than that of single D_6h C₃₆ cage except one, which may be in favor of kinetic stability of the C₃₆-rings.

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