chemia - Studia

M. MIRZARGAR, M.J. NADJAFI-ARANIA, A.R. ASHRAFIA
Suppose B 1 = {a 1 , a 2 }, B 2 = {b 1 , b 2 , b 3 }, B 3 = {b 4 , b 5 , b 6 , b 7 , b 8 ,
b 9 } and B 4 = {b 10 , b 11 , b 12 , b 13 , b 14 , b 15 , b 16 , b 17 , b 18 , b 19 , b 20 , b 21 }. Then
B 1 ∪ B 2 , B 1 ∪ B 2 ∪ B 3 and B 1 ∪ B 2 ∪ B 3 ∪ B 4 are generating sets of the
topological symmetry of D 1 [2], D 1 [3] and D 1 [4]. From these calculations, we
define permutations and , 2 ≤ i ≤ n, as follows:
Then B 1 ∪ B 2 ∪ B 3 ∪ … ∪ B n is a generating set for D 1 [n], where B i =
{ }.
We now consider the dendrimer molecule D 2 [n], Figure 2. The
topological symmetry group of the core of this dendrimer is isomorphic to
S 4 . This group can be generated by a 1 = (1,2), a 2 = (1,3) and a 3 = (1,4). In
order to characterize the symmetry of this molecule we note that each
dynamic symmetry operation of D 2 [n], considering the rotations of XY 2
groups in different generations of the whole molecule D 2 [n], is composed of
n sequential physical operations. We first have a physical symmetry of the
framework (as we have to map the XY 2 groups on XY 2 groups which are on
vertices of the framework). Such operations form the group G of order 24,
which as is well known to be isomorphic to S 4 or Sym(4). After accomplishing
the first framework symmetry operation we have to map each of the four
XY 2 group on itself in the first generation and so on. This is a group
isomorphic to H = ((…(Z 2 ∿ Z 2 ) ∿ Z 2 ) ∿ … )∿ Z 2 ) ∿ Z 2 with n – 1 components.
Therefore, the whole symmetry group is isomorphic to H ∿ G. This is a group
of order .
Suppose B 1 = {a 1 , a 2 , a 3 }, B 2 = {b 1 , b 2 , b 3 , b 4 }, B 3 = {b 5 , b 6 , b 7 , b 8
, b 9 b 10 , b 11 , b 12 } and B 4 = {b 13 , b 14 , b 15 , b 16 , b 17 , b 18 , b 19 , b 20 , b 21 , b 22 ,
b 23 , b 24 , b 25 , b 26 , b 27 , b 28 }, where b i 's are defined as follows:
Table 2. Generating Sets for D 2 [2], D 2 [3] and D 2 [4].
b 1 = (5,6) b 2 = (7,8) b 3 = (9,10) b 4 = (11,12) b 5 = (13,14) b 6 = (15,16)
b 7 = (17,18) b 8 = (19,20) b 9 = (21,22) b 10 = (23,24) b 11 = (25,26) b 12 = (27,28)
b 13 = (29,30) b 14 = (31,32) b 15 = (33,34) b 16 = (35,36) b 17 = (37,38) b 18 = (39,40)
b 19 = (41,42) b 20 = (43,44) b 21 = (45,46) b 22 = (47,48) b 23 = (49,50) b 24 = (51,52)
b 25 = (53,54) b 26 = (55,56) b 27 = (57,58) b 28 = (59,60)
.
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