Chemistry Reference
In-Depth Information
4.2.1 Notation of Permutation-Inversion Operators
Permutation operators may be written as cycles (123) which should be read as
(1
2
3
1), i.e., nucleus 1 is replaced by 2, 2 by 3, and 3 by 1 [
281
]. Like-
wise,
3), i.e., 1 and 2 are
interchanged, 3 remains in place. Atoms that do not change place (e.g., (3)) may
also be omitted for brevity. The identity operator is
E
and the laboratory-fixed
inversion is
E*
. Combinations of permutations and the inversion may be written,
e.g., (12)* [
246
,
247
,
281
].
Using the notation for permutation-inversion operators defined above and the
atom labeling in BAEs indicated in Fig.
1
, the 180
rotation about the horizontal
axis through the center of the overcrowded fjord regions may be written as
(12)(3) corresponds to (1
2
1) and (3
C
2
C
2
0
C
3
C
3
0
C
4
C
4
0
C
4a
C
4a
0
C
5
C
5
0
C
6
C
6
0
C
7
C
7
0
C
8
C
8
0
C
8a
C
8a
0
C
1
C
1
0
C
9
C
9
0
C
9a
C
9a
0
C
10a
C
10a
0
H
1
H
1
0
H
2
H
2
0
H
3
H
3
0
H
4
H
4
0
H
5
H
5
0
H
6
H
6
0
H
7
H
7
0
H
8
H
8
0
XY
ðÞ
This explicit notation of every atom in the molecule is only required when the
complete nuclear permutation-inversion group is considered and scrambling of
identical atoms is allowed. Considering as feasible only those permutations that
do not change the connectivity, i.e., do not break bonds [
247
], the element symbols
may be omitted since C
1
and H
1
necessarily have to undergo analogous permuta-
tions in order to conserve the bond C
1
-H
1
. Furthermore, the (non-hydrogen) atoms
of the molecule may be classified according to the topology into sets of atoms with
equivalent connectivity: {1, 8, 1
0
,8
0
}, {2, 7, 2
0
,7
0
}, {3, 6, 3
0
,6
0
}, {4, 5, 4
0
,5
0
}, {4a,
10a, 4a
0
, 10a
0
}, {8a, 9a, 8a
0
,9a
0
}, {9, 9
0
}, {X, Y}. Conserving the connectivity of
the molecule requires that the atoms in each of the six sets with four elements
(members in the set) undergo an analogous permutation. Likewise, the atoms in the
two sets with two elements undergo an analogous permutation. Thus, it is sufficient
to write only the permutations of one set of four and one set of two, e.g., the sets {1,
8, 1
0
,8
0
} and {9, 9
0
}, implying that, under the constraint of conserving the
connectivity, analogous permutations apply to all the other carbon, hydrogen, and
hetero atoms. The above-mentioned 180
rotation may thus be written in shorthand
as (11
0
)(88
0
)(99
0
).
Conserving bond connectivity imposes an additional constraint on the feasible
permutation operators: if atoms 9 and 9
0
are interchanged, all other atoms of the first
moiety also have to be interchanged with corresponding atoms of the second
moiety, i.e
.
, (99
0
) necessarily requires either (11
0
)(88
0
), or (18
0
)(81
0
), or (11
0
88
0
),
or (18
0
81
0
). It should be noted that the above considerations strictly apply only to
unsubstituted, homomerous BAEs. For heteromerous BAEs, and substituted BAEs,
the different constitution requires some modifications, which reflect the lower
symmetry. The corresponding molecular symmetry groups are subgroups of the
molecular symmetry group of unsubstituted homomerous BAEs (see below).
