Chemistry Reference
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contains at least one permutation of second order or a permutation of higher (only
even) order and its inverse [
271
,
272
].
3.5 Point Group Order and Number of Transition States
Bone et al. have derived two equations for the point group order and the number of
versions of the transition state [
253
]. The first equation relates the point group order
h
PG
of any conformation including transition states with the number of versions of
this conformation
n
and the order of the molecular symmetry group
h
MSG
[
253
]:
h
MSG
¼
n
h
PG
ð
3
Þ
Using the permutation operators corresponding to the various types of processes
and the versions of the conformations interconverted by them, connectivity net-
works may be generated. Based on these networks, Bone et al. derived an equation
which relates the number of versions of the transition states
n
TS
with the number of
versions of the interconverting minima
n
Min
, the connectivity
C
at the minima, and
the number of equivalent (parallel) pathways
p
effecting the interconversion of the
same two versions of the minimum [
253
]:
2
n
TS
¼
C
p
n
Min
ð
4
Þ
p
on the
left-hand-side and reflects the fact that a bona fide TS always connects two and only
two minima. Equation (
4
) does not apply in case of bifurcations along a pathway.
Assuming
p
The factor 2 on the right-hand-side of (
4
) corresponds to the term
C
1, the number of versions of the transition state
n
TS
may be inferred,
and using (
3
), the order of its point group
h
TS
. This assumption is equivalent to the
principle of maximum symmetry (PMS) of Metropoulos [
252
]. However, this is
only an upper limit for
h
TS
, since there may be more than one pathway (
p
¼
1)
[
251
,
253
,
273
]. Note that a higher number of parallel pathways leads to a lower
symmetry transition state. For recent publications on symmetry aspects of transition
states in degenerate reactions see [
261
,
262
].
>
3.6 Searching for Transition States
In summary, the symmetry of a transition state may include the operators of the
common subgroup of the educt and product point groups [
248
]. In the case of a
degenerate isomerization, these include all the symmetry operators of the confor-
mation in question. Furthermore, in self-inverse automerizations, the transition
state symmetry may also include permutation-inversion operators interchanging
the educt and product versions of the conformation. However, note that
the
