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be inverted first, giving rise to the two parallel pathways. The transient structures
along the pathways have
C
s
(
xz
) symmetry. The transition vector has B
u
symmetry.
Additional imaginary frequencies of vibrational modes with A
u
and B
g
symmetry
(in the case of a higher order saddle point) would suggest conformations ta-
C
2
(
y
)
and a-
C
i
as possible transition states.
Note the analogy of this mechanism with the inversion of the
anti
-folded
conformation via a
syn
-folded transition state (Fig.
24b
). Conformations of the
same symmetry and type are involved. However, their respective role as minima
and transition states along the pathways are interchanged. From a symmetry point
of view, both scenarios are possible and cannot be distinguished since they differ
only in the relative energy of the conformations and their character as minima or
transition states. Alternatively, a-
C
2h
(
y
) may be a local minimum. In this case, the
energy profile would be that of a two-step process as discussed in Sect.
4.3.6
.
E
,
Z
-Isomerization with Simultaneous Inversion of One Tricyclic Moiety
of the
syn
-Folded Conformation
Table
14
is constructed by combining the symmetry operators of the s-
C
2v
(
x
)
conformation,
E
, (18
0
)(81
0
)(99
0
), (18)(1
0
8
0
)*, and (11
0
)(88
0
)(99
0
)*, with operators
corresponding to the process of
E
,
Z
-isomerization with inversion of one moiety,
(18), (1
0
8
0
), (11
0
88
0
)(99
0
), (18
0
81
0
)(99
0
), (18)*, (1
0
8
0
)*, (11
0
88
0
)(99
0
)*, and (18
0
81
0
)
(99
0
)*, and adding the required operators to achieve closure of the resulting set of
operators. The possible transition state conformations and their point groups, point
group order
h
TS
, number of versions
n
TS
, connectivity
C
, and number of parallel
pathways
p
are listed in Table
14
. The symmetry along the pathways of steepest
descent connecting the transition state to the
syn
-folded conformation and the
symmetry species of the vibrational mode corresponding to the transition vector
is also given. Note that, as in the case of
E
,
Z
-isomerization of the
anti
-folded
conformation, the smallest group combining all symmetry elements of both sets
is
G
16
. The point group
D
2d
combines some symmetry operators and some operators
corresponding to the dynamic process plus some additions to give a closed set.
In the case of t
⊥
-
D
2d
,(
4
) has no integer solution for
p
, indicating that this is not a
valid mechanism. For the transition states t
⊥
-
D
2d
, t
⊥
-
S
4
, and t
⊥
-
C
2v
(
d
) the point
groups along the pathway (
C
2
(
x
),
C
1
, and
C
1
, respectively), and hence the sym-
metry properties of the transition vector do not correspond to any non-degenerate
irreducible representation of the point group of the transition state. Thus, these
conformations do not qualify as bona fide transition states for the
E
,
Z
-isomerization
with simultaneous inversion of one moiety. However, they may be intermediates
(local minima), transition states of multi-step pathways with other intermediates on
the way, or transition states of pathways with bifurcations.
For the conformations s-
C
2v
(
x
), ts-
C
2
(
x
), f-
C
s
(
xz
), and s-
C
s
(
xy
), symmetry
constraints do not allow an
E
,
Z
-isomerization without
leaving the respective
point group. Thus,
the only options for direct
(one step, no bifurcation)
