Chemistry Reference
In-Depth Information
The transition state plays a central role in calculating reaction rates [
265
]andin
understanding the mechanism of chemical reactions. It is a stationary point on the
potential energy hypersurface with zero first derivatives (gradients). The force con-
stant matrix of a transition state has exactly one negative eigenvalue [
265
,
269
]. The
corresponding eigenvector is defined as the transition vector of the reaction
[
270
]. There is only one reaction path through a bona fide transition state
[
269
]. Excluding bifurcations along the pathways and the unlikely case of a monkey
saddle, a transition state connects only one educt to one product [
269
,
270
]. Special
symmetry rules have been derived for the transition vector and the symmetry of the
transition state conformation. Stanton and McIver derived the following four theo-
rems using the physical principle that structures that can be interconverted by
symmetry have equivalent energies, group theory, and geometrical arguments [
268
,
270
].
Theorem I. The transition vector cannot belong to a degenerate representation of
the transition state point group.
Theorem II. The transition vector must be
anti
-symmetric under a transition state
symmetry operation, which converts educts into products.
Theorem III. The transition vector must be symmetric with respect to a symmetry
operation, which leaves either educts or products unchanged.
Theorem IV. If the transition vector for the reaction E
1
!
P
1
is symmetric under
a symmetry operation
ˆ
which converts educt E
1
into an equivalent educt E
2
and P
1
into P
2
, then there exist lower energy transition states for the reactions E
1
!
E
2
and
P
2
; if the transition vector is
anti
-symmetric under
ˆ
P
1
!
then there exists a lower
energy transition state for the reaction E
1
!
P
2
.
Pechukas has analyzed the pathways of steepest descent of a potential surface
(defined in terms of a mass-scaled coordinate system) from a transition state to the
product and the educt [
248
]. Using the fact that symmetry cannot change along a
path of steepest descent and cannot decrease at the endpoints, he showed that “the
point group of any point along a reaction pathway, including the transition state can
be no larger than the largest subgroup common to both educt and product point
groups” [
248
]. However, educts and products have to be directly linked to the
transition state by steepest descent paths without intermediate minima or bifurca-
tions [
248
]. Moreover, there is one exception: if educts and products are
interconverted by a symmetry operation, this may be an additional symmetry
element present in the transition state (but not along the steepest descent path)
[
248
]. Schaad and Hu have listed the allowed point groups for transition states in
common cases of degenerate reactions [
261
].
Nourse has further analyzed the case of degenerate isomerizations where educt
and product are symmetry related and found that additional symmetry operators can
only appear in the transition state of self-inverse degenerate isomerizations [
271
]. A
degenerate isomerization is self-inverse if the coset of permutation-inversion oper-
ators leading from the reference isomer to the symmetry equivalent products
