Chemistry Reference
In-Depth Information
Fig. 11 Scheme of a
bifurcating reaction path
M1
TS1
VRI
TS2
M2
state, TS2, rather than to a minimum as schematically outlined in Fig.
11
.This
situation leads to a bifurcation of the reaction paths and occurs, in particular, when
the two transition states TS1 and TS2 share symmetry elements that are not present
in the point groups of the minima M1 and M2, but interchange M1 and M2.
There are two viewpoints describing such a situation. The first opinion assumes
steepest descent paths in a mass-weighted coordinate system as a model for the
reaction path: a vibration-rotation-less motion of a reacting system is assumed [
248
,
254
,
262
,
275
-
277
]. According to this view, the path from TS1 to M1 and M2
proceeds in two steps according to TS1
M2)as
indicated by the straight reaction arrows in Fig.
11
. Note that the products of the
second step (the final products of the reactions) may be identified by following the
steepest descent paths starting at TS2. Symmetry is conserved along these steepest
descent paths, and the symmetry rules for reactions apply to each individual step
[
248
,
261
,
262
,
277
]. However, it is accepted that a transition state (TS2) may be the
end point of a steepest descent path, i.e., the product of a reaction step. Steepest
descent paths are useful mathematical tools for studying reactions on potential
energy surfaces [
261
]. However, this picture ignores the following peculiarity:
close to the bona fide transition state
TS1
, the reaction system follows the floor
of a valley. This is a stable pathway in the sense that there exists a restoring force
that tends to drive the trajectories of the system back to the floor. However, at a
certain point, the valley floor turns into the crest of a ridge and the symmetry
conserving steepest descent path continues along the crest of the ridge down to TS2.
The reaction path is expected to bifurcate close to the valley-ridge inflection point
(VRI), because any small deviation from symmetry (e.g., due to vibration) will
cause the system to veer away further and further [
278
]. Thus the system will head
towards the final products M1 or M2, taking a shortcut as indicated by the curved
arrows in Fig.
11
and not pass through TS2. The reacting molecule does not follow
the steepest descent path near and below a valley-ridge inflection point [
261
,
278
]. The valley-ridge inflection point is characterized by a Hessian matrix with a
zero eigenvalue, the corresponding eigenvector being orthogonal to the gradient at
that point [
278
]. Note that this is not a stationary point. The gradient does not
vanish. In conclusion, steepest descent paths are not good models for bifurcating
reaction paths below valley-ridge inflection points [
278
]. However, steepest descent
paths and the corresponding symmetry rules are still very useful tools in studying
TS2
M1 (or TS1
TS2
⇌
⇌
⇌
⇌
