Chemistry Reference
In-Depth Information
;
E
þ
RT
k
B
TQ
TS
ð
T
Þ
k
TST
ð
T
Þ¼
exp
(31)
hQ
R
ð
T
Þ
where
k
B
is Boltzmann's constant,
h
Planck's constant,
T
is the absolute temperature
and
E
+
is the difference in electronic energies between the transition state and the
reactant state, respectively. The partition functions of the transition state,
Q
TS
(
T
), and
the reactant state may be calculated, for example, like this:
!
o
el
;
p
s
0
:
5
3
n
6
Y
2
p
Mk
B
T
h
2
T
3
y
A
y
B
y
C
1
Q
¼
(32)
e
y
vj=T
1
j
where
M
is the molecular mass,
y
i
are the moments of inertia, the
y
vj
the normal
modes and
o
el
the electronic energy. A transmission coefficient can also be intro-
duced which has the general form like this [
19
]:
gð
T
Þ¼Gð
T
Þkð
T
Þ
g
ð
T
Þ;
(33)
(
T
) arises from dynamical recrossing. It takes into account that some
trajectories that cross the dividing surface in the direction of products recross and
return to the reactant region.
G
where
(
T
) arises from quantum
mechanical tunnelling.
k
(
T
) is greater or equal to one.
g
(
T
) takes deviations of the
equilibrium distribution in phase space into account.
g
(
T
) can be either less than or
greater than one. In conventional TST
G
(
T
) is smaller than one.
k
(
T
) is set equal to one. Further develop-
ments of TST may be found in papers by Truhlar's group [
7
-
9
].
There are important examples where the harmonic/rigid-rotor approximation to
TST fails in describing the reaction kinetics. Even worse, simulations based on the
static approach can sometimes lead to completely incorrect prediction of the
reaction mechanism. For example, in catalytic transformations of short alkanes,
entropy plays an important role. During the reaction the mobility of the reactants
varies according to the strength of their interactions with the zeolite, leading to a
substantial entropy contribution to the free-energy reaction barrier. Entropy can
even stabilise some otherwise unstable reaction intermediates, opening unexpected
alternative reaction channels competing with the mechanism deduced from a static
TST search. Therefore, one has to explore the free-energy surface and not just of the
PES in configuration space. Bucko and Hafner [
177
] have shown that the static
approach, corrected for dynamical effects within harmonic TST, is insufficient for
describing reactions including weakly bound adsorption complexes such as hydro-
carbon conversion reactions. The most important reasons for this failure were found
to be as follows. (1) An adsorption complex identified by static total-energy
minimisation is not a proper representation of the reactant state. Hence the work
needed to create an adsorption complex represents in some cases an important
contribution to the free-energy barrier. This contribution is not taken into account in
harmonic TST, which is based on the analysis of the energy surface in the vicinity
of stationary points only. (2) The static approach does not account for reaction
g
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