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Transition
s
tructure
A(g), B(g)
A(ads, O),
B(ads, O
)
C(g)
I
1
E
A(ads, H
+
),
B(ads, H
+
)
C(
ads, O)
-1
E
C(ads, H
+
)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Reaction coordinate
Fig. 7 Schematic energy diagram for a zeolite-catalysed one-step reaction A + B
!
C. (1)
Reactants in the gas phase. (2) Adsorption of A and B in the zeolite channels. (3) Co-adsorption
of A and B on the Brønsted acid site. (4) Formation of the transition state. (5) Product adsorption
on the Brønsted site. (6) Product adsorption in the channel system. (7) Product in the gas phase
reaction mechanism for the alkylation was elucidated using DFT applied to cluster
representations of the active site. Second, the MP2:DFT hybrid approach suggested
by Tuma and Sauer [
193
] was employed. Structure optimisation of all stationary
points within the full ZSM-5 unit cell using DFT with periodic boundary conditions
has been the first step in this approach. Stationary points were characterised by
harmonic frequencies obtained by diagonalisation of the full dynamical matrices
(absence of imaginary frequencies for minima and presence of exactly one imagi-
nary frequency for all transition structures). To confirm that the transition states
were connected to the correct energy minima, each transition state was perturbed
slightly along the reaction coordinate in the reactant and product direction. The per-
turbed geometries were used as starting geometries for energy minimisation. The
second step was the calculation of the high-level correction; that is the difference
between MP2 and DFT single-point adsorption energies and energy barriers, res-
pectively, for clusters of progressively larger size. These clusters were cut out from
the periodic DFT-optimised structure. The size-dependent high-level correction
[
187
]
D
E
ð
C
Þ
high
¼ D
E
ð
C
Þ
MP
2
D
E
ð
C
Þ
DFT
(34)
was then extrapolated to the periodic structure (S). This periodic model limit,
D E
ð
S
Þ
high
;
is added to the plane-wave DFT energy for the periodic structure,
D
E
(S)
DFT
, to get an estimate of the MP2 energy for the full periodic system,
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