Multiscale Modelling in Computational Heterogeneous Catalysis - Multiscale Molecular Methods in Applied Chemistry - page 91

as the charges at the QM/MM boundary are carefully treated [ 171 ]. Introduction of
dangling hydrogen bonds or treating the frontier functional group as a pseudo-atom
with an effective one-electron potential are the most common approaches.
Nowadays many QM and MM software packages offer QM/MM capabilities.
ChemShell ( www.chemshell.org ) software is an example of a modular QM/MM
implementation.
Finding the transition states in high-dimensional spaces is a challenging prob-
lem. Transition states are first-order saddle points. The algorithms for finding first-
order saddle points on one spin PES can be divided into two groups: (1) approaches
based on interpolation between a reactant and a product minimum and (2) those
using only local information. A combination of both algorithms is probably the
most efficient way of finding first-order saddle points. Interpolation methods
generate a sequence of approximate MEP by interpolating between a reactant and
a product state. The highest energy configuration along an MEP is a first-order
saddle point. Both reactant and product states must be known so that these methods
cannot reveal unexpected chemical pathways with multiple intermediates. Further-
more, if multiple pathways exist, only that nearest to the interpolated guess will be
found [ 10 ]. The interpolation algorithms convert a saddle point search in configu-
ration space to a minimisation problem in discretised path space. Minimisation
problems in path space can easily handle large numbers of low-frequency modes,
a significant challenge for most local surface walking algorithms. Interpolation
algorithms include, for example, nudged elastic band (NEB) [ 172 ] and the string
method [ 173 ]. These methods initiate the search for a transition state by assuming
that the MEP is a straight line in multidimensional space connecting the reactant
and product states. Peters et al. [ 10 ] have shown that the growing string method, an
interpolation method that does not require an initial guess for the initial pathway,
needs significantly fewer gradient calculations to find the saddle point than the NEB
and the string method.
Local surface-walking algorithms explore the PES using local gradient and
usually second derivative information. These methods can be initiated any-
where on the PES. These algorithms perform poorly for systems with several
low-frequency vibrational modes or for searches started far from a transition state.
Furthermore, even if a transition state is found it is possible that it does not
connect reactant and product states. Therefore, it is recommendable to employ
an interpolation algorithm like the growing string method to generate a starting
point for the local surface walking algorithm. Two of the most used algorithms
of this type are the P-RFO method by Baker [ 174 ] and the dimer method by
Henkelman and J´nsson [ 175 ] or its improved version by Heyden et al. [ 11 ]. The
latter method is available in some commercial program packages like VASP [ 65 ]
or QChem [ 13 ].
The reaction rate constants are mostly calculated based on the harmonic TST.
Comprehensive review of this subject was presented by H anggi et al. [ 176 ]. The

rate coefficients for elementary reactions on a catalyst surface are obtained by
conventional TST in the following way:

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