Chemistry Reference
In-Depth Information
From this probability density, the mean time between successive events is
R
1
. The time to an event is
h
t
i¼
ð
t
d
t
0
Re
Rt
0
e
Rt
T
ðtÞ¼
¼
1
(20)
0
e
Rt
is uniformly distributed
between [0, 1], and this relation allows one to obtain the real time
which lies between [1, 0]. So a random variable
U
¼
t
(in units of MC
steps) between successive events as
ln
U
R
t ¼
:
(21)
This random sampling of the Poisson time distribution for each chosen event
ensures that a direct and unambiguous relationship between a real time step and an
MC step is established. There are various algorithms for kMC simulations possible.
In connection with catalytic reactions on surfaces, Lukkien et al. [
126
] have deve-
loped an efficient time-order list algorithm. Kinetic MC simulations allow one to
bridge time scales over several orders of magnitude.
MD calculates the “real” dynamics of the system, from which time averages of
properties can be calculated. In contrast to MC, with MD non-equilibrium proper-
ties like, for example, transport diffusivities can also be calculated. This is impor-
tant for calculating diffusion of reactants and products in porous catalyst supports.
The position of the molecules as a function of time are obtained by integrating
Newton's equation of motion over several thousand or even million time steps,
typically up to a few femtoseconds (10
15
) per step. At each step, the forces on the
molecules are computed and combined with the current positions and velocities to
generate new positions and velocities a short time ahead. The force acting on each
molecule is assumed to be constant during the time interval. The molecules are then
moved to new positions, the forces are updated, and so on. By this approach,
trajectories of all molecules are generated.
A simplified MD algorithm is presented in Fig.
4b
.
The MD approach therefore provides information about the time dependence of
the properties of the system whereas there is no such information within the MC
scheme. In an MC simulation the outcome of each trial move depends only upon its
immediate predecessor, whereas in MD it is possible to predict the configuration of
the system at any time in the future. MD has a kinetic energy contribution to the
total energy whereas in an MC simulation the total energy is determined from the
potential energy function only. In general, MD simulations are executed for a
micro-canonical (N,V,E
const.) ensemble whereas MC simulations are executed
for a canonical (N,V,T) ensemble. MD and MC schemes can, however, be modified
for other ensembles.
Three aspects have to be considered with MD simulations: (1) the model
describing the intra- and inter-particle interactions, (2) the calculation of ener-
gies and forces from the model and (3) the algorithm employed to integrate the
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