Chemistry Reference
In-Depth Information
New configurations are usually generated by displacing, removing, or adding
individual molecules. The acceptance of new states is performed most commonly
according to the Metropolis criterion.
In the production phase of MC simulations, all configuration-dependent proper-
ties fluctuate around constant average values that correspond to the thermodynamic
equilibrium. Each state is thereby sampled with a frequency proportional to its
equilibrium probability density [
182
]. In the canonical ensemble the probability
density
NVT
m
r
is given by [
181
]:
ð
E
m
=ð
k
B
T
ÞÞ
exp
NVT
m
r
¼
P
ÞÞ
;
(28)
exp
ð
E
m
=ð
k
B
T
all states
where
k
B
is the Boltzmann constant and
E
m
is the potential energy of a state
m
.
An advantage of MC is that it can be readily adapted to any ensemble [
11
].
Therefore, many MC ensembles have been developed for the simulation of specific
systems or properties. A wide variety of MC simulation techniques can thus be
found in the literature. Reviews and detailed information about MC techniques are
presented, e.g., in [
11
,
181
-
185
].
5.3 Methods for Determining Phase Equilibria
The calculation of vapor-liquid equilibria by molecular simulation is a longstand-
ing and important task. In the last two decades a variety of methods for this purpose
have been presented. There are, among others, thermodynamic scaling [
186
],
histogram reweighting [
187
,
188
], Gibbs-Duhem integration [
189
],
NpT
plus test
particle method [
190
], grand canonical ensemble [
191
], grand equilibrium method
[
192
], or the Gibbs ensemble MC method [
193
]. Here, some of these simulation
methods will be briefly addressed. A comprehensive discussion of the different
approaches can be found, e.g., in [
181
,
182
,
194
,
195
].
The Gibbs ensemble MC method (GEMC) [
193
] was developed to sample two
homogeneous coexisting phases that are in thermodynamic equilibrium but not in
physical contact with each other. The pressure and chemical potential of the phases
are equated by allowing the volume and the number of molecules to fluctuate
between the phases, while keeping the total volume and total number of molecules
constant. This ensemble is widely employed to calculate phase equilibria [
18
], also
in combination with Gibbs-Duhem integration [
189
,
196
]. It is also used to
simulate chemical reactions in phase equilibrium [
197
,
198
]. In the literature,
some advanced methods related to this ensemble can be found, e.g., the thermody-
namic scaling Gibbs ensemble [
199
].
In the grand canonical (GC) ensemble, a system at constant temperature, vol-
ume, and chemical potential is considered. The number of molecules is therefore
allowed to fluctuate. In such simulations, molecule displacement, insertion, and
deletion are attempted. From a series of several GCMC simulations, the pressure
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