Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 475
In the MC methods the integration is done by re-writing in terms of the probabilities
P k of finding the configuration R k:
P
=∑
( )
f f R P
k
=
k
−β
e
e
∑
k
k k
V( Rk)
−βV(
Rk)
[8.105]
[8.106]
The search for the most probable configuration R k is done using Markov’s chain theory,
so that after setting the system at a given configuration R i, another configuration R j is
selected, making a random move in the coordinates. Not all configurations are accepted; the
decision on whether or not to accept a particular configuration is made in such a way as to
ensure that asymptotically the configuration space is sampled according to the probability
density appropriate to a chosen ensemble. In particular:
if V( R) < V( R)
j i
the move is accepted in the chain. Otherwise,
if V( R) > V( R)
j i
the move is subjected to a second screening and accepted
if e
−β( Vj−Vi) < γ
where γ is a random number between 0 and 1.
The original models used in MC were highly idealized representations of molecules,
such as hard spheres and disks, but nowadays MC simulations are carried out on the basis of
more reliable interaction potential. The use of realistic molecule-molecule interaction potential
makes it possible to compare data obtained from experiments with the computer generated
thermodynamic data derived from a model. The particle momenta do not enter the
calculation, there is no scale time involved, and the order in which the configurations occur
has no special significance.
It is worth noticing that, because only a finite number of states can be generated in a
simulation, the results obtained from a MC simulation are affected by the choice of initial
conditions, exactly as previously said for the MD results.
As we have already said, it is possible to extend the MC method, originally formulated
in the microcanonical ensemble, to other ensembles. Particularly important in the study of
liquid systems is the extension of the Monte Carlo method to the grand canonical μVT ensemble:
76 that is due to the fact that such an extension permits to calculate the free energy of
the system, a quantity of particular significance in the chemistry and physics of liquids. The
grand canonical MC involves a two-stage process: the first stage is exactly identical to what
is done in the conventional MC, with the particles moving and the move accepted or rejected
as previously said. The second stage involves an attempt either to insert a particle at a
randomly chosen point in the cell or to remove a particle that is already present. The deci-