Handbook of Solvents - George Wypych - ChemTech - Ventech!

476 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
sion whether to make an insertion or a removal is made randomly with equal probabilities.
The trial configuration is accepted if the pseudo-Boltzmann factor:
N
[ Nβμ−log N! −βVN
( r ) ]
W = e
N
increases. If it decreases, the change is accepted when equal to:
WN+
1 1 [ βμ−β( VN+ 1−VN]
= e
W N + 1
N
for the insertion, or to:
WN
W
N
[ βμ β( VN+ VN]
= Ne
−1 − − 1−
[8.107]
[8.108]
[8.109]
for the removal.
This method works very well at low and intermediate densities. As the density increases,
it becomes difficult to apply, because the probability of inserting or removing a particle
is very small.
As we have already said, the grand canonical Monte Carlo provides a mean to determining
the chemical potential, and hence, the free energy of the system. In other MC and
MD calculations a numerical value for the free energy can always be obtained by means of
an integration of thermodynamic relations along a path which links the state of interest to
one for which the free energy is already known, for example, the dilute gas or the low-temperature
solid. Such a procedure requires considerable computational effort, and it has a low
numerical stability. Several methods have been proposed and tested.
The difficulties in using MC and MD arise from the heavy computational cost, due to
the need of examining a large number of configurations of the system usually consisting of a
large number of particles. The size of the system one can study is limited by the available
storage on the computer and by the speed of execution of the program. The time taken to
evaluate the forces or the potential energy is proportional to N 2 .
Other difficulties in molecular simulations arise from the so-called quasi-ergodic
problem, 77 i.e., the possibility that the system becomes trapped in a small region of the phase
space. To avoid it, whatever the initial conditions, the system should be allowed to
equilibrate before starting the simulation, and during the calculation, the bulk properties
should be carefully monitored to detect any long-time drift.
As already said, apart from the initial conditions, the only input information in a computer
simulation are the details of the inter-particle potential, almost always assumed to be
pair-wise additive. Usually in practical simulations, in order to economize the computing
time, the interaction potential is truncated at a separations r c (the cut-off radius), typically of
the order of three molecular diameters. Obviously, the use of a cut-off sphere of small radius
is not acceptable when the inter-particle forces are very long ranged.
The truncation of the potential differently affects the calculation of bulk properties,
but the effect can be recovered by using appropriate “tail corrections”. For energy and pressure
for monoatomic fluids, for example, these “tail corrections” are obtained by evaluating