Handbook of Solvents - George Wypych - ChemTech - Ventech!

468 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
who used an expansion in powers of a softness parameter, n -1 , about n -1 =0 (corresponding to
a hard sphere fluid). This method gives good results for the repulsive, but not for the attractive,
part of the potential. Subsequently several researchers attempted to combine the advantages
of both approaches. The first successful method was that of Barker and Henderson
(BH), 60 who showed that a second-order theory gave quantitative results for the thermodynamic
properties of a Lennard-Jones liquid. Even more rapidly convergent results were
later obtained by Weeks, Chandler and Andersen (WCA) 61 using a somewhat different reference
system.
Given the great success of perturbation theories in treating the properties of atomic liquids,
a large effort has been devoted to extending the methods to deal with molecular systems.
The first rigorous application of these theories to molecular fluids seems to have been
made in 1951 by Barker, 62 who expanded the partition function for a polar fluid about that
for a fluid of isotropic molecules.
Roughly, the basic problem is the same as in the atomic case, but the practical difficulties
are much more severe. A possible approach is to choose a reference-system potential
v0(r) spherically symmetric so that the integrations over the orientations (absent in atomic
fluids) involve only the perturbation. The real system can be thus studied by a straightforward
generalization of the λ-expansion developed for atomic fluids.
Perturbation theories based on spherically symmetric reference potentials, however,
cannot be expected to work well when (as in the most real molecules) the short-range repulsive
forces are strongly anisotropic. The natural approach in such cases is to include the
strongly varying interactions in the specification of an isotropic reference potential to relate
the properties of the reference system to those of hard molecules having the same shape. 63
Calculation of this type have been made by Tildesley, 64 exploiting a generalization of the
WBA approach quoted above. In a similar way also, the BH perturbation theory has been
generalized with good results. 65 The main disadvantage of these methods is the large computational
effort that their implementation requires.
8.7.2 COMPUTER SIMULATIONS
In this section we will give a brief review of methodologies to perform computer simulations.
These approaches nowadays have a major role in the study of liquid-state physics:
they are developing so fast that it is impossible to give a complete view of all the different
methodologies used and of the overwhelmingly large number of applications. Therefore we
will only give a brief account of the basic principles of such methodologies, of how a computer
simulation can be carried out, and we will briefly discuss limitations and advantages
of the methods. The literature in the field is enormous: here we will refer interested readers
to some of the basic textbooks on this subject. 42,66,67
The microscopic state of a system may be specified in terms of the positions and
momenta of the set of particles (atoms or molecules). Making the approximation that a classical
description is adequate, the Hamiltonian H of a system of N particles can be written as
a sum of a kinetic K and a potential V energy functions of the set of coordinates qi and
momenta pi of each particle i, i.e.:
H( q, p) = K() p + V() q
[8.84]
In the equation the coordinates q can simply be the set of Cartesian coordinates of each
atom in the system, but in this case we treat the molecules as rigid bodies (as is usually done