Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 469
in most simulations); q will consist of the Cartesian coordinates of each molecular center of
mass together with a set of other parameters specifying the molecular orientation. In any
case p is the set of conjugate momenta.
Usually the kinetic energy K takes the form:
2
N piμ
K = ∑ 2m
∑
i = 1
μ
i
[8.85]
where the index μ runs over the components (x, y, z) of the momentum of molecule i and m i
is the molecular mass.
The potential energy V contains the interesting information regarding intermolecular
interactions. A detailed analysis on this subject has been given in the previous sections; here
we shall recall only some basic aspects.
It is possible to construct from H an equation of motion that governs the time-evolution
of the system, as well as to state the equilibrium distribution function for positions and
momenta. Thus H (or better V) is the basic input to a computer simulation program. Almost
universally, in computer simulation the potential energy is broken up into terms involving
pairs, triplets, etc. of molecules, i.e.:
∑ 1 2 3
i
i j> i
i j> i k>> j i
() i ∑∑(
i j) ∑∑∑(
i j k)
V = v r + v r, r + v r, r , r + � [8.86]
The notation indicates that the summation runs over all distinct pairs i and j or triplets
i, j and k, without counting any pair or triplet twice. In eq. [8.86], the first term represents the
effect of an external field on the system, while the remaining terms represent interactions
between the particles of the system. Among them, the second one, the pair potential, is the
most important. As said in the previous section, in the case of molecular systems the interaction
between nuclei and electronic charge clouds of a pair of molecules i and j is generally a
complicated function of the relative positions r i and r j, and orientationsΩ i andΩ j (for atomic
systems this term depends only on the pair separation, so that it may written as v 2(r ij)).
In the simplest approximation the total interaction is viewed as a sum of pair-wise contributions
from distinct sites a in molecule i, at position r ia, and b in molecule j, at position r jb,
i.e.:
( ij, Ωi, Ω j ) = ∑ ∑ ab( ab)
v r v r
[8.87]
In the equation, v ab is the potential acting between sites a and b, whose inter-separation is r ab.
The pair potential shows the typical features of intermolecular interactions as shown
in Figure 8.1: there is an attractive tail at large separation, due to correlation between the
electron clouds surrounding the atoms (‘van der Waals’ or ‘London’ dispersion), a negative
well, responsible for cohesion in condensed phases, and a steeply rising repulsive wall at
short distances, due to overlap between electron clouds.
Turning the attention to terms in eq. [8.86] involving triplets, they are usually significant
at liquid densities, while four-body and higher are expected to be small in comparison
with v 2 and v 3. Despite of their significance, only rarely triplet terms are included in computer
simulations: that is due to the fact that the calculation of quantities related to a sum
over triplets of molecules are very time-consuming. On the other hand, the average