Essentials of Computational Chemistry

3.5 ENSEMBLE AND DYNAMICAL PROPERTY EXAMPLES 85
We may thus interpret the l.h.s. of Eq. (3.39) as a probability function. That is, we may
express the probability of finding two atoms of A and B within some range r of distance
r from one another as
P {A, B,r,r}= 4πr2
V gAB(r)r (3.40)
where, in the limit of small r, we have approximated the integral as gAB(r) times the
volume of the thin spherical shell 4πr 2 r.
Note that its contribution to the probability function makes certain limiting behaviors
on gAB(r) intuitively obvious. For instance, the function should go to zero very rapidly
when r becomes less than the sum of the van der Waals radii of A and B. In addition, at
very large r, the function should be independent of r in homogeneous media, like fluids,
i.e., there should be an equal probability for any interatomic separation because the two
atoms no longer influence one another’s positions. In that case, we could move g outside
the integral on the l.h.s. of Eq. (3.39), and then the normalization makes it apparent that
g = 1 under such conditions. Values other than 1 thus indicate some kind of structuring in
a medium – values greater than 1 indicate preferred locations for surrounding atoms (e.g.,
a solvation shell) while values below 1 indicate underpopulated regions. A typical example
of a liquid solution r.d.f. is shown in Figure 3.4. Note that with increasing order, e.g., on
passing from a liquid to a solid phase, the peaks in g become increasingly narrow and the
valleys increasingly wide and near zero, until in the limit of a motionless, perfect crystal, g
would be a spectrum of Dirac δ functions positioned at the lattice spacings of the crystal.
It often happens that we consider one of our atoms A or B to be privileged, e.g., A
might be a sodium ion and B the oxygen atom of a water and our interests might focus
g
1
0
Figure 3.4 A radial distribution function showing preferred (g >1) and disfavored (g