Essentials of Computational Chemistry

86 3 SIMULATIONS OF MOLECULAR ENSEMBLES
on describing the solvation structure of water about sodium ions in general. Then, we can
define the total number of oxygen atoms nB within some distance range about any sodium
ion (atom A) as
nB{r, r} =NBP {A, B,r,r} (3.41)
We may then use Eq. (3.40) to write
nB{r, r} =4πr 2 ρBgAB(r)r (3.42)
where ρB is the number density of B in the total spherical volume. Thus, if instead of gAB(r)
we plot 4πr 2 ρBgAB(r), then the area under the latter curve provides the number of molecules
of B for arbitrary choices of r and r. Such an integration is typically performed for the
distinct peaks in g(r) so as to determine the number of molecules in the first, second, and
possibly higher solvation shells or the number of nearest neighbors, next-nearest neighbors,
etc., in a solid.
Determining g(r) from a simulation involves a procedure quite similar to that described
above for determining the continuous distribution of a scalar property. For each snapshot of
an MD or MC simulation, all A–B distances are computed, and each occurrence is added
to the appropriate bin of a histogram running from r = 0 to the maximum radius for the
system (e.g., one half the narrowest box dimension under periodic boundary conditions, vide
infra). Normalization now requires taking account not only of the total number of atoms A
and B, but also the number of snapshots, i.e.,
V
gAB(r) =
4πr2rMNANB M NA NB
Qm r; rAiBj
m=1 i=1 j=1
(3.43)
where r is the width of a histogram bin, M is the total number of snapshots, and Qm is
the counting function
Q
1 if r − r/2 ≤ rAiBj