158 Canonical ensemble
(^051015)
r (Å)
-200
-100
0
100
200
V
(r) (K)
(^051015)
r (Å)
0
0.5
1
1.5
2
2.5
3
g(r)100 K
200 K
300 K
400 K
(a) (b)Fig. 4.2(a) Potential as a function of the distancerbetween two particles withσ= 3.405
̊A andǫ= 119.8 K. (b) Radial distribution functions at four temperatures.
The potential between any two particles is shown in Fig. 4.2(a), where we can clearly
see an attractive well atr= 2^1 /^6 σof depthǫ. The radial distribution function for such
a system withσ= 3.405 ̊A,ǫ= 119.8 K,m=39.948 amu,ρ=0.02 ̊A−^3 (ρ∗= 0.8) and a
range of temperatures corresponding to liquid conditions is shown inFig. 4.2(b). In all
cases, the radial distribution functions show a pronounced peak inthe ranger=3.6-
3.7 ̊A depending on temperature, compared to the location of the potential energy
minimumr= 3.82 ̊A. The presence of such a peak in the radial distribution function
indicates a well-defined coordination structure in the liquid. Figure 4.2also shows
clear secondary peaks at larger distances, indicating second and third solvation shell
structures around each particle. We see, therefore, that spatial correlations survive out
to at least two solvation shells at the higher temperatures and three (or nearly 11 ̊A)
at the lower temperatures.
Note that the integral ofg(r) over all distances gives
4 πρ∫∞
0r^2 g(r) dr=N− 1 ≈N, (4.6.21)indicating that if we integrate over the correlation function, we must find all of the
particles. Eqn. (4.6.21) further suggests that the integration ofthe radial distribution
function under the first peak should yield the number of particles coordinating a given
particle in its first solvation shell. This number, known as thecoordination number,
can be written as
N 1 = 4πρ∫rmin0r^2 g(r) dr, (4.6.22)whererminis the location of the first minimum ofg(r). In fact, a more general “run-
ning” coordination number, defined as the average number of particles coordinating a
given particle out to a distancer, can be calculated via according to
N(r) = 4πρ∫r0̃r^2 g( ̃r) d ̃r. (4.6.23)