Biomedical Engineering Reference
In-Depth Information
10.3.3 Lennardjonesium
The first simulation of a 'real' chemical system was Rahman's (1964) study of liquid
argon. He studied a system comprising 864 Lennard-Jones (L-J) particles under conditions
appropriate to liquid argon at 84.4 K and a density of 1.374 g cm
−
3
. Once again, there is
much to be gained by studying the abstract, so here is the first part of it.
A system of 864 particles interacting with a Lennard-Jones potential and obeying classical
equations of motion has been studied on a digital computer (CDC 3600) to simulate molecular
dynamics in liquid argon at 94.4 K and a density of 1.374 g cm
−
3
. The pair correlation function
and the constant of self-diffusion are found to agree well with experiment; the latter is 15%
lower than the experimental value. The spectrumof the velocity autocorrelation function shows
a broad maximum in the frequency range ω
=
0.25(2π
k
B
T
/
h
). The shape of the Van Hove
function
G
s
(
r
,
t
) attains a maximum departure from a Gaussian at about
t
=
0.3
×
10
−
2
s and
becomes a Gaussian again at about 10
−
11
s ...
There are several interrelated problems. A sample size has to be chosen; this is usually
determined by the available computer resource and the complexity of the potential function,
because the potential function has to be calculated very many times during the simulation.
The number of particles and the density determine the size of the container. At the same
time we need to decide on a potential function; the natural choice for the inert gases is the
L-J potential, and we should note that the L-J potential is essentially short range.
So many early papers used the L-J potential that the noun
Lennardjonesium
was coined
to describe a nonexistent element whose atoms interacted via the L-J potential.
10.4 Algorithms for Time Dependence
Once we have calculated the potential and hence the force by differentiation, we have to
solve Newton's equation of motion. If
F
A
is the force on particle A, whose position vector
is
r
A
and whose mass is
m
A
then
m
A
d
2
r
A
d
t
2
F
A
=
(10.5)
=
m
A
a
A
This is a second-order differential equation that I can write equivalently as two first-order
differential equations for the particle position
r
A
and the velocity
v
A
d
v
A
d
t
F
A
=
m
A
(10.6)
d
r
A
d
t
=
v
A
10.4.1 Leapfrog Algorithm
A simple algorithm for integration of these two equations numerically in small time steps
t
can be found by considering the Taylor expansion for
v
(
t
):
v
A
t
d
v
A
d
t
d
2
v
A
d
t
2
t
2
2
t
2
t
2
+
1
2
+
=
v
A
(
t
)
+
+···
t
t
(10.7)
v
A
t
d
v
A
d
t
d
2
v
A
d
t
2
t
2
2
t
2
t
2
+
1
2
−
=
v
A
(
t
)
−
+···
t
t
