Biomedical Engineering Reference
In-Depth Information
σ
1
σ
2
0
Distance, R
U
0
Figure 10.5
Finite square well
If two particles A and B have initial positions
r
A,0
and
r
B,0
and velocities
v
A
and
v
B
then
the instantaneous positions at time t will be
r
A
=
r
A,0
+
u
A
t
r
B
=
r
B,0
+
u
B
t
giving
r
A
−
r
B
=
r
A,0
−
r
B,0
+
(
u
A
−
u
B
)
t
and so
=
r
A,0
−
r
B,0
2
2
t
r
A,0
−
r
B,0
. (
u
A
−
r
B
)
2
t
2
(
u
A
−
u
B
)
2
(
r
A
−
+
u
B
)
+
If we rewrite the last equation as a quadratic in
t
as
σ
α
where α takes values 1 or 2, we see that the time required for a repulsive or attractive
collision is
u
AB
t
2
+
2
b
AB
t
+
r
AB,0
=
b
AB
±
b
AB
−
u
AB
r
AB
−
σ
α
1/2
=
−
t
(α)
AB
(10.4)
u
AB
In order to find the first collision time, all pairs have to be analysed. All the particles are
then allowed to move for such time, and the velocities of the colliding pair are adjusted
according to the equations of motion.
The finite square well occupies an important place in the history of molecular modelling.
Real atomic and molecular systems have much more complicated mutual potential energy
functions, but the finite square well does at least show a minimum. On the other hand,
because of the finite square well potential, the equations of motion are particularly simple
and no complicated numerical techniques are needed. There are no accelerations until two
particles collide.
