14
1 From the Phenomenology of Chemical Reactions …
dependence of the vectors p r i and r i can be obtained using different classical formulations. Instead of the just mentioned popular Newton’s formulation, we shall use
the formulation of Hamilton
4 where one has a pair of first-order ordinary differential
equations of the conically conjugated variables p r iW and r i W (with W = X,Y,Z being
the set of chosen orthogonal coordinate system)
d p r iW
dt
= −
∂H
∂r i W
and
dr i W
dt
=
∂H
∂ p r iW
(1.29)
for a total of twelve equations. These equations can be integrated numerically with
standard techniques which we will mention later. In only a few special cases will the
equations of motion (1.29) have analytical solutions. In the vast majority of cases, no
analytical solutions are known. As we shall see later, analytical solutions, when they
are available, have been generated only after performing laborious analytical transformations (yet, they have the advantage of allowing useful decompositions of the
problem leading to both interesting insights and significant reduction of dimensionality of the problem). Very often, however, accurate approximations to the solution
can only be found using numerical techniques.
In the case of conservative systems, the potential V depends only on the relative
distance between the two particles. Then, the Hamiltonian of the system assumes a
form particularly convenient when one uses CM coordinates r C M and the internal
coordinates r AB (or more frequently simply r). Their definition is immediate:
r C M ≡
m A
M
r A +
m B
M
r B with M ≡ m A + m B
(1.30)
r AB ≡ r = r A − r B .
(1.31)
Expressing the Hamiltonian (1.28) in these coordinates, we have
H =
p
2
r C M
2M
+
p
2
r AB
2μ
+ V (r ) with μ =
m A m B
m A + m B
.
(1.32)
The first term of this Hamiltonian describes the motion of the CM. The second
and the third terms in the Hamiltonian describe instead the relative motion of the
two particles. Since V (r ) is independent of r C M , the motion of the CM relative
to the laboratory system (X, Y, Z) lab is that of a free particle which moves at a
constant velocity (inertial system). The CM coordinate system (x, y, z) CM differs
from the laboratory one in that it is not fixed in space but moves at a constant velocity
with respect to the laboratory coordinate system (X, Y, Z) lab . Accordingly, the CM
coordinate system (x, y, z) CM shown in Fig. 1.5 can also be used as the origin of an
4 Another popular formulation of the equations of motion is the Lagrange’s one
d
dt
∂ L
∂ ˙
r W
−
∂ L
∂r W
= 0,
where L = T − V is the Lagrangian of the system.
1 From the Phenomenology of Chemical Reactions …
dependence of the vectors p r i and r i can be obtained using different classical formulations. Instead of the just mentioned popular Newton’s formulation, we shall use
the formulation of Hamilton
4 where one has a pair of first-order ordinary differential
equations of the conically conjugated variables p r iW and r i W (with W = X,Y,Z being
the set of chosen orthogonal coordinate system)
d p r iW
dt
= −
∂H
∂r i W
and
dr i W
dt
=
∂H
∂ p r iW
(1.29)
for a total of twelve equations. These equations can be integrated numerically with
standard techniques which we will mention later. In only a few special cases will the
equations of motion (1.29) have analytical solutions. In the vast majority of cases, no
analytical solutions are known. As we shall see later, analytical solutions, when they
are available, have been generated only after performing laborious analytical transformations (yet, they have the advantage of allowing useful decompositions of the
problem leading to both interesting insights and significant reduction of dimensionality of the problem). Very often, however, accurate approximations to the solution
can only be found using numerical techniques.
In the case of conservative systems, the potential V depends only on the relative
distance between the two particles. Then, the Hamiltonian of the system assumes a
form particularly convenient when one uses CM coordinates r C M and the internal
coordinates r AB (or more frequently simply r). Their definition is immediate:
r C M ≡
m A
M
r A +
m B
M
r B with M ≡ m A + m B
(1.30)
r AB ≡ r = r A − r B .
(1.31)
Expressing the Hamiltonian (1.28) in these coordinates, we have
H =
p
2
r C M
2M
+
p
2
r AB
2μ
+ V (r ) with μ =
m A m B
m A + m B
.
(1.32)
The first term of this Hamiltonian describes the motion of the CM. The second
and the third terms in the Hamiltonian describe instead the relative motion of the
two particles. Since V (r ) is independent of r C M , the motion of the CM relative
to the laboratory system (X, Y, Z) lab is that of a free particle which moves at a
constant velocity (inertial system). The CM coordinate system (x, y, z) CM differs
from the laboratory one in that it is not fixed in space but moves at a constant velocity
with respect to the laboratory coordinate system (X, Y, Z) lab . Accordingly, the CM
coordinate system (x, y, z) CM shown in Fig. 1.5 can also be used as the origin of an
4 Another popular formulation of the equations of motion is the Lagrange’s one
d
dt
∂ L
∂ ˙
r W
−
∂ L
∂r W
= 0,
where L = T − V is the Lagrangian of the system.