Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

6.3 Trajectory Calculations 143
6.3 Trajectory Calculations
The “exact” dynamics of the molecules on a potential energy hypersurface can be
followed by classical trajectory calculations.
I Phase Space: 6-dimensional space spanned by the spatial coordinates ( )and
the conjugated momenta ( ):
↔ (6.12)
where
=
=
·
(6.13)
I
I
Hamilton function:
= + = 2
+ () (6.14)
2
Note that depends only on and depends only on .
Equations of motion:
• for a 1 particle system:
(1)
= =
= −
()
y
·
= −
= −
since contains only not .
(2)
y
y
• for an atomic system:
= + =
(6.15)
(6.16)
= 2
2 + (6.17)
= 1 × = 1 µ
×
·
3X
=1
·
=
= ·
(6.18)
(6.19)
2 2 + ( 1 2 3−6 ) (6.20)
·
= −
(6.21)
·
=
(6.22)
Thus we obtain a set of coupled differential equations which can be solved using
the Runge-Kutta or Gear methods.