Energy Transfer in Inelastic and Reactive Collisions using
Algebraic methods
Tim Wendler, Manuel Berrondo, Jean-Francois Van Huele
Physics of Collinear Inelastic Collision
Results of Inelastic Collision
Pif t 
x
 Dynamics for diatomic molecule reduced
to single coordinate with reduced mass
Collinear Reactive Collision
00
s, x
0 1
y
t
Remove center of mass and rescale energy
units
• Hamiltonian transformed to natural
coordinates and their conjugate momenta
• Diatomic state-to-state
transition probability over
t
• Initial: ground state
• Final: L.C. of time
dependent states
•
• Natural coordinates s and x smoothly connect reactants
and product reduced mass schemes
1  1 2
2
 2 Ps  Px   V s, x 
H
2m  

Pif t 
• Curvature function k (s) in kinetic
energy
 1   s x
t
• Probability
landscape
for transitions single initial
state
n
Classical Trajectories
Initial Superposition of States
•
•
Treat translation coordinate x classically
Treat vibrational coordinate y quantumly
y
py
x
Hamiltonian
y
t
1 2 1 2 1 2
   x  yˆ 
H
p x  pˆ y  yˆ  V0 e
2m
2
2
• Classical coordinate
s 2

 1 e
 s   e
s

 xˆ 2
2
s
s
• Harmonic oscillator for the quantum variable
• Landau-Teller model applied to classical
variable
V s, xˆ   e
• Classical reaction dynamics
simple to analyze in natural
coordinates
• Anharmonicity
in
quantum
coordinate
more
accurately
models the dynamics
x coupled to expectation
value y exhibits asymmetry (see also vib.
phase space)
x
x
Ensemble of Colliding Oscillators
Inelastic Collision
Internal Energy Lost
Lie Algebra
1

Pfin 
f
U

U
f

i
i
Z ( )
• Algebraic approach  both phase space
dynamics and transitions
• Both Hamiltonian and time evolution operator
constructed using four Lie Algebra basis elements:
a
U t   e

1 t a
• Equation of
Hamiltonian:

e
e
s
x
x
for
U(t)
e
follows
Equations of Motion for a’s
1
 x
i1  
V0e  1
2
1
 x
i 2  
V0e   2
2
• Algebraic approach
produces coupled ODEs:
solved numerically
Internal Energy Gained
n
from
   1
i U U  H
 t 
i3    1   V0e
s
 2 t a  3 t  N  4 t 
motion
2
e
 Ei
f
, a, N ,1

2
• Initial Boltzmann distribution of diatomic SHO’s
before and after colliding with atoms at 40K
•a non-equilibrium redistribution of states: temperature
undefined
Pfin t 
• Time evolution
dynamics at initial 40K
towards nonequilibrium
Pfin t 
t
n
 x
1
 x
 x
i 4  
V0e 1  V0e
2
Complete Dissociation
Quantum Effects
• Quantum harmonic oscillator in x
• Resonances appear with certain
initial velocities
• Quantum transition rates allude to
quanta being transferred to new bond
x
s
x
s
t
Pif
t
• Same at 300K
n
t
0 react  0 prod
t