 1
Tim Wendler and Manuel Berrondo BYU Physics
I am calculating the dynamics of a collinear
atom/diatomic molecule inelastic collision
o Jacobi coordinates
oQuantum-Classical coupling
oLie algebraic solution
o6 coupled equations
oCanonical ensemble
Collinear Configuration
Diatomic molecule
•Harmonic oscillator potential for BC
•Repulsive interaction for AB
•No AC interaction
•Energy is in units of 0
•Valid for Ee  Et
Atom
Jacobi Coordinates
1
2
m
m A mC
m
mB m A  mB  mC 
Potential Energy Surface
1 2
  x y 
y  V0 e
2
V0  1
 4
V0  1
 1
V0  4
 1
Equations of motion
pˆ
p
1 2
   x  yˆ 
H

 yˆ  V0 e
2m 2 2
2
x
2
d x
 x 
m 2  V0 e e
dt
2
y
yˆ t 
Classical for Translation
d
  H q  t 
i  tExpansion
dt
Quantum for vibration
Quantum Hamiltonian
V0 e
 x yˆ
e
 V0 e
 x
1 2 2


1  yˆ   yˆ  ... 
2


1 2 1 2 2
 x
 x
H q  pˆ y   yˆ  V0 e yˆ  V0 e
“dipole” term
2
2
Expanded and
Rearranged
d
i  t   H q  t 
dt
Quantum for vibration
Time Evolution Operator
 t   U t  0
d
i  t   H q  t 
dt
d
i U t   0  H qU t   0
dt
Quantum equation of
motion for vibration
Constant ket
d ˆ
i U t   H qUˆ t 
dt
Lie Algebraic Approach
Uˆ t   e
1aˆ   2 aˆ  3 Nˆ  4 1̂
n U 1aU n  1 t 
e e
e
1
1 t    2 t 
yˆ t  
2

1 t    2 t 
pˆ y t   i
2
Time dependence goes into “c” numbers
Nuances
Very general and useful
equation
 d ˆ  ˆ 1
i U U  H
 dt 
d ˆ
i U  HUˆ
dt
End up with terms like
1aˆ 
i 2e
aˆe
1aˆ 
Utilize Berrondo anti-symmetric product
A
e Be
A
A
 e B
6 coupled equations
x t   vt 
1
1  x 
vt   V0
e e
m
2
1 t   iV0
 2 t   iV0
 3 t   i
yˆ t 
Not operator
equations!
1  x
e  i1
2
1  x
e  i 2
2
 4 t   iV0 e  x  i

1  x
 iV0
e 1
2
2
  x   1   2V0e x
Transition Rates
Initial conditions
x0  20
v0  1.41
1 0  0
 2 0  0
 3 0  0
 4 0  0
U 0  1
02
Phase Space Calculations
Initial conditions
pˆ y t 
x0   20
v0   1.9
1   2Ve  x0
1 0  
2
1   2Ve  x0
 2 0   
2
 3 0   0
 4 0   0
ŷ t 
Intuitive plot of collision

 tv0 
 log cosh  
 2 

New Trajectory
Classical Comparison
pˆ y t 
ŷ t 
Quantum phase gained energy
p y t 
yt 
Classical phase lost energy
•Same until initial speed passes the max oscillator speed
•Certain phase relations result in opposite effects
Mixed States
A superposition is in both states
1
1  2
i 
2

A mixture is in perhaps one or perhaps the other
1
   1 1  2 2 
2
No interference
A density matrix can represent a statistical
mixture of pure states.
Quantum Liouville Equation

i
 H q , 
t



 0
t
Initial state in equilibrium at
temperature T
Just after the collision
 t   U  0U
Pi f  f UiU

1
f
Out of equilibrium, for the moment
1
Pif 
Zn
n

2
fUi e
 Ei
i 0
A canonical ensemble
of oscillators
Initial State of canonical
ensemble at Temperature T
Non-equilibrium
Just after collision, thermal
equilibrium is lost
Summary
•
•
No wave functions
A simple equation standard
o Phase Space
o Transition Rates
o Canonical Ensemble
 Infinite order transitions
Specific system input
xn
 d ˆ  ˆ 1
H  i U U
 dt 
General boson
algebra coefficients
already calculated!
Future – Reactive Collisions
oSN2 Reactions
oNuclear Reactions
oOxidation of methyl
esters