ALGEBRAIC SEMI-CLASSICAL MODEL
FOR REACTION DYNAMICS
Tim Wendler, PhD Defense Presentation
TOPICS:
Part
1 – Motivation
Part
2 – The Dipole Field Model
Part
3 – The Inelastic Molecular
Collision Model
Part
4 – The Reactive Molecular
Collision Model
PART 1 – THE MOTIVATION FOR THE MODEL
𝑑
𝑖ℏ 𝑈 𝑡 = 𝐻 𝑈 𝑡
𝑑𝑡
𝑑
−1
𝑖ℏ 𝑈 𝑡 𝑈 𝑡 = 𝐻
𝑑𝑡
(1)
A Computer Algebra System takes it from here
QUANTUM DYNAMICS WITH ALGEBRA
Find a Lie algebra,
with which a meaningful
Hamiltonian is constructed.
Find a decent ansatz for the
time-evolution operator.
𝑎† , 𝑎, 𝑁, 𝐼
1
𝐻 = ℏ𝜔 𝑁 +
+ 𝑓 𝑡 𝑎 + 𝑎†
2
𝑈 𝑡 =
𝛼1 𝑡 𝑎† 𝛼2 𝑡 𝑎 𝛼3 𝑡 𝑁 𝛼4 𝑡 𝐼
𝑒
𝑒
𝑒
𝑒
(Wei-Norman Ansatz: A time-evolution
operator group mapped to the Lie Algebra)
WHAT EXACTLY ARE WE DOING MATHEMATICALLY?
Computer produces
𝑑
−1
𝑖ℏ 𝑈 𝑡 𝑈 𝑡 = 𝐻
𝑑𝑡
We produce a model
Hamiltonian
1
𝐻 = ℏ𝜔 𝑁 +
+ 𝑓 𝑡 𝑎 + 𝑎†
2
Ca a  Ca a  CN N  CI  Ha a  Ha a  H N N  H I
WE’RE DERIVING AN EXPLICIT FORM †OF THE TIME𝛼1 𝑡 𝑎 𝛼2 𝑡 𝑎 𝛼3 𝑡 𝑁 𝛼4
𝑈
𝑡
=
𝑒
𝑒
𝑒
𝑒
EVOLUTION OPERATOR
𝑡 𝑎
,
1  i1  if t 
 2  i2  if t 
 3  i
 4  if t 1  2i 
n U aU n
n f U t  ni
Phase-space dynamics
Transition probabilities
1
Hats are now left off from here on out unless necessary
QUANTUM DYNAMICS WITH LIE ALGEBRA
n U aU n
n f U t  ni
Phase-space dynamics
Transition probabilities
1
PART 2 – THE DIPOLE-FIELD MODEL

1


H t    N    f t  a  a
2

Initial single state
    n

Final linear combination of
time-dependent states:
     an n
n 0
q
f t 
f t 
t  
t 
EXTERNAL FIELD PULSE, THEN ATOMIC COLLISION
f t 
Trajectories(Ehrenfest)
field
oscillator
Laser Pulse
Harmonic Oscillator Transition Probability
0 0
0 1
Atomic collision
t
t0
Single initial state
𝑥 −10s = 0
t0
t
PERSISTENCE PROBABILITIES FOR THE OSCILLATOR
(DIATOMIC MOLECULE) DURING THE EXTERNAL FIELD
PULSE (COLLISION)
5 5
4 4
n
3 3
2 2
1 1
0 0
External
dipole field
 sech2 t 
t
PART 2 – THE INELASTIC MOLECULAR COLLISION
Three hard spheres, same mass, perfectly elastic collisions
Three hard spheres, same mass, two of the three bound harmonically
COLLINEAR COORDINATES
 One-dimension
with 2 degrees of
freedom
RAB
A
B
RBC
No RAC interaction
C
LANDAU-TELLER MODEL HAMILTONIAN
[AB + C] inelastic collision with reduced coordinates
A
B
𝑦
classical
C
𝑥 − 𝑥0
quantum
1 2 1 2 1 2
   y  xˆ 
H
pˆ x  p y  xˆ  V0 e
2m
2
2
This is a semi-classical calculation because one
variable is classical and the other is quantum.
EXAMPLE OF INELASTIC COLLISIONS
𝑥 − 𝑥0
𝑦
INELASTIC COLLISION TRANSITION TIME
Reduced mass relative collinear distance
Trajectories
Molecule Transition Probability
molecule
x t 
atom
yt 
0 0
0 1
0 2
0 0
0 1
0 2
Single initial state
ti 0
t
ti  0
Initial single
ground state
t
INELASTIC COLLISION LANDSCAPE SINGLE
Collision Bath Landscape
Reduced mass relative collinear distance
Trajectories
atom yt 
molecu x t 
le
Pi f
t
With a zero expectation value we can
sum over final states from any initial state
of choice. For any single state n , x is
always zero.
t
RESONANCE: CLASSICAL WITH MORSE POTENTIAL
Anharmonic interatomic
potentials and different
masses result in resonance
t
mA : mB : mC  2 : 12 : 1
Actual video!
ALGEBRAIC CALCULATION APPLIED TO STATISTICAL
MECHANICS PRINCIPLES
Nuclear motion (Ehrenfest theorem)
Collision Bath Landscape
Pi f
Single initial state
•With a single initial state we can sum over
final states from any initial state of choice
•For any single state n , x is always zero for
the harmonic oscillator
t
METHANE/HYDROGEN COLLISION
Initial state
0
Transitions
0
2
1
0
Final state
  .94   .05   .01 
PART 3 – THE REACTIVE MOLECULAR COLLISION
REACTIVE COLLISIONS
Collinear triatomic reaction:
BG  R  B  RG
Reaction with a “Spectator”:
SBG  R  SB  RG
REACTIVE COLLISIONS
A
Transition state
or
Activated complex
RAB
B
RBC
C
POTENTIAL ENERGY SURFACE
A B
C
A
B
C
1. Reactants
2. Transition state
RAB
3. Products
A B C
3. Total dissociation
A
RBC
B C
CURVILINEAR COORDINATES – BASED ON MINIMUM
ENERGY PATHWAY OF POTENTIAL ENERGY SURFACE
Products
x
s
Transition state
x
quantum
s
classical
1  1 2
2
 2 ps  px   V s, x 
H
2m  

where  1  x
Reactants
CURVILINEAR COORDINATE OR “IRC”
Products
x
is perpendicular
distance to the red
line
xˆ
s0
The Frenet frame
Transition state
sˆ
Reactants
CURVILINEAR COORDINATES
Products
Transition state
Reactants
NATURAL COORDINATES
SKEWING IS NECESSARY FOR SINGLE MASS ANALYSIS
tan   
mb ma  mb  mc 
ma mc

mass scaled and skewed coordinates
CURVILINEAR COORDINATES
Products
x
quantum
s
classical
1  1 2
2
 2 ps  px   V s, x 
H
2m  

where   1   s x
s x
“curvature”
Transition state
Reactants
THE DEVELOPMENT OF A REACTION COORDINATE
Reaction Coordinate
Harmonic
Top view
Anharmonic
Top view
VISUALIZING THE SINGLE-MASS INTERPRETATION

LOOKING DOWN BOTH CHANNELS
Reduced mass relative
collinear distance
A FULL MODEL WOULD ACCOUNT FOR POSSIBLE
DISSOCIATION AS WELL- EXAMPLE: FESHBACH RESONANCE
s
t
x
s
x
Reduced mass relative
collinear distance
MATCH THE NUMBERS ON THE LEFT PLOT TO THE
ASSOCIATED POSITION ON THE RIGHT
s
D
t
1
B
3
x
s
2
1
3
2
A
B
x
A
C
Quantum Morse dissociation
Pi f
Could this the motion be related to the plot?
0 0
0 1
0 2
0 4
0 8
0  16
0  32
0  64
0  128
t
REACTIVE COLLISION LANDSCAPE BATH
Pi f
t
*Initial state of each collision is ground in a 1-indexed program*

CONCLUSION
How
do I know my calculations are
correct?

Manuel and I compare our derivation of the EOM done
by hand, twice over

The derivation is then compared with Manuel’s Lie
Algebra Coefficient Generator Program

We compare our trajectories via Ehrenfest theorem to
the classical limit model trajectories

When needed we classically bin the bound phase-space
motion to compare to quantum transitions

We watch 𝑈 † 𝑈 𝑡 to make sure it does not leave unity
What
could we do that we
CONCLUSION
couldn't
do before?
 Use
the Hamiltonian as a generalized algebraic
entity which has the potential to obviate
numerical error in quantum dynamics
 Simultaneously
analyze an oscillator’s motion
with its quantum dynamics continuously
throughout external interaction, with a more
unified model than what we’ve seen in the
literature
 Resolve
the quantum dynamic details of a bath
of collisions as they leave equilibrium
 Work
from a foundation of optimized [Algebraic
What
predictions have you made
CONCLUSION
that
need experimental
verification?
It’s
not that I have specific
predictions so much as the model is
generalized to be able to compare to
femtochemistry experiments, lasing,
and nuclear reactions by specifying
only a handful of parameters.
We
can predict state-to-state
transition probabilities of an
inelastic collision or a reaction from
Reference Slides Begin Here
CONCLUSION
What
experiments can we explain that
we couldn't before?
I’ve

yet to find the femtochemist!

The distribution of a fixed amount of energy among a number of identical particles depends
of identical particles depends upon the density of available energy states and the probability
energy states and the probability that a given state will be occupied. The probability that a given
occupied. The probability that a given energy state will be occupied is given by the distribution
occupied is given by the distribution function, but if there are more available energy states in a
more available energy states in a given energy interval, then that will give a greater weight to the
will give a greater weight to the probability for that energy interval.
Quantum Morse dissociation
Pi f
0 0
0 1
0 2
0 4
0 8
0  16
0  32
0  64
0  128
t
HARMONIC VS. ANHARMONIC
The Morse potential
x2
12th order
expansion of
Morse potential
6th order
expansion of
Morse potential
4th order
expansion of
Morse potential
CLASSICAL TRAJECTORY METHOD
 The
de Broglie wavelength associated with
motions of atoms and molecules is typically
short compared to the distances over which
these atoms and molecules move during a
scattering process.
 Exceptions
 in
the limits of low temperature and energy
 Separate
A
B
into classical and quantum variables
Mean free path >> “interaction region”
C
Reduced mass relative collinear distance
INELASTIC COLLISION TRANSITION
TIME SHOT 3
Amplitudes
atom yt 
molecu x t 
le
Molecule Transition Probability
Initial single ground
state
0 0
0 1
t  tf
t
n  t 
2
Being found in n at
t
0 2
t  tf
t
n f U t  ni
2
tf
Conditional t i
and
on
COMPARING DIFFERENT INITIAL STATES
Triatomic mass ratio 1:3:1
Initial states
Transitions
2
1
0
2
ni
1
0
Final states
  c1   c2   c3 
TYPICAL DIATOMIC MOLECULE STP
REFERENCES
a
v
tv
Relative velocity
during collision
h
Ev  h 
tv
aEv
v
 a
h
Diatomic molecule
vibrational
frequency
Typical molecule has vibrational
frequency of
  1013 s -1
Estimate for intermolecular force
range
Gas phase molecular speeds are v
about
o
a  2
 1km s -1
v  2km s-1
 vt  1
ROTATIONALLY ADIABATIC
h
Er 
tr
vrt  100m s-1
2 or 3 orders of
magnitude smaller than
vib. spacing
Gas phase molecular speeds are
about
tc
 rt 
tr
v  1km s -1
rt  1
TRANSITION PROBABILITIES FOR THE
DIATOMIC MOLECULE DURING THE
COLLISION
5 5
4 4
n
3 3
2 2
1 1
0 0
External
dipole field
 sech2 t 
t
INCREASING THE ATOMIC
COLLISION SPEED
5 5
4 4
n
3 3
2 2
1 1
0 0
External
dipole field
 sech2 t 
t
CANONICAL ENSEMBLE OF OSCILLATORS
A
“heat bath”
f t 
B
C
Canonical Ensemble
Microcanonical Ensemble
•The canonical ensemble is initially at a defined temperature, though
it can draw “infinite” amounts of energy from the heat bath, which
are the collisions or external fields.
•The microcanonical ensemble has a fixed energy
TYPICAL RELAXATION TIMES FOR AN
ENSEMBLE OF DIATOMIC MOLECULES
ns
N
VT
VR
VV
R T
RR
For external field
induced or collision
induced excitement of a
diatomic molecule
All other energy transfer
types are quickly relaxed
t
E VS. T IN EXTERNAL FIELD
t0
ns
N
VT
Canonical ensemble of
diatomic molecules initially
at 400K
Kcal/mol or Kelvin
t
f t 
Energy kcal/mol
Temperature K
t
Diatomic molecule(6 d.o.f.)
E vs. T in external field
ENERGY VS. TEMPERATURE
t1
ns
N
VT
Nonequilibrium
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
E vs. T in external field
TEMPERATURE UNDEFINED
t3
ns
N
VT
Nonequilibrium
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
E vs. T in external field
ENERGY-TEMPERATURE
t4
ns
N
VT
Thermal equilibrium is
reached again at 440K
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
E vs. T in external field
E VS. T IN INELASTIC COLLISION
t0
ns
N
VT
Temperature = 400K
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
Atom
E vs. T in inelastic collisions
ENERGY VS. TEMPERATURE
t1
ns
N
VT
Nonequilibrium
Temperature = ?
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
Atom
E vs. T in inelastic collisions
TEMPERATURE UNDEFINED
t3
ns
N
VT
Nonequilibrium
Temperature = ?
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
Atom
E vs. T in inelastic collisions
ENERGY-TEMPERATURE
t4
region of study
ns
N
VT
Thermal equilibrium is reached
again
Temperature = 440K
Kcal/mol or Kelvin
t
Energy kcal/mol
Temperature K
t
Diatomic molecule
Atom
E vs. T in inelastic collisions
CANONICAL PHASE-SPACE DENSITY
Thermal equilibrium is shown below
as a Boltzmann distribution of
oscillators
quantum
classical
Density of
states
e
 
 n  1
 kT 2
THERMAL NONEQUILIBRIUM
Initially a Boltzmann distribution
After collision, temperature undefined
(extreme
case)
p y t 
p y t 
y t 
y t 
Part 1 - Time-dependent
Hamiltonians
When the Hamiltonian is timeindependent the time-evolution is simply
When the Hamiltonian is time-dependent
the time-evolution is rougher
But what if the Hamiltonian does not
commute with itself at different times?
U t , t0   e
U t , t0   e
 i H t t0 
i t
  H t ' dt '
 t0
H t1 , H t2   0
TRADITIONAL QUANTUM DYNAMICS
•Differential equation approach
d
i  x, t   H t  x, t 
dt
H t   H 0  H1 t 
leads to large O.D.E. system
i E  0   E  0  t
d 0 
i
1
2 2 
0 
1
2 2 

a f  a f   a f  ...    an  an   an  ... e f n  f0 * H1 n0 
dt
 n




selection rules emerge when looking for time-dependent transitions…
a f t 
1
2
TRADITIONAL QUANTUM DYNAMICS
•Integral equation approach
d
i U t , t0   H t U t , t0 
dt
t
i
U t , t0   1   H t U t ' , t0 dt '
 t0
Iterative form leads
to
1. Dyson series
2. Volterra series
3. time-ordering
4. Magnus
expansion
NON-TRADITIONAL QUANTUM DYNAMICS
•Algebraic approach
A Lie algebra is a set of elements(operators) that is…
1. Closed under commutation
2. Linear
3. Satisfies Jacobi identity
Example: Heisenberg-Weyl algebra:
H2  I , x, p
x, p  i I
Exponential mapping to the Lie-Group, the Heisenberg group
GH  e
iaibxicp
NON-TRADITIONAL QUANTUM DYNAMICS
•algebraic approach
Boson algebra
Commutation relations

U2  a, a , N , I


a, a   I a, N   a

Wei-Norman result for time-evolution operator
U t   e
1 t a  2 t a  3 t N  4 t I
e
e
e
(exponential map to the Wei-Norman time-evolution operator group)
n f U t  ni
Transition probabilities
1
n U aU n
Phase-space dynamics
COMPUTERS EAT ALGEBRA IF FED
CORRECTLY
d
i U t , t0   H t U t , t0 
dt
d

1
i U t U t   H t 
 dt

Computer algebra
system solves for any
algebra U(N)
Any Hamiltonian that is
constructed of algebra
U(N)
EXAMPLE: HARMONIC OSCILLATOR IN A
TIME-DEPENDENT EXTERNAL FIELD
USING U(2)
Computer produces
Construct a Hamiltonian from
the boson algebra

d

1
i U t U t 
 dt


1

H t    N    f t  a  a 
2



Ca a  Ca a  CN N  CI  Ha a  Ha a  H N N  H I
THEN FIND THE EVOLUTION OPERATOR,
U t   e
1 t a  2 t a  3 t N  4 t 
e
e
e
,
1  i1  if t 
 2  i2  if t 
 3  i
 4  if t 1  2i 
INELASTIC COLLISION LANDSCAPE BATH
Amplitudes
Reduced mass relative collinear distance
atom yt 
Diatomic molecules leaving
thermal equilibrium
molecu x t 
le
Pi f
Single initial state
t
Density of
states
e
 
 n 1
 kT 2
HARMONIC VS. ANHARMONIC
The Morse potential
x2
12th order
expansion of
Morse potential
6th order
expansion of
Morse potential
4th order
expansion of
Morse potential