Defense Presentation
Download
Report
Transcript Defense Presentation
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 i1 if t
2 i2 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
t0
Single initial state
𝑥 −10s = 0
t0
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
yt
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 yt
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
s0
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 yt
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
aEv
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
VT
VR
VV
R T
RR
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
VT
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
VT
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
VT
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
VT
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
VT
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
VT
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
VT
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
VT
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 n0
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:
H2 I , x, p
x, p i I
Exponential mapping to the Lie-Group, the Heisenberg group
GH e
iaibxicp
NON-TRADITIONAL QUANTUM DYNAMICS
•algebraic approach
Boson algebra
Commutation relations
U2 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 i1 if t
2 i2 if t
3 i
4 if t 1 2i
INELASTIC COLLISION LANDSCAPE BATH
Amplitudes
Reduced mass relative collinear distance
atom yt
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