ALGEBRAIC SEMI-CLASSICAL MODEL FOR REACTION DYNAMICS Tim
Wendler, PhD Defense Presentation
Slide 2
TOPICS: Part 1 Motivation Part 2 The Dipole Field Model Part 3
The Inelastic Molecular Collision Model Part 4 The Reactive
Molecular Collision Model
Slide 3
A Computer Algebra System takes it from here PART 1 THE
MOTIVATION FOR THE MODEL (1)
Slide 4
Find a decent ansatz for the time-evolution operator.
(Wei-Norman Ansatz: A time-evolution operator group mapped to the
Lie Algebra) QUANTUM DYNAMICS WITH ALGEBRA Find a Lie algebra, with
which a meaningful Hamiltonian is constructed.
Slide 5
WHAT EXACTLY ARE WE DOING MATHEMATICALLY? Computer produces We
produce a model Hamiltonian
Slide 6
WERE DERIVING AN EXPLICIT FORM OF THE TIME- EVOLUTION OPERATOR
Transition probabilities Phase-space dynamics Hats are now left off
from here on out unless necessary
Slide 7
QUANTUM DYNAMICS WITH LIE ALGEBRA Transition probabilities
Phase-space dynamics
Slide 8
Initial single state Final linear combination of time-dependent
states: PART 2 THE DIPOLE-FIELD MODEL
Slide 9
EXTERNAL FIELD PULSE, THEN ATOMIC COLLISION Laser Pulse Atomic
collision Harmonic Oscillator Transition Probability
Trajectories(Ehrenfest) field oscillator Single initial state
Slide 10
External dipole field PERSISTENCE PROBABILITIES FOR THE
OSCILLATOR (DIATOMIC MOLECULE) DURING THE EXTERNAL FIELD PULSE
(COLLISION)
Slide 11
Three hard spheres, same mass, perfectly elastic collisions
Three hard spheres, same mass, two of the three bound harmonically
PART 2 THE INELASTIC MOLECULAR COLLISION
Slide 12
COLLINEAR COORDINATES One-dimension with 2 degrees of freedom A
BC No interaction
Slide 13
LANDAU-TELLER MODEL HAMILTONIAN [AB + C] inelastic collision
with reduced coordinates This is a semi-classical calculation
because one variable is classical and the other is quantum.
classical quantum A B C
Slide 14
EXAMPLE OF INELASTIC COLLISIONS
Slide 15
INELASTIC COLLISION TRANSITION TIME Molecule Transition
Probability Trajectories molecule atom Reduced mass relative
collinear distance Initial single ground state Single initial
state
Slide 16
INELASTIC COLLISION LANDSCAPE SINGLE Trajectories molecu le
atom Reduced mass relative collinear distance t distribution of
states Single initial state n = 2 With a zero expectation value we
can sum over final states from any initial state of choice. For any
single state, is always zero. t Collision Bath Landscape
Slide 17
RESONANCE: CLASSICAL WITH MORSE POTENTIAL Anharmonic
interatomic potentials and different masses result in resonance
Actual video!
Slide 18
ALGEBRAIC CALCULATION APPLIED TO STATISTICAL MECHANICS
PRINCIPLES Nuclear motion (Ehrenfest theorem) t distribution of
states Single initial state n = 2 With a single initial state we
can sum over final states from any initial state of choice For any
single state, is always zero for the harmonic oscillator Single
initial state Collision Bath Landscape
Slide 19
METHANE/HYDROGEN COLLISION Initial state Transitions Final
state
Slide 20
PART 3 THE REACTIVE MOLECULAR COLLISION
Slide 21
REACTIVE COLLISIONS Collinear triatomic reaction: Reaction with
a Spectator:
Slide 22
REACTIVE COLLISIONS Transition state or Activated complex A B
C
Slide 23
POTENTIAL ENERGY SURFACE ABC ABC A BC 1. Reactants 2.
Transition state 3. Products A B C 3. Total dissociation
Slide 24
CURVILINEAR COORDINATES BASED ON MINIMUM ENERGY PATHWAY OF
POTENTIAL ENERGY SURFACE Reactants Transition state Products
quantum classical where
Slide 25
CURVILINEAR COORDINATE OR IRC Reactants Transition state
Products The Frenet frame is perpendicular distance to the red
line
Slide 26
CURVILINEAR COORDINATES Reactants Transition state
Products
Slide 27
SKEWING IS NECESSARY FOR SINGLE MASS ANALYSIS mass scaled and
skewed coordinates NATURAL COORDINATES
Slide 28
CURVILINEAR COORDINATES quantum classical where Reactants
Products Transition state curvature
Slide 29
THE DEVELOPMENT OF A REACTION COORDINATE Top view Harmonic
Anharmonic Reaction Coordinate
Slide 30
VISUALIZING THE SINGLE-MASS INTERPRETATION
Slide 31
LOOKING DOWN BOTH CHANNELS
Slide 32
A FULL MODEL WOULD ACCOUNT FOR POSSIBLE DISSOCIATION AS WELL-
EXAMPLE: FESHBACH RESONANCE Reduced mass relative collinear
distance
Slide 33
MATCH THE NUMBERS ON THE LEFT PLOT TO THE ASSOCIATED POSITION
ON THE RIGHT Reduced mass relative collinear distance 2 1 3 B C 13
B A D 2 A
Slide 34
Quantum Morse dissociation Could this the motion be related to
the plot?
Slide 35
REACTIVE COLLISION LANDSCAPE BATH t distribution of states
*Initial state of each collision is ground in a 1-indexed
program*
Slide 36
CONCLUSION
Slide 37
What could we do that we 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 oscillators motion with its quantum
dynamics continuously throughout external interaction, with a more
unified model than what weve 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 and
Numeric] methods and move to a larger scale
Slide 38
CONCLUSION What predictions have you made that need
experimental verification? Its 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 classical trajectories.
Slide 39
Reference Slides Begin Here
Slide 40
CONCLUSION What experiments can we explain that we couldn't
before? Ive yet to find the femtochemist!
Slide 41
Distribution of energy The distribution of a fixed amount of
energy among a number of identical particles depends upon the
density of available energy states and the probability that a given
state will be occupied. The probability that a given energy state
will be occupied is given by the distribution function, but if
there are more available energy states in a given energy interval,
then that will give a greater weight to the probability for that
energy interval. Density of States the number of states per
interval of energy at each energy level that are available to be
occupied by electrons. The distribution of energy between identical
particles depends in part upon how many available states there are
in a given energy interval. This density of states as a function of
energy gives the number of states per unit volume in an energy
interval. The term "statistical weight" is sometimes used
synonymously, particularly in situations where the available states
are discrete. The physical constraints on the particles determine
the form of the density of states function. Density changes? Or the
occupation? Isnt the DOS independent of what the system is
doing?
Slide 42
Quantum Morse dissociation
Slide 43
HARMONIC VS. ANHARMONIC 12 th order expansion of Morse
potential 6 th order expansion of Morse potential 4 th order
expansion of Morse potential The Morse potential
Slide 44
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 into classical and quantum
variables Mean free path >> interaction region ABC
Slide 45
Reduced mass relative collinear distance INELASTIC COLLISION
TRANSITION TIME SHOT 3 Molecule Transition Probability Amplitudes
molecu le atomInitial single ground state Being found in n at t
Conditional on and
Slide 46
COMPARING DIFFERENT INITIAL STATES Triatomic mass ratio 1:3:1
Initial states Transitions Final states
Slide 47
TYPICAL DIATOMIC MOLECULE STP REFERENCES Typical molecule has
vibrational frequency of Estimate for intermolecular force range
Gas phase molecular speeds are about Relative velocity during
collision Diatomic molecule vibrational frequency
Slide 48
ROTATIONALLY ADIABATIC Gas phase molecular speeds are about 2
or 3 orders of magnitude smaller than vib. spacing
Slide 49
External dipole field TRANSITION PROBABILITIES FOR THE DIATOMIC
MOLECULE DURING THE COLLISION
Slide 50
INCREASING THE ATOMIC COLLISION SPEED External dipole
field
Slide 51
CANONICAL ENSEMBLE OF OSCILLATORS A BC 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 heat bath
Slide 52
TYPICAL RELAXATION TIMES FOR AN ENSEMBLE OF DIATOMIC MOLECULES
For external field induced or collision induced excitement of a
diatomic molecule All other energy transfer types are quickly
relaxed
Slide 53
E VS. T IN EXTERNAL FIELD Energy kcal/mol Temperature K
Diatomic molecule(6 d.o.f.) Canonical ensemble of diatomic
molecules initially at 400K E vs. T in external field Kcal/mol or
Kelvin
Slide 54
ENERGY VS. TEMPERATURE Energy kcal/mol Temperature K Diatomic
molecule E vs. T in external field Nonequilibrium Kcal/mol or
Kelvin
Slide 55
TEMPERATURE UNDEFINED Energy kcal/mol Temperature K Diatomic
molecule E vs. T in external field Nonequilibrium Kcal/mol or
Kelvin
Slide 56
ENERGY-TEMPERATURE Energy kcal/mol Temperature K Thermal
equilibrium is reached again at 440K Diatomic molecule E vs. T in
external field Kcal/mol or Kelvin
Slide 57
E VS. T IN INELASTIC COLLISION Energy kcal/mol Temperature K
Diatomic molecule Temperature = 400K Atom E vs. T in inelastic
collisions Kcal/mol or Kelvin
Slide 58
ENERGY VS. TEMPERATURE Energy kcal/mol Temperature K
Temperature = ? E vs. T in inelastic collisions Diatomic molecule
Atom Nonequilibrium Kcal/mol or Kelvin
Slide 59
TEMPERATURE UNDEFINED Energy kcal/mol Temperature K Temperature
= ? E vs. T in inelastic collisions Diatomic molecule Atom Kcal/mol
or Kelvin Nonequilibrium
Slide 60
ENERGY-TEMPERATURE Energy kcal/mol Temperature K Thermal
equilibrium is reached again Temperature = 440K E vs. T in
inelastic collisions Diatomic molecule Atom region of study
Kcal/mol or Kelvin
Slide 61
CANONICAL PHASE-SPACE DENSITY quantum classical Thermal
equilibrium is shown below as a Boltzmann distribution of
oscillators Density of states
Slide 62
THERMAL NONEQUILIBRIUM Initially a Boltzmann distributionAfter
collision, temperature undefined (extreme case)
Slide 63
Part 1 - Time-dependent Hamiltonians When the Hamiltonian is
time- independent 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?
Slide 64
TRADITIONAL QUANTUM DYNAMICS leads to large O.D.E. system
selection rules emerge when looking for time-dependent transitions
Differential equation approach
Slide 65
Integral equation approach Iterative form leads to 1.Dyson
series 2.Volterra series 3.time-ordering 4.Magnus expansion
TRADITIONAL QUANTUM DYNAMICS
Slide 66
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: Exponential mapping to the Lie-Group, the
Heisenberg group Algebraic approach NON-TRADITIONAL QUANTUM
DYNAMICS
Slide 67
Transition probabilitiesPhase-space dynamics Wei-Norman result
for time-evolution operator (exponential map to the Wei-Norman
time-evolution operator group) Boson algebra Commutation relations
algebraic approach NON-TRADITIONAL QUANTUM DYNAMICS
Slide 68
COMPUTERS EAT ALGEBRA IF FED CORRECTLY Computer algebra system
solves for any algebra U(N) Any Hamiltonian that is constructed of
algebra U(N)
Slide 69
EXAMPLE: HARMONIC OSCILLATOR IN A TIME-DEPENDENT EXTERNAL FIELD
USING U(2) Computer produces Construct a Hamiltonian from the boson
algebra
Slide 70
THEN FIND THE EVOLUTION OPERATOR,
Slide 71
INELASTIC COLLISION LANDSCAPE BATH Amplitudes molecu le atom
Reduced mass relative collinear distance Single initial state t
distribution of states Diatomic molecules leaving thermal
equilibrium Density of states
Slide 72
HARMONIC VS. ANHARMONIC 12 th order expansion of Morse
potential 6 th order expansion of Morse potential 4 th order
expansion of Morse potential The Morse potential