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

A Computer Algebra System takes it from here PART 1 – THE MOTIVATION FOR THE MODEL (1)

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.

WHAT EXACTLY ARE WE DOING MATHEMATICALLY? Computer produces We produce a model Hamiltonian

WE’RE 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

QUANTUM DYNAMICS WITH LIE ALGEBRA Transition probabilities Phase-space dynamics

Initial single state Final linear combination of time-dependent states: PART 2 – THE DIPOLE-FIELD MODEL

EXTERNAL FIELD PULSE, THEN ATOMIC COLLISION Laser Pulse Atomic collision Harmonic Oscillator Transition Probability Trajectories(Ehrenfest) field oscillator Single initial state

External dipole field PERSISTENCE PROBABILITIES FOR THE OSCILLATOR (DIATOMIC MOLECULE) DURING THE EXTERNAL FIELD PULSE (COLLISION)

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

COLLINEAR COORDINATES  One-dimension with 2 degrees of freedom A BC No interaction

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

EXAMPLE OF INELASTIC COLLISIONS

INELASTIC COLLISION TRANSITION TIME Molecule Transition Probability Trajectories molecule atom Reduced mass relative collinear distance Initial single ground state Single initial state

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

RESONANCE: CLASSICAL WITH MORSE POTENTIAL Anharmonic interatomic potentials and different masses result in resonance Actual video!

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

METHANE/HYDROGEN COLLISION Initial state Transitions Final state

PART 3 – THE REACTIVE MOLECULAR COLLISION

REACTIVE COLLISIONS Collinear triatomic reaction: Reaction with a “Spectator”:

REACTIVE COLLISIONS Transition state or Activated complex A B C

POTENTIAL ENERGY SURFACE ABC ABC A BC 1. Reactants 2. Transition state 3. Products A B C 3. Total dissociation

CURVILINEAR COORDINATES – BASED ON MINIMUM ENERGY PATHWAY OF POTENTIAL ENERGY SURFACE Reactants Transition state Products quantum classical where

CURVILINEAR COORDINATE OR “IRC” Reactants Transition state Products The Frenet frame is perpendicular distance to the red line

CURVILINEAR COORDINATES Reactants Transition state Products

SKEWING IS NECESSARY FOR SINGLE MASS ANALYSIS mass scaled and skewed coordinates NATURAL COORDINATES

CURVILINEAR COORDINATES quantum classical where Reactants Products Transition state “curvature”

THE DEVELOPMENT OF A REACTION COORDINATE Top view Harmonic Anharmonic Reaction Coordinate

VISUALIZING THE SINGLE-MASS INTERPRETATION

LOOKING DOWN BOTH CHANNELS

A FULL MODEL WOULD ACCOUNT FOR POSSIBLE DISSOCIATION AS WELL- EXAMPLE: FESHBACH RESONANCE Reduced mass relative collinear distance

MATCH THE NUMBERS ON THE LEFT PLOT TO THE ASSOCIATED POSITION ON THE RIGHT Reduced mass relative collinear distance B C 13 B A D 2 A

Quantum Morse dissociation Could this the motion be related to the plot?

REACTIVE COLLISION LANDSCAPE BATH t distribution of states *Initial state of each collision is ground in a 1-indexed program*

CONCLUSION

 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 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 and Numeric] methods and move to a larger scale

CONCLUSION  What predictions have you made 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 classical trajectories.

Reference Slides Begin Here

CONCLUSION  What experiments can we explain that we couldn't before?  I’ve yet to find the femtochemist! 

 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? Isn’t the DOS independent of what the system is doing?

Quantum Morse dissociation

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

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

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

COMPARING DIFFERENT INITIAL STATES Triatomic mass ratio 1:3:1 Initial states Transitions Final states

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

ROTATIONALLY ADIABATIC Gas phase molecular speeds are about 2 or 3 orders of magnitude smaller than vib. spacing

External dipole field TRANSITION PROBABILITIES FOR THE DIATOMIC MOLECULE DURING THE COLLISION

INCREASING THE ATOMIC COLLISION SPEED External dipole field

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”

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

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

ENERGY VS. TEMPERATURE Energy kcal/mol Temperature K Diatomic molecule E vs. T in external field Nonequilibrium Kcal/mol or Kelvin

TEMPERATURE UNDEFINED Energy kcal/mol Temperature K Diatomic molecule E vs. T in external field Nonequilibrium Kcal/mol or Kelvin

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

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

ENERGY VS. TEMPERATURE Energy kcal/mol Temperature K Temperature = ? E vs. T in inelastic collisions Diatomic molecule Atom Nonequilibrium Kcal/mol or Kelvin

TEMPERATURE UNDEFINED Energy kcal/mol Temperature K Temperature = ? E vs. T in inelastic collisions Diatomic molecule Atom Kcal/mol or Kelvin Nonequilibrium

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

CANONICAL PHASE-SPACE DENSITY quantum classical Thermal equilibrium is shown below as a Boltzmann distribution of oscillators Density of states

THERMAL NONEQUILIBRIUM Initially a Boltzmann distributionAfter collision, temperature undefined (extreme case)

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?

TRADITIONAL QUANTUM DYNAMICS leads to large O.D.E. system selection rules emerge when looking for time-dependent transitions… Differential equation approach

Integral equation approach Iterative form leads to 1.Dyson series 2.Volterra series 3.time-ordering 4.Magnus expansion TRADITIONAL QUANTUM DYNAMICS

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

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

COMPUTERS EAT ALGEBRA IF FED CORRECTLY 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

THEN FIND THE EVOLUTION OPERATOR,

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

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