Chemical Rate Process
Theory: from Electron Transfer to Ion Permeation
Chem.
3490 Spring 2009
Instructor:
Rob Coalson
coalson@pitt.edu
Eberly 321, (412)624-8261
Course Syllabus
Lecture Notes
1)
Notes, Jan. 11, 2009:
Problem of Random Flights, Markoff's method, with applications to polymer extension and swelling; the Central Limit Theorem; the Diffusion Equation.
2)
Notes, Jan. 13, 2009: Langevin Eq.; free particle velocity distribution, Einstein relation between friction and random force; position distribution for free particle: reduction to Diffusion Equation in high friction limit.
3)
Notes, Jan. 15, 2009: Langevin Eq. for particle in a constant force field; stochastic Langevin dynamics algorithm for motion in an arbitrary force field and its reduction in the high-friction limit.
4)
Notes, Jan. 20: Velocity auto-correlation function for a particle undergoing Langevin dynamics; correlation functions of the random force.
5)
Notes, Jan. 22, 2009: Connection of stochastic ordinary differential equations (ODEs) to deterministic particle differential equations (PDEs): equivalence of Langevin Eq. and phase space Fokker Planck Eq.; equivalence of high-friction Langevin Eq. and Smoluchowski Eq.
6)
Notes, Jan. 27, 2009: Thermally Activated Barrier Crossing: Transition State Theory, Kramer's Theory in the high and low friction limits.
7)
Notes, Feb. 5Transformation of the Smoluchowski Eq. to Schrodinger form.
8)
Notes, Feb. 10, 2009: Generalized Langevin Eq. (GLE): Phenomenology and Fluctuation-Dissipation Theorems.
9)
Notes, Feb. 17, 2009: Derivation of the Generalized Langevin Eq. from a microscopic Hamiltonian (a system coordinate coupled blinearly to a bath of harmonic oscillators).
10)
Notes,
Feb. 25, 2009: 1-D Drift-Diffusion Equations (steady state of the Smoluchowski Eq.): quadrature formulae for the steady state concentration profile and current; application to ion permeation through channel proteins (GHK theory).
11)
Notes,
Mar. 5, 2009: Computing mean first passage times in high-friction Brownian motion via the Adjoint Equation: reduction to quadrature for 1D motion; PDE formulation for multi-dimensional motion.
12)
Notes,
Mar. 24, 2009: Fermi's Golden Rule for state-to-state quantum mechanical transition rates: derivation using time-dependent perturbation theory; requirement of a dense manifold of accepting states; counterexample: quantum dynamics of an isolated two-level system.
13)
Notes,
Mar. 26, 2009: Decay of a Doorway State: Exact vs. Golden Rule quantum dynamics.
14)
Notes,
Apr. 9, 2009: Spectroscopy and the Golden Rule. Time-dependent perturbation theory with a monochromatic driving field: transition probabilities and cross sections for 1-photon absorption and emission; time-independent (sum over states) vs. time-dependent (correlation function) formulations; Application to vibronic absorption spectra: Franck-Condon factors, time-correlation function formulation for pure initial states and finite temperature systems.
15)
Notes,
July. 20, 2009: Simple Quantum Relaxation Theory: Population Relaxation of a System Coupled to a Bath. (Golden Rule for System Transitions; Detailed Balance; Pauli Master Equations; Semi-classical Evaluation of Heisenberg Correlation Functions.)
Problem Sets
1)
Problem Set 1,
Solution Key
2)
Problem Set 2,
Solution Key
3)
Problem Set 3,
Solution Key
Guest Lectures
1)
MC_feb09. Dr. Mary Cheng: Brownian Dynamics Simulation of Ion Permeation through Protein Channels.
2)
IK_apr09. Dr. Igor Kurnikov: Non-adiabatic Electron Transfer in Chemistry and Biology.
3)
KW_apr09. Dr. Kim Wong: Approaches for Calculating Non-adiabatic, Energy Transfer, and Chemical Reaction Rates.
Student Presentations
1)
JX_apr09. Jiawei Xu: Linear Response Theory and Its Applications.
2)
AS_apr09. Andrey Sharapov: Debye-Falkenhagen Theory and Its Computer Simulation.
3)
GZ_apr09. Guozhen Zhang: Poisson-Nernst-Planck Theory Approach to the Calculation of Ion Transport through Protein Channels.
4)
FY_apr09. Fangyong Yan: Graphical Rule-Based Modeling of Signal Transduction Systems.