MATH 597C: Introduction to Molecular Dynamics and Density-functional Theory
MWF 10:10AM - 11:00AM 201 EE West
Instructor: Xiantao Li
 Office: 219C McAllister
 Phone: 3-9081
 Email: xxl12@ucs.psu.edu
 Webpage: https://sites.psu.edu/intro2md/
 Office hours: Wednesdays and Fridays 1:00PM-2:00PM.
Course description:
This course will present a thorough introduction to molecular and quantum-mechanical
models. It has been motivated by problems in material science and computational
chemistry. We will discuss modeling principles, basic mathematical properties,
computational methods, and software. We will place a slight emphasis on the
computational methods, including methods for solving eigenvalue problems, solutions of
ODEs and PDEs, statistical averaging, fix-point problems, and absorbing boundary
conditions.
Topics to be covered:
Molecular models
Numerical methods for Hamiltonian ODEs
- Symplectic integrators
- time symmetry and energy conservation
- Operator-splitting methods
Modeling Ensembles using extended dynamics
- Canonical ensemble
- NPT ensemble
- Grand-canonical ensemble
- Operator splitting methods
Computing ensemble averages
- Monte Carlo methods
- The Liouville equation and the Koopman operators
- The equivalence of Schrodinger and Heissenberg definition of
ensemble averages
Methods for time averaging
- weighted average
- block averaging
- fast Fourier transform
Statistical mechanics-basis
- definition of entropy, free energy, etc
- law of thermodynamics
- definition of stress, pressure etc
Definition of local macroscopic quantities
- Irving-Kirkwood formalism
- formulas based on first law of thermodynamics
- quasi-local equilibrium
Methods for non-equilibrium simulations
- non-periodic boundary conditions
- quasi-local thermodynamic equilibrium
Quantum-mechanical models
The many-body Schrodinger equation
- Born-Oppenheimer
- analytical solutions
- Hartree-Fock
The Density functional theory
- Energetic formulation
- Eigen-value problem
- Self-consistent calculations
- local and nonlocal basis functions
Tight-binding approach
- Hamiltonian and overlap matrices
- two and three-center approximations
Time-dependent quantum-mechanical models
- Ab initio molecular dynamics
- time-dependent density functional theory
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References:
1. Statistical Mechanics of Nonequilibrium Liquids, Evans and Morriss, 2014.
2. Understanding Molecular Simulation, Second Edition: From Algorithms to
Applications, Frenkel and Smit, 2001.
3. Molecular Modeling and Simulation: An Interdisciplinary Guide, T. Schlick, 2010.
4. Electronic Structure: Basic Theory and Practical Methods, R.M. Martin, 2008.
Homework and projects: there will be eight homework assignments and each student is required
to complete four of them. Each homework assignment is worth 25% toward the final grade.
Grading scale: 90%-100% A; 80%-89% B; 70%-79% C; 60%-69% D; 59% or below: F.