To solve the full (linear) Schrödinger equation is computationally feasible only for some very simple molecules; the tiny time and space scales and the high dimension, 3(M+N)=d, of larger molecules, with M nuclei and N electrons sets severe computational limitations: already in water it means to solve a partial differential equation in d=39 dimensions. Approximations are therefore needed. The main two approximations, the Born-Oppenheimer-Hartree-Fock and the the Born-Oppenheimer-Kohn-Sham strategies, reduce the problem to a large system of nonlinear partial differential equations, in three dimensions. The main goal of the course is to understand accuracy and numerical complexity of the approximations that are made in order to arrive at some common simplified models.
The course will start with the Schrödinger equation and its simplifying approximations and continue with Dzugutov's lectures on molecular dynamics. The course ends with two invited guest lectures.
|Literature:||[JG], [OR], [AS]|
|[CLB] Cancès et al, Computational Chemistry - A Primer (chapter 1 and 5),|
|[BNS] sections I-IV.|
|14/10||1625||Hartree-Fock/Density functional theory||AS|
|28/10||4523|| (Non)adiabatic Born-Oppenheimer
approximation, QCMD, Car-Parrinello
| PDC seminar
Atomistic modelling of materials
(Part of the KCSE seminar series. )
|2/12||4523||Discretization of Schrödinger, KMC.||AS|
|9/12||4523||Quasi-continuum method||Björn Engquist|
|16/12||4523||Materials properties from DFT||Pavel Korzhavyi|
The exam will contain questions from the following list:
(A few more questions will be added later.)
The projects are made in groups of two and consist in presenting a scientific article in class. The presentations should be 30 minutes (max) and include problem formulation, theoretical background, results and any other interesting points you want to make. The material should be adapted to the audicence: your classmates. Make sure the level is such that they can understand everything and learn something new. You are encouraged to ask the teachers for input and comments during the preparations.
Note: Slides should be used. After the presentation they will be printed and distributed to the whole class. The slides will be part of the course literature.
Please use this opportunity to practice the difficult art of holding a seminar. Remember that presenting a material in a convincing and clear way is important and requires good preparation. For this we all need practice and constructive criticism, students as well as teachers.Suggested project articles:
|[CLB]||Claude Le Bris (ed), Computational Chemistry, North Holland, 2003.|
|[JG]||Jonathan Goodman, Quantum Mechanics Notes, lecture notes, 2001.|
|[TS]||Tamar Schlick, Molecular Modeling and Simulation, Springer, 2002.|
|[BNS]||F. A. Bornemann, P. Nettersheim and Ch. Schütte. Quantum-classical molecular dynamics as an approximation to full quantum dynamics. J. Chem. Phys. 105:1074-1083, 1996.|
|[MT]||R. E. Miller and E. B. Tadmor. The Quasicontinuum method: Overview, applications and current directions. J. Comput-Aided Mater. 9:203-239, 2002.|
|[MD]||Mikhail Dzugutov, Lecture notes.|
|[AS]||Anders Szepessy, Lecture notes (discretization of Schrödinger).|
|[OR]||Olof Runborg, Lecture notes (Schrödinger).|
|+ Presentation material from course projects|