F2D5249 Computational methods from micro to macro scales, 5 credits

Graduate course, fall 2004

Description

The course presents an overview on computational models from ab initio Schrödinger equation over molecular dynamics to related continuum partial differential equations, and something on their coupling.

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.

Contents

  1. Schrödinger equation
    1. Introduction, postulates
    2. Properties (conservation of L2-norm, symmetries, relation with classical mechanics, etc.)
    3. Approximations
      1. Born-Oppenheimer
      2. Hartree-Fock
      3. Density functional theory (Kohn-Sham)
      4. Time-dependent models (adiabatic/nonadiabatic,QCMD, Car-Parrinello)

    Literature: [JG], [OR], [AS]
    [CLB] Cancès et al, Computational Chemistry - A Primer (chapter 1 and 5),
    [BNS] sections I-IV.

  2. Molecular dynamics
    1. Thermodynamics and statistical mechanics
    2. Micro- / canonical ensemble
    3. Molecular dynamics simulation
    4. Illustration in solids and liquids: diffractograms, structure function, etc

    Literature: [MD]
    [TS] chapter 12
    [MT]

Schedule

Meetings are on Thursdays, 10.15-12.00, usually in room 4523.

Date Room Contents Teacher
23/9 1625 Introduction AS
30/9 No meeting
7/10 4523 Hartree-Fock/Schrödinger AS, OR
14/10 1625 Hartree-Fock/Density functional theory AS
21/10 4523 Schrödinger OR
28/10 4523 (Non)adiabatic Born-Oppenheimer
approximation, QCMD, Car-Parrinello
OR,AS
4/11 4523 MD MD
10/11
kl 15.30
PDC seminar
room
Atomistic modelling of materials
(Part of the KCSE seminar series. )
Börje Johansson
11/11 4523 MD MD
18/11 4523 MD MD
25/11 4523 MD MD
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

Teachers

The teachers will be Anders Szepessy, Olof Runborg and Mikhail Dzugutov.

Examination

There will be an exam at the end of the course. There will also be a project exercise. The project presentations will be around 25 Jan. 2005 and the exam ca a week after that. Precise times to be announced.

The exam will contain questions from the following list: Questionlist
(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:
  1. Cancès, Castella, Chartier, Faou, Le Bris, Legoll and Turinici. Higher-order averaging schemes with error bounds for thermodynamical properties calculations by MD simulations, INRIA report 4875, 2003. (To appear in J. Chem. Phys. 2004.)
  2. Katsoulakis, Majda and Vlachos. Coarse-grained stochastic processes and Monte Carlo simulations in lattice system. J. Comp. Phys., 186:250-278, 2003.
    Katsoulakis, Majda and Vlachos. Coarse-grained stochastic processes for microscopic lattice system. Proc. Nat. Acad. Sci., 100(3):782-787, 2003.
  3. Cancès, et al. Control of Molecular Systems. [CLB], 246-253.
    Bandrauk and Chelkowski. Assymetric electron-nuclear dynamics in two-color laser fields; laser phase directional control of photofragments in H2. Phys. Rev. Lett. 84:3562-3565, 2000.
    LeBris et al., Quantum Control, AMS.
  4. Bornemann and Schütte. A mathematical investigation of the Car-Parrinello method. Numer. Math. 78:359-376, 1998.
  5. Schütte and Huisinga. Biomolecular conformations can be identified as metastable sets of molecular dynamics, [CLB], 699-745.
    Huisinga, Schütte and Stuart. Extracting macroscopic stochastic dynamics: model problems. Comm. Pure. App. Math 57:234-269, 2003.
  6. Bornemann, Nettersheim and Schütte. Quantum-classical molecular dynamics as an approximation to full quantum dynamics. J. Chem. Phys. 105:1074-1083, 1996. (In particular section V.)

Literature

We will use parts of the following texts:

[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

Contact

Organizers are:

Olof Runborg, olofr@nada.kth.se
Anders Szepessy, szepessy@nada.kth.se