Corso di Laurea Magistrale Interateneo (Master) in Physics
Academic Year 2020-2021
Teacher: Paolo Giannozzi, room L1-2-BE, Department of Mathematics,
Computer Science and Physics, Via delle Scienze 206, Udine
tel. +39 0432 558216, fax +39 0432 558222
web:
http://www.fisica.uniud.it/~giannozz,
e-mail: paolo . giannozzi at uniud . it
Office hours: during the semester when possible,
I'll be in the Physics building Monday and Wednesday from 9:30 to 11:00.
Send me an e-mail or phone me
to get an appointment (in Trieste or in Udine) at a different time
when not possible.
Course Description
Goals:
this course provides an introduction to numerical methods and techniques
useful for the numerical solution of quantum mechanical problems, especially
in atomic and condensed-matter physics. The course is organized as a
series of theoretical lessons in which the physical problems and the
numerical concepts needed for their resolution are presented, followed
by practical sessions in which examples of implementatation for specific
simple problems are presented. The student will learn to use the concepts
and to practise scientific programming by modifying and extending the
examples presented during the course.
Syllabus:
Schroedinger equations in one dimension: techniques for numerical solutions.
Solution of the Schroedinger equations for a potential with spherical symmetry.
Scattering from a potential.
Variational method: expansion on a basis of functions, secular problem,
eigenvalues and eigenvectors.
Examples: gaussian basis, plane-wave basis.
Many-electron systems:
Hartree-Fock equations, self-consistent field, exchange
interaction. Numerical solution of Hartree-Fock equations in atoms
with radial integration and on a gaussian basis set. Introduction to
numerical solution of electronic states in molecules. Electronic states
in solids: solution of the Schroedinger equation for periodic potentials.
Introduction to exact diagonalization of spin systems. Introduction to
Density-Functional Theory.
Bibliography
Lecture Notes (updated):
Introduction,
Ch.1,
Ch.2,
Ch.3,
Ch.4,
Ch.5,
Ch.6,
Ch.7,
Ch.8,
Ch.9,
Ch.10,
Ch.11,
Appendix,
all.
See also:
J. M. Thijssen, Computational Physics, Cambridge University Press,
Cambridge, 1999, Ch.2-4, 5, 6.1-6.4, 6.7.
A rather detailed introduction to Density-Functional techniques can
be found in the first chapters of the lecture notes of my (now defunct)
course on
Numerical Methods in Electronic Structure.
Requirements: basic knowledge of Quantum Mechanics,
of Fortran or C programming, of an operating system (preferrably Linux).
Exam: personal project consisting in the numerical solution
of a problem, followed by oral examination (typically consisting in
the discussion of a subject, different from the one of the personal
project, chosen by the student). Contact me a few weeks before the
exam to receive the personal project (if not yet assigned) and to
set a date. A short written report on the personal project and the
related code(s) should be provided no later than the day before the exam.
Schedule
The course starts on March 8th.
Classes are held online on Teams (Link on
,
https://corsi.units.it/didattica-a-distanza,
send me an email if you cannot find it)
until further notice
- Monday 11-13,
Aula C - Wednesday 11-13,
Lab. T21
Deviation from the above schedule are marked in boldface in the detailed table
of arguments (subject to changes>/i>) below.
N.
|
Date
|
Subject
|
1.
|
8 March
|
One-dimensional Schroedinger equation:
Reminder: harmonic oscillator, analytical solution. Discretization,
Numerov algorithm, numerical stability, eigenvalue search using
stable outwards and inwards integrations. (Notes: Ch.1)
|
2.
|
10 March
practical session
|
Numerical solution of the one-dimensional Schroedinger equation:
examples for the harmonic oscillator (code harmonic0:
fortran,
C;
code harmonic1:
fortran,
C).
|
3.
|
15 March
|
Three-dimensional Schroedinger equation:
Central potentials, variable separation, logarithmic grids,
perturbative estimate to accelerate eigenvalue convergence.
(Notes: Ch.2). A glimpse on true three-dimensional problems on a grid.
(Notes: appendix A)
|
4.
|
17 March
practical session
|
Numerical solution for spherically symmetric potentials:
example for Hydrogen atom
(code hydrogen_radial:
fortran,
C)
|
5.
|
22 March
|
Scattering from a potential:
cross section, phase shifts, resonances.
(Notes: Ch.3; Thijssen: Ch.2)
|
6.
|
24 March
pratical session
|
Calculation of cross sections:
numerical solution for Lennard-Jones potential
(code crossection:
fortran,
C).
|
7.
|
29 March
|
Variational method:
Schroedinger equation as minimum problem,
expansion on a basis of functions, secular problem,
introduction to diagonalization algorithms.
(Notes: Ch.4; Thijssen: Ch.3)
|
8.
|
31 March
pratical session
|
Variational method using an orthonormal basis set:
example of a potential well in plane waves
(code pwell:
fortran,
C).
|
9.
|
12 April
|
Non-orthonormal basis sets:
gaussian functions.
(Notes: Ch.5)
|
10.
|
14 April
practical session
|
Variational method with gaussian basis set:
solution for Hydrogen atom
(code hydrogen_gauss:
fortran,
C)
|
11.
|
19 April
|
The Hartree-Fock method:
Slater determinants, Hartree-Fock equations, self-consistent field.
(Notes: Ch.6; Thijssen: Ch.4.1-4.5)
|
12.
|
21 April
practical session
|
He atom in Hartree-Fock approximation:
solution with radial integration and self-consistency (code
helium_hf_radial:
fortran,
C).
|
13.
|
26 April
|
Molecules:
Born-Oppenheimer approximation, potential energy surface,
diatomic molecules.
introduction to numerical solution for molecules.
(Notes: Ch.7; Thijssen: Ch.4.6-4.8)
|
14.
|
28 April
practical session
|
Molecules with gaussian basis:
solution of Hartree-Fock equations on a gaussian basis
for a H2 molecule (code h2_hf_gauss:
fortran,
C).
|
15.
|
3 May
|
Electronic states in crystals:
Bloch theorem, band structure.
(Notes: Ch.8; Thijssen: Ch.4.6-4.8)
|
16.
|
5 May
practical session
|
Periodic potentials:
numerical solution with plane waves of the Kronig-Penney model
(code periodicwell:
fortran,
C;
example of usage of FFT:
fortran,
C).
|
17.
|
10 May
|
Electronic states in crystals II:
three-dimensional case, methods of solution,
plane wave basis set, introduction to the
concept of pseudopotential.
(Notes: Ch.9; Thijssen: Ch.6.1-6.4, 6.7)
|
18.
|
12 May
pratical session
|
Pseudopotentials:
solution of the
Cohen-Bergstresser model
for Silicon
(code cohenbergstresser:
fortran,
C).
|
19.
|
17 May
|
Spin systems
Introduction to spin systems: Heisenberg model,
exact diagonalization, iterative methods for
diagonalization, sparseness.
(Notes: Ch.10)
|
20.
|
19 May
practical session
|
Exact Diagonalization
Solution of the Heisenberg model with Lanczos chains
(code heisenberg_exact:
fortran,
C).
|
21.
|
24 May
|
Density-Functional Theory
Hohenberg-Kohn theorem, Kohn-Sham equations
(Notes: Ch.11)
|
22.
|
26 May
semi-practical session
|
DFT with plane waves and pseudopotentials
Fast Fourier-Trasform and iterative techniques;
sample code ah (fortran,
C)
solving Si with
Appelbaum-Hamann
pseudopotentials).
|
23.
|
31 May
|
Density-Functional Theory II
|
24.
|
7 June
|
Assignment of exam problems
|
Last modified 23 May 2021
