Computing Methods for Physics

Giovanni Bachelet (giovanni.bachelet@roma1.infn.it, Fermi building, third floor, room 304, ext. 2-3474)

Saverio Moroni (saveriomoroni@gmail.com, Fermi building, third floor, room 304, ext. 2-3474)

Fermi building, third floor, room 304,

Monday 1 pm ‐ 4 pm (12 pm ‐ 4 pm, in certain weeks), Fermi building, second floor, Aula Calcolo

Friday 9 am ‐ 11 am, Fermi building, second floor, Aula Calcolo

The main objective of Computing Methods for Physics is to provide an introduction to up-to-date computational methods currently used in our fields of research; this is why four different courses (channels) are offered under one title.

This channel is mainly intended for students enrolled in the Condensed-Matter track. Its goal is to provide the students with the theoretical background and the hands-on experience of three numerical approaches within the field of condensed matter physics: (a) the Hartree-Fock method, historically the first mean-field approximation to electronic states of atoms and molecules; (b) the density-functional theory and the pseudopotential theory, two crucial ingredients for today's first-principles predictions of electronic states, structural energies and interatomic forces in real molecules and solids; (c) the quantum (variational, diffusion, path-integral) Monte Carlo methods, their applicability and the motivations of their use in the numerical study of quantum many-body systems (solid or liquid helium, the electron gas, electrons in atoms and molecules).

This course aims at the evaluation of macroscopic collective and average properties of many-body systems (up to ~10

- the Hartree-Fock method and the Density Functional theory, which reduce the many-electron problem to a self-consistent-field problem, and the pseudopotential theory, which further simplifies atoms by eliminating their inner-core electrons: two crucial ingredients for the first-principles prediction of the stability of a compound and of its lattice and molecular dynamics;
- quantum (variational, diffusion, path-integral) Monte Carlo methods, their applicability and the motivations of their use in the numerical study of quantum many-body interacting systems (like the electron gas, electrons in atoms and molecules, solid or liquid helium).

- have clear ideas on the theories and algorithms which presently allow the calculation from first principles of many properties of atoms, molecules, and solids, and also on their limits and on the directions of development of this field of research;
- be able to put into practice on a computer one or more of the methods learned in this course to a quantum many-body system (electrons in an atom, a molecule or a periodic crystal; a quantum liquid or solid).

Skills and knowledge learned within the basic courses of the BS program in Physics, in particular concerning: Computer science, Mathematical methods for Physics, Quantum Mechanics, Statistical Mechanics and Elementary atomic, molecular and solid-state Physics.

Along with the essential elements of the theory (frontal lectures), examples and applications will be both presented and tested in practice (computer lab). Hands-on sessions will represent a large portion of the course.

A numerical project based on the methods spanned by the course (implemented on the computer and presented with a short report or slide presentation) is foreseen for each individual or small group of students. The final examination consists of an individual discussion of (i) one or more subjects covered by the course, as listed in the short table of contents; and (ii) the numerical project to which the student has contributed. In alternative to the numerical project, three tests in progress ("prove di esonero", their dates may be found in the schedule) may be taken at the end of each part of the course (see also the lecture & exercises plan and the short table of contents below). The date of the final examination will be agreed upon with the students.

Self-consistent field for atoms and molecules: Hartree, Hartree-Fock (20 hours)

Density Functionals and Pseudopotentials for Crystalline solids (19 hours)

Quantum fluids: variational Monte Carlo, diffusion Monte Carlo, path-integral Monte Carlo (18 hours)

L.D. Landau e E.M. Lifsic,

G.B. Bachelet e V.D.P. Servedio:

Maria Rescigno & Giovanni B. Bachelet,

Simone Lo Franco,

N. Argaman and G. Makov,

R.G. Parr and W. Yang,

G.B. Bachelet, D.R. Hamann, and M. Schlüter,

M.C. Payne et al,

Microscopic Simulations in Physics, D.M. Ceperley, Rev. Mod. Phys. 71, S438-443 (1999)

QMC simulations of solids, W.M.C. Foulkes et al, Rev. Mod. Phys. 73, 33-83 (2001)

Applications of QMC methods in condensed systems, Jindrich Kolorenc and Lubos Mitas, Rep. Prog. Phys. 74, 026502 (28pp) (2011)

Schrödinger equation, variational principle

Interacting electrons

Hartree-Fock approximation

Density Functional Theory

Electrons in atoms: shell structure, Periodic Table, pseudopotentials

Electrons in crystals and plane waves: Bloch's theorem on the computer

Total energy and interatomic forces: the Hellmann-Feynman theorem

Variational Monte Carlo:

Stochastic integration, Metropolis algorithm

Correlated wavefunctions, local energy

Expectation values

Optimization by correlated sampling

Projection Monte Carlo:

Imaginary time evolution

Diffusion Monte Carlo, branching random walk, mixed and pure estimation

Fermion sign problema and Fixed Node Approximation

NB

Theoretical, methodological, and computational aspects of the above subject

list are addressed by the course. More details, references, notes, links,

and computer codes are incrementally supplied within the current academic year's e-learning pages;

last year's e-learning pages also contain the recording of most lectures of the lockdown time.