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)

Lilia Boeri (lilia.boeri@uniroma1.it, Fermi building, fourth floor, room 407, ext. 2-3148)

Fermi building, third floor, room 304, on Mondays 5-7 pm; otherwise by appointment

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

Friday 12 pm ‐ 2 pm, Fermi building, second floor, Aula 5 (excluding Friday, December 6 )

The main objective of Computing Methods for Physics is that of providing an introduction to up-to-date computational methods that are used in research areas of current interest. Three different courses (channels) are offered. This channel is intended for students enrolled in the Condensed-Matter track. Its goal is to provide the students with both the theoretical background and the hands-on experience of two state-of-the-art numerical approaches within the field of condensed matter physics: a) the density-functional theory and the pseudopotential theory, two crucial ingredients for first-principles predictions of electronic states, structural energies and interatomic forces in real molecules and solids; b) 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" of 4 hours each, 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).

Self-consistent field for atoms and molecules: Hartree, Hartree-Fock, Density Functionals and Pseudopotentials (23 hours)

Quantum fluids: variational Monte Carlo, diffusion Monte Carlo, path-integral Monte Carlo (22 hours)

Crystalline solids based on Density Functionals and Pseudopotentials (18 hours)

L.D. Landau e E.M. Lifsic, Fisica teorica. Vol. 3: Meccanica quantistica. Teoria non relativistica. Editori Riuniti 1999

G.B. Bachelet e V.D.P. Servedio, Elementi di fisica atomica, molecolare e dei solidi, Aracne 2017

M. Rescigno, Stato fondamentale dell'atomo di elio con il metodo di Hartree, teoria e calcolo numerico (BS dissertation, Sapienza 2019)

DFT: an introduction, N. Argaman and G. Makov, Am. J. Phys. 68, 69-79 (2000)

DFT of Atoms and Molecules, R.G. Parr and W. Yang, Oxford University Press 1989

Pseudopotentials that work: from hydrogen to plutonium,

G.B. Bachelet, D.R. Hamann, and M. Schlüter, Phys. Rev. B. 26, 4199-4228 (1982)

Iterative minimization techniques for ab initio calculations: molecular dynamics and conjugate gradients,

M.C. Payne et al, Rev. Mod. Phys. 64, 1046-1097 (1992)

Phonons and related crystal properties from density-functional perturbation theory,

S. Baroni et al, Rev. Mod. Phys. 73, 515-562, (2001)

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

Path integrals in the theory of condensed helium, D.M. Ceperley, Rev. Mod. Phys. 67, 279-355 (1995)

Worm algorithm and diagrammatic MC: A new approach to continuous-space PIMC simulations,

M. Boninsegni, N. V. Prokof'ev, and B. V. Svistunov, Phys. Rev. E 74, 036701 (2006)

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

Variational Monte Carlo:

Stochastic integration, Metropolis algorithm

Correlated wavefunctions, local energy

Expectation values

Optimization by correlated sampling

Projection Monte Carlo:

Imaginary time evolution

Variational Path Integral, mixed and pure estimation

Diffusion Monte Carlo, branching random walk

Fermion sign problema and Fixed Node Approximation

Path-Integral Monte Carlo:

The quantum-classical isomorphism

Bose condensation and superfluidity

The Worm Algorithm

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

Total energy and interatomic forces: the Hellmann-Feynman theorem

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 e-learning pages.