Course title:
Computing Methods for Physics

Giovanni Bachelet (, Fermi building, third floor, room 304, ext. 2-3474)
Saverio Moroni (, Fermi building, third floor, room 304, ext. 2-3474)

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

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*

* lectures, hands-on sessions and consulting sessions will be held in presence; in case of exceptional health problems, quarantine, etc, alternative remote procedures will be agreed upon with individual students, on a case-by-case basis

Subject and scope:
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 ~1022 particles) through the microscopic simulation (by e.g. mean-field theories or Monte Carlo methods) of very large samples of their constituent particles. In this context the course will focus on "realistic" many-body systems like electrons in atoms, molecules and solids, or atoms in classical and quantum solids and fluids. To attack them we will need tools common to other branches of Computational Physics (series expansions, matrix inversion and diagonalization, root-finding algorithms, numerical approaches to eigenvalue-eigenfunction systems), and specific tools (like Monte Carlo methods). This course will deal with some of these tools. On one hand, it will deal with the electronic structure theories which over the last 40 years, along with the ever-increasing computer power, have allowed a reliable description of many molecular and solid-state systems, like e.g. an extremely accurate prediction of their interatomic forces, and thus of their structural and dynamical properties "from first-principles". On the other hand, it will deal with quantum Monte Carlo methods, which over the last 40 years have proved a powerful, "numerically exact" simulation tool for realistic and model quantum many-body systems. For both groups of methods, the ambition of the course is to provide the attendants with both the theoretical background and the practice of state-of-the-art computer codes. In particular, the course will deal with:
  1. 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;
  2. 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).
At the end of the course the student should:
  1. 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;
  2. 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.

Teaching method:
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.

Evaluation method:
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.

Lecture & exercises plan:
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)

    prerequisites for attending the course:
      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
    self-consistent field, Density Functional theory (DFT):
      Maria Rescigno & Giovanni B. Bachelet, Ground-state Helium atom with the Hartree method: versione italiana, versione inglese (and references therein)
      Simone Lo Franco, Molecole biatomiche omonucleari con il metodo di Hartree-Fock (italian only, and references therein)
      N. Argaman and G. Makov, DFT, an introduction: Am. J. Phys. 68, 69-79 (2000)
      R.G. Parr and W. Yang, DFT of Atoms and Molecules, Oxford University Press 1989
      G.B. Bachelet, D.R. Hamann, and M. Schlüter, Pseudopotentials that work, from hydrogen to plutonium: Phys. Rev. B. 26, 4199-4228 (1982)
      M.C. Payne et al, Iterative minimization techniques for ab initio calculations: molecular dynamics and conjugate gradients: Rev. Mod. Phys. 64, 1046-1097 (1992)
    quantum Monte Carlo (QMC)
      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)

Short table of contents:
   Self-consistent field (Giovanni Bachelet)
      Schrödinger equation, variational principle
      Interacting electrons
      Hartree-Fock approximation
      Density Functional Theory
      Electrons in atoms: shell structure, Periodic Table, pseudopotentials
   Crystalline solids (Giovanni Bachelet)
      Electrons in crystals and plane waves: Bloch's theorem on the computer
      Total energy and interatomic forces: the Hellmann-Feynman theorem
   Quantum Monte Carlo (Saverio Moroni)
    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
   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.