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Computational Statistical Mechanics

Giovanni Bachelet (giovanni.bachelet@roma1.infn.it), Saverio Moroni (saveriomoroni@gmail.com)

Fermi building, room 304, ext. 2-3474

Computational Statistical Mechanics aims at the evaluation of macroscopic collective and average properties of many-body systems (up to ~10

- the Density Functional theory, which reduces 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 compund 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 implement on a computer at least one of the methods learned from this course to a quantum many-body system (electrons in an isolated atom or in a periodic crystal; a quantum liquid or solid).

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 larger portion of the Monte Carlo part of the course. A numerical project within the methods spanned by the course, conceived, implemented on the computer, summarized and presented (with a short report or slide presentation) by a single individual or a small group is an integral part of the student's responsibilities.

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 which the student has performed.

Hartree and Hartree-Fock method for interacting electrons (7 hours) / Periodic Table (2 hours) / Density Functional Theory (6 hours) / Electrons in solids and pseudopotentials (7 hours)

Variational Monte Carlo (6 hours) / Diffusion Monte Carlo (6 hours) / Path-Integral Monte Carlo (6 ore)

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**References:**

*prerequisites for attending the course*

N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston 1976*

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

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)

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**Short table of contents:**

Schrödinger equation, variational principle

Interacting electrons

Hartree-Fock approximation

Density Functional Theory

Electrons in atoms: shell structure, Periodic Table

Electrons in crystals: Bloch, plane waves, pseudopotentials

Total energy and interatomic forces: Hellmann-Feynman

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

Variational Path Integral, mixed and pure estimation

Diffusion Monte Carlo, branching random walk

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 on http://www.giovannibachelet.it/CSM-15-16/.