Università Roma Tre

Educational Goals

The course introduces the Density Functional Theory (DFT) for first principle quantum simulations aimed at the calculation of electronic properties in condensed matter. This theory goes beyond the one-electron approximation and the Hartree-Fock method. The course also proposes the study of the Car-Parrinello method for Quantum molecular dynamics. There will be practical exercises in lab on condensed matter problems, with the use of the well known program Quantum Espresso.



Program

Introduction. Functionals vs. functions. The Raleigh-Ritz variational principle. Hohemberg- Kohn theorems. Example: the He atom.

The construction of a functional: the Thomas-Fermi-Hartree functional. Exchange-correlation effects: the Local Density approximation. Searching for the solution via the functional minimization. The concept of effective one-particle potential.

The Kohn-Sham equations. Generalization to systems with net spin polarization.

How to solve the Kohn-Sham equations in practice: basis sets. The KS equations in plane

waves. Pseudo-potentials.

Finding the self-consistent solution of the Kohn-Sham equation. Sampling the Brillouin zone

with k-point grids. Fermi level smearing and input/output mixing. (Lab session: finding the

ground state for Si),

The Hellmann-Feynman theorem in the DFT. Generalized forces. Atomic forces and pressure.

Optimization methods: structure. Born-Oppenheimer first-principle (or DFT-based) molecular

dynamics.

The Car-Parrinello method. Car-Parrinello versus Born-Oppenheimer first-principles MD.

Generalization: Fictitious Lagrangean with slow and fast degrees of freedom.

Exchange-correlation functionals: the Generalized Gradient Approximation; hybrid functionals; the Jacob’s ladder to DFT.

Introduction to the Density Functional Perturbation Theory (DFPT). Perturbations to all orders.

Second-order perturbations. The Sternheimer equation within DFPT.

Phonons from DFPT. A short lab’s session will be organized and supervised by the teacher.

Electric perturbations. Dielectric constant, effective charges. Infra-red spectra from DFPT.

Beyond the ground state. The meaning of the Kohn-Sham eigenvalues. Connection between

DFT and many-body theories via the Dyson’s equation.

Lab’s session:

Using Quantum Espresso. Session 1: how to find the ground state for the diamond structure of Si. Sessions 2-6 will be used to run a simulation based on the DFT. Each student will choose his/her own project among the available ones, which span several problems in condensed matter physics.

Materiale del corso fornito dal docente: F.Finocchi “DFT for Beginners”, manuale in pdf e

Presentazioni. Quantum Espresso istallato PC con Sistema Operativo Linux.