lectures and exercises on DFT & related topics (G.B. Bachelet)

1^st hour

Introduction to this part of the course and expected computationa tools: home-made atomic codes, Quantum Espresso (QE), Gaussian.

2^nd & 3^rd hour

First part of a QE tutorial: self-consistent calculation of the electronic density and the total energy of crystalline silicon at the experimental lattice constant based on plane waves and pseudopotentials (input and output analysis and visualization via XCrysden).

4^th & 5^th hour

More self-consistent calculations (from the same tutorial) for different values of the lattice constant: total energies, bulk modulus, anharmonicity and zero-temperature thermal expansion are in good agreement with the corresponding experimental values.

6^th & 7^th hour

Theoretical and algorithmic basis for self-consistent calculations of the electronic structure of atoms, molecules and solids: start with the Hartree theory and its computer implementation for the (simplest) case of a neutral helium atom; with an attractive central potential (example: the hydrogen electron), for any energy there are two (non-normalizable) solutions of the radial Schödinger equation, one regular in the origin and the other regular at infinity, which become coincident and normalizable when the energy equals the eigenvalues of the discrete spectrum of bound states.

8^th & 9^th hour

More on the Hartree atom in the example of neutral helium: radial logarithmic grid (as opposed to evenly spaced grid), classical turning point, mismatch of the logarithmic derivative as a basis for the estimate of the error in the energy eigenvalue (based on notes and related fortran and C codes for the Hartree neutral helium atom.

10^th & 11^th hour

Hands-on computer exercises on the Hartree neutral helium atom.

12^th & 13^th hour

Density Functional Theory: Hohenberg-Kohn theorems (1964); Kohn-Sham equations (1965), first part.

14^th & 15^th hour

Density Functional Theory: second part, the homogeneous electron gas (jellium) model (Ashcroft-Mermin pag. 332-337), the local density approximation is bad for the entire HK functional (=Thomas-Fermi-Dirac approximation) but good if applied to the exchange-correlation part only (see e.g. this Introduction to DFT and exchange-correlation energy functionals by R.O. Jones, 2006).

16^th & 17^th hour

Quick recap of the Kohn-Sham theory and the local-density approximation of the exchange-correlation energy (7 slides).
Pseudopotential theory, first 27 out of 48 slides (~26 MB) NB: these slides are password protected and contain lots of useful links.

18^th & 19^th hour

Pseudopotential theory, slides 28-43 out of 48 slides (~26 MB) NB: these slides are password protected and contain lots of useful links.

20^th & 21^th hour

Pseudopotential theory, slides 44-48 out of 48 slides (~26 MB) NB: these slides are password protected and contain lots of useful links. Unscreening vs. core-valence overlap, problem and solution: discussion of the nonlinear core correction .
In aula: ripassi, bignamino DFT-LDA (1 facciata) e onde piane (1 facciata); teorema di Hellmann e Feynman (6 facciate); conseguenze per i calcoli da primi principi basati su DFT, pseudopotenziali e onde piane. Per la lettura a casa: "Unified approach for molecular dynamics and density functional theory", l'articolo originale di Car e Parrinello del 1985 e un'ottima rassegna sul metodo della risposta lineare per i fononi in ambito DFT di Baroni et al.

22^th & 23^th hour

Esercitazione hands-on sulle vibrazioni di HgI₂ con Quantum Espresso (non molto riuscita!)

24^th & 25^th hour

Esercitazione hands-on con Gaussian. Calcolo della struttura elettronica della molecola molecola H₂O nella geometria di equilibrio (cfr. dati sperimentali) e visualizzazione con Molden. Tutorial della EaStCHEM Research Computing Facility per geometria di equilibrio e vibrazioni. Qui e qui un paio di output tipici commentati.