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

1^st hour (Monday, February 25, 2019)

Introduction to this part of the course: theory and typical computational tools: home-made atomic codes, VESTA, Quantum Espresso (QE), possibly also Gaussian.

2^nd & 3^rd hour (Friday, March 1, 2019)

Pseudopotential theory, first 19 out of 48 slides (~26 MB) NB: these slides are password protected and contain lots of useful links.

4^nd & 5^th hour (Friday, March 8, 2019)

Getting familiar with periodic crystals: sc, fcc, bcc, CsCl, rocksalt, diamond and zincblende structures with VESTA.

6^th & 7^th hour (Friday, March 15, 2019)

Pseudopotential theory, slides 19-23 (~26 MB); a refresher on Bloch's theorem and plane-wave basis sets, which implies perfect control of Fourier series and transforms. [with F. Surra]

8^th & 9^th hour (Friday, March 22, 2019)

Pseudopotential theory, slides 19-23 (~26 MB); a refresher on Bloch's theorem and plane-wave basis sets, which implies perfect control of Fourier series and transforms. [with M. Lentini e M. Rozzi]

10^th & 11^th hour (Monday, March 25, 2019)

Pseudopotential theory, slides 24-27 (~26 MB); nonlocal pseudopotentials, model pseudopotentials. [with M. Lentini e M. Rozzi]

12^th & 13^th hour (Monday, March 29, 2019)

The homogeneous electron gas (summary: first lines of this page; underlying theory: here). Physical origin and mathematical expression of nonlocal (ℓ-dependent) ionic pseudopotentials (here an article with some useful formulae and references). [with M. Rozzi]

14^th & 15^th hour (Friday, April 5, 2019)

Plane-wave matrix elements of nonlocal (ℓ-dependent) ionic pseudopotentials (obtained by expanding each plane wave as an infinite sum of spherical waves ) no longer depend on the G‐G′ difference, but separately on k+G and on k+G′; the alternative of separable (aka "fully nonlocal" pseudopotentials) may be computationally convenient (see e.g. here). [with M. Rozzi]

16^th & 17^th hour (Monday, May 27, 2019)

Pseudopotential theory, slides 28-32 (~26 MB): from model pseudopotentials to transferable pseudopotentials. [with M. Lentini e M. Rozzi]

18^th‐19^th‐20^th hour (Friday, May 31, 2019)

Pseudopotential theory, slides 33-47 (~26 MB): from transferable pseudopotentials to norm-conserving pseudopotentials. [with M. Lentini e M. Rozzi]

21^th‐22^th‐23^th hour (Friday, June 7, 2019)

Recall lecture 12: the homogeneous electron gas (first lines of this page; underlying theory: here). The Thomas-Fermi(-Dirac) theory: at each point of space, approximate the (unknown) electronic energy density of a real system (atom, molecule or solid), whose electron spatial distribution is strongly inhomogeneous, with the (known) energy density of a homogeneous electron gas of the same local density, i.e. attribute to each point of space an electronic energy density ε(r)≃ε^HEG(n(r)), where ε^HEG(n) is the energy density of the HEG (Homogeneous Electron Gas) recalled at the beginning of the lecture. In this way the ground-state total electronic energy of the system becomes a functional of the electronic density E[n]≃∫ dr ε^HEG(n(r)) n(r) +∫ dr v^EXT(r) n(r)+ (1/2)∫ dr ∫ dr' n(r)n(r')/(|r-r'|), and the ground-state energy and electronic density are determined by minimizing such functional under the condition that the number of electrons N=∫ drn(r) remains fixed. The so-called Slater Xα approximation, instead, replaces only the Hartree-Fock (orbital-dependent) exchange energy with an average exchange energy which ends up in an exchange energy density proportional to the cube root of the electronic density like its HEG counterpart, but with a different coefficient (these and many other historical premises of DFT may be found in this Introduction to DFT and exchange-correlation energy functionals by R.O. Jones, 2006). Density Functional Theory aka exact theory of the inhomogeneous electron gas: the Hohenberg-Kohn theorems (1964), first part.
[with M. Lentini e M. Rozzi]

24^th‐25^th‐26^th hour (Thursday, June 13, 2019)

Density Functional Theory aka exact theory of the inhomogeneous electron gas: the Hohenberg-Kohn theorems (1964), second part; the Kohn-Sham equations; the local density approximation (LDA), bad for the entire HK functional (=Thomas-Fermi-Dirac approximation) but quite good when applied to the exchange-correlation part only (see e.g. this Introduction to DFT and exchange-correlation energy functionals by R.O. Jones, 2006). Quick recap of the Kohn-Sham theory and the local-density approximation of the exchange-correlation energy (7 slides).
[with M. Lentini e M. Rozzi]

27^th & 28^th hour (Friday, June 14, 2019)

The simplest example of a self-consistent-field calculation: laboratory exercise on the ground-state helium atom in the Hartree approximation (here a complete set of theoretical and numerical tools)
[with M. Lentini e M. Rozzi]