<div class="tex2jax_process">Density Functional Theory (DFT) is a widely used computational method for addressing the quantum-mechanical many-body problem, with applications across fields such as quantum chemistry, materials science, geology, astrophysics, and biophysics. DFT-based simulations are used to study materials important for key technological and societal applications, with particular relevance to major challenges in Africa. Because of its broad relevance, DFT is an essential part of the training of masterâs and PhD students. The widespread use of DFT is partly due to freely available computational codes. However, DFT practitioners often ignore the fundamentals of the theory, which hinders its appropriate utilisation or even leads to its misuse. With the organisation of this focused school addressing specifically the fundamental theory aspects of DFT we aim to fill this gap.</div>
External organizer
Local Organizer
Discussion on the many-body Schrödinger equation based on an assignment given to the participants before the school. Why is it difficult? (interactions and exponential scaling of the dimension of the many-electron wavefunction with the number of electrons). Why is it interesting? (emergent properties due to interactions)Discussion on the many-body Schrödinger equation based on an assignment given to the participants before the school. Why is it difficult? (interactions and exponential scaling of the dimension of the many-electron wavefunction with the number of electrons). Why is it interesting? (emergent properties due to interactions)
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Antisymmetry of many-electron wavefunctions for fermions (Pauli principle); Slater determinant for the many-body wavefunctions of non-interacting electrons; derivation of Slater Condon rules for the expectation values of one-body and two-body operators on Slater determinants
Speaker : Victor ODARI (Masinde Muriro University of Science and Technology,Kenya)
Discussion of the goals of the solution of the many-body Schrödinger equation (observables in the ground state; identification of the ground state as infimum of the energy functional). Derivation of the variational principle in quantum mechanics and discussion of its implications. Constrained optimisation: introduction to the method of Lagrange multipliers;
Speaker : Eleonora LUPPI (Sorbonne Université,France)
What is a functional? What is the definition of functional derivative? How to calculate a functional derivative in practice;
Derivation of the Hartree-Fock equations from the combined use of the variational principle, the assumption of a Slater determinant for the many-body wavefunction, the use of the Slater-Condon rules and of Lagrange multipliers for the constrained optimisation of the energy functional;
Physical and mathematical properties of the single-particle Hartree-Fock equations (Hartree and Fock exchange potentials; local and non-local operators; self-consistent solution of the equations; evaluation of the total energy; Koopmans theorem for the single-particle eigenvalues; calculation of neutral excitations);
Discussion on the motivations/goals and possible strategies (advantages and disadvantages of the calculation of observables as functional of the wavefunction or functionals of reduced quantities such as the density, the density matrix, or the single-particle Greenâs function; introduction to post-Hartree-Fock methods in quantum chemistry and Monte Carlo stochastic methods).
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Derivation of the theorems and discussion of their significance (one-to-one correspondence between the ground-state density and the local external potential; reformulation of the variational principle; observables as functionals of the density)
Speaker : Victor ODARI (Masinde Muriro University of Science and Technology,Kenya)
Introduction of the model hamiltonian (interacting many-body problem with uniform background and constant external potential; historical origin and physical significance of the model); Divergence and regularization of the Coulomb interaction in infinite systems
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Mathematical issues with the Hohenberg-Kohn functional (unicity vs. existence of the density-potential map). Generalizations (e.g. for degenerate ground-state). Discussion of the v-representability condition for the density (definition of functional domain and connection with the requirement for the definition of functional derivatives). Levy-Lieb functionals;
Speaker : George AMOLO (Technical University of Kenya,Kenya)
Discussion about DFT: what we have achieved so far (from known functional of complicated object to unknown functional of simple object). How to approximate the energy functional? Thomas-Fermi and density-only DFT (direct approximations to the energy functional; historical perspectives and modern realisations).
Speaker : Victor ODARI (Masinde Muriro University of Science and Technology,Kenya)
Flipped classroom on the concept of auxiliary system (After watching the short video from the MOOC on DFT, the goal of each group is to identify at least 3 general properties of an auxiliary system).
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Physical and mathematical properties of the single-particle Kohn-Sham equations (calculation of the density, calculation of the energy; self-consistent solution; exchange and correlation potentials; physical meaning of eigenvalues: janak theorem) ;
Speaker : Gladys Kingâori (Technical University of Kenya,Kenya),Victor ODARI (Masinde Muriro University of Science and Technology,Eleonora LUPPI (Sorbonne UniversitĂ©,France),George AMOLO (Technical University of Kenya,Miriam CHEPKO...
Questions and answers on the topics of the week;
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Generalisation to spin-dependent DFT and spin-dependent Kohn-Sham equations (magnetic field as external potential; differences with spinless DFT).
Speaker : Eleonora LUPPI (Sorbonne Université,France)
One-electron limiting case. Definition of the energy functional in terms of exchange-correlation holes. Adiabatic connection method to define the energy functional (formal connection between non-interacting and interacting systems). Properties of the energy functional from uniform coordinate scaling.
What âlocalâ means. Importing results from the homogeneous electron gas (historical perspective and modern revisit). Approximations of exchange-correlation energy functional and exchange-correlation potentials in terms of the local density
Speaker : Eleonora LUPPI (Sorbonne Université,France)
How to go beyond the LDA? (From the LDA to functionals of the gradient of the local density and/or kinetic energy density). Problems with a straightforward gradient expansion. Remedies and generalisations (analysis in terms of the exchange-correlation hole; use of exact constraints). Advantages and shortcomings of LDA and GGA. Extensions to meta-GGA functionals.
Conceptual generalisation of DFT to electronic systems with a fractional number of electrons. Piecewise linearity of the energy as a function of the number of electrons.
Kohn-Sham band gap vs. fundamental band gap. Derivative discontinuity of the exchange-correlation energy at integer electron numbers and development of steps in real space of the exchange-correlation potential in the dissociation limit.
Derivation of generalized Kohn-Sham equations and their physical and mathematical properties (use of non-local Fock exchange operator). Comparison with Kohn-Sham and Hartree-Fock (while Kohn-Sham defines the kinetic energy as orbital functional, generalised Kohn-Sham also takes the exchange orbital functional from Hartree-Fock).
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Introduction to hybrid functional approximations and their properties. Hybrid functionals as practical approximations of generalized Kohn-Sham. Performances, advantages and advantages of most common hybrid functional approximations.
Speaker : Victor ODARI (Masinde Muriro University of Science and Technology,Kenya)
Introduction to van der Waals physics (origin of the attraction between isolated neutral objects). Why non-local functionals of the density are needed for describing van der Waals physics. Review of most common approximations (phenomenological derivation).
Speaker : Victor ODARI (Masinde Muriro University of Science and Technology,Kenya)
Introduction to LDA+U scheme (Phenomenology of strong correlation. Definition of the Hubbard model on a lattice with a local Hubbard interaction U. How to merge the Kohn-Sham scheme and the Hubbard model. Different choices of the double counting correction). Comparison with hybrid functionals.
Speaker : George AMOLO (Technical University of Kenya,Kenya)
One-electron and many-electron self-interaction. (Limiting case of one-electron system: cancellation of Hartree and Fock exchange and errors in LDA/GGA; Generalization to the many-electron case and connection to piecewise linearity). Self-interaction correction.
Introduction to time-dependent density-functional theory as a generalisation of DFT to time-dependent external potentials. Differences and analogies (from observables in the ground state to their time evolution; Runge-Gross theorem; time-dependent Kohn-Sham equations);
Introduction to linear response theory (calculation of the first-order variation of an observable induced by an external perturbation from a Taylor expansion; introduction of response functions; general properties such as causality and Kramer-Kronig relations; relation to spectroscopy measurements) .
Speaker : Eleonora LUPPI (Sorbonne Université,France)
Introduction to linear response TDDFT (Linear response formalism within TDDFT; Derivation of the Dyson equation for the calculation of the density-density response function from the non-interacting one; Exchange-correlation kernel. Most popular approximations: random phase approximation and adiabatic local density approximation);
Speaker : George AMOLO (Technical University of Kenya,Kenya)
Scientific presentations (15 mins) by a selection of participants and Q&A session on participantsâ ongoing scientific projects. Discussion of possible project to be carried out after the school by the participants (e.g. The Hubbard dimer: exact solution vs. Kohn-Sham;) 2
How to participate
For registration and application to a CIMPA financial support, read carefully the instructions given here. If you already know what to do, you can also directly go to the application website, create an account (if necessary) and apply to the school of your choice. Be aware that you will be redirected to an external website.