Lectures Introduction to computational modelling and statistics1 Potential models2 Density Functional (quantum) 1 3 Density Functional 2 4 Or: Understanding the physical and chemical properties of materials from an understanding of the underlying atomic processes Computational Modelling of Materials Recent Advances in Contemporary Atomistic Simulation

Useful Material Books A chemist’s guide to density-functional theory Wolfram Koch and Max C. Holthausen (second edition, Wiley) The theory of the cohesive energies of solids G. P. Srivastava and D. Weaire Advances in Physics 36 (1987) Gulliver among the atoms Mike Gillan, New Scientist 138 (1993) 34 Web Some version compiled for windows, install and good tutorial

Outline: Part 1, The Framework of DFT DFT: the theory Schroedinger’s equation Hohenberg-Kohn Theorem Kohn-Sham Theorem Simplifying Schroedinger’s LDA, GGA Elements of Solid State Physics Reciprocal space Band structure Plane waves And then ? Forces (Hellmann-Feynman theorem) E.O., M.D., M.C. …

Outline: Part2 Using DFT Practical Issues Input File(s) Output files Configuration K-points mesh Pseudopotentials Control Parameters  LDA/GGA  ‘Diagonalisation’ Applications Isolated molecule Bulk Surface

The Basic Problem Dangerously classical representation Cores Electrons

Schroedinger’s Equation Hamiltonian operator Kinetic Energy Potential Energy Coulombic interaction External Fields Very Complex many body Problem !! (Because everything interacts) Wave function Energy levels

First approximations Adiabatic (or Born-Openheimer) Electrons are much lighter, and faster Decoupling in the wave function Nuclei are treated classically They go in the external potential

H.K. Theorem The ground state is unequivocally defined by the electronic density Universal functional Functional ?? Function of a function No more wave functions here But still too complex

K.S. Formulation Use an auxiliary system Non interacting electrons Same Density => Back to wave functions, but simpler this time (a lot more though) N K.S. equations (ONE particle in a box really) (KS3) (KS2) (KS1) Exchange correlation potential

Self consistent loop Solve the independents K.S. =>wave functions From density, work out Effective potential New density ‘=‘ input density ?? Deduce new density from w.f. Initial density Finita la musicaYES NO

DFT energy functional Exchange correlation funtional Contains:  Exchange  Correlation  Interacting part of K.E. Electrons are fermions (antisymmetric wave function)

Exchange correlation functional At this stage, the only thing we need is: Still a functional (way too many variables) #1 approximation, Local Density Approximation:  Homogeneous electron gas  Functional becomes function !! (see KS3)  Very good parameterisation for Generalised Gradient Approximation: GGA LDA

DFT: Summary The ground state energy depends only on the electronic density (H.K.) One can formally replace the SE for the system by a set of SE for non-interacting electrons (K.S.) Everything hard is dumped into E xc Simplistic approximations of E xc work ! LDA or GGA

And now, for something completely different: A little bit of Solid State Physics Crystal structurePeriodicity

Reciprocal space Real Space a i Reciprocal Space b i Brillouin Zone (Inverting effect) k-vector (or k-point) sin(k.r) See X-Ray diffraction for instance Also, Fourier transform and Bloch theorem

Band structure Molecule E Crystal Energy levels (eigenvalues of SE)

The k-point mesh Brillouin Zone (6x6) mesh Corresponds to a supercell 36 time bigger than the primitive cell Question: Which require a finer mesh, Metals or Insulators ??

Plane waves Project the wave functions on a basis set  Tricky integrals become linear algebra  Plane Wave for Solid State  Could be localised (ex: Gaussians) ++= Sum of plane waves of increasing frequency (or energy) One has to stop: E cut

Solid State: Summary Quantities can be calculated in the direct or reciprocal space k-point Mesh Plane wave basis set, E cut

Now what ? We have access to the energy of a system, without any empirical input With little efforts, the forces can be computed, Hellman-Feynman theorem Then, the methodologies discussed for atomistic potential can be used  Energy Optimisation  Monte Carlo  Molecular dynamics