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Computational Modelling of Materials

Computational Modelling of Materials. Recent Advances in Contemporary Atomistic Simulation. Or: Understanding the physical and chemical properties of materials from an understanding of the underlying atomic processes. http://secamlocal.ex.ac.uk/people/staff/ashm201/Atomistic/ Lectures

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Computational Modelling of Materials

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  1. Computational Modelling of Materials Recent Advances in Contemporary Atomistic Simulation Or: Understanding the physical and chemical properties of materials from an understanding of the underlying atomic processes http://secamlocal.ex.ac.uk/people/staff/ashm201/Atomistic/ Lectures Introduction to computational modelling and statistics 1 Potential models 2 Density Functional (quantum) 1 3 Density Functional 2 4

  2. Useful Material • Books • A chemist’s guide to density-functional theoryWolfram Koch and Max C. Holthausen (second edition, Wiley) • The theory of the cohesive energies of solidsG. P. Srivastava and D. WeaireAdvances in Physics 36 (1987) 463-517 • Gulliver among the atomsMike Gillan, New Scientist 138 (1993) 34 • Web • www.nobel.se/chemistry/laureates/1998/ • www.abinit.org Some version compiled for windows, install and good tutorial

  3. 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. …

  4. Outline: Part2Using DFT • Practical Issues • Input File(s) • Output files • Configuration • K-points mesh • Pseudopotentials • Control Parameters • LDA/GGA • ‘Diagonalisation’ • Applications • Isolated molecule • Bulk • Surface

  5. The Basic Problem Dangerously classical representation Cores Electrons

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

  7. 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

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

  9. 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) (KS1) Exchange correlation potential (KS2) (KS3)

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

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

  12. 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 LDA Generalised Gradient Approximation: GGA

  13. 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 Exc • Simplistic approximations of Exc work ! LDA or GGA

  14. And now, for something completely different:A little bit of Solid State Physics Crystal structure Periodicity

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

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

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

  18. 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: Ecut

  19. Solid State: Summary • Quantities can be calculated in the direct or reciprocal space • k-point Mesh • Plane wave basis set, Ecut

  20. 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

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