4 April 2008Suranaree Unversity of Technology Electronic Structure Calculations Stewart Clark Department of Physics University of Durham, UK The CASTEP code

4 April 2008Suranaree Unversity of Technology Where is Durham? Durham

4 April 2008Suranaree Unversity of Technology My Research Interests Developing ab initio methods for computational solution of electronic structure of materials Electronic structure leads to Material Properties Implementation of methods: an author of CASTEP electronic structure code ( Fortran 95 Massively parallel (MPI) Applications to many areas of physics, chemistry and materials science

4 April 2008Suranaree Unversity of Technology What would we like to achieve? Computers get cheaper and more powerful every year. Experiments tend to get more expensive each year. IF computer simulation offers acceptable accuracy then at some point it should become cheaper than experiment. This has already occurred in many branches of science and engineering. Possible to achieve this for properties of materials?

4 April 2008Suranaree Unversity of Technology Property Prediction Property calculation provided link with experimental measurements: -For analysis -For scientific/technological interest To enable interpretation of experimental results To predict properties over and above that of experimental measurements

4 April 2008Suranaree Unversity of Technology Computers Used in This Work Calculations performed on full range of computing platforms, including: Standard PC Beowulf Cluster Supercomputer Using distributed memory and fast interconnect Infiniband Myrinet Cray-Rainier

4 April 2008Suranaree Unversity of Technology Aim of ab initio calculations Atomic Numbers Solve the quantum mechanical equations for the electrons Predict physical and chemical properties of systems

4 April 2008Suranaree Unversity of Technology From First Principles Research output The equipment Application “Base Theory” (DFT) Implementation (the algorithms and program) Setup model, run the code Scientific problem- solving “Analysis Theory”

4 April 2008Suranaree Unversity of Technology The density functional plane wave approach Whole periodic table without bias. Periodic units containing thousands of atoms (on large enough computers). Structural optimisation. Finite temperature simulations (molecular dynamics) on pico-second timescales. Lots of others…if experiments can measure it, we try to calculate it – and then go further… Toolbox for material properties

4 April 2008Suranaree Unversity of Technology 3-Level Problem Need to know where the atomic nuclei are Need to know where the electrons are How do they vary with time? Genuine many-body problem: macroscopic materials contain > atoms Don’t want to do just a few atoms/molecules - want to do BULK materials

4 April 2008Suranaree Unversity of Technology Length and timescales Diffusion s s Bond motion Quantum mechanics s Intermolecular motion Atomistic Modelling s Molecular alignment Coarse-grained Polymers >>10 -7 s Electronic transition

4 April 2008Suranaree Unversity of Technology Electrons: the quantum mechanics A set of n one-electron equations that must be solved self-consistently Numerical methods represent variables and functions evaluate the terms iterate to self-consistency

4 April 2008Suranaree Unversity of Technology The nuclei: Model systems In this kind of first-principles calculation Are 3D-periodic From one atom to a few thousand atoms Supercells Periodic boundaries Bloch functions Bulk crystalSlab for surfaces Boundary conditions: periodic

4 April 2008Suranaree Unversity of Technology Electronic Structure Valence electron structure [Rb(anti-dchyl- 18c6)][Ni(dmit)2] Basics first: can get electronic structure for any arrangement of atoms in a solid (given enough computer power!) Robertson N; Clark, SJ; et al. Chem. Comm. Issue 25, 3204 (2005).

4 April 2008Suranaree Unversity of Technology Electron by electron Multiband molecular conductor

4 April 2008Suranaree Unversity of Technology Summary so far Rely only on quantum mechanics At first sight this just gives electronic structure Would like to calculate any property of a material without the need for experiment  Solids  Liquids  Surfaces  Molecules Limitations are finite speed and memory of computers

4 April 2008Suranaree Unversity of Technology Structure: where are the atoms?  Minimum energy corresponds to zero force (F=-dE/dR)  Plane wave methods get accurate forces for low cost  Much more efficient than just using energy alone  Equilibrium bond lengths, angles, etc.  Unit cell dimensions: Minimum enthalpy corresponds to zero force and stress  Can therefore minimise enthalpy w.r.t. supercell shape due to internal stress and external pressure  Pressure-driven phase transitions  Warning: nature does not always find the minimum energy!!!

4 April 2008Suranaree Unversity of Technology High Pressure Phases External pressure can be applied to determine high pressure structures and energy Volume Energy Phase I Phase II Common tangent gives transition pressure: P=-dE/dV V II VIVI

4 April 2008Suranaree Unversity of Technology Example: Silicon Structure is a multi-minimum problem Can obtain the order in which phases should appear The problem is transition barriers Hence (meta-)stability cannot be determined. Clark, SJ; et al. Phys. Rev. B 49, 5329 and Phys. Rev. B 49, 5341

4 April 2008Suranaree Unversity of Technology Surfaces Surface structure Catalysis Chemical reactions S. J. Clark, et al, Phys. Rev. B 50, 5728 V. Timon, S. J. Clark, et al, Phys. Rev. B 72, Movie, courtesy of M. J. Probert, University of York

4 April 2008Suranaree Unversity of Technology Structure prediction: case study Glycine (simple amino acid) Large range of bonding strengths  Covalent  Hydrogen-bonds  Van der Waals  Zwitterionic S. J. Clark, et al Crystal Growth and Design 5(4) 1437 and 5(4) 1443.

4 April 2008Suranaree Unversity of Technology Why choose glycine? “Simple” molecule (actually, it’s not!) Large range of bonding strengths. Good experimental results to compare to. Horrible things happen(!): Zwitterionic in crystal, not in gas phase. Difficult for empirical potentials to capture all of this in general Need quantum mechanics to get it right

4 April 2008Suranaree Unversity of Technology Prediction: what is the structure?

4 April 2008Suranaree Unversity of Technology How about something simpler? Hydrogen (how “difficult” can that be?) Structure of hydrogen under very high pressure C. J. Pickard, et al, Nature Physics 3, 473 (2007)

4 April 2008Suranaree Unversity of Technology Or something more complicated? TRP polypeptide (small protein) in water 1230 atoms per molecule + nH 2 O S. J. Clark, K. Refson and I. Kuprov, in press (2008)

4 April 2008Suranaree Unversity of Technology That’s the good news Note: this is an optimistic overview However structure prediction does not always work Amongst these successful cases, I could have reported some failures Sometimes nature is just too complicated (yet!) or needs too much CPU power!

4 April 2008Suranaree Unversity of Technology Finite temperature As noted, real materials do not have to stay in lowest energy state There are several ways of incorporating finite temperature: The two most useful are:  Molecular Dynamics  Phonon density of states (atomic vibrations)

4 April 2008Suranaree Unversity of Technology Molecular Dynamics  Can do dynamics of atoms using forces calculated from ab initio electronic structure  Copes with unusual geometry, bond-breaking, chemical reactions, catalysis, diffusion, etc  Incorporates effects of finite temperature of ions  Can generate thermodynamic information from ensemble averaging  Time dependent phenomena  Temperature driven phase transitions

4 April 2008Suranaree Unversity of Technology Structures without experiment? U(x) x start stop Simulated Annealing: Gets over barriers – however does not guarantee global minimum. A multi-minimum problem

4 April 2008Suranaree Unversity of Technology Example of Dynamics Radiation damage: breaking and making of chemical bonds Movie courtesy of M. Probert, University of York, UK

4 April 2008Suranaree Unversity of Technology We have the structure. Now what? I know of no experiment that measures total energy Want to make direct comparison to experiment Predict results of experimental measurements So how do we simulate experiments on condensed matter systems

4 April 2008Suranaree Unversity of Technology Experiments change the system! Experimentalists to perturbation theory (they just don’t realize they do!) Expand quantities (E, n, , v) Experiments often measure how a system responds to an external influence (light, x-ray, neutron, electron, etc)

4 April 2008Suranaree Unversity of Technology Some changes experiments make Perturb the external potential (from the ionic cores and any external field):  Ionic positions  phonons  Cell vectors  elastic constants  Electric fields  dielectric response STM Imaging  Magnetic fields  NMR But not only the potential, any perturbation to the Hamiltonian:  d/dk and d/dE  atomic charges  d/d(species)  alchemical change

4 April 2008Suranaree Unversity of Technology Property Prediction Atomic Vibrations Specific heats Bulk polarisabilities and Electric permittivities Scanning tunnelling microscopy (STM) Electron excitations Photon absorption and emission spectra Nuclear Magnetic Resonance (NMR) Excitons and Polarons Charge Transfer Infra Red Spectra Raman Spectra Incomplete list - some examples

4 April 2008Suranaree Unversity of Technology Bulk Elastic Constants Properties of minerals at lower mantle pressures (Mg x Fe 1-x SiO 3 ) Elastic constants and velocity of sound through minerals in the lower mantle of the Earth B. Karki, S. J. Clark, et al, Mineral. Mag. 62, 585 and Am. Mineral. 82, 635

4 April 2008Suranaree Unversity of Technology Detailed Electronic Structure Recent technologies in “generalised” DFT gets good band gaps GaN ZnO O Defects

4 April 2008Suranaree Unversity of Technology IR/Raman: Light emitting polymers P. R. Tulip and S. J. Clark, Phys. Rev. B 74, (2006)

4 April 2008Suranaree Unversity of Technology STM Imaging: example CO on Pd Theory gives full 3d image: perpendicular to surface gives experimental image 1x1 CO on Pd 2x1 CO on Pd: Tilted dimer Can also do electron spectroscopy: ELNES/EELS

4 April 2008Suranaree Unversity of Technology Solid State NMR Octafluoronaphthalene NMR Chemical Shifts Biot-Savart law: Induced currents in molecules D. B. Jochym, S. J. Clark, et al, Phys. Chem. Chem. Phys. 9, 2389 (2007)

4 April 2008Suranaree Unversity of Technology Conclusions Given a sensible starting point (often thanks to experiment, for now?!?) we can calculate the energy of a material and hence get:  Electronic electronic structure  Atomic positions  Phase transition information  Many properties of a material  Experimentally measured “results” (e.g. diffraction patterns, IR and Raman spectra, NMR, Electron Microscopy)  Many ‘unmeasurable’ quantities NOTE: I have skipped many details  I have occasionally given an over-optimistic review!  Some things are still VERY difficult even if given enough CPU cycles

4 April 2008Suranaree Unversity of Technology Acknowledgements CASTEP co-authors: Matt Probert, Phil Hasnip(University of York) Chris Pickard (University of St. Andrews) Mike Payne, Matt Segal (Cambridge) Keith Refson (Rutherford Labs) EPSRC (funding council) for the usual arguments required to get them to part with their cash ($$$).