Stewart Clark Department of Physics University of Durham

Stewart Clark Department of Physics University of Durham, UK Electronic Structure Calculations The CASTEP code s. j. clark@durham. ac. uk http: //cmt. dur. ac. uk/sjc/ 4 April 2008 Suranaree Unversity of Technology

Where is Durham? Durham 4 April 2008 Suranaree 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 (www. castep. org) • Fortran 95 • Massively parallel (MPI) • Applications to many areas of physics, chemistry and materials science 4 April 2008 Suranaree 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 2008 Suranaree 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 2008 Suranaree 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 2008 Suranaree 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 2008 Suranaree Unversity of Technology

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

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 2008 Suranaree 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? Don’t want to do just a few atoms/molecules - want to do BULK materials Genuine many-body problem: macroscopic materials contain > 10 23 atoms 4 April 2008 Suranaree Unversity of Technology

Length and timescales d e in a r g e- Polymers rs a o C tum s c i an h ec m n ua Q Electronic transition Diffusion Intermolecular motion Bond motion 10 -14 s 10 -15 s 4 April 2008 Molecular alignment 10 -8 s 10 -9 s ic t s i ng m o elli t A od M Suranaree Unversity of Technology >>10 -7 s

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 2008 Suranaree Unversity of Technology

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

Electronic Structure Basics first: can get electronic structure for any arrangement of atoms in a solid (given enough computer power!) [Rb(anti-dchyl 18 c 6)][Ni(dmit)2] Valence electron structure Robertson N; Clark, SJ; et al. Chem. Comm. Issue 25, 3204 (2005). 4 April 2008 Suranaree Unversity of Technology

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

Summary so far • Rely 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 2008 Suranaree Unversity of Technology

Structure: where are the atoms? § Minimum energy corresponds to zero force (F=-d. E/d. R) § 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 2008 Suranaree Unversity of Technology

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

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 2008 Suranaree 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, 35327 4 April 2008 Movie, courtesy of M. J. Probert, University of York Suranaree 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 2008 Suranaree 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 2008 Suranaree Unversity of Technology

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

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

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

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

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 2008 Suranaree Unversity of Technology

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

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

Property Prediction Incomplete list - some examples • • • 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 4 April 2008 Suranaree Unversity of Technology

Bulk Elastic Constants Properties of minerals at lower mantle pressures (Mgx. Fe 1 -x. Si. O 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 2008 Suranaree Unversity of Technology

Detailed Electronic Structure Ga. N O Defects Zn. O Recent technologies in “generalised” DFT gets good band gaps 4 April 2008 Suranaree Unversity of Technology

IR/Raman: Light emitting polymers P. R. Tulip and S. J. Clark, Phys. Rev. B 74, 064301 (2006) 4 April 2008 Suranaree Unversity of Technology

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

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