Electronic Structure Calculations

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

2. Durham Where is Durham? Suranaree Unversity of Technology

3. 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 Suranaree Unversity of Technology

4. 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? Suranaree Unversity of Technology

5. 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 Suranaree Unversity of Technology

6. 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 Suranaree Unversity of Technology

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

8. Scientificproblem-solving “BaseTheory”(DFT) Implementation(the algorithmsand program) Setup model,run the code “Analysis Theory” From First Principles The equipment Application Researchoutput Suranaree Unversity of Technology

9. 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 Suranaree Unversity of Technology

10. 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 > 1023 atoms Suranaree Unversity of Technology

11. Polymers Coarse-grained Molecular alignment >>10-7 s 10-8 s Diffusion Quantum mechanics Intermolecular motion 10-9 s Bond motion Electronic transition Atomistic Modelling 10-14 s 10-15 s Length and timescales Suranaree Unversity of Technology

12. 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 Suranaree Unversity of Technology

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

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

15. Electron by electron Multiband molecular conductor Suranaree Unversity of Technology

16. 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 Suranaree Unversity of Technology

17. 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!!! Suranaree Unversity of Technology

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

19. 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. B49, 5329 and Phys. Rev. B49, 5341 Suranaree Unversity of Technology

20. Surfaces • Surface structure • Catalysis • Chemical reactions S. J. Clark, et al, Phys. Rev. B50, 5728 V. Timon, S. J. Clark, et al, Phys. Rev. B72, 35327 Movie, courtesy of M. J. Probert, University of York Suranaree Unversity of Technology

21. 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 Design5(4) 1437 and 5(4) 1443. Suranaree Unversity of Technology

22. 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 Suranaree Unversity of Technology

23. Prediction: what is the structure? Suranaree Unversity of Technology

24. 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) Suranaree Unversity of Technology

25. Or something more complicated? TRP polypeptide (small protein) in water 1230 atoms per molecule + nH2O S. J. Clark, K. Refson and I. Kuprov, in press (2008) Suranaree Unversity of Technology

26. 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! Suranaree Unversity of Technology

27. 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) Suranaree Unversity of Technology

28. Molecular Dynamics • Can do dynamics of atoms using forces calculated from abinitio 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 Suranaree Unversity of Technology

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

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

31. 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 Suranaree Unversity of Technology

32. 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) Suranaree Unversity of Technology

33. 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 Suranaree Unversity of Technology

34. 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 Suranaree Unversity of Technology

35. Bulk Elastic Constants Properties of minerals at lower mantle pressures (MgxFe1-xSiO3) 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 Suranaree Unversity of Technology

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

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

38. 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 Suranaree Unversity of Technology

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

40. 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 Suranaree Unversity of Technology

41. 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 ($$$). Suranaree Unversity of Technology