Modelling Interatomic Forces – Progress and Challenges

Modelling Interatomic Forces – Progress and Challenges Mike Finnis Atomistic Simulation Centre School of Mathematics and Physics Queen’s University Belfast Belfast BT7 1NN Northern Ireland, UK http://titus.phy.qub.ac.uk CECAM 17 October 2005

Keep it simple Why make simple models at all when you have ABINIT, CASTEP, CONQUEST, NEWTEP, SIESTA, VASP, WIEN2K ….

Keep it simple Crack blunting in copper35 million atoms Courtesy of Brad Holian and Peter Lomdahl: http://bifrost.lanl.gov/

Keep it simple Size is less important to some people. You may want statistical sampling for time > 1nsto find equilibrium; free energy; Kinetic Monte-Carlo …

Keep it simple SCTB, Thermodynamic integration ZrO2: Free energy Fabris, S.; Paxton, A. T.; Finnis, M. W., Free energy and molecular dynamics calculations for the cubic-tetragonal phase transition in zirconia. Physical Review B 2001, 63 094101-1-13.

Keep it simple • Simple models can be more accurate than DFT – LDA • Band gaps • Correlated systems: van der Waals interaction, thin films of water

Keep it simple The too-big carpet problem “Give me n parameters…” Medieval – Renaissance – Baroque – Romantic

But not too simple! Look after the baby… • Morse and Lennard-Jones fail on defect energies and elastic constants • Non-self-consistent tight-binding fails at defects due to neglect of charge equilibration and three-centre integrals • Rigid ion/shell modelsfail on structural energies and defects

Models • Hybrid schemes • MGPT • - EMT • - EAM Depending on the structure of the DOS • Pair potentials • - empirical • ab initio DFT 2nd order The physics is in the functions • Ionic Models • Born • - shell • - variable charge • Tight-binding • empirical • self-consistent • BOPs

Variable Charge models Rappe, A. K.; Goddard, W. A., Charge Equilibration for Molecular Dynamics Simulations. Journal of Physical Chemistry 1991, 95, (8), 3358-3363. Streitz, F. H.; Mintmire, J. W., Electrostatic Potentials for Metal-Oxide Surfaces and Interfaces. Physical Review B 1994, 50, (16), 11996-12003. Campbell, T. J.; Aral, G.; Ogata, S.; Kalia, R. K.; Nakano, A.; Vashishta, P., Oxidation of aluminum nanoclusters. Physical Review B 2005, 71, (20), art. no.-205413.0

Bond Order Potentials Pettifor, D. G., New Many-Body Potential for the Bond Order. Physical Review Letters 1989, 63, (22), 2480-2483. Density matrix Partial bond order

Ry 0.005 Pair potentials for Al and Ga Ga Al Hafner, J.; Heine, V., Theory of the Atomic Interactions in (s,p)-bonded Metals. Journal of Physics F: Metal Physics 1986, 16, (10), 1429-1458.

Ingredients for DFT Free atom or ion densities LDA,GGA,…

The second-order functional Standard Kohn-Sham: Expand to second order about input charge density: Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G., Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Physical Review B 1998, 58, 7260-7268.

Kernel

Model Minimize quadratic form in charge transfers Variable charge ionic models

Model TB BOPs and all that

TB BOPs and all that Atomic charge neutrality Assume on-site repulsion keeps charge neutral on atoms: Enforce by iteration:

TB BOPs and all that Enforced neutral atoms Approximate the trace difference: Inter-site On-site

TB BOPs and all that Enforced neutral atoms The repulsive terms: (also dustbin for any other small terms) Bonding energy model:

The covalent energy For orthogonal orbitals this reduces to

k j i Environment-dependent BOPs Haas, H.; Wang, C. Z.; Fahnle, M.; Elsasser, C.; Ho, K. M., Environment-dependent tight-binding model for molybdenum. Physical Review B 1998, 57, (3), 1461-1470. Mrovec, M.; Nguyen-Manh, D.; Pettifor, D. G.; Vitek, V., Bond-order potential for molybdenum: Application to dislocation behavior. Physical Review B 2004, 69, (9), art. no.-094115.

k j i Screening functions Environment-dependent BOPs

Environment-dependent repulsion Environment-dependent BOPs

Environment-dependent BOPs Cawkwell, M. J.; Nguyen-Manh, D.; Woodward, C.; Pettifor, D. G.; Vitek, V., Origin of Brittle Cleavage in Iridium. Science 2005, 309, (5737), 1059-1062.

Where pair potentials come from jellium Non-local XC!

Conclusions • It is advantageous to define a functional both for understanding the ingredients (eg to avoid double counting) and to get consistent forces. • The second-order DFT functional is a universal starting point for most existing models.

Challenges • Is there a reliable universal recipe for oxides? • Include spin DFT - up and down spin density. • Build good non-local model functionals for correlated systems with variable valence • Weld-on non-local terms in C (Van der Waals) • Include self-consistency and excited electrons

harry potter and the interatomic forces Thank you! You’ve heard the talk…

Thank you! You’ve heard the talk…

Modelling Interatomic Forces – Progress and Challenges