The Nuts and Bolts of First-Principles Simulation

1. The Nuts and Bolts of First-Principles Simulation 2: The Modeller’s Perspective The philosophy and ingredients of atomic-scale modelling Durham, 6th-13th December 2001 CASTEP Developers’ Groupwith support from the ESF k Network

2. Outline • So why do we need computers? • What does “first principles” mean? • Potted history of simulation • Model systems • The horse before the cart • Taking advantage • Is it theory or experiment? The Equipment Applying it Lecture 2: the modeller's perspective

3. First principles: the whole picture The equipment Application Scientificproblem-solving “BaseTheory”(DFT) Implementation(the algorithmsand program) Setup model,run the code Researchoutput “Analysis Theory” Lecture 2: the modeller's perspective

4. So why do we need computers? • The “many-body problem”: atoms, molecules, electrons, nuclei... interact with each other • Example: equations of motionunder ionic interactions q2 F21 F23 • Two bodies: no problem • Three bodies: theHamiltonian yieldscoupled equations wecannot solve analytically F12 F13 F32 q1 F31 q3 Lecture 2: the modeller's perspective

5. Theory…exactly • In a simulation we solve coupled equations using numerical methods, e. g. • Equations of motion: molecular dynamics • Interacting electrons: “self-consistent field” • In principle we can do this with no additional approximations whatsoever • Contrast this with traditional theory: drastic approximations to allow solution • Note too the calculations have millions of variables numerical approach Lecture 2: the modeller's perspective

6. An aside: statistical mechanics • Pre-simulation days • Good theories of the liquid state, but solutions possible only when atomic interactions were simplified in the extreme • Experiments on the real liquid yield data with which to test these approximate theories • Using simulation • The “experiment” is done on the computer: exact answers for a model system, which may be the same model as in the analytic theory • There’s more: simulations the only way to find answers to the theory in 99% of cases The subject was revolutionised Lecture 2: the modeller's perspective

7. Computers and condensed matter • The Dark Ages: 1950’s • Before Computers (BC). Pencils and a slide rule • Enlightenment: 60’s, 70’s • Model systems, statistical mechanics, theory of liquids, simple band structure... • Revolution: 1980’s • Approximations persecuted — DFT implemented efficiently, QMC, functional development... • Superpower: 1990’s • Making it all useful: faster algorithms, supercomputers and parallel machines, scaleable calculations • Organisation: CDG, UKCP, Grand Challenge consortia, k... Lecture 2: the modeller's perspective

8. First-principles thinking • Use quantum mechanics to describe valence electrons: making and breaking of bonds • Don’t use adjustable parameters to fit to data • Make as few serious approximations as possible in arriving at the electronic solution Corollaries • Extract predictions (for a model system) • Don’t interfere! Accept all the results • Know your limits • What is the confidence limit in a calculated number? Lecture 2: the modeller's perspective

9. Electrons in condensed matter • H atom, 1e: undergraduate exam questionHe atom 2e: no analytic solutionCondensed matter 1023 e: hopeless? • Here’s what we do • Work with a few atoms (a model system) • Describe electronic interactions from first principles (DFT: simple, cheap, accurate, versatile) • Solve DFT equations numerically Lecture 2: the modeller's perspective

10. Glimpse of the DFT equations Numerical methods • represent variables and functions • evaluate the terms • iterate to self-consistency e-nuclear (external pot) Kinetic Hartree(Coulomb e-e) Exchange-correlation A set of n one-electron equations that must be solved self-consistently The one-electron “effective potential” Lecture 2: the modeller's perspective

11. Some key points about DFT • DFT is a description of interacting electrons in the ground state, including exchange and correlation • The basic variable is the density rather than the wavefunction • The theory is simple and the implementations efficient compared with other methods • Implementations scale at least as well as N2 • It offers an excellent balance between accuracy and scale of calculation Lecture 2: the modeller's perspective

12. Section summary • First principles: quantum mechanics for bonds, no adjustable parameters • Numerical solutions when we have coupled equations • Solutions may be exact but they are non-analytic • Must calculate on a small model system Lecture 2: the modeller's perspective

13. Model systems • In this kind of first-principles calculation • Are 3D-periodic • Are small: from one atom to a few hundred atoms • Supercells • Periodic boundaries • Bloch functions, k-point sampling Bulk crystal Slab for surfaces Lecture 2: the modeller's perspective

14. Modelling FP Simulation Make a model of a real system of interest Capture as much physics as possible Capture essential physics Make virtual matter Explore model properties and behaviour Gain insight, calculate real properties Gain insight Produce simple and transferable concepts Lecture 2: the modeller's perspective

15. Control and conditions • We can manipulate the model system: complete control • Move and place atoms • Apply strains • Try configurations • Any conditions and situations are accessible • High pressures and temperatures • Buried interfaces, porous media, nanostructures Lecture 2: the modeller's perspective

16. Horse before the cart • We can calculate experimental observables • But we can also can see the underlying model and all its details! • Contrast with the experimentalist, who must infer properties from obervables • Great power to interpret experiment Lecture 2: the modeller's perspective

17. Power to interpret The experimentalist sees... ...but we see this too Lecture 2: the modeller's perspective

18. Taking advantage • Calculate quantities for other theories • Transition states and barriers • Defect energies • Use unphysical routes, e.g. free energy calculations • Switch from reference system to full simulation • Transmute elements Lecture 2: the modeller's perspective

19. Approximations: where, how bad • The usually good: DFT within LDA, GGA • The not bad: plane waves and pseudopotnentials, k-point sampling, other parameters and tolerances • The frequently ugly: the model • Too small • Too simplistic • No relaxations • No entropy... Lecture 2: the modeller's perspective

20. Computer experiments? • Have to run the program to get the answer, just as have to do the experiment to get results • This is where a lot of the art of simulation lies • Very similar to experimental technique • Calibration, testing and validation • Sample preparation (model) • Analysis • Errors and precision Lecture 2: the modeller's perspective

21. Analysis • More theories applied to the raw data • Physical structure and energetics • Crystallography, defects, surfaces, phase stability • Electronic structure • STM • Optical properties • Positions and momenta • Statistical mechanics Lecture 2: the modeller's perspective

22. Is it theory or experiment? • Theory with high-quality, low approximation, non-analytic solutions for model systems • In its application, very much like experiment, giving high-quality, direct results for model systems! • Observables can be calculated, but we also have direct control at the atomistic level • It has ingredients of both, and more Lecture 2: the modeller's perspective

23. Further reading • A chemist’s guide to density-functional theoryWolfram Koch and Max C. Holthausen (second edition, Wiley. ISBN 3-52730372-3) • Understanding molecular simulationDaan Frenkel and Berend Smit(Academic press ISBN: 0122673700 • The theory of the cohesive energies of solidsG. P. Srivastava and D. WeaireAdvances in Physics 36 (1987) 463-517 • Gulliver among the atomsMike GillanNew Scientist 138 (1993) 34 • The Nobel prize in chemistry 1998John A. Pople and Walter Kohnhttp://www.nobel.se/chemistry/laureates/1998/ Lecture 2: the modeller's perspective