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

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 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

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 3 First principles: the whole picture “Base Theory” (DFT) Implementation (the algorithms and program) Setup model, run the code Scientific problem- solving “Analysis Theory” Research output The equipment Application

Nuts and Bolts 2001 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 motion under ionic interactions q1q1 q2q2 q3q3 F 12 F 13 F 21 F 32 F 23 F 31 Two bodies: no problem Three bodies: the Hamiltonian yields coupled equations we cannot solve analytically

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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...

Nuts and Bolts 2001 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?

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 9 Electrons in condensed matter  H atom, 1e: undergraduate exam question He atom 2e: no analytic solution Condensed matter 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

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 10 The one-electron “effective potential” A set of n one-electron equations that must be solved self-consistently e-nuclear (external pot) KineticHartree (Coulomb e-e) Exchange- correlation Glimpse of the DFT equations Numerical methods represent variables and functions evaluate the terms iterate to self-consistency

Nuts and Bolts 2001 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 N 2  It offers an excellent balance between accuracy and scale of calculation

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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 crystalSlab for surfaces

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 14 Modelling FP Simulation Make a model of a real system of interest Capture essential physics Capture as much physics as possible Explore model properties and behaviour Produce simple and transferable concepts Make virtual matter Gain insight, calculate real properties Gain insight

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 17 Power to interpret The experimentalist sees......but we see this too

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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...

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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

Nuts and Bolts 2001 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

Nuts and Bolts 2001 Lecture 2: the modeller's perspective 23 Further reading  A chemist’s guide to density-functional theory Wolfram Koch and Max C. Holthausen (second edition, Wiley. ISBN )  Understanding molecular simulation Daan Frenkel and Berend Smit (Academic press ISBN:  The theory of the cohesive energies of solids G. P. Srivastava and D. Weaire Advances in Physics 36 (1987)  Gulliver among the atoms Mike Gillan New Scientist 138 (1993) 34  The Nobel prize in chemistry 1998 John A. Pople and Walter Kohn