The Nuts and Bolts of First-Principles Simulation
Durham, 6th-13th December 2001
2: The Modeller’s Perspective The philosophy and ingredients of atomic-scale modelling
CASTEP Developers’ Groupwith support from the ESF k Network
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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?
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First principles: the whole picture
“BaseTheory”(DFT)
Implementation(the algorithmsand program)
Setup model,run the code
Scientificproblem-solving
“AnalysisTheory”
Researchoutput
The equipment Application
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So why do we need computers?
The “many-body problem”: atoms, molecules, electrons, nuclei... interact with each other
Example: equations of motionunder ionic interactions
q1
q2
q3
F12
F13
F21
F32
F23
F31
Two bodies: no problem Three bodies: the
Hamiltonian yieldscoupled equations wecannot solve analytically
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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
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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
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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...
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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 solutionCorollaries 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?
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Electrons in condensed matter H atom, 1e: undergraduate exam question
He 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
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2
2m2 Vext(r) VH (r)
EXC[(r)](r)
i i i
EE[(r)] drVext(r)(r) EKE[(r)] EH [(r)] EXC [(r)]
The one-electron “effective potential”
A set of n one-electron equations that must be solved self-consistently
effn
n
V
e-nuclear(external pot)
Kinetic Hartree(Coulomb e-e)
Exchange-correlation
Glimpse of the DFT equations
Numerical methods represent variables and functions evaluate the terms iterate to self-consistency
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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
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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
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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
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Modelling FP Simulation
Make a model of a real system of interest
Capture essential physicsCapture as much physics
as possible
Explore model properties and behaviour
Produce simple and transferable concepts
Make virtual matter
Gain insight, calculate real propertiesGain insight
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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
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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
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Power to interpret
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The experimentalist sees... ...but we see this too
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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
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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...
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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
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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
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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
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Further reading A chemist’s guide to density-functional theory
Wolfram 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/
