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Introduction to Computer Simulation: Edinburgh, May 2010
Set up the problem
Let’s start defining various quantities Assume that the nuclei (Mass Mi) are at: R1, R2, …, RN Assume that the electrons (mass me) are at: r1, r2, …, rm Now let’s put some details in the SE
ˆ R , R ,..., R ,r ,r ,...,r E R , R ,..., R ,r ,r ,...,r H 1 2 N 1 2 m 1 2 N 1 2 m
Introduction to Computer Simulation: Edinburgh, May 2010
Quantum Mechanics: Density Functional Theory and Practical Application to Alloys
Stewart Clark Condensed Matter Section Department of Physics University of Durham
Introduction to Computer Simulation: Edinburgh, May 2010
From first principles
The equipment
Application
Scientific problemsolving “Base Theory” (DFT) Implementation (the algorithms and program)
• The electronic charge density is given by
n(r )
* ... r1,r2 ,...,rn r1,r2 ,...,rn dr2dr3 ...drn
• so integrate over n-1 of the dimensions gives the probability, n(r), of finding an electron at r • This is (clearly!) a unique functional of the external potential, V • That is, fix V, solve SE (somehow) for and then get n(r).
Introduction to Computer Simulation: Edinburgh, May 2010
Property Prediction
•Property calculation of alloys provided link with experimental measurements:
ˆ H N ,e N ,e
R ,r r R
I i N ,e i N ,e I
• Nuclear problem is separable (and, as we know, the nucleus is merely a point charge!)
Summary of problem to solve
ˆ H N ,e N ,e
R ,r
I i
EN ,e N ,e
R ,r
I i
Where
ˆ T ˆ T ˆ V ˆ ˆ V ˆ H V N ,e N e N N N e ee
Introduction to Computer Simulation: Edinburgh, May 2010
Introduction to Computer Simulation: Edinburgh, May 2010
Density functional theory
• Let’s write the Hamiltonian operator in the following way:
– T is the kinetic energy terms – V is the potential terms external to the electrons – U is the electron-electron term
Supercells Periodic boundaries Bloch functions
Slab for surfaces
Introduction to Computer Simulation: Edinburgh, May 2010
First simplification
• The electron mass is much smaller than the nuclear mass • Electrons remain in a stationary state of the Hamiltonian wrt nuclear motion
•Why is this a hard problem? •Equation is not separable: genuine many-body problem •Interactions are all strong – perturbation won’t work •Must be Accurate --- Computation
Introduction to Computer Simulation: Edinburgh, May 2010
Electrons are difficult!
• The mathematical difficulty of solving the Schrodinger equation increases rapidly with N • It is an exponentially difficult problem • The number of computations scales as eN • With modern supercomputers we can solve this directly for a very small number of electrons (maybe 4 or 5 electrons) • Materials contain of the order of 1026 electrons
Introduction to Computer Simulation: Edinburgh, May 2010
The starting point
ˆ H E
As you can see, quantum mechanics is “simply” an eigenvalue problem
Introduction to Computer Simulation: Edinburgh, May 2010
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 alloys?
Outline
• Aim: To simulate real materials and experimental measurements • Method: Density functional theory and high performance computing • Results: Brief summary of capabilities and performing calculations
The full problem
h2 2 2 I i M m 2 I I e i e2 ZI e2 1 ZI ZJ e2 1 1 4 2 r r r R 2 R R I I J I J 0 i j i j i,I i R1, R2,...,RN , r1, r2 ,...,rm ER1, R2 ,...,RN , r1, r2,...,rm
For analysis For scientific/technological interest
•To enable interpretation of experimental results •To predict properties over and above that of experimental measurements
ˆ T ˆV ˆU ˆ H
• so we’ve just classified it into different ‘physical’ terms
Introduction to Computer Simulation: Edinburgh, May 2010
The electron density
Setup model, run the code
Reses Theory”
Introduction to Computer Simulation: Edinburgh, May 2010
