http://secamlocal.ex.ac.uk/people/staff/ashm201/Atomistic/

LecturesIntroduction to computational modelling and statistics 1Potential models 2Density Functional (quantum) 1 3Density Functional 2 4

Or: Understanding the physical and chemical properties of materials from an understanding of the underlying atomic processes

Computational Modelling of MaterialsRecent Advances in Contemporary Atomistic Simulation

Useful MaterialBooks A chemist’s guide to density-functional theory

Wolfram Koch and Max C. Holthausen (second edition, Wiley)

The theory of the cohesive energies of solidsG. P. Srivastava and D. WeaireAdvances in Physics 36 (1987) 463-517

Gulliver among the atomsMike Gillan, New Scientist 138 (1993) 34

Web www.nobel.se/chemistry/laureates/1998/ www.abinit.org

Some version compiled for windows, install and good tutorial

Outline: Part 1, The Framework of DFT

DFT: the theory Schroedinger’s equation Hohenberg-Kohn Theorem Kohn-Sham Theorem Simplifying Schroedinger’s LDA, GGA

Elements of Solid State Physics Reciprocal space Band structure Plane waves

And then ? Forces (Hellmann-Feynman theorem) E.O., M.D., M.C. …

Outline: Part2Using DFT

Practical Issues Input File(s) Output files Configuration K-points mesh Pseudopotentials Control Parameters

LDA/GGA ‘Diagonalisation’

Applications Isolated molecule Bulk Surface

The Basic ProblemDangerously classical representation

Cores

Electrons

Schroedinger’s Equation

iiii rRrRVm

,.,2

2

Hamiltonian operator

Kinetic EnergyPotential Energy

Coulombic interactionExternal Fields

Very Complex many body Problem !!(Because everything interacts)

Wave function

Energy levels

First approximations

Adiabatic (or Born-Openheimer) Electrons are much lighter, and faster Decoupling in the wave function

Nuclei are treated classically They go in the external potential

iiii rRrR .,

H.K. Theorem

The ground state is unequivocally defined by the electronic density

rrr dvFEv Universal functional

•Functional ?? Function of a function•No more wave functions here

•But still too complex

K.S. FormulationUse an auxiliary system Non interacting electrons Same Density => Back to wave functions, but simpler this time

(a lot more though)

rrVm iiieff .

22

rrrr

rrr

XCeff dVV

i

i

2rr

N K.S. equations(ONE particle in a box really)

(KS3)

(KS2)

(KS1)

Exchange correlation potential

Self consistent loop

Solve the independents K.S. =>wave functions

From density, work out Effective potential

New density ‘=‘ input density ??

Deduce new density from w.f.

Initial density

Finita la musica

YES

NO

DFT energy functional

XCNI EdddvTE

rrrr

rrrr

2

1

Exchange correlation funtionalContains:ExchangeCorrelationInteracting part of K.E.

Electrons are fermions (antisymmetric wave function)

Exchange correlation functional

At this stage, the only thing we need is: XCE

Still a functional (way too many variables)

#1 approximation, Local Density Approximation:Homogeneous electron gasFunctional becomes function !! (see KS3)Very good parameterisation for XCE

Generalised Gradient Approximation:

,XCEGGA

LDA

DFT: SummaryThe ground state energy depends only on the electronic density (H.K.)One can formally replace the SE for the system by a set of SE for non-interacting electrons (K.S.)Everything hard is dumped into Exc

Simplistic approximations of Exc work !LDA or GGA

And now, for something completely

different: A little bit of Solid State Physics

Crystal structure

Periodicity

Reciprocal space

Real Space

ai

ijji ba .2

Reciprocal Space

biBrillouin Zone

(Inverting effect)

k-vector (or k-point)

sin(k.r)

See X-Ray diffraction for instance

Also, Fourier transform and Bloch theorem

Band structure

Molecule

E

Crystal

Energy levels (eigenvalues of SE)

The k-point mesh

Brillouin Zone

(6x6) mesh

Corresponds to a supercell 36 time bigger than the primitive cell

Question:Which require a finer mesh, Metals or Insulators ??

Plane waves

Project the wave functions on a basis setTricky integrals become linear algebraPlane Wave for Solid StateCould be localised (ex: Gaussians)

+ + =

Sum of plane waves of increasing frequency (or energy)

One has to stop: Ecut

Solid State: Summary

Quantities can be calculated in the direct or reciprocal spacek-point Mesh Plane wave basis set, Ecut

Now what ?We have access to the energy of a system, without any empirical input

With little efforts, the forces can be computed, Hellman-Feynman theorem

Then, the methodologies discussed for atomistic potential can be used

Energy Optimisation

Monte Carlo

Molecular dynamics

rrrF dv iii