Outline of the lecture• Introduction• Density functional theory

– Born-Oppenheimer approximation– Hartree and Hartree-Fock approximation– Density functional theory and local density

approximation– Kohn-Sham equation

• Plane-wave Pseudopotential method– Ab-initio pseudopotentials– Plane-wave basis

• Applications of DFT to surface calculations.

Density functional theory

- Born-Oppenheimer approximation- Single-electron approximation- Hartree approximation- Hartree-Fock approximation- Density functional theory- Kohn-Sham equation

Quantum Mechanics of Crystals

• Materials are composed of nuclei {Materials are composed of nuclei {ZZαα , M, Mαα , R, Rαα} and electrons {} and electrons {rrii}}

the interactions are knownthe interactions are known

Kinetic energyKinetic energyof nucleiof nuclei

Kinetic energyKinetic energyof electronsof electrons

Nucleus-NucleusNucleus-Nucleusinteractioninteraction

Electron-NucleusElectron-Nucleusinteractioninteraction

Electron-ElectronElectron-Electroninteractioninteraction

Ab-initio (first principles) Method – Ab-initio (first principles) Method – ONLY Atomic Numbers {ZONLY Atomic Numbers {Zii} as input parameters} as input parameters

HHΨΨ = = EEΨΨ

Born-Oppenheimer Approximation I

• Hamiltonian of the coupled electron-ion system

• Many-body Schrödinger equation

• Difference in time scales of ionic and electronic motions– Electrons respond instantaneously to slow ionic motion1839

e

p

e

N

m

m

m

m

Born-Oppenheimer Approximation II

• Decoupled Schrödinger equations

• Adiabatic approximation– Electrons quickly respond to changes of nuclei, thus

allowing the electronic system to remain in its ground state

– Ions move on the potential-energy surface of the electronic ground state

Single-particle Approximation

• We still have to solve many-electron problem.– Exchange property: Pauli exclusion principle– Correlation property: interacting electrons

– XC effects are treated in an average way– Hartree approximation– Hartree-Fock approximation– Density functional theory

• Single-particle approach in an effective potential

Hartree Approximation

• Single-particle Hartree equation

• Electrons treated as non-interacting particles

• Total energy of the system

Variational Principle

• Variation in the many-body wavefunction will give zero variation in the energy.

• Since this has to be true for any

Hartree-Fock Approximation

• Single-particle HF equation

• Incorporation of fermionic nature of electrons

• Total energy with the HF wavefunction

Density Functional Theory (DFT)

• The functional E[n(r)] has its minimum at the equilibrium density n0(r).

• 1998 Nobel prize for W. Kohn with J.A. Popple

• Density instead of many-body wave functions

• The ground-state energy of a many-body system is a functional of the electron density, n(r).

Hohenberg-Kohn Theorem I

• The electron density n(r) is uniquely determined by the external potential V(r).

Hohenberg-Kohn Theorem II

• For a given V(r), the correct nGS(r) minimizes the ground-state

energy functional E[n(r)] and this minimum E[nGS(r)] is the ground

state energy.

where F[n(r)] = <Ψ|T+W| Ψ> must be a universal funtional of the density since T and W are common to all electronic systems.

GSGSGS GS GS

Kohn-Sham Hamiltonian

Derivation of Kohn-Sham Equation

0)(

)]([

)(0

rn

rn

rnE

• Kohn-Sham equation

• Variational procedure

subject to the condition

Nrdrn

)(

How to treat Exchange-Correlation

• Local density approximation (LDA)

• Generalized gradient approximation (GGA)

rdrnrnrdrnrnrnE xcxcxc

)]([)()]([)()]([ hom

Quantum Monte-Carlo simulations for homogeneous electron gas.

Algorithm of Self-Consistent Calculation

INPUT atomic number coordinates

Calculating QM force

OUTPUT final coordinates total energy energy eigenvalue electron density

Self-consistent ?

Converged ?yes

yes

no

no

New coordinates

solving Kohn-Sham equation

updating electron density

setting-up Hamiltonian

Implementations of the Kohn-Sham Method

Plane-wave pseudopotential method

- ab-initio pseudopotentials- Plane-wave basis- Energy cutoff- Brillouin zone and k-point sampling

FLAPW all-electron method

Pseudopotentials – Basic idea

• Core electrons are localized and therefore

chemically inactive (inert)

• Valence electrons determine chemical properties of atoms and SOLIDS

• Describe valence states by smooth wavefuctions

How to get smooth pseudo-valence-wavefunctions

from atomic valence wavefunctions?

• Within the core region (0 ≤ r ≤ rc)

• Outside the core region (r > rc)

Pseudopotentials – Philips-Kleinman Method

r

eZV s

ps

2

coreV

||)( cc

cccore

ps EEVV

vv

Norm Conserving Pseudopotentials

Norm-conserving Pseudopotentials

• Pseudopotential ⇒ shallow, nonlocal• Pseudo-wave function⇒ nodeless, slowly

varying

Benefits of Pseudopotential Approach

• Reduction of computational time– Only valence electrons Smaller number of eigenstates– Smoother potential and nodeless

wavefunctions Smaller plane-wave-basis set

Plane Waves Formalism

Bloch theorem Fourier series

• Kohn-Sham equation

• Wavefunction

Translational symmetry

Bloch theorem

Fourier expansion

How to solve eigenvalue problem

Eigenvalue equation

Plane-wave Basis

• Systematic improvement of the accuracy of the calculation with increasing cutoff energy Ecutoff

• Efficient use of fast Fourier transformation (FFT) in evaluating H

• Basis set is independent of atom positions (unbiased)

• Easy calculation of Hellmann-Feynman forces

Charge density & k-point sampling

• Charge density, (r)

• Example of k-point sampling– 2D square lattice– 4×4 k-point grid– Only 3 inequivalent k-points

Applications of DFT Applications of DFT to surface calculationsto surface calculations

Modeling Surface – Slab vs. Cluster

Cluster model Repeated slab model

Real system

Low index Surfaces of Si

(001) (110) (111)

Square lattice Rectangular lattice Hexagonal lattice

Relaxation and Reconstruction

a

2a

Relaxation Preserve (1x1)

Reconstruction (1x1) (2x1)

Reconstruction - Si(001) Surface

c(4x2)

ideal

(2x2) (2x1) asym

(2x1) symc(4x2)

Surface states at Si(001)-(2x1)

Potential-energy surface of Na on In/Si(111)

Atomic structure Potential-energy surface

Prediction of binding energy, diffusion barrier, binding site

Reaction pathways of acrylonitrile on Si(001)

Energy profiles for the three reaction pathways

(a) 0.30 eV, [2+2]

(b) 0.32 eV, [4+2]-1

(c) 0.09 eV, [4+2]-II

Formation of the [4+2]-II structure is kinetically favored over

energy barrier

the [2+2] or [4+2]-I structure by a factor of ~104 at 300 K.