DFT: Basic idea and Practical calculations

DFT: Basic idea and Practical calculationsPractical calculations

AP IRATH PHUS ITTRAKOOL

Large scale simulation research laboratory

National electronics and computer technology center

How the electronic structure calculation works

P t I DFTPart I: DFTUnderstanding the basic ideas

Li it tiLimitations

Part II: Practical methodsl h dComputational method

Calculation technique

2Apirath – DFT: basic and practice ‐ 6/5/07

Part I outline

C l l ti f M t i lCalculation of Materials

Hartree‐Fock method

Density Functional Theory

Exchange‐Correlationg

Solving Kohn‐Sham equation

Force calculationForce calculation

Limitation of DFT


Calculation of materials

Basic problem – many electrons in the presence of the nuclei

Fundamental Hamiltonian:

∑∑∑∇ +−

−= Ii

eeZm

H22

22

21

2ˆ

rrRrh electronic

part

∑∑∇ +−

−−

JII

ji jiIi Iii

eZZM

m2

22

,,

21

2

22

RR

rrRr

h

part

Born‐Oppenheimer approximation

∑∑∇ −JI JIIIM ,22 RR

pp ppNuclei are fixed – neglect the kinetic term of nuclei

The last term is a constant added to the electronic part


Many‐body interacting electron problemy y g p

Schrödinger eq ation for man electron s stemSchrödinger equation for many electron system

),,,(),,,(ˆ2121 NN EH rrrrrr KK Ψ=Ψ

If we could solve the Schrödinger equation, we could predict the behavior of any electronic systembehavior of any electronic system

Many electron wave functions Ψ(r1, r2, …, rN) is a function of 3Nvariables where N is the number of electronvariables, where N is the number of electron

Too many degree of freedom Can’t be solved


Approaching to many‐body problem pp g y y p

H t F kHartree‐FockIndependent electron approximation

Density Functional Theoryd l d d f h fConsider electron density instead of the wavefunction

Q M C lQuantum Monte CarloStatistical method


Hartree approximationpp

Neglecting the correlation ass mes the electrons areNeglecting the correlation – assumes the electrons are uncorrelated

Total wave function is written as the product of single electronTotal wave function is written as the product of single electron wave functions

)()()()(Ψ

Variational principle leads to Hartree equation

)()(),(),,,( 221121 NNN rrrrrr ψψψ KK =Ψ

Variational principle leads to Hartree equation

)()()](21[ Hion

2iiiiiii VV rrr ψεψ =++∇−

jij

jji dV r

rr

rr ∑∫ −

=

2

H

)(2)(

ψ Hartree potential,average Coulomb potential f th l t


ij ji rr≠ from other electrons

Hartree‐Fock (HF)( )

Electrons are fermions obe the Pa li e cl sion principle andElectrons are fermions; obey the Pauli exclusion principle and Fermi statistics

Wave function changes sign when the coordinates of two electrons areWave function changes sign when the coordinates of two electrons are interchanged

Wave function is zero if two electrons are in the same statef b d d ( l )Wave function is written to be an antisymmtrized product (Slater Determinant)

)()()( 12111 Nrrr ψψψ L

)()()()()()(

!1 22212

12111

N

N

Nrrrrrr

ψψψψψψ

MOMM

L=Ψ

)()()(!

21 NNNN

Nrrr ψψψ L

MOMM


Exchange interactiong

Minimi e the energ ith respect to Slater determinant leads toMinimize the energy with respect to Slater determinant leads to the Hartree‐Fock equations

1 2

)()()(1 *ijjijj rrr ψψψ

⎥⎤

⎢⎡∑ ∫

)()()]()(21[ XHion

2iiiiiiii VVV rrrr ψεψ =+++∇−

)()()()(

21)(

iX

i

ij

jj

ji

jijji dV

rr

rrr

rrr

ψψψψ

⎥⎥⎦⎢

⎢⎣ −

−= ∑ ∫

Adding Fermionic nature of electrons to the Hartree method give rise to an effective potential called exchange interaction

still neglect the correlation effect, so large deviation from the experiment results


Density Functional Theory (DFT)y y ( )

Hohenberg Kohn Sham proposed a ne approach to the manHohenberg‐Kohn‐Sham proposed a new approach to the many‐body interacting electron problem

All d t t ti d t i d b th d t tAll ground state properties are determined by the ground state density

H h b K h h (1964)Hohenberg‐Kohn theorems(1964)

Two statements constitute the basis of DFT

Do not offer a way of computing in practice

Kohn‐Sham ansatz (1965)

Turn DFT into practical method


Hohenberg‐Kohn theoremg

Theorem I densit as a basic ariableTheorem I: density as a basic variable

The total energy of a many‐body system is a unique functional of electron densityelectron density

The total energy functional can be written as

∫

Universal functional F[n] is independent of external potential

∫+= rrr dnvnFnE )()(][][ ext interaction with nucleiand any other field

Universal functional F[n] is independent of external potential –but unknown!

][][][ nUnTnF + ][][][ nUnTnF +=


Hohenberg‐Kohn theoremg

Theorem II ariational for the energ in term of densitTheorem II: variational for the energy in term of density

The functional E[n(r)] has its minimum when electron density is at the equilibrium electron density n0(r)

The minimum of the energy E0 is exactly equivalent to the true gy 0 y qground‐state energy

)]([)]([min 00 rr nEnEE ≡≡


The Kohn‐Sham ansatz

Replace original man bod problem ith an independentReplace original many‐body problem with an independent electron problem that can be solved

The unknown functional F[n] is cast in the formThe unknown functional F[n] is cast in the form

][''

)'()(][][ nEddnnnTnF xcs ++= ∫ rrrrrr

Ts[n] – non‐interacting kinetic energy'−∫ rr

2h

P t th t k i t E [ ]

rrr dm

nT ii

is )()(2

][ 2*2

ψψ ∇= ∑ ∫h

Put the rest unknown into Exc[n]Exact theory but unknown functional Exc[n]


Exchange‐correlation functionalg

E changeExchangeKeep electrons of same spin away from each other (Pauli exclusion)

CorrelationCorrelationKeep electrons away from each other due to Coulombic forces

i di E f i l i h h ll b iFinding Exc functional is the great challenge, but approximate functionals work:

LDALDA

GGA

Hybrid functionaly

EXX


Local (Spin) Density Approximation L(S)DA( p ) y pp ( )

The simplest b t s rprisingl good appro imationThe simplest but surprisingly good approximation

Assume that the charge density varies slowly; each small volume looks like a uniform electron gaslooks like a uniform electron gas

Exc[n] can be obtained by integrating uniform e‐ gas results

∫ε (n) is the exchange‐correlation energy per electron in homogenous electron

∫= rr dnnnE xcLDAxc )()(][ ε

εxc(n) is the exchange‐correlation energy per electron in homogenous electron gas at density n, which can be calculated

More generally, formulation for spin densityg y, p y

∫ ↓↑↓↑ = rr dnnnnnE xcLSDAxc ),()(],[ ε


Generalized‐Gradient Approximation (GGA)pp ( )

Densit gradient e pansionDensity gradient expansion

Take the value of the density at each point as well as the it d f th di t f th d itmagnitude of the gradient of the density

∫ ∇= rr dnnnnEGGA )()(][ ε

Make improvement over LDA for many cases

∫ ∇= rr dnnnnE xcxc ),()(][ ε

a e p o e e t o e o a y cases

E.g. improve predicted binding and dissociation energies


Hybrid functionaly

Combination of orbital dependent HF and an e plicit densitCombination of orbital‐dependent HF and an explicit density functional (LDA/GGA)

The most accurate functionals available (as far as energetic isThe most accurate functionals available (as far as energetic is concerned)

The method of choice in chemistryThe method of choice in chemistry

Example:

)( DFHF21half

xcxxc EEE +=

VWNc

LYPcxxxcxc EccEbEaEEaE )1()1( BeckeHFLDAB3LYP −++++−=


Orbital functionals ‐ OEP

F nctional e plicitl dependence on orbitalsFunctional explicitly dependence on orbitals

Optimized effective potential method (OEP) allows orbital d d t f ti l t b d i DFTdependent functionals to be used in DFT

The most common approach is to evaluate the HF exchange i hi KS bi l h (EXX) d l ienergy within KS orbitals , exact exchange (EXX), and solve using

OEP

Typically, OEP‐EXX methods have much better band gaps

But the computational cost of solving the OEP is a major drawback


Kohn‐Sham equationsq

Kohn Sham energ f nctionalKohn‐Sham energy functional:

∫∫ ++−

+= rrrrrrrrr dnvnEddnnnTnE xcs )()(][''

)'()(][][ ext

Minimizing the energy functional leads to the Kohn‐Sham equations (Schrödinger equations of single electron in effective

rr

equations (Schrödinger equations of single electron in effective potential)

)()()(22

rrrV ψεψ⎥⎤

⎢⎡

+∇h )()()(2 eff rrr iiiV

mψεψ =⎥

⎦⎢⎣

+∇−

)()()()( rrrr VVVV ++= )()()()( xcHexteff rrrr VVVV ++=

∑= in 2)()( rr ψ


∑i

Solving Kohn‐Sham equationg q

B th ff ti t ti l V d d ( ) dBecause the effective potential Veff depends upon n(r), and hence ψ(r), which is the solution of the equation

Veff

ψ(r)n(r)

The Kohn‐Sham equations must be solved by an iterative technique; self‐consistent field methodtechnique; self‐consistent field method


Self‐consistent calculation procedurep

Construct effective potentialGuess density )(rn ][][)()( xcHexteff nVnVVV ++= rr

)()()(2 eff

22

rrr iiiV ψεψ =⎥⎦

⎤⎢⎣

⎡+∇−

h

Solve KS equation

)()()(2 eff iiim

ψψ⎥⎦

⎢⎣

Calculate new density and Compare

∑=i

in 2)()( rr ψ

Calculate new density and Compare

Self‐consistent

Obtain output:Total energy, Force,

yesno


? Eigenvalues, …

Output from KS equationsp q

We ha e sol ed the Kohn Sham eq ations and fo nd the eigenWe have solved the Kohn‐Sham equations and found the eigen‐energies and eigen‐functions, what next?

We can obtainedWe can obtainedGround state total energy

Forces acting on atomg

Charge density

Band structure and density of states

Vibrational properties


Kohn‐Sham eigenvalues and orbitalsg

Kohn‐Sham eigenvalues and orbitals have no physical meaning

Only the ground state density and total energy can be trustedOnly the ground state density and total energy can be trusted

Except only highest occupied KS eigenvalues has a meaning as th fi t i i ti (b t l f t f ti l)the first ionization energy (but only for exact functional)

There is no straightforward relationship between KS eigen‐i d h i i f i i lenergies and the excitation energy of quasiparticles

But many band‐structure calculations in solid state physics is f KS i lfrom KS eigenvalues


Hellmann‐Feynman theorem: forces and stressy

Forces acting on ions can be e al ated anal ticallForces acting on ions can be evaluated analytically

ψψIHR

F∂∂

−=

This formula correct for position independent basis and complete basis set

IR∂

complete basis set

Forces are very sensitive to errors in total energy; the ground state must be determined very accuratelystate must be determined very accurately

This theorem can also be applied for stress on unit cells

∂ ψψσαβ

αβ tH

∂∂

−= derivative energy with respect to strain tensor


The accuracy of DFTy

Molec lar str ct resMolecular structuresbond lengths are accurate to within 1‐2%

Vibrational frequenciesVibrational frequencies within 5‐10% accuracy

Atomization energiesAtomization energies LDA < GGA < hybrid

desired accuracy for hybrid functionals

Ionization and affinity energiesAverage error around 0.2 eV for hybrid functionals


Limitations of DFT

Band gap problemBand‐gap problemHKS theorem is not valid for excited states

Band gap in semiconductors and insulators are always underestimatedBand gap in semiconductors and insulators are always underestimated

EXX and hybrid functionals lead to better gaps

OverbindingOverbindingLSDA calculations usually give too large cohesive energies, too high bulk moduli

GGA largely correct overbinding

The use of the GGA is mandatory for calculating adsorption energies


Limitations of DFT

Diffic lt in strong correlationsDifficulty in strong correlationsNo one knows a feasible approximation valid for all problems – especially for cases with strong electron‐electron correlation

Neglect of van‐der‐Waals interactionsNot included in any DFT functionalNot included in any DFT functional

Possible solution – approximate expression of dipole‐dipole vdW forces on the basis of local polarizabilities derived from DFT


Summaryy

Calc lation of electronic str ct re of materials b sol ing manCalculation of electronic structure of materials by solving many‐electron problem need approximations

HF – neglect electron correlation lead to large deviations fromHF – neglect electron correlation lead to large deviations from the experiment results

DFT – no attempt to compute many‐body wave function insteadDFT no attempt to compute many body wave function, instead energy is written in term of the electron density

KS – treat the kinetic energy properly, bring back the wave gy p p y, gfunction

Exc – approximation forms have proved to be very successful, but xcthere are failures

Requires care and understanding of limitation of the functionals


Part II outline

P i di lid d Bl h’ thPeriodic solid and Bloch’s theorem

Basis set

Plane wave method

Pseudopotentialp

Brillouin zone integration

Some practical aspectSome practical aspect


Finite vs. Extended systemsy

Fi it tFinite system Atom, molecules, clusters

B d iBoundary is open

Discrete energy level

E t d d tExtended systemCrystal, surface, …

Boundary condition is normally periodicBoundary condition is normally periodic

Lead to continuous band


Periodic solids

Cr stal str ct res Infinite repeated nit cellCrystal structures = Infinite repeated unit cell

Real (direct) spaceBravais lattice: aaaR nnn ++Bravais lattice:

Reciprocal spaceReciprocal lattice:

332211 aaaR nnn ++=

332211 bbbG ggg ++=Reciprocal lattice:

where real space and reciprocal space are related by

332211 bbbG ggg ++

2

1st Brillouin zone (BZ)

3212 1 aabRG ×Ω

=⇒=⋅ πie

1 Brillouin zone (BZ) primitive cell of reciprocal lattice


Bloch’s theorem

Bloch’s theorem The densit is periodic so the a e f nction isBloch’s theorem: The density is periodic so the wave function is

)exp()r()( rkr kk ⋅= iunnψ

The function u(r) has the periodicity of the unit cellWave function is repeat with a change of phaseWave function is repeat with a change of phase

Imply that it suffices to calculate ψkn in just one unit cell

All ibl t t ifi d b t k ithi th BZAll possible states are specified by wave vector k within the BZ and band index n

h l ( k) b b dThe eigenvalues ε(n,k) become continuous bands


Band structure

The whole electronic structure of periodic crystal are represented byThe whole electronic structure of periodic crystal are represented by the band structure

The band structure plot shows ε(n,k) only along the high symmetry directions in the BZ

1st Brillouin zone of FCC


1st Brillouin zone of FCC

Modeling non periodic systemg p y

Molecule cluster defect surfaceMolecule, cluster, defect, surface, …

As almost all plane wave codes impose periodic boundary conditions, the p y ,calculation of non‐periodic system must be performed using ’supercell approach’

The interaction between repeated images must be handled by a sufficiently large vacuum region

For surface calculation, the thickness of the slab must be large enough


Basis function

Expand wave function on the basis functionExpand wave function on the basis function

)()( rr ∑=i

iicφψ

We want to find the coefficients ci

Diagonalize H matrix to find the eigenvalues and eigenfunctions

i

There are three basic approaches to the calculation of independent‐particle electronic states in materials

Plane wavesLocalized atomic orbitalsAugmented methodsAugmented methods

Each method has its advantages and pitfalls


Plane wave

Wa e f nctions are ritten as s m of plane a esWave functions are written as sum of plane waves

K+++= ⋅⋅⋅ rGrGrGr 321321)( iii

i ecececψ

Historical reason: related to free electron picture for metal

K+++r 321)(i ecececψ

Historical reason: related to free electron picture for metal

Perfect for crystals (periodic systems)

Simple to implementSimple to implement

Calculation is simplicity and efficiency using FFT’s


Pros and Cons of the plane wave basis setp

PProsPlane wave basis set is complete and unbiased

Th i i l it iThere is single convergence criterion

Plane waves are mathematical simple, and their derivatives are product in k‐spaceproduct in k space

Plane waves do not depend on the atomic positions (force do not depend on the basis set; no Pulay forces)

ConsThe number of plane waves needed is quite large

Empty space is included in the calculation


Localized atomic orbital

Wave function are written as sum of atomic like orbitals (s p d f)Wave function are written as sum of atomic‐like orbitals (s, p, d, f)

K+++= pcscsci 221)( 321rψ

AdvantageThe intuitive appeal of atomic‐like statesThe intuitive appeal of atomic like states

Gaussian basis widely used in chemistry

Simplest interpretation in tight‐bind form

Small basis sets are required

DisadvantageNon orthogonalNon orthogonal

Depend on atomic position

Basis set super position errors (BSSE)


Augmented Plane Wave (APW)

Partition space into space around each atom and interstitial

g ( )

Partition space into space around each atom and interstitial region

Wave function outside sphere represent with plane waveWave function inside sphere represent with spherical harmonic functions

spherical harmonics

plane waves

Best of both region

harmonics

Best of both regionRequire matching inside and outside functions


Pseudopotential approximationp pp

For the plane a e basis set the n mber of plane a es o ldFor the plane wave basis set, the number of plane waves would be very large (>106)

Valence‐electron wave function is far from free‐electron like near atomic coresValence electron wave function is far from free electron like near atomic cores

The valence‐electron wave functions vary rapidly because the requirement for orthogonal to core‐electron wave functions

C l f i l li dCore‐electron wave functions are localized

To make plane wave basis set feasible; pseudopotential instead of exact pseudopotential must be appliedof exact pseudopotential must be applied

Why pseudopotential approximation work:Only valence electrons participate in chemical bonding and core electrons areOnly valence electrons participate in chemical bonding and core electrons are almost unaffected

The detail of valence wave functions near the atomic nuclei is unimportant


Pseudopotential conceptp p

Replace nucleus and core electrons by a fixed effective potentialReplace nucleus and core electrons by a fixed effective potential

Only valence electrons are taken into account in the calculation

R t t f th t

3 1 2 7 V valence 2p 1 2 7 eVoccupation, eigenvalues

Remove core state from the spectrum

2p 6 ‐69.8 eV

3s 2 ‐7.8 eV3p 1 ‐2.7 eV valence

states 1s 2 ‐7.8 eV2p 1 ‐2.7 eV

1s 2 1512 eV

2s 2 ‐108 eV core states

pseudo atomZ = 3

1s 2 ‐1512 eV

iiiV ψεψ =+∇− )21( eff

2 (ps)(ps)(ps)eff

(ps)eff

2 )21( iiV ψεψ =+∇−


2 2

Pseudopotential generationp g

1 Calculate exact all electron wave functions for a reference atom

PSPSAElll V⇒⇒ψψ

1. Calculate exact all electron wave functions for a reference atom

2. Replace the exact wave function by a node less pseudo‐wave function

3. Invert Schrödinger equation to obtain the pseudopotential

Pseudopotentials must conserve exactly the scattering properties of the original atom in the atomic configuration

Pseudopotentials must be generated with the same functional that will be later used in calculations.

Choice of pseudopotential is not unique; lot of freedom to construct


Transferability and softnessy

Transferabilit a pse dopotential s ch that can be sed inTransferability: a pseudopotential such that can be used in whatever environment (molecule, cluster, solid, surface, insulator, metal, …) is called transferableinsulator, metal, …) is called transferable

Softness: a pseudopotential is called soft when a few planeSoftness: a pseudopotential is called soft when a few plane waves are needed

Ultrasoft means very small amount of plane waves needed

The art of creating good pseudopotential is to find one that are both (ultra)soft and transferable


Norm‐conserving pseudopotentialg p p

The pse do a e f nctions generall do not ha e the sameThe pseudo‐wave functions, generally, do not have the same norm as the all electron wave functions inside the spheres

Ob i l ld lik t t i t ti th h i thObviously, one would like to get integrating the charge in the core region the same as the all‐electron one

RR

∫∫ =cc RR

dd0

2AE

0

2PS )()( rrrr ψψ

Key step to improve the accuracy, transferable of pseudopotentials

But not so soft as we desired (especially, first row elements and transition metals)


Ultrasolft pseudopotentialp p

Rela ing the norm conser ation constraint for the alence a eRelaxing the norm‐conservation constraint for the valence wave function and put more complexity in the core

D fi ( th) ili f ti dd t lDefine an (smooth) auxiliary wave function add to plane wave around each atom

C k f d i d b i iCan make as soft as desired because norm‐conservation is not required

B i t hl f t 2 3 llBasis sets are roughly a factor 2‐3 smaller

Accuracy can be better than norm‐conserving pseudopotential

l l l ( ) dClosely relate (approximation) to projector augmented wave method (PAW), which is as accurate as an all electron method


Projector augmented wave (PAW)j g ( )

Combination of the pse dopotential approach and theCombination of the pseudopotential approach and the augmented wave method

∑+= p ψφφψψ ~~)~(~ ∑ −+=m

mmm p ψφφψψ )(

Better method than pseudopotentialTotal density of the system is computed; no transferability problemTotal density of the system is computed; no transferability problem

Plane wave cutoff is equivalent to ultrasoft

PAW method is as accurate as all an electron method


Pseudopotential conclusionp

Use PAW or ultrasoft whenever possible

but it is not available in every code

Use provided pseudopotentials

almost plane wave codes come with well tested pseudopotential databases

create one by yourself is not a good idea, but you may have to

Always test your pseudopotentials


Plane wave cutoff

Wa e f nctions are ritten as s m of plane a esWave functions are written as sum of plane waves

∑ ⋅+= rGkGk r )(

, )( in ecψ

G is reciprocal lattice vectors

I i i l i fi it b f l i d

G

In principle, infinite number of plane waves are required

In practice, the number of plane waves is determined by the cutoff kinetic energy E tthe cutoff kinetic energy Ecut

Only reciprocal lattice vectors satisfied following condition are included in expansion

cut2 E≤+Gk

The quality of the plane wave basis set depend on this cutoff

cutE≤+Gk


Plane wave cutoff

2Gcut

h di f h

Gcut

The radius of the sphere is proportional to the square root of th t ff

Twice cutoff radius is needed for density expansion

b2

the cutoff energyy p

Onl reciprocal lattices inside the sphere are sed in e pansion

reciprocal space b1

2

Only reciprocal lattices inside the sphere are used in expansion

Basis set size is depend on cutoff and volume of box only


Convergence testg

E control the completeness ofEcut control the completeness of the basis set

T ti f b i t ill l dTruncation of basis set will lead to an error

d h bReduce the error by increasing the value of cutoff energy

Always check the convergence of physical properties against Ecut


Fast Fourier Transform (FFT)( )

FFT is essential for efficiencFFT is essential for efficiency: Transforming the wave functions and density between real and reciprocal space

e g V(r)ψ(r): convolution in G‐space scale as Nlog(N) instead of N2 for a directe.g., V(r)ψ(r): convolution in G space scale as Nlog(N) instead of N for a direct computation

))()((FFT)()( 1 GGrr ψψ VV −=

Fourier grid must contain


Fourier grid must contain all wave vectors up to Gc2

Brillouin zone integrationg

Man q antities req ire integration o er Brillo in one e gMany quantities require integration over Brillouin zone, e.g. charge density, total energy, …

∑∫occ d 32 k

To evaluate computationally the integrals transform to

∑∫ Ω=

Ωn

ndn

BZ

2

BZ

),()( krkr ψ

To evaluate computationally – the integrals transform to weighted sum over special k‐points

∑∑occ 2

)()( k

Sampling of k‐point within the Brillouin zone is crucial to the

∑∑≈n j

jnjwn , )()( rkr ψ

Sampling of k‐point within the Brillouin zone is crucial to the accuracy of the integration


Reducing the number of k‐pointsg p

Point group symmetry

Hamiltonian is invariant under any symmetry operator

6 k‐points

Irreducible Brillouin zone (IBZ), which is the smallest

fraction of the BZ that is sufficient to determine all the

information on the states in the crystal

Only k‐points within IBZ should be taken into account; the states at all other k points outside the IBZ are related by

non‐shift grid

states at all other k points outside the IBZ are related by the symmetry operations

Shift of k‐point

3 k‐points

Shift of k‐pointIt is possible to add a constant shift to all of the points in the set before symmetrization

53

y

Apirath – DFT: basic and practice ‐ 6/5/07

shift grid

Monkhorst‐Pack scheme

One of the most pop lar schemes for generating k pointsOne of the most popular schemes for generating k‐points.

A uniform grid of k‐points in BZ

E lExample:srpprs

qr

uuu

12321

−−

++= bbbkconstruction‐rule:

rr

rr qr

qqru ,2,1, ,

212

K==


Smearing methodg

Problem ith metallic s stemProblem with metallic systemDiscontinuities in occupation numbers (partially filled band )

High Fourier components are requiredHigh Fourier components are required

Large number of k point is necessary

In order to improve convergence with number of k‐pointsIn order to improve convergence with number of k points, replace step function by a smoother function (smearing)

Various schemes of smearingVarious schemes of smearingFermi‐Dirac smearing

Gaussian smearingg


Linear tetrahedron method

Another pop lar method for integrating o er BZAnother popular method for integrating over BZ

Main idea:Dividing up BZ into tetrahedraDividing up BZ into tetrahedra

Linear interpolation of the function to be integrated Xwithin these tetrahedra

Integration of the interpolated function

Good forDOS calculation

Very accurate total energyVery accurate total energy


Running a calculationg

PseudopotentialpChoose appropriate pseudopotential that model the science you want

hard, soft, semicore

Basis setTest convergence of physical property

If o r res lts are The whole setup mustg p y p p yyou are interested in against the basis

speed vs. accuracy

If your results are unphysical, then you may have to start again

The whole setup must be the same for comparison between calculations to be

k‐pointsTest convergence of physical property you are interested in against k‐points

start again calculations to be meaningful

y g pmetals!

Production run


Production run

Ionic relaxation

Conditions for eq ilibri mConditions for equilibriumForces on all atoms = 0

Stress = externally applied stressy pp

Method minimization algorithm (CG, BFGS)

by hand – minimum of total energy curve

WarningSymmetry could change during geometry optimization (may or may not be permitted)

Number of plane waves may change discontinuously when cell size changed –increase Ecut by 20‐30% to get a smooth E‐V curve


Speed up the SCF calculationp p

Number of iterations should be small (fast converge)Number of iterations should be small (fast converge)

Each iteration should be fast

Efficiency diagonalization

Mixing scheme‐ Linear

Efficiency diagonalizationmethod‐ Conjugate gradient minimizationLinear

‐ Anderson‐ Broyden

minimization


Computational time and Memoryp y

Con entional matri diagonali ation (standard LAPACK ro tine)Conventional matrix diagonalization (standard LAPACK routine)CPU time scales as O(N3

basis)

Memory storage scales as O(N2b i )Memory storage scales as O(N basis)

Conjugate gradient minimizationCPU time scales as O(N N lnN )CPU time scales as O(NelectronNbasislnNbasis)

k point samplingk point samplingCPU time and memory scale linearly with the number of k points


Other way to determine the KS ground statesy g

Direct minimization of KS functional

Damped second order

Preconditioned conjugate gradient

Di t I i f It ti S b (DIIS)Direct Inversion of Iterative Subspace (DIIS)

Car‐Parrinello molecular dynamics


Summaryy

Periodic solid consider onl one nit cellPeriodic solid – consider only one unit cell

Plane wave basis set – unbiased, no basis correction to forces, it h b t l d i l i FFT t lswitch between real space and reciprocal space via FFT, control

the accuracy with energy cutoff

P d i l d h b f l i dPseudopotentials – reduce the number of plane waves required

FFT – allow the calculation to scale well with system size

Brillouin zone integration – approximate by sum over special k points

Accuracy – always test the convergence of Ecut and k‐points


Documents

DFT: Basic idea and Practical calculations