vasp: basics (dft, pw, paw, … ) · vasp: basics (dft, pw, paw, … ) university of vienna,...

VASP:Basics(DFT,PW,PAW,…)

UniversityofVienna,FacultyofPhysicsandCenterforComputationalMaterialsScience,

Vienna,Austria

Outline

● Densityfunctionaltheory

● Translationalinvarianceandperiodicboundaryconditions

● Planewavebasisset

● TheProjector-Augmented-Wavemethod

● Electronicminimization

TheMany-BodySchrödingerequation

H (r1, ..., rN ) = E (r1, ..., rN )0

@�1

2

X

i

�i +X

i

V (ri) +X

i 6=j

1

|ri � rj |

1

A (r1, ..., rN ) = E (r1, ..., rN )

(r1, ..., rN ) ! { 1(r), 2(r), ..., N (r)}

Forinstance,many-bodyWFstoragedemandsareprohibitive:

Asolution:maponto“one-electron”theory:

5electronsona10×10×10grid~10PetaBytes !

(r1, ..., rN ) (#grid points)

N

Hohenberg-Kohn-ShamDFT

E[⇢] = Ts[{ i[⇢]}] + EH [⇢] + Exc

[⇢] + EZ [⇢] + U [Z]

(r1, ..., rN ) =NY

i

i(ri)

⇢(r) =NX

i

| i(r)|2

⇣�1

2�+ VZ(r) + VH [⇢](r) + V

xc

[⇢](r)⌘ i(r) = ✏i i(r)

(r1, ..., rN ) ! { 1(r), 2(r), ..., N (r)}

Maponto“one-electron”theory:

Totalenergyisafunctionalofthedensity:

Thedensityiscomputedusingtheone-electronorbitals:

Theone-electronorbitalsarethesolutionsoftheKohn-Shamequation:

BUT:E

xc

[⇢] =??? Vxc

[⇢](r) =???

Exchange-Correlation

Exc

[⇢] =??? Vxc

[⇢](r) =???

• Exchange-Correlationfunctionals aremodeledontheuniform-electron-gas(UEG):Thecorrelationenergy(andpotential)hasbeencalculatedbymeansofMonte-Carlomethodsforawiderangeofdensities,andhasbeenparametrized toyieldadensityfunctional.

• LDA:wesimplypretendthataninhomogeneouselectronicdensitylocallybehaveslikeahomogeneouselectrongas.

• Many,many,manydifferentfunctionals available:LDA,GGA,meta-GGA,van-der-Waalsfunctionals,etc etc

AnN-electronsystem:N=O(1023)

N ⇥ (#grid points)

(#grid points)

N

(r1, ..., rN ) ! { 1(r), 2(r), ..., N (r)}

Hohenberg-Kohn-ShamDFTtakesusalongway:

Niceforatomsandmolecules,butinarealisticpieceofsolidstatematerialN=O(1023)!

Translationalinvariance:PeriodicBoundaryConditions

nk(r+R) = nk(r)eikR

nk(r) = unk(r)eikr

unk(r+R) = unk(r)

Translationalinvarianceimplies:

and

AllstatesarelabeledbyBlochvector k andthebandindex n:

• TheBlochvectork isusuallyconstrainedtoliewithinthefirstBrillouin zoneofthereciprocalspacelattice.

• Thebandindexnisoftheorderifthenumberofelectronsperunitcell.

Reciprocalspace&thefirstBrillouin zone

b2b1

b3

z

y

x

C

b2 b1

b3

z

y

x

B

4pi/L

a2a1

a3 y

L

z

x

A

b1 =2⇡

⌦a2 ⇥ a3 b2 =

2⇡

⌦a3 ⇥ a1 b3 =

2⇡

⌦a1 ⇥ a2

⌦ = a1 · a2 ⇥ a3 ai · bj = 2⇡�ij

Samplingthe1st BZ

⇢(r) =1

⌦BZ

X

n

Z

BZfnk| nk(r)|2dk

⇢(r) =X

nk

wkfnk| nk(r)|2dk,

Theevaluationofmanykeyquantitiesinvolvesanintegraloverthe1st BZ.Forinstancethechargedensity:

WeexploitthefactthattheorbitalsatBlochvectorsk thatareclosetogetherarealmostidenticalandapproximatetheintegraloverthe1st BZbyaweightedsumoveradiscretesetofk-points:

Fazit:theintractabletaskofdeterminingwithN=1023,hasbeenreducedtocalculatingatadiscretesetofk-pointsinthe1st BZ,foranumberofbandsthatisoftheorderifthenumberofelectronsintheunitcell.

(r1, ..., rN ) nk(r)

b!

b!

BZ

IBZ

k1k1

k3

k2

0

4 4

4⇥ k1 = (1

8,1

8) ) !1 =

1

4

4⇥ k2 = (3

8,3

8) ) !2 =

1

4

8⇥ k3 = (3

8,1

8) ) !3 =

1

2

1

⌦BZ

Z

BZ

F (k)dk ) 1

4F (k1) +

1

4F (k2) +

1

2F (k3)

Idea:equallyspacedmeshinthe1st Brillouin zone

Example:aquadratic2Dlattice• q1=q2=4,i.e.,16pointsintotal• Only3symmetryinequivalent points:

Algorithm:

• Calculateequallyspacedmesh.• Shiftthemeshifdesired.• ApplyallsymmetryoperationsoftheBravais latticetoallk-points.• Extracttheirreduciblek-points(IBZ).• Calculatetheproperweighting.

Commonmeshes:Twodifferentchoicesforthecenterofthemesh.

• CenteredonΓ• CenteredaroundΓ (canbreakthesymmetry!)

!!

shifted to before after

symmetrization

• Incertaincellgeometries(e.g. hexagonalcells)evenmeshesbreakthesymmetry.

• Symmetrization resultsinnon-uniformdistributionsofk-points.

• Γ-pointcenteredmeshespreservethesymmetry.

E[⇢, {R, Z}] = Ts[{ nk[⇢]}] + EH [⇢, {R, Z}] + Exc

[⇢] + U({R, Z})

Ts[{ nk[⇢]}] =X

nk

wkfnkh nk|�1

2�| nki

EH[⇢, {R, Z}] = 1

2

ZZ⇢eZ(r)⇢eZ(r0)

|r� r0| dr0dr

⇢eZ(r) = ⇢(r) +X

i

Zi�(r�Ri) ⇢(r) =X

nk

wkfnk| nk(r)|2dk

⇣�1

2�+ VH [⇢eZ ](r) + V

xc

[⇢](r)⌘ nk(r) = ✏nk nk(r)

VH [⇢eZ ](r) =

Z⇢eZ(r0)

|r� r0|dr0

Thetotalenergy

Thekineticenergy

TheHartree energy

where

TheKohn-Shamequations

TheHartree potential

Aplanewavebasisset

unk(r+R) = unk(r) nk(r) = unk(r)eikr

unk(r) =1

⌦1/2

X

G

CGnkeiGr nk(r) =

1

⌦1/2

X

G

CGnkei(G+k)r

⇢(r) =X

G

⇢GeiGr

1

2|G+ k|2 < E

cuto↵

V (r) =X

G

VGeiGr

Allcell-periodicfunctionsareexpandedinplanewaves(Fourieranalysis):

Thebasissetincludesallplanewavesforwhich

Crnk =X

G

CGnkeiGr FFT �! CGnk =

1

NFFT

X

r

Crnke�iGr

TransformationbymeansofFFTbetween“real”spaceand“reciprocal”space:

Whyuseplanewaves?

• Historicalreason:Manyelementsexhibitaband-structurethatcanbeinterpretedinafreeelectronpicture(metallicsandpelements).Pseudopotential theorywasinitiallydevelopedtocopewiththeseelements(pseudopotential perturbationtheory).

• Practicalreason:ThetotalenergyexpressionsandtheHamiltonianareeasytoimplement.

• Computationalreason:TheactionoftheHamiltonianontheorbitalscanbeefficientlyevaluatedusingFFTs.

τ1

τ1 π / τ 1

τ2

b1

b2

real space reciprocal spaceFFT

0 1 2 3 0

1 1 x = n / N 1 1 g = n 2

N−1 −1−2−3−4 0 1 2 3 4 55

N/2−N/2+1

cutG

Crnk =X

G


1

NFFT

X

r

Crnke�iGr

Crnk =X

G


1

NFFT

X

r

Crnke�iGr

τ1

τ1 π / τ 1

τ2

b1

b2

real space reciprocal spaceFFT

0 1 2 3 0

1 1 x = n / N 1 1 g = n 2

N−1 −1−2−3−4 0 1 2 3 4 55

N/2−N/2+1

cutG

Crnk =X

G


1

NFFT

X

r

Crnke�iGr

Thechargedensity

ψrψG ψr

ρ rρG

2 Gcut

Gcut

FFT

FFT

TheactionoftheHamiltonianH| nki ✓

�1

2�+ V (r)

◆ nk(r)

hr|G+ ki = 1

⌦1/2ei(G+k)r �! hG+ k| nki = CGnk

NNPLWhG+ k|� 1

2�| nki =

1

2|G+ k|2CGnk

hG+ k|V | nki =1

NFFT

X

r

VrCrnke�iGr NFFT logNFFT

Theaction

Usingtheconvention

• Kineticenergy:

• Localpotential:• Exchange-correlation:easilyobtainedinrealspace• FFTtoreciprocalspace• Hartree potential:solvePoissoneq.inreciprocalspace• Addallcontributions• FFTtorealspaceTheaction

V = VH[⇢] + Vxc

[⇢] + Vext

Vxc,r = V

xc

[⇢r]

VH,G =4⇡

|G|2 ⇢GVG = VH,G + Vxc,G + Vext,G

{VG} �! {Vr}

{Vxc,r} �! {V

xc,G}

Theactionofthelocalpotential

4π e 2

G 2

VG Vr ψr

RG R (residual vector)r

ρG

ψG

2Gcut

3GcutGcut

FFT

FFTadd

TheProjector-Augmented-WavemethodThenumberofplanewavesneededtodescribe• tightlybound(spatiallystronglylocalized)states,• andrapidoscillations(nodalfeatures)oftheorbitalsnearthenucleusexceedsanypracticallimit,exceptmaybeforLiandH.

Thecommonsolution:• Introducethefrozencoreapproximation:

Coreelectronsarepre-calculatedinanatomicenvironmentandkeptfrozeninthecourseoftheremainingcalculations.

• Useofpseudopotentials insteadofexactpotentials:• Norm-conservingpseudopotentials• Ultra-softpseudopotentials• TheProjector-Augmented-Wave(PAW)method

[P.E.Blöchl,Phys.Rev.B 50,17953(1994)]

Ψ

Ψ~

V

V~

rc

Pseudopotentials:thegeneralidea

Pseudopotentials:cont.

wav

e-fu

nctio

n

R (a.u.)0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

2.5

s : E= -0.576 R c=1.9

p : E= -0.205 R c=1.9AlAl

PseudopotentialExactpotential(interstitial)

Al effectiveAlatom PAWAlatom3p 2p 3p3s 1s 3s

2p 2pand1sarenodeless nodalstructureretained2s12

ThePAWorbitals(cont.)

| ni = | e ni+X

i

(|�ii � |e�ii)hepi| e ni

(�1

2�+ ve↵)|�ii = ✏i|�ii

|�ii ! |e�ii ve↵ ! eve↵ hepi|e�ji = �ij

| e ni• isthepseudo(PS)orbital,expressedinaplanewavebasisset

|�ii |pii|�ii• ,,andareatom-centeredlocalizedfunctions

• Theall-electronpartialwavesaresolutionstotheradialscalarrelativisticnon-spinpolarized Schrödingerequation:

• Apseudization procedureyields:

ve↵ [⇢v] = vH [⇢v] + vH [⇢Zc] + vxc[⇢v + ⇢c] ⇢v(r) =X

i

ai|�i(r)|2

e⇢v(r) =X

i

ai|e�i(r)|2

• 1st s-channelinMn:ε14s“bound”state

• 2nd s-channelinMn:ε2“non-bound”state

eve↵ [e⇢v] = vH [e⇢v] + vH [e⇢Zc] + vxc[e⇢v + e⇢c]

• Andweusethefrozencoreapproximation:

Thekineticenergy

Ekin =X

n

fnh n|�1

2�| ni

| ni = | ni+X

i

(|�ii � |�ii)hpi| ni

Ekin = E � E1 + E1

X

n

fnh n|�1

2�| ni

| {z }E

�X

site

X

(i,j)

⇢ijh�i|�1

2�|�ji

| {z }E1

+X

site

X

(i,j)

⇢ijh�i|�1

2�|�ji

| {z }E1

⇢ij =X

n

fnh n|piihpj | ni

Forinstancethekineticenergy:

InsertingthePAWtransformation(wherei=lmε):

andassumingcompletenessoftheone-centerbasis,wehave

Where

aretheone-centeroccupancies,oron-sitedensitymatrix.

TheHartree energy

EH [⇢+ ⇢]�X

sites

EH [⇢1 + ⇢1] +X

sites

EH [⇢1]

EH = EH � E1H + E1

H

= +-

AE pseudo + compens. pseudo+comp. onsite AE-onsite

• ThePSorbitalsdonothavethesamenorm astheAEorbitalsinsideofthePAWspheres.

• Tocorrectlydescribethelong-rangeelectrostaticinteractionsbetweenthePAWspheres,asoftcompensation chargeisintroducedinthespheres(likeintheFLAPWmethod):

• ThiswaytheHartree energy(anon-localoperator!)decomposesinthesamemannerasa(semi)-localoperator:

ThePAWtotalenergy

E =X

n

fnh n|�1

2�| ni+ Exc[⇢+ ⇢+ ⇢c]+

EH [⇢+ ⇢] +

ZvH [⇢Zc] (⇢(r) + ⇢(r)) d3r+ U(R, Z

ion

)

E1 =X

sites

⇢X

(i,j)

⇢ijh�i|�1

2�|�ji+ Exc[⇢1 + ⇢+ ⇢c]+

EH [⇢1 + ⇢] +

Z

⌦r

vH [⇢Zc]�⇢1(r) + ⇢(r)

�d3r

�

E1 =X

sites

⇢X

(i,j)

⇢ijh�i|�1

2�|�ji+ Exc[⇢1 + ⇢c]+

EH [⇢1] +

Z

⌦r

vH [⇢Zc]⇢1(r) d3r

�

E = E � E1 + E1

Thesamethree-waydecompositionholdsforthetotalenergy

where

ThePAWtotalenergy(cont.)• isevaluatedonaregulargrid:

TheKohn-ShamfunctionalevaluatedinaplanewavebasissetwithadditionalcompensationchargetoaccountfortheincorrectnormofthePS-orbitalsandtocorrectlydescribelong-rangeelectrostatics

E

e⇢ =X

n

fn e ne ⇤n

⇢

PSchargedensity

Compensationcharges

E1 E1• andareevaluatedonatom-centeredradiallogarithmicgrids:TheKohn-Shamenergiesevaluatedusinglocalizedbasissets

Thesetermscorrectforthedifferenceintheshapeoftheall-electronandpseudoorbitals:

)AEnodalfeaturesnearthecore) Orthogonality betweencoreandvalencestates

TheessenceofthePAWmethod:therearenocross-termsbetweenquantitiesontheregulargrid(PWpart)andthequantitiesontheradialgrids(LCAOpart)!

AccuracyofthePAWmethod

Δ-evaluation(PAWvs.FLAPW)K.Lejaeghere etal.,CriticalReviewsinSolidStateandMaterialsSciences39,1(2014)

AccuracyofthePAWmethod(cont.)SubsetoftheG2-1testset ofsmallmolecules:deviationofPAWw.r.t.GTO(inkcal/mol)

Electronicminimization:Reachingthegroundstate

Directminimization oftheDFTfunctional(e.g. Car-Parrinello):startwithasetoftrialorbitals(randomnumbers)andminimizetheenergybypropagatingtheorbitalsalongthegradient:

Gradient: Fn(r) =

✓� ~22me

r2 + V e↵(r, { n(r0)})� ✏n

◆ n(r)

TheSelf-Consistency-Cycle:startwithatrialdensity,constructthecorrespondingHamiltonian.Solveittoobtainasetoforbitals:

✓� ~22me

r2 + V e↵(r, {⇢(r0)})◆ n(r) = ✏n n(r) n = 1, ..., Ne/2

Theseorbitalsdefineanewdensity,thatdefinesanewHamiltonian,…iteratetoself-consistency

iterationlo

g10

E-E

0

-8

-6

-4

-2

0

2

log

10 |F

-F0 |

-4

-3

-2

-1

0

1

0 10 20 30 40iteration

log

10 E

-E0

-8

-6

-4

-2

0

log

10 |F

-F0 |

-4

-3

-2

-1

0

1

0 5 10 15 20

n=8n=1

n=1

n=4

self.consistentn=8

n=1

Ln=1,2,4,8

forces

self.consistent

Car−Parrinellodirect

Car−Parrinellodirect

disordered fcc Fe, metaldisordered diamond, insulator

energy

Directminimizationvs.SCC

Directminimizationandchargesloshing

E

occupied

unoccupied

strong change in potentialπ4 e / G

2slowly varying charge

charge

potential

|gni = fn

⇣1�

X

m

| mih m|⌘H| ni+

X

m

1

2Hnm(fn � fm)| mi

Hnm = h m|H| ni

n = ei(kF��k)r m = ei(kF+�k)r

0n = n +�s m 0

m = m ��s n

�⇢(r) = 2�sRe ei2�k·r �VH(r) =2�s 4⇡e2

|2�k|2 Re ei2�k·r

Thegradientofthetotalenergywithrespecttoanorbitalisgivenby:

where

Considertwostates

andasmallsub-spacerotation(2nd comp.ofthegradient):

Thisleadstoalong-wavelengthchangeinthedensityandaverystrongchangeintheelectrostaticpotential(chargesloshing):

|�k| / 1/L �VH / L2 �s / 1/L2

StablestepsizeΔs (forasimulationboxwithlargestdimensionL):

TheSelf-Consistency-Cycle(cont.)

Twosub-problems:

• OptimizationofIterativeDiagonalizatione.g. RMM-DIISorBlockedDavidson

• ConstructionofDensityMixinge.g. Broyden mixer

{ n}

⇢in

Theself-Consistency-Cycle

H = hG| ˆH[⇢]|G0i ! diagonalize H ! { i, ✏i} i = 1, .., NFFT

⇢0 ! H0 ! ⇢0 ! ⇢1 = f(⇢0, ⇢0) ! H1 ! ...

⇢ = ⇢0

Anaïvealgorithm:expresstheHamiltonmatrixinaplanewavebasisanddiagonalize it:

Self-consistency-cycle:

Iterateuntil:

BUT: wedonotneedNFFT eigenvectorsoftheHamiltonian(atacostofO(NFFT3)).

ActuallyweonlytheNb lowesteigenstates ofH,whereNb isoftheorderofthenumberofelectronsperunitcell(Nb <<NFFT).

Solution:useiterativematrixdiagonalization techniquestofindtheNb lowestEigenvectoroftheHamiltonian:RMM-DIIS,blocked-Davidson,etc.

Keyingredients:SubspacediagonalizationandtheResidual

X

m

HnmBmk =X

m

✏appk SnmBmk

Hnm = h n|H| mi Snm = h n|S| mi

| ki =X

m

Bmk| mi

|R( n)i = (H � ✏appS)| ni

• Rayleigh-Ritz:diagonalization oftheNb xNb subspace

with

yieldsNb eigenvectorswitheigenvaluesεapp.

Theseeigenstates arethebestapproximationtotheexactNb lowesteigenstates ofH withinthesubspacespannedbythecurrentorbitals.

• TheResidual:

✏app =h n|H| nih n|S| ni

(itsnormismeasurefortheerrorintheeigenvector)

Blocked-Davidson• Takeasubsetofallbands: { n|n = 1, .., N} ) { 1

k|k = 1, .., n1}

{ 1k/g

1k = K(H� ✏appS)

1k|k = 1, .., n1}

{ 2k|k = 1, .., n1}

• Extendthissubsetbyaddingthe(preconditioned)residualvectorstothepresentlyconsideredsubspace:

• Rayleigh-Ritzoptimization(“sub-spacerotation”)inthe2n1 dimensionalsubspacetodeterminethen1 lowesteigenvectors:

diag{ 1k/g

1k}

• Extendsubspacewiththeresidualsof { 2k}

{ 1k/g

1k/g

2k = K(H� ✏appS)

2k|k = 1, .., n1}

• Rayleigh-Ritzoptimization ) { 3k|k = 1, .., n1}

• Etc …

{ mk |k = 1, .., n1} { n|n = 1, .., n1}

• Theoptimizedsetreplacestheoriginalsubset:

• Continuewithnextsubset:,etc,…{ 1k|k = n1 + 1, .., n2}

Aftertreatingallbands:Rayleigh-Ritzoptimizationof { n|n = 1, .., N}

Chargedensitymixing

R[⇢in

] = ⇢out

[⇢in

]� ⇢in

⇢out

(~r) =X

occupied

wkfnk| nk(~r)|2

R[⇢] = �J(⇢� ⇢sc) J = 1� � U|{z}4⇡e2

q2

R[⇢in

] = ⇢out

[⇢in

]� ⇢in

= �J(⇢in

� ⇢sc

)

Wewanttominimizeresidualvector

with

Linearizationoftheresidualaroundtheself-consistentdensity(linearresponsetheory):

⇢sc

whereJ isthechargedielectricfunction.Providedwehaveagoodapproximationforthechargedielectricfunction,minimizationoftheresidualistrivial:

⇢sc = ⇢in + J�1R[⇢in]

Thechargedielectricfunction

J�1 ⇡ G1q = max(

q2AMIX

q2 + BMIX, AMIN)

• Useamodeldielectricfunctionthatisagoodinitialapproximationformostsystems

• Thisiscombinedwithaconvergenceaccelerator

Theinitialdielectricfunctionisimprovedusingtheinformationaccumulatedineachelectronicmixingstep.

TheEnd

Thankyou!

vasp: basics (dft, pw, paw, … ) · vasp: basics (dft, pw, paw, … ) university of vienna,...

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