Tutorial:Octopus+BerkeleyGW

DavidA.StrubbeDepartmentofPhysics,

UniversityofCalifornia,Merced

FelipeHomrichdaJornadaDepartmentofPhysics,UniversityofCalifornia,Berkeley

MaterialsSciencesDivision,LawrenceBerkeleyNa?onalLaboratory

Benasque,Spain15September2016

Review:GWapproximaMon/Bethe-SalpeterStart with wavefunctions and energies from DFT as mean field

Add electronic correlation as a perturbation

GW self-energy: single-electron energy levels (band structure)

Bethe-Salpeter equation: electron-hole interaction for optical properties

widelyusedmassivelyparallelcode(~1000sofatoms)www.berkeleygw.org

J.Deslippe,G.Samsonidze,D.A.Strubbe,M.Jain,M.L.Cohen,andS.G.Louie,Comput.Phys.Comm.183,1269(2012)

Why Use BerkeleyGW

  SupportsalargesetofMean-Fieldcodes:PARATEC,QuantumESPRESSO,ABINIT,Octopus,PARSEC,SIESTA,EPM(TBPW)

  Supports3D,2D,1DandMolecularSystems.CoulombTruncaMon  SupportforSemiconductor,MetallicandSemi-MetallicSystems  EfficientAlgorithmsandUseofLibraries.(BLAS,FFTW3,LAPACK,

SCALAPACK,OpenMP,HDF5)  MassivelyParallel.Scalesto100,000CPUs,distributedMemory.  EfficientaccuratesoluMontoBSEviak-pointInterpolaMon

  SupportforLDA/GGA/Hybrid/HF/COHSEXstarMngpointsaswellasoff-diagonalΣcalculaMons

Website:www.berkeleygw.org

GW Method

Full-Frequency vs. GPP

Full-Frequencyvs.PlasmonPolecalculaMonsPlasmonpoleissignificantlyfaster.TheintegraloverfrequenciescanbeperformedanalyMcallyifassumethedielectricresponseisdominatedbyasingleplasmonpole.BerkeleyGWsupportsboth.Withfull-frequencyyoucancomputespectralfuncMons,lifeMmesandweights.

PracMcalissuesforGW

1.  ScreeningmodelsforEpsilon

2.  ConstrucMonofk-grids

3.  Symmetryanddegeneracy

4.  Realandcomplexversion

5.  SolvingDyson’sequaMon

6.  Convergence

Mean-FieldφMFnk , E

MFnk! "# $

WFN

, Vxc!"#$vxc.dat

, ρ!"#$RHO

epsilon

ϵ−1G,G′(q, E)! "# $eps0mat,epsmat

sigma

EQPnk!"#$

eqp.dat

kernel

Kvck,v′c′k′! "# $bsedmat,bsexmat

absorption

Asvck,Ω

s,! "# $eigenvectors,eigenvalues.dat

ϵ(ω), JDOS(ω)! "# $absorption eh.dat

kco kfi

Screeningmodels:Howdoweuseε?

Screeningmodels:Howdoweuseε?

Sigmaintegratesoverqwithε-1(q)

AbsorpMoninterpolateskerneloverqwithW(q)=ε-1(q)v(q)

Problem1:Non-smoothbehavior

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

ε−1 (q

)

q (AU)

(14,0) ε−1(q)(14,0)carbonnanotubewiretruncaMonGeneralfortruncaMon:seeBNtutorial

Problem2:Divergentbehavior

Head:G=0,G’=0Wing:G=0,G’≠0Wing’:G≠0,G’=0Body:G≠0,G’≠0

Cannotcalculateatq=0!

diverges

DOS inter/intra-bandtransiMons gap

SoluMon:ScreeningmodelsCalculateatq0≈0.001inperiodicdirecMonusetoparametrizescreeningmodelSigma:Integrateoverregionaroundq=0

head wing,wing’ bodyKernel:Interpolateinparts

Trunca>onfornon-orpar>allyperiodicsystemsPeriodicityin0,1,2,or3dimensions.EliminatespuriousimageinteracMons.

Cell(formolecule)

Slab(forgrapheneorsurface)

Wire(fornanotubeornanowire)

TruncaMonofCoulombpotenMal•  GWandBSEuMlizetheCoulombandscreenedCoulombinteracMon

•  Long-rangeinteracMonsmakeitcomputaMonallyinfeasibletoincreaselaocevectorsunMlperiodicimagesdonotinteract

TruncaMonSchemeswithinBerkeleyGW•  Cellbox: 0D•  Cellwire: 1D•  Cellslab: 2D•  Spherical: DefineradiusoftruncaMon

•  Celltrunca>on:athalflaocevectorlength–  AnalyMcalformforCoulombpotenMalink-space

•  Sphericaltrunca>on:convenient,availableinmanypackages

cVW 1−= ε

Example:BNsheets

Z

||)2/(

rr Lz −=θ)(VC ))2/(cos(1(

||4 )2/(

2 Lke zLkxy−−=

kπ)(VC k

QuasiparMcleCorrecMon SeparaMonlength(a.u.)

12x12x12k-mesh 10 14 18

NoTruncaMon 1.98 2.05 2.10

TruncaMon 2.53 2.55 2.58

Ismail-BeigiPRB73233103(2006)

• ConvergenceimprovedwithtruncaMon

Regulark-grids

Epsilon

Sigma

Kernel

k-gridsandbands

k-grid #bands Comments

SCF Uniform,0.5shir occupied asusualinDFT

WFN Uniform,0.5shir many

WFNq WFN+q-shir occupied

epsilon.inpq-points WFNbutnoshir,q0 many bandstosumover

WFN_inner WFNbutnoshir many bandstosumover

sigma.inpk-points subsetofWFN_inner few canchoosetocalculateSigmajustforbandsofinterest

WFN_co WFN_inner few

WFN_fi(absorpMon) Uniform,randomshir few

WFNq_fi WFN_fi+q-shir occupied

WFN_fi(inteqp) anything few whateverisofinterest

recommendedapproach

epsilon.inp

begin qpoints 0.000000 0.000000 0.005000 1.0 1 0.000000 0.000000 0.062500 1.0 0 0.000000 0.000000 0.125000 1.0 0 0.000000 0.000000 0.187500 1.0 0 … end

eps0mat:

epsmat:

Semiconductors

k-gridconstrucMon:4×4gridforgraphene

(0.5,0.5)Monkhorst-Packshirkgrid.xUniform->unfold->shirwithq->reduce

Maingrid(WFN)16infullBZReducedto6

Unfoldedto48infullBZ

AddiMonalq=(0.0,0.05)

Unfoldinggivesmorepoints!

(0.5,0.5)

k-gridconstrucMon:4×4gridforgraphene

kgrid.xUniform->unfold->shirwithq->reduce

Shiredgrid(WFNq)48infullBZReducedto26

AddiMonalq=(0.0,0.05)

Unfoldingandbreakingsymmetrygivesmorepoints!

Unfoldedto48infullBZ

Degeneracy

Summingoveronlysomeofadegeneratespacewillbreaksymmetry.Degeneracyinmean-field=>brokeninGW!ResultsdependsonarbitrarylinearcombinaMonsinmean-field.Notreproducible!IncorrectoscillatorstrengthsinabsorpMon!

Epsilon,Sigma:symmetryofHamiltonian

AbsorpMon:symmetryofe-hbasis

DegeneracycheckuMlity

$ degeneracy_check.x WFN Reading eigenvalues from file WFN Number of spins: 1 Number of bands: 35 Number of k-points: 8 == Degeneracy-allowed numbers of bands (for epsilon and sigma) == 4 8 14 18 20 32 Note: cannot assess whether or not highest band 35 is degenerate.

So,usenumber_bands 32 inEpsilon.

Realorcomplexflavor?

Real:onlywithinversionsymmetryabouttheoriginandMme-reversalsymmetry

e.g.bin/epsilon.real.x,bin/epsilon.cplx.x

WhatbreaksMme-reversal?MagneMcfields,spin-polarizaMon,spinorsPlane-wavecodesgenerallyjustusecomplexwavefuncMons.CondiMonsforrealitydependsonthebasis!Real-space:k=0,Mme-reversal.RealoutputnotimplementedinOctopusyet.

Complexisgeneral,butrealisfaster,useslessmemoryanddiskspace

SolvingDyson’sequaMoninSigma

Howcanwesolvewhenwedon’tknowEQPyet?(1) eqp0:evaluateatEMF.

(2)eqp1:solvelinearizedapproximaMon(Newton’sMethod)

Availableascolumnsinsigma_hp.log,andeqp0.datandeqp1.datfiles

QuasiparMclerenormalizaMonfactorZ

Between0and1WeightinQPpeak

TherearemanyconvergenceparametersinaGWcalculaMons:convergencewitheachmustbechecked

Screenedcutoff Emptybands(dielectricmatrix)

BandsinCHsummaMon(sigma)q-grid

WavefuncMoncutoff(matrixelements)

Coupledconvergenceparameters

B.Shihetal.,ZnO

Seeconvergenceand“Whenthingsgowrong”slidesonBerkeleyGW2015tutorialpage!

OctopusinterfacetoBerkeleyGW

Realspacetransformedtoplane-wavesforGW.CanonlyproducecomplexwavefuncMonscurrently.Periodicsystemsmustuseorthogonalunitcells(i.e.nothcp,fcc,…)sobuildasupercelltomatchthiscondiMon.Cantreatrigorouslyfinitesystems,unlikeplane-wavecodes.DomainparallelizaMonforreal-spacescalesbeerthanplanewaves.

Thetutorial

Threeexamples:(1)hexagonalboronnitridesheet(2)benzenemolecule(3)siliconToday:GWapproximaMonTomorrow:Bethe-SalpeterequaMonInstrucMonsathp://www.tddr.org/programs/octopus/wiki/index.php/Tutorial:BerkeleyGWGetyourCoriaccountifyoudidn’tyet!