VASP: Basics (DFT, PW, PAW, … ) · VASP: Basics (DFT, PW, PAW, … ) University of Vienna,...
Transcript of 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.
A B C
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
CGnkeiGr FFT �! CGnk =
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
CGnkeiGr FFT �! CGnk =
1
NFFT
X
r
Crnke�iGr
Crnk =X
G
CGnkeiGr FFT �! CGnk =
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
CGnkeiGr FFT �! CGnk =
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
= +-
pseudo-onsitepseudo AE-onsiteAE
PW PS-LCAO AE-LCAO
| ni = | e ni+X
i
(|�ii � |e�ii)hepi| e ni
| e ni | e ni�X
i
|e�iihepi| e ni | e ni �X
i
|e�iihepi| ni
+X
i
|�iihepi| e ni
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:
ThePAWorbitals(cont.)
⇣� 1
2�+ eve↵ +
X
ij
|epiiDijhepj |⌘|e�ki = ✏k
⇣1 +
X
ij
|epiiQijhepj |⌘|e�ki
• Thepseudopartialwavesobey:
Dij = h�i|�1
2�+ ve↵ |�ji � he�i|�
1
2�+ eve↵ |e�ji
Qij = h�i|�ji � he�i|e�ji
• withtheso-calledPAWstrengthparametersandaugmentationcharges:
Theall-electronandpseudoeigenvaluespectrumisidentical!All-electronscatteringpropertiesarereproducedoverawideenergy-range.
@e�l(r, ✏)
@r
1e�l(r, ✏)
�����r=rc
⇡ @�l(r, ✏)
@r
1
�l(r, ✏)
����r=rc
(�1
2�+ ve↵)|�ii = ✏i|�ii
⇣� 1
2�+ eve↵ +
X
ij
|epiiDijhepj |⌘|e�ki = ✏k
⇣1 +
X
ij
|epiiQijhepj |⌘|e�ki
Si:Troullier-Martins Si:PAW
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:
ThePAWorbitals(cont.)
= +-
pseudo-onsitepseudo AE-onsiteAE
PW PS-LCAO AE-LCAO
| ni | e ni �X
|�lm✏iclm✏ +X
|�lm✏iclm✏
where clm✏ = hplm✏| ni
Thisdecompositioninthreecontributionscanbeachievedforallrelevantquantities,e.g. orbitals,densities,andenergies.
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.
Localoperators
A = A+X
ij
|pii⇣h�i|A|�ji � h�i|A|�ji
⌘hpj |
h |A| i = h e |A| e i
|rihr|+X
ij
|pii⇣h�i|rihr|�ji � h�i|rihr|�ji
⌘hpj |
h |rihr| i = h e |rihr| e i+X
ij
h e |pii⇣h�i|rihr|�ji � h�i|rihr|�ji
⌘hpj | e i
= e⇢(r)� e⇢1(r) + ⇢1(r)
Toany(semi)-localoperatorA,thatactsonthetrueall-electronorbital,thePAWmethodassociatesapseudooperator:
thatactsonthepseudo-orbital,suchthat
ForinstancethePSoperatorassociatedwiththedensityoperator()|rihr|
andthedensity
Non-localoperatorsaremorecomplicated.
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)!
ThePAWtotalenergy(cont.)
⇣� 1
2�+ eVe↵ +
X
ij
|epii(Dij + ...)hepj |⌘| e ni = ✏n
⇣1 +
X
ij
|epiiQijhepj |⌘| e ni
Dij = h�i|�1
2�+ v1e↵ [⇢
1v]|�ji � he�i|�
1
2�+ ev1e↵ [e⇢1v]|e�ji
Qij = h�i|�ji � he�i|e�ji
⇢1v(r) =X
ij
⇢ijh�i|rihr|�jie⇢1v(r) =X
ij
⇢ijhe�i|rihr|e�ji
⇢ij =X
n
fnh e n|epiihepj | e ni
ThePSorbitals(planewaves!)aretheself-consistentsolutionsof
where
and
with
• ThePSorbitalsarethevariational quantityofthePAWmethod!• Ifthepartialwavesconstituteacomplete(enough)basisinsidethePAWspheres,
Theall-electronorbitalswillremainorthogonaltothecorestates.
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!