vasp: beyond dft the random phase approximation beyond dft the random phase approximation university...

VASP:beyondDFTTheRandomPhaseApproximation

UniversityofVienna,FacultyofPhysicsandCenterforComputationalMaterialsScience,

Vienna,Austria

PAST,PRESENT,FUTURE

● PAST

● TheWorkhorse:DFT

● Efficientandstablealgorithms

● PAWdatasets

● PRESENT

● BeyondDFT,andbeyondthegroundstate:Hybridfunctionals,linearresponse,RPA(GW&ACFTD),BSE

● FUTURE

● Nearfuture:cubic-scaling-RPA

● …

NeedtogobeyondDFTandHartree-Fock?Atomization energy

LDA PBE

MRE (%) 17.3 -1.9

MARE (%) 17.3 3.4

ME (eV) 0.76 0.14

Si

MgO

SiC C

ZnO

BN Ar

Ne

Egap

[eV] experimental

0

5

10

15

20

25

Eg

ap [eV

] th

eore

tical

Eg(HF)

Eg(DFT)

Band gaps

LiF

Ar

Ar

LiF

ZnS

(More)accuratetreatmentofelectroniccorrelationneededfor,e.g:

• Bandgaps(opticalproperties• Totalenergydifferenceswith

chemicalaccuracy(1kcal/mol ≈ 40meV)

• Atomizationandformationenergies• Reactionbarriers• VanderWaalsinteractions

Lattice constants and Bulk moduli:AlP, AlAs, BAs, BP, Si, C, SiC, MgO, LiF

LDA PBE HF�a0 �B0 �a0 �B0 �a0 �B0

MRE -1.4 3.5 0.8 -7.2 0.4 8.2MARE 1.4 7.9 0.8 7.2 0.7 8.2

(All in %)

Catalysis:dehydrogenationofpropaneinMordenite

Newdensityfunctionals (forsolids)

Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces

John P. Perdew,1 Adrienn Ruzsinszky,1 Gabor I. Csonka,2 Oleg A. Vydrov,3 Gustavo E. Scuseria,3 Lucian A. Constantin,4

Xiaolan Zhou,1 and Kieron Burke5

1Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA2Department of Chemistry, Budapest University of Technology and Economics, H-1521 Budapest, Hungary

3Department of Chemistry, Rice University, Houston, Texas 77005, USA4Donostia International Physics Center, E-20018, Donostia, Basque Country

5Departments of Chemistry and of Physics, University of California, Irvine, Irvine, California 92697, USA

PRL 100, 136406 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2008

Functional designed to include surface effects in self-consistent density functional theory

R. Armiento1,* and A. E. Mattsson2,†

1Department of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden2Computational Materials and Molecular Biology MS 1110, Sandia National Laboratories, Albuquerque,

New Mexico 87185-1110, USA

PHYSICAL REVIEW B 72, 085108 !2005"AM05

PBEsol

TABLE I. Statistical data for the equilibrium lattice constants !Å" of the 18 test solids of Ref. 38 at 0 K calculated from the SJEOS. TheMurnaghan EOS yields identical results within the reported number of decimal places. Experimental low temperature !5–50 K" latticeconstants are from Ref. 56 !Li", Ref. 57 !Na, K", Ref. 58 !Al, Cu, Rh, Pd, Ag", and Ref. 59 !NaCl". The rest are based on room temperaturevalues from Ref. 60 !C, Si, SiC, Ge, GaAs, NaF, LiF, MgO" and Ref. 57 !LiCl", corrected to the T=0 limit using the thermal expansion fromRef. 58. An estimate of the zero-point anharmonic expansion has been subtracted out from the experimental values !cf. Table II". !Thecalculated values are precise to within 0.001 Å for the given basis sets, although GAUSSIAN GTO1 and GTO2 basis-set incompleteness limitsthe accuracy to 0.02 Å." GTO1: the basis set used in Ref. 38. GTO2: For C, Si, SiC, Ge, GaAs, and MgO, the basis sets were taken fromRef. 41. For the rest of the solids, the GTO1 basis sets and effective core potentials from Ref. 38 were used. The best theoretical values arein boldface. The LDA, PBEsol, and PBE GTO2 results are from Ref. 14. The SOGGA GTO1 results are from Ref. 15.

LDA LDA PBEsol PBEsol PBEsol AM05 SOGGA PBE PBE PBE TPSS

GTO2 VASP GTO2 BAND VASP VASP GTO1 GTO2 VASP BAND BAND

MEa !Å" −0.047 −0.055 0.022 0.010 0.012 0.029 0.009 0.075 0.066 0.063 0.048MAEb !Å" 0.050 0.050 0.030 0.023 0.023 0.036 0.024 0.076 0.069 0.067 0.052MREc !%" −1.07 −1.29 0.45 0.19 0.24 0.58 0.19 1.62 1.42 1.35 0.99MAREd !%" 1.10 1.15 0.67 0.52 0.52 0.80 0.50 1.65 1.48 1.45 1.10

CSONKA et al. PHYSICAL REVIEW B 79, 155107 !2009"

AM05&PBEsol

Meta-GGAsLattice constant

MRE MARE

LDA -1.73 1.73

PBE 1.10 1.29

PBEsol -0.24 0.73

AM05 0.19 0.75

TPSS 0.73 0.90

revTPSS 0.29 0.68

Atomization energy (solids)

MRE MARE

LDA 16.5 16.5

PBE -3.68 4.23

PBEsol 5.97 6.52

TPSS -1.99 4.70

revTPSS 1.22 5.73

Atomization energy (AE6 mol.)

MRE MARE

PBE 3.2 4.2

PBEsol 8.1 8.1

TPSS 1.3 2.4

revTPSS 1.3 2.8

Exc

=

Zdrn✏

xc

(n,rn, ⌧) ⌧ =X

i

1/2|r i|2

J.Sunetal.,Phys.Rev.B83,12140(R)(2011).

VanderWaals-DFT

Enlc =

Z Z⇢(r)�(r, r0)⇢(r0)drdr0

Hybridfunctionals FazitLattice constant

MRE MAREPBE 0.8 1.0PBE0 0.1 0.5HSE 0.2 0.5B3LYP 1.0 1.2

Bulk modulusMRE MARE

PBE -9.8 9.4PBE0 -1.2 5.7HSE -3.1 6.4B3LYP -10.2 11.4

Atomization energyMRE MARE

PBE -1.9 3.4PBE0 -6.5 7.4HSE -5.1 6.3B3LYP -17.6 17.6

0.5 1 2 4 8 16Experiment (eV)

0.25

0.5

1

2

4

8

16

The

ory

(eV

)

PBEHSE03PBE0

PbSePbS

PbTe Si

GaAs

Ne

AlP

CdS

SiC

ZnO

GaNZnS

LiFAr

C BN MgO

Figure 8. Band gaps from PBE, PBE0, and HSE03 calculations,plotted against data from experiment.

Ehyb.xc = aEHF

X + (1� a)EDFTX + EDFT

c

COadsorptionond-metalsurfaces(cont.I)

CO @ top fcc hcp �Cu(111) PBE 0.709 0.874 0.862 �0.165

PBE0 0.606 0.579 0.565 0.027HSE03 0.561 0.555 0.535 0.006exp. 0.46-0.52

Rh(111) PBE 1.870 1.906 1.969 �0.099PBE0 2.109 2.024 2.104 0.005HSE03 2.012 1.913 1.996 0.016exp. 1.43-1.65

Pt(111) PBE 1.659 1.816 1.750 �0.157PBE0 1.941 1.997 1.944 �0.056HSE03 1.793 1.862 1.808 �0.069exp. 1.43-1.71

COadsorptionond-metalsurfaces(cont.II)

Hybridfunctionals reducethetendencytostabilizeadsorptionatthehollowsitesw.r.t.thetopsite.

ReducedCO2𝜋∗ ⟷metal-d interaction

• ImproveddescriptionoftheCOLUMO(2𝜋∗)w.r.t.theFermilevel(shiftedupwards).

• Downshiftofthemetald-bandcenterofgravityinCu(111).• But:Overestimationofthemetald-bandwidth.

A.Stroppa,K.Termentzidis,J.Paier,G.Kresse,andJ.Hafner,Phys.Rev.B76,195440(2007).A.Stroppa andG.Kresse,NewJournalofPhysics10,063020(2008).

One-electronpicture

✓�1

2�+ Vext(r) + VH(r) + Vxc(r)

◆ nk(r) = ✏nk nk(r)

✓�1

2�+ Vext(r) + VH(r)

◆ nk(r) +

ZVX(r, r

0) nk(r0)dr0 = ✏nk nk(r)

✓�1

2�+ Vext(r) + VH(r)

◆ nk(r) +

Z⌃(r, r0, Enk) nk(r

0)dr0 = Enk nk(r)

DFT:Kohn-Shameq.

DFT-HFhybridfunctionals:Roothaan eq.

GW:quasi-particleeq.

TheGreen’sfunction

1 = (! �H)G () G�1 = (! �H)

G0(r, r0,!) =

X

n

n(r) ⇤n(r

0)

! � ✏n + i⌘ sgn(✏n � µ)

TheGreen’sfunctionisthe“inverse”oftheHamiltonian:

TheGreen’sfunctionofaKohn-Sham(non-interacting)Hamiltonianisgivenby:

andtheGreen’sfunctionofaninteractingHamiltonian:

(H0 � !) + ⌃(!) = (H � !)

�G�10 (!) + ⌃(!) = �G�1(!)

G�1(!) = G�10 (!)� ⌃(!) () G(!) = G0(!) +G0(!)⌃(!)G(!)

✓� ~22me

�+ Vion(r) + VH(r)

◆+ ⌃(r, r0,!) = H(!) =) H0 + ⌃(!) = H(!)

“Dyson-equation”

energy/frequencydependentHamiltonian

non-int.

TheGreen’sfunction:physicalinterpretation

• TheGreen’sfunction𝐺(𝐫, 𝐫′, 𝑡 − 𝑡′) describesthepropagationofaparticlefrom(𝐫, 𝑡) to 𝐫., 𝑡 : i.e.,providedwehaveparticleatpositionr attimet,𝐺(𝐫, 𝐫′, 𝑡 − 𝑡′)isthechanceoffindingitatpositionr’attimet’.

G0(r, r0,!) =

allX

n

⇤n(r) n(r0)

! � ✏n + i⌘ sgn(✏n � µ)

G0(1, 2) =vir.X

n

n(r1)⇤ n(r2)e

�i(✏n�µ)(t2�t1)

G0

(1, 2) =occ.X

n

⇤n(r1) n(r2)e

�i(✏n�µ)(t1�t2)

(r1, t1) = 1 (r2, t2) = 2

(r2, t2) = 2 (r1, t1) = 1

• Particlepropagator:𝐺1 1,2 = 𝐺1(𝐫4, 𝐫5, 𝑡5−𝑡4) for𝑡5 > 𝑡4:

• Holepropagator:𝐺1 1,2 for𝑡4 > 𝑡5:

Perturbationtheory:Σ asafunctionofG• Theselfenergy Σ,ismadeupofallFeynmandiagramswithonein- andone

out-goingpropagatorline:

1st.order 2nd.order 3rd.order 4th.order

• Andmany,many,many,more

Perturbationtheory:Σ asafunctionofG

Hartree

Exchange

⌫(2, 1)G0(1, 1) =

Z⌫(r2, r1)n(r1)dr1 =

Zn(r1)

|r1 � r2|dr1

G0

(1, 1) =occ.X

n

n(r1)⇤ n(r1)e

�i(✏n�µ)(t1�t1) = n(r1

)

G0

(1, 2) =occ.X

n

n(r1)⇤ n(r2)e

�i(✏n�µ)(t1�t1) = �(r1

, r2

)

�G0(1, 2)⌫(1, 2) = �⌫(r1, r2)�(r1, r2) =�(r1, r2)

|r1 � r2|

2

1

2

2

1

1

Thetwofirst-orderdiagramsrepresenttheHartree andexchangeinteraction:

Perturbationtheory:Σ asafunctionofG

• Somediagramsareeasiertocalculatethanothers.Random-Phase-Approximation(RPA):

1st.order 2nd.order 3rd.order 4th.order

+…

ThescreenedCoulombinteraction:W• ThesediagramscanbeexpressedasascreenedCoulombinteraction,W:

RPA

W = ✏�1⌫ ✏�1 = 1 + ⌫� � = �0 + �0⌫�

TheIP-polarizability:𝜒9The“irreduciblepolarizabilityintheindependentparticlepicture”𝜒9 (or𝜒:;):

AdlerandWiserderivedexpressionsfor𝜒9

�0(r1, t1, r2, t2) = �(1, 2) = �G0(1, 2)G0(2, 1)

OrintermsofGreen’sfunctions(propagators):

�0(r, r0,!) :=@⇢ind(r,!)

@ve↵(r0,!)

�0

(r1

, r2

,!) =occ.X

i

virt.X

a

h a|r1| iih i|r2| ai✏i � ✏a � !

+occ.X

i

virt.X

a

h i|r1| aih a|r2| ii✏a � ✏i � !

𝐫4, 𝑡4 𝐫5, 𝑡5

a

i

𝐺9(1,2)

AndintermsofBlockfunctions𝜒9 canbewrittenas

�0G,G0(q,!) =

1

⌦

X

nn0k

2wk(fn0k+q � fn0k)

⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘

TheIP-polarizability:𝜒9

W = ⌫ + ⌫�0⌫ + ⌫�0⌫�0⌫ + ⌫�0⌫�0⌫�0⌫ + ... = ⌫ (1� �0⌫)�1

| {z }✏�1

Oncewehave𝜒9 thescreenedCoulombinteraction(intheRPA)iscomputedas:

1.ThebareCoulombinteractionbetweentwoparticles

2.Theelectronicenvironmentreactstothefieldgeneratedbyaparticle:inducedchangeinthedensity𝜒9𝜐,thatgivesrisetoachangeintheHartreepotential:𝜐𝜒9𝜐.

3.Theelectronsreacttotheinducedchangeinthepotential:additionalchangeinthedensity,𝜒9𝜐𝜒9𝜐,andcorrespondingchangeintheHartree potential:𝜐𝜒9𝜐𝜒9𝜐.

andsoon,andsoon…

geometricalseries

Expensive:computingtheIP-polarizabilityscalesasN4

GW

✓�1

2�+ Vext(r) + VH(r)

◆ nk(r) +

Z⌃(r, r0, Enk) nk(r

0)dr0 = Enk nk(r)

Thequasi-particleequation:

⌃ = iGW

The“self-energyisgivenby:

ormoreexplicitlyGreen’sfunction:G

screenedCoulombinteraction:W

ComparetoFock-exchange:

VX

(r, r0) = �occ.X

n

n(r) ⇤n(r

0)⇥ e2

|r� r0|

⌃(r, r0, E) =i

2⇡

Z 1

�1d!

allX

n

n(r) ⇤n(r

0)

! � E � En + i⌘ sgn(En � Efermi)⇥

⇥ e2Z

dr00✏�1(r, r00,!)

|r00 � r0|

bareCoulombinteraction:v

AnanalogybetweenGWandhybridfunctionals

• 𝜖?4 𝐺 :StrongscreeningforsmallG(staticscreeningproperties).

NoscreeningatlargeG.

Screeningissystemdependent,obviously.

• Hybrids:¼isacompromise,thatworkswellforsmall-to-mediumgapsystems.

Spectralrepresentationof𝜒9

�SG,G0(q,!0) =

1

⌦

X

nn0k

2wk sgn(!0)�(!0 + ✏nk � ✏n0k�q)(fnk � fn0k�q)⇥

⇥h nk|ei(q+G)r| n0k�qih n0k�q|e�i(q+G)r| nki

�SG,G0(q,!0) =

1

⇡=⇥�0G,G0(q,!)

⇤

�0G,G0(q,!) =

Z 1

0d!0�S

G,G0(q,!0)⇥✓

1

! � !0 � i⌘� 1

! + !0 + i⌘

◆

Itischeapertocalculatethepolarizabilityinitsspectralrepresentation

whichisrelatedtotheimaginarypartof𝜒9 through

Thepolarizability𝜒9 isthenobtainedfromitsspectralrepresentationthroughthefollowingHilberttransform

LSPECTRAL=.TRUE.|.FALSE. NOMEGA=[integer](DefaultforALGO=CHI|GW0|GW|scGW0|scGW,whenNOMEGA>2)

SolvingtheGWQP-equation

EN+1nk = <

h nk|�

1

2�+ Vext + VH + ⌃(EN

nk)| nki�

+ (EN+1nk � EN

nk)<h nk|

@⌃(!)

@!

���!=EN

nk

| nki�

= ENnk + ZN

nk<h nk|�

1

2�+ Vext + VH + ⌃(EN

nk)| nki � ENnk

�

Enk = <h nk|�

1

2�+ Vext + VH + ⌃(Enk)| nki

�

✓�1

2�+ Vext(r) + VH(r)

◆ nk(r) +

Z⌃(r, r0, Enk) nk(r

0)dr0 = Enk nk(r)

ZNnk =

✓1� h nk|

@⌃(!)

@!

���!=EN

nk

| nki◆�1

Thequasi-particleenergiesaregivenby

whichmaybesolvedbyiteration

where

SingleshotGW:𝐺9𝑊9

• CalculateDFTorbitals:✓�1

2�+ Vext(r) + VH(r) + Vxc(r)

◆ nk(r) = ✏nk nk(r)

Enk = <h nk|�

1

2�+ Vext + VH + ⌃(✏nk)| nki

�

ZNnk =

✓1� h nk|

@⌃(!)

@!

���!=EN

nk

| nki◆�1

Enk = ✏nk + Znk<h nk|�

1

2�+ Vext + VH + ⌃(✏nk)| nki � ✏nk

�

• Compute𝐺1,𝑊1,andΣ = 𝐺9𝑊9 fromtheDFTorbitalsandeigenenergies.• Determinethefirst-orderchangeintheeigenenergies:

• Actuallytheexpressionaboveislinearizedandinsingle-shotGW(𝐺9𝑊9)weevaluate:

where

The𝐺𝑊9 approximation• CalculateDFTorbitals:

✓�1

2�+ Vext(r) + VH(r) + Vxc(r)

◆ nk(r) = ✏nk nk(r)

Enk = <h nk|�

1

2�+ Vext + VH + ⌃(✏nk)| nki

�

• Compute𝐺1,𝑊1,andΣ = 𝐺9𝑊9 fromtheDFTorbitalsandeigenenergies.• Determinethefirst-orderchangeintheeigenenergies:

• Andrecompute theGreen’sfunctionusingtheQP-energiesofthepreviousstep:

GN (r, r0,!) =X

n

n(r) ⇤n(r

0)

! � ENn + i⌘ sgn(✏n � µ)

Enk = <h nk|�

1

2�+ Vext + VH + ⌃(Enk)| nki

�• ComputeΣ = 𝐺𝑊9 and:

• Thismayberepeatedanumberoftimes

𝐺9𝑊9 and 𝐺𝑊9 flowchart

ISMEAR=0;SIGMA=0.05EDIFF=1E-8

NBANDS=50-200peratomALGO=ExactISMEAR=0;SIGMA=0.05LOPTICS=.TRUE.

NBANDS=50-200peratomALGO=GW0ISMEAR=0;SIGMA=0.05NELM=1 →G0W0NELM=4-10 →convergedGW0

DFTgroundstate

GW0

DFTvirtual orbitals

Testyoucould(andshouldtryto)do

• ENCUT Planewaveenergycutoff fororbitals

• NBANDS Totalnumberofbands:Try(ifpossible)tosetNBANDStothetotalnumberofplanewaves.

• NOMEGA Numberoffrequencypoints:Default:50isprettygood,althoughweoftenuse100Smallgapsystemsmightneedmorefreq.pointslittleperformancepenalty(requiresmorememory).Tryusing100-200justtotest.

• ENCUTGW Planewaveenergycutoff forresponsefunctions:Default:2/3ENCUTisprettygood.

G0W0(PBE)andGW0QP-gaps

G0W0:MARE8.5%GW0:MARE4.5%

M.Shishkin,G.Kresse,PRB75,235102(2007).

M.Shishkin,M.Marsman,PRL95,246403(2007)

A.Grüneis,G.Kresse,PRL 112,096401(2014)

Updatingtheorbitals:sc-QPGW

(T + V ) + ⌃(E) = E

(T + V ) +

⌃(E0) +

d⌃(E0)

dE0(E � E0)

� = E

⌃Herm = ES () S�1/2⌃HermS�1/2 0 = E 0

• ConstructaHermitianone-electronHamiltonianapproximationtoΣ 𝐸anddiagonalize thatapproximateHamiltonian:

Tosolveforthequasi-particleorbitalswefollowthemethodproposedbyFaleev,vanSchilfgaarde,andKotani,Phys.Rev.Lett.93,126406(2004):

(T + V ) +

⌃(E0)�

d⌃(E0)

dE0E0

� = E

1� d⌃(E0)

dE0

�

𝑠𝑐𝑄𝑃𝐺𝑊9 flowchart

ISMEAR=0;SIGMA=0.05EDIFF=1E-8

NBANDS=50-200peratomALGO=ExactISMEAR=0;SIGMA=0.05LOPTICS=.TRUE.

NBANDS=50-200peratomALGO=QPGW0ISMEAR=0;SIGMA=0.05NELM=5-10 →convergedQPGW0

DFTgroundstate

sc-QPGW0

DFTvirtual orbitals

UpdatetheorbitalsinG:scGW0

• LittleimprovementoverGW0

• Onaverageslightlytoolargegaps

M.Shishkin,M.Marsman,PRL95,246403(2007)

Fullyself-consistentGWM.Shishkin,M.Marsman,G.Kresse,

PRL95,246403(2007)

UpdateGandW:vanSchilfgaarde &KotaniPRL96,226402(2006)

• Wellthisisdis-appointing,isn’tit?worsethanGW0

• Staticdielectricconstantsarenowtoosmallby20%

• ThisisalimitationoftheRPA!

Screeningbad

Fullyself-consistentGW

e-hinteraction:Nano-quantakernel(L.Reining)

• Excellentresultsacrossallmaterials:MARE:3.5%

• FurtherslightimprovementoverGW0(PBE)

• Tooexpensiveforlargescaleapplications,butfundamentallyimportant

M.Shishkin,M.Marsman,G.Kresse,PRL95,246403(2007)

Fullyself-consistentGW(𝜖F)

Vertexcorrectionincludee-hinteraction

Scaling𝑁H − 𝑁I

WhatdoweneglectintheRPAAlot!Wehaveevenneglectedonesecondorderdiagram,the“secondorder”exchangeInthirdorder,excitonic effectsandmanymorediagramshavebeenneglected

1st.order 2nd.order 3rd.order 4th.order

+…

+…

exitons2nd orderexchange

Particle-Holeladderdiagram:

• Electrostaticinteractionbetweenelectronsandholes

• Excitonic effects

• VertexcorrectionsinW

• Importanttoremoveself-screening

Secondorderexchange:

• InGW,vertexinself-energy

• Nosimple“physical”interpretation(asforexchange)

• Importanttoremoveself-interaction

WhatdoweneglectintheRPA

FAZIT

UseG0W0orGW0 orpossiblysc-QPGW0ontopofPBE,ifPBEyieldsreasonablescreening.

PossiblytryG0W0ontopofHSE,ifPBEisnotreasonable,slightlytoolargebandgapsbecauseRPAscreeningontopofHSEisnotgreat.

Stronglylocalizedstatesmightbewrong(toohigh)!

GWisanapproximatemethod:• VertexinW:Neglectofe-hinteraction.

• VertexinΣ:Notself-interactionfreeforlocalizedelectronsInprinciplethisissolvable,butverytimeconsuming.

Thebestpracticalapproachesrightnow:

TheGWpotentials:*_GWPOTCARfiles

RPAtotalenergies(ACFDT)

E[n] = TKS [{ i}] + EH [n] + EX [{ i}] + Eion�el

[n] + Ec

�0G,G0(q,!) =

1

⌦

X

nn0k

2wk(fn0k+q � fn0k)

⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘

The“RPA”totalenergyisgivenby:

withtheRPAcorrelation

Themaineffortis(again)computingtheIPpolarizability:

Ec =X

q

Z 1

0

d!

2⇡Tr{ln[1� �0(q, i!)⌫] + �0(q, i!)⌫}

Expensive:computingtheIP-polarizabilityscalesasN4

RPA:latticeconstantsJ.Harl etal.,PRB81,115126(2010)

Deviationsw.r.t.experiment(correctedforzero-pointvibrations)

MRE MAREPBE 1.2 1.2HF 1.1 1.1MP2 0.2 0.4

RPA 0.5 0.4

(in %)

Schimka,etal.,Phys.Rev.B87,214102(2013).

RPA:TMlatticeconstants

RPA:TMlatticeconstants(NCpotentials)

NC-potentials:Klimeš,etal.,PRB90,075125(2014).

RPA:atomizationenergiesJ.Harl etal.,PRB81,115126(2010)

Atomisation energiesMAE (eV) MARE (%)

HF 1.65MP2 0.27

PBE 0.17 5LDA 0.74 18RPA 0.30 7

Graphitevs.Diamond

QMC(Galli)

RPA EXP

d(Å) 3.426 3.34 3.34

C33 36 36-40

E(meV) 56 48 43-50

1/d4 behavioratshortdistances

J.Harl,G.Kresse,PRL103,056401(2009).S.Lebeque,etal.,PRL105,196401(2010).

RPA:noblegassolidsJ.Harl andG.Kresse,PRB77,045136(2008)

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

304050100 25

corr

elat

ion

ener

gy(e

V)

volume (ų)

Neon

Argon

Krypton

20 30 40 50 60 70 80 90Volume [ ų]

-150

-100

-50

0

50

Coh

esiv

e En

ergy

[meV

]

LDAPBEACFDTMP2

Argon

C6 coe�cients

RPA(LDA) RPA(PBE) Exp.

Ne 62 53 47

Ar 512 484 455

Kr 1030 980 895

RPA:heatsofformationJ.Harl andG.Kresse,PRL103,056401(2009)

PBE Hartree-Fock

RPA EXP

LiF 570 664 609 621NaF 522 607 567 576NaCl 355 433 405 413MgO 516 587 577 603MgH2 52 113 72 78AlN 262 350 291 321SiC 51 69 64 69

Example:Mg(bulk metal)+H2→MgH2

Heatsofformationw.r.t.normalstateatambientconditions(inkJ/mol)

RPA:heatsofformationJ.Harl andG.Kresse,PRL103,056401(2009)

RPA:CO@Pt(111)andRh(111)Schimka etal.,NatureMaterials9,741(2010)

RPA:• increasessurfaceenergy

and• decreasesadsorptionenergy

Schimka etal.,NatureMaterials9,741(2010)

RPA:

• Rightsightpreference

• Goodadsorptionenergies

• Excellentlatticeconstants

• Goodsurfaceenergies

Schimka etal.,NatureMaterials9,741(2010)

• DFTdoeswellforthemetallicsurface,butnotfortheCO:2𝜋∗ (LUMO)tooclosetotheFermilevel.

• HSEdoeswellfortheCO,butnotforthesurface:d-metalbandwith toolarge.

• GWgivesagooddescriptionofboththemetallicsurfaceaswellasoftheCO2𝜋∗ (LUMO).TheCO5𝜎 and1𝜋 areslightlyunderbound.

RPAflowchart

DFTgroundstate

HFenergy

DFTvirtual orbitals

RPAenergy

EDIFF=1E-8ISMEAR=0;SIGMA=0.1

NBANDS=maximum#ofplanewavesALGO=Exact;NELM=1ISMEAR=0;SIGMA=0.1LOPTICS=.TRUE.

ALGO=Eigenval ;NELM=1LWAVE=.FALSE.LHFCALC=.TRUE.;AEXX=1.0ISMEAR=0;SIGMA=0.1

NBANDS=maximum#ofplanewavesALGO=ACFDTorACFDTRNOMEGA=12-16

Asalways:test,test,test,…

• k-pointconvergenceismoredifficulttoattainthanforDFT,inparticularformetals.

J.Harl,G.Kresse,PRB81,115126(2010).

• ENCUTGWcontrolsbasissetforresponsefunctions.

• IncreaseENCUTbyabout25-30%andrecalculateeverything(noteNBANDSneedstobeincreasedaswell).

• NumberoffrequencypointsNOMEGA,…

FAZIT

• Well-balancedtotalenergyexpressionthatcapturesalltypesofbonding(equally)well,i.e.,metallic,covalent,ionic,andvan-der-Waals.

• Chemicalaccuracy?Unfortunatelyno..butadefiniteimprovementoverhybridfunctionals andDFT.

Cubic-scalingRPAM.Kaltak,J.Klimes,andG.Kresse,PRB90,054115(2014)

Nowtheworstscalingstepis

whichscalesasN3 duetothediagonalizationinvolvedinevaluatingthe“ln”

EvaluatetheGreen'sfunctionin“imaginary”time:

andthepolarizabilityas:

Followedbyacosine-transform:

ButstoringGandχ isexpensive! →weneedsmallsetsofcleverlychosen“τ”and“ω”[seeKaltaketal.,JCTC10,2498(2014)]

Scaling Sidefectcalculations:64-216atoms

NewRPAcode(comingsoon):

• Scaleslinearlyinthenumberofk-points(asDFT),insteadofquadratically asforconventionalRPAandhybridfunctionals

• Scalescubicallyinsystemsize(asDFT).

Prefactors aremuchlargerthaninDFT,butcalculationsfor200atomstakelessthan1hour(128cores)

DefectformationenergiesinSiPBE HSE HSE(+vdW) QMC RPA

Dumbbell X 3.56 4.43 4.41 4.4(1) 4.28

HollowH 3.62 4.49 4.40 4.7(1) 4.44

TetragonalT 3.79 4.74 4.51 5.1(1) 4.93

Vacancy 3.65 4.19 4.38 4.40

picturesandHSE+vdW:Gao,Tkatchenko,PRL111,45501

QMC:Parker,Wilkins,Hennig,Phys.StatusSolidiB248,267(2011).

Cubic-ScalingGW

TheEnd

Thankyou!