vasp: beyond dft the random phase approximation beyond dft the random phase approximation university...
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VASP:beyondDFTTheRandomPhaseApproximation
UniversityofVienna,FacultyofPhysicsandCenterforComputationalMaterialsScience,
Vienna,Austria
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PAST,PRESENT,FUTURE
● PAST
● TheWorkhorse:DFT
● Efficientandstablealgorithms
● PAWdatasets
● PRESENT
● BeyondDFT,andbeyondthegroundstate:Hybridfunctionals,linearresponse,RPA(GW&ACFTD),BSE
● FUTURE
● Nearfuture:cubic-scaling-RPA
● …
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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 %)
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Catalysis:dehydrogenationofpropaneinMordenite
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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
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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
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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).
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VanderWaals-DFT
Enlc =
Z Z⇢(r)�(r, r0)⇢(r0)drdr0
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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
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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
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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).
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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.
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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.
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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:
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Perturbationtheory:Σ asafunctionofG• Theselfenergy Σ,ismadeupofallFeynmandiagramswithonein- andone
out-goingpropagatorline:
1st.order 2nd.order 3rd.order 4th.order
• Andmany,many,many,more
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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:
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Perturbationtheory:Σ asafunctionofG
• Somediagramsareeasiertocalculatethanothers.Random-Phase-Approximation(RPA):
1st.order 2nd.order 3rd.order 4th.order
+…
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ThescreenedCoulombinteraction:W• ThesediagramscanbeexpressedasascreenedCoulombinteraction,W:
RPA
W = ✏�1⌫ ✏�1 = 1 + ⌫� � = �0 + �0⌫�
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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)
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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
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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
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AnanalogybetweenGWandhybridfunctionals
• 𝜖?4 𝐺 :StrongscreeningforsmallG(staticscreeningproperties).
NoscreeningatlargeG.
Screeningissystemdependent,obviously.
• Hybrids:¼isacompromise,thatworkswellforsmall-to-mediumgapsystems.
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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)
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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
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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
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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
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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
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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.
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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)
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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
�
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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
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UpdatetheorbitalsinG:scGW0
• LittleimprovementoverGW0
• Onaverageslightlytoolargegaps
M.Shishkin,M.Marsman,PRL95,246403(2007)
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Fullyself-consistentGWM.Shishkin,M.Marsman,G.Kresse,
PRL95,246403(2007)
UpdateGandW:vanSchilfgaarde &KotaniPRL96,226402(2006)
• Wellthisisdis-appointing,isn’tit?worsethanGW0
• Staticdielectricconstantsarenowtoosmallby20%
• ThisisalimitationoftheRPA!
Screeningbad
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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)
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Fullyself-consistentGW(𝜖F)
Vertexcorrectionincludee-hinteraction
Scaling𝑁H − 𝑁I
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WhatdoweneglectintheRPAAlot!Wehaveevenneglectedonesecondorderdiagram,the“secondorder”exchangeInthirdorder,excitonic effectsandmanymorediagramshavebeenneglected
1st.order 2nd.order 3rd.order 4th.order
+…
+…
exitons2nd orderexchange
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Particle-Holeladderdiagram:
• Electrostaticinteractionbetweenelectronsandholes
• Excitonic effects
• VertexcorrectionsinW
• Importanttoremoveself-screening
Secondorderexchange:
• InGW,vertexinself-energy
• Nosimple“physical”interpretation(asforexchange)
• Importanttoremoveself-interaction
WhatdoweneglectintheRPA
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FAZIT
UseG0W0orGW0 orpossiblysc-QPGW0ontopofPBE,ifPBEyieldsreasonablescreening.
PossiblytryG0W0ontopofHSE,ifPBEisnotreasonable,slightlytoolargebandgapsbecauseRPAscreeningontopofHSEisnotgreat.
Stronglylocalizedstatesmightbewrong(toohigh)!
GWisanapproximatemethod:• VertexinW:Neglectofe-hinteraction.
• VertexinΣ:Notself-interactionfreeforlocalizedelectronsInprinciplethisissolvable,butverytimeconsuming.
Thebestpracticalapproachesrightnow:
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TheGWpotentials:*_GWPOTCARfiles
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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
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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 %)
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Schimka,etal.,Phys.Rev.B87,214102(2013).
RPA:TMlatticeconstants
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RPA:TMlatticeconstants(NCpotentials)
NC-potentials:Klimeš,etal.,PRB90,075125(2014).
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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
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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).
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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
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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)
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RPA:heatsofformationJ.Harl andG.Kresse,PRL103,056401(2009)
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RPA:CO@Pt(111)andRh(111)Schimka etal.,NatureMaterials9,741(2010)
RPA:• increasessurfaceenergy
and• decreasesadsorptionenergy
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Schimka etal.,NatureMaterials9,741(2010)
RPA:
• Rightsightpreference
• Goodadsorptionenergies
• Excellentlatticeconstants
• Goodsurfaceenergies
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Schimka etal.,NatureMaterials9,741(2010)
• DFTdoeswellforthemetallicsurface,butnotfortheCO:2𝜋∗ (LUMO)tooclosetotheFermilevel.
• HSEdoeswellfortheCO,butnotforthesurface:d-metalbandwith toolarge.
• GWgivesagooddescriptionofboththemetallicsurfaceaswellasoftheCO2𝜋∗ (LUMO).TheCO5𝜎 and1𝜋 areslightlyunderbound.
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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
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Asalways:test,test,test,…
• k-pointconvergenceismoredifficulttoattainthanforDFT,inparticularformetals.
J.Harl,G.Kresse,PRB81,115126(2010).
• ENCUTGWcontrolsbasissetforresponsefunctions.
• IncreaseENCUTbyabout25-30%andrecalculateeverything(noteNBANDSneedstobeincreasedaswell).
• NumberoffrequencypointsNOMEGA,…
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FAZIT
• Well-balancedtotalenergyexpressionthatcapturesalltypesofbonding(equally)well,i.e.,metallic,covalent,ionic,andvan-der-Waals.
• Chemicalaccuracy?Unfortunatelyno..butadefiniteimprovementoverhybridfunctionals andDFT.
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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)]
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Scaling Sidefectcalculations:64-216atoms
NewRPAcode(comingsoon):
• Scaleslinearlyinthenumberofk-points(asDFT),insteadofquadratically asforconventionalRPAandhybridfunctionals
• Scalescubicallyinsystemsize(asDFT).
Prefactors aremuchlargerthaninDFT,butcalculationsfor200atomstakelessthan1hour(128cores)
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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).
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Cubic-ScalingGW
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TheEnd
Thankyou!