GW : AUTOMATION, PRECISION AND ACCURACY - YAMBO …...IMEC World-leading R&D and innovation hub in...

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GW : AUTOMATION, PRECISION AND ACCURACYMICHIEL VAN SETTEN

IMEC

▪ World-leading R&D and innovation hub in nanoelectronics and digital technologies.

▪ 4,000 researchers globally, 97 nationalities

▪ HQ in Leuven Belgium, 11 affiliated research groups at Flemish universities5 other sites in Europe, 3 in the USA and 5 in Asia.

▪ Modeling from ab initio to device level.

▪ Ab initio core group (4 permanent, 6 PhDs, 2 industrial residents)

▪ Around a dozen researchers in other groups involved in ab initio: quantum transport, qbits, materials science

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IMEC RESEARCH

▪ CMOS and beyond CMOS

▪ Image sensors and vision systems

▪ Silicon photonics

▪ Wearables

▪ Photovoltaics

▪ GaN

▪ Sensor solutions for IoT

▪ Wireless IoT communication

▪ Radar sensing systems

▪ Solid state batteries

▪ Data science and data security

▪ Large-Area Electronics

▪ Life sciences

▪ Artificial intelligence

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▪ Most advanced 300 mm (300 mm diameter wafers)cleanroom in the world

▪ Raman Spectroscopy and the related technique of Photo Luminescence (PL)

▪ Rutherford Backscattering (RBS) and related techniques of Elastic Recoil Detection (ERD) and Proton Induced X-ray Emission (PIXE)

▪ Scanning Probe Microscopy (SPM) inclusive of Atomic Force Microscopy (AFM), various electrical and physical variants thereof, as well as Scanning Tunneling Microscopy (STM)

▪ Secondary Ion Mass Spectrometry (SIMS)

▪ Transmission Electron Microscopy (TEM)

▪ X-ray Photoelectron Spectroscopy (XPS)

▪ AttoLab rt-Spectroscopies 20 as – 200 ps pump probe, up to 124 eV beam

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COMPUTATIONAL APPROACHES

▪ Density functional theory▪ Maps the many particle problem into a single particle problem

▪ Scalar effective potential Vxc(r)

▪ Applicable to large systems: > 2000 atoms

▪ Broadly used across many fields of sciences

▪ Description of electronic levels (charge transfer), long range interactions, no proper fundamental band gaps

▪ GW-method▪ Replace the potential by a self-energy matrix: Vxc(r) → Σ(r,r’,E)

▪ Available for bulk solids

▪ Becoming increasingly available for molecules and nano structures

▪ Significantly improved description of electronic levels (charge transfer), long range interactions, and band gaps

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HISTORICAL PERSPECTIVE

1965 New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas ProblemL. Hedin, Phys. Rev. 139, A796

1979 Many-Particle Effects in the Optical Excitations of a Semiconductor.

W. Hanke and L.J. Sham, Phys. Rev. Lett. 43, 387−390.

1985 First-Principles Theory of Quasiparticles: Calculation of Band Gaps in Semiconductors and Insulators (silicon and diamond)M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 1418

2005 A brief introduction to the ABINIT software packageX. Gonze et all. Zeit. Kristallogr. 220, 558-562

2006 Implementation and performance of the frequency-dependent GW method within the PAW framework (VASP)M. Shishkin and G. Kresse Phys. Rev. B 74, 035101

2006 Optical excitations in organic molecules, clusters, and defects studied by first-principles Green’s function methodsMurilo L. Tiago and James R. Chelikowsky Phys. Rev. B 73, 205334

2014 Predictive GW calculations using plane waves and pseudopotentialsJiří Klimeš, Merzuk Kaltak, and Georg Kresse Phys. Rev. B 90, 075125

2015 GW100: Benchmarking G0W0 for Molecular SystemsM. J. van Setten et all. J. Chem. Theory Comput., 2015, 11 (12), pp 5665–5687

2020 Reproducibility in G0W0 Calculations for Solids, 3 codes a few materialsRangel et all. in press

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GW FOR SOLIDS: BANDGAPS

VAN SCHILFGAARDE PRL 96, 226402 (2006) HEDIN J PHYS, COND MAT. 11 R489 (SHIRLEY) (1995)

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SCGW FOR SOLIDS: BANDGAPS

KRESSE PRB 75, 235102 (2007) VAN SCHILFGAARDE PRL 96, 226402 (2006)

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HOW PRECISE ARE THE RESULTS?

HOW ACCURATE IS GW IN PRACTICE?

IS THERE ANY HOPE TO TURN IT IN AN AUTOMATIC PROCEDURE?

Should we do G0W0?

Which starting point?

What level of self consistency?

Which GW integration method?

What do we compare to?

What basis type can we trust?

Which core valence partitioning?

Pseudopotenials?

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DO WE CARE TOO LITTLE?

Only recently DFT was truly

Benchmarked

For just elemental systems

For just a few ground state

properties

Maybe it is just too hard a problem….

FORMALISM

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KS V.S. QUASI-PARTICLE EQUATION

▪ Electron density parameterized by KS single particle states

▪ Greens function in spectral representation, expressed in terms of quasi-particle states leads to the quasi particle equation:

▪ The quasi-particle energies are the electron removal and addition energies

▪ 0th order correction to the KS energies:

)()()()()(2

1

)()()(

KSKSKS

XCextH

2

occ

*KSKS

rrrrr

rrr

nnn

n

nn

VVV

=

+++−

=

)()();,()()()(2

1 qp

,

qpqp

,

qpqp

,extH

2rrrrrrrr nrnnrnnr dVV =+

++−

G(r, ¢r ;z) =Yr,n

qp (r, z)Y l,n

qp†( ¢r , z)

z-enqp(z)+ ihsign(en

qp(z)-m)p

å

enG0W0 =en

KS + yn

KS S(enG0W0 )-VXC yn

KS

Vxc: exchange correlation potential

Σ: self-energy

HEDIN Phys Rev 139, A796, HYBERTSEN and LOUIE PRB 34, 539011

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HEDIN EQUATIONS

▪ Space time notation (the numbers indicate a contracted space, time and spin index)

+

+

−=

+=

+−−=

+=

=

)1,4()2,4,3()3,1()34()2,1(

)2,4()4,3()3,1()34()2,1()2,1(

)3,7,6()5,7()6,4()5,4(

)2,1()4567()32()21()3,2,1(

)2,4()4,3()3,1()34()2,1()2,1(

)4,3,2()4,1()3,1()34()2,1(

00

GGdiP

WPvdvW

GGG

d

GGdGG

WGdi

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HEDIN EQUATIONS

▪ Space time notation (the numbers indicate a contracted space, time and spin index)

▪ Neglecting the second term in the vertex function leads to the GW approximation for

the self-energy; Fourier transformed to frequency domain:

)1,2()2,1()2,1(

)2,4()4,3()3,1()34()2,1()2,1(

)2,4()4,3()3,1()34()2,1()2,1(

)2,1()2,1()2,1(

00

GGiP

WPvdvW

GGdGG

WGi

−=

+=

+=

=

+

dWEGei

E i )()(2

)( 0 −= +−

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SELF-ENERGY METHODS

▪ Exact Analytic▪ Solve Casida equations > spectral representation of the response function

▪ Analytic evaluation of the GW integral

▪ Self-energy expression as an explicit sum over poles

▪ Analytic continuation of the self energy▪ Calculate the GW integral on the imaginary axis

▪ Fit an n-pole expansion for Sigma in the entire complex plane (e.g. Pade)

▪ Plasmon pole▪ approximate the response in a single pole

▪ analytic evaluation of the GW convolution

▪ Multipole response expansion▪ approximate the response in multiple poles

▪ analytic evaluation of the GW convolution

▪ Real energy direct numerical integration▪ brute force numerical integration on the real axis

▪ Contour deformation▪ replace the integral on the real axis with an integral on the imaginary axis and a sum over a finite number of include poles

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dWEGei

E i )()(2

)( 0 −= +−

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FULL ANALYTIC

▪ Calculate response in spectral representation

▪ Close connection to TDDFT (actually TDH)

▪ Analytic expression of Sigma as a sum over

poles of G and W

▪ Calculate Sigma analytically

▪ Numerically exact except for finite basis

▪ Full analytic structure of Sigma

▪ Expensive

Re n Sc(en ) n( ) =

in rm( )2 en -ei +Wm

en -ei +Wm( )2+h2

i

occ

å

+ an rm( )2

a

unocc

åen -ea -Wm

en -ea -Wm( )2+h2

æ

è

ççççç

ö

ø

÷÷÷÷÷

m

å

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ANALYTIC CONTINUATION

▪ Calculate the GW integral on the imaginary axis

▪ Fit an n-pole expansion for Sigma in the entire complex plane

▪ Allows for a more detailed sigma

▪ Can be made exact

▪ May take many poles to converge

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PLASMON POLE

▪ Approximate the response by one single pole

▪ Calculate response at a few imaginary frequencies

▪ Analytic continuation to the real axis

▪ Calculate Sigma analytically

▪ Cheap

▪ Many different approaches

▪ Justification for complex systems

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CONTOUR DEFORMATION

▪ Calculate the GW integral on the imaginary axis

▪ Add pole contributions of G in the contour

▪ Allows for a more detailed sigma

▪ Can be made exact

▪ Integration grid parameters

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BENCHMARKING GW

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BENCHMARK SET FOR MOLECULES: GW100

▪ Original Collaboration: KIT Karlsruhe, FHI Berlin, and Berkeley Lab US DoE

▪ TURBOMOLE: Gaussian basis sets, spectral representation via Casida

▪ FHI-Aims: numerical local orbitals, analytic continuation

▪ BerkeleyGW: plane waves, plasmon pole and real frequency integration

▪ 5 different ways to evaluate the self-energy

▪ well converged all electron reference values for IP and EA

HOMO level data sets comparison

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BENCHMARK SET FOR MOLECULES: GW100

▪ Original Collaboration: KIT Karlsruhe, FHI Berlin, and Berkeley Lab US DoE

▪ TURBOMOLE: Gaussian basis sets, spectral representation via Casida

▪ FHI-Aims: numerical local orbitals, analytic continuation

▪ BerkeleyGW: plane waves, plasmon pole and real frequency integration

▪ 5 different ways to evaluate the self-energy

▪ well converged all electron reference values for IP and EA

▪ Follow ups

▪ CCSD(T) total energy reference, approximate GW (Klopper)

▪ Plane wave results by VASP (Kresse) and WEST (Galli)

▪ CP2k results (testing the O(3) GW implementation)

▪ (partial) Stochastic GW results (testing O(1) implementation)

▪ MolGW, Fiesta, Abinit, Yambo

▪ Evaluation of 6 types of (partial) self-consistency

▪ Several non-GW projects picking up GW100 as a test set

LDA ½ and Koopmans compliant functionals

▪ 2 more sets of CCSD on the way

HOMO level 48 data sets comparison

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GW100 TODAY

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GW100.WORDPRESS.COM

▪ online database with all contributed data sets with python tools on git

▪ unified data format

▪ web interface with interactive analysis tools

▪ two sets in full detail

▪ any combination of sets comparedto one reference

▪ all against all

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MOLECULES IN GW100

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He

Ne

Ar

Kr

Xe

C7H8

C8H10

C6F6

C6H5OH

C6H5NH2

C5H5N

C5H5N5O

C5H5N5O

C4H5N3O

C5H6N2O2

C4H4N2O2

CH4N2O

H2

Li2Na2

Na4

Na6

K2

Rb2

N2

P2

As2

F2

Cl2Br2

I2CH4

C2H6

C3H8

C4H10

C2H4

C2H2

C4

C3H6

C6H6

C8H8

C5H6

C2H3F

C2H3Cl

C2H3Br

C2H3I

CF4

CCl4CBr4

CI4

SiH4

GeH4

Si2H6

Si5H12

LiH

KH

BH3

B2H6

NH3

HN3

PH3

AsH3

SH2

FH

ClH

Ag2

Cu2

NCCu

LiF

F2Mg

TiF4

AlF3

BF

SF4

BrK

GaCl

NaCl

MgCl2AlI3

BN

NCH

PN

H2NNH2

H2CO

CH4O

C2H6O

C2H4O

C4H10O

CH2O2

HOOH

BeO

MgO

H2O

CO2

CS2

OCS

OCSe

CO

O3

SO2

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ELEMENTS IN GW100

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VIOLIN AND BOX PLOTS

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some quantity

regular box plot

median value

25% edge

25% edge

violin plot (kernel density)

whisker

beyond which lie statistical outlyers

REPRODUCIBILITY

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If I do the ‘exact’ same thing with two codes, do I get the same answer?

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FIRST SANITY CHECK

▪ MOLGW compared to TURBOMOLE

▪ same basis (def2-QZVP)

▪ same fully analytic GW approach

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TWO COMPLETELY DIFFERENT CODES BUT THE SAME APPROACH

BeO

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ALL QZVP RESULTS FROM LOCAL ORBITAL CODES

▪ G0W0@PBE

▪ Same local orbital basis set

▪ Most converged self energy

parameters

▪ (compared to full analytic)

▪ the box plot in the center collapses

for all comparisons:

▪ for more than 50% the deviations is

meV small

▪ we do have outliers

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THE PADE APPROXIMATION

▪ Analytic continuation of the self-energy

▪ QZVP

▪ G0W0@PBE

▪ Pade for the Homo does converge

▪ ‘automatic’ solving of the QPE can pick up wrong intersections.

▪ new AC method in FIESTA outperforms

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SOLVING THE QUASIPARTICLE EQUATION

▪ multiple intersections

▪ a solver may overshoot in the first

step

▪ most problematic in strongly polar

molecules

▪ check for slope

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enG0W0 =en

KS + yn

KS S(enG0W0 )-VXC yn

KS

PRECISION

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How big is the error in my final result due to numerical / mathematical approximations?

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BASIS SET EXTRAPOLATION

▪ Dunning basis sets and def2 basis sets converge similar.

▪ def2 are contracted gaussians, i.e., less actual functions

▪ Number of basis functions extrapolation and cardinal number extrapolation are similar

▪ Checked for all molecules individually

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SVP

TZVP

QZVP

T

Q

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CONVERGENCE WITH BASIS SET SIZE

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▪ Turbomole

▪ Cbas (auxiliary basis set)

▪ G0W0@PBE

▪ triple zeta is usually sufficient

for ground state

▪ quadruple still 0.1 eV of on average

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TYPE OF BASIS

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PLANE WAVE V.S. LOCAL ORBITALS

extrapolated PW FF v.s. extrapolated local basis set

PW numerical FF

▪ If done correctly and carefully

converged PW and Local

basis do lead to the same

results

PW CD

PW PP

WEST

BGW

BGW

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CD AND PLASMON POLE

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▪ If done correctly and carefully converged PW and Local basis do lead to the same results

▪ CD leads to the same as fully analytic, results could be under converged

▪ Numerical FF also ok, but again signs of under converged results

▪ (HL )Plasmon Pole is not able to describe the complexity of the molecule response function

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MOST CONVERGED G0W0@PBE PER CODE

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extrapolated

planewave nc (partial set)

gaussian + plane wave

plane wave nc

plane wave paw

Fiesta datasets are close to completion

Abinit and StochasticGW: only partial sets but decent agreement.

BGW numerical full frequency: reasonable agreement (under converged due to too heavy numerics)

local basis set

local basis set

realspace nc (partial set) not

extrapolated

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CONCLUDING NUMBERS FOR THE HOMO

▪ Self-energy

▪ 12 pole Pade approximation within 0.05 eV

▪ 128 plot Pade approximation within 0.01 eV

▪ CD and other FF methods converge to within 0.1 eV

▪ Plasmon pole model mean 0.5 (up to eV) error

▪ Basis

▪ TZVP mean 0.3 (up to 0.6 eV) error

▪ QZVP mean 0.1 (up to 0.3 eV) error

▪ auxiliary basis mean 0.04 (up to 0.2 eV) error

▪ in the converged limit extrapolated local basis and extrapolated pw align, 68% within 0.1 eV

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ACCURACY

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How good do my results represent experimental results or results from a higher level of theory?

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G0W0 STARTING POINTS

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100%

50%

25%

0%

CCSD(T) reference

monotonic

trend following

the amount

of exact exchange

in the starting point

CCSD(T) reference

def2-TZVPP

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STARTING POINT AND LEVELS OF SELF-CONSISTENCY

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qsGW

scGW

G0W0@BH-LYP

CCSD(T) reference

def2-TZVPP

partial self consistency

reduces starting point

dependence

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LEVELS OF SELF-CONSISTENCY

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the most self-consistent

is not the best match

CCSD(T) reference

def2-TZVPP

cheapest way to a close

match: 50% EX starting point

the most self-consistent

is not the best match

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NON-GW DEVELOPMENT IS ALSO STARTING TO USE GW100

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TAKE HOME MESSAGES ACCURACY

▪ Technical implementation details are under control

▪ Calculations with different types of basis sets converge to the same results

▪ G0W0 shifts ~ 1 eV starting from HF to PBE

▪ ~ 50% Exact exchange optimizes agreement with CCSD(T)

▪ qsGW underestimates

▪ scGW overestimates

▪ Partial self-consistency in G reduces starting point dependence by half

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Michiel J. van Setten, Fabio Caruso, Patrick Rinke,

Ferdinand Evers, Florian Weigend, Jeffrey B. Neaton,

Sahar Sharifzadeh, Xinguo Ren, Matthias Scheffler,

Fang Liu, Johannes Lischner, Lin Lin, Jack R. Deslippe,

Steven G. Louie, Chao Yang, Katharina Krause,

Michael E. Harding, Wim Klopper, Matthias Dauth,

Emanuele Maggio, Peitao Liu, Georg Kresse, Marco

Govoni, Giulia Galli, Jan Wilhelm, Jürg Hutter,

Christof Holzer, Rodrigues Pela, Andris Gulans,

Claudia Draxl, Nicola Colonna, Ngoc Linh Nguyen,

Andrea Ferretti, Nicola Marzari, Xavier Blase, Fabien

Bruneval

CONTRIBUTORS GW100

gw100.wordpress.com

github.com/setten/GW100

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AUTOMATION FOR SOLIDS

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HIGH-THROUGHPUT GW

from a structure

without any human intervention

to converged GW results

▪ Automatic calculations

▪ Screening for new compounds

▪ Database building

▪ Uniform results

▪ No human bias

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HIGH-THROUGHPUT GW

▪ additional difficulties as compared to DFT

▪ Pseudo potentials

▪ 4 step calculation

▪ N4 (at best N3) scaling

▪ More convergence parameters

▪ No ‘safe’ parameter set (converged results for all)

▪ No ‘safe’ computational settings (# cpu’s, memory, time, ...)

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BN

THE CONVERGENCE PROBLEM

▪ Coupled parameters:

▪ converging one at a low value of the

other and vice versa will lead to

under converged results

▪ Many systems converge similar, but

not all

▪ no safe works for all short cuts

▪ No way to perform a overcovered

calculations to obtain a reference

▪ System by system convergence testing

is needed

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THE MEMORY PROBLEM

GW tasks that finish at a given memory limit

▪ Many calculations will finish with

reasonable amount of computational

resources.

▪ During a convergence test, some,

however, will not.

▪ In automatic mode, needed for high

throughput, this needs to be fixed

automatically, i.e., no human

intervention.

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THE PSEUDO PROBLEMAuCl

Normal 19 electron Au potential

4f unfrozen Au potential

▪ DFT only cares about occupied states

▪ GW also cares about unoccupied

states

▪ also ‘no occupied states’ GW

methods care about the quality of the

Hilbert space

▪ occupied states needed down to

about 40 eV below EF

▪ And again: system dependend

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HIGH-THROUGHPUT GW








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HIGH-THROUGHPUT GW








DFT babysitting

problem

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HIGH-THROUGHPUT GW








GW babysitting

problem

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HIGH-THROUGHPUT GW

▪ additional difficulties







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3 XC-functionals / 3 formats (codes) / 70.000 test per table

Don Hamann PRB 88, 085117 (2013)

Computer Physics Communications 226, 39-54 (2018)

PSEUDO DOJO

▪ Optimized Norm Conserving Vanderbild Pseudo Potentials

▪ multiple projectors

▪ multiple gradient constraints

▪ Package to facilitate PSP

generation

▪ Graphical interface

▪ Automatized testing (6) www.pseudo-dojo.org

Log derivatives made to agree up to 8 Ha

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Flow generation:

GUI or scripted

based Python package

• Workflow management

• ABINIT I/O

• Post processing

AUTOMATING GW

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Scheduling the flow on the cluster

Error handling (‘custodian like’)

Convergence testing

SLURM

PBSPro

Torque

SGE

…

AUTOMATING GW

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Storing results:

query-able entries

+ netcdf data files

(via gridfs)

Interactive analysis

and presentation

of

results

AUTOMATING GW

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AUTOMATIC GW FLOW

▪ On a low k-point density (2x2x2):

▪ Set of single parameter ground state convergence studies

▪ Grid of nbands X encuteps

▪ For each nbands extrapolate in encuteps

▪ For the extrapolated encuteps find the converged nbands

▪ if not found > extend grid and retest

▪ On the final high k-point density:

▪ Test derivatives.

(in most cases the high density derivatives turn out smaller)

▪ Post process

(create scissor, apply scissor, plots, statistical analysis…)

(pw cutoff, …)

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CONVERGENCE STUDY










▪ Post process


(pw cutoff, …)

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CONVERGENCE STUDY










▪ Post process


(pw cutoff, …)

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Gold

2x2x2 kpoint mesh

14x14x14 kpoint mesh

0.35 eV

0.35 eV

shift in energy with k-grid

shape is maintained

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AUTOMATIC GW FLOW










▪ Post process


(pw cutoff, …)

WHAT CAN WE CONCLUDE FOR SOLIDS

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KS V.S. GW GAPS

▪ Close correlation between PBE and

G0W0@PBE gaps.

▪ no clear distinction between

compounds with transition metals

(TM) and those without (noTM)

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G0W0GN@PBE

▪ GW gaps should not reproduce

experiment:

▪ no ZPE renormalization

▪ no SO effects

▪ no finite temperature correction

▪ The 0K electronically exact theory

should overestimate the experimental

gaps, the blue lines indicate an

estimate.

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ERROR ANALYSIS

▪ remapping to error v.s. experimental

full values

▪ distinction between TM and noTM

▪ very light elements cause under

estimation (zpe renormalization)

▪ heavy elements, not so clear

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COMPARING TO EMPIRICAL MODELING

▪ Can we remove the linear component

of the error in PBE?

▪ to a large level we can

▪ but G0W0@PBE still slightly

outperforms corrected PBE

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CONCLUDING

▪ Can we automate GW sufficiently to go high-throughput?

▪ It is much harder than for DFT

▪ no safe computational settings: #CPUs, memory, etc. :

▪ no safe parameter sets: either noting will finish or noting will be converged

▪ But if we have

▪ automated input generation

▪ automated output processing

▪ automated job generation, submission, monitoring

▪ failure detection

▪ algorithms and rules to recover from problems

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GW : AUTOMATION, PRECISION AND ACCURACY - YAMBO …...IMEC World-leading R&D and innovation hub in...

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