Upload File

more basics of dftprimer in density functional theory, edited by c. fiolhais et al....

  • Home
  • Documents
  • More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,
31
More basics of DFT Kieron Burke and friends UC Irvine Physics and Chemistry 1 APS tutorial

Upload: others

Post on 28-Jun-2020

9 views

Category:

Documents


0 download

Report

TRANSCRIPT

Page 1: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

More basics of DFT

KieronBurkeandfriendsUCIrvinePhysicsandChemistry

1APStutorial

Page 2: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

References for ground-state DFT

– ABCofDFT,byKBandRudyMagyar,http://dft.uci.edu

– APrimerinDensityFunctionalTheory,editedbyC.Fiolhaisetal.(Springer‐Verlag,NY,2003)

– DensityFunctionalTheory,DreizlerandGross,(Springer‐Verlag,Berlin,1990)

– DensityFunctionalTheoryofAtomsandMolecules,ParrandYang(Oxford,NewYork,1989)

– AChemist’sGuidetoDensityFunctionalTheory,KochandHolthausen(Wiley‐VCH,Weinheim,2000)

2APStutorial

Page 3: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ll cover

•  Simplestpossibleexampleofafunctional•  EssentialsofKS‐DFT,andfunctionalzoo•  Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity

•  Exactexchange•  Quiz

3APStutorial

Page 4: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

4

Atomic units and particles in box •  Inatomicunits,allenergiesareinHartree(1H=

27.2eV)andalldistancesinBohr(1a0=0.529Å)

•  Towriteformulasinatomicunits,sete2=Ћ=me=1•  E.g.,usualformulaforenergylevelsofinfinitewell

ofwidthL:

•  Atomicunits,boxlengthL=1:

APStutorial

Page 5: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

Constructing your very first density functional

•  Let’slookatthekineticenergyofspinlessfermionsin1d:

•  IstheresomewaytogetTswithoutevaluatingallthosedamnorbitals?Yes!

•  Writeitasadensityfunctional,i.e.,anintegraloversomefunctionofn(x).

•  Simplestchoice:alocalapprox:

5APStutorial

Page 6: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

Particles in box

•  Accuracy

APStutorial 6

N Ts[0] Ts %err

1 4.112 4.934 -17

2 21.79 24.67 -12

3 62.92 69.09 -9

Page 7: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ve learned

•  Densityfunctionalsareapproximationsfortheenergyofmanyparticles

• WorkbestforlargeN,worstforsmallN

•  Localapproximationsarecrudelycorrect,butmissdetails

APStutorial 7

Page 8: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

Essence of Kohn-Sham DFT •  EvenwithexactExc[n],onlygetE0andn(r)(andI).Sootherpropertiesmaynotberight.

•  Resultsonlyasgoodasfunctionalused.•  VastamountofinformationfromE0alone,suchasgeometries,vibrations,bondenergies…

•  Well‐fittedfunctionalsareaccurateforlimitedset

•  Non‐empiricalfunctionalslessso,butmorereliableforabroaderrange,anderrorsunderstandable

APStutorial 8

Page 9: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

He atom in Kohn-Sham DFT

Dashed-line:

EXACT KS potential

Everything has (at most) one KS potential

9APStutorial

Page 10: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

10

Functionals in common use •  Localdensityapproximation(LDA)– Usesonlyn(r)atapoint.

•  Generalizedgradientapprox(GGA)– Usesbothn(r)and|∇n(r)|– Moreaccurate,correctsoverbindingofLDA– ExamplesarePBEandBLYP

•  Hybrid:– MixessomefractionofHF– ExamplesareB3LYPandPBE0

APStutorial

Page 11: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

11

Functional soup

•  Good:chooseonefunctionalofeachkindandstickwithit(e.g.,LDAorPBEorB3LYP).

•  Bad:Runseveralfunctionals,andpick‘best’answer.

•  Ugly:Designyourownfunctionalwith2300parameters.

APStutorial

Page 12: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

Functional Zoology

•  Empirical– GGA:BLYP– Hybrid:B3LYP

•  Names:– B=B88exchange– LYP=Lee‐Yang‐Parrcorelation

•  Non‐empirical– GGA:PBE– Meta‐GGA:TPSS– Hybrid:PBE0

APStutorial 12

Page 13: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ll cover

•  Simplestpossibleexampleofafunctional•  EssentialsofKS‐DFT,andfunctionalzoo•  Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity

•  Exactexchange•  Quiz

13APStutorial

Page 14: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ll cover

•  Simplestpossibleexampleofafunctional•  EssentialsofKS‐DFT,andfunctionalzoo•  Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity

•  Exactexchange•  Quiz

14APStutorial

Page 15: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

15

Simple conditions for Coulomb systems

•  Asymptoticdecayofthedensity

•  LeadstosevereconstraintonKSpotential

•  AnddeterminesKSHOMO:

APStutorial

Page 16: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

KS potential for He atom

16APStutorial

Page 17: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

17

Densities

APStutorial

Page 18: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

18

LDA potential

APStutorial

Page 19: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

19

Self interaction

•  Violatedbymostsemilocalfunctionals(unlessbuiltin)

APStutorial

Page 20: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

Energy as function of N

APStutorial 20

FromDreizler+Gross

Page 21: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

21

Derivative discontinuity

• Whenyouaddatinyfractionofanelectrontoasystem,theKSpotentialshiftsuniformly,sincebefore,εHOMO(N)=‐I,butnow,εHOMO(N+δ)=‐A

•  Thusvs(r)mustjumpbyΔxc=(I‐A)‐(εHOMO‐εLUMO)

APStutorial

Page 22: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

22

Ne Potentials

APStutorial

Page 23: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

23

Missing derivative discontinuity in LDA

LDAlookslikeexact,shiftedbyaboutI/2

APStutorial

Page 24: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ll cover

•  Simplestpossibleexampleofafunctional•  EssentialsofKS‐DFT,andfunctionalzoo•  Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity

•  Exactexchange•  Quiz

24APStutorial

Page 25: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ll cover

•  Simplestpossibleexampleofafunctional•  EssentialsofKS‐DFT,andfunctionalzoo•  Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity

•  Exactexchange•  Quiz

25APStutorial

Page 26: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What ever happened to HF?

• WeknowExisjust

•  Sowhycan’twejustputthatinKSequations?

•  Becausedon’tknowEx[n],somustapproximate

APStutorial 26

Page 27: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

27

OEP •  Waytohandleorbital‐dependentfunctionalsinKSscheme,i.e.,withsinglemultiplicativeKSpotential

•  Stilldensityfunctionals,sinceorbitalsuniquelydeterminedbydensity

•  OftencalledOPM•  Severalschemestoimplement,allmuchmoreexpensivethanregularKS‐DFT

•  Canimproveotherproperties:– Noself‐interactionerror– Potentialsandorbitalenergiesmuchbetter– Approximatesderivativediscontinuity

APStutorial

SeeRMP,KuemmelandKronik

Page 28: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

28

HF versus EXX

•  HFminimizesEx[{φi}]overallpossiblewavefunctions

•  EXXincludesadditionalconstraintofcommonpotential(i.e.,KS)

•  Yieldalmostidenticaltotalenergies,withHFaneenstybitlower.

•  Occupiedorbitalenergiesverysimilar,butbigdifferenceinunoccupiedorbitals

APStutorial

Page 29: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

29

A tale of three gaps

•  Fundamentalgap:–  Δ=I–A=24.6eVforHe

•  Kohn‐Shamgap:–  Δs=εHOMO‐εLUMO=21.16eV

•  Derivativediscontinuity:Δxc=Δ‐Δs

•  Lowestopticaltransition:– ωmin=E(1s,2p)‐E(1s2)=21.22eV

•  NOTE:Allsameifnon‐interacting,alldifferentwheninteracting

•  Of course, εHOMO(LDA)=15.5eVAPStutorial

Page 30: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

Quiz

1.  Dolocalfunctionalsdobetterfor:A.smallN,B.largeN?

2.  Howmanyempiricalparametersaretoomany?A.1;B.10.,C.100+

3.  GGA’shavenoself‐interactionerror,Trueorfalse?

4.  TheKohn‐Shamgapwouldequalthetruegapifonlywehadtheexactfunctional?

5.  WhynotuseExinsmallcalculationstoimprovegeometries,etc.?

APStutorial 30

Page 31: More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al. (Springer‐Verlag, NY, 2003) – Density Functional Theory , Dreizler and Gross, (Springer‐Verlag,

What we’ve learned, maybe •  Ground‐statedensitydeterminesallpropertiesofsystem,

inprinciple,butinpractice,onlyreallygetenergyanddensity(whichis90%ofwhatyouwant).

•  Localdensityfunctionaltheoriesgiveroughlycorrectanswers,butaretooinaccuratetobehelpfulinquantumchemistry.

•  Thecommonly‐usedfunctionalsinchemistryarewell‐foundedandhavefewparameters.

•  Thereareknownexactpropertiesofthedensityinrealatoms.

•  TherearesubtleandbizarreeffectsintheKSpotentialbecauserealelectronsdointeract.

•  Exactexchangeisexpensive,andwedon’thaveacorrelationfunctionaltogowithit,butitimprovessomeproperties.

31APStutorial


Springer-Verlag Berlin Heidelberg GmbH

Literatur: W. Demtröder, Experimentalphysik 1, Springer Lehrbuch P. A. Tipler, Physik, Spektrum Akad. Verlag Meschede: Gerthsen Physik, Springer Verlag

Springer verlag investment banking a guide to underwriting and advisory services

furPhySlkO Zeitschnfl Springer-Verlag

Introduction to Density Functional Theory · Density Functional Theory - An Advanced Course Eberhard Engel · Reiner M. Dreizler (2001) Springer-Verlag Berlin Heidelberg Kohn-Sham

9 by Springer-Verlag 1978 - uni-heidelberg.dedan/TMP/bernhard/LITERATUR/... · 9 by Springer-Verlag 1978 On the Segal-Shale-Weil Representations and Harmonic Polynomials M. Kashiwara

c Copyright 2010 Springer-Verlag London Ltd. Notice Please note … · 2021. 5. 17. · Springer-Verlag London Ltd, United Kingdom, pp. 743-749. This file was downloaded from:

Computational Mechanics 25 (2000) 87–98 Springer-Verlag

9 1989 Springer-Verlag New York Inc.gvc/Cybenko_MCSS.pdf · Math. Control Signals Systems (1989) 2:303-314 Mathematics of Control, Signals, and Systems 9 1989 Springer-Verlag New

Springer-Verlag Berlin Heidelberg GmbH978-3-662-04707... · 2017-08-28 · Springer-Verlag Berlin Heidelberg GmbH ONLINE LIBRARY ... [email protected] ... Originally published

Automatic fingerprint recognition systems N. Ratha, R. Bolle Springer Verlag 2004

Experiments in Fluids 28 (2000) 322–335 Springer-Verlag ...bsuther/papers/schlieren/reprint.pdf · density stratified systems. The techniques range from the introduction of passive

eBooks - Hochschule Koblenz · eBooks Springer Verlag Kohlhamrner Verlag Springer Veriaq stet-ten Ihnen die folgenden deutschsprachigen eaook-Pakete (Erscheinungsjahr 2013) zur Verfügung:

Top Organomet Chem (2006) 19: 1–48 Springer-Verlag Berlin

Lecture 24 See Section 9.3.3 in Springer-Verlag, London, 2014

Geometry Discrete & Computational 9 1994 Springer-Verlag ...Discrete Comput Geom 12:119-150 (1994) Geometry Discrete & Computational 9 1994 Springer-Verlag New York Inc. Castles in

9 Springer-Verlag 1985 - univie.ac.at

Springer-Verlag Berlin Heidelberg 2014

Springer-Verlag. 7 Promoting Development and Integration

c Springer Verlag. The copyright for this contribution is

Bibliography Springer Verlag

Springer-Verlag - Statistical Data Mining - Ripley Bd 2002.pdf

Density Functional Theory and the Family of (L)APW-methods ... · Density Functional Theory R.M. Dreizler and E.K.U. Gross, Springer-Verlag (1990), ISBN 3-540-51993-9 Density Functional

DOI 10.1007/b94405 Springer-Verlag 2004 Two-Photon ...nathan.instras.com/MyDocsDB/doc-712.pdf · APS (2004) 170:169--273 DOI 10.1007/b94405 Springer-Verlag 2004 Two-Photon Photopolymerization

Springer-Verlag London Ltd.978-1-4471-0101-7/1.pdf · Springer-Verlag London Ltd. ... Parallel Processingfor Real-time Signal Processing and Control M.a. Tokhi, ... concepts ofstochastic

9 Springer-Verlag 1991 - Carnegie Mellon School of …hgeyer/Teaching/R16-899B/Papers/...Biol. Cybern. 65, 147-159 (1991) Biological Cybernetics 9 Springer-Verlag 1991 Self-organized

9 Springer-Verlag New York Inc. 1994

Iu. I. Manin, A Course in Mathematical Logic, Springer-Verlag, 1977

Lecture 22 See Sections 9.2 and 9.3 in Springer-Verlag

9 Springer-Verlag 1981 · 2017. 8. 28. · Diabetologia (1981) 20:378-387 Diabetologia 9 Springer-Verlag 1981 Neural Pathways Involved in the Hypothalamic Integration of Autonomic

Springer-Verlag France S.A.R978-2-8178-0891-8/1.pdf · Springer-Verlag France ... 69003 Lyon France Christine Cuche 28, avenue Foch 07300 ... Michel Bretagnolle, Alain Fond, Denis

Copyright 1985 Springer-Verlag. All rights reserved

Experiments in Fluids 22 (1997) 199 Springer-Verlag …ceeserver.cee.cornell.edu/eac20/cee637/handouts/Cowen_Mo...Experiments in Fluids 22 (1997) 199—211 ( Springer-Verlag 1997 A

Mesalazin-Therapie mit © Springer Verlag GmbH günstigem

Linked Data Initiatives at Springer Verlag