AB-INITIO SIMULATIONS IN MATERIALS SCIENCE

Transcription

1 AB-INITIO SIMULATIONS IN MATEIALS SCIENCE J. Hafe Istitut fü Mateialphysik ad Cete fo Computatioal Mateial Sciece Uivesität Wie, Sesegasse 8/12, A-1090 Wie, Austia J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 1

2 Oveview I Levels of compute-simulatios i mateials sciece Bo-Oppeheime appoximatio Decouplig ios ad electos Hellma-Feyma theoem Ab-iitio electoic stuctue methods Hatee-Fock (HF) ad post-hf appoaches Desity-fuctioal theoy (DFT) Local desity appoximatio J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 2

3 Oveview II DFT methods - a oveview Desity-oly appoaches Thomas-Femi theoy Paametizatio of the desity i tems of obitals Koh-Sham theoy Choice of a basis-set Plae waves vs. local obitals Pseudopotetials vs. all-electo methods Solvig the Koh-Sham equatios Total-eegy miimizatio: Ca-Paiello dyamics Iteative diagoalizatio J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 3

4 Levels of mateials modellig Ab-iitio techiques Hatee-Fock ad post-hf techiques - Quatum chemisty Desity fuctioal techiques - Mateials sciece Tight-bidig techiques Foce-field simulatios Molecula dyamics Mote Calo J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 4

5 Bo-Oppeheime appoximatio I Hamiltoia of the coupled electo-io system: N ios, coodiates 1 N, mometa P 1 P N P, chages Z 1 ZN, masses M I M N e electos, coodiates 1 N, mometa p 1 p N p, mass m H N I 1 P 2 I 2M I N e 1 i T N T e p 2 i 2m ij Vee i e 2 j VNN IJ Z I Z J e 2 i J VNe i I Z I e 2 I i (1) Schödige equatio T N T e Vee VNN VNe x Φ EΦ x (2) x s full set of electoic positios ad spi vaiables J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 5

6 Bo-Oppeheime appoximatio II Diffeece i the time-scales of uclea ad electoic motios quasi-sepaable asatz Φ x Ψ x χ (3) Ψ x electoic wavefuctio, χ χ is moe localized tha Ψ x Iχ Ψ I x decoupled adiabatic Schödige equatios of electos ad uclei uclea wavefuctio Te Vee T N V NN VeN ε x Ψ χ ε Eχ Ψ x (4) Electoic eigevalue ε depeds paametically o the ioic positios J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 6

7 Bo-Oppeheime appoximatio III Adiabatic appoximatio: Ios move o the potetial-eegy suface of the electoic goud state oly. Te Vee T N V NN VeN ε x Ψ0 χ t ε0 i h t χ Ψ0 x t (5) Neglect quatum effects i ioic dyamics eplace time-depedet ioic Schödige equatio by classical Newtoia equatio of motio 2 P I E 0 t t 2 ε0 IE0 VNN (6) Foce IE0 cotais cotibutios fom the diect io-io iteactio ad a tem fom the gadiet of the electoic total eegy J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 7

8 Hellma-Feyma theoem I ε 0 I Ψ0 He I Ψ0 He Ψ0 I He Ψ0 He Ψ0 I He Ψ0 Ψ0 Ψ0 I Ψ0 Fist ad thid tems i the deivative vaish due to vaiatioal popety of the goud-state Foces actig o the ios ae give by the expectatio value of the gadiet of the electoic Hamiltoia i the goud-state Ψ0 (7) The electoic Schödige equatio ad the Newtoia equatios of motio of the ios, coupled via the Hellma-Feyma theoem ae the basis of the Ca-Paiello method. J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 8

9 q1 " " " " " " qn qn 1 " " qn Ab-iitio electoic stuctue - Hatee-Fock methods Quatum chemisty: Hatee-Fock ad post-hf techiques - May-electo wavefuctios Slate-detemiats Ψ a α 1 α N q1 qn 1 N φ α1 φ αn. q1 φ α1 φ αn. 1 N P P Pφ α1 q1 (8) "φαn - Vaiatioal coditio δ Ψ a Ψ a H Ψ a Ψa 0(9) Vaiatio with espect to the oe-electo obitals φ α J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 9

10 , ( % $ & ( Hatee-Fock methods II Hatee-Fock equatios # h 2 2m e 2 j j )+*i Ze 2 szi s z j& φ i e2 φ j ' ' φi ' i j d 3 φ j ' ' φ j 2 d 3 εiφi φ i (10) Poblems with Hatee-Fock calculatios Computatioal effot scales badly with the umbe of electos Neglect of coelatios - Too wide bad gaps, too small bad widths - Exchage-opeato fo metallic systems sigula at the Femi level J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 10

11 Post Hatee-Fock methods Expess wavefuctio as liea combiatio of Slate detemiats to iclude coelatio Cofiguatio iteactios - HF-CI Eve highe computatioal effot, scalig wose Covegece poblematic Metals???? J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 11

12 / 0. Ab-iitio electoic stuctue - Desity-fuctioal theoy Hohebeg-Koh-Sham theoem: - The goud-state eegy of a may-body system is a uique fuctioal of the paticle desity, E 0 E. - The fuctioal E has its miimum elative to vaiatios δ paticle desity at the equilibium desity 0, of the E E 0 δe δ mi o E - 0 (11) J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 12

13 Total-eegy fuctioal Desity-fuctioal theoy II E T E H E xc V d3(12) T kietic eegy, E H Hatee eegy (electo-electo epulsio), E xc exchage ad coelatio eegies, V exteal potetial - the exact fom of T ad E xc is ukow Local desity appoximatio - desity oly : - Appoximate the fuctioals T ad E xc by the coespodig eegies of a homogeeous electo gas of the same local desity Thomas-Femi theoy J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 13

14 1 Desity-fuctioal theoy III Local desity appoximatio - Koh-Sham theoy: - Paametize the paticle desity i tems of a set of oe-electo obitals epesetig a o-iteactig efeece system i φi - Calculate o-iteactig kietic eegy i tems of the φ i 2 s, (13) T i φ i h 2 2m 2 φ i d3(14) - Detemie the optimal oe-electo obitals usig the vaiatioal coditio Koh-Sham equatios δe δφ i 0(15) J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 14

15 2 ( 43 Exc - 5. Desity-fuctioal theoy IV E T E H V d3(16) with the exchage-coelatio eegy E xc ε xc d 3 (17) whee ε xc is the exchage-coelatio eegy of a homogeeous electo gas with the local desity Koh-Sham equatios: h 2 2m 2 V e 2 ( d 3 µ xc φ i εi φi (18) V e f f with the exchage-coelatio potetial µ xc δe xc δ δ δ ε xc (19) J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 15

16 Solvig the Koh-Sham equatios I Choice of a basis set Plae waves ad elated basis fuctios Plae waves (Lieaized) augmeted plae waves - (L)APW s (Lieaized) muffi-ti obitals - (L)MTO s Pojecto augmeted waves -PAW s Localized obitals Atomic obitals - LCAO s Gaussia obitals Mixed basis sets Discete vaiable epesetatios J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 16

17 Basis sets I Localized obitals Well localized obitals allow, at least i piciple, liea scalig of DFT calculatios with the system size. Loss of accuacy fo stog localizatio Basis depeds o ioic positios Pulay coectios have to be added to the Hellma-Feyma foces Basis-set completess ad supepositio eos ae difficult to cotol Fo Gaussias: may itegals appeaig i the DFT fuctioal ca be doe aalytically J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 17

18 Basis sets II Plae waves (PW s) Natual choice fo system with peiodic bouday coditios It is easy to pass fom eal- to ecipocal space epesetatio (ad vice vesa) by FFT No Pulay coectio to foces o atoms Basis set covegece easy to cotol Covegece slow - Electo-io iteactio must be epeseted by pseudopotetials o pojecto-augmeted wave (PAW) potetials - Use LAPW s o mixed basis sets J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 18

19 Pseudopotetials I Slow covegece of PW expasio caused by the ecessity to epoduce odal chaacte of valece obitals Nodes ae the cosequece of the othogoality to the tightly-boud coe-obitals Elimiate the tightly-boud coe states ad the stog potetial bidig these states: - Use foze-coe appoximatio - Poject Koh-Sham equatios oto sub-space othogoal to coe-states othogoalized plae waves..., o - eplace stog electo-io potetial by a weak pseudopotetial which has the same scatteig popeties as the all-electo potetial beyod a give cut-off adius J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 19

20 Pseudopotetials II Scatteig appoach to pseudopotetials Pefom all-electo calculatio fo atom o io at a efeece eegy ε Defie a cut-off adius c well outside the ode of the highest coe-state Costuct a pseudo valece-obital φ l that is idetical to the all-electo obital φ l fo but odeless fo 76c, c ad cotiuous ad cotiuously diffeetiable at c The scatteig phase-shifts fo electos agee (modulo 2π) if the logaithmic deivatives of φ l ad φ l agee o the suface of the cut-off sphee: logφ l ε log φ l ε at c(20) J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 20

21 Pseudopotetials III Mode pseudopotetials Nom-cosevig pseudopotetials (NC-PP) Nom-cosevatio: chage withi cut-off sphee fixed High cut-off eegies fo fist-ow ad tasitio elemets Ultasoft pseudopotetials - (US-PP) - Nom-cosevatio elaxed - moe feedom fo pseudizig 2p ad 3d states - Add augmetatio chages iside the cut-off sphee to coect chage - Multiple efeece eegies - impoved tasfeability - Lowe cut-off eegies J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 21

22 Pseudopotetials IV Pojecto-augmeted waves - PAW s - Pseudizatio as fo ultasoft potetials - ecostuctio of exact wavefuctio i the coe egio Decompositio of wavefuctios (ϕ lmε ϕ lmε - patial waves) φ φ atoms ϕ lmε c lmε atoms ϕlmε exact W F pseudo W F pseudo osite W F exact osite W F augmetatio c lmε compesatio (21) Pseudo-WF epeseted o FFT-gid, o-site tems o atom-ceted adial gids Same decompositio holds fo chage desities, kietic, Hatee, ad exchage-coelatio eegies ad potetials J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 22

23 Pseudopotetials vs. all-electo methods FLAPW Plae-wave expasio i itestitial egio Expasio i tems of spheical waves iside muffi-ti sphees (up to l 12) US-PP, PAW Plae-wave expasio thoughout etie cell Osite tems epeseted o adial gids (up to l PAW s combie the accuacy of all-electo methods such as FLAPW with the efficiecy of pseudopotetials 2 3 ) J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 23

24 . ( Solvig the Koh-Sham equatios I Diect miimizatio of the Koh-Sham total-eegy fuctioal Pecoditioed cojugate-gadiet miimizatio Gadiet : F l h 2 2m 2 Ve f f - φl ε l φ l (22) Ca-Paiello (CP) method: Use dyamical-simulated aealig appoach fo miimizatio pseudo-newtoia equatios of motio fo coupled electo-io system Difficulties with diect miimizatio appoaches: Difficult to keep wavefuctios othogoal Bad scalig fo metallic systems ( chage sloshig ) I CP calculatios: o adiabatic decouplig fo metals, the system difts away fom the Bo-Oppeheime suface J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 24

25 Solvig the Koh-Sham equatios II Iteative matix diagoalizatio ad mixig Geeal stategy: Stat with a set of tial vectos (wavefuctios) epesetig all occupied ad a few empty eigestates: 1 -φ N bads. Impove each wavefuctio by addig a factio of the esidual vecto, φ H φ ε app φ ε app φ Afte updatig all states, pefom subspace diagoalizatio H φ (23) Calculate ew chage desity ρ out Detemie optimal ew iput-chage desity (mixig old ρ i ad ρ out ) Iteate to selfcosistecy J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 25

26 Solvig the Koh-Sham equatios III Algoithms implemeted i VASP Updatig the wavefuctios - Blocked Davidso algoithm - MM-DIIS: esiduum miimizatio method - diect ivesio i the iteative subspace: miimize om to each eigestate (o othogoality costait) Mixig: - DIIS of esidual vecto J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 26

27 Ioic stuctue ad dyamics I Static optimizatio of cystal stuctue Atomic coodiates at fixed cell-shape:hellma-feyma foces Geomety of the uit cell: Hellma-Feyma stesses Algoithms implemeted i VASP: Cojugate gadiet techique Quasi-Newto scheme Damped molecula dyamics J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 27

28 Ioic stuctue ad dyamics II Ab-iitio molecula dyamics (AIMD) Ca-Paiello MD (ot implemeted i VASP): - Woks well fo isulatos ad semicoductos - Time-step cotolled by evolutio of eigestates - Fo metals, the systems teds to dift away fom the Bo-Oppeheime suface due to the couplig of electos ad ios - Must use Two-themostat appoach fo metals MD o the Bo-Oppeheime suface: Hellma-Feyma MD - Stable also fo metals, caoical esemble ealized usig Nosé themostat - Time-step cotolled by ioic dyamics J. HAFNE, AB-INITIO MATEIALS SIMULATIONS Page 28