FOUNDATIONS OF DENSITY-FUNCTIONAL THEORY

Transcription

1 FOUNDATIONS OF DENSITY-FUNCTIONAL THEORY J. Hafe Istitut fü Mateialphysik ad Cete fo Computatioal Mateial Sciece Uivesität Wie, Sesegasse 8/2, A-090 Wie, Austia J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page

2 Oveview I Hohebeg-Koh theoem Hellma-Feyma theoem Foces o atoms Stesses o uit cell Local-desity appoximatio: desity oly Thomas-Femi theoy ad beyod Local-desity appoximatio: Koh-Sham theoy Koh-Sham equatios Vaiatioal coditios Koh-Sham eigevalues J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 2

3 Oveview II Exchage-coelatio fuctioals Local-desity appoximatio (LDA) Geealized gadiet appoximatio (GGA), meta-gga Hybid-fuctioals Limitatios of DFT Bad-gap poblem Ovebidig Neglect of stog coelatios Neglect of va-de-waals iteactios Beyod LDA LDA+U GW, SIC, J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 3

4 Desity-fuctioal theoy - HKS theoem Hohebeg-Koh-Sham theoem: () The goud-state eegy of a may-body system is a uique fuctioal of the paticle desity, E 0 E. (2) 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 () J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 4

5 Vee Poof - HKS theoem Reductio ad absudum: H T V (2) H V is the Hamiltoia of a may-electo system i a exteal potetial V ad with a electo-electo iteactio V ee. I the goud-state this system has the eegy E 0, with E 0 ψ 0 ad the paticle desity 0 2. Let us assume that a diffeet exteal potetial leads to a diffeet goud-state ψ0, but to the same paticle desity: Accodig to the vaiatioal piciple it follows that E0 H V V V V (3) ψ0 ψ H ψ0 ψ0 ψ0 E0 ψ0 ψ0 ψ0 ψ0 J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 5

6 E 0 H Poof - HKS theoem E E0 0 0 V V d 3 (4) Statig fom ψ 0 ψ 0 (5) ad usig 0 0 it follows E 0 E E V V V V d 3 d 3 (6) i diect cotadictio to the esults obtaied above. Hece must be diffeet ad V is a uique fuctioal of. 0 ad 0 The vaiatioal popety of the Hohebeg-Koh-Sham fuctioal is a diect cosequece of the geeal vaiatioal piciple of quatum mechaics. J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 6

7 ψ0 T HKS theoem - Vaiatioal piciple With F E ψ T F V d3 Vee ψ (7) it follows E F F 0 V ψ T Vee Vee ψ0 ψ0 ψ ψ V d3 0 d3 V ψ V ψ0 E 0 (8) ad hece E0 E 0 δe δ mi 0 E 0 (9) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 7

8 I H H Hellma-Feyma theoem: Foces ad stesses The exteal potetial V is ceated by the ios located at the positios R I, V v. The goud-state eegy ad wavefuctio deped o I R I the ioic coodiates, R atom located at R I is give by R I as paametes. The foce F I actig o a F I I E0 I Ψ0 Ψ0 R R R I Ψ 0 I H H Ψ0 R R H Ψ0 Ψ0 R R Ψ0 Ψ 0 R Ψ0 I Ψ 0 (0) 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. The goud-state must be detemied vey accuately: eos i the total eegy ae 2d ode, eos i the foces ae st ode! J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 8

9 Hellma-Feyma theoem: Foces ad stesses The stess teso σ i j descibes the vaiatio of the total eegy ude a ifiitesimal distotio of the basis vectos a ude a stai t i j : k σ i j E a k t i j a k i j δ i j ti j a k j () σ i j Ψ0 t i j H a k Ψ0 J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 9

10 DFT fuctioal Total-eegy fuctioal E T E H E xc V d3(2) 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. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 0

11 Kietic eegy: T t Thomas-Femi theoy 3 h 2 0m whee t is the kietic eegy of a oiteactig homogeeous electo gas with the desity. t 3π d 3 Electo-electo iteactio: Coulomb epulsio oly 5 3 (3) E H e 2 2 d 3 d 3 (4) Add exchage-coelatio tem i mode vesios. Vaiatio of E leads to the Thomas-Femi equatio with 5 3 C 2 3 e 2 d 3 V 0(5) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page

12 %! " Koh-Sham theoy E V T d3 e 2 2 E xc! d 3 d 3 () Paametize the paticle desity i tems of a set of oe-electo obitals epesetig a o-iteactig efeece system i φ i 2 (6) (7) (2) Calculate o-iteactig kietic eegy i tems of the i φ T, $#T0 s, i.e. T 0 i φ i h 2 2m 2 i φ d3(8) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 2

13 Koh-Sham theoy II (3) Local-desity appoximatio fo exchage-coelatio eegy E xc ε xc d 3 (9) whee ε xc is the exchage-coelatio eegy of a homogeeous electo gas with the local desity. Fo the exchage-pat, a Hatee-Fock calculatio fo a homogeeous electo gas with the desity leads to 3e 2 3 ε x (20) 4π (4) Detemie the optimal oe-electo obitals usig the vaiatioal coditio ude the othoomality costait δ E i j ε i j 3π2 φi φ j φi δi j φ j δi j 0(2) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 3

14 ' ' )( φ i * Koh-Sham theoy III Koh-Sham equatios (afte diagoalizig εi j): h 2 2m 2 V e 2 d 3 µxc ε i φ i (22) with the exchage-coelatio potetial µ xc δe xc δ Total eegy: E ε i i 2 d 3 d 3 ε xc µxc d 3 () (2) ()* sum of oe-electo eegies double-coutig coectios 2 (23) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 4

15 Koh-Sham theoy IV Vaiatioal coditios Total eegy E δe δ 0 0(24) Koh-Sham eigevalues ε i δ φi HKS φi 0 with φi φ j 0 +εi ε j (25) Nom of esidual vecto Ri R φi H KS δ ε app i R φi φ i R φi ε app i 0 φi HKS φi (26) No othogoality costait! J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 5

16 % Koh-Sham theoy V Itepetatio of the oe-electo eegies ε i Hatee-Fock theoy - Koopma s theoem ε HF i EHF i EHF i 0 (27) ε HF i = Ioisatio eegy if elaxatio of the oe-electo obitals is eglected. Koh-Sham theoy: Total eegy is a oliea fuctioal of the desity ot valid. δe δ i εi i φ φ i Koopmas theoem (28) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 6

17 -,T Exchage-coelatio fuctioals I Defiitio of the exchage-coelatio fuctioal: E xc accouts fo the diffeece betwee the exact goud-state eegy ad the eegy calculated i a Hatee appoximatio ad usig the o-iteactig kietic eegy T 0, E xc T0 U xc (29) T exact ad o-iteactig kietic eegy fuctioal U xc iteactio of the electos with thei ow exchage-coelatio hole xc defied as (ρ 2 is the two-paticle desity matix) T0 ρ 2 s; s,s s xc s; s (30) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 7

18 "! δ s Exchage-coelatio fuctioals II Popeties of the exchage-coelatio hole Locality! lim xc s; s 0(3) Pauli piciple fo electos with paallel spi xc s; s s (32) Atisymmetic o-iteactig wavefuctio x s; s d3 (33) s Nomalizatio of two-paticle desity matix c s; s d3 O(34) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 8

19 . //D Exchage-coelatio fuctioals III Popeties of the exchage-coelatio fuctioal Adiabatic coectio fomula E xc 2 d 3 d3 0 dλ xc λ ; (35) Lieb-Oxfod boud E xc D 4 3 d (36) plus scalig elatios,... J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 9

20 0 2 4 ; : 6 3" Local (spi-)desity appoximatio - L(S)DA E xc ε xc d 3 (37) Exchage-fuctioal (fo spi-polaized systems, ) ε x εp x 3e 2 4π 3π 2 ε f x 3 ε p x " (38) with εx p εx 2fo the paamagetic (o-spipolaized) ad εx f εx 0fo the feomagetic (completely spi-polaized) limits of the fuctioal. Coelatio fuctioal ε c a homogeeous electo gas calculated usig quatum Mote Calo simulatios ad simila spi-itepolatios fitted to the goud-state eegy of J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 20

21 Semilocal fuctioals Geealized gadiet appoximatio - GGA E xc f d 3 (39) Thee ae two diffeet stategies fo detemiig the fuctio f : () Adjust f such that it satisfies all (o most) kow popeties of the exchage-coelatio hole ad eegy. (2) Fit f to a lage data-set ow exactly kow bidig eegies of atoms ad molecules. Stategy () is to be pefeed, but may diffeet vaiats: Pedew-Wag (PW), Becke-Pedew (BP), Lee-Yag-Pa (LYP), Pedew-Buke-Ezehof (PBE). J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 2

22 Semilocal fuctioals Meta-GGA Iclude i additio a depedece o the kietic eegy desity τ electos, τ occ i φ i 2 of the (40) I the meta-gga s, the exchage-coelatio potetial becomes obital-depedet! J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 22

23 aexexact # ceclyp Hybid fuctioals Geeal stategy: Mixig exact-exchage (i.e. Hatee-Fock) ad local-desity eegies, as suggested by the adiabatic coectio fomula E xc 0 U xc λ dλ 2 U 0 xc 2 U xc (4) U0 xc U xc olocal exchage eegy of Koh-Sham obitals potetial eegy fo exchage ad coelatio Example: B3LYP fuctioal E xc a Ex LSDA be x B88 c Ec VW N (42) whee EB88 x stad fo the exchage pat of the Becke88 GGA fuctioal, ELYP c fo the coelatio pat of the Lee-Youd-Pa local ad GGA fuctioal, ad EVWN c fo the local Vosko-Wilk-Nusai coelatio fuctioal. a ad c ae adjustable paametes. b J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 23

24 Limitatios of DFT I Bad-gap poblem: - HKS theoem ot valid fo excited states bad-gaps i semicoductos ad isulatos ae always udeestimated - Possible solutios: - Hybid-fuctioals lead to bette gaps - LDA+U, GW, SIC icease coelatio gaps Ovebidig: - LSDA: too small lattice costats, too lage cohesive eegies, too high bulk moduli - Possible solutios: - GGA: ovebidig lagely coected (tedecy too oveshoot fo the heaviest elemets) - The use of the GGA is madatoy fo calculatig adsoptio eegies, but the choice of the coect GGA is impotat. J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 24

25 Limitatios of DFT II Neglect of stog coelatios - Exchage-splittig udeestimated fo aow d- ad f -bads - May tasitio-metal compouds ae Mott-Hubbad o chage-tasfe isulatos, but DFT pedicts metallic state - Possible solutios: - Use LDA+U, GW, SIC, Neglect of va-de Waals iteactios - vdw foces aise fom mutual dyamical polaizatio of the iteactig atoms ot icluded i ay DFT fuctioal - Possible solutio: - Appoximate expessio of dipole-dipole vdw foces o the basis of local polaizabilities deived fom DFT?? J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 25

26 < Beyod DFT DFT+U Descibe o-site Coulomb-epulsio by Hubbad-Hamiltoia H U 2 m m whee m umbe m ad spi s, U exchage eegy. m s E s m " d 7 s E U d " J 2 m m m s s m s(43) s is the umbe opeato fo electos with the magetic quatum 2E d ad J a sceeed The DFT+U Hamiltoia icludes cotibutios aleady accouted fo i the DFT fuctioal subtact double-coutig, adopt otatioally ivaiat fomulatio E DFT 7U o-site desity matix ρ s i j EDFT U of the d electos J 2 T s ρs ρsρs (44) J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 26

27 " Beyod DFT II Calculate quasipaticle-excitatio eegies by low-ode may-body petubatio theoy. GW: Self-eegy appoximatio appoximated by Σ ;ω i 2π G ;ω W ;ω dω(45) whee G is the full iteactig Gee s fuctio ad W the dyamically sceed Coulomb iteactio, descibed by the ivese dielectic matix ε ad the bae Coulomb potetial v, " W ;ω ε " ;ω v d3 (46) I pactice, appoximate foms of G ad ε SIC: Self-iteactio coectios. have to be used. GW, SIC ae ot implemeted i VASP. Results lagely equivalet to LDA+U. J. HAFNER, AB-INITIO MATERIALS SIMULATIONS Page 27