Multiscale Modeling UCLA Prof. N. Ghoniem

Transcription

1 Interdscplnary graduate course Fall 003 Multscale Modelng UCLA Prof. N. Ghonem Introducton n Electronc Structure Calculatons Ncholas Kousss Department of Physcs Calforna State Unversty Northrdge

2 The Fundamental many-body Hamltonan Hˆ Ze e = + + m ZZe I I J I e, I r RI j r M rj I I I J RI RJ Atomc unts (e=, =, m e =) Electronc contrbuton Nuclear contrbuton Only one small term: The knetc energy of the nucle (M I /m e 0 3 ) If we omt ths term, the nucle are a fxed external potental actng on the electrons The fnal term s essental for charge neutralty - but s a classcal term that s added to the electronc part The fst three terms s the key problem for ab nto predcton of materals propertes The ground-state energy as a functon of the postons of the nucle determne: stable structures ---- phase transtons ---- mechancal deformatons ---- phonons, etc. The excted states determne propertes: --- Optcal propertes, photoemsson, etc.

3 Born Oppenhemer approxmaton Decouplng between the nuclear and electronc degrees of freedom due to dfferent dynamcal evoluton (electrons move n the feld of fxed nucle) Ψ ( rr,,, r, RR,,, R) =Ψ ( rr,,, r) Φ ( RR,,, R) el Ion N N { R } N N,, RN The electronc wavefuncton parametrcally depends on the onc poston varables The Schrodnger equaton decouples nto a tme-ndependent Schrodneger equaton of the e s n the constant feld of the fxed nucle, and the tme-dependent Newton-lke equaton of movement for the nucle. el H Ψ ( r,, r ) = E Ψ ( r,, r ) where el { R } {,,,, R N R N RN } { R,, RN} Hˆ el Z = + I, I r RI j r rj N

4 Devsng accurate schemes to approxmate the many-electron problem snce the early 900 s Thomas-Ferm theory (late 90 s) employ the electron densty n r) rather than r,, r N ) Hartree-Fock (930) whch bulds upon the sngle-partcle approxmaton proposed earler by Hartree; r,s,, r N, s N ) s a determnant of sngle-partcle wavefunctons Densty Functonal Theory Hohenberg and Kohn (964) & Kohn and Sham (965)

5 Hˆ = m Free electron gas ˆ (,, ) ( HΨ r r = EΨ r,, r ) N Independent electron approxmaton N H r r εψ r m ˆ ψ( ) = ψ( ) = ( ) ψ ( r) = e k V k r k ε = k m Fll the sngle-electron energy levels consstent wth the Paul prncple Perodc Boundary condtons ψ ( r+ R) = ψ ( r) k π k = n L k k x k z Total energy F L L k N = d k= 4π π π 3 k F k 3 y N k 3 3 F = π = ( 3π n) k E kf F = m V 3 ( 3π n) 3 kf 4 5 L k dk V k F V E nkεk (4 π) = = = = m m m ε < ε ( ) 5 3π n 3 π π 5 π 5 0

6 Thomas-Ferm equatons Earlest schemes for solvng the many-electron problem n terms of n(r) rather than Knetc energy of non-nteractng e s n a homogenous electron gas Classcal electrostatc nteracton between nucle and electrons Classcal Coulomb repulson between e s (Hartree) t 0 [n(r)] s summed over occuped free-electron states where

7 Varatonal calculaton usng Lagrange multplers subject to constrant Self-consstent Thomas-Ferm equatons for n(r) Defcences Does not predct bondng between atoms, so molecules and solds can not from Ignores the quantum nature of the e-e nteractons and hence the exchange nteracton

8 Orthonormal set of one-partcle Hartree and Hartree-Fock Approxmaton φ ( r, s ), φ ( r, s ),, φ ( r, s ) N N N * 3 ( r) ( r) d r, j r s = r < φ φ >= φ φ = δ φ (, ) φ ( ) χ () φ () Hˆ = Z + = H + I j, I r j R r I rj U Hartree Approxmaton Ψ ( r, s, r, s, r, s ) = φ ( r, s ), φ ( r, s ),, φ ( r, s ) (tral functon) N N N N N Varatonal calculaton < H > =< Ψ H Ψ >= < ϕ H ϕ > + < ϕ ϕ U ϕ ϕ > j j j, j = d r ϕ ( r ) H ϕ ( r ) + d r d r ϕ ( r ) ϕ ( r ) U ϕ ( r ) ϕ ( r )? Sngle partcle wavefunctons mnmze <H> < H>=<Ψ H Ψ>= dr drψ (,, NH ) Ψ(,, N) s s N 3 3 * N 3 * 3 3 * * j j j j j j, j

9 Mnmze <H> wth respect to varatons of *, wth the constrant < δϕ ϕ >= δϕ ϕ = * 3 ( r) ( r) d r 0 < δ H > = < δϕ H ϕ > + < δϕ ϕ U ϕ ϕ > j j, j * 3 * 3 δ ϕ ( r ) H d rjϕ j ( rj) U jϕ j( rj) = + ϕ ( r ) d r = 0 j Multplyng each of the equatons ε δϕ ( r) ϕ ( r ) d r = 0 * 3 and subtractng * 3 * 3 δh δϕ ( r ) H d rjϕ j ( rj) U jϕ j( rj) ε < >= + ϕ ( r ) d r = 0 j j Snce the varatons * are ndependent, the coeffcents of each vansh ndependently 3 H * d rjϕ j ( rj) U jϕ j( rj) + ϕ = εϕ j ρ( r) = ϕj ( r) j Hartree equatons to be solved self-consstently

10 Slater determnant Hartree-Fock Approxmaton φ () φ () φ () φ() φ() φn () Ψ (,,, N) = N! φ ( N) φ ( N) φ ( N) N N < H > =< Ψ H Ψ >= d r ϕ ( r ) H ϕ ( r ) + d r d r ϕ ( r ) ϕ ( r ) U ϕ ( r ) ϕ ( r ) 3 * 3 3 * * j j j j j j, j, j δ s s 3 d r j d r ϕ ( r ) ϕ ( r ) U ϕ ( r ) ϕ ( r ) 3 * * j j j j j j Paul excluson prncple Exchange term (Integral operator whch s non-local) Mnmze <H> wth respect to varatons of *, wth the constrant < δϕ ϕ >= δϕ ϕ = * 3 ( r) ( r) d r 0

11 H ϕ ( r ) + d r ϕ ( r ) U ϕ ( r ) ϕ ( r ) δ H >= δϕ ( r d r = * 3 * j j j j j j ) j * δ s ( ) ( ) ( ) s d r j jϕ j rj U jϕ r ϕ j r j j Hartree-Fock Equatons 3 * 3 * H + d rjϕ j ( rj) U jϕ j( r ) j ϕ ( r ) δs ( ) ( ) ( ) ( ) s d r j jϕ j r j U jϕ r j ϕ j r = εϕ r j j E HF = E H + E exch < E H Defnton of the correlaton energy E exact = E HF + E correl < E HF E correl E exch

12 Hartree-Fock Theory of Free Electrons Free electron gas wth unform postve background (overall electrcal neutralty wth postve onc charges smeared out unformly ψ ( r) = e k V k r Unform densty (/V) Cancellaton of the Hartree term and the electron-postve background nteracton k 4πe k e k ε( k) k F F = = m V k' < k k k m π k x + x F F where, F( x) = + ln 4x x Total energy 3 3 ek F E= ε( k ) = N EF = N where k< k 5 4 π ( r / 0) ( / 0) F s a rs a r s 3 = 4πn 3 F(x) ε( k) E F x k k F

13 D() ε dn dn dk = = V dε V dk dε D( ε ) = F k π F dk dε At k=k F, dk/de vanshes and hence D(e F )=0 nsulatng?? Orgn

14 E = ε( k) = N k< k ( r / 0) ( / 0) F s a rs a For small values of r s (hgh densty) the KE domnates As r s ncreases (densty decreases), exchange term contrbutes more and more and eventually gves bndng In the low-densty lmt (r s large) exchange contrbuton domnates In ths lmt correlaton effects become domnant and lead to the formaton of Wgner lattces Metal r s L 3. Na 3.9 K 4.9 Cu.7 Metal r s Be.9 Al. Sn. Pb.3

15 Many-Body Electron Problem Hˆ = Tˆ+ Vˆ + Vˆ + E ext nt II T s the knetc energy of the electrons V ext s the potental actng on the electrons due to nucle V nt s the many-body electron-electron nteracton E II s the classcal on-on nteracton ˆ V = V r R ( ) ext I I I, The total energy s the expectaton value < Ψ Hˆ Ψ> E = < H >=< T >+< V >+ d rv r n r < Ψ Ψ> ˆ ˆ ˆ 3 ˆ nt ext ( ) ( ) The ground-state wavefuncton Y 0 s the state wth lowest energy; that obeys the symmetres of the partcles and all conservaton laws

16 Hohenberg-Kohn Theorems Theorem I: For any system of electrons n an external potental V ext (r), that potental s determned unquely, except for a constant, by the ground state n(r) Proof: Suppose that there were two dfferent external potentals V () ext (r) and V ext (r) wth the same GS densty n(r). The two external potentals lead to two dfferent Hamltonans, H () and H (), whch have dfferent GS wavefunctons, and whch are hypotheszed to have the same densty, n(r). Then: () () ˆ () () () ˆ () () E =< Ψ H Ψ > < < Ψ H Ψ > ˆ () () = ˆ+ ˆ ˆ ext + nt H T V V Hˆ ˆ () Tˆ Vˆ () = + ext + ˆ nt () () () () H = H + V ext + V ext ˆ Vˆ ˆ () () () () () () E < < Ψ Hˆ + V ext V ext Ψ > () () 3 () () E < E + d r{ V ext ( r) V ext ( r) } n( r) Interchangng labels leads to () () 3 () () E < E + d r V ext ( r) V ext ( r) n( r) { } whch s a contradcton () () () () E + E < E + E Snce V ext (r) fxes H r,.r N )

17 Theorem II: A unversal functonal for the energy E[n] of the densty n(r) can be defned for all electron systems. The exact GS energy s the global mnumum for a gven Snce s a functonal of n(r), so s the knetc and nteracton Fnr T Vnt [ ( )] < Ψ ( + ) Ψ > 3 Ev[ n] d rvnt( r) n( r) + F[ n] 3 nrnr ( ) ( ') drdr' F[ nr ( )] = Gn [ ] r r' nrnr ( ) ( ') drdr' EV [ n] = Vnt ( r) n( r) d r+ + G[ n] r r' Gnr [ ( )] T[ n] + E [ n] Wth the ad of F[n] the energy functonal can be wrtten It s convenent to separate out Knetc energy of a system of nonnteractng electrons wth densty n(r) E xc s xc Exchange correlaton energy of an nteractng system wth densty n(r) [ Tn [ ] T n] ] + [ V n] V n ] [ n] = [ [ [ ] s nt H

18 j r rj Ψ( r, r,..., r ) N vh( r) + vxc( r) φ ( r ) KS Interactng electrons +real potental Non-nteractng, fcttous partcles + effectve potental N fcttous partcles non-nteractng partcles movng n an effectve potental > s a sngle-partcle wave functon N Densty: n ( ) ( ) Knetc energy: S r = φ r K S = < φ φ > 3 The functonal E [ ] ( ) ( ) [ ] v n d rvnt r n r + F n attans ts mnmum value wth respect to all allowed denstes f an only f the nput densty s the true GS densty.e. ns ( r) = n( r) The explct form of the functonal F HK [n] s the major challenge of DFT =

19 The Kohn-Sham Ansatz The Kohn-Sham approach s to replace the orgnal dffcult nteractng-partcle Hamltonan wth a dfferent Hamltonan whch could be solved more easly KS Hamltonan for non-nteractng electrons assumed to have the same densty as the true nteractng system H eff = + V eff ( r ) m where e N neff ( r) = φ ( r) and Keff = < φ φ > The GS can be found my mnmzng the KS functonal wth respect to the ndependentelectron wavefunctons (r) subject to the constrant < ϕ ϕ >= δ ε < δϕ ϕ >= 0 δe δφ δ, j KS 3 = ( ) ( ) [ ] [ ] 0 * Teff + Vext r neff r d r+ EH neff + Exc neff = δφ δe T E n ( r) = + = 0 δφ ( r) φ ( r) n ( r) φ ( r) KS eff other eff * * * eff =

20 and where Kohn-Sham sngle-electron Schrodnger equaton ( H ε ) φ( r) = 0 eff H r V r eff ( ) eff ( ) me EH Exc Veff ( r ) = Vext ( r ) + + = Vext ( r ) + VH [ neff ] + Vxc[ neff ] n ( r) n ( r) eff eff xc eff eff n ( r ) 3 δ E [ n( r)] V ( r, n ( r)) = Vext ( r) + e d r+ r r δ n( r) where n ( r) = φ ( r) eff eff The above equatons need to be solved self-consstently

21 ( ) ( ) ( ) eff + V r φk r = εkφk r m e E F neff ( r) = φk ( r) k, ε < E k eff xc eff eff n ( r ) 3 δ E [ n( r)] V ( r, n ( r)) = Vext ( r) + e d r+ r r δ n( r) F Compare eff Vnput ( r ) and eff V ( ) nput r Total Energy N 3 3 E= ε VH( rn ) eff( rdr ) + Exc[ n] Vn xc eff ( rdr ) =

22 The local densty approxmaton (LDA) The LDA approxmaton s the bass of all approxmate exchange-correlaton functonals. At the center of ths model s the dea of a unform electron gas (e s movng n a postve background charge dstrbuton such that the total ensemble s neutral) Central assumpton LDA 3 Exc n( r) εxc ( n( r)) d r = xc s the exchange-correlaton energy per partcle of a unform electron gas of densty n(r) The exchange correlaton energy per partcle can be further splt nto exchange and correlaton contrbutons Exchange part ε x ε ( nr ( )) = ε ( nr ( )) + ε ( nr ( )) [ nr ( )] xc x c 3 3 nr ( ) = = 4 π r δe [ ( )] x n r Vx[ n( r) ] [ εx[ n( r)] n( r) ] εx[ n( r)] = = = = [ n( r)] δnr ( ) nr ( ) 3 π s 3

23 Correlaton energy RPA approxmaton -good at hgh denstes Interpolaton between low and hgh densty - Wgner (934), Lndberg and Rosen (970),. Wgner Exact Monte Carlo by Ceperly and Alder (980) ε c n ( r ) = A ( B + r ) [ ] s A=0.884; B=7.8 Perdew Zunger r s << ε n ( r ) = A + A r + [ A + A r ] ln ( r ) [ ] 3 4 c s s s A =-0.096, A =-0.03, A 3 = 0.006, A 4 = r s ε n ( r ) = B [ + B r + B r ] [ ] c s 3 s B =-0.846, B =.059, B 3 =

24 ψ k Choosng a Bass ( r) = αnφn( r) n φ ( r ) n Plane waves G r e Atomc orbtals(lcao) Gaussan Combnaton of atomc orbtals and plane waves (LAPW, LMTO, )

25 Pseudopotentals Chemcal propertes of a molecule (or sold) depend predomnantly on the valence electrons Core electrons occupy flled nner shells of the atoms and ther man effect s to screen the nuclear charge from the valence electrons Elmnate the core electrons and replace them by an effectve potental actng on valence electrons If one keeps the core electrons (all-electron calculatons), the valence electrons are orthogonal to them the valence orbtals vary rapdly n the core regon When core electrons are not consdered, such as an orthogonalzaton requrement dsappears Replace a true valence orbtal by a pseudoorbtal vares smoothly near the nucleus n the outer regon the pseudoorbtal agrees wth the true orbtal The potental actng on the pseudoorbtal s called pseudopotental whch prevents the pseudoorbtal from collapsng nto the nucleus.

26 Constructon of the pseudopotental for an solated atom Let n be a sngle-partcle state whch are separated nto valence and core states denoted by v n and c n resepctvely H H sp sp ( v) ( v) ( v) n n ψ >= ε ψ > ( c) ( c) ( c) n n ψ >= ε ψ > H Defne a new set of sngle-partcle states v n ( v) ( v) ( c) ( v) ( c) ψ >= φ > < ψ φ > ψ > H ψ >= ε ψ > sp ( v) ( v) ( v) c φ > < ψ φ > H ψ >= ε φ > < ψ φ > ψ > c c sp ( c) ( c) ( c) ( v) ( v) ( c) ( c) ( v) H ε ψ >< ψ φ >= ε ψ >< ψ φ > c c sp ( v) ( c) ( c) ( c) ( v) ( v) ( v) H ( ε ε ) ψ >< ψ φ >= ε φ > c sp ( v ) ( c ) ( v ) sp ( c ) ( v ) ( v ) ( c ) ( v ) ( c )

27 ( v) ( v) ( v) n n pseudo H φ >= ε φ > V = V ε ε ψ >< ψ pseudo sp ( v) ( c) ( c) ( c) ( ) c >0 Repulsve and tends to push the correspondng v n outsde the core The pseudo wavefunctons experence an attractve Coulomb potental whch s shelded near the nucleus by the core electrons, so t should be much smoother potental wthout the /r sngularty Created a new set of valence states whch experence a weaker potental near the atomc nucleus, but the proper onc potental away from the core regon

28 Typcal R eal Space (D rect L attce) Pseudopotental Pseudo wavefuncton True wavefuncton

29 Typcal k-space (Recprocal Lattce) Pseudopotental

30 Conventonal cell Prmtve cell () surface for fcc BaTO3

31 Perodc Boundary Condtons (PBC) Atoms n the unt cell are perodcally repeated throughout space along the lattce vectors Perodc systems and crystallne solds: Aperodc systems: Supercell approxmaton Defects Molecules Surfaces M. C. Payne et al, Rev.Mod.Phys., 64, 045 (9)

32 Samplng of k-ponts n the frst Brlloun zone k-samplng Many magntudes requre ntegraton of Bloch functons over Brlloun zone (BZ) r dk r n k r r k In practce: ntegral BZ sum over a fnte unform grd Small systems Essental for: Metals Magnetc systems Good descrpton of the Bloch states at the Ferm level Brlloun Zones: FCC Real space Recprocal space Even n same nsulators: Perovskte oxdes

33 Energy cutt-off convergence Total Energy Energy cut-off (ev) 9X9X9 k-ponts K-pont convergence Total Energy (ev) Mesh Perameter Energy cut-off 50eV

34 Alumnum Energy as a functon of lattce constant Energy (ev) y =.354x x a (Å) Theory Expermental a (Å) B (N/m) *0 0.7*0

35 Some comments: The accuracy of the LDA for the exchange energy s typcally wthn 0%, whle the normally smaller correlaton energy s generally overestmated by up to a factor of. The two errors typcally cancel partally Experence has shown that the LDA gves onzaton energes of atoms, dssocaton energes of molecules and cohesve energes wthn a far accuracy of typcally 0-0%. However, the LDA gves bond lengths of molecules and solds wth an astonshng accuracy of about %. Ths moderately accuracy that LDA delvers s certanly nsuffcent for most applcatons n chemstry LDA can also fal, lke Heavy Fermon systems, domnated by the e-e nteracton effects