Carlo Adamo. Density Functional Theory

Transcription

1 Carlo Adamo Densty Functonal Theory Concepts and models Équpe de Modélsaton des Systèmes Complexes Laboratore d Électrochme et Chme Analytque CNRS, UMR 7575 cole Natonale Supéreure de Chme de Pars 1 The ngredents (1): Densty : It provdes us nformaton about how somethng s dstrbuted/spread on a gven space (volume) lectron Densty : It tells us where the electrons are lkely l to exst * ( ( ( ( r ) lectron densty s an «observable» exp bs(dmnosuccnontrlo)nckel DFT 1

2 The ngredents (): A functon (f) maps a set of numbers to another set of numbers Y=f(X) X Y=X Y A functonal (F) s a functon of a functon A functon whch maps a set of functons to a set of numbers x. F[A(X),B(X),C(X),.] xample : F xc [(,(] X A(X) Y B(X) 013 C(X) D(X) Y 3 Why DFT? From a pragmatc pont of vew:. Brght Wlson, 1965 The knowledge of the densty s all that t s necessary for a complete determnaton of all molecular propertes. If one knows the exact electron densty, (, then the cusps of ths densty would occur at the postons of the nucle. Furthermore, a knowledge of ( at the nucle would gve ther nuclear charges. Thus the full Schrödnger Hamltonan was known because t s completely defned once the poston and charge of the nucle are gven. In prncple, the wavefuncton and are known, and thus everythng s known. 4

3 Can the total energy be expressed as a functon of the densty? Hˆ Hˆ ( r1, r1 ) 1 ( Vext ( ( r1, r ) 1 ' r r1 r1 1 ' 1 ( r1, r1 ) N... ( r1 1, x...x N ) * ( r1 1, x... x N ) d 1dx... dx N N( N 1) ( r1, r )... ( x1, x... x N ) d 1d dx3... dx N Crtcal ngredents: knetc energy and electron-electron nteracton 5 arly attempts: Thomas-Ferm (197) Hˆ T [ ] ( V ( [ ] ext ee 1) xact kn. en. (T[]) s substtuted by the kn. en. of a homogenous electrons gas ) The external potental (V ext []) s that generated by the nucle 3) The electron-electron nteracton s the Coulomb repulson TF T [ ] ( v( [ ] TF ee 5/ 3 c f c F =constante de Ferm rrd Nucle-electrons nteracton (Coulomb) 1 Z A r A r R A e - /e - (Coulomb) ( r1 ) ( r r 1 ) 1 J[ ] 6 3

4 Thomas-Ferm (197): how does t works? Not very well : n TF theory no molecular system s stable relatve to dssocaton nto constutents fragments non bondng theorem (Teller (196) / Balazs (1967) / Leb (1973) / Smon (1977)) TF T Z A 1 ( r1 ) ( r ) [ ] ( 1 r r r TF A R A 1 What s mssng? xchange Correlaton and furthermore T TF s local classcal Coulomb nteractons 7 Add exchange : Thomas-Ferm-Drac model (TFD) TF TF T c TF f [ ] ( ( v( J[ ] K[ ] 5/3 ( v( J[ ] c x ( c f = 3/10(3 ) / 3 c x = 3/4(3/) 1/ 3 4/3 Add gradent correctons to knetc energy : Thomas-Ferm-Drac-Wezsacker model (TFDW or TFD-W) TF c f 5/3 1 ( ( r ) r v r J cx r r ( ) ( ) [ ] ( ) 8 ( ) 4/3 verythng s expressed as a functon of the densty (or gradent of the densty) 8 4

5 A dfferent approach: Slater and the X model (or HFS model) HF SLATR 1 Z J (1) K (1) (1) (1) A 1 a a A r R A a1, N / 1 Z A V (1) (1) (1) (1) 1 coulomb VXHF A r RA C x = constant 075< 0.75 < 1 V XHF c x r 1/ 3 xchange only; no correlaton Works XTRMLY well (and t s stll used especally for sold state) 9 X performances 10 5

6 The Hohenberg and Kohn theorems (1964) v ext ( r ) 0 r ( ( 0 All propertes of the many-body system are determned by the ground state densty 0 ( ach property s a functonal of the ground state densty 0 ( whch s wrtten as f [ 0 ] 11 Hohenberg and Kohn: theorem I r 0 ( defne the external potental V ext ( and thus all propertes (except for constant) 1 6

7 r r r r 13 r r In prncple, one can fnd all other propertes and they are functonals of 0 (. HK F T V V r HK V ext nt r r r d ext 14 7

8 QC v ext ( r ( ( DFT- HK 0 0 All observables How to get the wf from the densty? How do I know, gven an arbtrary functon (, that t s a densty comng from an antsymmetrc N-body wavefuncton Ψ(r 1,...,r N )? N-representablty Solved: any square-ntegrable nonnegatve functon satsfes t There s an nfnte number of w.f. yeldng How to get the ground state densty (? How do I know, gven an arbtrary functon (, that t s the ground state densty of a local potental v(? V-representablty Constraned-search formalsm (Levy-Leb) Defnton of the unversal functonal 15 Constraned search formalsm (Levy and Leb (1977)) Double mnmzaton procedure: 0 mn mn Tˆ Vˆ ext Vˆ ee 1 Wavefunctons gvng densty ( r ) Tˆ Vˆ 0 mn mn ee ( r ) V F mn Tˆ ˆ V ee ext Defnton of the unversal functonal : ˆ ˆ F mn T V ee 16 8

9 Double mnmsaton or how to fnd the tallest chld n a school? 1. Fnd the tallest n each classroom mn. Fnd the tallest of the tallest mn 17 From Parr&Yang Varous extensons have been proposed: xtensons to spn dependent systems (Barth, Hedn, 197) [ n, n ] xtenson to relatvstc systems (Vgnale, Kohn, 1988) [ jr ( )] xtenson to fnte temperatures Fn [ ] n [ ] TSn [ ] Tme-Dependent DFT (Runge, Gross, 1984) 18 9

10 What have we ganed so far? Apparently Nothng: The only result s that the densty determnes the potental We are stll left wth the orgnal many-body problem If you don t lke the answer, change the queston Kohn and Sham ansatz (1965) Replace the nteractng-partcles hamltonan wth one that t can be solved more easly KS hamltonan: an hamltonan descrbng N non-nteractng partcles assumed to have the same densty as the true nteractng system. Hˆ KS 1 veff r 1/ det... N! 1 3 N 19 Kohn and Sham ansatz vsualzed Hˆ 1 KS v eff r As f non-nteractng electrons n an effectve (self-consstent) potental From. Artacho 0 10

11 Let us start agan wth the defnton of the energy : HK HK TVnt Vext ( ( xpress the energy usng the KS ansatz TS J XC Vext ( ( And get the defnton of exchange correlaton energy : XC HK TS J TT V J S ee Knetc All non classcal contrbutons (.e. non Coulomb, J) to electron-electron nteracton 1 Hˆ HK KS TS J XC Vext ( ( 1 ˆ KS H Hˆ KS v eff r 1 ( r' ) ' vxc r r r' r vext 1 v Hartree r r, 1, N, r v xc r v ext r HF lke equatons v xc xc r 11

12 Intal guess: and Calculate the effectve potental (v eff ) r v r v r v r How does t works n practse v Smlartes wth HF: eff Hartree xc ext A bass set s stll needed, but can be more flexble (numercal bass functons) Soluton of secular equaton Solve KS equatons (v eff ) SCF procedure s stll used Hˆ 1 KS v eff r Calculate electron densty ( f Dfferences wth HF: e - correlaton s mplctly ncluded The soluton of the secular equaton s computatonally more effcent formally scales as N 3 as opposed to N 4. No Yes Happy end Test convergence Output quanttes 3 Advantages of KS equatons All dffcult terms to be computed are collected n xc If v xc s exact the KS soluton s exact - exchange s ncluded - correlaton s ncluded - knetc energy s computed from orbtals (contrary to TF theory) - scalng s O(N 3 ) (1). The wave functon of an N-electron system ncludes 3N varables, whle the densty, no matter how large the system s, has only three varables x, y, and z. Movngfrom[] to[] n computatonal chemstry sgnfcantly reduces the computatonal effort needed to understand electronc propertes of atoms, molecules, and solds. (). Formulaton along ths lne provdes the possblty of the lnear scalng algorthm, whose computatonal complexty goes lke O(NlogN), essentally lnear n N when N s very large. (3). The other advantage of DFT s that t provdes some chemcally mportant concepts, such as electronegatvty (chemcal potental), hardness (softness), Fuku functon, response functon, etc.. 4 1

13 Meanng of KS genvalues ( f ( f= occupaton number (could be fractonal) The onzaton potental s I k N 1 k N k 1 v d fk k f k fk d fk Usng the mean-value approxmaton I 0.5 k k Ths s the so-called Janak-(Slate transton state theorem It could be consdered as the equvalent of the Koopman s theorem 5 The problem: v xc exact s unknown Dfferent approxmate forms of vxc have been proposed. The theory s exact, the functonals are approxmate 6 13

14 Consequences of the use of approxmate v xc : self nteracton error (SI) TS J XC Vext ( ( HK TS J T T V J HK XC HF H 1 1,N S, j1,n ee In exact KS as n HF: there s no Coulomb nteracton of one electron wth tself SI(N) J j K j But n approxmate DFT ths s not the case: J xc,0 7 ffects of self nteracton error He + BLYP LDA (He + +He) He+ PW91 CCSD(T) (He +0.5 ) 8 14

15 How to get an approxmate xc functonal? Contans nformaton on the many-body system of nteractng electrons The easest way: Local Densty Approxmaton LDA Assume the functonal s the same as a model problem the homogeneous electron gas Separate the exchange and correlaton contrbutons: xc = x + c xc can be calculated as a functon of the densty only 9 LDA model problem : the homogeneous electron gas LDA XC [ ] ( XC ( ( ) ( X ( ( ) C ( ( ) probablty of fndng the partcle at r homogeneous electron gas exchangecorrelaton energy per partcle (at the pont The value of the xc energy depends only on the local densty. The e- densty () may vary as a functon of r, but s sngle-valued, and the fluctuatons n away from r do not affect the value of xc at r

16 Is the Local Densty Approxmaton physcally soundng? around each electron other electrons tend to be excluded Defnton of x-c hole : ( r, r') xc xcs the nteracton of the electron wth the hole : t nvolves only a sphercal average ( r,r' ) [ ] r xc ' Sphercal average xc r - r' around electron xchange hole n Ne atom Gunnarsson, et. al. nucleus Very non-sphercal! electron Sphercal average very close to the hole n a homogeneous electron gas! 50 n x (r, r ) xact r r r r r r o = LD (r /o 0 n x(r, r ) + xact r = 0.4 o 0 LD (r /o xact xact LD + LD r n x 5.0 (r, r ) r r r = 0.09 o (a) r /o n r n x 3.0 (r, r ) r = 0.4 o (b) r xchange-correlaton (x-c) hole n slcon Calculated by Monte Carlo methods xchange Correlaton (a) (b) Hole s reasonably well localzed near the electron Supports a local approxmaton Hood et al 3 16

17 LDA model problem : the homogeneous electron gas LDA XC [ ] ( XC ( ( ) ( X ( ( ) C ( ( ) Get an expresson for them 33 LDA : xchange part LDA C x( X (per partcle) 1/ 3 Derved by Bloch et Drac (199/1930) for homogeneous electron gas Functonal form dentcal to that of Slater (HFS) Usually called Slater exchange functonal LDA X [ ] ( X ( ( ) LDA : Correlaton part No explct formulaton Approxmate analytcal expresson to reproduce accurate quantum Monte-Carlo (Ceperly & Alder, 1980) results for a homogeneous electron gas Most used LDA approxmaton for correlaton Volsko, Wlk et Nusar (1980): VWN. LDA A x b Q bx0 ( b x0 ) Q C ln arctan ln( x x0 ) arctan X ( x) Q x b X ( x0 ) Q x b34 17

18 LDA: how t works v xc Dscontnuty of the potental for the fllng of a electronc shell xc r xc xc r From A. V. Morozov 35 LDA: how t works A: Hgh densty, large knetc energy, LDA approxmaton unmportant B: Small densty gradent, LDA s good C:large gradent, LDA fals 36 18

19 Open shell systems and Local Spn Densty Approxmaton (LSDA) Dfferent denstes for dfferent spns : splt the total densty ( ( ( LSDA XC [, ] ( XC ( (, ( ) Measure of spn polarzaton : ( r ) ( r ) ( Remark : n prncple, snce the external potental s spn ndependent there s no need to splt the dfferent spn denstes 37 How to amelorate the LDA? Atoms, molecules or solds are not a homogeneous electron gas : Include non local effects GA (Gradent xpanson Approxmaton): F(( GA XC [, ] ( XC (, ), ' Taylor expanson C, XC (, ) / 3 / 3... Note: mathematcally speakng GAs are stll local How do they work? Not a great mprovement Reason : xchange correlaton hole propertes not satsfed 38 19

20 GGA (Generalzed Gradent Approxmaton) Impose the fulfllment of the propertes of the exchange-correlaton hole GGA XC [, ] f (,,, ) GGA XC GGA X GGA C GGA LDA X X F( s ) ( where s s the reduced densty Measure local nhomogenty s 4 / 3 ( 4 / 3 ( r ) ( 3 S hgh for hgh gradent or small densty regons (far from nucle) S small for small gradents (bondng regon) S ntermedate for hgh gradent and small densty (near the nucle) 39 xample of commonly used exchange functonals Becke, 1988 (B ou B88) s F B s snh s Perdew, 1986 (P ou P86) 86 F P s (4 ) 1/ 3 s 14 (4 ) 1/ 3 4 s 0. (4 ) 1/ 3 6 1/15 Functonal form can get extremely complex (especally for correlaton functonals) Problem: How to get a GGA? 40 0

21 Parametrzed functonals sem-emprcaluse adjusted parameters to reproduced exact or expermental data (ex. atomc energes) Over parametrsaton Sem-emprcal Good propertes Non parametrzed functonals mpose physcal constrans (unform electron gas lmt, asymptotc behavo «unversal» (worse) chemcal propertes Bref (and extremely non exhaustve) lst of commonly used GGA functonals F authors x F authors c B PW91 PB mpw HCTH B97 Becke (1988) Perdew et Wang (1991) Perdew, Burke, rnzerhof (1996) Adamo, Barone (1997) Handy et al. (1999) Becke P86 LYP PW91 PB Perdew (1986) Parr et al. (1988) Perdew et Wang (1991) Perdew, Burke, rnzerhof (1996) 41 Some theoretcal contrants Sze consstency : (AB)=(A)+(B) Vral Theorem Self-nteracton error Janak Theorem Leb-Oxfod bound Coordnate scalng Hrao JCP

22 Leb-Oxford Lmt Bondng regon Adamo JCP Atomsaton energes (n ev) of several molecules: theory vs experment. HF LDA GGA xp. H H O HF O F CH Solds : damond Property HF LDA GGA xp. a 0, Å a, ev K 0, GPa

23 How to mprove GGA results? HAVN (chemcal accuracy) rung 5 rung 4 rung 3 rung + explct dependence on unoccuped orbtals + explct dependence on occuped orbtals + explct dependence on knetc energy densty +explct dependence on gradents of the densty John Perdew Jacob Ladder* fully nonlocal hybrd functonals meta-ggas GGAs rung 1 local densty only LDA ARTH (Hartree theory) *DFT conference Menton 000 Update 007: Rung 4bs: Hyper GGAs 45 Phlppe Ratner meta-gga or dependent F xc 1, occ knetc energy densty Introduce quas-local nformaton 46 3

24 How to mprove the exchange energy : Hybrd Functonals Idea: HF exchange s exact. Therefore we could thnk to combne exact (HF) exchange wth a (GGA ) correlaton functonal (Le & Clement 1974) XC exact X KS C How does t work? Very bad! Mean average error 3 kcal/mole for the G ensemble (50 molecules) whle a MA of 5-7 kcal/mol s obtaned wth standard GGA Actually combnng a percentage of HF exchange wth a GGA exchange works much better!!! Theoretcal justfcaton : adabatc connecton (Becke 1993 Half and Half) In practse : HF% between 0 and 30% 47 B3LYP B3 xc a x0 LSD x (1 a xo ) HF x a x1 B x LSD c a c PW 91 c (Becke, JCP 1993) 3 parameters ftted on G (onzaton and atomzaton energes) PB0 PB0 xc 1 4 HF x 3 4 (Adamo, Scusera JCP 1999) No ftted parameters PB x PB c Normally hybrd functonals outperform all GGA and meta-gga functonals and they are the reference for chemcal applcatons 48 4

25 Method Dstance (Å) D 0 (kcal/mol) Dpole moment (D) Harmonc freq. (cm -1 ) HF & post-hf HF MP CCSD[T] Performance of selected functonals datomc molecules LDA & GGA LSDA BPW BLYP LGLYP PWPW mpwpw Hybrd 3 parameters B3LYP B3PW mpw3pw Hybrd ab-nto B1LYP B1PW LG1LYP Performance of selected functonals MA for harmonc frequences (cm -1 ) (G set,>50 organc molecules) Level of thoery rrorr HF and post-hf HF/6-311G(3df,p) 144 MP/6-31G(d,p) 99 CCSD/6-311G(3df,p) 31 LSDA SVWN/6-31G(d,p) 75 GGA BLYP/6-311G(d,p) 59 BPW91/6-311G(d,p) 69 PWPW91/6-311G(d,p) 66 mpwpw91/6-311g(d,p) 66 Hybrd functonals B1LYP/6-311G(d,p) 33 B1PW91/6-311G(d,p) 48 mpw1pw91/6-311g(d,p) 39 B3LYP/6-311G(d,p) 31 B3PW91/6-311G(d,p) 45 mpw3pw91/6-311g(d,p)

26 Rutle - GTO 51 The XC functonal can be crucal for chemcal understandng Homogeneous catalyss of ethylene R growng chan R 1 N N Al R TS 13 P14 R The catalyst termnaton R 1 = H, R = so-propyl p py ; R 1 = tert-butyl, R = soprop R=growng chan BHT TS 15 G. Talarco, P. H. M. Budzelaar, V. Barone and C. Adamo Chem. Phys. Lett. 39, 99, (000). G. Talarco, V. Barone, P. H. M. Budzelaar and C. Adamo, J. Phys. Chem. A, 105 (001) 9014 G. Talarco, V. Barone, L. Joubert and C. Adamo Int. J. Quantum Chem. 91 (003)

27 functonal f best est. MP VSXC B1Bc95 PB0 B98 mpw0 B1LYP B3PW91 B3LYP BP86 BLYP termnaton channel Dfferent functonals dfferent CHMICAL answers. best est. MP VSXC B1Bc95 PB0 B98 mpw0 B1LYP B3PW91 # ns B3LYP BP86 BLYP fu unctonal nserton channel (kcal/mol) # BHT (kcal/mol) BP86 termnaton most probable (low m.w.) VSXC nserton most probable (hgher m.w.) 53 Known breakdowns of DFT Many efforts to assess the relablty of DFT by a tral-and-error approach among others reacton barrers, CT complexes, Rydberg exctatons, IP from Koopmans, band gap, bond length alternaton, magnetc propertes, p vdw nteractons, (Un)Known causes? Localzaton Self Interacton rror (SI) Non-dynamc correlaton effects All problems comng from the approxmate nature of the XC contrbuton The theory s exact, the functonals are approxmate 54 7

28 Challenges n DFT Better functonals (e.g. CR, MR-05) rror correctons (e.g. SIC, ad-hoc parametrzaton) Statc correlaton (e.g. MC-DFT, RSH) Localzaton vs delocalzaton In chemcal language 55 8