Density functional theory

Transcription

1 Densty functonal theoy Tadtonal ab nto: fndng the N-electon wavefunton Ψ(1,,,N) dependng on 4N cood. DFT: fndng the total electon spn-denstes dependng on 8 coodnates Hohenbeg & Kohn: E0[ ρ0] = ρ0( ) VNed + T[ ρ0] + Eee[ ρ0] system dependent unvesally vald E [ ρ ] = ρ ( ) V d + F [ ρ ] Ne HK 0 Hohenbeg-Kohn functonal: Knetc enegy of electon Coulomb epulson Non-classcal nteacton (self-nteacton, exchange, and coelaton) All popetes (defned by V ext ) ae detemned by the gound state densty H&K only poofed that F HK exst, howeve, we do not know t H&K do not gve a decton how can we fnd densty H&K theoems allow us to constuct the goous many-body theoy usng densty as a fundamental popetes F[ ρ( )] = T[ ρ( )] + J[ ρ( )] + E [ ρ( )] ncl An old Thomas-Fem-Dac model s a DFT model wth appoxmate xc functonal

2 Thomas-Fem Model T TF ρ [ ()] = 3 10 (3π ) / 3 ρ 5 / 3 ()d 1/ /3 X [ ρ( )] ρ ( ) E = d 4 π 4/3 [ ( )] ρ ( ) EC ρ = d /3 + ρ ( ) Thomas-Fem knetc enegy (C k =.871 a.u.) Dac (1930) exchange (C X =0.739 a.u.) => Thomas-Fem-Dac model (TFD) Local appoxmaton to homogeneous electon gas Wgne (1938) 1 ρ( ) ρ( ) E [ ρ( )] = C ρ ( ) d ρ( ) v ( ) d + d d C ρ ( ) d + E [ ρ( )] 5/3 1 4/3 TFD K ext 1 X C 1 Thomas-Fem-Dac equaton Soluton vaatonal pncple; addng the constant of a fxed numbe of electons δ δρ( ) ( E THD ρ µ ρ d) [ ] ( ) = 0 5 ρ( ) 4 µ = C ρ ( ) + v ( ) + d C ρ + µ [ ρ( )] 3 3 /3 1/3 K ext 1 X C 1 Rgoously poved that molecules ae less stable than the fagments! Man poblem comes fom knetc enegy descpton. δec[ ρ( )] δρ( ) Coelaton potental

3 Thomas-Fem Type Models T TF eplaced by moe ealstc evaluaton fo one-electon obtals => Iteatve soluton Intoducng the efeence system of non-nteactng electons => Reques efeence potental V S Sgnfcant mpovement => descbes chemcal bond! Kohn-Sham Type Models

4 Kohn-Sham Appoach Adoptng a bette expesson fo knetc enegy: Usng exact knetc enegy of the non-nteactng efeence system that has the same densty as a eal one. Such knetc enegy cannot be the same as a tue one; t s expected to be close. Resdual pat of knetc enegy (T C ) s shfted to the functonal. Kohn-Sham functonal s then: E XC [ ρ] T S 1 = T S T N ( T[ ρ] T [ ρ] ) + ( E [ ρ] J[ ρ] ) = T [ ρ] + E [ ρ] S ee C ϕ F[ ρ()] = T [ ρ()] + J[ ρ()] + E S XC ϕ ncl [ ρ()] Knetc enegy of non-nteactng efeence system Coulomb epulson of uncoelated denstes HK theoems ( ρ + ρ F[ ] ()V d ) E = 0 mn Ne ρ N F[ ρ()] = T[ ρ()] + J[ ρ()] + E ncl [ ρ()] Exchange-coelaton functonal Includes: Electon exchange Electon coelaton Resdual pat of knetc en.

5 Puttng thngs togethe: E[ ρ()] = T [ ρ] + J[ ρ] + E S XC [ ρ] + E Ne [ ρ] 1 ρ(1 ) ρ( ) = TS[ ρ] + d1 d + E XC[ ρ] + VNeρ()d 1 N N N = ϕ ϕ + ϕ ( x1 ) ϕ j ( x ) dx1dx j 1 + E XC [ ρ()] N M ZA ϕ ( x ) A 1A 1 dx 1 Applyng vatatonal pncple Contans all poblematc tems 1 ρ( ) + d + VXC (1 ) 1 1 = + Veff (1 ) ϕ = εϕ V S () V eff M A Z A 1A ϕ = ε ϕ Satsfyng the condtons stated fo non-nteactng efeence system () = ρ( ) d + VXC (1 ) 1 M A Z A 1A Iteatve soluton. What s V XC? If V XC s know => Exact theoy V XC δe δρ XC

6 Kohn-Sham vs. Hatee-Fock 1. SIC poblem not pesent n HF (J =K ).. Effectve KS potental V KS ncludes also elekton coelaton. 3. Wave functon n the fom of Slate detemnant s an exact wave functon fo the KS efeence system (defnton). It s not an exact wave functon of the tue nteactng system. Exact wave functon s not known (HK theoems). Densty can expessed fom one-electon functons consttutng sngle SD: YES non-nteactng pue-state-v S epesentable NO - non-nteactng enesmbe-v S epesentable => Poblem! (TS mostly not pue-state-v S epesentable; shown fo exact wf) 4. Slate detemnant n HF s not exact wave functon t s only a consequence of model of ndependent electons. 5. HF vs. KS obtals. 6. Koopman s theoem does not hold fo KS obtals. 7. Janak s theoem vald fo KS: negatve of HOMO enegy coesponds to 1 st onzaton potental. 8. V S s local => V XC must be local! Contay to V eff (HF) that s non-local. 9. KS equaton fomally less complcated that HF equaton, nevetheless, they ae (n pncple) exact. 10. Unestcted fomulaton. 11. Impovng the HF descpton => accountng fo electon coelaton: systematc way. Impovng the appoxmate V ks (E XC )? 1. Relablty fo small/lage systems. M ρ( ) Z A VS () Veff () = d + VXC (1 ) 1 A 1A

7 Sngle SD based densty: Spn o spatal degeneacy (most atoms) => appoxmate xc functonals have poblem!

8 Sze-consstency poblem!

9 Kohn-Sham vs. Hatee-Fock 1. SIC poblem not pesent n HF (J =K ).. Effectve KS potental V KS ncludes also elekton coelaton. 3. Wave functon n the fom of Slate detemnant s an exact wave functon fo the KS efeence system (defnton). It s not an exact wave functon of the tue nteactng system. Exact wave functon s not known (HK theoems). Densty can expessed fom one-electon functons consttutng sngle SD: YES non-nteactng pue-state-v S epesentable NO - non-nteactng enesmbe-v S epesentable => Poblem! (TS mostly not pue-state-v S epesentable; shown fo exact wf) 4. Slate detemnant n HF s not exact wave functon t s only a consequence of model of ndependent electons. 5. HF vs. KS obtals. 6. Koopman s theoem does not hold fo KS obtals. 7. Janak s theoem vald fo KS: negatve of HOMO enegy coesponds to 1 st onzaton potental. 8. V S s local => V XC must be local! Contay to V eff (HF) that s non-local. 9. KS equaton fomally less complcated that HF equaton, nevetheless, they ae (n pncple) exact. 10. Unestcted fomulaton. 11. Impovng the HF descpton => accountng fo electon coelaton: systematc way. Impovng the appoxmate V ks (E XC )? 1. Relablty fo small/lage systems. M ρ( ) Z A VS () Veff () = d + VXC (1 ) 1 A 1A

10 CCSD(T) Statonay Schödnge equaton H Ψ = EΨ MP Electon coelaton Expanson ove Slate det. Φ= C0Ψ 0 + CSΨ S + CDΨ D + Non-elatvstc Hamltonan Bon-Oppenheme appoxmaon occ Electon Densty ρ( ) ϕ ( ) One-el. Functons Tadtonal Ab nto = ϕ(1) = c µ χ µ (1) DFT µ Hybd functonals B3LYP, B3PW91,... Genealzed gadent appoxmaton (GGA) E E[ ρ, ρ] PW91, BP86, BLYP, PBE,... Post-HF methods Model of ndependent electons ˆ el eff H (, j) V () Non-nteactng efeence system Kohn-Sham obtals Hatee-Fock method φ (1) HF obtals 1 Ψ (1,,..., n) = det ϕ1(1) ϕ()... ϕn( n) n! Electon coelaton neglected Local densty appoxmaton LDA (LSD, SVWN) E E[ ρ]

11 Appoxmate Exchange-Coelaton Functonals Kohn-Sham equatons (geneal, does not depend on the fom of patcula funtonal) 1 V ( ) ϕ εϕ + eff 1 = V S () V eff () = ρ( ) d + VXC (1 ) 1 M A Z A 1A V XC δ EXC δρ Local Densty Appoxmaton E XC components deved fom the unfom electon gas LDA E [ ρ] = ρ() ε ( ρ()) d ε XC exchange-coelaton enegy pe patcle of UEG ε XC XC XC ( ρ()) = εx( ρ()) + εc( ρ()) Splttng nto exchange and coelaton pat (T C neglected) 3 3 ρ( ) ε [ ] 3 X ρ = 4 π S used by Slate (Dac, 1930) ε PZ C Aln S + B+ CS ln S + DS, S 1 [ ρ] = γ /1 ( + β1 S + βs), S > 1 PZ (Pedew-Zunge, 1981) VWN(1-5) (Vosk et al., 1980) => Ft to QMC denstes Constant fom QMC ft Hgh- and low-denstes

12 LDA uses fo E XC expessons fom UEG s ths acceptable fo molecules? LDA woks bette than expected - h XC satsfed most of the ules. LDA holes ae easonable fo small dstances between efeence and the othe electon. LDA holes ae poblematc mostly at the egons wth hghly ansotopc densty (atoms).

13 Tends wthn the LDA/LSDA It favous electonc denstes that ae moe homogeneous than the exact ones (exchange hole s sphecally symmetcal) (Consequently) t oveestmates the bndng n molecules. (Consequently) t gves too shot bond lengths. It does not account fo dspeson nteacton, nevetheless t can stll bnd van de Waals complesxes; fo a wong eason! Numbe of applcatons, n patcula n physcs Untll today used fo metals, gaphte, o even weakly bounded molecula cystals. Geneal lmtatons of LDA 1. Poo denstes n a coe egon. (Insuffcent cancellaton of self-nteacton n localzed dense coe.. Too hgh atomc eneges (hghe than HF and expement). e 3. Incoect decay of exchange-coelaton potental exponental (as the densty) whle t should be Poblematc fo all fnte systems, ncludng sufaces. 4. Fals to descbe negatvely chaged ons (due to too fast decay). 5. H-bonds pooly descbed. Possble mpovments () Consdeng nhomogeneous denstes. () Impovng the self-nteacton poblem. () Accountng fo non-local exchange and coelaton. GGA, hybd, meta- functonals

14 Genealzed Gadent Appoxmaton: (GGA) E XC depends not only on densty but also on the densty gadents E [ ρ, ρ ] = f ( ρ, ρ, ρ, ρ ) d GGA XC α β α β α β Patcula foms of GGA functonals should be athe vewed as mathematcal concept GGA LSD 4/3 ρσ ( ) EX = EX F( sσ) ρσ ( ) d sσ ( ) = 4/3 σ ρσ ( ) Denomnato makes GGA coecton mpotant also fo Reduced densty gadent egon wth small densty (local nhomogenety) (extended valence egon) Example: E X deved by Becke B (=B88) β=0.004 (empcal) F B β sσ = 1 + 6β s snh σ 1 s σ Example: E X deved by Pedew P86 paamete fee F P86 4 sσ sσ sσ = / 3 1/ 3 (4π ) (4π ) (4π ) 1/ 3 6 1/15

15 PBE Pedew, Buke, Enzehof Functonal that satsfes most of the condtons equed (bounday condtons, popetes of holes) Meta-GGA Consdeng the fouth ode gadent expanson ncludes second devatves of denstes ( knetc enegy densty ) TPSS OEP Optmzed effectve potental methods Includes the exact exchange no need fo h X, only h C constacted Hybd Densty Functonals: HF exchange s mxed nto the functonal fom Example: B3PW91 (Becke) a, b, and c - ftted paametes E = E + a( E E ) + be + ce B3 LSD λ= 0 LSD B PW 91 XC XC XC X X C Non-local functonals: vdw-df1 van de Waals densty functonal (Don, Rydbeg, Schode, Langeth, Lundqvst, 004) vdw-df mpoved long-ange assymptote Román-Péez & Sole effcent mplementaton ecpocal space N speeds up to NlogN

16 vdw-df: DF1 vs. DF

17

18 Seveal most commonly used functonals Abbevaton Type Exchange Coelat Authos pat on pat S LDA + - Slate (Dac) VWN LDA - + Vosko, Wlk, Nusa (1980) B, B88 GGA + - Becke (1988) LYP GGA - + Lee, Young, Pa (1988) PW91 GGA + + Pedew, Wang (199) P86 (P) GGA - + Pedew (1986) PBE GGA + + Pedew, Buke, Enzehof (1996) B3 Hybd + - Becke

19 4.5 Themocheme Enfluss von Gadentenkoektufunktonalen auf de Dchte m F ρ MP -ρ HF ρ SVWN -ρ HF ρ MP -ρ BLYP Isopyknen be 0.00 e - ρ SVWN -ρ BVWN ρ SVWN -ρ SLYP ρ SVWN -ρ BLYP Theoetsche Cheme III D. Max Holthausen

20 Dpole moments fo selected molecules [n D, 1 D = a.u.] Molecule HF MP SVWN SVWN BLYP BLYP BLYP B3LYP B3LYP Exp. POL a POL a numecal b TZVP-FIP c TZVP-FIP c POL a 6-31G(d) d cc-pvtz POL a CO H O H S HF HCl NH PH SO a taken fom Cohen and Tantungotecha, 1999; b taken fom Dckson and Becke, 1996 c taken fom Calamnc, Jug and Köste, 1998, d taken fom Johnson, Gll and Pople, 1993 Bae heghts of H + H H + H [n kcal/mol] Method bae wthout SIC bae wth SIC LSD BLYP BPW B3LYP exp. 9.7 Taken fom Johnson, 1995 and Csonka and Johnson, 1998

21 SIC

22 Complaton of mean absolute devatons fo bond lengths [Å] / bond angles [degees] fo small man goup molecules fom dffeent souces. 3 fst ow speces, 6-31G(d) bass set, Johnson, Gll, and Pople, 1993 HF 0.00 /.0 SVWN 0.01 / 1.9 MP / 1.8 BLYP 0.00 /.3 QCISD / speces, Matn, El-Yazal, and Fanços, 1995a CCSD(T)/cc-pVDZ /. B3LYP/cc-pVDZ / 1.7 CCSD(T)/cc-pVTZ / 0.6 B3LYP/cc-pVTZ / 0.3 CCSD(T)/cc-pVQZ 0.00 / 0.4 B3LYP/cc-pVQZ / speces cont. thd ow elements, 6-31G(d) bass set, Redfen, Blaudeau and Cutss, 1997 MP 0.0 / 0.4 B3LYP / 0.5 BLYP / 1.0 B3PW / 0.5 BPW / 0.5 a uncontacted aug-cc-pvtz bass

23 Devatons between computed atomzaton eneges and expement fo the JGP test set employng the 6-31G(d) bass set [n kcal/mol]. Taken fom Johnson, Gll and Pople, HF MP QCISD SVWN SLYP BVWN BLYP mean abs. dev. a (40) a 38 4 (4) a 6 mean dev (40) a 38 0 (4) a 1 a Bass set fee esults taken fom Becke, 199.