Density Functional Theory I

Transcription

1 Densty Functonal Theoy I cholas M. Hason Depatment of Chemsty Impeal College Lonon & Computatonal Mateals Scence Daesbuy Laboatoy ncholas.hason@c.ac.uk

2 Densty Functonal Theoy I The Many Electon Schönge Equaton Hatee-Fock Theoy Solvng the Schönge Equaton Avong Solvng the Schönge Equaton Densty Matces The basc eas of DFT Kohn-Sham an the non-nteactng system The local ensty appoxmaton LDA Conclusons Densty Functonal Theoy II

3 The Schönge Equaton fo Many Electons Tme nepenent non-elatvstc Bon-Oppenheme A lnea equaton n 3 vaables < j j ext V H E H Ψ Ψ Thee tems knetc enegy extenal potental an electon-electon.

4 The Extenal Potental The nteacton wth the atomc nucle s: V ext at Zα R α α The extenal potental an the numbe of electons completely etemne the Hamltonan.

5 The Vaatonal Pncples o any legal wavefuncton antsymmetc nomalse e enegy s E * [ Ψ] Ψ HΨ Ψ H Ψ - the enegy s a functonal of Ψ E[ Ψ] E 0 - Seach all Ψ to mnmse E > the goun state

6 Hatee Fock Theoy An ansatz fo the stuctue of the wavefuncton [ ] HF... et! 3 Ψ j j j j j j ext HF V E * * * * *

7 The Hatee Fock Equatons non-nteactng electons une the nfluence of a mean fel potental consstng of the classcal Coulomb potental an a nonlocal exchange potental. X ext v V ε j j j X v *

8 Beyon Hatee-Fock Many methos/appoxmatons applcable to small systems. Expensve & scalng wth poblem sze s feocous Eg: MP MP3 MP4 ~ CISD ~ 6 CCSD ~ 6 CCSDT ~ 7

9 Avong Solvng the Schonge Equaton Is t necessay to solve the Schönge equaton an etemne the 3 mensonal wavefuncton n oe to compute the goun state enegy?.. o!

10 The Pa Densty The secon oe ensty matx s efne as P ; 4 3 * Ψ Ψ The agonal elements ae the two patcle ensty matx o pa ensty functon ; P P Ths object n 6 mensons s all we nee to compute the exact total enegy!

11 The Enegy an the Densty Matces The fst oe ensty matx s H only contans one-electon an two-electon opeatos - the enegy can be wtten exactly n tems of P an P P P ; ; R P P Z E at α α α

12 The enegy s a functonal of P : E[P ] Pehaps solve usng by mnmsng E[P ]? Vey ffcult poblem A legal P must be constuctble fom an antsymmetc Ψ. Applyng ths constant n pactce s non tval. Lfe woul be much ease f thee was a way of ong t..!

13 Do we eally nee to know P? o To fn the exact total enegy knowlege of the chage ensty s enough!

14 DFT the theoems Theoem. The extenal potental s unquely etemne by the ensty - - so the total enegy s a unque functonal of the ensty - E[]! Theoem. The ensty whch mnmses the enegy s the goun state ensty an the mnmum enegy s the goun state enegy. [ ] E0 Mn E Hohenbug-Kohn 964 Mel Levy E B Wlson

15 Theoem Wlson s poof The chage ensty has a cusp at the nucleus of any atom such that 0 0 α α α α Z The chage ensty unquely etemnes the Hamltonan thus the wavefuncton an all mateal popetes!!!

16 Densty Functonal Theoy The funamental statement of DFT s δ [ ] E[ ] µ 0 thee s a unvesal functonal E[] whch coul be nsete nto the above equaton an mnmse to obtan the exact goun state ensty an enegy.

17 What s the functonal? Thee ae thee tems n the Hamltonan V ext [] s tval E [ ] T[ ] V ext [ ] V [ ] ee V [ ] V ext ext T an V ee ae vey ffcult to appoxmate!

18 E xc [] - Popetes Does not epen on V ext the specfc system: t s a unvesal functonal. The exact epenence on s unknown E xc can be exactly etemne fo any specfc ensty but the effot s geate than fo usual many-boy calculatons E xc must be appoxmate n applcatons

19 The Homogeneous Electon Gas Fo the non-nteactng gas the knetc an exchange enegy pe patcle can be compute the sngle patcle wavefunctons ae smply plane waves. Pehaps ntegate these enegy enstes fo an nhomogeneous system? 3 T [ ] o chemcal bonng! 3 E x 0.74 E x s OK but what about T? 4 Thomas 97 Fem 98 Dac 930

20 A local functon of the ensty Thomas-Fem-Dac suggests: [ ] Exc ε xc ε ε ε xc x c ε x 3 C J. Slate 95 The functonal s a local functon of the ensty What to o about ε c?

21 E[] The Kohn Sham Appoach Wte the ensty n tems of a set of non-nteactng obtals The non nteactng knetc enegy an the classcal Coulomb nteacton s T ] [ ] [ V H Allow us to ecast the enegy functonal as: ] [ ] [ ] [ ] [ ] [ xc H ext s E V V T E ] [ ] [ ] [ ] [ ] [ H ee s xc V V T T E Whee we have ntouce

22 Vaaton Theoem > Kohn Sham Equatons Vay the enegy wth espect to the obtals an. o appoxmatons So If we knew E xc [] we coul solve fo the exact goun state enegy an ensty! Cost 3.. In pncple. ] [ xc xc E V ext V xc V ε

23 The on-nteactng system Thee exsts an effectve mean fel potental whch when apple to a system of non-nteactng femons wll geneate the exact goun state enegy an chage ensty!!! Ψ j... V xc E[]

24 Hatee Fock s a ensty functonal theoy Hatee-Fock theoy s a ensty functonal theoy! - but wth a non-local potental. j j j xc V V

25 Quantum Monte Calo Smulatons Fo the Homogeneous electon gas the exact epenence of ε xc can be compute. ε xc Cepeley an Ale 980

26 The Local Densty Appoxmaton - LDA LDA Exc [ ] ε xc ε xc Pctue coutesy of Aneas Savn

27 Conclusons I Fo the goun state enegy an ensty thee s an exact mappng between the many boy system an a fcttous nonnteactng system. - DFT-people stuy the fcttous system! The fcttous system s subject to an unknown potental eve fom the exchange-coelaton functonal The enegy functonal may be appoxmate as a local functon of the ensty!

28 Densty Functonal Theoy II Why oes the LDA wok? The exchange coelaton hole Compason wth exact exchange an coelaton enegy enstes Genealse gaent appoxmatons GGA s Sem-local nteactons: Meta-GGA s Hyb-exchange functonals Pefomance n molecules an sols