Outline. GW approximation. Electrons in solids. The Green Function. Total energy---well solved Single particle excitation---under developing

Transcription

1 Peenaon fo Theoecal Condened Mae Phyc n TU Beln Geen-Funcon and GW appoxmaon Xnzheng L Theoy Depamen FHI May.8h 2005

2 Elecon n old Oulne Toal enegy---well olved Sngle pacle excaon---unde developng The Geen Funcon GW appoxmaon 2

3 Pa I: Elecon n old Two majo phyc we wan o know fo elecon n old: Toal Enegy; gound ae----geomey of he cyal he elac popey fo magnec yem: magnec popey In expemen we ue he phooemon and nvee phooemon o pobe he occuped ae and unoccuped ae of old. Excaon Specum. exced ae----sngle pacle excaon n h alk 3

4 4

5 Exac oluon fo he wave funcon mpoble. Two way fo olvng h poblem. Appoxmae he wave funcon: Haee and Haee-Fock appoxmaon. Go aound he poblem: ue anohe quany nead of wave funcon o olve he poblem: moe accuae Pa one: Deny Funconal Theoy ---deny Pa wo: Geen Funcon mehod ---popagao Geen Funcon 5

6 Pa one: 6

7 DFT whn LDA aleady vey good n he oal enegy calculaon. In pecum calculaon hee ae ome poblem. 7

8 F-pncple calculaon woe han h. 8

9 Fom: Mebaek Alouan IPCMS hp://www-pcm.uabg.f/gemm/people/alouan/ The eo n LDA-DFT coeced by GW appoxmaon. Snce all DFT ae LDA-lke baed lack a eaonable decpon of exchange and coelaon. GW appoxmaon moe eaonable on h apec. Pa wo: Geen funcon mehod Pa one: DFT whn LDA In pncple Geen funcon mehod could alo be ued o calculae he oal enegy and he expecaon value of one-pacle opeao. Bu n h alk I wll focu on well behavo on he ngle pacle excaon. 9

10 Summay fo Pa I Wave funcon of many elecon yem no poble o be ge. DFT and Geen funcon mehod amed a go aound h poblem wh anohe quany. DFT wokng vey well on he oal enegy calculaon. Bu fo he ngle pacle excaon. o ye good enough. Geen funcon mehod wokng bee on ngle pacle excaon now. 0

11 Pa II: Geen Funcon Repeenaon n Quanum mechanc: Schödnge epeenaon. ˆ H H e 0 Wave funcon---me dependen Opeao lke Hamlona---me ndependen Heenbeg epeenaon. O [ O Hˆ ] O e H ˆ Wave funcon---me ndependen Opeao lke Hamlona---me dependen O 0 e Hˆ

12 Geneao nead of wave funcon The many-body wave funcon no poble o be ge. : v ˆ ˆ gound ae of a - elecon yem : Annhla on of one elecon a place v : Ceaon of one elecon a place me me v ˆ ˆ : ake one elecon a place me ou fom he gound ae of - v : pu one elecon a place me n o he gound ae of - elecon elecon ˆ [ ˆ Hˆ ] ˆ e H ˆ ˆ 0 e Hˆ 2

13 G ; ˆ ˆ ˆ ˆ fo > fo elecon < hole Heenbeg pcue: Wave funcon ae ndependen of me and opeao ae me dependen. h Hˆ ˆ d ˆ 2 [ ˆ Hˆ ] h d 0 d ˆ ˆ ˆ v ˆ ˆ 3

14 [ ] ˆ ˆ ˆ ˆ ; 3 0 T v d G h δ A wo-pacle Geen funcon appea hee o olve he wo-pacle Geen funcon we wll alo ge a hee pacle one n un. To beak h heachy we noduce he ma opeao M Haee poenal elf-enegy: [ ] T v d G M d ˆ ˆ ˆ ˆ 3 Then we ge: ; 3 0 G M d G h δ 4

15 v M H Σ δ Take: ; 3 0 G d G H Σ δ ; 0 0 G H δ on-neacng yem: The meanng of elf-enegy n analogy o one pacle yem: 0 H H H 0 0 E H Whou peubaon: [ ] 0 E H H Wh peubaon: H Σ 5 Peubaon Many body neacon

16 So fa we gave he defnon of Geen funcon and equaon fo. Wha he ue fo he Geen funcon? Many many n pncple can be ued o ge: The expecaon value of any ngle-pacle opeao n he gound ae The gound ae enegy The excaon pecum------in h alk only h. Refe o: La Hedn Phycal Revew vol 39 A

17 Infomaon fom he Geen funcon: G ; ˆ ˆ ˆ ˆ fo > fo elecon < hole 7

18 ˆ ˆ Relax ew concep: Quapacle ε Phyc meanng of quapacle nk Fo non-neacng yem: o E E E E ε Fo neacng yem: ˆ o ˆ o E E E E ε 8

19 9

20 20

21 2

22 So fa we ge he meanng of Popee of ε ε Fo non-neacng yem: nk ε E E o E E Fo neacng yem: ε ˆ o ˆ E E o E E Dffeence: Σ......no Heman ew concep: Quapacle And we need o know how o ge hem 22

23 > < ; * * unocc occ e e G ε ε ; 3 0 G M d G h δ Then we ge: 3 0 M d h ε v M H Σ δ Take: d H ε Σ

24 So fa: 3 0 d H ε Σ All he complcy of many body yem ae conaned hee. Poblem: Unknown Can be exended n em of V bu he convegence exemely bad. ; Σ GW appoxmaon 24

25 Pa III: GW appoxmaon o Quapacle neacon Elecon n old Beak when collon happen and fom agan afe ha. When he nex collon happen follow he ame poce agan and agan Elecon polazaon cloud 25

26 o o When he collon neval longe han he me needed o fom he cloud we can aume he elecon ae quapacle neacng wh each ohe hough a ceened Coulomb neacon. Th mean he elecon ae can no be oo localzed. F ode exenon of elf-enegy accodng o he ceened Coulomb neacon GW appoxmaon Refe o: La Hedn Phycal Revew vol 39 A

27 o GW appoxmaon Hedn GW elf - enegy: Σ ω 2π dω e ω δ G ; ω ω W ; ω Sceenedneacon : W ; ω d" ε "; ω v " Deleccfunon: ε ; ω δ d" v " P0 " ; ω Polazably : P0 " ; ω dω G ; ω ω G ; ω 2π Quapacle enegy: ε qp ε H qp Σ ; ε 27

28 Haee appoxmaon: Σ 0 Haee-Fock appoxmaon: Σ G ; v " δ " ac only exchange Kohn-Sham whn LDAGGA: XC Σ ; " " V δ " δ " coelaon paly decbed bu local and ac We need h whch phycally meanngful dynamc and non-local o expe he elf-enegy GW appoxmaon 28

29 Elecon n old V XC Kohn-Sham whn LDAGGA: Haee-Fock GW appoxmaon 29

30 Fom: Mebaek Alouan IPCMS hp://www-pcm.uabg.f/gemm/people/alouan/ So fa GW appoxmaon manly ued o coec he band ucue of old. Some ae gong beyond o udy he fuhe ode exenon of he elf-enegy. Peonally I hnk vey pey phyc whch decbe he complex many-body naue wh a eaonable effo. 30

31 Thank fo aenon!!! Thank o Rcado Iaac Gomez Abal Pack Rnke and Choph Feyold fo helpful dcuon! 3