Select/Special Topics in Atomic Physics

Transcription

1 Select/Specal Topcs n Atomc Physcs P. C. Deshmukh Depatment of Physcs Indan Insttute of Technology Madas Chenna pcd@physcs.tm.ac.n Unt 4() Lectue 9 Many-Electon Atoms: Hatee-Fock Self-Consstent Feld fomalsm -

2 Leanng Goals: Poblems of cuent nteest n the physcs of atoms, molecules and othe foms of condensed matte eque a thoough undestandng of electon nteactons and electon coelatons. coelatons exchange coulomb

3 Quantum Theoy of Many Electon Atoms els Boh Albet Ensten D.R.Hatee Vladm Fock 3

4 Pmay Refeences fo HF SCF method Intemedate quantum mechancs Hans A. Bethe and Roman W. Jackw Addson-Wesley, 997 Physcs of atoms and molecules B. H. Bansden and C. J. Joachan Publshe Pentce Hall, 003 P. C. Deshmukh, Alak Bank and Dlp Angom Hatee-Fock Self-Consstent Feld Method fo Many-Electon Systems Invted atcle n DST-SERC-School publcaton (aosa, ovembe 0); collecton of atcles based on lectue couse gven at the DST-SERC School at the Bla Insttute of Technology, Plan, Januay 9-8,

5 The Many Electon Atom: H ( ) ( ) ( ) ( ) ψ = E ψ ew dffcultes! The poblem can be posed fomally, but the vey conceptualzaton of the electon poblem leads to an mmedate CATCH stuaton 5

6 Catch- : novel by Joseph Helle. A stuaton nvolvng an absud ntenal nconsstency - - awkwad stuaton whose soluton s uled out by a constant ntnsc to the stuaton. 6

7 H H ( ) ( ) ( ) ( ) ψ Z ( ) = + = < = = E ψ H D.R.Hatee Cambdge, England Futhemoe: SPI! ψ = E ψ ( ) ( ) ( ) ( ) Appoxmate umecal Solutons Self- Consstent-Feld V.A.Fock

8 The Many Electon Atom: ew dffcultes! H ( ) ( ) ( ) ( ) ψ = E ψ the soluton to the electon Schodnge equaton eques the coespondng Hamltonan, but settng up the Hamltonan tself eques the vey same solutons! 8

9 H ( q,.., q ) ψ ( q,.., q ) = E ψ ( q,.., q ) ( ) ( ) ( ) ( ) Z ( ) (,,.., ) = + = < = H q q q = h ( q ) + 0 = = ; = = F + G = H + H } meely a matte of notaton 9

10 H ( q,.., q ) ψ ( q,.., q ) = E ψ ( q,.., q ) ( ) ( ) ( ) ( ) Exact Soluton? Havng no body at all s aleady too many G. E. Bown even f t wee possble to get an exact soluton, how much space, nk, stoage would be needed to wte the soluton? 0

11 even f t wee possble to get an exact soluton, how much space, nk, stoage would be needed to wte the soluton? Hatee/ Hemann-Skllman/Johnson: Fo electons descbed by only the 3 space coodnates: 3 vaables 3 Coase 0-pont gd: 0 numbes to tabulate! Estmate fo =, 0, 80 wll you? q = (, ζ ), space a nd 'spn' coodnate

12 A lage dffeental analyze, desgned by Hatee (935) Hatee s ntal nteest: ant-acaft gunney n Pototype fo ths: small-scale machne bult fom peces of chlden's Meccano - actually solved useful equatons concened wth Boh s lectue couse atomc theoy n 934. at Cambdge (9) nfluenced Hatee Hatee s fathe: Wllam Hatee taught Engneeng at Cambdge

13 When John Ecket set up EIAC, Hatee was asked to go to the USA to advse on ts use. Electonc umecal Integato and Compute Hatee showed how to use EIAC to calculate taectoes of poectles. It may well be that the hgh-speed dgtal compute wll have as geat an nfluence on cvlzaton as the advent of nuclea powe. - Hatee (Cambdge), n 946 3

14 Appox. umecal Solutons: Self- Consstent-Feld H ( ) ( ) ( ) ( ) ψ = E ψ SCF ( ) (,,.., ) = + = < = H q q q STRATEGY 0 = = ; = δψ Z = h ( q ) + = H + H ( ) ( ) ( ) H ψ = 0 COSTRAITS We need: = δ Ψ Ω Ψ wth Ω = F; Ω = G 4

15 Hatee and Hatee: self-consstent feld Fock: ncluson of spn 5

16 Goudsmt : One of the thngs whch stuck to me s that n Paschen's expements on the helum lne,.. thee was a fobdden component whch was obvously pesent.... but you know how theoetcans ae... they then say: Poo expements. Geoge Uhlenbeck and Samuel Goudsmt Hendk Kames, anothe of Ehenfest's students. abacadaba - Uhlenbeck knd of numeology - Goudsmt 6

17 97: Paul ntoduced the matces as a bass of spn opeatos. (nonelatvstc/ad-hoc theoy of spn). Paul Aden Mauce Dac 98: Dac Relatvstc Quantum Mechancs Povded fomal bass fo electon s spn 940: Paul poved the spn-statstcs theoem of quantum feld theoy - patcles wth half-ntege spn ae femons, whle patcles wth ntege spn ae bosons. 7

18 Electon spn has ts ogn n Relatvstc QM P.A.M.Dac 98 The quantum theoy of the electon Poc. R. Soc. (London) A The quantum theoy of the electon Pat II Poc. R. Soc. (London) A

19 How does STATISTICS ente classcal mechancs, and how does t ente quantum mechancs? Equpatton Theoem: Each degee of feedom n the classcal expesson fo the Hamltonan contbutes to the AVERAGE enegy. kt B In Quantum Theoy, STATISTICS entes though TWO channels: [] Uncetanty Pncple [] SPI ; Identty of Patcles 9

20 I I Ψ ( q, q ) =Ψ( q, q ) { } Intechange opeato actng TWICE on a gemnal wavefuncton 0

21 Two-electon state: gemnal ψ ( q, q )

22 Intechange opeato I { I Ψ ( q }, q) =Ψ( q, q) actng TWICE on a gemnal wavefuncton I Ψ ( q, q ) = e Ψ( q, q ) I Ψ ( q, q ) =±Ψ( q, q ) e = Femons: spn,,,... =, e =± 0 o Bosons o Femons Bosons : spn 0,,,3, 4,... π

23 I Ψ ( q, q ) = e Ψ( q, q ) I Ψ ( q, q ) =±Ψ( q, q ) Femons: spn,,,... e spn =, e =± = 0 o π Bosons o Femons Bosons : spn 0,,,3,4,... sgn of the wavefuncton on ntechange statstcs 3

24 RELATIO BETWEE SPI & STATISTICS IS APPARET, BUT HARD TO UDERSTAD. - Tomonaga It appeas to be one of the few places n physcs whee thee s a ule whch can be stated vey smply, but fo whch no one has found a smple and easy explanaton. The explanaton s down deep n elatvstc quantum mechancs.. - Feynman, Vol.3 p4-3 Recommended efeence: The Theoy of Spn by Sn-Ito Tomonaga, tanslated n Englsh by Takesh Oka (The Unv. of Chcago Pess, 997) 4

25 Fo electons: I Ψ ( q, q ) =Ψ ( q, q ) = Ψ( q, q ) sepaablty n ' two electon ' coodnates : [ ] Ψ ( q, q ) = u ( q ) u ( q ) u ( q ) u ( q ) u ( q ) = =, ζ n, l, m, m l s ndstngushable elementay patcles 5

26 u ( q ) = =, ζ n, l, m, m l s [ ] Ψ ( q, q ) = u ( q ) u ( q ) u ( q ) u ( q ) = u( q) u( q) u ( q ) u ( q ) Rows: occuped sngle patcle states (labeled by a set of 4 quantum numbes) n the many-electon system. Columns: set of (space, spn) coodnates John C. SLATER DETERMIAT Paul excluson pncple; Antsymmety of the wavefuncton 6

27 u( q) = =, ζ n, l, m, m ( ) ψ = l s u() u() u(3) u(4)! u ( ) electon confguaton Occupancy of sngle patcle states Rows: occuped of sngle patcle states (labeled by a set of 4 quantum numbes) n the many-electon system. Columns: set of (space, spn) coodnates SLATER DETERMIAT Paul excluson pncple; Antsymmety of the wavefuncton : column (space-spn coodnate) ndex; =,,3,. : (occuped) quantum state ndex; =,,3, 7

28 Matx elements of the SLATER DETERMIAT One-electon SPI-ORBITALS u( q) = u ( q ) =, ζ n, l, m, m l s u( q) = n, l, ml m Spn ζ s pat Obtal u( q) = ψ n,, ( ) ( ) l m χ ζ l m s pat 8

29 SLATER DETERMIAT th ψ ( ) ow u () u ( ) u() = = u( q)! u () u ( ) u ( q ) = ψ ( ) χ ( ζ ) Pobablty ampltude that an electon at space-spn coodnate n, l, m m q n, l, m, m l s s n the quantum state th column l s 9

30 () ( ) ψ = = u( q)! u () u ( ) u u () u ( ) u( q) = n, l, ml ζ ms = u ( ) χ ( ζ ) n, l, m m l s : column (space-spn coodnate) ndex; =,,3,. : (occuped) quantum state ndex; =,,3,! ways of pemutng the ndstngushable electons n the quantum states ψ (,.., ) ( ) ( ) ( ).. ( )! ( ),,... = P q q P u q u q u q! P =! P A = ( ) P! P = Antsymmetse opeato: Questons: pcd@physcs.tm.ac.n [ ] 30

31 Select/Specal Topcs n Atomc Physcs P. C. Deshmukh Depatment of Physcs Indan Insttute of Technology Madas Chenna pcd@physcs.tm.ac.n Unt 4() Lectue 0 Many-Electon Atoms: Hatee-Fock Self-Consstent Feld fomalsm - 3

32 u ( q ) = n, l, m ζ m l s = u ( ) χ ( ζ ) Spn-Obtal Thee s no classcal analog to the spn pat, no to the obtal pat! obtal n, l, m m l s spn 3

33 () ( ) ψ = = u( q)! u () u ( ) u u () u ( ) u( q) = n, l, ml ζ ms = u ( ) χ ( ζ ) n, l, m m l s : column (space-spn coodnate) ndex; =,,3,. : (occuped) quantum state ndex; =,,3,! ways of pemutng the ndstngushable electons n the quantum states ψ (,.., ) ( ) ( ) ( ).. ( )! ( ),,... = p q q P u q u q u q! P = Antsymmetse opeato: A [ ]! =! P = p ( ) P 33

34 l l = l l, l = l x y z l l, l m = l( l+ ), l m l, l m = m, l m z l l l s = : fxed ntenal popety of an electon ms = s,..., s=, + l s s = s l = 0,,,3,... m = l,..., l l sx, s y = sz s sm, s = ss ( + ) sm, s s s, m = m s, m z s s s 34

35 Befoe we take up a detaled dscusson on the HF/SCF method..a quck ecaptulaton of the smallest many-electon system -electons system. 35

36 Two-electon (gemnal) state: ψ ( q, q ) = φ(, ) χ( ζ, ζ ) ψ ( q, q) = ψ ( q, q) = { φ( }, ) χζ (, ζ) ψ ( q { }, q) = φ(, ) χ( ζ, ζ) = φ(, ) χ( ζ, ζ ) whch pat, space pat o spn pat, { } s antsymmetc, and whch s symmetc? 36

37 Two-electon (gemnal) state: ψ ( q, q ) = φ(, ) χ( ζ, ζ ) ψ ( q, q) = ψ( q, q) = φ(, ) χ( ζ, ζ) ψ ( q { }, q) = φ(, ) χ( ζ, ζ) = φ(, ) χζ (, ζ ) S = s+ s S = 0, S = 0; M = 0 S S = ; M =, 0, S { } { } φ(, ) =+ φ(, ) and χ( ζ, ζ ) = χ( ζ, ζ ) o Snglet State Tplet State φ(, ) = φ(, ) and χ( ζ, ζ ) =+ χ( ζ, ζ ) 37

38 S = s+ s S = 0, S S = 0; M = 0 S = ; M =, 0, S Snglet State Tplet State φ(, ) =+ φ(, ) χζ (, ζ) = χ( ζ, ζ ) φ(, ) = φ(, ) χζ (, ζ) =+ χ( ζ, ζ ) Whch state has lowe enegy? Snglet o Tplet? Snglet : χζ (, ζ) = χ( ζ, ζ) ant-symmetc spn pat φ(, ) =+ φ(, ) = ( ) ( ) + ( ) ( ) ϕ ϕ ϕ ϕ Tplet : χζ (, ζ) = + χ( ζ, ζ) ant-symmetc space pat φ(, ) = φ(, ) = ( ) ( ) ( ) ( ) ϕ ϕ ϕ ϕ 38

39 S = s+ s S = 0, S S = 0; M = 0 S = ; M =, 0, S Snglet State Tplet State Snglet : χ( ζ, ζ) = χ( ζ, ζ) ant-symmetc spn pat φ(, ) =+ φ(, ) = ( ) ( ) + ( ) ( ) ϕ ϕ ϕ ϕ Tplet : χζ (, ζ) = + χ( ζ, ζ) ant-symmetc space pat φ(, ) = φ(, ) = ( ) ( ) ( ) ( ) ϕ ϕ ϕ ϕ Coulomb epulson? Tplet State s less punshed by the coulomb nteacton - Landau & Lfshtz 39

40 Coulomb and Exchange ntegals electon -dmensonal obtal bass : {, } = { ϕ } ( ) ϕ( ), ϕ( ) ϕ( ) U = = Ω = Ω dv f () f () = Ω U U U U U = = U U U U 40

41 Coulomb and Exchange ntegals electon -dmensonal obtal bass : {, } = { ϕ } ( ) ϕ( ), ϕ( ) ϕ( ) U = = U U U = U U 3 3 U = d d ϕ( ) ϕ( ) ϕ( ) ϕ( ) = J U = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = K U 3 3 = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = K 3 3 U = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = J 3 3 4

42 Coulomb and Exchange ntegals U = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = J U U U = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = K 3 3 = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = K U = d d ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) = J U U J K = = U U K J 3 3 J K = K J J: Coulomb K: Exchange o : I : J 4

43 In the bass ( ) ( ), ( ) ( ), { ϕ } ϕ ϕ ϕ T Dagonalzaton: U J K = K J s not dagonal. J K J K K + J J + K K J T = K J J K K J J K K J J K 0 = 0 + J K T = T U T = U dagonal Snglet J + K K Tplet J K 43

44 In the bass ( ) ( ), ( ) ( ), { ϕ } ϕ ϕ ϕ Dagonalzaton: T = T T U J K = K J Snglet J + K K Tplet J K s not dagonal. J K J K 0 T = 0 + K J J K ϕ( ) ϕ( ) ϕ( ) ϕ( ) ϕ( ) ϕ( ) = ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ) + ϕ ( ) ϕ ( ) Tplet φ = Sngle φ 44

45 Ou nteest: SCF of the many electon system Why dd we need coulomb & exchange ntegals n advance? δψ ( ) ( ) ( ) H ψ = 0 Ψ Ω Ψ Ω= Ω= H H 45

46 δψ ( ) H ( q, q,.., q ) = H + H ( ) ( ) ( ) H ψ = 0 = δ Ψ Ω Ψ wth Ω = H and Ω= H ψ ( )! p = ( ) P { u ( q ) u ( q )... u ( q )}! P= ( ) ( ) Ω = ψ ψ... dq dq.. dq ( q, q,.. q ) ( ) P { u ( q ) u ( q )... u ( q )} ATI-SYMMETRISER Opeato! p ψ Ω! P=! tems too much wok to do! 46

47 ( ) ( ) ψ Ω ψ =!..... (,.., q ) { ( q )... ( q! = dq dqψ q Ω ( ) p P u u P= )}! p = ( )..... (,.., ) Ω { ( )... ( )}! dq dqψ q q P u q u q P=! p = ( )..... (,.., ) Ω{ ( )... ( )}! dq dqψ q q P u q u q P=! p = ( ) P... dq.. dq (,.., ) Ω P ψ q q { u... u ( q )}! P = 47

48 ( ) ( ) ψ Ω ψ =! p = ( ) P..... (,.., ) {... ( )}! dq dq Ω P ψ q q u u q P=! p p = ( ) P..... ( ) (,.., ) Ω{... ( )}! dq dq ψ q q u u q P=! = p ( ) P... dq.. dqψ ( q,.., q) Ω{ u ( q )... u ( q )}! P =! = P... dq.. dqψ ( q,.., q ) Ω{ u ( q )... u ( q )}! P= =!... dq.. dq ψ ( q,.., q ) Ω{ u ( q )... u ( q )} 48

49 ( ) ( ) ψ Ω ψ = =!... dq.. dq ψ ( q,.., q ) Ω{ u ( q )... u ( q )} ψ H ψ = ( ) ( ) = q f q ) u =!... dq.. dq ψ ( q,.., ) ( { u ( q )... ( q )} ψ ( q,.., q ) ( ) P { u ( q ) u ( q )... u ( q )}! = p! P = ψ H ψ = ( ) ( ) =.. dq.. dq u q q { ( q ).. u ( q )}! p ( ) P { ( ).. u ( )} ( ) f q u P= = Ω H,H 49

50 ψ H ψ = ( ) ( )! p d f q P= = =.. dq.. q ( ) P { u ( q ).. u ( q )} ( ){ u ( q ).. u ( q )} { u ( q ) u ( q )... u ( q )} ( ) f q = =... dq.. dq { u ( q )... u ( q )} + Tem fo Identty pemutaton! p ( ) P { u ( q ) u ( q )... u ( q )} +... dq.. dq f( q ) { u ( q )... u ( q )} P= Rest of the (! ) tems = dq d { u ( q ) u ( q )... u ( q )} ( ) f q u =..... q { ( q )... u ( q )} + R Fst, we show that the REMAIDER tems R=0 = 50

51 R =! p dq dq ( ) P { u ( q ) u ( q )... u ( q ) f q u q u =.... } ( ){ ( )... ( q )}! P= = p =... dq.. dq ( ) P { u ( q ) u ( q )... u ( q )} f( q ){ u ( q )... u ( q )} + P= OTE!! p P u q u q u q P= = +... dq.. dq ( ) { ( ) ( )... ( )} f( q ){ u ( q )... u ( q )} OTE! COSIDER now the tem wth ust one ntechange ( )... dq.. dq { u ( q ) u ( q )... u ( q )} f( q ){ u ( q )... u ( q )} = dq u ( q ) f ( q ) u ( q ) dq u ( q ) u ( q ) dq... = 0 3 q q R = 0 5

52 ψ H ψ = ( ) ( )! p d f q P= = =.. dq.. q ( ) P { u ( q ).. u ( q )} ( ){ u ( q ).. u ( q )} dq d { u ( q ) u ( q )... u ( q )} ( ) f q u =..... q { ( q )... u ( q )} + R = R = zeo 5

53 ψ H ψ = ( ) ( ) dq d q { u ( q ) u ( q ).. u ( q )} ( ) u q =.... f q { ( ).. u ( q )} = ψ H ψ = ( ) ( ) = q ( ) ( ) ( q ) d u q f q u dqu ( q) u ( q) dq u ( q ) u ( q ) dq.. dq { u ( q ) u ( q )... u ( q )} f ( q ){ u ( q )... u ( q )} = we get d u ( q ) f( q ) u ( q ) q multpled by nomalzaton ntegals... Plus smla tems fo =,3,4,. 53

54 ( ) ( ) ψ Ω ψ = =!... dq.. dq ψ ( q,.., q ) Ω{ u ( q )... u ( q )} ψ H ψ = ( ) ( ) = =... dq.. dq { u ( q ) u ( q )... u ( q )} f( q ){ u ( q )... u ( q )} ( ) ( ) ψ q = ψ H = du ( qfqu ) ( ) ( q) Questons: pcd@physcs.tm.ac.n ote the dummy label 54

55 Select/Specal Topcs n Atomc Physcs P. C. Deshmukh Depatment of Physcs Indan Insttute of Technology Madas Chenna pcd@physcs.tm.ac.n Unt 4() Lectue Many-Electon Atoms: Hatee-Fock Self-Consstent Feld fomalsm

56 ψ H ψ = ( ) ( ) = =... dq.. dq { u ( q ) u ( q )... u ( q )} f( q ){ u ( q )... u ( q )} ( ) ( ) ψ q = ψ H = du ( qfqu ) ( ) ( q) ote the dummy label 56

57 ( ) ( ) ψ d = ψ = H qu ( q) f( q) u ( q) dq dv ζ = dvu () () () f u ζ ζ = ζ 57

58 ( ) ( ) ψ qu = ψ = H d ( q) f( q) u ( q) = dvu () () () f u ζ ζ = ζ ζ ζ ζ = ζ ζ = ζ ( ) ( ) ψ H ψ = f = f = = 58

59 Matx elements of two-electon pat of the many-electon Hamltonan H = =, = 59

60 ( ) ( ) ψ Ω ψ = =!... dq.. dq ψ ( q,.., q ) Ω{ u ( q )... u ( q )} ψ H ψ = ( ) ( ) =!..... (,.., ) dq dqψ q q { u ( q )... u ( q )} =, = ψ ( q,.., q ) ( ) P { u ( q ) u ( q )... u ( q )}! p =! P = PCD_QTMEA/L3 60

61 ψ H ψ = ( ) ( ) =!..... (,.., ) dq dqψ q q { u ( q )... u ( q )} =, = ψ H ψ = ( ) ( )!!.. dq.. dq p ( ) P { u ( q ).. u ( q )} { ( ).. ( )}! P= =, = u q u q = Cancel the facto! ψ H ψ = = ( ) ( )!.... { u ( q ).. u ( q )} P= =, = p dq dq ( ) P { u ( q ).. u ( q )} 6

62 ψ H ψ = = ( ) ( ) ψ H ψ = = ( ) ( )! p.. dq.. dq ( ) P { u ( q ).. u ( q )} { ( ).. ( )} u q u q P= =, = Sepaate the! pemutatons n dentty, one-exchange, and the est of the pemutatons... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} + u q u q =, = dentty +... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} + u q u q =, =!... dq.. dq { ( P= =, = one ntechange p ( ) P { u ( q ) u ( q )... u ( q )}. u q u est of the pemutatons ).. ( q )} 6

63 ( ) ( ) ψ H ψ = =... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} + u q u q =, =... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} + u q u q =, = +!... dq.. dq { ( P= =, = p ( ) P { u ( q ) u ( q )... u ( q )}. u q u ).. ( q )} =... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} + u q u q =, =..... { ( ) ( )... ( )} dq dq u q u q u q { ( )... ( )} + u q u q R =, = R = zeo 63

64 ψ H ψ = ( ) ( ) =... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} + u q u q =, =... dq.. dq{ u ( q ) u ( q )... u ( q )} { ( )... ( )} u q u q =, = = =, = = = = = J K = = dqdq u ( q ) ( ) ( ) ( ) u q u q u q dq ( ) ( ) ( ) ( ) dq u q u q u q u q [ g g ] do you have to woy about? 64

65 ( ) ( ) ψ H ψ = f = f = = ψ H ψ = g g ( ) ( ) = = ( ) ( ) ψ H ψ = f + = [ ] + = = [ g g ] 65

66 Coulomb ntegal dq dq u ( q ) u ( q ) u ( q ) u ( q ) = u u = dvdv ( ) ( ) ( ) ( ) u u ζ m ζ m ζ m ζ m ζ ζ s s s s dq dq u ( q ) u ( q ) u ( q ) u ( q ) = ( ) ( ) = dvdvu u ( ) ( ) u u m ζ m ζ ζ m ζ m ζ ζ s s s s 66

67 Coulomb ntegal dq dq u ( q ) u ( q ) u ( q ) u ( q ) = = dvdvu ( ) u ( ) u ( ) u ( ) m ζ m ζ ζ m ζ m ζ ζ s s s s dq dq u ( q ) u ( q ) u ( q ) u ( q ) = = dv dv u ( ) u ( ) u ( ) u ( ) m ζ ζ m m ζ ζ m = ζ s s s s ζ 67

68 Exchange ntegal dq dq u ( q ) u ( q ) u ( q ) u ( q ) = = ( ) ( ) ( ) ( ) dqdqu q u q u q u q Integaton ove contnuous, summaton ove dscete vaables = dvdv u ( ) u ( ) ( ) ( ) u u ζ m ζ m ζ m ζ m ζ ζ s s s s 68

69 Exchange ntegal = ( ) ( ) dvdv u u ( ) ( ) u u ζ m ζ m ζ m ζ m ζ ζ s s s s Is thee any physcal ovelap between the spatal pats of the obtals? Exchange effects do not exst between two spn-obtals whose spatal pats do not ovelap. -Bethe & Jackew -Intemedate Quantum Mechancs 69

70 dq dq u ( q ) u ( q ) u ( q ) u ( q ) = = = dvdvu ( ) u ( ) u ( ) u ( ) = ( ) ( ) ( ) ( dv dv u u u u ) dv dv ζ m ζ m ζ m ζ m ζ m ζ m ζ ζ m ζ m ζ ζ ζ u u u ( m, m ) ( ) ( ) ( ) ( ) u δ s s s s s s s s s s 70

71 ( ) ( ) ψ H ψ = f + ( ) ( ) H ψ dvu = = = = ψ = () f() u () + [ g g ] The ntegals to be detemned fom atomc wavefunctons ae: + dvdv u ( ) u ( ) ( ) ( ) u u ( ms, m ) ( ) ( ) ( ) ( δ ) s dvdvu u u u 7

72 Mnmse: ψ H = dvu ( ) f( ) u ( ) + ( ) ( ) ψ = + ( ) ( ) H ψ = δψ 0 = dv u u = ( ) ( ) ( ) ( dvdv u u u u ) δ ( ) ( ) 0 ( ms, m ) ( ) ( ) ( ) ( ) s dvdvu u u u constants dv u () u () dv u () = = = = dv u u = () () 0 7

73 ' Extemum' of f ( x, y), subect to constant gxy (, ) = k At the ' Extemum' of f ( x, y), df = 0. df = 0 f dx x f + dy = 0 y f f = 0 & = 0 x y only f thee ae no constants. 73

74 f f df = 0 dx + dy = 0 x y g g constant g( x, y) = k 0 = dg = dx + dy x y f f x = = y λ g g x y UDETERMIED MULTIPLER f x f y g λ = x g λ = y 0 0 Lagange s vaatonal multple Extemum of the functon: f( x, y) λg( x, y) 74

75 Extemum of the functon: Ψ(x,y)= f ( xy, ) λgxy (, ) 0 = δψ( xy, ) f g f g = λ δx + λ δ x x y y y 75

76 Extemum of the functon: Ψ(x,y)= f ( x, y) λg( x, y) {} 0 = δ = 0 = δψ( xy, ) = δf ( xy, ) λδgxy (, ) { f ( xy, ) λgxy (, )} = δ ( ) ( ) ψ H ψ + δ = + λ () () dv u u + = δ( ms, m ) + s λ dv u u + λ dv u u λ = λ λ λ () () () () = Hemtan Matx 76

77 0 δ {} = Vaaton: one obtal at a tme. ( ) th k : u HF-SCF: Fozen Obtal Appoxmaton Othe obtals kept fozen, not vaed. Appoxmaton : Independent Patcle Appoxmaton Ignoe Coulomb coelatons Howeve, Exchange coelatons ncluded k 77

78 Many-electon coelatons Coulomb coelatons Many-Body theoy Quantum Feld nd Quantzaton methods Confguaton nteactons exchange coelatons Fem (Dac) Statstcal coelatons 78

79 ( ) ( ) ψ H ψ + 0 δ = + λ dv u () u () + = δ( ms, m ) + s λ dv u u + λ dv u u ( ) ( ) ψ H ψ = dvu () f() u () + = () () () () + ( ) ( ) ( ) ( dvdv u u ) u u δ ( ms, m ) ( ) ( ) ( ) ( ) s dvdvu u u u 79

80 δψ ( ) ( ) H ψ = dvu () f () u() + = = δ ( ) ( ) ( ) ( dvdv u u u u ) + (, ) ( ) ( ) ( ) ( δ ms m ) s dvdvu u u u th Vaaton of only the k obtal : u () k δ u = k ( ) 0 : fozen obtals 80

81 ψ H = dvu ( ) f( ) u ( ) +... VARIATIO n one-electon obtals ( ) ( ) ψ = δψ H ψ Vaaton of only the th k obtal: u { dv δu ()} k f () uk() ( ) ( ) = k ( ) dvu () (){ ()} + k f δuk

82 Mnmse: Vaaton of only the ψ H = dvu ( ) f( ) u ( ) + () th k obtal : u k ( ) ( ) ψ = + ( ) ( ) 0 ( ) ( ) ( ) ( dvdv u u u u ) δ ( ms, m ) ( ) ( ) ( ) ( ) s dvdvu u u u ( ) ( ) δψ H ψ = 0 dv u () u () dv u () = = = = dv u u = = dv u u = () () 0 Questons: pcd@physcs.tm.ac.n 8

83 Select/Specal Topcs n Atomc Physcs P. C. Deshmukh Depatment of Physcs Indan Insttute of Technology Madas Chenna pcd@physcs.tm.ac.n Unt 4(v) Lectue Many-Electon Atoms: Hatee-Fock Self-Consstent Feld fomalsm

84 Mnmse: Vaaton of only the δψ ψ H = dvu ( ) f( ) u ( ) + () th k obtal : u ( ) ( ) ψ = + ( ) ( ) H ψ = k 0 ( ) ( ) ( ) ( dvdv u u u u ) δ ( ms, m ) ( ) ( ) ( ) ( ) s dvdvu u u u Constants: dv u () u () dv u () = = = = dv u ( ) u ( ) 0 = = dv u u = ( ) ( ) 0 84

85 λ : ( ) ( ) ψ H ψ + 0 δ multples = + λ dv u () u () + = δ( ms, m ) + s λ dv u u + λ dv u u ( ) ( ) ψ H ψ = dvu () f() u () + = Lagange vaatonal () () () () + ( ) ( ) ( ) ( dvdv u u ) u u δ ( ms, m ) ( ) ( ) ( ) ( ) s dvdvu u u u 85

86 δψ ( ) ( ) H ψ = dvu () f () u() + = = δ ( ) ( ) ( ) ( dvdv u u u u ) + (, ) ( ) ( ) ( ) ( δ ms m ) s dvdvu u u u th Vaaton of only the k obtal : u () k δ u = k ( ) 0 : fozen obtals 86

87 ψ H = dvu ( ) f( ) u ( ) +... VARIATIO n one-electon obtals ( ) ( ) ψ = δψ H ψ Vaaton of only the th k obtal: u { dv δu ()} k f () uk() ( ) ( ) = k ( ) dvu () (){ ()} + k f δuk

88 H ψ ( ) ( ) = δψ { dv δu } k() f () uk() + { } dvdv δu k( ) u ( ) uk( ) u( ) + { } δ m m dvdv δu u u u ( s, ) ( ) ( ) ( ) ( ) k s k k 88

89 ( ) ( ) = δψ H ψ dv { δu k() } f () uk() + dvdv { δu k( ) } u ( ) uk( ) u( ) + δ( ms, m ) { ( ) } ( ) ( ) ( ) k s dvdv δu k u uk u Examne ths tem + dvu k() f ( ) { δuk ( ) } + dvdv u k( ) u ( ) { δuk( ) } u( ) + δ( ms, m ) ( ) ( ) { ( ) } ( ) k s dvdv u k u δuk u 89

90 f ( ) : Hemtan Opeato dv u f u k { } ( ) ( ) ( ) δ k = = dv u f u { ( )} ( ) δ ( ) k k { } 90

91 δψ Facto out δu ( ) k H ψ ( ) ( ) = Facto out δu k ( ) f : Hemtan { } dv δu k( ) f ( ) uk( ) + { } ( ) ( ) ( ) ( dvdv δu k u uk u ) + { } (, ) ( ) ( ) ( ) ( δ ms m ) k s dvdv δu k u uk u + { ( )} dv δuk { ( ) ( )} f u k + { } ( ) ( ) ( ) ( dvdv u k u δuk u ) + (, ) { } ( ) ( ) ( ) ( δ ms m ) k s dvdv u k u δuk u 9

92 fu ( ) k( ) + { ( ) } dv δu k ( ) ( ) ( ) ( ) ( ) u uk u δψ H ψ = + dv δ ( m, ) ( ) ( ) s m k s u k u + { () dv δuk }( ) 9

93 ( ) ( ) ψ H ψ + 0 δ = + λ () () dv u u + = δ( ms, m ) + s λ dv u u + λ dv u u 0 = δ H + () () () () λ ( ) ( ) dv u u + = + δ ( ) ( ) δ( m, ) ( ) ( ) s m + s λ dv u u + λ dv u u 93

94 0 ( ) ( ) λ dv u u + = = δ H + δ δ( ms, m ) + ( ) ( ) ( ) ( ) s λ dv u u + λ dv u u δ u = k ( ) 0 λkk dv ( δuk ( ) ) uk ( ) λkk dv uk ( ) ( δuk ( + ) ) 0 = δ H + + δ( ms, m ) ( ( ) ) ( ) ( ) ( ( ) k s λ k dv δuk u + λk dv u δuk ) uˆ duˆ= 0 0 = δ H + δ( ms, m ) k s λ k dv δuk u + λk dv u δuk ( ( )) ( ) ( ) ( ) ( ) 94

95 fu ( ) ( ) k + { } ( ) ( ) ( ) ( ) u k u dv δu k u + + dv (, ) ( ) ( δ m ) s m k s u k u 0= + (, ) ( ) δ ms m k s λ ku { + dv δuk () }( ) + δ ( m ( ( )) ( ) ( ) ( ( )), ) s m + k s λ k dv δuk u λk dv u δuk 95

96 fu ( ) ( ) k + { } ( ) ( ) ( ) ( ) u k u dv δu k u = dv (, ) ( ) ( ) δ ms m k s u k u + (, ) ( ) δ ms m k s λ ku { + dv δuk () }( ) The ntegals must go to zeo egadless of the (abtay) vaaton n the sngle patcle obtal Ths facto must be zeo. also: ( ) = 0 96

97 fu ( ) ( ) k + ( ) ( ) ( ) { ( ) u k u dv δu k u } + + dv (, ) ( ) ( δ ) ms m k s u k u + δ( ms, m ) ( ) k s λ ku 0= + dv{ δuk () }( ) The ntegals must go to zeo egadless of the (abtay) vaaton n the sngle patcle obtals 97

98 fu ( ) ( ) k + ( ) ( ) ( ) { ( ) u k u dv δu k u } + + dv (, ) ( ) ( δ ) ms m k s u k u + δ( ms, m ) ( ) k s λ ku 0= + dv{ δuk () }( ) k + u ( ) ( ) dv uk( ) u( ) δ ( ms, m ) ( ) ( ) k s u k u ( m, ) ( ) s m k s λ ku also ( ) = f( ) u ( ) δ = 0 : 0 98

99 f( ) uk ( ) + u ( ) + ( ) ( ) ( ) (, ) ( ) ( dv uk u δ ms m ) = k s u k u = δ( m, m ) λ u s s k k ( ) Coupled ntego-dffeental equatons: teatve numecal solutons HF Equaton: CODITIO that must be satsfed fo ( ) ( ) δψ H ψ = 0 subect to the constants: omalzaton and Othogonalty 99

100 f( ) uk ( ) + u ( ) + ( ) ( ) ( ) (, ) ( ) ( dv uk u δ ms m ) = k s u k u ( m, m ) λ u λ k = δ k : vaatonal paametes [ C] [ ][ C] k [ ] = [ ] λ = λ λ λ k s s k ( ) λ = λ dagonal λ δ = (, ) C = C δ m m s s [ C] = [ C] : Untay 00

101 u ( ) f( ) uk( ) + dv ( uk( ) u( ) ( ms, m ) ( ) ( δ )) k s u k u = = δ( m, m ) λ u [ C] [ ][ C] s s k dagonal λ = λ = λδ u () c u () ' = k k k = = k = c u () k u c u ' () = l l() l= k k ( ) [ C] = [ C] ' u c c.... u ' u u =

102 ' f( ) u ( ) + ' u ( ) ( ' ' ' ' dv ) u( ) u( ) δ ( ms, m ) ( ) ( ) s u u = = λ u ' ' ( ) Hatee-Fock-Dac coupled ntego-dffeental equatons : dagonal fom f( ) u ( ) + u ( ) dv u u m m u u = λ u ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) 0

103 f( ) u ( ) + u ( ) dv u u m m u u = λ u ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) ε = f( ) u ( ) + u ( ) dv u u m m u u = ε u λ ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) 03

104 HF Equaton: CODITIO that must be satsfed fo ( ) ( ) f( ) u ( ) + ε = λ u ( ) dv u u m m u u = ε u δψ H ψ = 0 subect to the constants: omalzaton and Othogonalty ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) Questons: pcd@physcs.tm.ac.n What s the connecton wth obsevables? Is ths an egenvalue equaton? 04

105 Select/Specal Topcs n Atomc Physcs P. C. Deshmukh Depatment of Physcs Indan Insttute of Technology Madas Chenna pcd@physcs.tm.ac.n Unt 4(v) Lectue 3 Many-Electon Atoms: Hatee-Fock Self-Consstent Feld fomalsm

106 HF Equaton: CODITIO that must be ( ) ( ) satsfed fo δψ H ψ = 0 subect to the constants: omalzaton and Othogonalty f( ) u ( ) + u ( ) dv u u m m u u = ε u ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) Is ths an egen-value equaton? whee ε = λ What s the physcal meanng of the Lagange multples? How does the HF SCF fomalsm connect to obsevables? 06

107 f( ) u ( ) + u ( ) dv u u m m u u = ε u ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) dvu ( ) f ( ) u( ) + u ( ) u ( ) dv dv u u m m u u = ε dv u ( ) u u ( ) Multply by thoughout and ntegate ove ( ) ( ) ( ) δ ( s, s ) ( ) ( ) ( ) = 07

108 dvu ( ) f ( ) u( ) + u ( ) ( ) u dv dv u u m m u u = ε One electon ntegals ( ) ( ) ( ) δ ( s, s ) ( ) ( ) Two electon exchange ntegals Two electon coulomb ntegals 08

109 dvu () f () u () f = One electon ntegals Two electon coulomb dq ntegals dqu ( q) u ( ) ( ) ( ) = q u q u q = dvdvu ( ) u ( ) u ( ) u ( ) = g dq dq u ( q ) u ( q ) u ( q ) u ( q ) = = δ = ( m, ) s m ( ) ( ) ( ) ( ) s dv dv u u u u = g Two electon exchange ntegals 09

110 dvu ( ) f ( ) u( ) + u ( ) ( ) u dv dv u u m m u u = ε = ( ) ( ) ( ) δ ( s, s ) ( ) ( ) [ ] f + g g = ε [ ] f + g g = ε = = = = 0

111 [ ] f + g g = ε = = = = ψ H ψ = f = f ( ) ( ) ( ) ( ) f + [ g g ] = ψ H ψ = = = = = ( ) ( ) ψ H ψ = g g = = ( ) ( ) ( ) ( ) [ ] ψ H ψ + ψ H ψ = ε =

112 (3) E( ψ ) =< f >+< f >+< 3 f 3>+ + ( J K ) + ( J K ) + ( J K ) ( ) E f f J K () ( ψ n = 0) =< >+< 3 3>+ 3 3 ( ) ( ) E E f J K J K E Example: 3-electon system: (3) () ( ψ ) ( ψ n = 0) =< > = ε ( ) ( ( )) ( ) E ψ ψ = ε [ ] snce: k f k + k g k k g k = ε k Fozen obtal appoxmaton ( n = 0) k k = λ kk

113 ( ) ( ) ( ) ψ H ψ = f + n n g g = ( ) ( ψ ) = + = E f n n J K wth n [ ] [ ] k f k + k g k k g k = ε k [ ] ( ) ( ) ( ) ψ H ψ = f + n n g g E( ψ k ( ) = 0 (and fozen obtal appoxmaton) ) = f + n n J K E ( ( )) ( ) E ( ) ψ ψ = ε n: occupaton numbes ( n = 0) k summaton ove - sngle-patcle states k 3

114 E (3) () ( ψ ) E( ψ n = 0) = ε = λ E ( ) ( ( )) ( ) E ψ ψ = ε = λ ( n = 0) k k kk Tallng C.Koopmans Physca 04 (934) Hans Kames s student The Sveges Rksbank Pze n Economc Scences n Memoy of Alfed obel 975 4

115 Does the HF equaton look lke an egen-value equaton? Is t and egenvalue poblem? f( ) u ( ) + u ( ) dv u u m m u u = ε u HF equaton ( ) ( ) ( ) δ ( s, s ) ( ) ( ) u ( ) + f( ) u ( ) dv u ( ) u ( ) δ u ( ) u ( ) ( ) ( ms, m ) ( ) s dv u = ε u ( ) 5

116 u ( ) + J f( ) u ( ) dv u ( ) u ( ) u ( ) u ( ) δ( m, m ) dv u ( ) = u s s ε Multply by χ ζ, m s u ( ) f( ) u ( q ) dv u ( ) u ( q ) + J ( ) ( ) u ( ) u ( ) δ( m, m ) dv u ( ) χ ζ = ε u q s s s, m ( ) ( ) 6

117 u ( ) f( ) u ( q ) dv u ( ) u ( q ) + u ( ) u ( ) δ( m, m ) dv u ( ) χ ζ = ε u q s s s, m ( ) ( ) dect u ( ) ( ) q u q coulomb m V ( q) = dq s ζ ζ ms = ζ u ( ) ( ) u dect V ( q) u( q) = dv u( q) dect f( ) u ( q ) + V ( q ) u ( q ) u ( ) u ( ) δ( m, m ) dv u ( ) χ ζ = ε u q s s s, m ( ) ( ) 7

118 f( ) u ( q ) + V ( q ) u ( q ) dect u ( ) u ( ) δ( m, m ) dv u ( ) χ ζ = ε u q s s s, m u ( ) ( ) q φ q exchange V ( q) φ ( q) = dq u ( q) u ( ) ( ) q u q exchange V ( q) u( q) = dq u ( q) ζ m ζ ζ m = δ m, s s s s ( ) ( ) u ( ) ( ) u ex V ( q) u( q) = δ m, ( ) ( ) s m dv u χ ζ s, m s m 8

119 f( ) u ( q ) + V ( q ) u ( q ) dect u ( ) ( ) u δ( ms, m ) ( ) ( ) = ( ) s dv u χ ζ ε u q, m s u ( ) ( ) u ex V ( q) u( q) = δ m, ( ) ( ) s m dv u χ ζ s, m s u ( ) ( ) u ex V ( q) u( q) = δ m, ( ) ( ) s m dv u χ ζ s, m s f ( ) u ( q ) + V ( q ) u ( q ) V ( q ) u ( q ) = ε u q ( ) dect ex f ( ) u ( q ) + V u ( q ) V u ( q ) = ε u q ( ) d ex f ( ) u ( q ) + V u ( q ) = ε u q ( ) HF deceptvely smple - Bansden & Joachan 9

120 T C.Koopmans Physca 04 (934) Fozen Obtal Appoxmaton E ( ) ( ( )) ( ) E ψ ψ = ε = λ ( n = 0) What s the physcal meanng of the Lagange multples? How does the HF SCF fomalsm connect to obsevables? Ionzaton enegy : Electon spectoscopy fo chemcal analyss ESCA Chemcal shfts Ka Manne Böe Segbahn 98 obel pze fo fo ESCA (XPS/PES/UVPES) - wth Bloembegen and Schawlow fo Lase Spectoscopy k k kk 0

121 Beyond the Hatee-Fock. Relatvstc effects. Begn wth Dac eq., athe than Schodnge Dac-Hatee-Fock. - Johnson, Desclaux, Gant Beyong the IPA: - ndependent patcle appoxmaton -sngle patcle models - Include many body (many-electon) coelatons

122 Electon Coelatons [] Exchange / Fem-Dac / Statstcal HF / DF [] Coulomb Coelatons MCHF / MCDF / CI / MBPT / RPA / RRPA / RRPA-R.. nd Quantzaton Methods: most convenent E(HF/DHF) E(Gound State) Coelaton Enegy

123 Hatee and Hatee Fock / Slate 3

124 f ( ) u ( q ) + V u ( q ) = ε u q Hatee-Fock-Dac SCF coupled ntego-dffeental equatons Two component : Fou u ( ) ( ) q = u χ ( ζ), m u s ( q) 0 ( ) o ( ) Ω ˆ κm u 0 u ( ) HF m m ' ' s, m' = m s s = Component G () () ˆ nκ Ωκm = () () ˆ Fnκ Ω κm ( ) : supepostons of two-component Paul spnos χζ ( ) defnton ˆ Ω = Y () χ ( ζ) m m m m Unt 3, Slde numbes 76-8 Relatvstc SCF: DHF Sphecal Hamonc Spnos Ωm 4

125 Sphecal Hamonc Spnos fo = Ωm fo = + + m Y () ˆ =, m ' = m Ω m = m Y () ˆ =, m ' = m+ m+ Y () ˆ + =+, m ' = m Ω m = + m+ Y () ˆ + =+, m ' = m+ August Septembe 0 PCD STAP Unt 3 slde no.80 5

126 f( ) u ( q ) + V ( q ) u ( q ) dect u ( ) u ( ) δ( m, m ) dv u ( ) χ ζ = ε u q s s s, m f ( ) u ( q ) + V u ( q ) V u ( q ) = ε u q ( ) d ex ( ) ( ) u ( ) ( ) u dect V ( q) u( q) = dv u( q) u ( ) ( ) ex u V ( q) u( q) = δ m, ( ) ( ) s m dv u χ ζ s, m s Local densty Local Densty Appoxmaton to exchange - John C. Slate (LDA) LDA=? X appox 6

127 Alex Dalgaano Havad-Smthsonan Cente fo Astophyscs Ugo Fano 9-00 ext: Unt 5 SPECTROSCOPY... M.J.Seaton Much of atomc and molecula physcs today s about the study of ELECTRO CORRELATIOS, RELATIVISTIC EFFECTS n spectoscopy, collson dynamcs,. Questons: pcd@physcs.tm.ac.c Jan Lndebeg Unvesty of Aahus Walte R Johnson Unv. of ote Dame 7