THE MANY-BODY PROBLEM

Transcription

1 3.30: Lectue 5 Feb THE MANY-BODY PROBLEM Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

2 When s a patcle lke a wave? Wavelength momentum = Planck λ p = h h = 6.6 x 0-34 J s Ψ = Ψ, t Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

3 Tme-dependent Schödnge s equaton Newton s nd law fo quantum obects t t t t V t m Ψ = Ψ + Ψ,,,, h h 95-onwads: E. Schödnge wave equaton, W. Hesenbeg matx fomulaton, P.A.M. Dac elatvstc Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

4 Statonay Schödnge s Equaton I t t t t V t m Ψ = Ψ + Ψ,,,, h h * Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

5 Statonay Schödnge s Equaton II t E f t f dt d = h E V m h = + Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

6 d h f t = E f t dt f t E = exp t h Fee patcle Ψx,t=φxft h m x = E x x me = exp x h Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

7 Intepetaton of the Quantum Wavefuncton Copenhagen Ψ xt, s the pobablty of fndng an electon n x and t xexp Et = x h Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

8 A Tavelng Plane Wave Ψ xt, exp[ kx ωt] Dagam of plane wave emoved fo copyght easons. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

9 Metal Sufaces I Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

10 Metal Sufaces II Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

11 Infnte Squae Well 8ma π h E 6 n=4 ψx ψ n=3 ψ n= ψ 0 n= ψ -a 0 a x Fgue by MIT OCW. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

12 Fnte Squae Well aψ x aψ x 0.5 x/a 0.5 x/a aψ 3 x aψ 4 x x/a x/a Fgue by MIT OCW. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

13 A Cental Potental e.g. the Nucleus ˆ h H = + = + + m x y z V ˆ h H = sn ϑ V m snϑ ϑ ϑ sn ϑ ψ Elm = R Y ϑ, Elm lm h d d l l+ h + V R + + El = EREl m d d µ Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

14 Solutons n a Coulomb Potental: the Peodc Table Coutesy of Davd Manthey. Used wth pemsson. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

15 Othogonalty, Expectaton Values, and Dac s <ba kets> ψ ψ ψ = = d δ ψ ψ ψ ψ = = * E H d V m = = + ψ ψ ψ ψ ˆ * h Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

16 Matx Fomulaton I ψ ψ ψ ψ E H E H = = ˆ ˆ { } functons k othogonal, n k n n n c ψ = = ψ ψ m m E H = ˆ m n k n m n Ec H c = = ˆ, Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

17 Matx Fomulaton II m n m k n n Ec H c = = ˆ, m k n n mn Ec c H = =, = k k kk k k c c E c c H H H H Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

18 Vaatonal Pncple E < Φ Hˆ Φ > Φ = < Φ Φ > [ ] E [ Φ] E0 [ Φ ] = E0 If, then Φ s the gound E state wavefuncton, and vcevesa Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

19 Enegy of an Hydogen Atom E α = Ψ α Ψ α Ĥ Ψ Ψ α α Ψ = Cexp α α Ψ Ψ = C Ψ Ψ = C Ψ Ψ = C α α α α α π, 3 α α π α α π Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

20 Two-electon atom Z Z +, Eelψ, ψ = Many-electon atom Z + ψ,..., n = Eelψ,..., n > Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

21 Enegy of a collecton of atoms N e N N e e N e V V V T T H = ˆ ˆ ˆ ˆ ˆ ˆ T e : quantum knetc enegy of the electons V e-e : electon-electon nteactons V N-N : electostatc nucleus-nucleus epulson V e-n : electostatc electon-nucleus attacton electons n the feld of all the nucle > = = = e e I I N e e V R V V T ˆ ˆ ˆ Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

22 Electons and Nucle ˆ Hψ,...,, R,..., R = E,...,,,..., n N tot n N ψ R R We teat only the electons as quantum patcles, n the feld of the fxed o slowly vayng nucle Ths s genecally called the adabatc o Bon- Oppenheme appoxmaton Adabatc means that thee s no couplng between dffeent electonc sufaces; B-O no nfluence of the onc moton on one electonc suface. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

23 Complexty of the many-body Ψ Some fom of appoxmaton s essental, and ths would mean the constucton of tables. The tabulaton functon of one vaable eques a page, of two vaables a volume and of thee vaables a lbay; but the full specfcaton of a sngle wave functon of neutal on s a functon of 78 vaables. It would be athe cude to estct to 0 the numbe of values of each vaable at whch to tabulate ths functon, but even so, full tabulaton would eque 0 78 entes. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

24 Mean-feld appoach Independent patcle model Hatee: each electon moves n an effectve potental, epesentng the attacton of the nucle and the aveage effect of the epulsve nteactons of the othe electons Ths aveage epulson s the electostatc epulson of the aveage chage densty of all othe electons Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

25 Hatee Equatons The Hatee equatons can be obtaned dectly fom the vaatonal pncple, once the seach s estcted to the many-body wavefunctons that ae wtten as the poduct of sngle obtals.e. we ae wokng wth ndependent electons,..., n n n L ψ = I I d R V ε = + + Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

26 The self-consstent feld The sngle-patcle Hatee opeato s selfconsstent! I.e., t depends n tself on the obtals that ae the soluton of all othe Hatee equatons We have n smultaneous ntego-dffeental equatons fo the n obtals Soluton s acheved teatvely V R + + I I d = ε Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

27 Iteatons to self-consstency Intal guess at the obtals Constucton of all the opeatos Soluton of the sngle-patcle pseudo- Schodnge equatons Wth ths new set of obtals, constuct the Hatee opeatos agan Iteate the pocedue untl t hopefully conveges Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

28 Dffeental Analyze Vanneva Bush and the Dffeental Analyze. Coutesy of the MIT Museum. Used wth pemsson. Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

29 What s mssng It does not nclude coelaton The wavefuncton s not antsymmetc It does emove nl accdental degeneacy of the hydogenod atoms Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

30 Spn-Statstcs All elementay patcles ae ethe femons half-ntege spns o bosons ntege A set of dentcal ndstngushable femons has a wavefuncton that s antsymmetc by exchange,,...,,...,,..., =,,...,,...,,..., ψ ψ k n k n Fo bosons t s symmetc Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

31 Slate detemnant An antsymmetc wavefuncton s constucted va a Slate detemnant of the ndvdual obtals nstead of ust a poduct, as n the Hatee appoach!,...,, n n n n n L M O M M L L ν β α ν β α ν β α ψ = Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

32 Paul pncple If two states ae dentcal, the detemnant vanshes.e. we can t have two electons n the same quantum state Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

33 Hatee-Fock Equatons The Hatee-Fock equatons ae, agan, obtaned fom the vaatonal pncple: we look fo the mnmum of the many-electon Schoednge equaton n the class of all wavefunctons that ae wtten as a sngle Slate detemnant * * I I d d R V λ µ µ λ µ λ µ µ µ λ ε = + + Slate n =,..., ψ Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

34 Shell stuctue of atoms Self-nteacton fee Good fo atomc popetes Stat hghe-ode petubaton theoy Exchange s n, coelaton stll out Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

35 Faste, o bette The exchange ntegals ae the hdden cost fouth powe. Lnea-scalng effots undeway Sem-empcal methods ZDO, NDDO, INDO, CNDO, MINDO: neglect cetan mult-cente ntegals Confguaton nteacton, Mǿlle-Plesset Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

36 Restcted vs. Unestcted Spnobtals n the Slate detemnant: spatal obtal tmes a spn functon Unestcted: dffeent obtals fo dffeent spns Restcted: same obtal pat Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

37 Koopmans Theoems Total enegy s nvaant unde untay tansfomatons It s not the sum of the canoncal MO obtal eneges Ionzaton enegy, electon affnty ae gven by the egenvalue of the espectve MO, n the fozen obtals appoxmaton Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

38 Atomc Unts and Conveson Factos see handout a.u. = Ry = Ha Ry = ev ev = 3.05 kcal/mol Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza

39 Softwae Gaussan Cystal Refeences F. Jensen, Intoducton to Computatonal Chemsty J. M. Thssen, Computatonal Physcs B. H. Bansden and C. J. Joachm, Quantum Mechancs, and also Physcs of Atoms and Molecules Feb Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza