Chapter 3. Many-Electron Atoms

Transcription

1 Capte. May-ecto Atom Readg: Bade & Joaca Capte 8

2 Ceta ed Appoxmato Ampto: eac of te atomc eecto move a effectve pecay ymmetc poteta ceated by te ce ad a te ote eecto. amtoa of te -eecto atom m 4πε e > > e 4πε a.. o p-obt teacto p-p teacto eatvtc effect cea coecto etc. cödge eato Ψ > Tota wave fcto: atymmetc... Ψ Ψ Ψ Ψ...

3 cödge eato fo pata wave fcto ψ ψ > Ceta fed appoxmato depedet patce mode: ac eecto move a effectve pecay ymmetc poteta ceated by te ce ad a te ote eecto. > ceeg? at temedate dtace Toma-em ad atee-ock metod

4 Upetbed amtoa > > > << c c cödge eato fo ceta fed wave fcto... m m a a a c c c c c c ψ ψ ψ ψ epaabe to eato oe-eecto o ceta fed obta Y R m m m ψ φ θ : ydoge wave fcto pecay ymmetc ae depedet of m deped o bot ad.

5 p-obta: p-obta ad ate etemat χ R Y θ φ χ mm m m m ate detemat Tota -eecto ceta fed wave fcto : atymmetc a exco pcpe. Ψ c...! α α α M / m m β β β M m m Ψ O c... ν ν ν M / m Qatm mbe α β... ν m m decbg a atom wc oe eecto ate α aote tate β ad o o. gevae: c..... Ψ Ψ... α β α Ψ β a exco pcpe ν getate ν : γ β α ecto

6 Te ate detemat w va f two eecto ave te ame vae of te fo atm mbe m ad m two com ae ea te detemat. a exco pcpe: o two eecto a atom ca ave te ame et of te fo atm mbe m ad m. Te ate detemat cde! pemtato of. Ψ c : eve pemtato : odd pemtato... α β Kν! emtato of te eecto coodate e.g. e god tate : m ove a pemtato / / χ χ Ψ c! α α [ χ χ χ χ ] [ ] β β χ χ χ χ

7 getate gevae ecto tate a Ceta ed Ψc...! c α β Kν α β m m R Ym θ φ χ m / d d R R d d ν R Odeg of dvda eegy eve 4 5 p d p 4f 4d 4p 4 pectocopc otato 5p 5 p d f : egeeacy g fo gve fo gve moe ceeg becae of te age aga mometm 4 < d dffeet fom atom >

8 e ad be Aveage ad of eecto obt a ydogec atom m ~ a ecto wc ave te ame vae of te pcpa atm mbe ae ad to beog to te ame e. - Maxmm mbe of eecto a e coed o fed e ecto avg te ame vae of ad ae ad to beog to te ame be. - Maxmm mbe of eecto a be degeeacy - Coed o fed be: a aemby of evaet eecto egeeacy d δ! ν! δ ν! ν : mbe of eecto occpyg a eve δ : degeeacy of te eve c.f. d fo a coed be. e.g. God tate of te cabo atom p : : p : ν ν ν δ δ δ 6 d d d 5 g 5

9 eodc ytem of te emet God tate of eta atom a exco pcpe: eecto : atomc mbe cea cage occpy te owet dvda eegy eve. aece eecto: eecto ae te te be of get eegy eat tgty bod eecto ffcet mbe to fom aote coed be Re-ade otato J T. p AM T. obta AM e.g. T. AM J e /

10 ecto cofgato: fg p e ad be tcte K-e emet. Cofg. e 4 5 p d p 4f 4d 4p 4 pectocopc otato 5p 5 p d f : -e emet. Cofg. [e] 4 Be [e] 5 B [e] p e [e] p 6 M-e emet. Cofg. a [e] 8 A [e] p 6 ce 4 < d ceeg by [A] te added eecto K 9 ad Ca go to te 4 ate ta te d be.

11 4 -e emet. Cofg. 9 K [A]4 Ca [A] 4 c [A] 4 d 4 C [A] 4d 5 5 M [A] 4 d 5 9 C [A] 4d [A] 4 d Ga [A] 4 d 4p 6 K [A] 4 d 4p 6 4 ~ d t tato o gop d ecod tato paadm gop 4d fom 9 to 48 Td tato patm gop 5d fom 7 to 8 Rae-eat eemet atade 4f ~ 5d fom 57 to 7 Actde 5f ~ 6d fom 89 to o age eatvtc effect e.g. p-obt copg become mpotat ad pevet te mpe decopg of te pace ad p pat of te wave fcto.

12 Iozato oteta obe et gae: coed e aka meta: [e] -e e [e] p 6 a [e] K-e m ~ a

13 Cemca opete ad eodc Tabe aoge et ga aka meta a exco pcpe dvety of cemca ad matea popete

14 Mode fo Ceta ed Appoxmato Ceta fed appoxmato depedet patce mode: ac eecto move a effectve pecay ymmetc poteta ceated by te ce ad a te ote eecto. ceeg? > at temedate dtace Toma-em ad atee-ock metod

15 em ecto Ga em eecto ga: A ytem cotg of a age mbe of fee eecto cofed to a ceta ego of pace age cbe of de : fte poteta we. ee patce cödge eato: z y x m ψ ψ z y x z y x z y x π π π ψ 8 v pata gefcto z y x vag at te boday degeeated m m z y x z y x π π egy egevae z y x z y x m m / χ ψ ψ v Tota wave fcto

16 ety of tate : mbe of eecto atm tate p-obta pe t eegy age x z d - mbe of eecto tate betwee ad d π m x y z m π / Tota mbe of tate fo eege p to vome of te pece of te pee y m / d d d π 8 π 4 π π π m π m / / / / m π /

17 / / m π em eegy god tate at T fed vacat / / / / m d m d π π / / ρ π ρ ρ π k m k m tot m d m d 5 5 5/ / / / π π Toma-em teoy tot 5

18 Toma-em Teoy -eecto atom: em eecto ga te god tate cofed to a ego of pace by a ceta poteta.? ρ? Tota eegy of a eecto p max m < depedet of k m m max k max ρ k π π > max m / [ ] max [ ] k π ρ / / max at : boday max

19 ectotatc poteta e φ [ ] < / / max e m e π ρ φ φ φ oo eato [ ] e d d ρ ε [ ] [ ] < / / e m e d d ε π 4 m πε e Boday codto 4 d ρ π omazato

20 meoe vaabe x ad fcto χ / e π / bx χ x wee b a 7 / 4πε / / d χ x x χ x χ dx χ < Boday codto fo eta atom : χ χ Toma-em eato Uvea fcto χx fo a eta atom meca tegato Ceta poteta e χ 4πε b e 4πε e χ 4πε ceeg: mootocay deceag a ceae eta atom b

21 atee-ock Metod: ef-cotet ed Oe eecto cödge eato: k k Ta fcto fo -eecto atom: Ψ... c m m m m teato pecay ymmetc eectotatc poteta eegy: k k k Ψc... ef-cotet fed Oe eecto cödge eato: k k k k m m m m < δ? k k o: k k Ye f k Ψ f c... f f

22 o-eatvtc amtoa fo -eecto atom > aatoa metod: ta fcto fo -eecto atom ate detemat xpectato vae of amtoa eegy fo a ta wave fcto φ [ ]!... ν β α ν β α ν β α M O M M... m m ν β α Qatm mbe δ d * Otooma p-obta

23 A!!... ν β α K... ν β α K A! emta ad poecto opeato wee ad A A ad ae vaat de pemtato of te eecto coodate [ ] [ ] A A > egy expectato vae fo te ta wave fcto [ ]!!! I I A A A A δ

24 [ ] ν β α!!! K > > K J A A A A K J ect tem xcage tem

25 [ ] [ ] K J I egy expectato vae fo a ta wave fcto [] tatoay wt epect to vaato of te p-obta αβ ν. aatoa eato p-6: δ ε δ δ ε δ agage mtpe dagoazato atee-ock eato ν β α * * K d d

26 ν β α δ * * K m m d d p-obta: χ m / d : dect opeato ex : excage opeato [ ] k k ex m m d δ k ex d

27 Coecto to te Ceta ed Appoxmato Coeato effect co becae exact < [ ] p-obt teacto ξ ξ m c d d eatvtc coecto Tota amtoa >> << - o Re-ade copg cae: ma ad temedate - copg cae: age

28 - Copg Upetbed amtoa wee >> ce egy egetate of γ γ : eve cofgato of c degeeated. e.g. cabo atom C p g5 ft pat of te degeeacy bt ot competey: tate. [ ] [ ] wee eegy egetate ca be wtte a γ M M egy egevae ae depedet of M ad M. γ Tem: eegy eve coepodg to defte vae of ad Mtpcty: get dobet tpet 4 Coed be:

29 Icompete be: optcay actve eecto. o-evaet eecto: eecto beogg to dffeet be a exco pcpe atomatcay atfed. Addto of aga mometm ad Cofgato p p / Tem: Cofgato p d / Tem:

30 Cofgato p p d cofgato p p / cofgato d G G G / / 4 / / / / / 4 / / Tem: mbe of detca tem: G G

31 . vaet eecto: eecto beogg to ame be a exco pcpe e ot ceta vae of ad. Cofgato Coed be: M M tem / Tem: ot aowed becae of te a exco pcpe Cofgato p g 5 / Tem: ot aowed becae of te a exco pcpe

32 M mbe m m m m M m m m m ½ ½ ½ - ½ ½ - ½ - 4 -½ -½ - 5 -½ - -½ - 6 -½ - -½ ½ -½ 8 ½ -½ 9 ½ - -½ ½ -½ ½ -½ ½ - -½ - - ½ -½ 4 - ½ -½ ½ - -½ - m

33 Tem: M M mg : o M M M M : tem wt mt ext. degeeacy 5 M M M M... : tem wt mt ext. degeeacy 9 M M 594 : emate ad degeeacy

34 Tota amtoa e tcte of tem - copg egy egetate Coeato effect ξ ξ m c p-obt teacto d d >> [ ] [ ] - o Re-ade copg cae [ J ] bt wee J A A J ft pat of te degeeacy of a tate epeeted by a tem J... J

35 egeeacy g g J J J Cofgato p p / Tem: d e. o gve tem aget owet eegy. o gve aget owet eegy

36 - Copg d d c m ξ ξ << ~ ~ ~ J ξ ξ amtoa [ ] J ~ egy