Complex atoms and the Periodic System of the elements

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1 Complex atoms and the Peodc System of the elements Non-cental foces due to electon epulson Cental feld appoxmaton electonc obtals lft degeneacy of l E n l = R( hc) ( n δ ) l Aufbau pncple Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

2 Complex atoms and the cental feld appoxmaton The potental enegy n a mult-electon atom: Z V = h m Ze + 4πε0 = 4πε 0 < j j attacton to nucleus e Schödnge equaton epulson between electons V s a non-cental potental [ ] Ψ + VΨ = EΨ The wave functon depends on 3Z spatal coodnates Ψ (,,, ) Z Z CFA: The oveall effect of V s centally dected towad the nucleus V c ( ) Ze fo 0 4πε 0 e fo 4πε Then V can be wtten n tems of an effectve sceened nuclea chage V cfa () Z = eff 0 ( ) e 4πε NC Note: V atom = Vcfa () + Vee' 0 Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

3 Sepaaton of vaables n the cental feld appoxmaton Poduct wave functon (,, ) = ψ ( ) ψ ( ) ψ ( ) Ψ, Z Potental and egenenegy Z = ( ) V = V c E = Z = E Inseton of tal yelds a set of equatons: h m Ψ ( ) + V ( ) Ψ( ) = E Ψ( ) c Z Z. Angula pat of the wave functon s the same, hence angula functons Υ l m ( θ, φ ). Radal pat of the wave functon h m d d R nl ( ) h l( l ) Zeff m Enegy E nl depends on l R Enegy E nl does not depend on m nl = E nl R nl The potental functon s not Coulombc.e. s not / but Z eff ( ) Wave functons fo sngle obtal ψ ( ) = R ( ) Υ ( θ φ ), nl l m, Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

Coulomb attacton L Moe bndng enegy Lowe l states lowe enegy

4 Sceenng n the cental feld appoxmaton Hghl Low l l Fo low values (and same n) electon comes close to the nucleus Moe Coulomb attacton L Moe bndng enegy Lowe l states lowe enegy Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

5 Sceenng n the cental feld appoxmaton Hghl Low l l Fo low values (and same n) electon comes close to the nucleus Moe Coulomb attacton L Moe bndng enegy Lowe l states lowe enegy Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

6 Loweng of low l quantum states as an effect of sceenng Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

7 Sceenng and the quantum defect Levels descbed wth: E nl = R Na ( n ) δ l Wth quantum defects: δ s =.35 δ p = 0.86 δ d = 0.0 δ f = 0.00 Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

8 Aufbau pncple fo mult-electon atoms Egenfunctons n mult-electon atom (,, ) = ψ ( ) ψ ( ) ψ ( ) Ψ, Z Z Z Degeneacy n stuctues the Peodc System ) Electons fll the one-electon obtals ψ ( ) = Rnl ( ) Υl m ( θ, φ ), nto a confguaton : ( ) ) Paul pncple dctates: sngle occupancy ψ 3) Fo flled shells ml = L tot shell sl = S tot shell 0 = 0 0 = 0 Note: ths s about gound states of the atoms Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

9 Gound state obtal confguatons fo mult-electon atoms Noble gases He confguaton( ) Ne conf. ( ) ( ) ( ) 6 s ; tem S 0 s s ; tem S 0 s s p 6 3s 3 p A conf. ( ) ( ) ( ) ( ) ( p) 6 Alkal metals L ( s) ( s) tem S / 6 s s ( p) 3s tem S / 6 6 Na ( ) ( ) ( ) K ( s) ( s) ( p) ( 3s) ( 3p) ( 4s) Eath alkal metals Mg ( ) ( ) 6 ( ) s s ( p) 3 tem S 0 s Ca ( ) ( ) 6 ( ) ( ) 6 ( ) s ( p) 3s 3p 4 s s Bndng eneges of the one-electon obtals vay wth Z: Sceenng effects Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

10 Iegulates and degeneaces Tanston metals; effect of 3d obtals: gound tem Ca ( ) ( ) 6 ( ) ( ) 6 ( ) s ( p) 3s 3p s s Sc ( s) ( s) ( p) ( 3s) ( 3p) ( 4s) ( 3d ) T ( ) ( ) 6 ( ) ( ) 6 ( ) ( ) s ( p) 3s 3p 4s 3 s ( p) 6 3s 3p 6 4s 3 s ( p) 6 3s 3p 6 4s 3 s ( p) 6 3s 3p 6 4s 3 s d V ( s) ( ) ( ) ( ) ( ) ( d ) 3 C ( s) ( ) ( ) ( ) ( )( d ) 5 Mn ( s) ( ) ( ) ( ) ( ) ( d ) 5 Fe ( s) ( s) ( p) 6 ( 3s) ( 3p) 6 ( 4s) ( d ) 6 Co ( s) ( s) ( p) 6 ( 3s) ( 3p) 6 ( 4s) ( d ) 7 N ( s) ( s) ( p) 6 ( 3s) ( 3p) 6 ( 4s) ( d ) 8 Cu ( s) ( s) ( p) 6 ( 3s) ( 3p) 6 ( 4s)( 3 d ) 0 Zn ( s) ( s) ( p) 6 ( 3s) ( 3p) 6 ( 4s) ( d ) 0 S 0 D 3/ 3 F 4 F 3/ 7 S 3 6 S 5/ 5 D 4 4 F 9/ 3 F 4 S / S 0 Nea degeneacy of 3d and 4s obtals Competton n allocaton of electons Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

11 Ionzaton Potentals vay ove the peodc stuctues -Shell closng fo the noble gases; -Alkal metals oute electon least bndng enegy Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

12 Alkal atoms Angula momentum couplng L = l + l l z S = s + s s z J = L + S M L = m l + m l... + m M = m + m... + m S s s l s z Flled shells have L = S = 0 S 0 state z Alkal = hydogenlke wth Z eff ~ v Ze S L V SL = 4 3 wth πε 0 m c S L v = Selecton ules: Δl = ± Δj = 0, ± Δs = 0 np ( ) J L S P 3/ P / Flled shells: sphecal chage dstbuton V CFA () s OK ns S / Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

13 Devaton fom cental feld appoxmaton V ee Z Ze e ' = + 4πε 4πε 0 = 0 < j j V c ( ) V ee epesents the non-cental foces Causes a dstncton between Ψ ( ) and ( ) Smla s n case of He Ψ + ( ) space Ψ has lowe enegy Hund ule (): States of hghest multplcty have lowest enegy Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

14 Tems and multplets: Hund s ules confguatons k ( n l ) ( n l ) j j multplets S+ L J Z V c = ( ) V ee ' V LS Hund ule (): States of hghest multplcty have lowest enegy Hund ule (): States of hghest L le lowe Empcal, but some sem-classcal agument Hghe L composed of hghest m l, so less ovelap Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

15 Hund ule (3): Odeng of spn-obt states V SL Ah = ξ s ( ) S L = A S L = [ j( j + ) l( l + ) s( + ) ] Fo: j=j max =l+s V SL = A ls If A>0 hgh J up h Spn-obt nveson fo less than half flled shell V SL = n ξ electons N N n ( ) S L = ξ ( ) S L ξ ( ) S L = ξ ( ) electons holes N n holes S L J=L=S=0 fo full shell nveson fo less than half flled shell A>0 So A=0 fo half-flled shell A<0 Why A<0? Holes do the job (postve chage) Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

16 Example: stuctue of the cabon atom Z=6 : Gound state confguaton (s) (s) (p) Excted states: (s) (s) (p)(3s) (s) (s) (p)(3p), etc Angula momenta fo the (p) confguaton So possble tems: S 0 () P (3) D (5) 3 S (3) 3 P 0,, (9) 3 D,,3 (5) 36 states { } l =, s = l' =, s' = = 36space l l =, s =, s = = L S = = l s + l + s = 0,, = 0, But ths does not yet nclude the Paul excluson pncple Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

17 Sot out what substates exst fo the (p) confguaton M S Wte (m l ) m s M L Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

18 Sot out what substates exst fo the (p) confguaton; nvoke the Paul pncple M S Wte (m l ) m s M L Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

19 Sot out what substates ae allowed by the Paul pncple M S 0 - Wte (m l ) m s State wth the hghest M S M S = So thee s a state wth S= M L Hghest M L =, so thee exsts a state wth L= So 3 P s a Paul allowed state Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

20 Fnd the exstng multplets 3 = + Take out the nne states fomng the 3 P D S 0 Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

21 Gound state confguaton of the Cabon atom Fom Hund s ules Fst excted state (p)(3s) P 3 P 3 P 3 P 0 (p) S 0 D 3 P 3 P 3 P 0 Vefy the tanstons fom selecton ules and consde also the oxygen atom Lectue Notes Stuctue of Matte: Atoms and Molecules; W. Ubachs

Complex Atoms; The Exclusion Principle and the Periodic System

Complex Atoms; The Exclusion Principle and the Periodic System Complex Atoms; The Excluson Prncple and the Perodc System In order to understand the electron dstrbutons n atoms, another prncple s needed. Ths s the Paul excluson prncple: No two electrons n an atom can