# The Pauli principle says, that we have to build an antisymmetric combination, i.e. Ψ a,b (1,2) = 1 2 [ψ a(1)ψ b (2) ψ a (2)ψ b (1)] = - Ψ a,b (2,1)

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1 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age Shell tructure of the atom, eriodic ytem The Hydrogen atom i the mot imle atomic ytem (1 electron roblem) but it contain eentially all the element which we need to know for building u the eriodic ytem (Aufbaurinzi alo in Englih). Thu the zero order aroximation for any atom i to aume all electron can be treated indeendently in a imilar manner a in the hydrogen cae. Only afterward we conider the interaction of the electron with each other. We then tart with a et of quantum number n,,m,m (n a a m a m a, n b b m b m b to ditinguih different et) for each electron (1,,...N) Pauli Pincile In addition, we have to oberve the Pauli rinicle (Wolfgang Pauli: Nobel Prize 1946, Biograhy). The baic have already been dicued in the context of the free electron ga in a metal: In a quantum ytem of identical fermion (= in 1/,3/ etc. article) no two electron can have the ame et of quantum number (each article mut differ from any other one by at leat one quantum number). {n a a m a m a a} {n b b m b m b b}. More general: The total wave function of a ytem of identical fermion mut be antyymmetric with reect to exchanging any two of the article. We how that thi imlie the former verion of the Pauli rincile by one examle: Wave function of two electron ytem with two et of quantum number n a a m a m a and n b b m b m b : electron 1: ψ na amama (r 1, ϑ 1, ϕ 1 ; 1 ) electron : ψ nb b m b m b (r, ϑ, ϕ ; ) The total wave function i written a the roduct of both: ψ na amama (r 1, ϑ 1, ϕ 1 ; 1 )ψ nb b m b m b (r, ϑ, ϕ ; ) ψ a (1)ψ b (). However, ince both article are inditinguihable we cannot ay which of the article (1 or ) ha the et n a a m a m a or n b b m b m b : Hence the wave function can alo be ψ na amama (r, ϑ, ϕ, )ψ nb b m b m b (r 1, ϑ 1, ϕ 1, 1 ) ψ a ()ψ b (1) The Pauli rincile ay, that we have to build an antiymmetric combination, i.e. Ψ a,b (1,) = 1 [ψ a(1)ψ b () ψ a ()ψ b (1)] = - Ψ a,b (,1) 8.4. Slater determinant For a ytem of N-fermion with ψ n a amama(r i, ϑ i, ϕ i ; i ) ψ a (i) Ψ a,b,c... (1,,3,...) = 1 N! ψ a (1) ψ a () ψ a (3)... ψ b (1) ψ b () ψ b (3)... ψ c (1) ψ c () ψ c (3) The electron ytem with Singlet and Trilet tate Note: The Slater detarminant i not the only oible form which fullfill the Pauli rincile. Alternatively, for a electron ytem we may tart with:

2 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 13 ψ na ama (r 1 ) ψ nb b m b (r ) χ 1 ± 1 (1) χ 1 ± 1 () to build antiymmetric combination from the atial and in art earately: Satial wave function Ψ ym,anti (r 1,r ) = 1 and in tate: ψ na ama (r 1 ) ψ nb b m b (r ) ± ψ na ama (r ) ψ nb b m b (r 1 ) T 11 ) = χ 1 1 (1) χ 1 1 () = M =1 ymmetric trilet S = 1 T 10 ) = 1 χ 1 1 (1) χ 1 1 () + χ 1 1 (1) χ 1 1 () = M = 0 T 1 1 ) = χ 1 1 (1) χ 1 1 () = M = -1 antiymmetric inglet S = 0 S 00 ) = 1 χ 1 total wave function mut be antyymmetric: thu 1 (1) χ 1 1 () χ 1 1 (1) χ 1 1 () Ψ a,b (1,) = Ψ ym (r 1,r ) S 00 ) or = Ψ anti (1,)r 1,r ) T 10 ) or = Ψ anti (1,)r 1,r ) T 1±1 ) How the hell are filled hell tate n # of 1 1, 8 M 3,3,3d 3 18 N 4,,d,f 4 3 tate = M = 0

3 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 14 Z Atom ground tate configuration hell 1 H 1 He 1 3 i 1 4 Be 1 5 B Ne Na = [Ne]3 M See alo table in the web, e.g.: htt://www.neubert.net/psetable.html, htt://xlor.cb.yale.edu/web-element-i/ and a nice animation (David Wizzy Table): htt://www.colorado.edu/hyic/000/alet/a.html N N O O F F Xe Xe i i Be Be B B C C H H He He Comlete overview

4 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 15 Z Element State M d 1 H S 1/ 1 He 1 S 0 3 i S 1/ 1 4 Be 1 S 0 5 B P 1/ 1 6 C 3 P 0 7 N 4 S 3/ 3 8 O 3 P 4 9 F P 3/ 5 10 Ne 1 S Na S 1/ Mg S Al P 1/ Si 3 P P 4 S 3/ S 3 P Cl P 3/ Ar 1 S 0 6 6

5 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 16 Z Element State M N d Ar 1 S S 1/ Ca 1 S Sc D 3/ Ti 3 F V 4 F 3/ Cr 7 S Mn 6 S 3/ Fe 5 D Co 4 F 1/ Ni 8 F Cu S 1/ Zn 1 S Ga P 1/ Ge 3 P A 4 S 3/ Se 3 P Br P 3/ r 1 S

6 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 17

7 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 18 Z Element State M N O P d 4 4 4d 4f 5 5 5d 5f 6 6 6d 37 Rb S 1/ Sr 1 S Y D 3/ Zr 3 F Nb 6 D 1/ Mo 7 S Tc 6 S 5/ Ru 5 F Rh 4 F 3/ Pd 1 S Ag S 1/ Cd 1 S In P 1/ Sn 3 P Sb 4 S 3/ Te 3 P J P 3/ Xe S C S 1/ Ba 1 S a D 3/ Ce ( 3 H 4 ) ? 59 Pr ? 60 Nd Pm ? 6 Sm 7 F Eu 8 S 7/ Gd 9 D Tb ? 66 Dy ? 67 Ho ? 68 Er ? 69 Tm F 7/ Yb 1 S u D 3/ Hf 3 F

8 Ingolf Hertel, WS01/0, Phyik 3, Chater 8 age 19 Z Element State M N O P d 4 4 4d 4f 5 5 5d 5f 6 6 6d 73 Ta 4 F 3/ W 5 D Re 6 S 5/ O 5 D Ir 4 F Pt ( 3 D) ? 79 Au S 1/ Hg 1 S Tl P 1/ Pb 3P Bi 4 P 3/ Po 3P At P 3/ Rn 1 P Fr S 1/ Ra 1 P Ac ( D 3/ ) Th (3P ) Pa (4F 3/ ) U ( 5 D 0 ) N? Pu? Am? Cm? Bk? Cf? E? Fm? Mv? No?

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