By Boca, Roman (Auth.)

ISBN-10: 012416014X

ISBN-13: 9780124160149

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6 , T1 ! Γ4 S 5 1=2 ! Γ6 , T2 ! Γ5 S 5 1 ! Γ4 , T1 ! Γ4 S 5 1 ! Γ4 , T2 ! Γ5 S 5 3=2 ! Γ8 , T1 ! Γ4 S 5 3=2 ! Γ8 , T2 ! Γ5 S 5 2 ! fΓ3 1 Γ5 g, T1 ! Γ4 S 5 2 ! fΓ3 1 Γ5 g, T2 ! Γ5 S 5 0 ! Γ1 , Eg ! Γ3 S 5 1=2 ! Γ6 , Eg ! Γ3 S 5 3=2 ! Γ8 , Eg ! Γ3 S 5 2 ! fΓ3 1 Γ5 g, Eg ! Γ3 S 5 1 ! Γ4 , A2g ! Γ2 S 5 3=2 ! Γ8 , A2g ! Γ2 S 5 0 ! Γ1 , A1g ! Γ1 S 5 1=2 ! Γ6 , A1g ! Γ1 S 5 3=2 ! Γ8 , A1g ! Γ1 S 5 5=2 ! fΓ7 1 Γ8 g, A1g ! 19b 2 S 5 1=2 ! Γ6 , Eg ! Γ5 Γ6 3 Γ5 5 Γ6 1 Γ7 S 5 1 ! fΓ2 1 Γ5 g, Eg ! Γ5 fΓ2 1 Γ5 g 3 Γ5 5 fΓ5 g 1 fΓ1 1 Γ2 1 Γ3 1 Γ4 g S 5 3=2 !

A@ jΓ1 γ 1 Γ2 γ 2 i A ? 6 Overview of the Vector Coupling Coefficients for the R3 Group Coupling Coefficient ClebschÀGordan coefficient h j1 j2m1m2jJMi Racah V-coefficient V( j1 j2J; m1m2M)  3j-Symbol j1 m1 Relationship X j j1 m1 ij j2 m2 iUh j1 m1 j2 m2 jJMi jJMi 5 j3 m3  Coupling of two angular momenta jJMi UÀ À jj1 m1 ij j2 m2 i m1 ;m2 Vð j1 j2 j3 ; m1 m2 m3 Þ 5 ð21Þ j1 2 j2 2 j3 j2 m2 Usage  j1 m1 j2 m2 J 2M  j1 m1 j2 m2 j3 m3   5 ð21Þ j1 2 j2 1 M ð2J 1 1Þ21=2 hj1 j2 m1 m2 jJMi 3-Momenta recoupling h j1 j2 j12 j3Jj j1 j2 j3 j23Ji & 6j-Symbol j1 j4 j2 j5 j3 j6 ' j j1 j2 j12 j3 JMi X 5 j j1 j2 j3 j23 JMiU hj1 j2 j12 j3 Jj j1 j2 j3 j23 Ji & j23 ' j1 j2 j12 j3 J j23 j1 1 j2 1 j3 1 J ½ð2j12 1 1Þð2j23 1 1ފ 21=2 5 ð21Þ 3 h j1 j2 j12 j3 Jj j1 j2 j3 j23 Ji Racah W-coefficient W( j1 j2 j5 j4; j3 j6) & j1 j4 j2 j5 j3 j6 ' 5 ð21Þ j1 1 j2 1 j4 1 j5 Wð j1 j2 j5 j4 ; j3 j6 Þ Coupling of three angular momenta ÀÀ jj1 m1 ij j2 m2 ijj3 m3 i jJMi UÀ À jj12 m12 ij j3 m3 i U 4-Momenta recoupling h j1 j2 j12 j3 j4 j34 Jj j1 j3 j13 j2 j4 j24Ji j j1 j2 j12 j3 j4 j34 JMi 5 XX j13 jj1 j3 j13 j2 j4 j24 JMi Coupling of four angular momenta j24 3 hj1 j2 j12 j3 j4 j34 Jjj1 j3 j13 j2 j4 j24 Ji 8 < j11 9j-Symbol j21 : j31 9 j13 = j23 ; j33 j12 j22 j32 h j1 j2 j12 j3 j4 j34 Jj j1 j3 j13 j2 j4 j24 Ji 5 ½ð2j12 1 1Þð2j34 1 1Þð2j13 1 1Þð2j24 1 1ފ1=2 9 8 > = < j1 j2 j12 > 3 j3 j4 j34 > > ; : j13 j24 J 12j-, 15j-, etc.

M. The above general formula reduces to two important cases as follows: 1. For the fully symmetric representation [λ] 5 [N] the dimension is È É ðn 1 N 2 1Þ! ðn 2 1Þ! ð1:127Þ and this matches the formula of the BoseÀEinstein statistics for the number of ways of distributing N particles among n single-particle states; 2. For the fully antisymmetric representation [λ] 5 [1N] the dimension is n o dim Γ½1N Š ðnÞ 5 n! ðn 2 NÞ! which refers to the formula of the FermiÀDirac statistics. ð1:128Þ 44 A Handbook of Magnetochemical Formulae In practice, the dimension of IRs in Un matches the simple formula È É f ðnÞ dim Γλ ðnÞ 5 jYj ð1:129Þ Here f(n) is a polynomial in n obtained from the Young diagram by multiplying the numbers written in the boxes according to the following rules: 1.

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A Handbook of Magnetochemical Formulae by Boca, Roman (Auth.)


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