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3 Irreducible tensor operator techniques in atomic spectroscopy 44̂π 26 Ô α β ([2 2 1 2 ]12) = ∑α 13 α 36ϖ 36 ɛ (26)(6) ɛ (236)(6)̂π 23̂π 36 Ô α β ([2 2 1 2 ]12),̂π 26 Ô α β ([2 2 1 2 ]12) =ϖ 16̂π 12̂π 16 Ô α β ([2 2 1 2 ]12).In the language of commutative diagrams, the coefficients ε 26 and ε 16 are found from the diagramswith ν s = ν 8 = 17 and ν s = ν 9 = 19 compositions, respectively. These numbers pointto an ocular demonstration of the advantage of diagrammatic method presented here.To conclude, it is worth to mention that most of commutative diagrams studied in this sectioncan be drawn in several different ways with the same number ν s . For instance,p 25 = τ6 −1 ◦ p ′ 35 ◦ τ 6 ◦ p ′ 24 ◦ τ −1ξ◦ p ′ 25 ◦ τ ξ ◦ τ6 −1 ◦ p ′ 45 ◦ p ′ 23 ◦ τ 6is also a composition of ν s = ν 5 = 11 maps, thus it is equivalent to the diagram characterisedby the composition presented in Eq. (3.41b). If, for instance, φ i ◦ φ j = φ j ◦ φ i , then it ought tobe possible to construct the classes C i that contain equivalent elements φ i = φ −1j ◦ φ i ◦ φ j withall possible j including j = i. Such classification is still to be clarified.3.1.4 Equivalent permutationsIn the present section, the following statement will be proved.3.1.8 Theorem. Let ̂π be a permutation representation of S l . If the irreducible tensor operatorÔ α ([λ]κ) on H q contains a set of equal irreducible representations α s , s = 1, 2, . . . , t < l, thenthere exists a permutation representation ̂π min which represents a cycle of the smallest possiblelength or a product of cycles of the smallest possible length such that the correspondenceE π ξ ′ ξ = E π minξ ′ ξ(3.45)is satisfied, though the map τ ξ ◦ p πmin ◦ τ −1ξ: T [λ′ ]′ κ−→ (̂π ′min T ) [λ]κ does not exist.An incitement to find the smallest possible length ̂π min associated to ̂π is caused by aprospect to reduce the number of intermediate irreps that appear due to transformations: theless number of transpositions, the less number of indices of summation.To avoid further misleading, it is more suitable to label irreps α s with s ≤ t by ς x , wherex = i, j, k, l, p, q, . . .. That is, α s designates the irrep of the sth operator a αs ; ς x denotes itsvalue, though.Suppose there is operator Ôα [21]([21]2) associated to the scheme T 2 (see Tab. 3) with irrepsα 1 = α 2 = ς i , α 3 = ς j , α = ς, β = ι. In accordance with Eq. (3.17d), α 12 = ς ii is odd. Thenϖ 12 = (−1) α 1+α 2 +α 12 +1 = (−1) 2ς i+ς ii +1 = 1 since (−1) 2ς i= (−1) 1+2λ i= 1 (see Eq. (3.1)):(−1) 2λ i= (−1) 1+2l i= −1 (l i = 0, 1, . . .) for LS-coupling and (−1) 2λ i= (−1) 2j i= −1(j i = 1 /2, 3 /2, . . .) for jj-coupling.Consider two operators: ̂π 132 Ôι ς ([21]2), ̂π 13 Ôι ς ([21]2). The permutation representation ̂π 132is realised by the permutation (132) = ( 1 2 33 1 2); the permutation representation ̂π13 acts on irrepsα s , s = 1, 2, 3 by transposing them with (13) = ( 1 2 33 2 1). Consequently, the anticommutationrule in Eq. (3.2) enables us to writêπ 132 [W ς ii(λ i λ i ) × a ς j] ς ι = [W ς ij(λ j λ i ) × a ς i] ς ι ≡ [W α 13(λ 3 λ 1 ) × a α 2] α β= ∑C 132 [W ς ii(λ i λ i ) × a ς j] ς ι ≡∑C 132 [W α 12(λ 1 λ 2 ) × a α 3] α β,ς ii =oddα 12 =odd(3.46a)defC 132 = ∑ 〈ς j ι j ς i ι i1 |ς ij ι ij 〉〈ς ij ι ij ς i ι i2 |ςι〉〈ς i ι i1 ς i ι i2 |ς ii ι ii 〉〈ς ii ι ii ς j ι j |ςι〉. (3.46b)ι i1 ι i2ι j ι ij ι iiThe summation over indices causes the appearance of 6j–symbols. However, it is unimportantin the present case. The transposition (13) permutes operators with the same value ς i . Thus
3 Irreducible tensor operator techniques in atomic spectroscopy 45̂π 13 [W ς ii(λ i λ i ) × a ς j] ς ι = [W ς ij(λ j λ i ) × a ς i] ς ι ≡ [W α 23(λ 3 λ 2 ) × a α 1] α β= ∑C 13 [W ς ii(λ i λ i ) × a ς j] ς ι + δ(ς j , ς)(−1) ς i+ς ij −ς+1 [ς ij ] 1/2 [ς] −1/2 a ς ι≡ς ii =odd∑α 12 =oddC 13 [W α 12(λ 1 λ 2 ) × a α 3] α β + δ(α 3 , α)(−1) α 1+α 13 −α+1 [α 13 ] 1/2 [α] −1/2 a α β,(3.47a)defC 13 = − ∑ 〈ς j ι j ς i ι i2 |ς ij ι ij 〉〈ς ij ι ij ς i ι i1 |ςι〉〈ς i ι i1 ς i ι i2 |ς ii ι ii 〉〈ς ii ι ii ς j ι j |ςι〉. (3.47b)ι i1 ι i2ι j ι ij ι iiIn Eq. (3.47b), the third CGC equals to 〈ς i ι i1 ς i ι i2 |ς ii ι ii 〉 = −ϖ 12 〈ς i ι i2 ς i ι i1 |ς ii ι ii 〉. But ϖ 12 = 1.Replace ς i ι i1 with ς i ι i2 in Eq. (3.46b). ThenC 132 = C 13 . (3.48)Eq. (3.48) has a significant meaning that makes sense for any operator Ôα ([λ]κ). A generalisationis straightforward due to the structure of coefficients C 13 , C 132 : it is a product of CGCs.Consequently, it is a manner of quantity of transpositions that permute the same value irrepsonly, but for both even and odd permutations, the coefficients coincide with each other becausefor any equal irreps α s , α s ′, the phase multiplier ϖ ss ′ = 1. Thus for the map τ ξ ◦ p 13 ◦ p 23 ◦ τ −1ξ, ′Eξ 132 ′ ξ = E ξ 13 ′ ξ . Indeed, (132) = (13)(23) and∑̂π 132 Ôβ α ([2 2 1 2 ]12) =̂π 13 ε 23 Ôβ α ([2 2 1 2 ]12) = ∑ ε (13)(23) ε 13 Ôβ α ([2 2 1 2 ]12)α 12 α 12= ∑ α 12ε 132 Ô α β ([2 2 1 2 ]12),where ε (13)(23) ≡ ̂π 13 ε 23 , ξ = ξ ′ = 14. Write ε 132 = ε 13 ε (13)(23) . But ε 23 = ∑ α 23ϖ 23 ɛ 6 ɛ (23)(6)(see Eq. (3.34)). Thus ε (13)(23) = ∑ α 12ɛ (13)(6) ɛ (132)(6) , as ̂π 13 ϖ 23 = ϖ 12 = 1. Now refer to Eq.(A.6a) in Appendix A. This implies ɛ (13)(6) = ɛ (132)(6) = ɛ 6 . Then ε (13)(23) = ∑ α 12ɛ 2 6 = 1 (seeEq. (3.13)), recalling that α 1 = α 2 and thus α 13 = α 23 . The result is ε 132 = ε 13 .The map τ ξ ◦ p 13 ◦ p 23 ◦ τ −1ξ: T [λ′ ]′ κ ′Eq. (3.24))E 132ξ ′ ξ = ∑−→ (̂π 132 T ) [λ]κα η∈Nξ ′ ξε 132 ɛ ξ ′ɛ (132)(ξ) .is realised through the coefficient (seeIt has been already pointed out that ε 132 = ε 13 . The coefficient ɛ (132)(ξ) = ɛ (13)(ξ) : for ̂π 13 ,α 1 ↔ α 3 ; for ̂π 132 , α 1 → α 3 , α 2 → α 1 , α 3 → α 2 , therefore (α 1 = α 2 ) α 1 → α 3 , α 3 → α 1 ⇒α 1 ↔ α 3 . ThenEξ 132 ′ ξ = ∑ε 13 ɛ ξ ′ɛ (13)(ξ) = Eξ 13ξ. ′α η∈Nξ ′ ξIt turns out that although a one-to-one correspondence τ ξ ◦ p 13 ◦ τ −1ξ: T [λ′ ]′ κ−→ (̂π ′13 T ) [λ]κdoes not exist (the permutation of equal representations causes additional terms due to non-zero−→ (̂π 132 T ) [λ]κKronecker deltas), but the realisation of a bijective map τ ξ ◦p 13 ◦p 23 ◦τ −1ξ: T [λ′ ]′ κ ′is actualised by the coefficient Eξ 13 ′ ξ that relates operators ̂π 13Ôα ([λ]κ) and Ôα ([λ ′ ]κ ′ ) if onlyα 1 ≠ α 2 , that is, if only the map τ ξ ◦ p 13 ◦ τ −1ξis valid.′A generalisation is plain due to the peculiarity of method based on commutative diagrams(Sec. 3.1.3). That is, each permutation operator ̂π realises a permutation π of irreps α 1 , α 2 , . . .by making the transformations so that only the irreps α s and α s ′ within W α ss ′ (λ s λ s ′) are permuted(the maps τ ξ , τ −1ξ). If the irreducible tensor operator Ôα ([λ]κ) associated to the schemecontains a set of several irreps α s that are equal, then a minimal cardinality of such set isT [λ]κ
Page 1 and 2: INSTITUTE OF THEORETICAL PHYSICS AN
Page 3 and 4: VILNIAUS UNIVERSITETOTEORINĖS FIZI
Page 5 and 6: 5Contents1 Introduction . . . . . .
Page 7 and 8: 75 The generation of ĥ(3) 11;1 ter
Page 9 and 10: 1 Introduction 9lation, the multi-c
Page 11 and 12: 1 Introduction 111.3 The scientific
Page 13 and 14: 2 Partitioning of function space an
Page 25: 2 Partitioning of function space an
Page 34 and 35: 3 Irreducible tensor operator techn
Page 59 and 60: 4 Applications to the third-order M
Page 71 and 72: 5 Prime results and conclusions 71A
Page 73 and 74: A Basis coefficients 73(α1 α 2 (
Page 79 and 80: B The classification of three-parti
Page 85 and 86: C SU(2)-invariant part of the secon
Page 87 and 88: C SU(2)-invariant part of the secon
Page 89 and 90: D Symbolic computations with NCoper
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D Symbolic computations with NCoper
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