Algebraic development of many-body perturbation theory in ...

3 Irreducible tensor operator techniques in atomic spectroscopy 42which is equivalent for several schemes with [λ] = [2 3 ], κ = 1, 2 and [λ] = [2 2 1 2 ], κ = 1, 2, 3.ThenT [22 1 2 ] τ 6 12 T [22 1 2 ]≡∑ɛ46 Ôβ α ([2 2 1 2 ]12), Υ\Υ 6 = {12},α η∈Υ\Υ6T [22 1 2 ]4p ′ 23(̂π 23 T ) [22 1 2 ]4≡ ∑ α 12ϖ 23 ɛ 6 Ô α β ([2 2 1 2 ]12),(̂π 23 T ) [22 1 2 ]4(̂π 23 T ) [22 1 2 ]12(̂π 23 T ) [λ]κ(̂π 234 T ) [λ]κτ −16(̂π 23 T ) [22 1 2 ]12τ ξp ′ 24τ −1ξ≡ ∑ α 12 (̂π 23 T ) κ [λ] ≡ ∑(̂π 234 T ) [λ]κ ≡ ∑(̂π 234 T ) [22 1 2 ]12∑α η∈bπ23 (Υ 6 \Υ)∑α 12 α 23 α η∈bπ23 (Υ\Υ ξ )∑α 12 α 23 α η∈bπ23 (Υ\Υ ξ )≡∑α 12 α 23∑ϖ 23 ɛ 6 ɛ (23)(6) Ô α β ([2 2 1 2 ]12), Υ 6 \Υ = {23},ϖ 23 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) Ô α β ([2 2 1 2 ]12),ϖ 23 ϖ 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) Ô α β ([2 2 1 2 ]12),α η∈Aξϖ 23 ϖ 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ) Ô α β ([2 2 1 2 ]12),(̂π 234 T ) [22 1 2 ]12τ 6 (̂π 234 T ) [22 1 2 ]4≡= ( (̂π 23 (Υ\Υ ξ )) ⋃ (̂π 234 (Υ ξ \Υ)) ) ,∑ ∑ ∑ϖ 23 ϖ 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ)A ξdefα 12 α 23 α η∈Aξ α η ′ ∈bπ234 (Υ\Υ 6 )(̂π 234 T ) [22 1 2 ]4(̂π 24 T ) [22 1 2 ]4p ′ 34(̂π 24 T ) [22 1 2 ]4τ −16(̂π 24 T ) [22 1 2 ]12≡× ɛ (234)(6) Ô α β ([2 2 1 2 ]12),∑α 12 α 23 α 13∑α η∈Aξϖ 23 ϖ 24 ϖ 34 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ)× ɛ (234)(6) Ôβ α ([2 2 1 2 ]12),≡∑ ∑ ∑α 12 α 23 α 13 α η∈Aξ α η ′ ∈bπ24 (Υ 6 \Υ)ϖ 23 ϖ 24 ϖ 34 ɛ 6 ɛ (23)(6) ɛ (23)(ξ)Thus× ɛ (234)(ξ) ɛ (234)(6) ɛ (24)(6) Ôβ α ([2 2 1 2 ]12).ε 24 =∑ ∑f 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ) ɛ (234)(6) ɛ (24)(6) , (3.35)α 13 α 23 α 34 α η∈Aξ \Υwhere the multiplier f ij denotes the following productdeff ij =j−i∏k=1j−i−1∏l=1ϖ i i+k ϖ i+l j . (3.36)Other coefficients ε i i+2 are found from the commutative diagrams with ν s = ν 2 = 5 so thatp ′ tu ◦ τ −1ξ◦ p ′ i i+2 ◦ τ ξ ◦ p ′ rs = p i i+2 . A composition is realised for the numbers r, s, t, u such that̂π i i+2 = ̂π tûπ i i+2̂π rs . It is to be whether (rs) = (i i + 1), (tu) = (i + 1 i + 2) or (tu) = (i i + 1),(rs) = (i + 1 i + 2). Then