C.1 Clebsch–Gordon expansions

l = 1 with l = 1

Y 1Y 0= √i-𝜖ijkY 1 + 1-Y 1, 1i 1j 2 1k 2 2ij 1 i√2- 1 Y 11iY1−j1= -δij − ----𝜖ijkY10k − ---Y02ij, 3 4 12 Y 0Y 0= 2δ + 1Y 0 1i 1j 3 ij 3 2ij

l = 1 with l = 2

Y 11iY22ij=Y 33ijk, ( ) Y 0Y 0 =− 4-δkjY 0 + 6- δijY 0 + δikY0 + 1Y 0 , 1i 2jk 5 1i 5 1k 1j 5 3ijk 1 0 2- 1 3- 1 3- 1 -i--( 1 1 ) 2- 1 Y1iY2jk= 5Y1iδjk − 5 Y1jδik − 5 Y1kδij + √2-- 𝜖iklY2jl + 𝜖ijlY2kl + 5 Y3ijk, ( ) Y11iY21jk=− 1-∂ Y 11iY20jk , 6 √ -- − 1 1 3 0 3 0 1 0 i 2 ( 0 0) 1 0 Y2ij Y1k= --Y1iδjk +---Y1jδik − --Y1kδij + ---- 𝜖jklY2il + 𝜖iklY2lj − ---Y3ijk, 102 103 53 1i2 ( ) 304 Y10iY21jk=− --Y11iδjk + --Y11jδik + -Y 11kδij − -√--- 𝜖iklY12jl + 𝜖ijlY12kl + ---Y13ijk, 5 5 5 3 2- 15 3 3 1 i√ 2 ( ) 1 Y22ijY−1k1= --Y10iδjk +---Y10jδik − --Y10kδij − ---- 𝜖jklY20il + 𝜖iklY20lj − ---Y30ijk, 2 0 10( 2 − 11)0 5 12 30 Y2ijY1k=∂ Y2ijY1k

l = 2 with l = 2

The Clebsch–Gordon expansions involving two l = 2 harmonics have been used in the text. They are fairly long and are not given here but can be found in [43].


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