Mathematica Can't Compute Sum?

I am trying to calculate a complicated Clebsch-Gordan sum in which appear also a dot product of vectors. However, when I run the code, Mathematica just won't compute anything, even if I let it run for several hours. Below is my code.

Clear["Global*"]
Off[ClebschGordan::phy]

Subscript[e, -1] = {1/Sqrt[2], -I/Sqrt[2], 0}
Subscript[e, 1] = {-1/Sqrt[2], -I/Sqrt[2], 0}
Subscript[e, 0] = {0, 0, 1}
Subscript[\[Chi], 1/2] = {1, 0}
Subscript[\[Chi], -1/2] = {0, 1}
Subscript[\[Eta], 1/2] = {0, 1}
Subscript[\[Eta], -1/2] = {-1, 0}
Subscript[\[Sigma], 1] = {{0, 1}, {1, 0}}
Subscript[\[Sigma], 2] = {{0, -I}, {I, 0}}
Subscript[\[Sigma], 3] = {{1, 0}, {0, -1}}

HybridM[S1_, S2_, l1_, l2_, L1_, L2_, J1_, J2_] :=
Sum[(4 Pi/(2 G5 + 1))*(Sqrt[
3/((2 l1 + 1)*(2 l2 + 1))])*(\[Lambda]1*\[Lambda]2/
2)*(-1)^(\[Lambda]1 + \[Lambda]2)
ClebschGordan[{1/2, m1}, {1/2, mbar1}, {S1,
MS1}] ClebschGordan[{l1, ml1}, {1, mg1}, {L1,
ML1}] ClebschGordan[{S1, MS1}, {L1, ML1}, {J1,
0}] ClebschGordan[{1/2, m2}, {1/2, mbar1}, {S2,
MS2}] ClebschGordan[{l2, ml2}, {1, mg2}, {L2,
ML2}] ClebschGordan[{S2, MS2}, {L2, ML2}, {J2,
0}] ClebschGordan[{1, mg1}, {1, -a}, {G1,
b}] ClebschGordan[{1, \[Lambda]1}, {1, -\[Lambda]1}, {G1,
0}] ClebschGordan[{G1, b}, {1, \[Alpha]1}, {G3,
d}] ClebschGordan[{G1, 0}, {1, 0}, {G3, 0}] ClebschGordan[{1,
mg2}, {1, -a}, {G2,
c}] ClebschGordan[{1, \[Lambda]2}, {1, -\[Lambda]2}, {G2,
0}] ClebschGordan[{G2, c}, {1, \[Alpha]2}, {G4,
e}] ClebschGordan[{G2, 0}, {1, 0}, {G4, 0}] ClebschGordan[{l1,
ml1}, {G3, d}, {G5, f}] ClebschGordan[{l1, 0}, {G3, 0}, {G5,
0}] ClebschGordan[{l2, ml2}, {G4, e}, {G5, f}] ClebschGordan[{l2,
0}, {G4, 0}, {G5,
0}] Dot[{Subscript[\[Chi], m1] . Subscript[\[Sigma], 1] .
Subscript[\[Chi], m2],
Subscript[\[Chi], m1] . Subscript[\[Sigma], 2] .
Subscript[\[Chi], m2],
Subscript[\[Chi], m1] . Subscript[\[Sigma], 3] .
Subscript[\[Chi], m2]},
Cross[Subscript[e, \[Alpha]1],
Conjugate[Subscript[e, \[Alpha]2]]]], {\[Lambda]1, -1, 1,
2}, {\[Lambda]2, -1, 1, 2}, {m1, -1/2, 1/2, 1}, {mbar1, -1/2, 1/2,
1}, {MS1, -S1, S1}, {ml1, -l1, l1, 1}, {mg1, -1, 1, 1}, {ML1, -L1,
L1, 1}, {m2, -1/2, 1/2, 1}, {MS2, -S2, S2}, {ml2, -l2, l2,
1}, {mg2, -1, 1, 1}, {ML2, -L2, L2, 1}, {a, -1, 1, 1}, {G1, 0, 2,
1}, {b, -G1, G1, 1}, {\[Alpha]1, -1, 1, 1}, {G3, Abs[G1 - 1],
G1 + 1}, {d, -G3, G3}, {G2, 0, 2, 1}, {c, -G2, G2,
1}, {\[Alpha]2, -1, 1, 1}, {G4, Abs[G2 - 1], G2 + 1}, {e, -G4,
G4}, {G5, Abs[l1 - G3], l1 + G3}, {f, -G5, G5, 1}]


When I run this for different values of the variables, like, for example,

HybridM[1, 1, 0, 0, 0, 0, 0, 0]
`

it just stalls and does not give any output. Is it just too difficult to compute for Mathematica, or is it the way I am defining things that makes it harder to compute? I have run similar but slightly simpler sums before and Mathematica computed them in second, no problem.

Thank you.

• This sum has just too many summands. The most of them will probably be zero, so you might still have a chance to evaluate this provide you are able to substantially simplify it. Commented Jul 14, 2022 at 6:40
• I get the error message: "ClebschGordan::tri: ThreeJSymbol[{0,0},{1,-1},{0,0}] is not triangular." You may note that m1+m2 != m3 as requested by ClebGordan coefficients. Commented Jul 14, 2022 at 7:47