This blog is the theory for https://github.com/0382/CGcoefficient.jl .
It is mainly inspired by Ref[1]. See Jensen’s code for details.
The idea is to simplify 3nj Symbols to sum combinations of binominal coefficients. We can calculate binominal coefficients by Pascal’s Triangle, and store them first. Then we calculate 3nj Symbols using the stored binominal coefficients.
My code is just a toy model, using Julia’s own binominal function, and use BigInt to get absolute results. If you want to get the exact result, and you don’t use it for numeric computing, it is really a good package.
Ref[2], P240, Section 8.2.4, Formula (20).
C j 1 m 1 j 2 m 2 j 3 m 3 = δ m 3 , m 1 + m 2 [ ( 2 j 1 J − 2 j 3 ) ( 2 j 2 J − 2 j 3 ) ( J + 1 J − 2 j 3 ) ( 2 j 1 j 1 − m 1 ) ( 2 j 2 j 2 − m 2 ) ( 2 j 3 j 3 − m 3 ) ] 1 / 2 ∑ z ( − 1 ) z ( J − 2 j 3 z ) ( J − 2 j 2 j 1 − m 1 − z ) ( J − 2 j 1 j 2 + m 2 − z ) C_{j_1m_1j_2m_2}^{j_3m_3} = \delta_{m_3, m_1+m_2}\left[\dfrac{\begin{pmatrix}2j_1 \\ J-2j_3\end{pmatrix}\begin{pmatrix}2j_2\\J-2j_3\end{pmatrix}}{\begin{pmatrix}J+1\\J-2j_3\end{pmatrix}\begin{pmatrix}2j_1\\j_1-m_1\end{pmatrix}\begin{pmatrix}2j_2 \\ j_2 - m_2\end{pmatrix}\begin{pmatrix}2j_3 \\ j_3-m_3\end{pmatrix}}\right]^{1/2} \sum_{z} (-1)^{z}\begin{pmatrix}J-2j_3 \\ z\end{pmatrix}\begin{pmatrix}J-2j_2 \\ j_1-m_1-z\end{pmatrix}\begin{pmatrix}J-2j_1 \\ j_2 + m_2 - z\end{pmatrix}
C j 1 m 1 j 2 m 2 j 3 m 3 = δ m 3 , m 1 + m 2 ⎣ ⎢ ⎢ ⎡ ( J + 1 J − 2 j 3 ) ( 2 j 1 j 1 − m 1 ) ( 2 j 2 j 2 − m 2 ) ( 2 j 3 j 3 − m 3 ) ( 2 j 1 J − 2 j 3 ) ( 2 j 2 J − 2 j 3 ) ⎦ ⎥ ⎥ ⎤ 1 / 2 z ∑ ( − 1 ) z ( J − 2 j 3 z ) ( J − 2 j 2 j 1 − m 1 − z ) ( J − 2 j 1 j 2 + m 2 − z )
where, J = j 1 + j 2 + j 3 J = j_1+j_2+j_3 J = j 1 + j 2 + j 3
Ref[2], P236, Section 8.1.2, Formula (11).
( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( − 1 ) j 3 + m 3 + 2 j 1 1 2 j 3 + 1 C j 1 − m 1 j 2 − m 2 j 3 m 3 \begin{pmatrix}j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3\end{pmatrix} = (-1)^{j_3+m_3+2j_1}\dfrac{1}{\sqrt{2j_3+1}}C_{j_1-m_1j_2-m_2}^{j_3m_3}
( j 1 m 1 j 2 m 2 j 3 m 3 ) = ( − 1 ) j 3 + m 3 + 2 j 1 2 j 3 + 1 1 C j 1 − m 1 j 2 − m 2 j 3 m 3
In my code, I use CG coefficient to calculate 3j symbol.
Ref [2], P293, Section 9.2.1, Formula (1).
{ j 1 j 2 j 3 j 4 j 5 j 6 } = Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 3 ) Δ ( j 1 j 5 j 6 ) Δ ( j 4 j 2 j 6 ) × ∑ n ( − 1 ) n ( n + 1 ) ! ( n − j 1 j 2 j 3 ) ! ( n − j 4 j 5 j 3 ) ! ( n − j 1 j 5 j 6 ) ! ( n − j 4 j 2 j 6 ) ! ( j 1 j 2 j 4 j 5 − n ) ! ( j 1 j 3 j 4 j 6 − n ) ! ( j 2 j 3 j 5 j 6 − n ) ! \begin{Bmatrix}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6\end{Bmatrix} = \Delta(j_1j_2j_3)\Delta(j_4j_5j_3)\Delta(j_1j_5j_6)\Delta(j_4j_2j_6) \\
\times \sum\limits_n\dfrac{(-1)^n(n+1)!}{(n-j_1j_2j_3)!(n-j_4j_5j_3)!(n-j_1j_5j_6)!(n-j_4j_2j_6)!(j_1j_2j_4j_5-n)!(j_1j_3j_4j_6-n)!(j_2j_3j_5j_6-n)!}
{ j 1 j 4 j 2 j 5 j 3 j 6 } = Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 3 ) Δ ( j 1 j 5 j 6 ) Δ ( j 4 j 2 j 6 ) × n ∑ ( n − j 1 j 2 j 3 ) ! ( n − j 4 j 5 j 3 ) ! ( n − j 1 j 5 j 6 ) ! ( n − j 4 j 2 j 6 ) ! ( j 1 j 2 j 4 j 5 − n ) ! ( j 1 j 3 j 4 j 6 − n ) ! ( j 2 j 3 j 5 j 6 − n ) ! ( − 1 ) n ( n + 1 ) !
Here, I use j 1 j 2 j 3 j_1j_2j_3 j 1 j 2 j 3 j 1 + j 2 + j 2 j_1+j_2+j_2 j 1 + j 2 + j 2 Δ ( a b c ) \Delta(abc) Δ ( a b c )
Δ ( a b c ) = [ ( a + b − c ) ! ( a − b + c ) ! ( − a + b + c ) ! ( a + b + c + 1 ) ! ] 1 2 . \Delta(abc) = \left[\dfrac{(a+b-c)!(a-b+c)!(-a+b+c)!}{(a+b+c+1)!}\right]^{\frac{1}{2}}.
Δ ( a b c ) = [ ( a + b + c + 1 ) ! ( a + b − c ) ! ( a − b + c ) ! ( − a + b + c ) ! ] 2 1 .
We can find that
( j 1 + j 2 − j 3 n − j 4 j 5 j 3 ) = ( j 1 + j 2 − j 3 ) ! ( n − j 4 j 5 j 3 ) ! ( j 1 j 2 j 4 j 5 − n ) ! ( j 1 − j 2 + j 3 n − j 4 j 2 j 6 ) = ( j 1 − j 2 + j 3 ) ! ( n − j 4 j 2 j 6 ) ! ( j 1 j 3 j 4 j 6 − n ) ! ( j 2 + j 3 − j 1 n − j 1 j 5 j 6 ) = ( j 2 + j 3 − j 1 ) ! ( n − j 1 j 5 j 6 ) ! ( j 2 j 3 j 5 j 6 − n ) ! ( n + 1 j 1 j 2 j 3 + 1 ) = ( n + 1 ) ! ( n − j 1 j 2 j 3 ) ( j 1 j 2 j 3 + 1 ) ! . \begin{pmatrix}j_1+j_2-j_3 \\ n - j_4j_5j_3\end{pmatrix} = \dfrac{(j_1+j_2-j_3)!}{(n-j_4j_5j_3)!(j_1j_2j_4j_5-n)!} \\
\begin{pmatrix}j_1-j_2+j_3 \\ n - j_4j_2j_6\end{pmatrix} = \dfrac{(j_1-j_2+j_3)!}{(n-j_4j_2j_6)!(j_1j_3j_4j_6-n)!} \\
\begin{pmatrix}j_2+j_3-j_1 \\ n - j_1j_5j_6\end{pmatrix} = \dfrac{(j_2+j_3-j_1)!}{(n-j_1j_5j_6)!(j_2j_3j_5j_6-n)!} \\
\begin{pmatrix}n+1 \\ j_1j_2j_3+1\end{pmatrix} = \dfrac{(n+1)!}{(n-j_1j_2j_3)(j_1j_2j_3+1)!}.
( j 1 + j 2 − j 3 n − j 4 j 5 j 3 ) = ( n − j 4 j 5 j 3 ) ! ( j 1 j 2 j 4 j 5 − n ) ! ( j 1 + j 2 − j 3 ) ! ( j 1 − j 2 + j 3 n − j 4 j 2 j 6 ) = ( n − j 4 j 2 j 6 ) ! ( j 1 j 3 j 4 j 6 − n ) ! ( j 1 − j 2 + j 3 ) ! ( j 2 + j 3 − j 1 n − j 1 j 5 j 6 ) = ( n − j 1 j 5 j 6 ) ! ( j 2 j 3 j 5 j 6 − n ) ! ( j 2 + j 3 − j 1 ) ! ( n + 1 j 1 j 2 j 3 + 1 ) = ( n − j 1 j 2 j 3 ) ( j 1 j 2 j 3 + 1 ) ! ( n + 1 ) ! .
So, we have
{ j 1 j 2 j 3 j 4 j 5 j 6 } = Δ ( j 4 j 5 j 3 ) Δ ( j 1 j 5 j 6 ) Δ ( j 4 j 2 j 6 ) Δ ( j 1 j 2 j 3 ) × ∑ n ( − 1 ) n ( n + 1 j 1 j 2 j 3 + 1 ) ( j 1 + j 2 − j 3 n − j 4 j 5 j 3 ) ( j 1 − j 2 + j 3 n − j 4 j 2 j 6 ) ( j 2 + j 3 − j 1 n − j 1 j 5 j 6 ) . \begin{Bmatrix}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6\end{Bmatrix} = \dfrac{\Delta(j_4j_5j_3)\Delta(j_1j_5j_6)\Delta(j_4j_2j_6)}{\Delta(j_1j_2j_3)} \\
\times \sum\limits_n (-1)^n \begin{pmatrix}n+1 \\ j_1j_2j_3+1\end{pmatrix} \begin{pmatrix}j_1+j_2-j_3 \\ n - j_4j_5j_3\end{pmatrix} \begin{pmatrix}j_1-j_2+j_3 \\ n - j_4j_2j_6\end{pmatrix} \begin{pmatrix}j_2+j_3-j_1 \\ n - j_1j_5j_6\end{pmatrix}.
{ j 1 j 4 j 2 j 5 j 3 j 6 } = Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 3 ) Δ ( j 1 j 5 j 6 ) Δ ( j 4 j 2 j 6 ) × n ∑ ( − 1 ) n ( n + 1 j 1 j 2 j 3 + 1 ) ( j 1 + j 2 − j 3 n − j 4 j 5 j 3 ) ( j 1 − j 2 + j 3 n − j 4 j 2 j 6 ) ( j 2 + j 3 − j 1 n − j 1 j 5 j 6 ) .
Rewrite Δ ( a b c ) \Delta(abc) Δ ( a b c )
Δ ( a b c ) = [ 1 ( a + b + c + 1 2 a + 1 ) ( 2 a a + b − c ) ( 2 a + 1 ) ] 1 2 . \Delta(abc) = \left[\dfrac{1}{\begin{pmatrix}a+b+c+1 \\ 2a + 1\end{pmatrix} \begin{pmatrix}2a \\ a + b - c\end{pmatrix}(2a+1)}\right]^{\frac{1}{2}}.
Δ ( a b c ) = ⎣ ⎢ ⎢ ⎡ ( a + b + c + 1 2 a + 1 ) ( 2 a a + b − c ) ( 2 a + 1 ) 1 ⎦ ⎥ ⎥ ⎤ 2 1 .
So
Δ ( j 4 j 5 j 3 ) Δ ( j 1 j 5 j 6 ) Δ ( j 4 j 2 j 6 ) Δ ( j 1 j 2 j 3 ) = 1 2 j 4 + 1 [ ( j 1 j 2 j 3 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 2 − j 3 ) ( j 1 j 5 j 6 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 5 − j 6 ) ( j 4 j 5 j 3 + 1 2 j 4 + 1 ) ( 2 j 4 j 4 + j 5 − j 3 ) ( j 4 j 2 j 6 + 1 2 j 4 + 1 ) ( 2 j 4 j 4 + j 2 − j 6 ) ] 1 2 \dfrac{\Delta(j_4j_5j_3)\Delta(j_1j_5j_6)\Delta(j_4j_2j_6)}{\Delta(j_1j_2j_3)} \\
= \dfrac{1}{2j_4+1} \left[\dfrac{\begin{pmatrix}j_1j_2j_3+1 \\ 2j_1+1\end{pmatrix}\begin{pmatrix}2j_1 \\ j_1 + j_2 - j_3\end{pmatrix}}{\begin{pmatrix}j_1j_5j_6+1 \\ 2j_1+1\end{pmatrix}\begin{pmatrix}2j_1 \\ j_1+j_5-j_6\end{pmatrix}\begin{pmatrix}j_4j_5j_3+1\\2j_4+1\end{pmatrix}\begin{pmatrix}2j_4\\j_4+j_5-j_3\end{pmatrix}\begin{pmatrix}j_4j_2j_6+1 \\ 2j_4+1\end{pmatrix}\begin{pmatrix}2j_4 \\ j_4+j_2-j_6\end{pmatrix}}\right]^{\frac{1}{2}}
Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 3 ) Δ ( j 1 j 5 j 6 ) Δ ( j 4 j 2 j 6 ) = 2 j 4 + 1 1 ⎣ ⎢ ⎢ ⎡ ( j 1 j 5 j 6 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 5 − j 6 ) ( j 4 j 5 j 3 + 1 2 j 4 + 1 ) ( 2 j 4 j 4 + j 5 − j 3 ) ( j 4 j 2 j 6 + 1 2 j 4 + 1 ) ( 2 j 4 j 4 + j 2 − j 6 ) ( j 1 j 2 j 3 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 2 − j 3 ) ⎦ ⎥ ⎥ ⎤ 2 1
Ref[2], P340, Section 10.2.4, Formula (20)
{ j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = ∑ t ( − 1 ) 2 t ( 2 t + 1 ) { j 1 j 2 j 3 j 6 j 9 t } { j 4 j 5 j 6 j 2 t j 8 } { j 7 j 8 j 9 t j 1 j 4 } \begin{Bmatrix}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9\end{Bmatrix} = \sum\limits_{t}(-1)^{2t}(2t+1)\begin{Bmatrix}j_1 & j_2 & j_3 \\ j_6 & j_9 & t\end{Bmatrix} \begin{Bmatrix}j_4 & j_5 & j_6 \\ j_2 & t & j_8\end{Bmatrix} \begin{Bmatrix}j_7 & j_8 & j_9 \\ t & j_1 & j_4\end{Bmatrix}
⎩ ⎨ ⎧ j 1 j 4 j 7 j 2 j 5 j 8 j 3 j 6 j 9 ⎭ ⎬ ⎫ = t ∑ ( − 1 ) 2 t ( 2 t + 1 ) { j 1 j 6 j 2 j 9 j 3 t } { j 4 j 2 j 5 t j 6 j 8 } { j 7 t j 8 j 1 j 9 j 4 }
Use 6j symbol result above, we get
Δ ( j 1 j 9 t ) Δ ( j 6 j 9 j 3 ) Δ ( j 6 j 2 t ) Δ ( j 1 j 2 j 3 ) Δ ( j 4 t j 8 ) Δ ( j 2 t j 6 ) Δ ( j 2 j 5 j 8 ) Δ ( j 4 j 5 j 6 ) Δ ( j 7 j 1 j 4 ) Δ ( t j 1 j 9 ) Δ ( t j 8 j 4 ) Δ ( j 7 j 8 j 9 ) = Δ ( j 3 j 6 j 9 ) Δ ( j 2 j 5 j 8 ) Δ ( j 1 j 4 j 7 ) Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 6 ) Δ ( j 7 j 8 j 9 ) Δ 2 ( j 1 j 9 t ) Δ 2 ( j 2 j 6 t ) Δ 2 ( j 4 j 8 t ) . \dfrac{\Delta(j_1j_9t)\Delta(j_6j_9j_3)\Delta(j_6j_2t)}{\Delta(j_1j_2j_3)} \dfrac{\Delta(j_4tj_8)\Delta(j_2tj_6)\Delta(j_2j_5j_8)}{\Delta(j_4j_5j_6)} \dfrac{\Delta(j_7j_1j_4)\Delta(tj_1j_9)\Delta(tj_8j_4)}{\Delta(j_7j_8j_9)} \\
= \dfrac{\Delta(j_3j_6j_9)\Delta(j_2j_5j_8)\Delta(j_1j_4j_7)}{\Delta(j_1j_2j_3)\Delta(j_4j_5j_6)\Delta(j_7j_8j_9)} \Delta^2(j_1j_9t)\Delta^2(j_2j_6t)\Delta^2(j_4j_8t).
Δ ( j 1 j 2 j 3 ) Δ ( j 1 j 9 t ) Δ ( j 6 j 9 j 3 ) Δ ( j 6 j 2 t ) Δ ( j 4 j 5 j 6 ) Δ ( j 4 t j 8 ) Δ ( j 2 t j 6 ) Δ ( j 2 j 5 j 8 ) Δ ( j 7 j 8 j 9 ) Δ ( j 7 j 1 j 4 ) Δ ( t j 1 j 9 ) Δ ( t j 8 j 4 ) = Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 6 ) Δ ( j 7 j 8 j 9 ) Δ ( j 3 j 6 j 9 ) Δ ( j 2 j 5 j 8 ) Δ ( j 1 j 4 j 7 ) Δ 2 ( j 1 j 9 t ) Δ 2 ( j 2 j 6 t ) Δ 2 ( j 4 j 8 t ) .
Define j 123 ≡ j 1 + j 2 + j 3 j_{123} \equiv j_1+j_2+j_3 j 1 2 3 ≡ j 1 + j 2 + j 3
P 0 ≡ Δ ( j 3 j 6 j 9 ) Δ ( j 2 j 5 j 8 ) Δ ( j 1 j 4 j 7 ) Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 6 ) Δ ( j 7 j 8 j 9 ) = [ ( j 123 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 2 − j 3 ) ( j 456 + 1 2 j 5 + 1 ) ( 2 j 5 j 4 + j 5 − j 6 ) ( j 789 + 1 2 j 9 + 1 ) ( 2 j 9 j 7 + j 9 − j 8 ) ( j 147 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 4 − j 7 ) ( j 258 + 1 2 j 5 + 1 ) ( 2 j 5 j 2 + j 5 − j 8 ) ( j 369 + 1 2 j 9 + 1 ) ( 2 j 9 j 3 + j 9 − j 6 ) ] 1 / 2 . P_0 \equiv \dfrac{\Delta(j_3j_6j_9)\Delta(j_2j_5j_8)\Delta(j_1j_4j_7)}{\Delta(j_1j_2j_3)\Delta(j_4j_5j_6)\Delta(j_7j_8j_9)} \\
= \left[\dfrac{\begin{pmatrix}j_{123} + 1 \\ 2j_1+1\end{pmatrix}\begin{pmatrix}2j_1\\j_1+j_2-j_3\end{pmatrix}\begin{pmatrix}j_{456}+1\\2j_5+1\end{pmatrix}\begin{pmatrix}2j_5 \\ j_4+j_5-j_6\end{pmatrix}\begin{pmatrix}j_{789}+1 \\ 2j_9+1\end{pmatrix}\begin{pmatrix}2j_9 \\ j_7 + j_9 - j_8\end{pmatrix}}{\begin{pmatrix}j_{147} + 1\\ 2j_1 + 1\end{pmatrix}\begin{pmatrix}2j_1 \\ j_1+j_4-j_7\end{pmatrix}\begin{pmatrix}j_{258}+1 \\ 2j_5+1\end{pmatrix}\begin{pmatrix}2j_5 \\ j_2+j_5 - j_8\end{pmatrix}\begin{pmatrix}j_{369}+1 \\ 2j_9+1\end{pmatrix}\begin{pmatrix}2j_9 \\ j_3+j_9 - j_6\end{pmatrix}}\right]^{1/2}.
P 0 ≡ Δ ( j 1 j 2 j 3 ) Δ ( j 4 j 5 j 6 ) Δ ( j 7 j 8 j 9 ) Δ ( j 3 j 6 j 9 ) Δ ( j 2 j 5 j 8 ) Δ ( j 1 j 4 j 7 ) = ⎣ ⎢ ⎢ ⎡ ( j 1 4 7 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 4 − j 7 ) ( j 2 5 8 + 1 2 j 5 + 1 ) ( 2 j 5 j 2 + j 5 − j 8 ) ( j 3 6 9 + 1 2 j 9 + 1 ) ( 2 j 9 j 3 + j 9 − j 6 ) ( j 1 2 3 + 1 2 j 1 + 1 ) ( 2 j 1 j 1 + j 2 − j 3 ) ( j 4 5 6 + 1 2 j 5 + 1 ) ( 2 j 5 j 4 + j 5 − j 6 ) ( j 7 8 9 + 1 2 j 9 + 1 ) ( 2 j 9 j 7 + j 9 − j 8 ) ⎦ ⎥ ⎥ ⎤ 1 / 2 .
Define
P ( t ) ≡ ( − 1 ) 2 t ( 2 t + 1 ) Δ 2 ( j 1 j 9 t ) Δ 2 ( j 2 j 6 t ) Δ 2 ( j 4 j 8 t ) = ( − 1 ) 2 t 1 ( j 1 + j 9 + t + 1 2 t + 1 ) ( 2 t j 1 + t − j 9 ) ( j 2 + j 6 + t + 1 2 t + 1 ) ( 2 t j 2 + t − j 6 ) ( j 4 + j 8 + t 2 t + 1 ) ( 2 t j 4 + t − j 8 ) ( 2 t + 1 ) 2 . P(t) \equiv (-1)^{2t}(2t+1)\Delta^2(j_1j_9t)\Delta^2(j_2j_6t)\Delta^2(j_4j_8t) \\
= (-1)^{2t} \dfrac{1}{\begin{pmatrix}j_1+j_9+t+1 \\ 2t+1\end{pmatrix}\begin{pmatrix}2t \\ j_1+t-j_9\end{pmatrix}\begin{pmatrix}j_2+j_6+t+1 \\ 2t+1\end{pmatrix}\begin{pmatrix}2t \\ j_2+t-j_6\end{pmatrix}\begin{pmatrix}j_4+j_8+t \\ 2t+1\end{pmatrix}\begin{pmatrix}2t \\ j_4+t-j_8\end{pmatrix}(2t+1)^2}.
P ( t ) ≡ ( − 1 ) 2 t ( 2 t + 1 ) Δ 2 ( j 1 j 9 t ) Δ 2 ( j 2 j 6 t ) Δ 2 ( j 4 j 8 t ) = ( − 1 ) 2 t ( j 1 + j 9 + t + 1 2 t + 1 ) ( 2 t j 1 + t − j 9 ) ( j 2 + j 6 + t + 1 2 t + 1 ) ( 2 t j 2 + t − j 6 ) ( j 4 + j 8 + t 2 t + 1 ) ( 2 t j 4 + t − j 8 ) ( 2 t + 1 ) 2 1 .
A ( t , x ) ≡ ( − 1 ) x ( x + 1 j 123 + 1 ) ( j 1 + j 2 − j 3 x − ( j 6 + j 9 + j 3 ) ) ( j 1 + j 3 − j 2 x − ( j 6 + j 2 + t ) ) ( j 2 + j 3 − j 1 x − ( j 1 + j 9 + t ) ) . A(t,x) \equiv (-1)^x\begin{pmatrix}x+1 \\ j_{123} + 1\end{pmatrix}\begin{pmatrix}j_1+j_2-j_3 \\ x - (j_6+j_9+j_3)\end{pmatrix} \begin{pmatrix}j_1+j_3-j_2 \\ x - (j_6+j_2+t)\end{pmatrix}\begin{pmatrix}j_2+j_3-j_1 \\ x - (j_1 + j_9 + t)\end{pmatrix}.
A ( t , x ) ≡ ( − 1 ) x ( x + 1 j 1 2 3 + 1 ) ( j 1 + j 2 − j 3 x − ( j 6 + j 9 + j 3 ) ) ( j 1 + j 3 − j 2 x − ( j 6 + j 2 + t ) ) ( j 2 + j 3 − j 1 x − ( j 1 + j 9 + t ) ) .
B ( t , y ) ≡ ( − 1 ) y ( y + 1 j 456 + 1 ) ( j 4 + j 5 − j 6 y − ( j 2 + t + j 6 ) ) ( j 4 + j 6 − j 5 y − ( j 2 + j 5 + j 8 ) ) ( j 5 + j 6 − j 4 y − ( j 4 + t + j 8 ) ) . B(t,y) \equiv (-1)^y\begin{pmatrix}y+1 \\ j_{456} + 1\end{pmatrix}\begin{pmatrix}j_4+j_5-j_6 \\ y - (j_2+t+j_6)\end{pmatrix}\begin{pmatrix}j_4+j_6-j_5 \\ y - (j_2+j_5+j_8)\end{pmatrix}\begin{pmatrix}j_5+j_6-j_4 \\ y - (j_4+t+j_8)\end{pmatrix}.
B ( t , y ) ≡ ( − 1 ) y ( y + 1 j 4 5 6 + 1 ) ( j 4 + j 5 − j 6 y − ( j 2 + t + j 6 ) ) ( j 4 + j 6 − j 5 y − ( j 2 + j 5 + j 8 ) ) ( j 5 + j 6 − j 4 y − ( j 4 + t + j 8 ) ) .
C ( t , z ) ≡ ( − 1 ) z ( z + 1 j 789 + 1 ) ( j 7 + j 8 − j 9 z − ( t + j 1 + j 9 ) ) ( j 7 + j 9 − j 8 z − ( t + j 8 + j 4 ) ) ( j 8 + j 9 − j 7 z − ( j 7 + j 1 + j 4 ) ) . C(t,z) \equiv (-1)^z \begin{pmatrix}z+1 \\ j_{789} + 1\end{pmatrix}\begin{pmatrix}j_7+j_8-j_9 \\ z - (t+j_1+j_9)\end{pmatrix}\begin{pmatrix}j_7+j_9-j_8 \\ z - (t+j_8+j_4)\end{pmatrix}\begin{pmatrix}j_8+j_9- j_7 \\ z - (j_7 + j_1 + j_4)\end{pmatrix}.
C ( t , z ) ≡ ( − 1 ) z ( z + 1 j 7 8 9 + 1 ) ( j 7 + j 8 − j 9 z − ( t + j 1 + j 9 ) ) ( j 7 + j 9 − j 8 z − ( t + j 8 + j 4 ) ) ( j 8 + j 9 − j 7 z − ( j 7 + j 1 + j 4 ) ) .
At last, we get
{ j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = P 0 ∑ t P ( t ) ( ∑ x A ( t , x ) ) ( ∑ y B ( t , y ) ) ( ∑ z C ( t , z ) ) \begin{Bmatrix}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9\end{Bmatrix} = P_0\sum_t P(t) \left(\sum_x A(t,x)\right) \left(\sum_y B(t,y)\right) \left(\sum_z C(t,z)\right)
⎩ ⎨ ⎧ j 1 j 4 j 7 j 2 j 5 j 8 j 3 j 6 j 9 ⎭ ⎬ ⎫ = P 0 t ∑ P ( t ) ( x ∑ A ( t , x ) ) ( y ∑ B ( t , y ) ) ( z ∑ C ( t , z ) )
It deserves to be mentioned that, although the formula has 4 ∑ \sum ∑ ∑ \sum ∑ x , y , z x,y,z x , y , z for loops respectively, which means the depth of for loop is not 4 but 2.
Reference
T. Engeland and M. Hjorth-Jensen, the Oslo-FCI code. https://github.com/ManyBodyPhysics/CENS .
A. N. Moskalev D. A. Varshalovich and V. K. Khersonskii, Quantum theory of angular momentum .