Wigner's $9$-$j$ symbol - four electrons orbital angular momentum coupling

Question

In constructing the total angular momentum operator, that is the sum of 4 independent angular momentum operators:

$$J=J_1+J_2+J_3+J_4 $$ one has the following set of commuting operators and eigenvectors in case of the uncoupled configuration:

$$ {\textbf{J}_1^2,J_{1z},\textbf{J}_2^2,J_{2z},\textbf{J}_3^2,J_{3z},\textbf{J}_4^2,J_{4z}},$$

$$|j_1m_1\rangle|j_2m_2\rangle|j_3m_3\rangle|j_4m_4\rangle.$$

Now in the coupled representation, if one couple pairwise there are different choices. Most commons I have seen in books are:

$$|(j_1j_2)J_{12}(j_3j_4)J_{34};J\rangle \ and\ |(j_1j_3)J_{13}(j_2j_4)J_{24};J\rangle $$

Now the Wigner 9j symbols are within a constant the coefficients that allow us to go from one basis to another, thus

$$\langle (j_1j_2)J_{12}(j_3j_4)J_{34};J|{(j_1j_3)J_{13}(j_2j_4)J_{24};J}\rangle \propto \Bigg\{ \begin{matrix} j_1 & j_2 & J_{12}\\ j_3 & j_4 & J_{34}\\ J_{13} & J_{24} & J\\ \end{matrix} \Bigg\}.$$

I am more interested in

extend coupling to n angular momenta by successive coupling of an extra angular momentum to the former n - 1 system [1]

In this case we have [2,3]

$$\langle [(j_1j_2)J_{12},j_3]J_{123},j_4;J|{[(j_4j_2)J_{42},j_3]J_{423},j_1;J}\rangle \propto \Bigg\{ \begin{matrix} j_2 & J_{12} & j_{1}\\ J_{42} & j_3 & J_{423}\\ j_{4} & J_{123} & J\\ \end{matrix} \Bigg\}.$$

and here my question and my doubt:

Is it possible to relate the following coupling scheme through a Wigner symbol ?

1. $$\langle [(j_1j_2)J_{12},j_3]J_{123},j_4;J|{[(j_2j_3)J_{23},j_4]J_{234},j_1;J}\rangle $$
2. $$\langle [(j_1j_2)J_{12},j_3]J_{123},j_4;J|{[(j_1j_2)J_{12},j_4]J_{124},j_3;J}\rangle $$

If yes, how can I build the Wigner 9j symbol (i.e. the positions of the $j$)? Is there any symbolic calculator or table where I can look for? It will really help me since I would like to extend the same also to the Wigner 12j and so on.

References

[1] Professor Dr. Kris L. G. Heyde - The Nuclear Shell Model - Study Edition (1994). pp 26

[2] Edmonds - Angular momentum in quantum mechanics-Princeton, N.J., Princeton University Press (1957). pp 104

[3] Albert Messiah - Quantum Mechanics. 2 - John Wiley and Sons, Inc. (1961). pp 1067

[4] A. P. Yutsis - Mathematical Apparatus of the Theory of Angular Momentum.

Answer 1

I do not think your overlaps reduce to the usual 9j symbols through symmetry operations.

In your first coupling scheme, you are forced by the scheme on the left to have $j_1,j_2$ and $J_{12}$ on the same row, and then $J_{12}, j_3$ and $J_{123}$ in the same column, and then $J_{123}, j_4$ and $J$ on the same row, i.e something like \begin{align} \left\{\begin{array}{ccc} j_1&j_2&J_{12}\\ &&j_3\\ J&j_4&J_{123}\\ \end{array}\right\} \tag{1} \end{align} But then for your scheme on the right you have $j_2,j_3\to J_{23}$ so $j_2,j_3$ and $J_{23}$ must be either on the same row or column. However, you can see this is not possible since $j_3$ must already be in the same column as $J_{12}$ and $J_{123}$, and cannot be in the same row as $j_2$ by your left coupling.

The same logic applies to your second coupling scheme, where from the left you'd have again (1) but this time you need to fit $J_{12}$ on the same line or column as $j_4$ when it's already in a given full line or column.

There are calculators but I don't think they will be useful since I don't think what you want is equivalent to a $9j$. You probably have to build a custom function using the same idea of writing the standard 9j as a sum of 6j's. Moreover, in your second scheme, you can probably reorganize \begin{align} [J_{12}j_4]J_{124}j_3 \to [J_{12}j_3]J_{123}j_4 \end{align} using a $6j$ symbol and make mileage there but again you'd have to do this as a custom function.

Note that it should not come as a surprise to get a negative answer. Historically the 9j was introduced to deal with the undoing of $jj$ coupling back to an LS coupling to compute matrix elements of - say - the spin-orbit coupling or some tensor operator acting on angular momenta only. Hence by design it is not constructed for recursive coupling of the type you want.