Multiple counting#
Multiple counting occurs in some terms of the Hamiltonian. In those terms the same set of magnetic centers enters the sum multiple times with different order of spin operators.
For the sake of the choosing the primary version of the same bond we define the comparison of two spin operators based on their positions as (where \(\mu = (\mu^1, \mu^2, \mu^3)\))
\(\boldsymbol{S}_{\mu_1,\alpha_1} < \boldsymbol{S}_{\mu_2,\alpha_2}\) if one of the conditions is met
\(\mu_2^1 - \mu_1^1 > 0\)
\(\mu_2^1 - \mu_1^1 = 0, \mu_2^2 - \mu_1^2 > 0\)
\(\mu_2^1 - \mu_1^1 = \mu_2^2 - \mu_1^2 = 0, \mu_2^3 - \mu_1^3 > 0\)
\(\mu_2^1 - \mu_1^1 = \mu_2^2 - \mu_1^2 = \mu_2^3 - \mu_1^3 = 0, \alpha_2 - \alpha_1 > 0\)
\(\boldsymbol{S}_{\mu_1,\alpha_1} > \boldsymbol{S}_{\mu_2,\alpha_2}\) otherwise.
Moreover, we use in this page a short notation for the terms of the Hamiltonian
With two sites:
\[\begin{split}(\alpha, \beta, \nu) &\rightarrow J_{2,2}^{ij}(\boldsymbol{r}_{\nu,\alpha\beta}) S_{\mu,\alpha}^i S_{\mu+\nu,\beta}^j \\ (\alpha, \beta, \nu) &\rightarrow J_{3, 2}^{iju}(\boldsymbol{r}_{\nu,\alpha\beta}) S_{\mu,\alpha}^i S_{\mu,\alpha}^j S_{\mu+\nu,\beta}^u \\ (\alpha, \beta, \nu) &\rightarrow J_{4, 2, 1}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}) S_{\mu,\alpha}^i S_{\mu,\alpha}^j S_{\mu,\alpha}^u S_{\mu+\nu,\beta}^v \\ (\alpha, \beta, \nu) &\rightarrow J_{4, 2, 2}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}) S_{\mu,\alpha}^i S_{\mu,\alpha}^j S_{\mu+\nu,\beta}^u S_{\mu+\nu,\beta}^v\end{split}\]With three sites
\[\begin{split}(\alpha, \beta, \gamma, \nu, \lambda) &\rightarrow J_{3, 3}^{iju}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}) S_{\mu,\alpha}^i S_{\mu+\nu,\beta}^j S_{\mu+\lambda,\gamma}^u \\ (\alpha, \beta, \gamma, \nu, \lambda) &\rightarrow J_{4, 3}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}) S_{\mu,\alpha}^i S_{\mu,\alpha}^j S_{\mu+\nu,\beta}^u S_{\mu+\lambda,\gamma}^v\end{split}\]With four sites
\[(\alpha, \beta, \gamma, \varepsilon, \nu, \lambda, \rho) \rightarrow J_{4, 3}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon}) S_{\mu,\alpha}^i S_{\mu+\nu,\beta}^j S_{\mu+\lambda,\gamma}^u S_{\mu+\rho,\varepsilon}^v\]
Two spins & two sites#
For the given term of the Hamiltonian
two cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \beta, \nu)\) is a primary version with the parameter
\[J_{2,2}^{ij}(\boldsymbol{r}_{\nu,\alpha\beta})\]If \(\boldsymbol{S}_{\mu,\alpha} > \boldsymbol{S}_{\mu+\nu,\beta}\), then \((\beta, \alpha, -\nu)\) is a primary version with the parameter
\[J_{2,2}^{ij}(\boldsymbol{r}_{-\nu,\beta\alpha}) = J_{2,2}^{ji}(\boldsymbol{r}_{\nu,\alpha\beta})\]
Three spins & two sites#
For the given term of the Hamiltonian
two cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \beta, \nu)\) is a primary version with the parameter
\[J_{3, 2}^{iju}(\boldsymbol{r}_{\nu,\alpha\beta})\]If \(\boldsymbol{S}_{\mu,\alpha} > \boldsymbol{S}_{\mu+\nu,\beta}\), then \((\beta, \alpha, -\nu)\) is a primary version with the parameter
\[J_{3,2}^{iju}(\boldsymbol{r}_{-\nu,\beta\alpha}) = \dfrac{S_{\alpha}}{S_{\beta}} J_{3,2}^{uji}(\boldsymbol{r}_{\nu,\alpha\beta})\]
Three spins & three sites#
For the given term of the Hamiltonian
six cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\alpha, \beta, \gamma, \nu, \lambda)\) is a primary version with the parameter
\[J_{3, 3}^{iju}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \gamma, \beta, \lambda, \nu)\) is a primary version with the parameter
\[J_{3, 3}^{iju}(\boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\nu,\alpha\beta}) = J_{3, 3}^{iuj}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\beta, \alpha, \gamma, -\nu, \lambda - \nu)\) is a primary version with the parameter
\[J_{3, 3}^{iju}(\boldsymbol{r}_{-\nu,\beta\alpha}, \boldsymbol{r}_{\lambda - \nu,\beta\gamma}) = J_{3, 3}^{jiu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha}\) then \((\beta, \gamma, \alpha, \lambda - \nu, -\nu)\) is a primary version with the parameter
\[J_{3, 3}^{iju}(\boldsymbol{r}_{\lambda - \nu,\beta\gamma}, \boldsymbol{r}_{-\nu,\beta\alpha}) = J_{3, 3}^{uij}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\gamma, \alpha, \beta, -\lambda, \nu - \lambda)\) is a primary version with the parameter
\[J_{3, 3}^{iju}(\boldsymbol{r}_{-\lambda, \gamma\alpha}, \boldsymbol{r}_{\nu - \lambda,\gamma\beta}) = J_{3, 3}^{jui}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha}\) then \((\gamma, \beta, \alpha, \nu - \lambda, -\lambda)\) is a primary version with the parameter
\[J_{3, 3}^{iju}(\boldsymbol{r}_{\nu - \lambda,\gamma\beta}, \boldsymbol{r}_{-\lambda, \gamma\alpha}) = J_{3, 3}^{uji}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]
Four spins & two sites (1+3)#
For the given term of the Hamiltonian
two cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \beta, \nu)\) is a primary version with the parameter
\[J_{4, 2, 1}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta})\]If \(\boldsymbol{S}_{\mu,\alpha} > \boldsymbol{S}_{\mu+\nu,\beta}\), then \((\beta, \alpha, -\nu)\) is a primary version with the parameter
\[J_{4, 2, 1}^{ijuv}(\boldsymbol{r}_{-\nu,\beta\alpha}) = \left(\dfrac{S_{\alpha}}{S_{\beta}}\right)^2 J_{4, 2, 1}^{vjui}(\boldsymbol{r}_{\nu,\alpha\beta})\]
Four spins & two sites (2+2)#
For the given term of the Hamiltonian
two cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \beta, \nu)\) is a primary version with the parameter
\[J_{4, 2, 2}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta})\]If \(\boldsymbol{S}_{\mu,\alpha} > \boldsymbol{S}_{\mu+\nu,\beta}\), then \((\beta, \alpha, -\nu)\) is a primary version with the parameter
\[J_{4, 2, 2}^{ijuv}(\boldsymbol{r}_{-\nu,\beta\alpha}) = J_{4, 2, 2}^{uvij}(\boldsymbol{r}_{\nu,\alpha\beta})\]
Four spins & three sites#
For the given term of the Hamiltonian
six cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\alpha, \beta, \gamma, \nu, \lambda)\) is a primary version with the parameter
\[J_{4, 3}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \gamma, \beta, \lambda, \nu)\) is a primary version with the parameter
\[J_{4, 3}^{ijuv}(\boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\nu,\alpha\beta}) = J_{4, 3}^{ijvu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\beta, \alpha, \gamma, -\nu, \lambda - \nu)\) is a primary version with the parameter
\[J_{4, 3}^{ijuv}(\boldsymbol{r}_{-\nu,\beta\alpha}, \boldsymbol{r}_{\lambda - \nu,\beta\gamma}) = \dfrac{S_{\alpha}}{S_{\beta}} J_{4, 3}^{ujiv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha}\) then \((\beta, \gamma, \alpha, \lambda - \nu, -\nu)\) is a primary version with the parameter
\[J_{4, 3}^{ijuv}(\boldsymbol{r}_{\lambda - \nu,\beta\gamma}, \boldsymbol{r}_{-\nu,\beta\alpha}) = \dfrac{S_{\alpha}}{S_{\beta}} J_{4, 3}^{vjiu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\gamma, \alpha, \beta, -\lambda, \nu - \lambda)\) is a primary version with the parameter
\[J_{4, 3}^{ijuv}(\boldsymbol{r}_{-\lambda, \gamma\alpha}, \boldsymbol{r}_{\nu - \lambda,\gamma\beta}) = \dfrac{S_{\alpha}}{S_{\gamma}} J_{4, 3}^{ujvi}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha}\) then \((\gamma, \beta, \alpha, \nu - \lambda, -\lambda)\) is a primary version with the parameter
\[J_{4, 3}^{ijuv}(\boldsymbol{r}_{\nu - \lambda,\gamma\beta}, \boldsymbol{r}_{-\lambda, \gamma\alpha}) = \dfrac{S_{\alpha}}{S_{\gamma}} J_{4, 3}^{vjui}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})\]
Four spins & four sites#
For the given term of the Hamiltonian
twenty four cases are possible
If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\rho,\varepsilon}\) then \((\alpha, \beta, \gamma, \varepsilon, \nu, \lambda, \rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\alpha, \beta, \varepsilon, \gamma, \nu, \rho, \lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\rho,\alpha\varepsilon}, \boldsymbol{r}_{\lambda,\alpha\gamma}) = J_{4, 4}^{ijvu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\rho,\varepsilon}\) then \((\alpha, \gamma, \beta, \varepsilon, \lambda, \nu, \rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\rho,\alpha\varepsilon}) = J_{4, 4}^{iujv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \gamma, \varepsilon, \beta, \lambda, \rho, \nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon}, \boldsymbol{r}_{\nu,\alpha\beta}) = J_{4, 4}^{ivju}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\alpha, \varepsilon, \beta, \gamma, \rho, \nu, \lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\rho,\alpha\varepsilon}, \boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}) = J_{4, 4}^{iuvj}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\alpha, \varepsilon, \gamma, \beta, \rho, \lambda, \nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\rho,\alpha\varepsilon}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\nu,\alpha\beta}) = J_{4, 4}^{ivuj}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\rho,\varepsilon}\) then \((\beta, \alpha, \gamma, \varepsilon, -\nu, \lambda-\nu, \rho-\nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{-\nu,\beta\alpha}, \boldsymbol{r}_{\lambda-\nu,\beta\gamma}, \boldsymbol{r}_{\rho-\nu,\beta\varepsilon}) = J_{4, 4}^{jiuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\beta, \alpha, \gamma, \varepsilon, -\nu, \rho-\nu, \lambda-\nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{-\nu,\beta\alpha}, \boldsymbol{r}_{\rho-\nu,\beta\varepsilon}, \boldsymbol{r}_{\lambda-\nu,\beta\gamma}) = J_{4, 4}^{jivu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\rho,\varepsilon}\) then \((\beta, \gamma, \alpha, \varepsilon, \lambda-\nu, -\nu, \rho-\nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\lambda-\nu,\beta\gamma}, \boldsymbol{r}_{-\nu,\beta\alpha}, \boldsymbol{r}_{\rho-\nu,\beta\varepsilon}) = J_{4, 4}^{uijv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu,\alpha}\) then \((\beta, \gamma, \varepsilon, \alpha, \lambda-\nu, \rho-\nu, -\nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\lambda-\nu,\beta\gamma}, \boldsymbol{r}_{\rho-\nu,\beta\varepsilon}, \boldsymbol{r}_{-\nu,\beta\alpha}) = J_{4, 4}^{viju}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\beta, \varepsilon, \alpha, \gamma, \rho-\nu, -\nu, \lambda-\nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\rho-\nu,\beta\varepsilon}, \boldsymbol{r}_{-\nu,\beta\alpha}, \boldsymbol{r}_{\lambda-\nu,\beta\gamma}) = J_{4, 4}^{uivj}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha}\) then \((\beta, \varepsilon, \gamma, \alpha, \rho-\nu, \lambda-\nu, -\nu)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\rho-\nu,\beta\varepsilon}, \boldsymbol{r}_{\lambda-\nu,\beta\gamma}, \boldsymbol{r}_{-\nu,\beta\alpha}) = J_{4, 4}^{viuj}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\rho,\varepsilon}\) then \((\gamma, \alpha, \beta, \varepsilon, -\lambda, \nu-\lambda, \rho-\lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{-\lambda,\gamma\alpha}, \boldsymbol{r}_{\nu-\lambda,\gamma\beta}, \boldsymbol{r}_{\rho-\lambda,\gamma\varepsilon}) = J_{4, 4}^{juiv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\gamma, \alpha, \varepsilon, \beta, -\lambda, \rho-\lambda, \nu-\lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{-\lambda,\gamma\alpha}, \boldsymbol{r}_{\rho-\lambda,\gamma\varepsilon}, \boldsymbol{r}_{\nu-\lambda,\gamma\beta}) = J_{4, 4}^{jviu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\rho,\varepsilon}\) then \((\gamma, \beta, \alpha, \varepsilon, \nu-\lambda, -\lambda, \rho-\lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu-\lambda,\gamma\beta}, \boldsymbol{r}_{-\lambda,\gamma\alpha}, \boldsymbol{r}_{\rho-\lambda,\gamma\varepsilon}) = J_{4, 4}^{ujiv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu,\alpha}\) then \((\gamma, \beta, \varepsilon, \alpha, \nu-\lambda, \rho-\lambda, -\lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu-\lambda,\gamma\beta}, \boldsymbol{r}_{\rho-\lambda,\gamma\varepsilon}, \boldsymbol{r}_{-\lambda,\gamma\alpha}) = J_{4, 4}^{vjiu}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\gamma, \varepsilon, \alpha, \beta, \rho-\lambda, -\lambda, \nu-\lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\rho-\lambda,\gamma\varepsilon}, \boldsymbol{r}_{-\lambda,\gamma\alpha}, \boldsymbol{r}_{\nu-\lambda,\gamma\beta}) = J_{4, 4}^{uvij}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha}\) then \((\gamma, \varepsilon, \beta, \alpha, \rho-\lambda, \nu-\lambda, -\lambda)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\rho-\lambda,\gamma\varepsilon}, \boldsymbol{r}_{\nu-\lambda,\gamma\beta}, \boldsymbol{r}_{-\lambda,\gamma\alpha}) = J_{4, 4}^{vuij}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\varepsilon, \alpha, \beta, \gamma, -\rho, \nu-\rho, \lambda-\rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{-\rho,\varepsilon,\alpha}, \boldsymbol{r}_{\nu-\rho,\varepsilon,\beta}, \boldsymbol{r}_{\lambda-\rho,\varepsilon,\gamma}) = J_{4, 4}^{juvi}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\varepsilon, \alpha, \gamma, \beta, -\rho, \lambda-\rho, \nu-\rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{-\rho,\varepsilon,\alpha}, \boldsymbol{r}_{\lambda-\rho,\varepsilon,\gamma}, \boldsymbol{r}_{\nu-\rho,\varepsilon,\beta}) = J_{4, 4}^{jvui}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\lambda,\gamma}\) then \((\varepsilon, \beta, \alpha, \gamma, \nu-\rho, -\rho, \lambda-\rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu-\rho,\varepsilon,\beta}, \boldsymbol{r}_{-\rho,\varepsilon,\alpha}, \boldsymbol{r}_{\lambda-\rho,\varepsilon,\gamma}) = J_{4, 4}^{ujvi}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha}\) then \((\varepsilon, \beta, \gamma, \alpha, \nu-\rho, \lambda-\rho, -\rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu-\rho,\varepsilon,\beta}, \boldsymbol{r}_{\lambda-\rho,\varepsilon,\gamma}, \boldsymbol{r}_{-\rho,\varepsilon,\alpha}) = J_{4, 4}^{vjui}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu,\alpha} < \boldsymbol{S}_{\mu+\nu,\beta}\) then \((\varepsilon, \gamma, \alpha, \beta, \lambda-\rho, -\rho, \nu-\rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\lambda-\rho,\varepsilon,\gamma}, \boldsymbol{r}_{-\rho,\varepsilon,\alpha}, \boldsymbol{r}_{\nu-\rho,\varepsilon,\beta}) = J_{4, 4}^{uvji}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]If \(\boldsymbol{S}_{\mu+\rho,\varepsilon} < \boldsymbol{S}_{\mu+\lambda,\gamma} < \boldsymbol{S}_{\mu+\nu,\beta} < \boldsymbol{S}_{\mu,\alpha}\) then \((\varepsilon, \gamma, \beta, \alpha, \lambda-\rho, \nu-\rho, -\rho)\) is a primary version with the parameter
\[J_{4, 4}^{ijuv}(\boldsymbol{r}_{\lambda-\rho,\varepsilon,\gamma}, \boldsymbol{r}_{\nu-\rho,\varepsilon,\beta}, \boldsymbol{r}_{-\rho,\varepsilon,\alpha}) = J_{4, 4}^{vuji}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})\]