Exchange striction¶
Exchange striction involves three entities. Therefore, it can be expressed as a \(\mathcal{H}_3\) term.
\[\begin{split}\mathcal{H}_3
=
C_3
\sum_{\substack{\mu_1, \mu_2, \mu_3\\ \alpha_1, \alpha_2, \alpha_3\\ i_1, i_2, i_3}}
V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3}
X_{\mu_1; \alpha_1}^{i_1}
X_{\mu_2; \alpha_2}^{i_2}
X_{\mu_3; \alpha_3}^{i_3}\end{split}\]
The summary of the mapping for each code is given in the table below.
Code |
|
|---|---|
\(\mathcal{H}\) |
\[\begin{split}\hat{\mathcal{H}}
=
-
\dfrac{1}{2}
\sum_{\substack{nn^{\prime},\alpha\beta\alpha^{\prime}\gamma=1,2,3\\\beta^{\prime}=1-6}}
\Biggl(
\dfrac{\partial \mathcal{J}_{\alpha\beta}}{\partial \epsilon_{\beta^{\prime}}}
+
\dfrac{\partial \mathcal{J}_{\alpha\beta}}{\partial R_{nn^{\prime}}^{\alpha^{\prime}}}
\dfrac{\partial \epsilon_{\alpha^{\prime}\gamma} R_{nn^{\prime}}^{\gamma}}{\partial \epsilon_{\beta^{\prime}}}
\Biggr)
\epsilon_{\beta^{\prime}}
\hat{\mathcal{I}}_{\alpha}^n
\hat{\mathcal{I}}_{\beta}^{n^{\prime}}\end{split}\]
|
Renaming of indices |
\(n \rightarrow (\mu_2, \alpha_2)\), \(n^{\prime} \rightarrow (\mu_3, \alpha_3)\), \(\beta^{\prime} \rightarrow i_1\), \(\alpha \rightarrow i_2\), \(\beta \rightarrow i_3\) |
\(C_3\) |
\(-\dfrac{1}{2}\) |
\(V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3}\) |
\[\sum_{\alpha^{\prime}\gamma=1,2,3}
\Biggl(
\dfrac{\partial \mathcal{J}_{i_2,i_3}}{\partial \epsilon_{i_1}}
+
\dfrac{\partial \mathcal{J}_{i_2,i_3}}{\partial R_{\mu_2,\mu_3;\alpha_2,\alpha_3}^{\alpha^{\prime}}}
\dfrac{\partial \epsilon_{\alpha^{\prime}\gamma} R_{\mu_2,\mu_3;\alpha_2,\alpha_3}^{\gamma}}{\partial \epsilon_{i_1}}
\Biggr)
\delta_{\mu_1, \mu_2} \delta_{\alpha_1, \alpha_2}\]
|
\(X_{\mu_1, \alpha_1}^{i_1}\) |
\(\epsilon_{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(\hat{\mathcal{I}}_{i_2}^{\mu_2,\alpha_2}\) |
\(X_{\mu_3; \alpha_3}^{i_3}\) |
\(\hat{\mathcal{I}}_{i_3}^{\mu_3,\alpha_3}\) |