Crystal field¶
Crystal field terms involves either one entity. Therefore, it can be expressed as a \(\mathcal{H}_1\) term.
The summary of the mapping for each code is given in the table below.
Code |
McPhase (v1) |
McPhase (v2) |
|
|---|---|---|---|
\(\mathcal{H}\) |
\[\hat{\mathcal{H}}
=
\sum_n
\sum_{lm}
B_l^m
\hat{O}_{lm}(\mathbf{J}^n)\]
|
\[\hat{\mathcal{H}}
=
\sum_n
\sum_{lm}
L_l^m(n)
\hat{T}_{lm}^{n}\]
|
\[\mathcal{H}
=
\sum_{i}\sum_{k,q}
c_{i,k,q}
\mathcal{O}_{k,q}
(\mathbf{S}_{i})\]
|
Renaming of indices |
\(n \rightarrow (\mu_1, \alpha_1)\), \((l,m) \rightarrow i_1\) |
\(n \rightarrow (\mu_1, \alpha_1)\), \((l,m) \rightarrow i_1\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \((k,q) \rightarrow i_1\) |
\(C_1\) |
1 |
1 |
1 |
\(V_{\mu_1; \alpha_1}^{i_1}\) |
\(B^{i_1}\) |
\(L^{i_1}(\mu_1, \alpha_1)\) |
\(c_{\mu_1, \alpha_1, i_1}\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(\hat{O}_{i_1}(\mathbf{J}^{\mu_1, \alpha_1})\) |
\(\hat{T}_{i_1}^{\mu_1, \alpha_1}\) |
\(\mathcal{O}_{i_1}(\mathbf{S}_{\mu_1, \alpha_1})\) |
Note
Pairs of indices \((l, m)\) or \((k, q)\) run over the finite set of the integer pairs. Therefore, they can be enumerated with a single integer and mapped to the index \(i_1\).