Intermediate Coupling

For some rare earth ions and for transition metals or actinides it is necessary to include more singleion ion states with differen L,S into the calculation. This can be done in intermediate coupling using the module ic1ion, which explicitely includes electrostatic and spin orbit interactions for each ion:

$${\mathcal H} = \sum_n \left \{ \sum_{i_n=1}^{\nu_n} \left [ \frac{p_{i_n}^2}{2m_... ...n}\frac{e^2}{4\pi\epsilon_0\vert\mathbf r_{i_n}-\mathbf r_{j_n}\vert} \right \}$$

$$- \sum_{n} \mu_B (2\mathbf S^n+\mathbf L^n) {\mathbf H}$$

$$-\frac{1}{2} \sum_{nn'} \left[ ({\hat \mathbf L}^n,{\hat \mathbf ... ...'} \sum_{qq'} \mathcal{K}_{kk'}^{qq'}(nn') \hat{T}_{kq}^n T_{k'q'}^{n'} \right]$$ (4)

Here $\nu_n$,$Z_n$ and $\mathbf R_n$ denote the number of electrons, the charge of the nucleus and the position of the ion number $n$,respectively, for each electron being $p$ the momentum, $m_e$ the mass, $e$ the charge and $\mathbf r$ the location. Spin orbit coupling is written in terms of the orbital momentum $\mathbf l$ and spin $\mathbf s$ of the individual electrons. The Zeman interaction and two ion interaction are written in terms of the (inverse) total spin $\mathbf S_n$ and (inverse) total orbital momentum $\mathbf L_n$ of ion number $n$. The crystal field in intermediate coupling is written in terms of Wybourne parameters $L_l^m$ and Wybourne operators $T_{lm}^n$, operator equivalents of real valued spherical harmonic functions $T_{l0}=\sqrt{4\pi/(2l+1)}\sum_iY_{l0}(\Omega_{i_n})$, $T_{l,\pm\vert m\vert}=\sqrt{4\pi/(2l+1)}\sum_i \sqrt{\pm1}[Y_{l,-\vert m\vert}(\Omega_{i_n})\pm (-1)^m Y_{l,\vert m\vert}(\Omega_{i_n})]$ for the ion $n$, for details on crystal field parameter conventions see appendix E.