Stevens Operators

In the single ion module so1ion the operators are identified as shown below with $I_{\alpha}$ .

$\displaystyle X=J(J+1)$
$\displaystyle O_{00}$	$\textstyle =$	$\displaystyle 1$
$\displaystyle \hline O_{11}$	$\textstyle =$	$\displaystyle \frac{1}{2}[J_++J_-]=J_x=I_1$
$\displaystyle O^s_{11}$	$\textstyle =$	$\displaystyle \frac{-i}{2}[J_+-J_-]=J_y=I_2$
$\displaystyle O_{10}$	$\textstyle =$	$\displaystyle J_z=I_3$
$\displaystyle \hline O^s_{22}$	$\textstyle =$	$\displaystyle \frac{-i}{2}[J_+^2-J_-^2]=J_xJ_y+J_yJ_x=2P_{xy}=I_4$
$\displaystyle O^s_{21}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[J_z(J_+-J_-)+(J_+-J_-)J_z]=\frac{1}{2}[J_yJ_z+J_zJ_y]=P_{yz}=I_5$
$\displaystyle O_{20}$	$\textstyle =$	$\displaystyle [3J_z^2-X]=I_6$
$\displaystyle O_{21}$	$\textstyle =$	$\displaystyle \frac{1}{4}[J_z(J_++J_-)+(J_++J_-)J_z]=\frac{1}{2}[J_xJ_z+J_zJ_x]=P_{xz}=I_7$
$\displaystyle O_{22}$	$\textstyle =$	$\displaystyle \frac{1}{2}[J_+^2+J_-^2]=J_x^2-J_y^2 =I_8$
$\displaystyle \hline O^s_{33}$	$\textstyle =$	$\displaystyle \frac{-i}{2}[J_+^3-J_-^3]=I_9$
$\displaystyle O^s_{32}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^2-J_-^2)J_z+J_z(J_+^2-J_-^2)]=I_{10}$
$\displaystyle O^s_{31}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+-J_-)(5J_z^2-X-1/2)+(5J_z^2-X-1/2)(J_+-J_-)]=I_{11}$
$\displaystyle O_{30}$	$\textstyle =$	$\displaystyle [5J_z^3-(3X-1)J_z]=I_{12}$
$\displaystyle O_{31}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_++J_-)(5J_z^2-X-1/2)+(5J_z^2-X-1/2)(J_++J_-)] =I_{13}$
$\displaystyle O_{32}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^2+J_-^2)J_z+J_z(J_+^2+J_-^2)]=I_{14}$
$\displaystyle O_{33}$	$\textstyle =$	$\displaystyle \frac{1}{2}[J_+^3+J_-^3]=I_{15}$
$\displaystyle \hline O^s_{44}$	$\textstyle =$	$\displaystyle \frac{-i}{2}[(J_+^4-J_-^4]=I_{16}$
$\displaystyle O^s_{43}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^3-J_-^3)J_z+J_z(J_+^3-J_-^3)] =I_{17}$
$\displaystyle O^s_{42}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^2-J_-^2)(7J_z^2-X-5)+(7J_z^2-X-5)(J_+^2-J_-^2)]=I_{18}$
$\displaystyle O^s_{41}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+-J_-)(7J_z^3-(3X+1)J_z)+(7J_z^3-(3X+1)J_z)(J_+-J_-)]=I_{19}$
$\displaystyle O_{40}$	$\textstyle =$	$\displaystyle [35J_z^4-(30X-25)J_z^2+3X^2-6X]=I_{20}$
$\displaystyle O_{41}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_++J_-)(7J_z^3-(3X+1)J_z)+(7J_z^3-(3X+1)J_z)(J_++J_-)]=I_{21}$
$\displaystyle O_{42}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^2+J_-^2)(7J_z^2-X-5)+(7J_z^2-X-5)(J_+^2+J_-^2)] =I_{22}$
$\displaystyle O_{43}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^3+J_-^3)J_z+J_z(J_+^3+J_-^3)]=I_{23}$
$\displaystyle O_{44}$	$\textstyle =$	$\displaystyle \frac{1}{2}[(J_+^4+J_-^4]=I_{24}$
$\displaystyle \hline O^s_{55}$	$\textstyle =$	$\displaystyle \frac{-i}{2}[J_+^5-J_-^5]=I_{25}$
$\displaystyle O^s_{54}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^4-J_-^4)J_z+J_z(J_+^4-J_-^4)]=I_{26}$
$\displaystyle O^s_{53}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^3-J_-^3)(9J_z^2-X-33/2)+(9J_z^2-X-33/2)(J_+^3-J_-^3)]=I_{27}$
$\displaystyle O^s_{52}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^2-J_-^2)(3J_z^3-(X+6)J_z)+(3J_z^3-(X+6)J_z)(J_+^2-J_-^2)]=I_{28}$
$\displaystyle O^s_{51}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+-J_-)\{21J_z^4-14J_z^2X+X^2-X+3/2\}+\{\dots\}(J_+-J_-)]=I_{29}$
$\displaystyle O_{50}$	$\textstyle =$	$\displaystyle [63J_z^5-(70X-105)J_z^3+(15X^2-50X+12)J_z]=I_{30}$
$\displaystyle O_{51}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_++J_-)(21J_z^4-14J_z^2X+X^2-X+3/2)+(21J_z^4-14J_z^2X+X^2-X+3/2)(J_++J_-)]=I_{31}$
$\displaystyle O_{52}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^2+J_-^2)(3J_z^3-(X+6)J_z)+(3J_z^3-(X+6)J_z)(J_+^2+J_-^2)] =I_{32}$
$\displaystyle O_{53}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^3+J_-^3)(9J_z^2-X-33/2)+(9J_z^2-X-33/2)(J_+^3+J_-^3)]=I_{33}$
$\displaystyle O_{54}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^4+J_-^4)J_z+J_z(J_+^4+J_-^4)]=I_{34}$
$\displaystyle O_{55}$	$\textstyle =$	$\displaystyle \frac{1}{2}[J_+^5+J_-^5]=I_{35}$
$\displaystyle \hline O^s_{66}$	$\textstyle =$	$\displaystyle \frac{-i}{2}[J_+^6-J_-^6]=I_{36}$
$\displaystyle O^s_{65}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^5-J_-^5)J_z+J_z(J_+^5-J_-^5)] =I_{37}$
$\displaystyle O^s_{64}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^4-J_-^4)(11J_z^2-X-38)+(11J_z^2-X-38)(J_+^4-J_-^4)]=I_{38}$
$\displaystyle O^s_{63}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^3-J_-^3)(11J_z^3-(3X+59)J_z)+(11J_z^3-(3X+59)J_z)(J_+^3-J_-^3)]=I_{39}$
$\displaystyle O^s_{62}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+^2-J_-^2)\{33J_z^4-(18X+123)J_z^2+X^2+10X+102\}+\{\dots\}(J_+^2-J_-^2)]=I_{40}$
$\displaystyle O^s_{61}$	$\textstyle =$	$\displaystyle \frac{-i}{4}[(J_+-J_-)\{33J_z^5-(30X-15)J_z^3+(5X^2-10X+12)J_z\}+\{\dots\}(J_+-J_-)]=I_{41}$
$\displaystyle O_{60}$	$\textstyle =$	$\displaystyle [231J_z^6-(315X-735)J_z^4+(105X^2-525X+294)J_z^2-5X^3+40X^2-60X]=I_{42}$
$\displaystyle O_{61}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_++J_-)\{33J_z^5-(30X-15)J_z^3+(5X^2-10X+12)J_z\}+\{\dots\}(J_++J_-)]=I_{43}$
$\displaystyle O_{62}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^2+J_-^2)\{33J_z^4-(18X+123)J_z^2+X^2+10X+102\}+\{\dots\}(J_+^2+J_-^2)]=I_{44}$
$\displaystyle O_{63}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^3+J_-^3)(11J_z^3-(3X+59)J_z)+(11J_z^3-(3X+59)J_z)(J_+^3+J_-^3)]=I_{45}$
$\displaystyle O_{64}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^4+J_-^4)(11J_z^2-X-38)+(11J_z^2-X-38)(J_+^4+J_-^4)]=I_{46}$
$\displaystyle O_{65}$	$\textstyle =$	$\displaystyle \frac{1}{4}[(J_+^5+J_-^5)J_z+J_z(J_+^5+J_-^5)]=I_{47}$
$\displaystyle O_{66}$	$\textstyle =$	$\displaystyle \frac{1}{2}[J_+^6+J_-^6]=I_{48}$