Table 3.

Relationship between irreducible spherical tensor (ISTO) and Cartesian operators (spin-independent).

ISTO	Cartesian decomposition	Description
T00	1	Identity
T10	Iz	Longitudinal magnetization
T1±1	$\mp \frac{1}{\sqrt{2}}{I}_{\pm }$	Rank 1 SQC
T20	$\frac{1}{\sqrt{6}}\left(3I_z^2-I(I+1)\right)$	Quadrupolar order
T2±1	$\mp \frac{1}{2}{[I_z,I_{\pm }]}_{+}$	Rank 2 SQC
T2±2	$\frac{1}{2}{I}_{\pm }^2$	Rank 2 DQC
T30	$\frac{1}{\sqrt{10}}\left(5I_z^3-(3I(I+1)-1)I_z\right)$	Octupolar order
T3±1	$\mp \frac{1}{4}\sqrt{\frac{3}{10}}{\left[5I_z^3-I(I+1)-\frac{1}{2},I_{\pm }\right]}_{+}$	Rank 3 SQC
T3±2	$\frac{1}{2}\sqrt{\frac{3}{4}}{\left[I_z,I_{\pm }^2\right]}_{+}$	Rank 3 DQC
T3±3	$\mp \frac{1}{2\sqrt{2}}{I}_{\pm }^3$	Rank 3 TQC

SQC, DQC and TQC stand for single, double, and triple quantum coherences. The anticommutator for the operators A and B is defined as [A, B]₊ = AB + BA.