Table 1 Rank-2 Spherical tensors depending on choice of lab or rotating frame
\({\hat{T}}_{\mathrm{2,0}}\) | \({\hat{T}}_{2,\pm 1}\) | \({\hat{T}}_{2,\pm 2}\) | |
|---|---|---|---|
Spin-Field (LF) | \(\sqrt{\frac{2}{3}}{\hat{S}}_{z}\) | \(\frac{1}{2}{\hat{S}}^{\pm }\) | 0 |
Spin-Field (RF) | \(\sqrt{\frac{2}{3}}{\hat{S}}_{z}\) | 0 | 0 |
Spin-Spin (LF/LF) | \(\frac{1}{\sqrt{6}}\left[3{\hat{{\rm{S}}}}_{1{\rm{z}}}{\hat{{\rm{S}}}}_{2{\rm{z}}}-{\vec{\hat{{\rm{S}}}}}_{1}\cdot {\vec{\hat{{\rm{S}}}}}_{2}\right]\) | \(\mp \frac{1}{2}\left[{\hat{S}}_{1}^{\pm }{\hat{S}}_{2z}+{\hat{S}}_{1z}{\hat{S}}_{2}^{\pm }\right]\) | \(\frac{1}{2}{\hat{S}}_{1}^{\pm }{\hat{S}}_{2}^{\pm }\) |
Spin-Spin (LF/RF) | \(\frac{2}{\sqrt{6}}{\hat{{\rm{S}}}}_{1{\rm{z}}}{\hat{{\rm{S}}}}_{2{\rm{z}}}\) | \(\mp \frac{1}{2}{\hat{S}}_{1}^{\pm }{\hat{S}}_{2z}\) | 0 |
Spin-Spin (RF/LF) | \(\frac{2}{\sqrt{6}}{\hat{{\rm{S}}}}_{1{\rm{z}}}{\hat{{\rm{S}}}}_{2{\rm{z}}}\) | \(\mp \frac{1}{2}{{\hat{S}}_{1z}\hat{S}}_{2}^{\pm }\) | 0 |
Spin-Spin (RF/RF, homonuclear) | \(\frac{1}{\sqrt{6}}\left[3{\hat{{\rm{S}}}}_{1{\rm{z}}}{\hat{{\rm{S}}}}_{2{\rm{z}}}-{\vec{\hat{{\rm{S}}}}}_{1}\cdot {\vec{\hat{{\rm{S}}}}}_{2}\right]\) | 0 | 0 |
Spin-Spin (RF/RF, heteronuclear) | \(\frac{2}{\sqrt{6}}{\hat{{\rm{S}}}}_{1{\rm{z}}}{\hat{{\rm{S}}}}_{2{\rm{z}}}\) | 0 | 0 |
