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--- physics_chemistry:orbitals:k [2026/06/20 16:09] – Maurits W. Haverkort
+++ physics_chemistry:orbitals:k [2026/06/20 16:13] (current) – Maurits W. Haverkort
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-The relation between the kubic harmonics and the tesseral harmonics is given by the matrix $T_{Z\,\mathrm{to}\,K}$, whereby the vector $\{Z_{l}^{(m=-l)},K_{l}^{(m=-l+1)},\dots,K_{l}^{(m=l)}\}$ is given as $T_{Z\,\mathrm{to}\,K} \cdot \{Y_{l}^{(m=-l)},Y_{l}^{(m=-l+1)},\dots,Y_{l}^{(m=l)}\}$. Explicit forms of $T_{Z\,\mathrm{to}\,K}$ for $l$ equal or smaller to 6 are:
+The relation between the kubic harmonics and the tesseral harmonics is given by the matrix $T_{Z\,\mathrm{to}\,K}$, whereby the vector $\{K_{l}^{(m=-l)},K_{l}^{(m=-l+1)},\dots,K_{l}^{(m=l)}\}$ is given as $T_{Z\,\mathrm{to}\,K} \cdot \{Z_{l}^{(m=-l)},Z_{l}^{(m=-l+1)},\dots,Z_{l}^{(m=l)}\}$. Explicit forms of $T_{Z\,\mathrm{to}\,K}$ for $l$ equal or smaller to 6 are:
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 \end{align}
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 The following table shows the kubic harmonics up to $l=6$. We list the explicit function in terms of the directional cosines $x$, $y$ and $z$. The plots show the surface defined by the equation $r={K_l^{(m)}}^* K_l^{(m)}$. The color of the surface is according to the phase with red for positive and cyan for negative. We show a 3D image as well as a projection along the $x$, $y$ and $z$ direction.
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