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--- physics_chemistry:point_groups:ih:orientation_xyz [2018/03/21 18:51] – created Stefano Agrestini
+++ physics_chemistry:point_groups:ih:orientation_xyz [2018/04/05 10:36] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
+~~CLOSETOC~~
 ====== Orientation xyz ======
+===== Symmetry Operations =====
 ###
-alligned paragraph text
+In the Ih Point Group, with orientation xyz there are the following symmetry operations
 ###
-===== Example =====
+###
+{{:physics_chemistry:pointgroup:ih_xyz.png}}
 ###
-description text
 ###
-==== Input ====
+^ Operator ^ Orientation ^
-<code Quanty Example.Quanty>
+^ $\text{E}$ | $\{0,0,0\}$ , |
--- some example code
+^ $C_5$ | $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),1,0\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),1\right\}$ , $\left\{1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , |
+^ $C_5^2$ | $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),1,0\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),1\right\}$ , $\left\{1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , |
+^ $C_3$ | $\{-1,-1,-1\}$ , $\left\{0,-1,\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{0,1,\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\{1,1,1\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),0,-1\right\}$ , $\left\{-1,\frac{1}{2} \left(-3-\sqrt{5}\right),0\right\}$ , $\left\{1,\frac{1}{2} \left(3+\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),0,1\right\}$ , $\left\{-1,\frac{1}{2} \left(3+\sqrt{5}\right),0\right\}$ , $\{-1,1,1\}$ , $\{1,-1,-1\}$ , $\left\{1,\frac{1}{2} \left(-3-\sqrt{5}\right),0\right\}$ , $\left\{0,1,\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{0,-1,\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\{-1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,-1,1\}$ , $\{1,1,-1\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),0,1\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),0,-1\right\}$ , |
+^ $C_2$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),1,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),1,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),1,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),1,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right),1\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right),1\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right),1\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right),1\right\}$ , |
+^ $\text{i}$ | $\{0,0,0\}$ , |
+^ $S_{10}$ | $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),1,0\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),1\right\}$ , $\left\{1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , |
+^ $S_{10}^3$ | $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),-1,0\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),1,0\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(1+\sqrt{5}\right),-1\right\}$ , $\left\{0,\frac{1}{2} \left(-1-\sqrt{5}\right),1\right\}$ , $\left\{1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{-1,0,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{1,0,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , |
+^ $S_6$ | $\{-1,-1,-1\}$ , $\left\{0,-1,\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{0,1,\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\{1,1,1\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),0,-1\right\}$ , $\left\{-1,\frac{1}{2} \left(-3-\sqrt{5}\right),0\right\}$ , $\left\{1,\frac{1}{2} \left(3+\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),0,1\right\}$ , $\left\{-1,\frac{1}{2} \left(3+\sqrt{5}\right),0\right\}$ , $\{-1,1,1\}$ , $\{1,-1,-1\}$ , $\left\{1,\frac{1}{2} \left(-3-\sqrt{5}\right),0\right\}$ , $\left\{0,1,\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{0,-1,\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\{-1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,-1,1\}$ , $\{1,1,-1\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),0,1\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),0,-1\right\}$ , |
+^ $\sigma _h$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),1,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(-3-\sqrt{5}\right),1,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),1,\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(3+\sqrt{5}\right),1,\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{1,\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right),1\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right),1\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(3+\sqrt{5}\right),1\right\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(-3-\sqrt{5}\right),1\right\}$ , |
+###
+===== Different Settings =====
+###
+  * [[physics_chemistry:point_groups:ih:orientation_xyz|Point Group Ih with orientation xyz]]
+###
+===== Character Table =====
+###
+|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_5 \,{\text{(12)}} $  ^  $ C_5^2{} \,{\text{(12)}} $  ^  $ C_3 \,{\text{(20)}} $  ^  $ C_2 \,{\text{(15)}} $  ^  $ \text{i} \,{\text{(1)}} $  ^  $ S_{10} \,{\text{(12)}} $  ^  $ S_{10}^3{} \,{\text{(12)}} $  ^  $ S_6 \,{\text{(20)}} $  ^  $ \sigma_h \,{\text{(15)}} $  ^
+^ $ A_g $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
+^ $ T_{1g} $ |  $ 3 $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ 0 $ |  $ -1 $ |  $ 3 $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ 0 $ |  $ -1 $ |
+^ $ T_{2g} $ |  $ 3 $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ 0 $ |  $ -1 $ |  $ 3 $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ 0 $ |  $ -1 $ |
+^ $ G_g $ |  $ 4 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ 0 $ |  $ 4 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ 0 $ |
+^ $ H_g $ |  $ 5 $ |  $ 0 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ 5 $ |  $ 0 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |
+^ $ A_u $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |
+^ $ T_{1u} $ |  $ 3 $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ 0 $ |  $ -1 $ |  $ -3 $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ 0 $ |  $ 1 $ |
+^ $ T_{2u} $ |  $ 3 $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ 0 $ |  $ -1 $ |  $ -3 $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ 0 $ |  $ 1 $ |
+^ $ G_u $ |  $ 4 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ 0 $ |  $ -4 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 0 $ |
+^ $ H_u $ |  $ 5 $ |  $ 0 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ -5 $ |  $ 0 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |
+###
+===== Product Table =====
+###
+|  $  $  ^  $ A_g $  ^  $ T_{1g} $  ^  $ T_{2g} $  ^  $ G_g $  ^  $ H_g $  ^  $ A_u $  ^  $ T_{1u} $  ^  $ T_{2u} $  ^  $ G_u $  ^  $ H_u $  ^
+^ $ A_g $  | $ A_g $  | $ T_{1g} $  | $ T_{2g} $  | $ G_g $  | $ H_g $  | $ A_u $  | $ T_{1u} $  | $ T_{2u} $  | $ G_u $  | $ H_u $  |
+^ $ T_{1g} $  | $ T_{1g} $  | $ A_g+H_g+T_{1g} $  | $ G_g+H_g $  | $ G_g+H_g+T_{2g} $  | $ G_g+H_g+T_{1g}+T_{2g} $  | $ T_{1u} $  | $ A_u+H_u+T_{1u} $  | $ G_u+H_u $  | $ G_u+H_u+T_{2u} $  | $ G_u+H_u+T_{1u}+T_{2u} $  |
+^ $ T_{2g} $  | $ T_{2g} $  | $ G_g+H_g $  | $ A_g+H_g+T_{2g} $  | $ G_g+H_g+T_{1g} $  | $ G_g+H_g+T_{1g}+T_{2g} $  | $ T_{2u} $  | $ G_u+H_u $  | $ A_u+H_u+T_{2u} $  | $ G_u+H_u+T_{1u} $  | $ G_u+H_u+T_{1u}+T_{2u} $  |
+^ $ G_g $  | $ G_g $  | $ G_g+H_g+T_{2g} $  | $ G_g+H_g+T_{1g} $  | $ A_g+G_g+H_g+T_{1g}+T_{2g} $  | $ G_g+2 H_g+T_{1g}+T_{2g} $  | $ G_u $  | $ G_u+H_u+T_{2u} $  | $ G_u+H_u+T_{1u} $  | $ A_u+G_u+H_u+T_{1u}+T_{2u} $  | $ G_u+2 H_u+T_{1u}+T_{2u} $  |
+^ $ H_g $  | $ H_g $  | $ G_g+H_g+T_{1g}+T_{2g} $  | $ G_g+H_g+T_{1g}+T_{2g} $  | $ G_g+2 H_g+T_{1g}+T_{2g} $  | $ A_g+2 G_g+2 H_g+T_{1g}+T_{2g} $  | $ H_u $  | $ G_u+H_u+T_{1u}+T_{2u} $  | $ G_u+H_u+T_{1u}+T_{2u} $  | $ G_u+2 H_u+T_{1u}+T_{2u} $  | $ A_u+2 G_u+2 H_u+T_{1u}+T_{2u} $  |
+^ $ A_u $  | $ A_u $  | $ T_{1u} $  | $ T_{2u} $  | $ G_u $  | $ H_u $  | $ A_g $  | $ T_{1g} $  | $ T_{2g} $  | $ G_g $  | $ H_g $  |
+^ $ T_{1u} $  | $ T_{1u} $  | $ A_u+H_u+T_{1u} $  | $ G_u+H_u $  | $ G_u+H_u+T_{2u} $  | $ G_u+H_u+T_{1u}+T_{2u} $  | $ T_{1g} $  | $ A_g+H_g+T_{1g} $  | $ G_g+H_g $  | $ G_g+H_g+T_{2g} $  | $ G_g+H_g+T_{1g}+T_{2g} $  |
+^ $ T_{2u} $  | $ T_{2u} $  | $ G_u+H_u $  | $ A_u+H_u+T_{2u} $  | $ G_u+H_u+T_{1u} $  | $ G_u+H_u+T_{1u}+T_{2u} $  | $ T_{2g} $  | $ G_g+H_g $  | $ A_g+H_g+T_{2g} $  | $ G_g+H_g+T_{1g} $  | $ G_g+H_g+T_{1g}+T_{2g} $  |
+^ $ G_u $  | $ G_u $  | $ G_u+H_u+T_{2u} $  | $ G_u+H_u+T_{1u} $  | $ A_u+G_u+H_u+T_{1u}+T_{2u} $  | $ G_u+2 H_u+T_{1u}+T_{2u} $  | $ G_g $  | $ G_g+H_g+T_{2g} $  | $ G_g+H_g+T_{1g} $  | $ A_g+G_g+H_g+T_{1g}+T_{2g} $  | $ G_g+2 H_g+T_{1g}+T_{2g} $  |
+^ $ H_u $  | $ H_u $  | $ G_u+H_u+T_{1u}+T_{2u} $  | $ G_u+H_u+T_{1u}+T_{2u} $  | $ G_u+2 H_u+T_{1u}+T_{2u} $  | $ A_u+2 G_u+2 H_u+T_{1u}+T_{2u} $  | $ H_g $  | $ G_g+H_g+T_{1g}+T_{2g} $  | $ G_g+H_g+T_{1g}+T_{2g} $  | $ G_g+2 H_g+T_{1g}+T_{2g} $  | $ A_g+2 G_g+2 H_g+T_{1g}+T_{2g} $  |
+###
+===== Sub Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
+###
+===== Super Groups with compatible settings =====
+###
+###
+===== Invariant Potential expanded on renormalized spherical Harmonics =====
+###
+Any potential (function) can be written as a sum over spherical harmonics.
+$$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$
+Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$
+The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Ih Point group with orientation xyz the form of the expansion coefficients is:
+###
+==== Expansion ====
+###
+ $$A_{k,m} = \begin{cases}
+ A(0,0) & k=0\land m=0 \\
+ \frac{1}{2} \sqrt{\frac{105}{11}} A(6,0) & k=6\land (m=-6\lor m=6) \\
+ -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\
+ -\frac{1}{2} \sqrt{21} A(6,0) & k=6\land (m=-2\lor m=2) \\
+ A(6,0) & k=6\land m=0
+\end{cases}$$
+###
+==== Input format suitable for Mathematica (Quanty.nb) ====
+###
+<code Quanty Akm_Ih_xyz.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(Sqrt[105/11]*A[6, 0])/2, k == 6 && (m == -6 || m == 6)}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {-(Sqrt[21]*A[6, 0])/2, k == 6 && (m == -2 || m == 2)}, {A[6, 0], k == 6 && m == 0}}, 0]
 </code>
-==== Result ====
+###
-<WRAP center box 100%>
-text produced as output
-</WRAP>
-===== Table of contents =====
+==== Input format suitable for Quanty ====
-{{indexmenu>.#1}}
+###
+<code Quanty Akm_Ih_xyz.Quanty>
+Akm = {{0, 0, A(0,0)} ,
+       {6, 0, A(6,0)} ,
+       {6,-2, (-1/2)*((sqrt(21))*(A(6,0)))} ,
+       {6, 2, (-1/2)*((sqrt(21))*(A(6,0)))} ,
+       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} ,
+       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} ,
+       {6,-6, (1/2)*((sqrt(105/11))*(A(6,0)))} ,
+       {6, 6, (1/2)*((sqrt(105/11))*(A(6,0)))} }
+</code>
+###
+==== One particle coupling on a basis of spherical harmonics ====
+###
+The operator representing the potential in second quantisation is given as:
+$$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$
+For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter.
+$$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$
+Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$
+###
+###
+we can express the operator as
+$$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$
+###
+###
+The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter.
+###
+###
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
+^$ {Y_{0}^{(0)}} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ {Y_{0}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{5}{429} \text{Aff}(6,0) $|$ 0 $|$ \frac{35 \text{Aff}(6,0)}{143 \sqrt{3}} $|$ 0 $|$ \frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ -\frac{35}{143} \sqrt{5} \text{Aff}(6,0) $|
+^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{70}{143} \sqrt{\frac{2}{3}} \text{Aff}(6,0) $|$ 0 $|$ -\frac{70}{143} \text{Aff}(6,0) $|$ 0 $|
+^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{35 \text{Aff}(6,0)}{143 \sqrt{3}} $|$ 0 $|$ \text{Aff}(0,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ \frac{35}{143} \sqrt{5} \text{Aff}(6,0) $|$ 0 $|$ \frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|
+^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{70}{143} \sqrt{\frac{2}{3}} \text{Aff}(6,0) $|$ 0 $|$ \text{Aff}(0,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ -\frac{70}{143} \sqrt{\frac{2}{3}} \text{Aff}(6,0) $|$ 0 $|
+^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ \frac{35}{143} \sqrt{5} \text{Aff}(6,0) $|$ 0 $|$ \text{Aff}(0,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ \frac{35 \text{Aff}(6,0)}{143 \sqrt{3}} $|
+^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{70}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{70}{143} \sqrt{\frac{2}{3}} \text{Aff}(6,0) $|$ 0 $|$ \text{Aff}(0,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|
+^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{35}{143} \sqrt{5} \text{Aff}(6,0) $|$ 0 $|$ \frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ \frac{35 \text{Aff}(6,0)}{143 \sqrt{3}} $|$ 0 $|$ \text{Aff}(0,0)-\frac{5}{429} \text{Aff}(6,0) $|
+###
+==== Rotation matrix to symmetry adapted functions (choice is not unique) ====
+###
+Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field
+###
+###
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
+^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ f_{x\left(5x^2-3r^2+3\sqrt{5}\left(y^2-z^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{8} \left(\sqrt{5}-3\right) $|$ 0 $|$ -\frac{1}{8} \sqrt{3} \left(1+\sqrt{5}\right) $|$ 0 $|$ \frac{1}{8} \left(\sqrt{3}+\sqrt{15}\right) $|$ 0 $|$ \frac{1}{8} \left(3-\sqrt{5}\right) $|
+^$ f_{y\left(5y^2-3r^2+3\sqrt{5}\left(z^2-x^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{8} i \left(3+\sqrt{5}\right) $|$ 0 $|$ \frac{1}{8} i \sqrt{3} \left(\sqrt{5}-1\right) $|$ 0 $|$ \frac{1}{8} i \sqrt{3} \left(\sqrt{5}-1\right) $|$ 0 $|$ -\frac{1}{8} i \left(3+\sqrt{5}\right) $|
+^$ f_{z\left(5z^2-3r^2+3\sqrt{5}\left(x^2-y^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|$ \frac{1}{2} $|$ 0 $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|
+^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
+^$ f_{x\left(5x^2-3r^2-\sqrt{5}\left(y^2-z^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{8} \left(\sqrt{3}+\sqrt{15}\right) $|$ 0 $|$ \frac{1}{8} \left(\sqrt{5}-3\right) $|$ 0 $|$ \frac{1}{8} \left(3-\sqrt{5}\right) $|$ 0 $|$ -\frac{1}{8} \sqrt{3} \left(1+\sqrt{5}\right) $|
+^$ f_{y\left(5y^2-3r^2-\sqrt{5}\left(z^2-x^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{8} i \sqrt{3} \left(\sqrt{5}-1\right) $|$ 0 $|$ -\frac{1}{8} i \left(3+\sqrt{5}\right) $|$ 0 $|$ -\frac{1}{8} i \left(3+\sqrt{5}\right) $|$ 0 $|$ -\frac{1}{8} i \sqrt{3} \left(\sqrt{5}-1\right) $|
+^$ f_{z\left(5z^2-3r^2-\sqrt{5}\left(x^2-y^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|$ \frac{\sqrt{3}}{2} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|
+###
+==== One particle coupling on a basis of symmetry adapted functions ====
+###
+After rotation we find
+###
+###
+|  $  $  ^  $ \text{s} $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^  $ f_{x\left(5x^2-3r^2+3\sqrt{5}\left(y^2-z^2\right)\right)} $  ^  $ f_{y\left(5y^2-3r^2+3\sqrt{5}\left(z^2-x^2\right)\right)} $  ^  $ f_{z\left(5z^2-3r^2+3\sqrt{5}\left(x^2-y^2\right)\right)} $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-3r^2-\sqrt{5}\left(y^2-z^2\right)\right)} $  ^  $ f_{y\left(5y^2-3r^2-\sqrt{5}\left(z^2-x^2\right)\right)} $  ^  $ f_{z\left(5z^2-3r^2-\sqrt{5}\left(x^2-y^2\right)\right)} $  ^
+^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ p_x $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
+^$ f_{x\left(5x^2-3r^2+3\sqrt{5}\left(y^2-z^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{320}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{y\left(5y^2-3r^2+3\sqrt{5}\left(z^2-x^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{320}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{z\left(5z^2-3r^2+3\sqrt{5}\left(x^2-y^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{320}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{80}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(5x^2-3r^2-\sqrt{5}\left(y^2-z^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{80}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|
+^$ f_{y\left(5y^2-3r^2-\sqrt{5}\left(z^2-x^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{80}{143} \text{Aff}(6,0) $|$ 0 $|
+^$ f_{z\left(5z^2-3r^2-\sqrt{5}\left(x^2-y^2\right)\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{80}{143} \text{Aff}(6,0) $|
+###
+===== Coupling for a single shell =====
+###
+Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$.
+###
+###
+Click on one of the subsections to expand it or <hiddenSwitch expand all>
+###
+==== Potential for s orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Eag} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Ih_xyz.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Eag, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Ih_xyz.Quanty>
+Akm = {{0, 0, Eag} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
+^$ {Y_{0}^{(0)}} $|$ \text{Eag} $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ \text{s} $  ^
+^$ \text{s} $|$ \text{Eag} $|
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
+^$ \text{s} $|$ 1 $|
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^$$\text{Eag}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_0_1.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
+###
+</hidden>
+==== Potential for p orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Et1u} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Ih_xyz.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Ih_xyz.Quanty>
+Akm = {{0, 0, Et1u} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
+^$ {Y_{-1}^{(1)}} $|$ \text{Et1u} $|$ 0 $|$ 0 $|
+^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Et1u} $|$ 0 $|
+^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Et1u} $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^
+^$ p_x $|$ \text{Et1u} $|$ 0 $|$ 0 $|
+^$ p_y $|$ 0 $|$ \text{Et1u} $|$ 0 $|
+^$ p_z $|$ 0 $|$ 0 $|$ \text{Et1u} $|
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
+^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|
+^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
+^$ p_z $|$ 0 $|$ 1 $|$ 0 $|
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_1_1.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
+^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_1_2.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
+^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_1_3.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
+###
+</hidden>
+==== Potential for d orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Ehu} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Ih_xyz.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Ehu, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Ih_xyz.Quanty>
+Akm = {{0, 0, Ehu} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
+^$ {Y_{-2}^{(2)}} $|$ \text{Ehu} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Ehu} $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ehu} $|$ 0 $|$ 0 $|
+^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ehu} $|$ 0 $|
+^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ehu} $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^
+^$ d_{x^2-y^2} $|$ \text{Ehu} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{3z^2-r^2} $|$ 0 $|$ \text{Ehu} $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Ehu} $|$ 0 $|$ 0 $|
+^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ehu} $|$ 0 $|
+^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ehu} $|
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
+^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|
+^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
+^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|
+^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|
+^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^$$\text{Ehu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_2_1.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |
+^ ^$$\text{Ehu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_2_2.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
+^ ^$$\text{Ehu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_2_3.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
+^ ^$$\text{Ehu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_2_4.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
+^ ^$$\text{Ehu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_2_5.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |
+###
+</hidden>
+==== Potential for f orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{7} (4 \text{Egu}+3 \text{Et2u}) & k=0\land m=0 \\
+ \frac{39}{32} \sqrt{\frac{33}{35}} (\text{Egu}-\text{Et2u}) & k=6\land (m=-6\lor m=6) \\
+ -\frac{429 (\text{Egu}-\text{Et2u})}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\
+ -\frac{429}{160} \sqrt{\frac{3}{7}} (\text{Egu}-\text{Et2u}) & k=6\land (m=-2\lor m=2) \\
+ \frac{429 (\text{Egu}-\text{Et2u})}{560} & k=6\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Ih_xyz.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(4*Egu + 3*Et2u)/7, k == 0 && m == 0}, {(39*Sqrt[33/35]*(Egu - Et2u))/32, k == 6 && (m == -6 || m == 6)}, {(-429*(Egu - Et2u))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(-429*Sqrt[3/7]*(Egu - Et2u))/160, k == 6 && (m == -2 || m == 2)}, {(429*(Egu - Et2u))/560, k == 6 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Ih_xyz.Quanty>
+Akm = {{0, 0, (1/7)*((4)*(Egu) + (3)*(Et2u))} ,
+       {6, 0, (429/560)*(Egu + (-1)*(Et2u))} ,
+       {6,-2, (-429/160)*((sqrt(3/7))*(Egu + (-1)*(Et2u)))} ,
+       {6, 2, (-429/160)*((sqrt(3/7))*(Egu + (-1)*(Et2u)))} ,
+       {6,-4, (-429/80)*((1/(sqrt(14)))*(Egu + (-1)*(Et2u)))} ,
+       {6, 4, (-429/80)*((1/(sqrt(14)))*(Egu + (-1)*(Et2u)))} ,
+       {6,-6, (39/32)*((sqrt(33/35))*(Egu + (-1)*(Et2u)))} ,
+       {6, 6, (39/32)*((sqrt(33/35))*(Egu + (-1)*(Et2u)))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
+^$ {Y_{-3}^{(3)}} $|$ \frac{1}{16} (9 \text{Egu}+7 \text{Et2u}) $|$ 0 $|$ \frac{1}{16} \sqrt{3} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} \sqrt{15} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ -\frac{3}{16} \sqrt{5} (\text{Egu}-\text{Et2u}) $|
+^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{1}{8} (5 \text{Egu}+3 \text{Et2u}) $|$ 0 $|$ \frac{1}{4} \sqrt{\frac{3}{2}} (\text{Et2u}-\text{Egu}) $|$ 0 $|$ -\frac{3}{8} (\text{Egu}-\text{Et2u}) $|$ 0 $|
+^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \sqrt{3} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} (7 \text{Egu}+9 \text{Et2u}) $|$ 0 $|$ \frac{3}{16} \sqrt{5} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} \sqrt{15} (\text{Egu}-\text{Et2u}) $|
+^$ {Y_{0}^{(3)}} $|$ 0 $|$ \frac{1}{4} \sqrt{\frac{3}{2}} (\text{Et2u}-\text{Egu}) $|$ 0 $|$ \frac{1}{4} (3 \text{Egu}+\text{Et2u}) $|$ 0 $|$ \frac{1}{4} \sqrt{\frac{3}{2}} (\text{Et2u}-\text{Egu}) $|$ 0 $|
+^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \sqrt{15} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{3}{16} \sqrt{5} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} (7 \text{Egu}+9 \text{Et2u}) $|$ 0 $|$ \frac{1}{16} \sqrt{3} (\text{Egu}-\text{Et2u}) $|
+^$ {Y_{2}^{(3)}} $|$ 0 $|$ -\frac{3}{8} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{4} \sqrt{\frac{3}{2}} (\text{Et2u}-\text{Egu}) $|$ 0 $|$ \frac{1}{8} (5 \text{Egu}+3 \text{Et2u}) $|$ 0 $|
+^$ {Y_{3}^{(3)}} $|$ -\frac{3}{16} \sqrt{5} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} \sqrt{15} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} \sqrt{3} (\text{Egu}-\text{Et2u}) $|$ 0 $|$ \frac{1}{16} (9 \text{Egu}+7 \text{Et2u}) $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ f_{x\left(5x^2-3r^2+3\sqrt{5}\left(y^2-z^2\right)\right)} $  ^  $ f_{y\left(5y^2-3r^2+3\sqrt{5}\left(z^2-x^2\right)\right)} $  ^  $ f_{z\left(5z^2-3r^2+3\sqrt{5}\left(x^2-y^2\right)\right)} $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-3r^2-\sqrt{5}\left(y^2-z^2\right)\right)} $  ^  $ f_{y\left(5y^2-3r^2-\sqrt{5}\left(z^2-x^2\right)\right)} $  ^  $ f_{z\left(5z^2-3r^2-\sqrt{5}\left(x^2-y^2\right)\right)} $  ^
+^$ f_{x\left(5x^2-3r^2+3\sqrt{5}\left(y^2-z^2\right)\right)} $|$ \text{Et2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{y\left(5y^2-3r^2+3\sqrt{5}\left(z^2-x^2\right)\right)} $|$ 0 $|$ \text{Et2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{z\left(5z^2-3r^2+3\sqrt{5}\left(x^2-y^2\right)\right)} $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{\text{xyz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Egu} $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(5x^2-3r^2-\sqrt{5}\left(y^2-z^2\right)\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Egu} $|$ 0 $|$ 0 $|
+^$ f_{y\left(5y^2-3r^2-\sqrt{5}\left(z^2-x^2\right)\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Egu} $|$ 0 $|
+^$ f_{z\left(5z^2-3r^2-\sqrt{5}\left(x^2-y^2\right)\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Egu} $|
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
+^$ f_{x\left(5x^2-3r^2+3\sqrt{5}\left(y^2-z^2\right)\right)} $|$ \frac{\sqrt{5}}{8}-\frac{3}{8} $|$ 0 $|$ -\frac{\sqrt{3}}{8}-\frac{\sqrt{15}}{8} $|$ 0 $|$ \frac{\sqrt{3}}{8}+\frac{\sqrt{15}}{8} $|$ 0 $|$ \frac{3}{8}-\frac{\sqrt{5}}{8} $|
+^$ f_{y\left(5y^2-3r^2+3\sqrt{5}\left(z^2-x^2\right)\right)} $|$ -\frac{3 i}{8}-\frac{i \sqrt{5}}{8} $|$ 0 $|$ \frac{i \sqrt{15}}{8}-\frac{i \sqrt{3}}{8} $|$ 0 $|$ \frac{i \sqrt{15}}{8}-\frac{i \sqrt{3}}{8} $|$ 0 $|$ -\frac{3 i}{8}-\frac{i \sqrt{5}}{8} $|
+^$ f_{z\left(5z^2-3r^2+3\sqrt{5}\left(x^2-y^2\right)\right)} $|$ 0 $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|$ \frac{1}{2} $|$ 0 $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|
+^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
+^$ f_{x\left(5x^2-3r^2-\sqrt{5}\left(y^2-z^2\right)\right)} $|$ \frac{\sqrt{3}}{8}+\frac{\sqrt{15}}{8} $|$ 0 $|$ \frac{\sqrt{5}}{8}-\frac{3}{8} $|$ 0 $|$ \frac{3}{8}-\frac{\sqrt{5}}{8} $|$ 0 $|$ -\frac{\sqrt{3}}{8}-\frac{\sqrt{15}}{8} $|
+^$ f_{y\left(5y^2-3r^2-\sqrt{5}\left(z^2-x^2\right)\right)} $|$ \frac{i \sqrt{3}}{8}-\frac{i \sqrt{15}}{8} $|$ 0 $|$ -\frac{3 i}{8}-\frac{i \sqrt{5}}{8} $|$ 0 $|$ -\frac{3 i}{8}-\frac{i \sqrt{5}}{8} $|$ 0 $|$ \frac{i \sqrt{3}}{8}-\frac{i \sqrt{15}}{8} $|
+^$ f_{z\left(5z^2-3r^2-\sqrt{5}\left(x^2-y^2\right)\right)} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|$ \frac{\sqrt{3}}{2} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_1.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{64} \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \sin (\theta ) \left(\left(5-3 \sqrt{5}\right) \left(1+e^{6 i \phi }\right) \sin ^2(\theta )-15 \left(1+\sqrt{5}\right) \left(e^{2 i \phi }+e^{4 i \phi }\right) \cos ^2(\theta )+6 \left(1+\sqrt{5}\right) e^{3 i \phi } \cos (\phi )\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{7}{\pi }} x \left(\left(5-3 \sqrt{5}\right) x^2+3 \left(\left(3 \sqrt{5}-5\right) y^2-\left(1+\sqrt{5}\right) \left(5 z^2-1\right)\right)\right)$$ | ::: |
+^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_2.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{64} \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \sin (\theta ) \left(i \left(5+3 \sqrt{5}\right) \left(-1+e^{6 i \phi }\right) \sin ^2(\theta )+3 \left(\sqrt{5}-1\right) e^{3 i \phi } (5 \cos (2 \theta )+3) \sin (\phi )\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{7}{\pi }} y \left(-3 \left(5+3 \sqrt{5}\right) x^2+\left(5+3 \sqrt{5}\right) y^2+3 \left(\sqrt{5}-1\right) \left(5 z^2-1\right)\right)$$ | ::: |
+^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_3.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \cos (\theta ) \left(6 \sqrt{5} \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )-1\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} z \left(3 \sqrt{5} x^2-3 \sqrt{5} y^2+5 z^2-3\right)$$ | ::: |
+^ ^$$\text{Egu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_4.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
+^ ^$$\text{Egu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_5.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{21}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \left(5+\sqrt{5}\right) \sin ^2(\theta ) \cos (2 \phi )+\left(3 \sqrt{5}-5\right) \cos (2 \theta )+\sqrt{5}-7\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{21}{\pi }} x \left(\left(5+\sqrt{5}\right) x^2-3 \left(5+\sqrt{5}\right) y^2+\left(\sqrt{5}-3\right) \left(5 z^2-1\right)\right)$$ | ::: |
+^ ^$$\text{Egu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_6.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{64} \sqrt{\frac{21}{\pi }} e^{-3 i \phi } \sin (\theta ) \left(-i \left(\sqrt{5}-5\right) \left(-1+e^{6 i \phi }\right) \sin ^2(\theta )-\left(3+\sqrt{5}\right) e^{3 i \phi } (5 \cos (2 \theta )+3) \sin (\phi )\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{32} \sqrt{\frac{21}{\pi }} y \left(-3 \left(\sqrt{5}-5\right) x^2+\left(\sqrt{5}-5\right) y^2+\left(3+\sqrt{5}\right) \left(5 z^2-1\right)\right)$$ | ::: |
+^ ^$$\text{Egu}$$ | {{:physics_chemistry:pointgroup:ih_xyz_orb_3_7.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{21}{\pi }} \left(\cos (\theta ) \left(3-4 \sqrt{5} \sin ^2(\theta ) \cos (2 \phi )\right)+5 \cos (3 \theta )\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{21}{\pi }} z \left(-\sqrt{5} x^2+\sqrt{5} y^2+5 z^2-3\right)$$ | ::: |
+###
+</hidden>
+===== Coupling between two shells =====
+###
+Click on one of the subsections to expand it or <hiddenSwitch expand all>
+###
+===== Table of several point groups =====
+###
+[[physics_chemistry:point_groups|Return to Main page on Point Groups]]
+###
+###
+^Nonaxial groups      | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> | | | | |
+^C<sub>n</sub> groups | [[physics_chemistry:point_groups:c2|C]]<sub>[[physics_chemistry:point_groups:c2|2]]</sub> | [[physics_chemistry:point_groups:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> |
+^D<sub>n</sub> groups | [[physics_chemistry:point_groups:d2|D]]<sub>[[physics_chemistry:point_groups:d2|2]]</sub> | [[physics_chemistry:point_groups:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub> | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub> | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub> | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub> | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub> | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub> |
+^C<sub>nv</sub> groups | [[physics_chemistry:point_groups:c2v|C]]<sub>[[physics_chemistry:point_groups:c2v|2v]]</sub> | [[physics_chemistry:point_groups:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> |
+^C<sub>nh</sub> groups | [[physics_chemistry:point_groups:c2h|C]]<sub>[[physics_chemistry:point_groups:c2h|2h]]</sub> | [[physics_chemistry:point_groups:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> | | |
+^D<sub>nh</sub> groups | [[physics_chemistry:point_groups:d2h|D]]<sub>[[physics_chemistry:point_groups:d2h|2h]]</sub> | [[physics_chemistry:point_groups:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> |
+^D<sub>nd</sub> groups | [[physics_chemistry:point_groups:d2d|D]]<sub>[[physics_chemistry:point_groups:d2d|2d]]</sub> | [[physics_chemistry:point_groups:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> |
+^S<sub>n</sub> groups | [[physics_chemistry:point_groups:S2|S]]<sub>[[physics_chemistry:point_groups:S2|2]]</sub> | [[physics_chemistry:point_groups:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> |  |
+^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> |
+^Linear groups      | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | |
+###

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