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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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