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--- physics_chemistry:point_groups:oh:orientation_xyz [2018/03/21 18:47] – created Stefano Agrestini
+++ physics_chemistry:point_groups:oh:orientation_xyz [2018/09/06 12:48] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
+~~CLOSETOC~~
 ====== Orientation XYZ ======
+===== Symmetry Operations =====
 ###
-alligned paragraph text
+In the Oh Point Group, with orientation XYZ there are the following symmetry operations
 ###
-===== Example =====
+###
+{{:physics_chemistry:pointgroup:oh_xyz.png}}
 ###
-description text
 ###
-==== Input ====
+^ Operator ^ Orientation ^
-<code Quanty Example.Quanty>
+^ $\text{E}$ | $\{0,0,0\}$ , |
--- some example code
+^ $C_3$ | $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ , |
+^ $C_2$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ , |
+^ $C_4$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ , |
+^ $C_2$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , |
+^ $\text{i}$ | $\{0,0,0\}$ , |
+^ $S_4$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ , |
+^ $S_6$ | $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ , |
+^ $\sigma _h$ | $\{1,0,0\}$ , $\{0,1,0\}$ , $\{0,0,1\}$ , |
+^ $\sigma _d$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ , |
+###
+===== Different Settings =====
+###
+  * [[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Point Group Oh with orientation 0sqrt2-1z]]
+  * [[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Point Group Oh with orientation 0sqrt21z]]
+  * [[physics_chemistry:point_groups:oh:orientation_11-1z|Point Group Oh with orientation 11-1z]]
+  * [[physics_chemistry:point_groups:oh:orientation_111z|Point Group Oh with orientation 111z]]
+  * [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]]
+  * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]]
+  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
+###
+===== Character Table =====
+###
+|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_3 \,{\text{(8)}} $  ^  $ C_2 \,{\text{(6)}} $  ^  $ C_4 \,{\text{(6)}} $  ^  $ C_2 \,{\text{(3)}} $  ^  $ \text{i} \,{\text{(1)}} $  ^  $ S_4 \,{\text{(6)}} $  ^  $ S_6 \,{\text{(8)}} $  ^  $ \sigma_h \,{\text{(3)}} $  ^  $ \sigma_d \,{\text{(6)}} $  ^
+^ $ A_{1g} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
+^ $ A_{2g} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |
+^ $ E_g $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 0 $ |  $ 2 $ |  $ 2 $ |  $ 0 $ |  $ -1 $ |  $ 2 $ |  $ 0 $ |
+^ $ T_{1g} $ |  $ 3 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ 3 $ |  $ 1 $ |  $ 0 $ |  $ -1 $ |  $ -1 $ |
+^ $ T_{2g} $ |  $ 3 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 3 $ |  $ -1 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |
+^ $ A_{1u} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |
+^ $ A_{2u} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |
+^ $ E_u $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 0 $ |  $ 2 $ |  $ -2 $ |  $ 0 $ |  $ 1 $ |  $ -2 $ |  $ 0 $ |
+^ $ T_{1u} $ |  $ 3 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ -3 $ |  $ -1 $ |  $ 0 $ |  $ 1 $ |  $ 1 $ |
+^ $ T_{2u} $ |  $ 3 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -3 $ |  $ 1 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |
+###
+===== Product Table =====
+###
+|  $  $  ^  $ A_{1g} $  ^  $ A_{2g} $  ^  $ E_g $  ^  $ T_{1g} $  ^  $ T_{2g} $  ^  $ A_{1u} $  ^  $ A_{2u} $  ^  $ E_u $  ^  $ T_{1u} $  ^  $ T_{2u} $  ^
+^ $ A_{1g} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ T_{1g} $  | $ T_{2g} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ T_{1u} $  | $ T_{2u} $  |
+^ $ A_{2g} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ T_{2g} $  | $ T_{1g} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ T_{2u} $  | $ T_{1u} $  |
+^ $ E_g $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ T_{1g}+T_{2g} $  | $ T_{1g}+T_{2g} $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ T_{1u}+T_{2u} $  | $ T_{1u}+T_{2u} $  |
+^ $ T_{1g} $  | $ T_{1g} $  | $ T_{2g} $  | $ T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  | $ T_{1u} $  | $ T_{2u} $  | $ T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  |
+^ $ T_{2g} $  | $ T_{2g} $  | $ T_{1g} $  | $ T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  | $ T_{2u} $  | $ T_{1u} $  | $ T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  |
+^ $ A_{1u} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ T_{1u} $  | $ T_{2u} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ T_{1g} $  | $ T_{2g} $  |
+^ $ A_{2u} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ T_{2u} $  | $ T_{1u} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ T_{2g} $  | $ T_{1g} $  |
+^ $ E_u $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ T_{1u}+T_{2u} $  | $ T_{1u}+T_{2u} $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ T_{1g}+T_{2g} $  | $ T_{1g}+T_{2g} $  |
+^ $ T_{1u} $  | $ T_{1u} $  | $ T_{2u} $  | $ T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  | $ T_{1g} $  | $ T_{2g} $  | $ T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  |
+^ $ T_{2u} $  | $ T_{2u} $  | $ T_{1u} $  | $ T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  | $ T_{2g} $  | $ T_{1g} $  | $ T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  |
+###
+===== Sub Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
+  * [[physics_chemistry:point_groups:c2h:orientation_z|Point Group C2h with orientation Z]]
+  * [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]]
+  * [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]]
+  * [[physics_chemistry:point_groups:c2:orientation_y|Point Group C2 with orientation Y]]
+  * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]]
+  * [[physics_chemistry:point_groups:c4h:orientation_z|Point Group C4h with orientation Z]]
+  * [[physics_chemistry:point_groups:c4v:orientation_zxy|Point Group C4v with orientation Zxy]]
+  * [[physics_chemistry:point_groups:c4:orientation_x|Point Group C4 with orientation X]]
+  * [[physics_chemistry:point_groups:c4:orientation_y|Point Group C4 with orientation Y]]
+  * [[physics_chemistry:point_groups:c4:orientation_z|Point Group C4 with orientation Z]]
+  * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]]
+  * [[physics_chemistry:point_groups:cs:orientation_x|Point Group Cs with orientation X]]
+  * [[physics_chemistry:point_groups:cs:orientation_y|Point Group Cs with orientation Y]]
+  * [[physics_chemistry:point_groups:cs:orientation_z|Point Group Cs with orientation Z]]
+  * [[physics_chemistry:point_groups:d2d:orientation_zxy|Point Group D2d with orientation Zxy]]
+  * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]]
+  * [[physics_chemistry:point_groups:d2:orientation_xyz|Point Group D2 with orientation XYZ]]
+  * [[physics_chemistry:point_groups:d3d:orientation_111|Point Group D3d with orientation 111]]
+  * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
+  * [[physics_chemistry:point_groups:d4:orientation_zxy|Point Group D4 with orientation Zxy]]
+  * [[physics_chemistry:point_groups:o:orientation_xyz|Point Group O with orientation XYZ]]
+  * [[physics_chemistry:point_groups:s4:orientation_z|Point Group S4 with orientation Z]]
+  * [[physics_chemistry:point_groups:td:orientation_xyz|Point Group Td with orientation xyz]]
+  * [[physics_chemistry:point_groups:th:orientation_xyz|Point Group Th with orientation xyz]]
+  * [[physics_chemistry:point_groups:t:orientation_xyz|Point Group T with orientation xyz]]
+###
+===== Super Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:ih:orientation_xyz|Point Group Ih with orientation xyz]]
+###
+===== 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 Oh Point group with orientation XYZ the form of the expansion coefficients is:
+###
+==== Expansion ====
+###
+ $$A_{k,m} = \begin{cases}
+ A(0,0) & k=0\land m=0 \\
+ \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\
+ A(4,0) & k=4\land m=0 \\
+ -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\
+ A(6,0) & k=6\land m=0
+\end{cases}$$
+###
+==== Input format suitable for Mathematica (Quanty.nb) ====
+###
+<code Quanty Akm_Oh_XYZ.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {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_Oh_XYZ.Quanty>
+Akm = {{0, 0, A(0,0)} ,
+       {4, 0, A(4,0)} ,
+       {4,-4, (sqrt(5/14))*(A(4,0))} ,
+       {4, 4, (sqrt(5/14))*(A(4,0))} ,
+       {6, 0, A(6,0)} ,
+       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} ,
+       {6, 4, (-1)*((sqrt(7/2))*(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 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|
+^$ {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 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 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 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|
+^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{21} \text{Add}(4,0) $|$\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)-\frac{4}{21} \text{Add}(4,0) $|$ 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)+\frac{2}{7} \text{Add}(4,0) $|$ 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)-\frac{4}{21} \text{Add}(4,0) $|$ 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 }$|$ \frac{5}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$\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 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ 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{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $|$ 0 $|
+^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|
+^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 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 $|$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|
+^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\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_{\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-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
+^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
+^$ f_{z\left(5z^2-r^2\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 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
+^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
+^$ f_{z\left(x^2-y^2\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}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\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_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\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 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 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 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 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 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$ 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)+\frac{2}{7} \text{Add}(4,0) $|$ 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)-\frac{4}{21} \text{Add}(4,0) $|$ 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)-\frac{4}{21} \text{Add}(4,0) $|$ 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)-\frac{4}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 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 }$|$ \text{Aff}(0,0)-\frac{4}{11} \text{Aff}(4,0)+\frac{80}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(y^2-z^2\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{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|
+^$ f_{y\left(z^2-x^2\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{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0) $|$ 0 $|
+^$ f_{z\left(x^2-y^2\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{2}{33} \text{Aff}(4,0)-\frac{60}{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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Ea1g} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty>
+Akm = {{0, 0, Ea1g} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
+^$ {Y_{0}^{(0)}} $|$ \text{Ea1g} $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ \text{s} $  ^
+^$ \text{s} $|$ \text{Ea1g} $|
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
+^$ \text{s} $|$ 1 $|
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:oh_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_Oh_XYZ.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_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:oh_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:oh_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:oh_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{2 \text{Eeg}}{5}+\frac{3 \text{Et2g}}{5} & k=0\land m=0 \\
+ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eeg}-\text{Et2g}) & k=4\land (m=-4\lor m=4) \\
+ \frac{21 (\text{Eeg}-\text{Et2g})}{10} & k=4\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(2*Eeg)/5 + (3*Et2g)/5, k == 0 && m == 0}, {(3*Sqrt[7/10]*(Eeg - Et2g))/2, k == 4 && (m == -4 || m == 4)}, {(21*(Eeg - Et2g))/10, k == 4 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty>
+Akm = {{0, 0, (2/5)*(Eeg) + (3/5)*(Et2g)} ,
+       {4, 0, (21/10)*(Eeg + (-1)*(Et2g))} ,
+       {4,-4, (3/2)*((sqrt(7/10))*(Eeg + (-1)*(Et2g)))} ,
+       {4, 4, (3/2)*((sqrt(7/10))*(Eeg + (-1)*(Et2g)))} }
+</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)}} $|$ \frac{\text{Eeg}+\text{Et2g}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eeg}-\text{Et2g}}{2} $|
+^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|
+^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|
+^$ {Y_{2}^{(2)}} $|$ \frac{\text{Eeg}-\text{Et2g}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eeg}+\text{Et2g}}{2} $|
+###
+</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{Eeg} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{3z^2-r^2} $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $|
+^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $|
+^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|
+^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $|
+###
+</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** >
+###
+^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_2_1.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |
+^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_2_2.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
+^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_2_3.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
+^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_2_4.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
+^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\
+ -\frac{3}{4} \sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land (m=-4\lor m=4) \\
+ -\frac{3}{4} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\
+ -\frac{39 (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\
+ \frac{39}{280} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {(-3*Sqrt[5/14]*(2*Ea2u - 3*Et1u + Et2u))/4, k == 4 && (m == -4 || m == 4)}, {(-3*(2*Ea2u - 3*Et1u + Et2u))/4, k == 4 && m == 0}, {(-39*(4*Ea2u + 5*Et1u - 9*Et2u))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(39*(4*Ea2u + 5*Et1u - 9*Et2u))/280, k == 6 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty>
+Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} ,
+       {4, 0, (-3/4)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} ,
+       {4,-4, (-3/4)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} ,
+       {4, 4, (-3/4)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} ,
+       {6, 0, (39/280)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} ,
+       {6,-4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} ,
+       {6, 4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(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}{8} (5 \text{Et1u}+3 \text{Et2u}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|
+^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Ea2u}+\text{Et2u}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Et2u}-\text{Ea2u}}{2} $|$ 0 $|
+^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} (3 \text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u}) $|
+^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} (3 \text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|
+^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{\text{Et2u}-\text{Ea2u}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ea2u}+\text{Et2u}}{2} $|$ 0 $|
+^$ {Y_{3}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} (5 \text{Et1u}+3 \text{Et2u}) $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
+^$ f_{\text{xyz}} $|$ \text{Ea2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|$ 0 $|
+^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|
+^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|
+###
+</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_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
+^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
+^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
+^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
+^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
+^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
+^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_1.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
+^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_2.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: |
+^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_3.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: |
+^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_4.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
+^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_5.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: |
+^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_6.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: |
+^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_xyz_orb_3_7.png?150}} |
+|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
+|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |
+###
+</hidden>
+===== Coupling between two shells =====
+###
+Click on one of the subsections to expand it or <hiddenSwitch expand all>
+###
+==== Potential for p-f orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\
+ \frac{3}{4} \sqrt{\frac{15}{2}} \text{Mt1u} & k=4\land (m=-4\lor m=4) \\
+ \frac{3 \sqrt{21} \text{Mt1u}}{4} & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -4 && m != 0 && m != 4)}, {(3*Sqrt[15/2]*Mt1u)/4, k == 4 && (m == -4 || m == 4)}}, (3*Sqrt[21]*Mt1u)/4]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_XYZ.Quanty>
+Akm = {{4, 0, (3/4)*((sqrt(21))*(Mt1u))} ,
+       {4,-4, (3/4)*((sqrt(15/2))*(Mt1u))} ,
+       {4, 4, (3/4)*((sqrt(15/2))*(Mt1u))} }
+</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_{-1}^{(1)}} $|$ 0 $|$ 0 $|$ -\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt1u} $|
+^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|
+^$ {Y_{1}^{(1)}} $|$ -\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt1u} $|$ 0 $|$ 0 $|
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
+^$ p_x $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ p_y $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
+^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|
+###
+</hidden>
+===== 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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