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physics_chemistry:point_groups:oh:orientation_sqrt201z [2018/03/21 18:50] – created Stefano Agrestiniphysics_chemistry:point_groups:oh:orientation_sqrt201z [2018/09/06 12:53] (current) – Maurits W. Haverkort
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 +~~CLOSETOC~~
 +
 ====== Orientation sqrt201z ====== ====== Orientation sqrt201z ======
 +
 +===== Symmetry Operations =====
  
 ### ###
-alligned paragraph text+ 
 +In the Oh Point Group, with orientation sqrt201z there are the following symmetry operations 
 ### ###
  
-===== Example =====+### 
 + 
 +{{:physics_chemistry:pointgroup:oh_sqrt201z.png}}
  
 ### ###
-description text+
 ### ###
  
-==== Input ==== +^ Operator ^ Orientation ^ 
-<code Quanty Example.Quanty> +^ $\text{E}$ | $\{0,0,0\}$ , | 
--- some example code+^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\sqrt{2},-\sqrt{6},1\right\}$ , $\left\{2 \sqrt{2},0,-1\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},-1\right\}$ , $\left\{-\sqrt{2},\sqrt{6},-1\right\}$ , $\left\{-2 \sqrt{2},0,1\right\}$ , $\left\{\sqrt{2},\sqrt{6},1\right\}$ , | 
 +^ $C_2$ | $\{0,1,0\}$ , $\left\{-\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{1,0,-\sqrt{2}\right\}$ , $\left\{1,-\sqrt{3},2 \sqrt{2}\right\}$ , $\left\{1,\sqrt{3},2 \sqrt{2}\right\}$ , | 
 +^ $C_4$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{-\sqrt{2},0,-1\right\}$ , $\left\{-1,-\sqrt{3},\sqrt{2}\right\}$ , $\left\{-1,\sqrt{3},\sqrt{2}\right\}$ , $\left\{1,\sqrt{3},-\sqrt{2}\right\}$ , $\left\{1,-\sqrt{3},-\sqrt{2}\right\}$ , | 
 +^ $C_2$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{1,\sqrt{3},-\sqrt{2}\right\}$ , $\left\{1,-\sqrt{3},-\sqrt{2}\right\}$ , | 
 +^ $\text{i}$ | $\{0,0,0\}$ , | 
 +^ $S_4$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{-\sqrt{2},0,-1\right\}$ , $\left\{-1,-\sqrt{3},\sqrt{2}\right\}$ , $\left\{-1,\sqrt{3},\sqrt{2}\right\}$ , $\left\{1,\sqrt{3},-\sqrt{2}\right\}$ , $\left\{1,-\sqrt{3},-\sqrt{2}\right\}$ , | 
 +^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\sqrt{2},-\sqrt{6},1\right\}$ , $\left\{2 \sqrt{2},0,-1\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},-1\right\}$ , $\left\{-\sqrt{2},\sqrt{6},-1\right\}$ , $\left\{-2 \sqrt{2},0,1\right\}$ , $\left\{\sqrt{2},\sqrt{6},1\right\}$ , | 
 +^ $\sigma _h$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{1,\sqrt{3},-\sqrt{2}\right\}$ , $\left\{1,-\sqrt{3},-\sqrt{2}\right\}$ , | 
 +^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{-\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{1,0,-\sqrt{2}\right\}$ , $\left\{1,-\sqrt{3},2 \sqrt{2}\right\}$ , $\left\{1,\sqrt{3},2 \sqrt{2}\right\}$ , | 
 + 
 +### 
 + 
 +===== 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:c2:orientation_y|Point Group C2 with orientation Y]] 
 +  * [[physics_chemistry:point_groups:c3v:orientation_zy|Point Group C3v with orientation Zy]] 
 +  * [[physics_chemistry:point_groups:c3:orientation_z|Point Group C3 with orientation Z]] 
 +  * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]] 
 +  * [[physics_chemistry:point_groups:cs:orientation_y|Point Group Cs with orientation Y]] 
 +  * [[physics_chemistry:point_groups:d3d:orientation_zy|Point Group D3d with orientation Zy]] 
 +  * [[physics_chemistry:point_groups:d3d:orientation_zy_a|Point Group D3d with orientation Zy_A]] 
 +  * [[physics_chemistry:point_groups:d3d:orientation_zy_b|Point Group D3d with orientation Zy_B]] 
 +  * [[physics_chemistry:point_groups:d3:orientation_zy|Point Group D3 with orientation Zy]] 
 +  * [[physics_chemistry:point_groups:s6:orientation_z|Point Group S6 with orientation Z]] 
 + 
 +### 
 + 
 +===== 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 Oh Point group with orientation sqrt201z the form of the expansion coefficients is: 
 + 
 +### 
 + 
 +==== Expansion ==== 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + A(0,0) & k=0\land m=0 \\ 
 + -\sqrt{\frac{10}{7}} A(4,0) & k=4\land m=-3 \\ 
 + A(4,0) & k=4\land m=0 \\ 
 + \sqrt{\frac{10}{7}} A(4,0) & k=4\land m=3 \\ 
 + \frac{1}{8} \sqrt{\frac{77}{3}} A(6,0) & k=6\land (m=-6\lor m=6) \\ 
 + \frac{1}{4} \sqrt{\frac{35}{6}} A(6,0) & k=6\land m=-3 \\ 
 + A(6,0) & k=6\land m=0 \\ 
 + -\frac{1}{4} \sqrt{\frac{35}{6}} A(6,0) & k=6\land m=3 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +==== Input format suitable for Mathematica (Quanty.nb) ==== 
 + 
 +### 
 + 
 +<code Quanty Akm_Oh_sqrt201z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-(Sqrt[10/7]*A[4, 0]), k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {Sqrt[10/7]*A[4, 0], k == 4 && m == 3}, {(Sqrt[77/3]*A[6, 0])/8, k == 6 && (m == -6 || m == 6)}, {(Sqrt[35/6]*A[6, 0])/4, k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {-(Sqrt[35/6]*A[6, 0])/4, k == 6 && m == 3}}, 0] 
 </code> </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_sqrt201z.Quanty>
 +
 +Akm = {{0, 0, A(0,0)} , 
 +       {4, 0, A(4,0)} , 
 +       {4,-3, (-1)*((sqrt(10/7))*(A(4,0)))} , 
 +       {4, 3, (sqrt(10/7))*(A(4,0))} , 
 +       {6, 0, A(6,0)} , 
 +       {6, 3, (-1/4)*((sqrt(35/6))*(A(6,0)))} , 
 +       {6,-3, (1/4)*((sqrt(35/6))*(A(6,0)))} , 
 +       {6,-6, (1/8)*((sqrt(77/3))*(A(6,0)))} , 
 +       {6, 6, (1/8)*((sqrt(77/3))*(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 $|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$ 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 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|
 +^$ {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 $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$ 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 $|$ \frac{5}{21} \sqrt{2} \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_{-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 $|$ -\frac{5}{21} \sqrt{2} \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_{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 }$|$ \frac{5}{21} \sqrt{2} \text{Add}(4,0) $|$ 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 }$|$ 0 $|$ -\frac{5}{21} \sqrt{2} \text{Add}(4,0) $|$ 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 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 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 $|$ \frac{1}{11} \sqrt{10} \text{Aff}(4,0)+\frac{35}{858} \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{35}{156} \text{Aff}(6,0) $|
 +^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$\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 $|$ \frac{2}{33} \sqrt{5} \text{Aff}(4,0)-\frac{35}{572} \sqrt{5} \text{Aff}(6,0) $|$ 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 $|$ \frac{35}{572} \sqrt{5} \text{Aff}(6,0)-\frac{2}{33} \sqrt{5} \text{Aff}(4,0) $|$ 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 }$|$ \frac{1}{11} \sqrt{10} \text{Aff}(4,0)+\frac{35}{858} \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{1}{11} \sqrt{10} \text{Aff}(4,0)-\frac{35}{858} \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|
 +^$ {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 }$|$ 0 $|$ \frac{2}{33} \sqrt{5} \text{Aff}(4,0)-\frac{35}{572} \sqrt{5} \text{Aff}(6,0) $|$ 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 }$|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{35}{572} \sqrt{5} \text{Aff}(6,0)-\frac{2}{33} \sqrt{5} \text{Aff}(4,0) $|$ 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 }$|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{35}{156} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{1}{11} \sqrt{10} \text{Aff}(4,0)-\frac{35}{858} \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 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_{\text{xy}-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$ 0 $|$ -\frac{i}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \frac{1}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $|$ 0 $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$\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 }$|
 +^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{2}}{3} $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\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{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\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} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|$ \frac{i}{2 \sqrt{3}} $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{\frac{5}{2}}}{3} $|$ 0 $|$ 0 $|$ -\frac{2}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{\frac{5}{2}}}{3} $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash 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 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\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{i}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
 +
 +
 +###
 +
 +==== One particle coupling on a basis of symmetry adapted functions ====
 +
 +###
 +
 +After rotation we find
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ \text{s} $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^  $ d_{\text{xy}-\sqrt{2}\text{yz}} $  ^  $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $  ^  $ d_{\text{yz}+\sqrt{2}\text{xy}} $  ^  $ d_{x^2-y^2-\sqrt{2}\text{xz}} $  ^  $ d_{3z^2-r^2} $  ^  $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $  ^  $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $  ^  $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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 $|$ -\frac{2 \text{Apf}(4,0)}{\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{2 \text{Apf}(4,0)}{\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{2 \text{Apf}(4,0)}{\sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{3}{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_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)-\frac{3}{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}+\sqrt{2}\text{xy}} $|$ 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 }$|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \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_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\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{6}{11} \text{Aff}(4,0)+\frac{45}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,0)}{\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{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{2 \text{Apf}(4,0)}{\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{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{2 \text{Apf}(4,0)}{\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{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash 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{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\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{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0) $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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{1}{11} \text{Aff}(4,0)-\frac{135}{572} \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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \text{Ea1g} & k=0\land m=0 \\
 + 0 & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.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_sqrt201z_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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \text{Et1u} & k=0\land m=0 \\
 + 0 & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.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_sqrt201z_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_sqrt201z_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_sqrt201z_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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \frac{1}{5} (2 \text{Eeg}+3 \text{Et2g}) & k=0\land m=0 \\
 + \sqrt{\frac{14}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land m=-3 \\
 + \frac{7 (\text{Et2g}-\text{Eeg})}{5} & k=4\land m=0 \\
 + \sqrt{\frac{14}{5}} (\text{Et2g}-\text{Eeg}) & k=4\land m=3
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{(2*Eeg + 3*Et2g)/5, k == 0 && m == 0}, {Sqrt[14/5]*(Eeg - Et2g), k == 4 && m == -3}, {(7*(-Eeg + Et2g))/5, k == 4 && m == 0}, {Sqrt[14/5]*(-Eeg + Et2g), k == 4 && m == 3}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty>
 +
 +Akm = {{0, 0, (1/5)*((2)*(Eeg) + (3)*(Et2g))} , 
 +       {4, 0, (7/5)*((-1)*(Eeg) + Et2g)} , 
 +       {4, 3, (sqrt(14/5))*((-1)*(Eeg) + Et2g)} , 
 +       {4,-3, (sqrt(14/5))*(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{1}{3} (\text{Eeg}+2 \text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{2} (\text{Et2g}-\text{Eeg}) $|$ 0 $|
 +^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{2} (\text{Eeg}-\text{Et2g}) $|
 +^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $|
 +^$ {Y_{1}^{(2)}} $|$ \frac{1}{3} \sqrt{2} (\text{Et2g}-\text{Eeg}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $|
 +^$ {Y_{2}^{(2)}} $|$ 0 $|$ \frac{1}{3} \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (\text{Eeg}+2 \text{Et2g}) $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ d_{\text{xy}-\sqrt{2}\text{yz}} $  ^  $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $  ^  $ d_{\text{yz}+\sqrt{2}\text{xy}} $  ^  $ d_{x^2-y^2-\sqrt{2}\text{xz}} $  ^  $ d_{3z^2-r^2} $  ^
 +^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|
 +^$ d_{3z^2-r^2} $|$ 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_{\text{xy}-\sqrt{2}\text{yz}} $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$ 0 $|$ -\frac{i}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|
 +^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \frac{1}{\sqrt{6}} $|
 +^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ \frac{i}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $|$ 0 $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|
 +^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_2_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \sin (\phi ) \left(\sin (\theta ) \cos (\phi )-\sqrt{2} \cos (\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} y \left(x-\sqrt{2} z\right)$$ | ::: |
 +^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_2_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(2 \sqrt{2} \cos (\theta ) \cos (\phi )+\sin (\theta ) \cos (2 \phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(x^2+2 \sqrt{2} x z-y^2\right)$$ | ::: |
 +^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_2_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \sin (\phi ) \left(\sqrt{2} \sin (\theta ) \cos (\phi )+\cos (\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} y \left(\sqrt{2} x+z\right)$$ | ::: |
 +^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_2_4.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(\sqrt{2} \sin ^2(\theta ) \cos (2 \phi )-\sin (2 \theta ) \cos (\phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(\sqrt{2} x^2-2 x z-\sqrt{2} y^2\right)$$ | ::: |
 +^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_2_5.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
 +
 +
 +###
 +
 +</hidden>
 +==== Potential for f orbitals ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\
 + -\sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=-3 \\
 + \frac{1}{2} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\
 + \sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=3 \\
 + \frac{13}{60} \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land (m=-6\lor m=6) \\
 + \frac{13 (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{6 \sqrt{210}} & k=6\land m=-3 \\
 + \frac{26}{105} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0 \\
 + -\frac{13 (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{6 \sqrt{210}} & k=6\land m=3
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {-(Sqrt[5/14]*(2*Ea2u - 3*Et1u + Et2u)), k == 4 && m == -3}, {(2*Ea2u - 3*Et1u + Et2u)/2, k == 4 && m == 0}, {Sqrt[5/14]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && m == 3}, {(13*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u))/60, k == 6 && (m == -6 || m == 6)}, {(13*(4*Ea2u + 5*Et1u - 9*Et2u))/(6*Sqrt[210]), k == 6 && m == -3}, {(26*(4*Ea2u + 5*Et1u - 9*Et2u))/105, k == 6 && m == 0}, {(-13*(4*Ea2u + 5*Et1u - 9*Et2u))/(6*Sqrt[210]), k == 6 && m == 3}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty>
 +
 +Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} , 
 +       {4, 0, (1/2)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} , 
 +       {4,-3, (-1)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
 +       {4, 3, (sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} , 
 +       {6, 0, (26/105)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} , 
 +       {6, 3, (-13/6)*((1/(sqrt(210)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
 +       {6,-3, (13/6)*((1/(sqrt(210)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
 +       {6,-6, (13/60)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
 +       {6, 6, (13/60)*((sqrt(11/21))*((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}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} (-4 \text{Ea2u}-5 \text{Et1u}+9 \text{Et2u}) $|
 +^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} \sqrt{5} (\text{Et2u}-\text{Et1u}) $|$ 0 $|$ 0 $|
 +^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|
 +^$ {Y_{0}^{(3)}} $|$ \frac{1}{9} \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} (5 \text{Ea2u}+4 \text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} \sqrt{10} (\text{Et1u}-\text{Ea2u}) $|
 +^$ {Y_{1}^{(3)}} $|$ 0 $|$ \frac{1}{6} \sqrt{5} (\text{Et2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|
 +^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $|
 +^$ {Y_{3}^{(3)}} $|$ \frac{1}{18} (-4 \text{Ea2u}-5 \text{Et1u}+9 \text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} \sqrt{10} (\text{Et1u}-\text{Ea2u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u}) $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $  ^  $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $  ^  $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $  ^
 +^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$ \text{Ea2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$ \frac{\sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{2}}{3} $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|$ \frac{i}{2 \sqrt{3}} $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$ \frac{\sqrt{\frac{5}{2}}}{3} $|$ 0 $|$ 0 $|$ -\frac{2}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{\frac{5}{2}}}{3} $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(\sqrt{2} \left(1+e^{6 i \phi }\right) \sin ^3(\theta )+e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^3-3 \sqrt{2} x y^2+5 z^3-3 z\right)$$ | ::: |
 +^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \left((5 \cos (2 \theta )+3) \cos (\phi )+5 \sqrt{2} \sin (2 \theta ) \cos (2 \phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 \sqrt{2} x^2 z+x \left(5 z^2-1\right)-5 \sqrt{2} y^2 z\right)$$ | ::: |
 +^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(-10 \sqrt{2} \sin (2 \theta ) \cos (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{7}{\pi }} y \left(10 \sqrt{2} x z-5 z^2+1\right)$$ | ::: |
 +^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_4.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} \sin ^3(\theta ) \cos (3 \phi )+3 \cos (\theta )+5 \cos (3 \theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 \sqrt{2} x^3-15 \sqrt{2} x y^2+4 z \left(3-5 z^2\right)\right)$$ | ::: |
 +^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_5.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (2 \theta ) \cos (2 \phi )-(5 \cos (2 \theta )+3) \cos (\phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^2 z-5 x z^2+x-\sqrt{2} y^2 z\right)$$ | ::: |
 +^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_6.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \sin (\phi ) \left(2 \sqrt{2} \sin (2 \theta ) \cos (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{35}{\pi }} y \left(2 \sqrt{2} x z+5 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_sqrt201z_orb_3_7.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$$ | ::: |
 +
 +
 +###
 +
 +</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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + 0 & k\neq 4\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
 + \sqrt{\frac{15}{2}} \text{Mt1u} & k=4\land m=-3 \\
 + -\frac{\sqrt{21} \text{Mt1u}}{2} & k=4\land m=0 \\
 + -\sqrt{\frac{15}{2}} \text{Mt1u} & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -3 && m != 0 && m != 3)}, {Sqrt[15/2]*Mt1u, k == 4 && m == -3}, {-(Sqrt[21]*Mt1u)/2, k == 4 && m == 0}}, -(Sqrt[15/2]*Mt1u)]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_sqrt201z.Quanty>
 +
 +Akm = {{4, 0, (-1/2)*((sqrt(21))*(Mt1u))} , 
 +       {4, 3, (-1)*((sqrt(15/2))*(Mt1u))} , 
 +       {4,-3, (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{\text{Mt1u}}{\sqrt{6}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{5}{6}} \text{Mt1u} $|$ 0 $|
 +^$ {Y_{0}^{(1)}} $|$ \frac{1}{3} \sqrt{\frac{5}{2}} \text{Mt1u} $|$ 0 $|$ 0 $|$ -\frac{2 \text{Mt1u}}{3} $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{2}} \text{Mt1u} $|
 +^$ {Y_{1}^{(1)}} $|$ 0 $|$ -\sqrt{\frac{5}{6}} \text{Mt1u} $|$ 0 $|$ 0 $|$ \frac{\text{Mt1u}}{\sqrt{6}} $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $  ^  $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $  ^  $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\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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