Differences

This shows you the differences between two versions of the page.

Link to this comparison view

physics_chemistry:point_groups:oh:orientation_111z [2018/03/21 18:48] – created Stefano Agrestiniphysics_chemistry:point_groups:oh:orientation_111z [2018/09/06 12:52] (current) Maurits W. Haverkort
Line 1: Line 1:
 +~~CLOSETOC~~
 +
 ====== Orientation 111z ====== ====== Orientation 111z ======
 +
 +===== Symmetry Operations =====
  
 ### ###
-alligned paragraph text+ 
 +In the Oh Point Group, with orientation 111z there are the following symmetry operations 
 ### ###
  
-===== Example =====+### 
 + 
 +{{:physics_chemistry:pointgroup:oh_111z.png}}
  
 ### ###
-description text+
 ### ###
  
-==== Input ==== +^ Operator ^ Orientation ^ 
-<code Quanty Example.Quanty> +^ $\text{E}$ | $\{0,0,0\}$ , | 
--- some example code+^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{2+\sqrt{3},-1,\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ , $\{2,2,-1\}$ , $\left\{1,-2-\sqrt{3},\frac{1}{1-\sqrt{3}}\right\}$ , $\left\{-2-\sqrt{3},1,\frac{1}{1-\sqrt{3}}\right\}$ , $\{-2,-2,1\}$ , $\left\{-1,2+\sqrt{3},\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ , | 
 +^ $C_2$ | $\{1,-1,0\}$ , $\left\{2+\sqrt{3},1,0\right\}$ , $\left\{1,2+\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{-2-\sqrt{3},1,-2 \left(1+\sqrt{3}\right)\right\}$ , $\left\{1,-2-\sqrt{3},-2 \left(1+\sqrt{3}\right)\right\}$ , | 
 +^ $C_4$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ , $\left\{-1,2+\sqrt{3},-1-\sqrt{3}\right\}$ , $\left\{2+\sqrt{3},-1,-1-\sqrt{3}\right\}$ , | 
 +^ $C_2$ | $\{1,1,1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ , | 
 +^ $\text{i}$ | $\{0,0,0\}$ , | 
 +^ $S_4$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ , $\left\{-1,2+\sqrt{3},-1-\sqrt{3}\right\}$ , $\left\{2+\sqrt{3},-1,-1-\sqrt{3}\right\}$ , | 
 +^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{2+\sqrt{3},-1,\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ , $\{2,2,-1\}$ , $\left\{1,-2-\sqrt{3},\frac{1}{1-\sqrt{3}}\right\}$ , $\left\{-2-\sqrt{3},1,\frac{1}{1-\sqrt{3}}\right\}$ , $\{-2,-2,1\}$ , $\left\{-1,2+\sqrt{3},\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ , | 
 +^ $\sigma _h$ | $\{1,1,1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ , | 
 +^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{2+\sqrt{3},1,0\right\}$ , $\left\{1,2+\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{-2-\sqrt{3},1,-2 \left(1+\sqrt{3}\right)\right\}$ , $\left\{1,-2-\sqrt{3},-2 \left(1+\sqrt{3}\right)\right\}$ , | 
 + 
 +### 
 + 
 +===== 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:c3:orientation_z|Point Group C3 with orientation Z]] 
 +  * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]] 
 +  * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)|Point Group D3d with orientation Z(x-y)]] 
 +  * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a|Point Group D3d with orientation Z(x-y)_A]] 
 +  * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b|Point Group D3d with orientation Z(x-y)_B]] 
 +  * [[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 111z the form of the expansion coefficients is: 
 + 
 +### 
 + 
 +==== Expansion ==== 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + A(0,0) & k=0\land m=0 \\ 
 + (1-i) \sqrt{\frac{5}{7}} A(4,0) & k=4\land m=-3 \\ 
 + A(4,0) & k=4\land m=0 \\ 
 + (-1-i) \sqrt{\frac{5}{7}} A(4,0) & k=4\land m=3 \\ 
 + -\frac{1}{8} i \sqrt{\frac{77}{3}} A(6,0) & k=6\land m=-6 \\ 
 + \left(-\frac{1}{8}+\frac{i}{8}\right) \sqrt{\frac{35}{3}} A(6,0) & k=6\land m=-3 \\ 
 + A(6,0) & k=6\land m=0 \\ 
 + \left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\frac{35}{3}} A(6,0) & k=6\land m=3 \\ 
 + \frac{1}{8} i \sqrt{\frac{77}{3}} A(6,0) & k=6\land m=6 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +==== Input format suitable for Mathematica (Quanty.nb) ==== 
 + 
 +### 
 + 
 +<code Quanty Akm_Oh_111z.Quanty.nb
 + 
 +Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(1 I)*Sqrt[5/7]*A[4, 0], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {(-1 - I)*Sqrt[5/7]*A[4, 0], k == 4 && m == 3}, {(-I/8)*Sqrt[77/3]*A[6, 0], k == 6 && m == -6}, {(-1/8 + I/8)*Sqrt[35/3]*A[6, 0], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {(1/8 + I/8)*Sqrt[35/3]*A[6, 0], k == 6 && m == 3}, {(I/8)*Sqrt[77/3]*A[6, 0], k == 6 && m == 6}}, 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_111z.Quanty>
 +
 +Akm = {{0, 0, A(0,0)} , 
 +       {4, 0, A(4,0)} , 
 +       {4, 3, (-1+-1*I)*((sqrt(5/7))*(A(4,0)))} , 
 +       {4,-3, (1+-1*I)*((sqrt(5/7))*(A(4,0)))} , 
 +       {6, 0, A(6,0)} , 
 +       {6,-3, (-1/8+1/8*I)*((sqrt(35/3))*(A(6,0)))} , 
 +       {6, 3, (1/8+1/8*I)*((sqrt(35/3))*(A(6,0)))} , 
 +       {6,-6, (-1/8*I)*((sqrt(77/3))*(A(6,0)))} , 
 +       {6, 6, (1/8*I)*((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 $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{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 }$|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{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 $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{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 $|$ \left(-\frac{5}{21}+\frac{5 i}{21}\right) \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 $|$ \left(\frac{5}{21}-\frac{5 i}{21}\right) \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 }$|$ \left(-\frac{5}{21}-\frac{5 i}{21}\right) \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 $|$ \left(\frac{5}{21}+\frac{5 i}{21}\right) \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 $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{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 $|$ \left(-\frac{1}{11}+\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)-\left(\frac{35}{1716}-\frac{35 i}{1716}\right) \sqrt{5} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{35}{156} i \text{Aff}(6,0) $|
 +^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{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 $|$ \left(\frac{35}{572}-\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0)-\left(\frac{1}{33}-\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0) $|$ 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 $|$ \left(\frac{1}{33}-\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0)-\left(\frac{35}{572}-\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 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 }$|$ \left(-\frac{1}{11}-\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)-\left(\frac{35}{1716}+\frac{35 i}{1716}\right) \sqrt{5} \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 $|$ \left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)+\left(\frac{35}{1716}-\frac{35 i}{1716}\right) \sqrt{5} \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 $|$ \left(\frac{35}{572}+\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0)-\left(\frac{1}{33}+\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0) $|$ 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 }$|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \left(\frac{1}{33}+\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0)-\left(\frac{35}{572}+\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 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 $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{21}} \text{Apf}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{35}{156} i \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)+\left(\frac{35}{1716}+\frac{35 i}{1716}\right) \sqrt{5} \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_{(x-y)(x+y-2z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{6}} $|$ -\frac{1-i}{\sqrt{6}} $|$ 0 $|$ \frac{1+i}{\sqrt{6}} $|$ \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}+\text{xz}+\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ \frac{1+i}{\sqrt{6}} $|$ 0 $|$ -\frac{1-i}{\sqrt{6}} $|$ -\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-y)(x+y+z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \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_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\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_{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_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\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{1}{3}-\frac{i}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{1}{3}-\frac{i}{3} $|
 +^$ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\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 $|$ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
 +^$ f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\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 $|$ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{i}{2 \sqrt{3}} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
 +^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{6}-\frac{i}{6}\right) \sqrt{5} $|$ 0 $|$ 0 $|$ \frac{2}{3} $|$ 0 $|$ 0 $|$ \left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt{5} $|
 +^$ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\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{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
 +^$ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\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{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
 +^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+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{1}{2}+\frac{i}{2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|
 +
 +
 +###
 +
 +==== One particle coupling on a basis of symmetry adapted functions ====
 +
 +###
 +
 +After rotation we find
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ \text{s} $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^  $ d_{(x-y)(x+y-2z)} $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^  $ d_{(x-y)(x+y+z)} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{3z^2-r^2} $  ^  $ f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $  ^  $ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $  ^  $ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+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_{(x-y)(x+y-2z)} $|$ 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_{\text{yz}+\text{xz}+\text{xy}} $|$ 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_{(x-y)(x+y+z)} $|$ 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_{2\text{xy}-\text{xz}-\text{yz}} $|$ 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_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\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 }$|$ \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(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\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_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\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_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\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_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\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 $|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|
 +^$ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\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_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+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_111z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.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_111z_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_111z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.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_111z_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_111z_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_111z_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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \frac{1}{5} (2 \text{Eeg}+3 \text{Et2g}) & k=0\land m=0 \\
 + (-1+i) \sqrt{\frac{7}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land m=-3 \\
 + -\frac{7}{5} (\text{Eeg}-\text{Et2g}) & k=4\land m=0 \\
 + (1+i) \sqrt{\frac{7}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land m=3
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{(2*Eeg + 3*Et2g)/5, k == 0 && m == 0}, {(-1 + I)*Sqrt[7/5]*(Eeg - Et2g), k == 4 && m == -3}, {(-7*(Eeg - Et2g))/5, k == 4 && m == 0}, {(1 + I)*Sqrt[7/5]*(Eeg - Et2g), k == 4 && m == 3}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.Quanty>
 +
 +Akm = {{0, 0, (1/5)*((2)*(Eeg) + (3)*(Et2g))} , 
 +       {4, 0, (-7/5)*(Eeg + (-1)*(Et2g))} , 
 +       {4,-3, (-1+1*I)*((sqrt(7/5))*(Eeg + (-1)*(Et2g)))} , 
 +       {4, 3, (1+1*I)*((sqrt(7/5))*(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 $|$ \left(\frac{1}{3}-\frac{i}{3}\right) (\text{Eeg}-\text{Et2g}) $|$ 0 $|
 +^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $|$ 0 $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) (\text{Eeg}-\text{Et2g}) $|
 +^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $|
 +^$ {Y_{1}^{(2)}} $|$ \left(\frac{1}{3}+\frac{i}{3}\right) (\text{Eeg}-\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $|
 +^$ {Y_{2}^{(2)}} $|$ 0 $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) (\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_{(x-y)(x+y-2z)} $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^  $ d_{(x-y)(x+y+z)} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{3z^2-r^2} $  ^
 +^$ d_{(x-y)(x+y-2z)} $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{(x-y)(x+y+z)} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $|
 +^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 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_{(x-y)(x+y-2z)} $|$ \frac{1}{\sqrt{6}} $|$ -\frac{1-i}{\sqrt{6}} $|$ 0 $|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{6}} $|
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ \frac{i}{\sqrt{6}} $|$ \frac{1+i}{\sqrt{6}} $|$ 0 $|$ -\frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $|
 +^$ d_{(x-y)(x+y+z)} $|$ \frac{1}{\sqrt{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \frac{1}{\sqrt{3}} $|
 +^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ \frac{i}{\sqrt{3}} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\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_111z_orb_2_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta ))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} (x-y) (x+y-2 z)$$ | ::: |
 +^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_2_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$$ | ::: |
 +^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_2_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta ))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (x-y) (x+y+z)$$ | ::: |
 +^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_2_4.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$$ | ::: |
 +^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_111z_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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\
 + \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{7}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=-3 \\
 + \frac{1}{2} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\
 + \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{7}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=3 \\
 + -\frac{13}{60} i \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=-6 \\
 + -\frac{\left(\frac{13}{12}-\frac{13 i}{12}\right) (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{\sqrt{105}} & k=6\land m=-3 \\
 + \frac{26}{105} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0 \\
 + \frac{\left(\frac{13}{12}+\frac{13 i}{12}\right) (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{\sqrt{105}} & k=6\land m=3 \\
 + \frac{13}{60} i \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=6
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {(1/2 - I/2)*Sqrt[5/7]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && m == -3}, {(2*Ea2u - 3*Et1u + Et2u)/2, k == 4 && m == 0}, {(-1/2 - I/2)*Sqrt[5/7]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && m == 3}, {((-13*I)/60)*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u), k == 6 && m == -6}, {((-13/12 + (13*I)/12)*(4*Ea2u + 5*Et1u - 9*Et2u))/Sqrt[105], k == 6 && m == -3}, {(26*(4*Ea2u + 5*Et1u - 9*Et2u))/105, k == 6 && m == 0}, {((13/12 + (13*I)/12)*(4*Ea2u + 5*Et1u - 9*Et2u))/Sqrt[105], k == 6 && m == 3}, {((13*I)/60)*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u), k == 6 && m == 6}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.Quanty>
 +
 +Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} , 
 +       {4, 0, (1/2)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} , 
 +       {4, 3, (-1/2+-1/2*I)*((sqrt(5/7))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
 +       {4,-3, (1/2+-1/2*I)*((sqrt(5/7))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
 +       {6, 0, (26/105)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} , 
 +       {6,-3, (-13/12+13/12*I)*((1/(sqrt(105)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
 +       {6, 3, (13/12+13/12*I)*((1/(sqrt(105)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
 +       {6,-6, (-13/60*I)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
 +       {6, 6, (13/60*I)*((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 $|$ \left(-\frac{1}{9}+\frac{i}{9}\right) \sqrt{5} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} i (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} (-1)^{3/4} \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} (-1)^{3/4} \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|
 +^$ {Y_{0}^{(3)}} $|$ \left(\frac{1}{9}+\frac{i}{9}\right) \sqrt{5} (\text{Et1u}-\text{Ea2u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} (5 \text{Ea2u}+4 \text{Et1u}) $|$ 0 $|$ 0 $|$ \left(\frac{1}{9}-\frac{i}{9}\right) \sqrt{5} (\text{Ea2u}-\text{Et1u}) $|
 +^$ {Y_{1}^{(3)}} $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{\frac{5}{2}} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|
 +^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{\frac{5}{2}} (\text{Et2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $|
 +^$ {Y_{3}^{(3)}} $|$ -\frac{1}{18} i (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) $|$ 0 $|$ 0 $|$ \left(\frac{1}{9}+\frac{i}{9}\right) \sqrt{5} (\text{Ea2u}-\text{Et1u}) $|$ 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_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $  ^  $ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $  ^  $ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $  ^
 +^$ f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $|$ \text{Ea2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|$ 0 $|
 +^$ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|
 +^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+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_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $|$ \frac{1}{3}-\frac{i}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{1}{3}-\frac{i}{3} $|
 +^$ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
 +^$ f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{i}{2 \sqrt{3}} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
 +^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $|$ \left(\frac{1}{6}-\frac{i}{6}\right) \sqrt{5} $|$ 0 $|$ 0 $|$ \frac{2}{3} $|$ 0 $|$ 0 $|$ \left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt{5} $|
 +^$ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
 +^$ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
 +^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $|$ \frac{1}{2}+\frac{i}{2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{24}+\frac{i}{24}\right) \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(e^{6 i \phi } \sin ^3(\theta )-(1-i) e^{3 i \phi } \cos (\theta ) \left(5 \cos ^2(\theta )-3\right)-i \sin ^3(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(x^3-3 x^2 y-3 x y^2+y^3-5 z^3+3 z\right)$$ | ::: |
 +^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )-\cos (2 \phi )))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 x^2 z-10 x y z-5 x z^2+x-5 y^2 z\right)$$ | ::: |
 +^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \sin (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi )))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 x^2 z-10 x y z+5 y^2 z-5 y z^2+y\right)$$ | ::: |
 +^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_4.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{48}+\frac{i}{48}\right) \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \left(5 \left(e^{6 i \phi }-i\right) \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 x^3-15 x^2 y-15 x y^2+5 y^3+4 z \left(5 z^2-3\right)\right)$$ | ::: |
 +^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_5.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+\sin (2 \theta ) (\cos (2 \phi )-\sin (2 \phi )))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 (-z)+2 x y z-5 x z^2+x+y^2 z\right)$$ | ::: |
 +^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_6.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi ))-(5 \cos (2 \theta )+3) \sin (\phi ))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 z+2 x y z-y^2 z-5 y z^2+y\right)$$ | ::: |
 +^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_111z_orb_3_7.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin ^3(\theta ) (\sin (3 \phi )+\cos (3 \phi ))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^3+3 x^2 y-3 x y^2-y^3\right)$$ | ::: |
 +
 +
 +###
 +
 +</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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + 0 & k\neq 4\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
 + \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{15} \text{Mt1u} & k=4\land m=-3 \\
 + \frac{\sqrt{21} \text{Mt1u}}{2} & k=4\land m=0 \\
 + \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{15} \text{Mt1u} & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -3 && m != 0 && m != 3)}, {(1/2 - I/2)*Sqrt[15]*Mt1u, k == 4 && m == -3}, {(Sqrt[21]*Mt1u)/2, k == 4 && m == 0}}, (-1/2 - I/2)*Sqrt[15]*Mt1u]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_Oh_111z.Quanty>
 +
 +Akm = {{4, 0, (1/2)*((sqrt(21))*(Mt1u))} , 
 +       {4, 3, (-1/2+-1/2*I)*((sqrt(15))*(Mt1u))} , 
 +       {4,-3, (1/2+-1/2*I)*((sqrt(15))*(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 $|$ \text{Mt1u} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right] $|$ 0 $|
 +^$ {Y_{0}^{(1)}} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} \text{Mt1u} $|$ 0 $|$ 0 $|$ \frac{2 \text{Mt1u}}{3} $|$ 0 $|$ 0 $|$ \left(-\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} \text{Mt1u} $|
 +^$ {Y_{1}^{(1)}} $|$ 0 $|$ \text{Mt1u} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right] $|$ 0 $|$ 0 $|$ -\frac{\text{Mt1u}}{\sqrt{6}} $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $  ^  $ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $  ^  $ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $  ^
 +^$ p_x $|$ 0 $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \text{Mt1u} \left(i \sqrt{15} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]+\sqrt{15} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]+(1-i)\right) $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} \text{Mt1u} \left(\text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]\right) $|$ 0 $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \text{Mt1u} \left(\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]+(1+i) \sqrt{5}\right) $|$ -\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \text{Mt1u} \left(\text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]\right)}{\sqrt{3}} $|$ 0 $|
 +^$ p_y $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} \text{Mt1u} \left(\text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]\right) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \text{Mt1u} \left(i \sqrt{15} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]+\sqrt{15} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]+(1-i)\right) $|$ 0 $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \text{Mt1u} \left(\text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]\right)}{\sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \text{Mt1u} \left(\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+25\$|$,3\right]+(1+i) \sqrt{5}\right) $|$ 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> | | | | | |
 +
 +###
Print/export