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physics_chemistry:point_groups:d3d:orientation_111 [2018/03/21 18:36]
Stefano Agrestini created
physics_chemistry:point_groups:d3d:orientation_111 [2018/09/06 12:59] (current)
Maurits W. Haverkort
Line 1: Line 1:
 +~~CLOSETOC~~
 +
 ====== Orientation 111 ====== ====== Orientation 111 ======
  
 ### ###
-alligned paragraph text+ 
 +This orientation is non-standard,​ but related to the orientation of the Oh pointgroup, which normally would be orrientated with the C3 axes in the 111 direction. We only show one of the options of the D3d subgroups of the Oh group with orientation XYZ. 
 ### ###
  
-===== Example ​=====+===== Symmetry Operations ​=====
  
 ### ###
-description text+ 
 +In the D3d Point Group, with orientation 111 there are the following symmetry operations 
 ### ###
  
-==== Input ==== +### 
-<code Quanty ​Example.Quanty>​ + 
--- some example code+{{:​physics_chemistry:​pointgroup:​d3d_111.png}} 
 + 
 +### 
 + 
 +### 
 + 
 +^ Operator ^ Orientation ^ 
 +^ $\text{E}$ | $\{0,0,0\}$ , | 
 +^ $C_3$ | $\{1,1,1\}$ , $\{-1,​-1,​-1\}$ , | 
 +^ $C_2$ | $\{1,​-1,​0\}$ , $\{0,​1,​-1\}$ , $\{1,​0,​-1\}$ , | 
 +^ $\text{i}$ | $\{0,0,0\}$ , | 
 +^ $S_6$ | $\{1,1,1\}$ , $\{-1,​-1,​-1\}$ , | 
 +^ $\sigma _d$ | $\{1,​-1,​0\}$ , $\{0,​1,​-1\}$ , $\{1,​0,​-1\}$ , | 
 + 
 +### 
 + 
 +===== Different Settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_111|Point Group D3d with orientation 111]] 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zx|Point Group D3d with orientation Zx]] 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zx_a|Point Group D3d with orientation Zx_A]] 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zx_b|Point Group D3d with orientation Zx_B]] 
 +  * [[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:​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]] 
 + 
 +### 
 + 
 +===== Character Table ===== 
 + 
 +### 
 + 
 +|  $  $  ^  $ \text{E} \,​{\text{(1)}} $  ^  $ C_3 \,​{\text{(2)}} $  ^  $ C_2 \,​{\text{(3)}} $  ^  $ \text{i} \,​{\text{(1)}} $  ^  $ S_6 \,​{\text{(2)}} $  ^  $ \sigma_d \,​{\text{(3)}} $  ^ 
 +^ $ A_{1g} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ | 
 +^ $ A_{2g} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ | 
 +^ $ E_g $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 2 $ |  $ -1 $ |  $ 0 $ | 
 +^ $ A_{1u} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ | 
 +^ $ A_{2u} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ | 
 +^ $ E_u $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ -2 $ |  $ 1 $ |  $ 0 $ | 
 + 
 +### 
 + 
 +===== Product Table ===== 
 + 
 +### 
 + 
 +|  $  $  ^  $ A_{1g} $  ^  $ A_{2g} $  ^  $ E_g $  ^  $ A_{1u} $  ^  $ A_{2u} $  ^  $ E_u $  ^ 
 +^ $ A_{1g} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | 
 +^ $ A_{2g} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | 
 +^ $ E_g $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | 
 +^ $ A_{1u} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | 
 +^ $ A_{2u} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | 
 +^ $ E_u $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | 
 + 
 +### 
 + 
 +===== Sub Groups with compatible settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:​point_groups:​c1:​orientation_1|Point Group C1 with orientation 1]] 
 +  * [[physics_chemistry:​point_groups:​ci:​orientation_|Point Group Ci with orientation ]] 
 + 
 +### 
 + 
 +===== Super Groups with compatible settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:​point_groups:​oh:​orientation_xyz|Point Group Oh with orientation XYZ]] 
 + 
 +### 
 + 
 +===== Invariant Potential expanded on renormalized spherical Harmonics ===== 
 + 
 +### 
 + 
 +Any potential (function) can be written as a sum over spherical harmonics. 
 +$$V(r,​\theta,​\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,​\phi)$$ 
 +Here $A_{k,​m}(r)$ is a radial function and $C^{(m)}_k(\theta,​\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,​\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,​\phi)$$ 
 +The presence of symmetry induces relations between the expansion coefficients such that $V(r,​\theta,​\phi)$ is invariant under all symmetry operations. For the D3d Point group with orientation 111 the form of the expansion coefficients is: 
 + 
 +### 
 + 
 +==== Expansion ==== 
 + 
 +### 
 + 
 + ​$$A_{k,​m} = \begin{cases} 
 + ​A(0,​0) & k=0\land m=0 \\ 
 + -i A(2,1) & k=2\land m=-2 \\ 
 + ​(-1-i) A(2,1) & k=2\land m=-1 \\ 
 + (1-i) A(2,1) & k=2\land m=1 \\ 
 + i A(2,1) & k=2\land m=2 \\ 
 + ​\sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ 
 + ​(-1+i) \sqrt{7} A(4,1) & k=4\land m=-3 \\ 
 + 2 i \sqrt{2} A(4,1) & k=4\land m=-2 \\ 
 + ​(-1-i) A(4,1) & k=4\land m=-1 \\ 
 + ​A(4,​0) & k=4\land m=0 \\ 
 + (1-i) A(4,1) & k=4\land m=1 \\ 
 + -2 i \sqrt{2} A(4,1) & k=4\land m=2 \\ 
 + (1+i) \sqrt{7} A(4,1) & k=4\land m=3 \\ 
 + ​-\frac{1}{33} i \left(8 \sqrt{22} A(6,​1)-\sqrt{55} B(6,​2)\right) & k=6\land m=-6 \\ 
 + ​-\frac{(1+i) \left(A(6,​1)+2 \sqrt{10} B(6,​2)\right)}{\sqrt{66}} & k=6\land m=-5 \\ 
 + ​-\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ 
 + ​\left(\frac{1}{6}-\frac{i}{6}\right) \left(\sqrt{10} A(6,1)-4 B(6,​2)\right) & k=6\land m=-3 \\ 
 + -i B(6,2) & k=6\land m=-2 \\ 
 + ​(-1-i) A(6,1) & k=6\land m=-1 \\ 
 + ​A(6,​0) & k=6\land m=0 \\ 
 + (1-i) A(6,1) & k=6\land m=1 \\ 
 + i B(6,2) & k=6\land m=2 \\ 
 + ​\left(-\frac{1}{6}-\frac{i}{6}\right) \left(\sqrt{10} A(6,1)-4 B(6,​2)\right) & k=6\land m=3 \\ 
 + ​\frac{(1-i) \left(A(6,​1)+2 \sqrt{10} B(6,​2)\right)}{\sqrt{66}} & k=6\land m=5 \\ 
 + ​\frac{1}{33} i \left(8 \sqrt{22} A(6,​1)-\sqrt{55} B(6,​2)\right) & k=6\land m=6 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +==== Input format suitable for Mathematica (Quanty.nb) ​==== 
 + 
 +### 
 + 
 +<code Quanty ​Akm_D3d_111.Quanty.nb
 + 
 +Akm[k_,​m_]:​=Piecewise[{{A[0,​ 0], k == 0 && m == 0}, {(-I)*A[2, 1], k == 2 && m == -2}, {(-1 - I)*A[2, 1], k == 2 && m == -1}, {(1 - I)*A[2, 1], k == 2 && m == 1}, {I*A[2, 1], k == 2 && m == 2}, {Sqrt[5/​14]*A[4,​ 0], k == 4 && (m == -4 || m == 4)}, {(-1 + I)*Sqrt[7]*A[4,​ 1], k == 4 && m == -3}, {(2*I)*Sqrt[2]*A[4,​ 1], k == 4 && m == -2}, {(-1 - I)*A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {(1 - I)*A[4, 1], k == 4 && m == 1}, {(-2*I)*Sqrt[2]*A[4,​ 1], k == 4 && m == 2}, {(1 + I)*Sqrt[7]*A[4,​ 1], k == 4 && m == 3}, {(-I/​33)*(8*Sqrt[22]*A[6,​ 1] - Sqrt[55]*B[6,​ 2]), k == 6 && m == -6}, {((-1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6,​ 2]))/​Sqrt[66],​ k == 6 && m == -5}, {-(Sqrt[7/​2]*A[6,​ 0]), k == 6 && (m == -4 || m == 4)}, {(1/6 - I/​6)*(Sqrt[10]*A[6,​ 1] - 4*B[6, 2]), k == 6 && m == -3}, {(-I)*B[6, 2], k == 6 && m == -2}, {(-1 - I)*A[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {(1 - I)*A[6, 1], k == 6 && m == 1}, {I*B[6, 2], k == 6 && m == 2}, {(-1/6 - I/​6)*(Sqrt[10]*A[6,​ 1] - 4*B[6, 2]), k == 6 && m == 3}, {((1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6,​ 2]))/​Sqrt[66],​ k == 6 && m == 5}, {(I/​33)*(8*Sqrt[22]*A[6,​ 1] - Sqrt[55]*B[6,​ 2]), 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_D3d_111.Quanty>​
 +
 +Akm = {{0, 0, A(0,0)} , 
 +       ​{2,​-1,​ (-1+-1*I)*(A(2,​1))} , 
 +       {2, 1, (1+-1*I)*(A(2,​1))} , 
 +       ​{2,​-2,​ (-I)*(A(2,​1))} , 
 +       {2, 2, (I)*(A(2,​1))} , 
 +       {4, 0, A(4,0)} , 
 +       ​{4,​-1,​ (-1+-1*I)*(A(4,​1))} , 
 +       {4, 1, (1+-1*I)*(A(4,​1))} , 
 +       {4, 2, (-2*I)*((sqrt(2))*(A(4,​1)))} , 
 +       ​{4,​-2,​ (2*I)*((sqrt(2))*(A(4,​1)))} , 
 +       ​{4,​-3,​ (-1+1*I)*((sqrt(7))*(A(4,​1)))} , 
 +       {4, 3, (1+1*I)*((sqrt(7))*(A(4,​1)))} , 
 +       ​{4,​-4,​ (sqrt(5/​14))*(A(4,​0))} , 
 +       {4, 4, (sqrt(5/​14))*(A(4,​0))} , 
 +       {6, 0, A(6,0)} , 
 +       ​{6,​-1,​ (-1+-1*I)*(A(6,​1))} , 
 +       {6, 1, (1+-1*I)*(A(6,​1))} , 
 +       ​{6,​-2,​ (-I)*(B(6,​2))} , 
 +       {6, 2, (I)*(B(6,​2))} , 
 +       {6, 3, (-1/​6+-1/​6*I)*((sqrt(10))*(A(6,​1)) + (-4)*(B(6,​2)))} , 
 +       ​{6,​-3,​ (1/​6+-1/​6*I)*((sqrt(10))*(A(6,​1)) + (-4)*(B(6,​2)))} , 
 +       ​{6,​-4,​ (-1)*((sqrt(7/​2))*(A(6,​0)))} , 
 +       {6, 4, (-1)*((sqrt(7/​2))*(A(6,​0)))} , 
 +       ​{6,​-5,​ (-1+-1*I)*((1/​(sqrt(66)))*(A(6,​1) + (2)*((sqrt(10))*(B(6,​2)))))} , 
 +       {6, 5, (1+-1*I)*((1/​(sqrt(66)))*(A(6,​1) + (2)*((sqrt(10))*(B(6,​2)))))} , 
 +       ​{6,​-6,​ (-1/​33*I)*((8)*((sqrt(22))*(A(6,​1))) + (-1)*((sqrt(55))*(B(6,​2))))} , 
 +       {6, 6, (1/​33*I)*((8)*((sqrt(22))*(A(6,​1))) + (-1)*((sqrt(55))*(B(6,​2))))} }
 +
 +</​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 }$|$ \frac{i \text{Asd}(2,​1)}{\sqrt{5}} $|$ -\frac{(1-i) \text{Asd}(2,​1)}{\sqrt{5}} $|$ 0 $|$ \frac{(1+i) \text{Asd}(2,​1)}{\sqrt{5}} $|$ -\frac{i \text{Asd}(2,​1)}{\sqrt{5}} $|$\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) $|$ \left(-\frac{1}{5}-\frac{i}{5}\right) \sqrt{3} \text{App}(2,​1) $|$ \frac{1}{5} i \sqrt{6} \text{App}(2,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{3 i \text{Apf}(2,​1)}{\sqrt{35}}+\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,​1) $|$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,​1)}{\sqrt{7}}-(1-i) \sqrt{\frac{6}{35}} \text{Apf}(2,​1) $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0) $|$ \frac{\left(\frac{3}{5}+\frac{3 i}{5}\right) \text{Apf}(2,​1)}{\sqrt{7}}-\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,​1) $|$ -\frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,​1)-\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,​1) $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,​1) $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,​0) $|
 +^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{5}+\frac{i}{5}\right) \sqrt{3} \text{App}(2,​1) $|$ \text{App}(0,​0) $|$ \left(\frac{1}{5}+\frac{i}{5}\right) \sqrt{3} \text{App}(2,​1) $|$\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{7}{3}} \text{Apf}(4,​1) $|$ i \sqrt{\frac{3}{35}} \text{Apf}(2,​1)-\frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,​1) $|$ \left(-\frac{2}{5}+\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,​1)-\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,​1) $|$ \frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}} $|$ \left(\frac{2}{5}+\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,​1)+\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,​1) $|$ \frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,​1)-i \sqrt{\frac{3}{35}} \text{Apf}(2,​1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,​1) $|
 +^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} i \sqrt{6} \text{App}(2,​1) $|$ \left(\frac{1}{5}-\frac{i}{5}\right) \sqrt{3} \text{App}(2,​1) $|$ \text{App}(0,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,​0) $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,​1) $|$ \frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,​1)+\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,​1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,​1)-\frac{\left(\frac{3}{5}-\frac{3 i}{5}\right) \text{Apf}(2,​1)}{\sqrt{7}} $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0) $|$ (1+i) \sqrt{\frac{6}{35}} \text{Apf}(2,​1)-\frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,​1)}{\sqrt{7}} $|$ -\frac{3 i \text{Apf}(2,​1)}{\sqrt{35}}-\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,​1) $|
 +^$ {Y_{-2}^{(2)}} $|$ -\frac{i \text{Asd}(2,​1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,​0)+\frac{1}{21} \text{Add}(4,​0) $|$ \left(\frac{1}{21}+\frac{i}{21}\right) \sqrt{5} \text{Add}(4,​1)-\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{6} \text{Add}(2,​1) $|$ \frac{2}{7} i \text{Add}(2,​1)+\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{5} \text{Add}(4,​1) $|$ \frac{5}{21} \text{Add}(4,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{-1}^{(2)}} $|$ -\frac{(1+i) \text{Asd}(2,​1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{21}-\frac{i}{21}\right) \sqrt{5} \text{Add}(4,​1)-\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{6} \text{Add}(2,​1) $|$ \text{Add}(0,​0)-\frac{4}{21} \text{Add}(4,​0) $|$ \left(-\frac{1}{7}-\frac{i}{7}\right) \text{Add}(2,​1)-\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \frac{1}{7} i \sqrt{6} \text{Add}(2,​1)-\frac{8}{21} i \sqrt{5} \text{Add}(4,​1) $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{5} \text{Add}(4,​1) $|$\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 }$|$ -\frac{2}{7} i \text{Add}(2,​1)-\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \left(-\frac{1}{7}+\frac{i}{7}\right) \text{Add}(2,​1)-\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \text{Add}(0,​0)+\frac{2}{7} \text{Add}(4,​0) $|$ \left(\frac{1}{7}+\frac{i}{7}\right) \text{Add}(2,​1)+\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \frac{2}{7} i \text{Add}(2,​1)+\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{1}^{(2)}} $|$ \frac{(1-i) \text{Asd}(2,​1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{5} \text{Add}(4,​1) $|$ \frac{8}{21} i \sqrt{5} \text{Add}(4,​1)-\frac{1}{7} i \sqrt{6} \text{Add}(2,​1) $|$ \left(\frac{1}{7}-\frac{i}{7}\right) \text{Add}(2,​1)+\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \text{Add}(0,​0)-\frac{4}{21} \text{Add}(4,​0) $|$ \left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{6} \text{Add}(2,​1)-\left(\frac{1}{21}+\frac{i}{21}\right) \sqrt{5} \text{Add}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{2}^{(2)}} $|$ \frac{i \text{Asd}(2,​1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{5}{21} \text{Add}(4,​0) $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{5} \text{Add}(4,​1) $|$ -\frac{2}{7} i \text{Add}(2,​1)-\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,​1) $|$ \left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{6} \text{Add}(2,​1)-\left(\frac{1}{21}-\frac{i}{21}\right) \sqrt{5} \text{Add}(4,​1) $|$ \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 }$|$ -\frac{3 i \text{Apf}(2,​1)}{\sqrt{35}}-\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,​1) $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,​1) $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,​0)+\frac{1}{11} \text{Aff}(4,​0)-\frac{5}{429} \text{Aff}(6,​0) $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \text{Aff}(2,​1)+\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,​1)-\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,​1) $|$ \frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,​1)+\frac{4}{11} i \sqrt{3} \text{Aff}(4,​1)+\frac{10}{429} i \sqrt{7} \text{Bff}(6,​2) $|$ \left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right)+\left(\frac{7}{11}-\frac{7 i}{11}\right) \text{Aff}(4,​1) $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,​0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,​0) $|$ \left(-\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,​1)+2 \sqrt{10} \text{Bff}(6,​2)\right) $|$ \frac{10}{429} i \sqrt{\frac{7}{33}} \left(8 \sqrt{22} \text{Aff}(6,​1)-\sqrt{55} \text{Bff}(6,​2)\right) $|
 +^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,​1)}{\sqrt{7}}-(1+i) \sqrt{\frac{6}{35}} \text{Apf}(2,​1) $|$ \frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,​1)-i \sqrt{\frac{3}{35}} \text{Apf}(2,​1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \text{Aff}(2,​1)+\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,​1)-\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,​1) $|$ \text{Aff}(0,​0)-\frac{7}{33} \text{Aff}(4,​0)+\frac{10}{143} \text{Aff}(6,​0) $|$ -\frac{(1+i) \text{Aff}(2,​1)}{\sqrt{15}}-\left(\frac{4}{33}+\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,​1)+\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,​1) $|$ \frac{2 i \text{Aff}(2,​1)}{3 \sqrt{5}}-\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,​1)-\frac{20}{429} i \sqrt{14} \text{Bff}(6,​2) $|$ \left(\frac{7}{33}-\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,​1)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right) $|$ \frac{5}{33} \text{Aff}(4,​0)-\frac{70}{143} \text{Aff}(6,​0) $|$ \left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,​1)+2 \sqrt{10} \text{Bff}(6,​2)\right) $|
 +^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0) $|$ \left(-\frac{2}{5}-\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,​1)-\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,​1) $|$ -\frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,​1)-\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,​1)-\frac{4}{11} i \sqrt{3} \text{Aff}(4,​1)-\frac{10}{429} i \sqrt{7} \text{Bff}(6,​2) $|$ -\frac{(1-i) \text{Aff}(2,​1)}{\sqrt{15}}-\left(\frac{4}{33}-\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,​1)+\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,​1) $|$ \text{Aff}(0,​0)+\frac{1}{33} \text{Aff}(4,​0)-\frac{25}{143} \text{Aff}(6,​0) $|$ \left(-\frac{1}{15}-\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,​1)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,​1)-\left(\frac{25}{429}+\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,​1) $|$ \frac{2}{5} i \sqrt{\frac{2}{3}} \text{Aff}(2,​1)-\frac{8}{33} i \sqrt{5} \text{Aff}(4,​1)+\frac{10}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ \left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right)-\left(\frac{7}{33}-\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,​1) $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,​0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,​0) $|
 +^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{\left(\frac{3}{5}-\frac{3 i}{5}\right) \text{Apf}(2,​1)}{\sqrt{7}}-\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,​1) $|$ \frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}} $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,​1)-\frac{\left(\frac{3}{5}+\frac{3 i}{5}\right) \text{Apf}(2,​1)}{\sqrt{7}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right)+\left(\frac{7}{11}+\frac{7 i}{11}\right) \text{Aff}(4,​1) $|$ -\frac{2 i \text{Aff}(2,​1)}{3 \sqrt{5}}+\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,​1)+\frac{20}{429} i \sqrt{14} \text{Bff}(6,​2) $|$ \left(-\frac{1}{15}+\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,​1)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,​1)-\left(\frac{25}{429}-\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,​1) $|$ \text{Aff}(0,​0)+\frac{2}{11} \text{Aff}(4,​0)+\frac{100}{429} \text{Aff}(6,​0) $|$ \left(\frac{1}{15}+\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,​1)+\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,​1)+\left(\frac{25}{429}+\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,​1) $|$ \frac{2 i \text{Aff}(2,​1)}{3 \sqrt{5}}-\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,​1)-\frac{20}{429} i \sqrt{14} \text{Bff}(6,​2) $|$ \left(-\frac{7}{11}+\frac{7 i}{11}\right) \text{Aff}(4,​1)-\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right) $|
 +^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,​1)+\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,​1) $|$ \left(\frac{2}{5}-\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,​1)+\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,​1) $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,​0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,​0) $|$ \left(\frac{7}{33}+\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,​1)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right) $|$ -\frac{2}{5} i \sqrt{\frac{2}{3}} \text{Aff}(2,​1)+\frac{8}{33} i \sqrt{5} \text{Aff}(4,​1)-\frac{10}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ \left(\frac{1}{15}-\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,​1)+\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,​1)+\left(\frac{25}{429}-\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,​1) $|$ \text{Aff}(0,​0)+\frac{1}{33} \text{Aff}(4,​0)-\frac{25}{143} \text{Aff}(6,​0) $|$ \frac{(1+i) \text{Aff}(2,​1)}{\sqrt{15}}+\left(\frac{4}{33}+\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,​1)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,​1) $|$ \frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,​1)+\frac{4}{11} i \sqrt{3} \text{Aff}(4,​1)+\frac{10}{429} i \sqrt{7} \text{Bff}(6,​2) $|
 +^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,​1) $|$ i \sqrt{\frac{3}{35}} \text{Apf}(2,​1)-\frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,​1) $|$ (1-i) \sqrt{\frac{6}{35}} \text{Apf}(2,​1)-\frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,​1)}{\sqrt{7}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,​1)+2 \sqrt{10} \text{Bff}(6,​2)\right) $|$ \frac{5}{33} \text{Aff}(4,​0)-\frac{70}{143} \text{Aff}(6,​0) $|$ \left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right)-\left(\frac{7}{33}+\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,​1) $|$ -\frac{2 i \text{Aff}(2,​1)}{3 \sqrt{5}}+\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,​1)+\frac{20}{429} i \sqrt{14} \text{Bff}(6,​2) $|$ \frac{(1-i) \text{Aff}(2,​1)}{\sqrt{15}}+\left(\frac{4}{33}-\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,​1)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,​1) $|$ \text{Aff}(0,​0)-\frac{7}{33} \text{Aff}(4,​0)+\frac{10}{143} \text{Aff}(6,​0) $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \text{Aff}(2,​1)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,​1)+\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,​1) $|
 +^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,​0) $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,​1) $|$ \frac{3 i \text{Apf}(2,​1)}{\sqrt{35}}+\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{429} i \sqrt{\frac{7}{33}} \left(8 \sqrt{22} \text{Aff}(6,​1)-\sqrt{55} \text{Bff}(6,​2)\right) $|$ \left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,​1)+2 \sqrt{10} \text{Bff}(6,​2)\right) $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,​0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,​0) $|$ \left(-\frac{7}{11}-\frac{7 i}{11}\right) \text{Aff}(4,​1)-\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,​1)-4 \text{Bff}(6,​2)\right) $|$ -\frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,​1)-\frac{4}{11} i \sqrt{3} \text{Aff}(4,​1)-\frac{10}{429} i \sqrt{7} \text{Bff}(6,​2) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \text{Aff}(2,​1)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,​1)+\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,​1) $|$ \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+y+z} $|$\color{darkred}{ 0 }$|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1-i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_{x-y} $|$\color{darkred}{ 0 }$|$ \frac{1}{2}-\frac{i}{2} $|$ 0 $|$ -\frac{1}{2}-\frac{i}{2} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_{3z-r} $|$\color{darkred}{ 0 }$|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \sqrt{\frac{2}{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 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_{\text{yz}-\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|$ 0 $|$ \frac{1}{2}+\frac{i}{2} $|$ 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 }$|$ \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_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
 +^$ f_{x^3+y^3+z^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{3}} $|$ \frac{1}{4}-\frac{i}{4} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|
 +^$ f_{x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|
 +^$ f_{2z^3-x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \sqrt{\frac{2}{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|
 +^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{6}} $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{4}-\frac{i}{4} $|
 +^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \frac{1}{\sqrt{3}} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $|
 +^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|
 +
 +
 +###
 +
 +==== One particle coupling on a basis of symmetry adapted functions ====
 +
 +###
 +
 +After rotation we find
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ \text{s} $  ^  $ p_{x+y+z} $  ^  $ p_{x-y} $  ^  $ p_{3z-r} $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^  $ d_{\text{yz}-\text{xz}} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ f_{\text{xyz}} $  ^  $ f_{x^3+y^3+z^3} $  ^  $ f_{x^3-y^3} $  ^  $ f_{2z^3-x^3-y^3} $  ^  $ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $  ^  $ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $  ^  $ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $  ^
 +^$ \text{s} $|$ \text{Ass}(0,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{5}} \text{Asd}(2,​1) $|$ 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+y+z} $|$\color{darkred}{ 0 }$|$ \text{App}(0,​0)-\frac{2}{5} \sqrt{6} \text{App}(2,​1) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{8 \text{Apf}(4,​1)}{\sqrt{21}}-3 \sqrt{\frac{2}{35}} \text{Apf}(2,​1) $|$ \frac{6}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​1)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}-\frac{4}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​1) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_{x-y} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,​0)+\frac{1}{5} \sqrt{6} \text{App}(2,​1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​1)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​1) $|$ 0 $|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,​1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,​1) $|$ 0 $|
 +^$ p_{3z-r} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,​0)+\frac{1}{5} \sqrt{6} \text{App}(2,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​1)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​1) $|$ 0 $|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,​1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,​1) $|
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ -\sqrt{\frac{6}{5}} \text{Asd}(2,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,​0)-\frac{2}{7} \sqrt{6} \text{Add}(2,​1)-\frac{4}{21} \text{Add}(4,​0)+\frac{16}{21} \sqrt{5} \text{Add}(4,​1) $|$ 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}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,​0)+\frac{1}{7} \sqrt{6} \text{Add}(2,​1)-\frac{4}{21} \text{Add}(4,​0)-\frac{8}{21} \sqrt{5} \text{Add}(4,​1) $|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,​1)+\frac{2}{7} \sqrt{10} \text{Add}(4,​1) $|$ 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 $|$ \text{Add}(0,​0)+\frac{1}{7} \sqrt{6} \text{Add}(2,​1)-\frac{4}{21} \text{Add}(4,​0)-\frac{8}{21} \sqrt{5} \text{Add}(4,​1) $|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,​1)+\frac{2}{7} \sqrt{10} \text{Add}(4,​1) $|$\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} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,​1)+\frac{2}{7} \sqrt{10} \text{Add}(4,​1) $|$ 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 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,​1)+\frac{2}{7} \sqrt{10} \text{Add}(4,​1) $|$ 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_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ \frac{8 \text{Apf}(4,​1)}{\sqrt{21}}-3 \sqrt{\frac{2}{35}} \text{Apf}(2,​1) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,​0)-\frac{4}{11} \text{Aff}(4,​0)+\frac{80}{143} \text{Aff}(6,​0) $|$ 2 \sqrt{\frac{2}{15}} \text{Aff}(2,​1)-\frac{4}{11} \text{Aff}(4,​1)-\frac{40}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x^3+y^3+z^3} $|$\color{darkred}{ 0 }$|$ \frac{6}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​1)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}-\frac{4}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​1) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 2 \sqrt{\frac{2}{15}} \text{Aff}(2,​1)-\frac{4}{11} \text{Aff}(4,​1)-\frac{40}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,​2) $|$ \text{Aff}(0,​0)+\frac{1}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,​1)+\frac{2}{11} \text{Aff}(4,​0)+\frac{8}{11} \sqrt{5} \text{Aff}(4,​1)+\frac{100}{429} \text{Aff}(6,​0)+\frac{100}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,​1)-\frac{20}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​1)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,​0)-\frac{\text{Aff}(2,​1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,​0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,​1)+\frac{100}{429} \text{Aff}(6,​0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,​1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,​1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,​1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,​1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,​2) $|$ 0 $|
 +^$ f_{2z^3-x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​1)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,​0)-\frac{\text{Aff}(2,​1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,​0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,​1)+\frac{100}{429} \text{Aff}(6,​0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,​1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,​1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,​1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,​1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,​2) $|
 +^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$\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)+\sqrt{\frac{2}{3}} \text{Aff}(2,​1)-\frac{2}{33} \text{Aff}(4,​0)+\frac{8}{33} \sqrt{5} \text{Aff}(4,​1)-\frac{60}{143} \text{Aff}(6,​0)-\frac{20}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,​1)+\frac{20}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|
 +^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,​1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,​1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,​1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,​1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,​1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,​0)-\frac{\text{Aff}(2,​1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,​0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,​1)-\frac{60}{143} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,​1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|$ 0 $|
 +^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,​1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,​1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,​1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,​1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,​1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,​2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,​0)-\frac{\text{Aff}(2,​1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,​0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,​1)-\frac{60}{143} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,​1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,​2) $|
 +
 +
 +###
 +
 +===== 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_D3d_111.Quanty.nb>​
 +
 +Akm[k_,​m_]:​=Piecewise[{{Ea1g,​ k == 0 && m == 0}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.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:​d3d_111_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** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + ​\frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\
 + ​\frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=-2 \\
 + ​\frac{\left(\frac{5}{3}+\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=-1 \\
 + ​-\frac{\left(\frac{5}{3}-\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=1 \\
 + ​-\frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=2
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty.nb>​
 +
 +Akm[k_,​m_]:​=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(((5*I)/​3)*(Ea2u - Eeu))/​Sqrt[6],​ k == 2 && m == -2}, {((5/3 + (5*I)/​3)*(Ea2u - Eeu))/​Sqrt[6],​ k == 2 && m == -1}, {((-5/3 + (5*I)/​3)*(Ea2u - Eeu))/​Sqrt[6],​ k == 2 && m == 1}, {(((-5*I)/​3)*(Ea2u - Eeu))/​Sqrt[6],​ k == 2 && m == 2}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty>​
 +
 +Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , 
 +       {2, 1, (-5/​3+5/​3*I)*((1/​(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
 +       ​{2,​-1,​ (5/​3+5/​3*I)*((1/​(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
 +       {2, 2, (-5/​3*I)*((1/​(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
 +       ​{2,​-2,​ (5/​3*I)*((1/​(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} }
 +
 +</​code>​
 +
 +###
 +
 +</​hidden>​
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
 +^$ {Y_{-1}^{(1)}} $|$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $|$ \frac{1}{3} \sqrt[4]{-1} (\text{Ea2u}-\text{Eeu}) $|$ -\frac{1}{3} i (\text{Ea2u}-\text{Eeu}) $|
 +^$ {Y_{0}^{(1)}} $|$ \frac{1}{3} (-1)^{3/4} (\text{Eeu}-\text{Ea2u}) $|$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $|$ \frac{1}{3} \sqrt[4]{-1} (\text{Eeu}-\text{Ea2u}) $|
 +^$ {Y_{1}^{(1)}} $|$ \frac{1}{3} i (\text{Ea2u}-\text{Eeu}) $|$ \frac{1}{3} (-1)^{3/4} (\text{Ea2u}-\text{Eeu}) $|$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ p_{x+y+z} $  ^  $ p_{x-y} $  ^  $ p_{3z-r} $  ^
 +^$ p_{x+y+z} $|$ \text{Ea2u} $|$ 0 $|$ 0 $|
 +^$ p_{x-y} $|$ 0 $|$ \text{Eeu} $|$ 0 $|
 +^$ p_{3z-r} $|$ 0 $|$ 0 $|$ \text{Eeu} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
 +^$ p_{x+y+z} $|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1-i}{\sqrt{6}} $|
 +^$ p_{x-y} $|$ \frac{1}{2}-\frac{i}{2} $|$ 0 $|$ -\frac{1}{2}-\frac{i}{2} $|
 +^$ p_{3z-r} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \sqrt{\frac{2}{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_1_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta )}{2 \sqrt{\pi }}$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{x+y+z}{2 \sqrt{\pi }}$$ | ::: |
 +^ ^$$\text{Eeu}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_1_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi ))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{2 \pi }} (x-y)$$ | ::: |
 +^ ^$$\text{Eeu}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_1_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta )}{2 \sqrt{2 \pi }}$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{x+y-2 z}{2 \sqrt{2 \pi }}$$ | ::: |
 +
 +
 +###
 +
 +</​hidden>​
 +==== Potential for d orbitals ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + ​\frac{1}{5} (\text{Ea1g}+2 (\text{Eeg$\pi $}+\text{Eeg$\sigma $})) & k=0\land m=0 \\
 + ​\frac{1}{6} i \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg$\pi $}-4 \sqrt{3} \text{Meg}\right) & k=2\land m=-2 \\
 + ​\left(\frac{1}{6}+\frac{i}{6}\right) \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg$\pi $}-4 \sqrt{3} \text{Meg}\right) & k=2\land m=-1 \\
 + ​\left(\frac{1}{6}-\frac{i}{6}\right) \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg$\pi $}+4 \sqrt{3} \text{Meg}\right) & k=2\land m=1 \\
 + ​\frac{1}{6} i \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg$\pi $}+4 \sqrt{3} \text{Meg}\right) & k=2\land m=2 \\
 + ​-\frac{1}{2} \sqrt{\frac{7}{10}} (\text{Ea1g}+2 \text{Eeg$\pi $}-3 \text{Eeg$\sigma $}) & k=4\land (m=-4\lor m=4) \\
 + ​\left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) & k=4\land m=-3 \\
 + ​\frac{i \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=-2 \\
 + ​-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=-1 \\
 + ​-\frac{7}{10} (\text{Ea1g}+2 \text{Eeg$\pi $}-3 \text{Eeg$\sigma $}) & k=4\land m=0 \\
 + ​\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=1 \\
 + ​-\frac{i \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=2 \\
 + ​\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) & k=4\land m=3
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty.nb>​
 +
 +Akm[k_,​m_]:​=Piecewise[{{(Ea1g + 2*(Eeg\[Pi] + Eeg\[Sigma]))/​5,​ k == 0 && m == 0}, {(I/​6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg),​ k == 2 && m == -2}, {(1/6 + I/​6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg),​ k == 2 && m == -1}, {(1/6 - I/​6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg),​ k == 2 && m == 1}, {(I/​6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg),​ k == 2 && m == 2}, {-(Sqrt[7/​10]*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/​2,​ k == 4 && (m == -4 || m == 4)}, {(-1/4 + I/​4)*Sqrt[7/​5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg),​ k == 4 && m == -3}, {(I*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/​Sqrt[10],​ k == 4 && m == -2}, {((-1/4 - I/​4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/​Sqrt[5],​ k == 4 && m == -1}, {(-7*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/​10,​ k == 4 && m == 0}, {((1/4 - I/​4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/​Sqrt[5],​ k == 4 && m == 1}, {((-I)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/​Sqrt[10],​ k == 4 && m == 2}, {(1/4 + I/​4)*Sqrt[7/​5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg),​ k == 4 && m == 3}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty>​
 +
 +Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg\[Pi] + Eeg\[Sigma]))} , 
 +       {2, 1, (1/​6+-1/​6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , 
 +       ​{2,​-1,​ (1/​6+1/​6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , 
 +       {2, 2, (1/​6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , 
 +       ​{2,​-2,​ (1/​6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , 
 +       {4, 0, (-7/​10)*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma]))} , 
 +       ​{4,​-1,​ (-1/​4+-1/​4*I)*((1/​(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
 +       {4, 1, (1/​4+-1/​4*I)*((1/​(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
 +       {4, 2, (-I)*((1/​(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
 +       ​{4,​-2,​ (I)*((1/​(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
 +       ​{4,​-3,​ (-1/​4+1/​4*I)*((sqrt(7/​5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
 +       {4, 3, (1/​4+1/​4*I)*((sqrt(7/​5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
 +       ​{4,​-4,​ (-1/​2)*((sqrt(7/​10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} , 
 +       {4, 4, (-1/​2)*((sqrt(7/​10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} }
 +
 +</​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}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) $|$ \frac{i \text{Meg}}{\sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $|
 +^$ {Y_{-1}^{(2)}} $|$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) $|$ \text{Meg} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right] $|$ -\frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) $|$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|
 +^$ {Y_{0}^{(2)}} $|$ -\frac{i \text{Meg}}{\sqrt{3}} $|$ \frac{(-1)^{3/​4} \text{Meg}}{\sqrt{6}} $|$ \text{Eeg$\sigma $} $|$ \frac{\sqrt[4]{-1} \text{Meg}}{\sqrt{6}} $|$ \frac{i \text{Meg}}{\sqrt{3}} $|
 +^$ {Y_{1}^{(2)}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) $|$ \text{Meg} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right] $|$ \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|
 +^$ {Y_{2}^{(2)}} $|$ \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) $|$ -\frac{i \text{Meg}}{\sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^  $ d_{\text{yz}-\text{xz}} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ \text{Ea1g} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Meg} \left(\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(1+i)\right)}{\sqrt{2}} $|
 +^$ d_{\text{yz}-\text{xz}} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ \text{Meg} $|$ \left(-\frac{1}{2}-\frac{i}{2}\right) \text{Meg} \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right) $|
 +^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \text{Meg} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(2-2 i)\right) $|
 +^$ d_{x^2-y^2} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $|
 +^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\sigma $} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
 +^$ 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_{\text{yz}-\text{xz}} $|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|$ 0 $|$ \frac{1}{2}+\frac{i}{2} $|$ 0 $|
 +^$ 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_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|
 +^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea1g}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_2_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$$ | ::: |
 +^ ^$$\text{Eeg$\pi $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_2_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{2 \pi }} \sin (2 \theta ) (\sin (\phi )-\cos (\phi ))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{2 \pi }} z (y-x)$$ | ::: |
 +^ ^$$\text{Eeg$\pi $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_2_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$$ | ::: |
 +^ ^$$\text{Eeg$\sigma $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_2_4.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |
 +^ ^$$\text{Eeg$\sigma $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_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{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 (\text{Eeu1}+\text{Eeu2})) & k=0\land m=0 \\
 + ​-\frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=-2 \\
 + ​-\frac{\left(\frac{5}{28}+\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=-1 \\
 + ​\frac{\left(\frac{5}{28}-\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=1 \\
 + ​\frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=2 \\
 + ​-\frac{1}{4} \sqrt{\frac{5}{14}} (\text{Ea1u}+6 \text{Ea2u1}-3 \text{Ea2u2}-6 \text{Eeu1}+2 \text{Eeu2}) & k=4\land (m=-4\lor m=4) \\
 + ​-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=-3 \\
 + ​\frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=-2 \\
 + ​\left(-\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=-1 \\
 + ​\frac{1}{4} (-\text{Ea1u}-6 \text{Ea2u1}+3 \text{Ea2u2}+6 \text{Eeu1}-2 \text{Eeu2}) & k=4\land m=0 \\
 + ​\left(\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=1 \\
 + ​-\frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=2 \\
 + ​\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=3 \\
 + ​\frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=-6 \\
 + ​\left(-\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=-5 \\
 + ​\frac{13 (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\
 + ​-\frac{\left(\frac{13}{40}-\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=-3 \\
 + ​-\frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=-2 \\
 + ​\frac{\left(\frac{13}{20}+\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=-1 \\
 + ​-\frac{13}{280} (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2}) & k=6\land m=0 \\
 + ​-\frac{\left(\frac{13}{20}-\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=1 \\
 + ​\frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=2 \\
 + ​\frac{\left(\frac{13}{40}+\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=3 \\
 + ​\left(\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=5 \\
 + ​-\frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=6
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty.nb>​
 +
 +Akm[k_,​m_]:​=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*(Eeu1 + Eeu2))/7, k == 0 && m == 0}, {(((-5*I)/​28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/​Sqrt[6],​ k == 2 && m == -2}, {((-5/28 - (5*I)/​28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/​Sqrt[6],​ k == 2 && m == -1}, {((5/28 - (5*I)/​28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/​Sqrt[6],​ k == 2 && m == 1}, {(((5*I)/​28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/​Sqrt[6],​ k == 2 && m == 2}, {-(Sqrt[5/​14]*(Ea1u + 6*Ea2u1 - 3*Ea2u2 - 6*Eeu1 + 2*Eeu2))/4, k == 4 && (m == -4 || m == 4)}, {((-1/4 + I/​4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/​Sqrt[7],​ k == 4 && m == -3}, {((I/​7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/​Sqrt[2],​ k == 4 && m == -2}, {(-1/28 - I/​28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == -1}, {(-Ea1u - 6*Ea2u1 + 3*Ea2u2 + 6*Eeu1 - 2*Eeu2)/4, k == 4 && m == 0}, {(1/28 - I/​28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == 1}, {((-I/​7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/​Sqrt[2],​ k == 4 && m == 2}, {((1/4 + I/​4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/​Sqrt[7],​ k == 4 && m == 3}, {((13*I)/​80)*Sqrt[11/​21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu),​ k == 6 && m == -6}, {(-13/40 - (13*I)/​40)*Sqrt[11/​7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == -5}, {(13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/​(40*Sqrt[14]),​ k == 6 && (m == -4 || m == 4)}, {((-13/40 + (13*I)/​40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/​Sqrt[21],​ k == 6 && m == -3}, {(((-13*I)/​80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/​Sqrt[21],​ k == 6 && m == -2}, {((13/20 + (13*I)/​20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/​Sqrt[42],​ k == 6 && m == -1}, {(-13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/​280,​ k == 6 && m == 0}, {((-13/20 + (13*I)/​20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/​Sqrt[42],​ k == 6 && m == 1}, {(((13*I)/​80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/​Sqrt[21],​ k == 6 && m == 2}, {((13/40 + (13*I)/​40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/​Sqrt[21],​ k == 6 && m == 3}, {(13/40 - (13*I)/​40)*Sqrt[11/​7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == 5}, {((-13*I)/​80)*Sqrt[11/​21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu),​ k == 6 && m == 6}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty>​
 +
 +Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1 + Eeu2))} , 
 +       ​{2,​-1,​ (-5/​28+-5/​28*I)*((1/​(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
 +       {2, 1, (5/​28+-5/​28*I)*((1/​(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
 +       ​{2,​-2,​ (-5/​28*I)*((1/​(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
 +       {2, 2, (5/​28*I)*((1/​(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
 +       {4, 0, (1/​4)*((-1)*(Ea1u) + (-6)*(Ea2u1) + (3)*(Ea2u2) + (6)*(Eeu1) + (-2)*(Eeu2))} , 
 +       ​{4,​-1,​ (-1/​28+-1/​28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , 
 +       {4, 1, (1/​28+-1/​28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , 
 +       {4, 2, (-1/​7*I)*((1/​(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
 +       ​{4,​-2,​ (1/​7*I)*((1/​(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
 +       ​{4,​-3,​ (-1/​4+1/​4*I)*((1/​(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
 +       {4, 3, (1/​4+1/​4*I)*((1/​(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
 +       ​{4,​-4,​ (-1/​4)*((sqrt(5/​14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , 
 +       {4, 4, (-1/​4)*((sqrt(5/​14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , 
 +       {6, 0, (-13/​280)*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2))} , 
 +       {6, 1, (-13/​20+13/​20*I)*((1/​(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , 
 +       ​{6,​-1,​ (13/​20+13/​20*I)*((1/​(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , 
 +       ​{6,​-2,​ (-13/​80*I)*((1/​(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , 
 +       {6, 2, (13/​80*I)*((1/​(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , 
 +       ​{6,​-3,​ (-13/​40+13/​40*I)*((1/​(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , 
 +       {6, 3, (13/​40+13/​40*I)*((1/​(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , 
 +       ​{6,​-4,​ (13/​40)*((1/​(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , 
 +       {6, 4, (13/​40)*((1/​(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , 
 +       ​{6,​-5,​ (-13/​40+-13/​40*I)*((sqrt(11/​7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , 
 +       {6, 5, (13/​40+-13/​40*I)*((sqrt(11/​7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , 
 +       {6, 6, (-13/​80*I)*((sqrt(11/​21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} , 
 +       ​{6,​-6,​ (13/​80*I)*((sqrt(11/​21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} }
 +
 +</​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}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Meu}-\text{Ma2u})\right)}{\sqrt{6}} $|$ \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|$ -\frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $|
 +^$ {Y_{-2}^{(3)}} $|$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) $|$ -\frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{i \text{Ma2u}}{\sqrt{6}} $|$ \frac{1}{12} (-1)^{3/4} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) $|$ \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|
 +^$ {Y_{-1}^{(3)}} $|$ -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|$ \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{1}{24} i \left(5 \text{Ea1u}-3 \text{Ea2u2}+3 \text{Eeu1}-5 \text{Eeu2}-6 \sqrt{5} \text{Meu}\right) $|$ \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|
 +^$ {Y_{0}^{(3)}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|$ -\frac{i \text{Ma2u}}{\sqrt{6}} $|$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{1}{3} (\text{Ea2u2}+2 \text{Eeu1}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea2u2}-\text{Eeu1}-\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{i \text{Ma2u}}{\sqrt{6}} $|$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|
 +^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|$ \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) $|$ \frac{1}{24} i \left(-5 \text{Ea1u}+3 \text{Ea2u2}-3 \text{Eeu1}+5 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $|$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) $|$ \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|
 +^$ {Y_{2}^{(3)}} $|$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) $|$ \frac{1}{12} \sqrt[4]{-1} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) $|$ -\frac{i \text{Ma2u}}{\sqrt{6}} $|$ -\frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $|
 +^$ {Y_{3}^{(3)}} $|$ \frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $|$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|$ \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|$ -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x^3+y^3+z^3} $  ^  $ f_{x^3-y^3} $  ^  $ f_{2z^3-x^3-y^3} $  ^  $ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $  ^  $ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $  ^  $ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $  ^
 +^$ f_{\text{xyz}} $|$ \text{Ea2u1} $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x^3+y^3+z^3} $|$ \text{Ma2u} $|$ \text{Ea2u2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x^3-y^3} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|
 +^$ f_{2z^3-x^3-y^3} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|
 +^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1u} $|$ 0 $|$ 0 $|
 +^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|
 +^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
 +^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
 +^$ f_{x^3+y^3+z^3} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{3}} $|$ \frac{1}{4}-\frac{i}{4} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|
 +^$ f_{x^3-y^3} $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|
 +^$ f_{2z^3-x^3-y^3} $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \sqrt{\frac{2}{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|
 +^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{6}} $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{4}-\frac{i}{4} $|
 +^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \frac{1}{\sqrt{3}} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $|
 +^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2u1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
 +^ ^$$\text{Ea2u2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{32}+\frac{i}{32}\right) \sqrt{\frac{7}{3 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{3 \pi }} \left(5 x^3-15 x^2 y-3 x \left(5 y^2+5 z^2-1\right)+5 y^3+y \left(3-15 z^2\right)+4 z \left(5 z^2-3\right)\right)$$ | ::: |
 +^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(5 \sin ^2(\theta ) \sin (2 \phi )-5 \cos (2 \theta )-1\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{2 \pi }} (x-y) \left(5 x^2+20 x y+5 y^2-15 z^2+3\right)$$ | ::: |
 +^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_4.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(-\frac{1}{32}-\frac{i}{32}\right) \sqrt{\frac{7}{6 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )-(4-4 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{6 \pi }} \left(-5 x^3+15 x^2 y+3 x \left(5 y^2+5 z^2-1\right)-5 y^3+3 y \left(5 z^2-1\right)+8 z \left(5 z^2-3\right)\right)$$ | ::: |
 +^ ^$$\text{Ea1u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_5.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(-\sin ^2(\theta ) \sin (2 \phi )+\sin (2 \theta ) (\sin (\phi )+\cos (\phi ))-2 \cos ^2(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} (x-y) \left(x^2+4 x (y-z)+y^2-4 y z+5 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_6.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )+2 \sin (2 \theta ) (\sin (\phi )+\cos (\phi ))+2 \cos ^2(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{2 \pi }} (x-y) \left(x^2+4 x (y+2 z)+y^2+8 y z+5 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_111_orb_3_7.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{105}{2 \pi }} \sin (\theta ) (\sin (\phi )+\cos (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )-2 \cos ^2(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{2 \pi }} (x+y) \left(x^2-4 x y+y^2+5 z^2-1\right)$$ | ::: |
 +
 +
 +###
 +
 +</​hidden>​
 +===== Coupling between two shells =====
 +
 +
 +
 +###
 +
 +Click on one of the subsections to expand it or <​hiddenSwitch expand all> ​
 +
 +###
 +
 +==== Potential for s-d orbital mixing ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + 0 & k=0\land m=0 \\
 + i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-2 \\
 + (1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-1 \\
 + ​(-1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=1 \\
 + -i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=2
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty.nb>​
 +
 +Akm[k_,​m_]:​=Piecewise[{{0,​ k == 0 && m == 0}, {I*Sqrt[5/​6]*Ma1g,​ k == 2 && m == -2}, {(1 + I)*Sqrt[5/​6]*Ma1g,​ k == 2 && m == -1}, {(-1 + I)*Sqrt[5/​6]*Ma1g,​ k == 2 && m == 1}, {(-I)*Sqrt[5/​6]*Ma1g,​ k == 2 && m == 2}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty>​
 +
 +Akm = {{2, 1, (-1+1*I)*((sqrt(5/​6))*(Ma1g))} , 
 +       ​{2,​-1,​ (1+1*I)*((sqrt(5/​6))*(Ma1g))} , 
 +       {2, 2, (-I)*((sqrt(5/​6))*(Ma1g))} , 
 +       ​{2,​-2,​ (I)*((sqrt(5/​6))*(Ma1g))} }
 +
 +</​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_{0}^{(0)}} $|$ -\frac{i \text{Ma1g}}{\sqrt{6}} $|$ \frac{(1-i) \text{Ma1g}}{\sqrt{6}} $|$ 0 $|$ -\frac{(1+i) \text{Ma1g}}{\sqrt{6}} $|$ \frac{i \text{Ma1g}}{\sqrt{6}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^  $ d_{\text{yz}-\text{xz}} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^
 +^$ \text{s} $|$ \text{Ma1g} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</​hidden>​
 +==== Potential for p-f orbital mixing ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\
 + ​\frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-2 \\
 + ​\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-1 \\
 + ​\left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=1 \\
 + ​-\frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=2 \\
 + ​-\frac{1}{4} \sqrt{\frac{3}{10}} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land (m=-4\lor m=4) \\
 + ​\left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-3 \\
 + i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-2 \\
 + ​\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-1 \\
 + ​-\frac{1}{20} \sqrt{21} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land m=0 \\
 + ​\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=1 \\
 + -i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=2 \\
 + ​\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty.nb>​
 +
 +Akm[k_,​m_]:​=Piecewise[{{0,​ (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(I/​3)*Sqrt[5/​14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -2}, {(1/3 + I/​3)*Sqrt[5/​14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -1}, {(-1/3 + I/​3)*Sqrt[5/​14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 1}, {(-I/​3)*Sqrt[5/​14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 2}, {-(Sqrt[3/​10]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/​4,​ k == 4 && (m == -4 || m == 4)}, {(-1/2 + I/​2)*Sqrt[3]*(Ma2u1 - Meu2), k == 4 && m == -3}, {I*Sqrt[6/​7]*(Ma2u1 - Meu2), k == 4 && m == -2}, {(-1/2 - I/​2)*Sqrt[3/​7]*(Ma2u1 - Meu2), k == 4 && m == -1}, {-(Sqrt[21]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/​20,​ k == 4 && m == 0}, {(1/2 - I/​2)*Sqrt[3/​7]*(Ma2u1 - Meu2), k == 4 && m == 1}, {(-I)*Sqrt[6/​7]*(Ma2u1 - Meu2), k == 4 && m == 2}}, (1/2 + I/​2)*Sqrt[3]*(Ma2u1 - Meu2)]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_111.Quanty>​
 +
 +Akm = {{2, 1, (-1/​3+1/​3*I)*((sqrt(5/​14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
 +       ​{2,​-1,​ (1/​3+1/​3*I)*((sqrt(5/​14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
 +       {2, 2, (-1/​3*I)*((sqrt(5/​14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
 +       ​{2,​-2,​ (1/​3*I)*((sqrt(5/​14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
 +       {4, 0, (-1/​20)*((sqrt(21))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , 
 +       ​{4,​-1,​ (-1/​2+-1/​2*I)*((sqrt(3/​7))*(Ma2u1 + (-1)*(Meu2)))} , 
 +       {4, 1, (1/​2+-1/​2*I)*((sqrt(3/​7))*(Ma2u1 + (-1)*(Meu2)))} , 
 +       {4, 2, (-I)*((sqrt(6/​7))*(Ma2u1 + (-1)*(Meu2)))} , 
 +       ​{4,​-2,​ (I)*((sqrt(6/​7))*(Ma2u1 + (-1)*(Meu2)))} , 
 +       ​{4,​-3,​ (-1/​2+1/​2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , 
 +       {4, 3, (1/​2+1/​2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , 
 +       ​{4,​-4,​ (-1/​4)*((sqrt(3/​10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , 
 +       {4, 4, (-1/​4)*((sqrt(3/​10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} }
 +
 +</​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)}} $|$ -\frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} $|$ (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right] $|$ \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} $|$ (2 \text{Ma2u1}+\text{Meu2}) \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right] $|$ -\frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} $|$ \frac{(-1)^{3/​4} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} $|$ \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} $|
 +^$ {Y_{0}^{(1)}} $|$ \left(-\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) $|$ -\frac{i \text{Ma2u1}}{\sqrt{6}} $|$ (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right] $|$ -\frac{2 \text{Ma2u1}}{3 \sqrt{5}}+\text{Meu1}-\frac{\text{Meu2}}{3 \sqrt{5}} $|$ (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right] $|$ \frac{i \text{Ma2u1}}{\sqrt{6}} $|$ \left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) $|
 +^$ {Y_{1}^{(1)}} $|$ \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} $|$ \frac{\sqrt[4]{-1} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} $|$ \frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} $|$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) (2 \text{Ma2u1}+\text{Meu2})}{\sqrt{10}} $|$ \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} $|$ (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right] $|$ \frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x^3+y^3+z^3} $  ^  $ f_{x^3-y^3} $  ^  $ f_{2z^3-x^3-y^3} $  ^  $ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $  ^  $ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $  ^  $ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $  ^
 +^$ p_{x+y+z} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(2-2 i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(-1+i)\right)\right) $|$ \left(-\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(-15 \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+15 i \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]-60 \sqrt{2} \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+(29+29 i) \sqrt{5}\right)+\text{Meu2} \left(-45 \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+45 i \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]-30 \sqrt{2} \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+(7+7 i) \sqrt{5}\right)-(90+90 i) \text{Meu1}\right) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)}{\sqrt{2}} $|$ \frac{\text{Ma2u1} \left(-(3-3 i) \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+(3+3 i) \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+(24-24 i) \sqrt{2} \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+2 \sqrt{5}\right)+(1+i) \text{Meu2} \left(9 i \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+9 \sqrt{3} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]-12 i \sqrt{2} \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+(-1+i) \sqrt{5}\right)}{36 \sqrt{2}} $|$ \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+2 i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+2 i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)\right) $|$ -\frac{\left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(4 \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+4 i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)+\text{Meu2} \left(4 \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+4 i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)\right)}{\sqrt{2}} $|$ \frac{\left(\frac{1}{12}+\frac{i}{12}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(i \sqrt{15} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{15} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+(-1+i)\right)}{\sqrt{2}} $|
 +^$ p_{x-y} $|$ \frac{\left(\frac{1}{2}+\frac{i}{2}\right) (\text{Ma2u1}+\text{Meu2}) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)}{\sqrt{2}} $|$ \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+i \sqrt{10}\right)}{30 \sqrt{3}} $|$ \text{Meu1} $|$ \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+i \sqrt{10}\right)}{15 \sqrt{6}} $|$ \frac{\left(\frac{1}{6}+\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(-1+i)\right)}{\sqrt{2}} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(-1+i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(2-2 i)\right)\right) $|$ 0 $|
 +^$ p_{3z-r} $|$ \frac{\left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(\sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+(1+i)\right)}{\sqrt{2}} $|$ \left(\frac{1}{180}-\frac{i}{180}\right) \left(\text{Ma2u1} \left(-4 \text{Root}\left[\text{$\#​$1}^4+25\$|$,​1\right]+15 \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]-15 i \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+(1+i) \sqrt{10}\right)+\text{Meu2} \left(-2 \text{Root}\left[\text{$\#​$1}^4+25\$|$,​1\right]+45 \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]-45 i \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+(8+8 i) \sqrt{10}\right)\right) $|$ \left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right) $|$ \frac{\left(\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(15 i \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+15 \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+120 i \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+(-7+7 i) \sqrt{10}\right)+\text{Meu2} \left(45 i \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+45 \sqrt{6} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+60 i \text{Root}\left[2025 \text{$\#​$1}^4+1\$|$,​1\right]+(-11+11 i) \sqrt{10}\right)+(90-90 i) \sqrt{2} \text{Meu1}\right)}{\sqrt{2}} $|$ \frac{\left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)+\text{Meu2} \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)\right)}{\sqrt{2}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+2 i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+2 i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]\right)\right)\right) $|$ \frac{1}{12} \left(\text{Ma2u1} \left(-(1-i) \sqrt{15} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+(1+i) \sqrt{15} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]-2\right)+(3+3 i) \text{Meu2} \left(i \sqrt{15} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​1\right]+\sqrt{15} \text{Root}\left[900 \text{$\#​$1}^4+1\$|$,​3\right]+(1-i)\right)\right) $|
 +
 +
 +###
 +
 +</​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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