~~CLOSETOC~~

====== Orientation  ======

===== Symmetry Operations =====

###

In the Ci Point Group, with orientation  there are the following symmetry operations

###

###

{{:physics_chemistry:pointgroup:ci.png}}

###

###

^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |

###

===== Different Settings =====

###

  * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]]

###

===== Character Table =====

###

|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ \text{i} \,{\text{(1)}} $  ^
^ $ A_g $ |  $ 1 $ |  $ 1 $ |
^ $ A_u $ |  $ 1 $ |  $ -1 $ |

###

===== Product Table =====

###

|  $  $  ^  $ A_g $  ^  $ A_u $  ^
^ $ A_g $  | $ A_g $  | $ A_u $  |
^ $ A_u $  | $ A_u $  | $ A_g $  |

###

===== Sub Groups with compatible settings =====

###

  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]

###

===== Super Groups with compatible settings =====

###

  * [[physics_chemistry:point_groups:c2h:orientation_z|Point Group C2h with orientation Z]]
  * [[physics_chemistry:point_groups:c4h:orientation_z|Point Group C4h with orientation Z]]
  * [[physics_chemistry:point_groups:c6h:orientation_z|Point Group C6h with orientation Z]]
  * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]]
  * [[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]]
  * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
  * [[physics_chemistry:point_groups:d5d:orientation_zx|Point Group D5d with orientation Zx]]
  * [[physics_chemistry:point_groups:d5d:orientation_zy|Point Group D5d with orientation Zy]]
  * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]]
  * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]]
  * [[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Point Group Oh with orientation 0sqrt2-1z]]
  * [[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Point Group Oh with orientation 0sqrt21z]]
  * [[physics_chemistry:point_groups:oh:orientation_111z|Point Group Oh with orientation 111z]]
  * [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]]
  * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]]
  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
  * [[physics_chemistry:point_groups:s6:orientation_z|Point Group S6 with orientation Z]]
  * [[physics_chemistry:point_groups:th:orientation_xyz|Point Group Th 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 Ci Point group with orientation  the form of the expansion coefficients is:

###

==== Expansion ====

###

 $$A_{k,m} = \begin{cases}
 A(0,0) & k=0\land m=0 \\
 A(2,2)-i B(2,2) & k=2\land m=-2 \\
 -A(2,1)+i B(2,1) & k=2\land m=-1 \\
 A(2,0) & k=2\land m=0 \\
 A(2,1)+i B(2,1) & k=2\land m=1 \\
 A(2,2)+i B(2,2) & k=2\land m=2 \\
 A(4,4)-i B(4,4) & k=4\land m=-4 \\
 -A(4,3)+i B(4,3) & k=4\land m=-3 \\
 A(4,2)-i B(4,2) & k=4\land m=-2 \\
 -A(4,1)+i B(4,1) & k=4\land m=-1 \\
 A(4,0) & k=4\land m=0 \\
 A(4,1)+i B(4,1) & k=4\land m=1 \\
 A(4,2)+i B(4,2) & k=4\land m=2 \\
 A(4,3)+i B(4,3) & k=4\land m=3 \\
 A(4,4)+i B(4,4) & k=4\land m=4 \\
 A(6,6)-i B(6,6) & k=6\land m=-6 \\
 -A(6,5)+i B(6,5) & k=6\land m=-5 \\
 A(6,4)-i B(6,4) & k=6\land m=-4 \\
 -A(6,3)+i B(6,3) & k=6\land m=-3 \\
 A(6,2)-i B(6,2) & k=6\land m=-2 \\
 -A(6,1)+i B(6,1) & k=6\land m=-1 \\
 A(6,0) & k=6\land m=0 \\
 A(6,1)+i B(6,1) & k=6\land m=1 \\
 A(6,2)+i B(6,2) & k=6\land m=2 \\
 A(6,3)+i B(6,3) & k=6\land m=3 \\
 A(6,4)+i B(6,4) & k=6\land m=4 \\
 A(6,5)+i B(6,5) & k=6\land m=5 \\
 A(6,6)+i B(6,6) & k=6\land m=6
\end{cases}$$

###

==== Input format suitable for Mathematica (Quanty.nb) ====

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {-A[4, 1] + I*B[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1] + I*B[4, 1], k == 4 && m == 1}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 3] + I*B[4, 3], k == 4 && m == 3}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {-A[6, 5] + I*B[6, 5], k == 6 && m == -5}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {-A[6, 3] + I*B[6, 3], k == 6 && m == -3}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {-A[6, 1] + I*B[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {A[6, 1] + I*B[6, 1], k == 6 && m == 1}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 3] + I*B[6, 3], k == 6 && m == 3}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 5] + I*B[6, 5], k == 6 && m == 5}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]

</code>

###

==== Input format suitable for Quanty ====

###

<code Quanty Akm_Ci.Quanty>

Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , 
       {2, 1, A(2,1) + (I)*(B(2,1))} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1)*(A(4,1)) + (I)*(B(4,1))} , 
       {4, 1, A(4,1) + (I)*(B(4,1))} , 
       {4,-2, A(4,2) + (-I)*(B(4,2))} , 
       {4, 2, A(4,2) + (I)*(B(4,2))} , 
       {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , 
       {4, 3, A(4,3) + (I)*(B(4,3))} , 
       {4,-4, A(4,4) + (-I)*(B(4,4))} , 
       {4, 4, A(4,4) + (I)*(B(4,4))} , 
       {6, 0, A(6,0)} , 
       {6,-1, (-1)*(A(6,1)) + (I)*(B(6,1))} , 
       {6, 1, A(6,1) + (I)*(B(6,1))} , 
       {6,-2, A(6,2) + (-I)*(B(6,2))} , 
       {6, 2, A(6,2) + (I)*(B(6,2))} , 
       {6,-3, (-1)*(A(6,3)) + (I)*(B(6,3))} , 
       {6, 3, A(6,3) + (I)*(B(6,3))} , 
       {6,-4, A(6,4) + (-I)*(B(6,4))} , 
       {6, 4, A(6,4) + (I)*(B(6,4))} , 
       {6,-5, (-1)*(A(6,5)) + (I)*(B(6,5))} , 
       {6, 5, A(6,5) + (I)*(B(6,5))} , 
       {6,-6, A(6,6) + (-I)*(B(6,6))} , 
       {6, 6, A(6,6) + (I)*(B(6,6))} }

</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{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $|$ -\frac{\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ -\frac{-\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\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)-\frac{1}{5} \text{App}(2,0) $|$ \frac{1}{5} \sqrt{3} (-\text{App}(2,1)+i \text{Bpp}(2,1)) $|$ -\frac{1}{5} \sqrt{6} (\text{App}(2,2)-i \text{Bpp}(2,2)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} $|$ \frac{\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} (\text{Apf}(2,1)+i \text{Bpf}(2,1)) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{1}{3} \sqrt{\frac{10}{21}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1))-\frac{3 (-\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}} $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $|$ \frac{1}{3} (-\text{Apf}(4,3)+i \text{Bpf}(4,3)) $|$ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $|
^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{3} (\text{App}(2,1)+i \text{Bpp}(2,1)) $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ -\frac{1}{5} \sqrt{3} (-\text{App}(2,1)+i \text{Bpp}(2,1)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ -\frac{2}{5} \sqrt{\frac{6}{7}} (\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,1)+i \text{Bpf}(4,1)) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ -\frac{2}{5} \sqrt{\frac{6}{7}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1)) $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ -\frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|
^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} (\text{App}(2,2)+i \text{Bpp}(2,2)) $|$ \frac{1}{5} \sqrt{3} (\text{App}(2,1)+i \text{Bpp}(2,1)) $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$ \frac{1}{3} (\text{Apf}(4,3)+i \text{Bpf}(4,3)) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) $|$ \frac{1}{3} \sqrt{\frac{10}{21}} (\text{Apf}(4,1)+i \text{Bpf}(4,1))-\frac{3 (\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}} $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{-\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1)) $|$ \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $|
^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ \frac{1}{7} \sqrt{6} (-\text{Add}(2,1)+i \text{Bdd}(2,1))-\frac{1}{21} \sqrt{5} (-\text{Add}(4,1)+i \text{Bdd}(4,1)) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4)) $|$\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{-\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{21} \sqrt{5} (\text{Add}(4,1)+i \text{Bdd}(4,1))-\frac{1}{7} \sqrt{6} (\text{Add}(2,1)+i \text{Bdd}(2,1)) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ \frac{1}{7} (-\text{Add}(2,1)+i \text{Bdd}(2,1))+\frac{1}{7} \sqrt{\frac{10}{3}} (-\text{Add}(4,1)+i \text{Bdd}(4,1)) $|$ -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)-i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)-i \text{Bdd}(4,2)) $|$ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) $|$ \frac{1}{7} (-\text{Add}(2,1)-i \text{Bdd}(2,1))-\frac{1}{7} \sqrt{\frac{10}{3}} (\text{Add}(4,1)+i \text{Bdd}(4,1)) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ \frac{1}{7} (\text{Add}(2,1)-i \text{Bdd}(2,1))-\frac{1}{7} \sqrt{\frac{10}{3}} (-\text{Add}(4,1)+i \text{Bdd}(4,1)) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $|$\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{\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)+i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)+i \text{Bdd}(4,2)) $|$ \frac{1}{7} (\text{Add}(2,1)+i \text{Bdd}(2,1))+\frac{1}{7} \sqrt{\frac{10}{3}} (\text{Add}(4,1)+i \text{Bdd}(4,1)) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ \frac{1}{21} \sqrt{5} (-\text{Add}(4,1)+i \text{Bdd}(4,1))-\frac{1}{7} \sqrt{6} (-\text{Add}(2,1)+i \text{Bdd}(2,1)) $|$\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{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4)) $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) $|$ \frac{1}{7} \sqrt{6} (\text{Add}(2,1)+i \text{Bdd}(2,1))-\frac{1}{21} \sqrt{5} (\text{Add}(4,1)+i \text{Bdd}(4,1)) $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\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 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $|$ \frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ \frac{1}{3} (-\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{10}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{429} \sqrt{7} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ \frac{10}{143} \sqrt{\frac{7}{3}} (-\text{Aff}(6,3)+i \text{Bff}(6,3))-\frac{1}{11} \sqrt{7} (-\text{Aff}(4,3)+i \text{Bff}(4,3)) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $|$ \frac{5}{13} \sqrt{\frac{14}{33}} (-\text{Aff}(6,5)+i \text{Bff}(6,5)) $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $|
^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \sqrt{\frac{6}{35}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{-\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}} $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ \frac{1}{3} (\text{Apf}(4,3)-i \text{Bpf}(4,3)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} (-\text{Aff}(2,1)-i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{10}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{429} \sqrt{7} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \frac{-\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} \sqrt{2} (-\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{143} \sqrt{\frac{35}{3}} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ -\frac{1}{33} \sqrt{14} (-\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{5}{143} \sqrt{42} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $|$ -\frac{5}{13} \sqrt{\frac{14}{33}} (-\text{Aff}(6,5)+i \text{Bff}(6,5)) $|
^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{2}{5} \sqrt{\frac{6}{7}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1))+\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1)) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ -\frac{\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} \sqrt{2} (\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ \frac{1}{15} \sqrt{2} (-\text{Aff}(2,1)+i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{5}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{25}{429} \sqrt{14} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)-i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ \frac{1}{33} \sqrt{14} (-\text{Aff}(4,3)+i \text{Bff}(4,3))+\frac{5}{143} \sqrt{42} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $|
^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3 (\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}}-\frac{1}{3} \sqrt{\frac{10}{21}} (\text{Apf}(4,1)+i \text{Bpf}(4,1)) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \frac{3 (-\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}}-\frac{1}{3} \sqrt{\frac{10}{21}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{7} (\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{10}{143} \sqrt{\frac{7}{3}} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ -\frac{1}{15} \sqrt{2} (\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{5}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{25}{429} \sqrt{14} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ -\frac{1}{15} \sqrt{2} (-\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{5}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{25}{429} \sqrt{14} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ \frac{1}{11} \sqrt{7} (-\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{10}{143} \sqrt{\frac{7}{3}} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $|
^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) $|$ \frac{2}{5} \sqrt{\frac{6}{7}} (\text{Apf}(2,1)+i \text{Bpf}(2,1))+\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,1)+i \text{Bpf}(4,1)) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $|$ \frac{1}{33} \sqrt{14} (\text{Aff}(4,3)+i \text{Bff}(4,3))+\frac{5}{143} \sqrt{42} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)+i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ \frac{1}{15} \sqrt{2} (\text{Aff}(2,1)+i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{5}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{25}{429} \sqrt{14} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ -\frac{-\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} \sqrt{2} (-\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{143} \sqrt{\frac{35}{3}} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|
^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} (-\text{Apf}(4,3)-i \text{Bpf}(4,3)) $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ \sqrt{\frac{6}{35}} (\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{5}{13} \sqrt{\frac{14}{33}} (\text{Aff}(6,5)+i \text{Bff}(6,5)) $|$ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $|$ -\frac{1}{33} \sqrt{14} (\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{5}{143} \sqrt{42} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ \frac{\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} \sqrt{2} (\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \frac{1}{3} (\text{Aff}(2,1)-i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{10}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{429} \sqrt{7} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $|
^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$ \frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) $|$ \frac{5}{13} \sqrt{\frac{14}{33}} (\text{Aff}(6,5)+i \text{Bff}(6,5)) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $|$ \frac{10}{143} \sqrt{\frac{7}{3}} (\text{Aff}(6,3)+i \text{Bff}(6,3))-\frac{1}{11} \sqrt{7} (\text{Aff}(4,3)+i \text{Bff}(4,3)) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ \frac{1}{3} (\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{10}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{429} \sqrt{7} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|


###

==== Rotation matrix to symmetry adapted functions (choice is not unique) ====

###


Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field

###



###

|  $  $  ^  $ {Y_{0}^{(0)}} $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|


###

==== One particle coupling on a basis of symmetry adapted functions ====

###

After rotation we find

###



###

|  $  $  ^  $ \text{s} $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $|$ -\sqrt{\frac{2}{5}} \text{Asd}(2,1) $|$ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ p_x $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $|$ -\frac{1}{5} \sqrt{6} \text{App}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) $|
^$ p_y $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|
^$ p_z $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} \text{App}(2,1) $|$ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|
^$ d_{x^2-y^2} $|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,1)+\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$\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} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ \frac{1}{7} \sqrt{2} \text{Bdd}(2,1)+\frac{2}{7} \sqrt{\frac{5}{3}} \text{Bdd}(4,1) $|$ -\frac{1}{7} \sqrt{2} \text{Add}(2,1)-\frac{2}{7} \sqrt{\frac{5}{3}} \text{Add}(4,1) $|$ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{yz}} $|$ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,1)+\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ \frac{1}{7} \sqrt{2} \text{Bdd}(2,1)+\frac{2}{7} \sqrt{\frac{5}{3}} \text{Bdd}(4,1) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xz}} $|$ -\sqrt{\frac{2}{5}} \text{Asd}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ -\frac{1}{7} \sqrt{2} \text{Add}(2,1)-\frac{2}{7} \sqrt{\frac{5}{3}} \text{Add}(4,1) $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xy}} $|$ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\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 }$|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $|$ -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|$ -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $|
^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|
^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) $|$ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|
^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $|$ \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) $|$ \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|
^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $|$ \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|
^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|$ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) $|$ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $|$ -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|$ \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|$ -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|


###

===== Coupling for a single shell =====



###

Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$.

###



###

Click on one of the subsections to expand it or <hiddenSwitch expand all> 

###

==== Potential for s orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 \text{Eag} & k=0\land m=0 \\
 0 & \text{True}
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{Eag, k == 0 && m == 0}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_Ci.Quanty>

Akm = {{0, 0, Eag} }

</code>

###

</hidden>
<hidden **The Hamiltonian on a basis of spherical Harmonics** >

###

|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
^$ {Y_{0}^{(0)}} $|$ \text{Eag} $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ \text{s} $  ^
^$ \text{s} $|$ \text{Eag} $|


###

</hidden>
<hidden **Rotation matrix used** >

###

|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
^$ \text{s} $|$ 1 $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Eag}$$ | {{:physics_chemistry:pointgroup:ci_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** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{3} (\text{Epxpx}+\text{Epypy}+\text{Epzpz}) & k=0\land m=0 \\
 \frac{5 (\text{Epxpx}+2 i \text{Epxpy}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=-2 \\
 \frac{5 (\text{Epxpz}+i \text{Epypz})}{\sqrt{6}} & k=2\land m=-1 \\
 -\frac{5}{6} (\text{Epxpx}+\text{Epypy}-2 \text{Epzpz}) & k=2\land m=0 \\
 -\frac{5 (\text{Epxpz}-i \text{Epypz})}{\sqrt{6}} & k=2\land m=1 \\
 \frac{5 (\text{Epxpx}-2 i \text{Epxpy}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=2
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Epxpx + Epypy + Epzpz)/3, k == 0 && m == 0}, {(5*(Epxpx + (2*I)*Epxpy - Epypy))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(Epxpz + I*Epypz))/Sqrt[6], k == 2 && m == -1}, {(-5*(Epxpx + Epypy - 2*Epzpz))/6, k == 2 && m == 0}, {(-5*(Epxpz - I*Epypz))/Sqrt[6], k == 2 && m == 1}, {(5*(Epxpx - (2*I)*Epxpy - Epypy))/(2*Sqrt[6]), k == 2 && m == 2}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_Ci.Quanty>

Akm = {{0, 0, (1/3)*(Epxpx + Epypy + Epzpz)} , 
       {2, 0, (-5/6)*(Epxpx + Epypy + (-2)*(Epzpz))} , 
       {2, 1, (-5)*((1/(sqrt(6)))*(Epxpz + (-I)*(Epypz)))} , 
       {2,-1, (5)*((1/(sqrt(6)))*(Epxpz + (I)*(Epypz)))} , 
       {2, 2, (5/2)*((1/(sqrt(6)))*(Epxpx + (-2*I)*(Epxpy) + (-1)*(Epypy)))} , 
       {2,-2, (5/2)*((1/(sqrt(6)))*(Epxpx + (2*I)*(Epxpy) + (-1)*(Epypy)))} }

</code>

###

</hidden>
<hidden **The Hamiltonian on a basis of spherical Harmonics** >

###

|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
^$ {Y_{-1}^{(1)}} $|$ \frac{\text{Epxpx}+\text{Epypy}}{2} $|$ \frac{\text{Epxpz}+i \text{Epypz}}{\sqrt{2}} $|$ \frac{1}{2} (-\text{Epxpx}-2 i \text{Epxpy}+\text{Epypy}) $|
^$ {Y_{0}^{(1)}} $|$ \frac{\text{Epxpz}-i \text{Epypz}}{\sqrt{2}} $|$ \text{Epzpz} $|$ -\frac{\text{Epxpz}+i \text{Epypz}}{\sqrt{2}} $|
^$ {Y_{1}^{(1)}} $|$ \frac{1}{2} (-\text{Epxpx}+2 i \text{Epxpy}+\text{Epypy}) $|$ -\frac{\text{Epxpz}-i \text{Epypz}}{\sqrt{2}} $|$ \frac{\text{Epxpx}+\text{Epypy}}{2} $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^
^$ p_x $|$ \text{Epxpx} $|$ \text{Epxpy} $|$ \text{Epxpz} $|
^$ p_y $|$ \text{Epxpy} $|$ \text{Epypy} $|$ \text{Epypz} $|
^$ p_z $|$ \text{Epxpz} $|$ \text{Epypz} $|$ \text{Epzpz} $|


###

</hidden>
<hidden **Rotation matrix used** >

###

|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|
^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
^$ p_z $|$ 0 $|$ 1 $|$ 0 $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Epxpx}$$ | {{:physics_chemistry:pointgroup:ci_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
^ ^$$\text{Epypy}$$ | {{:physics_chemistry:pointgroup:ci_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Epzpz}$$ | {{:physics_chemistry:pointgroup:ci_orb_1_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |


###

</hidden>
==== Potential for d orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{5} (\text{Edx2y2dx2y2}+\text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+\text{Edz2dz2}) & k=0\land m=0 \\
 -\frac{4 \text{Edx2y2dz2}-\sqrt{3} \text{Edxzdxz}-2 i \sqrt{3} \text{Edyzdxz}+\sqrt{3} \text{Edyzdyz}+4 i \text{Edz2dxy}}{2 \sqrt{2}} & k=2\land m=-2 \\
 \frac{\sqrt{3} \text{Edx2y2dxz}-i \sqrt{3} \text{Edx2y2dyz}+i \sqrt{3} \text{Edxzdxy}+\sqrt{3} \text{Edyzdxy}+\text{Edz2dxz}+i \text{Edz2dyz}}{\sqrt{2}} & k=2\land m=-1 \\
 \frac{1}{2} (-2 \text{Edx2y2dx2y2}-2 \text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+2 \text{Edz2dz2}) & k=2\land m=0 \\
 -\frac{\sqrt{3} \text{Edx2y2dxz}+i \sqrt{3} \text{Edx2y2dyz}-i \sqrt{3} \text{Edxzdxy}+\sqrt{3} \text{Edyzdxy}+\text{Edz2dxz}-i \text{Edz2dyz}}{\sqrt{2}} & k=2\land m=1 \\
 -\frac{4 \text{Edx2y2dz2}-\sqrt{3} \text{Edxzdxz}+2 i \sqrt{3} \text{Edyzdxz}+\sqrt{3} \text{Edyzdyz}-4 i \text{Edz2dxy}}{2 \sqrt{2}} & k=2\land m=2 \\
 \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}+2 i \text{Edx2y2dxy}-\text{Edxydxy}) & k=4\land m=-4 \\
 \frac{3}{2} \sqrt{\frac{7}{5}} (\text{Edx2y2dxz}+i (\text{Edx2y2dyz}+\text{Edxzdxy}+i \text{Edyzdxy})) & k=4\land m=-3 \\
 \frac{3 \left(\sqrt{3} \text{Edx2y2dz2}+\text{Edxzdxz}+2 i \text{Edyzdxz}-\text{Edyzdyz}+i \sqrt{3} \text{Edz2dxy}\right)}{\sqrt{10}} & k=4\land m=-2 \\
 -\frac{3 \left(\text{Edx2y2dxz}-i \text{Edx2y2dyz}+i \text{Edxzdxy}+\text{Edyzdxy}-2 \sqrt{3} \text{Edz2dxz}-2 i \sqrt{3} \text{Edz2dyz}\right)}{2 \sqrt{5}} & k=4\land m=-1 \\
 \frac{3}{10} (\text{Edx2y2dx2y2}+\text{Edxydxy}-4 \text{Edxzdxz}-4 \text{Edyzdyz}+6 \text{Edz2dz2}) & k=4\land m=0 \\
 \frac{3 \left(\text{Edx2y2dxz}+i \text{Edx2y2dyz}-i \text{Edxzdxy}+\text{Edyzdxy}-2 \sqrt{3} \text{Edz2dxz}+2 i \sqrt{3} \text{Edz2dyz}\right)}{2 \sqrt{5}} & k=4\land m=1 \\
 \frac{3 \left(\sqrt{3} \text{Edx2y2dz2}+\text{Edxzdxz}-2 i \text{Edyzdxz}-\text{Edyzdyz}-i \sqrt{3} \text{Edz2dxy}\right)}{\sqrt{10}} & k=4\land m=2 \\
 \frac{3}{2} \sqrt{\frac{7}{5}} (-\text{Edx2y2dxz}+i \text{Edx2y2dyz}+i \text{Edxzdxy}+\text{Edyzdxy}) & k=4\land m=3 \\
 \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}-2 i \text{Edx2y2dxy}-\text{Edxydxy}) & k=4\land m=4
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)/5, k == 0 && m == 0}, {-(4*Edx2y2dz2 - Sqrt[3]*Edxzdxz - (2*I)*Sqrt[3]*Edyzdxz + Sqrt[3]*Edyzdyz + (4*I)*Edz2dxy)/(2*Sqrt[2]), k == 2 && m == -2}, {(Sqrt[3]*Edx2y2dxz - I*Sqrt[3]*Edx2y2dyz + I*Sqrt[3]*Edxzdxy + Sqrt[3]*Edyzdxy + Edz2dxz + I*Edz2dyz)/Sqrt[2], k == 2 && m == -1}, {(-2*Edx2y2dx2y2 - 2*Edxydxy + Edxzdxz + Edyzdyz + 2*Edz2dz2)/2, k == 2 && m == 0}, {-((Sqrt[3]*Edx2y2dxz + I*Sqrt[3]*Edx2y2dyz - I*Sqrt[3]*Edxzdxy + Sqrt[3]*Edyzdxy + Edz2dxz - I*Edz2dyz)/Sqrt[2]), k == 2 && m == 1}, {-(4*Edx2y2dz2 - Sqrt[3]*Edxzdxz + (2*I)*Sqrt[3]*Edyzdxz + Sqrt[3]*Edyzdyz - (4*I)*Edz2dxy)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Edx2y2dx2y2 + (2*I)*Edx2y2dxy - Edxydxy))/2, k == 4 && m == -4}, {(3*Sqrt[7/5]*(Edx2y2dxz + I*(Edx2y2dyz + Edxzdxy + I*Edyzdxy)))/2, k == 4 && m == -3}, {(3*(Sqrt[3]*Edx2y2dz2 + Edxzdxz + (2*I)*Edyzdxz - Edyzdyz + I*Sqrt[3]*Edz2dxy))/Sqrt[10], k == 4 && m == -2}, {(-3*(Edx2y2dxz - I*Edx2y2dyz + I*Edxzdxy + Edyzdxy - 2*Sqrt[3]*Edz2dxz - (2*I)*Sqrt[3]*Edz2dyz))/(2*Sqrt[5]), k == 4 && m == -1}, {(3*(Edx2y2dx2y2 + Edxydxy - 4*Edxzdxz - 4*Edyzdyz + 6*Edz2dz2))/10, k == 4 && m == 0}, {(3*(Edx2y2dxz + I*Edx2y2dyz - I*Edxzdxy + Edyzdxy - 2*Sqrt[3]*Edz2dxz + (2*I)*Sqrt[3]*Edz2dyz))/(2*Sqrt[5]), k == 4 && m == 1}, {(3*(Sqrt[3]*Edx2y2dz2 + Edxzdxz - (2*I)*Edyzdxz - Edyzdyz - I*Sqrt[3]*Edz2dxy))/Sqrt[10], k == 4 && m == 2}, {(3*Sqrt[7/5]*(-Edx2y2dxz + I*Edx2y2dyz + I*Edxzdxy + Edyzdxy))/2, k == 4 && m == 3}, {(3*Sqrt[7/10]*(Edx2y2dx2y2 - (2*I)*Edx2y2dxy - Edxydxy))/2, k == 4 && m == 4}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_Ci.Quanty>

Akm = {{0, 0, (1/5)*(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)} , 
       {2, 0, (1/2)*((-2)*(Edx2y2dx2y2) + (-2)*(Edxydxy) + Edxzdxz + Edyzdyz + (2)*(Edz2dz2))} , 
       {2, 1, (-1)*((1/(sqrt(2)))*((sqrt(3))*(Edx2y2dxz) + (I)*((sqrt(3))*(Edx2y2dyz)) + (-I)*((sqrt(3))*(Edxzdxy)) + (sqrt(3))*(Edyzdxy) + Edz2dxz + (-I)*(Edz2dyz)))} , 
       {2,-1, (1/(sqrt(2)))*((sqrt(3))*(Edx2y2dxz) + (-I)*((sqrt(3))*(Edx2y2dyz)) + (I)*((sqrt(3))*(Edxzdxy)) + (sqrt(3))*(Edyzdxy) + Edz2dxz + (I)*(Edz2dyz))} , 
       {2,-2, (-1/2)*((1/(sqrt(2)))*((4)*(Edx2y2dz2) + (-1)*((sqrt(3))*(Edxzdxz)) + (-2*I)*((sqrt(3))*(Edyzdxz)) + (sqrt(3))*(Edyzdyz) + (4*I)*(Edz2dxy)))} , 
       {2, 2, (-1/2)*((1/(sqrt(2)))*((4)*(Edx2y2dz2) + (-1)*((sqrt(3))*(Edxzdxz)) + (2*I)*((sqrt(3))*(Edyzdxz)) + (sqrt(3))*(Edyzdyz) + (-4*I)*(Edz2dxy)))} , 
       {4, 0, (3/10)*(Edx2y2dx2y2 + Edxydxy + (-4)*(Edxzdxz) + (-4)*(Edyzdyz) + (6)*(Edz2dz2))} , 
       {4,-1, (-3/2)*((1/(sqrt(5)))*(Edx2y2dxz + (-I)*(Edx2y2dyz) + (I)*(Edxzdxy) + Edyzdxy + (-2)*((sqrt(3))*(Edz2dxz)) + (-2*I)*((sqrt(3))*(Edz2dyz))))} , 
       {4, 1, (3/2)*((1/(sqrt(5)))*(Edx2y2dxz + (I)*(Edx2y2dyz) + (-I)*(Edxzdxy) + Edyzdxy + (-2)*((sqrt(3))*(Edz2dxz)) + (2*I)*((sqrt(3))*(Edz2dyz))))} , 
       {4, 2, (3)*((1/(sqrt(10)))*((sqrt(3))*(Edx2y2dz2) + Edxzdxz + (-2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (-I)*((sqrt(3))*(Edz2dxy))))} , 
       {4,-2, (3)*((1/(sqrt(10)))*((sqrt(3))*(Edx2y2dz2) + Edxzdxz + (2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (I)*((sqrt(3))*(Edz2dxy))))} , 
       {4, 3, (3/2)*((sqrt(7/5))*((-1)*(Edx2y2dxz) + (I)*(Edx2y2dyz) + (I)*(Edxzdxy) + Edyzdxy))} , 
       {4,-3, (3/2)*((sqrt(7/5))*(Edx2y2dxz + (I)*(Edx2y2dyz + Edxzdxy + (I)*(Edyzdxy))))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (-2*I)*(Edx2y2dxy) + (-1)*(Edxydxy)))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (2*I)*(Edx2y2dxy) + (-1)*(Edxydxy)))} }

</code>

###

</hidden>
<hidden **The Hamiltonian on a basis of spherical Harmonics** >

###

|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $|$ \frac{1}{2} (\text{Edx2y2dxz}-i \text{Edx2y2dyz}+i \text{Edxzdxy}+\text{Edyzdxy}) $|$ \frac{\text{Edx2y2dz2}+i \text{Edz2dxy}}{\sqrt{2}} $|$ \frac{1}{2} (-\text{Edx2y2dxz}-i (\text{Edx2y2dyz}+\text{Edxzdxy})+\text{Edyzdxy}) $|$ \frac{1}{2} (\text{Edx2y2dx2y2}+2 i \text{Edx2y2dxy}-\text{Edxydxy}) $|
^$ {Y_{-1}^{(2)}} $|$ \frac{1}{2} (\text{Edx2y2dxz}+i (\text{Edx2y2dyz}-\text{Edxzdxy})+\text{Edyzdxy}) $|$ \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} $|$ \frac{\text{Edz2dxz}+i \text{Edz2dyz}}{\sqrt{2}} $|$ \frac{1}{2} (-\text{Edxzdxz}-2 i \text{Edyzdxz}+\text{Edyzdyz}) $|$ \frac{1}{2} (\text{Edx2y2dxz}+i (\text{Edx2y2dyz}+\text{Edxzdxy}+i \text{Edyzdxy})) $|
^$ {Y_{0}^{(2)}} $|$ \frac{\text{Edx2y2dz2}-i \text{Edz2dxy}}{\sqrt{2}} $|$ \frac{\text{Edz2dxz}-i \text{Edz2dyz}}{\sqrt{2}} $|$ \text{Edz2dz2} $|$ -\frac{\text{Edz2dxz}+i \text{Edz2dyz}}{\sqrt{2}} $|$ \frac{\text{Edx2y2dz2}+i \text{Edz2dxy}}{\sqrt{2}} $|
^$ {Y_{1}^{(2)}} $|$ \frac{1}{2} (-\text{Edx2y2dxz}+i (\text{Edx2y2dyz}+\text{Edxzdxy})+\text{Edyzdxy}) $|$ \frac{1}{2} (-\text{Edxzdxz}+2 i \text{Edyzdxz}+\text{Edyzdyz}) $|$ -\frac{\text{Edz2dxz}-i \text{Edz2dyz}}{\sqrt{2}} $|$ \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} $|$ \frac{1}{2} (-\text{Edx2y2dxz}+i (\text{Edx2y2dyz}-\text{Edxzdxy}+i \text{Edyzdxy})) $|
^$ {Y_{2}^{(2)}} $|$ \frac{1}{2} (\text{Edx2y2dx2y2}-2 i \text{Edx2y2dxy}-\text{Edxydxy}) $|$ \frac{1}{2} (\text{Edx2y2dxz}-i (\text{Edx2y2dyz}+\text{Edxzdxy})-\text{Edyzdxy}) $|$ \frac{\text{Edx2y2dz2}-i \text{Edz2dxy}}{\sqrt{2}} $|$ \frac{1}{2} (-\text{Edx2y2dxz}-i \text{Edx2y2dyz}+i \text{Edxzdxy}-\text{Edyzdxy}) $|$ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^
^$ d_{x^2-y^2} $|$ \text{Edx2y2dx2y2} $|$ \text{Edx2y2dz2} $|$ \text{Edx2y2dyz} $|$ \text{Edx2y2dxz} $|$ \text{Edx2y2dxy} $|
^$ d_{3z^2-r^2} $|$ \text{Edx2y2dz2} $|$ \text{Edz2dz2} $|$ \text{Edz2dyz} $|$ \text{Edz2dxz} $|$ \text{Edz2dxy} $|
^$ d_{\text{yz}} $|$ \text{Edx2y2dyz} $|$ \text{Edz2dyz} $|$ \text{Edyzdyz} $|$ \text{Edyzdxz} $|$ \text{Edyzdxy} $|
^$ d_{\text{xz}} $|$ \text{Edx2y2dxz} $|$ \text{Edz2dxz} $|$ \text{Edyzdxz} $|$ \text{Edxzdxz} $|$ \text{Edxzdxy} $|
^$ d_{\text{xy}} $|$ \text{Edx2y2dxy} $|$ \text{Edz2dxy} $|$ \text{Edyzdxy} $|$ \text{Edxzdxy} $|$ \text{Edxydxy} $|


###

</hidden>
<hidden **Rotation matrix used** >

###

|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|
^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|
^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Edx2y2dx2y2}$$ | {{:physics_chemistry:pointgroup:ci_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |
^ ^$$\text{Edz2dz2}$$ | {{:physics_chemistry:pointgroup:ci_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
^ ^$$\text{Edyzdyz}$$ | {{:physics_chemistry:pointgroup:ci_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
^ ^$$\text{Edxzdxz}$$ | {{:physics_chemistry:pointgroup:ci_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
^ ^$$\text{Edxydxy}$$ | {{:physics_chemistry:pointgroup:ci_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |


###

</hidden>
==== Potential for f orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{7} (\text{Efx3fx3}+\text{Efxy2z2fxy2z2}+\text{Efxyzfxyz}+\text{Efy3fy3}+\text{Efyz2x2fyz2x2}+\text{Efz3fz3}+\text{Efzx2y2fzx2y2}) & k=0\land m=0 \\
 \frac{5}{56} \left(2 \left(\sqrt{6} \text{Efx3fx3}+\sqrt{10} \text{Efx3fxy2z2}-\sqrt{6} \text{Efy3fy3}+\sqrt{10} \text{Efy3fyz2x2}-2 \sqrt{10} \text{Efz3fzx2y2}\right)-i \left(\sqrt{6} \text{Efx3fy3}+\sqrt{10} \text{Efx3fyz2x2}+5 \sqrt{6} \text{Efxy2z2fyz2x2}+4 \sqrt{10} \text{Efxyzfz3}-\sqrt{10} \text{Efy3fxy2z2}\right)\right) & k=2\land m=-2 \\
 \frac{5}{56} \left(-\sqrt{6} \text{Efx3fz3}+\sqrt{10} \text{Efx3fzx2y2}-5 \sqrt{6} \text{Efxy2z2fzx2y2}-i \left(4 \sqrt{10} \text{Efxyzfx3}+\sqrt{6} \text{Efy3fz3}+\sqrt{10} \text{Efy3fzx2y2}+5 \sqrt{6} \text{Efyz2x2fzx2y2}-\sqrt{10} \text{Efz3fyz2x2}\right)-4 \sqrt{10} \text{Efxyzfy3}-\sqrt{10} \text{Efz3fxy2z2}\right) & k=2\land m=-1 \\
 -\frac{5}{14} \left(\text{Efx3fx3}-\sqrt{15} \text{Efx3fxy2z2}+\text{Efy3fy3}+\sqrt{15} \text{Efy3fyz2x2}-2 \text{Efz3fz3}\right) & k=2\land m=0 \\
 \frac{5}{56} \left(\sqrt{6} \text{Efx3fz3}-\sqrt{10} \text{Efx3fzx2y2}+5 \sqrt{6} \text{Efxy2z2fzx2y2}-i \left(4 \sqrt{10} \text{Efxyzfx3}+\sqrt{6} \text{Efy3fz3}+\sqrt{10} \text{Efy3fzx2y2}+5 \sqrt{6} \text{Efyz2x2fzx2y2}-\sqrt{10} \text{Efz3fyz2x2}\right)+4 \sqrt{10} \text{Efxyzfy3}+\sqrt{10} \text{Efz3fxy2z2}\right) & k=2\land m=1 \\
 \frac{5}{56} \left(2 \left(\sqrt{6} \text{Efx3fx3}+\sqrt{10} \text{Efx3fxy2z2}-\sqrt{6} \text{Efy3fy3}+\sqrt{10} \text{Efy3fyz2x2}-2 \sqrt{10} \text{Efz3fzx2y2}\right)+i \left(\sqrt{6} \text{Efx3fy3}+\sqrt{10} \text{Efx3fyz2x2}+5 \sqrt{6} \text{Efxy2z2fyz2x2}+4 \sqrt{10} \text{Efxyzfz3}-\sqrt{10} \text{Efy3fxy2z2}\right)\right) & k=2\land m=2 \\
 \frac{3 \left(3 \sqrt{5} \text{Efx3fx3}+2 \sqrt{3} \text{Efx3fxy2z2}-8 i \sqrt{3} \text{Efx3fyz2x2}-3 \sqrt{5} \text{Efxy2z2fxy2z2}-4 \sqrt{5} \text{Efxyzfxyz}+8 i \sqrt{5} \text{Efxyzfzx2y2}-8 i \sqrt{3} \text{Efy3fxy2z2}+3 \sqrt{5} \text{Efy3fy3}-2 \sqrt{3} \text{Efy3fyz2x2}-3 \sqrt{5} \text{Efyz2x2fyz2x2}+4 \sqrt{5} \text{Efzx2y2fzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=-4 \\
 -\frac{3 \left(3 \sqrt{5} \text{Efx3fz3}+\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \sqrt{3} \text{Efxyzfx3}+i \sqrt{5} \text{Efxyzfxy2z2}-\sqrt{3} \text{Efxyzfy3}+\sqrt{5} \text{Efxyzfyz2x2}-3 i \sqrt{5} \text{Efy3fz3}+i \sqrt{3} \text{Efy3fzx2y2}-i \sqrt{5} \text{Efyz2x2fzx2y2}-3 \sqrt{3} \text{Efz3fxy2z2}-3 i \sqrt{3} \text{Efz3fyz2x2}\right)}{4 \sqrt{7}} & k=4\land m=-3 \\
 \frac{3}{56} \left(-3 \sqrt{10} \text{Efx3fx3}+2 \sqrt{6} \text{Efx3fxy2z2}+4 i \left(3 \sqrt{10} \text{Efx3fy3}-2 \sqrt{6} \text{Efx3fyz2x2}+\sqrt{10} \text{Efxy2z2fyz2x2}-\sqrt{6} \text{Efxyzfz3}+2 \sqrt{6} \text{Efy3fxy2z2}\right)+7 \sqrt{10} \text{Efxy2z2fxy2z2}+3 \sqrt{10} \text{Efy3fy3}+2 \sqrt{6} \text{Efy3fyz2x2}-7 \sqrt{10} \text{Efyz2x2fyz2x2}-4 \sqrt{6} \text{Efz3fzx2y2}\right) & k=4\land m=-2 \\
 \frac{3}{28} \left(-3 \sqrt{5} \text{Efx3fz3}-9 \sqrt{3} \text{Efx3fzx2y2}-\sqrt{5} \text{Efxy2z2fzx2y2}+i \left(\sqrt{3} \text{Efxyzfx3}-7 \sqrt{5} \text{Efxyzfxy2z2}-3 \sqrt{5} \text{Efy3fz3}+9 \sqrt{3} \text{Efy3fzx2y2}-\sqrt{5} \text{Efyz2x2fzx2y2}+5 \sqrt{3} \text{Efz3fyz2x2}\right)+\sqrt{3} \text{Efxyzfy3}+7 \sqrt{5} \text{Efxyzfyz2x2}-5 \sqrt{3} \text{Efz3fxy2z2}\right) & k=4\land m=-1 \\
 \frac{3}{56} \left(9 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+7 \text{Efxy2z2fxy2z2}-28 \text{Efxyzfxyz}+9 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+7 \text{Efyz2x2fyz2x2}+24 \text{Efz3fz3}-28 \text{Efzx2y2fzx2y2}\right) & k=4\land m=0 \\
 \frac{3}{28} \left(3 \sqrt{5} \text{Efx3fz3}+9 \sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \left(\sqrt{3} \text{Efxyzfx3}-7 \sqrt{5} \text{Efxyzfxy2z2}-3 \sqrt{5} \text{Efy3fz3}+9 \sqrt{3} \text{Efy3fzx2y2}-\sqrt{5} \text{Efyz2x2fzx2y2}+5 \sqrt{3} \text{Efz3fyz2x2}\right)-\sqrt{3} \text{Efxyzfy3}-7 \sqrt{5} \text{Efxyzfyz2x2}+5 \sqrt{3} \text{Efz3fxy2z2}\right) & k=4\land m=1 \\
 \frac{3}{56} \left(-3 \sqrt{10} \text{Efx3fx3}+2 \sqrt{6} \text{Efx3fxy2z2}-4 i \left(3 \sqrt{10} \text{Efx3fy3}-2 \sqrt{6} \text{Efx3fyz2x2}+\sqrt{10} \text{Efxy2z2fyz2x2}-\sqrt{6} \text{Efxyzfz3}+2 \sqrt{6} \text{Efy3fxy2z2}\right)+7 \sqrt{10} \text{Efxy2z2fxy2z2}+3 \sqrt{10} \text{Efy3fy3}+2 \sqrt{6} \text{Efy3fyz2x2}-7 \sqrt{10} \text{Efyz2x2fyz2x2}-4 \sqrt{6} \text{Efz3fzx2y2}\right) & k=4\land m=2 \\
 \frac{3 \left(3 \sqrt{5} \text{Efx3fz3}+\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}-i \left(\sqrt{3} \text{Efxyzfx3}+\sqrt{5} \text{Efxyzfxy2z2}-3 \sqrt{5} \text{Efy3fz3}+\sqrt{3} \text{Efy3fzx2y2}-\sqrt{5} \text{Efyz2x2fzx2y2}-3 \sqrt{3} \text{Efz3fyz2x2}\right)-\sqrt{3} \text{Efxyzfy3}+\sqrt{5} \text{Efxyzfyz2x2}-3 \sqrt{3} \text{Efz3fxy2z2}\right)}{4 \sqrt{7}} & k=4\land m=3 \\
 \frac{3 \left(3 \sqrt{5} \text{Efx3fx3}+2 \sqrt{3} \text{Efx3fxy2z2}+8 i \sqrt{3} \text{Efx3fyz2x2}-3 \sqrt{5} \text{Efxy2z2fxy2z2}-4 \sqrt{5} \text{Efxyzfxyz}-8 i \sqrt{5} \text{Efxyzfzx2y2}+8 i \sqrt{3} \text{Efy3fxy2z2}+3 \sqrt{5} \text{Efy3fy3}-2 \sqrt{3} \text{Efy3fyz2x2}-3 \sqrt{5} \text{Efyz2x2fyz2x2}+4 \sqrt{5} \text{Efzx2y2fzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=4 \\
 \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Efx3fx3}-6 \sqrt{5} \text{Efx3fxy2z2}-10 i \sqrt{3} \text{Efx3fy3}-6 i \sqrt{5} \text{Efx3fyz2x2}+3 \sqrt{3} \text{Efxy2z2fxy2z2}+6 i \sqrt{3} \text{Efxy2z2fyz2x2}+6 i \sqrt{5} \text{Efy3fxy2z2}-5 \sqrt{3} \text{Efy3fy3}-6 \sqrt{5} \text{Efy3fyz2x2}-3 \sqrt{3} \text{Efyz2x2fyz2x2}\right) & k=6\land m=-6 \\
 \frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}+i \sqrt{15} \text{Efxyzfx3}-3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}-i \sqrt{15} \text{Efy3fzx2y2}-3 i \text{Efyz2x2fzx2y2}\right) & k=6\land m=-5 \\
 -\frac{13 \left(15 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}-8 i \sqrt{15} \text{Efx3fyz2x2}-15 \text{Efxy2z2fxy2z2}+24 \text{Efxyzfxyz}-48 i \text{Efxyzfzx2y2}-8 i \sqrt{15} \text{Efy3fxy2z2}+15 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}-15 \text{Efyz2x2fyz2x2}-24 \text{Efzx2y2fzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=-4 \\
 \frac{13 \left(2 \sqrt{15} \text{Efx3fz3}-9 \text{Efx3fzx2y2}-3 \sqrt{15} \text{Efxy2z2fzx2y2}-9 i \text{Efxyzfx3}-3 i \sqrt{15} \text{Efxyzfxy2z2}+9 \text{Efxyzfy3}-3 \sqrt{15} \text{Efxyzfyz2x2}-2 i \sqrt{15} \text{Efy3fz3}-9 i \text{Efy3fzx2y2}+3 i \sqrt{15} \text{Efyz2x2fzx2y2}-6 \text{Efz3fxy2z2}-6 i \text{Efz3fyz2x2}\right)}{40 \sqrt{7}} & k=6\land m=-3 \\
 \frac{13 \left(5 \sqrt{15} \text{Efx3fx3}+34 \text{Efx3fxy2z2}+2 i \sqrt{15} \text{Efx3fy3}-26 i \text{Efx3fyz2x2}+3 \sqrt{15} \text{Efxy2z2fxy2z2}-14 i \sqrt{15} \text{Efxy2z2fyz2x2}+64 i \text{Efxyzfz3}+26 i \text{Efy3fxy2z2}-5 \sqrt{15} \text{Efy3fy3}+34 \text{Efy3fyz2x2}-3 \sqrt{15} \text{Efyz2x2fyz2x2}+64 \text{Efz3fzx2y2}\right)}{160 \sqrt{7}} & k=6\land m=-2 \\
 \frac{13}{280} \left(-5 \sqrt{42} \text{Efx3fz3}+2 \sqrt{70} \text{Efx3fzx2y2}+2 \sqrt{42} \text{Efxy2z2fzx2y2}+i \left(\sqrt{70} \text{Efxyzfx3}+3 \sqrt{42} \text{Efxyzfxy2z2}-5 \sqrt{42} \text{Efy3fz3}-2 \sqrt{70} \text{Efy3fzx2y2}+2 \sqrt{42} \text{Efyz2x2fzx2y2}+5 \sqrt{70} \text{Efz3fyz2x2}\right)+\sqrt{70} \text{Efxyzfy3}-3 \sqrt{42} \text{Efxyzfyz2x2}-5 \sqrt{70} \text{Efz3fxy2z2}\right) & k=6\land m=-1 \\
 -\frac{13}{560} \left(25 \text{Efx3fx3}+14 \sqrt{15} \text{Efx3fxy2z2}+39 \text{Efxy2z2fxy2z2}-24 \text{Efxyzfxyz}+25 \text{Efy3fy3}-14 \sqrt{15} \text{Efy3fyz2x2}+39 \text{Efyz2x2fyz2x2}-80 \text{Efz3fz3}-24 \text{Efzx2y2fzx2y2}\right) & k=6\land m=0 \\
 \frac{13}{280} \left(5 \sqrt{42} \text{Efx3fz3}-2 \sqrt{70} \text{Efx3fzx2y2}-2 \sqrt{42} \text{Efxy2z2fzx2y2}+i \left(\sqrt{70} \text{Efxyzfx3}+3 \sqrt{42} \text{Efxyzfxy2z2}-5 \sqrt{42} \text{Efy3fz3}-2 \sqrt{70} \text{Efy3fzx2y2}+2 \sqrt{42} \text{Efyz2x2fzx2y2}+5 \sqrt{70} \text{Efz3fyz2x2}\right)-\sqrt{70} \text{Efxyzfy3}+3 \sqrt{42} \text{Efxyzfyz2x2}+5 \sqrt{70} \text{Efz3fxy2z2}\right) & k=6\land m=1 \\
 \frac{13 \left(5 \sqrt{15} \text{Efx3fx3}+34 \text{Efx3fxy2z2}-2 i \sqrt{15} \text{Efx3fy3}+26 i \text{Efx3fyz2x2}+3 \sqrt{15} \text{Efxy2z2fxy2z2}+14 i \sqrt{15} \text{Efxy2z2fyz2x2}-64 i \text{Efxyzfz3}-26 i \text{Efy3fxy2z2}-5 \sqrt{15} \text{Efy3fy3}+34 \text{Efy3fyz2x2}-3 \sqrt{15} \text{Efyz2x2fyz2x2}+64 \text{Efz3fzx2y2}\right)}{160 \sqrt{7}} & k=6\land m=2 \\
 -\frac{13 \left(2 \sqrt{15} \text{Efx3fz3}-9 \text{Efx3fzx2y2}-3 \sqrt{15} \text{Efxy2z2fzx2y2}+9 i \text{Efxyzfx3}+3 i \sqrt{15} \text{Efxyzfxy2z2}+9 \text{Efxyzfy3}-3 \sqrt{15} \text{Efxyzfyz2x2}+2 i \sqrt{15} \text{Efy3fz3}+9 i \text{Efy3fzx2y2}-3 i \sqrt{15} \text{Efyz2x2fzx2y2}-6 \text{Efz3fxy2z2}+6 i \text{Efz3fyz2x2}\right)}{40 \sqrt{7}} & k=6\land m=3 \\
 -\frac{13 \left(15 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+8 i \sqrt{15} \text{Efx3fyz2x2}-15 \text{Efxy2z2fxy2z2}+24 \text{Efxyzfxyz}+48 i \text{Efxyzfzx2y2}+8 i \sqrt{15} \text{Efy3fxy2z2}+15 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}-15 \text{Efyz2x2fyz2x2}-24 \text{Efzx2y2fzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=4 \\
 -\frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}-i \sqrt{15} \text{Efxyzfx3}+3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}+i \sqrt{15} \text{Efy3fzx2y2}+3 i \text{Efyz2x2fzx2y2}\right) & k=6\land m=5 \\
 \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Efx3fx3}-6 \sqrt{5} \text{Efx3fxy2z2}+10 i \sqrt{3} \text{Efx3fy3}+6 i \sqrt{5} \text{Efx3fyz2x2}+3 \sqrt{3} \text{Efxy2z2fxy2z2}-6 i \sqrt{3} \text{Efxy2z2fyz2x2}-6 i \sqrt{5} \text{Efy3fxy2z2}-5 \sqrt{3} \text{Efy3fy3}-6 \sqrt{5} \text{Efy3fyz2x2}-3 \sqrt{3} \text{Efyz2x2fyz2x2}\right) & k=6\land m=6
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Efx3fx3 + Efxy2z2fxy2z2 + Efxyzfxyz + Efy3fy3 + Efyz2x2fyz2x2 + Efz3fz3 + Efzx2y2fzx2y2)/7, k == 0 && m == 0}, {(5*((-I)*(Sqrt[6]*Efx3fy3 + Sqrt[10]*Efx3fyz2x2 + 5*Sqrt[6]*Efxy2z2fyz2x2 + 4*Sqrt[10]*Efxyzfz3 - Sqrt[10]*Efy3fxy2z2) + 2*(Sqrt[6]*Efx3fx3 + Sqrt[10]*Efx3fxy2z2 - Sqrt[6]*Efy3fy3 + Sqrt[10]*Efy3fyz2x2 - 2*Sqrt[10]*Efz3fzx2y2)))/56, k == 2 && m == -2}, {(5*(-(Sqrt[6]*Efx3fz3) + Sqrt[10]*Efx3fzx2y2 - 5*Sqrt[6]*Efxy2z2fzx2y2 - 4*Sqrt[10]*Efxyzfy3 - Sqrt[10]*Efz3fxy2z2 - I*(4*Sqrt[10]*Efxyzfx3 + Sqrt[6]*Efy3fz3 + Sqrt[10]*Efy3fzx2y2 + 5*Sqrt[6]*Efyz2x2fzx2y2 - Sqrt[10]*Efz3fyz2x2)))/56, k == 2 && m == -1}, {(-5*(Efx3fx3 - Sqrt[15]*Efx3fxy2z2 + Efy3fy3 + Sqrt[15]*Efy3fyz2x2 - 2*Efz3fz3))/14, k == 2 && m == 0}, {(5*(Sqrt[6]*Efx3fz3 - Sqrt[10]*Efx3fzx2y2 + 5*Sqrt[6]*Efxy2z2fzx2y2 + 4*Sqrt[10]*Efxyzfy3 + Sqrt[10]*Efz3fxy2z2 - I*(4*Sqrt[10]*Efxyzfx3 + Sqrt[6]*Efy3fz3 + Sqrt[10]*Efy3fzx2y2 + 5*Sqrt[6]*Efyz2x2fzx2y2 - Sqrt[10]*Efz3fyz2x2)))/56, k == 2 && m == 1}, {(5*(I*(Sqrt[6]*Efx3fy3 + Sqrt[10]*Efx3fyz2x2 + 5*Sqrt[6]*Efxy2z2fyz2x2 + 4*Sqrt[10]*Efxyzfz3 - Sqrt[10]*Efy3fxy2z2) + 2*(Sqrt[6]*Efx3fx3 + Sqrt[10]*Efx3fxy2z2 - Sqrt[6]*Efy3fy3 + Sqrt[10]*Efy3fyz2x2 - 2*Sqrt[10]*Efz3fzx2y2)))/56, k == 2 && m == 2}, {(3*(3*Sqrt[5]*Efx3fx3 + 2*Sqrt[3]*Efx3fxy2z2 - (8*I)*Sqrt[3]*Efx3fyz2x2 - 3*Sqrt[5]*Efxy2z2fxy2z2 - 4*Sqrt[5]*Efxyzfxyz + (8*I)*Sqrt[5]*Efxyzfzx2y2 - (8*I)*Sqrt[3]*Efy3fxy2z2 + 3*Sqrt[5]*Efy3fy3 - 2*Sqrt[3]*Efy3fyz2x2 - 3*Sqrt[5]*Efyz2x2fyz2x2 + 4*Sqrt[5]*Efzx2y2fzx2y2))/(8*Sqrt[14]), k == 4 && m == -4}, {(-3*(3*Sqrt[5]*Efx3fz3 + Sqrt[3]*Efx3fzx2y2 + Sqrt[5]*Efxy2z2fzx2y2 + I*Sqrt[3]*Efxyzfx3 + I*Sqrt[5]*Efxyzfxy2z2 - Sqrt[3]*Efxyzfy3 + Sqrt[5]*Efxyzfyz2x2 - (3*I)*Sqrt[5]*Efy3fz3 + I*Sqrt[3]*Efy3fzx2y2 - I*Sqrt[5]*Efyz2x2fzx2y2 - 3*Sqrt[3]*Efz3fxy2z2 - (3*I)*Sqrt[3]*Efz3fyz2x2))/(4*Sqrt[7]), k == 4 && m == -3}, {(3*(-3*Sqrt[10]*Efx3fx3 + 2*Sqrt[6]*Efx3fxy2z2 + 7*Sqrt[10]*Efxy2z2fxy2z2 + (4*I)*(3*Sqrt[10]*Efx3fy3 - 2*Sqrt[6]*Efx3fyz2x2 + Sqrt[10]*Efxy2z2fyz2x2 - Sqrt[6]*Efxyzfz3 + 2*Sqrt[6]*Efy3fxy2z2) + 3*Sqrt[10]*Efy3fy3 + 2*Sqrt[6]*Efy3fyz2x2 - 7*Sqrt[10]*Efyz2x2fyz2x2 - 4*Sqrt[6]*Efz3fzx2y2))/56, k == 4 && m == -2}, {(3*(-3*Sqrt[5]*Efx3fz3 - 9*Sqrt[3]*Efx3fzx2y2 - Sqrt[5]*Efxy2z2fzx2y2 + Sqrt[3]*Efxyzfy3 + 7*Sqrt[5]*Efxyzfyz2x2 - 5*Sqrt[3]*Efz3fxy2z2 + I*(Sqrt[3]*Efxyzfx3 - 7*Sqrt[5]*Efxyzfxy2z2 - 3*Sqrt[5]*Efy3fz3 + 9*Sqrt[3]*Efy3fzx2y2 - Sqrt[5]*Efyz2x2fzx2y2 + 5*Sqrt[3]*Efz3fyz2x2)))/28, k == 4 && m == -1}, {(3*(9*Efx3fx3 - 2*Sqrt[15]*Efx3fxy2z2 + 7*Efxy2z2fxy2z2 - 28*Efxyzfxyz + 9*Efy3fy3 + 2*Sqrt[15]*Efy3fyz2x2 + 7*Efyz2x2fyz2x2 + 24*Efz3fz3 - 28*Efzx2y2fzx2y2))/56, k == 4 && m == 0}, {(3*(3*Sqrt[5]*Efx3fz3 + 9*Sqrt[3]*Efx3fzx2y2 + Sqrt[5]*Efxy2z2fzx2y2 - Sqrt[3]*Efxyzfy3 - 7*Sqrt[5]*Efxyzfyz2x2 + 5*Sqrt[3]*Efz3fxy2z2 + I*(Sqrt[3]*Efxyzfx3 - 7*Sqrt[5]*Efxyzfxy2z2 - 3*Sqrt[5]*Efy3fz3 + 9*Sqrt[3]*Efy3fzx2y2 - Sqrt[5]*Efyz2x2fzx2y2 + 5*Sqrt[3]*Efz3fyz2x2)))/28, k == 4 && m == 1}, {(3*(-3*Sqrt[10]*Efx3fx3 + 2*Sqrt[6]*Efx3fxy2z2 + 7*Sqrt[10]*Efxy2z2fxy2z2 - (4*I)*(3*Sqrt[10]*Efx3fy3 - 2*Sqrt[6]*Efx3fyz2x2 + Sqrt[10]*Efxy2z2fyz2x2 - Sqrt[6]*Efxyzfz3 + 2*Sqrt[6]*Efy3fxy2z2) + 3*Sqrt[10]*Efy3fy3 + 2*Sqrt[6]*Efy3fyz2x2 - 7*Sqrt[10]*Efyz2x2fyz2x2 - 4*Sqrt[6]*Efz3fzx2y2))/56, k == 4 && m == 2}, {(3*(3*Sqrt[5]*Efx3fz3 + Sqrt[3]*Efx3fzx2y2 + Sqrt[5]*Efxy2z2fzx2y2 - Sqrt[3]*Efxyzfy3 + Sqrt[5]*Efxyzfyz2x2 - 3*Sqrt[3]*Efz3fxy2z2 - I*(Sqrt[3]*Efxyzfx3 + Sqrt[5]*Efxyzfxy2z2 - 3*Sqrt[5]*Efy3fz3 + Sqrt[3]*Efy3fzx2y2 - Sqrt[5]*Efyz2x2fzx2y2 - 3*Sqrt[3]*Efz3fyz2x2)))/(4*Sqrt[7]), k == 4 && m == 3}, {(3*(3*Sqrt[5]*Efx3fx3 + 2*Sqrt[3]*Efx3fxy2z2 + (8*I)*Sqrt[3]*Efx3fyz2x2 - 3*Sqrt[5]*Efxy2z2fxy2z2 - 4*Sqrt[5]*Efxyzfxyz - (8*I)*Sqrt[5]*Efxyzfzx2y2 + (8*I)*Sqrt[3]*Efy3fxy2z2 + 3*Sqrt[5]*Efy3fy3 - 2*Sqrt[3]*Efy3fyz2x2 - 3*Sqrt[5]*Efyz2x2fyz2x2 + 4*Sqrt[5]*Efzx2y2fzx2y2))/(8*Sqrt[14]), k == 4 && m == 4}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Efx3fx3 - 6*Sqrt[5]*Efx3fxy2z2 - (10*I)*Sqrt[3]*Efx3fy3 - (6*I)*Sqrt[5]*Efx3fyz2x2 + 3*Sqrt[3]*Efxy2z2fxy2z2 + (6*I)*Sqrt[3]*Efxy2z2fyz2x2 + (6*I)*Sqrt[5]*Efy3fxy2z2 - 5*Sqrt[3]*Efy3fy3 - 6*Sqrt[5]*Efy3fyz2x2 - 3*Sqrt[3]*Efyz2x2fyz2x2))/160, k == 6 && m == -6}, {(13*Sqrt[11/7]*(Sqrt[15]*Efx3fzx2y2 - 3*Efxy2z2fzx2y2 + I*Sqrt[15]*Efxyzfx3 - (3*I)*Efxyzfxy2z2 + Sqrt[15]*Efxyzfy3 + 3*Efxyzfyz2x2 - I*Sqrt[15]*Efy3fzx2y2 - (3*I)*Efyz2x2fzx2y2))/40, k == 6 && m == -5}, {(-13*(15*Efx3fx3 + 2*Sqrt[15]*Efx3fxy2z2 - (8*I)*Sqrt[15]*Efx3fyz2x2 - 15*Efxy2z2fxy2z2 + 24*Efxyzfxyz - (48*I)*Efxyzfzx2y2 - (8*I)*Sqrt[15]*Efy3fxy2z2 + 15*Efy3fy3 - 2*Sqrt[15]*Efy3fyz2x2 - 15*Efyz2x2fyz2x2 - 24*Efzx2y2fzx2y2))/(80*Sqrt[14]), k == 6 && m == -4}, {(13*(2*Sqrt[15]*Efx3fz3 - 9*Efx3fzx2y2 - 3*Sqrt[15]*Efxy2z2fzx2y2 - (9*I)*Efxyzfx3 - (3*I)*Sqrt[15]*Efxyzfxy2z2 + 9*Efxyzfy3 - 3*Sqrt[15]*Efxyzfyz2x2 - (2*I)*Sqrt[15]*Efy3fz3 - (9*I)*Efy3fzx2y2 + (3*I)*Sqrt[15]*Efyz2x2fzx2y2 - 6*Efz3fxy2z2 - (6*I)*Efz3fyz2x2))/(40*Sqrt[7]), k == 6 && m == -3}, {(13*(5*Sqrt[15]*Efx3fx3 + 34*Efx3fxy2z2 + (2*I)*Sqrt[15]*Efx3fy3 - (26*I)*Efx3fyz2x2 + 3*Sqrt[15]*Efxy2z2fxy2z2 - (14*I)*Sqrt[15]*Efxy2z2fyz2x2 + (64*I)*Efxyzfz3 + (26*I)*Efy3fxy2z2 - 5*Sqrt[15]*Efy3fy3 + 34*Efy3fyz2x2 - 3*Sqrt[15]*Efyz2x2fyz2x2 + 64*Efz3fzx2y2))/(160*Sqrt[7]), k == 6 && m == -2}, {(13*(-5*Sqrt[42]*Efx3fz3 + 2*Sqrt[70]*Efx3fzx2y2 + 2*Sqrt[42]*Efxy2z2fzx2y2 + Sqrt[70]*Efxyzfy3 - 3*Sqrt[42]*Efxyzfyz2x2 - 5*Sqrt[70]*Efz3fxy2z2 + I*(Sqrt[70]*Efxyzfx3 + 3*Sqrt[42]*Efxyzfxy2z2 - 5*Sqrt[42]*Efy3fz3 - 2*Sqrt[70]*Efy3fzx2y2 + 2*Sqrt[42]*Efyz2x2fzx2y2 + 5*Sqrt[70]*Efz3fyz2x2)))/280, k == 6 && m == -1}, {(-13*(25*Efx3fx3 + 14*Sqrt[15]*Efx3fxy2z2 + 39*Efxy2z2fxy2z2 - 24*Efxyzfxyz + 25*Efy3fy3 - 14*Sqrt[15]*Efy3fyz2x2 + 39*Efyz2x2fyz2x2 - 80*Efz3fz3 - 24*Efzx2y2fzx2y2))/560, k == 6 && m == 0}, {(13*(5*Sqrt[42]*Efx3fz3 - 2*Sqrt[70]*Efx3fzx2y2 - 2*Sqrt[42]*Efxy2z2fzx2y2 - Sqrt[70]*Efxyzfy3 + 3*Sqrt[42]*Efxyzfyz2x2 + 5*Sqrt[70]*Efz3fxy2z2 + I*(Sqrt[70]*Efxyzfx3 + 3*Sqrt[42]*Efxyzfxy2z2 - 5*Sqrt[42]*Efy3fz3 - 2*Sqrt[70]*Efy3fzx2y2 + 2*Sqrt[42]*Efyz2x2fzx2y2 + 5*Sqrt[70]*Efz3fyz2x2)))/280, k == 6 && m == 1}, {(13*(5*Sqrt[15]*Efx3fx3 + 34*Efx3fxy2z2 - (2*I)*Sqrt[15]*Efx3fy3 + (26*I)*Efx3fyz2x2 + 3*Sqrt[15]*Efxy2z2fxy2z2 + (14*I)*Sqrt[15]*Efxy2z2fyz2x2 - (64*I)*Efxyzfz3 - (26*I)*Efy3fxy2z2 - 5*Sqrt[15]*Efy3fy3 + 34*Efy3fyz2x2 - 3*Sqrt[15]*Efyz2x2fyz2x2 + 64*Efz3fzx2y2))/(160*Sqrt[7]), k == 6 && m == 2}, {(-13*(2*Sqrt[15]*Efx3fz3 - 9*Efx3fzx2y2 - 3*Sqrt[15]*Efxy2z2fzx2y2 + (9*I)*Efxyzfx3 + (3*I)*Sqrt[15]*Efxyzfxy2z2 + 9*Efxyzfy3 - 3*Sqrt[15]*Efxyzfyz2x2 + (2*I)*Sqrt[15]*Efy3fz3 + (9*I)*Efy3fzx2y2 - (3*I)*Sqrt[15]*Efyz2x2fzx2y2 - 6*Efz3fxy2z2 + (6*I)*Efz3fyz2x2))/(40*Sqrt[7]), k == 6 && m == 3}, {(-13*(15*Efx3fx3 + 2*Sqrt[15]*Efx3fxy2z2 + (8*I)*Sqrt[15]*Efx3fyz2x2 - 15*Efxy2z2fxy2z2 + 24*Efxyzfxyz + (48*I)*Efxyzfzx2y2 + (8*I)*Sqrt[15]*Efy3fxy2z2 + 15*Efy3fy3 - 2*Sqrt[15]*Efy3fyz2x2 - 15*Efyz2x2fyz2x2 - 24*Efzx2y2fzx2y2))/(80*Sqrt[14]), k == 6 && m == 4}, {(-13*Sqrt[11/7]*(Sqrt[15]*Efx3fzx2y2 - 3*Efxy2z2fzx2y2 - I*Sqrt[15]*Efxyzfx3 + (3*I)*Efxyzfxy2z2 + Sqrt[15]*Efxyzfy3 + 3*Efxyzfyz2x2 + I*Sqrt[15]*Efy3fzx2y2 + (3*I)*Efyz2x2fzx2y2))/40, k == 6 && m == 5}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Efx3fx3 - 6*Sqrt[5]*Efx3fxy2z2 + (10*I)*Sqrt[3]*Efx3fy3 + (6*I)*Sqrt[5]*Efx3fyz2x2 + 3*Sqrt[3]*Efxy2z2fxy2z2 - (6*I)*Sqrt[3]*Efxy2z2fyz2x2 - (6*I)*Sqrt[5]*Efy3fxy2z2 - 5*Sqrt[3]*Efy3fy3 - 6*Sqrt[5]*Efy3fyz2x2 - 3*Sqrt[3]*Efyz2x2fyz2x2))/160, k == 6 && m == 6}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_Ci.Quanty>

Akm = {{0, 0, (1/7)*(Efx3fx3 + Efxy2z2fxy2z2 + Efxyzfxyz + Efy3fy3 + Efyz2x2fyz2x2 + Efz3fz3 + Efzx2y2fzx2y2)} , 
       {2, 0, (-5/14)*(Efx3fx3 + (-1)*((sqrt(15))*(Efx3fxy2z2)) + Efy3fy3 + (sqrt(15))*(Efy3fyz2x2) + (-2)*(Efz3fz3))} , 
       {2,-1, (5/56)*((-1)*((sqrt(6))*(Efx3fz3)) + (sqrt(10))*(Efx3fzx2y2) + (-5)*((sqrt(6))*(Efxy2z2fzx2y2)) + (-4)*((sqrt(10))*(Efxyzfy3)) + (-1)*((sqrt(10))*(Efz3fxy2z2)) + (-I)*((4)*((sqrt(10))*(Efxyzfx3)) + (sqrt(6))*(Efy3fz3) + (sqrt(10))*(Efy3fzx2y2) + (5)*((sqrt(6))*(Efyz2x2fzx2y2)) + (-1)*((sqrt(10))*(Efz3fyz2x2))))} , 
       {2, 1, (5/56)*((sqrt(6))*(Efx3fz3) + (-1)*((sqrt(10))*(Efx3fzx2y2)) + (5)*((sqrt(6))*(Efxy2z2fzx2y2)) + (4)*((sqrt(10))*(Efxyzfy3)) + (sqrt(10))*(Efz3fxy2z2) + (-I)*((4)*((sqrt(10))*(Efxyzfx3)) + (sqrt(6))*(Efy3fz3) + (sqrt(10))*(Efy3fzx2y2) + (5)*((sqrt(6))*(Efyz2x2fzx2y2)) + (-1)*((sqrt(10))*(Efz3fyz2x2))))} , 
       {2,-2, (5/56)*((-I)*((sqrt(6))*(Efx3fy3) + (sqrt(10))*(Efx3fyz2x2) + (5)*((sqrt(6))*(Efxy2z2fyz2x2)) + (4)*((sqrt(10))*(Efxyzfz3)) + (-1)*((sqrt(10))*(Efy3fxy2z2))) + (2)*((sqrt(6))*(Efx3fx3) + (sqrt(10))*(Efx3fxy2z2) + (-1)*((sqrt(6))*(Efy3fy3)) + (sqrt(10))*(Efy3fyz2x2) + (-2)*((sqrt(10))*(Efz3fzx2y2))))} , 
       {2, 2, (5/56)*((I)*((sqrt(6))*(Efx3fy3) + (sqrt(10))*(Efx3fyz2x2) + (5)*((sqrt(6))*(Efxy2z2fyz2x2)) + (4)*((sqrt(10))*(Efxyzfz3)) + (-1)*((sqrt(10))*(Efy3fxy2z2))) + (2)*((sqrt(6))*(Efx3fx3) + (sqrt(10))*(Efx3fxy2z2) + (-1)*((sqrt(6))*(Efy3fy3)) + (sqrt(10))*(Efy3fyz2x2) + (-2)*((sqrt(10))*(Efz3fzx2y2))))} , 
       {4, 0, (3/56)*((9)*(Efx3fx3) + (-2)*((sqrt(15))*(Efx3fxy2z2)) + (7)*(Efxy2z2fxy2z2) + (-28)*(Efxyzfxyz) + (9)*(Efy3fy3) + (2)*((sqrt(15))*(Efy3fyz2x2)) + (7)*(Efyz2x2fyz2x2) + (24)*(Efz3fz3) + (-28)*(Efzx2y2fzx2y2))} , 
       {4,-1, (3/28)*((-3)*((sqrt(5))*(Efx3fz3)) + (-9)*((sqrt(3))*(Efx3fzx2y2)) + (-1)*((sqrt(5))*(Efxy2z2fzx2y2)) + (sqrt(3))*(Efxyzfy3) + (7)*((sqrt(5))*(Efxyzfyz2x2)) + (-5)*((sqrt(3))*(Efz3fxy2z2)) + (I)*((sqrt(3))*(Efxyzfx3) + (-7)*((sqrt(5))*(Efxyzfxy2z2)) + (-3)*((sqrt(5))*(Efy3fz3)) + (9)*((sqrt(3))*(Efy3fzx2y2)) + (-1)*((sqrt(5))*(Efyz2x2fzx2y2)) + (5)*((sqrt(3))*(Efz3fyz2x2))))} , 
       {4, 1, (3/28)*((3)*((sqrt(5))*(Efx3fz3)) + (9)*((sqrt(3))*(Efx3fzx2y2)) + (sqrt(5))*(Efxy2z2fzx2y2) + (-1)*((sqrt(3))*(Efxyzfy3)) + (-7)*((sqrt(5))*(Efxyzfyz2x2)) + (5)*((sqrt(3))*(Efz3fxy2z2)) + (I)*((sqrt(3))*(Efxyzfx3) + (-7)*((sqrt(5))*(Efxyzfxy2z2)) + (-3)*((sqrt(5))*(Efy3fz3)) + (9)*((sqrt(3))*(Efy3fzx2y2)) + (-1)*((sqrt(5))*(Efyz2x2fzx2y2)) + (5)*((sqrt(3))*(Efz3fyz2x2))))} , 
       {4, 2, (3/56)*((-3)*((sqrt(10))*(Efx3fx3)) + (2)*((sqrt(6))*(Efx3fxy2z2)) + (7)*((sqrt(10))*(Efxy2z2fxy2z2)) + (-4*I)*((3)*((sqrt(10))*(Efx3fy3)) + (-2)*((sqrt(6))*(Efx3fyz2x2)) + (sqrt(10))*(Efxy2z2fyz2x2) + (-1)*((sqrt(6))*(Efxyzfz3)) + (2)*((sqrt(6))*(Efy3fxy2z2))) + (3)*((sqrt(10))*(Efy3fy3)) + (2)*((sqrt(6))*(Efy3fyz2x2)) + (-7)*((sqrt(10))*(Efyz2x2fyz2x2)) + (-4)*((sqrt(6))*(Efz3fzx2y2)))} , 
       {4,-2, (3/56)*((-3)*((sqrt(10))*(Efx3fx3)) + (2)*((sqrt(6))*(Efx3fxy2z2)) + (7)*((sqrt(10))*(Efxy2z2fxy2z2)) + (4*I)*((3)*((sqrt(10))*(Efx3fy3)) + (-2)*((sqrt(6))*(Efx3fyz2x2)) + (sqrt(10))*(Efxy2z2fyz2x2) + (-1)*((sqrt(6))*(Efxyzfz3)) + (2)*((sqrt(6))*(Efy3fxy2z2))) + (3)*((sqrt(10))*(Efy3fy3)) + (2)*((sqrt(6))*(Efy3fyz2x2)) + (-7)*((sqrt(10))*(Efyz2x2fyz2x2)) + (-4)*((sqrt(6))*(Efz3fzx2y2)))} , 
       {4, 3, (3/4)*((1/(sqrt(7)))*((3)*((sqrt(5))*(Efx3fz3)) + (sqrt(3))*(Efx3fzx2y2) + (sqrt(5))*(Efxy2z2fzx2y2) + (-1)*((sqrt(3))*(Efxyzfy3)) + (sqrt(5))*(Efxyzfyz2x2) + (-3)*((sqrt(3))*(Efz3fxy2z2)) + (-I)*((sqrt(3))*(Efxyzfx3) + (sqrt(5))*(Efxyzfxy2z2) + (-3)*((sqrt(5))*(Efy3fz3)) + (sqrt(3))*(Efy3fzx2y2) + (-1)*((sqrt(5))*(Efyz2x2fzx2y2)) + (-3)*((sqrt(3))*(Efz3fyz2x2)))))} , 
       {4,-3, (-3/4)*((1/(sqrt(7)))*((3)*((sqrt(5))*(Efx3fz3)) + (sqrt(3))*(Efx3fzx2y2) + (sqrt(5))*(Efxy2z2fzx2y2) + (I)*((sqrt(3))*(Efxyzfx3)) + (I)*((sqrt(5))*(Efxyzfxy2z2)) + (-1)*((sqrt(3))*(Efxyzfy3)) + (sqrt(5))*(Efxyzfyz2x2) + (-3*I)*((sqrt(5))*(Efy3fz3)) + (I)*((sqrt(3))*(Efy3fzx2y2)) + (-I)*((sqrt(5))*(Efyz2x2fzx2y2)) + (-3)*((sqrt(3))*(Efz3fxy2z2)) + (-3*I)*((sqrt(3))*(Efz3fyz2x2))))} , 
       {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Efx3fx3)) + (2)*((sqrt(3))*(Efx3fxy2z2)) + (-8*I)*((sqrt(3))*(Efx3fyz2x2)) + (-3)*((sqrt(5))*(Efxy2z2fxy2z2)) + (-4)*((sqrt(5))*(Efxyzfxyz)) + (8*I)*((sqrt(5))*(Efxyzfzx2y2)) + (-8*I)*((sqrt(3))*(Efy3fxy2z2)) + (3)*((sqrt(5))*(Efy3fy3)) + (-2)*((sqrt(3))*(Efy3fyz2x2)) + (-3)*((sqrt(5))*(Efyz2x2fyz2x2)) + (4)*((sqrt(5))*(Efzx2y2fzx2y2))))} , 
       {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Efx3fx3)) + (2)*((sqrt(3))*(Efx3fxy2z2)) + (8*I)*((sqrt(3))*(Efx3fyz2x2)) + (-3)*((sqrt(5))*(Efxy2z2fxy2z2)) + (-4)*((sqrt(5))*(Efxyzfxyz)) + (-8*I)*((sqrt(5))*(Efxyzfzx2y2)) + (8*I)*((sqrt(3))*(Efy3fxy2z2)) + (3)*((sqrt(5))*(Efy3fy3)) + (-2)*((sqrt(3))*(Efy3fyz2x2)) + (-3)*((sqrt(5))*(Efyz2x2fyz2x2)) + (4)*((sqrt(5))*(Efzx2y2fzx2y2))))} , 
       {6, 0, (-13/560)*((25)*(Efx3fx3) + (14)*((sqrt(15))*(Efx3fxy2z2)) + (39)*(Efxy2z2fxy2z2) + (-24)*(Efxyzfxyz) + (25)*(Efy3fy3) + (-14)*((sqrt(15))*(Efy3fyz2x2)) + (39)*(Efyz2x2fyz2x2) + (-80)*(Efz3fz3) + (-24)*(Efzx2y2fzx2y2))} , 
       {6, 1, (13/280)*((5)*((sqrt(42))*(Efx3fz3)) + (-2)*((sqrt(70))*(Efx3fzx2y2)) + (-2)*((sqrt(42))*(Efxy2z2fzx2y2)) + (-1)*((sqrt(70))*(Efxyzfy3)) + (3)*((sqrt(42))*(Efxyzfyz2x2)) + (5)*((sqrt(70))*(Efz3fxy2z2)) + (I)*((sqrt(70))*(Efxyzfx3) + (3)*((sqrt(42))*(Efxyzfxy2z2)) + (-5)*((sqrt(42))*(Efy3fz3)) + (-2)*((sqrt(70))*(Efy3fzx2y2)) + (2)*((sqrt(42))*(Efyz2x2fzx2y2)) + (5)*((sqrt(70))*(Efz3fyz2x2))))} , 
       {6,-1, (13/280)*((-5)*((sqrt(42))*(Efx3fz3)) + (2)*((sqrt(70))*(Efx3fzx2y2)) + (2)*((sqrt(42))*(Efxy2z2fzx2y2)) + (sqrt(70))*(Efxyzfy3) + (-3)*((sqrt(42))*(Efxyzfyz2x2)) + (-5)*((sqrt(70))*(Efz3fxy2z2)) + (I)*((sqrt(70))*(Efxyzfx3) + (3)*((sqrt(42))*(Efxyzfxy2z2)) + (-5)*((sqrt(42))*(Efy3fz3)) + (-2)*((sqrt(70))*(Efy3fzx2y2)) + (2)*((sqrt(42))*(Efyz2x2fzx2y2)) + (5)*((sqrt(70))*(Efz3fyz2x2))))} , 
       {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Efx3fx3)) + (34)*(Efx3fxy2z2) + (-2*I)*((sqrt(15))*(Efx3fy3)) + (26*I)*(Efx3fyz2x2) + (3)*((sqrt(15))*(Efxy2z2fxy2z2)) + (14*I)*((sqrt(15))*(Efxy2z2fyz2x2)) + (-64*I)*(Efxyzfz3) + (-26*I)*(Efy3fxy2z2) + (-5)*((sqrt(15))*(Efy3fy3)) + (34)*(Efy3fyz2x2) + (-3)*((sqrt(15))*(Efyz2x2fyz2x2)) + (64)*(Efz3fzx2y2)))} , 
       {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Efx3fx3)) + (34)*(Efx3fxy2z2) + (2*I)*((sqrt(15))*(Efx3fy3)) + (-26*I)*(Efx3fyz2x2) + (3)*((sqrt(15))*(Efxy2z2fxy2z2)) + (-14*I)*((sqrt(15))*(Efxy2z2fyz2x2)) + (64*I)*(Efxyzfz3) + (26*I)*(Efy3fxy2z2) + (-5)*((sqrt(15))*(Efy3fy3)) + (34)*(Efy3fyz2x2) + (-3)*((sqrt(15))*(Efyz2x2fyz2x2)) + (64)*(Efz3fzx2y2)))} , 
       {6, 3, (-13/40)*((1/(sqrt(7)))*((2)*((sqrt(15))*(Efx3fz3)) + (-9)*(Efx3fzx2y2) + (-3)*((sqrt(15))*(Efxy2z2fzx2y2)) + (9*I)*(Efxyzfx3) + (3*I)*((sqrt(15))*(Efxyzfxy2z2)) + (9)*(Efxyzfy3) + (-3)*((sqrt(15))*(Efxyzfyz2x2)) + (2*I)*((sqrt(15))*(Efy3fz3)) + (9*I)*(Efy3fzx2y2) + (-3*I)*((sqrt(15))*(Efyz2x2fzx2y2)) + (-6)*(Efz3fxy2z2) + (6*I)*(Efz3fyz2x2)))} , 
       {6,-3, (13/40)*((1/(sqrt(7)))*((2)*((sqrt(15))*(Efx3fz3)) + (-9)*(Efx3fzx2y2) + (-3)*((sqrt(15))*(Efxy2z2fzx2y2)) + (-9*I)*(Efxyzfx3) + (-3*I)*((sqrt(15))*(Efxyzfxy2z2)) + (9)*(Efxyzfy3) + (-3)*((sqrt(15))*(Efxyzfyz2x2)) + (-2*I)*((sqrt(15))*(Efy3fz3)) + (-9*I)*(Efy3fzx2y2) + (3*I)*((sqrt(15))*(Efyz2x2fzx2y2)) + (-6)*(Efz3fxy2z2) + (-6*I)*(Efz3fyz2x2)))} , 
       {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Efx3fx3) + (2)*((sqrt(15))*(Efx3fxy2z2)) + (-8*I)*((sqrt(15))*(Efx3fyz2x2)) + (-15)*(Efxy2z2fxy2z2) + (24)*(Efxyzfxyz) + (-48*I)*(Efxyzfzx2y2) + (-8*I)*((sqrt(15))*(Efy3fxy2z2)) + (15)*(Efy3fy3) + (-2)*((sqrt(15))*(Efy3fyz2x2)) + (-15)*(Efyz2x2fyz2x2) + (-24)*(Efzx2y2fzx2y2)))} , 
       {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Efx3fx3) + (2)*((sqrt(15))*(Efx3fxy2z2)) + (8*I)*((sqrt(15))*(Efx3fyz2x2)) + (-15)*(Efxy2z2fxy2z2) + (24)*(Efxyzfxyz) + (48*I)*(Efxyzfzx2y2) + (8*I)*((sqrt(15))*(Efy3fxy2z2)) + (15)*(Efy3fy3) + (-2)*((sqrt(15))*(Efy3fyz2x2)) + (-15)*(Efyz2x2fyz2x2) + (-24)*(Efzx2y2fzx2y2)))} , 
       {6, 5, (-13/40)*((sqrt(11/7))*((sqrt(15))*(Efx3fzx2y2) + (-3)*(Efxy2z2fzx2y2) + (-I)*((sqrt(15))*(Efxyzfx3)) + (3*I)*(Efxyzfxy2z2) + (sqrt(15))*(Efxyzfy3) + (3)*(Efxyzfyz2x2) + (I)*((sqrt(15))*(Efy3fzx2y2)) + (3*I)*(Efyz2x2fzx2y2)))} , 
       {6,-5, (13/40)*((sqrt(11/7))*((sqrt(15))*(Efx3fzx2y2) + (-3)*(Efxy2z2fzx2y2) + (I)*((sqrt(15))*(Efxyzfx3)) + (-3*I)*(Efxyzfxy2z2) + (sqrt(15))*(Efxyzfy3) + (3)*(Efxyzfyz2x2) + (-I)*((sqrt(15))*(Efy3fzx2y2)) + (-3*I)*(Efyz2x2fzx2y2)))} , 
       {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Efx3fx3)) + (-6)*((sqrt(5))*(Efx3fxy2z2)) + (-10*I)*((sqrt(3))*(Efx3fy3)) + (-6*I)*((sqrt(5))*(Efx3fyz2x2)) + (3)*((sqrt(3))*(Efxy2z2fxy2z2)) + (6*I)*((sqrt(3))*(Efxy2z2fyz2x2)) + (6*I)*((sqrt(5))*(Efy3fxy2z2)) + (-5)*((sqrt(3))*(Efy3fy3)) + (-6)*((sqrt(5))*(Efy3fyz2x2)) + (-3)*((sqrt(3))*(Efyz2x2fyz2x2))))} , 
       {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Efx3fx3)) + (-6)*((sqrt(5))*(Efx3fxy2z2)) + (10*I)*((sqrt(3))*(Efx3fy3)) + (6*I)*((sqrt(5))*(Efx3fyz2x2)) + (3)*((sqrt(3))*(Efxy2z2fxy2z2)) + (-6*I)*((sqrt(3))*(Efxy2z2fyz2x2)) + (-6*I)*((sqrt(5))*(Efy3fxy2z2)) + (-5)*((sqrt(3))*(Efy3fy3)) + (-6)*((sqrt(5))*(Efy3fyz2x2)) + (-3)*((sqrt(3))*(Efyz2x2fyz2x2))))} }

</code>

###

</hidden>
<hidden **The Hamiltonian on a basis of spherical Harmonics** >

###

|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ {Y_{-3}^{(3)}} $|$ \frac{1}{16} \left(5 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+3 \text{Efxy2z2fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{8} \left(\sqrt{10} \text{Efx3fzx2y2}-\sqrt{2} \left(\sqrt{3} \text{Efxy2z2fzx2y2}+i \sqrt{5} \text{Efxyzfx3}-i \sqrt{3} \text{Efxyzfxy2z2}+\sqrt{5} \text{Efxyzfy3}+\sqrt{3} \text{Efxyzfyz2x2}+i \left(\sqrt{5} \text{Efy3fzx2y2}+\sqrt{3} \text{Efyz2x2fzx2y2}\right)\right)\right) $|$ \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}+2 i \sqrt{15} \text{Efx3fy3}-2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}+2 i \sqrt{15} \text{Efxy2z2fyz2x2}+2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{4} \left(\sqrt{5} \text{Efx3fz3}-i \sqrt{5} \text{Efy3fz3}-\sqrt{3} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}-8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}-8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|$ \frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}+i \sqrt{15} \text{Efxyzfx3}-3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}-i \left(\sqrt{15} \text{Efy3fzx2y2}+3 \text{Efyz2x2fzx2y2}\right)}{4 \sqrt{6}} $|$ \frac{1}{16} \left(-5 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+10 i \text{Efx3fy3}+2 i \sqrt{15} \text{Efx3fyz2x2}-3 \text{Efxy2z2fxy2z2}-6 i \text{Efxy2z2fyz2x2}-2 i \sqrt{15} \text{Efy3fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right) $|
^$ {Y_{-2}^{(3)}} $|$ \frac{1}{8} \left(\sqrt{10} \text{Efx3fzx2y2}-\sqrt{6} \text{Efxy2z2fzx2y2}+i \left(\sqrt{10} \text{Efxyzfx3}-\sqrt{6} \text{Efxyzfxy2z2}+i \sqrt{10} \text{Efxyzfy3}+i \sqrt{6} \text{Efxyzfyz2x2}+\sqrt{10} \text{Efy3fzx2y2}+\sqrt{6} \text{Efyz2x2fzx2y2}\right)\right) $|$ \frac{\text{Efxyzfxyz}+\text{Efzx2y2fzx2y2}}{2} $|$ -\frac{\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \sqrt{3} \text{Efxyzfx3}+i \sqrt{5} \text{Efxyzfxy2z2}+\sqrt{3} \text{Efxyzfy3}-\sqrt{5} \text{Efxyzfyz2x2}-i \sqrt{3} \text{Efy3fzx2y2}+i \sqrt{5} \text{Efyz2x2fzx2y2}}{4 \sqrt{2}} $|$ \frac{\text{Efz3fzx2y2}+i \text{Efxyzfz3}}{\sqrt{2}} $|$ \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}-\sqrt{6} \text{Efxyzfy3}+\sqrt{10} \text{Efxyzfyz2x2}+i \sqrt{6} \text{Efy3fzx2y2}-i \sqrt{10} \text{Efyz2x2fzx2y2}\right) $|$ \frac{1}{2} (-\text{Efxyzfxyz}+2 i \text{Efxyzfzx2y2}+\text{Efzx2y2fzx2y2}) $|$ -\frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}+i \sqrt{15} \text{Efxyzfx3}-3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}-i \left(\sqrt{15} \text{Efy3fzx2y2}+3 \text{Efyz2x2fzx2y2}\right)}{4 \sqrt{6}} $|
^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}-2 i \sqrt{15} \text{Efx3fy3}+2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}-2 i \sqrt{15} \text{Efxy2z2fyz2x2}-2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{8} \left(-\sqrt{6} \text{Efx3fzx2y2}-\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}-\sqrt{6} \text{Efxyzfy3}+\sqrt{10} \text{Efxyzfyz2x2}-i \sqrt{6} \text{Efy3fzx2y2}+i \sqrt{10} \text{Efyz2x2fzx2y2}\right) $|$ \frac{1}{16} \left(3 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+5 \text{Efxy2z2fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{4} \left(-\sqrt{3} \text{Efx3fz3}-i \sqrt{3} \text{Efy3fz3}-\sqrt{5} (\text{Efz3fxy2z2}-i \text{Efz3fyz2x2})\right) $|$ \frac{1}{16} \left(-3 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}-6 i \text{Efx3fy3}+2 i \sqrt{15} \text{Efx3fyz2x2}-5 \text{Efxy2z2fxy2z2}+10 i \text{Efxy2z2fyz2x2}-2 i \sqrt{15} \text{Efy3fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) $|$ -\frac{\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \sqrt{3} \text{Efxyzfx3}+i \sqrt{5} \text{Efxyzfxy2z2}-\sqrt{3} \text{Efxyzfy3}+\sqrt{5} \text{Efxyzfyz2x2}+i \sqrt{3} \text{Efy3fzx2y2}-i \sqrt{5} \text{Efyz2x2fzx2y2}}{4 \sqrt{2}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}-8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}-8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|
^$ {Y_{0}^{(3)}} $|$ \frac{1}{4} \left(\sqrt{5} \text{Efx3fz3}+i \left(\sqrt{5} \text{Efy3fz3}+\sqrt{3} (\text{Efz3fyz2x2}+i \text{Efz3fxy2z2})\right)\right) $|$ \frac{\text{Efz3fzx2y2}-i \text{Efxyzfz3}}{\sqrt{2}} $|$ \frac{1}{4} \left(-\sqrt{3} \text{Efx3fz3}+i \sqrt{3} \text{Efy3fz3}-\sqrt{5} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) $|$ \text{Efz3fz3} $|$ \frac{1}{4} \left(\sqrt{3} \text{Efx3fz3}+i \sqrt{3} \text{Efy3fz3}+\sqrt{5} (\text{Efz3fxy2z2}-i \text{Efz3fyz2x2})\right) $|$ \frac{\text{Efz3fzx2y2}+i \text{Efxyzfz3}}{\sqrt{2}} $|$ \frac{1}{4} \left(-\sqrt{5} \text{Efx3fz3}+i \sqrt{5} \text{Efy3fz3}+\sqrt{3} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) $|
^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}+8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}+8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}-i \sqrt{6} \text{Efxyzfx3}-i \sqrt{10} \text{Efxyzfxy2z2}-\sqrt{6} \text{Efxyzfy3}+\sqrt{10} \text{Efxyzfyz2x2}-i \sqrt{6} \text{Efy3fzx2y2}+i \sqrt{10} \text{Efyz2x2fzx2y2}\right) $|$ \frac{1}{16} \left(-3 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+6 i \text{Efx3fy3}-2 i \sqrt{15} \text{Efx3fyz2x2}-5 \text{Efxy2z2fxy2z2}-10 i \text{Efxy2z2fyz2x2}+2 i \sqrt{15} \text{Efy3fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{4} \left(\sqrt{3} \text{Efx3fz3}-i \sqrt{3} \text{Efy3fz3}+\sqrt{5} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) $|$ \frac{1}{16} \left(3 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+5 \text{Efxy2z2fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}+\sqrt{6} \text{Efxyzfy3}-\sqrt{10} \text{Efxyzfyz2x2}-i \sqrt{6} \text{Efy3fzx2y2}+i \sqrt{10} \text{Efyz2x2fzx2y2}\right) $|$ \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}+2 i \sqrt{15} \text{Efx3fy3}-2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}+2 i \sqrt{15} \text{Efxy2z2fyz2x2}+2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|
^$ {Y_{2}^{(3)}} $|$ \frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}-i \sqrt{15} \text{Efxyzfx3}+3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}+i \sqrt{15} \text{Efy3fzx2y2}+3 i \text{Efyz2x2fzx2y2}}{4 \sqrt{6}} $|$ \frac{1}{2} (-\text{Efxyzfxyz}-2 i \text{Efxyzfzx2y2}+\text{Efzx2y2fzx2y2}) $|$ \frac{1}{8} \left(-\sqrt{6} \text{Efx3fzx2y2}-\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}+\sqrt{6} \text{Efxyzfy3}-\sqrt{10} \text{Efxyzfyz2x2}+i \sqrt{6} \text{Efy3fzx2y2}-i \sqrt{10} \text{Efyz2x2fzx2y2}\right) $|$ \frac{\text{Efz3fzx2y2}-i \text{Efxyzfz3}}{\sqrt{2}} $|$ \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}-i \sqrt{6} \text{Efxyzfx3}-i \sqrt{10} \text{Efxyzfxy2z2}+\sqrt{6} \text{Efxyzfy3}-\sqrt{10} \text{Efxyzfyz2x2}+i \sqrt{6} \text{Efy3fzx2y2}-i \sqrt{10} \text{Efyz2x2fzx2y2}\right) $|$ \frac{\text{Efxyzfxyz}+\text{Efzx2y2fzx2y2}}{2} $|$ \frac{1}{8} \left(-\sqrt{10} \text{Efx3fzx2y2}+\sqrt{6} \text{Efxy2z2fzx2y2}+i \sqrt{10} \text{Efxyzfx3}-i \sqrt{6} \text{Efxyzfxy2z2}+\sqrt{10} \text{Efxyzfy3}+\sqrt{6} \text{Efxyzfyz2x2}+i \left(\sqrt{10} \text{Efy3fzx2y2}+\sqrt{6} \text{Efyz2x2fzx2y2}\right)\right) $|
^$ {Y_{3}^{(3)}} $|$ \frac{1}{16} \left(-5 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}-10 i \text{Efx3fy3}-2 i \sqrt{15} \text{Efx3fyz2x2}-3 \text{Efxy2z2fxy2z2}+6 i \text{Efxy2z2fyz2x2}+2 i \sqrt{15} \text{Efy3fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right) $|$ -\frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}-i \sqrt{15} \text{Efxyzfx3}+3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}+i \sqrt{15} \text{Efy3fzx2y2}+3 i \text{Efyz2x2fzx2y2}}{4 \sqrt{6}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}+8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}+8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{4} \left(-\sqrt{5} \text{Efx3fz3}-i \sqrt{5} \text{Efy3fz3}+\sqrt{3} (\text{Efz3fxy2z2}-i \text{Efz3fyz2x2})\right) $|$ \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}-2 i \sqrt{15} \text{Efx3fy3}+2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}-2 i \sqrt{15} \text{Efxy2z2fyz2x2}-2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) $|$ \frac{1}{8} \left(-\sqrt{10} \text{Efx3fzx2y2}+\sqrt{6} \text{Efxy2z2fzx2y2}-i \sqrt{10} \text{Efxyzfx3}+i \sqrt{6} \text{Efxyzfxy2z2}+\sqrt{10} \text{Efxyzfy3}+\sqrt{6} \text{Efxyzfyz2x2}-i \left(\sqrt{10} \text{Efy3fzx2y2}+\sqrt{6} \text{Efyz2x2fzx2y2}\right)\right) $|$ \frac{1}{16} \left(5 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+3 \text{Efxy2z2fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right) $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
^$ f_{\text{xyz}} $|$ \text{Efxyzfxyz} $|$ \text{Efxyzfx3} $|$ \text{Efxyzfy3} $|$ \text{Efxyzfz3} $|$ \text{Efxyzfxy2z2} $|$ \text{Efxyzfyz2x2} $|$ \text{Efxyzfzx2y2} $|
^$ f_{x\left(5x^2-r^2\right)} $|$ \text{Efxyzfx3} $|$ \text{Efx3fx3} $|$ \text{Efx3fy3} $|$ \text{Efx3fz3} $|$ \text{Efx3fxy2z2} $|$ \text{Efx3fyz2x2} $|$ \text{Efx3fzx2y2} $|
^$ f_{y\left(5y^2-r^2\right)} $|$ \text{Efxyzfy3} $|$ \text{Efx3fy3} $|$ \text{Efy3fy3} $|$ \text{Efy3fz3} $|$ \text{Efy3fxy2z2} $|$ \text{Efy3fyz2x2} $|$ \text{Efy3fzx2y2} $|
^$ f_{z\left(5z^2-r^2\right)} $|$ \text{Efxyzfz3} $|$ \text{Efx3fz3} $|$ \text{Efy3fz3} $|$ \text{Efz3fz3} $|$ \text{Efz3fxy2z2} $|$ \text{Efz3fyz2x2} $|$ \text{Efz3fzx2y2} $|
^$ f_{x\left(y^2-z^2\right)} $|$ \text{Efxyzfxy2z2} $|$ \text{Efx3fxy2z2} $|$ \text{Efy3fxy2z2} $|$ \text{Efz3fxy2z2} $|$ \text{Efxy2z2fxy2z2} $|$ \text{Efxy2z2fyz2x2} $|$ \text{Efxy2z2fzx2y2} $|
^$ f_{y\left(z^2-x^2\right)} $|$ \text{Efxyzfyz2x2} $|$ \text{Efx3fyz2x2} $|$ \text{Efy3fyz2x2} $|$ \text{Efz3fyz2x2} $|$ \text{Efxy2z2fyz2x2} $|$ \text{Efyz2x2fyz2x2} $|$ \text{Efyz2x2fzx2y2} $|
^$ f_{z\left(x^2-y^2\right)} $|$ \text{Efxyzfzx2y2} $|$ \text{Efx3fzx2y2} $|$ \text{Efy3fzx2y2} $|$ \text{Efz3fzx2y2} $|$ \text{Efxy2z2fzx2y2} $|$ \text{Efyz2x2fzx2y2} $|$ \text{Efzx2y2fzx2y2} $|


###

</hidden>
<hidden **Rotation matrix used** >

###

|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Efxyzfxyz}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
^ ^$$\text{Efx3fx3}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: |
^ ^$$\text{Efy3fy3}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: |
^ ^$$\text{Efz3fz3}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
^ ^$$\text{Efxy2z2fxy2z2}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: |
^ ^$$\text{Efyz2x2fyz2x2}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: |
^ ^$$\text{Efzx2y2fzx2y2}$$ | {{:physics_chemistry:pointgroup:ci_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |


###

</hidden>
===== Coupling between two shells =====



###

Click on one of the subsections to expand it or <hiddenSwitch expand all> 

###

==== Potential for s-d orbital mixing ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 0 & k=0\land m=0 \\
 A(2,2)-i B(2,2) & k=2\land m=-2 \\
 -A(2,1)+i B(2,1) & k=2\land m=-1 \\
 A(2,0) & k=2\land m=0 \\
 A(2,1)+i B(2,1) & k=2\land m=1 \\
 A(2,2)+i B(2,2) & k=2\land m=2
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_Ci.Quanty>

Akm = {{2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , 
       {2, 1, A(2,1) + (I)*(B(2,1))} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} }

</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{A(2,2)+i B(2,2)}{\sqrt{5}} $|$ -\frac{A(2,1)+i B(2,1)}{\sqrt{5}} $|$ \frac{A(2,0)}{\sqrt{5}} $|$ \frac{A(2,1)-i B(2,1)}{\sqrt{5}} $|$ \frac{A(2,2)-i B(2,2)}{\sqrt{5}} $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^
^$ \text{s} $|$ \sqrt{\frac{2}{5}} A(2,2) $|$ \frac{A(2,0)}{\sqrt{5}} $|$ \sqrt{\frac{2}{5}} B(2,1) $|$ -\sqrt{\frac{2}{5}} A(2,1) $|$ -\sqrt{\frac{2}{5}} B(2,2) $|


###

</hidden>
==== Potential for p-f orbital mixing ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 0 & k=0\land m=0 \\
 A(2,2)-i B(2,2) & k=2\land m=-2 \\
 -A(2,1)+i B(2,1) & k=2\land m=-1 \\
 A(2,0) & k=2\land m=0 \\
 A(2,1)+i B(2,1) & k=2\land m=1 \\
 A(2,2)+i B(2,2) & k=2\land m=2 \\
 A(4,4)-i B(4,4) & k=4\land m=-4 \\
 -A(4,3)+i B(4,3) & k=4\land m=-3 \\
 A(4,2)-i B(4,2) & k=4\land m=-2 \\
 -A(4,1)+i B(4,1) & k=4\land m=-1 \\
 A(4,0) & k=4\land m=0 \\
 A(4,1)+i B(4,1) & k=4\land m=1 \\
 A(4,2)+i B(4,2) & k=4\land m=2 \\
 A(4,3)+i B(4,3) & k=4\land m=3 \\
 A(4,4)+i B(4,4) & k=4\land m=4
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_Ci.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {-A[4, 1] + I*B[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1] + I*B[4, 1], k == 4 && m == 1}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 3] + I*B[4, 3], k == 4 && m == 3}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_Ci.Quanty>

Akm = {{2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , 
       {2, 1, A(2,1) + (I)*(B(2,1))} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1)*(A(4,1)) + (I)*(B(4,1))} , 
       {4, 1, A(4,1) + (I)*(B(4,1))} , 
       {4,-2, A(4,2) + (-I)*(B(4,2))} , 
       {4, 2, A(4,2) + (I)*(B(4,2))} , 
       {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , 
       {4, 3, A(4,3) + (I)*(B(4,3))} , 
       {4,-4, A(4,4) + (-I)*(B(4,4))} , 
       {4, 4, A(4,4) + (I)*(B(4,4))} }

</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{3 (A(2,2)+i B(2,2))}{\sqrt{35}}-\frac{A(4,2)+i B(4,2)}{3 \sqrt{21}} $|$ \frac{A(4,1)+i B(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} (A(2,1)+i B(2,1)) $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ \frac{27 A(2,1)-5 \sqrt{30} A(4,1)-27 i B(2,1)+5 i \sqrt{30} B(4,1)}{45 \sqrt{7}} $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i B(4,2)) $|$ \frac{1}{3} (-A(4,3)+i B(4,3)) $|$ -\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}} $|
^$ {Y_{0}^{(1)}} $|$ -\frac{A(4,3)+i B(4,3)}{3 \sqrt{3}} $|$ \sqrt{\frac{3}{35}} (A(2,2)+i B(2,2))+\frac{2 (A(4,2)+i B(4,2))}{3 \sqrt{7}} $|$ -\frac{2}{5} \sqrt{\frac{6}{7}} (A(2,1)+i B(2,1))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,1)+i B(4,1)) $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ \frac{2}{5} \sqrt{\frac{6}{7}} (A(2,1)-i B(2,1))+\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,1)-i B(4,1)) $|$ \sqrt{\frac{3}{35}} (A(2,2)-i B(2,2))+\frac{2 (A(4,2)-i B(4,2))}{3 \sqrt{7}} $|$ \frac{A(4,3)-i B(4,3)}{3 \sqrt{3}} $|
^$ {Y_{1}^{(1)}} $|$ -\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}} $|$ \frac{1}{3} (A(4,3)+i B(4,3)) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i B(4,2)) $|$ \frac{5 \sqrt{30} (A(4,1)+i B(4,1))-27 (A(2,1)+i B(2,1))}{45 \sqrt{7}} $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ \frac{1}{105} \left(3 \sqrt{210} (A(2,1)-i B(2,1))-5 \sqrt{7} (A(4,1)-i B(4,1))\right) $|$ \frac{3 (A(2,2)-i B(2,2))}{\sqrt{35}}-\frac{A(4,2)-i B(4,2)}{3 \sqrt{21}} $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
^$ p_x $|$ \sqrt{\frac{6}{35}} B(2,1)-\frac{B(4,1)}{3 \sqrt{7}}+\frac{1}{3} B(4,3) $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)+7 B(4,4)\right)\right) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} A(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} A(4,1) $|$ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ \sqrt{\frac{6}{35}} B(2,2)-\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} $|$ \frac{1}{21} \left(\sqrt{7} A(4,1)-7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) $|
^$ p_y $|$ \frac{1}{21} \left(\sqrt{7} A(4,1)+7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) $|$ \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)-7 B(4,4)\right)\right) $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ \frac{1}{45} \sqrt{\frac{2}{7}} \left(5 \sqrt{30} B(4,1)-27 B(2,1)\right) $|$ -\sqrt{\frac{6}{35}} B(2,2)+\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} $|$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ \frac{1}{21} \left(\sqrt{7} B(4,1)+7 B(4,3)\right)-\sqrt{\frac{6}{35}} B(2,1) $|
^$ p_z $|$ -\sqrt{\frac{6}{35}} B(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} B(4,2) $|$ \frac{1}{630} \left(54 \sqrt{14} A(2,1)+5 \sqrt{15} \left(3 \sqrt{7} A(4,1)-7 A(4,3)\right)\right) $|$ \frac{1}{630} \left(-54 \sqrt{14} B(2,1)-5 \sqrt{15} \left(3 \sqrt{7} B(4,1)+7 B(4,3)\right)\right) $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ \sqrt{\frac{6}{35}} A(2,1)+\frac{5 A(4,1)}{6 \sqrt{7}}+\frac{1}{6} A(4,3) $|$ \sqrt{\frac{6}{35}} B(2,1)+\frac{5 B(4,1)}{6 \sqrt{7}}-\frac{1}{6} B(4,3) $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $|


###

</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> | | | | | |

###