~~CLOSETOC~~

====== Orientation Z ======

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

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In the S6 Point Group, with orientation Z there are the following symmetry operations

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{{:physics_chemistry:pointgroup:s6_z.png}}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |

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

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  * [[physics_chemistry:point_groups:s6:orientation_z|Point Group S6 with orientation Z]]

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

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|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_3 \,{\text{(2)}} $  ^  $ \text{i} \,{\text{(1)}} $  ^  $ S_6 \,{\text{(2)}} $  ^
^ $ A_g $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
^ $ E_g $ |  $ 2 $ |  $ -1 $ |  $ 2 $ |  $ -1 $ |
^ $ A_u $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |
^ $ E_u $ |  $ 2 $ |  $ -1 $ |  $ -2 $ |  $ 1 $ |

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

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|  $  $  ^  $ A_g $  ^  $ E_g $  ^  $ A_u $  ^  $ E_u $  ^
^ $ A_g $  | $ A_g $  | $ E_g $  | $ A_u $  | $ E_u $  |
^ $ E_g $  | $ E_g $  | $ 2 A_g+E_g $  | $ E_u $  | $ 2 A_u+E_u $  |
^ $ A_u $  | $ A_u $  | $ E_u $  | $ A_g $  | $ E_g $  |
^ $ E_u $  | $ E_u $  | $ 2 A_u+E_u $  | $ E_g $  | $ 2 A_g+E_g $  |

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

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  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
  * [[physics_chemistry:point_groups:c3:orientation_z|Point Group C3 with orientation Z]]
  * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]]

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

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  * [[physics_chemistry:point_groups:c6h:orientation_z|Point Group C6h with orientation Z]]
  * [[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: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_11-1z|Point Group Oh with orientation 11-1z]]
  * [[physics_chemistry:point_groups:oh:orientation_111z|Point Group Oh with orientation 111z]]
  * [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]]
  * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]]

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===== Invariant Potential expanded on renormalized spherical Harmonics =====

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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 S6 Point group with orientation Z the form of the expansion coefficients is:

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==== Expansion ====

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

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

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<code Quanty Akm_S6_Z.Quanty.nb>

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

</code>

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

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<code Quanty Akm_S6_Z.Quanty>

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

</code>

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==== One particle coupling on a basis of spherical harmonics ====

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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 $A_{n''l'',n'l'}(k,m)$


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we can express the operator as 
$$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$


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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.

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|  $  $  ^  $ {Y_{0}^{(0)}} $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ {Y_{0}^{(0)}} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{1}{3} (-\text{Apf}(4,3)+i \text{Bpf}(4,3)) $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\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}} $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ -\frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|
^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ \frac{1}{3} (\text{Apf}(4,3)+i \text{Bpf}(4,3)) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|
^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ \frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$\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) $|$ 0 $|$ 0 $|$ \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)) $|$ 0 $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $|
^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\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)) $|$ 0 $|$ 0 $|
^$ {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) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \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)) $|$ 0 $|
^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{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)) $|$ 0 $|$ 0 $|$ \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) $|$ 0 $|$ 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)) $|
^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ \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)) $|$ 0 $|$ 0 $|$ \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) $|$ 0 $|$ 0 $|
^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} (-\text{Apf}(4,3)-i \text{Bpf}(4,3)) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\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)) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|
^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$\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)) $|$ 0 $|$ 0 $|$ \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)) $|$ 0 $|$ 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) $|


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

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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

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|  $  $  ^  $ {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_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 $|
^$ 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 $|
^$ 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 }$|
^$ 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_{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{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_{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 }$|
^$ f_{y\left(3x^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 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
^$ 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_{y\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 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|
^$ f_{z\left(5z^2-3r^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(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 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|
^$ f_{x\left(x^2-3y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|


###

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

###

After rotation we find

###



###

|  $  $  ^  $ \text{s} $  ^  $ p_y $  ^  $ p_z $  ^  $ p_x $  ^  $ d_{\text{xy}} $  ^  $ d_{\text{yz}} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{xz}} $  ^  $ d_{x^2-y^2} $  ^  $ f_{y\left(3x^2-y^2\right)} $  ^  $ f_{\text{xyz}} $  ^  $ f_{y\left(5z^2-r^2\right)} $  ^  $ f_{z\left(5z^2-3r^2\right)} $  ^  $ f_{x\left(5z^2-r^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^  $ f_{x\left(x^2-3y^2\right)} $  ^
^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ p_y $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{3} \text{Apf}(4,3) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|
^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{2}{3}} \text{Bpf}(4,3) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{3}} \text{Apf}(4,3) $|
^$ p_x $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ \frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|$ 0 $|$ \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} \text{Apf}(4,3) $|$ 0 $|
^$ d_{\text{xy}} $|$ 0 $|$\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{5}{7}} \text{Add}(4,3) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 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 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \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_{3z^2-r^2} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ -\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_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \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 }$|
^$ f_{y\left(3x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{3} \sqrt{\frac{2}{3}} \text{Bpf}(4,3) $|$ 0 $|$\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{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)-\frac{1}{11} \sqrt{14} \text{Bff}(4,3) $|$ 0 $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|
^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} \text{Apf}(4,3) $|$ 0 $|$ \frac{1}{3} \text{Bpf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,3)+\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|$ 0 $|
^$ f_{y\left(5z^2-r^2\right)} $|$\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) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,3)+\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ \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) $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|
^$ f_{z\left(5z^2-3r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)-\frac{1}{11} \sqrt{14} \text{Bff}(4,3) $|$ 0 $|$ 0 $|$ \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) $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{14} \text{Aff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,3) $|
^$ f_{x\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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 }$|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|$ 0 $|$ \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}{33} \sqrt{14} \text{Aff}(4,3)-\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|$ -\frac{1}{3} \text{Apf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,3)-\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|
^$ f_{x\left(x^2-3y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{3}} \text{Apf}(4,3) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{14} \text{Aff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,3) $|$ 0 $|$ 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{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|


###

===== 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_S6_Z.Quanty.nb>

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

</code>

###

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

###

<code Quanty Akm_S6_Z.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:s6_z_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** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{3} (\text{Eau}+2 \text{Eeu}) & k=0\land m=0 \\
 \frac{5 (\text{Eau}-\text{Eeu})}{3} & k=2\land m=0
\end{cases}$$

###

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

###

<code Quanty Akm_S6_Z.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Eau + 2*Eeu)/3, k == 0 && m == 0}, {(5*(Eau - Eeu))/3, k == 2 && m == 0}}, 0]

</code>

###

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

###

<code Quanty Akm_S6_Z.Quanty>

Akm = {{0, 0, (1/3)*(Eau + (2)*(Eeu))} , 
       {2, 0, (5/3)*(Eau + (-1)*(Eeu))} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
^$ {Y_{-1}^{(1)}} $|$ \text{Eeu} $|$ 0 $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Eau} $|$ 0 $|
^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Eeu} $|


###

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

###

|  $  $  ^  $ p_y $  ^  $ p_z $  ^  $ p_x $  ^
^$ p_y $|$ \text{Eeu} $|$ 0 $|$ 0 $|
^$ p_z $|$ 0 $|$ \text{Eau} $|$ 0 $|
^$ p_x $|$ 0 $|$ 0 $|$ \text{Eeu} $|


###

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

###

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


###

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

###

^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Eau}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_1_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |


###

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

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

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{5} (\text{Eag}+2 (\text{Eeu1}+\text{Eeu2})) & k=0\land m=0 \\
 \text{Eag}+\text{Eeu1}-2 \text{Eeu2} & k=2\land m=0 \\
 -3 \sqrt{\frac{7}{5}} (\text{Meu1}-i \text{Meu2}) & k=4\land m=-3 \\
 \frac{3}{5} (3 \text{Eag}-4 \text{Eeu1}+\text{Eeu2}) & k=4\land m=0 \\
 3 \sqrt{\frac{7}{5}} (\text{Meu1}+i \text{Meu2}) & k=4\land m=3
\end{cases}$$

###

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

###

<code Quanty Akm_S6_Z.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Eag + 2*(Eeu1 + Eeu2))/5, k == 0 && m == 0}, {Eag + Eeu1 - 2*Eeu2, k == 2 && m == 0}, {-3*Sqrt[7/5]*(Meu1 - I*Meu2), k == 4 && m == -3}, {(3*(3*Eag - 4*Eeu1 + Eeu2))/5, k == 4 && m == 0}, {3*Sqrt[7/5]*(Meu1 + I*Meu2), k == 4 && m == 3}}, 0]

</code>

###

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

###

<code Quanty Akm_S6_Z.Quanty>

Akm = {{0, 0, (1/5)*(Eag + (2)*(Eeu1 + Eeu2))} , 
       {2, 0, Eag + Eeu1 + (-2)*(Eeu2)} , 
       {4, 0, (3/5)*((3)*(Eag) + (-4)*(Eeu1) + Eeu2)} , 
       {4,-3, (-3)*((sqrt(7/5))*(Meu1 + (-I)*(Meu2)))} , 
       {4, 3, (3)*((sqrt(7/5))*(Meu1 + (I)*(Meu2)))} }

</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)}} $|$ \text{Eeu2} $|$ 0 $|$ 0 $|$ \text{Meu1}-i \text{Meu2} $|$ 0 $|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ -\text{Meu1}+i \text{Meu2} $|
^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Eag} $|$ 0 $|$ 0 $|
^$ {Y_{1}^{(2)}} $|$ \text{Meu1}+i \text{Meu2} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|
^$ {Y_{2}^{(2)}} $|$ 0 $|$ -\text{Meu1}-i \text{Meu2} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|


###

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

###

|  $  $  ^  $ d_{\text{xy}} $  ^  $ d_{\text{yz}} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{xz}} $  ^  $ d_{x^2-y^2} $  ^
^$ d_{\text{xy}} $|$ \text{Eeu2} $|$ \text{Meu1} $|$ 0 $|$ \text{Meu2} $|$ 0 $|
^$ d_{\text{yz}} $|$ \text{Meu1} $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu2} $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Eag} $|$ 0 $|$ 0 $|
^$ d_{\text{xz}} $|$ \text{Meu2} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ -\text{Meu1} $|
^$ d_{x^2-y^2} $|$ 0 $|$ \text{Meu2} $|$ 0 $|$ -\text{Meu1} $|$ \text{Eeu2} $|


###

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

###

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


###

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

###

^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
^ ^$$\text{Eag}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |


###

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

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

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{7} (\text{Eau1}+\text{Eau2}+\text{Eau3}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\
 -\frac{5}{28} (5 \text{Eau1}-4 \text{Eau2}+5 \text{Eau3}-6 \text{Eeu1}) & k=2\land m=0 \\
 \frac{-9 i \text{Mau12}-9 \text{Mau23}-6 \text{Meu1}+6 i \text{Meu2}}{\sqrt{14}} & k=4\land m=-3 \\
 \frac{3}{14} (3 \text{Eau1}+6 \text{Eau2}+3 \text{Eau3}+2 \text{Eeu1}-14 \text{Eeu2}) & k=4\land m=0 \\
 \frac{-9 i \text{Mau12}+9 \text{Mau23}+6 \text{Meu1}+6 i \text{Meu2}}{\sqrt{14}} & k=4\land m=3 \\
 -\frac{13}{20} \sqrt{\frac{33}{7}} (\text{Eau1}-\text{Eau3}-2 i \text{Mau13}) & k=6\land m=-6 \\
 \frac{13}{5} \sqrt{\frac{3}{14}} (i \text{Mau12}+\text{Mau23}-3 \text{Meu1}+3 i \text{Meu2}) & k=6\land m=-3 \\
 -\frac{13}{140} (\text{Eau1}-20 \text{Eau2}+\text{Eau3}+30 \text{Eeu1}-12 \text{Eeu2}) & k=6\land m=0 \\
 \frac{13}{5} \sqrt{\frac{3}{14}} (i \text{Mau12}-\text{Mau23}+3 \text{Meu1}+3 i \text{Meu2}) & k=6\land m=3 \\
 -\frac{13}{20} \sqrt{\frac{33}{7}} (\text{Eau1}-\text{Eau3}+2 i \text{Mau13}) & k=6\land m=6
\end{cases}$$

###

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

###

<code Quanty Akm_S6_Z.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Eau1 + Eau2 + Eau3 + 2*Eeu1 + 2*Eeu2)/7, k == 0 && m == 0}, {(-5*(5*Eau1 - 4*Eau2 + 5*Eau3 - 6*Eeu1))/28, k == 2 && m == 0}, {((-9*I)*Mau12 - 9*Mau23 - 6*Meu1 + (6*I)*Meu2)/Sqrt[14], k == 4 && m == -3}, {(3*(3*Eau1 + 6*Eau2 + 3*Eau3 + 2*Eeu1 - 14*Eeu2))/14, k == 4 && m == 0}, {((-9*I)*Mau12 + 9*Mau23 + 6*Meu1 + (6*I)*Meu2)/Sqrt[14], k == 4 && m == 3}, {(-13*Sqrt[33/7]*(Eau1 - Eau3 - (2*I)*Mau13))/20, k == 6 && m == -6}, {(13*Sqrt[3/14]*(I*Mau12 + Mau23 - 3*Meu1 + (3*I)*Meu2))/5, k == 6 && m == -3}, {(-13*(Eau1 - 20*Eau2 + Eau3 + 30*Eeu1 - 12*Eeu2))/140, k == 6 && m == 0}, {(13*Sqrt[3/14]*(I*Mau12 - Mau23 + 3*Meu1 + (3*I)*Meu2))/5, k == 6 && m == 3}, {(-13*Sqrt[33/7]*(Eau1 - Eau3 + (2*I)*Mau13))/20, k == 6 && m == 6}}, 0]

</code>

###

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

###

<code Quanty Akm_S6_Z.Quanty>

Akm = {{0, 0, (1/7)*(Eau1 + Eau2 + Eau3 + (2)*(Eeu1) + (2)*(Eeu2))} , 
       {2, 0, (-5/28)*((5)*(Eau1) + (-4)*(Eau2) + (5)*(Eau3) + (-6)*(Eeu1))} , 
       {4, 0, (3/14)*((3)*(Eau1) + (6)*(Eau2) + (3)*(Eau3) + (2)*(Eeu1) + (-14)*(Eeu2))} , 
       {4,-3, (1/(sqrt(14)))*((-9*I)*(Mau12) + (-9)*(Mau23) + (-6)*(Meu1) + (6*I)*(Meu2))} , 
       {4, 3, (1/(sqrt(14)))*((-9*I)*(Mau12) + (9)*(Mau23) + (6)*(Meu1) + (6*I)*(Meu2))} , 
       {6, 0, (-13/140)*(Eau1 + (-20)*(Eau2) + Eau3 + (30)*(Eeu1) + (-12)*(Eeu2))} , 
       {6, 3, (13/5)*((sqrt(3/14))*((I)*(Mau12) + (-1)*(Mau23) + (3)*(Meu1) + (3*I)*(Meu2)))} , 
       {6,-3, (13/5)*((sqrt(3/14))*((I)*(Mau12) + Mau23 + (-3)*(Meu1) + (3*I)*(Meu2)))} , 
       {6,-6, (-13/20)*((sqrt(33/7))*(Eau1 + (-1)*(Eau3) + (-2*I)*(Mau13)))} , 
       {6, 6, (-13/20)*((sqrt(33/7))*(Eau1 + (-1)*(Eau3) + (2*I)*(Mau13)))} }

</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{\text{Eau1}+\text{Eau3}}{2} $|$ 0 $|$ 0 $|$ \frac{\text{Mau23}+i \text{Mau12}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{2} (\text{Eau1}-\text{Eau3}-2 i \text{Mau13}) $|
^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \text{Eeu2} $|$ 0 $|$ 0 $|$ \text{Meu1}-i \text{Meu2} $|$ 0 $|$ 0 $|
^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ -\text{Meu1}+i \text{Meu2} $|$ 0 $|
^$ {Y_{0}^{(3)}} $|$ \frac{\text{Mau23}-i \text{Mau12}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \text{Eau2} $|$ 0 $|$ 0 $|$ \frac{-\text{Mau23}-i \text{Mau12}}{\sqrt{2}} $|
^$ {Y_{1}^{(3)}} $|$ 0 $|$ \text{Meu1}+i \text{Meu2} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|
^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ -\text{Meu1}-i \text{Meu2} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|
^$ {Y_{3}^{(3)}} $|$ \frac{1}{2} (\text{Eau1}-\text{Eau3}+2 i \text{Mau13}) $|$ 0 $|$ 0 $|$ \frac{i (\text{Mau12}+i \text{Mau23})}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{\text{Eau1}+\text{Eau3}}{2} $|


###

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

###

|  $  $  ^  $ f_{y\left(3x^2-y^2\right)} $  ^  $ f_{\text{xyz}} $  ^  $ f_{y\left(5z^2-r^2\right)} $  ^  $ f_{z\left(5z^2-3r^2\right)} $  ^  $ f_{x\left(5z^2-r^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^  $ f_{x\left(x^2-3y^2\right)} $  ^
^$ f_{y\left(3x^2-y^2\right)} $|$ \text{Eau1} $|$ 0 $|$ 0 $|$ \text{Mau12} $|$ 0 $|$ 0 $|$ \text{Mau13} $|
^$ f_{\text{xyz}} $|$ 0 $|$ \text{Eeu2} $|$ \text{Meu1} $|$ 0 $|$ \text{Meu2} $|$ 0 $|$ 0 $|
^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ \text{Meu1} $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu2} $|$ 0 $|
^$ f_{z\left(5z^2-3r^2\right)} $|$ \text{Mau12} $|$ 0 $|$ 0 $|$ \text{Eau2} $|$ 0 $|$ 0 $|$ \text{Mau23} $|
^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ \text{Meu2} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ -\text{Meu1} $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ \text{Meu2} $|$ 0 $|$ -\text{Meu1} $|$ \text{Eeu2} $|$ 0 $|
^$ f_{x\left(x^2-3y^2\right)} $|$ \text{Mau13} $|$ 0 $|$ 0 $|$ \text{Mau23} $|$ 0 $|$ 0 $|$ \text{Eau3} $|


###

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


###

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

###

^ ^$$\text{Eau1}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$$ | ::: |
^ ^$$\text{Eau2}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |
^ ^$$\text{Eau3}$$ | {{:physics_chemistry:pointgroup:s6_z_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$$ | ::: |


###

</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** >

###

 $$A_{k,m} = \begin{cases}
 0 & k\neq 2\lor m\neq 0 \\
 \sqrt{5} \text{Mag} & \text{True}
\end{cases}$$

###

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

###

<code Quanty Akm_S6_Z.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, Sqrt[5]*Mag]

</code>

###

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

###

<code Quanty Akm_S6_Z.Quanty>

Akm = {{2, 0, (sqrt(5))*(Mag)} }

</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)}} $|$ 0 $|$ 0 $|$ \text{Mag} $|$ 0 $|$ 0 $|


###

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

###

|  $  $  ^  $ d_{\text{xy}} $  ^  $ d_{\text{yz}} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{xz}} $  ^  $ d_{x^2-y^2} $  ^
^$ \text{s} $|$ 0 $|$ 0 $|$ \text{Mag} $|$ 0 $|$ 0 $|


###

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

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

###

 $$A_{k,m} = \begin{cases}
 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
 \frac{5}{21} \left(\sqrt{21} \text{Mau2}+2 \sqrt{14} \text{Meu}\right) & k=2\land m=0 \\
 3 \sqrt{\frac{3}{2}} (\text{Mau3}+i \text{Mau1}) & k=4\land m=-3 \\
 3 \sqrt{\frac{3}{7}} \text{Mau2}-\frac{9 \text{Meu}}{\sqrt{14}} & k=4\land m=0 \\
 3 i \sqrt{\frac{3}{2}} (\text{Mau1}+i \text{Mau3}) & \text{True}
\end{cases}$$

###

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

###

<code Quanty Akm_S6_Z.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {(5*(Sqrt[21]*Mau2 + 2*Sqrt[14]*Meu))/21, k == 2 && m == 0}, {3*Sqrt[3/2]*(I*Mau1 + Mau3), k == 4 && m == -3}, {3*Sqrt[3/7]*Mau2 - (9*Meu)/Sqrt[14], k == 4 && m == 0}}, (3*I)*Sqrt[3/2]*(Mau1 + I*Mau3)]

</code>

###

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

###

<code Quanty Akm_S6_Z.Quanty>

Akm = {{2, 0, (5/21)*((sqrt(21))*(Mau2) + (2)*((sqrt(14))*(Meu)))} , 
       {4, 0, (3)*((sqrt(3/7))*(Mau2)) + (-9)*((1/(sqrt(14)))*(Meu))} , 
       {4, 3, (3*I)*((sqrt(3/2))*(Mau1 + (I)*(Mau3)))} , 
       {4,-3, (3)*((sqrt(3/2))*((I)*(Mau1) + Mau3))} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ {Y_{-1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \sqrt{\frac{3}{2}} (\text{Mau3}+i \text{Mau1}) $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$ \frac{\text{Mau3}-i \text{Mau1}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \text{Mau2} $|$ 0 $|$ 0 $|$ -\frac{\text{Mau3}+i \text{Mau1}}{\sqrt{2}} $|
^$ {Y_{1}^{(1)}} $|$ 0 $|$ i \sqrt{\frac{3}{2}} (\text{Mau1}+i \text{Mau3}) $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|


###

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

###

|  $  $  ^  $ f_{y\left(3x^2-y^2\right)} $  ^  $ f_{\text{xyz}} $  ^  $ f_{y\left(5z^2-r^2\right)} $  ^  $ f_{z\left(5z^2-3r^2\right)} $  ^  $ f_{x\left(5z^2-r^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^  $ f_{x\left(x^2-3y^2\right)} $  ^
^$ p_y $|$ 0 $|$ -\sqrt{\frac{3}{2}} \text{Mau3} $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \sqrt{\frac{3}{2}} \text{Mau1} $|$ 0 $|
^$ p_z $|$ \text{Mau1} $|$ 0 $|$ 0 $|$ \text{Mau2} $|$ 0 $|$ 0 $|$ \text{Mau3} $|
^$ p_x $|$ 0 $|$ \sqrt{\frac{3}{2}} \text{Mau1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ \sqrt{\frac{3}{2}} \text{Mau3} $|$ 0 $|


###

</hidden>

===== Table of several point groups =====

###

[[physics_chemistry:point_groups|Return to Main page on Point Groups]]

###

###

^Nonaxial groups      | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> | | | | |
^C<sub>n</sub> groups | [[physics_chemistry:point_groups:c2|C]]<sub>[[physics_chemistry:point_groups:c2|2]]</sub> | [[physics_chemistry:point_groups:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> | 
^D<sub>n</sub> groups | [[physics_chemistry:point_groups:d2|D]]<sub>[[physics_chemistry:point_groups:d2|2]]</sub> | [[physics_chemistry:point_groups:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub> | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub> | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub> | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub> | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub> | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub> | 
^C<sub>nv</sub> groups | [[physics_chemistry:point_groups:c2v|C]]<sub>[[physics_chemistry:point_groups:c2v|2v]]</sub> | [[physics_chemistry:point_groups:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> | 
^C<sub>nh</sub> groups | [[physics_chemistry:point_groups:c2h|C]]<sub>[[physics_chemistry:point_groups:c2h|2h]]</sub> | [[physics_chemistry:point_groups:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> | | | 
^D<sub>nh</sub> groups | [[physics_chemistry:point_groups:d2h|D]]<sub>[[physics_chemistry:point_groups:d2h|2h]]</sub> | [[physics_chemistry:point_groups:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> | 
^D<sub>nd</sub> groups | [[physics_chemistry:point_groups:d2d|D]]<sub>[[physics_chemistry:point_groups:d2d|2d]]</sub> | [[physics_chemistry:point_groups:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> | 
^S<sub>n</sub> groups | [[physics_chemistry:point_groups:S2|S]]<sub>[[physics_chemistry:point_groups:S2|2]]</sub> | [[physics_chemistry:point_groups:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> |  | 
^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> | 
^Linear groups      | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | |

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