~~CLOSETOC~~

====== Orientation Z(x-y)_B ======

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The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry eg$\sigma$ and the eg orbitals that descend from the t2g irreducible representation eg$\pi$ orbitals. (The mixing is given by the parameter Meg.)
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As one can see in the list of supergroups of D3d, there are two different orientations of Oh that are a supergroup of this orientation of D3d. The different orientations of Oh with respect to D3d do however change the definitions of the eg$\pi$ and eg$\sigma$ orbitals. We include three different representations of the orbitals and potentials for each setting of D3d symmetry. The orientation without additional letter takes the tesseral harmonics as a basis. This basis does not relate to the states in Oh symmetry. The orientation with an additional A or B relate to the two different supergroup representations of the Oh point group.
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The parameterization B of the orientation Z(x-y) is related to the orientation 11-1z of the Oh pointgroup.

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

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In the D3d Point Group, with orientation Z(x-y)_B there are the following symmetry operations

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{{:physics_chemistry:pointgroup:d3d_z(x-y)_b.png}}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,-1,0\}$ , $\left\{2+\sqrt{3},1,0\right\}$ , $\left\{1,2+\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{2+\sqrt{3},1,0\right\}$ , $\left\{1,2+\sqrt{3},0\right\}$ , |

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

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  * [[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]]

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

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|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_3 \,{\text{(2)}} $  ^  $ C_2 \,{\text{(3)}} $  ^  $ \text{i} \,{\text{(1)}} $  ^  $ S_6 \,{\text{(2)}} $  ^  $ \sigma_d \,{\text{(3)}} $  ^
^ $ A_{1g} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
^ $ A_{2g} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |
^ $ E_g $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |
^ $ A_{1u} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |
^ $ A_{2u} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |
^ $ E_u $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ -2 $ |  $ 1 $ |  $ 0 $ |

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

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|  $  $  ^  $ A_{1g} $  ^  $ A_{2g} $  ^  $ E_g $  ^  $ A_{1u} $  ^  $ A_{2u} $  ^  $ E_u $  ^
^ $ A_{1g} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  |
^ $ A_{2g} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  |
^ $ E_g $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  |
^ $ A_{1u} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  |
^ $ A_{2u} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  |
^ $ E_u $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  |

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

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

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  * [[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:oh:orientation_11-1z|Point Group Oh with orientation 11-1z]]
  * [[physics_chemistry:point_groups:oh:orientation_111z|Point Group Oh with orientation 111z]]

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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 D3d Point group with orientation Z(x-y)_B the form of the expansion coefficients is:

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==== Expansion ====

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

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

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<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {(-1 + I)*A[4, 3], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {(1 + I)*A[4, 3], k == 4 && m == 3}, {(-I)*B[6, 6], k == 6 && m == -6}, {(-1 + I)*A[6, 3], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {(1 + I)*A[6, 3], k == 6 && m == 3}, {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_D3d_Z(x-y)_B.Quanty>

Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {4, 0, A(4,0)} , 
       {4,-3, (-1+1*I)*(A(4,3))} , 
       {4, 3, (1+1*I)*(A(4,3))} , 
       {6, 0, A(6,0)} , 
       {6,-3, (-1+1*I)*(A(6,3))} , 
       {6, 3, (1+1*I)*(A(6,3))} , 
       {6,-6, (-I)*(B(6,6))} , 
       {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 $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(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{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,3)}{\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{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,3)}{\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 $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(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 $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(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 $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \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 }$|
^$ {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 }$|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(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 $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(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{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,3)}{\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 $|$ \left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,3)-\left(\frac{10}{143}-\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|$ \frac{10}{13} i \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|
^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(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 $|$ \left(\frac{1}{33}-\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,3)+\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(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 $|$ \left(-\frac{1}{33}+\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,3)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(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 }$|$ \left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,3)-\left(\frac{10}{143}+\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(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 $|$ \left(\frac{10}{143}-\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,3)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,3) $|
^$ {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 $|$ \left(\frac{1}{33}+\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,3)+\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(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 }$|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,3) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \left(-\frac{1}{33}-\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,3)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(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{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,3)}{\sqrt{3}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} i \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ \left(\frac{10}{143}+\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,3)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{7} \text{Aff}(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_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{-\text{yz}-\text{xz}+\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ -\frac{1+i}{\sqrt{6}} $|$ 0 $|$ \frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{(x-y)(x+y+2z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{6}} $|$ \frac{1-i}{\sqrt{6}} $|$ 0 $|$ -\frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{2\text{xy}+\text{xz}+\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{(x-y)(x+y-z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{3}} $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \frac{1}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3}-\frac{i}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{1}{3}-\frac{i}{3} $|
^$ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
^$ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{i}{2 \sqrt{3}} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} $|$ 0 $|$ 0 $|$ \frac{2}{3} $|$ 0 $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} $|
^$ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
^$ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{2}+\frac{i}{2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|


###

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

###

After rotation we find

###



###

|  $  $  ^  $ \text{s} $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^  $ d_{-\text{yz}-\text{xz}+\text{xy}} $  ^  $ d_{(x-y)(x+y+2z)} $  ^  $ d_{2\text{xy}+\text{xz}+\text{yz}} $  ^  $ d_{(x-y)(x+y-z)} $  ^  $ d_{3z^2-r^2} $  ^  $ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $  ^  $ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $  ^  $ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $  ^
^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\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) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ 0 $|$ 0 $|$ -\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|
^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ 0 $|$ 0 $|$ -\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ 0 $|
^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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{3}{35}} \text{Apf}(2,0)+\frac{4}{9} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{4 \text{Apf}(4,3)}{9 \sqrt{3}} $|$ 0 $|$ 0 $|$ \frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{8 \text{Apf}(4,0)}{9 \sqrt{21}}+\frac{2}{9} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{-\text{yz}-\text{xz}+\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{1}{9} \text{Add}(4,0)-\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ 0 $|$ -\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,3) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{(x-y)(x+y+2z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)-\frac{1}{9} \text{Add}(4,0)-\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ 0 $|$ -\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(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_{2\text{xy}+\text{xz}+\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,3) $|$ 0 $|$ \text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)-\frac{2}{63} \text{Add}(4,0)+\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{(x-y)(x+y-z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,3) $|$ 0 $|$ \text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)-\frac{2}{63} \text{Add}(4,0)+\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(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_{3z^2-r^2} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{4}{9} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{4 \text{Apf}(4,3)}{9 \sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)+\frac{14}{99} \text{Aff}(4,0)+\frac{8}{99} \sqrt{35} \text{Aff}(4,3)+\frac{160 \text{Aff}(6,0)}{1287}-\frac{80 \sqrt{\frac{35}{3}} \text{Aff}(6,3)}{1287}+\frac{40}{117} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ \frac{2 \text{Aff}(2,0)}{3 \sqrt{5}}+\frac{2}{99} \sqrt{5} \text{Aff}(4,0)-\frac{2}{99} \sqrt{7} \text{Aff}(4,3)+\frac{70 \sqrt{5} \text{Aff}(6,0)}{1287}+\frac{20 \sqrt{\frac{7}{3}} \text{Aff}(6,3)}{1287}-\frac{20}{117} \sqrt{\frac{35}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{30} \text{Aff}(2,0)-\frac{17}{99} \text{Aff}(4,0)-\frac{2}{99} \sqrt{35} \text{Aff}(4,3)+\frac{25}{858} \text{Aff}(6,0)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{4}{99} \sqrt{7} \text{Aff}(4,3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|
^$ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{30} \text{Aff}(2,0)-\frac{17}{99} \text{Aff}(4,0)-\frac{2}{99} \sqrt{35} \text{Aff}(4,3)+\frac{25}{858} \text{Aff}(6,0)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{4}{99} \sqrt{7} \text{Aff}(4,3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ 0 $|
^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{8 \text{Apf}(4,0)}{9 \sqrt{21}}+\frac{2}{9} \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 \text{Aff}(2,0)}{3 \sqrt{5}}+\frac{2}{99} \sqrt{5} \text{Aff}(4,0)-\frac{2}{99} \sqrt{7} \text{Aff}(4,3)+\frac{70 \sqrt{5} \text{Aff}(6,0)}{1287}+\frac{20 \sqrt{\frac{7}{3}} \text{Aff}(6,3)}{1287}-\frac{20}{117} \sqrt{\frac{35}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{15} \text{Aff}(2,0)+\frac{13}{99} \text{Aff}(4,0)-\frac{8}{99} \sqrt{35} \text{Aff}(4,3)+\frac{125 \text{Aff}(6,0)}{1287}+\frac{80 \sqrt{\frac{35}{3}} \text{Aff}(6,3)}{1287}+\frac{50}{117} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{4}{99} \sqrt{7} \text{Aff}(4,3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{6} \text{Aff}(2,0)-\frac{1}{99} \text{Aff}(4,0)+\frac{2}{99} \sqrt{35} \text{Aff}(4,3)-\frac{115}{858} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|
^$ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{4}{99} \sqrt{7} \text{Aff}(4,3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{6} \text{Aff}(2,0)-\frac{1}{99} \text{Aff}(4,0)+\frac{2}{99} \sqrt{35} \text{Aff}(4,3)-\frac{115}{858} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ 0 $|
^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{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{Bff}(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** >

###

 $$A_{k,m} = \begin{cases}
 \text{Ea1g} & k=0\land m=0 \\
 0 & \text{True}
\end{cases}$$

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]

</code>

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty>

Akm = {{0, 0, Ea1g} }

</code>

###

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

###

|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
^$ {Y_{0}^{(0)}} $|$ \text{Ea1g} $|


###

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

###

|  $  $  ^  $ \text{s} $  ^
^$ \text{s} $|$ \text{Ea1g} $|


###

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

###

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


###

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

###

^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_0_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |


###

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

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

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\
 \frac{5 (\text{Ea2u}-\text{Eeu})}{3} & k=2\land m=0
\end{cases}$$

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(5*(Ea2u - Eeu))/3, k == 2 && m == 0}}, 0]

</code>

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty>

Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , 
       {2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
^$ {Y_{-1}^{(1)}} $|$ \text{Eeu} $|$ 0 $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea2u} $|$ 0 $|
^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Eeu} $|


###

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

###

|  $  $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^
^$ p_x $|$ \text{Eeu} $|$ 0 $|$ 0 $|
^$ p_y $|$ 0 $|$ \text{Eeu} $|$ 0 $|
^$ p_z $|$ 0 $|$ 0 $|$ \text{Ea2u} $|


###

</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** >

###

^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_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** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\
 \text{Ea1g}-\text{Eeg}\pi -2 \sqrt{2} \text{Meg} & k=2\land m=0 \\
 \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{7}{5}} \left(2 \text{Eeg}\pi -2 \text{Eeg}\sigma -\sqrt{2} \text{Meg}\right) & k=4\land m=-3 \\
 \frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg}\pi -7 \text{Eeg}\sigma +10 \sqrt{2} \text{Meg}\right) & k=4\land m=0 \\
 \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{7}{5}} \left(2 \text{Eeg}\pi -2 \text{Eeg}\sigma -\sqrt{2} \text{Meg}\right) & k=4\land m=3
\end{cases}$$

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Ea1g + 2*(Eeg\[Pi] + Eeg\[Sigma]))/5, k == 0 && m == 0}, {Ea1g - Eeg\[Pi] - 2*Sqrt[2]*Meg, k == 2 && m == 0}, {(-1/2 + I/2)*Sqrt[7/5]*(2*Eeg\[Pi] - 2*Eeg\[Sigma] - Sqrt[2]*Meg), k == 4 && m == -3}, {(9*Ea1g - 2*Eeg\[Pi] - 7*Eeg\[Sigma] + 10*Sqrt[2]*Meg)/5, k == 4 && m == 0}, {(1/2 + I/2)*Sqrt[7/5]*(2*Eeg\[Pi] - 2*Eeg\[Sigma] - Sqrt[2]*Meg), k == 4 && m == 3}}, 0]

</code>

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty>

Akm = {{0, 0, (1/5)*(Ea1g + (2)*(EegPi + EegSigma))} , 
       {2, 0, Ea1g + (-1)*(EegPi) + (-2)*((sqrt(2))*(Meg))} , 
       {4, 0, (1/5)*((9)*(Ea1g) + (-2)*(EegPi) + (-7)*(EegSigma) + (10)*((sqrt(2))*(Meg)))} , 
       {4,-3, (-1/2+1/2*I)*((sqrt(7/5))*((2)*(EegPi) + (-2)*(EegSigma) + (-1)*((sqrt(2))*(Meg))))} , 
       {4, 3, (1/2+1/2*I)*((sqrt(7/5))*((2)*(EegPi) + (-2)*(EegSigma) + (-1)*((sqrt(2))*(Meg))))} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
^$ {Y_{-2}^{(2)}} $|$ \frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right) $|$ 0 $|$ 0 $|$ \left(-\frac{1}{6}+\frac{i}{6}\right) \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right) $|$ 0 $|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right) $|$ 0 $|$ 0 $|$ \left(\frac{1}{6}-\frac{i}{6}\right) \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right) $|
^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea1g} $|$ 0 $|$ 0 $|
^$ {Y_{1}^{(2)}} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(2 \text{Eeg$\pi $}-2 \text{Eeg$\sigma $}-\sqrt{2} \text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right) $|$ 0 $|
^$ {Y_{2}^{(2)}} $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right) $|


###

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

###

|  $  $  ^  $ d_{-\text{yz}-\text{xz}+\text{xy}} $  ^  $ d_{(x-y)(x+y+2z)} $  ^  $ d_{2\text{xy}+\text{xz}+\text{yz}} $  ^  $ d_{(x-y)(x+y-z)} $  ^  $ d_{3z^2-r^2} $  ^
^$ d_{-\text{yz}-\text{xz}+\text{xy}} $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ 0 $|
^$ d_{(x-y)(x+y+2z)} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|
^$ d_{2\text{xy}+\text{xz}+\text{yz}} $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ 0 $|
^$ d_{(x-y)(x+y-z)} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1g} $|


###

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

###

|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
^$ d_{-\text{yz}-\text{xz}+\text{xy}} $|$ \frac{i}{\sqrt{6}} $|$ -\frac{1+i}{\sqrt{6}} $|$ 0 $|$ \frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $|
^$ d_{(x-y)(x+y+2z)} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1-i}{\sqrt{6}} $|$ 0 $|$ -\frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{6}} $|
^$ d_{2\text{xy}+\text{xz}+\text{yz}} $|$ \frac{i}{\sqrt{3}} $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\sqrt{3}} $|
^$ d_{(x-y)(x+y-z)} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \frac{1}{\sqrt{3}} $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|


###

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

###

^ ^$$\text{Eeg$\sigma $}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y-z)-y z)$$ | ::: |
^ ^$$\text{Eeg$\sigma $}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))+2 \cos (\theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} (x-y) (x+y+2 z)$$ | ::: |
^ ^$$\text{Eeg$\pi $}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (x (2 y+z)+y z)$$ | ::: |
^ ^$$\text{Eeg$\pi $}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))-\cos (\theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (x-y) (x+y-z)$$ | ::: |
^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |


###

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

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

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\
 -\frac{5}{28} \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}-4 \sqrt{5} \text{Ma2u}-2 \sqrt{5} \text{Meu}\right) & k=2\land m=0 \\
 -\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}-\text{Ma2u}-4 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=-3 \\
 \frac{1}{14} \left(9 \text{Ea1u}+14 \text{Ea2u1}+13 \text{Ea2u2}-34 \text{Eeu1}-2 \text{Eeu2}+4 \sqrt{5} \text{Ma2u}+16 \sqrt{5} \text{Meu}\right) & k=4\land m=0 \\
 \frac{\left(\frac{1}{2}+\frac{i}{2}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}-\text{Ma2u}-4 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=3 \\
 \frac{13}{60} i \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) & k=6\land m=-6 \\
 \frac{\left(\frac{13}{60}-\frac{13 i}{60}\right) \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}-2 \text{Ma2u}+36 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=-3 \\
 -\frac{13}{420} \left(3 \text{Ea1u}-32 \text{Ea2u1}-25 \text{Ea2u2}-15 \text{Eeu1}+69 \text{Eeu2}-28 \sqrt{5} \text{Ma2u}+42 \sqrt{5} \text{Meu}\right) & k=6\land m=0 \\
 -\frac{\left(\frac{13}{60}+\frac{13 i}{60}\right) \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}-2 \text{Ma2u}+36 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=3 \\
 -\frac{13}{60} i \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) & k=6\land m=6
\end{cases}$$

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*Eeu1 + 2*Eeu2)/7, k == 0 && m == 0}, {(-5*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 - 4*Sqrt[5]*Ma2u - 2*Sqrt[5]*Meu))/28, k == 2 && m == 0}, {((-1/2 + I/2)*(2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 - Ma2u - 4*Meu))/Sqrt[7], k == 4 && m == -3}, {(9*Ea1u + 14*Ea2u1 + 13*Ea2u2 - 34*Eeu1 - 2*Eeu2 + 4*Sqrt[5]*Ma2u + 16*Sqrt[5]*Meu)/14, k == 4 && m == 0}, {((1/2 + I/2)*(2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 - Ma2u - 4*Meu))/Sqrt[7], k == 4 && m == 3}, {((13*I)/60)*Sqrt[11/21]*(9*Ea1u - 4*Ea2u1 - 5*Ea2u2 + 4*Sqrt[5]*Ma2u), k == 6 && m == -6}, {((13/60 - (13*I)/60)*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 - 2*Ma2u + 36*Meu))/Sqrt[21], k == 6 && m == -3}, {(-13*(3*Ea1u - 32*Ea2u1 - 25*Ea2u2 - 15*Eeu1 + 69*Eeu2 - 28*Sqrt[5]*Ma2u + 42*Sqrt[5]*Meu))/420, k == 6 && m == 0}, {((-13/60 - (13*I)/60)*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 - 2*Ma2u + 36*Meu))/Sqrt[21], k == 6 && m == 3}, {((-13*I)/60)*Sqrt[11/21]*(9*Ea1u - 4*Ea2u1 - 5*Ea2u2 + 4*Sqrt[5]*Ma2u), k == 6 && m == 6}}, 0]

</code>

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty>

Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1) + (2)*(Eeu2))} , 
       {2, 0, (-5/28)*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (-4)*((sqrt(5))*(Ma2u)) + (-2)*((sqrt(5))*(Meu)))} , 
       {4, 0, (1/14)*((9)*(Ea1u) + (14)*(Ea2u1) + (13)*(Ea2u2) + (-34)*(Eeu1) + (-2)*(Eeu2) + (4)*((sqrt(5))*(Ma2u)) + (16)*((sqrt(5))*(Meu)))} , 
       {4,-3, (-1/2+1/2*I)*((1/(sqrt(7)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + (-1)*(Ma2u) + (-4)*(Meu)))} , 
       {4, 3, (1/2+1/2*I)*((1/(sqrt(7)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + (-1)*(Ma2u) + (-4)*(Meu)))} , 
       {6, 0, (-13/420)*((3)*(Ea1u) + (-32)*(Ea2u1) + (-25)*(Ea2u2) + (-15)*(Eeu1) + (69)*(Eeu2) + (-28)*((sqrt(5))*(Ma2u)) + (42)*((sqrt(5))*(Meu)))} , 
       {6, 3, (-13/60+-13/60*I)*((1/(sqrt(21)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (-2)*(Ma2u) + (36)*(Meu)))} , 
       {6,-3, (13/60+-13/60*I)*((1/(sqrt(21)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (-2)*(Ma2u) + (36)*(Meu)))} , 
       {6, 6, (-13/60*I)*((sqrt(11/21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (4)*((sqrt(5))*(Ma2u))))} , 
       {6,-6, (13/60*I)*((sqrt(11/21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (4)*((sqrt(5))*(Ma2u))))} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ {Y_{-3}^{(3)}} $|$ \frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) $|$ 0 $|$ 0 $|$ \left(\frac{1}{18}-\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right) $|$ 0 $|$ 0 $|$ \frac{1}{18} i \left(-9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) $|
^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}-2 \sqrt{5} \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} (-1)^{3/4} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right) $|$ 0 $|$ 0 $|
^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}+2 \sqrt{5} \text{Meu}\right) $|$ 0 $|$ 0 $|$ -\frac{1}{6} (-1)^{3/4} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right) $|$ 0 $|
^$ {Y_{0}^{(3)}} $|$ \left(\frac{1}{18}+\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right) $|$ 0 $|$ 0 $|$ \frac{1}{9} \left(5 \text{Ea2u1}+4 \left(\text{Ea2u2}+\sqrt{5} \text{Ma2u}\right)\right) $|$ 0 $|$ 0 $|$ \left(-\frac{1}{18}+\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right) $|
^$ {Y_{1}^{(3)}} $|$ 0 $|$ -\frac{1}{6} \sqrt[4]{-1} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}+2 \sqrt{5} \text{Meu}\right) $|$ 0 $|$ 0 $|
^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} \sqrt[4]{-1} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}-2 \sqrt{5} \text{Meu}\right) $|$ 0 $|
^$ {Y_{3}^{(3)}} $|$ \frac{1}{18} i \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) $|$ 0 $|$ 0 $|$ \left(-\frac{1}{18}-\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right) $|$ 0 $|$ 0 $|$ \frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) $|


###

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

###

|  $  $  ^  $ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $  ^  $ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $  ^  $ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $  ^
^$ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $|$ \text{Ea2u1} $|$ 0 $|$ 0 $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|
^$ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|
^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ \text{Ea2u2} $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|$ 0 $|
^$ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|
^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1u} $|


###

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

###

|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $|$ \frac{1}{3}-\frac{i}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{1}{3}-\frac{i}{3} $|
^$ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
^$ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ -\frac{i}{2 \sqrt{3}} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}} $|$ 0 $|
^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $|$ \left(-\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} $|$ 0 $|$ 0 $|$ \frac{2}{3} $|$ 0 $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} $|
^$ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
^$ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ -\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}} $|$ 0 $|
^$ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $|$ \frac{1}{2}+\frac{i}{2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|


###

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

###

^ ^$$\text{Ea2u1}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{24}+\frac{i}{24}\right) \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(e^{6 i \phi } \sin ^3(\theta )+(1-i) e^{3 i \phi } \cos (\theta ) \left(5 \cos ^2(\theta )-3\right)-i \sin ^3(\theta )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(x^3-3 x^2 y-3 x y^2+y^3+5 z^3-3 z\right)$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+5 \sin (2 \theta ) (\cos (2 \phi )-\sin (2 \phi )))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 x^2 z+10 x y z-5 x z^2+x+5 y^2 z\right)$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) (5 \sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi ))-(5 \cos (2 \theta )+3) \sin (\phi ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 x^2 z+10 x y z-5 y^2 z-5 y z^2+y\right)$$ | ::: |
^ ^$$\text{Ea2u2}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(-\frac{1}{48}-\frac{i}{48}\right) \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \left(5 \left(e^{6 i \phi }-i\right) \sin ^3(\theta )-(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-5 x^3+15 x^2 y+15 x y^2-5 y^3+4 z \left(5 z^2-3\right)\right)$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+\sin (2 \theta ) (\sin (2 \phi )-\cos (2 \phi )))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 z-2 x y z-5 x z^2+x-y^2 z\right)$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \sin (\phi )+\sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi )))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 (-z)-2 x y z+y^2 z-5 y z^2+y\right)$$ | ::: |
^ ^$$\text{Ea1u}$$ | {{:physics_chemistry:pointgroup:d3d_z(x-y)_b_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin ^3(\theta ) (\sin (3 \phi )+\cos (3 \phi ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^3+3 x^2 y-3 x y^2-y^3\right)$$ | ::: |


###

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



###

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

###

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

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

###

 $$A_{k,m} = \begin{cases}
 0 & k\neq 2\lor m\neq 0 \\
 \sqrt{5} \text{Ma1g} & \text{True}
\end{cases}$$

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, Sqrt[5]*Ma1g]

</code>

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty>

Akm = {{2, 0, (sqrt(5))*(Ma1g)} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
^$ {Y_{0}^{(0)}} $|$ 0 $|$ 0 $|$ \text{Ma1g} $|$ 0 $|$ 0 $|


###

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

###

|  $  $  ^  $ d_{-\text{yz}-\text{xz}+\text{xy}} $  ^  $ d_{(x-y)(x+y+2z)} $  ^  $ d_{2\text{xy}+\text{xz}+\text{yz}} $  ^  $ d_{(x-y)(x+y-z)} $  ^  $ d_{3z^2-r^2} $  ^
^$ \text{s} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma1g} $|


###

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

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

###

 $$A_{k,m} = \begin{cases}
 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
 -\frac{5 \left(\sqrt{5} \text{Ma2u1}-4 \text{Ma2u2}+4 \text{Meu1}\right)}{\sqrt{21}} & k=2\land m=0 \\
 \left(-\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{15} \text{Ma2u2}-2 \sqrt{3} \text{Ma2u1}\right) & k=4\land m=-3 \\
 \frac{1}{2} \sqrt{\frac{3}{7}} \left(8 \sqrt{5} \text{Ma2u1}-11 \text{Ma2u2}+18 \text{Meu1}\right) & k=4\land m=0 \\
 \left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{15} \text{Ma2u2}-2 \sqrt{3} \text{Ma2u1}\right) & \text{True}
\end{cases}$$

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {(-5*(Sqrt[5]*Ma2u1 - 4*Ma2u2 + 4*Meu1))/Sqrt[21], k == 2 && m == 0}, {(-1/2 + I/2)*(-2*Sqrt[3]*Ma2u1 + Sqrt[15]*Ma2u2), k == 4 && m == -3}, {(Sqrt[3/7]*(8*Sqrt[5]*Ma2u1 - 11*Ma2u2 + 18*Meu1))/2, k == 4 && m == 0}}, (1/2 + I/2)*(-2*Sqrt[3]*Ma2u1 + Sqrt[15]*Ma2u2)]

</code>

###

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

###

<code Quanty Akm_D3d_Z(x-y)_B.Quanty>

Akm = {{2, 0, (-5)*((1/(sqrt(21)))*((sqrt(5))*(Ma2u1) + (-4)*(Ma2u2) + (4)*(Meu1)))} , 
       {4, 0, (1/2)*((sqrt(3/7))*((8)*((sqrt(5))*(Ma2u1)) + (-11)*(Ma2u2) + (18)*(Meu1)))} , 
       {4,-3, (-1/2+1/2*I)*((-2)*((sqrt(3))*(Ma2u1)) + (sqrt(15))*(Ma2u2))} , 
       {4, 3, (1/2+1/2*I)*((-2)*((sqrt(3))*(Ma2u1)) + (sqrt(15))*(Ma2u2))} }

</code>

###

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

###

|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
^$ {Y_{-1}^{(1)}} $|$ 0 $|$ 0 $|$ \frac{-2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}} $|$ 0 $|$ 0 $|$ \left(-\frac{1}{6}+\frac{i}{6}\right) \left(\sqrt{15} \text{Ma2u2}-2 \sqrt{3} \text{Ma2u1}\right) $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(\sqrt{5} \text{Ma2u1}+2 \text{Ma2u2}\right) $|$ 0 $|$ 0 $|$ \left(-\frac{1}{6}+\frac{i}{6}\right) \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right) $|
^$ {Y_{1}^{(1)}} $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\sqrt{15} \text{Ma2u2}-2 \sqrt{3} \text{Ma2u1}\right) $|$ 0 $|$ 0 $|$ \frac{-2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}} $|$ 0 $|$ 0 $|


###

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

###

|  $  $  ^  $ f_{(x+y-z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2-2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.} $  ^  $ f_{\left(x^3-\left.5\left\backslash x^2\right.\right\backslash z+\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2+10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(\left.-x^2\right\backslash (y+5\backslash z)-10\backslash x\backslash y\backslash z-y\left\backslash \left(y^2-5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.+\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z-8\left\backslash z^3\right.\right)} $  ^  $ f_{\left(x^3+\left.x^2\right\backslash z-\left.y^2\right\backslash z+x\left\backslash \left(y^2-2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{\left(-2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y-z)+y\left\backslash \left(y^2+y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $  ^  $ f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.} $  ^
^$ p_x $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ 2 \text{Ma2u1}+\sqrt{5} (\text{Meu1}-\text{Ma2u2}) $|$ 0 $|$ 0 $|
^$ p_y $|$ 0 $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ 2 \text{Ma2u1}+\sqrt{5} (\text{Meu1}-\text{Ma2u2}) $|$ 0 $|
^$ p_z $|$ \text{Ma2u1} $|$ 0 $|$ 0 $|$ \text{Ma2u2} $|$ 0 $|$ 0 $|$ 0 $|


###

</hidden>

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

###

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

###

###

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

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