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physics_chemistry:point_groups:d3d:orientation_z_x-y_a [2018/09/06 14:00] (current)
Maurits W. Haverkort created
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 +~~CLOSETOC~~
  
 +====== Orientation Z(x-y)_A ======
 +
 +###
 +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.)
 +###
 +
 +###
 +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.
 +###
 +
 +###
 +The parameterization A of the orientation Z(x-y) is related to the orientation 111z of the Oh pointgroup.
 +
 +###
 +
 +===== Symmetry Operations =====
 +
 +###
 +
 +In the D3d Point Group, with orientation Z(x-y)_A there are the following symmetry operations
 +
 +###
 +
 +###
 +
 +{{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a.png}}
 +
 +###
 +
 +###
 +
 +^ Operator ^ Orientation ^
 +^ $\text{E}$ | $\{0,0,0\}$ , |
 +^ $C_3$ | $\{0,0,1\}$ , $\{0,​0,​-1\}$ , |
 +^ $C_2$ | $\{1,​-1,​0\}$ , $\left\{2+\sqrt{3},​1,​0\right\}$ , $\left\{1,​2+\sqrt{3},​0\right\}$ , |
 +^ $\text{i}$ | $\{0,0,0\}$ , |
 +^ $S_6$ | $\{0,0,1\}$ , $\{0,​0,​-1\}$ , |
 +^ $\sigma _d$ | $\{1,​-1,​0\}$ , $\left\{2+\sqrt{3},​1,​0\right\}$ , $\left\{1,​2+\sqrt{3},​0\right\}$ , |
 +
 +###
 +
 +===== Different Settings =====
 +
 +###
 +
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_111|Point Group D3d with orientation 111]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zx|Point Group D3d with orientation Zx]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zx_a|Point Group D3d with orientation Zx_A]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zx_b|Point Group D3d with orientation Zx_B]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_z(x-y)|Point Group D3d with orientation Z(x-y)]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_z(x-y)_a|Point Group D3d with orientation Z(x-y)_A]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_z(x-y)_b|Point Group D3d with orientation Z(x-y)_B]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy|Point Group D3d with orientation Zy]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy_a|Point Group D3d with orientation Zy_A]]
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy_b|Point Group D3d with orientation Zy_B]]
 +
 +###
 +
 +===== Character Table =====
 +
 +###
 +
 +|  $  $  ^  $ \text{E} \,​{\text{(1)}} $  ^  $ C_3 \,​{\text{(2)}} $  ^  $ C_2 \,​{\text{(3)}} $  ^  $ \text{i} \,​{\text{(1)}} $  ^  $ S_6 \,​{\text{(2)}} $  ^  $ \sigma_d \,​{\text{(3)}} $  ^
 +^ $ A_{1g} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
 +^ $ A_{2g} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |
 +^ $ E_g $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |
 +^ $ A_{1u} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |
 +^ $ A_{2u} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |
 +^ $ E_u $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ -2 $ |  $ 1 $ |  $ 0 $ |
 +
 +###
 +
 +===== Product Table =====
 +
 +###
 +
 +|  $  $  ^  $ A_{1g} $  ^  $ A_{2g} $  ^  $ E_g $  ^  $ A_{1u} $  ^  $ A_{2u} $  ^  $ E_u $  ^
 +^ $ A_{1g} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  |
 +^ $ A_{2g} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  |
 +^ $ E_g $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  |
 +^ $ A_{1u} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  |
 +^ $ A_{2u} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  |
 +^ $ E_u $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  |
 +
 +###
 +
 +===== Sub Groups with compatible settings =====
 +
 +###
 +
 +  * [[physics_chemistry:​point_groups:​c1:​orientation_1|Point Group C1 with orientation 1]]
 +  * [[physics_chemistry:​point_groups:​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)_b|Point Group D3d with orientation Z(x-y)_B]]
 +  * [[physics_chemistry:​point_groups:​s6:​orientation_z|Point Group S6 with orientation Z]]
 +
 +###
 +
 +===== Super Groups with compatible settings =====
 +
 +###
 +
 +  * [[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)_b|Point Group D3d with orientation Z(x-y)_B]]
 +  * [[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]]
 +
 +###
 +
 +===== Invariant Potential expanded on renormalized spherical Harmonics =====
 +
 +###
 +
 +Any potential (function) can be written as a sum over spherical harmonics.
 +$$V(r,​\theta,​\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,​\phi)$$
 +Here $A_{k,​m}(r)$ is a radial function and $C^{(m)}_k(\theta,​\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,​\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,​\phi)$$
 +The presence of symmetry induces relations between the expansion coefficients such that $V(r,​\theta,​\phi)$ is invariant under all symmetry operations. For the D3d Point group with orientation Z(x-y)_A the form of the expansion coefficients is:
 +
 +###
 +
 +==== Expansion ====
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + ​A(0,​0) & k=0\land m=0 \\
 + ​A(2,​0) & k=2\land m=0 \\
 + ​(-1+i) A(4,3) & k=4\land m=-3 \\
 + ​A(4,​0) & k=4\land m=0 \\
 + (1+i) A(4,3) & k=4\land m=3 \\
 + -i B(6,6) & k=6\land m=-6 \\
 + ​(-1+i) A(6,3) & k=6\land m=-3 \\
 + ​A(6,​0) & k=6\land m=0 \\
 + (1+i) A(6,3) & k=6\land m=3 \\
 + i B(6,6) & k=6\land m=6
 +\end{cases}$$
 +
 +###
 +
 +==== Input format suitable for Mathematica (Quanty.nb) ====
 +
 +###
 +
 +<code Quanty Akm_D3d_Z(x-y)_A.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>​
 +
 +###
 +
 +==== Input format suitable for Quanty ====
 +
 +###
 +
 +<code Quanty Akm_D3d_Z(x-y)_A.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>​
 +
 +###
 +
 +==== One particle coupling on a basis of spherical harmonics ====
 +
 +###
 +
 +The operator representing the potential in second quantisation is given as:
 +$$ O = \sum_{n'',​l'',​m'',​n',​l',​m'​} \left\langle \psi_{n'',​l'',​m''​}(r,​\theta,​\phi) \left| V(r,​\theta,​\phi) \right| \psi_{n',​l',​m'​}(r,​\theta,​\phi) \right\rangle a^{\dagger}_{n'',​l'',​m''​}a^{\phantom{\dagger}}_{n',​l',​m'​}$$
 +For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,​l,​m}(r,​\theta,​\phi)=R_{n,​l}(r)Y_{m}^{(l)}(\theta,​\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter.
 +$$ A_{n''​l'',​n'​l'​}(k,​m) = \left\langle R_{n'',​l''​} \left| A_{k,m}(r) \right| R_{n',​l'​} \right\rangle $$
 +Note the difference between the function $A_{k,m}$ and the parameter $A_{n''​l'',​n'​l'​}(k,​m)$
 +
 +
 +###
 +
 +
 +
 +###
 +
 +
 +we can express the operator as 
 +$$ O = \sum_{n'',​l'',​m'',​n',​l',​m',​k,​m} A_{n''​l'',​n'​l'​}(k,​m) \left\langle Y_{l''​}^{(m''​)}(\theta,​\phi) \left| C_{k}^{(m)}(\theta,​\phi) \right| Y_{l'​}^{(m'​)}(\theta,​\phi) \right\rangle a^{\dagger}_{n'',​l'',​m''​}a^{\phantom{\dagger}}_{n',​l',​m'​}$$
 +
 +
 +###
 +
 +
 +
 +###
 +
 +
 +The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',​l'​}(k,​m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',​l'​}(k,​m) + \mathrm{I}\,​ B_{l'',​l'​}(k,​m)$ (with both A and B real) as the expansion parameter.
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ {Y_{0}^{(0)}} $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
 +^$ {Y_{0}^{(0)}} $|$ \text{Ass}(0,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \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) $|
 +
 +
 +###
 +
 +==== Rotation matrix to symmetry adapted functions (choice is not unique) ====
 +
 +###
 +
 +
 +Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ {Y_{0}^{(0)}} $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
 +^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ p_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 $|
 +^$ 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 $|
 +^$ 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_{(x-y)(x+y+z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \frac{1}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{(x-y)(x+y-2z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{6}} $|$ -\frac{1-i}{\sqrt{6}} $|$ 0 $|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ \frac{1+i}{\sqrt{6}} $|$ 0 $|$ -\frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ 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} $|
 +^$ 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(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.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(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 $|
 +
 +
 +###
 +
 +==== One particle coupling on a basis of symmetry adapted functions ====
 +
 +###
 +
 +After rotation we find
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ \text{s} $  ^  $ p_z $  ^  $ p_x $  ^  $ p_y $  ^  $ d_{3z^2-r^2} $  ^  $ d_{(x-y)(x+y+z)} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{(x-y)(x+y-2z)} $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^  $ 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.} $  ^  $ 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.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(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)} $  ^
 +^$ \text{s} $|$ \text{Ass}(0,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\text{Asd}(2,​0)}{\sqrt{5}} $|$ 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_z $|$\color{darkred}{ 0 }$|$ \text{App}(0,​0)+\frac{2}{5} \text{App}(2,​0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 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}} $|$ \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 $|$ 0 $|
 +^$ p_x $|$\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 $|$ 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 $|$ -\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_y $|$\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 $|$ 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 $|$ -\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}} $|
 +^$ d_{3z^2-r^2} $|$ \frac{\text{Asd}(2,​0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,​0)+\frac{2}{7} \text{Add}(2,​0)+\frac{2}{7} \text{Add}(4,​0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{(x-y)(x+y+z)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 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 $|$ -\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 }$|$ 0 $|$ 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 $|$ -\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) $|$\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 $|$ -\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}{9} \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_{\text{yz}+\text{xz}+\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 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}{9} \text{Add}(4,​0)+\frac{4}{9} \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 }$|
 +^$ 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 }$|$ \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) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 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 }$|$ -\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 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 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) $|$ -\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 $|$ 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 }$|$ \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 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 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) $|$ \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 $|$ 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 }$|$ 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 $|$ 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 $|$ \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(\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 $|$ -\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) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,​0)+\frac{1}{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 $|$ \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) $|
 +^$ 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 $|$ -\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 $|$ 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 $|$ \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_{\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 $|$ -\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}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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 $|$ \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) $|
 +
 +
 +###
 +
 +===== 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)_A.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)_A.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)_a_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)_A.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)_A.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_z $  ^  $ p_x $  ^  $ p_y $  ^
 +^$ p_z $|$ \text{Ea2u} $|$ 0 $|$ 0 $|
 +^$ p_x $|$ 0 $|$ \text{Eeu} $|$ 0 $|
 +^$ p_y $|$ 0 $|$ 0 $|$ \text{Eeu} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
 +^$ p_z $|$ 0 $|$ 1 $|$ 0 $|
 +^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|
 +^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_1_1.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:​d3d_z(x-y)_a_orb_1_2.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)_a_orb_1_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
 +
 +
 +###
 +
 +</​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)_A.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)_A.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_{3z^2-r^2} $  ^  $ d_{(x-y)(x+y+z)} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{(x-y)(x+y-2z)} $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^
 +^$ d_{3z^2-r^2} $|$ \text{Ea1g} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{(x-y)(x+y+z)} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ \text{Meg} $|$ 0 $|
 +^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ \text{Meg} $|
 +^$ d_{(x-y)(x+y-2z)} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $|
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ 0 $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\sigma $} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
 +^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
 +^$ d_{(x-y)(x+y+z)} $|$ \frac{1}{\sqrt{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \frac{1}{\sqrt{3}} $|
 +^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ \frac{i}{\sqrt{3}} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\sqrt{3}} $|
 +^$ d_{(x-y)(x+y-2z)} $|$ \frac{1}{\sqrt{6}} $|$ -\frac{1-i}{\sqrt{6}} $|$ 0 $|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{6}} $|
 +^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ \frac{i}{\sqrt{6}} $|$ \frac{1+i}{\sqrt{6}} $|$ 0 $|$ -\frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea1g}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Eeg$\pi $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta ))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (x-y) (x+y+z)$$ | ::: |
 +^ ^$$\text{Eeg$\pi $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$$ | ::: |
 +^ ^$$\text{Eeg$\sigma $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_4.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta ))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} (x-y) (x+y-2 z)$$ | ::: |
 +^ ^$$\text{Eeg$\sigma $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_5.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$$ | ::: |
 +
 +
 +###
 +
 +</​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)_A.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)_A.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) + 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) + 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 \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) $|
 +^$ {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)\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.} $  ^  $ 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.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(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{Ea1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 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.} $|$ 0 $|$ \text{Ea2u1} $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)} $|$ 0 $|$ \text{Ma2u} $|$ \text{Ea2u2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ \text{Meu} $|$ 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 $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ \text{Meu} $|
 +^$ f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ \text{Eeu2} $|$ 0 $|
 +^$ f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ \text{Eeu2} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-3}^{(3)}} $  ^  $ {Y_{-2}^{(3)}} $  ^  $ {Y_{-1}^{(3)}} $  ^  $ {Y_{0}^{(3)}} $  ^  $ {Y_{1}^{(3)}} $  ^  $ {Y_{2}^{(3)}} $  ^  $ {Y_{3}^{(3)}} $  ^
 +^$ f_{(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} $|
 +^$ 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(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.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(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 $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea1u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin ^3(\theta ) (\sin (3 \phi )+\cos (3 \phi ))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^3+3 x^2 y-3 x y^2-y^3\right)$$ | ::: |
 +^ ^$$\text{Ea2u1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{24}+\frac{i}{24}\right) \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(e^{6 i \phi } \sin ^3(\theta )-(1-i) e^{3 i \phi } \cos (\theta ) \left(5 \cos ^2(\theta )-3\right)-i \sin ^3(\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(x^3-3 x^2 y-3 x y^2+y^3-5 z^3+3 z\right)$$ | ::: |
 +^ ^$$\text{Ea2u2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{48}+\frac{i}{48}\right) \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \left(5 \left(e^{6 i \phi }-i\right) \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 x^3-15 x^2 y-15 x y^2+5 y^3+4 z \left(5 z^2-3\right)\right)$$ | ::: |
 +^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_4.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )-\cos (2 \phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 x^2 z-10 x y z-5 x z^2+x-5 y^2 z\right)$$ | ::: |
 +^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_5.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \sin (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 x^2 z-10 x y z+5 y^2 z-5 y z^2+y\right)$$ | ::: |
 +^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_6.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+\sin (2 \theta ) (\cos (2 \phi )-\sin (2 \phi )))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 (-z)+2 x y z-5 x z^2+x+y^2 z\right)$$ | ::: |
 +^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_7.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi ))-(5 \cos (2 \theta )+3) \sin (\phi ))$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 z+2 x y z-y^2 z-5 y z^2+y\right)$$ | ::: |
 +
 +
 +###
 +
 +</​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)_A.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)_A.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_{3z^2-r^2} $  ^  $ d_{(x-y)(x+y+z)} $  ^  $ d_{2\text{xy}-\text{xz}-\text{yz}} $  ^  $ d_{(x-y)(x+y-2z)} $  ^  $ d_{\text{yz}+\text{xz}+\text{xy}} $  ^
 +^$ \text{s} $|$ \text{Ma1g} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</​hidden>​
 +==== Potential for p-f orbital mixing ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
 + ​\frac{5 \left(\sqrt{5} \text{Ma2u1}+4 \text{Ma2u2}-4 \text{Meu1}\right)}{\sqrt{21}} & k=2\land m=0 \\
 + ​\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) & k=4\land m=-3 \\
 + ​-\frac{1}{2} \sqrt{\frac{3}{7}} \left(8 \sqrt{5} \text{Ma2u1}+11 \text{Ma2u2}-18 \text{Meu1}\right) & k=4\land m=0 \\
 + ​\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Z(x-y)_A.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)*Sqrt[3]*(2*Ma2u1 + Sqrt[5]*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)*Sqrt[3]*(2*Ma2u1 + Sqrt[5]*Ma2u2)]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Z(x-y)_A.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)*((sqrt(3))*((2)*(Ma2u1) + (sqrt(5))*(Ma2u2)))} , 
 +       ​{4,​-3,​ (1/​2+-1/​2*I)*((sqrt(3))*((2)*(Ma2u1) + (sqrt(5))*(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(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\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(2 \text{Ma2u2}-\sqrt{5} \text{Ma2u1}\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(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\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)\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.} $  ^  $ 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.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(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)} $  ^
 +^$ p_z $|$ 0 $|$ \text{Ma2u1} $|$ \text{Ma2u2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_x $|$ 0 $|$ 0 $|$ 0 $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+i \sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right) $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-\sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-i \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \sqrt{5} \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right) $|$ -\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)}{\sqrt{3}} $|
 +^$ p_y $|$ 0 $|$ 0 $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+i \sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)}{\sqrt{3}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-\sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-i \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \sqrt{5} \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right) $|
 +
 +
 +###
 +
 +</​hidden>​
 +
 +===== Table of several point groups =====
 +
 +###
 +
 +[[physics_chemistry:​point_groups|Return to Main page on Point Groups]]
 +
 +###
 +
 +###
 +
 +^Nonaxial groups ​     | [[physics_chemistry:​point_groups:​c1|C]]<​sub>​[[physics_chemistry:​point_groups:​c1|1]]</​sub>​ | [[physics_chemistry:​point_groups:​cs|C]]<​sub>​[[physics_chemistry:​point_groups:​cs|s]]</​sub>​ | [[physics_chemistry:​point_groups:​ci|C]]<​sub>​[[physics_chemistry:​point_groups:​ci|i]]</​sub>​ | | | | |
 +^C<​sub>​n</​sub>​ groups | [[physics_chemistry:​point_groups:​c2|C]]<​sub>​[[physics_chemistry:​point_groups:​c2|2]]</​sub>​ | [[physics_chemistry:​point_groups:​c3|C]]<​sub>​[[physics_chemistry:​point_groups:​c3|3]]</​sub>​ | [[physics_chemistry:​point_groups:​c4|C]]<​sub>​[[physics_chemistry:​point_groups:​c4|4]]</​sub>​ | [[physics_chemistry:​point_groups:​c5|C]]<​sub>​[[physics_chemistry:​point_groups:​c5|5]]</​sub>​ | [[physics_chemistry:​point_groups:​c6|C]]<​sub>​[[physics_chemistry:​point_groups:​c6|6]]</​sub>​ | [[physics_chemistry:​point_groups:​c7|C]]<​sub>​[[physics_chemistry:​point_groups:​c7|7]]</​sub>​ | [[physics_chemistry:​point_groups:​c8|C]]<​sub>​[[physics_chemistry:​point_groups:​c8|8]]</​sub>​ | 
 +^D<​sub>​n</​sub>​ groups | [[physics_chemistry:​point_groups:​d2|D]]<​sub>​[[physics_chemistry:​point_groups:​d2|2]]</​sub>​ | [[physics_chemistry:​point_groups:​d3|D]]<​sub>​[[physics_chemistry:​point_groups:​d3|3]]</​sub>​ | [[physics_chemistry:​point_groups:​d4|D]]<​sub>​[[physics_chemistry:​point_groups:​d4|4]]</​sub>​ | [[physics_chemistry:​point_groups:​d5|D]]<​sub>​[[physics_chemistry:​point_groups:​d5|5]]</​sub>​ | [[physics_chemistry:​point_groups:​d6|D]]<​sub>​[[physics_chemistry:​point_groups:​d6|6]]</​sub>​ | [[physics_chemistry:​point_groups:​d7|D]]<​sub>​[[physics_chemistry:​point_groups:​d7|7]]</​sub>​ | [[physics_chemistry:​point_groups:​d8|D]]<​sub>​[[physics_chemistry:​point_groups:​d8|8]]</​sub>​ | 
 +^C<​sub>​nv</​sub>​ groups | [[physics_chemistry:​point_groups:​c2v|C]]<​sub>​[[physics_chemistry:​point_groups:​c2v|2v]]</​sub>​ | [[physics_chemistry:​point_groups:​c3v|C]]<​sub>​[[physics_chemistry:​point_groups:​c3v|3v]]</​sub>​ | [[physics_chemistry:​point_groups:​c4v|C]]<​sub>​[[physics_chemistry:​point_groups:​c4v|4v]]</​sub>​ | [[physics_chemistry:​point_groups:​c5v|C]]<​sub>​[[physics_chemistry:​point_groups:​c5v|5v]]</​sub>​ | [[physics_chemistry:​point_groups:​c6v|C]]<​sub>​[[physics_chemistry:​point_groups:​c6v|6v]]</​sub>​ | [[physics_chemistry:​point_groups:​c7v|C]]<​sub>​[[physics_chemistry:​point_groups:​c7v|7v]]</​sub>​ | [[physics_chemistry:​point_groups:​c8v|C]]<​sub>​[[physics_chemistry:​point_groups:​c8v|8v]]</​sub>​ | 
 +^C<​sub>​nh</​sub>​ groups | [[physics_chemistry:​point_groups:​c2h|C]]<​sub>​[[physics_chemistry:​point_groups:​c2h|2h]]</​sub>​ | [[physics_chemistry:​point_groups:​c3h|C]]<​sub>​[[physics_chemistry:​point_groups:​c3h|3h]]</​sub>​ | [[physics_chemistry:​point_groups:​c4h|C]]<​sub>​[[physics_chemistry:​point_groups:​c4h|4h]]</​sub>​ | [[physics_chemistry:​point_groups:​c5h|C]]<​sub>​[[physics_chemistry:​point_groups:​c5h|5h]]</​sub>​ | [[physics_chemistry:​point_groups:​c6h|C]]<​sub>​[[physics_chemistry:​point_groups:​c6h|6h]]</​sub>​ | | | 
 +^D<​sub>​nh</​sub>​ groups | [[physics_chemistry:​point_groups:​d2h|D]]<​sub>​[[physics_chemistry:​point_groups:​d2h|2h]]</​sub>​ | [[physics_chemistry:​point_groups:​d3h|D]]<​sub>​[[physics_chemistry:​point_groups:​d3h|3h]]</​sub>​ | [[physics_chemistry:​point_groups:​d4h|D]]<​sub>​[[physics_chemistry:​point_groups:​d4h|4h]]</​sub>​ | [[physics_chemistry:​point_groups:​d5h|D]]<​sub>​[[physics_chemistry:​point_groups:​d5h|5h]]</​sub>​ | [[physics_chemistry:​point_groups:​d6h|D]]<​sub>​[[physics_chemistry:​point_groups:​d6h|6h]]</​sub>​ | [[physics_chemistry:​point_groups:​d7h|D]]<​sub>​[[physics_chemistry:​point_groups:​d7h|7h]]</​sub>​ | [[physics_chemistry:​point_groups:​d8h|D]]<​sub>​[[physics_chemistry:​point_groups:​d8h|8h]]</​sub>​ | 
 +^D<​sub>​nd</​sub>​ groups | [[physics_chemistry:​point_groups:​d2d|D]]<​sub>​[[physics_chemistry:​point_groups:​d2d|2d]]</​sub>​ | [[physics_chemistry:​point_groups:​d3d|D]]<​sub>​[[physics_chemistry:​point_groups:​d3d|3d]]</​sub>​ | [[physics_chemistry:​point_groups:​d4d|D]]<​sub>​[[physics_chemistry:​point_groups:​d4d|4d]]</​sub>​ | [[physics_chemistry:​point_groups:​d5d|D]]<​sub>​[[physics_chemistry:​point_groups:​d5d|5d]]</​sub>​ | [[physics_chemistry:​point_groups:​d6d|D]]<​sub>​[[physics_chemistry:​point_groups:​d6d|6d]]</​sub>​ | [[physics_chemistry:​point_groups:​d7d|D]]<​sub>​[[physics_chemistry:​point_groups:​d7d|7d]]</​sub>​ | [[physics_chemistry:​point_groups:​d8d|D]]<​sub>​[[physics_chemistry:​point_groups:​d8d|8d]]</​sub>​ | 
 +^S<​sub>​n</​sub>​ groups | [[physics_chemistry:​point_groups:​S2|S]]<​sub>​[[physics_chemistry:​point_groups:​S2|2]]</​sub>​ | [[physics_chemistry:​point_groups:​S4|S]]<​sub>​[[physics_chemistry:​point_groups:​S4|4]]</​sub>​ | [[physics_chemistry:​point_groups:​S6|S]]<​sub>​[[physics_chemistry:​point_groups:​S6|6]]</​sub>​ | [[physics_chemistry:​point_groups:​S8|S]]<​sub>​[[physics_chemistry:​point_groups:​S8|8]]</​sub>​ | [[physics_chemistry:​point_groups:​S10|S]]<​sub>​[[physics_chemistry:​point_groups:​S10|10]]</​sub>​ | [[physics_chemistry:​point_groups:​S12|S]]<​sub>​[[physics_chemistry:​point_groups:​S12|12]]</​sub>​ |  | 
 +^Cubic groups | [[physics_chemistry:​point_groups:​T|T]] | [[physics_chemistry:​point_groups:​Th|T]]<​sub>​[[physics_chemistry:​point_groups:​Th|h]]</​sub>​ | [[physics_chemistry:​point_groups:​Td|T]]<​sub>​[[physics_chemistry:​point_groups:​Td|d]]</​sub>​ | [[physics_chemistry:​point_groups:​O|O]] | [[physics_chemistry:​point_groups:​Oh|O]]<​sub>​[[physics_chemistry:​point_groups:​Oh|h]]</​sub>​ | [[physics_chemistry:​point_groups:​I|I]] | [[physics_chemistry:​point_groups:​Ih|I]]<​sub>​[[physics_chemistry:​point_groups:​Ih|h]]</​sub>​ | 
 +^Linear groups ​     | [[physics_chemistry:​point_groups:​cinfv|C]]<​sub>​[[physics_chemistry:​point_groups:​cinfv|$\infty$v]]</​sub>​ | [[physics_chemistry:​point_groups:​cinfv|D]]<​sub>​[[physics_chemistry:​point_groups:​dinfh|$\infty$h]]</​sub>​ | | | | | |
 +
 +###
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