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physics_chemistry:point_groups:d3d:orientation_zy_a [2018/03/21 18:40]
Stefano Agrestini created
physics_chemistry:point_groups:d3d:orientation_zy_a [2018/09/06 13:56] (current)
Maurits W. Haverkort
Line 1: Line 1:
 +~~CLOSETOC~~
 +
 ====== Orientation Zy_A ====== ====== Orientation Zy_A ======
  
 ### ###
-alligned paragraph text+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.)
 ### ###
  
-===== Example =====+### 
 +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. 
 +###
  
 ### ###
-description text+The parameterization A of the orientation Zy is related to the orientation Sqrt[2]01z of the Oh pointgroup.
 ### ###
  
-==== Input ==== +===== Symmetry Operations ===== 
-<code Quanty ​Example.Quanty>​ + 
--- some example code+### 
 + 
 +In the D3d Point Group, with orientation Zy_A there are the following symmetry operations 
 + 
 +### 
 + 
 +### 
 + 
 +{{:​physics_chemistry:​pointgroup:​d3d_zy_a.png}} 
 + 
 +### 
 + 
 +### 
 + 
 +^ Operator ^ Orientation ^ 
 +^ $\text{E}$ | $\{0,0,0\}$ , | 
 +^ $C_3$ | $\{0,0,1\}$ , $\{0,​0,​-1\}$ , | 
 +^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},​1,​0\right\}$ , $\left\{-\sqrt{3},​1,​0\right\}$ , | 
 +^ $\text{i}$ | $\{0,0,0\}$ , | 
 +^ $S_6$ | $\{0,0,1\}$ , $\{0,​0,​-1\}$ , | 
 +^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},​1,​0\right\}$ , $\left\{-\sqrt{3},​1,​0\right\}$ , | 
 + 
 +### 
 + 
 +===== 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:​c2:​orientation_y|Point Group C2 with orientation Y]] 
 +  * [[physics_chemistry:​point_groups:​c3v:​orientation_zy|Point Group C3v with orientation Zy]] 
 +  * [[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:​cs:​orientation_y|Point Group Cs with orientation Y]] 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy|Point Group D3d with orientation Zy]] 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy_b|Point Group D3d with orientation Zy_B]] 
 +  * [[physics_chemistry:​point_groups:​d3:​orientation_zy|Point Group D3 with orientation Zy]] 
 +  * [[physics_chemistry:​point_groups:​s6:​orientation_z|Point Group S6 with orientation Z]] 
 + 
 +### 
 + 
 +===== Super Groups with compatible settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy|Point Group D3d with orientation Zy]] 
 +  * [[physics_chemistry:​point_groups:​d3d:​orientation_zy_b|Point Group D3d with orientation Zy_B]] 
 +  * [[physics_chemistry:​point_groups:​d6h:​orientation_zx|Point Group D6h with orientation Zx]] 
 +  * [[physics_chemistry:​point_groups:​d6h:​orientation_zy|Point Group D6h with orientation Zy]] 
 +  * [[physics_chemistry:​point_groups:​oh:​orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]] 
 +  * [[physics_chemistry:​point_groups:​oh:​orientation_sqrt201z|Point Group Oh with orientation sqrt201z]] 
 + 
 +### 
 + 
 +===== 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 Zy_A the form of the expansion coefficients is: 
 + 
 +### 
 + 
 +==== Expansion ==== 
 + 
 +### 
 + 
 + ​$$A_{k,​m} = \begin{cases} 
 + ​A(0,​0) & k=0\land m=0 \\ 
 + ​A(2,​0) & k=2\land m=0 \\ 
 + ​-A(4,​3) & k=4\land m=-3 \\ 
 + ​A(4,​0) & k=4\land m=0 \\ 
 + ​A(4,​3) & k=4\land m=3 \\ 
 + ​A(6,​6) & k=6\land (m=-6\lor m=6) \\ 
 + ​-A(6,​3) & k=6\land m=-3 \\ 
 + ​A(6,​0) & k=6\land m=0 \\ 
 + ​A(6,​3) & k=6\land m=3 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +==== Input format suitable for Mathematica (Quanty.nb) ​==== 
 + 
 +### 
 + 
 +<code Quanty ​Akm_D3d_Zy_A.Quanty.nb
 + 
 +Akm[k_,​m_]:​=Piecewise[{{A[0,​ 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {-A[4, 3], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {A[4, 3], k == 4 && m == 3}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {-A[6, 3], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {A[6, 3], k == 6 && m == 3}}, 0] 
 </​code>​ </​code>​
  
-==== Result ==== +###
-<WRAP center box 100%> +
-text produced as output +
-</​WRAP>​+
  
-===== Table of contents ​===== +==== Input format suitable for Quanty ​====
-{{indexmenu>​.#​1}}+
  
 +###
 +
 +<code Quanty Akm_D3d_Zy_A.Quanty>​
 +
 +Akm = {{0, 0, A(0,0)} , 
 +       {2, 0, A(2,0)} , 
 +       {4, 0, A(4,0)} , 
 +       ​{4,​-3,​ (-1)*(A(4,​3))} , 
 +       {4, 3, A(4,3)} , 
 +       {6, 0, A(6,0)} , 
 +       ​{6,​-3,​ (-1)*(A(6,​3))} , 
 +       {6, 3, A(6,3)} , 
 +       ​{6,​-6,​ A(6,6)} , 
 +       {6, 6, A(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 $|$ -\frac{1}{3} \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{\text{Apf}(4,​3)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \frac{\text{Apf}(4,​3)}{3 \sqrt{3}} $|
 +^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,​0)-\frac{1}{5} \text{App}(2,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{3} \text{Apf}(4,​3) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0) $|$ 0 $|$ 0 $|
 +^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,​0)-\frac{2}{7} \text{Add}(2,​0)+\frac{1}{21} \text{Add}(4,​0) $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,​3) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,​0)+\frac{1}{7} \text{Add}(2,​0)-\frac{4}{21} \text{Add}(4,​0) $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,​3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2,​0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,​0)+\frac{2}{7} \text{Add}(2,​0)+\frac{2}{7} \text{Add}(4,​0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,​3) $|$ 0 $|$ 0 $|$ \text{Add}(0,​0)+\frac{1}{7} \text{Add}(2,​0)-\frac{4}{21} \text{Add}(4,​0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,​3) $|$ 0 $|$ 0 $|$ \text{Add}(0,​0)-\frac{2}{7} \text{Add}(2,​0)+\frac{1}{21} \text{Add}(4,​0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{\text{Apf}(4,​3)}{3 \sqrt{3}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,​0)-\frac{1}{3} \text{Aff}(2,​0)+\frac{1}{11} \text{Aff}(4,​0)-\frac{5}{429} \text{Aff}(6,​0) $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{7} \text{Aff}(4,​3)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3) $|$ 0 $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,​6) $|
 +^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{3} \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 $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,​3)+\frac{5}{143} \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 $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,​3)-\frac{5}{143} \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 }$|$ \frac{1}{11} \sqrt{7} \text{Aff}(4,​3)-\frac{10}{143} \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 $|$ \frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3)-\frac{1}{11} \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 $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,​3)+\frac{5}{143} \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 }$|$ -\frac{1}{3} \text{Apf}(4,​3) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,​3)-\frac{5}{143} \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{\text{Apf}(4,​3)}{3 \sqrt{3}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,​6) $|$ 0 $|$ 0 $|$ \frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3)-\frac{1}{11} \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_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{xy}-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$ 0 $|$ -\frac{i}{\sqrt{3}} $|$ -\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^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \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}+\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $|$ 0 $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{2}}{3} $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\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 }$|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\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 }$|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|$ \frac{i}{2 \sqrt{3}} $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\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{\sqrt{\frac{5}{2}}}{3} $|$ 0 $|$ 0 $|$ -\frac{2}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{\frac{5}{2}}}{3} $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\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 }$|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{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{xy}-\sqrt{2}\text{yz}} $  ^  $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $  ^  $ d_{\text{yz}+\sqrt{2}\text{xy}} $  ^  $ d_{x^2-y^2-\sqrt{2}\text{xz}} $  ^  $ d_{3z^2-r^2} $  ^  $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $  ^  $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $  ^  $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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}{6}} \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{6}} $|$ 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}{6}} \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{6}} $|$ 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{2}{9} \sqrt{\frac{2}{3}} \text{Apf}(4,​3) $|$ 0 $|$ 0 $|$ -\frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)-\frac{8 \text{Apf}(4,​0)}{9 \sqrt{21}}-\frac{1}{9} \sqrt{\frac{10}{3}} \text{Apf}(4,​3) $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,​0)-\frac{1}{9} \text{Add}(4,​0)-\frac{2}{9} \sqrt{\frac{10}{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{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^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,​0)-\frac{1}{9} \text{Add}(4,​0)-\frac{2}{9} \sqrt{\frac{10}{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{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}+\sqrt{2}\text{xy}} $|$ 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{5}{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{2}{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^2-y^2-\sqrt{2}\text{xz}} $|$ 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{5}{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{2}{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_{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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\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{2}{9} \sqrt{\frac{2}{3}} \text{Apf}(4,​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{4}{99} \sqrt{70} \text{Aff}(4,​3)+\frac{160 \text{Aff}(6,​0)}{1287}-\frac{40 \sqrt{\frac{70}{3}} \text{Aff}(6,​3)}{1287}+\frac{40}{117} \sqrt{\frac{7}{33}} \text{Aff}(6,​6) $|$ 0 $|$ 0 $|$ -\frac{2 \text{Aff}(2,​0)}{3 \sqrt{5}}-\frac{2}{99} \sqrt{5} \text{Aff}(4,​0)+\frac{1}{99} \sqrt{14} \text{Aff}(4,​3)-\frac{70 \sqrt{5} \text{Aff}(6,​0)}{1287}-\frac{10 \sqrt{\frac{14}{3}} \text{Aff}(6,​3)}{1287}+\frac{20}{117} \sqrt{\frac{35}{33}} \text{Aff}(6,​6) $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\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}{6}} \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{1}{99} \sqrt{70} \text{Aff}(4,​3)+\frac{25}{858} \text{Aff}(6,​0)-\frac{5}{143} \sqrt{\frac{70}{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{2}{99} \sqrt{14} \text{Aff}(4,​3)+\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,​3) $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\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}{6}} \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{1}{99} \sqrt{70} \text{Aff}(4,​3)+\frac{25}{858} \text{Aff}(6,​0)-\frac{5}{143} \sqrt{\frac{70}{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{2}{99} \sqrt{14} \text{Aff}(4,​3)+\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,​3) $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\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{1}{9} \sqrt{\frac{10}{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{1}{99} \sqrt{14} \text{Aff}(4,​3)-\frac{70 \sqrt{5} \text{Aff}(6,​0)}{1287}-\frac{10 \sqrt{\frac{14}{3}} \text{Aff}(6,​3)}{1287}+\frac{20}{117} \sqrt{\frac{35}{33}} \text{Aff}(6,​6) $|$ 0 $|$ 0 $|$ \text{Aff}(0,​0)-\frac{1}{15} \text{Aff}(2,​0)+\frac{13}{99} \text{Aff}(4,​0)-\frac{4}{99} \sqrt{70} \text{Aff}(4,​3)+\frac{125 \text{Aff}(6,​0)}{1287}+\frac{40 \sqrt{\frac{70}{3}} \text{Aff}(6,​3)}{1287}+\frac{50}{117} \sqrt{\frac{7}{33}} \text{Aff}(6,​6) $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\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{6}} $|$ 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{2}{99} \sqrt{14} \text{Aff}(4,​3)+\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{14}{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{1}{99} \sqrt{70} \text{Aff}(4,​3)-\frac{115}{858} \text{Aff}(6,​0)+\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,​3) $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\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{6}} $|$ 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{2}{99} \sqrt{14} \text{Aff}(4,​3)+\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{14}{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{1}{99} \sqrt{70} \text{Aff}(4,​3)-\frac{115}{858} \text{Aff}(6,​0)+\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,​3) $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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{Aff}(6,​6) $|
 +
 +
 +###
 +
 +===== Coupling for a single shell =====
 +
 +
 +
 +###
 +
 +Although the parameters $A_{l'',​l'​}(k,​m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',​l'​}(k,​m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''​$ and $l'$.
 +
 +###
 +
 +
 +
 +###
 +
 +Click on one of the subsections to expand it or <​hiddenSwitch expand all> ​
 +
 +###
 +
 +==== Potential for s orbitals ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + ​\text{Ea1g} & k=0\land m=0 \\
 + 0 & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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_Zy_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_zy_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_Zy_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_Zy_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_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_zy_a_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_zy_a_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_zy_a_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** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + ​\frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\
 + ​\text{Ea1g}-\text{Eeg}\pi -2 \sqrt{2} \text{Meg} & k=2\land m=0 \\
 + ​\sqrt{\frac{7}{5}} \left(-\sqrt{2} \text{Eeg}\pi +\sqrt{2} \text{Eeg}\sigma +\text{Meg}\right) & k=4\land m=-3 \\
 + ​\frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg}\pi -7 \text{Eeg}\sigma +10 \sqrt{2} \text{Meg}\right) & k=4\land m=0 \\
 + ​\sqrt{\frac{7}{5}} \left(\sqrt{2} \text{Eeg}\pi -\sqrt{2} \text{Eeg}\sigma -\text{Meg}\right) & k=4\land m=3
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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}, {Sqrt[7/​5]*(-(Sqrt[2]*Eeg\[Pi]) + Sqrt[2]*Eeg\[Sigma] + Meg), k == 4 && m == -3}, {(9*Ea1g - 2*Eeg\[Pi] - 7*Eeg\[Sigma] + 10*Sqrt[2]*Meg)/​5,​ k == 4 && m == 0}, {Sqrt[7/​5]*(Sqrt[2]*Eeg\[Pi] - Sqrt[2]*Eeg\[Sigma] - Meg), k == 4 && m == 3}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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,​ (sqrt(7/​5))*((-1)*((sqrt(2))*(EegPi)) + (sqrt(2))*(EegSigma) + Meg)} , 
 +       {4, 3, (sqrt(7/​5))*((sqrt(2))*(EegPi) + (-1)*((sqrt(2))*(EegSigma)) + (-1)*(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 $|$ \frac{1}{3} \left(\sqrt{2} \text{Eeg$\pi $}-\sqrt{2} \text{Eeg$\sigma $}-\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 $|$ \frac{1}{3} \left(-\sqrt{2} \text{Eeg$\pi $}+\sqrt{2} \text{Eeg$\sigma $}+\text{Meg}\right) $|
 +^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea1g} $|$ 0 $|$ 0 $|
 +^$ {Y_{1}^{(2)}} $|$ \frac{1}{3} \left(\sqrt{2} \text{Eeg$\pi $}-\sqrt{2} \text{Eeg$\sigma $}-\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 $|$ \frac{1}{3} \left(-\sqrt{2} \text{Eeg$\pi $}+\sqrt{2} \text{Eeg$\sigma $}+\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{xy}-\sqrt{2}\text{yz}} $  ^  $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $  ^  $ d_{\text{yz}+\sqrt{2}\text{xy}} $  ^  $ d_{x^2-y^2-\sqrt{2}\text{xz}} $  ^  $ d_{3z^2-r^2} $  ^
 +^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ 0 $|
 +^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|
 +^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ 0 $|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 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{xy}-\sqrt{2}\text{yz}} $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$ 0 $|$ -\frac{i}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|
 +^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \frac{1}{\sqrt{6}} $|
 +^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ \frac{i}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $|$ 0 $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|
 +^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \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_zy_a_orb_2_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \sin (\phi ) \left(\sin (\theta ) \cos (\phi )-\sqrt{2} \cos (\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} y \left(x-\sqrt{2} z\right)$$ | ::: |
 +^ ^$$\text{Eeg$\sigma $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_2_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(2 \sqrt{2} \cos (\theta ) \cos (\phi )+\sin (\theta ) \cos (2 \phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(x^2+2 \sqrt{2} x z-y^2\right)$$ | ::: |
 +^ ^$$\text{Eeg$\pi $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_2_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \sin (\phi ) \left(\sqrt{2} \sin (\theta ) \cos (\phi )+\cos (\theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} y \left(\sqrt{2} x+z\right)$$ | ::: |
 +^ ^$$\text{Eeg$\pi $}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_2_4.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(\sqrt{2} \sin ^2(\theta ) \cos (2 \phi )-\sin (2 \theta ) \cos (\phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(\sqrt{2} x^2-2 x z-\sqrt{2} y^2\right)$$ | ::: |
 +^ ^$$\text{Ea1g}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_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** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + ​\frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\
 + ​-\frac{5}{28} \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+4 \sqrt{5} \text{Ma2u}+2 \sqrt{5} \text{Meu}\right) & k=2\land m=0 \\
 + ​-\frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}+\text{Ma2u}+4 \text{Meu}}{\sqrt{14}} & k=4\land m=-3 \\
 + ​\frac{1}{14} \left(9 \text{Ea1u}+14 \text{Ea2u1}+13 \text{Ea2u2}-34 \text{Eeu1}-2 \text{Eeu2}-4 \sqrt{5} \text{Ma2u}-16 \sqrt{5} \text{Meu}\right) & k=4\land m=0 \\
 + ​\frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}+\text{Ma2u}+4 \text{Meu}}{\sqrt{14}} & k=4\land m=3 \\
 + ​-\frac{13}{60} \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) & k=6\land (m=-6\lor m=6) \\
 + ​\frac{13 \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}+2 \text{Ma2u}-36 \text{Meu}\right)}{30 \sqrt{42}} & k=6\land m=-3 \\
 + ​-\frac{13}{420} \left(3 \text{Ea1u}-32 \text{Ea2u1}-25 \text{Ea2u2}-15 \text{Eeu1}+69 \text{Eeu2}+28 \sqrt{5} \text{Ma2u}-42 \sqrt{5} \text{Meu}\right) & k=6\land m=0 \\
 + ​-\frac{13 \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}+2 \text{Ma2u}-36 \text{Meu}\right)}{30 \sqrt{42}} & k=6\land m=3
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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}, {-((2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 + Ma2u + 4*Meu)/​Sqrt[14]),​ 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}, {(2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 + Ma2u + 4*Meu)/​Sqrt[14],​ k == 4 && m == 3}, {(-13*Sqrt[11/​21]*(9*Ea1u - 4*Ea2u1 - 5*Ea2u2 - 4*Sqrt[5]*Ma2u))/​60,​ k == 6 && (m == -6 || m == 6)}, {(13*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 + 2*Ma2u - 36*Meu))/​(30*Sqrt[42]),​ 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*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 + 2*Ma2u - 36*Meu))/​(30*Sqrt[42]),​ k == 6 && m == 3}}, 0]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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)*((1/​(sqrt(14)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + Ma2u + (4)*(Meu)))} , 
 +       {4, 3, (1/​(sqrt(14)))*((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/​30)*((1/​(sqrt(42)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (2)*(Ma2u) + (-36)*(Meu)))} , 
 +       ​{6,​-3,​ (13/​30)*((1/​(sqrt(42)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (2)*(Ma2u) + (-36)*(Meu)))} , 
 +       ​{6,​-6,​ (-13/​60)*((sqrt(11/​21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (-4)*((sqrt(5))*(Ma2u))))} , 
 +       {6, 6, (-13/​60)*((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 $|$ \frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{18} \left(9 \text{Ea1u}-4 \left(\text{Ea2u1}+\sqrt{5} \text{Ma2u}\right)-5 \text{Ea2u2}\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} \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} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-4 \text{Meu}\right) $|$ 0 $|
 +^$ {Y_{0}^{(3)}} $|$ \frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{9} \left(5 \text{Ea2u1}+4 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) $|$ 0 $|$ 0 $|$ -\frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|
 +^$ {Y_{1}^{(3)}} $|$ 0 $|$ \frac{1}{6} \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} \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} \left(9 \text{Ea1u}-4 \left(\text{Ea2u1}+\sqrt{5} \text{Ma2u}\right)-5 \text{Ea2u2}\right) $|$ 0 $|$ 0 $|$ -\frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|$ 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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $  ^  $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $  ^  $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $  ^
 +^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$ \text{Ea2u1} $|$ 0 $|$ 0 $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ \text{Ea2u2} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$ \frac{\sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{2}}{3} $|
 +^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|$ \frac{i}{2 \sqrt{3}} $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|
 +^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$ \frac{\sqrt{\frac{5}{2}}}{3} $|$ 0 $|$ 0 $|$ -\frac{2}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{\frac{5}{2}}}{3} $|
 +^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|
 +^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
 +
 +
 +###
 +
 +</​hidden>​
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2u1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_1.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(\sqrt{2} \left(1+e^{6 i \phi }\right) \sin ^3(\theta )+e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^3-3 \sqrt{2} x y^2+5 z^3-3 z\right)$$ | ::: |
 +^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_2.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \left((5 \cos (2 \theta )+3) \cos (\phi )+5 \sqrt{2} \sin (2 \theta ) \cos (2 \phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 \sqrt{2} x^2 z+x \left(5 z^2-1\right)-5 \sqrt{2} y^2 z\right)$$ | ::: |
 +^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_3.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(-10 \sqrt{2} \sin (2 \theta ) \cos (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{7}{\pi }} y \left(10 \sqrt{2} x z-5 z^2+1\right)$$ | ::: |
 +^ ^$$\text{Ea2u2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_4.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} \sin ^3(\theta ) \cos (3 \phi )+3 \cos (\theta )+5 \cos (3 \theta )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 \sqrt{2} x^3-15 \sqrt{2} x y^2+4 z \left(3-5 z^2\right)\right)$$ | ::: |
 +^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_5.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (2 \theta ) \cos (2 \phi )-(5 \cos (2 \theta )+3) \cos (\phi )\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^2 z-5 x z^2+x-\sqrt{2} y^2 z\right)$$ | ::: |
 +^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_6.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \sin (\phi ) \left(2 \sqrt{2} \sin (2 \theta ) \cos (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{35}{\pi }} y \left(2 \sqrt{2} x z+5 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Ea1u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_zy_a_orb_3_7.png?​150}} |
 +|$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$$ | ::: |
 +|$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$$ | ::: |
 +
 +
 +###
 +
 +</​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_Zy_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_Zy_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_{\text{xy}-\sqrt{2}\text{yz}} $  ^  $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $  ^  $ d_{\text{yz}+\sqrt{2}\text{xy}} $  ^  $ d_{x^2-y^2-\sqrt{2}\text{xz}} $  ^  $ 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** >
 +
 +###
 +
 + ​$$A_{k,​m} = \begin{cases}
 + 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
 + ​-\frac{5 \left(\sqrt{5} \text{Ma2u1}+4 \text{Ma2u2}-4 \text{Meu1}\right)}{\sqrt{21}} & k=2\land m=0 \\
 + ​\sqrt{6} \text{Ma2u1}+\sqrt{\frac{15}{2}} \text{Ma2u2} & k=4\land m=-3 \\
 + ​\frac{1}{2} \sqrt{\frac{3}{7}} \left(8 \sqrt{5} \text{Ma2u1}+11 \text{Ma2u2}-18 \text{Meu1}\right) & k=4\land m=0 \\
 + ​-\sqrt{\frac{3}{2}} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</​hidden>​
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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}, {Sqrt[6]*Ma2u1 + Sqrt[15/​2]*Ma2u2,​ k == 4 && m == -3}, {(Sqrt[3/​7]*(8*Sqrt[5]*Ma2u1 + 11*Ma2u2 - 18*Meu1))/​2,​ k == 4 && m == 0}}, -(Sqrt[3/​2]*(2*Ma2u1 + Sqrt[5]*Ma2u2))]
 +
 +</​code>​
 +
 +###
 +
 +</​hidden><​hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_D3d_Zy_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)*((sqrt(3/​2))*((2)*(Ma2u1) + (sqrt(5))*(Ma2u2)))} , 
 +       ​{4,​-3,​ (sqrt(6))*(Ma2u1) + (sqrt(15/​2))*(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 $|$ \frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{\sqrt{6}} $|$ 0 $|
 +^$ {Y_{0}^{(1)}} $|$ \frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{3 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(\sqrt{5} \text{Ma2u1}-2 \text{Ma2u2}\right) $|$ 0 $|$ 0 $|$ -\frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{3 \sqrt{2}} $|
 +^$ {Y_{1}^{(1)}} $|$ 0 $|$ -\frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{\sqrt{6}} $|$ 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_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $  ^  $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $  ^  $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $  ^  $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $  ^
 +^$ p_x $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ 2 \text{Ma2u1}+\sqrt{5} (\text{Ma2u2}-\text{Meu1}) $|$ 0 $|$ 0 $|
 +^$ p_y $|$ 0 $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ 2 \text{Ma2u1}+\sqrt{5} (\text{Ma2u2}-\text{Meu1}) $|$ 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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