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--- physics_chemistry:point_groups:oh:orientation_0sqrt2-1z [2018/03/21 18:48] – created Stefano Agrestini
+++ physics_chemistry:point_groups:oh:orientation_0sqrt2-1z [2018/09/06 12:49] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
+~~CLOSETOC~~
 ====== Orientation 0sqrt2-1z ======
+===== Symmetry Operations =====
 ###
-alligned paragraph text
+In the Oh Point Group, with orientation 0sqrt2-1z there are the following symmetry operations
 ###
-===== Example =====
+###
+{{:physics_chemistry:pointgroup:oh_0sqrt2-1z.png}}
 ###
-description text
 ###
-==== Input ====
+^ Operator ^ Orientation ^
-<code Quanty Example.Quanty>
+^  $\text{E}$  |  $\{0,0,0\}$  , |
--- some example code
+^  $C_3$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{-\sqrt{6},-\sqrt{2},1\right\}$  ,  $\left\{0,-2 \sqrt{2},-1\right\}$  ,  $\left\{-\sqrt{6},\sqrt{2},-1\right\}$  ,  $\left\{\sqrt{6},\sqrt{2},-1\right\}$  ,  $\left\{0,2 \sqrt{2},1\right\}$  ,  $\left\{\sqrt{6},-\sqrt{2},1\right\}$  , |
+^  $C_2$  |  $\{1,0,0\}$  ,  $\left\{1,\sqrt{3},0\right\}$  ,  $\left\{1,-\sqrt{3},0\right\}$  ,  $\left\{0,1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},1,-2 \sqrt{2}\right\}$  ,  $\left\{-\sqrt{3},1,-2 \sqrt{2}\right\}$  , |
+^  $C_4$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{0,\sqrt{2},-1\right\}$  ,  $\left\{-\sqrt{3},1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},-1,-\sqrt{2}\right\}$  ,  $\left\{-\sqrt{3},-1,-\sqrt{2}\right\}$  , |
+^  $C_2$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{-\sqrt{3},1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},1,\sqrt{2}\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{0,\sqrt{2},-1\right\}$  ,  $\left\{-\sqrt{3},1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},-1,-\sqrt{2}\right\}$  ,  $\left\{-\sqrt{3},-1,-\sqrt{2}\right\}$  , |
+^  $S_6$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{-\sqrt{6},-\sqrt{2},1\right\}$  ,  $\left\{0,-2 \sqrt{2},-1\right\}$  ,  $\left\{-\sqrt{6},\sqrt{2},-1\right\}$  ,  $\left\{\sqrt{6},\sqrt{2},-1\right\}$  ,  $\left\{0,2 \sqrt{2},1\right\}$  ,  $\left\{\sqrt{6},-\sqrt{2},1\right\}$  , |
+^  $\sigma _h$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{-\sqrt{3},1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},1,\sqrt{2}\right\}$  , |
+^  $\sigma _d$  |  $\{1,0,0\}$  ,  $\left\{1,\sqrt{3},0\right\}$  ,  $\left\{1,-\sqrt{3},0\right\}$  ,  $\left\{0,1,\sqrt{2}\right\}$  ,  $\left\{\sqrt{3},1,-2 \sqrt{2}\right\}$  ,  $\left\{-\sqrt{3},1,-2 \sqrt{2}\right\}$  , |
+###
+===== Different Settings =====
+###
+  * [[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Point Group Oh with orientation 0sqrt2-1z]]
+  * [[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Point Group Oh with orientation 0sqrt21z]]
+  * [[physics_chemistry:point_groups:oh:orientation_11-1z|Point Group Oh with orientation 11-1z]]
+  * [[physics_chemistry:point_groups:oh:orientation_111z|Point Group Oh with orientation 111z]]
+  * [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]]
+  * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]]
+  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
+###
+===== Character Table =====
+###
+|    ^   $\text{E} \,{\text{(1)}}$   ^   $C_3 \,{\text{(8)}}$   ^   $C_2 \,{\text{(6)}}$   ^   $C_4 \,{\text{(6)}}$   ^   $C_2 \,{\text{(3)}}$   ^   $\text{i} \,{\text{(1)}}$   ^   $S_4 \,{\text{(6)}}$   ^   $S_6 \,{\text{(8)}}$   ^   $\sigma_h \,{\text{(3)}}$   ^   $\sigma_d \,{\text{(6)}}$   ^
+^  $A_{1g}$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |
+^  $A_{2g}$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |
+^  $E_g$  |   $2$  |   $-1$  |   $0$  |   $0$  |   $2$  |   $2$  |   $0$  |   $-1$  |   $2$  |   $0$  |
+^  $T_{1g}$  |   $3$  |   $0$  |   $-1$  |   $1$  |   $-1$  |   $3$  |   $1$  |   $0$  |   $-1$  |   $-1$  |
+^  $T_{2g}$  |   $3$  |   $0$  |   $1$  |   $-1$  |   $-1$  |   $3$  |   $-1$  |   $0$  |   $-1$  |   $1$  |
+^  $A_{1u}$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |
+^  $A_{2u}$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |
+^  $E_u$  |   $2$  |   $-1$  |   $0$  |   $0$  |   $2$  |   $-2$  |   $0$  |   $1$  |   $-2$  |   $0$  |
+^  $T_{1u}$  |   $3$  |   $0$  |   $-1$  |   $1$  |   $-1$  |   $-3$  |   $-1$  |   $0$  |   $1$  |   $1$  |
+^  $T_{2u}$  |   $3$  |   $0$  |   $1$  |   $-1$  |   $-1$  |   $-3$  |   $1$  |   $0$  |   $1$  |   $-1$  |
+###
+===== Product Table =====
+###
+|    ^   $A_{1g}$   ^   $A_{2g}$   ^   $E_g$   ^   $T_{1g}$   ^   $T_{2g}$   ^   $A_{1u}$   ^   $A_{2u}$   ^   $E_u$   ^   $T_{1u}$   ^   $T_{2u}$   ^
+^  $A_{1g}$   |  $A_{1g}$   |  $A_{2g}$   |  $E_g$   |  $T_{1g}$   |  $T_{2g}$   |  $A_{1u}$   |  $A_{2u}$   |  $E_u$   |  $T_{1u}$   |  $T_{2u}$   |
+^  $A_{2g}$   |  $A_{2g}$   |  $A_{1g}$   |  $E_g$   |  $T_{2g}$   |  $T_{1g}$   |  $A_{2u}$   |  $A_{1u}$   |  $E_u$   |  $T_{2u}$   |  $T_{1u}$   |
+^  $E_g$   |  $E_g$   |  $E_g$   |  $A_{1g}+A_{2g}+E_g$   |  $T_{1g}+T_{2g}$   |  $T_{1g}+T_{2g}$   |  $E_u$   |  $E_u$   |  $A_{1u}+A_{2u}+E_u$   |  $T_{1u}+T_{2u}$   |  $T_{1u}+T_{2u}$   |
+^  $T_{1g}$   |  $T_{1g}$   |  $T_{2g}$   |  $T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |  $T_{1u}$   |  $T_{2u}$   |  $T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |
+^  $T_{2g}$   |  $T_{2g}$   |  $T_{1g}$   |  $T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |  $T_{2u}$   |  $T_{1u}$   |  $T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |
+^  $A_{1u}$   |  $A_{1u}$   |  $A_{2u}$   |  $E_u$   |  $T_{1u}$   |  $T_{2u}$   |  $A_{1g}$   |  $A_{2g}$   |  $E_g$   |  $T_{1g}$   |  $T_{2g}$   |
+^  $A_{2u}$   |  $A_{2u}$   |  $A_{1u}$   |  $E_u$   |  $T_{2u}$   |  $T_{1u}$   |  $A_{2g}$   |  $A_{1g}$   |  $E_g$   |  $T_{2g}$   |  $T_{1g}$   |
+^  $E_u$   |  $E_u$   |  $E_u$   |  $A_{1u}+A_{2u}+E_u$   |  $T_{1u}+T_{2u}$   |  $T_{1u}+T_{2u}$   |  $E_g$   |  $E_g$   |  $A_{1g}+A_{2g}+E_g$   |  $T_{1g}+T_{2g}$   |  $T_{1g}+T_{2g}$   |
+^  $T_{1u}$   |  $T_{1u}$   |  $T_{2u}$   |  $T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |  $T_{1g}$   |  $T_{2g}$   |  $T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |
+^  $T_{2u}$   |  $T_{2u}$   |  $T_{1u}$   |  $T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |  $T_{2g}$   |  $T_{1g}$   |  $T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |
+###
+===== Sub Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
+  * [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]]
+  * [[physics_chemistry:point_groups:c3v:orientation_zx|Point Group C3v with orientation Zx]]
+  * [[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_x|Point Group Cs with orientation X]]
+  * [[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:d3:orientation_zx|Point Group D3 with orientation Zx]]
+  * [[physics_chemistry:point_groups:s6:orientation_z|Point Group S6 with orientation Z]]
+###
+===== Super Groups with compatible settings =====
+###
+###
+===== 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 Oh Point group with orientation 0sqrt2-1z the form of the expansion coefficients is:
+###
+==== Expansion ====
+###
+ $$A_{k,m} = \begin{cases}
+ A(0,0) & k=0\land m=0 \\
+ -i \sqrt{\frac{10}{7}} A(4,0) & k=4\land (m=-3\lor m=3) \\
+ A(4,0) & k=4\land m=0 \\
+ -\frac{1}{8} \sqrt{\frac{77}{3}} A(6,0) & k=6\land (m=-6\lor m=6) \\
+ \frac{1}{4} i \sqrt{\frac{35}{6}} A(6,0) & k=6\land (m=-3\lor m=3) \\
+ A(6,0) & k=6\land m=0
+\end{cases}$$
+###
+==== Input format suitable for Mathematica (Quanty.nb) ====
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(-I)*Sqrt[10/7]*A[4, 0], k == 4 && (m == -3 || m == 3)}, {A[4, 0], k == 4 && m == 0}, {-(Sqrt[77/3]*A[6, 0])/8, k == 6 && (m == -6 || m == 6)}, {(I/4)*Sqrt[35/6]*A[6, 0], k == 6 && (m == -3 || m == 3)}, {A[6, 0], k == 6 && m == 0}}, 0]
 </code>
-==== Result ====
+###
-<WRAP center box 100%>
-text produced as output
-</WRAP>
-===== Table of contents =====
+==== Input format suitable for Quanty ====
-{{indexmenu>.#1}}
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty>
+Akm = {{0, 0, A(0,0)} ,
+       {4, 0, A(4,0)} ,
+       {4,-3, (-I)*((sqrt(10/7))*(A(4,0)))} ,
+       {4, 3, (-I)*((sqrt(10/7))*(A(4,0)))} ,
+       {6, 0, A(6,0)} ,
+       {6,-3, (1/4*I)*((sqrt(35/6))*(A(6,0)))} ,
+       {6, 3, (1/4*I)*((sqrt(35/6))*(A(6,0)))} ,
+       {6,-6, (-1/8)*((sqrt(77/3))*(A(6,0)))} ,
+       {6, 6, (-1/8)*((sqrt(77/3))*(A(6,0)))} }
+</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.
+###
+###
+|    ^   ${Y_{0}^{(0)}}$   ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $\text{Ass}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-1}^{(1)}}$ | $\color{darkred}{ 0 }$ | $\text{App}(0,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $-\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0)$ | $0$ |
+^ ${Y_{0}^{(1)}}$ | $\color{darkred}{ 0 }$ | $0$ | $\text{App}(0,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0)$ | $0$ | $0$ | $\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $0$ | $\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0)$ |
+^ ${Y_{1}^{(1)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{App}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ |
+^ ${Y_{-2}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0)$ | $0$ | $0$ | $\frac{5}{21} i \sqrt{2} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $0$ | $-\frac{5}{21} i \sqrt{2} \text{Add}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{0}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{1}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{5}{21} i \sqrt{2} \text{Add}(4,0)$ | $0$ | $0$ | $\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{2}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{5}{21} i \sqrt{2} \text{Add}(4,0)$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-3}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $\frac{1}{11} i \sqrt{10} \text{Aff}(4,0)+\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0)$ | $0$ | $0$ | $\frac{35}{156} \text{Aff}(6,0)$ |
+^ ${Y_{-2}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $\frac{2}{33} i \sqrt{5} \text{Aff}(4,0)-\frac{35}{572} i \sqrt{5} \text{Aff}(6,0)$ | $0$ | $0$ |
+^ ${Y_{-1}^{(3)}}$ | $\color{darkred}{ 0 }$ | $-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $\frac{35}{572} i \sqrt{5} \text{Aff}(6,0)-\frac{2}{33} i \sqrt{5} \text{Aff}(4,0)$ | $0$ |
+^ ${Y_{0}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{1}{11} i \sqrt{10} \text{Aff}(4,0)-\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0)$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $-\frac{1}{11} i \sqrt{10} \text{Aff}(4,0)-\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0)$ |
+^ ${Y_{1}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{35}{572} i \sqrt{5} \text{Aff}(6,0)-\frac{2}{33} i \sqrt{5} \text{Aff}(4,0)$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ |
+^ ${Y_{2}^{(3)}}$ | $\color{darkred}{ 0 }$ | $\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{2}{33} i \sqrt{5} \text{Aff}(4,0)-\frac{35}{572} i \sqrt{5} \text{Aff}(6,0)$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$ | $0$ |
+^ ${Y_{3}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{35}{156} \text{Aff}(6,0)$ | $0$ | $0$ | $\frac{1}{11} i \sqrt{10} \text{Aff}(4,0)+\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0)$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ |
+###
+==== 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
+###
+###
+|    ^   ${Y_{0}^{(0)}}$   ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ $\text{s}$ | $1$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $p_x$ | $\color{darkred}{ 0 }$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_y$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_z$ | $\color{darkred}{ 0 }$ | $0$ | $1$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $d_{\text{xy}+\sqrt{2}\text{xz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{6}}$ | $\frac{1}{\sqrt{3}}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $-\frac{i}{\sqrt{6}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{-x^2+y^2-2\sqrt{2}\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{1}{\sqrt{6}}$ | $-\frac{i}{\sqrt{3}}$ | $0$ | $-\frac{i}{\sqrt{3}}$ | $-\frac{1}{\sqrt{6}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xz}-\sqrt{2}\text{xy}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{3}}$ | $-\frac{1}{\sqrt{6}}$ | $0$ | $\frac{1}{\sqrt{6}}$ | $-\frac{i}{\sqrt{3}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{-x^2+y^2+\sqrt{2}\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{1}{\sqrt{3}}$ | $\frac{i}{\sqrt{6}}$ | $0$ | $\frac{i}{\sqrt{6}}$ | $-\frac{1}{\sqrt{3}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{3z^2-r^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $1$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{i \sqrt{2}}{3}$ | $0$ | $0$ | $\frac{\sqrt{5}}{3}$ | $0$ | $0$ | $\frac{i \sqrt{2}}{3}$ |
+^ $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $-\frac{1}{2 \sqrt{3}}$ | $0$ | $\frac{1}{2 \sqrt{3}}$ | $\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $0$ |
+^ $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{\sqrt{\frac{5}{3}}}{2}$ | $-\frac{i}{2 \sqrt{3}}$ | $0$ | $-\frac{i}{2 \sqrt{3}}$ | $-\frac{\sqrt{\frac{5}{3}}}{2}$ | $0$ |
+^ $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{1}{3} i \sqrt{\frac{5}{2}}$ | $0$ | $0$ | $\frac{2}{3}$ | $0$ | $0$ | $-\frac{1}{3} i \sqrt{\frac{5}{2}}$ |
+^ $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{i}{2 \sqrt{3}}$ | $\frac{\sqrt{\frac{5}{3}}}{2}$ | $0$ | $-\frac{\sqrt{\frac{5}{3}}}{2}$ | $\frac{i}{2 \sqrt{3}}$ | $0$ |
+^ $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{1}{2 \sqrt{3}}$ | $\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $0$ | $\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $-\frac{1}{2 \sqrt{3}}$ | $0$ |
+^ $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ |
+###
+==== One particle coupling on a basis of symmetry adapted functions ====
+###
+After rotation we find
+###
+###
+|    ^   $\text{s}$   ^   $p_x$   ^   $p_y$   ^   $p_z$   ^   $d_{\text{xy}+\sqrt{2}\text{xz}}$   ^   $d_{-x^2+y^2-2\sqrt{2}\text{yz}}$   ^   $d_{\text{xz}-\sqrt{2}\text{xy}}$   ^   $d_{-x^2+y^2+\sqrt{2}\text{yz}}$   ^   $d_{3z^2-r^2}$   ^   $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$   ^   $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$   ^   $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$   ^   $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$   ^   $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$   ^   $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$   ^   $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$   ^
+^ $\text{s}$ | $\text{Ass}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $p_x$ | $\color{darkred}{ 0 }$ | $\text{App}(0,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_y$ | $\color{darkred}{ 0 }$ | $0$ | $\text{App}(0,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_z$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{App}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ $d_{\text{xy}+\sqrt{2}\text{xz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Add}(0,0)-\frac{3}{7} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{-x^2+y^2-2\sqrt{2}\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Add}(0,0)-\frac{3}{7} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xz}-\sqrt{2}\text{xy}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{-x^2+y^2+\sqrt{2}\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{3z^2-r^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Aff}(0,0)+\frac{6}{11} \text{Aff}(4,0)+\frac{45}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ |
+^ $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0)$ | $0$ | $0$ |
+^ $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0)$ | $0$ |
+^ $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0)$ |
+###
+===== 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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Ea1g} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.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:oh_0sqrt2-1z_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Et1u} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty>
+Akm = {{0, 0, Et1u} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^
+^ ${Y_{-1}^{(1)}}$ | $\text{Et1u}$ | $0$ | $0$ |
+^ ${Y_{0}^{(1)}}$ | $0$ | $\text{Et1u}$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $0$ | $0$ | $\text{Et1u}$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $p_x$   ^   $p_y$   ^   $p_z$   ^
+^ $p_x$ | $\text{Et1u}$ | $0$ | $0$ |
+^ $p_y$ | $0$ | $\text{Et1u}$ | $0$ |
+^ $p_z$ | $0$ | $0$ | $\text{Et1u}$ |
+###
+</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** >
+###
+^ ^ $\text{Et1u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_1_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} x$  | ::: |
+^ ^ $\text{Et1u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_1_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} y$  | ::: |
+^ ^ $\text{Et1u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{5} (2 \text{Eeg}+3 \text{Et2g}) & k=0\land m=0 \\
+ i \sqrt{\frac{14}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land (m=-3\lor m=3) \\
+ -\frac{7}{5} (\text{Eeg}-\text{Et2g}) & k=4\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(2*Eeg + 3*Et2g)/5, k == 0 && m == 0}, {I*Sqrt[14/5]*(Eeg - Et2g), k == 4 && (m == -3 || m == 3)}, {(-7*(Eeg - Et2g))/5, k == 4 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty>
+Akm = {{0, 0, (1/5)*((2)*(Eeg) + (3)*(Et2g))} ,
+       {4, 0, (-7/5)*(Eeg + (-1)*(Et2g))} ,
+       {4,-3, (I)*((sqrt(14/5))*(Eeg + (-1)*(Et2g)))} ,
+       {4, 3, (I)*((sqrt(14/5))*(Eeg + (-1)*(Et2g)))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ ${Y_{-2}^{(2)}}$ | $\frac{1}{3} (\text{Eeg}+2 \text{Et2g})$ | $0$ | $0$ | $-\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g})$ | $0$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $\frac{1}{3} (2 \text{Eeg}+\text{Et2g})$ | $0$ | $0$ | $\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g})$ |
+^ ${Y_{0}^{(2)}}$ | $0$ | $0$ | $\text{Et2g}$ | $0$ | $0$ |
+^ ${Y_{1}^{(2)}}$ | $\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g})$ | $0$ | $0$ | $\frac{1}{3} (2 \text{Eeg}+\text{Et2g})$ | $0$ |
+^ ${Y_{2}^{(2)}}$ | $0$ | $-\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g})$ | $0$ | $0$ | $\frac{1}{3} (\text{Eeg}+2 \text{Et2g})$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $d_{\text{xy}+\sqrt{2}\text{xz}}$   ^   $d_{-x^2+y^2-2\sqrt{2}\text{yz}}$   ^   $d_{\text{xz}-\sqrt{2}\text{xy}}$   ^   $d_{-x^2+y^2+\sqrt{2}\text{yz}}$   ^   $d_{3z^2-r^2}$   ^
+^ $d_{\text{xy}+\sqrt{2}\text{xz}}$ | $\text{Eeg}$ | $0$ | $0$ | $0$ | $0$ |
+^ $d_{-x^2+y^2-2\sqrt{2}\text{yz}}$ | $0$ | $\text{Eeg}$ | $0$ | $0$ | $0$ |
+^ $d_{\text{xz}-\sqrt{2}\text{xy}}$ | $0$ | $0$ | $\text{Et2g}$ | $0$ | $0$ |
+^ $d_{-x^2+y^2+\sqrt{2}\text{yz}}$ | $0$ | $0$ | $0$ | $\text{Et2g}$ | $0$ |
+^ $d_{3z^2-r^2}$ | $0$ | $0$ | $0$ | $0$ | $\text{Et2g}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ $d_{\text{xy}+\sqrt{2}\text{xz}}$ | $\frac{i}{\sqrt{6}}$ | $\frac{1}{\sqrt{3}}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $-\frac{i}{\sqrt{6}}$ |
+^ $d_{-x^2+y^2-2\sqrt{2}\text{yz}}$ | $-\frac{1}{\sqrt{6}}$ | $-\frac{i}{\sqrt{3}}$ | $0$ | $-\frac{i}{\sqrt{3}}$ | $-\frac{1}{\sqrt{6}}$ |
+^ $d_{\text{xz}-\sqrt{2}\text{xy}}$ | $\frac{i}{\sqrt{3}}$ | $-\frac{1}{\sqrt{6}}$ | $0$ | $\frac{1}{\sqrt{6}}$ | $-\frac{i}{\sqrt{3}}$ |
+^ $d_{-x^2+y^2+\sqrt{2}\text{yz}}$ | $-\frac{1}{\sqrt{3}}$ | $\frac{i}{\sqrt{6}}$ | $0$ | $\frac{i}{\sqrt{6}}$ | $-\frac{1}{\sqrt{3}}$ |
+^ $d_{3z^2-r^2}$ | $0$ | $0$ | $1$ | $0$ | $0$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Eeg}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sin (\theta ) \sin (\phi )+\sqrt{2} \cos (\theta )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(y+\sqrt{2} z\right)$  | ::: |
+^ ^ $\text{Eeg}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sin (\theta ) \cos (2 \phi )+2 \sqrt{2} \cos (\theta ) \sin (\phi )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(x^2-y \left(y-2 \sqrt{2} z\right)\right)$  | ::: |
+^ ^ $\text{Et2g}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sqrt{2} \sin (\theta ) \sin (\phi )-\cos (\theta )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(\sqrt{2} y-z\right)$  | ::: |
+^ ^ $\text{Et2g}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_4.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (\theta ) \cos (2 \phi )-2 \cos (\theta ) \sin (\phi )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(y \left(\sqrt{2} y+2 z\right)-\sqrt{2} x^2\right)$  | ::: |
+^ ^ $\text{Et2g}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\
+ -i \sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land (m=-3\lor m=3) \\
+ \frac{1}{2} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\
+ -\frac{13}{60} \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land (m=-6\lor m=6) \\
+ \frac{13 i (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{6 \sqrt{210}} & k=6\land (m=-3\lor m=3) \\
+ \frac{26}{105} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {(-I)*Sqrt[5/14]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && (m == -3 || m == 3)}, {(2*Ea2u - 3*Et1u + Et2u)/2, k == 4 && m == 0}, {(-13*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u))/60, k == 6 && (m == -6 || m == 6)}, {(((13*I)/6)*(4*Ea2u + 5*Et1u - 9*Et2u))/Sqrt[210], k == 6 && (m == -3 || m == 3)}, {(26*(4*Ea2u + 5*Et1u - 9*Et2u))/105, k == 6 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty>
+Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} ,
+       {4, 0, (1/2)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} ,
+       {4,-3, (-I)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} ,
+       {4, 3, (-I)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} ,
+       {6, 0, (26/105)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} ,
+       {6,-3, (13/6*I)*((1/(sqrt(210)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} ,
+       {6, 3, (13/6*I)*((1/(sqrt(210)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} ,
+       {6,-6, (-13/60)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} ,
+       {6, 6, (-13/60)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-3}^{(3)}}$ | $\frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u})$ | $0$ | $0$ | $\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u})$ | $0$ | $0$ | $\frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})$ |
+^ ${Y_{-2}^{(3)}}$ | $0$ | $\frac{1}{6} (5 \text{Et1u}+\text{Et2u})$ | $0$ | $0$ | $-\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u})$ | $0$ | $0$ |
+^ ${Y_{-1}^{(3)}}$ | $0$ | $0$ | $\frac{1}{6} (\text{Et1u}+5 \text{Et2u})$ | $0$ | $0$ | $\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u})$ | $0$ |
+^ ${Y_{0}^{(3)}}$ | $-\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u})$ | $0$ | $0$ | $\frac{1}{9} (5 \text{Ea2u}+4 \text{Et1u})$ | $0$ | $0$ | $-\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u})$ |
+^ ${Y_{1}^{(3)}}$ | $0$ | $\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u})$ | $0$ | $0$ | $\frac{1}{6} (\text{Et1u}+5 \text{Et2u})$ | $0$ | $0$ |
+^ ${Y_{2}^{(3)}}$ | $0$ | $0$ | $-\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u})$ | $0$ | $0$ | $\frac{1}{6} (5 \text{Et1u}+\text{Et2u})$ | $0$ |
+^ ${Y_{3}^{(3)}}$ | $\frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})$ | $0$ | $0$ | $\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u})$ | $0$ | $0$ | $\frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u})$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$   ^   $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$   ^   $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$   ^   $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$   ^   $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$   ^   $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$   ^   $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$   ^
+^ $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$ | $\text{Ea2u}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$ | $0$ | $\text{Et1u}$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$ | $0$ | $0$ | $\text{Et1u}$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$ | $0$ | $0$ | $0$ | $\text{Et1u}$ | $0$ | $0$ | $0$ |
+^ $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$ | $0$ | $0$ | $0$ | $0$ | $\text{Et2u}$ | $0$ | $0$ |
+^ $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\text{Et2u}$ | $0$ |
+^ $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\text{Et2u}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$ | $\frac{i \sqrt{2}}{3}$ | $0$ | $0$ | $\frac{\sqrt{5}}{3}$ | $0$ | $0$ | $\frac{i \sqrt{2}}{3}$ |
+^ $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$ | $0$ | $-\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $-\frac{1}{2 \sqrt{3}}$ | $0$ | $\frac{1}{2 \sqrt{3}}$ | $\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $0$ |
+^ $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$ | $0$ | $-\frac{\sqrt{\frac{5}{3}}}{2}$ | $-\frac{i}{2 \sqrt{3}}$ | $0$ | $-\frac{i}{2 \sqrt{3}}$ | $-\frac{\sqrt{\frac{5}{3}}}{2}$ | $0$ |
+^ $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$ | $-\frac{1}{3} i \sqrt{\frac{5}{2}}$ | $0$ | $0$ | $\frac{2}{3}$ | $0$ | $0$ | $-\frac{1}{3} i \sqrt{\frac{5}{2}}$ |
+^ $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$ | $0$ | $-\frac{i}{2 \sqrt{3}}$ | $\frac{\sqrt{\frac{5}{3}}}{2}$ | $0$ | $-\frac{\sqrt{\frac{5}{3}}}{2}$ | $\frac{i}{2 \sqrt{3}}$ | $0$ |
+^ $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$ | $0$ | $-\frac{1}{2 \sqrt{3}}$ | $\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $0$ | $\frac{1}{2} i \sqrt{\frac{5}{3}}$ | $-\frac{1}{2 \sqrt{3}}$ | $0$ |
+^ $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$ | $-\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Ea2u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{24} \sqrt{\frac{35}{\pi }} \left(2 \sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+\cos (\theta ) (5 \cos (2 \theta )-1)\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(3 \sqrt{2} x^2 y-\sqrt{2} y^3+5 z^3-3 z\right)$  | ::: |
+^ ^ $\text{Et1u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{8} \sqrt{\frac{7}{\pi }} x \left(10 \sqrt{2} y z+5 z^2-1\right)$  | ::: |
+^ ^ $\text{Et1u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \left(5 \sqrt{2} \sin (2 \theta ) \cos (2 \phi )+(5 \cos (2 \theta )+3) \sin (\phi )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} x^2 z+5 \sqrt{2} y^2 z-5 y z^2+y\right)$  | ::: |
+^ ^ $\text{Et1u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_4.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+3 \cos (\theta )+5 \cos (3 \theta )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-15 \sqrt{2} x^2 y+5 \sqrt{2} y^3+4 z \left(5 z^2-3\right)\right)$  | ::: |
+^ ^ $\text{Et2u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_5.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \cos (\phi ) \left(-2 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{8} \sqrt{\frac{35}{\pi }} x \left(2 \sqrt{2} y z-5 z^2+1\right)$  | ::: |
+^ ^ $\text{Et2u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_6.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \left((5 \cos (2 \theta )+3) \sin (\phi )-\sqrt{2} \sin (2 \theta ) \cos (2 \phi )\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^2 z-\sqrt{2} y^2 z-5 y z^2+y\right)$  | ::: |
+^ ^ $\text{Et2u}$  | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_7.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$  | ::: |
+###
+</hidden>
+===== Coupling between two shells =====
+###
+Click on one of the subsections to expand it or <hiddenSwitch expand all>
+###
+==== Potential for p-f orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& k\neq 4\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
+ -i \sqrt{\frac{15}{2}} \text{Mt1u} & k=4\land (m=-3\lor m=3) \\
+ \frac{\sqrt{21} \text{Mt1u}}{2} & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -3 && m != 0 && m != 3)}, {(-I)*Sqrt[15/2]*Mt1u, k == 4 && (m == -3 || m == 3)}}, (Sqrt[21]*Mt1u)/2]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_Oh_0sqrt2-1z.Quanty>
+Akm = {{4, 0, (1/2)*((sqrt(21))*(Mt1u))} ,
+       {4,-3, (-I)*((sqrt(15/2))*(Mt1u))} ,
+       {4, 3, (-I)*((sqrt(15/2))*(Mt1u))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-1}^{(1)}}$ | $0$ | $0$ | $-\frac{\text{Mt1u}}{\sqrt{6}}$ | $0$ | $0$ | $-i \sqrt{\frac{5}{6}} \text{Mt1u}$ | $0$ |
+^ ${Y_{0}^{(1)}}$ | $\frac{1}{3} i \sqrt{\frac{5}{2}} \text{Mt1u}$ | $0$ | $0$ | $\frac{2 \text{Mt1u}}{3}$ | $0$ | $0$ | $\frac{1}{3} i \sqrt{\frac{5}{2}} \text{Mt1u}$ |
+^ ${Y_{1}^{(1)}}$ | $0$ | $-i \sqrt{\frac{5}{6}} \text{Mt1u}$ | $0$ | $0$ | $-\frac{\text{Mt1u}}{\sqrt{6}}$ | $0$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{\left.3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$   ^   $f_{-x\left\backslash \left(-1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$   ^   $f_{y-\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z+\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$   ^   $f_{\left.-15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$   ^   $f_{-x\left\backslash \left(1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.}$   ^   $f_{-\left(y+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.\right)}$   ^   $f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$   ^
+^ $p_x$ | $0$ | $\text{Mt1u}$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_y$ | $0$ | $0$ | $\text{Mt1u}$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_z$ | $0$ | $0$ | $0$ | $\text{Mt1u}$ | $0$ | $0$ | $0$ |
+###
+</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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