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--- physics_chemistry:point_groups:d4h:orientation_zxy [2018/03/21 18:32] – created Stefano Agrestini
+++ physics_chemistry:point_groups:d4h:orientation_zxy [2018/04/06 09:03] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
+~~CLOSETOC~~
 ====== Orientation Zxy ======
+===== Symmetry Operations =====
 ###
-alligned paragraph text
+In the D4h Point Group, with orientation Zxy there are the following symmetry operations
 ###
-===== Example =====
+###
+{{:physics_chemistry:pointgroup:d4h_zxy.png}}
 ###
-description text
 ###
-==== Input ====
+^ Operator ^ Orientation ^
-<code Quanty Example.Quanty>
+^  $\text{E}$  |  $\{0,0,0\}$  , |
--- some example code
+^  $C_4$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  , |
+^  $C_2$  |  $\{0,0,1\}$  , |
+^  $C_2$  |  $\{0,1,0\}$  ,  $\{1,0,0\}$  , |
+^  $C_2$  |  $\{1,1,0\}$  ,  $\{1,-1,0\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  , |
+^  $\sigma _h$  |  $\{0,0,1\}$  , |
+^  $\sigma _v$  |  $\{1,0,0\}$  ,  $\{0,1,0\}$  , |
+^  $\sigma _d$  |  $\{1,1,0\}$  ,  $\{1,-1,0\}$  , |
+###
+===== Different Settings =====
+###
+  * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
+###
+===== Character Table =====
+###
+|    ^   $\text{E} \,{\text{(1)}}$   ^   $C_4 \,{\text{(2)}}$   ^   $C_2 \,{\text{(1)}}$   ^   $C_2 \,{\text{(2)}}$   ^   $C_2 \,{\text{(2)}}$   ^   $\text{i} \,{\text{(1)}}$   ^   $S_4 \,{\text{(2)}}$   ^   $\sigma_h \,{\text{(1)}}$   ^   $\sigma_v \,{\text{(2)}}$   ^   $\sigma_d \,{\text{(2)}}$   ^
+^  $A_{1g}$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |
+^  $A_{2g}$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |
+^  $B_{1g}$  |   $1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |
+^  $B_{2g}$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |
+^  $E_g$  |   $2$  |   $0$  |   $-2$  |   $0$  |   $0$  |   $2$  |   $0$  |   $-2$  |   $0$  |   $0$  |
+^  $A_{1u}$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |
+^  $A_{2u}$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |   $1$  |   $1$  |
+^  $B_{1u}$  |   $1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |
+^  $B_{2u}$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |
+^  $E_u$  |   $2$  |   $0$  |   $-2$  |   $0$  |   $0$  |   $-2$  |   $0$  |   $2$  |   $0$  |   $0$  |
+###
+===== Product Table =====
+###
+|    ^   $A_{1g}$   ^   $A_{2g}$   ^   $B_{1g}$   ^   $B_{2g}$   ^   $E_g$   ^   $A_{1u}$   ^   $A_{2u}$   ^   $B_{1u}$   ^   $B_{2u}$   ^   $E_u$   ^
+^  $A_{1g}$   |  $A_{1g}$   |  $A_{2g}$   |  $B_{1g}$   |  $B_{2g}$   |  $E_g$   |  $A_{1u}$   |  $A_{2u}$   |  $B_{1u}$   |  $B_{2u}$   |  $E_u$   |
+^  $A_{2g}$   |  $A_{2g}$   |  $A_{1g}$   |  $B_{2g}$   |  $B_{1g}$   |  $E_g$   |  $A_{2u}$   |  $A_{1u}$   |  $B_{2u}$   |  $B_{1u}$   |  $E_u$   |
+^  $B_{1g}$   |  $B_{1g}$   |  $B_{2g}$   |  $A_{1g}$   |  $A_{2g}$   |  $E_g$   |  $B_{1u}$   |  $B_{2u}$   |  $A_{1u}$   |  $A_{2u}$   |  $E_u$   |
+^  $B_{2g}$   |  $B_{2g}$   |  $B_{1g}$   |  $A_{2g}$   |  $A_{1g}$   |  $E_g$   |  $B_{2u}$   |  $B_{1u}$   |  $A_{2u}$   |  $A_{1u}$   |  $E_u$   |
+^  $E_g$   |  $E_g$   |  $E_g$   |  $E_g$   |  $E_g$   |  $A_{1g}+A_{2g}+B_{1g}+B_{2g}$   |  $E_u$   |  $E_u$   |  $E_u$   |  $E_u$   |  $A_{1u}+A_{2u}+B_{1u}+B_{2u}$   |
+^  $A_{1u}$   |  $A_{1u}$   |  $A_{2u}$   |  $B_{1u}$   |  $B_{2u}$   |  $E_u$   |  $A_{1g}$   |  $A_{2g}$   |  $B_{1g}$   |  $B_{2g}$   |  $E_g$   |
+^  $A_{2u}$   |  $A_{2u}$   |  $A_{1u}$   |  $B_{2u}$   |  $B_{1u}$   |  $E_u$   |  $A_{2g}$   |  $A_{1g}$   |  $B_{2g}$   |  $B_{1g}$   |  $E_g$   |
+^  $B_{1u}$   |  $B_{1u}$   |  $B_{2u}$   |  $A_{1u}$   |  $A_{2u}$   |  $E_u$   |  $B_{1g}$   |  $B_{2g}$   |  $A_{1g}$   |  $A_{2g}$   |  $E_g$   |
+^  $B_{2u}$   |  $B_{2u}$   |  $B_{1u}$   |  $A_{2u}$   |  $A_{1u}$   |  $E_u$   |  $B_{2g}$   |  $B_{1g}$   |  $A_{2g}$   |  $A_{1g}$   |  $E_g$   |
+^  $E_u$   |  $E_u$   |  $E_u$   |  $E_u$   |  $E_u$   |  $A_{1u}+A_{2u}+B_{1u}+B_{2u}$   |  $E_g$   |  $E_g$   |  $E_g$   |  $E_g$   |  $A_{1g}+A_{2g}+B_{1g}+B_{2g}$   |
+###
+===== Sub Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
+  * [[physics_chemistry:point_groups:c2h:orientation_z|Point Group C2h with orientation Z]]
+  * [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]]
+  * [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]]
+  * [[physics_chemistry:point_groups:c2:orientation_y|Point Group C2 with orientation Y]]
+  * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]]
+  * [[physics_chemistry:point_groups:c4h:orientation_z|Point Group C4h with orientation Z]]
+  * [[physics_chemistry:point_groups:c4v:orientation_zxy|Point Group C4v with orientation Zxy]]
+  * [[physics_chemistry:point_groups:c4:orientation_z|Point Group C4 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:cs:orientation_y|Point Group Cs with orientation Y]]
+  * [[physics_chemistry:point_groups:cs:orientation_z|Point Group Cs with orientation Z]]
+  * [[physics_chemistry:point_groups:d2d:orientation_zxy|Point Group D2d with orientation Zxy]]
+  * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]]
+  * [[physics_chemistry:point_groups:d2:orientation_xyz|Point Group D2 with orientation XYZ]]
+  * [[physics_chemistry:point_groups:d4:orientation_zxy|Point Group D4 with orientation Zxy]]
+  * [[physics_chemistry:point_groups:s4:orientation_z|Point Group S4 with orientation Z]]
+###
+===== Super Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
+###
+===== 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 D4h Point group with orientation Zxy the form of the expansion coefficients is:
+###
+==== Expansion ====
+###
+ $$A_{k,m} = \begin{cases}
+ A(0,0) & k=0\land m=0 \\
+ A(2,0) & k=2\land m=0 \\
+ A(4,4) & k=4\land (m=-4\lor m=4) \\
+ A(4,0) & k=4\land m=0 \\
+ A(6,4) & k=6\land (m=-4\lor m=4) \\
+ A(6,0) & k=6\land m=0
+\end{cases}$$
+###
+==== Input format suitable for Mathematica (Quanty.nb) ====
+###
+<code Quanty Akm_D4h_Zxy.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {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_D4h_Zxy.Quanty>
+Akm = {{0, 0, A(0,0)} ,
+       {2, 0, A(2,0)} ,
+       {4, 0, A(4,0)} ,
+       {4,-4, A(4,4)} ,
+       {4, 4, A(4,4)} ,
+       {6, 0, A(6,0)} ,
+       {6,-4, A(6,4)} ,
+       {6, 4, A(6,4)} }
+</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.
+###
+###
+|    ^   ${Y_{0}^{(0)}}$   ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $\text{Ass}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-1}^{(1)}}$ | $\color{darkred}{ 0 }$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $0$ | $-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ |
+^ ${Y_{0}^{(1)}}$ | $\color{darkred}{ 0 }$ | $0$ | $\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ |
+^ ${Y_{-2}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{0}^{(2)}}$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{1}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{2}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-3}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$ | $0$ | $0$ |
+^ ${Y_{-2}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$ | $0$ |
+^ ${Y_{-1}^{(3)}}$ | $\color{darkred}{ 0 }$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$ |
+^ ${Y_{0}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ |
+^ ${Y_{2}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$ | $0$ |
+^ ${Y_{3}^{(3)}}$ | $\color{darkred}{ 0 }$ | $-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ |
+###
+==== 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
+###
+###
+|    ^   ${Y_{0}^{(0)}}$   ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ $\text{s}$ | $1$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $p_x$ | $\color{darkred}{ 0 }$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_y$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_z$ | $\color{darkred}{ 0 }$ | $0$ | $1$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $d_{x^2-y^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{3z^2-r^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $1$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xy}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $f_{\text{xyz}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ | $0$ |
+^ $f_{x\left(5x^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{\sqrt{5}}{4}$ | $0$ | $-\frac{\sqrt{3}}{4}$ | $0$ | $\frac{\sqrt{3}}{4}$ | $0$ | $-\frac{\sqrt{5}}{4}$ |
+^ $f_{y\left(5y^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{i \sqrt{5}}{4}$ | $0$ | $-\frac{i \sqrt{3}}{4}$ | $0$ | $-\frac{i \sqrt{3}}{4}$ | $0$ | $-\frac{i \sqrt{5}}{4}$ |
+^ $f_{z\left(5z^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $1$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(y^2-z^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{\sqrt{3}}{4}$ | $0$ | $-\frac{\sqrt{5}}{4}$ | $0$ | $\frac{\sqrt{5}}{4}$ | $0$ | $\frac{\sqrt{3}}{4}$ |
+^ $f_{y\left(z^2-x^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{i \sqrt{3}}{4}$ | $0$ | $\frac{i \sqrt{5}}{4}$ | $0$ | $\frac{i \sqrt{5}}{4}$ | $0$ | $-\frac{i \sqrt{3}}{4}$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ |
+###
+==== One particle coupling on a basis of symmetry adapted functions ====
+###
+After rotation we find
+###
+###
+|    ^   $\text{s}$   ^   $p_x$   ^   $p_y$   ^   $p_z$   ^   $d_{x^2-y^2}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{yz}}$   ^   $d_{\text{xz}}$   ^   $d_{\text{xy}}$   ^   $f_{\text{xyz}}$   ^   $f_{x\left(5x^2-r^2\right)}$   ^   $f_{y\left(5y^2-r^2\right)}$   ^   $f_{z\left(5z^2-r^2\right)}$   ^   $f_{x\left(y^2-z^2\right)}$   ^   $f_{y\left(z^2-x^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^
+^ $\text{s}$ | $\text{Ass}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $p_x$ | $\color{darkred}{ 0 }$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$ | $0$ | $0$ | $-\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$ | $0$ | $0$ |
+^ $p_y$ | $\color{darkred}{ 0 }$ | $0$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$ | $0$ | $0$ | $\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$ | $0$ |
+^ $p_z$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ $d_{x^2-y^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{3z^2-r^2}$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xy}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $f_{\text{xyz}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(5x^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)$ | $0$ | $0$ | $\frac{\text{Aff}(2,0)}{\sqrt{15}}-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)$ | $0$ | $0$ |
+^ $f_{y\left(5y^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)$ | $0$ | $0$ | $-\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)$ | $0$ |
+^ $f_{z\left(5z^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(y^2-z^2\right)}$ | $\color{darkred}{ 0 }$ | $-\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{\text{Aff}(2,0)}{\sqrt{15}}-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)$ | $0$ | $0$ |
+^ $f_{y\left(z^2-x^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $-\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)$ | $0$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$ |
+###
+===== 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_D4h_Zxy.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_D4h_Zxy.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:d4h_zxy_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_D4h_Zxy.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_D4h_Zxy.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:d4h_zxy_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:d4h_zxy_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:d4h_zxy_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{5} (\text{Ea1g}+\text{Eb1g}+\text{Eb2g}+2 \text{Eeg}) & k=0\land m=0 \\
+ \text{Ea1g}-\text{Eb1g}-\text{Eb2g}+\text{Eeg} & k=2\land m=0 \\
+ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eb1g}-\text{Eb2g}) & k=4\land (m=-4\lor m=4) \\
+ \frac{3}{10} (6 \text{Ea1g}+\text{Eb1g}+\text{Eb2g}-8 \text{Eeg}) & k=4\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea1g + Eb1g + Eb2g + 2*Eeg)/5, k == 0 && m == 0}, {Ea1g - Eb1g - Eb2g + Eeg, k == 2 && m == 0}, {(3*Sqrt[7/10]*(Eb1g - Eb2g))/2, k == 4 && (m == -4 || m == 4)}, {(3*(6*Ea1g + Eb1g + Eb2g - 8*Eeg))/10, k == 4 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty>
+Akm = {{0, 0, (1/5)*(Ea1g + Eb1g + Eb2g + (2)*(Eeg))} ,
+       {2, 0, Ea1g + (-1)*(Eb1g) + (-1)*(Eb2g) + Eeg} ,
+       {4, 0, (3/10)*((6)*(Ea1g) + Eb1g + Eb2g + (-8)*(Eeg))} ,
+       {4,-4, (3/2)*((sqrt(7/10))*(Eb1g + (-1)*(Eb2g)))} ,
+       {4, 4, (3/2)*((sqrt(7/10))*(Eb1g + (-1)*(Eb2g)))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ ${Y_{-2}^{(2)}}$ | $\frac{\text{Eb1g}+\text{Eb2g}}{2}$ | $0$ | $0$ | $0$ | $\frac{\text{Eb1g}-\text{Eb2g}}{2}$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $\text{Eeg}$ | $0$ | $0$ | $0$ |
+^ ${Y_{0}^{(2)}}$ | $0$ | $0$ | $\text{Ea1g}$ | $0$ | $0$ |
+^ ${Y_{1}^{(2)}}$ | $0$ | $0$ | $0$ | $\text{Eeg}$ | $0$ |
+^ ${Y_{2}^{(2)}}$ | $\frac{\text{Eb1g}-\text{Eb2g}}{2}$ | $0$ | $0$ | $0$ | $\frac{\text{Eb1g}+\text{Eb2g}}{2}$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $d_{x^2-y^2}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{yz}}$   ^   $d_{\text{xz}}$   ^   $d_{\text{xy}}$   ^
+^ $d_{x^2-y^2}$ | $\text{Eb1g}$ | $0$ | $0$ | $0$ | $0$ |
+^ $d_{3z^2-r^2}$ | $0$ | $\text{Ea1g}$ | $0$ | $0$ | $0$ |
+^ $d_{\text{yz}}$ | $0$ | $0$ | $\text{Eeg}$ | $0$ | $0$ |
+^ $d_{\text{xz}}$ | $0$ | $0$ | $0$ | $\text{Eeg}$ | $0$ |
+^ $d_{\text{xy}}$ | $0$ | $0$ | $0$ | $0$ | $\text{Eb2g}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ $d_{x^2-y^2}$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ |
+^ $d_{3z^2-r^2}$ | $0$ | $0$ | $1$ | $0$ | $0$ |
+^ $d_{\text{yz}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ |
+^ $d_{\text{xz}}$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $0$ |
+^ $d_{\text{xy}}$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Eb1g}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$  | ::: |
+^ ^ $\text{Ea1g}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$  | ::: |
+^ ^ $\text{Eeg}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$  | ::: |
+^ ^ $\text{Eeg}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_4.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$  | ::: |
+^ ^ $\text{Eb2g}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_5.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$  | ::: |
+###
+</hidden>
+==== Potential for f orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{7} (\text{Ea2u}+\text{Eb1u}+\text{Eb2u}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\
+ \frac{5}{7} \left(\text{Ea2u}-\text{Eeu1}+\sqrt{15} \text{Meu}\right) & k=2\land m=0 \\
+ -\frac{3}{56} \left(2 \sqrt{70} \text{Eb1u}-2 \sqrt{70} \text{Eb2u}-3 \sqrt{70} \text{Eeu1}+3 \sqrt{70} \text{Eeu2}-2 \sqrt{42} \text{Meu}\right) & k=4\land (m=-4\lor m=4) \\
+ \frac{3}{28} \left(12 \text{Ea2u}-14 \text{Eb1u}-14 \text{Eb2u}+9 \text{Eeu1}+7 \text{Eeu2}-2 \sqrt{15} \text{Meu}\right) & k=4\land m=0 \\
+ -\frac{13}{560} \sqrt{3} \left(4 \sqrt{42} \text{Eb1u}-4 \sqrt{42} \text{Eb2u}+5 \sqrt{42} \text{Eeu1}-5 \sqrt{42} \text{Eeu2}+2 \sqrt{70} \text{Meu}\right) & k=6\land (m=-4\lor m=4) \\
+ \frac{13}{280} \left(40 \text{Ea2u}+12 \text{Eb1u}+12 \text{Eb2u}-25 \text{Eeu1}-39 \text{Eeu2}-14 \sqrt{15} \text{Meu}\right) & k=6\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea2u + Eb1u + Eb2u + 2*Eeu1 + 2*Eeu2)/7, k == 0 && m == 0}, {(5*(Ea2u - Eeu1 + Sqrt[15]*Meu))/7, k == 2 && m == 0}, {(-3*(2*Sqrt[70]*Eb1u - 2*Sqrt[70]*Eb2u - 3*Sqrt[70]*Eeu1 + 3*Sqrt[70]*Eeu2 - 2*Sqrt[42]*Meu))/56, k == 4 && (m == -4 || m == 4)}, {(3*(12*Ea2u - 14*Eb1u - 14*Eb2u + 9*Eeu1 + 7*Eeu2 - 2*Sqrt[15]*Meu))/28, k == 4 && m == 0}, {(-13*Sqrt[3]*(4*Sqrt[42]*Eb1u - 4*Sqrt[42]*Eb2u + 5*Sqrt[42]*Eeu1 - 5*Sqrt[42]*Eeu2 + 2*Sqrt[70]*Meu))/560, k == 6 && (m == -4 || m == 4)}, {(13*(40*Ea2u + 12*Eb1u + 12*Eb2u - 25*Eeu1 - 39*Eeu2 - 14*Sqrt[15]*Meu))/280, k == 6 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty>
+Akm = {{0, 0, (1/7)*(Ea2u + Eb1u + Eb2u + (2)*(Eeu1) + (2)*(Eeu2))} ,
+       {2, 0, (5/7)*(Ea2u + (-1)*(Eeu1) + (sqrt(15))*(Meu))} ,
+       {4, 0, (3/28)*((12)*(Ea2u) + (-14)*(Eb1u) + (-14)*(Eb2u) + (9)*(Eeu1) + (7)*(Eeu2) + (-2)*((sqrt(15))*(Meu)))} ,
+       {4,-4, (-3/56)*((2)*((sqrt(70))*(Eb1u)) + (-2)*((sqrt(70))*(Eb2u)) + (-3)*((sqrt(70))*(Eeu1)) + (3)*((sqrt(70))*(Eeu2)) + (-2)*((sqrt(42))*(Meu)))} ,
+       {4, 4, (-3/56)*((2)*((sqrt(70))*(Eb1u)) + (-2)*((sqrt(70))*(Eb2u)) + (-3)*((sqrt(70))*(Eeu1)) + (3)*((sqrt(70))*(Eeu2)) + (-2)*((sqrt(42))*(Meu)))} ,
+       {6, 0, (13/280)*((40)*(Ea2u) + (12)*(Eb1u) + (12)*(Eb2u) + (-25)*(Eeu1) + (-39)*(Eeu2) + (-14)*((sqrt(15))*(Meu)))} ,
+       {6,-4, (-13/560)*((sqrt(3))*((4)*((sqrt(42))*(Eb1u)) + (-4)*((sqrt(42))*(Eb2u)) + (5)*((sqrt(42))*(Eeu1)) + (-5)*((sqrt(42))*(Eeu2)) + (2)*((sqrt(70))*(Meu))))} ,
+       {6, 4, (-13/560)*((sqrt(3))*((4)*((sqrt(42))*(Eb1u)) + (-4)*((sqrt(42))*(Eb2u)) + (5)*((sqrt(42))*(Eeu1)) + (-5)*((sqrt(42))*(Eeu2)) + (2)*((sqrt(70))*(Meu))))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-3}^{(3)}}$ | $\frac{1}{8} \left(5 \text{Eeu1}+3 \text{Eeu2}-2 \sqrt{15} \text{Meu}\right)$ | $0$ | $0$ | $0$ | $\frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right)$ | $0$ | $0$ |
+^ ${Y_{-2}^{(3)}}$ | $0$ | $\frac{\text{Eb1u}+\text{Eb2u}}{2}$ | $0$ | $0$ | $0$ | $\frac{\text{Eb2u}-\text{Eb1u}}{2}$ | $0$ |
+^ ${Y_{-1}^{(3)}}$ | $0$ | $0$ | $\frac{1}{8} \left(3 \text{Eeu1}+5 \text{Eeu2}+2 \sqrt{15} \text{Meu}\right)$ | $0$ | $0$ | $0$ | $\frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right)$ |
+^ ${Y_{0}^{(3)}}$ | $0$ | $0$ | $0$ | $\text{Ea2u}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(3)}}$ | $\frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right)$ | $0$ | $0$ | $0$ | $\frac{1}{8} \left(3 \text{Eeu1}+5 \text{Eeu2}+2 \sqrt{15} \text{Meu}\right)$ | $0$ | $0$ |
+^ ${Y_{2}^{(3)}}$ | $0$ | $\frac{\text{Eb2u}-\text{Eb1u}}{2}$ | $0$ | $0$ | $0$ | $\frac{\text{Eb1u}+\text{Eb2u}}{2}$ | $0$ |
+^ ${Y_{3}^{(3)}}$ | $0$ | $0$ | $\frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right)$ | $0$ | $0$ | $0$ | $\frac{1}{8} \left(5 \text{Eeu1}+3 \text{Eeu2}-2 \sqrt{15} \text{Meu}\right)$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{\text{xyz}}$   ^   $f_{x\left(5x^2-r^2\right)}$   ^   $f_{y\left(5y^2-r^2\right)}$   ^   $f_{z\left(5z^2-r^2\right)}$   ^   $f_{x\left(y^2-z^2\right)}$   ^   $f_{y\left(z^2-x^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^
+^ $f_{\text{xyz}}$ | $\text{Eb1u}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(5x^2-r^2\right)}$ | $0$ | $\text{Eeu1}$ | $0$ | $0$ | $\text{Meu}$ | $0$ | $0$ |
+^ $f_{y\left(5y^2-r^2\right)}$ | $0$ | $0$ | $\text{Eeu1}$ | $0$ | $0$ | $-\text{Meu}$ | $0$ |
+^ $f_{z\left(5z^2-r^2\right)}$ | $0$ | $0$ | $0$ | $\text{Ea2u}$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(y^2-z^2\right)}$ | $0$ | $\text{Meu}$ | $0$ | $0$ | $\text{Eeu2}$ | $0$ | $0$ |
+^ $f_{y\left(z^2-x^2\right)}$ | $0$ | $0$ | $-\text{Meu}$ | $0$ | $0$ | $\text{Eeu2}$ | $0$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\text{Eb2u}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ $f_{\text{xyz}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ | $0$ |
+^ $f_{x\left(5x^2-r^2\right)}$ | $\frac{\sqrt{5}}{4}$ | $0$ | $-\frac{\sqrt{3}}{4}$ | $0$ | $\frac{\sqrt{3}}{4}$ | $0$ | $-\frac{\sqrt{5}}{4}$ |
+^ $f_{y\left(5y^2-r^2\right)}$ | $-\frac{i \sqrt{5}}{4}$ | $0$ | $-\frac{i \sqrt{3}}{4}$ | $0$ | $-\frac{i \sqrt{3}}{4}$ | $0$ | $-\frac{i \sqrt{5}}{4}$ |
+^ $f_{z\left(5z^2-r^2\right)}$ | $0$ | $0$ | $0$ | $1$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(y^2-z^2\right)}$ | $-\frac{\sqrt{3}}{4}$ | $0$ | $-\frac{\sqrt{5}}{4}$ | $0$ | $\frac{\sqrt{5}}{4}$ | $0$ | $\frac{\sqrt{3}}{4}$ |
+^ $f_{y\left(z^2-x^2\right)}$ | $-\frac{i \sqrt{3}}{4}$ | $0$ | $\frac{i \sqrt{5}}{4}$ | $0$ | $\frac{i \sqrt{5}}{4}$ | $0$ | $-\frac{i \sqrt{3}}{4}$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Eb1u}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$  | ::: |
+^ ^ $\text{Eeu1}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$  | ::: |
+^ ^ $\text{Eeu1}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$  | ::: |
+^ ^ $\text{Ea2u}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_4.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$  | ::: |
+^ ^ $\text{Eeu2}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_5.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$  | ::: |
+^ ^ $\text{Eeu2}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_6.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$  | ::: |
+^ ^ $\text{Eb2u}$  | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_7.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$  | ::: |
+###
+</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** >
+###
+ $$A_{k,m} = \begin{cases}
+& k\neq 2\lor m\neq 0 \\
+ A(2,0) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, A[2, 0]]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty>
+Akm = {{2, 0, A(2,0)} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $0$ | $0$ | $\frac{A(2,0)}{\sqrt{5}}$ | $0$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $d_{x^2-y^2}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{yz}}$   ^   $d_{\text{xz}}$   ^   $d_{\text{xy}}$   ^
+^ $\text{s}$ | $0$ | $\frac{A(2,0)}{\sqrt{5}}$ | $0$ | $0$ | $0$ |
+###
+</hidden>
+==== Potential for p-f orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& k=0\land m=0 \\
+ A(2,0) & k=2\land m=0 \\
+ A(4,4) & k=4\land (m=-4\lor m=4) \\
+ A(4,0) & k=4\land m=0
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_D4h_Zxy.Quanty>
+Akm = {{2, 0, A(2,0)} ,
+       {4, 0, A(4,0)} ,
+       {4,-4, A(4,4)} ,
+       {4, 4, A(4,4)} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-1}^{(1)}}$ | $0$ | $0$ | $\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$ | $0$ | $0$ | $0$ | $-\frac{2 A(4,4)}{3 \sqrt{3}}$ |
+^ ${Y_{0}^{(1)}}$ | $0$ | $0$ | $0$ | $\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $-\frac{2 A(4,4)}{3 \sqrt{3}}$ | $0$ | $0$ | $0$ | $\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$ | $0$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{\text{xyz}}$   ^   $f_{x\left(5x^2-r^2\right)}$   ^   $f_{y\left(5y^2-r^2\right)}$   ^   $f_{z\left(5z^2-r^2\right)}$   ^   $f_{x\left(y^2-z^2\right)}$   ^   $f_{y\left(z^2-x^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^
+^ $p_x$ | $0$ | $\frac{5 A(4,0)-9 A(2,0)}{10 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} A(4,4)$ | $0$ | $0$ | $\frac{5 A(4,0)-9 A(2,0)}{6 \sqrt{35}}-\frac{A(4,4)}{3 \sqrt{2}}$ | $0$ | $0$ |
+^ $p_y$ | $0$ | $0$ | $\frac{5 A(4,0)-9 A(2,0)}{10 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} A(4,4)$ | $0$ | $0$ | $\frac{9 A(2,0)-5 A(4,0)}{6 \sqrt{35}}+\frac{A(4,4)}{3 \sqrt{2}}$ | $0$ |
+^ $p_z$ | $0$ | $0$ | $0$ | $\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+###
+</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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