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--- physics_chemistry:point_groups:c4:orientation_z [2018/03/21 15:16] – Stefano Agrestini
+++ physics_chemistry:point_groups:c4:orientation_z [2018/04/06 09:11] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
+~~CLOSETOC~~
 ====== Orientation Z ======
+===== Symmetry Operations =====
 ###
-alligned paragraph text
+In the C4 Point Group, with orientation Z there are the following symmetry operations
 ###
-===== Example =====
+###
+{{:physics_chemistry:pointgroup:c4_z.png}}
 ###
-description text
 ###
-==== Input ====
+^ Operator ^ Orientation ^
-<code Quanty Example.Quanty>
+^  $\text{E}$  |  $\{0,0,0\}$  , |
--- some example code
+^  $C_4$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  , |
+^  $C_2$  |  $\{0,0,1\}$  , |
+###
+===== Different Settings =====
+###
+  * [[physics_chemistry:point_groups:c4:orientation_x|Point Group C4 with orientation X]]
+  * [[physics_chemistry:point_groups:c4:orientation_y|Point Group C4 with orientation Y]]
+  * [[physics_chemistry:point_groups:c4:orientation_z|Point Group C4 with orientation Z]]
+###
+===== Character Table =====
+###
+|    ^   $\text{E} \,{\text{(1)}}$   ^   $C_4 \,{\text{(2)}}$   ^   $C_2 \,{\text{(1)}}$   ^
+^  $\text{A}$  |   $1$  |   $1$  |   $1$  |
+^  $\text{B}$  |   $1$  |   $-1$  |   $1$  |
+^  $\text{E}$  |   $2$  |   $0$  |   $-2$  |
+###
+===== Product Table =====
+###
+|    ^   $\text{A}$   ^   $\text{B}$   ^   $\text{E}$   ^
+^  $\text{A}$   |  $\text{A}$   |  $\text{B}$   |  $\text{E}$   |
+^  $\text{B}$   |  $\text{B}$   |  $\text{A}$   |  $\text{E}$   |
+^  $\text{E}$   |  $\text{E}$   |  $\text{E}$   |  $2 \text{A}+2 \text{B}$   |
+###
+===== Sub Groups with compatible settings =====
+###
+  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
+  * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]]
+###
+===== Super Groups with compatible settings =====
+###
+  * [[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:d4d:orientation_zxy|Point Group D4d with orientation Zxy]]
+  * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
+  * [[physics_chemistry:point_groups:d4:orientation_zxy|Point Group D4 with orientation Zxy]]
+  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
+  * [[physics_chemistry:point_groups:o:orientation_xyz|Point Group O 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 C4 Point group with orientation Z the form of the expansion coefficients is:
+###
+==== Expansion ====
+###
+ $$A_{k,m} = \begin{cases}
+ A(0,0) & k=0\land m=0 \\
+ A(1,0) & k=1\land m=0 \\
+ A(2,0) & k=2\land m=0 \\
+ A(3,0) & k=3\land m=0 \\
+ A(4,4)-i B(4,4) & k=4\land m=-4 \\
+ A(4,0) & k=4\land m=0 \\
+ A(4,4)+i B(4,4) & k=4\land m=4 \\
+ A(5,4)-i B(5,4) & k=5\land m=-4 \\
+ A(5,0) & k=5\land m=0 \\
+ A(5,4)+i B(5,4) & k=5\land m=4 \\
+ A(6,4)-i B(6,4) & k=6\land m=-4 \\
+ A(6,0) & k=6\land m=0 \\
+ A(6,4)+i B(6,4) & k=6\land m=4
+\end{cases}$$
+###
+==== Input format suitable for Mathematica (Quanty.nb) ====
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[3, 0], k == 3 && m == 0}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 0], k == 4 && m == 0}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 0], k == 5 && m == 0}, {A[5, 4] + I*B[5, 4], k == 5 && m == 4}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 0], k == 6 && m == 0}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}}, 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_C4_Z.Quanty>
+Akm = {{0, 0, A(0,0)} ,
+       {1, 0, A(1,0)} ,
+       {2, 0, A(2,0)} ,
+       {3, 0, A(3,0)} ,
+       {4, 0, A(4,0)} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
+       {5, 0, A(5,0)} ,
+       {5,-4, A(5,4) + (-I)*(B(5,4))} ,
+       {5, 4, A(5,4) + (I)*(B(5,4))} ,
+       {6, 0, A(6,0)} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(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.
+###
+###
+|    ^   ${Y_{0}^{(0)}}$   ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $\text{Ass}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-1}^{(1)}}$ | $\color{darkred}{ 0 }$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $0$ | $-\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}}$ |
+^ ${Y_{0}^{(1)}}$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $0$ | $\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $-\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}}$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ |
+^ ${Y_{-2}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4))$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ |
+^ ${Y_{0}^{(2)}}$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{1}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{2}^{(2)}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4))$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ |
+^ ${Y_{-3}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $-\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ | $\color{darkred}{ 0 }$ | $\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4))$ | $0$ | $0$ |
+^ ${Y_{-2}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4))$ | $0$ |
+^ ${Y_{-1}^{(3)}}$ | $\color{darkred}{ 0 }$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4))$ |
+^ ${Y_{0}^{(3)}}$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4))$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ |
+^ ${Y_{2}^{(3)}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $0$ | $\frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4))$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$ | $0$ |
+^ ${Y_{3}^{(3)}}$ | $\color{darkred}{ 0 }$ | $-\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4))$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ |
+###
+==== 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
+###
+###
+|    ^   ${Y_{0}^{(0)}}$   ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ $\text{s}$ | $1$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $p_y$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_z$ | $\color{darkred}{ 0 }$ | $0$ | $1$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $p_x$ | $\color{darkred}{ 0 }$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
+^ $d_{\text{xy}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{yz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{3z^2-r^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $1$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{x^2-y^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $f_{y\left(3x^2-y^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\frac{i}{\sqrt{2}}$ |
+^ $f_{\text{xyz}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ | $0$ |
+^ $f_{y\left(5z^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ |
+^ $f_{z\left(5z^2-3r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $1$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(5z^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $0$ | $0$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ |
+^ $f_{x\left(x^2-3y^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-\frac{1}{\sqrt{2}}$ |
+###
+==== One particle coupling on a basis of symmetry adapted functions ====
+###
+After rotation we find
+###
+###
+|    ^   $\text{s}$   ^   $p_y$   ^   $p_z$   ^   $p_x$   ^   $d_{\text{xy}}$   ^   $d_{\text{yz}}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{xz}}$   ^   $d_{x^2-y^2}$   ^   $f_{y\left(3x^2-y^2\right)}$   ^   $f_{\text{xyz}}$   ^   $f_{y\left(5z^2-r^2\right)}$   ^   $f_{z\left(5z^2-3r^2\right)}$   ^   $f_{x\left(5z^2-r^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^   $f_{x\left(x^2-3y^2\right)}$   ^
+^ $\text{s}$ | $\text{Ass}(0,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $p_y$ | $\color{darkred}{ 0 }$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $0$ | $-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$ |
+^ $p_z$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $0$ | $\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ $p_x$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$ | $0$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ |
+^ $d_{\text{xy}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$ | $0$ | $0$ | $0$ | $-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{yz}}$ | $0$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $0$ | $0$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$ |
+^ $d_{3z^2-r^2}$ | $\frac{\text{Asd}(2,0)}{\sqrt{5}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
+^ $d_{\text{xz}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$ | $0$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$ |
+^ $d_{x^2-y^2}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4)$ | $0$ | $0$ | $0$ | $\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $\color{darkred}{ 0 }$ |
+^ $f_{y\left(3x^2-y^2\right)}$ | $\color{darkred}{ 0 }$ | $-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ | $0$ | $-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$ | $0$ | $0$ |
+^ $f_{\text{xyz}}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$ | $0$ | $0$ | $0$ | $-\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)$ | $0$ |
+^ $f_{y\left(5z^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $0$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$ | $0$ | $\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$ |
+^ $f_{z\left(5z^2-3r^2\right)}$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $0$ | $\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$ | $0$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(5z^2-r^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$ | $0$ | $\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)-\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $0$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $0$ | $-\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)$ | $0$ | $0$ | $0$ | $\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$ | $0$ |
+^ $f_{x\left(x^2-3y^2\right)}$ | $\color{darkred}{ 0 }$ | $-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$ | $0$ | $\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$ | $\color{darkred}{ 0 }$ | $0$ | $0$ | $\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$ | $0$ | $\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)-\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)$ | $0$ | $\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$ |
+###
+===== 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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \text{Ea} & k=0\land m=0 \\
+& \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Ea, k == 0 && m == 0}}, 0]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{0, 0, Ea} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{0}^{(0)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $\text{Ea}$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $\text{s}$   ^
+^ $\text{s}$ | $\text{Ea}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{0}^{(0)}}$   ^
+^ $\text{s}$ | $1$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Ea}$  | {{:physics_chemistry:pointgroup:c4_z_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** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{3} (\text{Ea}+2 \text{Ee}) & k=0\land m=0 \\
+& k\neq 2\lor m\neq 0 \\
+ \frac{5 (\text{Ea}-\text{Ee})}{3} & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea + 2*Ee)/3, k == 0 && m == 0}, {0, k != 2 || m != 0}}, (5*(Ea - Ee))/3]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{0, 0, (1/3)*(Ea + (2)*(Ee))} ,
+       {2, 0, (5/3)*(Ea + (-1)*(Ee))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^
+^ ${Y_{-1}^{(1)}}$ | $\text{Ee}$ | $0$ | $0$ |
+^ ${Y_{0}^{(1)}}$ | $0$ | $\text{Ea}$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $0$ | $0$ | $\text{Ee}$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $p_y$   ^   $p_z$   ^   $p_x$   ^
+^ $p_y$ | $\text{Ee}$ | $0$ | $0$ |
+^ $p_z$ | $0$ | $\text{Ea}$ | $0$ |
+^ $p_x$ | $0$ | $0$ | $\text{Ee}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^
+^ $p_y$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ |
+^ $p_z$ | $0$ | $1$ | $0$ |
+^ $p_x$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Ee}$  | {{:physics_chemistry:pointgroup:c4_z_orb_1_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} y$  | ::: |
+^ ^ $\text{Ea}$  | {{:physics_chemistry:pointgroup:c4_z_orb_1_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} z$  | ::: |
+^ ^ $\text{Ee}$  | {{:physics_chemistry:pointgroup:c4_z_orb_1_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} x$  | ::: |
+###
+</hidden>
+==== Potential for d orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{5} (\text{Ea}+\text{Ebx2y2}+\text{Ebxy}+2 \text{Ee}) & k=0\land m=0 \\
+& (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\
+ \text{Ea}-\text{Ebx2y2}-\text{Ebxy}+\text{Ee} & k=2\land m=0 \\
+ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ebx2y2}-\text{Ebxy}+2 i \text{Mb}) & k=4\land m=-4 \\
+ \frac{3}{10} (6 \text{Ea}+\text{Ebx2y2}+\text{Ebxy}-8 \text{Ee}) & k=4\land m=0 \\
+ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ebx2y2}-\text{Ebxy}-2 i \text{Mb}) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea + Ebx2y2 + Ebxy + 2*Ee)/5, k == 0 && m == 0}, {0, (k != 4 && (k != 2 || m != 0)) || (m != -4 && m != 0 && m != 4)}, {Ea - Ebx2y2 - Ebxy + Ee, k == 2 && m == 0}, {(3*Sqrt[7/10]*(Ebx2y2 - Ebxy + (2*I)*Mb))/2, k == 4 && m == -4}, {(3*(6*Ea + Ebx2y2 + Ebxy - 8*Ee))/10, k == 4 && m == 0}}, (3*Sqrt[7/10]*(Ebx2y2 - Ebxy - (2*I)*Mb))/2]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{0, 0, (1/5)*(Ea + Ebx2y2 + Ebxy + (2)*(Ee))} ,
+       {2, 0, Ea + (-1)*(Ebx2y2) + (-1)*(Ebxy) + Ee} ,
+       {4, 0, (3/10)*((6)*(Ea) + Ebx2y2 + Ebxy + (-8)*(Ee))} ,
+       {4, 4, (3/2)*((sqrt(7/10))*(Ebx2y2 + (-1)*(Ebxy) + (-2*I)*(Mb)))} ,
+       {4,-4, (3/2)*((sqrt(7/10))*(Ebx2y2 + (-1)*(Ebxy) + (2*I)*(Mb)))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ ${Y_{-2}^{(2)}}$ | $\frac{\text{Ebx2y2}+\text{Ebxy}}{2}$ | $0$ | $0$ | $0$ | $\frac{1}{2} (\text{Ebx2y2}-\text{Ebxy}+2 i \text{Mb})$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $\text{Ee}$ | $0$ | $0$ | $0$ |
+^ ${Y_{0}^{(2)}}$ | $0$ | $0$ | $\text{Ea}$ | $0$ | $0$ |
+^ ${Y_{1}^{(2)}}$ | $0$ | $0$ | $0$ | $\text{Ee}$ | $0$ |
+^ ${Y_{2}^{(2)}}$ | $\frac{1}{2} (\text{Ebx2y2}-\text{Ebxy}-2 i \text{Mb})$ | $0$ | $0$ | $0$ | $\frac{\text{Ebx2y2}+\text{Ebxy}}{2}$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $d_{\text{xy}}$   ^   $d_{\text{yz}}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{xz}}$   ^   $d_{x^2-y^2}$   ^
+^ $d_{\text{xy}}$ | $\text{Ebxy}$ | $0$ | $0$ | $0$ | $\text{Mb}$ |
+^ $d_{\text{yz}}$ | $0$ | $\text{Ee}$ | $0$ | $0$ | $0$ |
+^ $d_{3z^2-r^2}$ | $0$ | $0$ | $\text{Ea}$ | $0$ | $0$ |
+^ $d_{\text{xz}}$ | $0$ | $0$ | $0$ | $\text{Ee}$ | $0$ |
+^ $d_{x^2-y^2}$ | $\text{Mb}$ | $0$ | $0$ | $0$ | $\text{Ebx2y2}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ $d_{\text{xy}}$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ |
+^ $d_{\text{yz}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ |
+^ $d_{3z^2-r^2}$ | $0$ | $0$ | $1$ | $0$ | $0$ |
+^ $d_{\text{xz}}$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $0$ |
+^ $d_{x^2-y^2}$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Ebxy}$  | {{:physics_chemistry:pointgroup:c4_z_orb_2_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$  | ::: |
+^ ^ $\text{Ee}$  | {{:physics_chemistry:pointgroup:c4_z_orb_2_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$  | ::: |
+^ ^ $\text{Ea}$  | {{:physics_chemistry:pointgroup:c4_z_orb_2_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$  | ::: |
+^ ^ $\text{Ee}$  | {{:physics_chemistry:pointgroup:c4_z_orb_2_4.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$  | ::: |
+^ ^ $\text{Ebx2y2}$  | {{:physics_chemistry:pointgroup:c4_z_orb_2_5.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$  | ::: |
+###
+</hidden>
+==== Potential for f orbitals ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+ \frac{1}{7} (\text{Ea}+\text{Ebxyz}+\text{Ebzx2y2}+2 \text{Ee1}+2 \text{Ee3}) & k=0\land m=0 \\
+& (k\neq 4\land k\neq 6\land (k\neq 2\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\
+ \frac{5}{14} (2 \text{Ea}+3 \text{Ee1}-5 \text{Ee3}) & k=2\land m=0 \\
+ \frac{3}{28} \left(-\sqrt{70} \text{Ebxyz}+\sqrt{70} \text{Ebzx2y2}+2 i \sqrt{70} \text{Mb}-4 i \sqrt{42} \text{MeIm}+4 \sqrt{42} \text{MeRe}\right) & k=4\land m=-4 \\
+ \frac{3}{14} (6 \text{Ea}-7 \text{Ebxyz}-7 \text{Ebzx2y2}+2 \text{Ee1}+6 \text{Ee3}) & k=4\land m=0 \\
+ \frac{3}{28} \left(-\sqrt{70} \text{Ebxyz}+\sqrt{70} \text{Ebzx2y2}-2 i \sqrt{70} \text{Mb}+4 i \sqrt{42} \text{MeIm}+4 \sqrt{42} \text{MeRe}\right) & k=4\land m=4 \\
+ -\frac{13 \left(3 \text{Ebxyz}-3 \text{Ebzx2y2}-6 i \text{Mb}-2 i \sqrt{15} \text{MeIm}+2 \sqrt{15} \text{MeRe}\right)}{10 \sqrt{14}} & k=6\land m=-4 \\
+ \frac{13}{70} (10 \text{Ea}+3 \text{Ebxyz}+3 \text{Ebzx2y2}-15 \text{Ee1}-\text{Ee3}) & k=6\land m=0 \\
+ -\frac{13 \left(3 \text{Ebxyz}-3 \text{Ebzx2y2}+6 i \text{Mb}+2 i \sqrt{15} \text{MeIm}+2 \sqrt{15} \text{MeRe}\right)}{10 \sqrt{14}} & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{(Ea + Ebxyz + Ebzx2y2 + 2*Ee1 + 2*Ee3)/7, k == 0 && m == 0}, {0, (k != 4 && k != 6 && (k != 2 || m != 0)) || (m != -4 && m != 0 && m != 4)}, {(5*(2*Ea + 3*Ee1 - 5*Ee3))/14, k == 2 && m == 0}, {(3*(-(Sqrt[70]*Ebxyz) + Sqrt[70]*Ebzx2y2 + (2*I)*Sqrt[70]*Mb - (4*I)*Sqrt[42]*MeIm + 4*Sqrt[42]*MeRe))/28, k == 4 && m == -4}, {(3*(6*Ea - 7*Ebxyz - 7*Ebzx2y2 + 2*Ee1 + 6*Ee3))/14, k == 4 && m == 0}, {(3*(-(Sqrt[70]*Ebxyz) + Sqrt[70]*Ebzx2y2 - (2*I)*Sqrt[70]*Mb + (4*I)*Sqrt[42]*MeIm + 4*Sqrt[42]*MeRe))/28, k == 4 && m == 4}, {(-13*(3*Ebxyz - 3*Ebzx2y2 - (6*I)*Mb - (2*I)*Sqrt[15]*MeIm + 2*Sqrt[15]*MeRe))/(10*Sqrt[14]), k == 6 && m == -4}, {(13*(10*Ea + 3*Ebxyz + 3*Ebzx2y2 - 15*Ee1 - Ee3))/70, k == 6 && m == 0}}, (-13*(3*Ebxyz - 3*Ebzx2y2 + (6*I)*Mb + (2*I)*Sqrt[15]*MeIm + 2*Sqrt[15]*MeRe))/(10*Sqrt[14])]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{0, 0, (1/7)*(Ea + Ebxyz + Ebzx2y2 + (2)*(Ee1) + (2)*(Ee3))} ,
+       {2, 0, (5/14)*((2)*(Ea) + (3)*(Ee1) + (-5)*(Ee3))} ,
+       {4, 0, (3/14)*((6)*(Ea) + (-7)*(Ebxyz) + (-7)*(Ebzx2y2) + (2)*(Ee1) + (6)*(Ee3))} ,
+       {4, 4, (3/28)*((-1)*((sqrt(70))*(Ebxyz)) + (sqrt(70))*(Ebzx2y2) + (-2*I)*((sqrt(70))*(Mb)) + (4*I)*((sqrt(42))*(MeIm)) + (4)*((sqrt(42))*(MeRe)))} ,
+       {4,-4, (3/28)*((-1)*((sqrt(70))*(Ebxyz)) + (sqrt(70))*(Ebzx2y2) + (2*I)*((sqrt(70))*(Mb)) + (-4*I)*((sqrt(42))*(MeIm)) + (4)*((sqrt(42))*(MeRe)))} ,
+       {6, 0, (13/70)*((10)*(Ea) + (3)*(Ebxyz) + (3)*(Ebzx2y2) + (-15)*(Ee1) + (-1)*(Ee3))} ,
+       {6,-4, (-13/10)*((1/(sqrt(14)))*((3)*(Ebxyz) + (-3)*(Ebzx2y2) + (-6*I)*(Mb) + (-2*I)*((sqrt(15))*(MeIm)) + (2)*((sqrt(15))*(MeRe))))} ,
+       {6, 4, (-13/10)*((1/(sqrt(14)))*((3)*(Ebxyz) + (-3)*(Ebzx2y2) + (6*I)*(Mb) + (2*I)*((sqrt(15))*(MeIm)) + (2)*((sqrt(15))*(MeRe))))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-3}^{(3)}}$ | $\text{Ee3}$ | $0$ | $0$ | $0$ | $\text{MeRe}-i \text{MeIm}$ | $0$ | $0$ |
+^ ${Y_{-2}^{(3)}}$ | $0$ | $\frac{\text{Ebxyz}+\text{Ebzx2y2}}{2}$ | $0$ | $0$ | $0$ | $\frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}+2 i \text{Mb})$ | $0$ |
+^ ${Y_{-1}^{(3)}}$ | $0$ | $0$ | $\text{Ee1}$ | $0$ | $0$ | $0$ | $\text{MeRe}-i \text{MeIm}$ |
+^ ${Y_{0}^{(3)}}$ | $0$ | $0$ | $0$ | $\text{Ea}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(3)}}$ | $\text{MeRe}+i \text{MeIm}$ | $0$ | $0$ | $0$ | $\text{Ee1}$ | $0$ | $0$ |
+^ ${Y_{2}^{(3)}}$ | $0$ | $\frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}-2 i \text{Mb})$ | $0$ | $0$ | $0$ | $\frac{\text{Ebxyz}+\text{Ebzx2y2}}{2}$ | $0$ |
+^ ${Y_{3}^{(3)}}$ | $0$ | $0$ | $\text{MeRe}+i \text{MeIm}$ | $0$ | $0$ | $0$ | $\text{Ee3}$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{y\left(3x^2-y^2\right)}$   ^   $f_{\text{xyz}}$   ^   $f_{y\left(5z^2-r^2\right)}$   ^   $f_{z\left(5z^2-3r^2\right)}$   ^   $f_{x\left(5z^2-r^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^   $f_{x\left(x^2-3y^2\right)}$   ^
+^ $f_{y\left(3x^2-y^2\right)}$ | $\text{Ee3}$ | $0$ | $\text{MeRe}$ | $0$ | $\text{MeIm}$ | $0$ | $0$ |
+^ $f_{\text{xyz}}$ | $0$ | $\text{Ebxyz}$ | $0$ | $0$ | $0$ | $\text{Mb}$ | $0$ |
+^ $f_{y\left(5z^2-r^2\right)}$ | $\text{MeRe}$ | $0$ | $\text{Ee1}$ | $0$ | $0$ | $0$ | $\text{MeIm}$ |
+^ $f_{z\left(5z^2-3r^2\right)}$ | $0$ | $0$ | $0$ | $\text{Ea}$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(5z^2-r^2\right)}$ | $\text{MeIm}$ | $0$ | $0$ | $0$ | $\text{Ee1}$ | $0$ | $-\text{MeRe}$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $0$ | $\text{Mb}$ | $0$ | $0$ | $0$ | $\text{Ebzx2y2}$ | $0$ |
+^ $f_{x\left(x^2-3y^2\right)}$ | $0$ | $0$ | $\text{MeIm}$ | $0$ | $-\text{MeRe}$ | $0$ | $\text{Ee3}$ |
+###
+</hidden>
+<hidden **Rotation matrix used** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ $f_{y\left(3x^2-y^2\right)}$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\frac{i}{\sqrt{2}}$ |
+^ $f_{\text{xyz}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $-\frac{i}{\sqrt{2}}$ | $0$ |
+^ $f_{y\left(5z^2-r^2\right)}$ | $0$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $\frac{i}{\sqrt{2}}$ | $0$ | $0$ |
+^ $f_{z\left(5z^2-3r^2\right)}$ | $0$ | $0$ | $0$ | $1$ | $0$ | $0$ | $0$ |
+^ $f_{x\left(5z^2-r^2\right)}$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $-\frac{1}{\sqrt{2}}$ | $0$ | $0$ |
+^ $f_{z\left(x^2-y^2\right)}$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $\frac{1}{\sqrt{2}}$ | $0$ |
+^ $f_{x\left(x^2-3y^2\right)}$ | $\frac{1}{\sqrt{2}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-\frac{1}{\sqrt{2}}$ |
+###
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
+###
+^ ^ $\text{Ee3}$  | {{:physics_chemistry:pointgroup:c4_z_orb_3_1.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$  | ::: |
+^ ^ $\text{Ebxyz}$  | {{:physics_chemistry:pointgroup:c4_z_orb_3_2.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$  | ::: |
+^ ^ $\text{Ee1}$  | {{:physics_chemistry:pointgroup:c4_z_orb_3_3.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$  | ::: |
+^ ^ $\text{Ea}$  | {{:physics_chemistry:pointgroup:c4_z_orb_3_4.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$  | ::: |
+^ ^ $\text{Ee1}$  | {{:physics_chemistry:pointgroup:c4_z_orb_3_5.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$  | ::: |
+^ ^ $\text{Ebzx2y2}$  | {{:physics_chemistry:pointgroup:c4_z_orb_3_6.png?150}} |
+| $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$  | ::: |
+| $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$  | ::: |
+^ ^ $\text{Ee3}$  | {{:physics_chemistry:pointgroup:c4_z_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 s-p orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& k\neq 1\lor m\neq 0 \\
+ A(1,0) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, k != 1 || m != 0}}, A[1, 0]]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{1, 0, A(1,0)} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-1}^{(1)}}$   ^   ${Y_{0}^{(1)}}$   ^   ${Y_{1}^{(1)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $0$ | $\frac{A(1,0)}{\sqrt{3}}$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $p_y$   ^   $p_z$   ^   $p_x$   ^
+^ $\text{s}$ | $0$ | $\frac{A(1,0)}{\sqrt{3}}$ | $0$ |
+###
+</hidden>
+==== 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_C4_Z.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_C4_Z.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** >
+###
+|    ^   $d_{\text{xy}}$   ^   $d_{\text{yz}}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{xz}}$   ^   $d_{x^2-y^2}$   ^
+^ $\text{s}$ | $0$ | $0$ | $\frac{A(2,0)}{\sqrt{5}}$ | $0$ | $0$ |
+###
+</hidden>
+==== Potential for s-f orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& k\neq 3\lor m\neq 0 \\
+ A(3,0) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, k != 3 || m != 0}}, A[3, 0]]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{3, 0, A(3,0)} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{0}^{(0)}}$ | $0$ | $0$ | $0$ | $\frac{A(3,0)}{\sqrt{7}}$ | $0$ | $0$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{y\left(3x^2-y^2\right)}$   ^   $f_{\text{xyz}}$   ^   $f_{y\left(5z^2-r^2\right)}$   ^   $f_{z\left(5z^2-3r^2\right)}$   ^   $f_{x\left(5z^2-r^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^   $f_{x\left(x^2-3y^2\right)}$   ^
+^ $\text{s}$ | $0$ | $0$ | $0$ | $\frac{A(3,0)}{\sqrt{7}}$ | $0$ | $0$ | $0$ |
+###
+</hidden>
+==== Potential for p-d orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& (k\neq 1\land k\neq 3)\lor m\neq 0 \\
+ A(1,0) & k=1\land m=0 \\
+ A(3,0) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, (k != 1 && k != 3) || m != 0}, {A[1, 0], k == 1 && m == 0}}, A[3, 0]]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{1, 0, A(1,0)} ,
+       {3, 0, A(3,0)} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-2}^{(2)}}$   ^   ${Y_{-1}^{(2)}}$   ^   ${Y_{0}^{(2)}}$   ^   ${Y_{1}^{(2)}}$   ^   ${Y_{2}^{(2)}}$   ^
+^ ${Y_{-1}^{(1)}}$ | $0$ | $\frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}}$ | $0$ | $0$ | $0$ |
+^ ${Y_{0}^{(1)}}$ | $0$ | $0$ | $\frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}}$ | $0$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $0$ | $0$ | $0$ | $\frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}}$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $d_{\text{xy}}$   ^   $d_{\text{yz}}$   ^   $d_{3z^2-r^2}$   ^   $d_{\text{xz}}$   ^   $d_{x^2-y^2}$   ^
+^ $p_y$ | $0$ | $\frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}}$ | $0$ | $0$ | $0$ |
+^ $p_z$ | $0$ | $0$ | $\frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}}$ | $0$ | $0$ |
+^ $p_x$ | $0$ | $0$ | $0$ | $\frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}}$ | $0$ |
+###
+</hidden>
+==== Potential for p-f orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\
+ A(2,0) & k=2\land m=0 \\
+ A(4,4)-i B(4,4) & k=4\land m=-4 \\
+ A(4,0) & k=4\land m=0 \\
+ A(4,4)+i B(4,4) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || m != 0)) || (m != -4 && m != 0 && m != 4)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 0], k == 4 && m == 0}}, A[4, 4] + I*B[4, 4]]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{2, 0, A(2,0)} ,
+       {4, 0, A(4,0)} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-1}^{(1)}}$ | $0$ | $0$ | $\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$ | $0$ | $0$ | $0$ | $-\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}}$ |
+^ ${Y_{0}^{(1)}}$ | $0$ | $0$ | $0$ | $\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(1)}}$ | $-\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}}$ | $0$ | $0$ | $0$ | $\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$ | $0$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{y\left(3x^2-y^2\right)}$   ^   $f_{\text{xyz}}$   ^   $f_{y\left(5z^2-r^2\right)}$   ^   $f_{z\left(5z^2-3r^2\right)}$   ^   $f_{x\left(5z^2-r^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^   $f_{x\left(x^2-3y^2\right)}$   ^
+^ $p_y$ | $-\frac{2 A(4,4)}{3 \sqrt{3}}$ | $0$ | $\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$ | $0$ | $0$ | $0$ | $-\frac{2 B(4,4)}{3 \sqrt{3}}$ |
+^ $p_z$ | $0$ | $0$ | $0$ | $\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$ | $0$ | $0$ | $0$ |
+^ $p_x$ | $-\frac{2 B(4,4)}{3 \sqrt{3}}$ | $0$ | $0$ | $0$ | $\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$ | $0$ | $\frac{2 A(4,4)}{3 \sqrt{3}}$ |
+###
+</hidden>
+==== Potential for d-f orbital mixing ====
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
+###
+ $$A_{k,m} = \begin{cases}
+& (k\neq 5\land ((k\neq 1\land k\neq 3)\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\
+ A(1,0) & k=1\land m=0 \\
+ A(3,0) & k=3\land m=0 \\
+ A(5,4)-i B(5,4) & k=5\land m=-4 \\
+ A(5,0) & k=5\land m=0 \\
+ A(5,4)+i B(5,4) & \text{True}
+\end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_C4_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{0, (k != 5 && ((k != 1 && k != 3) || m != 0)) || (m != -4 && m != 0 && m != 4)}, {A[1, 0], k == 1 && m == 0}, {A[3, 0], k == 3 && m == 0}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 0], k == 5 && m == 0}}, A[5, 4] + I*B[5, 4]]
+</code>
+###
+</hidden><hidden **Input format suitable for Quanty** >
+###
+<code Quanty Akm_C4_Z.Quanty>
+Akm = {{1, 0, A(1,0)} ,
+       {3, 0, A(3,0)} ,
+       {5, 0, A(5,0)} ,
+       {5,-4, A(5,4) + (-I)*(B(5,4))} ,
+       {5, 4, A(5,4) + (I)*(B(5,4))} }
+</code>
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
+###
+|    ^   ${Y_{-3}^{(3)}}$   ^   ${Y_{-2}^{(3)}}$   ^   ${Y_{-1}^{(3)}}$   ^   ${Y_{0}^{(3)}}$   ^   ${Y_{1}^{(3)}}$   ^   ${Y_{2}^{(3)}}$   ^   ${Y_{3}^{(3)}}$   ^
+^ ${Y_{-2}^{(2)}}$ | $0$ | $\frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}}$ | $0$ | $0$ | $0$ | $\frac{1}{11} \sqrt{10} (A(5,4)-i B(5,4))$ | $0$ |
+^ ${Y_{-1}^{(2)}}$ | $0$ | $0$ | $\frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0))$ | $0$ | $0$ | $0$ | $-\frac{2}{11} \sqrt{\frac{5}{3}} (A(5,4)-i B(5,4))$ |
+^ ${Y_{0}^{(2)}}$ | $0$ | $0$ | $0$ | $\frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}}$ | $0$ | $0$ | $0$ |
+^ ${Y_{1}^{(2)}}$ | $-\frac{2}{11} \sqrt{\frac{5}{3}} (A(5,4)+i B(5,4))$ | $0$ | $0$ | $0$ | $\frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0))$ | $0$ | $0$ |
+^ ${Y_{2}^{(2)}}$ | $0$ | $\frac{1}{11} \sqrt{10} (A(5,4)+i B(5,4))$ | $0$ | $0$ | $0$ | $\frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}}$ | $0$ |
+###
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
+###
+|    ^   $f_{y\left(3x^2-y^2\right)}$   ^   $f_{\text{xyz}}$   ^   $f_{y\left(5z^2-r^2\right)}$   ^   $f_{z\left(5z^2-3r^2\right)}$   ^   $f_{x\left(5z^2-r^2\right)}$   ^   $f_{z\left(x^2-y^2\right)}$   ^   $f_{x\left(x^2-3y^2\right)}$   ^
+^ $d_{\text{xy}}$ | $0$ | $\frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right)$ | $0$ | $0$ | $0$ | $-\frac{1}{11} \sqrt{10} B(5,4)$ | $0$ |
+^ $d_{\text{yz}}$ | $-\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4)$ | $0$ | $\frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0))$ | $0$ | $0$ | $0$ | $-\frac{2}{11} \sqrt{\frac{5}{3}} B(5,4)$ |
+^ $d_{3z^2-r^2}$ | $0$ | $0$ | $0$ | $\frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}}$ | $0$ | $0$ | $0$ |
+^ $d_{\text{xz}}$ | $-\frac{2}{11} \sqrt{\frac{5}{3}} B(5,4)$ | $0$ | $0$ | $0$ | $\frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0))$ | $0$ | $\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4)$ |
+^ $d_{x^2-y^2}$ | $0$ | $-\frac{1}{11} \sqrt{10} B(5,4)$ | $0$ | $0$ | $0$ | $\frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)+21 \sqrt{10} A(5,4)\right)$ | $0$ |
+###
+</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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