~~CLOSETOC~~
====== Orientation X ======
===== Symmetry Operations =====
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In the Cs Point Group, with orientation X there are the following symmetry operations
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{{:physics_chemistry:pointgroup:cs_x.png}}
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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $\sigma _h$ | $\{1,0,0\}$ , |
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===== Different Settings =====
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* [[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]]
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===== Character Table =====
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| $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ \sigma_h \,{\text{(1)}} $ ^
^ $ \text{A'} $ | $ 1 $ | $ 1 $ |
^ $ \text{A''} $ | $ 1 $ | $ -1 $ |
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===== Product Table =====
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| $ $ ^ $ \text{A'} $ ^ $ \text{A''} $ ^
^ $ \text{A'} $ | $ \text{A'} $ | $ \text{A''} $ |
^ $ \text{A''} $ | $ \text{A''} $ | $ \text{A'} $ |
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===== Sub Groups with compatible settings =====
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* [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
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===== Super Groups with compatible settings =====
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* [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]]
* [[physics_chemistry:point_groups:c3v:orientation_zx|Point Group C3v with orientation Zx]]
* [[physics_chemistry:point_groups:c4v:orientation_zxy|Point Group C4v with orientation Zxy]]
* [[physics_chemistry:point_groups:c6v:orientation_zx|Point Group C6v with orientation Zx]]
* [[physics_chemistry:point_groups:c6v:orientation_zy|Point Group C6v with orientation Zy]]
* [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]]
* [[physics_chemistry:point_groups:d3d:orientation_zx|Point Group D3d with orientation Zx]]
* [[physics_chemistry:point_groups:d3d:orientation_zx_a|Point Group D3d with orientation Zx_A]]
* [[physics_chemistry:point_groups:d3d:orientation_zx_b|Point Group D3d with orientation Zx_B]]
* [[physics_chemistry:point_groups:d3h:orientation_zy|Point Group D3h with orientation Zy]]
* [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
* [[physics_chemistry:point_groups:d5d:orientation_zx|Point Group D5d with orientation Zx]]
* [[physics_chemistry:point_groups:d5h:orientation_zy|Point Group D5h with orientation Zy]]
* [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]]
* [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]]
* [[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Point Group Oh with orientation 0sqrt2-1z]]
* [[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Point Group Oh with orientation 0sqrt21z]]
* [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]]
* [[physics_chemistry:point_groups:th:orientation_xyz|Point Group Th with orientation xyz]]
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===== Invariant Potential expanded on renormalized spherical Harmonics =====
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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 Cs Point group with orientation X the form of the expansion coefficients is:
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==== Expansion ====
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$$A_{k,m} = \begin{cases}
A(0,0) & k=0\land m=0 \\
i B(1,1) & k=1\land (m=-1\lor m=1) \\
A(1,0) & k=1\land m=0 \\
A(2,2) & k=2\land (m=-2\lor m=2) \\
i B(2,1) & k=2\land (m=-1\lor m=1) \\
A(2,0) & k=2\land m=0 \\
i B(3,3) & k=3\land (m=-3\lor m=3) \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
i B(3,1) & k=3\land (m=-1\lor m=1) \\
A(3,0) & k=3\land m=0 \\
A(4,4) & k=4\land (m=-4\lor m=4) \\
i B(4,3) & k=4\land (m=-3\lor m=3) \\
A(4,2) & k=4\land (m=-2\lor m=2) \\
i B(4,1) & k=4\land (m=-1\lor m=1) \\
A(4,0) & k=4\land m=0 \\
i B(5,5) & k=5\land (m=-5\lor m=5) \\
A(5,4) & k=5\land (m=-4\lor m=4) \\
i B(5,3) & k=5\land (m=-3\lor m=3) \\
A(5,2) & k=5\land (m=-2\lor m=2) \\
i B(5,1) & k=5\land (m=-1\lor m=1) \\
A(5,0) & k=5\land m=0 \\
A(6,6) & k=6\land (m=-6\lor m=6) \\
i B(6,5) & k=6\land (m=-5\lor m=5) \\
A(6,4) & k=6\land (m=-4\lor m=4) \\
i B(6,3) & k=6\land (m=-3\lor m=3) \\
A(6,2) & k=6\land (m=-2\lor m=2) \\
i B(6,1) & k=6\land (m=-1\lor m=1) \\
A(6,0) & k=6\land m=0
\end{cases}$$
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==== Input format suitable for Mathematica (Quanty.nb) ====
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Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {I*B[1, 1], k == 1 && (m == -1 || m == 1)}, {A[1, 0], k == 1 && m == 0}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}, {A[2, 0], k == 2 && m == 0}, {I*B[3, 3], k == 3 && (m == -3 || m == 3)}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {I*B[3, 1], k == 3 && (m == -1 || m == 1)}, {A[3, 0], k == 3 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {I*B[4, 1], k == 4 && (m == -1 || m == 1)}, {A[4, 0], k == 4 && m == 0}, {I*B[5, 5], k == 5 && (m == -5 || m == 5)}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {I*B[5, 3], k == 5 && (m == -3 || m == 3)}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {I*B[5, 1], k == 5 && (m == -1 || m == 1)}, {A[5, 0], k == 5 && m == 0}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {I*B[6, 5], k == 6 && (m == -5 || m == 5)}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {I*B[6, 3], k == 6 && (m == -3 || m == 3)}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {I*B[6, 1], k == 6 && (m == -1 || m == 1)}, {A[6, 0], k == 6 && m == 0}}, 0]
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==== Input format suitable for Quanty ====
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Akm = { 0, A(0,0)} ,
{1, 0, A(1,0)} ,
{1,-1, (I)*(B(1,1))} ,
{1, 1, (I)*(B(1,1))} ,
{2, 0, A(2,0)} ,
{2,-1, (I)*(B(2,1))} ,
{2, 1, (I)*(B(2,1))} ,
{2,-2, A(2,2)} ,
{2, 2, A(2,2)} ,
{3, 0, A(3,0)} ,
{3,-1, (I)*(B(3,1))} ,
{3, 1, (I)*(B(3,1))} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} ,
{3,-3, (I)*(B(3,3))} ,
{3, 3, (I)*(B(3,3))} ,
{4, 0, A(4,0)} ,
{4,-1, (I)*(B(4,1))} ,
{4, 1, (I)*(B(4,1))} ,
{4,-2, A(4,2)} ,
{4, 2, A(4,2)} ,
{4,-3, (I)*(B(4,3))} ,
{4, 3, (I)*(B(4,3))} ,
{4,-4, A(4,4)} ,
{4, 4, A(4,4)} ,
{5, 0, A(5,0)} ,
{5,-1, (I)*(B(5,1))} ,
{5, 1, (I)*(B(5,1))} ,
{5,-2, A(5,2)} ,
{5, 2, A(5,2)} ,
{5,-3, (I)*(B(5,3))} ,
{5, 3, (I)*(B(5,3))} ,
{5,-4, A(5,4)} ,
{5, 4, A(5,4)} ,
{5,-5, (I)*(B(5,5))} ,
{5, 5, (I)*(B(5,5))} ,
{6, 0, A(6,0)} ,
{6,-1, (I)*(B(6,1))} ,
{6, 1, (I)*(B(6,1))} ,
{6,-2, A(6,2)} ,
{6, 2, A(6,2)} ,
{6,-3, (I)*(B(6,3))} ,
{6, 3, (I)*(B(6,3))} ,
{6,-4, A(6,4)} ,
{6, 4, A(6,4)} ,
{6,-5, (I)*(B(6,5))} ,
{6, 5, (I)*(B(6,5))} ,
{6,-6, A(6,6)} ,
{6, 6, A(6,6)} }
###
==== One particle coupling on a basis of spherical harmonics ====
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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 $A_{n''l'',n'l'}(k,m)$
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we can express the operator as
$$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$
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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.
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| $ $ ^ $ {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}{ -\frac{i \text{Bsp}(1,1)}{\sqrt{3}} }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ -\frac{i \text{Bsp}(1,1)}{\sqrt{3}} }$|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$ -\frac{i \text{Bsd}(2,1)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ -\frac{i \text{Bsd}(2,1)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ -\frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ -\frac{i \text{Bsf}(3,1)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ -\frac{i \text{Bsf}(3,1)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ -\frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$|
^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ \frac{i \text{Bsp}(1,1)}{\sqrt{3}} }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ \frac{1}{5} i \sqrt{3} \text{Bpp}(2,1) $|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$\color{darkred}{ \frac{1}{7} i \sqrt{\frac{3}{5}} \text{Bpd}(3,1)-i \sqrt{\frac{2}{5}} \text{Bpd}(1,1) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ \frac{3}{7} i \sqrt{\frac{2}{5}} \text{Bpd}(3,1)-\frac{i \text{Bpd}(1,1)}{\sqrt{15}} }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{3}{7} i \text{Bpd}(3,3) }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ \frac{i \text{Bpf}(4,1)}{3 \sqrt{7}}-i \sqrt{\frac{6}{35}} \text{Bpf}(2,1) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1)-\frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}} $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ \frac{1}{3} i \text{Bpf}(4,3) $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|
^$ {Y_{0}^{(1)}} $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ -\frac{1}{5} i \sqrt{3} \text{Bpp}(2,1) $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ -\frac{1}{5} i \sqrt{3} \text{Bpp}(2,1) $|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} i \sqrt{\frac{6}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ -\frac{i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} i \sqrt{\frac{6}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$ -\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ -\frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ -\frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1) $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ -\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}} $|
^$ {Y_{1}^{(1)}} $|$\color{darkred}{ \frac{i \text{Bsp}(1,1)}{\sqrt{3}} }$|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ \frac{1}{5} i \sqrt{3} \text{Bpp}(2,1) $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ \frac{3}{7} i \text{Bpd}(3,3) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{3}{7} i \sqrt{\frac{2}{5}} \text{Bpd}(3,1)-\frac{i \text{Bpd}(1,1)}{\sqrt{15}} }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ \frac{1}{7} i \sqrt{\frac{3}{5}} \text{Bpd}(3,1)-i \sqrt{\frac{2}{5}} \text{Bpd}(1,1) }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ \frac{1}{3} i \text{Bpf}(4,3) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ \frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1)-\frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}} $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{i \text{Bpf}(4,1)}{3 \sqrt{7}}-i \sqrt{\frac{6}{35}} \text{Bpf}(2,1) $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|
^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ i \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} i \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{3}{7} i \text{Bpd}(3,3) }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ \frac{1}{7} i \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} i \sqrt{5} \text{Bdd}(4,1) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ -i \sqrt{\frac{3}{7}} \text{Bdf}(1,1)+\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,1)-\frac{1}{33} i \sqrt{\frac{5}{7}} \text{Bdf}(5,1) }$|$\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}{ -\frac{i \text{Bdf}(1,1)}{\sqrt{35}}+2 i \sqrt{\frac{2}{105}} \text{Bdf}(3,1)-\frac{5 i \text{Bdf}(5,1)}{11 \sqrt{21}} }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3)-\frac{5}{33} i \sqrt{2} \text{Bdf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ -\frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|
^$ {Y_{-1}^{(2)}} $|$ \frac{i \text{Bsd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ \frac{i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} i \sqrt{\frac{6}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ \frac{1}{21} i \sqrt{5} \text{Bdd}(4,1)-\frac{1}{7} i \sqrt{6} \text{Bdd}(2,1) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ \frac{1}{7} i \text{Bdd}(2,1)+\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ -i \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} i \sqrt{\frac{10}{21}} \text{Bdf}(5,1) }$|$\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}{ -i \sqrt{\frac{3}{35}} \text{Bdf}(1,1)+\frac{1}{3} i \sqrt{\frac{2}{35}} \text{Bdf}(3,1)+\frac{20 i \text{Bdf}(5,1)}{33 \sqrt{7}} }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|
^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ \frac{i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} i \sqrt{\frac{2}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ \frac{i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} i \sqrt{\frac{2}{5}} \text{Bpd}(3,1) }$|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ -\frac{1}{7} i \text{Bdd}(2,1)-\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ -\frac{1}{7} i \text{Bdd}(2,1)-\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{33} i \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ -i \sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} i \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\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}{ -i \sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} i \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{33} i \sqrt{5} \text{Bdf}(5,3) }$|
^$ {Y_{1}^{(2)}} $|$ \frac{i \text{Bsd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} i \sqrt{\frac{6}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ \frac{1}{7} i \text{Bdd}(2,1)+\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ \frac{1}{21} i \sqrt{5} \text{Bdd}(4,1)-\frac{1}{7} i \sqrt{6} \text{Bdd}(2,1) $|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ -i \sqrt{\frac{3}{35}} \text{Bdf}(1,1)+\frac{1}{3} i \sqrt{\frac{2}{35}} \text{Bdf}(3,1)+\frac{20 i \text{Bdf}(5,1)}{33 \sqrt{7}} }$|$\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}{ -i \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} i \sqrt{\frac{10}{21}} \text{Bdf}(5,1) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|
^$ {Y_{2}^{(2)}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ -\frac{3}{7} i \text{Bpd}(3,3) }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ i \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} i \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ \frac{1}{7} i \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} i \sqrt{5} \text{Bdd}(4,1) $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ -\frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3)-\frac{5}{33} i \sqrt{2} \text{Bdf}(5,3) }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ -\frac{i \text{Bdf}(1,1)}{\sqrt{35}}+2 i \sqrt{\frac{2}{105}} \text{Bdf}(3,1)-\frac{5 i \text{Bdf}(5,1)}{11 \sqrt{21}} }$|$\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}{ -i \sqrt{\frac{3}{7}} \text{Bdf}(1,1)+\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,1)-\frac{1}{33} i \sqrt{\frac{5}{7}} \text{Bdf}(5,1) }$|
^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ \frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ \frac{i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$\color{darkred}{ i \sqrt{\frac{3}{7}} \text{Bdf}(1,1)-\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,1)+\frac{1}{33} i \sqrt{\frac{5}{7}} \text{Bdf}(5,1) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \frac{2}{33} i \sqrt{5} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3) }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ \frac{1}{3} i \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)+\frac{5}{429} i \sqrt{7} \text{Bff}(6,1) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ \frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)-\frac{1}{11} i \sqrt{7} \text{Bff}(4,3) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ \frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|
^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$ i \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{i \text{Bpf}(4,1)}{3 \sqrt{7}} $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ -\frac{1}{3} i \text{Bpf}(4,3) $|$\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}{ i \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} i \sqrt{\frac{10}{21}} \text{Bdf}(5,1) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$ -\frac{1}{3} i \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)-\frac{5}{429} i \sqrt{7} \text{Bff}(6,1) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \frac{i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)-\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ -\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)-\frac{5}{143} i \sqrt{42} \text{Bff}(6,3) $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ -\frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|
^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ \frac{i \text{Bsf}(3,1)}{\sqrt{7}} }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)+\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$\color{darkred}{ \frac{i \text{Bdf}(1,1)}{\sqrt{35}}-2 i \sqrt{\frac{2}{105}} \text{Bdf}(3,1)+\frac{5 i \text{Bdf}(5,1)}{11 \sqrt{21}} }$|$\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}{ i \sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} i \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ \frac{5}{33} i \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ -\frac{i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)+\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1) $|$ \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) $|$ \frac{1}{15} i \sqrt{2} \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)+\frac{25}{429} i \sqrt{14} \text{Bff}(6,1) $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $|$ \frac{1}{33} i \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} i \sqrt{42} \text{Bff}(6,3) $|$ \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}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ \frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}}-\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}}-\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1) $|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ i \sqrt{\frac{3}{35}} \text{Bdf}(1,1)-\frac{1}{3} i \sqrt{\frac{2}{35}} \text{Bdf}(3,1)-\frac{20 i \text{Bdf}(5,1)}{33 \sqrt{7}} }$|$\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}{ i \sqrt{\frac{3}{35}} \text{Bdf}(1,1)-\frac{1}{3} i \sqrt{\frac{2}{35}} \text{Bdf}(3,1)-\frac{20 i \text{Bdf}(5,1)}{33 \sqrt{7}} }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$ \frac{1}{11} i \sqrt{7} \text{Bff}(4,3)-\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ -\frac{1}{15} i \sqrt{2} \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)-\frac{25}{429} i \sqrt{14} \text{Bff}(6,1) $|$ \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) $|$ -\frac{1}{15} i \sqrt{2} \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)-\frac{25}{429} i \sqrt{14} \text{Bff}(6,1) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ \frac{1}{11} i \sqrt{7} \text{Bff}(4,3)-\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3) $|
^$ {Y_{1}^{(3)}} $|$\color{darkred}{ \frac{i \text{Bsf}(3,1)}{\sqrt{7}} }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ \frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)+\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ \frac{5}{33} i \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ i \sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} i \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\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}{ \frac{i \text{Bdf}(1,1)}{\sqrt{35}}-2 i \sqrt{\frac{2}{105}} \text{Bdf}(3,1)+\frac{5 i \text{Bdf}(5,1)}{11 \sqrt{21}} }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ \frac{1}{33} i \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} i \sqrt{42} \text{Bff}(6,3) $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $|$ \frac{1}{15} i \sqrt{2} \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)+\frac{25}{429} i \sqrt{14} \text{Bff}(6,1) $|$ \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) $|$ -\frac{i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)+\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|
^$ {Y_{2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$ -\frac{1}{3} i \text{Bpf}(4,3) $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ i \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{i \text{Bpf}(4,1)}{3 \sqrt{7}} $|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ i \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} i \sqrt{\frac{10}{21}} \text{Bdf}(5,1) }$|$\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{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ -\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)-\frac{5}{143} i \sqrt{42} \text{Bff}(6,3) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ \frac{i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)-\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ -\frac{1}{3} i \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)-\frac{5}{429} i \sqrt{7} \text{Bff}(6,1) $|
^$ {Y_{3}^{(3)}} $|$\color{darkred}{ \frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ \frac{i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$\color{darkred}{ \frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{2}{33} i \sqrt{5} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ i \sqrt{\frac{3}{7}} \text{Bdf}(1,1)-\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,1)+\frac{1}{33} i \sqrt{\frac{5}{7}} \text{Bdf}(5,1) }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ \frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ \frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)-\frac{1}{11} i \sqrt{7} \text{Bff}(4,3) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ \frac{1}{3} i \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)+\frac{5}{429} i \sqrt{7} \text{Bff}(6,1) $|$ \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_{y^2-z^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|$ -\frac{\sqrt{3}}{2} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{3x^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|$ -\frac{1}{2} $|$ 0 $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$\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 }$|
^$ 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 }$|
^$ 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(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(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(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 $|
^$ 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} $|
###
==== One particle coupling on a basis of symmetry adapted functions ====
###
After rotation we find
###
###
| $ $ ^ $ \text{s} $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ $ d_{y^2-z^2} $ ^ $ d_{3x^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^
^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ 0 }$|$ -\frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} $|$ \sqrt{\frac{3}{10}} \text{Asd}(2,2)-\frac{\text{Asd}(2,0)}{2 \sqrt{5}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$\color{darkred}{ 0 }$|
^$ p_y $|$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $|$ 0 $|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)-\frac{3}{14} \text{Bpd}(3,3) }$|$\color{darkred}{ -\sqrt{\frac{2}{15}} \text{Bpd}(1,1)-\frac{3 \text{Bpd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Bpd}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) $|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|
^$ p_z $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)+\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$ 0 $|
^$ p_x $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$|$\color{darkred}{ 0 }$|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|
^$ d_{y^2-z^2} $|$ -\frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} $|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)-\frac{3}{14} \text{Bpd}(3,3) }$|$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{19}{84} \text{Add}(4,0)+\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{6} \sqrt{\frac{5}{14}} \text{Add}(4,4) $|$ \frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)+\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)-\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) $|$ 0 $|$ 0 $|$ -\frac{1}{6} \sqrt{5} \text{Bdd}(4,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,3)+\frac{5 \text{Bdf}(5,1)}{6 \sqrt{14}}+\frac{5}{44} \sqrt{3} \text{Bdf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ -\frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)-\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{17}{22} \sqrt{\frac{5}{42}} \text{Bdf}(5,1)+\frac{1}{132} \sqrt{5} \text{Bdf}(5,3)+\frac{5}{44} \text{Bdf}(5,5) }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|
^$ d_{3x^2-r^2} $|$ \sqrt{\frac{3}{10}} \text{Asd}(2,2)-\frac{\text{Asd}(2,0)}{2 \sqrt{5}} $|$\color{darkred}{ -\sqrt{\frac{2}{15}} \text{Bpd}(1,1)-\frac{3 \text{Bpd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Bpd}(3,3) }$|$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)+\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)+\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)-\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) $|$ \text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{3}{28} \text{Add}(4,0)-\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{14}} \text{Add}(4,4) $|$ 0 $|$ 0 $|$ -\frac{2}{7} \sqrt{2} \text{Bdd}(2,1)-\frac{1}{14} \sqrt{\frac{5}{3}} \text{Bdd}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdd}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{7}{15}} \text{Bdf}(3,1)-\frac{\text{Bdf}(3,3)}{6 \sqrt{7}}+\frac{5 \text{Bdf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{5} \text{Bdf}(5,5) }$|$\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}-\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)+\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{14}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)-\frac{3}{22} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)+\frac{7}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{3} \text{Bdf}(5,5) }$|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{\sqrt{21}}+\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }$|$\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}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 0 $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|
^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$|$ 0 $|$ 0 $|$ \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ \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 $|$\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}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$|
^$ d_{\text{yz}} $|$ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ 0 }$|$ -\frac{1}{6} \sqrt{5} \text{Bdd}(4,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ -\frac{2}{7} \sqrt{2} \text{Bdd}(2,1)-\frac{1}{14} \sqrt{\frac{5}{3}} \text{Bdd}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdd}(4,3) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|
^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\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 }$|$ \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 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ 0 $|$ 0 $|$ -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|
^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) $|$ 0 $|$\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,3)+\frac{5 \text{Bdf}(5,1)}{6 \sqrt{14}}+\frac{5}{44} \sqrt{3} \text{Bdf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{7}{15}} \text{Bdf}(3,1)-\frac{\text{Bdf}(3,3)}{6 \sqrt{7}}+\frac{5 \text{Bdf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{5} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ 0 $|
^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ -\frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)-\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}-\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)+\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$ 0 $|$ -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) $|$ \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 $|$ \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|
^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) $|$ 0 $|$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{17}{22} \sqrt{\frac{5}{42}} \text{Bdf}(5,1)+\frac{1}{132} \sqrt{5} \text{Bdf}(5,3)+\frac{5}{44} \text{Bdf}(5,5) }$|$\color{darkred}{ -\sqrt{\frac{3}{14}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)-\frac{3}{22} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)+\frac{7}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{3} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$ 0 $|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{\sqrt{21}}+\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$ 0 $|$ -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|$ \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(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$ -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|
###
===== Coupling for a single shell =====
###
Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$.
###
###
Click on one of the subsections to expand it or
###
==== Potential for s orbitals ====
###
$$A_{k,m} = \begin{cases}
\text{Eap} & k=0\land m=0 \\
0 & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{Eap, k == 0 && m == 0}}, 0]
###
###
Akm = {{0, 0, Eap} }
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^
^$ {Y_{0}^{(0)}} $|$ \text{Eap} $|
###
###
| $ $ ^ $ \text{s} $ ^
^$ \text{s} $|$ \text{Eap} $|
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^
^$ \text{s} $|$ 1 $|
###
###
^ ^$$\text{Eap}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_0_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
###
==== Potential for p orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{3} (\text{Eapp}+\text{Eapy}+\text{Eapz}) & k=0\land m=0 \\
0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\
\frac{5 (\text{Eapp}-\text{Eapy})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\
\frac{5 i \text{Mapyz}}{\sqrt{6}} & k=2\land (m=-1\lor m=1) \\
-\frac{5}{6} (\text{Eapp}+\text{Eapy}-2 \text{Eapz}) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{(Eapp + Eapy + Eapz)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {(5*(Eapp - Eapy))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}, {((5*I)*Mapyz)/Sqrt[6], k == 2 && (m == -1 || m == 1)}}, (-5*(Eapp + Eapy - 2*Eapz))/6]
###
###
Akm = {{0, 0, (1/3)*(Eapp + Eapy + Eapz)} ,
{2, 0, (-5/6)*(Eapp + Eapy + (-2)*(Eapz))} ,
{2,-1, (5*I)*((1/(sqrt(6)))*(Mapyz))} ,
{2, 1, (5*I)*((1/(sqrt(6)))*(Mapyz))} ,
{2,-2, (5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapy)))} ,
{2, 2, (5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapy)))} }
###
###
| $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^
^$ {Y_{-1}^{(1)}} $|$ \frac{\text{Eapp}+\text{Eapy}}{2} $|$ \frac{i \text{Mapyz}}{\sqrt{2}} $|$ \frac{\text{Eapy}-\text{Eapp}}{2} $|
^$ {Y_{0}^{(1)}} $|$ -\frac{i \text{Mapyz}}{\sqrt{2}} $|$ \text{Eapz} $|$ -\frac{i \text{Mapyz}}{\sqrt{2}} $|
^$ {Y_{1}^{(1)}} $|$ \frac{\text{Eapy}-\text{Eapp}}{2} $|$ \frac{i \text{Mapyz}}{\sqrt{2}} $|$ \frac{\text{Eapp}+\text{Eapy}}{2} $|
###
###
| $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^
^$ p_y $|$ \text{Eapy} $|$ \text{Mapyz} $|$ 0 $|
^$ p_z $|$ \text{Mapyz} $|$ \text{Eapz} $|$ 0 $|
^$ p_x $|$ 0 $|$ 0 $|$ \text{Eapp} $|
###
###
| $ $ ^ $ {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}} $|
###
###
^ ^$$\text{Eapy}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Eapz}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
^ ^$$\text{Eapp}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_1_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
###
==== Potential for d orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{5} (\text{Eappxy}+\text{Eappxz}+\text{Eapx2}+\text{Eapy2z2}+\text{Eapyz}) & k=0\land m=0 \\
0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\
\frac{1}{4} \left(\sqrt{6} \text{Eappxz}+\sqrt{6} \text{Eapx2}-\sqrt{6} \text{Eapy2z2}-\sqrt{6} \text{Eapyz}+2 \sqrt{2} \text{Mapy2z2x2}\right) & k=2\land (m=-2\lor m=2) \\
\frac{1}{2} i \left(\sqrt{6} \text{Mappxzxy}-2 \sqrt{2} \text{Mapx2yz}\right) & k=2\land (m=-1\lor m=1) \\
\frac{1}{2} \left(-2 \text{Eappxy}+\text{Eappxz}-\text{Eapx2}+\text{Eapy2z2}+\text{Eapyz}+2 \sqrt{3} \text{Mapy2z2x2}\right) & k=2\land m=0 \\
-\frac{3}{8} \sqrt{\frac{7}{10}} \left(4 \text{Eappxy}-3 \text{Eapx2}-\text{Eapy2z2}+2 \sqrt{3} \text{Mapy2z2x2}\right) & k=4\land (m=-4\lor m=4) \\
\frac{3}{4} i \sqrt{\frac{7}{5}} \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) & k=4\land (m=-3\lor m=3) \\
\frac{3 \left(4 \text{Eappxz}-3 \text{Eapx2}+3 \text{Eapy2z2}-4 \text{Eapyz}-2 \sqrt{3} \text{Mapy2z2x2}\right)}{4 \sqrt{10}} & k=4\land (m=-2\lor m=2) \\
-\frac{3 i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}+7 \text{Mapy2z2yz}\right)}{4 \sqrt{5}} & k=4\land (m=-1\lor m=1) \\
\frac{3}{40} \left(4 \text{Eappxy}-16 \text{Eappxz}+9 \text{Eapx2}+19 \text{Eapy2z2}-16 \text{Eapyz}+10 \sqrt{3} \text{Mapy2z2x2}\right) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{(Eappxy + Eappxz + Eapx2 + Eapy2z2 + Eapyz)/5, k == 0 && m == 0}, {0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(Sqrt[6]*Eappxz + Sqrt[6]*Eapx2 - Sqrt[6]*Eapy2z2 - Sqrt[6]*Eapyz + 2*Sqrt[2]*Mapy2z2x2)/4, k == 2 && (m == -2 || m == 2)}, {(I/2)*(Sqrt[6]*Mappxzxy - 2*Sqrt[2]*Mapx2yz), k == 2 && (m == -1 || m == 1)}, {(-2*Eappxy + Eappxz - Eapx2 + Eapy2z2 + Eapyz + 2*Sqrt[3]*Mapy2z2x2)/2, k == 2 && m == 0}, {(-3*Sqrt[7/10]*(4*Eappxy - 3*Eapx2 - Eapy2z2 + 2*Sqrt[3]*Mapy2z2x2))/8, k == 4 && (m == -4 || m == 4)}, {((3*I)/4)*Sqrt[7/5]*(2*Mappxzxy + Sqrt[3]*Mapx2yz - Mapy2z2yz), k == 4 && (m == -3 || m == 3)}, {(3*(4*Eappxz - 3*Eapx2 + 3*Eapy2z2 - 4*Eapyz - 2*Sqrt[3]*Mapy2z2x2))/(4*Sqrt[10]), k == 4 && (m == -2 || m == 2)}, {(((-3*I)/4)*(2*Mappxzxy + Sqrt[3]*Mapx2yz + 7*Mapy2z2yz))/Sqrt[5], k == 4 && (m == -1 || m == 1)}}, (3*(4*Eappxy - 16*Eappxz + 9*Eapx2 + 19*Eapy2z2 - 16*Eapyz + 10*Sqrt[3]*Mapy2z2x2))/40]
###
###
Akm = {{0, 0, (1/5)*(Eappxy + Eappxz + Eapx2 + Eapy2z2 + Eapyz)} ,
{2, 0, (1/2)*((-2)*(Eappxy) + Eappxz + (-1)*(Eapx2) + Eapy2z2 + Eapyz + (2)*((sqrt(3))*(Mapy2z2x2)))} ,
{2,-1, (1/2*I)*((sqrt(6))*(Mappxzxy) + (-2)*((sqrt(2))*(Mapx2yz)))} ,
{2, 1, (1/2*I)*((sqrt(6))*(Mappxzxy) + (-2)*((sqrt(2))*(Mapx2yz)))} ,
{2,-2, (1/4)*((sqrt(6))*(Eappxz) + (sqrt(6))*(Eapx2) + (-1)*((sqrt(6))*(Eapy2z2)) + (-1)*((sqrt(6))*(Eapyz)) + (2)*((sqrt(2))*(Mapy2z2x2)))} ,
{2, 2, (1/4)*((sqrt(6))*(Eappxz) + (sqrt(6))*(Eapx2) + (-1)*((sqrt(6))*(Eapy2z2)) + (-1)*((sqrt(6))*(Eapyz)) + (2)*((sqrt(2))*(Mapy2z2x2)))} ,
{4, 0, (3/40)*((4)*(Eappxy) + (-16)*(Eappxz) + (9)*(Eapx2) + (19)*(Eapy2z2) + (-16)*(Eapyz) + (10)*((sqrt(3))*(Mapy2z2x2)))} ,
{4,-1, (-3/4*I)*((1/(sqrt(5)))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (7)*(Mapy2z2yz)))} ,
{4, 1, (-3/4*I)*((1/(sqrt(5)))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (7)*(Mapy2z2yz)))} ,
{4,-2, (3/4)*((1/(sqrt(10)))*((4)*(Eappxz) + (-3)*(Eapx2) + (3)*(Eapy2z2) + (-4)*(Eapyz) + (-2)*((sqrt(3))*(Mapy2z2x2))))} ,
{4, 2, (3/4)*((1/(sqrt(10)))*((4)*(Eappxz) + (-3)*(Eapx2) + (3)*(Eapy2z2) + (-4)*(Eapyz) + (-2)*((sqrt(3))*(Mapy2z2x2))))} ,
{4,-3, (3/4*I)*((sqrt(7/5))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (-1)*(Mapy2z2yz)))} ,
{4, 3, (3/4*I)*((sqrt(7/5))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (-1)*(Mapy2z2yz)))} ,
{4,-4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapx2) + (-1)*(Eapy2z2) + (2)*((sqrt(3))*(Mapy2z2x2))))} ,
{4, 4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapx2) + (-1)*(Eapy2z2) + (2)*((sqrt(3))*(Mapy2z2x2))))} }
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ {Y_{-2}^{(2)}} $|$ \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $|$ \frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $|$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $|$ -\frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $|$ \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $|
^$ {Y_{-1}^{(2)}} $|$ -\frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $|$ \frac{\text{Eappxz}+\text{Eapyz}}{2} $|$ -\frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $|$ \frac{\text{Eapyz}-\text{Eappxz}}{2} $|$ \frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $|
^$ {Y_{0}^{(2)}} $|$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $|$ \frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $|$ \frac{1}{4} \left(\text{Eapx2}+3 \text{Eapy2z2}+2 \sqrt{3} \text{Mapy2z2x2}\right) $|$ \frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $|$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $|
^$ {Y_{1}^{(2)}} $|$ \frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $|$ \frac{\text{Eapyz}-\text{Eappxz}}{2} $|$ -\frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $|$ \frac{\text{Eappxz}+\text{Eapyz}}{2} $|$ -\frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $|
^$ {Y_{2}^{(2)}} $|$ \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $|$ -\frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $|$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $|$ \frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $|$ \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $|
###
###
| $ $ ^ $ d_{y^2-z^2} $ ^ $ d_{3x^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^
^$ d_{y^2-z^2} $|$ \text{Eapy2z2} $|$ \text{Mapy2z2x2} $|$ 0 $|$ 0 $|$ \text{Mapy2z2yz} $|
^$ d_{3x^2-r^2} $|$ \text{Mapy2z2x2} $|$ \text{Eapx2} $|$ 0 $|$ 0 $|$ \text{Mapx2yz} $|
^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ \text{Eappxz} $|$ \text{Mappxzxy} $|$ 0 $|
^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ \text{Mappxzxy} $|$ \text{Eappxy} $|$ 0 $|
^$ d_{\text{yz}} $|$ \text{Mapy2z2yz} $|$ \text{Mapx2yz} $|$ 0 $|$ 0 $|$ \text{Eapyz} $|
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ d_{y^2-z^2} $|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|$ -\frac{\sqrt{3}}{2} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|
^$ d_{3x^2-r^2} $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|$ -\frac{1}{2} $|$ 0 $|$ \frac{\sqrt{\frac{3}{2}}}{2} $|
^$ 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}} $|
^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|
###
###
^ ^$$\text{Eapy2z2}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{15}{\pi }} \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{15}{\pi }} \left(-x^2+y^2-3 z^2+1\right)$$ | ::: |
^ ^$$\text{Eapx2}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{5}{\pi }} \left(-6 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} \left(3 x^2-3 y^2-3 z^2+1\right)$$ | ::: |
^ ^$$\text{Eappxz}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
^ ^$$\text{Eappxy}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |
^ ^$$\text{Eapyz}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
###
==== Potential for f orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{7} (\text{Eappx3}+\text{Eappxy2z2}+\text{Eappxyz}+\text{Eapy3}+\text{Eapyz2x2}+\text{Eapz3}+\text{Eapzx2y2}) & k=0\land m=0 \\
0 & (k\neq 2\land k\neq 4\land k\neq 6)\lor (k\neq 4\land k\neq 6\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (k\neq 6\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4)\lor (m\neq -6\land m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5\land m\neq 6) \\
\frac{5}{28} \left(\sqrt{6} \text{Eappx3}-\sqrt{6} \text{Eapy3}+\sqrt{10} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2}-2 \text{Mapz3zx2y2})\right) & k=2\land (m=-2\lor m=2) \\
-\frac{5}{56} i \left(4 \sqrt{10} \text{Mappxyzx3}+\sqrt{6} \text{Mapy3z3}+\sqrt{10} \text{Mapy3zx2y2}+5 \sqrt{6} \text{Mapyz2x2zx2y2}-\sqrt{10} \text{Mapz3yz2x2}\right) & k=2\land (m=-1\lor m=1) \\
-\frac{5}{14} \left(\text{Eappx3}+\text{Eapy3}-2 \text{Eapz3}-\sqrt{15} \text{Mappx3xy2z2}+\sqrt{15} \text{Mapy3yz2x2}\right) & k=2\land m=0 \\
\frac{3 \left(3 \sqrt{5} \text{Eappx3}-3 \sqrt{5} \text{Eappxy2z2}-4 \sqrt{5} \text{Eappxyz}+3 \sqrt{5} \text{Eapy3}-3 \sqrt{5} \text{Eapyz2x2}+4 \sqrt{5} \text{Eapzx2y2}+2 \sqrt{3} \text{Mappx3xy2z2}-2 \sqrt{3} \text{Mapy3yz2x2}\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\
-\frac{3 i \left(\sqrt{3} \text{Mappxyzx3}+\sqrt{5} \text{Mappxyzxy2z2}-3 \sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapy3zx2y2}-\sqrt{5} \text{Mapyz2x2zx2y2}-3 \sqrt{3} \text{Mapz3yz2x2}\right)}{4 \sqrt{7}} & k=4\land (m=-3\lor m=3) \\
-\frac{3}{56} \left(3 \sqrt{10} \text{Eappx3}-7 \sqrt{10} \text{Eappxy2z2}-3 \sqrt{10} \text{Eapy3}+7 \sqrt{10} \text{Eapyz2x2}-2 \sqrt{6} \text{Mappx3xy2z2}-2 \sqrt{6} \text{Mapy3yz2x2}+4 \sqrt{6} \text{Mapz3zx2y2}\right) & k=4\land (m=-2\lor m=2) \\
\frac{3}{28} i \left(\sqrt{3} \text{Mappxyzx3}-7 \sqrt{5} \text{Mappxyzxy2z2}-3 \sqrt{5} \text{Mapy3z3}+9 \sqrt{3} \text{Mapy3zx2y2}-\sqrt{5} \text{Mapyz2x2zx2y2}+5 \sqrt{3} \text{Mapz3yz2x2}\right) & k=4\land (m=-1\lor m=1) \\
\frac{3}{56} \left(9 \text{Eappx3}+7 \text{Eappxy2z2}-28 \text{Eappxyz}+9 \text{Eapy3}+7 \text{Eapyz2x2}+24 \text{Eapz3}-28 \text{Eapzx2y2}-2 \sqrt{15} \text{Mappx3xy2z2}+2 \sqrt{15} \text{Mapy3yz2x2}\right) & k=4\land m=0 \\
\frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappx3}+3 \sqrt{3} \text{Eappxy2z2}-5 \sqrt{3} \text{Eapy3}-3 \sqrt{3} \text{Eapyz2x2}-6 \sqrt{5} \text{Mappx3xy2z2}-6 \sqrt{5} \text{Mapy3yz2x2}\right) & k=6\land (m=-6\lor m=6) \\
\frac{13}{40} i \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right) & k=6\land (m=-5\lor m=5) \\
-\frac{13 \left(15 \text{Eappx3}-15 \text{Eappxy2z2}+24 \text{Eappxyz}+15 \text{Eapy3}-15 \text{Eapyz2x2}-24 \text{Eapzx2y2}+2 \sqrt{15} \text{Mappx3xy2z2}-2 \sqrt{15} \text{Mapy3yz2x2}\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\
-\frac{13 i \left(9 \text{Mappxyzx3}+3 \sqrt{15} \text{Mappxyzxy2z2}+2 \sqrt{15} \text{Mapy3z3}+9 \text{Mapy3zx2y2}-3 \sqrt{15} \text{Mapyz2x2zx2y2}+6 \text{Mapz3yz2x2}\right)}{40 \sqrt{7}} & k=6\land (m=-3\lor m=3) \\
\frac{13 \left(5 \sqrt{15} \text{Eappx3}+3 \sqrt{15} \text{Eappxy2z2}-5 \sqrt{15} \text{Eapy3}-3 \sqrt{15} \text{Eapyz2x2}+34 \text{Mappx3xy2z2}+34 \text{Mapy3yz2x2}+64 \text{Mapz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\
\frac{13}{280} i \left(\sqrt{70} \text{Mappxyzx3}+3 \sqrt{42} \text{Mappxyzxy2z2}-5 \sqrt{42} \text{Mapy3z3}-2 \sqrt{70} \text{Mapy3zx2y2}+2 \sqrt{42} \text{Mapyz2x2zx2y2}+5 \sqrt{70} \text{Mapz3yz2x2}\right) & k=6\land (m=-1\lor m=1) \\
-\frac{13}{560} \left(25 \text{Eappx3}+39 \text{Eappxy2z2}-24 \text{Eappxyz}+25 \text{Eapy3}+39 \text{Eapyz2x2}-80 \text{Eapz3}-24 \text{Eapzx2y2}+14 \sqrt{15} \text{Mappx3xy2z2}-14 \sqrt{15} \text{Mapy3yz2x2}\right) & \text{True}
\end{cases}$$
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Akm[k_,m_]:=Piecewise[{{(Eappx3 + Eappxy2z2 + Eappxyz + Eapy3 + Eapyz2x2 + Eapz3 + Eapzx2y2)/7, k == 0 && m == 0}, {0, (k != 2 && k != 4 && k != 6) || (k != 4 && k != 6 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (k != 6 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4) || (m != -6 && m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5 && m != 6)}, {(5*(Sqrt[6]*Eappx3 - Sqrt[6]*Eapy3 + Sqrt[10]*(Mappx3xy2z2 + Mapy3yz2x2 - 2*Mapz3zx2y2)))/28, k == 2 && (m == -2 || m == 2)}, {((-5*I)/56)*(4*Sqrt[10]*Mappxyzx3 + Sqrt[6]*Mapy3z3 + Sqrt[10]*Mapy3zx2y2 + 5*Sqrt[6]*Mapyz2x2zx2y2 - Sqrt[10]*Mapz3yz2x2), k == 2 && (m == -1 || m == 1)}, {(-5*(Eappx3 + Eapy3 - 2*Eapz3 - Sqrt[15]*Mappx3xy2z2 + Sqrt[15]*Mapy3yz2x2))/14, k == 2 && m == 0}, {(3*(3*Sqrt[5]*Eappx3 - 3*Sqrt[5]*Eappxy2z2 - 4*Sqrt[5]*Eappxyz + 3*Sqrt[5]*Eapy3 - 3*Sqrt[5]*Eapyz2x2 + 4*Sqrt[5]*Eapzx2y2 + 2*Sqrt[3]*Mappx3xy2z2 - 2*Sqrt[3]*Mapy3yz2x2))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(((-3*I)/4)*(Sqrt[3]*Mappxyzx3 + Sqrt[5]*Mappxyzxy2z2 - 3*Sqrt[5]*Mapy3z3 + Sqrt[3]*Mapy3zx2y2 - Sqrt[5]*Mapyz2x2zx2y2 - 3*Sqrt[3]*Mapz3yz2x2))/Sqrt[7], k == 4 && (m == -3 || m == 3)}, {(-3*(3*Sqrt[10]*Eappx3 - 7*Sqrt[10]*Eappxy2z2 - 3*Sqrt[10]*Eapy3 + 7*Sqrt[10]*Eapyz2x2 - 2*Sqrt[6]*Mappx3xy2z2 - 2*Sqrt[6]*Mapy3yz2x2 + 4*Sqrt[6]*Mapz3zx2y2))/56, k == 4 && (m == -2 || m == 2)}, {((3*I)/28)*(Sqrt[3]*Mappxyzx3 - 7*Sqrt[5]*Mappxyzxy2z2 - 3*Sqrt[5]*Mapy3z3 + 9*Sqrt[3]*Mapy3zx2y2 - Sqrt[5]*Mapyz2x2zx2y2 + 5*Sqrt[3]*Mapz3yz2x2), k == 4 && (m == -1 || m == 1)}, {(3*(9*Eappx3 + 7*Eappxy2z2 - 28*Eappxyz + 9*Eapy3 + 7*Eapyz2x2 + 24*Eapz3 - 28*Eapzx2y2 - 2*Sqrt[15]*Mappx3xy2z2 + 2*Sqrt[15]*Mapy3yz2x2))/56, k == 4 && m == 0}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eappx3 + 3*Sqrt[3]*Eappxy2z2 - 5*Sqrt[3]*Eapy3 - 3*Sqrt[3]*Eapyz2x2 - 6*Sqrt[5]*Mappx3xy2z2 - 6*Sqrt[5]*Mapy3yz2x2))/160, k == 6 && (m == -6 || m == 6)}, {((13*I)/40)*Sqrt[11/7]*(Sqrt[15]*Mappxyzx3 - 3*Mappxyzxy2z2 - Sqrt[15]*Mapy3zx2y2 - 3*Mapyz2x2zx2y2), k == 6 && (m == -5 || m == 5)}, {(-13*(15*Eappx3 - 15*Eappxy2z2 + 24*Eappxyz + 15*Eapy3 - 15*Eapyz2x2 - 24*Eapzx2y2 + 2*Sqrt[15]*Mappx3xy2z2 - 2*Sqrt[15]*Mapy3yz2x2))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(((-13*I)/40)*(9*Mappxyzx3 + 3*Sqrt[15]*Mappxyzxy2z2 + 2*Sqrt[15]*Mapy3z3 + 9*Mapy3zx2y2 - 3*Sqrt[15]*Mapyz2x2zx2y2 + 6*Mapz3yz2x2))/Sqrt[7], k == 6 && (m == -3 || m == 3)}, {(13*(5*Sqrt[15]*Eappx3 + 3*Sqrt[15]*Eappxy2z2 - 5*Sqrt[15]*Eapy3 - 3*Sqrt[15]*Eapyz2x2 + 34*Mappx3xy2z2 + 34*Mapy3yz2x2 + 64*Mapz3zx2y2))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {((13*I)/280)*(Sqrt[70]*Mappxyzx3 + 3*Sqrt[42]*Mappxyzxy2z2 - 5*Sqrt[42]*Mapy3z3 - 2*Sqrt[70]*Mapy3zx2y2 + 2*Sqrt[42]*Mapyz2x2zx2y2 + 5*Sqrt[70]*Mapz3yz2x2), k == 6 && (m == -1 || m == 1)}}, (-13*(25*Eappx3 + 39*Eappxy2z2 - 24*Eappxyz + 25*Eapy3 + 39*Eapyz2x2 - 80*Eapz3 - 24*Eapzx2y2 + 14*Sqrt[15]*Mappx3xy2z2 - 14*Sqrt[15]*Mapy3yz2x2))/560]
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Akm = {{0, 0, (1/7)*(Eappx3 + Eappxy2z2 + Eappxyz + Eapy3 + Eapyz2x2 + Eapz3 + Eapzx2y2)} ,
{2, 0, (-5/14)*(Eappx3 + Eapy3 + (-2)*(Eapz3) + (-1)*((sqrt(15))*(Mappx3xy2z2)) + (sqrt(15))*(Mapy3yz2x2))} ,
{2,-1, (-5/56*I)*((4)*((sqrt(10))*(Mappxyzx3)) + (sqrt(6))*(Mapy3z3) + (sqrt(10))*(Mapy3zx2y2) + (5)*((sqrt(6))*(Mapyz2x2zx2y2)) + (-1)*((sqrt(10))*(Mapz3yz2x2)))} ,
{2, 1, (-5/56*I)*((4)*((sqrt(10))*(Mappxyzx3)) + (sqrt(6))*(Mapy3z3) + (sqrt(10))*(Mapy3zx2y2) + (5)*((sqrt(6))*(Mapyz2x2zx2y2)) + (-1)*((sqrt(10))*(Mapz3yz2x2)))} ,
{2,-2, (5/28)*((sqrt(6))*(Eappx3) + (-1)*((sqrt(6))*(Eapy3)) + (sqrt(10))*(Mappx3xy2z2 + Mapy3yz2x2 + (-2)*(Mapz3zx2y2)))} ,
{2, 2, (5/28)*((sqrt(6))*(Eappx3) + (-1)*((sqrt(6))*(Eapy3)) + (sqrt(10))*(Mappx3xy2z2 + Mapy3yz2x2 + (-2)*(Mapz3zx2y2)))} ,
{4, 0, (3/56)*((9)*(Eappx3) + (7)*(Eappxy2z2) + (-28)*(Eappxyz) + (9)*(Eapy3) + (7)*(Eapyz2x2) + (24)*(Eapz3) + (-28)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappx3xy2z2)) + (2)*((sqrt(15))*(Mapy3yz2x2)))} ,
{4,-1, (3/28*I)*((sqrt(3))*(Mappxyzx3) + (-7)*((sqrt(5))*(Mappxyzxy2z2)) + (-3)*((sqrt(5))*(Mapy3z3)) + (9)*((sqrt(3))*(Mapy3zx2y2)) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (5)*((sqrt(3))*(Mapz3yz2x2)))} ,
{4, 1, (3/28*I)*((sqrt(3))*(Mappxyzx3) + (-7)*((sqrt(5))*(Mappxyzxy2z2)) + (-3)*((sqrt(5))*(Mapy3z3)) + (9)*((sqrt(3))*(Mapy3zx2y2)) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (5)*((sqrt(3))*(Mapz3yz2x2)))} ,
{4,-2, (-3/56)*((3)*((sqrt(10))*(Eappx3)) + (-7)*((sqrt(10))*(Eappxy2z2)) + (-3)*((sqrt(10))*(Eapy3)) + (7)*((sqrt(10))*(Eapyz2x2)) + (-2)*((sqrt(6))*(Mappx3xy2z2)) + (-2)*((sqrt(6))*(Mapy3yz2x2)) + (4)*((sqrt(6))*(Mapz3zx2y2)))} ,
{4, 2, (-3/56)*((3)*((sqrt(10))*(Eappx3)) + (-7)*((sqrt(10))*(Eappxy2z2)) + (-3)*((sqrt(10))*(Eapy3)) + (7)*((sqrt(10))*(Eapyz2x2)) + (-2)*((sqrt(6))*(Mappx3xy2z2)) + (-2)*((sqrt(6))*(Mapy3yz2x2)) + (4)*((sqrt(6))*(Mapz3zx2y2)))} ,
{4,-3, (-3/4*I)*((1/(sqrt(7)))*((sqrt(3))*(Mappxyzx3) + (sqrt(5))*(Mappxyzxy2z2) + (-3)*((sqrt(5))*(Mapy3z3)) + (sqrt(3))*(Mapy3zx2y2) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (-3)*((sqrt(3))*(Mapz3yz2x2))))} ,
{4, 3, (-3/4*I)*((1/(sqrt(7)))*((sqrt(3))*(Mappxyzx3) + (sqrt(5))*(Mappxyzxy2z2) + (-3)*((sqrt(5))*(Mapy3z3)) + (sqrt(3))*(Mapy3zx2y2) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (-3)*((sqrt(3))*(Mapz3yz2x2))))} ,
{4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eapy3)) + (-3)*((sqrt(5))*(Eapyz2x2)) + (4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (-2)*((sqrt(3))*(Mapy3yz2x2))))} ,
{4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eapy3)) + (-3)*((sqrt(5))*(Eapyz2x2)) + (4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (-2)*((sqrt(3))*(Mapy3yz2x2))))} ,
{6, 0, (-13/560)*((25)*(Eappx3) + (39)*(Eappxy2z2) + (-24)*(Eappxyz) + (25)*(Eapy3) + (39)*(Eapyz2x2) + (-80)*(Eapz3) + (-24)*(Eapzx2y2) + (14)*((sqrt(15))*(Mappx3xy2z2)) + (-14)*((sqrt(15))*(Mapy3yz2x2)))} ,
{6,-1, (13/280*I)*((sqrt(70))*(Mappxyzx3) + (3)*((sqrt(42))*(Mappxyzxy2z2)) + (-5)*((sqrt(42))*(Mapy3z3)) + (-2)*((sqrt(70))*(Mapy3zx2y2)) + (2)*((sqrt(42))*(Mapyz2x2zx2y2)) + (5)*((sqrt(70))*(Mapz3yz2x2)))} ,
{6, 1, (13/280*I)*((sqrt(70))*(Mappxyzx3) + (3)*((sqrt(42))*(Mappxyzxy2z2)) + (-5)*((sqrt(42))*(Mapy3z3)) + (-2)*((sqrt(70))*(Mapy3zx2y2)) + (2)*((sqrt(42))*(Mapyz2x2zx2y2)) + (5)*((sqrt(70))*(Mapz3yz2x2)))} ,
{6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eapy3)) + (-3)*((sqrt(15))*(Eapyz2x2)) + (34)*(Mappx3xy2z2) + (34)*(Mapy3yz2x2) + (64)*(Mapz3zx2y2)))} ,
{6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eapy3)) + (-3)*((sqrt(15))*(Eapyz2x2)) + (34)*(Mappx3xy2z2) + (34)*(Mapy3yz2x2) + (64)*(Mapz3zx2y2)))} ,
{6,-3, (-13/40*I)*((1/(sqrt(7)))*((9)*(Mappxyzx3) + (3)*((sqrt(15))*(Mappxyzxy2z2)) + (2)*((sqrt(15))*(Mapy3z3)) + (9)*(Mapy3zx2y2) + (-3)*((sqrt(15))*(Mapyz2x2zx2y2)) + (6)*(Mapz3yz2x2)))} ,
{6, 3, (-13/40*I)*((1/(sqrt(7)))*((9)*(Mappxyzx3) + (3)*((sqrt(15))*(Mappxyzxy2z2)) + (2)*((sqrt(15))*(Mapy3z3)) + (9)*(Mapy3zx2y2) + (-3)*((sqrt(15))*(Mapyz2x2zx2y2)) + (6)*(Mapz3yz2x2)))} ,
{6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eapy3) + (-15)*(Eapyz2x2) + (-24)*(Eapzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (-2)*((sqrt(15))*(Mapy3yz2x2))))} ,
{6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eapy3) + (-15)*(Eapyz2x2) + (-24)*(Eapzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (-2)*((sqrt(15))*(Mapy3yz2x2))))} ,
{6,-5, (13/40*I)*((sqrt(11/7))*((sqrt(15))*(Mappxyzx3) + (-3)*(Mappxyzxy2z2) + (-1)*((sqrt(15))*(Mapy3zx2y2)) + (-3)*(Mapyz2x2zx2y2)))} ,
{6, 5, (13/40*I)*((sqrt(11/7))*((sqrt(15))*(Mappxyzx3) + (-3)*(Mappxyzxy2z2) + (-1)*((sqrt(15))*(Mapy3zx2y2)) + (-3)*(Mapyz2x2zx2y2)))} ,
{6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eapy3)) + (-3)*((sqrt(3))*(Eapyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (-6)*((sqrt(5))*(Mapy3yz2x2))))} ,
{6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eapy3)) + (-3)*((sqrt(3))*(Eapyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (-6)*((sqrt(5))*(Mapy3yz2x2))))} }
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| $ $ ^ $ {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}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mapy3yz2x2}-\text{Mappx3xy2z2})\right) $|$ -\frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|$ -\frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $|$ \frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $|$ \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|
^$ {Y_{-2}^{(3)}} $|$ \frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $|$ \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} $|$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} $|$ -\frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $|
^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mapy3yz2x2})\right) $|$ -\frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $|$ \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}-2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $|
^$ {Y_{0}^{(3)}} $|$ \frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $|$ \text{Eapz3} $|$ \frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $|
^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $|$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}-2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|$ -\frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $|$ \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mapy3yz2x2})\right) $|$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|
^$ {Y_{2}^{(3)}} $|$ -\frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $|$ \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} $|$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $|$ \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} $|$ \frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $|
^$ {Y_{3}^{(3)}} $|$ \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|$ \frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $|$ -\frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $|$ -\frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $|$ \frac{1}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mapy3yz2x2}-\text{Mappx3xy2z2})\right) $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^
^$ f_{\text{xyz}} $|$ \text{Eappxyz} $|$ 0 $|$ 0 $|$ \text{Mappxyzx3} $|$ 0 $|$ 0 $|$ \text{Mappxyzxy2z2} $|
^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ \text{Eapy3} $|$ \text{Mapy3z3} $|$ 0 $|$ \text{Mapy3yz2x2} $|$ \text{Mapy3zx2y2} $|$ 0 $|
^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ \text{Mapy3z3} $|$ \text{Eapz3} $|$ 0 $|$ \text{Mapz3yz2x2} $|$ \text{Mapz3zx2y2} $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$ \text{Mappxyzx3} $|$ 0 $|$ 0 $|$ \text{Eappx3} $|$ 0 $|$ 0 $|$ \text{Mappx3xy2z2} $|
^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ \text{Mapy3yz2x2} $|$ \text{Mapz3yz2x2} $|$ 0 $|$ \text{Eapyz2x2} $|$ \text{Mapyz2x2zx2y2} $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \text{Mapy3zx2y2} $|$ \text{Mapz3zx2y2} $|$ 0 $|$ \text{Mapyz2x2zx2y2} $|$ \text{Eapzx2y2} $|$ 0 $|
^$ f_{x\left(y^2-z^2\right)} $|$ \text{Mappxyzxy2z2} $|$ 0 $|$ 0 $|$ \text{Mappx3xy2z2} $|$ 0 $|$ 0 $|$ \text{Eappxy2z2} $|
###
###
| $ $ ^ $ {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_{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(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(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 $|
^$ 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} $|
###
###
^ ^$$\text{Eappxyz}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
^ ^$$\text{Eapy3}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: |
^ ^$$\text{Eapz3}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
^ ^$$\text{Eappx3}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: |
^ ^$$\text{Eapyz2x2}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: |
^ ^$$\text{Eapzx2y2}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |
^ ^$$\text{Eappxy2z2}$$ | {{:physics_chemistry:pointgroup:cs_x_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: |
###
===== Coupling between two shells =====
###
Click on one of the subsections to expand it or
###
==== Potential for s-p orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 1\lor (m\neq -1\land m\neq 0\land m\neq 1) \\
i B(1,1) & k=1\land (m=-1\lor m=1) \\
A(1,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 1 || (m != -1 && m != 0 && m != 1)}, {I*B[1, 1], k == 1 && (m == -1 || m == 1)}}, A[1, 0]]
###
###
Akm = {{1, 0, A(1,0)} ,
{1,-1, (I)*(B(1,1))} ,
{1, 1, (I)*(B(1,1))} }
###
###
| $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^
^$ {Y_{0}^{(0)}} $|$ -\frac{i B(1,1)}{\sqrt{3}} $|$ \frac{A(1,0)}{\sqrt{3}} $|$ -\frac{i B(1,1)}{\sqrt{3}} $|
###
###
| $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^
^$ \text{s} $|$ \sqrt{\frac{2}{3}} B(1,1) $|$ \frac{A(1,0)}{\sqrt{3}} $|$ 0 $|
###
==== Potential for s-d orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\
A(2,2) & k=2\land (m=-2\lor m=2) \\
i B(2,1) & k=2\land (m=-1\lor m=1) \\
A(2,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}}, A[2, 0]]
###
###
Akm = {{2, 0, A(2,0)} ,
{2,-1, (I)*(B(2,1))} ,
{2, 1, (I)*(B(2,1))} ,
{2,-2, A(2,2)} ,
{2, 2, A(2,2)} }
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ {Y_{0}^{(0)}} $|$ \frac{A(2,2)}{\sqrt{5}} $|$ -\frac{i B(2,1)}{\sqrt{5}} $|$ \frac{A(2,0)}{\sqrt{5}} $|$ -\frac{i B(2,1)}{\sqrt{5}} $|$ \frac{A(2,2)}{\sqrt{5}} $|
###
###
| $ $ ^ $ d_{y^2-z^2} $ ^ $ d_{3x^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^
^$ \text{s} $|$ -\frac{1}{2} \sqrt{\frac{3}{5}} A(2,0)-\frac{A(2,2)}{\sqrt{10}} $|$ -\frac{A(2,0)-\sqrt{6} A(2,2)}{2 \sqrt{5}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{5}} B(2,1) $|
###
==== Potential for s-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 3\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3) \\
i B(3,3) & k=3\land (m=-3\lor m=3) \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
i B(3,1) & k=3\land (m=-1\lor m=1) \\
A(3,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3)}, {I*B[3, 3], k == 3 && (m == -3 || m == 3)}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {I*B[3, 1], k == 3 && (m == -1 || m == 1)}}, A[3, 0]]
###
###
Akm = {{3, 0, A(3,0)} ,
{3,-1, (I)*(B(3,1))} ,
{3, 1, (I)*(B(3,1))} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} ,
{3,-3, (I)*(B(3,3))} ,
{3, 3, (I)*(B(3,3))} }
###
###
| $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ {Y_{0}^{(0)}} $|$ -\frac{i B(3,3)}{\sqrt{7}} $|$ \frac{A(3,2)}{\sqrt{7}} $|$ -\frac{i B(3,1)}{\sqrt{7}} $|$ \frac{A(3,0)}{\sqrt{7}} $|$ -\frac{i B(3,1)}{\sqrt{7}} $|$ \frac{A(3,2)}{\sqrt{7}} $|$ -\frac{i B(3,3)}{\sqrt{7}} $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^
^$ \text{s} $|$ 0 $|$ \frac{1}{14} \left(-\sqrt{21} B(3,1)-\sqrt{35} B(3,3)\right) $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ \frac{1}{14} \left(\sqrt{35} B(3,1)-\sqrt{21} B(3,3)\right) $|$ \sqrt{\frac{2}{7}} A(3,2) $|$ 0 $|
###
==== Potential for p-d orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 3\land (k\neq 1\lor (m\neq -1\land m\neq 0\land m\neq 1)))\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3) \\
i B(1,1) & k=1\land (m=-1\lor m=1) \\
A(1,0) & k=1\land m=0 \\
i B(3,3) & k=3\land (m=-3\lor m=3) \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
i B(3,1) & k=3\land (m=-1\lor m=1) \\
A(3,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || (m != -1 && m != 0 && m != 1))) || (m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3)}, {I*B[1, 1], k == 1 && (m == -1 || m == 1)}, {A[1, 0], k == 1 && m == 0}, {I*B[3, 3], k == 3 && (m == -3 || m == 3)}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {I*B[3, 1], k == 3 && (m == -1 || m == 1)}}, A[3, 0]]
###
###
Akm = {{1, 0, A(1,0)} ,
{1,-1, (I)*(B(1,1))} ,
{1, 1, (I)*(B(1,1))} ,
{3, 0, A(3,0)} ,
{3,-1, (I)*(B(3,1))} ,
{3, 1, (I)*(B(3,1))} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} ,
{3,-3, (I)*(B(3,3))} ,
{3, 3, (I)*(B(3,3))} }
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ {Y_{-1}^{(1)}} $|$ -\frac{1}{35} i \left(7 \sqrt{10} B(1,1)-\sqrt{15} B(3,1)\right) $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ \frac{3}{7} i \sqrt{\frac{2}{5}} B(3,1)-\frac{i B(1,1)}{\sqrt{15}} $|$ -\frac{1}{7} \sqrt{6} A(3,2) $|$ \frac{3}{7} i B(3,3) $|
^$ {Y_{0}^{(1)}} $|$ \frac{1}{7} \sqrt{3} A(3,2) $|$ -\frac{i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ -\frac{i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} $|$ \frac{1}{7} \sqrt{3} A(3,2) $|
^$ {Y_{1}^{(1)}} $|$ \frac{3}{7} i B(3,3) $|$ -\frac{1}{7} \sqrt{6} A(3,2) $|$ \frac{3}{7} i \sqrt{\frac{2}{5}} B(3,1)-\frac{i B(1,1)}{\sqrt{15}} $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ -\frac{1}{35} i \left(7 \sqrt{10} B(1,1)-\sqrt{15} B(3,1)\right) $|
###
###
| $ $ ^ $ d_{y^2-z^2} $ ^ $ d_{3x^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^
^$ p_y $|$ \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} B(3,1)-\frac{3}{14} B(3,3) $|$ \frac{1}{210} \left(-14 \sqrt{30} B(1,1)-9 \sqrt{5} B(3,1)+45 \sqrt{3} B(3,3)\right) $|$ 0 $|$ 0 $|$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) $|
^$ p_z $|$ \frac{1}{70} \left(-\sqrt{5} (14 A(1,0)+9 A(3,0))-5 \sqrt{6} A(3,2)\right) $|$ \frac{3 A(3,2)}{7 \sqrt{2}}-\frac{14 A(1,0)+9 A(3,0)}{14 \sqrt{15}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{5}} B(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} B(3,1) $|
^$ p_x $|$ 0 $|$ 0 $|$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right) $|$ \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} B(3,1)+\frac{3}{7} B(3,3) $|$ 0 $|
###
==== Potential for p-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\
A(2,2) & k=2\land (m=-2\lor m=2) \\
i B(2,1) & k=2\land (m=-1\lor m=1) \\
A(2,0) & k=2\land m=0 \\
A(4,4) & k=4\land (m=-4\lor m=4) \\
i B(4,3) & k=4\land (m=-3\lor m=3) \\
A(4,2) & k=4\land (m=-2\lor m=2) \\
i B(4,1) & k=4\land (m=-1\lor m=1) \\
A(4,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {I*B[4, 1], k == 4 && (m == -1 || m == 1)}}, A[4, 0]]
###
###
Akm = {{2, 0, A(2,0)} ,
{2,-1, (I)*(B(2,1))} ,
{2, 1, (I)*(B(2,1))} ,
{2,-2, A(2,2)} ,
{2, 2, A(2,2)} ,
{4, 0, A(4,0)} ,
{4,-1, (I)*(B(4,1))} ,
{4, 1, (I)*(B(4,1))} ,
{4,-2, A(4,2)} ,
{4, 2, A(4,2)} ,
{4,-3, (I)*(B(4,3))} ,
{4, 3, (I)*(B(4,3))} ,
{4,-4, A(4,4)} ,
{4, 4, A(4,4)} }
###
###
| $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ {Y_{-1}^{(1)}} $|$ \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $|$ -\frac{i \left(3 \sqrt{30} B(2,1)-5 B(4,1)\right)}{15 \sqrt{7}} $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ -\frac{i \left(27 B(2,1)-5 \sqrt{30} B(4,1)\right)}{45 \sqrt{7}} $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ \frac{1}{3} i B(4,3) $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|
^$ {Y_{0}^{(1)}} $|$ -\frac{i B(4,3)}{3 \sqrt{3}} $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ -\frac{1}{105} i \left(6 \sqrt{42} B(2,1)+5 \sqrt{35} B(4,1)\right) $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ -\frac{1}{105} i \left(6 \sqrt{42} B(2,1)+5 \sqrt{35} B(4,1)\right) $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ -\frac{i B(4,3)}{3 \sqrt{3}} $|
^$ {Y_{1}^{(1)}} $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|$ \frac{1}{3} i B(4,3) $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ -\frac{i \left(27 B(2,1)-5 \sqrt{30} B(4,1)\right)}{45 \sqrt{7}} $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ -\frac{i \left(3 \sqrt{30} B(2,1)-5 B(4,1)\right)}{15 \sqrt{7}} $|$ \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^
^$ p_y $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ \frac{1}{45} \sqrt{\frac{2}{7}} \left(5 \sqrt{30} B(4,1)-27 B(2,1)\right) $|$ 0 $|$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ \frac{1}{21} \left(\sqrt{7} B(4,1)+7 B(4,3)\right)-\sqrt{\frac{6}{35}} B(2,1) $|$ 0 $|
^$ p_z $|$ 0 $|$ \frac{1}{630} \left(-54 \sqrt{14} B(2,1)-5 \sqrt{15} \left(3 \sqrt{7} B(4,1)+7 B(4,3)\right)\right) $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{6}{35}} B(2,1)+\frac{5 B(4,1)}{6 \sqrt{7}}-\frac{1}{6} B(4,3) $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $|$ 0 $|
^$ p_x $|$ \sqrt{\frac{6}{35}} B(2,1)-\frac{B(4,1)}{3 \sqrt{7}}+\frac{1}{3} B(4,3) $|$ 0 $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ 0 $|$ 0 $|$ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|
###
==== Potential for d-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 1\land k\neq 3\land k\neq 5)\lor (k\neq 3\land k\neq 5\land m\neq -1\land m\neq 0\land m\neq 1)\lor (k\neq 5\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3)\lor (m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5) \\
i B(1,1) & k=1\land (m=-1\lor m=1) \\
A(1,0) & k=1\land m=0 \\
i B(3,3) & k=3\land (m=-3\lor m=3) \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
i B(3,1) & k=3\land (m=-1\lor m=1) \\
A(3,0) & k=3\land m=0 \\
i B(5,5) & k=5\land (m=-5\lor m=5) \\
A(5,4) & k=5\land (m=-4\lor m=4) \\
i B(5,3) & k=5\land (m=-3\lor m=3) \\
A(5,2) & k=5\land (m=-2\lor m=2) \\
i B(5,1) & k=5\land (m=-1\lor m=1) \\
A(5,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, (k != 1 && k != 3 && k != 5) || (k != 3 && k != 5 && m != -1 && m != 0 && m != 1) || (k != 5 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3) || (m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5)}, {I*B[1, 1], k == 1 && (m == -1 || m == 1)}, {A[1, 0], k == 1 && m == 0}, {I*B[3, 3], k == 3 && (m == -3 || m == 3)}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {I*B[3, 1], k == 3 && (m == -1 || m == 1)}, {A[3, 0], k == 3 && m == 0}, {I*B[5, 5], k == 5 && (m == -5 || m == 5)}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {I*B[5, 3], k == 5 && (m == -3 || m == 3)}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {I*B[5, 1], k == 5 && (m == -1 || m == 1)}}, A[5, 0]]
###
###
Akm = {{1, 0, A(1,0)} ,
{1,-1, (I)*(B(1,1))} ,
{1, 1, (I)*(B(1,1))} ,
{3, 0, A(3,0)} ,
{3,-1, (I)*(B(3,1))} ,
{3, 1, (I)*(B(3,1))} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} ,
{3,-3, (I)*(B(3,3))} ,
{3, 3, (I)*(B(3,3))} ,
{5, 0, A(5,0)} ,
{5,-1, (I)*(B(5,1))} ,
{5, 1, (I)*(B(5,1))} ,
{5,-2, A(5,2)} ,
{5, 2, A(5,2)} ,
{5,-3, (I)*(B(5,3))} ,
{5, 3, (I)*(B(5,3))} ,
{5,-4, A(5,4)} ,
{5, 4, A(5,4)} ,
{5,-5, (I)*(B(5,5))} ,
{5, 5, (I)*(B(5,5))} }
###
###
| $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ {Y_{-2}^{(2)}} $|$ -\frac{1}{231} i \left(33 \sqrt{21} B(1,1)-11 \sqrt{14} B(3,1)+\sqrt{35} B(5,1)\right) $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ -\frac{i \left(33 \sqrt{35} B(1,1)-22 \sqrt{210} B(3,1)+25 \sqrt{21} B(5,1)\right)}{1155} $|$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $|$ \frac{1}{231} i \sqrt{2} \left(11 \sqrt{7} B(3,3)-35 B(5,3)\right) $|$ \frac{1}{11} \sqrt{10} A(5,4) $|$ -\frac{5}{11} i \sqrt{\frac{2}{3}} B(5,5) $|
^$ {Y_{-1}^{(2)}} $|$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $|$ -\frac{1}{231} i \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)-2 \sqrt{210} B(5,1)\right) $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ -\frac{i \left(33 \sqrt{15} B(1,1)-11 \sqrt{10} B(3,1)-100 B(5,1)\right)}{165 \sqrt{7}} $|$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $|$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)+28 B(5,3)\right) $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $|
^$ {Y_{0}^{(2)}} $|$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)-14 B(5,3)\right) $|$ \frac{1}{11} \sqrt{5} A(5,2) $|$ -\frac{1}{385} i \left(11 \sqrt{210} B(1,1)+11 \sqrt{35} B(3,1)+25 \sqrt{14} B(5,1)\right) $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ -\frac{1}{385} i \left(11 \sqrt{210} B(1,1)+11 \sqrt{35} B(3,1)+25 \sqrt{14} B(5,1)\right) $|$ \frac{1}{11} \sqrt{5} A(5,2) $|$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)-14 B(5,3)\right) $|
^$ {Y_{1}^{(2)}} $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $|$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)+28 B(5,3)\right) $|$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $|$ -\frac{i \left(33 \sqrt{15} B(1,1)-11 \sqrt{10} B(3,1)-100 B(5,1)\right)}{165 \sqrt{7}} $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ -\frac{1}{231} i \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)-2 \sqrt{210} B(5,1)\right) $|$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $|
^$ {Y_{2}^{(2)}} $|$ -\frac{5}{11} i \sqrt{\frac{2}{3}} B(5,5) $|$ \frac{1}{11} \sqrt{10} A(5,4) $|$ \frac{1}{231} i \sqrt{2} \left(11 \sqrt{7} B(3,3)-35 B(5,3)\right) $|$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $|$ -\frac{i \left(33 \sqrt{35} B(1,1)-22 \sqrt{210} B(3,1)+25 \sqrt{21} B(5,1)\right)}{1155} $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ -\frac{1}{231} i \left(33 \sqrt{21} B(1,1)-11 \sqrt{14} B(3,1)+\sqrt{35} B(5,1)\right) $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^
^$ d_{y^2-z^2} $|$ 0 $|$ 3 \sqrt{\frac{3}{70}} B(1,1)-\frac{B(3,1)}{6 \sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{7}} B(3,3)+\frac{5 B(5,1)}{6 \sqrt{14}}+\frac{5}{44} \sqrt{3} B(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} B(5,5) $|$ \frac{5 \sqrt{2} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)-\sqrt{105} (99 A(1,0)+44 A(3,0)+50 A(5,0))}{2310} $|$ 0 $|$ \frac{1}{924} \left(-66 \sqrt{14} B(1,1)-22 \sqrt{21} B(3,1)-22 \sqrt{35} B(3,3)-17 \sqrt{210} B(5,1)+7 \sqrt{5} B(5,3)+105 B(5,5)\right) $|$ \frac{\sqrt{14} (-33 A(1,0)+22 A(3,0)-5 A(5,0))-42 \sqrt{15} A(5,2)-42 \sqrt{5} A(5,4)}{462 \sqrt{2}} $|$ 0 $|
^$ d_{3x^2-r^2} $|$ 0 $|$ \frac{-198 \sqrt{70} B(1,1)+154 \sqrt{105} B(3,1)-5 \left(22 \sqrt{7} B(3,3)-5 \sqrt{42} B(5,1)+35 \left(B(5,3)+3 \sqrt{5} B(5,5)\right)\right)}{4620} $|$ \frac{-\sqrt{35} (99 A(1,0)+44 A(3,0)+50 A(5,0))-5 \sqrt{6} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)}{2310} $|$ 0 $|$ \frac{1}{924} \left(-66 \sqrt{42} B(1,1)-66 \sqrt{7} B(3,1)-22 \sqrt{105} B(3,3)-9 \sqrt{70} B(5,1)+49 \sqrt{15} B(5,3)-105 \sqrt{3} B(5,5)\right) $|$ \frac{\sqrt{42} (33 A(1,0)-22 A(3,0)+5 A(5,0))-42 \sqrt{5} A(5,2)+42 \sqrt{15} A(5,4)}{462 \sqrt{2}} $|$ 0 $|
^$ d_{\text{xz}} $|$ \frac{1}{231} \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)-2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $|$ 0 $|$ 0 $|$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $|$ 0 $|$ 0 $|$ \frac{1}{462} \left(-66 \sqrt{7} A(1,0)-11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)+25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)-21 \sqrt{10} A(5,4)\right) $|
^$ d_{\text{xy}} $|$ \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 $|$ \frac{-66 \sqrt{210} B(1,1)-11 \sqrt{35} B(3,1)+5 \left(11 \sqrt{21} B(3,3)+5 \sqrt{14} B(5,1)-35 \sqrt{3} B(5,3)+35 \sqrt{15} B(5,5)\right)}{2310} $|$ 0 $|$ 0 $|$ \frac{1}{462} \left(66 \sqrt{14} B(1,1)-33 \sqrt{21} B(3,1)+11 \sqrt{35} B(3,3)+3 \sqrt{210} B(5,1)-35 \sqrt{5} B(5,3)-105 B(5,5)\right) $|
^$ d_{\text{yz}} $|$ 0 $|$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $|$ -\sqrt{\frac{6}{35}} B(1,1)+\frac{2 B(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} B(5,1) $|$ 0 $|$ \frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right) $|$ \frac{1}{231} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)+2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $|$ 0 $|
###
===== Table of several point groups =====
###
[[physics_chemistry:point_groups|Return to Main page on Point Groups]]
###
###
^Nonaxial groups | [[physics_chemistry:point_groups:c1|C]][[physics_chemistry:point_groups:c1|1]] | [[physics_chemistry:point_groups:cs|C]][[physics_chemistry:point_groups:cs|s]] | [[physics_chemistry:point_groups:ci|C]][[physics_chemistry:point_groups:ci|i]] | | | | |
^Cn groups | [[physics_chemistry:point_groups:c2|C]][[physics_chemistry:point_groups:c2|2]] | [[physics_chemistry:point_groups:c3|C]][[physics_chemistry:point_groups:c3|3]] | [[physics_chemistry:point_groups:c4|C]][[physics_chemistry:point_groups:c4|4]] | [[physics_chemistry:point_groups:c5|C]][[physics_chemistry:point_groups:c5|5]] | [[physics_chemistry:point_groups:c6|C]][[physics_chemistry:point_groups:c6|6]] | [[physics_chemistry:point_groups:c7|C]][[physics_chemistry:point_groups:c7|7]] | [[physics_chemistry:point_groups:c8|C]][[physics_chemistry:point_groups:c8|8]] |
^Dn groups | [[physics_chemistry:point_groups:d2|D]][[physics_chemistry:point_groups:d2|2]] | [[physics_chemistry:point_groups:d3|D]][[physics_chemistry:point_groups:d3|3]] | [[physics_chemistry:point_groups:d4|D]][[physics_chemistry:point_groups:d4|4]] | [[physics_chemistry:point_groups:d5|D]][[physics_chemistry:point_groups:d5|5]] | [[physics_chemistry:point_groups:d6|D]][[physics_chemistry:point_groups:d6|6]] | [[physics_chemistry:point_groups:d7|D]][[physics_chemistry:point_groups:d7|7]] | [[physics_chemistry:point_groups:d8|D]][[physics_chemistry:point_groups:d8|8]] |
^Cnv groups | [[physics_chemistry:point_groups:c2v|C]][[physics_chemistry:point_groups:c2v|2v]] | [[physics_chemistry:point_groups:c3v|C]][[physics_chemistry:point_groups:c3v|3v]] | [[physics_chemistry:point_groups:c4v|C]][[physics_chemistry:point_groups:c4v|4v]] | [[physics_chemistry:point_groups:c5v|C]][[physics_chemistry:point_groups:c5v|5v]] | [[physics_chemistry:point_groups:c6v|C]][[physics_chemistry:point_groups:c6v|6v]] | [[physics_chemistry:point_groups:c7v|C]][[physics_chemistry:point_groups:c7v|7v]] | [[physics_chemistry:point_groups:c8v|C]][[physics_chemistry:point_groups:c8v|8v]] |
^Cnh groups | [[physics_chemistry:point_groups:c2h|C]][[physics_chemistry:point_groups:c2h|2h]] | [[physics_chemistry:point_groups:c3h|C]][[physics_chemistry:point_groups:c3h|3h]] | [[physics_chemistry:point_groups:c4h|C]][[physics_chemistry:point_groups:c4h|4h]] | [[physics_chemistry:point_groups:c5h|C]][[physics_chemistry:point_groups:c5h|5h]] | [[physics_chemistry:point_groups:c6h|C]][[physics_chemistry:point_groups:c6h|6h]] | | |
^Dnh groups | [[physics_chemistry:point_groups:d2h|D]][[physics_chemistry:point_groups:d2h|2h]] | [[physics_chemistry:point_groups:d3h|D]][[physics_chemistry:point_groups:d3h|3h]] | [[physics_chemistry:point_groups:d4h|D]][[physics_chemistry:point_groups:d4h|4h]] | [[physics_chemistry:point_groups:d5h|D]][[physics_chemistry:point_groups:d5h|5h]] | [[physics_chemistry:point_groups:d6h|D]][[physics_chemistry:point_groups:d6h|6h]] | [[physics_chemistry:point_groups:d7h|D]][[physics_chemistry:point_groups:d7h|7h]] | [[physics_chemistry:point_groups:d8h|D]][[physics_chemistry:point_groups:d8h|8h]] |
^Dnd groups | [[physics_chemistry:point_groups:d2d|D]][[physics_chemistry:point_groups:d2d|2d]] | [[physics_chemistry:point_groups:d3d|D]][[physics_chemistry:point_groups:d3d|3d]] | [[physics_chemistry:point_groups:d4d|D]][[physics_chemistry:point_groups:d4d|4d]] | [[physics_chemistry:point_groups:d5d|D]][[physics_chemistry:point_groups:d5d|5d]] | [[physics_chemistry:point_groups:d6d|D]][[physics_chemistry:point_groups:d6d|6d]] | [[physics_chemistry:point_groups:d7d|D]][[physics_chemistry:point_groups:d7d|7d]] | [[physics_chemistry:point_groups:d8d|D]][[physics_chemistry:point_groups:d8d|8d]] |
^Sn groups | [[physics_chemistry:point_groups:S2|S]][[physics_chemistry:point_groups:S2|2]] | [[physics_chemistry:point_groups:S4|S]][[physics_chemistry:point_groups:S4|4]] | [[physics_chemistry:point_groups:S6|S]][[physics_chemistry:point_groups:S6|6]] | [[physics_chemistry:point_groups:S8|S]][[physics_chemistry:point_groups:S8|8]] | [[physics_chemistry:point_groups:S10|S]][[physics_chemistry:point_groups:S10|10]] | [[physics_chemistry:point_groups:S12|S]][[physics_chemistry:point_groups:S12|12]] | |
^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]][[physics_chemistry:point_groups:Th|h]] | [[physics_chemistry:point_groups:Td|T]][[physics_chemistry:point_groups:Td|d]] | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]][[physics_chemistry:point_groups:Oh|h]] | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]][[physics_chemistry:point_groups:Ih|h]] |
^Linear groups | [[physics_chemistry:point_groups:cinfv|C]][[physics_chemistry:point_groups:cinfv|$\infty$v]] | [[physics_chemistry:point_groups:cinfv|D]][[physics_chemistry:point_groups:dinfh|$\infty$h]] | | | | | |
###