~~CLOSETOC~~
====== Orientation Zxy ======
===== Symmetry Operations =====
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In the C2v Point Group, with orientation Zxy there are the following symmetry operations
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{{:physics_chemistry:pointgroup:c2v_zxy.png}}
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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_2$ | $\{0,0,1\}$ , |
^ $\sigma _v\text{(xz)}$ | $\{0,1,0\}$ , |
^ $\sigma _v\text{(yz)}$ | $\{1,0,0\}$ , |
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===== Different Settings =====
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* [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]]
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===== Character Table =====
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| $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ $ \sigma_v\text{(xz)} \,{\text{(1)}} $ ^ $ \sigma_v\text{(yz)} \,{\text{(1)}} $ ^
^ $ A_1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ |
^ $ A_2 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ |
^ $ B_1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ |
^ $ B_2 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ |
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===== Product Table =====
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| $ $ ^ $ A_1 $ ^ $ A_2 $ ^ $ B_1 $ ^ $ B_2 $ ^
^ $ A_1 $ | $ A_1 $ | $ A_2 $ | $ B_1 $ | $ B_2 $ |
^ $ A_2 $ | $ A_2 $ | $ A_1 $ | $ B_2 $ | $ B_1 $ |
^ $ B_1 $ | $ B_1 $ | $ B_2 $ | $ A_1 $ | $ A_2 $ |
^ $ B_2 $ | $ B_2 $ | $ B_1 $ | $ A_2 $ | $ A_1 $ |
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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]]
* [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]]
* [[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]]
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===== Super Groups with compatible settings =====
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* [[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:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
* [[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_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 C2v Point group with orientation Zxy the form of the expansion coefficients is:
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==== Expansion ====
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$$A_{k,m} = \begin{cases}
A(0,0) & k=0\land m=0 \\
A(1,0) & k=1\land m=0 \\
A(2,2) & k=2\land (m=-2\lor m=2) \\
A(2,0) & k=2\land m=0 \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
A(3,0) & k=3\land m=0 \\
A(4,4) & k=4\land (m=-4\lor m=4) \\
A(4,2) & k=4\land (m=-2\lor m=2) \\
A(4,0) & k=4\land m=0 \\
A(5,4) & k=5\land (m=-4\lor m=4) \\
A(5,2) & k=5\land (m=-2\lor m=2) \\
A(5,0) & k=5\land m=0 \\
A(6,6) & k=6\land (m=-6\lor m=6) \\
A(6,4) & k=6\land (m=-4\lor m=4) \\
A(6,2) & k=6\land (m=-2\lor m=2) \\
A(6,0) & k=6\land m=0
\end{cases}$$
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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}, {A[1, 0], k == 1 && m == 0}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {A[2, 0], k == 2 && m == 0}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {A[3, 0], k == 3 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {A[4, 0], k == 4 && m == 0}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {A[5, 0], k == 5 && m == 0}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {A[6, 0], k == 6 && m == 0}}, 0]
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==== Input format suitable for Quanty ====
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Akm = {{0, 0, A(0,0)} ,
{1, 0, A(1,0)} ,
{2, 0, A(2,0)} ,
{2,-2, A(2,2)} ,
{2, 2, A(2,2)} ,
{3, 0, A(3,0)} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} ,
{4, 0, A(4,0)} ,
{4,-2, A(4,2)} ,
{4, 2, A(4,2)} ,
{4,-4, A(4,4)} ,
{4, 4, A(4,4)} ,
{5, 0, A(5,0)} ,
{5,-2, A(5,2)} ,
{5, 2, A(5,2)} ,
{5,-4, A(5,4)} ,
{5, 4, A(5,4)} ,
{6, 0, A(6,0)} ,
{6,-2, A(6,2)} ,
{6, 2, A(6,2)} ,
{6,-4, A(6,4)} ,
{6, 4, A(6,4)} ,
{6,-6, A(6,6)} ,
{6, 6, A(6,6)} }
###
==== One particle coupling on a basis of spherical harmonics ====
###
The operator representing the potential in second quantisation is given as:
$$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$
For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter.
$$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$
Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$
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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}{ 0 }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ 0 }$|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|
^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ 0 $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|
^$ {Y_{0}^{(1)}} $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|
^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{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}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\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}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|
^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{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}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|
^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ 0 $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ 0 $|$ -\frac{1}{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) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|
^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ -\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) $|$ 0 $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|
^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$ -\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) $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{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) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|
^$ {Y_{0}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$ 0 $|$ -\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) $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ -\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) $|$ 0 $|
^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ -\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) $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{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}} }$|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$ 0 $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ -\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) $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|
^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ 0 $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ -\frac{1}{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) $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|
###
==== Rotation matrix to symmetry adapted functions (choice is not unique) ====
###
Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|
###
==== One particle coupling on a basis of symmetry adapted functions ====
###
After rotation we find
###
###
| $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^
^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|
^$ p_x $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 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}} $|$ 0 $|$ 0 $|
^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$\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) $|$ 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}} $|$ 0 $|
^$ p_z $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ 0 $|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|
^$ d_{x^2-y^2} $|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ \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) $|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|
^$ d_{3z^2-r^2} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$|
^$ d_{\text{yz}} $|$ 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}{ 0 }$|$ 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) $|$ 0 $|$ 0 $|$\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) }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xz}} $|$ 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}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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) $|$ 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) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 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}{ \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) }$|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{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 $|$\color{darkred}{ 0 }$|$\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 }$|$ 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) $|$ 0 $|$ 0 $|
^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{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 $|$\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 }$|$ 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) $|$ 0 $|
^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\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) $|
^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 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}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\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}{ 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) $|$ 0 $|$ 0 $|
^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 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}} $|$ 0 $|$\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}{ 0 }$|$\color{darkred}{ 0 }$|$ 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) $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$\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}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ -\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 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|
###
===== Coupling for a single shell =====
###
Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$.
###
###
Click on one of the subsections to expand it or
###
==== Potential for s orbitals ====
###
$$A_{k,m} = \begin{cases}
\text{Ea1} & k=0\land m=0 \\
0 & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{Ea1, k == 0 && m == 0}}, 0]
###
###
Akm = {{0, 0, Ea1} }
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^
^$ {Y_{0}^{(0)}} $|$ \text{Ea1} $|
###
###
| $ $ ^ $ \text{s} $ ^
^$ \text{s} $|$ \text{Ea1} $|
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^
^$ \text{s} $|$ 1 $|
###
###
^ ^$$\text{Ea1}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_0_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
###
==== Potential for p orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{3} (\text{Ea1}+\text{Eb1}+\text{Eb2}) & k=0\land m=0 \\
0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\
\frac{5 (\text{Eb1}-\text{Eb2})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\
\frac{5}{6} (2 \text{Ea1}-\text{Eb1}-\text{Eb2}) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{(Ea1 + Eb1 + Eb2)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != 0 && m != 2)}, {(5*(Eb1 - Eb2))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}}, (5*(2*Ea1 - Eb1 - Eb2))/6]
###
###
Akm = {{0, 0, (1/3)*(Ea1 + Eb1 + Eb2)} ,
{2, 0, (5/6)*((2)*(Ea1) + (-1)*(Eb1) + (-1)*(Eb2))} ,
{2,-2, (5/2)*((1/(sqrt(6)))*(Eb1 + (-1)*(Eb2)))} ,
{2, 2, (5/2)*((1/(sqrt(6)))*(Eb1 + (-1)*(Eb2)))} }
###
###
| $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^
^$ {Y_{-1}^{(1)}} $|$ \frac{\text{Eb1}+\text{Eb2}}{2} $|$ 0 $|$ \frac{\text{Eb2}-\text{Eb1}}{2} $|
^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea1} $|$ 0 $|
^$ {Y_{1}^{(1)}} $|$ \frac{\text{Eb2}-\text{Eb1}}{2} $|$ 0 $|$ \frac{\text{Eb1}+\text{Eb2}}{2} $|
###
###
| $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^
^$ p_x $|$ \text{Eb1} $|$ 0 $|$ 0 $|
^$ p_y $|$ 0 $|$ \text{Eb2} $|$ 0 $|
^$ p_z $|$ 0 $|$ 0 $|$ \text{Ea1} $|
###
###
| $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^
^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|
^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
^$ p_z $|$ 0 $|$ 1 $|$ 0 $|
###
###
^ ^$$\text{Eb1}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
^ ^$$\text{Eb2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Ea1}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_1_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
###
==== Potential for d orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{5} (\text{Ea1x2y2}+\text{Ea1z2}+\text{Ea2}+\text{Eb1}+\text{Eb2}) & k=0\land m=0 \\
0 & k=1\land m=0 \\
\frac{1}{4} \left(\sqrt{6} \text{Eb1}-\sqrt{6} \text{Eb2}-4 \sqrt{2} \text{Ma1}\right) & k=2\land (m=-2\lor m=2) \\
\frac{1}{2} (-2 \text{Ea1x2y2}+2 \text{Ea1z2}-2 \text{Ea2}+\text{Eb1}+\text{Eb2}) & k=2\land m=0 \\
0 & (k=3\land (m=-2\lor m=2))\lor (k=3\land m=0) \\
\frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ea1x2y2}-\text{Ea2}) & k=4\land (m=-4\lor m=4) \\
\frac{3 \left(\text{Eb1}-\text{Eb2}+\sqrt{3} \text{Ma1}\right)}{\sqrt{10}} & k=4\land (m=-2\lor m=2) \\
\frac{3}{10} (\text{Ea1x2y2}+6 \text{Ea1z2}+\text{Ea2}-4 (\text{Eb1}+\text{Eb2})) & k=4\land m=0
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{(Ea1x2y2 + Ea1z2 + Ea2 + Eb1 + Eb2)/5, k == 0 && m == 0}, {0, k == 1 && m == 0}, {(Sqrt[6]*Eb1 - Sqrt[6]*Eb2 - 4*Sqrt[2]*Ma1)/4, k == 2 && (m == -2 || m == 2)}, {(-2*Ea1x2y2 + 2*Ea1z2 - 2*Ea2 + Eb1 + Eb2)/2, k == 2 && m == 0}, {0, (k == 3 && (m == -2 || m == 2)) || (k == 3 && m == 0)}, {(3*Sqrt[7/10]*(Ea1x2y2 - Ea2))/2, k == 4 && (m == -4 || m == 4)}, {(3*(Eb1 - Eb2 + Sqrt[3]*Ma1))/Sqrt[10], k == 4 && (m == -2 || m == 2)}, {(3*(Ea1x2y2 + 6*Ea1z2 + Ea2 - 4*(Eb1 + Eb2)))/10, k == 4 && m == 0}}, 0]
###
###
Akm = {{0, 0, (1/5)*(Ea1x2y2 + Ea1z2 + Ea2 + Eb1 + Eb2)} ,
{2, 0, (1/2)*((-2)*(Ea1x2y2) + (2)*(Ea1z2) + (-2)*(Ea2) + Eb1 + Eb2)} ,
{2,-2, (1/4)*((sqrt(6))*(Eb1) + (-1)*((sqrt(6))*(Eb2)) + (-4)*((sqrt(2))*(Ma1)))} ,
{2, 2, (1/4)*((sqrt(6))*(Eb1) + (-1)*((sqrt(6))*(Eb2)) + (-4)*((sqrt(2))*(Ma1)))} ,
{4, 0, (3/10)*(Ea1x2y2 + (6)*(Ea1z2) + Ea2 + (-4)*(Eb1 + Eb2))} ,
{4,-2, (3)*((1/(sqrt(10)))*(Eb1 + (-1)*(Eb2) + (sqrt(3))*(Ma1)))} ,
{4, 2, (3)*((1/(sqrt(10)))*(Eb1 + (-1)*(Eb2) + (sqrt(3))*(Ma1)))} ,
{4,-4, (3/2)*((sqrt(7/10))*(Ea1x2y2 + (-1)*(Ea2)))} ,
{4, 4, (3/2)*((sqrt(7/10))*(Ea1x2y2 + (-1)*(Ea2)))} }
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Ea1x2y2}+\text{Ea2}}{2} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Ea1x2y2}-\text{Ea2}}{2} $|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{\text{Eb1}+\text{Eb2}}{2} $|$ 0 $|$ \frac{\text{Eb2}-\text{Eb1}}{2} $|$ 0 $|
^$ {Y_{0}^{(2)}} $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|$ \text{Ea1z2} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|
^$ {Y_{1}^{(2)}} $|$ 0 $|$ \frac{\text{Eb2}-\text{Eb1}}{2} $|$ 0 $|$ \frac{\text{Eb1}+\text{Eb2}}{2} $|$ 0 $|
^$ {Y_{2}^{(2)}} $|$ \frac{\text{Ea1x2y2}-\text{Ea2}}{2} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Ea1x2y2}+\text{Ea2}}{2} $|
###
###
| $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^
^$ d_{x^2-y^2} $|$ \text{Ea1x2y2} $|$ \text{Ma1} $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{3z^2-r^2} $|$ \text{Ma1} $|$ \text{Ea1z2} $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Eb2} $|$ 0 $|$ 0 $|
^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb1} $|$ 0 $|
^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea2} $|
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|
^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|
^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|
###
###
^ ^$$\text{Ea1x2y2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |
^ ^$$\text{Ea1z2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
^ ^$$\text{Eb2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
^ ^$$\text{Eb1}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
^ ^$$\text{Ea2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |
###
==== Potential for f orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{7} (\text{Ea1z3}+\text{Ea1zx2y2}+\text{Ea2}+\text{Eb1x3}+\text{Eb1xy2z2}+\text{Eb2y3}+\text{Eb2yz2x2}) & k=0\land m=0 \\
0 & k=1\land m=0 \\
\frac{5}{28} \left(\sqrt{6} \text{Eb1x3}-\sqrt{6} \text{Eb2y3}+\sqrt{10} (-2 \text{Ma1}+\text{Mb1}+\text{Mb2})\right) & k=2\land (m=-2\lor m=2) \\
-\frac{5}{14} \left(-2 \text{Ea1z3}+\text{Eb1x3}+\text{Eb2y3}+\sqrt{15} (\text{Mb2}-\text{Mb1})\right) & k=2\land m=0 \\
0 & (k=3\land (m=-2\lor m=2))\lor (k=3\land m=0) \\
\frac{3 \left(4 \sqrt{5} \text{Ea1zx2y2}-4 \sqrt{5} \text{Ea2}+3 \sqrt{5} \text{Eb1x3}-3 \sqrt{5} \text{Eb1xy2z2}+3 \sqrt{5} \text{Eb2y3}-3 \sqrt{5} \text{Eb2yz2x2}+2 \sqrt{3} (\text{Mb1}-\text{Mb2})\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\
\frac{3}{56} \left(-3 \sqrt{10} \text{Eb1x3}+7 \sqrt{10} \text{Eb1xy2z2}+3 \sqrt{10} \text{Eb2y3}-7 \sqrt{10} \text{Eb2yz2x2}+2 \sqrt{6} (-2 \text{Ma1}+\text{Mb1}+\text{Mb2})\right) & k=4\land (m=-2\lor m=2) \\
\frac{3}{56} \left(24 \text{Ea1z3}-28 \text{Ea1zx2y2}-28 \text{Ea2}+9 \text{Eb1x3}+7 \text{Eb1xy2z2}+9 \text{Eb2y3}+7 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right) & k=4\land m=0 \\
0 & (k=5\land (m=-4\lor m=4))\lor (k=5\land (m=-2\lor m=2))\lor (k=5\land m=0) \\
\frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eb1x3}+3 \sqrt{3} \text{Eb1xy2z2}-5 \sqrt{3} \text{Eb2y3}-3 \sqrt{3} \text{Eb2yz2x2}-6 \sqrt{5} (\text{Mb1}+\text{Mb2})\right) & k=6\land (m=-6\lor m=6) \\
\frac{13 \left(24 \text{Ea1zx2y2}-24 \text{Ea2}-15 \text{Eb1x3}+15 \text{Eb1xy2z2}-15 \text{Eb2y3}+15 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\
\frac{13 \left(5 \sqrt{15} \text{Eb1x3}+3 \sqrt{15} \text{Eb1xy2z2}-5 \sqrt{15} \text{Eb2y3}-3 \sqrt{15} \text{Eb2yz2x2}+64 \text{Ma1}+34 (\text{Mb1}+\text{Mb2})\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\
-\frac{13}{560} \left(-80 \text{Ea1z3}-24 \text{Ea1zx2y2}-24 \text{Ea2}+25 \text{Eb1x3}+39 \text{Eb1xy2z2}+25 \text{Eb2y3}+39 \text{Eb2yz2x2}+14 \sqrt{15} (\text{Mb1}-\text{Mb2})\right) & k=6\land m=0
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{(Ea1z3 + Ea1zx2y2 + Ea2 + Eb1x3 + Eb1xy2z2 + Eb2y3 + Eb2yz2x2)/7, k == 0 && m == 0}, {0, k == 1 && m == 0}, {(5*(Sqrt[6]*Eb1x3 - Sqrt[6]*Eb2y3 + Sqrt[10]*(-2*Ma1 + Mb1 + Mb2)))/28, k == 2 && (m == -2 || m == 2)}, {(-5*(-2*Ea1z3 + Eb1x3 + Eb2y3 + Sqrt[15]*(-Mb1 + Mb2)))/14, k == 2 && m == 0}, {0, (k == 3 && (m == -2 || m == 2)) || (k == 3 && m == 0)}, {(3*(4*Sqrt[5]*Ea1zx2y2 - 4*Sqrt[5]*Ea2 + 3*Sqrt[5]*Eb1x3 - 3*Sqrt[5]*Eb1xy2z2 + 3*Sqrt[5]*Eb2y3 - 3*Sqrt[5]*Eb2yz2x2 + 2*Sqrt[3]*(Mb1 - Mb2)))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(3*(-3*Sqrt[10]*Eb1x3 + 7*Sqrt[10]*Eb1xy2z2 + 3*Sqrt[10]*Eb2y3 - 7*Sqrt[10]*Eb2yz2x2 + 2*Sqrt[6]*(-2*Ma1 + Mb1 + Mb2)))/56, k == 4 && (m == -2 || m == 2)}, {(3*(24*Ea1z3 - 28*Ea1zx2y2 - 28*Ea2 + 9*Eb1x3 + 7*Eb1xy2z2 + 9*Eb2y3 + 7*Eb2yz2x2 + 2*Sqrt[15]*(-Mb1 + Mb2)))/56, k == 4 && m == 0}, {0, (k == 5 && (m == -4 || m == 4)) || (k == 5 && (m == -2 || m == 2)) || (k == 5 && m == 0)}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eb1x3 + 3*Sqrt[3]*Eb1xy2z2 - 5*Sqrt[3]*Eb2y3 - 3*Sqrt[3]*Eb2yz2x2 - 6*Sqrt[5]*(Mb1 + Mb2)))/160, k == 6 && (m == -6 || m == 6)}, {(13*(24*Ea1zx2y2 - 24*Ea2 - 15*Eb1x3 + 15*Eb1xy2z2 - 15*Eb2y3 + 15*Eb2yz2x2 + 2*Sqrt[15]*(-Mb1 + Mb2)))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(13*(5*Sqrt[15]*Eb1x3 + 3*Sqrt[15]*Eb1xy2z2 - 5*Sqrt[15]*Eb2y3 - 3*Sqrt[15]*Eb2yz2x2 + 64*Ma1 + 34*(Mb1 + Mb2)))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {(-13*(-80*Ea1z3 - 24*Ea1zx2y2 - 24*Ea2 + 25*Eb1x3 + 39*Eb1xy2z2 + 25*Eb2y3 + 39*Eb2yz2x2 + 14*Sqrt[15]*(Mb1 - Mb2)))/560, k == 6 && m == 0}}, 0]
###
###
Akm = {{0, 0, (1/7)*(Ea1z3 + Ea1zx2y2 + Ea2 + Eb1x3 + Eb1xy2z2 + Eb2y3 + Eb2yz2x2)} ,
{2, 0, (-5/14)*((-2)*(Ea1z3) + Eb1x3 + Eb2y3 + (sqrt(15))*((-1)*(Mb1) + Mb2))} ,
{2,-2, (5/28)*((sqrt(6))*(Eb1x3) + (-1)*((sqrt(6))*(Eb2y3)) + (sqrt(10))*((-2)*(Ma1) + Mb1 + Mb2))} ,
{2, 2, (5/28)*((sqrt(6))*(Eb1x3) + (-1)*((sqrt(6))*(Eb2y3)) + (sqrt(10))*((-2)*(Ma1) + Mb1 + Mb2))} ,
{4, 0, (3/56)*((24)*(Ea1z3) + (-28)*(Ea1zx2y2) + (-28)*(Ea2) + (9)*(Eb1x3) + (7)*(Eb1xy2z2) + (9)*(Eb2y3) + (7)*(Eb2yz2x2) + (2)*((sqrt(15))*((-1)*(Mb1) + Mb2)))} ,
{4,-2, (3/56)*((-3)*((sqrt(10))*(Eb1x3)) + (7)*((sqrt(10))*(Eb1xy2z2)) + (3)*((sqrt(10))*(Eb2y3)) + (-7)*((sqrt(10))*(Eb2yz2x2)) + (2)*((sqrt(6))*((-2)*(Ma1) + Mb1 + Mb2)))} ,
{4, 2, (3/56)*((-3)*((sqrt(10))*(Eb1x3)) + (7)*((sqrt(10))*(Eb1xy2z2)) + (3)*((sqrt(10))*(Eb2y3)) + (-7)*((sqrt(10))*(Eb2yz2x2)) + (2)*((sqrt(6))*((-2)*(Ma1) + Mb1 + Mb2)))} ,
{4,-4, (3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Ea1zx2y2)) + (-4)*((sqrt(5))*(Ea2)) + (3)*((sqrt(5))*(Eb1x3)) + (-3)*((sqrt(5))*(Eb1xy2z2)) + (3)*((sqrt(5))*(Eb2y3)) + (-3)*((sqrt(5))*(Eb2yz2x2)) + (2)*((sqrt(3))*(Mb1 + (-1)*(Mb2)))))} ,
{4, 4, (3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Ea1zx2y2)) + (-4)*((sqrt(5))*(Ea2)) + (3)*((sqrt(5))*(Eb1x3)) + (-3)*((sqrt(5))*(Eb1xy2z2)) + (3)*((sqrt(5))*(Eb2y3)) + (-3)*((sqrt(5))*(Eb2yz2x2)) + (2)*((sqrt(3))*(Mb1 + (-1)*(Mb2)))))} ,
{6, 0, (-13/560)*((-80)*(Ea1z3) + (-24)*(Ea1zx2y2) + (-24)*(Ea2) + (25)*(Eb1x3) + (39)*(Eb1xy2z2) + (25)*(Eb2y3) + (39)*(Eb2yz2x2) + (14)*((sqrt(15))*(Mb1 + (-1)*(Mb2))))} ,
{6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eb1x3)) + (3)*((sqrt(15))*(Eb1xy2z2)) + (-5)*((sqrt(15))*(Eb2y3)) + (-3)*((sqrt(15))*(Eb2yz2x2)) + (64)*(Ma1) + (34)*(Mb1 + Mb2)))} ,
{6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eb1x3)) + (3)*((sqrt(15))*(Eb1xy2z2)) + (-5)*((sqrt(15))*(Eb2y3)) + (-3)*((sqrt(15))*(Eb2yz2x2)) + (64)*(Ma1) + (34)*(Mb1 + Mb2)))} ,
{6,-4, (13/80)*((1/(sqrt(14)))*((24)*(Ea1zx2y2) + (-24)*(Ea2) + (-15)*(Eb1x3) + (15)*(Eb1xy2z2) + (-15)*(Eb2y3) + (15)*(Eb2yz2x2) + (2)*((sqrt(15))*((-1)*(Mb1) + Mb2))))} ,
{6, 4, (13/80)*((1/(sqrt(14)))*((24)*(Ea1zx2y2) + (-24)*(Ea2) + (-15)*(Eb1x3) + (15)*(Eb1xy2z2) + (-15)*(Eb2y3) + (15)*(Eb2yz2x2) + (2)*((sqrt(15))*((-1)*(Mb1) + Mb2))))} ,
{6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eb1x3)) + (3)*((sqrt(3))*(Eb1xy2z2)) + (-5)*((sqrt(3))*(Eb2y3)) + (-3)*((sqrt(3))*(Eb2yz2x2)) + (-6)*((sqrt(5))*(Mb1 + Mb2))))} ,
{6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eb1x3)) + (3)*((sqrt(3))*(Eb1xy2z2)) + (-5)*((sqrt(3))*(Eb2y3)) + (-3)*((sqrt(3))*(Eb2yz2x2)) + (-6)*((sqrt(5))*(Mb1 + Mb2))))} }
###
###
| $ $ ^ $ {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{Eb1x3}+3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right) $|$ 0 $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right) $|$ 0 $|$ \frac{1}{16} \left(-5 \text{Eb1x3}-3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right) $|
^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Ea1zx2y2}+\text{Ea2}}{2} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Ea1zx2y2}-\text{Ea2}}{2} $|$ 0 $|
^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eb1x3}+5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}-\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(-3 \text{Eb1x3}-5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}-2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right) $|
^$ {Y_{0}^{(3)}} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|$ \text{Ea1z3} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|
^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right) $|$ 0 $|$ \frac{1}{16} \left(-3 \text{Eb1x3}-5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}-2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eb1x3}+5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}-\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right) $|
^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{\text{Ea1zx2y2}-\text{Ea2}}{2} $|$ 0 $|$ \frac{\text{Ma1}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Ea1zx2y2}+\text{Ea2}}{2} $|$ 0 $|
^$ {Y_{3}^{(3)}} $|$ \frac{1}{16} \left(-5 \text{Eb1x3}-3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right) $|$ 0 $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right) $|$ 0 $|$ \frac{1}{16} \left(5 \text{Eb1x3}+3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right) $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^
^$ f_{\text{xyz}} $|$ \text{Ea2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Eb1x3} $|$ 0 $|$ 0 $|$ \text{Mb1} $|$ 0 $|$ 0 $|
^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Eb2y3} $|$ 0 $|$ 0 $|$ \text{Mb2} $|$ 0 $|
^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1z3} $|$ 0 $|$ 0 $|$ \text{Ma1} $|
^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \text{Mb1} $|$ 0 $|$ 0 $|$ \text{Eb1xy2z2} $|$ 0 $|$ 0 $|
^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ 0 $|$ \text{Mb2} $|$ 0 $|$ 0 $|$ \text{Eb2yz2x2} $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma1} $|$ 0 $|$ 0 $|$ \text{Ea1zx2y2} $|
###
###
| $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|
###
###
^ ^$$\text{Ea2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
^ ^$$\text{Eb1x3}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: |
^ ^$$\text{Eb2y3}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: |
^ ^$$\text{Ea1z3}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
^ ^$$\text{Eb1xy2z2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: |
^ ^$$\text{Eb2yz2x2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: |
^ ^$$\text{Ea1zx2y2}$$ | {{:physics_chemistry:pointgroup:c2v_zxy_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |
###
===== Coupling between two shells =====
###
Click on one of the subsections to expand it or
###
==== Potential for s-p orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 1\lor m\neq 0 \\
A(1,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 1 || m != 0}}, A[1, 0]]
###
###
Akm = {{1, 0, A(1,0)} }
###
###
| $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^
^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{A(1,0)}{\sqrt{3}} $|$ 0 $|
###
###
| $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^
^$ \text{s} $|$ 0 $|$ 0 $|$ \frac{A(1,0)}{\sqrt{3}} $|
###
==== Potential for s-d orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\
A(2,2) & k=2\land (m=-2\lor m=2) \\
A(2,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != 0 && m != 2)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}}, A[2, 0]]
###
###
Akm = {{2, 0, A(2,0)} ,
{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}} $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,2)}{\sqrt{5}} $|
###
###
| $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^
^$ \text{s} $|$ \sqrt{\frac{2}{5}} A(2,2) $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $|
###
==== Potential for s-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 3\lor (m\neq -2\land m\neq 0\land m\neq 2) \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
A(3,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 0 && m != 2)}, {A[3, 2], k == 3 && (m == -2 || m == 2)}}, A[3, 0]]
###
###
Akm = {{3, 0, A(3,0)} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} }
###
###
| $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{A(3,2)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,2)}{\sqrt{7}} $|$ 0 $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^
^$ \text{s} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{7}} A(3,2) $|
###
==== Potential for p-d orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 3\land (k\neq 1\lor m\neq 0))\lor (m\neq -2\land m\neq 0\land m\neq 2) \\
A(1,0) & k=1\land m=0 \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
A(3,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || m != 0)) || (m != -2 && m != 0 && m != 2)}, {A[1, 0], k == 1 && m == 0}, {A[3, 2], k == 3 && (m == -2 || m == 2)}}, A[3, 0]]
###
###
Akm = {{1, 0, A(1,0)} ,
{3, 0, A(3,0)} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} }
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ {Y_{-1}^{(1)}} $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ -\frac{1}{7} \sqrt{6} A(3,2) $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$ \frac{1}{7} \sqrt{3} A(3,2) $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ \frac{1}{7} \sqrt{3} A(3,2) $|
^$ {Y_{1}^{(1)}} $|$ 0 $|$ -\frac{1}{7} \sqrt{6} A(3,2) $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|
###
###
| $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^
^$ p_x $|$ 0 $|$ 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) $|$ 0 $|
^$ p_y $|$ 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) $|$ 0 $|$ 0 $|
^$ p_z $|$ \frac{1}{7} \sqrt{6} A(3,2) $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $|$ 0 $|
###
==== Potential for p-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\
A(2,2) & k=2\land (m=-2\lor m=2) \\
A(2,0) & k=2\land m=0 \\
A(4,4) & k=4\land (m=-4\lor m=4) \\
A(4,2) & k=4\land (m=-2\lor m=2) \\
A(4,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}}, A[4, 0]]
###
###
Akm = {{2, 0, A(2,0)} ,
{2,-2, A(2,2)} ,
{2, 2, A(2,2)} ,
{4, 0, A(4,0)} ,
{4,-2, A(4,2)} ,
{4, 2, A(4,2)} ,
{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}} $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ 0 $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|
^$ {Y_{0}^{(1)}} $|$ 0 $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ 0 $|
^$ {Y_{1}^{(1)}} $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^
^$ p_x $|$ 0 $|$ \frac{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) $|$ 0 $|$ 0 $|
^$ p_y $|$ 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) $|$ 0 $|
^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $|
###
==== Potential for d-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 5\land (((k\neq 1\lor m\neq 0)\land k\neq 3)\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\
A(1,0) & k=1\land m=0 \\
A(3,2) & k=3\land (m=-2\lor m=2) \\
A(3,0) & k=3\land m=0 \\
A(5,4) & k=5\land (m=-4\lor m=4) \\
A(5,2) & k=5\land (m=-2\lor m=2) \\
A(5,0) & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, (k != 5 && (((k != 1 || m != 0) && k != 3) || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {A[1, 0], k == 1 && m == 0}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {A[3, 0], k == 3 && m == 0}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {A[5, 2], k == 5 && (m == -2 || m == 2)}}, A[5, 0]]
###
###
Akm = {{1, 0, A(1,0)} ,
{3, 0, A(3,0)} ,
{3,-2, A(3,2)} ,
{3, 2, A(3,2)} ,
{5, 0, A(5,0)} ,
{5,-2, A(5,2)} ,
{5, 2, A(5,2)} ,
{5,-4, A(5,4)} ,
{5, 4, A(5,4)} }
###
###
| $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^
^$ {Y_{-2}^{(2)}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $|$ 0 $|$ \frac{1}{11} \sqrt{10} A(5,4) $|$ 0 $|
^$ {Y_{-1}^{(2)}} $|$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $|$ 0 $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $|
^$ {Y_{0}^{(2)}} $|$ 0 $|$ \frac{1}{11} \sqrt{5} A(5,2) $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ \frac{1}{11} \sqrt{5} A(5,2) $|$ 0 $|
^$ {Y_{1}^{(2)}} $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $|$ 0 $|$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $|
^$ {Y_{2}^{(2)}} $|$ 0 $|$ \frac{1}{11} \sqrt{10} A(5,4) $|$ 0 $|$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|
###
###
| $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^
^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ \sqrt{2} \left(\frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}}\right) $|$ 0 $|$ 0 $|$ \frac{\sqrt{14} (33 A(1,0)-22 A(3,0)+5 A(5,0))+42 \sqrt{5} A(5,4)}{231 \sqrt{2}} $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{10} A(5,2) $|
^$ d_{\text{yz}} $|$ 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) $|$ 0 $|
^$ d_{\text{xz}} $|$ 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) $|$ 0 $|$ 0 $|
^$ 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 $|$ 0 $|$ 0 $|$ 0 $|$ 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]] | | | | | |
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