Table of Contents
Orientation 1
Symmetry Operations
In the C1 Point Group, with orientation 1 there are the following symmetry operations
| Operator | Orientation |
|---|---|
| $\text{E}$ | $\{0,0,0\}$ , |
Different Settings
Character Table
| $ $ | $ \text{E} \,{\text{(1)}} $ |
|---|---|
| $ \text{A} $ | $ 1 $ |
Product Table
| $ $ | $ \text{A} $ |
|---|---|
| $ \text{A} $ | $ \text{A} $ |
Sub Groups with compatible settings
Super Groups with compatible settings
Invariant Potential expanded on renormalized spherical Harmonics
Any potential (function) can be written as a sum over spherical harmonics. $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the C1 Point group with orientation 1 the form of the expansion coefficients is:
Expansion
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i B(1,1) & k=1\land m=-1 \\ A(1,0) & k=1\land m=0 \\ A(1,1)+i B(1,1) & k=1\land m=1 \\ A(2,2)-i B(2,2) & k=2\land m=-2 \\ -A(2,1)+i B(2,1) & k=2\land m=-1 \\ A(2,0) & k=2\land m=0 \\ A(2,1)+i B(2,1) & k=2\land m=1 \\ A(2,2)+i B(2,2) & k=2\land m=2 \\ -A(3,3)+i B(3,3) & k=3\land m=-3 \\ A(3,2)-i B(3,2) & k=3\land m=-2 \\ -A(3,1)+i B(3,1) & k=3\land m=-1 \\ A(3,0) & k=3\land m=0 \\ A(3,1)+i B(3,1) & k=3\land m=1 \\ A(3,2)+i B(3,2) & k=3\land m=2 \\ A(3,3)+i B(3,3) & k=3\land m=3 \\ A(4,4)-i B(4,4) & k=4\land m=-4 \\ -A(4,3)+i B(4,3) & k=4\land m=-3 \\ A(4,2)-i B(4,2) & k=4\land m=-2 \\ -A(4,1)+i B(4,1) & k=4\land m=-1 \\ A(4,0) & k=4\land m=0 \\ A(4,1)+i B(4,1) & k=4\land m=1 \\ A(4,2)+i B(4,2) & k=4\land m=2 \\ A(4,3)+i B(4,3) & k=4\land m=3 \\ A(4,4)+i B(4,4) & k=4\land m=4 \\ -A(5,5)+i B(5,5) & k=5\land m=-5 \\ A(5,4)-i B(5,4) & k=5\land m=-4 \\ -A(5,3)+i B(5,3) & k=5\land m=-3 \\ A(5,2)-i B(5,2) & k=5\land m=-2 \\ -A(5,1)+i B(5,1) & k=5\land m=-1 \\ A(5,0) & k=5\land m=0 \\ A(5,1)+i B(5,1) & k=5\land m=1 \\ A(5,2)+i B(5,2) & k=5\land m=2 \\ A(5,3)+i B(5,3) & k=5\land m=3 \\ A(5,4)+i B(5,4) & k=5\land m=4 \\ A(5,5)+i B(5,5) & k=5\land m=5 \\ A(6,6)-i B(6,6) & k=6\land m=-6 \\ -A(6,5)+i B(6,5) & k=6\land m=-5 \\ A(6,4)-i B(6,4) & k=6\land m=-4 \\ -A(6,3)+i B(6,3) & k=6\land m=-3 \\ A(6,2)-i B(6,2) & k=6\land m=-2 \\ -A(6,1)+i B(6,1) & k=6\land m=-1 \\ A(6,0) & k=6\land m=0 \\ A(6,1)+i B(6,1) & k=6\land m=1 \\ A(6,2)+i B(6,2) & k=6\land m=2 \\ A(6,3)+i B(6,3) & k=6\land m=3 \\ A(6,4)+i B(6,4) & k=6\land m=4 \\ A(6,5)+i B(6,5) & k=6\land m=5 \\ A(6,6)+i B(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Mathematica (Quanty.nb)
- Akm_C1.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {-A[4, 1] + I*B[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1] + I*B[4, 1], k == 4 && m == 1}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 3] + I*B[4, 3], k == 4 && m == 3}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 0], k == 5 && m == 0}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 4] + I*B[5, 4], k == 5 && m == 4}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {-A[6, 5] + I*B[6, 5], k == 6 && m == -5}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {-A[6, 3] + I*B[6, 3], k == 6 && m == -3}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {-A[6, 1] + I*B[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {A[6, 1] + I*B[6, 1], k == 6 && m == 1}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 3] + I*B[6, 3], k == 6 && m == 3}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 5] + I*B[6, 5], k == 6 && m == 5}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]
Input format suitable for Quanty
- Akm_C1.Quanty
Akm = { 0, A(0,0)} , {1, 0, A(1,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , {1, 1, A(1,1) + (I)*(B(1,1))} , {2, 0, A(2,0)} , {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , {2, 1, A(2,1) + (I)*(B(2,1))} , {2,-2, A(2,2) + (-I)*(B(2,2))} , {2, 2, A(2,2) + (I)*(B(2,2))} , {3, 0, A(3,0)} , {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , {3, 1, A(3,1) + (I)*(B(3,1))} , {3,-2, A(3,2) + (-I)*(B(3,2))} , {3, 2, A(3,2) + (I)*(B(3,2))} , {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , {3, 3, A(3,3) + (I)*(B(3,3))} , {4, 0, A(4,0)} , {4,-1, (-1)*(A(4,1)) + (I)*(B(4,1))} , {4, 1, A(4,1) + (I)*(B(4,1))} , {4,-2, A(4,2) + (-I)*(B(4,2))} , {4, 2, A(4,2) + (I)*(B(4,2))} , {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , {4, 3, A(4,3) + (I)*(B(4,3))} , {4,-4, A(4,4) + (-I)*(B(4,4))} , {4, 4, A(4,4) + (I)*(B(4,4))} , {5, 0, A(5,0)} , {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , {5, 1, A(5,1) + (I)*(B(5,1))} , {5,-2, A(5,2) + (-I)*(B(5,2))} , {5, 2, A(5,2) + (I)*(B(5,2))} , {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , {5, 3, A(5,3) + (I)*(B(5,3))} , {5,-4, A(5,4) + (-I)*(B(5,4))} , {5, 4, A(5,4) + (I)*(B(5,4))} , {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} , {5, 5, A(5,5) + (I)*(B(5,5))} , {6, 0, A(6,0)} , {6,-1, (-1)*(A(6,1)) + (I)*(B(6,1))} , {6, 1, A(6,1) + (I)*(B(6,1))} , {6,-2, A(6,2) + (-I)*(B(6,2))} , {6, 2, A(6,2) + (I)*(B(6,2))} , {6,-3, (-1)*(A(6,3)) + (I)*(B(6,3))} , {6, 3, A(6,3) + (I)*(B(6,3))} , {6,-4, A(6,4) + (-I)*(B(6,4))} , {6, 4, A(6,4) + (I)*(B(6,4))} , {6,-5, (-1)*(A(6,5)) + (I)*(B(6,5))} , {6, 5, A(6,5) + (I)*(B(6,5))} , {6,-6, A(6,6) + (-I)*(B(6,6))} , {6, 6, A(6,6) + (I)*(B(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 $A_{n''l'',n'l'}(k,m)$
we can express the operator as $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$
The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter.
| $ $ | $ {Y_{0}^{(0)}} $ | $ {Y_{-1}^{(1)}} $ | $ {Y_{0}^{(1)}} $ | $ {Y_{1}^{(1)}} $ | $ {Y_{-2}^{(2)}} $ | $ {Y_{-1}^{(2)}} $ | $ {Y_{0}^{(2)}} $ | $ {Y_{1}^{(2)}} $ | $ {Y_{2}^{(2)}} $ | $ {Y_{-3}^{(3)}} $ | $ {Y_{-2}^{(3)}} $ | $ {Y_{-1}^{(3)}} $ | $ {Y_{0}^{(3)}} $ | $ {Y_{1}^{(3)}} $ | $ {Y_{2}^{(3)}} $ | $ {Y_{3}^{(3)}} $ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $ {Y_{0}^{(0)}} $ | $ \text{Ass}(0,0) $ | $\color{darkred}{ -\frac{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $\color{darkred}{ -\frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }$ | $ \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $ | $ -\frac{\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $ -\frac{-\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $ | $ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} $ | $\color{darkred}{ -\frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ | $\color{darkred}{ \frac{\text{Asf}(3,2)+i \text{Bsf}(3,2)}{\sqrt{7}} }$ | $\color{darkred}{ -\frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $\color{darkred}{ -\frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $\color{darkred}{ \frac{\text{Asf}(3,2)-i \text{Bsf}(3,2)}{\sqrt{7}} }$ | $\color{darkred}{ -\frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ |
| $ {Y_{-1}^{(1)}} $ | $\color{darkred}{ \frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }$ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $ \frac{1}{5} \sqrt{3} (-\text{App}(2,1)+i \text{Bpp}(2,1)) $ | $ -\frac{1}{5} \sqrt{6} (\text{App}(2,2)-i \text{Bpp}(2,2)) $ | $\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1)) }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}} }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$ | $\color{darkred}{ \frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $ \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} $ | $ \frac{\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} (\text{Apf}(2,1)+i \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} \sqrt{\frac{10}{21}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1))-\frac{3 (-\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}} $ | $ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $ | $ \frac{1}{3} (-\text{Apf}(4,3)+i \text{Bpf}(4,3)) $ | $ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $ |
| $ {Y_{0}^{(1)}} $ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $ -\frac{1}{5} \sqrt{3} (\text{App}(2,1)+i \text{Bpp}(2,1)) $ | $ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $ | $ -\frac{1}{5} \sqrt{3} (-\text{App}(2,1)+i \text{Bpp}(2,1)) $ | $\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$ | $\color{darkred}{ -\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ -\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$ | $ -\frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $ | $ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} $ | $ -\frac{2}{5} \sqrt{\frac{6}{7}} (\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,1)+i \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} \sqrt{\frac{6}{7}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1)) $ | $ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $ | $ -\frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $ |
| $ {Y_{1}^{(1)}} $ | $\color{darkred}{ \frac{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }$ | $ -\frac{1}{5} \sqrt{6} (\text{App}(2,2)+i \text{Bpp}(2,2)) $ | $ \frac{1}{5} \sqrt{3} (\text{App}(2,1)+i \text{Bpp}(2,1)) $ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $\color{darkred}{ \frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$ | $\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}} }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1)) }$ | $ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $ | $ \frac{1}{3} (\text{Apf}(4,3)+i \text{Bpf}(4,3)) $ | $ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) $ | $ \frac{1}{3} \sqrt{\frac{10}{21}} (\text{Apf}(4,1)+i \text{Bpf}(4,1))-\frac{3 (\text{Apf}(2,1)+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{-\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1)) $ | $ \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $ |
| $ {Y_{-2}^{(2)}} $ | $ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} $ | $\color{darkred}{ \sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$ | $\color{darkred}{ -\frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $ \frac{1}{7} \sqrt{6} (-\text{Add}(2,1)+i \text{Bdd}(2,1))-\frac{1}{21} \sqrt{5} (-\text{Add}(4,1)+i \text{Bdd}(4,1)) $ | $ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $ | $ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $ | $ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4)) $ | $\color{darkred}{ -\sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ -\frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)-i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)-i \text{Bdf}(3,2))}{3 \sqrt{7}} }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ | $\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ |
| $ {Y_{-1}^{(2)}} $ | $ \frac{-\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ \frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$ | $ \frac{1}{21} \sqrt{5} (\text{Add}(4,1)+i \text{Bdd}(4,1))-\frac{1}{7} \sqrt{6} (\text{Add}(2,1)+i \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} (-\text{Add}(2,1)+i \text{Bdd}(2,1))+\frac{1}{7} \sqrt{\frac{10}{3}} (-\text{Add}(4,1)+i \text{Bdd}(4,1)) $ | $ -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)-i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)-i \text{Bdd}(4,2)) $ | $ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)+i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ -\frac{\text{Adf}(3,2)-i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)-i \text{Bdf}(5,2))}{11 \sqrt{3}} }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ |
| $ {Y_{0}^{(2)}} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $\color{darkred}{ \frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ \frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) $ | $ \frac{1}{7} (-\text{Add}(2,1)-i \text{Bdd}(2,1))-\frac{1}{7} \sqrt{\frac{10}{3}} (\text{Add}(4,1)+i \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} (\text{Add}(2,1)-i \text{Bdd}(2,1))-\frac{1}{7} \sqrt{\frac{10}{3}} (-\text{Add}(4,1)+i \text{Bdd}(4,1)) $ | $ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ |
| $ {Y_{1}^{(2)}} $ | $ \frac{\text{Asd}(2,1)+i \text{Bsd}(2,1)}{\sqrt{5}} $ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$ | $\color{darkred}{ \frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $ | $ -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)+i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)+i \text{Bdd}(4,2)) $ | $ \frac{1}{7} (\text{Add}(2,1)+i \text{Bdd}(2,1))+\frac{1}{7} \sqrt{\frac{10}{3}} (\text{Add}(4,1)+i \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} \sqrt{5} (-\text{Add}(4,1)+i \text{Bdd}(4,1))-\frac{1}{7} \sqrt{6} (-\text{Add}(2,1)+i \text{Bdd}(2,1)) $ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ -\frac{\text{Adf}(3,2)+i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)+i \text{Bdf}(5,2))}{11 \sqrt{3}} }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)-i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$ |
| $ {Y_{2}^{(2)}} $ | $ \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $ | $\color{darkred}{ -\frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4)) $ | $ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $ | $ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) $ | $ \frac{1}{7} \sqrt{6} (\text{Add}(2,1)+i \text{Bdd}(2,1))-\frac{1}{21} \sqrt{5} (\text{Add}(4,1)+i \text{Bdd}(4,1)) $ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)+i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)+i \text{Bdf}(3,2))}{3 \sqrt{7}} }$ | $\color{darkred}{ -\frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ -\sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ |
| $ {Y_{-3}^{(3)}} $ | $\color{darkred}{ \frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ | $ \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $ | $ \frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $ | $ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $ | $\color{darkred}{ \sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)-i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$ | $\color{darkred}{ \frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ | $\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ | $ \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} (-\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{10}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{429} \sqrt{7} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $ | $ \frac{10}{143} \sqrt{\frac{7}{3}} (-\text{Aff}(6,3)+i \text{Bff}(6,3))-\frac{1}{11} \sqrt{7} (-\text{Aff}(4,3)+i \text{Bff}(4,3)) $ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $ | $ \frac{5}{13} \sqrt{\frac{14}{33}} (-\text{Aff}(6,5)+i \text{Bff}(6,5)) $ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $ |
| $ {Y_{-2}^{(3)}} $ | $\color{darkred}{ \frac{\text{Asf}(3,2)-i \text{Bsf}(3,2)}{\sqrt{7}} }$ | $ \sqrt{\frac{6}{35}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{-\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}} $ | $ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $ | $ \frac{1}{3} (\text{Apf}(4,3)-i \text{Bpf}(4,3)) $ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$ | $\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$ | $ \frac{1}{3} (-\text{Aff}(2,1)-i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{10}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{429} \sqrt{7} (\text{Aff}(6,1)+i \text{Bff}(6,1)) $ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $ | $ \frac{-\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} \sqrt{2} (-\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{143} \sqrt{\frac{35}{3}} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $ | $ -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $ | $ -\frac{1}{33} \sqrt{14} (-\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{5}{143} \sqrt{42} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $ | $ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $ | $ -\frac{5}{13} \sqrt{\frac{14}{33}} (-\text{Aff}(6,5)+i \text{Bff}(6,5)) $ |
| $ {Y_{-1}^{(3)}} $ | $\color{darkred}{ \frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $ \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} \sqrt{\frac{6}{7}} (-\text{Apf}(2,1)+i \text{Bpf}(2,1))+\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1)) $ | $ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $ | $\color{darkred}{ \frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ -\frac{\text{Adf}(3,2)-i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)-i \text{Bdf}(5,2))}{11 \sqrt{3}} }$ | $\color{darkred}{ \frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $ | $ -\frac{\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} \sqrt{2} (\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,1)+i \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} \sqrt{2} (-\text{Aff}(2,1)+i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{5}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{25}{429} \sqrt{14} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $ | $ -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)-i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $ | $ \frac{1}{33} \sqrt{14} (-\text{Aff}(4,3)+i \text{Bff}(4,3))+\frac{5}{143} \sqrt{42} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $ |
| $ {Y_{0}^{(3)}} $ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $ \frac{3 (\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}}-\frac{1}{3} \sqrt{\frac{10}{21}} (\text{Apf}(4,1)+i \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 (-\text{Apf}(2,1)+i \text{Bpf}(2,1))}{5 \sqrt{7}}-\frac{1}{3} \sqrt{\frac{10}{21}} (-\text{Apf}(4,1)+i \text{Bpf}(4,1)) $ | $\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)+i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)+i \text{Bdf}(3,2))}{3 \sqrt{7}} }$ | $\color{darkred}{ \sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ \sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)-i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)-i \text{Bdf}(3,2))}{3 \sqrt{7}} }$ | $ \frac{1}{11} \sqrt{7} (\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{10}{143} \sqrt{\frac{7}{3}} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $ | $ -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $ | $ -\frac{1}{15} \sqrt{2} (\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{5}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{25}{429} \sqrt{14} (\text{Aff}(6,1)+i \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} \sqrt{2} (-\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{5}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{25}{429} \sqrt{14} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $ | $ -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $ | $ \frac{1}{11} \sqrt{7} (-\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{10}{143} \sqrt{\frac{7}{3}} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $ |
| $ {Y_{1}^{(3)}} $ | $\color{darkred}{ \frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) $ | $ \frac{2}{5} \sqrt{\frac{6}{7}} (\text{Apf}(2,1)+i \text{Bpf}(2,1))+\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,1)+i \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) $ | $\color{darkred}{ \frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $\color{darkred}{ -\frac{\text{Adf}(3,2)+i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)+i \text{Bdf}(5,2))}{11 \sqrt{3}} }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ \frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $ | $ \frac{1}{33} \sqrt{14} (\text{Aff}(4,3)+i \text{Bff}(4,3))+\frac{5}{143} \sqrt{42} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $ | $ -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)+i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $ | $ \frac{1}{15} \sqrt{2} (\text{Aff}(2,1)+i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{5}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{25}{429} \sqrt{14} (\text{Aff}(6,1)+i \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{-\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} \sqrt{2} (-\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{143} \sqrt{\frac{35}{3}} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $ |
| $ {Y_{2}^{(3)}} $ | $\color{darkred}{ \frac{\text{Asf}(3,2)+i \text{Bsf}(3,2)}{\sqrt{7}} }$ | $ \frac{1}{3} (-\text{Apf}(4,3)-i \text{Bpf}(4,3)) $ | $ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} $ | $ \sqrt{\frac{6}{35}} (\text{Apf}(2,1)+i \text{Bpf}(2,1))-\frac{\text{Apf}(4,1)+i \text{Bpf}(4,1)}{3 \sqrt{7}} $ | $\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $ -\frac{5}{13} \sqrt{\frac{14}{33}} (\text{Aff}(6,5)+i \text{Bff}(6,5)) $ | $ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $ | $ -\frac{1}{33} \sqrt{14} (\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{5}{143} \sqrt{42} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $ | $ -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $ | $ \frac{\text{Aff}(2,1)+i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} \sqrt{2} (\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,1)+i \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} (\text{Aff}(2,1)-i \text{Bff}(2,1))+\frac{1}{11} \sqrt{\frac{10}{3}} (-\text{Aff}(4,1)+i \text{Bff}(4,1))-\frac{5}{429} \sqrt{7} (-\text{Aff}(6,1)+i \text{Bff}(6,1)) $ |
| $ {Y_{3}^{(3)}} $ | $\color{darkred}{ \frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ | $ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $ | $ \frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $ | $ \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} $ | $\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ | $\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$ | $\color{darkred}{ \frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)+i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$ | $\color{darkred}{ \sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) $ | $ \frac{5}{13} \sqrt{\frac{14}{33}} (\text{Aff}(6,5)+i \text{Bff}(6,5)) $ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $ | $ \frac{10}{143} \sqrt{\frac{7}{3}} (\text{Aff}(6,3)+i \text{Bff}(6,3))-\frac{1}{11} \sqrt{7} (\text{Aff}(4,3)+i \text{Bff}(4,3)) $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $ | $ \frac{1}{3} (\text{Aff}(2,1)+i \text{Bff}(2,1))-\frac{1}{11} \sqrt{\frac{10}{3}} (\text{Aff}(4,1)+i \text{Bff}(4,1))+\frac{5}{429} \sqrt{7} (\text{Aff}(6,1)+i \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_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) $ | $\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$ | $\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $ | $ -\sqrt{\frac{2}{5}} \text{Asd}(2,1) $ | $ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$ | $\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$ |
| $ p_x $ | $\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$ | $ \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}(2,2) $ | $ -\frac{1}{5} \sqrt{6} \text{App}(2,1) $ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$ | $\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $ | $ -\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}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $ | $ -\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}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $ | $ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) $ |
| $ p_y $ | $\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$ | $ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $ | $ \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) $ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$ | $ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $ | $ -\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) $ | $ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $ | $ \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) $ |
| $ p_z $ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $ -\frac{1}{5} \sqrt{6} \text{App}(2,1) $ | $ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $ | $ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $ | $\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$ | $ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $ | $ -\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}} $ | $ \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) $ | $ \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) $ |
| $ d_{x^2-y^2} $ | $ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$ | $ \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) $ | $ -\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) $ | $ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $ | $ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$ | $\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$ | $\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ |
| $ d_{3z^2-r^2} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$ | $\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $ \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) $ | $ \frac{1}{7} \sqrt{2} \text{Bdd}(2,1)+\frac{2}{7} \sqrt{\frac{5}{3}} \text{Bdd}(4,1) $ | $ -\frac{1}{7} \sqrt{2} \text{Add}(2,1)-\frac{2}{7} \sqrt{\frac{5}{3}} \text{Add}(4,1) $ | $ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,2) }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$ | $\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$ |
| $ d_{\text{yz}} $ | $ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$ | $ -\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) $ | $ \frac{1}{7} \sqrt{2} \text{Bdd}(2,1)+\frac{2}{7} \sqrt{\frac{5}{3}} \text{Bdd}(4,1) $ | $ \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}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $ | $ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ | $\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}-\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ |
| $ d_{\text{xz}} $ | $ -\sqrt{\frac{2}{5}} \text{Asd}(2,1) $ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$ | $ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $ | $ -\frac{1}{7} \sqrt{2} \text{Add}(2,1)-\frac{2}{7} \sqrt{\frac{5}{3}} \text{Add}(4,1) $ | $ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $ | $ \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) $ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}+\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$ | $\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ |
| $ d_{\text{xy}} $ | $ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$ | $\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$ | $ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $ | $ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $ | $ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $ | $ \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) $ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ |
| $ f_{\text{xyz}} $ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$ | $ \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}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) $ | $ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,2) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $ \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) $ | $ -\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) $ | $ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $ | $ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $ | $ -\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) $ | $ -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $ | $ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $ |
| $ f_{x\left(5x^2-r^2\right)} $ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$ | $ -\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}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$ | $\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}-\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $ -\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) $ | $ \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}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $ | $ \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $ | $ \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}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $ | $ -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(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}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $ | $ -\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) $ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}+\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $ | $ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $ | $ \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) $ | $ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $ | $ -\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) $ |
| $ f_{z\left(5z^2-r^2\right)} $ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $ | $ \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}} $ | $\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$ | $\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) }$ | $ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $ | $ \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $ | $ -\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) $ | $ \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) $ | $ \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) $ |
| $ f_{x\left(y^2-z^2\right)} $ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$ | $ -\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}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $ | $ \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) $ | $\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$ | $\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $ -\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) $ | $ \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}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $ | $ \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) $ | $ \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}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $ | $ \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $ |
| $ f_{y\left(z^2-x^2\right)} $ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$ | $ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $ | $ \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) $ | $\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$ | $ -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $ | $ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $ | $ -\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) $ | $ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $ | $ \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) $ |
| $ f_{z\left(x^2-y^2\right)} $ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$ | $ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) $ | $ -\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) $ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$ | $\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ | $\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$ | $ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $ | $ -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $ | $ -\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) $ | $ \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $ | $ -\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) $ |
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
Potential for p orbitals
Potential for d orbitals
Potential for f orbitals
Coupling between two shells
Click on one of the subsections to expand it or
Potential for s-p orbital mixing
Potential for s-d orbital mixing
Potential for s-f orbital mixing
Potential for p-d orbital mixing
Potential for p-f orbital mixing
Potential for d-f orbital mixing
Table of several point groups
Return to Main page on Point Groups
| Nonaxial groups | C1 | Cs | Ci | ||||
|---|---|---|---|---|---|---|---|
| Cn groups | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
| Dn groups | D2 | D3 | D4 | D5 | D6 | D7 | D8 |
| Cnv groups | C2v | C3v | C4v | C5v | C6v | C7v | C8v |
| Cnh groups | C2h | C3h | C4h | C5h | C6h | ||
| Dnh groups | D2h | D3h | D4h | D5h | D6h | D7h | D8h |
| Dnd groups | D2d | D3d | D4d | D5d | D6d | D7d | D8d |
| Sn groups | S2 | S4 | S6 | S8 | S10 | S12 | |
| Cubic groups | T | Th | Td | O | Oh | I | Ih |
| Linear groups | C$\infty$v | D$\infty$h |
