Table of Contents
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Orientation Z
Symmetry Operations
In the Cs Point Group, with orientation Z there are the following symmetry operations
| Operator | Orientation |
|---|---|
| $\text{E}$ | $\{0,0,0\}$ , |
| $\sigma _h$ | $\{0,0,1\}$ , |
Different Settings
Character Table
| $ $ | $ \text{E} \,{\text{(1)}} $ | $ \sigma_h \,{\text{(1)}} $ |
|---|---|---|
| $ \text{A'} $ | $ 1 $ | $ 1 $ |
| $ \text{A''} $ | $ 1 $ | $ -1 $ |
Product Table
| $ $ | $ \text{A'} $ | $ \text{A''} $ |
|---|---|---|
| $ \text{A'} $ | $ \text{A'} $ | $ \text{A''} $ |
| $ \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 Cs Point group with orientation Z the form of the expansion coefficients is:
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(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) $ | $\color{darkred}{ -\frac{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }$ | $ \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $ | $ 0 $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $ 0 $ | $ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} $ | $\color{darkred}{ -\frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ |
| $ {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) $ | $ 0 $ | $ -\frac{1}{5} \sqrt{6} (\text{App}(2,2)-i \text{Bpp}(2,2)) $ | $\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $ \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}} $ | $ 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)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $ | $ 0 $ | $ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $ |
| $ {Y_{0}^{(1)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \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}} $ | $ 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)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $ | $ 0 $ |
| $ {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)) $ | $ 0 $ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $\color{darkred}{ \frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1)) }$ | $ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $ | $ 0 $ | $ \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)) $ | $ 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)-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}} $ | $\color{darkred}{ \sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $ 0 $ | $ \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)) $ | $ 0 $ | $ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4)) $ | $\color{darkred}{ -\sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ |
| $ {Y_{-1}^{(2)}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ 0 }$ | $ 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)-i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)-i \text{Bdd}(4,2)) $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ |
| $ {Y_{0}^{(2)}} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $\color{darkred}{ \frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $ \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)) $ | $ 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)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ |
| $ {Y_{1}^{(2)}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ -\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)) $ | $ 0 $ | $ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ |
| $ {Y_{2}^{(2)}} $ | $ \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $ | $\color{darkred}{ -\frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }$ | $ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4)) $ | $ 0 $ | $ \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)) $ | $ 0 $ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ |
| $ {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}} $ | $ 0 $ | $ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $ | $\color{darkred}{ \sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ | $ \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) $ | $ 0 $ | $ -\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)) $ | $ 0 $ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $ | $ 0 $ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $ |
| $ {Y_{-2}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \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}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ | $ 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)-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)) $ | $ 0 $ | $ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $ | $ 0 $ |
| $ {Y_{-1}^{(3)}} $ | $\color{darkred}{ \frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }$ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $ 0 $ | $ \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)) $ | $\color{darkred}{ \frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $ -\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)) $ | $ 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)-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)) $ | $ 0 $ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $ |
| $ {Y_{0}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ -\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)) $ | $ 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)-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)) $ | $ 0 $ |
| $ {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)) $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $\color{darkred}{ \frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }$ | $ \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)) $ | $ 0 $ | $ -\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)) $ | $ 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)-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)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \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}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $ | $ 0 $ | $ -\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)) $ | $ 0 $ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $ | $ 0 $ |
| $ {Y_{3}^{(3)}} $ | $\color{darkred}{ \frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$ | $ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $ | $ 0 $ | $ \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}} $ | $\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) $ | $ 0 $ | $ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $ | $ 0 $ | $ -\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)) $ | $ 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_{x\left(5z^2-r^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 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_{x\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) $ | $\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$ | $\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$ | $\color{darkred}{ 0 }$ | $ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $ 0 $ | $ 0 $ | $ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$ | $\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$ | $\color{darkred}{ 0 }$ |
| $ 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) $ | $ 0 $ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$ | $\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $ 0 $ | $ -\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) $ | $ 0 $ | $ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $ | $ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $ | $ 0 $ |
| $ 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) $ | $ 0 $ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$ | $ 0 $ | $ \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) $ | $ 0 $ | $ -\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}} $ | $ 0 $ |
| $ p_z $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$ | $\color{darkred}{ 0 }$ | $ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $ | $ 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) $ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $\color{darkred}{ 0 }$ | $ \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) $ | $ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $ | $ 0 $ | $ 0 $ | $ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$ | $\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $\color{darkred}{ 0 }$ |
| $ d_{3z^2-r^2} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$ | $\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$ | $\color{darkred}{ 0 }$ | $ \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 $ | $ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$ | $\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$ | $\color{darkred}{ 0 }$ |
| $ d_{\text{yz}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $ | $ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $ | $ 0 $ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ |
| $ d_{\text{xz}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$ | $ 0 $ | $ 0 $ | $ -\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) $ | $ 0 $ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ |
| $ d_{\text{xy}} $ | $ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $ | $\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$ | $\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$ | $\color{darkred}{ 0 }$ | $ -\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) $ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$ | $\color{darkred}{ 0 }$ |
| $ f_{\text{xyz}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ | $\color{darkred}{ 0 }$ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $ | $ 0 $ | $ 0 $ | $ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $ | $ 0 $ | $ 0 $ | $ -\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) $ | $ 0 $ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $ 0 $ | $ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $ | $ \frac{\text{Bff}(2,2)}{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) $ | $ 0 $ | $ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $ | $ \frac{\text{Bff}(2,2)}{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) $ | $ 0 $ |
| $ 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) $ | $ 0 $ | $\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$ | $\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$ | $ 0 $ | $ \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) $ | $ 0 $ | $ -\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) $ | $ 0 $ |
| $ f_{x\left(5z^2-r^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$ | $\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$ | $\color{darkred}{ 0 }$ | $ \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) $ | $ 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)} $ | $\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$ | $ -\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}} $ | $ 0 $ | $\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $ 0 $ | $ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $ | $ -\frac{\text{Bff}(2,2)}{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) $ | $ 0 $ | $ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $ | $ \frac{\text{Bff}(2,2)}{\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) $ | $ 0 $ |
| $ 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}} $ | $ 0 $ | $\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$ | $\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$ | $ 0 $ | $ \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) $ | $ 0 $ | $ \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) $ | $ 0 $ |
| $ f_{z\left(x^2-y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$ | $\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$ | $\color{darkred}{ 0 }$ | $ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $ | $ 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 $l'$.
Potential for s orbitals
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} \text{Eag} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, Eag} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{0}^{(0)}} $ |
|---|---|
| $ {Y_{0}^{(0)}} $ | $ \text{Eag} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ \text{s} $ |
|---|---|
| $ \text{s} $ | $ \text{Eag} $ |
Potential for p orbitals
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Epxpx}+\text{Epypy}+\text{Epzpz}) & k=0\land m=0 \\ \frac{5 (\text{Epxpx}+2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=-2 \\ \frac{5 (\text{Epzpx}+i \text{Epypz})}{\sqrt{6}} & k=2\land m=-1 \\ -\frac{5}{6} (\text{Epxpx}+\text{Epypy}-2 \text{Epzpz}) & k=2\land m=0 \\ \frac{5 i (\text{Epypz}+i \text{Epzpx})}{\sqrt{6}} & k=2\land m=1 \\ \frac{5 (\text{Epxpx}-2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=2 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, (1/3)*(Epxpx + Epypy + Epzpz)} , {2, 0, (-5/6)*(Epxpx + Epypy + (-2)*(Epzpz))} , {2,-1, (5)*((1/(sqrt(6)))*((I)*(Epypz) + Epzpx))} , {2, 1, (5*I)*((1/(sqrt(6)))*(Epypz + (I)*(Epzpx)))} , {2, 2, (5/2)*((1/(sqrt(6)))*(Epxpx + (-2*I)*(Epypx) + (-1)*(Epypy)))} , {2,-2, (5/2)*((1/(sqrt(6)))*(Epxpx + (2*I)*(Epypx) + (-1)*(Epypy)))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-1}^{(1)}} $ | $ {Y_{0}^{(1)}} $ | $ {Y_{1}^{(1)}} $ |
|---|---|---|---|
| $ {Y_{-1}^{(1)}} $ | $ \frac{\text{Epxpx}+\text{Epypy}}{2} $ | $ \frac{\text{Epzpx}+i \text{Epypz}}{\sqrt{2}} $ | $ \frac{1}{2} (-\text{Epxpx}-2 i \text{Epypx}+\text{Epypy}) $ |
| $ {Y_{0}^{(1)}} $ | $ \frac{\text{Epzpx}-i \text{Epypz}}{\sqrt{2}} $ | $ \text{Epzpz} $ | $ -\frac{\text{Epzpx}+i \text{Epypz}}{\sqrt{2}} $ |
| $ {Y_{1}^{(1)}} $ | $ \frac{1}{2} (-\text{Epxpx}+2 i \text{Epypx}+\text{Epypy}) $ | $ \frac{i (\text{Epypz}+i \text{Epzpx})}{\sqrt{2}} $ | $ \frac{\text{Epxpx}+\text{Epypy}}{2} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ p_x $ | $ p_y $ | $ p_z $ |
|---|---|---|---|
| $ p_x $ | $ \text{Epxpx} $ | $ \text{Epypx} $ | $ \text{Epzpx} $ |
| $ p_y $ | $ \text{Epypx} $ | $ \text{Epypy} $ | $ \text{Epypz} $ |
| $ p_z $ | $ \text{Epzpx} $ | $ \text{Epypz} $ | $ \text{Epzpz} $ |
Potential for d orbitals
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Edx2y2dx2y2}+\text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+\text{Edz2dz2}) & k=0\land m=0 \\ \frac{-4 i \text{Edxydz2}+\sqrt{3} \text{Edxzdxz}+2 i \sqrt{3} \text{Edyzdxz}-\sqrt{3} \text{Edyzdyz}-4 \text{Edz2dx2y2}}{2 \sqrt{2}} & k=2\land m=-2 \\ \frac{i \sqrt{3} \text{Edxydxz}+\sqrt{3} \text{Edxydyz}+\sqrt{3} \text{Edxzdx2y2}-i \sqrt{3} \text{Edyzdx2y2}+i \text{Edyzdz2}+\text{Edz2dxz}}{\sqrt{2}} & k=2\land m=-1 \\ -\text{Edx2y2dx2y2}-\text{Edxydxy}+\frac{\text{Edxzdxz}+\text{Edyzdyz}}{2}+\text{Edz2dz2} & k=2\land m=0 \\ \frac{i \left(\sqrt{3} \text{Edxydxz}+i \sqrt{3} \text{Edxydyz}+i \sqrt{3} \text{Edxzdx2y2}-\sqrt{3} \text{Edyzdx2y2}+\text{Edyzdz2}+i \text{Edz2dxz}\right)}{\sqrt{2}} & k=2\land m=1 \\ \frac{4 i \text{Edxydz2}+\sqrt{3} \text{Edxzdxz}-2 i \sqrt{3} \text{Edyzdxz}-\sqrt{3} \text{Edyzdyz}-4 \text{Edz2dx2y2}}{2 \sqrt{2}} & k=2\land m=2 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}+2 i \text{Edxydx2y2}-\text{Edxydxy}) & k=4\land m=-4 \\ \frac{3}{2} \sqrt{\frac{7}{5}} (i \text{Edxydxz}-\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}) & k=4\land m=-3 \\ \frac{3 \left(i \sqrt{3} \text{Edxydz2}+\text{Edxzdxz}+2 i \text{Edyzdxz}-\text{Edyzdyz}+\sqrt{3} \text{Edz2dx2y2}\right)}{\sqrt{10}} & k=4\land m=-2 \\ \frac{3 \left(-i \text{Edxydxz}-\text{Edxydyz}-\text{Edxzdx2y2}+i \text{Edyzdx2y2}+2 i \sqrt{3} \text{Edyzdz2}+2 \sqrt{3} \text{Edz2dxz}\right)}{2 \sqrt{5}} & k=4\land m=-1 \\ \frac{3}{10} (\text{Edx2y2dx2y2}+\text{Edxydxy}-4 (\text{Edxzdxz}+\text{Edyzdyz})+6 \text{Edz2dz2}) & k=4\land m=0 \\ \frac{3 \left(-i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}+2 i \sqrt{3} \text{Edyzdz2}-2 \sqrt{3} \text{Edz2dxz}\right)}{2 \sqrt{5}} & k=4\land m=1 \\ \frac{3 \left(-i \sqrt{3} \text{Edxydz2}+\text{Edxzdxz}-2 i \text{Edyzdxz}-\text{Edyzdyz}+\sqrt{3} \text{Edz2dx2y2}\right)}{\sqrt{10}} & k=4\land m=2 \\ \frac{3}{2} \sqrt{\frac{7}{5}} (i \text{Edxydxz}+\text{Edxydyz}-\text{Edxzdx2y2}+i \text{Edyzdx2y2}) & k=4\land m=3 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}-2 i \text{Edxydx2y2}-\text{Edxydxy}) & k=4\land m=4 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, (1/5)*(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)} , {2, 0, (-1)*(Edx2y2dx2y2) + (-1)*(Edxydxy) + (1/2)*(Edxzdxz + Edyzdyz) + Edz2dz2} , {2,-1, (1/(sqrt(2)))*((I)*((sqrt(3))*(Edxydxz)) + (sqrt(3))*(Edxydyz) + (sqrt(3))*(Edxzdx2y2) + (-I)*((sqrt(3))*(Edyzdx2y2)) + (I)*(Edyzdz2) + Edz2dxz)} , {2, 1, (I)*((1/(sqrt(2)))*((sqrt(3))*(Edxydxz) + (I)*((sqrt(3))*(Edxydyz)) + (I)*((sqrt(3))*(Edxzdx2y2)) + (-1)*((sqrt(3))*(Edyzdx2y2)) + Edyzdz2 + (I)*(Edz2dxz)))} , {2,-2, (1/2)*((1/(sqrt(2)))*((-4*I)*(Edxydz2) + (sqrt(3))*(Edxzdxz) + (2*I)*((sqrt(3))*(Edyzdxz)) + (-1)*((sqrt(3))*(Edyzdyz)) + (-4)*(Edz2dx2y2)))} , {2, 2, (1/2)*((1/(sqrt(2)))*((4*I)*(Edxydz2) + (sqrt(3))*(Edxzdxz) + (-2*I)*((sqrt(3))*(Edyzdxz)) + (-1)*((sqrt(3))*(Edyzdyz)) + (-4)*(Edz2dx2y2)))} , {4, 0, (3/10)*(Edx2y2dx2y2 + Edxydxy + (-4)*(Edxzdxz + Edyzdyz) + (6)*(Edz2dz2))} , {4,-1, (3/2)*((1/(sqrt(5)))*((-I)*(Edxydxz) + (-1)*(Edxydyz) + (-1)*(Edxzdx2y2) + (I)*(Edyzdx2y2) + (2*I)*((sqrt(3))*(Edyzdz2)) + (2)*((sqrt(3))*(Edz2dxz))))} , {4, 1, (3/2)*((1/(sqrt(5)))*((-I)*(Edxydxz) + Edxydyz + Edxzdx2y2 + (I)*(Edyzdx2y2) + (2*I)*((sqrt(3))*(Edyzdz2)) + (-2)*((sqrt(3))*(Edz2dxz))))} , {4, 2, (3)*((1/(sqrt(10)))*((-I)*((sqrt(3))*(Edxydz2)) + Edxzdxz + (-2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (sqrt(3))*(Edz2dx2y2)))} , {4,-2, (3)*((1/(sqrt(10)))*((I)*((sqrt(3))*(Edxydz2)) + Edxzdxz + (2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (sqrt(3))*(Edz2dx2y2)))} , {4,-3, (3/2)*((sqrt(7/5))*((I)*(Edxydxz) + (-1)*(Edxydyz) + Edxzdx2y2 + (I)*(Edyzdx2y2)))} , {4, 3, (3/2)*((sqrt(7/5))*((I)*(Edxydxz) + Edxydyz + (-1)*(Edxzdx2y2) + (I)*(Edyzdx2y2)))} , {4, 4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (-2*I)*(Edxydx2y2) + (-1)*(Edxydxy)))} , {4,-4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (2*I)*(Edxydx2y2) + (-1)*(Edxydxy)))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-2}^{(2)}} $ | $ {Y_{-1}^{(2)}} $ | $ {Y_{0}^{(2)}} $ | $ {Y_{1}^{(2)}} $ | $ {Y_{2}^{(2)}} $ |
|---|---|---|---|---|---|
| $ {Y_{-2}^{(2)}} $ | $ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $ | $ \frac{1}{2} (i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}-i \text{Edyzdx2y2}) $ | $ \frac{\text{Edz2dx2y2}+i \text{Edxydz2}}{\sqrt{2}} $ | $ -\frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}-\text{Edxzdx2y2})+\text{Edyzdx2y2}) $ | $ \frac{1}{2} (\text{Edx2y2dx2y2}+2 i \text{Edxydx2y2}-\text{Edxydxy}) $ |
| $ {Y_{-1}^{(2)}} $ | $ \frac{1}{2} (-i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}) $ | $ \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} $ | $ \frac{\text{Edz2dxz}+i \text{Edyzdz2}}{\sqrt{2}} $ | $ \frac{1}{2} (-\text{Edxzdxz}-2 i \text{Edyzdxz}+\text{Edyzdyz}) $ | $ \frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}-\text{Edxzdx2y2})+\text{Edyzdx2y2}) $ |
| $ {Y_{0}^{(2)}} $ | $ \frac{\text{Edz2dx2y2}-i \text{Edxydz2}}{\sqrt{2}} $ | $ \frac{\text{Edz2dxz}-i \text{Edyzdz2}}{\sqrt{2}} $ | $ \text{Edz2dz2} $ | $ \frac{-\text{Edz2dxz}-i \text{Edyzdz2}}{\sqrt{2}} $ | $ \frac{\text{Edz2dx2y2}+i \text{Edxydz2}}{\sqrt{2}} $ |
| $ {Y_{1}^{(2)}} $ | $ \frac{1}{2} i (\text{Edxydxz}-i \text{Edxydyz}+i \text{Edxzdx2y2}+\text{Edyzdx2y2}) $ | $ \frac{1}{2} (-\text{Edxzdxz}+2 i \text{Edyzdxz}+\text{Edyzdyz}) $ | $ \frac{i (\text{Edyzdz2}+i \text{Edz2dxz})}{\sqrt{2}} $ | $ \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} $ | $ -\frac{1}{2} i (\text{Edxydxz}-i (\text{Edxydyz}+\text{Edxzdx2y2})-\text{Edyzdx2y2}) $ |
| $ {Y_{2}^{(2)}} $ | $ \frac{1}{2} (\text{Edx2y2dx2y2}-2 i \text{Edxydx2y2}-\text{Edxydxy}) $ | $ -\frac{1}{2} i (\text{Edxydxz}-i \text{Edxydyz}+i \text{Edxzdx2y2}+\text{Edyzdx2y2}) $ | $ \frac{\text{Edz2dx2y2}-i \text{Edxydz2}}{\sqrt{2}} $ | $ \frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2})) $ | $ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ d_{x^2-y^2} $ | $ d_{3z^2-r^2} $ | $ d_{\text{yz}} $ | $ d_{\text{xz}} $ | $ d_{\text{xy}} $ |
|---|---|---|---|---|---|
| $ d_{x^2-y^2} $ | $ \text{Edx2y2dx2y2} $ | $ \text{Edz2dx2y2} $ | $ \text{Edyzdx2y2} $ | $ \text{Edxzdx2y2} $ | $ \text{Edxydx2y2} $ |
| $ d_{3z^2-r^2} $ | $ \text{Edz2dx2y2} $ | $ \text{Edz2dz2} $ | $ \text{Edyzdz2} $ | $ \text{Edz2dxz} $ | $ \text{Edxydz2} $ |
| $ d_{\text{yz}} $ | $ \text{Edyzdx2y2} $ | $ \text{Edyzdz2} $ | $ \text{Edyzdyz} $ | $ \text{Edyzdxz} $ | $ \text{Edxydyz} $ |
| $ d_{\text{xz}} $ | $ \text{Edxzdx2y2} $ | $ \text{Edz2dxz} $ | $ \text{Edyzdxz} $ | $ \text{Edxzdxz} $ | $ \text{Edxydxz} $ |
| $ d_{\text{xy}} $ | $ \text{Edxydx2y2} $ | $ \text{Edxydz2} $ | $ \text{Edxydyz} $ | $ \text{Edxydxz} $ | $ \text{Edxydxy} $ |
Potential for f orbitals
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-3}^{(3)}} $ | $ {Y_{-2}^{(3)}} $ | $ {Y_{-1}^{(3)}} $ | $ {Y_{0}^{(3)}} $ | $ {Y_{1}^{(3)}} $ | $ {Y_{2}^{(3)}} $ | $ {Y_{3}^{(3)}} $ |
|---|---|---|---|---|---|---|---|
| $ {Y_{-3}^{(3)}} $ | $ A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) $ | $ 0 $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)-i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i \text{Ap}(6,2)) $ | $ 0 $ | $ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)-i \text{Ap}(6,4))\right) $ | $ 0 $ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)-i \text{Ap}(6,6)) $ |
| $ {Y_{-2}^{(3)}} $ | $ 0 $ | $ A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) $ | $ 0 $ | $ -\frac{2 (A(2,2)-i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i \text{Ap}(6,2)) $ | $ 0 $ | $ \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)-i \text{Ap}(4,4))+30 (A(6,4)-i \text{Ap}(6,4))\right) $ | $ 0 $ |
| $ {Y_{-1}^{(3)}} $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i \text{Ap}(6,2)) $ | $ 0 $ | $ A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0) $ | $ 0 $ | $ \frac{2 \left(-143 \sqrt{6} (A(2,2)-i \text{Ap}(2,2))-65 \sqrt{10} (A(4,2)-i \text{Ap}(4,2))-25 \sqrt{105} (A(6,2)-i \text{Ap}(6,2))\right)}{2145} $ | $ 0 $ | $ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)-i \text{Ap}(6,4))\right) $ |
| $ {Y_{0}^{(3)}} $ | $ 0 $ | $ -\frac{2 (A(2,2)+i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i \text{Ap}(6,2)) $ | $ 0 $ | $ A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0)) $ | $ 0 $ | $ -\frac{2 (A(2,2)-i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i \text{Ap}(6,2)) $ | $ 0 $ |
| $ {Y_{1}^{(3)}} $ | $ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)+i \text{Ap}(6,4))\right) $ | $ 0 $ | $ \frac{2 \left(-143 \sqrt{6} (A(2,2)+i \text{Ap}(2,2))-65 \sqrt{10} (A(4,2)+i \text{Ap}(4,2))-25 \sqrt{105} (A(6,2)+i \text{Ap}(6,2))\right)}{2145} $ | $ 0 $ | $ A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0) $ | $ 0 $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)-i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i \text{Ap}(6,2)) $ |
| $ {Y_{2}^{(3)}} $ | $ 0 $ | $ \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)+i \text{Ap}(4,4))+30 (A(6,4)+i \text{Ap}(6,4))\right) $ | $ 0 $ | $ -\frac{2 (A(2,2)+i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i \text{Ap}(6,2)) $ | $ 0 $ | $ A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) $ | $ 0 $ |
| $ {Y_{3}^{(3)}} $ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)+i \text{Ap}(6,6)) $ | $ 0 $ | $ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)+i \text{Ap}(6,4))\right) $ | $ 0 $ | $ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i \text{Ap}(6,2)) $ | $ 0 $ | $ A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ f_{\text{xyz}} $ | $ f_{x\left(5x^2-r^2\right)} $ | $ f_{y\left(5y^2-r^2\right)} $ | $ f_{x\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}} $ | $ \frac{1}{429} \left(429 A(0,0)-91 A(4,0)-13 \sqrt{70} A(4,4)+30 A(6,0)-30 \sqrt{14} A(6,4)\right) $ | $ 0 $ | $ 0 $ | $ \frac{286 \sqrt{10} \text{Ap}(2,2)+65 \sqrt{6} \text{Ap}(4,2)-200 \sqrt{7} \text{Ap}(6,2)}{2145} $ | $ 0 $ | $ 0 $ | $ -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} \text{Ap}(4,4)+30 \text{Ap}(6,4)\right) $ |
| $ f_{x\left(5x^2-r^2\right)} $ | $ 0 $ | $ \frac{8580 A(0,0)-1144 A(2,0)+1144 \sqrt{6} A(2,2)+585 A(4,0)-390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)+125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)+125 \sqrt{231} A(6,6)}{8580} $ | $ \frac{286 \sqrt{6} \text{Ap}(2,2)-780 \sqrt{10} \text{Ap}(4,2)-25 \sqrt{105} \text{Ap}(6,2)+125 \sqrt{231} \text{Ap}(6,6)}{8580} $ | $ 0 $ | $ \frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)-26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)-85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)+15 \sqrt{385} A(6,6)\right)}{8580} $ | $ \frac{286 \sqrt{10} \text{Ap}(2,2)+5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)-20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} $ | $ 0 $ |
| $ f_{y\left(5y^2-r^2\right)} $ | $ 0 $ | $ \frac{286 \sqrt{6} \text{Ap}(2,2)-780 \sqrt{10} \text{Ap}(4,2)-25 \sqrt{105} \text{Ap}(6,2)+125 \sqrt{231} \text{Ap}(6,6)}{8580} $ | $ \frac{8580 A(0,0)-1144 A(2,0)-1144 \sqrt{6} A(2,2)+585 A(4,0)+390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)-125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)-125 \sqrt{231} A(6,6)}{8580} $ | $ 0 $ | $ \frac{-286 \sqrt{10} \text{Ap}(2,2)-5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(-52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)+20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} $ | $ \frac{-572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)+5 \left(13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{8580} $ | $ 0 $ |
| $ f_{x\left(5z^2-r^2\right)} $ | $ \frac{286 \sqrt{10} \text{Ap}(2,2)+65 \sqrt{6} \text{Ap}(4,2)-200 \sqrt{7} \text{Ap}(6,2)}{2145} $ | $ 0 $ | $ 0 $ | $ A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0)) $ | $ 0 $ | $ 0 $ | $ \frac{-286 \sqrt{10} A(2,2)-65 \sqrt{6} A(4,2)+200 \sqrt{7} A(6,2)}{2145} $ |
| $ f_{x\left(y^2-z^2\right)} $ | $ 0 $ | $ \frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)-26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)-85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)+15 \sqrt{385} A(6,6)\right)}{8580} $ | $ \frac{-286 \sqrt{10} \text{Ap}(2,2)-5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(-52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)+20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} $ | $ 0 $ | $ \frac{1716 A(0,0)+91 A(4,0)+182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)+15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)+15 \sqrt{231} A(6,6)}{1716} $ | $ \frac{286 \sqrt{6} \text{Ap}(2,2)-52 \sqrt{10} \text{Ap}(4,2)+35 \sqrt{105} \text{Ap}(6,2)-15 \sqrt{231} \text{Ap}(6,6)}{1716} $ | $ 0 $ |
| $ f_{y\left(z^2-x^2\right)} $ | $ 0 $ | $ \frac{286 \sqrt{10} \text{Ap}(2,2)+5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)-20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} $ | $ \frac{-572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)+5 \left(13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{8580} $ | $ 0 $ | $ \frac{286 \sqrt{6} \text{Ap}(2,2)-52 \sqrt{10} \text{Ap}(4,2)+35 \sqrt{105} \text{Ap}(6,2)-15 \sqrt{231} \text{Ap}(6,6)}{1716} $ | $ \frac{1716 A(0,0)+91 A(4,0)-182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)-15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)-15 \sqrt{231} A(6,6)}{1716} $ | $ 0 $ |
| $ f_{z\left(x^2-y^2\right)} $ | $ -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} \text{Ap}(4,4)+30 \text{Ap}(6,4)\right) $ | $ 0 $ | $ 0 $ | $ \frac{-286 \sqrt{10} A(2,2)-65 \sqrt{6} A(4,2)+200 \sqrt{7} A(6,2)}{2145} $ | $ 0 $ | $ 0 $ | $ \frac{1}{429} \left(429 A(0,0)-91 A(4,0)+13 \sqrt{70} A(4,4)+30 A(6,0)+30 \sqrt{14} A(6,4)\right) $ |
Coupling between two shells
Potential for s-p orbital mixing
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-1}^{(1)}} $ | $ {Y_{0}^{(1)}} $ | $ {Y_{1}^{(1)}} $ |
|---|---|---|---|
| $ {Y_{0}^{(0)}} $ | $ -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{3}} $ | $ 0 $ | $ \frac{A(1,1)-i \text{Ap}(1,1)}{\sqrt{3}} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ p_x $ | $ p_y $ | $ p_z $ |
|---|---|---|---|
| $ \text{s} $ | $ -\sqrt{\frac{2}{3}} A(1,1) $ | $ \sqrt{\frac{2}{3}} \text{Ap}(1,1) $ | $ 0 $ |
Potential for s-d orbital mixing
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-2}^{(2)}} $ | $ {Y_{-1}^{(2)}} $ | $ {Y_{0}^{(2)}} $ | $ {Y_{1}^{(2)}} $ | $ {Y_{2}^{(2)}} $ |
|---|---|---|---|---|---|
| $ {Y_{0}^{(0)}} $ | $ \frac{A(2,2)+i \text{Ap}(2,2)}{\sqrt{5}} $ | $ 0 $ | $ \frac{A(2,0)}{\sqrt{5}} $ | $ 0 $ | $ \frac{A(2,2)-i \text{Ap}(2,2)}{\sqrt{5}} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ d_{x^2-y^2} $ | $ d_{3z^2-r^2} $ | $ d_{\text{yz}} $ | $ d_{\text{xz}} $ | $ d_{\text{xy}} $ |
|---|---|---|---|---|---|
| $ \text{s} $ | $ \sqrt{\frac{2}{5}} A(2,2) $ | $ \frac{A(2,0)}{\sqrt{5}} $ | $ 0 $ | $ 0 $ | $ -\sqrt{\frac{2}{5}} \text{Ap}(2,2) $ |
Potential for s-f orbital mixing
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-3}^{(3)}} $ | $ {Y_{-2}^{(3)}} $ | $ {Y_{-1}^{(3)}} $ | $ {Y_{0}^{(3)}} $ | $ {Y_{1}^{(3)}} $ | $ {Y_{2}^{(3)}} $ | $ {Y_{3}^{(3)}} $ |
|---|---|---|---|---|---|---|---|
| $ {Y_{0}^{(0)}} $ | $ -\frac{A(3,3)+i \text{Ap}(3,3)}{\sqrt{7}} $ | $ 0 $ | $ -\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{7}} $ | $ 0 $ | $ \frac{A(3,1)-i \text{Ap}(3,1)}{\sqrt{7}} $ | $ 0 $ | $ \frac{A(3,3)-i \text{Ap}(3,3)}{\sqrt{7}} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ f_{\text{xyz}} $ | $ f_{x\left(5x^2-r^2\right)} $ | $ f_{y\left(5y^2-r^2\right)} $ | $ f_{x\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 $ | $ \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) $ | $ -\frac{\sqrt{3} \text{Ap}(3,1)+\sqrt{5} \text{Ap}(3,3)}{2 \sqrt{7}} $ | $ 0 $ | $ \frac{\sqrt{5} A(3,1)+\sqrt{3} A(3,3)}{2 \sqrt{7}} $ | $ \frac{1}{14} \left(\sqrt{35} \text{Ap}(3,1)-\sqrt{21} \text{Ap}(3,3)\right) $ | $ 0 $ |
Potential for p-d orbital mixing
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-2}^{(2)}} $ | $ {Y_{-1}^{(2)}} $ | $ {Y_{0}^{(2)}} $ | $ {Y_{1}^{(2)}} $ | $ {Y_{2}^{(2)}} $ |
|---|---|---|---|---|---|
| $ {Y_{-1}^{(1)}} $ | $ \frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)+i \text{Ap}(3,1))-\sqrt{\frac{2}{5}} (A(1,1)+i \text{Ap}(1,1)) $ | $ 0 $ | $ \frac{1}{105} \left(7 \sqrt{15} (A(1,1)-i \text{Ap}(1,1))-9 \sqrt{10} (A(3,1)-i \text{Ap}(3,1))\right) $ | $ 0 $ | $ -\frac{3}{7} (A(3,3)-i \text{Ap}(3,3)) $ |
| $ {Y_{0}^{(1)}} $ | $ 0 $ | $ -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (A(3,1)+i \text{Ap}(3,1)) $ | $ 0 $ | $ \frac{7 A(1,1)+2 \sqrt{6} A(3,1)-i \left(7 \text{Ap}(1,1)+2 \sqrt{6} \text{Ap}(3,1)\right)}{7 \sqrt{5}} $ | $ 0 $ |
| $ {Y_{1}^{(1)}} $ | $ \frac{3}{7} (A(3,3)+i \text{Ap}(3,3)) $ | $ 0 $ | $ \frac{3}{7} \sqrt{\frac{2}{5}} (A(3,1)+i \text{Ap}(3,1))-\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{15}} $ | $ 0 $ | $ \sqrt{\frac{2}{5}} (A(1,1)-i \text{Ap}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)-i \text{Ap}(3,1)) $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ d_{x^2-y^2} $ | $ d_{3z^2-r^2} $ | $ d_{\text{yz}} $ | $ d_{\text{xz}} $ | $ d_{\text{xy}} $ |
|---|---|---|---|---|---|
| $ p_x $ | $ \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)-15 A(3,3)\right) $ | $ \sqrt{\frac{2}{15}} A(1,1)-\frac{6 A(3,1)}{7 \sqrt{5}} $ | $ 0 $ | $ 0 $ | $ \sqrt{\frac{2}{5}} \text{Ap}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Ap}(3,1)+\frac{3}{7} \text{Ap}(3,3) $ |
| $ p_y $ | $ \frac{1}{35} \left(-7 \sqrt{10} \text{Ap}(1,1)+\sqrt{15} \text{Ap}(3,1)+15 \text{Ap}(3,3)\right) $ | $ \frac{6 \text{Ap}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Ap}(1,1) $ | $ 0 $ | $ 0 $ | $ \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) $ |
| $ p_z $ | $ 0 $ | $ 0 $ | $ \sqrt{\frac{2}{5}} \text{Ap}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Ap}(3,1) $ | $ -\frac{1}{7} \sqrt{\frac{2}{5}} \left(7 A(1,1)+2 \sqrt{6} A(3,1)\right) $ | $ 0 $ |
Potential for p-f orbital mixing
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-3}^{(3)}} $ | $ {Y_{-2}^{(3)}} $ | $ {Y_{-1}^{(3)}} $ | $ {Y_{0}^{(3)}} $ | $ {Y_{1}^{(3)}} $ | $ {Y_{2}^{(3)}} $ | $ {Y_{3}^{(3)}} $ |
|---|---|---|---|---|---|---|---|
| $ {Y_{-1}^{(1)}} $ | $ \frac{3 (A(2,2)+i \text{Ap}(2,2))}{\sqrt{35}}-\frac{A(4,2)+i \text{Ap}(4,2)}{3 \sqrt{21}} $ | $ 0 $ | $ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $ | $ 0 $ | $ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i \text{Ap}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i \text{Ap}(4,2)) $ | $ 0 $ | $ -\frac{2 (A(4,4)-i \text{Ap}(4,4))}{3 \sqrt{3}} $ |
| $ {Y_{0}^{(1)}} $ | $ 0 $ | $ \sqrt{\frac{3}{35}} (A(2,2)+i \text{Ap}(2,2))+\frac{2 (A(4,2)+i \text{Ap}(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)-i \text{Ap}(2,2))+\frac{2 (A(4,2)-i \text{Ap}(4,2))}{3 \sqrt{7}} $ | $ 0 $ |
| $ {Y_{1}^{(1)}} $ | $ -\frac{2 (A(4,4)+i \text{Ap}(4,4))}{3 \sqrt{3}} $ | $ 0 $ | $ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i \text{Ap}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i \text{Ap}(4,2)) $ | $ 0 $ | $ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $ | $ 0 $ | $ \frac{3 (A(2,2)-i \text{Ap}(2,2))}{\sqrt{35}}-\frac{A(4,2)-i \text{Ap}(4,2)}{3 \sqrt{21}} $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ f_{\text{xyz}} $ | $ f_{x\left(5x^2-r^2\right)} $ | $ f_{y\left(5y^2-r^2\right)} $ | $ f_{x\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) $ | $ \frac{1}{630} \left(54 \sqrt{14} \text{Ap}(2,2)+5 \sqrt{30} \left(\sqrt{7} \text{Ap}(4,2)+7 \text{Ap}(4,4)\right)\right) $ | $ 0 $ | $ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $ | $ \sqrt{\frac{6}{35}} \text{Ap}(2,2)-\frac{\text{Ap}(4,2)}{\sqrt{14}}+\frac{\text{Ap}(4,4)}{3 \sqrt{2}} $ | $ 0 $ |
| $ p_y $ | $ 0 $ | $ \frac{1}{630} \left(54 \sqrt{14} \text{Ap}(2,2)+5 \sqrt{30} \left(\sqrt{7} \text{Ap}(4,2)-7 \text{Ap}(4,4)\right)\right) $ | $ \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 $ | $ -\sqrt{\frac{6}{35}} \text{Ap}(2,2)+\frac{\text{Ap}(4,2)}{\sqrt{14}}+\frac{\text{Ap}(4,4)}{3 \sqrt{2}} $ | $ \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 $ | $ -\sqrt{\frac{6}{35}} \text{Ap}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Ap}(4,2) $ | $ 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
Input format suitable for Mathematica (Quanty.nb)
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , {1, 1, A(1,1) + (I)*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(Ap(2,2))} , {2, 2, A(2,2) + (I)*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , {3, 1, A(3,1) + (I)*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , {3, 3, A(3,3) + (I)*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(Ap(4,2))} , {4, 2, A(4,2) + (I)*(Ap(4,2))} , {4,-4, A(4,4) + (-I)*(Ap(4,4))} , {4, 4, A(4,4) + (I)*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , {5, 1, A(5,1) + (I)*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , {5, 3, A(5,3) + (I)*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , {5, 5, A(5,5) + (I)*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(Ap(6,2))} , {6, 2, A(6,2) + (I)*(Ap(6,2))} , {6,-4, A(6,4) + (-I)*(Ap(6,4))} , {6, 4, A(6,4) + (I)*(Ap(6,4))} , {6,-6, A(6,6) + (-I)*(Ap(6,6))} , {6, 6, A(6,6) + (I)*(Ap(6,6))} }
The Hamiltonian on a basis of spherical Harmonics
| $ $ | $ {Y_{-3}^{(3)}} $ | $ {Y_{-2}^{(3)}} $ | $ {Y_{-1}^{(3)}} $ | $ {Y_{0}^{(3)}} $ | $ {Y_{1}^{(3)}} $ | $ {Y_{2}^{(3)}} $ | $ {Y_{3}^{(3)}} $ |
|---|---|---|---|---|---|---|---|
| $ {Y_{-2}^{(2)}} $ | $ -\sqrt{\frac{3}{7}} (A(1,1)+i \text{Ap}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)+i \text{Ap}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)+i \text{Ap}(5,1)) $ | $ 0 $ | $ \frac{33 \sqrt{35} (A(1,1)-i \text{Ap}(1,1))-22 \sqrt{210} (A(3,1)-i \text{Ap}(3,1))+25 \sqrt{21} (A(5,1)-i \text{Ap}(5,1))}{1155} $ | $ 0 $ | $ \frac{5}{33} \sqrt{2} (A(5,3)-i \text{Ap}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)-i \text{Ap}(3,3)) $ | $ 0 $ | $ \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i \text{Ap}(5,5)) $ |
| $ {Y_{-1}^{(2)}} $ | $ 0 $ | $ -\sqrt{\frac{2}{7}} (A(1,1)+i \text{Ap}(1,1))-\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (A(5,1)+i \text{Ap}(5,1)) $ | $ 0 $ | $ \frac{33 \sqrt{105} (A(1,1)-i \text{Ap}(1,1))-11 \sqrt{70} (A(3,1)-i \text{Ap}(3,1))-100 \sqrt{7} (A(5,1)-i \text{Ap}(5,1))}{1155} $ | $ 0 $ | $ -\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i \text{Ap}(3,3))-\frac{4}{33} \sqrt{5} (A(5,3)-i \text{Ap}(5,3)) $ | $ 0 $ |
| $ {Y_{0}^{(2)}} $ | $ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i \text{Ap}(3,3))-\frac{2}{33} \sqrt{5} (A(5,3)+i \text{Ap}(5,3)) $ | $ 0 $ | $ -\sqrt{\frac{6}{35}} (A(1,1)+i \text{Ap}(1,1))-\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (A(5,1)+i \text{Ap}(5,1)) $ | $ 0 $ | $ \frac{1}{385} \left(11 \sqrt{210} (A(1,1)-i \text{Ap}(1,1))+11 \sqrt{35} (A(3,1)-i \text{Ap}(3,1))+25 \sqrt{14} (A(5,1)-i \text{Ap}(5,1))\right) $ | $ 0 $ | $ \frac{2}{33} \sqrt{5} (A(5,3)-i \text{Ap}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i \text{Ap}(3,3)) $ |
| $ {Y_{1}^{(2)}} $ | $ 0 $ | $ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i \text{Ap}(3,3))+\frac{4}{33} \sqrt{5} (A(5,3)+i \text{Ap}(5,3)) $ | $ 0 $ | $ -\sqrt{\frac{3}{35}} (A(1,1)+i \text{Ap}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (A(3,1)+i \text{Ap}(3,1))+\frac{20 (A(5,1)+i \text{Ap}(5,1))}{33 \sqrt{7}} $ | $ 0 $ | $ \frac{1}{231} \left(33 \sqrt{14} (A(1,1)-i \text{Ap}(1,1))+11 \sqrt{21} (A(3,1)-i \text{Ap}(3,1))-2 \sqrt{210} (A(5,1)-i \text{Ap}(5,1))\right) $ | $ 0 $ |
| $ {Y_{2}^{(2)}} $ | $ -\frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)+i \text{Ap}(5,5)) $ | $ 0 $ | $ \frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)+i \text{Ap}(3,3))-\frac{5}{33} \sqrt{2} (A(5,3)+i \text{Ap}(5,3)) $ | $ 0 $ | $ -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (A(3,1)+i \text{Ap}(3,1))-\frac{5 (A(5,1)+i \text{Ap}(5,1))}{11 \sqrt{21}} $ | $ 0 $ | $ \sqrt{\frac{3}{7}} (A(1,1)-i \text{Ap}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)-i \text{Ap}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)-i \text{Ap}(5,1)) $ |
The Hamiltonian on a basis of symmetric functions
| $ $ | $ f_{\text{xyz}} $ | $ f_{x\left(5x^2-r^2\right)} $ | $ f_{y\left(5y^2-r^2\right)} $ | $ f_{x\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 $ | $ \frac{-99 \sqrt{210} A(1,1)+121 \sqrt{35} A(3,1)-5 \left(11 \sqrt{21} A(3,3)+10 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)+35 \sqrt{15} A(5,5)\right)}{2310} $ | $ \frac{-99 \sqrt{210} \text{Ap}(1,1)+121 \sqrt{35} \text{Ap}(3,1)+55 \sqrt{21} \text{Ap}(3,3)-50 \sqrt{14} \text{Ap}(5,1)-175 \sqrt{3} \text{Ap}(5,3)-175 \sqrt{15} \text{Ap}(5,5)}{2310} $ | $ 0 $ | $ \frac{1}{462} \left(33 \sqrt{14} A(1,1)+11 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)-2 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)+105 A(5,5)\right) $ | $ \frac{1}{462} \left(-33 \sqrt{14} \text{Ap}(1,1)-11 \sqrt{21} \text{Ap}(3,1)-11 \sqrt{35} \text{Ap}(3,3)+2 \sqrt{210} \text{Ap}(5,1)+35 \sqrt{5} \text{Ap}(5,3)-105 \text{Ap}(5,5)\right) $ | $ 0 $ |
| $ d_{3z^2-r^2} $ | $ 0 $ | $ \frac{99 \sqrt{70} A(1,1)+33 \sqrt{105} A(3,1)+275 \sqrt{7} A(3,3)+75 \sqrt{42} A(5,1)-350 A(5,3)}{2310} $ | $ \frac{-99 \sqrt{70} \text{Ap}(1,1)-33 \sqrt{105} \text{Ap}(3,1)+275 \sqrt{7} \text{Ap}(3,3)-75 \sqrt{42} \text{Ap}(5,1)-350 \text{Ap}(5,3)}{2310} $ | $ 0 $ | $ \frac{1}{462} \left(33 \sqrt{42} A(1,1)+33 \sqrt{7} A(3,1)-11 \sqrt{105} A(3,3)+15 \sqrt{70} A(5,1)+14 \sqrt{15} A(5,3)\right) $ | $ \frac{1}{462} \left(33 \sqrt{42} \text{Ap}(1,1)+33 \sqrt{7} \text{Ap}(3,1)+11 \sqrt{105} \text{Ap}(3,3)+15 \sqrt{70} \text{Ap}(5,1)-14 \sqrt{15} \text{Ap}(5,3)\right) $ | $ 0 $ |
| $ d_{\text{yz}} $ | $ \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)\right) $ | $ 0 $ | $ 0 $ | $ -\sqrt{\frac{6}{35}} \text{Ap}(1,1)+\frac{2 \text{Ap}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Ap}(5,1) $ | $ 0 $ | $ 0 $ | $ \frac{1}{231} \left(-33 \sqrt{14} \text{Ap}(1,1)-11 \sqrt{21} \text{Ap}(3,1)+\sqrt{5} \left(11 \sqrt{7} \text{Ap}(3,3)+2 \sqrt{42} \text{Ap}(5,1)+28 \text{Ap}(5,3)\right)\right) $ |
| $ d_{\text{xz}} $ | $ \frac{1}{231} \left(33 \sqrt{14} \text{Ap}(1,1)+11 \sqrt{21} \text{Ap}(3,1)+\sqrt{5} \left(11 \sqrt{7} \text{Ap}(3,3)-2 \sqrt{42} \text{Ap}(5,1)+28 \text{Ap}(5,3)\right)\right) $ | $ 0 $ | $ 0 $ | $ \sqrt{\frac{6}{35}} A(1,1)-\frac{2 A(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} A(5,1) $ | $ 0 $ | $ 0 $ | $ \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(-11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)-28 A(5,3)\right)\right) $ |
| $ d_{\text{xy}} $ | $ 0 $ | $ \frac{-66 \sqrt{210} \text{Ap}(1,1)-11 \sqrt{35} \text{Ap}(3,1)+5 \left(11 \sqrt{21} \text{Ap}(3,3)+5 \sqrt{14} \text{Ap}(5,1)-35 \sqrt{3} \text{Ap}(5,3)+35 \sqrt{15} \text{Ap}(5,5)\right)}{2310} $ | $ \frac{66 \sqrt{210} A(1,1)+11 \sqrt{35} A(3,1)+5 \left(11 \sqrt{21} A(3,3)-5 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)-35 \sqrt{15} A(5,5)\right)}{2310} $ | $ 0 $ | $ \frac{1}{462} \left(66 \sqrt{14} \text{Ap}(1,1)-33 \sqrt{21} \text{Ap}(3,1)+11 \sqrt{35} \text{Ap}(3,3)+3 \sqrt{210} \text{Ap}(5,1)-35 \sqrt{5} \text{Ap}(5,3)-105 \text{Ap}(5,5)\right) $ | $ \frac{1}{462} \left(66 \sqrt{14} A(1,1)-33 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)+3 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)-105 A(5,5)\right) $ | $ 0 $ |
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 |
