+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

Point Group C1 with orientation 1

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)) + ((+1*I))*(Ap(1,1))} , 
       {1, 1, A(1,1) + ((+1*I))*(Ap(1,1))} , 
       {2, 0, A(2,0)} , 
       {2,-2, A(2,2) + ((+-1*I))*(Ap(2,2))} , 
       {2, 2, A(2,2) + ((+1*I))*(Ap(2,2))} , 
       {3,-1, (-1)*(A(3,1)) + ((+1*I))*(Ap(3,1))} , 
       {3, 1, A(3,1) + ((+1*I))*(Ap(3,1))} , 
       {3,-3, (-1)*(A(3,3)) + ((+1*I))*(Ap(3,3))} , 
       {3, 3, A(3,3) + ((+1*I))*(Ap(3,3))} , 
       {4, 0, A(4,0)} , 
       {4,-2, A(4,2) + ((+-1*I))*(Ap(4,2))} , 
       {4, 2, A(4,2) + ((+1*I))*(Ap(4,2))} , 
       {4,-4, A(4,4) + ((+-1*I))*(Ap(4,4))} , 
       {4, 4, A(4,4) + ((+1*I))*(Ap(4,4))} , 
       {5,-1, (-1)*(A(5,1)) + ((+1*I))*(Ap(5,1))} , 
       {5, 1, A(5,1) + ((+1*I))*(Ap(5,1))} , 
       {5,-3, (-1)*(A(5,3)) + ((+1*I))*(Ap(5,3))} , 
       {5, 3, A(5,3) + ((+1*I))*(Ap(5,3))} , 
       {5,-5, (-1)*(A(5,5)) + ((+1*I))*(Ap(5,5))} , 
       {5, 5, A(5,5) + ((+1*I))*(Ap(5,5))} , 
       {6, 0, A(6,0)} , 
       {6,-2, A(6,2) + ((+-1*I))*(Ap(6,2))} , 
       {6, 2, A(6,2) + ((+1*I))*(Ap(6,2))} , 
       {6,-4, A(6,4) + ((+-1*I))*(Ap(6,4))} , 
       {6, 4, A(6,4) + ((+1*I))*(Ap(6,4))} , 
       {6,-6, A(6,6) + ((+-1*I))*(Ap(6,6))} , 
       {6, 6, A(6,6) + ((+1*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 we can choose a basis of spherical harmonics times some radial function $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$ . With this choice we can separate the radial part from the angular part for the evaluation of the operator. Where the angular part has an analytical solution

With the definition $A_{l'',l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle$ we can express the operator as $O = \sum_{n'',l'',m'',n',l',m',k,m} A_{l'',l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$ The coefficient in front of the creation and annihilation operators $\sum_{k,m} A_{l'',l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle$ is shown in the table below. 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)

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$\text{s}$	$1$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$1$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{x^2-y^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$\frac{1}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$1$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{\text{xyz}}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$	$0$
$f_{x\left(5x^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{\sqrt{5}}{4}$	$0$	$-\frac{\sqrt{3}}{4}$	$0$	$\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{5}}{4}$
$f_{y\left(5y^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i \sqrt{5}}{4}$	$0$	$-\frac{i \sqrt{3}}{4}$	$0$	$-\frac{i \sqrt{3}}{4}$	$0$	$-\frac{i \sqrt{5}}{4}$
$f_{x\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$1$	$0$	$0$	$0$
$f_{x\left(y^2-z^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{5}}{4}$	$0$	$\frac{\sqrt{5}}{4}$	$0$	$\frac{\sqrt{3}}{4}$
$f_{y\left(z^2-x^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i \sqrt{3}}{4}$	$0$	$\frac{i \sqrt{5}}{4}$	$0$	$\frac{i \sqrt{5}}{4}$	$0$	$-\frac{i \sqrt{3}}{4}$
$f_{z\left(x^2-y^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$\frac{1}{\sqrt{2}}$	$0$

One particle coupling on a basis of symmetry adapted functions

	$\text{s}$	$p_x$	$p_y$	$p_z$	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$	$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$	$\color{darkred}{ 0 }$	$\sqrt{\frac{2}{5}} \text{Asd}(2,2)$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$0$	$0$	$-\sqrt{\frac{2}{5}} \text{Bsd}(2,2)$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$	$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2)$	$-\frac{1}{5} \sqrt{6} \text{Bpp}(2,2)$	$0$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$	$\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$	$0$	$-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4)$	$0$	$-\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$	$\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}}$	$0$
$p_y$	$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$	$-\frac{1}{5} \sqrt{6} \text{Bpp}(2,2)$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2)$	$0$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$	$\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4)$	$-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$	$0$	$-\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}}$	$\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ 0 }$	$-\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2)$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$\sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)$
$d_{x^2-y^2}$	$\sqrt{\frac{2}{5}} \text{Asd}(2,2)$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$\frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2)$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$	$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$	$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$	$\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$	$\color{darkred}{ 0 }$	$\frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2)$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2)$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$	$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$	$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$
$d_{\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2)$	$-\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2)$	$0$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$	$0$	$0$	$-\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2)$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2)$	$0$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$
$d_{\text{xy}}$	$-\sqrt{\frac{2}{5}} \text{Bsd}(2,2)$	$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4)$	$\frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2)$	$0$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$	$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$
$f_{\text{xyz}}$	$\color{darkred}{ 0 }$	$0$	$0$	$-\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$	$0$	$0$	$\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2)$	$0$	$0$	$-\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)$
$f_{x\left(5x^2-r^2\right)}$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$	$-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4)$	$0$	$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$	$\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$	$0$	$\text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$\frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6)$	$0$	$\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$\frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6)$	$0$
$f_{y\left(5y^2-r^2\right)}$	$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4)$	$-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$	$0$	$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$	$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$	$0$	$\frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6)$	$\text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$0$	$-\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6)$	$-\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$0$
$f_{x\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$	$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2)$	$0$	$0$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$-\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2)$
$f_{x\left(y^2-z^2\right)}$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$	$-\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$	$-\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}}$	$0$	$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$	$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$	$0$	$\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$-\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6)$	$0$	$\text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6)$	$\frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6)$	$0$
$f_{y\left(z^2-x^2\right)}$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$	$\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}}$	$\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$	$0$	$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$	$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$	$0$	$\frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6)$	$-\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$0$	$\frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6)$	$\text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6)$	$0$
$f_{z\left(x^2-y^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$\sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$-\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)$	$0$	$0$	$-\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2)$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$

Potential for s orbitals

Potential for p orbitals

Potential for d orbitals

Potential for f orbitals

Potential for s-p orbital mixing

Potential for s-d orbital mixing

Potential for s-f orbital mixing

Potential for p-d orbital mixing

Potential for p-f orbital mixing

Potential for d-f orbital mixing

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups	C₁	C_s	C_i
C_n groups	C₂	C₃	C₄	C₅	C₆	C₇	C₈
D_n groups	D₂	D₃	D₄	D₅	D₆	D₇	D₈
C_nv groups	C_2v	C_3v	C_4v	C_5v	C_6v	C_7v	C_8v
C_nh groups	C_2h	C_3h	C_4h	C_5h	C_6h
D_nh groups	D_2h	D_3h	D_4h	D_5h	D_6h	D_7h	D_8h
D_nd groups	D_2d	D_3d	D_4d	D_5d	D_6d	D_7d	D_8d
S_n groups	S₂	S₄	S₆	S₈	S₁₀	S₁₂
Cubic groups	T	T_h	T_d	O	O_h	I	I_h
Linear groups	C_{$\infty$ v}	D_{$\infty$ h}

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