+Table of Contents

Orientation X

Symmetry Operations

In the C2 Point Group, with orientation X there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_2$	$\{1,0,0\}$ ,

Different Settings

Point Group C2 with orientation X

Character Table

	$\text{E} \,{\text{(1)}}$	$C_2 \,{\text{(1)}}$
$\text{A}$	$1$	$1$
$\text{B}$	$1$	$-1$

Product Table

	$\text{A}$	$\text{B}$
$\text{A}$	$\text{A}$	$\text{B}$
$\text{B}$	$\text{B}$	$\text{A}$

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 C2 Point group with orientation X the form of the expansion coefficients is:

Expansion

$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1) & k=1\land m=-1 \\ A(1,1) & k=1\land m=1 \\ A(2,2) & k=2\land (m=-2\lor m=2) \\ i B(2,1) & k=2\land (m=-1\lor m=1) \\ A(2,0) & k=2\land m=0 \\ -A(3,3) & k=3\land m=-3 \\ -i B(3,2) & k=3\land m=-2 \\ -A(3,1) & k=3\land m=-1 \\ A(3,1) & k=3\land m=1 \\ i B(3,2) & k=3\land m=2 \\ A(3,3) & k=3\land m=3 \\ A(4,4) & k=4\land (m=-4\lor m=4) \\ i B(4,3) & k=4\land (m=-3\lor m=3) \\ A(4,2) & k=4\land (m=-2\lor m=2) \\ i B(4,1) & k=4\land (m=-1\lor m=1) \\ A(4,0) & k=4\land m=0 \\ -A(5,5) & k=5\land m=-5 \\ -i B(5,4) & k=5\land m=-4 \\ -A(5,3) & k=5\land m=-3 \\ -i B(5,2) & k=5\land m=-2 \\ -A(5,1) & k=5\land m=-1 \\ A(5,1) & k=5\land m=1 \\ i B(5,2) & k=5\land m=2 \\ A(5,3) & k=5\land m=3 \\ i B(5,4) & k=5\land m=4 \\ A(5,5) & k=5\land m=5 \\ A(6,6) & k=6\land (m=-6\lor m=6) \\ i B(6,5) & k=6\land (m=-5\lor m=5) \\ A(6,4) & k=6\land (m=-4\lor m=4) \\ i B(6,3) & k=6\land (m=-3\lor m=3) \\ A(6,2) & k=6\land (m=-2\lor m=2) \\ i B(6,1) & k=6\land (m=-1\lor m=1) \\ A(6,0) & k=6\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1], k == 1 && m == -1}, {A[1, 1], k == 1 && m == 1}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}, {A[2, 0], k == 2 && m == 0}, {-A[3, 3], k == 3 && m == -3}, {(-I)*B[3, 2], k == 3 && m == -2}, {-A[3, 1], k == 3 && m == -1}, {A[3, 1], k == 3 && m == 1}, {I*B[3, 2], k == 3 && m == 2}, {A[3, 3], k == 3 && m == 3}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {I*B[4, 1], k == 4 && (m == -1 || m == 1)}, {A[4, 0], k == 4 && m == 0}, {-A[5, 5], k == 5 && m == -5}, {(-I)*B[5, 4], k == 5 && m == -4}, {-A[5, 3], k == 5 && m == -3}, {(-I)*B[5, 2], k == 5 && m == -2}, {-A[5, 1], k == 5 && m == -1}, {A[5, 1], k == 5 && m == 1}, {I*B[5, 2], k == 5 && m == 2}, {A[5, 3], k == 5 && m == 3}, {I*B[5, 4], k == 5 && m == 4}, {A[5, 5], k == 5 && m == 5}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {I*B[6, 5], k == 6 && (m == -5 || m == 5)}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {I*B[6, 3], k == 6 && (m == -3 || m == 3)}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {I*B[6, 1], k == 6 && (m == -1 || m == 1)}, {A[6, 0], k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{0, 0, A(0,0)} , 
       {1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} , 
       {2, 0, A(2,0)} , 
       {2,-1, (I)*(B(2,1))} , 
       {2, 1, (I)*(B(2,1))} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, (-I)*(B(3,2))} , 
       {3, 2, (I)*(B(3,2))} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} , 
       {4, 0, A(4,0)} , 
       {4,-1, (I)*(B(4,1))} , 
       {4, 1, (I)*(B(4,1))} , 
       {4,-2, A(4,2)} , 
       {4, 2, A(4,2)} , 
       {4,-3, (I)*(B(4,3))} , 
       {4, 3, (I)*(B(4,3))} , 
       {4,-4, A(4,4)} , 
       {4, 4, A(4,4)} , 
       {5,-1, (-1)*(A(5,1))} , 
       {5, 1, A(5,1)} , 
       {5,-2, (-I)*(B(5,2))} , 
       {5, 2, (I)*(B(5,2))} , 
       {5,-3, (-1)*(A(5,3))} , 
       {5, 3, A(5,3)} , 
       {5,-4, (-I)*(B(5,4))} , 
       {5, 4, (I)*(B(5,4))} , 
       {5,-5, (-1)*(A(5,5))} , 
       {5, 5, A(5,5)} , 
       {6, 0, A(6,0)} , 
       {6,-1, (I)*(B(6,1))} , 
       {6, 1, (I)*(B(6,1))} , 
       {6,-2, A(6,2)} , 
       {6, 2, A(6,2)} , 
       {6,-3, (I)*(B(6,3))} , 
       {6, 3, (I)*(B(6,3))} , 
       {6,-4, A(6,4)} , 
       {6, 4, A(6,4)} , 
       {6,-5, (I)*(B(6,5))} , 
       {6, 5, (I)*(B(6,5))} , 
       {6,-6, A(6,6)} , 
       {6, 6, A(6,6)} }

One particle coupling on a basis of spherical harmonics

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.

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$\text{Ass}(0,0)$	$\color{darkred}{ -\frac{\text{Asp}(1,1)}{\sqrt{3}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Asp}(1,1)}{\sqrt{3}} }$	$\frac{\text{Asd}(2,2)}{\sqrt{5}}$	$-\frac{i \text{Bsd}(2,1)}{\sqrt{5}}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$-\frac{i \text{Bsd}(2,1)}{\sqrt{5}}$	$\frac{\text{Asd}(2,2)}{\sqrt{5}}$	$\color{darkred}{ -\frac{\text{Asf}(3,3)}{\sqrt{7}} }$	$\color{darkred}{ \frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$\color{darkred}{ -\frac{\text{Asf}(3,1)}{\sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Asf}(3,1)}{\sqrt{7}} }$	$\color{darkred}{ -\frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$\color{darkred}{ \frac{\text{Asf}(3,3)}{\sqrt{7}} }$
${Y_{-1}^{(1)}}$	$\color{darkred}{ -\frac{\text{Asp}(1,1)}{\sqrt{3}} }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\frac{1}{5} i \sqrt{3} \text{Bpp}(2,1)$	$-\frac{1}{5} \sqrt{6} \text{App}(2,2)$	$\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\sqrt{\frac{2}{5}} \text{Apd}(1,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ \frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ -\frac{3}{7} \text{Apd}(3,3) }$	$\frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}}$	$\frac{i \text{Bpf}(4,1)}{3 \sqrt{7}}-i \sqrt{\frac{6}{35}} \text{Bpf}(2,1)$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1)-\frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}}$	$\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2)$	$\frac{1}{3} i \text{Bpf}(4,3)$	$-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$
${Y_{0}^{(1)}}$	$\color{darkred}{ 0 }$	$-\frac{1}{5} i \sqrt{3} \text{Bpp}(2,1)$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$-\frac{1}{5} i \sqrt{3} \text{Bpp}(2,1)$	$\color{darkred}{ \frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$\color{darkred}{ -\frac{\text{Apd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ -\frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$-\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$	$\sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}}$	$-\frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1)$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$-\frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1)$	$\sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}}$	$-\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$
${Y_{1}^{(1)}}$	$\color{darkred}{ \frac{\text{Asp}(1,1)}{\sqrt{3}} }$	$-\frac{1}{5} \sqrt{6} \text{App}(2,2)$	$\frac{1}{5} i \sqrt{3} \text{Bpp}(2,1)$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\color{darkred}{ \frac{3}{7} \text{Apd}(3,3) }$	$\color{darkred}{ -\frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1)-\frac{\text{Apd}(1,1)}{\sqrt{15}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$	$-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$	$\frac{1}{3} i \text{Bpf}(4,3)$	$\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2)$	$\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1)-\frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}}$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\frac{i \text{Bpf}(4,1)}{3 \sqrt{7}}-i \sqrt{\frac{6}{35}} \text{Bpf}(2,1)$	$\frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}}$
${Y_{-2}^{(2)}}$	$\frac{\text{Asd}(2,2)}{\sqrt{5}}$	$\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\sqrt{\frac{2}{5}} \text{Apd}(1,1) }$	$\color{darkred}{ -\frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$\color{darkred}{ \frac{3}{7} \text{Apd}(3,3) }$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$\frac{1}{7} i \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} i \sqrt{5} \text{Bdd}(4,1)$	$\frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2)$	$-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$\color{darkred}{ -\sqrt{\frac{3}{7}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)-\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)+\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$	$\color{darkred}{ \frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}}-\frac{5}{33} i \text{Bdf}(5,2) }$	$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3) }$	$\color{darkred}{ -\frac{1}{11} i \sqrt{10} \text{Bdf}(5,4) }$	$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$
${Y_{-1}^{(2)}}$	$\frac{i \text{Bsd}(2,1)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{\text{Apd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ \frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\frac{1}{21} i \sqrt{5} \text{Bdd}(4,1)-\frac{1}{7} i \sqrt{6} \text{Bdd}(2,1)$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$\frac{1}{7} i \text{Bdd}(2,1)+\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1)$	$-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2)$	$\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2)-\frac{1}{33} i \sqrt{5} \text{Bdf}(5,2) }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{3}{35}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)-\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$	$\color{darkred}{ \frac{i \text{Bdf}(3,2)}{\sqrt{21}}+\frac{5 i \text{Bdf}(5,2)}{11 \sqrt{3}} }$	$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ \frac{2}{11} i \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$
${Y_{0}^{(2)}}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1)-\frac{\text{Apd}(1,1)}{\sqrt{15}} }$	$\frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2)$	$-\frac{1}{7} i \text{Bdd}(2,1)-\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1)$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$-\frac{1}{7} i \text{Bdd}(2,1)-\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1)$	$\frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2)$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{2}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ \frac{1}{11} i \sqrt{5} \text{Bdf}(5,2) }$	$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ -\frac{1}{11} i \sqrt{5} \text{Bdf}(5,2) }$	$\color{darkred}{ \frac{2}{33} \sqrt{5} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3) }$
${Y_{1}^{(2)}}$	$\frac{i \text{Bsd}(2,1)}{\sqrt{5}}$	$\color{darkred}{ -\frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ 0 }$	$\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2)$	$\frac{1}{7} i \text{Bdd}(2,1)+\frac{1}{7} i \sqrt{\frac{10}{3}} \text{Bdd}(4,1)$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$\frac{1}{21} i \sqrt{5} \text{Bdd}(4,1)-\frac{1}{7} i \sqrt{6} \text{Bdd}(2,1)$	$\color{darkred}{ -\frac{2}{11} i \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ -\frac{i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 i \text{Bdf}(5,2)}{11 \sqrt{3}} }$	$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)+\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$	$\color{darkred}{ \frac{1}{33} i \sqrt{5} \text{Bdf}(5,2)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2) }$
${Y_{2}^{(2)}}$	$\frac{\text{Asd}(2,2)}{\sqrt{5}}$	$\color{darkred}{ -\frac{3}{7} \text{Apd}(3,3) }$	$\color{darkred}{ \frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$	$\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$\frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2)$	$\frac{1}{7} i \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} i \sqrt{5} \text{Bdd}(4,1)$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$	$\color{darkred}{ \frac{1}{11} i \sqrt{10} \text{Bdf}(5,4) }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$	$\color{darkred}{ \frac{5}{33} i \text{Bdf}(5,2)-\frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}} }$	$\color{darkred}{ -\frac{\text{Adf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)-\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \sqrt{\frac{3}{7}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)+\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$
${Y_{-3}^{(3)}}$	$\color{darkred}{ -\frac{\text{Asf}(3,3)}{\sqrt{7}} }$	$\frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}}$	$\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$	$-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$	$\color{darkred}{ -\sqrt{\frac{3}{7}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)-\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ \frac{1}{33} i \sqrt{5} \text{Bdf}(5,2)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{2}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ \frac{2}{11} i \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$\frac{1}{3} i \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)+\frac{5}{429} i \sqrt{7} \text{Bff}(6,1)$	$-\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2)$	$\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)-\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$	$\frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5)$	$-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$
${Y_{-2}^{(3)}}$	$\color{darkred}{ -\frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$i \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{i \text{Bpf}(4,1)}{3 \sqrt{7}}$	$\sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}}$	$-\frac{1}{3} i \text{Bpf}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$	$\color{darkred}{ -\frac{1}{11} i \sqrt{5} \text{Bdf}(5,2) }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ -\frac{1}{11} i \sqrt{10} \text{Bdf}(5,4) }$	$-\frac{1}{3} i \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)-\frac{5}{429} i \sqrt{7} \text{Bff}(6,1)$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$\frac{i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)-\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1)$	$-\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2)$	$-\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)-\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$	$-\frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5)$
${Y_{-1}^{(3)}}$	$\color{darkred}{ -\frac{\text{Asf}(3,1)}{\sqrt{7}} }$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)+\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1)$	$\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2)$	$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)+\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ \frac{i \text{Bdf}(3,2)}{\sqrt{21}}+\frac{5 i \text{Bdf}(5,2)}{11 \sqrt{3}} }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$	$-\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2)$	$-\frac{i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)+\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1)$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$\frac{1}{15} i \sqrt{2} \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)+\frac{25}{429} i \sqrt{14} \text{Bff}(6,1)$	$-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2)$	$\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$
${Y_{0}^{(3)}}$	$\color{darkred}{ 0 }$	$\frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}}-\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1)$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$\frac{3 i \text{Bpf}(2,1)}{5 \sqrt{7}}-\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Bpf}(4,1)$	$\color{darkred}{ \frac{5}{33} i \text{Bdf}(5,2)-\frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}} }$	$\color{darkred}{ \sqrt{\frac{3}{35}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)-\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)+\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$	$\color{darkred}{ \frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}}-\frac{5}{33} i \text{Bdf}(5,2) }$	$\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)-\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)$	$-\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2)$	$-\frac{1}{15} i \sqrt{2} \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)-\frac{25}{429} i \sqrt{14} \text{Bff}(6,1)$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$-\frac{1}{15} i \sqrt{2} \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)-\frac{25}{429} i \sqrt{14} \text{Bff}(6,1)$	$-\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2)$	$\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)-\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)$
${Y_{1}^{(3)}}$	$\color{darkred}{ \frac{\text{Asf}(3,1)}{\sqrt{7}} }$	$\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2)$	$\frac{2}{5} i \sqrt{\frac{6}{7}} \text{Bpf}(2,1)+\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bpf}(4,1)$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3) }$	$\color{darkred}{ -\frac{i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 i \text{Bdf}(5,2)}{11 \sqrt{3}} }$	$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{\text{Adf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)-\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$	$\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2)$	$\frac{1}{15} i \sqrt{2} \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{5}{3}} \text{Bff}(4,1)+\frac{25}{429} i \sqrt{14} \text{Bff}(6,1)$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$-\frac{i \text{Bff}(2,1)}{\sqrt{15}}-\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)+\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1)$	$-\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2)$
${Y_{2}^{(3)}}$	$\color{darkred}{ \frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$-\frac{1}{3} i \text{Bpf}(4,3)$	$\sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}}$	$i \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{i \text{Bpf}(4,1)}{3 \sqrt{7}}$	$\color{darkred}{ \frac{1}{11} i \sqrt{10} \text{Bdf}(5,4) }$	$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ \frac{1}{11} i \sqrt{5} \text{Bdf}(5,2) }$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$	$\color{darkred}{ 0 }$	$-\frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5)$	$\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$	$-\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)-\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$-\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2)$	$\frac{i \text{Bff}(2,1)}{\sqrt{15}}+\frac{4}{33} i \sqrt{2} \text{Bff}(4,1)-\frac{5}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,1)$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$-\frac{1}{3} i \text{Bff}(2,1)+\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)-\frac{5}{429} i \sqrt{7} \text{Bff}(6,1)$
${Y_{3}^{(3)}}$	$\color{darkred}{ \frac{\text{Asf}(3,3)}{\sqrt{7}} }$	$-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$	$\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$	$\frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}}$	$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$	$\color{darkred}{ -\frac{2}{11} i \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ \frac{2}{33} \sqrt{5} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3) }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2)-\frac{1}{33} i \sqrt{5} \text{Bdf}(5,2) }$	$\color{darkred}{ \sqrt{\frac{3}{7}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)+\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$	$-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$\frac{5}{13} i \sqrt{\frac{14}{33}} \text{Bff}(6,5)$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$	$\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)-\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)$	$-\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2)$	$\frac{1}{3} i \text{Bff}(2,1)-\frac{1}{11} i \sqrt{\frac{10}{3}} \text{Bff}(4,1)+\frac{5}{429} i \sqrt{7} \text{Bff}(6,1)$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

Rotation matrix to symmetry adapted functions (choice is not unique)

Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$\text{s}$	$1$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_z$	$\color{darkred}{ 0 }$	$0$	$1$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_x$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3x^2-r^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{\sqrt{\frac{3}{2}}}{2}$	$0$	$-\frac{1}{2}$	$0$	$\frac{\sqrt{\frac{3}{2}}}{2}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{y^2-z^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{1}{2 \sqrt{2}}$	$0$	$-\frac{\sqrt{3}}{2}$	$0$	$-\frac{1}{2 \sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{z\left(3y^2-z^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{\frac{5}{2}}}{2}$	$0$	$-\frac{\sqrt{3}}{4}$	$0$
$f_{\text{xyz}}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$	$0$
$f_{z\left(5x^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{\sqrt{5}}{4}$	$0$	$-\frac{\sqrt{\frac{3}{2}}}{2}$	$0$	$\frac{\sqrt{5}}{4}$	$0$
$f_{x\left(5x^2-3r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{\sqrt{5}}{4}$	$0$	$-\frac{\sqrt{3}}{4}$	$0$	$\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{5}}{4}$
$f_{y\left(5x^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{4} i \sqrt{\frac{15}{2}}$	$0$	$-\frac{i}{4 \sqrt{2}}$	$0$	$-\frac{i}{4 \sqrt{2}}$	$0$	$\frac{1}{4} i \sqrt{\frac{15}{2}}$
$f_{x\left(y^2-z^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{5}}{4}$	$0$	$\frac{\sqrt{5}}{4}$	$0$	$\frac{\sqrt{3}}{4}$
$f_{y\left(y^2-3z^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i}{4 \sqrt{2}}$	$0$	$-\frac{1}{4} i \sqrt{\frac{15}{2}}$	$0$	$-\frac{1}{4} i \sqrt{\frac{15}{2}}$	$0$	$-\frac{i}{4 \sqrt{2}}$

One particle coupling on a basis of symmetry adapted functions

After rotation we find

	$\text{s}$	$p_z$	$p_x$	$p_y$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{3x^2-r^2}$	$d_{\text{xz}}$	$d_{y^2-z^2}$	$f_{z\left(3y^2-z^2\right)}$	$f_{\text{xyz}}$	$f_{z\left(5x^2-r^2\right)}$	$f_{x\left(5x^2-3r^2\right)}$	$f_{y\left(5x^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(y^2-3z^2\right)}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$	$\color{darkred}{ 0 }$	$\sqrt{\frac{2}{5}} \text{Bsd}(2,1)$	$0$	$\sqrt{\frac{3}{10}} \text{Asd}(2,2)-\frac{\text{Asd}(2,0)}{2 \sqrt{5}}$	$0$	$-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$	$\color{darkred}{ 0 }$
$p_z$	$\color{darkred}{ 0 }$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$0$	$\frac{1}{5} \sqrt{6} \text{Bpp}(2,1)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$-\frac{3}{2} \sqrt{\frac{3}{70}} \text{Apf}(2,0)-\frac{3 \text{Apf}(2,2)}{2 \sqrt{35}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0)-\frac{\text{Apf}(4,2)}{\sqrt{21}}$	$0$	$-\frac{9 \text{Apf}(2,0)}{10 \sqrt{14}}+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2)$	$0$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Bpf}(2,1)-\frac{1}{6} \sqrt{\frac{5}{14}} \text{Bpf}(4,1)+\frac{1}{6} \sqrt{\frac{5}{2}} \text{Bpf}(4,3)$	$0$	$-\frac{3 \text{Bpf}(2,1)}{\sqrt{35}}-\frac{5 \text{Bpf}(4,1)}{2 \sqrt{42}}-\frac{\text{Bpf}(4,3)}{6 \sqrt{6}}$
$p_x$	$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2)$	$0$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -2 \sqrt{\frac{2}{15}} \text{Apd}(1,1)+\frac{9 \text{Apd}(3,1)}{14 \sqrt{5}}-\frac{3}{14} \sqrt{3} \text{Apd}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{14} \sqrt{15} \text{Apd}(3,1)+\frac{3}{14} \text{Apd}(3,3) }$	$0$	$\sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3)$	$0$	$-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$	$0$	$-\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$	$0$
$p_y$	$\color{darkred}{ 0 }$	$\frac{1}{5} \sqrt{6} \text{Bpp}(2,1)$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$	$\color{darkred}{ 0 }$	$\frac{3 \text{Bpf}(2,1)}{\sqrt{35}}-\frac{13 \text{Bpf}(4,1)}{6 \sqrt{42}}-\frac{\text{Bpf}(4,3)}{2 \sqrt{6}}$	$0$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Bpf}(2,1)-\frac{1}{6} \sqrt{\frac{5}{14}} \text{Bpf}(4,1)+\frac{1}{6} \sqrt{\frac{5}{2}} \text{Bpf}(4,3)$	$0$	$-\frac{3 \text{Apf}(2,0)}{10 \sqrt{14}}+\frac{1}{10} \sqrt{21} \text{Apf}(2,2)+\frac{\text{Apf}(4,0)}{6 \sqrt{14}}-\frac{1}{6} \sqrt{5} \text{Apf}(4,4)$	$0$	$-\frac{3}{2} \sqrt{\frac{3}{70}} \text{Apf}(2,0)-\frac{3 \text{Apf}(2,2)}{2 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Apf}(4,0)+\frac{4 \text{Apf}(4,2)}{3 \sqrt{21}}+\frac{\text{Apf}(4,4)}{6 \sqrt{3}}$
$d_{\text{yz}}$	$\sqrt{\frac{2}{5}} \text{Bsd}(2,1)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2)$	$0$	$-\frac{2}{7} \sqrt{2} \text{Bdd}(2,1)-\frac{1}{14} \sqrt{\frac{5}{3}} \text{Bdd}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdd}(4,3)$	$0$	$-\frac{1}{6} \sqrt{5} \text{Bdd}(4,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}-\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2)$	$0$	$\frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\color{darkred}{ \frac{5 \text{Adf}(3,1)}{6 \sqrt{14}}+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Adf}(3,3)+\frac{1}{33} \sqrt{35} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{4 \text{Adf}(1,1)}{\sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{70}} \text{Adf}(3,1)-\frac{5 \text{Adf}(3,3)}{6 \sqrt{14}}+\frac{5}{11} \sqrt{\frac{3}{7}} \text{Adf}(5,1)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{\text{Bdf}(3,2)}{\sqrt{21}}+\frac{5 \text{Bdf}(5,2)}{22 \sqrt{3}}-\frac{5}{22} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,2)+\frac{7}{66} \sqrt{5} \text{Bdf}(5,2)+\frac{1}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$
$d_{3x^2-r^2}$	$\sqrt{\frac{3}{10}} \text{Asd}(2,2)-\frac{\text{Asd}(2,0)}{2 \sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ -2 \sqrt{\frac{2}{15}} \text{Apd}(1,1)+\frac{9 \text{Apd}(3,1)}{14 \sqrt{5}}-\frac{3}{14} \sqrt{3} \text{Apd}(3,3) }$	$\color{darkred}{ 0 }$	$-\frac{2}{7} \sqrt{2} \text{Bdd}(2,1)-\frac{1}{14} \sqrt{\frac{5}{3}} \text{Bdd}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdd}(4,3)$	$0$	$\text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{3}{28} \text{Add}(4,0)-\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{14}} \text{Add}(4,4)$	$0$	$\frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)+\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)-\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,2)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -3 \sqrt{\frac{2}{35}} \text{Adf}(1,1)+\frac{2 \text{Adf}(3,1)}{\sqrt{105}}-\frac{2 \text{Adf}(3,3)}{3 \sqrt{7}}-\frac{25 \text{Adf}(5,1)}{22 \sqrt{42}}+\frac{25}{132} \text{Adf}(5,3)-\frac{5}{44} \sqrt{5} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{22} \sqrt{\frac{35}{2}} \text{Adf}(5,1)+\frac{1}{44} \sqrt{15} \text{Adf}(5,3)+\frac{5}{44} \sqrt{3} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$	$0$	$\frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$0$	$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,2)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,2)+\frac{1}{22} \sqrt{15} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{\text{Bdf}(3,2)}{\sqrt{21}}+\frac{5 \text{Bdf}(5,2)}{22 \sqrt{3}}-\frac{5}{22} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{4 \text{Adf}(1,1)}{\sqrt{35}}+\frac{1}{2} \sqrt{\frac{7}{30}} \text{Adf}(3,1)+\frac{\text{Adf}(3,3)}{6 \sqrt{14}}-\frac{5 \text{Adf}(5,1)}{22 \sqrt{21}}-\frac{5 \text{Adf}(5,3)}{66 \sqrt{2}}+\frac{5}{22} \sqrt{\frac{5}{2}} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5 \text{Adf}(3,1)}{6 \sqrt{14}}+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Adf}(3,3)-\frac{1}{66} \sqrt{35} \text{Adf}(5,1)-\frac{5}{22} \sqrt{\frac{5}{6}} \text{Adf}(5,3)-\frac{5 \text{Adf}(5,5)}{22 \sqrt{6}} }$
$d_{y^2-z^2}$	$-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{14} \sqrt{15} \text{Apd}(3,1)+\frac{3}{14} \text{Apd}(3,3) }$	$\color{darkred}{ 0 }$	$-\frac{1}{6} \sqrt{5} \text{Bdd}(4,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)+\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)-\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4)$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{19}{84} \text{Add}(4,0)+\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{6} \sqrt{\frac{5}{14}} \text{Add}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,1)-\frac{\text{Adf}(3,3)}{\sqrt{21}}-\frac{5}{66} \sqrt{\frac{7}{2}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{44 \sqrt{3}}+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{13}{22} \sqrt{\frac{5}{42}} \text{Adf}(5,1)-\frac{1}{12} \sqrt{5} \text{Adf}(5,3)-\frac{5}{44} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$
$f_{z\left(3y^2-z^2\right)}$	$\color{darkred}{ 0 }$	$-\frac{3}{2} \sqrt{\frac{3}{70}} \text{Apf}(2,0)-\frac{3 \text{Apf}(2,2)}{2 \sqrt{35}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0)-\frac{\text{Apf}(4,2)}{\sqrt{21}}$	$0$	$\frac{3 \text{Bpf}(2,1)}{\sqrt{35}}-\frac{13 \text{Bpf}(4,1)}{6 \sqrt{42}}-\frac{\text{Bpf}(4,3)}{2 \sqrt{6}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5 \text{Adf}(3,1)}{6 \sqrt{14}}+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Adf}(3,3)+\frac{1}{33} \sqrt{35} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,2)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,2)+\frac{1}{22} \sqrt{15} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)+\frac{1}{6} \text{Aff}(2,0)-\frac{\text{Aff}(2,2)}{\sqrt{6}}+\frac{3}{88} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{44} \sqrt{\frac{35}{2}} \text{Aff}(4,4)+\frac{295 \text{Aff}(6,0)}{1716}+\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2)+\frac{15}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)$	$0$	$\frac{\text{Aff}(2,0)}{2 \sqrt{15}}+\frac{\text{Aff}(2,2)}{3 \sqrt{10}}+\frac{13}{88} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{22 \sqrt{6}}-\frac{5}{44} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2)-\frac{5}{286} \sqrt{\frac{105}{2}} \text{Aff}(6,4)$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{2 \text{Bff}(4,1)}{11 \sqrt{3}}+\frac{2}{11} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{25 \sqrt{\frac{35}{2}} \text{Bff}(6,1)}{1716}-\frac{35 \sqrt{7} \text{Bff}(6,3)}{1144}-\frac{5}{104} \sqrt{\frac{105}{11}} \text{Bff}(6,5)$	$0$	$\frac{5}{52} \sqrt{\frac{21}{2}} \text{Bff}(6,1)+\frac{5}{104} \sqrt{\frac{35}{3}} \text{Bff}(6,3)+\frac{5}{104} \sqrt{\frac{7}{11}} \text{Bff}(6,5)$
$f_{\text{xyz}}$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$	$0$	$\sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3)$	$0$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,2)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$	$0$	$-\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5)$	$0$	$-\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5)$	$0$
$f_{z\left(5x^2-r^2\right)}$	$\color{darkred}{ 0 }$	$-\frac{9 \text{Apf}(2,0)}{10 \sqrt{14}}+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2)$	$0$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Bpf}(2,1)-\frac{1}{6} \sqrt{\frac{5}{14}} \text{Bpf}(4,1)+\frac{1}{6} \sqrt{\frac{5}{2}} \text{Bpf}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{4 \text{Adf}(1,1)}{\sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{70}} \text{Adf}(3,1)-\frac{5 \text{Adf}(3,3)}{6 \sqrt{14}}+\frac{5}{11} \sqrt{\frac{3}{7}} \text{Adf}(5,1)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{\text{Bdf}(3,2)}{\sqrt{21}}+\frac{5 \text{Bdf}(5,2)}{22 \sqrt{3}}-\frac{5}{22} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\frac{\text{Aff}(2,0)}{2 \sqrt{15}}+\frac{\text{Aff}(2,2)}{3 \sqrt{10}}+\frac{13}{88} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{22 \sqrt{6}}-\frac{5}{44} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2)-\frac{5}{286} \sqrt{\frac{105}{2}} \text{Aff}(6,4)$	$0$	$\text{Aff}(0,0)+\frac{1}{10} \text{Aff}(2,0)+\frac{\text{Aff}(2,2)}{\sqrt{6}}-\frac{17}{264} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{5}{132} \sqrt{\frac{35}{2}} \text{Aff}(4,4)+\frac{75}{572} \text{Aff}(6,0)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)$	$0$	$\frac{2}{5} \sqrt{\frac{2}{3}} \text{Bff}(2,1)-\frac{1}{33} \sqrt{5} \text{Bff}(4,1)+\frac{1}{33} \sqrt{35} \text{Bff}(4,3)+\frac{25}{572} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{15 \sqrt{105} \text{Bff}(6,3)}{1144}+\frac{25}{104} \sqrt{\frac{7}{11}} \text{Bff}(6,5)$	$0$	$\frac{1}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{5 \text{Bff}(4,1)}{11 \sqrt{3}}-\frac{1}{11} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{35 \sqrt{\frac{35}{2}} \text{Bff}(6,1)}{1716}-\frac{5}{88} \sqrt{7} \text{Bff}(6,3)-\frac{5}{104} \sqrt{\frac{35}{33}} \text{Bff}(6,5)$
$f_{x\left(5x^2-3r^2\right)}$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$	$0$	$-\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4)$	$0$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}-\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -3 \sqrt{\frac{2}{35}} \text{Adf}(1,1)+\frac{2 \text{Adf}(3,1)}{\sqrt{105}}-\frac{2 \text{Adf}(3,3)}{3 \sqrt{7}}-\frac{25 \text{Adf}(5,1)}{22 \sqrt{42}}+\frac{25}{132} \text{Adf}(5,3)-\frac{5}{44} \sqrt{5} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,1)-\frac{\text{Adf}(3,3)}{\sqrt{21}}-\frac{5}{66} \sqrt{\frac{7}{2}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{44 \sqrt{3}}+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$	$0$	$-\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5)$	$0$	$\text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$0$	$\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$0$
$f_{y\left(5x^2-r^2\right)}$	$\color{darkred}{ 0 }$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Bpf}(2,1)-\frac{1}{6} \sqrt{\frac{5}{14}} \text{Bpf}(4,1)+\frac{1}{6} \sqrt{\frac{5}{2}} \text{Bpf}(4,3)$	$0$	$-\frac{3 \text{Apf}(2,0)}{10 \sqrt{14}}+\frac{1}{10} \sqrt{21} \text{Apf}(2,2)+\frac{\text{Apf}(4,0)}{6 \sqrt{14}}-\frac{1}{6} \sqrt{5} \text{Apf}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{\text{Bdf}(3,2)}{\sqrt{21}}+\frac{5 \text{Bdf}(5,2)}{22 \sqrt{3}}-\frac{5}{22} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{4 \text{Adf}(1,1)}{\sqrt{35}}+\frac{1}{2} \sqrt{\frac{7}{30}} \text{Adf}(3,1)+\frac{\text{Adf}(3,3)}{6 \sqrt{14}}-\frac{5 \text{Adf}(5,1)}{22 \sqrt{21}}-\frac{5 \text{Adf}(5,3)}{66 \sqrt{2}}+\frac{5}{22} \sqrt{\frac{5}{2}} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{2 \text{Bff}(4,1)}{11 \sqrt{3}}+\frac{2}{11} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{25 \sqrt{\frac{35}{2}} \text{Bff}(6,1)}{1716}-\frac{35 \sqrt{7} \text{Bff}(6,3)}{1144}-\frac{5}{104} \sqrt{\frac{105}{11}} \text{Bff}(6,5)$	$0$	$\frac{2}{5} \sqrt{\frac{2}{3}} \text{Bff}(2,1)-\frac{1}{33} \sqrt{5} \text{Bff}(4,1)+\frac{1}{33} \sqrt{35} \text{Bff}(4,3)+\frac{25}{572} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{15 \sqrt{105} \text{Bff}(6,3)}{1144}+\frac{25}{104} \sqrt{\frac{7}{11}} \text{Bff}(6,5)$	$0$	$\text{Aff}(0,0)-\frac{3}{10} \text{Aff}(2,0)+\frac{\text{Aff}(2,2)}{5 \sqrt{6}}+\frac{23}{264} \text{Aff}(4,0)-\frac{5}{66} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{44} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{25 \text{Aff}(6,0)}{1144}+\frac{5 \sqrt{\frac{35}{3}} \text{Aff}(6,2)}{1144}+\frac{25}{572} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{104} \sqrt{\frac{21}{11}} \text{Aff}(6,6)$	$0$	$\frac{\text{Aff}(2,0)}{2 \sqrt{15}}+\frac{\text{Aff}(2,2)}{3 \sqrt{10}}-\frac{1}{88} \sqrt{\frac{5}{3}} \text{Aff}(4,0)-\frac{13 \text{Aff}(4,2)}{22 \sqrt{6}}-\frac{7}{44} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35 \sqrt{\frac{5}{3}} \text{Aff}(6,0)}{1144}-\frac{5 \sqrt{7} \text{Aff}(6,2)}{3432}+\frac{35}{572} \sqrt{\frac{35}{6}} \text{Aff}(6,4)+\frac{5}{104} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$
$f_{x\left(y^2-z^2\right)}$	$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$	$0$	$-\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}}$	$0$	$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{22} \sqrt{\frac{35}{2}} \text{Adf}(5,1)+\frac{1}{44} \sqrt{15} \text{Adf}(5,3)+\frac{5}{44} \sqrt{3} \text{Adf}(5,5) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{13}{22} \sqrt{\frac{5}{42}} \text{Adf}(5,1)-\frac{1}{12} \sqrt{5} \text{Adf}(5,3)-\frac{5}{44} \text{Adf}(5,5) }$	$0$	$-\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5)$	$0$	$\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$0$	$\text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6)$	$0$
$f_{y\left(y^2-3z^2\right)}$	$\color{darkred}{ 0 }$	$-\frac{3 \text{Bpf}(2,1)}{\sqrt{35}}-\frac{5 \text{Bpf}(4,1)}{2 \sqrt{42}}-\frac{\text{Bpf}(4,3)}{6 \sqrt{6}}$	$0$	$-\frac{3}{2} \sqrt{\frac{3}{70}} \text{Apf}(2,0)-\frac{3 \text{Apf}(2,2)}{2 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Apf}(4,0)+\frac{4 \text{Apf}(4,2)}{3 \sqrt{21}}+\frac{\text{Apf}(4,4)}{6 \sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,2)+\frac{7}{66} \sqrt{5} \text{Bdf}(5,2)+\frac{1}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5 \text{Adf}(3,1)}{6 \sqrt{14}}+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Adf}(3,3)-\frac{1}{66} \sqrt{35} \text{Adf}(5,1)-\frac{5}{22} \sqrt{\frac{5}{6}} \text{Adf}(5,3)-\frac{5 \text{Adf}(5,5)}{22 \sqrt{6}} }$	$\color{darkred}{ 0 }$	$\frac{5}{52} \sqrt{\frac{21}{2}} \text{Bff}(6,1)+\frac{5}{104} \sqrt{\frac{35}{3}} \text{Bff}(6,3)+\frac{5}{104} \sqrt{\frac{7}{11}} \text{Bff}(6,5)$	$0$	$\frac{1}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{5 \text{Bff}(4,1)}{11 \sqrt{3}}-\frac{1}{11} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{35 \sqrt{\frac{35}{2}} \text{Bff}(6,1)}{1716}-\frac{5}{88} \sqrt{7} \text{Bff}(6,3)-\frac{5}{104} \sqrt{\frac{35}{33}} \text{Bff}(6,5)$	$0$	$\frac{\text{Aff}(2,0)}{2 \sqrt{15}}+\frac{\text{Aff}(2,2)}{3 \sqrt{10}}-\frac{1}{88} \sqrt{\frac{5}{3}} \text{Aff}(4,0)-\frac{13 \text{Aff}(4,2)}{22 \sqrt{6}}-\frac{7}{44} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35 \sqrt{\frac{5}{3}} \text{Aff}(6,0)}{1144}-\frac{5 \sqrt{7} \text{Aff}(6,2)}{3432}+\frac{35}{572} \sqrt{\frac{35}{6}} \text{Aff}(6,4)+\frac{5}{104} \sqrt{\frac{35}{11}} \text{Aff}(6,6)$	$0$	$\text{Aff}(0,0)+\frac{1}{6} \text{Aff}(2,0)-\frac{\text{Aff}(2,2)}{\sqrt{6}}+\frac{3}{88} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{44} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{565 \text{Aff}(6,0)}{3432}-\frac{85 \sqrt{\frac{35}{3}} \text{Aff}(6,2)}{1144}-\frac{25}{572} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{104} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$

Coupling for a single shell

Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$ .

Click on one of the subsections to expand it or

Potential for s orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Ea} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Ea, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{0, 0, Ea} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{0}^{(0)}}$
${Y_{0}^{(0)}}$	$\text{Ea}$

The Hamiltonian on a basis of symmetric functions

	$\text{s}$
$\text{s}$	$\text{Ea}$

Rotation matrix used

	${Y_{0}^{(0)}}$
$\text{s}$	$1$

Irriducible representations and their onsite energy

	$\text{Ea}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea}+\text{Eby}+\text{Ebz}) & k=0\land m=0 \\ 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ \frac{5 (\text{Ea}-\text{Eby})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\ \frac{5 i \text{Mb}}{\sqrt{6}} & k=2\land (m=-1\lor m=1) \\ -\frac{5}{6} (\text{Ea}+\text{Eby}-2 \text{Ebz}) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea + Eby + Ebz)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {(5*(Ea - Eby))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}, {((5*I)*Mb)/Sqrt[6], k == 2 && (m == -1 || m == 1)}}, (-5*(Ea + Eby - 2*Ebz))/6]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{0, 0, (1/3)*(Ea + Eby + Ebz)} , 
       {2, 0, (-5/6)*(Ea + Eby + (-2)*(Ebz))} , 
       {2,-1, (5*I)*((1/(sqrt(6)))*(Mb))} , 
       {2, 1, (5*I)*((1/(sqrt(6)))*(Mb))} , 
       {2,-2, (5/2)*((1/(sqrt(6)))*(Ea + (-1)*(Eby)))} , 
       {2, 2, (5/2)*((1/(sqrt(6)))*(Ea + (-1)*(Eby)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\frac{\text{Ea}+\text{Eby}}{2}$	$\frac{i \text{Mb}}{\sqrt{2}}$	$\frac{\text{Eby}-\text{Ea}}{2}$
${Y_{0}^{(1)}}$	$-\frac{i \text{Mb}}{\sqrt{2}}$	$\text{Ebz}$	$-\frac{i \text{Mb}}{\sqrt{2}}$
${Y_{1}^{(1)}}$	$\frac{\text{Eby}-\text{Ea}}{2}$	$\frac{i \text{Mb}}{\sqrt{2}}$	$\frac{\text{Ea}+\text{Eby}}{2}$

The Hamiltonian on a basis of symmetric functions

	$p_z$	$p_x$	$p_y$
$p_z$	$\text{Ebz}$	$0$	$\text{Mb}$
$p_x$	$0$	$\text{Ea}$	$0$
$p_y$	$\text{Mb}$	$0$	$\text{Eby}$

Rotation matrix used

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
$p_z$	$0$	$1$	$0$
$p_x$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$
$p_y$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$

Irriducible representations and their onsite energy

	$\text{Ebz}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$
	$\text{Ea}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$
	$\text{Eby}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Eax2}+\text{Eay2z2}+\text{Eayz}+\text{Ebxy}+\text{Ebxz}) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ \frac{1}{4} \left(\sqrt{6} \text{Eax2}-\sqrt{6} \text{Eay2z2}-\sqrt{6} \text{Eayz}+\sqrt{6} \text{Ebxz}+2 \sqrt{2} \text{May2z2x2}\right) & k=2\land (m=-2\lor m=2) \\ \frac{1}{2} i \left(\sqrt{6} \text{Mb}-2 \sqrt{2} \text{Max2yz}\right) & k=2\land (m=-1\lor m=1) \\ \frac{1}{2} \left(-\text{Eax2}+\text{Eay2z2}+\text{Eayz}-2 \text{Ebxy}+\text{Ebxz}+2 \sqrt{3} \text{May2z2x2}\right) & k=2\land m=0 \\ \frac{3}{8} \sqrt{\frac{7}{10}} \left(3 \text{Eax2}+\text{Eay2z2}-4 \text{Ebxy}-2 \sqrt{3} \text{May2z2x2}\right) & k=4\land (m=-4\lor m=4) \\ \frac{3}{4} i \sqrt{\frac{7}{5}} \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}+2 \text{Mb}\right) & k=4\land (m=-3\lor m=3) \\ -\frac{3 \left(3 \text{Eax2}-3 \text{Eay2z2}+4 \text{Eayz}-4 \text{Ebxz}+2 \sqrt{3} \text{May2z2x2}\right)}{4 \sqrt{10}} & k=4\land (m=-2\lor m=2) \\ -\frac{3 i \left(\sqrt{3} \text{Max2yz}+7 \text{May2z2yz}+2 \text{Mb}\right)}{4 \sqrt{5}} & k=4\land (m=-1\lor m=1) \\ \frac{3}{40} \left(9 \text{Eax2}+19 \text{Eay2z2}+2 \left(-8 \text{Eayz}+2 \text{Ebxy}-8 \text{Ebxz}+5 \sqrt{3} \text{May2z2x2}\right)\right) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Eax2 + Eay2z2 + Eayz + Ebxy + Ebxz)/5, k == 0 && m == 0}, {0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(Sqrt[6]*Eax2 - Sqrt[6]*Eay2z2 - Sqrt[6]*Eayz + Sqrt[6]*Ebxz + 2*Sqrt[2]*May2z2x2)/4, k == 2 && (m == -2 || m == 2)}, {(I/2)*(-2*Sqrt[2]*Max2yz + Sqrt[6]*Mb), k == 2 && (m == -1 || m == 1)}, {(-Eax2 + Eay2z2 + Eayz - 2*Ebxy + Ebxz + 2*Sqrt[3]*May2z2x2)/2, k == 2 && m == 0}, {(3*Sqrt[7/10]*(3*Eax2 + Eay2z2 - 4*Ebxy - 2*Sqrt[3]*May2z2x2))/8, k == 4 && (m == -4 || m == 4)}, {((3*I)/4)*Sqrt[7/5]*(Sqrt[3]*Max2yz - May2z2yz + 2*Mb), k == 4 && (m == -3 || m == 3)}, {(-3*(3*Eax2 - 3*Eay2z2 + 4*Eayz - 4*Ebxz + 2*Sqrt[3]*May2z2x2))/(4*Sqrt[10]), k == 4 && (m == -2 || m == 2)}, {(((-3*I)/4)*(Sqrt[3]*Max2yz + 7*May2z2yz + 2*Mb))/Sqrt[5], k == 4 && (m == -1 || m == 1)}}, (3*(9*Eax2 + 19*Eay2z2 + 2*(-8*Eayz + 2*Ebxy - 8*Ebxz + 5*Sqrt[3]*May2z2x2)))/40]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{0, 0, (1/5)*(Eax2 + Eay2z2 + Eayz + Ebxy + Ebxz)} , 
       {2, 0, (1/2)*((-1)*(Eax2) + Eay2z2 + Eayz + (-2)*(Ebxy) + Ebxz + (2)*((sqrt(3))*(May2z2x2)))} , 
       {2,-1, (1/2*I)*((-2)*((sqrt(2))*(Max2yz)) + (sqrt(6))*(Mb))} , 
       {2, 1, (1/2*I)*((-2)*((sqrt(2))*(Max2yz)) + (sqrt(6))*(Mb))} , 
       {2,-2, (1/4)*((sqrt(6))*(Eax2) + (-1)*((sqrt(6))*(Eay2z2)) + (-1)*((sqrt(6))*(Eayz)) + (sqrt(6))*(Ebxz) + (2)*((sqrt(2))*(May2z2x2)))} , 
       {2, 2, (1/4)*((sqrt(6))*(Eax2) + (-1)*((sqrt(6))*(Eay2z2)) + (-1)*((sqrt(6))*(Eayz)) + (sqrt(6))*(Ebxz) + (2)*((sqrt(2))*(May2z2x2)))} , 
       {4, 0, (3/40)*((9)*(Eax2) + (19)*(Eay2z2) + (2)*((-8)*(Eayz) + (2)*(Ebxy) + (-8)*(Ebxz) + (5)*((sqrt(3))*(May2z2x2))))} , 
       {4,-1, (-3/4*I)*((1/(sqrt(5)))*((sqrt(3))*(Max2yz) + (7)*(May2z2yz) + (2)*(Mb)))} , 
       {4, 1, (-3/4*I)*((1/(sqrt(5)))*((sqrt(3))*(Max2yz) + (7)*(May2z2yz) + (2)*(Mb)))} , 
       {4,-2, (-3/4)*((1/(sqrt(10)))*((3)*(Eax2) + (-3)*(Eay2z2) + (4)*(Eayz) + (-4)*(Ebxz) + (2)*((sqrt(3))*(May2z2x2))))} , 
       {4, 2, (-3/4)*((1/(sqrt(10)))*((3)*(Eax2) + (-3)*(Eay2z2) + (4)*(Eayz) + (-4)*(Ebxz) + (2)*((sqrt(3))*(May2z2x2))))} , 
       {4,-3, (3/4*I)*((sqrt(7/5))*((sqrt(3))*(Max2yz) + (-1)*(May2z2yz) + (2)*(Mb)))} , 
       {4, 3, (3/4*I)*((sqrt(7/5))*((sqrt(3))*(Max2yz) + (-1)*(May2z2yz) + (2)*(Mb)))} , 
       {4,-4, (3/8)*((sqrt(7/10))*((3)*(Eax2) + Eay2z2 + (-4)*(Ebxy) + (-2)*((sqrt(3))*(May2z2x2))))} , 
       {4, 4, (3/8)*((sqrt(7/10))*((3)*(Eax2) + Eay2z2 + (-4)*(Ebxy) + (-2)*((sqrt(3))*(May2z2x2))))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\frac{1}{8} \left(3 \text{Eax2}+\text{Eay2z2}+4 \text{Ebxy}-2 \sqrt{3} \text{May2z2x2}\right)$	$-\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}-2 \text{Mb}\right)$	$\frac{1}{8} \left(-\sqrt{6} \text{Eax2}+\sqrt{6} \text{Eay2z2}-2 \sqrt{2} \text{May2z2x2}\right)$	$-\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}+2 \text{Mb}\right)$	$\frac{1}{8} \left(3 \text{Eax2}+\text{Eay2z2}-4 \text{Ebxy}-2 \sqrt{3} \text{May2z2x2}\right)$
${Y_{-1}^{(2)}}$	$\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}-2 \text{Mb}\right)$	$\frac{\text{Eayz}+\text{Ebxz}}{2}$	$-\frac{i \left(\text{Max2yz}+\sqrt{3} \text{May2z2yz}\right)}{2 \sqrt{2}}$	$\frac{\text{Eayz}-\text{Ebxz}}{2}$	$\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}+2 \text{Mb}\right)$
${Y_{0}^{(2)}}$	$\frac{1}{8} \left(-\sqrt{6} \text{Eax2}+\sqrt{6} \text{Eay2z2}-2 \sqrt{2} \text{May2z2x2}\right)$	$\frac{i \left(\text{Max2yz}+\sqrt{3} \text{May2z2yz}\right)}{2 \sqrt{2}}$	$\frac{1}{4} \left(\text{Eax2}+3 \text{Eay2z2}+2 \sqrt{3} \text{May2z2x2}\right)$	$\frac{i \left(\text{Max2yz}+\sqrt{3} \text{May2z2yz}\right)}{2 \sqrt{2}}$	$\frac{1}{8} \left(-\sqrt{6} \text{Eax2}+\sqrt{6} \text{Eay2z2}-2 \sqrt{2} \text{May2z2x2}\right)$
${Y_{1}^{(2)}}$	$\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}+2 \text{Mb}\right)$	$\frac{\text{Eayz}-\text{Ebxz}}{2}$	$-\frac{i \left(\text{Max2yz}+\sqrt{3} \text{May2z2yz}\right)}{2 \sqrt{2}}$	$\frac{\text{Eayz}+\text{Ebxz}}{2}$	$\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}-2 \text{Mb}\right)$
${Y_{2}^{(2)}}$	$\frac{1}{8} \left(3 \text{Eax2}+\text{Eay2z2}-4 \text{Ebxy}-2 \sqrt{3} \text{May2z2x2}\right)$	$-\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}+2 \text{Mb}\right)$	$\frac{1}{8} \left(-\sqrt{6} \text{Eax2}+\sqrt{6} \text{Eay2z2}-2 \sqrt{2} \text{May2z2x2}\right)$	$-\frac{1}{4} i \left(\sqrt{3} \text{Max2yz}-\text{May2z2yz}-2 \text{Mb}\right)$	$\frac{1}{8} \left(3 \text{Eax2}+\text{Eay2z2}+4 \text{Ebxy}-2 \sqrt{3} \text{May2z2x2}\right)$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{3x^2-r^2}$	$d_{\text{xz}}$	$d_{y^2-z^2}$
$d_{\text{yz}}$	$\text{Eayz}$	$0$	$\text{Max2yz}$	$0$	$\text{May2z2yz}$
$d_{\text{xz}}$	$0$	$\text{Ebxz}$	$0$	$\text{Mb}$	$0$
$d_{3x^2-r^2}$	$\text{Max2yz}$	$0$	$\text{Eax2}$	$0$	$\text{May2z2x2}$
$d_{\text{xz}}$	$0$	$\text{Mb}$	$0$	$\text{Ebxy}$	$0$
$d_{y^2-z^2}$	$\text{May2z2yz}$	$0$	$\text{May2z2x2}$	$0$	$\text{Eay2z2}$

Rotation matrix used

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
$d_{\text{yz}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$
$d_{\text{xz}}$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$0$
$d_{3x^2-r^2}$	$\frac{\sqrt{\frac{3}{2}}}{2}$	$0$	$-\frac{1}{2}$	$0$	$\frac{\sqrt{\frac{3}{2}}}{2}$
$d_{\text{xz}}$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$
$d_{y^2-z^2}$	$-\frac{1}{2 \sqrt{2}}$	$0$	$-\frac{\sqrt{3}}{2}$	$0$	$-\frac{1}{2 \sqrt{2}}$

Irriducible representations and their onsite energy

	$\text{Eayz}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$
	$\text{Ebxz}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$
	$\text{Eax2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{5}{\pi }} \left(-6 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} \left(3 x^2-3 y^2-3 z^2+1\right)$
	$\text{Ebxy}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$
	$\text{Eay2z2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{15}{\pi }} \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{15}{\pi }} \left(-x^2+y^2-3 z^2+1\right)$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4\land k\neq 6)\lor (k\neq 4\land k\neq 6\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (k\neq 6\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4)\lor (m\neq -6\land m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5\land m\neq 6) \\ A(2,2) & k=2\land (m=-2\lor m=2) \\ i B(2,1) & k=2\land (m=-1\lor m=1) \\ A(2,0) & k=2\land m=0 \\ A(4,4) & k=4\land (m=-4\lor m=4) \\ i B(4,3) & k=4\land (m=-3\lor m=3) \\ A(4,2) & k=4\land (m=-2\lor m=2) \\ i B(4,1) & k=4\land (m=-1\lor m=1) \\ A(4,0) & k=4\land m=0 \\ A(6,6) & k=6\land (m=-6\lor m=6) \\ i B(6,5) & k=6\land (m=-5\lor m=5) \\ A(6,4) & k=6\land (m=-4\lor m=4) \\ i B(6,3) & k=6\land (m=-3\lor m=3) \\ A(6,2) & k=6\land (m=-2\lor m=2) \\ i B(6,1) & k=6\land (m=-1\lor m=1) \\ A(6,0) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {0, (k != 2 && k != 4 && k != 6) || (k != 4 && k != 6 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (k != 6 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4) || (m != -6 && m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5 && m != 6)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {I*B[4, 1], k == 4 && (m == -1 || m == 1)}, {A[4, 0], k == 4 && m == 0}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {I*B[6, 5], k == 6 && (m == -5 || m == 5)}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {I*B[6, 3], k == 6 && (m == -3 || m == 3)}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {I*B[6, 1], k == 6 && (m == -1 || m == 1)}}, A[6, 0]]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {2,-1, (I)*(B(2,1))} , 
       {2, 1, (I)*(B(2,1))} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} , 
       {4, 0, A(4,0)} , 
       {4,-1, (I)*(B(4,1))} , 
       {4, 1, (I)*(B(4,1))} , 
       {4,-2, A(4,2)} , 
       {4, 2, A(4,2)} , 
       {4,-3, (I)*(B(4,3))} , 
       {4, 3, (I)*(B(4,3))} , 
       {4,-4, A(4,4)} , 
       {4, 4, A(4,4)} , 
       {6, 0, A(6,0)} , 
       {6,-1, (I)*(B(6,1))} , 
       {6, 1, (I)*(B(6,1))} , 
       {6,-2, A(6,2)} , 
       {6, 2, A(6,2)} , 
       {6,-3, (I)*(B(6,3))} , 
       {6, 3, (I)*(B(6,3))} , 
       {6,-4, A(6,4)} , 
       {6, 4, A(6,4)} , 
       {6,-5, (I)*(B(6,5))} , 
       {6, 5, (I)*(B(6,5))} , 
       {6,-6, A(6,6)} , 
       {6, 6, A(6,6)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0)$	$\frac{1}{429} i \left(143 B(2,1)-13 \sqrt{30} B(4,1)+5 \sqrt{7} B(6,1)\right)$	$-\frac{1}{3} \sqrt{\frac{2}{5}} A(2,2)+\frac{1}{11} \sqrt{6} A(4,2)-\frac{10}{429} \sqrt{7} A(6,2)$	$-\frac{1}{429} i \sqrt{7} \left(39 B(4,3)-10 \sqrt{3} B(6,3)\right)$	$\frac{1}{143} \sqrt{\frac{14}{3}} \left(13 A(4,4)-5 \sqrt{5} A(6,4)\right)$	$\frac{5}{13} i \sqrt{\frac{14}{33}} B(6,5)$	$-\frac{10}{13} \sqrt{\frac{7}{33}} A(6,6)$
${Y_{-2}^{(3)}}$	$-\frac{1}{429} i \left(143 B(2,1)-13 \sqrt{30} B(4,1)+5 \sqrt{7} B(6,1)\right)$	$A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0)$	$\frac{i B(2,1)}{\sqrt{15}}+\frac{1}{429} i \left(52 \sqrt{2} B(4,1)-5 \sqrt{105} B(6,1)\right)$	$\frac{-286 \sqrt{5} A(2,2)-65 \sqrt{3} A(4,2)+100 \sqrt{14} A(6,2)}{2145}$	$-\frac{1}{429} i \sqrt{14} \left(13 B(4,3)+15 \sqrt{3} B(6,3)\right)$	$\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} A(4,4)+30 A(6,4)\right)$	$-\frac{5}{13} i \sqrt{\frac{14}{33}} B(6,5)$
${Y_{-1}^{(3)}}$	$-\frac{1}{3} \sqrt{\frac{2}{5}} A(2,2)+\frac{1}{11} \sqrt{6} A(4,2)-\frac{10}{429} \sqrt{7} A(6,2)$	$-\frac{i \left(143 \sqrt{15} B(2,1)+260 \sqrt{2} B(4,1)-25 \sqrt{105} B(6,1)\right)}{2145}$	$A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0)$	$\frac{i \left(143 \sqrt{2} B(2,1)+65 \sqrt{15} B(4,1)+125 \sqrt{14} B(6,1)\right)}{2145}$	$-\frac{2 \left(143 \sqrt{6} A(2,2)+65 \sqrt{10} A(4,2)+25 \sqrt{105} A(6,2)\right)}{2145}$	$\frac{1}{429} i \sqrt{14} \left(13 B(4,3)+15 \sqrt{3} B(6,3)\right)$	$\frac{1}{143} \sqrt{\frac{14}{3}} \left(13 A(4,4)-5 \sqrt{5} A(6,4)\right)$
${Y_{0}^{(3)}}$	$\frac{1}{429} i \sqrt{7} \left(39 B(4,3)-10 \sqrt{3} B(6,3)\right)$	$\frac{-286 \sqrt{5} A(2,2)-65 \sqrt{3} A(4,2)+100 \sqrt{14} A(6,2)}{2145}$	$-\frac{i \left(143 \sqrt{2} B(2,1)+65 \sqrt{15} B(4,1)+125 \sqrt{14} B(6,1)\right)}{2145}$	$A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0))$	$-\frac{i \left(143 \sqrt{2} B(2,1)+65 \sqrt{15} B(4,1)+125 \sqrt{14} B(6,1)\right)}{2145}$	$\frac{-286 \sqrt{5} A(2,2)-65 \sqrt{3} A(4,2)+100 \sqrt{14} A(6,2)}{2145}$	$\frac{1}{429} i \sqrt{7} \left(39 B(4,3)-10 \sqrt{3} B(6,3)\right)$
${Y_{1}^{(3)}}$	$\frac{1}{143} \sqrt{\frac{14}{3}} \left(13 A(4,4)-5 \sqrt{5} A(6,4)\right)$	$\frac{1}{429} i \sqrt{14} \left(13 B(4,3)+15 \sqrt{3} B(6,3)\right)$	$-\frac{2 \left(143 \sqrt{6} A(2,2)+65 \sqrt{10} A(4,2)+25 \sqrt{105} A(6,2)\right)}{2145}$	$\frac{i \left(143 \sqrt{2} B(2,1)+65 \sqrt{15} B(4,1)+125 \sqrt{14} B(6,1)\right)}{2145}$	$A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0)$	$-\frac{i \left(143 \sqrt{15} B(2,1)+260 \sqrt{2} B(4,1)-25 \sqrt{105} B(6,1)\right)}{2145}$	$-\frac{1}{3} \sqrt{\frac{2}{5}} A(2,2)+\frac{1}{11} \sqrt{6} A(4,2)-\frac{10}{429} \sqrt{7} A(6,2)$
${Y_{2}^{(3)}}$	$-\frac{5}{13} i \sqrt{\frac{14}{33}} B(6,5)$	$\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} A(4,4)+30 A(6,4)\right)$	$-\frac{1}{429} i \sqrt{14} \left(13 B(4,3)+15 \sqrt{3} B(6,3)\right)$	$\frac{-286 \sqrt{5} A(2,2)-65 \sqrt{3} A(4,2)+100 \sqrt{14} A(6,2)}{2145}$	$\frac{i B(2,1)}{\sqrt{15}}+\frac{1}{429} i \left(52 \sqrt{2} B(4,1)-5 \sqrt{105} B(6,1)\right)$	$A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0)$	$-\frac{1}{429} i \left(143 B(2,1)-13 \sqrt{30} B(4,1)+5 \sqrt{7} B(6,1)\right)$
${Y_{3}^{(3)}}$	$-\frac{10}{13} \sqrt{\frac{7}{33}} A(6,6)$	$\frac{5}{13} i \sqrt{\frac{14}{33}} B(6,5)$	$\frac{1}{143} \sqrt{\frac{14}{3}} \left(13 A(4,4)-5 \sqrt{5} A(6,4)\right)$	$-\frac{1}{429} i \sqrt{7} \left(39 B(4,3)-10 \sqrt{3} B(6,3)\right)$	$-\frac{1}{3} \sqrt{\frac{2}{5}} A(2,2)+\frac{1}{11} \sqrt{6} A(4,2)-\frac{10}{429} \sqrt{7} A(6,2)$	$\frac{1}{429} i \left(143 B(2,1)-13 \sqrt{30} B(4,1)+5 \sqrt{7} B(6,1)\right)$	$A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0)$

The Hamiltonian on a basis of symmetric functions

	$f_{z\left(3y^2-z^2\right)}$	$f_{\text{xyz}}$	$f_{z\left(5x^2-r^2\right)}$	$f_{x\left(5x^2-3r^2\right)}$	$f_{y\left(5x^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(y^2-3z^2\right)}$
$f_{z\left(3y^2-z^2\right)}$	$\frac{3432 A(0,0)+572 A(2,0)-572 \sqrt{6} A(2,2)+117 A(4,0)-78 \sqrt{10} A(4,2)+39 \sqrt{70} A(4,4)+590 A(6,0)+80 \sqrt{105} A(6,2)+90 \sqrt{14} A(6,4)}{3432}$	$0$	$\frac{A(2,0)}{2 \sqrt{15}}+\frac{A(2,2)}{3 \sqrt{10}}+\frac{169 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-65 \sqrt{42} A(4,4)+70 \sqrt{15} A(6,0)-80 \sqrt{7} A(6,2)-30 \sqrt{210} A(6,4)}{3432}$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{5}} B(2,1)+\frac{2 B(4,1)}{11 \sqrt{3}}+\frac{\sqrt{7} \left(208 \sqrt{3} B(4,3)+25 \sqrt{10} B(6,1)-15 \left(7 B(6,3)+\sqrt{165} B(6,5)\right)\right)}{3432}$	$0$	$\frac{5 \left(66 \sqrt{21} B(6,1)+11 \sqrt{210} B(6,3)+3 \sqrt{154} B(6,5)\right)}{3432 \sqrt{2}}$
$f_{\text{xyz}}$	$0$	$\frac{1}{429} \left(429 A(0,0)-91 A(4,0)-13 \sqrt{70} A(4,4)+30 A(6,0)-30 \sqrt{14} A(6,4)\right)$	$0$	$\frac{-572 \sqrt{10} B(2,1)+65 \sqrt{3} B(4,1)+5 \sqrt{7} \left(-13 \sqrt{3} B(4,3)+5 \sqrt{10} B(6,1)-45 B(6,3)+5 \sqrt{165} B(6,5)\right)}{4290}$	$0$	$\frac{1}{858} \left(-91 \sqrt{5} B(4,1)-13 \sqrt{35} B(4,3)+15 \left(\sqrt{42} B(6,1)-\sqrt{105} B(6,3)-\sqrt{77} B(6,5)\right)\right)$	$0$
$f_{z\left(5x^2-r^2\right)}$	$\frac{A(2,0)}{2 \sqrt{15}}+\frac{A(2,2)}{3 \sqrt{10}}+\frac{169 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-65 \sqrt{42} A(4,4)+70 \sqrt{15} A(6,0)-80 \sqrt{7} A(6,2)-30 \sqrt{210} A(6,4)}{3432}$	$0$	$A(0,0)+\frac{1}{10} A(2,0)+\frac{572 \sqrt{6} A(2,2)-221 A(4,0)+78 \sqrt{10} A(4,2)+65 \sqrt{70} A(4,4)+450 A(6,0)-80 \sqrt{105} A(6,2)+150 \sqrt{14} A(6,4)}{3432}$	$0$	$\frac{2}{5} \sqrt{\frac{2}{3}} B(2,1)+\frac{-104 \sqrt{5} B(4,1)+104 \sqrt{35} B(4,3)+25 \sqrt{42} B(6,1)-45 \sqrt{105} B(6,3)+75 \sqrt{77} B(6,5)}{3432}$	$0$	$\frac{1}{3} \sqrt{\frac{2}{5}} B(2,1)+\frac{5 B(4,1)}{11 \sqrt{3}}-\frac{\sqrt{7} \left(104 \sqrt{3} B(4,3)+5 \left(-7 \sqrt{10} B(6,1)+39 B(6,3)+\sqrt{165} B(6,5)\right)\right)}{3432}$
$f_{x\left(5x^2-3r^2\right)}$	$0$	$\frac{-1144 \sqrt{5} B(2,1)+65 \sqrt{6} B(4,1)-65 \sqrt{42} B(4,3)+50 \sqrt{35} B(6,1)-225 \sqrt{14} B(6,3)+25 \sqrt{2310} B(6,5)}{4290 \sqrt{2}}$	$0$	$\frac{8580 A(0,0)-1144 A(2,0)+1144 \sqrt{6} A(2,2)+585 A(4,0)-390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)+125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)+125 \sqrt{231} A(6,6)}{8580}$	$0$	$\frac{A(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} A(2,2)+\frac{-13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)+13 \sqrt{42} A(4,4)-35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)-5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)}{1716}$	$0$
$f_{y\left(5x^2-r^2\right)}$	$-\frac{1}{3} \sqrt{\frac{2}{5}} B(2,1)+\frac{2 B(4,1)}{11 \sqrt{3}}+\frac{\sqrt{7} \left(208 \sqrt{3} B(4,3)+25 \sqrt{10} B(6,1)-15 \left(7 B(6,3)+\sqrt{165} B(6,5)\right)\right)}{3432}$	$0$	$\frac{2}{5} \sqrt{\frac{2}{3}} B(2,1)+\frac{-104 \sqrt{5} B(4,1)+104 \sqrt{35} B(4,3)+25 \sqrt{42} B(6,1)-45 \sqrt{105} B(6,3)+75 \sqrt{77} B(6,5)}{3432}$	$0$	$\frac{17160 A(0,0)-5148 A(2,0)+572 \sqrt{6} A(2,2)+1495 A(4,0)-650 \sqrt{10} A(4,2)-195 \sqrt{70} A(4,4)-375 A(6,0)+25 \sqrt{105} A(6,2)+375 \sqrt{14} A(6,4)-375 \sqrt{231} A(6,6)}{17160}$	$0$	$\frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)+338 \sqrt{6} A(4,2)+91 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+5 \sqrt{7} A(6,2)-35 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{17160}$
$f_{x\left(y^2-z^2\right)}$	$0$	$-\frac{91 \sqrt{10} B(4,1)+13 \sqrt{70} B(4,3)-30 \sqrt{21} B(6,1)+15 \sqrt{210} B(6,3)+15 \sqrt{154} B(6,5)}{858 \sqrt{2}}$	$0$	$\frac{A(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} A(2,2)+\frac{-13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)+13 \sqrt{42} A(4,4)-35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)-5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)}{1716}$	$0$	$\frac{1716 A(0,0)+91 A(4,0)+182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)+15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)+15 \sqrt{231} A(6,6)}{1716}$	$0$
$f_{y\left(y^2-3z^2\right)}$	$\frac{5 \left(33 \sqrt{42} B(6,1)+11 \sqrt{105} B(6,3)+3 \sqrt{77} B(6,5)\right)}{3432}$	$0$	$\frac{1}{3} \sqrt{\frac{2}{5}} B(2,1)+\frac{5 B(4,1)}{11 \sqrt{3}}-\frac{\sqrt{7} \left(104 \sqrt{3} B(4,3)+5 \left(-7 \sqrt{10} B(6,1)+39 B(6,3)+\sqrt{165} B(6,5)\right)\right)}{3432}$	$0$	$\frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)+338 \sqrt{6} A(4,2)+91 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+5 \sqrt{7} A(6,2)-35 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{17160}$	$0$	$\frac{3432 A(0,0)+572 A(2,0)-572 \sqrt{6} A(2,2)+117 A(4,0)-78 \sqrt{10} A(4,2)+39 \sqrt{70} A(4,4)-565 A(6,0)-85 \sqrt{105} A(6,2)-75 \sqrt{14} A(6,4)-5 \sqrt{231} A(6,6)}{3432}$

Rotation matrix used

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$f_{z\left(3y^2-z^2\right)}$	$0$	$-\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{\frac{5}{2}}}{2}$	$0$	$-\frac{\sqrt{3}}{4}$	$0$
$f_{\text{xyz}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$	$0$
$f_{z\left(5x^2-r^2\right)}$	$0$	$\frac{\sqrt{5}}{4}$	$0$	$-\frac{\sqrt{\frac{3}{2}}}{2}$	$0$	$\frac{\sqrt{5}}{4}$	$0$
$f_{x\left(5x^2-3r^2\right)}$	$\frac{\sqrt{5}}{4}$	$0$	$-\frac{\sqrt{3}}{4}$	$0$	$\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{5}}{4}$
$f_{y\left(5x^2-r^2\right)}$	$\frac{1}{4} i \sqrt{\frac{15}{2}}$	$0$	$-\frac{i}{4 \sqrt{2}}$	$0$	$-\frac{i}{4 \sqrt{2}}$	$0$	$\frac{1}{4} i \sqrt{\frac{15}{2}}$
$f_{x\left(y^2-z^2\right)}$	$-\frac{\sqrt{3}}{4}$	$0$	$-\frac{\sqrt{5}}{4}$	$0$	$\frac{\sqrt{5}}{4}$	$0$	$\frac{\sqrt{3}}{4}$
$f_{y\left(y^2-3z^2\right)}$	$-\frac{i}{4 \sqrt{2}}$	$0$	$-\frac{1}{4} i \sqrt{\frac{15}{2}}$	$0$	$-\frac{1}{4} i \sqrt{\frac{15}{2}}$	$0$	$-\frac{i}{4 \sqrt{2}}$

Irriducible representations and their onsite energy

	$\frac{3432 A(0,0)+572 A(2,0)-572 \sqrt{6} A(2,2)+117 A(4,0)-78 \sqrt{10} A(4,2)+39 \sqrt{70} A(4,4)+590 A(6,0)+80 \sqrt{105} A(6,2)+90 \sqrt{14} A(6,4)}{3432}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{35}{2 \pi }} \cos (\theta ) \left(6 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )-1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{35}{2 \pi }} z \left(-3 x^2+3 y^2-5 z^2+3\right)$
	$\frac{1}{429} \left(429 A(0,0)-91 A(4,0)-13 \sqrt{70} A(4,4)+30 A(6,0)-30 \sqrt{14} A(6,4)\right)$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$
	$A(0,0)+\frac{1}{10} A(2,0)+\frac{572 \sqrt{6} A(2,2)-221 A(4,0)+78 \sqrt{10} A(4,2)+65 \sqrt{70} A(4,4)+450 A(6,0)-80 \sqrt{105} A(6,2)+150 \sqrt{14} A(6,4)}{3432}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} \cos (\theta ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} z \left(5 x^2-5 y^2-5 z^2+3\right)$
	$\frac{8580 A(0,0)-1144 A(2,0)+1144 \sqrt{6} A(2,2)+585 A(4,0)-390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)+125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)+125 \sqrt{231} A(6,6)}{8580}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$
	$\frac{17160 A(0,0)-5148 A(2,0)+572 \sqrt{6} A(2,2)+1495 A(4,0)-650 \sqrt{10} A(4,2)-195 \sqrt{70} A(4,4)-375 A(6,0)+25 \sqrt{105} A(6,2)+375 \sqrt{14} A(6,4)-375 \sqrt{231} A(6,6)}{17160}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} y \left(15 x^2-5 y^2-5 z^2+1\right)$
	$\frac{1716 A(0,0)+91 A(4,0)+182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)+15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)+15 \sqrt{231} A(6,6)}{1716}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$
	$\frac{3432 A(0,0)+572 A(2,0)-572 \sqrt{6} A(2,2)+117 A(4,0)-78 \sqrt{10} A(4,2)+39 \sqrt{70} A(4,4)-565 A(6,0)-85 \sqrt{105} A(6,2)-75 \sqrt{14} A(6,4)-5 \sqrt{231} A(6,6)}{3432}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{35}{2 \pi }} \sin (\theta ) \sin (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+7 \cos (2 \theta )+5\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{35}{2 \pi }} y \left(-3 x^2+y^2-15 z^2+3\right)$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-p orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 1\lor (m\neq -1\land m\neq 1) \\ \sqrt{\frac{3}{2}} \text{Ma} & k=1\land m=-1 \\ -\sqrt{\frac{3}{2}} \text{Ma} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 1 || (m != -1 && m != 1)}, {Sqrt[3/2]*Ma, k == 1 && m == -1}}, -(Sqrt[3/2]*Ma)]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{1, 1, (-1)*((sqrt(3/2))*(Ma))} , 
       {1,-1, (sqrt(3/2))*(Ma)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{0}^{(0)}}$	$\frac{\text{Ma}}{\sqrt{2}}$	$0$	$-\frac{\text{Ma}}{\sqrt{2}}$

The Hamiltonian on a basis of symmetric functions

	$p_z$	$p_x$	$p_y$
$\text{s}$	$0$	$\text{Ma}$	$0$

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ A(2,2) & k=2\land (m=-2\lor m=2) \\ i B(2,1) & k=2\land (m=-1\lor m=1) \\ A(2,0) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}}, A[2, 0]]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{2, 0, A(2,0)} , 
       {2,-1, (I)*(B(2,1))} , 
       {2, 1, (I)*(B(2,1))} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{0}^{(0)}}$	$\frac{A(2,2)}{\sqrt{5}}$	$-\frac{i B(2,1)}{\sqrt{5}}$	$\frac{A(2,0)}{\sqrt{5}}$	$-\frac{i B(2,1)}{\sqrt{5}}$	$\frac{A(2,2)}{\sqrt{5}}$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{3x^2-r^2}$	$d_{\text{xz}}$	$d_{y^2-z^2}$
$\text{s}$	$\sqrt{\frac{2}{5}} B(2,1)$	$0$	$-\frac{A(2,0)-\sqrt{6} A(2,2)}{2 \sqrt{5}}$	$0$	$-\frac{1}{2} \sqrt{\frac{3}{5}} A(2,0)-\frac{A(2,2)}{\sqrt{10}}$

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2\land m\neq 3) \\ -A(3,3) & k=3\land m=-3 \\ -i B(3,2) & k=3\land m=-2 \\ -A(3,1) & k=3\land m=-1 \\ A(3,1) & k=3\land m=1 \\ i B(3,2) & k=3\land m=2 \\ A(3,3) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != -2 && m != -1 && m != 1 && m != 2 && m != 3)}, {-A[3, 3], k == 3 && m == -3}, {(-I)*B[3, 2], k == 3 && m == -2}, {-A[3, 1], k == 3 && m == -1}, {A[3, 1], k == 3 && m == 1}, {I*B[3, 2], k == 3 && m == 2}}, A[3, 3]]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, (-I)*(B(3,2))} , 
       {3, 2, (I)*(B(3,2))} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$-\frac{A(3,3)}{\sqrt{7}}$	$\frac{i B(3,2)}{\sqrt{7}}$	$-\frac{A(3,1)}{\sqrt{7}}$	$0$	$\frac{A(3,1)}{\sqrt{7}}$	$-\frac{i B(3,2)}{\sqrt{7}}$	$\frac{A(3,3)}{\sqrt{7}}$

The Hamiltonian on a basis of symmetric functions

	$f_{z\left(3y^2-z^2\right)}$	$f_{\text{xyz}}$	$f_{z\left(5x^2-r^2\right)}$	$f_{x\left(5x^2-3r^2\right)}$	$f_{y\left(5x^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(y^2-3z^2\right)}$
$\text{s}$	$0$	$-\sqrt{\frac{2}{7}} B(3,2)$	$0$	$\frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right)$	$0$	$\frac{1}{14} \left(\sqrt{35} A(3,1)+\sqrt{21} A(3,3)\right)$	$0$

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & (k\neq 3\land (k\neq 1\lor (m\neq -1\land m\neq 1)))\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2\land m\neq 3) \\ -A(1,1) & k=1\land m=-1 \\ A(1,1) & k=1\land m=1 \\ -A(3,3) & k=3\land m=-3 \\ -i B(3,2) & k=3\land m=-2 \\ -A(3,1) & k=3\land m=-1 \\ A(3,1) & k=3\land m=1 \\ i B(3,2) & k=3\land m=2 \\ A(3,3) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || (m != -1 && m != 1))) || (m != -3 && m != -2 && m != -1 && m != 1 && m != 2 && m != 3)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 1], k == 1 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {(-I)*B[3, 2], k == 3 && m == -2}, {-A[3, 1], k == 3 && m == -1}, {A[3, 1], k == 3 && m == 1}, {I*B[3, 2], k == 3 && m == 2}}, A[3, 3]]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, (-I)*(B(3,2))} , 
       {3, 2, (I)*(B(3,2))} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-1}^{(1)}}$	$\frac{1}{35} \left(\sqrt{15} A(3,1)-7 \sqrt{10} A(1,1)\right)$	$0$	$\frac{A(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} A(3,1)$	$\frac{1}{7} i \sqrt{6} B(3,2)$	$-\frac{3}{7} A(3,3)$
${Y_{0}^{(1)}}$	$\frac{1}{7} i \sqrt{3} B(3,2)$	$-\frac{7 A(1,1)+2 \sqrt{6} A(3,1)}{7 \sqrt{5}}$	$0$	$\frac{A(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} A(3,1)$	$-\frac{1}{7} i \sqrt{3} B(3,2)$
${Y_{1}^{(1)}}$	$\frac{3}{7} A(3,3)$	$-\frac{1}{7} i \sqrt{6} B(3,2)$	$\frac{3}{7} \sqrt{\frac{2}{5}} A(3,1)-\frac{A(1,1)}{\sqrt{15}}$	$0$	$\sqrt{\frac{2}{5}} A(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} A(3,1)$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{3x^2-r^2}$	$d_{\text{xz}}$	$d_{y^2-z^2}$
$p_z$	$0$	$-\sqrt{\frac{2}{5}} A(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} A(3,1)$	$0$	$-\frac{1}{7} \sqrt{6} B(3,2)$	$0$
$p_x$	$-\frac{1}{7} \sqrt{6} B(3,2)$	$0$	$\frac{1}{210} \left(-28 \sqrt{30} A(1,1)+27 \sqrt{5} A(3,1)-45 \sqrt{3} A(3,3)\right)$	$0$	$\frac{1}{14} \left(\sqrt{15} A(3,1)+3 A(3,3)\right)$
$p_y$	$0$	$-\frac{1}{7} \sqrt{6} B(3,2)$	$0$	$\frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right)$	$0$

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ A(2,2) & k=2\land (m=-2\lor m=2) \\ i B(2,1) & k=2\land (m=-1\lor m=1) \\ A(2,0) & k=2\land m=0 \\ A(4,4) & k=4\land (m=-4\lor m=4) \\ i B(4,3) & k=4\land (m=-3\lor m=3) \\ A(4,2) & k=4\land (m=-2\lor m=2) \\ i B(4,1) & k=4\land (m=-1\lor m=1) \\ A(4,0) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {I*B[2, 1], k == 2 && (m == -1 || m == 1)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {I*B[4, 1], k == 4 && (m == -1 || m == 1)}}, A[4, 0]]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{2, 0, A(2,0)} , 
       {2,-1, (I)*(B(2,1))} , 
       {2, 1, (I)*(B(2,1))} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} , 
       {4, 0, A(4,0)} , 
       {4,-1, (I)*(B(4,1))} , 
       {4, 1, (I)*(B(4,1))} , 
       {4,-2, A(4,2)} , 
       {4, 2, A(4,2)} , 
       {4,-3, (I)*(B(4,3))} , 
       {4, 3, (I)*(B(4,3))} , 
       {4,-4, A(4,4)} , 
       {4, 4, A(4,4)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-1}^{(1)}}$	$\frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}}$	$-\frac{i \left(3 \sqrt{30} B(2,1)-5 B(4,1)\right)}{15 \sqrt{7}}$	$\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$	$-\frac{i \left(27 B(2,1)-5 \sqrt{30} B(4,1)\right)}{45 \sqrt{7}}$	$\frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right)$	$\frac{1}{3} i B(4,3)$	$-\frac{2 A(4,4)}{3 \sqrt{3}}$
${Y_{0}^{(1)}}$	$-\frac{i B(4,3)}{3 \sqrt{3}}$	$\sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}}$	$-\frac{1}{105} i \left(6 \sqrt{42} B(2,1)+5 \sqrt{35} B(4,1)\right)$	$\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$	$-\frac{1}{105} i \left(6 \sqrt{42} B(2,1)+5 \sqrt{35} B(4,1)\right)$	$\sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}}$	$-\frac{i B(4,3)}{3 \sqrt{3}}$
${Y_{1}^{(1)}}$	$-\frac{2 A(4,4)}{3 \sqrt{3}}$	$\frac{1}{3} i B(4,3)$	$\frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right)$	$-\frac{i \left(27 B(2,1)-5 \sqrt{30} B(4,1)\right)}{45 \sqrt{7}}$	$\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$	$-\frac{i \left(3 \sqrt{30} B(2,1)-5 B(4,1)\right)}{15 \sqrt{7}}$	$\frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}}$

The Hamiltonian on a basis of symmetric functions

	$f_{z\left(3y^2-z^2\right)}$	$f_{\text{xyz}}$	$f_{z\left(5x^2-r^2\right)}$	$f_{x\left(5x^2-3r^2\right)}$	$f_{y\left(5x^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(y^2-3z^2\right)}$
$p_z$	$-\frac{27 \sqrt{10} A(2,0)+18 \sqrt{15} A(2,2)+20 \sqrt{10} A(4,0)+60 A(4,2)}{60 \sqrt{21}}$	$0$	$\frac{1}{420} \left(-27 \sqrt{14} A(2,0)+30 \sqrt{21} A(2,2)-20 \sqrt{14} A(4,0)+20 \sqrt{35} A(4,2)\right)$	$0$	$\frac{1}{420} \left(-12 \sqrt{21} B(2,1)-5 \sqrt{70} B(4,1)+35 \sqrt{10} B(4,3)\right)$	$0$	$-\frac{3 B(2,1)}{\sqrt{35}}-\frac{15 \sqrt{7} B(4,1)+7 B(4,3)}{42 \sqrt{6}}$
$p_x$	$0$	$\sqrt{\frac{6}{35}} B(2,1)-\frac{B(4,1)}{3 \sqrt{7}}+\frac{1}{3} B(4,3)$	$0$	$\frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right)$	$0$	$\frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right)$	$0$
$p_y$	$\frac{3 B(2,1)}{\sqrt{35}}-\frac{13 \sqrt{7} B(4,1)+21 B(4,3)}{42 \sqrt{6}}$	$0$	$\frac{1}{420} \left(-12 \sqrt{21} B(2,1)-5 \sqrt{70} B(4,1)+35 \sqrt{10} B(4,3)\right)$	$0$	$\frac{1}{420} \left(-9 \sqrt{14} A(2,0)+42 \sqrt{21} A(2,2)+5 \sqrt{14} A(4,0)-70 \sqrt{5} A(4,4)\right)$	$0$	$\frac{-27 \sqrt{210} A(2,0)-54 \sqrt{35} A(2,2)+5 \sqrt{3} \left(3 \sqrt{70} A(4,0)+16 \sqrt{7} A(4,2)+14 A(4,4)\right)}{1260}$

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & (k\neq 1\land k\neq 3\land k\neq 5)\lor (k\neq 3\land k\neq 5\land m\neq -1\land m\neq 1)\lor (k\neq 5\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2\land m\neq 3)\lor (m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5) \\ -A(1,1) & k=1\land m=-1 \\ A(1,1) & k=1\land m=1 \\ -A(3,3) & k=3\land m=-3 \\ -i B(3,2) & k=3\land m=-2 \\ -A(3,1) & k=3\land m=-1 \\ A(3,1) & k=3\land m=1 \\ i B(3,2) & k=3\land m=2 \\ A(3,3) & k=3\land m=3 \\ -A(5,5) & k=5\land m=-5 \\ -i B(5,4) & k=5\land m=-4 \\ -A(5,3) & k=5\land m=-3 \\ -i B(5,2) & k=5\land m=-2 \\ -A(5,1) & k=5\land m=-1 \\ A(5,1) & k=5\land m=1 \\ i B(5,2) & k=5\land m=2 \\ A(5,3) & k=5\land m=3 \\ i B(5,4) & k=5\land m=4 \\ A(5,5) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_X.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 1 && k != 3 && k != 5) || (k != 3 && k != 5 && m != -1 && m != 1) || (k != 5 && m != -3 && m != -2 && m != -1 && m != 1 && m != 2 && m != 3) || (m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 1], k == 1 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {(-I)*B[3, 2], k == 3 && m == -2}, {-A[3, 1], k == 3 && m == -1}, {A[3, 1], k == 3 && m == 1}, {I*B[3, 2], k == 3 && m == 2}, {A[3, 3], k == 3 && m == 3}, {-A[5, 5], k == 5 && m == -5}, {(-I)*B[5, 4], k == 5 && m == -4}, {-A[5, 3], k == 5 && m == -3}, {(-I)*B[5, 2], k == 5 && m == -2}, {-A[5, 1], k == 5 && m == -1}, {A[5, 1], k == 5 && m == 1}, {I*B[5, 2], k == 5 && m == 2}, {A[5, 3], k == 5 && m == 3}, {I*B[5, 4], k == 5 && m == 4}}, A[5, 5]]

Input format suitable for Quanty

Akm_C2_X.Quanty

Akm = {{1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, (-I)*(B(3,2))} , 
       {3, 2, (I)*(B(3,2))} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} , 
       {5,-1, (-1)*(A(5,1))} , 
       {5, 1, A(5,1)} , 
       {5,-2, (-I)*(B(5,2))} , 
       {5, 2, (I)*(B(5,2))} , 
       {5,-3, (-1)*(A(5,3))} , 
       {5, 3, A(5,3)} , 
       {5,-4, (-I)*(B(5,4))} , 
       {5, 4, (I)*(B(5,4))} , 
       {5,-5, (-1)*(A(5,5))} , 
       {5, 5, A(5,5)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-2}^{(2)}}$	$\frac{1}{231} \left(-33 \sqrt{21} A(1,1)+11 \sqrt{14} A(3,1)-\sqrt{35} A(5,1)\right)$	$0$	$\frac{A(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} A(3,1)+\frac{5 A(5,1)}{11 \sqrt{21}}$	$\frac{1}{231} i \left(22 \sqrt{7} B(3,2)-35 B(5,2)\right)$	$\frac{5}{33} \sqrt{2} A(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)$	$-\frac{1}{11} i \sqrt{10} B(5,4)$	$\frac{5}{11} \sqrt{\frac{2}{3}} A(5,5)$
${Y_{-1}^{(2)}}$	$\frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,2)-7 B(5,2)\right)$	$-\sqrt{\frac{2}{7}} A(1,1)-\frac{A(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} A(5,1)$	$0$	$\sqrt{\frac{3}{35}} A(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} A(3,1)-\frac{20 A(5,1)}{33 \sqrt{7}}$	$\frac{i \left(11 \sqrt{7} B(3,2)+35 B(5,2)\right)}{77 \sqrt{3}}$	$-\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+28 A(5,3)\right)$	$\frac{2}{11} i \sqrt{\frac{5}{3}} B(5,4)$
${Y_{0}^{(2)}}$	$\frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)-\frac{2}{33} \sqrt{5} A(5,3)$	$\frac{1}{11} i \sqrt{5} B(5,2)$	$-\sqrt{\frac{6}{35}} A(1,1)-\frac{A(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} A(5,1)$	$0$	$\sqrt{\frac{6}{35}} A(1,1)+\frac{A(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} A(5,1)$	$-\frac{1}{11} i \sqrt{5} B(5,2)$	$\frac{2}{33} \sqrt{5} A(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)$
${Y_{1}^{(2)}}$	$-\frac{2}{11} i \sqrt{\frac{5}{3}} B(5,4)$	$\frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)+\frac{4}{33} \sqrt{5} A(5,3)$	$-\frac{i \left(11 \sqrt{7} B(3,2)+35 B(5,2)\right)}{77 \sqrt{3}}$	$-\sqrt{\frac{3}{35}} A(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} A(3,1)+\frac{20 A(5,1)}{33 \sqrt{7}}$	$0$	$\sqrt{\frac{2}{7}} A(1,1)+\frac{A(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} A(5,1)$	$-\frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,2)-7 B(5,2)\right)$
${Y_{2}^{(2)}}$	$-\frac{5}{11} \sqrt{\frac{2}{3}} A(5,5)$	$\frac{1}{11} i \sqrt{10} B(5,4)$	$\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)-\frac{5}{33} \sqrt{2} A(5,3)$	$\frac{5}{33} i B(5,2)-\frac{2 i B(3,2)}{3 \sqrt{7}}$	$-\frac{A(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} A(3,1)-\frac{5 A(5,1)}{11 \sqrt{21}}$	$0$	$\frac{1}{231} \left(33 \sqrt{21} A(1,1)-11 \sqrt{14} A(3,1)+\sqrt{35} A(5,1)\right)$

The Hamiltonian on a basis of symmetric functions

	$f_{z\left(3y^2-z^2\right)}$	$f_{\text{xyz}}$	$f_{z\left(5x^2-r^2\right)}$	$f_{x\left(5x^2-3r^2\right)}$	$f_{y\left(5x^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(y^2-3z^2\right)}$
$d_{\text{yz}}$	$0$	$-\sqrt{\frac{2}{7}} A(1,1)-\frac{A(3,1)}{\sqrt{21}}+\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)$	$0$	$\frac{2}{3} \sqrt{\frac{2}{7}} B(3,2)+\frac{5 \left(B(5,2)-\sqrt{3} B(5,4)\right)}{33 \sqrt{2}}$	$0$	$\frac{1}{11} \sqrt{\frac{5}{6}} \left(3 B(5,2)+\sqrt{3} B(5,4)\right)$	$0$
$d_{\text{xz}}$	$\frac{1}{924} \left(55 \sqrt{14} A(3,1)+11 \sqrt{210} A(3,3)+28 \left(\sqrt{35} A(5,1)+\sqrt{30} A(5,3)\right)\right)$	$0$	$-\frac{4 A(1,1)}{\sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{70}} A(3,1)+\frac{5 \left(-11 \sqrt{7} A(3,3)+6 \sqrt{42} A(5,1)-28 A(5,3)\right)}{462 \sqrt{2}}$	$0$	$\frac{5}{66} \left(\sqrt{3} B(5,2)-3 B(5,4)\right)-\frac{B(3,2)}{\sqrt{21}}$	$0$	$\frac{1}{462} \sqrt{5} \left(22 \sqrt{7} B(3,2)+49 B(5,2)+7 \sqrt{3} B(5,4)\right)$
$d_{3x^2-r^2}$	$0$	$\frac{1}{11} \sqrt{\frac{5}{2}} \left(B(5,2)-\sqrt{3} B(5,4)\right)$	$0$	$-3 \sqrt{\frac{2}{35}} A(1,1)+\frac{2 A(3,1)}{\sqrt{105}}+\frac{1}{924} \left(-88 \sqrt{7} A(3,3)-25 \sqrt{42} A(5,1)+175 A(5,3)-105 \sqrt{5} A(5,5)\right)$	$0$	$\frac{1}{44} \left(-\sqrt{70} A(5,1)+\sqrt{15} A(5,3)+5 \sqrt{3} A(5,5)\right)$	$0$
$d_{\text{xz}}$	$\frac{1}{462} \sqrt{5} \left(-22 \sqrt{7} B(3,2)+35 B(5,2)+21 \sqrt{3} B(5,4)\right)$	$0$	$\frac{5}{66} \left(\sqrt{3} B(5,2)-3 B(5,4)\right)-\frac{B(3,2)}{\sqrt{21}}$	$0$	$\frac{-264 \sqrt{70} A(1,1)+77 \sqrt{105} A(3,1)+55 \sqrt{7} A(3,3)-25 \sqrt{42} A(5,1)-175 A(5,3)+525 \sqrt{5} A(5,5)}{2310 \sqrt{2}}$	$0$	$\frac{55 \sqrt{7} A(3,1)+11 \sqrt{105} A(3,3)-7 \left(\sqrt{70} A(5,1)+5 \sqrt{3} \left(\sqrt{5} A(5,3)+A(5,5)\right)\right)}{462 \sqrt{2}}$
$d_{y^2-z^2}$	$0$	$\frac{1}{11} \sqrt{\frac{5}{2}} \left(\sqrt{3} B(5,2)+B(5,4)\right)$	$0$	$-\frac{1}{3} \sqrt{\frac{5}{7}} A(3,1)-\frac{A(3,3)}{\sqrt{21}}+\frac{5}{132} \left(\sqrt{3} \left(A(5,3)+\sqrt{5} A(5,5)\right)-\sqrt{14} A(5,1)\right)$	$0$	$\frac{1}{924} \left(-132 \sqrt{14} A(1,1)-44 \sqrt{21} A(3,1)+44 \sqrt{35} A(3,3)-13 \sqrt{210} A(5,1)-77 \sqrt{5} A(5,3)-105 A(5,5)\right)$	$0$

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups	C₁	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}

You are here: Quanty - a quantum many body script language » Physics, Chemistry and Mathematics » Point groups » Point Group C2 » Orientation X

+Table of Contents

Orientation X

Symmetry Operations

Different Settings

Character Table

Product Table

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Expansion

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Quanty

One particle coupling on a basis of spherical harmonics

Rotation matrix to symmetry adapted functions (choice is not unique)

One particle coupling on a basis of symmetry adapted functions

Coupling for a single shell

Potential for s orbitals

Potential for p orbitals

Potential for d orbitals

Potential for f orbitals

Coupling between two shells

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

Search

Navigation

Print/export

Tools