−Table of Contents

Orientation Zxy

Orientation Zxy

Symmetry Operations

In the C2v Point Group, with orientation Zxy there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_2$	$\{0,0,1\}$ ,
$\sigma _v\text{(xz)}$	$\{0,1,0\}$ ,
$\sigma _v\text{(yz)}$	$\{1,0,0\}$ ,

Different Settings

Point Group C2v with orientation Zxy

Character Table

	$\text{E} \,{\text{(1)}}$	$C_2 \,{\text{(1)}}$	$\sigma_v\text{(xz)} \,{\text{(1)}}$	$\sigma_v\text{(yz)} \,{\text{(1)}}$
$A_1$	$1$	$1$	$1$	$1$
$A_2$	$1$	$1$	$-1$	$-1$
$B_1$	$1$	$-1$	$1$	$-1$
$B_2$	$1$	$-1$	$-1$	$1$

Product Table

	$A_1$	$A_2$	$B_1$	$B_2$
$A_1$	$A_1$	$A_2$	$B_1$	$B_2$
$A_2$	$A_2$	$A_1$	$B_2$	$B_1$
$B_1$	$B_1$	$B_2$	$A_1$	$A_2$
$B_2$	$B_2$	$B_1$	$A_2$	$A_1$

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 C2v Point group with orientation Zxy the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

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

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

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

One particle coupling on a basis of spherical harmonics

The operator representing the potential in second quantisation is given as: $O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$ For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$ . With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. $A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle$ Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$

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.

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

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

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

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

One particle coupling on a basis of symmetry adapted functions

After rotation we find

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

Coupling for a single shell

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

Click on one of the subsections to expand it or

Potential for s orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Ea1} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Ea1, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{0, 0, Ea1} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{0}^{(0)}}$
${Y_{0}^{(0)}}$	$\text{Ea1}$

The Hamiltonian on a basis of symmetric functions

	$\text{s}$
$\text{s}$	$\text{Ea1}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ea1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea1}+\text{Eb1}+\text{Eb2}) & k=0\land m=0 \\ 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ \frac{5 (\text{Eb1}-\text{Eb2})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\ \frac{5}{6} (2 \text{Ea1}-\text{Eb1}-\text{Eb2}) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1 + Eb1 + Eb2)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != 0 && m != 2)}, {(5*(Eb1 - Eb2))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}}, (5*(2*Ea1 - Eb1 - Eb2))/6]

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{0, 0, (1/3)*(Ea1 + Eb1 + Eb2)} , 
       {2, 0, (5/6)*((2)*(Ea1) + (-1)*(Eb1) + (-1)*(Eb2))} , 
       {2,-2, (5/2)*((1/(sqrt(6)))*(Eb1 + (-1)*(Eb2)))} , 
       {2, 2, (5/2)*((1/(sqrt(6)))*(Eb1 + (-1)*(Eb2)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\frac{\text{Eb1}+\text{Eb2}}{2}$	$0$	$\frac{\text{Eb2}-\text{Eb1}}{2}$
${Y_{0}^{(1)}}$	$0$	$\text{Ea1}$	$0$
${Y_{1}^{(1)}}$	$\frac{\text{Eb2}-\text{Eb1}}{2}$	$0$	$\frac{\text{Eb1}+\text{Eb2}}{2}$

The Hamiltonian on a basis of symmetric functions

	$p_x$	$p_y$	$p_z$
$p_x$	$\text{Eb1}$	$0$	$0$
$p_y$	$0$	$\text{Eb2}$	$0$
$p_z$	$0$	$0$	$\text{Ea1}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Eb1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$
	$\text{Eb2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$
	$\text{Ea1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea1x2y2}+\text{Ea1z2}+\text{Ea2}+\text{Eb1}+\text{Eb2}) & k=0\land m=0 \\ 0 & k=1\land m=0 \\ \frac{1}{4} \left(\sqrt{6} \text{Eb1}-\sqrt{6} \text{Eb2}-4 \sqrt{2} \text{Ma1}\right) & k=2\land (m=-2\lor m=2) \\ \frac{1}{2} (-2 \text{Ea1x2y2}+2 \text{Ea1z2}-2 \text{Ea2}+\text{Eb1}+\text{Eb2}) & k=2\land m=0 \\ 0 & (k=3\land (m=-2\lor m=2))\lor (k=3\land m=0) \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ea1x2y2}-\text{Ea2}) & k=4\land (m=-4\lor m=4) \\ \frac{3 \left(\text{Eb1}-\text{Eb2}+\sqrt{3} \text{Ma1}\right)}{\sqrt{10}} & k=4\land (m=-2\lor m=2) \\ \frac{3}{10} (\text{Ea1x2y2}+6 \text{Ea1z2}+\text{Ea2}-4 (\text{Eb1}+\text{Eb2})) & k=4\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1x2y2 + Ea1z2 + Ea2 + Eb1 + Eb2)/5, k == 0 && m == 0}, {0, k == 1 && m == 0}, {(Sqrt[6]*Eb1 - Sqrt[6]*Eb2 - 4*Sqrt[2]*Ma1)/4, k == 2 && (m == -2 || m == 2)}, {(-2*Ea1x2y2 + 2*Ea1z2 - 2*Ea2 + Eb1 + Eb2)/2, k == 2 && m == 0}, {0, (k == 3 && (m == -2 || m == 2)) || (k == 3 && m == 0)}, {(3*Sqrt[7/10]*(Ea1x2y2 - Ea2))/2, k == 4 && (m == -4 || m == 4)}, {(3*(Eb1 - Eb2 + Sqrt[3]*Ma1))/Sqrt[10], k == 4 && (m == -2 || m == 2)}, {(3*(Ea1x2y2 + 6*Ea1z2 + Ea2 - 4*(Eb1 + Eb2)))/10, k == 4 && m == 0}}, 0]

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{0, 0, (1/5)*(Ea1x2y2 + Ea1z2 + Ea2 + Eb1 + Eb2)} , 
       {2, 0, (1/2)*((-2)*(Ea1x2y2) + (2)*(Ea1z2) + (-2)*(Ea2) + Eb1 + Eb2)} , 
       {2,-2, (1/4)*((sqrt(6))*(Eb1) + (-1)*((sqrt(6))*(Eb2)) + (-4)*((sqrt(2))*(Ma1)))} , 
       {2, 2, (1/4)*((sqrt(6))*(Eb1) + (-1)*((sqrt(6))*(Eb2)) + (-4)*((sqrt(2))*(Ma1)))} , 
       {4, 0, (3/10)*(Ea1x2y2 + (6)*(Ea1z2) + Ea2 + (-4)*(Eb1 + Eb2))} , 
       {4,-2, (3)*((1/(sqrt(10)))*(Eb1 + (-1)*(Eb2) + (sqrt(3))*(Ma1)))} , 
       {4, 2, (3)*((1/(sqrt(10)))*(Eb1 + (-1)*(Eb2) + (sqrt(3))*(Ma1)))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Ea1x2y2 + (-1)*(Ea2)))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Ea1x2y2 + (-1)*(Ea2)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\frac{\text{Ea1x2y2}+\text{Ea2}}{2}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$	$\frac{\text{Ea1x2y2}-\text{Ea2}}{2}$
${Y_{-1}^{(2)}}$	$0$	$\frac{\text{Eb1}+\text{Eb2}}{2}$	$0$	$\frac{\text{Eb2}-\text{Eb1}}{2}$	$0$
${Y_{0}^{(2)}}$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$	$\text{Ea1z2}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$
${Y_{1}^{(2)}}$	$0$	$\frac{\text{Eb2}-\text{Eb1}}{2}$	$0$	$\frac{\text{Eb1}+\text{Eb2}}{2}$	$0$
${Y_{2}^{(2)}}$	$\frac{\text{Ea1x2y2}-\text{Ea2}}{2}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$	$\frac{\text{Ea1x2y2}+\text{Ea2}}{2}$

The Hamiltonian on a basis of symmetric functions

	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$
$d_{x^2-y^2}$	$\text{Ea1x2y2}$	$\text{Ma1}$	$0$	$0$	$0$
$d_{3z^2-r^2}$	$\text{Ma1}$	$\text{Ea1z2}$	$0$	$0$	$0$
$d_{\text{yz}}$	$0$	$0$	$\text{Eb2}$	$0$	$0$
$d_{\text{xz}}$	$0$	$0$	$0$	$\text{Eb1}$	$0$
$d_{\text{xy}}$	$0$	$0$	$0$	$0$	$\text{Ea2}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ea1x2y2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$
	$\text{Ea1z2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$
	$\text{Eb2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$
	$\text{Eb1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$
	$\text{Ea2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea1z3}+\text{Ea1zx2y2}+\text{Ea2}+\text{Eb1x3}+\text{Eb1xy2z2}+\text{Eb2y3}+\text{Eb2yz2x2}) & k=0\land m=0 \\ 0 & k=1\land m=0 \\ \frac{5}{28} \left(\sqrt{6} \text{Eb1x3}-\sqrt{6} \text{Eb2y3}+\sqrt{10} (-2 \text{Ma1}+\text{Mb1}+\text{Mb2})\right) & k=2\land (m=-2\lor m=2) \\ -\frac{5}{14} \left(-2 \text{Ea1z3}+\text{Eb1x3}+\text{Eb2y3}+\sqrt{15} (\text{Mb2}-\text{Mb1})\right) & k=2\land m=0 \\ 0 & (k=3\land (m=-2\lor m=2))\lor (k=3\land m=0) \\ \frac{3 \left(4 \sqrt{5} \text{Ea1zx2y2}-4 \sqrt{5} \text{Ea2}+3 \sqrt{5} \text{Eb1x3}-3 \sqrt{5} \text{Eb1xy2z2}+3 \sqrt{5} \text{Eb2y3}-3 \sqrt{5} \text{Eb2yz2x2}+2 \sqrt{3} (\text{Mb1}-\text{Mb2})\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\ \frac{3}{56} \left(-3 \sqrt{10} \text{Eb1x3}+7 \sqrt{10} \text{Eb1xy2z2}+3 \sqrt{10} \text{Eb2y3}-7 \sqrt{10} \text{Eb2yz2x2}+2 \sqrt{6} (-2 \text{Ma1}+\text{Mb1}+\text{Mb2})\right) & k=4\land (m=-2\lor m=2) \\ \frac{3}{56} \left(24 \text{Ea1z3}-28 \text{Ea1zx2y2}-28 \text{Ea2}+9 \text{Eb1x3}+7 \text{Eb1xy2z2}+9 \text{Eb2y3}+7 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right) & k=4\land m=0 \\ 0 & (k=5\land (m=-4\lor m=4))\lor (k=5\land (m=-2\lor m=2))\lor (k=5\land m=0) \\ \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eb1x3}+3 \sqrt{3} \text{Eb1xy2z2}-5 \sqrt{3} \text{Eb2y3}-3 \sqrt{3} \text{Eb2yz2x2}-6 \sqrt{5} (\text{Mb1}+\text{Mb2})\right) & k=6\land (m=-6\lor m=6) \\ \frac{13 \left(24 \text{Ea1zx2y2}-24 \text{Ea2}-15 \text{Eb1x3}+15 \text{Eb1xy2z2}-15 \text{Eb2y3}+15 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ \frac{13 \left(5 \sqrt{15} \text{Eb1x3}+3 \sqrt{15} \text{Eb1xy2z2}-5 \sqrt{15} \text{Eb2y3}-3 \sqrt{15} \text{Eb2yz2x2}+64 \text{Ma1}+34 (\text{Mb1}+\text{Mb2})\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\ -\frac{13}{560} \left(-80 \text{Ea1z3}-24 \text{Ea1zx2y2}-24 \text{Ea2}+25 \text{Eb1x3}+39 \text{Eb1xy2z2}+25 \text{Eb2y3}+39 \text{Eb2yz2x2}+14 \sqrt{15} (\text{Mb1}-\text{Mb2})\right) & k=6\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1z3 + Ea1zx2y2 + Ea2 + Eb1x3 + Eb1xy2z2 + Eb2y3 + Eb2yz2x2)/7, k == 0 && m == 0}, {0, k == 1 && m == 0}, {(5*(Sqrt[6]*Eb1x3 - Sqrt[6]*Eb2y3 + Sqrt[10]*(-2*Ma1 + Mb1 + Mb2)))/28, k == 2 && (m == -2 || m == 2)}, {(-5*(-2*Ea1z3 + Eb1x3 + Eb2y3 + Sqrt[15]*(-Mb1 + Mb2)))/14, k == 2 && m == 0}, {0, (k == 3 && (m == -2 || m == 2)) || (k == 3 && m == 0)}, {(3*(4*Sqrt[5]*Ea1zx2y2 - 4*Sqrt[5]*Ea2 + 3*Sqrt[5]*Eb1x3 - 3*Sqrt[5]*Eb1xy2z2 + 3*Sqrt[5]*Eb2y3 - 3*Sqrt[5]*Eb2yz2x2 + 2*Sqrt[3]*(Mb1 - Mb2)))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(3*(-3*Sqrt[10]*Eb1x3 + 7*Sqrt[10]*Eb1xy2z2 + 3*Sqrt[10]*Eb2y3 - 7*Sqrt[10]*Eb2yz2x2 + 2*Sqrt[6]*(-2*Ma1 + Mb1 + Mb2)))/56, k == 4 && (m == -2 || m == 2)}, {(3*(24*Ea1z3 - 28*Ea1zx2y2 - 28*Ea2 + 9*Eb1x3 + 7*Eb1xy2z2 + 9*Eb2y3 + 7*Eb2yz2x2 + 2*Sqrt[15]*(-Mb1 + Mb2)))/56, k == 4 && m == 0}, {0, (k == 5 && (m == -4 || m == 4)) || (k == 5 && (m == -2 || m == 2)) || (k == 5 && m == 0)}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eb1x3 + 3*Sqrt[3]*Eb1xy2z2 - 5*Sqrt[3]*Eb2y3 - 3*Sqrt[3]*Eb2yz2x2 - 6*Sqrt[5]*(Mb1 + Mb2)))/160, k == 6 && (m == -6 || m == 6)}, {(13*(24*Ea1zx2y2 - 24*Ea2 - 15*Eb1x3 + 15*Eb1xy2z2 - 15*Eb2y3 + 15*Eb2yz2x2 + 2*Sqrt[15]*(-Mb1 + Mb2)))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(13*(5*Sqrt[15]*Eb1x3 + 3*Sqrt[15]*Eb1xy2z2 - 5*Sqrt[15]*Eb2y3 - 3*Sqrt[15]*Eb2yz2x2 + 64*Ma1 + 34*(Mb1 + Mb2)))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {(-13*(-80*Ea1z3 - 24*Ea1zx2y2 - 24*Ea2 + 25*Eb1x3 + 39*Eb1xy2z2 + 25*Eb2y3 + 39*Eb2yz2x2 + 14*Sqrt[15]*(Mb1 - Mb2)))/560, k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{0, 0, (1/7)*(Ea1z3 + Ea1zx2y2 + Ea2 + Eb1x3 + Eb1xy2z2 + Eb2y3 + Eb2yz2x2)} , 
       {2, 0, (-5/14)*((-2)*(Ea1z3) + Eb1x3 + Eb2y3 + (sqrt(15))*((-1)*(Mb1) + Mb2))} , 
       {2,-2, (5/28)*((sqrt(6))*(Eb1x3) + (-1)*((sqrt(6))*(Eb2y3)) + (sqrt(10))*((-2)*(Ma1) + Mb1 + Mb2))} , 
       {2, 2, (5/28)*((sqrt(6))*(Eb1x3) + (-1)*((sqrt(6))*(Eb2y3)) + (sqrt(10))*((-2)*(Ma1) + Mb1 + Mb2))} , 
       {4, 0, (3/56)*((24)*(Ea1z3) + (-28)*(Ea1zx2y2) + (-28)*(Ea2) + (9)*(Eb1x3) + (7)*(Eb1xy2z2) + (9)*(Eb2y3) + (7)*(Eb2yz2x2) + (2)*((sqrt(15))*((-1)*(Mb1) + Mb2)))} , 
       {4,-2, (3/56)*((-3)*((sqrt(10))*(Eb1x3)) + (7)*((sqrt(10))*(Eb1xy2z2)) + (3)*((sqrt(10))*(Eb2y3)) + (-7)*((sqrt(10))*(Eb2yz2x2)) + (2)*((sqrt(6))*((-2)*(Ma1) + Mb1 + Mb2)))} , 
       {4, 2, (3/56)*((-3)*((sqrt(10))*(Eb1x3)) + (7)*((sqrt(10))*(Eb1xy2z2)) + (3)*((sqrt(10))*(Eb2y3)) + (-7)*((sqrt(10))*(Eb2yz2x2)) + (2)*((sqrt(6))*((-2)*(Ma1) + Mb1 + Mb2)))} , 
       {4,-4, (3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Ea1zx2y2)) + (-4)*((sqrt(5))*(Ea2)) + (3)*((sqrt(5))*(Eb1x3)) + (-3)*((sqrt(5))*(Eb1xy2z2)) + (3)*((sqrt(5))*(Eb2y3)) + (-3)*((sqrt(5))*(Eb2yz2x2)) + (2)*((sqrt(3))*(Mb1 + (-1)*(Mb2)))))} , 
       {4, 4, (3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Ea1zx2y2)) + (-4)*((sqrt(5))*(Ea2)) + (3)*((sqrt(5))*(Eb1x3)) + (-3)*((sqrt(5))*(Eb1xy2z2)) + (3)*((sqrt(5))*(Eb2y3)) + (-3)*((sqrt(5))*(Eb2yz2x2)) + (2)*((sqrt(3))*(Mb1 + (-1)*(Mb2)))))} , 
       {6, 0, (-13/560)*((-80)*(Ea1z3) + (-24)*(Ea1zx2y2) + (-24)*(Ea2) + (25)*(Eb1x3) + (39)*(Eb1xy2z2) + (25)*(Eb2y3) + (39)*(Eb2yz2x2) + (14)*((sqrt(15))*(Mb1 + (-1)*(Mb2))))} , 
       {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eb1x3)) + (3)*((sqrt(15))*(Eb1xy2z2)) + (-5)*((sqrt(15))*(Eb2y3)) + (-3)*((sqrt(15))*(Eb2yz2x2)) + (64)*(Ma1) + (34)*(Mb1 + Mb2)))} , 
       {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eb1x3)) + (3)*((sqrt(15))*(Eb1xy2z2)) + (-5)*((sqrt(15))*(Eb2y3)) + (-3)*((sqrt(15))*(Eb2yz2x2)) + (64)*(Ma1) + (34)*(Mb1 + Mb2)))} , 
       {6,-4, (13/80)*((1/(sqrt(14)))*((24)*(Ea1zx2y2) + (-24)*(Ea2) + (-15)*(Eb1x3) + (15)*(Eb1xy2z2) + (-15)*(Eb2y3) + (15)*(Eb2yz2x2) + (2)*((sqrt(15))*((-1)*(Mb1) + Mb2))))} , 
       {6, 4, (13/80)*((1/(sqrt(14)))*((24)*(Ea1zx2y2) + (-24)*(Ea2) + (-15)*(Eb1x3) + (15)*(Eb1xy2z2) + (-15)*(Eb2y3) + (15)*(Eb2yz2x2) + (2)*((sqrt(15))*((-1)*(Mb1) + Mb2))))} , 
       {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eb1x3)) + (3)*((sqrt(3))*(Eb1xy2z2)) + (-5)*((sqrt(3))*(Eb2y3)) + (-3)*((sqrt(3))*(Eb2yz2x2)) + (-6)*((sqrt(5))*(Mb1 + Mb2))))} , 
       {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eb1x3)) + (3)*((sqrt(3))*(Eb1xy2z2)) + (-5)*((sqrt(3))*(Eb2y3)) + (-3)*((sqrt(3))*(Eb2yz2x2)) + (-6)*((sqrt(5))*(Mb1 + Mb2))))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\frac{1}{16} \left(5 \text{Eb1x3}+3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right)$	$0$	$\frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right)$	$0$	$\frac{1}{16} \left(-5 \text{Eb1x3}-3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right)$
${Y_{-2}^{(3)}}$	$0$	$\frac{\text{Ea1zx2y2}+\text{Ea2}}{2}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$	$\frac{\text{Ea1zx2y2}-\text{Ea2}}{2}$	$0$
${Y_{-1}^{(3)}}$	$\frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(3 \text{Eb1x3}+5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}-\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(-3 \text{Eb1x3}-5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}-2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right)$
${Y_{0}^{(3)}}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$	$\text{Ea1z3}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$
${Y_{1}^{(3)}}$	$\frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right)$	$0$	$\frac{1}{16} \left(-3 \text{Eb1x3}-5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}-2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(3 \text{Eb1x3}+5 \text{Eb1xy2z2}+3 \text{Eb2y3}+5 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}-\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right)$
${Y_{2}^{(3)}}$	$0$	$\frac{\text{Ea1zx2y2}-\text{Ea2}}{2}$	$0$	$\frac{\text{Ma1}}{\sqrt{2}}$	$0$	$\frac{\text{Ea1zx2y2}+\text{Ea2}}{2}$	$0$
${Y_{3}^{(3)}}$	$\frac{1}{16} \left(-5 \text{Eb1x3}-3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb1}+\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(\sqrt{15} \text{Eb1x3}-\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}+2 \text{Mb1}-2 \text{Mb2}\right)$	$0$	$\frac{1}{16} \left(-\sqrt{15} \text{Eb1x3}+\sqrt{15} \text{Eb1xy2z2}+\sqrt{15} \text{Eb2y3}-\sqrt{15} \text{Eb2yz2x2}-2 (\text{Mb1}+\text{Mb2})\right)$	$0$	$\frac{1}{16} \left(5 \text{Eb1x3}+3 \text{Eb1xy2z2}+5 \text{Eb2y3}+3 \text{Eb2yz2x2}+2 \sqrt{15} (\text{Mb2}-\text{Mb1})\right)$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$f_{\text{xyz}}$	$\text{Ea2}$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{x\left(5x^2-r^2\right)}$	$0$	$\text{Eb1x3}$	$0$	$0$	$\text{Mb1}$	$0$	$0$
$f_{y\left(5y^2-r^2\right)}$	$0$	$0$	$\text{Eb2y3}$	$0$	$0$	$\text{Mb2}$	$0$
$f_{z\left(5z^2-r^2\right)}$	$0$	$0$	$0$	$\text{Ea1z3}$	$0$	$0$	$\text{Ma1}$
$f_{x\left(y^2-z^2\right)}$	$0$	$\text{Mb1}$	$0$	$0$	$\text{Eb1xy2z2}$	$0$	$0$
$f_{y\left(z^2-x^2\right)}$	$0$	$0$	$\text{Mb2}$	$0$	$0$	$\text{Eb2yz2x2}$	$0$
$f_{z\left(x^2-y^2\right)}$	$0$	$0$	$0$	$\text{Ma1}$	$0$	$0$	$\text{Ea1zx2y2}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ea2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$
	$\text{Eb1x3}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$
	$\text{Eb2y3}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$
	$\text{Ea1z3}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$
	$\text{Eb1xy2z2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$
	$\text{Eb2yz2x2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$
	$\text{Ea1zx2y2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-p orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 1 || m != 0}}, A[1, 0]]

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{1, 0, A(1,0)} }

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

	$p_x$	$p_y$	$p_z$
$\text{s}$	$0$	$0$	$\frac{A(1,0)}{\sqrt{3}}$

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

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

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{2, 0, A(2,0)} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} }

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

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

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{3, 0, A(3,0)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} }

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

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

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{1, 0, A(1,0)} , 
       {3, 0, A(3,0)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-1}^{(1)}}$	$0$	$\frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}}$	$0$	$-\frac{1}{7} \sqrt{6} A(3,2)$	$0$
${Y_{0}^{(1)}}$	$\frac{1}{7} \sqrt{3} A(3,2)$	$0$	$\frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}}$	$0$	$\frac{1}{7} \sqrt{3} A(3,2)$
${Y_{1}^{(1)}}$	$0$	$-\frac{1}{7} \sqrt{6} A(3,2)$	$0$	$\frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}}$	$0$

The Hamiltonian on a basis of symmetric functions

	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$
$p_x$	$0$	$0$	$0$	$\frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right)$	$0$
$p_y$	$0$	$0$	$\frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right)$	$0$	$0$
$p_z$	$\frac{1}{7} \sqrt{6} A(3,2)$	$\frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}}$	$0$	$0$	$0$

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

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

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{2, 0, A(2,0)} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} , 
       {4, 0, A(4,0)} , 
       {4,-2, A(4,2)} , 
       {4, 2, A(4,2)} , 
       {4,-4, A(4,4)} , 
       {4, 4, A(4,4)} }

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$p_x$	$0$	$\frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right)$	$0$	$0$	$\frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right)$	$0$	$0$
$p_y$	$0$	$0$	$\frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right)$	$0$	$0$	$\frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right)$	$0$
$p_z$	$0$	$0$	$0$	$\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$	$0$	$0$	$\sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2)$

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2v_Zxy.Quanty.nb

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

Input format suitable for Quanty

Akm_C2v_Zxy.Quanty

Akm = {{1, 0, A(1,0)} , 
       {3, 0, A(3,0)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} , 
       {5, 0, A(5,0)} , 
       {5,-2, A(5,2)} , 
       {5, 2, A(5,2)} , 
       {5,-4, A(5,4)} , 
       {5, 4, A(5,4)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-2}^{(2)}}$	$0$	$\frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}}$	$0$	$\frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}}$	$0$	$\frac{1}{11} \sqrt{10} A(5,4)$	$0$
${Y_{-1}^{(2)}}$	$\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right)$	$0$	$\frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0))$	$0$	$-\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}}$	$0$	$-\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4)$
${Y_{0}^{(2)}}$	$0$	$\frac{1}{11} \sqrt{5} A(5,2)$	$0$	$\frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}}$	$0$	$\frac{1}{11} \sqrt{5} A(5,2)$	$0$
${Y_{1}^{(2)}}$	$-\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4)$	$0$	$-\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}}$	$0$	$\frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0))$	$0$	$\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right)$
${Y_{2}^{(2)}}$	$0$	$\frac{1}{11} \sqrt{10} A(5,4)$	$0$	$\frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}}$	$0$	$\frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}}$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$d_{x^2-y^2}$	$0$	$0$	$0$	$\sqrt{2} \left(\frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}}\right)$	$0$	$0$	$\frac{\sqrt{14} (33 A(1,0)-22 A(3,0)+5 A(5,0))+42 \sqrt{5} A(5,4)}{231 \sqrt{2}}$
$d_{3z^2-r^2}$	$0$	$0$	$0$	$\frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}}$	$0$	$0$	$\frac{1}{11} \sqrt{10} A(5,2)$
$d_{\text{yz}}$	$0$	$0$	$\frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310}$	$0$	$0$	$\frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right)$	$0$
$d_{\text{xz}}$	$0$	$\frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310}$	$0$	$0$	$\frac{1}{462} \left(-66 \sqrt{7} A(1,0)-11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)+25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)-21 \sqrt{10} A(5,4)\right)$	$0$	$0$
$d_{\text{xy}}$	$\frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right)$	$0$	$0$	$0$	$0$	$0$	$0$

Table of several point groups

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}