−Table of Contents

Orientation Z

Orientation Z

Symmetry Operations

In the C4 Point Group, with orientation Z there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_4$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{0,0,1\}$ ,

Different Settings

Character Table

	$\text{E} \,{\text{(1)}}$	$C_4 \,{\text{(2)}}$	$C_2 \,{\text{(1)}}$
$\text{A}$	$1$	$1$	$1$
$\text{B}$	$1$	$-1$	$1$
$\text{E}$	$2$	$0$	$-2$

Product Table

	$\text{A}$	$\text{B}$	$\text{E}$
$\text{A}$	$\text{A}$	$\text{B}$	$\text{E}$
$\text{B}$	$\text{B}$	$\text{A}$	$\text{E}$
$\text{E}$	$\text{E}$	$\text{E}$	$2 \text{A}+2 \text{B}$

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 C4 Point group with orientation Z the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.Quanty

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

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.

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

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

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

One particle coupling on a basis of symmetry adapted functions

After rotation we find

	$\text{s}$	$p_y$	$p_z$	$p_x$	$d_{\text{xy}}$	$d_{\text{yz}}$	$d_{3z^2-r^2}$	$d_{\text{xz}}$	$d_{x^2-y^2}$	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_y$	$\color{darkred}{ 0 }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$0$	$-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$
$p_z$	$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$
$p_x$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$	$\color{darkred}{ 0 }$	$-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$	$0$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$
$d_{\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$0$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$
$d_{\text{yz}}$	$0$	$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$
$d_{3z^2-r^2}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$
$d_{x^2-y^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4)$	$0$	$0$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$	$\color{darkred}{ 0 }$
$f_{y\left(3x^2-y^2\right)}$	$\color{darkred}{ 0 }$	$-\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$	$0$	$-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$	$0$	$0$
$f_{\text{xyz}}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$	$0$	$0$	$0$	$-\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)$	$0$
$f_{y\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$
$f_{z\left(5z^2-3r^2\right)}$	$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$
$f_{x\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$	$\color{darkred}{ 0 }$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)-\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)$
$f_{z\left(x^2-y^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$	$0$	$-\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)$	$0$
$f_{x\left(x^2-3y^2\right)}$	$\color{darkred}{ 0 }$	$-\frac{2 \text{Bpf}(4,4)}{3 \sqrt{3}}$	$0$	$\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,4) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)$	$0$	$\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4)-\frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)$	$0$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

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_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.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

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea + 2*Ee)/3, k == 0 && m == 0}, {0, k != 2 || m != 0}}, (5*(Ea - Ee))/3]

Input format suitable for Quanty

Akm_C4_Z.Quanty

Akm = {{0, 0, (1/3)*(Ea + (2)*(Ee))} , 
       {2, 0, (5/3)*(Ea + (-1)*(Ee))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\text{Ee}$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$\text{Ea}$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$\text{Ee}$

The Hamiltonian on a basis of symmetric functions

	$p_y$	$p_z$	$p_x$
$p_y$	$\text{Ee}$	$0$	$0$
$p_z$	$0$	$\text{Ea}$	$0$
$p_x$	$0$	$0$	$\text{Ee}$

Rotation matrix used

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

Irriducible representations and their onsite energy

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

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea}+\text{Ebx2y2}+\text{Ebxy}+2 \text{Ee}) & k=0\land m=0 \\ 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \text{Ea}-\text{Ebx2y2}-\text{Ebxy}+\text{Ee} & k=2\land m=0 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ebx2y2}-\text{Ebxy}+2 i \text{Mb}) & k=4\land m=-4 \\ \frac{3}{10} (6 \text{Ea}+\text{Ebx2y2}+\text{Ebxy}-8 \text{Ee}) & k=4\land m=0 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ebx2y2}-\text{Ebxy}-2 i \text{Mb}) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea + Ebx2y2 + Ebxy + 2*Ee)/5, k == 0 && m == 0}, {0, (k != 4 && (k != 2 || m != 0)) || (m != -4 && m != 0 && m != 4)}, {Ea - Ebx2y2 - Ebxy + Ee, k == 2 && m == 0}, {(3*Sqrt[7/10]*(Ebx2y2 - Ebxy + (2*I)*Mb))/2, k == 4 && m == -4}, {(3*(6*Ea + Ebx2y2 + Ebxy - 8*Ee))/10, k == 4 && m == 0}}, (3*Sqrt[7/10]*(Ebx2y2 - Ebxy - (2*I)*Mb))/2]

Input format suitable for Quanty

Akm_C4_Z.Quanty

Akm = {{0, 0, (1/5)*(Ea + Ebx2y2 + Ebxy + (2)*(Ee))} , 
       {2, 0, Ea + (-1)*(Ebx2y2) + (-1)*(Ebxy) + Ee} , 
       {4, 0, (3/10)*((6)*(Ea) + Ebx2y2 + Ebxy + (-8)*(Ee))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Ebx2y2 + (-1)*(Ebxy) + (-2*I)*(Mb)))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Ebx2y2 + (-1)*(Ebxy) + (2*I)*(Mb)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\frac{\text{Ebx2y2}+\text{Ebxy}}{2}$	$0$	$0$	$0$	$\frac{1}{2} (\text{Ebx2y2}-\text{Ebxy}+2 i \text{Mb})$
${Y_{-1}^{(2)}}$	$0$	$\text{Ee}$	$0$	$0$	$0$
${Y_{0}^{(2)}}$	$0$	$0$	$\text{Ea}$	$0$	$0$
${Y_{1}^{(2)}}$	$0$	$0$	$0$	$\text{Ee}$	$0$
${Y_{2}^{(2)}}$	$\frac{1}{2} (\text{Ebx2y2}-\text{Ebxy}-2 i \text{Mb})$	$0$	$0$	$0$	$\frac{\text{Ebx2y2}+\text{Ebxy}}{2}$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{xy}}$	$d_{\text{yz}}$	$d_{3z^2-r^2}$	$d_{\text{xz}}$	$d_{x^2-y^2}$
$d_{\text{xy}}$	$\text{Ebxy}$	$0$	$0$	$0$	$\text{Mb}$
$d_{\text{yz}}$	$0$	$\text{Ee}$	$0$	$0$	$0$
$d_{3z^2-r^2}$	$0$	$0$	$\text{Ea}$	$0$	$0$
$d_{\text{xz}}$	$0$	$0$	$0$	$\text{Ee}$	$0$
$d_{x^2-y^2}$	$\text{Mb}$	$0$	$0$	$0$	$\text{Ebx2y2}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ebxy}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$
	$\text{Ee}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$
	$\text{Ea}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$
	$\text{Ee}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$
	$\text{Ebx2y2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea}+\text{Ebxyz}+\text{Ebzx2y2}+2 \text{Ee1}+2 \text{Ee3}) & k=0\land m=0 \\ 0 & (k\neq 4\land k\neq 6\land (k\neq 2\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \frac{5}{14} (2 \text{Ea}+3 \text{Ee1}-5 \text{Ee3}) & k=2\land m=0 \\ \frac{3}{28} \left(-\sqrt{70} \text{Ebxyz}+\sqrt{70} \text{Ebzx2y2}+2 i \sqrt{70} \text{Mb}-4 i \sqrt{42} \text{MeIm}+4 \sqrt{42} \text{MeRe}\right) & k=4\land m=-4 \\ \frac{3}{14} (6 \text{Ea}-7 \text{Ebxyz}-7 \text{Ebzx2y2}+2 \text{Ee1}+6 \text{Ee3}) & k=4\land m=0 \\ \frac{3}{28} \left(-\sqrt{70} \text{Ebxyz}+\sqrt{70} \text{Ebzx2y2}-2 i \sqrt{70} \text{Mb}+4 i \sqrt{42} \text{MeIm}+4 \sqrt{42} \text{MeRe}\right) & k=4\land m=4 \\ -\frac{13 \left(3 \text{Ebxyz}-3 \text{Ebzx2y2}-6 i \text{Mb}-2 i \sqrt{15} \text{MeIm}+2 \sqrt{15} \text{MeRe}\right)}{10 \sqrt{14}} & k=6\land m=-4 \\ \frac{13}{70} (10 \text{Ea}+3 \text{Ebxyz}+3 \text{Ebzx2y2}-15 \text{Ee1}-\text{Ee3}) & k=6\land m=0 \\ -\frac{13 \left(3 \text{Ebxyz}-3 \text{Ebzx2y2}+6 i \text{Mb}+2 i \sqrt{15} \text{MeIm}+2 \sqrt{15} \text{MeRe}\right)}{10 \sqrt{14}} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea + Ebxyz + Ebzx2y2 + 2*Ee1 + 2*Ee3)/7, k == 0 && m == 0}, {0, (k != 4 && k != 6 && (k != 2 || m != 0)) || (m != -4 && m != 0 && m != 4)}, {(5*(2*Ea + 3*Ee1 - 5*Ee3))/14, k == 2 && m == 0}, {(3*(-(Sqrt[70]*Ebxyz) + Sqrt[70]*Ebzx2y2 + (2*I)*Sqrt[70]*Mb - (4*I)*Sqrt[42]*MeIm + 4*Sqrt[42]*MeRe))/28, k == 4 && m == -4}, {(3*(6*Ea - 7*Ebxyz - 7*Ebzx2y2 + 2*Ee1 + 6*Ee3))/14, k == 4 && m == 0}, {(3*(-(Sqrt[70]*Ebxyz) + Sqrt[70]*Ebzx2y2 - (2*I)*Sqrt[70]*Mb + (4*I)*Sqrt[42]*MeIm + 4*Sqrt[42]*MeRe))/28, k == 4 && m == 4}, {(-13*(3*Ebxyz - 3*Ebzx2y2 - (6*I)*Mb - (2*I)*Sqrt[15]*MeIm + 2*Sqrt[15]*MeRe))/(10*Sqrt[14]), k == 6 && m == -4}, {(13*(10*Ea + 3*Ebxyz + 3*Ebzx2y2 - 15*Ee1 - Ee3))/70, k == 6 && m == 0}}, (-13*(3*Ebxyz - 3*Ebzx2y2 + (6*I)*Mb + (2*I)*Sqrt[15]*MeIm + 2*Sqrt[15]*MeRe))/(10*Sqrt[14])]

Input format suitable for Quanty

Akm_C4_Z.Quanty

Akm = {{0, 0, (1/7)*(Ea + Ebxyz + Ebzx2y2 + (2)*(Ee1) + (2)*(Ee3))} , 
       {2, 0, (5/14)*((2)*(Ea) + (3)*(Ee1) + (-5)*(Ee3))} , 
       {4, 0, (3/14)*((6)*(Ea) + (-7)*(Ebxyz) + (-7)*(Ebzx2y2) + (2)*(Ee1) + (6)*(Ee3))} , 
       {4, 4, (3/28)*((-1)*((sqrt(70))*(Ebxyz)) + (sqrt(70))*(Ebzx2y2) + (-2*I)*((sqrt(70))*(Mb)) + (4*I)*((sqrt(42))*(MeIm)) + (4)*((sqrt(42))*(MeRe)))} , 
       {4,-4, (3/28)*((-1)*((sqrt(70))*(Ebxyz)) + (sqrt(70))*(Ebzx2y2) + (2*I)*((sqrt(70))*(Mb)) + (-4*I)*((sqrt(42))*(MeIm)) + (4)*((sqrt(42))*(MeRe)))} , 
       {6, 0, (13/70)*((10)*(Ea) + (3)*(Ebxyz) + (3)*(Ebzx2y2) + (-15)*(Ee1) + (-1)*(Ee3))} , 
       {6,-4, (-13/10)*((1/(sqrt(14)))*((3)*(Ebxyz) + (-3)*(Ebzx2y2) + (-6*I)*(Mb) + (-2*I)*((sqrt(15))*(MeIm)) + (2)*((sqrt(15))*(MeRe))))} , 
       {6, 4, (-13/10)*((1/(sqrt(14)))*((3)*(Ebxyz) + (-3)*(Ebzx2y2) + (6*I)*(Mb) + (2*I)*((sqrt(15))*(MeIm)) + (2)*((sqrt(15))*(MeRe))))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\text{Ee3}$	$0$	$0$	$0$	$\text{MeRe}-i \text{MeIm}$	$0$	$0$
${Y_{-2}^{(3)}}$	$0$	$\frac{\text{Ebxyz}+\text{Ebzx2y2}}{2}$	$0$	$0$	$0$	$\frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}+2 i \text{Mb})$	$0$
${Y_{-1}^{(3)}}$	$0$	$0$	$\text{Ee1}$	$0$	$0$	$0$	$\text{MeRe}-i \text{MeIm}$
${Y_{0}^{(3)}}$	$0$	$0$	$0$	$\text{Ea}$	$0$	$0$	$0$
${Y_{1}^{(3)}}$	$\text{MeRe}+i \text{MeIm}$	$0$	$0$	$0$	$\text{Ee1}$	$0$	$0$
${Y_{2}^{(3)}}$	$0$	$\frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}-2 i \text{Mb})$	$0$	$0$	$0$	$\frac{\text{Ebxyz}+\text{Ebzx2y2}}{2}$	$0$
${Y_{3}^{(3)}}$	$0$	$0$	$\text{MeRe}+i \text{MeIm}$	$0$	$0$	$0$	$\text{Ee3}$

The Hamiltonian on a basis of symmetric functions

	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$f_{y\left(3x^2-y^2\right)}$	$\text{Ee3}$	$0$	$\text{MeRe}$	$0$	$\text{MeIm}$	$0$	$0$
$f_{\text{xyz}}$	$0$	$\text{Ebxyz}$	$0$	$0$	$0$	$\text{Mb}$	$0$
$f_{y\left(5z^2-r^2\right)}$	$\text{MeRe}$	$0$	$\text{Ee1}$	$0$	$0$	$0$	$\text{MeIm}$
$f_{z\left(5z^2-3r^2\right)}$	$0$	$0$	$0$	$\text{Ea}$	$0$	$0$	$0$
$f_{x\left(5z^2-r^2\right)}$	$\text{MeIm}$	$0$	$0$	$0$	$\text{Ee1}$	$0$	$-\text{MeRe}$
$f_{z\left(x^2-y^2\right)}$	$0$	$\text{Mb}$	$0$	$0$	$0$	$\text{Ebzx2y2}$	$0$
$f_{x\left(x^2-3y^2\right)}$	$0$	$0$	$\text{MeIm}$	$0$	$-\text{MeRe}$	$0$	$\text{Ee3}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ee3}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$
	$\text{Ebxyz}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$
	$\text{Ee1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$
	$\text{Ea}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$
	$\text{Ee1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$
	$\text{Ebzx2y2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$
	$\text{Ee3}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$

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_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.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

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

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.Quanty

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

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || m != 0}}, A[3, 0]]

Input format suitable for Quanty

Akm_C4_Z.Quanty

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

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.Quanty

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

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.Quanty

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

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C4_Z.Quanty.nb

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

Input format suitable for Quanty

Akm_C4_Z.Quanty

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

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

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

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}