Orientation 111z

Symmetry Operations

In the Oh Point Group, with orientation 111z there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{2+\sqrt{3},-1,\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ , $\{2,2,-1\}$ , $\left\{1,-2-\sqrt{3},\frac{1}{1-\sqrt{3}}\right\}$ , $\left\{-2-\sqrt{3},1,\frac{1}{1-\sqrt{3}}\right\}$ , $\{-2,-2,1\}$ , $\left\{-1,2+\sqrt{3},\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ ,
$C_2$	$\{1,-1,0\}$ , $\left\{2+\sqrt{3},1,0\right\}$ , $\left\{1,2+\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{-2-\sqrt{3},1,-2 \left(1+\sqrt{3}\right)\right\}$ , $\left\{1,-2-\sqrt{3},-2 \left(1+\sqrt{3}\right)\right\}$ ,
$C_4$	$\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ , $\left\{-1,2+\sqrt{3},-1-\sqrt{3}\right\}$ , $\left\{2+\sqrt{3},-1,-1-\sqrt{3}\right\}$ ,
$C_2$	$\{1,1,1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ , $\left\{-1,2+\sqrt{3},-1-\sqrt{3}\right\}$ , $\left\{2+\sqrt{3},-1,-1-\sqrt{3}\right\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{2+\sqrt{3},-1,\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ , $\{2,2,-1\}$ , $\left\{1,-2-\sqrt{3},\frac{1}{1-\sqrt{3}}\right\}$ , $\left\{-2-\sqrt{3},1,\frac{1}{1-\sqrt{3}}\right\}$ , $\{-2,-2,1\}$ , $\left\{-1,2+\sqrt{3},\frac{1}{2} \left(1+\sqrt{3}\right)\right\}$ ,
$\sigma _h$	$\{1,1,1\}$ , $\left\{1,-2-\sqrt{3},1+\sqrt{3}\right\}$ , $\left\{-2-\sqrt{3},1,1+\sqrt{3}\right\}$ ,
$\sigma _d$	$\{1,-1,0\}$ , $\left\{2+\sqrt{3},1,0\right\}$ , $\left\{1,2+\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{-2-\sqrt{3},1,-2 \left(1+\sqrt{3}\right)\right\}$ , $\left\{1,-2-\sqrt{3},-2 \left(1+\sqrt{3}\right)\right\}$ ,

Different Settings

Character Table

	$\text{E} \,{\text{(1)}}$	$C_3 \,{\text{(8)}}$	$C_2 \,{\text{(6)}}$	$C_4 \,{\text{(6)}}$	$C_2 \,{\text{(3)}}$	$\text{i} \,{\text{(1)}}$	$S_4 \,{\text{(6)}}$	$S_6 \,{\text{(8)}}$	$\sigma_h \,{\text{(3)}}$	$\sigma_d \,{\text{(6)}}$
$A_{1g}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$A_{2g}$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
$E_g$	$2$	$-1$	$0$	$0$	$2$	$2$	$0$	$-1$	$2$	$0$
$T_{1g}$	$3$	$0$	$-1$	$1$	$-1$	$3$	$1$	$0$	$-1$	$-1$
$T_{2g}$	$3$	$0$	$1$	$-1$	$-1$	$3$	$-1$	$0$	$-1$	$1$
$A_{1u}$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$
$A_{2u}$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
$E_u$	$2$	$-1$	$0$	$0$	$2$	$-2$	$0$	$1$	$-2$	$0$
$T_{1u}$	$3$	$0$	$-1$	$1$	$-1$	$-3$	$-1$	$0$	$1$	$1$
$T_{2u}$	$3$	$0$	$1$	$-1$	$-1$	$-3$	$1$	$0$	$1$	$-1$

Product Table

	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$
$A_{1g}$	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$
$A_{2g}$	$A_{2g}$	$A_{1g}$	$E_g$	$T_{2g}$	$T_{1g}$	$A_{2u}$	$A_{1u}$	$E_u$	$T_{2u}$	$T_{1u}$
$E_g$	$E_g$	$E_g$	$A_{1g}+A_{2g}+E_g$	$T_{1g}+T_{2g}$	$T_{1g}+T_{2g}$	$E_u$	$E_u$	$A_{1u}+A_{2u}+E_u$	$T_{1u}+T_{2u}$	$T_{1u}+T_{2u}$
$T_{1g}$	$T_{1g}$	$T_{2g}$	$T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$T_{1u}$	$T_{2u}$	$T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$
$T_{2g}$	$T_{2g}$	$T_{1g}$	$T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$T_{2u}$	$T_{1u}$	$T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$
$A_{1u}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$
$A_{2u}$	$A_{2u}$	$A_{1u}$	$E_u$	$T_{2u}$	$T_{1u}$	$A_{2g}$	$A_{1g}$	$E_g$	$T_{2g}$	$T_{1g}$
$E_u$	$E_u$	$E_u$	$A_{1u}+A_{2u}+E_u$	$T_{1u}+T_{2u}$	$T_{1u}+T_{2u}$	$E_g$	$E_g$	$A_{1g}+A_{2g}+E_g$	$T_{1g}+T_{2g}$	$T_{1g}+T_{2g}$
$T_{1u}$	$T_{1u}$	$T_{2u}$	$T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$T_{1g}$	$T_{2g}$	$T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$
$T_{2u}$	$T_{2u}$	$T_{1u}$	$T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$T_{2g}$	$T_{1g}$	$T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$

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 Oh Point group with orientation 111z the form of the expansion coefficients is:

Expansion

$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ (1-i) \sqrt{\frac{5}{7}} A(4,0) & k=4\land m=-3 \\ A(4,0) & k=4\land m=0 \\ (-1-i) \sqrt{\frac{5}{7}} A(4,0) & k=4\land m=3 \\ -\frac{1}{8} i \sqrt{\frac{77}{3}} A(6,0) & k=6\land m=-6 \\ \left(-\frac{1}{8}+\frac{i}{8}\right) \sqrt{\frac{35}{3}} A(6,0) & k=6\land m=-3 \\ A(6,0) & k=6\land m=0 \\ \left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\frac{35}{3}} A(6,0) & k=6\land m=3 \\ \frac{1}{8} i \sqrt{\frac{77}{3}} A(6,0) & k=6\land m=6 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_111z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(1 - I)*Sqrt[5/7]*A[4, 0], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {(-1 - I)*Sqrt[5/7]*A[4, 0], k == 4 && m == 3}, {(-I/8)*Sqrt[77/3]*A[6, 0], k == 6 && m == -6}, {(-1/8 + I/8)*Sqrt[35/3]*A[6, 0], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {(1/8 + I/8)*Sqrt[35/3]*A[6, 0], k == 6 && m == 3}, {(I/8)*Sqrt[77/3]*A[6, 0], k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_Oh_111z.Quanty

Akm = {{0, 0, A(0,0)} , 
       {4, 0, A(4,0)} , 
       {4, 3, (-1+-1*I)*((sqrt(5/7))*(A(4,0)))} , 
       {4,-3, (1+-1*I)*((sqrt(5/7))*(A(4,0)))} , 
       {6, 0, A(6,0)} , 
       {6,-3, (-1/8+1/8*I)*((sqrt(35/3))*(A(6,0)))} , 
       {6, 3, (1/8+1/8*I)*((sqrt(35/3))*(A(6,0)))} , 
       {6,-6, (-1/8*I)*((sqrt(77/3))*(A(6,0)))} , 
       {6, 6, (1/8*I)*((sqrt(77/3))*(A(6,0)))} }

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.

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(1)}}$	$\color{darkred}{ 0 }$	$\text{App}(0,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,0)$	$0$
${Y_{0}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{21}} \text{Apf}(4,0)$	$0$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$\left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{21}} \text{Apf}(4,0)$
${Y_{1}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,0)$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$
${Y_{-2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0)$	$0$	$0$	$\left(-\frac{5}{21}+\frac{5 i}{21}\right) \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$\left(\frac{5}{21}-\frac{5 i}{21}\right) \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{0}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\left(-\frac{5}{21}-\frac{5 i}{21}\right) \text{Add}(4,0)$	$0$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\left(\frac{5}{21}+\frac{5 i}{21}\right) \text{Add}(4,0)$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-3}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{21}} \text{Apf}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$0$	$0$	$\left(-\frac{1}{11}+\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)-\left(\frac{35}{1716}-\frac{35 i}{1716}\right) \sqrt{5} \text{Aff}(6,0)$	$0$	$0$	$\frac{35}{156} i \text{Aff}(6,0)$
${Y_{-2}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$	$0$	$\left(\frac{35}{572}-\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0)-\left(\frac{1}{33}-\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0)$	$0$	$0$
${Y_{-1}^{(3)}}$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$	$\left(\frac{1}{33}-\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0)-\left(\frac{35}{572}-\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0)$	$0$
${Y_{0}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\left(-\frac{1}{11}-\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)-\left(\frac{35}{1716}+\frac{35 i}{1716}\right) \sqrt{5} \text{Aff}(6,0)$	$0$	$0$	$\text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)+\left(\frac{35}{1716}-\frac{35 i}{1716}\right) \sqrt{5} \text{Aff}(6,0)$
${Y_{1}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\left(\frac{35}{572}+\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0)-\left(\frac{1}{33}+\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0)$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$
${Y_{2}^{(3)}}$	$\color{darkred}{ 0 }$	$\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\left(\frac{1}{33}+\frac{i}{33}\right) \sqrt{10} \text{Aff}(4,0)-\left(\frac{35}{572}+\frac{35 i}{572}\right) \sqrt{\frac{5}{2}} \text{Aff}(6,0)$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$
${Y_{3}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{21}} \text{Apf}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{35}{156} i \text{Aff}(6,0)$	$0$	$0$	$\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{5} \text{Aff}(4,0)+\left(\frac{35}{1716}+\frac{35 i}{1716}\right) \sqrt{5} \text{Aff}(6,0)$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

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

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$\text{s}$	$1$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$1$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{(x-y)(x+y-2z)}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{6}}$	$-\frac{1-i}{\sqrt{6}}$	$0$	$\frac{1+i}{\sqrt{6}}$	$\frac{1}{\sqrt{6}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{yz}+\text{xz}+\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{6}}$	$\frac{1+i}{\sqrt{6}}$	$0$	$-\frac{1-i}{\sqrt{6}}$	$-\frac{i}{\sqrt{6}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{(x-y)(x+y+z)}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{3}}$	$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$	$0$	$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$	$\frac{1}{\sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{2\text{xy}-\text{xz}-\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{3}}$	$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$	$0$	$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$	$-\frac{i}{\sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$1$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{3}-\frac{i}{3}$	$0$	$0$	$-\frac{\sqrt{5}}{3}$	$0$	$0$	$-\frac{1}{3}-\frac{i}{3}$
$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$-\frac{1}{2 \sqrt{3}}$	$0$	$\frac{1}{2 \sqrt{3}}$	$\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$0$
$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$-\frac{i}{2 \sqrt{3}}$	$0$	$-\frac{i}{2 \sqrt{3}}$	$\left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$0$
$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\left(\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$	$0$	$0$	$\frac{2}{3}$	$0$	$0$	$\left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$
$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$	$-\frac{\sqrt{\frac{5}{3}}}{2}$	$0$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$	$0$
$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$0$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$	$0$
$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{2}+\frac{i}{2}$	$0$	$0$	$0$	$0$	$0$	$-\frac{1}{2}+\frac{i}{2}$

One particle coupling on a basis of symmetry adapted functions

After rotation we find

	$\text{s}$	$p_x$	$p_y$	$p_z$	$d_{(x-y)(x+y-2z)}$	$d_{\text{yz}+\text{xz}+\text{xy}}$	$d_{(x-y)(x+y+z)}$	$d_{2\text{xy}-\text{xz}-\text{yz}}$	$d_{3z^2-r^2}$	$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ 0 }$	$\text{App}(0,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$	$0$	$0$	$0$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$	$0$	$0$	$0$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$	$0$	$0$	$0$
$d_{(x-y)(x+y-2z)}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)-\frac{3}{7} \text{Add}(4,0)$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{yz}+\text{xz}+\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)-\frac{3}{7} \text{Add}(4,0)$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{(x-y)(x+y+z)}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{2\text{xy}-\text{xz}-\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)+\frac{6}{11} \text{Aff}(4,0)+\frac{45}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$
$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$
$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{2 \text{Apf}(4,0)}{\sqrt{21}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0)$	$0$	$0$	$0$
$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0)$	$0$	$0$
$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0)$	$0$
$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0)$

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

$A_{k,m} = \begin{cases} \text{Ea1g} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_111z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Oh_111z.Quanty

Akm = {{0, 0, Ea1g} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{0}^{(0)}}$
${Y_{0}^{(0)}}$	$\text{Ea1g}$

The Hamiltonian on a basis of symmetric functions

	$\text{s}$
$\text{s}$	$\text{Ea1g}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ea1g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Et1u} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_111z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Oh_111z.Quanty

Akm = {{0, 0, Et1u} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\text{Et1u}$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$\text{Et1u}$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$\text{Et1u}$

The Hamiltonian on a basis of symmetric functions

	$p_x$	$p_y$	$p_z$
$p_x$	$\text{Et1u}$	$0$	$0$
$p_y$	$0$	$\text{Et1u}$	$0$
$p_z$	$0$	$0$	$\text{Et1u}$

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

	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (2 \text{Eeg}+3 \text{Et2g}) & k=0\land m=0 \\ (-1+i) \sqrt{\frac{7}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land m=-3 \\ -\frac{7}{5} (\text{Eeg}-\text{Et2g}) & k=4\land m=0 \\ (1+i) \sqrt{\frac{7}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land m=3 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_111z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(2*Eeg + 3*Et2g)/5, k == 0 && m == 0}, {(-1 + I)*Sqrt[7/5]*(Eeg - Et2g), k == 4 && m == -3}, {(-7*(Eeg - Et2g))/5, k == 4 && m == 0}, {(1 + I)*Sqrt[7/5]*(Eeg - Et2g), k == 4 && m == 3}}, 0]

Input format suitable for Quanty

Akm_Oh_111z.Quanty

Akm = {{0, 0, (1/5)*((2)*(Eeg) + (3)*(Et2g))} , 
       {4, 0, (-7/5)*(Eeg + (-1)*(Et2g))} , 
       {4,-3, (-1+1*I)*((sqrt(7/5))*(Eeg + (-1)*(Et2g)))} , 
       {4, 3, (1+1*I)*((sqrt(7/5))*(Eeg + (-1)*(Et2g)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\frac{1}{3} (\text{Eeg}+2 \text{Et2g})$	$0$	$0$	$\left(\frac{1}{3}-\frac{i}{3}\right) (\text{Eeg}-\text{Et2g})$	$0$
${Y_{-1}^{(2)}}$	$0$	$\frac{1}{3} (2 \text{Eeg}+\text{Et2g})$	$0$	$0$	$\left(-\frac{1}{3}+\frac{i}{3}\right) (\text{Eeg}-\text{Et2g})$
${Y_{0}^{(2)}}$	$0$	$0$	$\text{Et2g}$	$0$	$0$
${Y_{1}^{(2)}}$	$\left(\frac{1}{3}+\frac{i}{3}\right) (\text{Eeg}-\text{Et2g})$	$0$	$0$	$\frac{1}{3} (2 \text{Eeg}+\text{Et2g})$	$0$
${Y_{2}^{(2)}}$	$0$	$\left(-\frac{1}{3}-\frac{i}{3}\right) (\text{Eeg}-\text{Et2g})$	$0$	$0$	$\frac{1}{3} (\text{Eeg}+2 \text{Et2g})$

The Hamiltonian on a basis of symmetric functions

	$d_{(x-y)(x+y-2z)}$	$d_{\text{yz}+\text{xz}+\text{xy}}$	$d_{(x-y)(x+y+z)}$	$d_{2\text{xy}-\text{xz}-\text{yz}}$	$d_{3z^2-r^2}$
$d_{(x-y)(x+y-2z)}$	$\text{Eeg}$	$0$	$0$	$0$	$0$
$d_{\text{yz}+\text{xz}+\text{xy}}$	$0$	$\text{Eeg}$	$0$	$0$	$0$
$d_{(x-y)(x+y+z)}$	$0$	$0$	$\text{Et2g}$	$0$	$0$
$d_{2\text{xy}-\text{xz}-\text{yz}}$	$0$	$0$	$0$	$\text{Et2g}$	$0$
$d_{3z^2-r^2}$	$0$	$0$	$0$	$0$	$\text{Et2g}$

Rotation matrix used

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
$d_{(x-y)(x+y-2z)}$	$\frac{1}{\sqrt{6}}$	$-\frac{1-i}{\sqrt{6}}$	$0$	$\frac{1+i}{\sqrt{6}}$	$\frac{1}{\sqrt{6}}$
$d_{\text{yz}+\text{xz}+\text{xy}}$	$\frac{i}{\sqrt{6}}$	$\frac{1+i}{\sqrt{6}}$	$0$	$-\frac{1-i}{\sqrt{6}}$	$-\frac{i}{\sqrt{6}}$
$d_{(x-y)(x+y+z)}$	$\frac{1}{\sqrt{3}}$	$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$	$0$	$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$	$\frac{1}{\sqrt{3}}$
$d_{2\text{xy}-\text{xz}-\text{yz}}$	$\frac{i}{\sqrt{3}}$	$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$	$0$	$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$	$-\frac{i}{\sqrt{3}}$
$d_{3z^2-r^2}$	$0$	$0$	$1$	$0$	$0$

Irriducible representations and their onsite energy

	$\text{Eeg}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} (x-y) (x+y-2 z)$
	$\text{Eeg}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$
	$\text{Et2g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (x-y) (x+y+z)$
	$\text{Et2g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$
	$\text{Et2g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{7}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=-3 \\ \frac{1}{2} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{7}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=3 \\ -\frac{13}{60} i \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=-6 \\ -\frac{\left(\frac{13}{12}-\frac{13 i}{12}\right) (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{\sqrt{105}} & k=6\land m=-3 \\ \frac{26}{105} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0 \\ \frac{\left(\frac{13}{12}+\frac{13 i}{12}\right) (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{\sqrt{105}} & k=6\land m=3 \\ \frac{13}{60} i \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=6 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_111z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {(1/2 - I/2)*Sqrt[5/7]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && m == -3}, {(2*Ea2u - 3*Et1u + Et2u)/2, k == 4 && m == 0}, {(-1/2 - I/2)*Sqrt[5/7]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && m == 3}, {((-13*I)/60)*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u), k == 6 && m == -6}, {((-13/12 + (13*I)/12)*(4*Ea2u + 5*Et1u - 9*Et2u))/Sqrt[105], k == 6 && m == -3}, {(26*(4*Ea2u + 5*Et1u - 9*Et2u))/105, k == 6 && m == 0}, {((13/12 + (13*I)/12)*(4*Ea2u + 5*Et1u - 9*Et2u))/Sqrt[105], k == 6 && m == 3}, {((13*I)/60)*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u), k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_Oh_111z.Quanty

Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} , 
       {4, 0, (1/2)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} , 
       {4, 3, (-1/2+-1/2*I)*((sqrt(5/7))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
       {4,-3, (1/2+-1/2*I)*((sqrt(5/7))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
       {6, 0, (26/105)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} , 
       {6,-3, (-13/12+13/12*I)*((1/(sqrt(105)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
       {6, 3, (13/12+13/12*I)*((1/(sqrt(105)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
       {6,-6, (-13/60*I)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
       {6, 6, (13/60*I)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u})$	$0$	$0$	$\left(-\frac{1}{9}+\frac{i}{9}\right) \sqrt{5} (\text{Ea2u}-\text{Et1u})$	$0$	$0$	$\frac{1}{18} i (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})$
${Y_{-2}^{(3)}}$	$0$	$\frac{1}{6} (5 \text{Et1u}+\text{Et2u})$	$0$	$0$	$\frac{1}{6} (-1)^{3/4} \sqrt{5} (\text{Et2u}-\text{Et1u})$	$0$	$0$
${Y_{-1}^{(3)}}$	$0$	$0$	$\frac{1}{6} (\text{Et1u}+5 \text{Et2u})$	$0$	$0$	$\frac{1}{6} (-1)^{3/4} \sqrt{5} (\text{Et1u}-\text{Et2u})$	$0$
${Y_{0}^{(3)}}$	$\left(\frac{1}{9}+\frac{i}{9}\right) \sqrt{5} (\text{Et1u}-\text{Ea2u})$	$0$	$0$	$\frac{1}{9} (5 \text{Ea2u}+4 \text{Et1u})$	$0$	$0$	$\left(\frac{1}{9}-\frac{i}{9}\right) \sqrt{5} (\text{Ea2u}-\text{Et1u})$
${Y_{1}^{(3)}}$	$0$	$\left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{\frac{5}{2}} (\text{Et1u}-\text{Et2u})$	$0$	$0$	$\frac{1}{6} (\text{Et1u}+5 \text{Et2u})$	$0$	$0$
${Y_{2}^{(3)}}$	$0$	$0$	$\left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{\frac{5}{2}} (\text{Et2u}-\text{Et1u})$	$0$	$0$	$\frac{1}{6} (5 \text{Et1u}+\text{Et2u})$	$0$
${Y_{3}^{(3)}}$	$-\frac{1}{18} i (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})$	$0$	$0$	$\left(\frac{1}{9}+\frac{i}{9}\right) \sqrt{5} (\text{Ea2u}-\text{Et1u})$	$0$	$0$	$\frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u})$

The Hamiltonian on a basis of symmetric functions

	$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$
$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$\text{Ea2u}$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$\text{Et1u}$	$0$	$0$	$0$	$0$	$0$
$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$0$	$\text{Et1u}$	$0$	$0$	$0$	$0$
$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$0$	$0$	$0$	$\text{Et1u}$	$0$	$0$	$0$
$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$0$	$0$	$0$	$\text{Et2u}$	$0$	$0$
$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$0$	$0$	$0$	$0$	$\text{Et2u}$	$0$
$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Et2u}$

Rotation matrix used

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$\frac{1}{3}-\frac{i}{3}$	$0$	$0$	$-\frac{\sqrt{5}}{3}$	$0$	$0$	$-\frac{1}{3}-\frac{i}{3}$
$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$-\frac{1}{2 \sqrt{3}}$	$0$	$\frac{1}{2 \sqrt{3}}$	$\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$0$
$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$-\frac{i}{2 \sqrt{3}}$	$0$	$-\frac{i}{2 \sqrt{3}}$	$\left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$	$0$
$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$\left(\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$	$0$	$0$	$\frac{2}{3}$	$0$	$0$	$\left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$
$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$-\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$	$-\frac{\sqrt{\frac{5}{3}}}{2}$	$0$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$	$0$
$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$0$	$\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$0$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$	$0$
$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$	$\frac{1}{2}+\frac{i}{2}$	$0$	$0$	$0$	$0$	$0$	$-\frac{1}{2}+\frac{i}{2}$

Irriducible representations and their onsite energy

	$\text{Ea2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\left(\frac{1}{24}+\frac{i}{24}\right) \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(e^{6 i \phi } \sin ^3(\theta )-(1-i) e^{3 i \phi } \cos (\theta ) \left(5 \cos ^2(\theta )-3\right)-i \sin ^3(\theta )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(x^3-3 x^2 y-3 x y^2+y^3-5 z^3+3 z\right)$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )-\cos (2 \phi )))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 x^2 z-10 x y z-5 x z^2+x-5 y^2 z\right)$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \sin (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi )))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 x^2 z-10 x y z+5 y^2 z-5 y z^2+y\right)$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\left(\frac{1}{48}+\frac{i}{48}\right) \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \left(5 \left(e^{6 i \phi }-i\right) \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 x^3-15 x^2 y-15 x y^2+5 y^3+4 z \left(5 z^2-3\right)\right)$
	$\text{Et2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+\sin (2 \theta ) (\cos (2 \phi )-\sin (2 \phi )))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 (-z)+2 x y z-5 x z^2+x+y^2 z\right)$
	$\text{Et2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi ))-(5 \cos (2 \theta )+3) \sin (\phi ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 z+2 x y z-y^2 z-5 y z^2+y\right)$
	$\text{Et2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin ^3(\theta ) (\sin (3 \phi )+\cos (3 \phi ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^3+3 x^2 y-3 x y^2-y^3\right)$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 4\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{15} \text{Mt1u} & k=4\land m=-3 \\ \frac{\sqrt{21} \text{Mt1u}}{2} & k=4\land m=0 \\ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{15} \text{Mt1u} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_111z.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -3 && m != 0 && m != 3)}, {(1/2 - I/2)*Sqrt[15]*Mt1u, k == 4 && m == -3}, {(Sqrt[21]*Mt1u)/2, k == 4 && m == 0}}, (-1/2 - I/2)*Sqrt[15]*Mt1u]

Input format suitable for Quanty

Akm_Oh_111z.Quanty

Akm = {{4, 0, (1/2)*((sqrt(21))*(Mt1u))} , 
       {4, 3, (-1/2+-1/2*I)*((sqrt(15))*(Mt1u))} , 
       {4,-3, (1/2+-1/2*I)*((sqrt(15))*(Mt1u))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$-\frac{\text{Mt1u}}{\sqrt{6}}$	$0$	$0$	$\text{Mt1u} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]$	$0$
${Y_{0}^{(1)}}$	$\left(\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} \text{Mt1u}$	$0$	$0$	$\frac{2 \text{Mt1u}}{3}$	$0$	$0$	$\left(-\frac{1}{6}+\frac{i}{6}\right) \sqrt{5} \text{Mt1u}$
${Y_{1}^{(1)}}$	$0$	$\text{Mt1u} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]$	$0$	$0$	$-\frac{\text{Mt1u}}{\sqrt{6}}$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$	$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$	$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$	$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$
$p_x$	$0$	$\left(\frac{1}{12}+\frac{i}{12}\right) \text{Mt1u} \left(i \sqrt{15} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]+\sqrt{15} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]+(1-i)\right)$	$\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} \text{Mt1u} \left(\text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]+i \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]\right)$	$0$	$\left(\frac{1}{12}-\frac{i}{12}\right) \text{Mt1u} \left(\sqrt{3} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]+(1+i) \sqrt{5}\right)$	$-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \text{Mt1u} \left(\text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]+i \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]\right)}{\sqrt{3}}$	$0$
$p_y$	$0$	$\left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} \text{Mt1u} \left(\text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]+i \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]\right)$	$\left(\frac{1}{12}+\frac{i}{12}\right) \text{Mt1u} \left(i \sqrt{15} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]+\sqrt{15} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]+(1-i)\right)$	$0$	$\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \text{Mt1u} \left(\text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]+i \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]\right)}{\sqrt{3}}$	$\left(\frac{1}{12}-\frac{i}{12}\right) \text{Mt1u} \left(\sqrt{3} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$ \#$1}^4+25$	$,3\right]+(1+i) \sqrt{5}\right)$	$0$
$p_z$	$0$	$0$	$0$	$\text{Mt1u}$	$0$	$0$	$0$

Table of several point groups

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

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Orientation 111z

Symmetry Operations

Different Settings

Character Table

Product Table

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Expansion

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One particle coupling on a basis of spherical harmonics

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

One particle coupling on a basis of symmetry adapted functions

Coupling for a single shell

Potential for s orbitals

Potential for p orbitals

Potential for d orbitals

Potential for f orbitals

Coupling between two shells

Potential for p-f orbital mixing

Table of several point groups

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